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Poor Charlie’s Almanack: The Wit and Wisdom of Charles T. Munger

Being well read and having many different mental models and vast knowledge is a major asset in life. Most people fall prey to “man with a hammer” syndrome, where if your only tool is a hammer, every problem looks like a nail. A person trained only in economics will see every issue through that single lens, leading to distorted thinking and poor decisions. To overcome this, we must acquire and use the big, reliable ideas from the world’s major disciplines. You don’t need to become an expert in each field, but you must understand their fundamental principles so they can work together in your mind. Wise judgment in investing requires a latticework of mental models and some of the most important ones are provided below:

• Compound Interest: A small force, sustained over time, creates an overwhelming result (applies to finance, knowledge, trust).

• Permutations and Combinations: Understanding how probabilities work (e.g., knowing that a chain of small risks leads to a large total risk).

• Fermat’s Principle: Looking at the logical extreme of a situation to understand its constraints or limitations.

• Critical Mass (physics + chemistry): Many systems require a threshold of input to trigger a massive, non-linear output.

• Incentives: Show me the incentive, and I will show you the outcome. Never underestimate the power of rewards/punishments to drive behavior.

• Confirmation Bias: The tendency to seek out and interpret information that confirms one’s pre-existing beliefs.

• Lollapalooza Effect: When multiple biases (e.g., social proof, authority) combine, they create extreme and irrational outcomes.

• Evolution: Systems that adapt through variation, selection, and retention are incredibly robust.

• Everything is connected: A change in one part of a system (a company, a market) will inevitably lead to predictable and unpredictable changes elsewhere.

• Comparative Advantage: Do what you do best relative to others and trade for the rest. Specialization is a fundamental driver of prosperity.

• Economies of Scale: The massive benefits that accrue to size (e.g., economies of scale in production or distribution, and informational advantages).

• Margin of Safety (engineering): Build in a buffer for error or unforeseen events. This is central to value investing, engineering, and risk management.

• Inversion: Think backward—instead of asking “how can I succeed?” ask “what must I absolutely avoid doing to fail?” By focusing on guaranteed failure, you can simplify the path to success.

The profound importance of having many models lies in their synthesis. Real-world problems are not labeled “economics” or “psychology”; they are complex puzzles that require multiple angles of attack. It is important to use a mental checklist based on these models to avoid blind spots.

Prometheus Rising (by Robert Anton Wilson)

We all see reality through very narrow tunnels and we actively ignore the things that don’t match our own reality tunnel while actively looking for evidence to reinforce the tunnel that we believe in. This “reality” that we experience isn’t reality at all: it’s mostly our own nervous system, imprinted and conditioned into rigid patterns. We have both the burden and the possibility of loosening those patterns and re-educating ourselves.

Like Prometheus stealing fire from the gods, the individual who learns to meta-program their own brain is seizing a power that was previously reserved for accident, tradition, and authority. Humans possess the neurological potential to intentionally transcend their current limitations and become self-directed agents of their own psychological and evolutionary development, and they can do that by forever testing hypotheses and dismantling the apparatus of perception to see what, if anything, lies beneath its gears. Curiosity and continual questioning and discovery are the true key to our own liberation.

The Millionaire Fastlane (by MJ DeMarco)

There is the slow way of obtaining wealth (by working for someone else, investing, saving) and a fastlane approach (producing something that has value). Slowlaners buy depreciating assets like cars and electronics; fastlaners buy appreciating assets like patents, businesses, and cash flows. Fastlaners view time as their most important asset and making something of value as their primary means of wealth accumulation.

A few examples of fastlane projects: write a book (lots of work, but then it makes money forever without you having to put in more work), make an invention (lots of work, but then you get royalties for a long time), or generally obtain an income generating asset or assets. The key to the fastlane is producing instead of consuming. Don’t be the guy who buys a franchise, be the guy who sells franchises.

The key to creating and having wealth is by working hard today on something that people value in the future. Around 80% of millionaires make their wealth by either starting their own company or working for a small company that has explosive growth.

Some keys to success in starting a business:

• The amount of money a business earns is a reflection of the amount of value it has provided.

• The harder it is to enter your business, the better you business will be.

• There are levels of business scale: community, city, state, region, national, and international. The larger your scale, the bigger your leverage. Profit = units sold * profit/unit. Maximize your scale / leverage as much as possible.

• Investment of time: can the business be automated? Are margins large enough to hire others to do your work? Could you (eventually) get the business to operate without spending much of your time on it? Your business should allow you to eventually have abundance of your most precious resource (time) – if it doesn’t do that, then it is not a great pathway to wealth.

• Don’t guess on whether there is demand for your product. The best way to figure out where to go is to listen to the rest of the world. You should come out with a minimal version of your product and then see how customers react. Their reactions will guide you during your next iteration and so on and so on.

Being wealthy is not about money, fancy cars, expensive vacations, or vacation homes in Fiji. Being wealthy means being healthy, being surrounded by great friends and family, and the freedom to live life how you want to live it. Faux wealth is the illusion of wealth. It’s having nice, flashy things that you can’t afford which take away your freedom and make you even more dependent on your existing sources of income. Faux wealth is not real wealth.

Your time is precious, don’t waste it and don’t trade it away for pennies. Your time is finite and always decreasing — it is your most precious and most valuable resource. Treat it as such.

Daily Rituals (by Mason Currey)

Consistency matters. A solid routine fosters a well-worn groove for one’s mental energies and helps stave off the tyranny of moods.

Stephen King writes every day of the year, including his birthday and holidays, and he almost never lets himself quit before he reaches his daily quota of two thousand words.

“A modern stoic,” he observed, “knows that the surest way to discipline passion is to discipline time: decide what you want or ought to do during the day, then always do it at exactly the same moment every day, and passion will give you no trouble.”

In Everybody’s Autobiography, Stein confirmed that she had never been able to write much more than half an hour a day—but added, “If you write a half hour a day it makes a lot of writing year by year. To be sure all day and every day you are waiting around to write that half hour a day.”

How the World Really Works (by Vaclav Smil)

Everything that constitutes our modern world—its food, its infrastructure, its technology—is a product of the conversion of energy. The history of human progress is, in large part, the history of harnessing more powerful and concentrated energy sources.

A useful way to grasp our energy consumption is to equate it to human labor. The annual energy consumption of an average person in 2020 is equivalent to having about 60 adult servants working non-stop. For a person in an affluent country, that number rises to 200 or even 240 servants. This illustrates the sheer scale of the work done by inanimate energy converters on our behalf.

There are 4 pillars that the modern world is built on:

1. Ammonia is the “gas that feeds the world.” It is the critical ingredient in synthetic nitrogen fertilizers, and without it, we simply could not grow enough food to feed our current global population. It’s estimated that ammonia-based fertilizers are responsible for sustaining roughly half the world’s people. Almost all industrial ammonia today is made from hydrogen plus nitrogen, and that hydrogen usually comes from natural gas (methane) or sometimes coal. Ammonia accounts for a noticeable chunk of global CO₂ emissions.

2. Plastics: These versatile materials, derived from fossil fuel feed-stocks, are ubiquitous in packaging, medicine, construction, and consumer goods. Plastic production depends very heavily on fossil fuels, in two ways: as a raw material and as an energy source. Most conventional plastics are made from petrochemicals derived from oil and natural gas. Their problem is not utility but disposal, creating a massive waste management challenge.

3. Steel: This alloy is the skeleton of modern civilization. It defines the look of our world, from skyscrapers and bridges to cars and appliances. Its production in blast furnaces is a process of extreme heat, typically generated by burning coal. Steel production today still depends a lot on fossil fuels, though the amount varies by process and country.

4. Cement: The key ingredient in concrete, cement is the most widely used human-made material; it is the backbone of modern construction and infrastructure. Cement production still depends heavily on fossil fuels because making it requires extremely high kiln temperatures, which are usually produced by burning coal, petroleum coke, or natural gas, and also because heating limestone releases CO₂ as a chemical by-product.

Despite decades of discussion about transition, the global energy system remains overwhelmingly dependent on fossil fuels. Their share of the world’s primary energy supply has decreased only marginally, from about 87% in 2000 to 84% in 2020. This is not for a lack of trying, but because of the sheer scale of the existing infrastructure. Replacing it is the work of generations, not decades – and we’re just getting started.

Economics in One Lesson (by Henry Hazlitt)

Almost all bad economics comes from focusing only on what is immediately seen—the short-run effect on one visible group—while ignoring what is unseen: the longer-run effects on everyone. The “one lesson” is that the art of economics is to trace all consequences of a policy, not just the first ones and not just for the most visible beneficiaries.

As an example, government expenditure is often lauded for the jobs it creates, but where does the money for that spending come from? Yes, it comes from taxes. The funds taken through taxation are money that would have been used by private citizens for investment, consumption, or savings, all of which would have also created jobs. The unseen consequence is the lost private investment and production that never occurs because the funds were seized by the state.

Another example are tariffs: they protect a small number of domestic producers by allowing them to charge a higher price. The unseen harm is that all domestic consumers must pay a higher price, effectively being taxed to subsidize the protected industry. Furthermore, it prevents efficient foreign producers from earning dollars they would have used to buy American exports, thus harming domestic export industries.

Another example are minimum wages. They clearly benefits workers who can keep their jobs at a higher rate. This benefit is visible. The unseen consequence is that businesses, unable to afford the higher cost, will hire fewer workers or replace some with automation, leading to unemployment for the least-skilled workers. The policy intended to help all low-wage workers instead helps some at the expense of others.

The final, indispensable takeaway is that the art of economics lies in training oneself to see not just the immediate, visible consequences of a policy or action, but to trace out its longer-term effects on all groups. It requires looking beyond the obvious beneficiaries—the subsidized industry, the hired workers, the funded project—to recognize the unseen victims: the consumers who pay higher prices, the taxpayers whose wealth is diverted, the entrepreneurs whose capital is unavailable, and the opportunities that are never born. Good economics is therefore an exercise in disciplined empathy and imagination, demanding we count all the costs, not just the loudest or most politically convenient ones, and remember that society loses when we favor the seen at the expense of the unseen.

QED – The Strange Theory of Light and Matter (by Richard Feynman)

The book is a guided tour of quantum electrodynamics—the theory that explains how light and matter interact—told with almost no formal math. Instead of equations, Feynman uses little clocks to represent probability amplitudes.

To calculate the probability of where a particle ends up, we assign each possible path a “probability amplitude” (visualized as a tiny spinning arrow, or clock hand). The arrow’s direction is determined by the path’s phase, which is related to the time taken. We then sum the arrows for all paths. The paths whose arrows point in roughly the same direction reinforce each other while those pointing in opposite directions cancel out. This “sum over histories” explains why, in the aggregate, light appears to take a straight line or a classical reflection path; the neighborhoods of paths of least time cooperate while others which take longer cancel each other out. Really though, there is no ‘one path’ that light really takes. Nature at the quantum level doesn’t behave like tiny ‘billiard balls’ or ‘smooth waves’ or tiny things ‘bumping’ into each other; instead, photons actually take all possible paths from point A to point B into account and move according to the path of least energy / time.

Feynman’s genius was to decode this algorithm not as a wall of equations, but as a lucid, almost playful procedure—giving us not just the how of light and matter, but a humbling glimpse into the foundational software of reality. The theory’s immense success (it predicts measurements with unmatched precision) forces us to accept that this is not a model of reality, but a reliable description of how reality actually behaves. It provides us with a picture of what reality actually is: a very complex system which can only be modeled through a huge degree of computational complexity and not a simple dynamical one with simple rules.

The Hard Thing About Hard Things (by Ben Horowitz)

This is a brutally honest and practical guide to building and running a startup. Drawing from his tumultuous experiences as a founder and CEO, Horowitz delves into the “hard things”—like firing executives, demoting friends, managing your own psychology, and making agonizing decisions with no good options—that most business books ignore.

Some key lessons and quotes from the book:

• You must believe that there is an answer no matter how much the odds may be against you. During his time with LoudCloud, they were close to losing a major customer (called EDS which accounted for 90% of their revenue) and talks about how he won them back: found out that the executive from EDS was not happy with their software and preferred something called Tangram. Ben found out that Tangram was public and was worth around 6 million dollars at the time, so decided to acquire them for 10 million which made EDS’s executive happy and played a huge role in saving the deal and company.

• Almost as soon as the EDS crisis was resolved, they hit more trouble yet again from competitors: three new clients were fading away and excellent new competitor was eating their lunch. Told employees that they had 6 months to really work hard and play catch up with BladeLogic or else they were dead. This meant working over-time and weekends and talks about the time and stress and toll this took on the team, although some team-members ended up having a great experience from this.

• Had to really decide whether to prioritize requirements from existing customers or whether to pursue something else that they thought would help. “It turns out that is exactly what product strategy is all about—figuring out the right product is the innovator’s job, not the customer’s job. The customer only knows what she thinks she wants based on her experience with the current product. The innovator can take into account everything that’s possible, but often must go against what she knows to be true. As a result, innovation requires a combination of knowledge, skill, and courage.” His answer: “I don’t care about any of the existing requirements; I need you to reinvent the product and we need to win.”

• Lead bullets: talks about facing challenges with Netscape (and with Microsoft challenging them) as well as with his issues with OpsWare (their number one competitor was dominating them). Many people suggested they shift business strategies or go into a different market. He refused to do so: with Netscape, they made their servers faster than Microsoft and ended up beating BladeLogic. Instead of running from problems, they faced them head-on and made the product better.

• His general advice: “There comes a time in every company’s life where it must fight for its life. If you find yourself running when you should be fighting, you need to ask yourself, “If our company isn’t good enough to win, then do we need to exist at all?”. In other words, face your competitors and issues head on instead of running from them.

• Embracing the struggle: there are lots of struggles and many young companies go through them. Ben went through major struggles at almost every point of his life, but he prevailed. “Every great entrepreneur from Steve Jobs to Mark Zuckerberg went through the Struggle and struggle they did, so you are not alone. But that does not mean that you will make it. You may not make it. That is why it is the Struggle. The struggle is where greatness comes from.”

The keys to over-coming a struggle:

• Don’t put it all on your own shoulders (share the struggles with the team): “You won’t be able to share every burden, but share every burden that you can. Get the maximum number of brains on the problems even if the problems represent existential threats.”

• Persistence: the technology business is complex and there are always moves you can make to save your company / rescue the situation. He took a company bleeding money and IPO-ed at the worst time ever and made it, so you can too. “Play long enough and you might get lucky. In the technology game, tomorrow looks nothing like today. If you survive long enough to see tomorrow, it may bring you the answer that seems so impossible today.”

• Take care of the people, the products and the profits, in that order. It’s a simple saying, but it’s deep. “Taking care of the people” is the most difficult of the three by far and if you don’t do it, the other two won’t matter. Taking care of the people means that your company is a good place to work. Most workplaces are far from good.

The main takeaway is that there is no secret formula for success during crises; instead, true leadership is about mustering the courage to make tough choices, persevering through sheer will when all seems lost, and accepting that struggle and doubt are not signs of failure but the core, unavoidable components of the entrepreneurial journey. The secret to winning it seems is simply being determined and not giving up.

Build (by Tony Fadell)

Build products which people want (and follow with the correct technology after) – don’t put technology before product – the product must always come first. If you’re not solving a real-world problem, there is no need for your product. Also, the whole customer journey from start to finish is vitally important:

• Design at Nest: every minute, from opening the box to reading the instructions to getting it on their wall to turning on the heat for the first time had to be incredibly smooth.

• Customer panels don’t do well in design. People can’t articulate what they really want clearly enough definitely point in one direction or another. Customers will always be more comfortable what exists already despite it being terrible.

• Story Telling: the story of your company, your product and your team should drive everything you do. The story isn’t there just to sell your product: it’s there to help you define it, understand it, and understand your customers – and it all starts with why? Why does the product exist? What problem does it solve? Use great analogies in telling your story (people can relate to them) and tell a story that others can connect with.

• Your product isn’t your only focus: it’s the whole user experience. Don’t just build a prototype of your product; map our the entire user experience. Map out and visualize how your customer discovers, considers, installs, uses, fixes and ever returns your product. It all matters. Make models. Draw pictures. Sketch out the bones of the process in wire-frames. Write imaginary press releases. Created detailed mock ups. Put it all out and make it visible: how can you delight your customers in every possible way.

• Always seek out new ways to disrupt yourself. If you make something truly disruptive, your competition should not be able to easily replicate it.

• Companies that become too big and don’t innovate eventually die.

No matter what you build – it’ll take some time. The iPad took 3 generations and around 3 years to reach profitability. Google also wasn’t remotely profitable for 3 years and Facebook initially focused on growth rather than what their business model was going to be. The first version of the product should always focus on the customer journey and everything else will come later.

In Cold Blood (by Truman Capote)

If there’s one key lesson that I obtained from reading this, it’s that 1) small things matter and 2) chaos rules our everyday lives. The murders at the heart of In Cold Blood are not the result of some grand plan or deep personal vendetta; they come from a string of minor choices, misunderstandings, and bits of bad information. A rumor about a safe full of money, an offhand comment in a prison cell, a moment of idle curiosity—these small details, which could easily have led nowhere, instead set in motion a sequence of events that destroys an entire family. Capote shows how the line between “ordinary life” and catastrophe is much thinner than we like to admit.

At the same time, the book keeps circling back to how random and unfair this all is. The Clutters are not targeted because of anything they did; they just happen to be the family that fits a half-remembered description. Perry and Dick, too, are shaped by accidents of birth, trauma, and circumstance as much as by their own choices. This doesn’t excuse what they do, but it complicates the story. There is no neat moral accounting where the “bad” are punished and the “good” are spared. Instead, we see a world in which chance meetings, childhood injuries, timing, and geography all combine into something that feels almost senseless.

Capote’s style reinforces this sense of fragile order sitting on top of chaos. He gives us painstaking details about the town, the family’s routines, the layout of the house, the highways the killers drive along, as if documenting everything could make it understandable, controllable, safe. But the more we know, the more we see that knowledge does not prevent randomness; it only highlights how much had to align for this tragedy to happen. The book suggests that we live with an illusion of stability, and that behind our schedules and habits there is always a degree of unpredictability we cannot fully master.

In the end, In Cold Blood is not just about a crime; it’s about how we try to make sense of a world that often doesn’t make sense at all. Small actions ripple outward in ways no one can foresee, and lives intersect in patterns that feel almost arbitrary. The key message, for me, is both unsettling and sobering: we should never underestimate the weight of small decisions, and we should recognize that much of what happens to us—good and bad—is shaped by forces we don’t see and can’t fully control.

The Boy Who Was Raised as a Dog (by Bruce Perry & Maia Szalavitz)

Written by child psychiatrist Dr. Bruce Perry and Maia Szalavitz, the book is a collection of harrowing yet hopeful case studies that explore how extreme trauma affects the developing brains of children. It takes readers through Dr. Perry’s clinical experiences with children who have faced unimaginable neglect and abuse—including a boy raised in a cage, children from the Branch Davidian cult, and survivors of genocide. Rather than viewing these children as damaged goods, Perry uses these stories to illustrate how traumatic experiences physically alter brain biology, shifting the focus from “what is wrong with this child?” to “what happened to this child?”

The key message of the book is that the brain develops based on repeated experiences. Trauma disrupts this development, often locking children into a persistent state of “fight or flight” that hinders their ability to learn and connect. As an example, one of the chapter’s discusses a 16 year old boy called Leon who committed 2 murders. Leon feels no remorse for his murder other than the remorse of being caught. He has two loving parents and an older brother, but Dr. Perry figures out that Leon’s parents would would regularly leave him alone in his crib and leave to go out for walks for hours at a time when he was a young baby; leaving him alone and crying in his crib for many hours. After some time, his mother recounted that Leon stopped crying so much (a huge red flag), so she believed that this regimen had helped him. In reality, her neglect, however innocent, had deprived Leon of the important loving and physical affection experiences an infant needs to form family bonds and later maintain healthy relationships. This isn’t the first story where this type of neglect occurs: in another story, a couple recounts hiring a baby-sitter who would come in during the morning and leave the baby alone for hours at a time, only to come back to change his diaper. Both children ended up with major development disorders, with Leon getting the worst of the bunch.

Even though many of the recounted stories of neglect are absolutely awful, Perry argues that because the brain is plastic, it can be healed. The most powerful tool for this healing is not medication or complex medical procedures, but patterned, repetitive human connection. The book emphasizes that recovery requires meeting a child at their developmental age rather than their chronological age and that healthy, consistent relationships are the ultimate engine of healing.

As an example, Perry and his team learned many valuable lessons from the children in Waco. He would usually get the question: why do some children overcome trauma with minimal or no long-lasting emotional scars, while others do not move past it, or move past it but continue to bear the weight of it, unable to live “normal” lives. His answer: relationships were the key to their ability to move forward and claim a “normal” life for themselves, or not. Children who were placed into stable, loving homes, where consistency and support were constants and they were able to develop deep supportive relationships, whether with blood relatives or foster families, were much more likely to lead normal adult lives. Those children who were less fortunate in their placements and did not develop supportive relationships, were much less likely to lead normal adult lives.

150 years ago most families lived in communities of 20-150 people, many of them family, where everyone knew everyone else, and supported everyone else, but by the 1950’s this was closer to 10 people in community and now the average family is 4 people and 25% of Americans live alone. This lack of connections coupled with our propensity to spend more time watching TV (or our cell phones) than interacting with one another even when we are together lowers both the number and quality of the relationships we are in compared to 100 years ago; this lack of deep relationships in turn heightens our children’s risk for bad and dangerous relationships while lowering their ability to rebound after trauma happens.

Crime and Punishment (by Fyodor Dostoevsky)

The book changed the way I see the world. You look at a man or woman across from you and you might not give too much thought to the fact that inside his or her head, they are making sense of the world just like you are. Yes, everyone you meet has an inner life as crowded and haunted as your own. The drunk in the tavern, the inspector in the office; even killers have their own stories and mental models which they live by. The book shows a murderer’s stream of consciousness. Through it, it also shows that you cannot simply just call someone a monster and walk away. To understand someone, you must know what story he is telling himself, what lies he believes in and what wounds he is hiding. Since this is impossible in the real world, the act of judging someone is possibly the greatest arrogance of all. Can you assume that if you wore the same boots and embarked on the same road that you would have walked a better path? Dostoevsky tries to show you that we are all made of the same stuff and that the line that separates us is very thin one. It taught me to judge less and to observe more, because in the darkness of our own minds, we are all stumbling toward the same light.

The Demon-Haunted World: Science as a Candle in the Dark (by Carl Sagan)

As the world increasingly dominated by technology but plagued by scientific illiteracy, humanity risks sliding back into the darkness of superstition and ignorance. The scientific method—characterized by skepticism, observation, and the demand for evidence—is the only reliable tool we have to illuminate reality and protect ourselves from the “demons” of pseudoscience, whether they take the form of faith healing or political propaganda.

There is a baloney detection kit we all can use:

• Independent Confirmation: Wherever possible, there must be independent confirmation of the “facts.”

• Quantification: Wherever possible, use numbers and data. Vague claims are hard to evaluate; quantified claims can be checked.

• Substantive Debate: Encourage knowledgeable proponents of all points of view to engage in substantive debate on the evidence.

• Occam’s Razor: When faced with competing hypotheses, prefer the simpler one that requires the fewest new assumptions. It’s not a guarantee of truth, but a guiding principle.

• In science, there are no authorities; at most, there are experts. Arguments from authority carry little weight—”authorities” have made mistakes in the past and will do so again.

• Every step in an argument must be solid. A single weak link can break the entire chain of reasoning.

Constantly test hypotheses and question authority. Skepticism is not a negative force, but a constructive one that shields us from manipulation. A society dependent on science and technology in which hardly anyone understands science and technology is a prescription for disaster. A future where the public is unable to distinguish between genuine expertise and charlatanism, leaving them vulnerable to demagogues who can exploit their ignorance. Scientific literacy is not just an academic luxury, but a fundamental pillar of a free society.

The values of science—open debate, the lack of ultimate authority, and the willingness to be proven wrong—are identical to the values of a healthy democracy. Skepticism is a patriotic duty. Just as science requires us to question our hypotheses to find the truth, democracy requires citizens to question their leaders to maintain their freedom. The Bill of Rights exists for a reason; the freedom of speech and the freedom to question are essential checks against the errors of government. Without a skeptical, educated citizenry willing to ask difficult questions, the democratic experiment itself is in jeopardy. The right to question is the most precious freedom we have, and exercising it is the only way to keep the candle burning.

The Fiddler in the Subway: The Story of the World-Class Violinist Who Played for Handouts. . . And Other Virtuoso Performances by America’s Foremost Feature Writer (by Gene Weingarten)

One cold January morning, the world’s best violinist (Joshua Bell)—fresh from sold-out concert halls where tickets cost hundreds of dollars—stood disguised by the escalators of a Washington metro station and played Bach and other virtuoso pieces on a multi-million-dollar Stradivarius violin. For about three quarters of an hour, more than a thousand commuters passed within a few steps of him. Fewer than a dozen stopped to listen to Bell; most hurried by without even turning their heads. The few people who did stop; the children who instinctively wanted to listen to him were pulled away by their parents; and the few who did actually recognize him could not believe what they were witnessing. For the most part, his performance pretty much went unnoticed and he earned about 30 dollars in change from the experience.

The experiment shows how much our sense of value depends on context and how little attention people pay to the things going on around them. If we are unable to stop, even briefly, to hear one of the finest musicians in the world playing some of the finest music ever written, what else, in our haste, do we fail to see? How much beauty going on around us are we ready to miss due to being preoccupied by life’s diversions, and what does that say about us?

The Old Man and the Sea (by Ernest Hemingway)

The story is a simple one. It is about an old man who fishes alone. He has gone many days without taking a fish. He prepares his gear. He rows out into the dark. He goes out farther than anyone else dares to go. He hooks a giant marlin. The fish is strong and he is old, but he holds the line. For three days they are locked together. The line cuts his back and his hands cramp and he is hungry. He eats raw tuna to keep his strength. He talks to his hand. He talks to the bird. He respects the fish. He calls him brother. He loves him, but he has to kill him. That is the way of the world. It is the pride of the man against the dignity of the fish. Then the sharks came. They came to take what was his. He fights them for a long time, but in the end they win and the great fisherman is left with nothing but fish bones.

The story doesn’t have a great plot. It doesn’t have any outstanding twists and turns. The stream of words that Hemingway produces though are beautiful and will leave you stunned. The story taught me that victory and defeat are not as simple as they look. Although the old man came back with almost nothing, the boy sees the marlin’s great spine and tail and admires the old man. The real win is not made in an achievement or what it produces, it is the fact that the old man went far out, fought well, did not break, and came back with great hope. A man can be destroyed but not defeated. Your body can give out. Your catch can be taken. Your luck can run dry. What is left is how you carry yourself when you are alone—doing what you were born to do, even if it breaks you.

Happy 2026 everyone and once again a big thank you to all of the paid subscribers to this Substack!!!

Frontiers of Microeconomics

In the first two parts of this series, we mapped the fundamental landscape of economics: supply and demand, the elegance of markets, and the forces that can cause them to stumble, like monopolies and externalities. We pictured the economy as a complex, self-organizing structure guided by an invisible hand -- but this model makes a large assumption: that everyone involved has reasonably good information. What happens when that isn’t true? What are the consequences when the economic game is played in a fog, where some players can see clearly and others are nearly blind?

The Economics of Information Asymmetry

In many situations in life, knowledge is not distributed equally. This asymmetry creates two of the most powerful and pervasive concepts in modern economics: moral hazards and adverse selection.

Moral Hazards

• Imagine you rent a car. When you drive your own car, you are careful. You avoid potholes and you brake gently — but when you’re in a rented car and you’ve paid for the full insurance coverage, your behavior changes. You take that dirt road shortcut you normally wouldn’t and you end up being a lot less vigilant. This is Moral Hazard. It arises when one person (the agent) is performing a task on behalf of another (the principal), but the principal cannot perfectly monitor the agent’s behavior.

• The problem is everywhere. It’s the corporate executive who takes giant risks with company money because their bonus is tied to short-term gains, not long-term stability; it’s the employee who spends hours browsing social media because the manager can’t see their screen. The principal’s challenge is to design a system—like a deductible on insurance, or a performance-based bonus—to align the agent’s incentives with their own.

Adverse Selection

• Imagine a used car lot. You’re looking at two identical-looking sedans. One is a meticulously maintained by a former mechanic while the other is a lemon (a car with lots of issues) that a problematic seller is desperate to unload. You, the buyer, can’t tell the difference, but the sellers knows the truth. The seller of the good car knows it’s worth a high price, while the seller of the lemon knows it’s worth a low one. Since you can’t tell them apart, you’ll only be willing to pay an average price—somewhere in the middle. What happens?

• The owner of the good car, unwilling to sell his gem for an average price, pulls it off the market. The owner of the lemon, thrilled to get more than his car is worth, gladly sells. The result? The market becomes flooded with lemons. The “selection” of cars available for purchase has become “adverse.” The uninformed party (the buyer) gets a disproportionate number of undesirable trading partners (sellers of low-quality goods) because the informed party uses their private information to their own advantage. The market collapses. The only cars left for sale are the Lemons.

• This exact logic applies to health insurance. Who knows more about your health than you do? People who know they have hidden health problems are the most likely to buy expensive, comprehensive insurance. This drives up the average cost for everyone. Healthy people then look at the high price, decide it’s not worth it, and leave the market. The “insurance pool” gets sicker and sicker, and prices spiral upward. This is a clear case where the invisible hand fails to produce a good outcome for society.

Fighting the Fog: Signaling and Screening

So how do markets cope with this fog of information? They develop clever tools.

• Signaling: The informed party takes an action to reveal their private information. A job applicant gets a university degree. It’s not just about the knowledge; it’s a costly, visible signal that says, “I am a high-quality, dedicated worker.” The used car dealer offering a solid warranty is signaling, “I am so confident this isn’t a lemon that I’ll bet my own money on it.” Degrees, certifications, brand reputation, long warranties, generous return policies are all examples of signaling. A good signal is costly for low-quality sellers to imitate (e.g., a lengthy warranty that would be expensive if your product failed often).

• Screening: The uninformed party takes an action to uncover the private information. An insurance company doesn’t just charge one premium; it screens applicants by offering different plans with different deductibles and co-pays. The choices you reveal help them sort you into risk categories. One example of this might be an insurance company offering 2 plans:

  • A very cheap plan with a massive $10,000 deductible, or

  • A very expensive plan with a $0 deductible.

• This is a screen: healthy people (who know they’re healthy) will grab the cheap plan; people who know they’ll need care will pay extra for the expensive plan. The trick? Design choices so each customer picks the deal meant for them without you ever having to intervene.

The Political Marketplace: Why Governments Stumble, Too

• We often look to government to fix market failures. But governments are not run by omniscient angels; they are made of people and are subject to their own, often more perplexing, failures.

• Collective decisions have their own physics: majority voting can cycle, no voting rule is perfect, and the median voter often pulls policy toward the middle—while interest groups tug at the edges.

• The Condorcet Paradox: This shows that majority-rule voting doesn’t always give you a clear, logical answer. Imagine three roommates (Ann, Bob, Chris) are voting on one pizza topping:

  • Ann’s vote: 1st: Pepperoni, 2nd: Mushroom, 3rd: Onion

  • Bob’s vote: 1st: Mushroom, 2nd: Onion, 3rd: Pepperoni

  • Chris’s vote: 1st: Onion, 2nd: Pepperoni, 3rd: Mushroom

• Now, let’s have them vote head-to-head:

  • Pepperoni vs. Mushroom? Ann and Chris vote Pepperoni. Pepperoni wins.

  • Mushroom vs. Onion? Ann and Bob vote Mushroom. Mushroom wins.

  • Onion vs. Pepperoni? Bob and Chris vote Onion. Onion wins.

• The group “prefers” Pepperoni over Mushroom, and Mushroom over Onion... but they also prefer Onion over Pepperoni! It’s a never-ending loop (A>B>C>A). Arrow’s impossibility theorem is the formal proof that every voting system has a potential flaw like this. There is no “perfect” way to vote. Every method (ranked-choice, majority rule, etc.) will always violate at least one seemingly fair and reasonable criterion. There is no perfect way to translate individual preferences into a collective will.

• The Median Voter Theorem: This explains why politicians often sound the same. Let’s imagine a scenario where there’s a 1-mile-long beach with sunbathers spread out perfectly evenly. Two competing ice cream carts (A and B) want to sell the most ice cream. Where do they set up?

• If A is at the 1/4-mile mark and B is at the 3/4-mile mark, they each get half the beach. But then A thinks, “If I move a little closer to the center, I’ll keep all my customers to the left, plus I’ll steal some of B’s.” B sees this and does the same. Where do they end up? Right next to each other, exactly at the 1/2-mile mark (the median). This is why political candidates often “run to the center” to capture the median voter, ignoring the people at the extremes.

Behavioral Economics: We Are Human, Not Robots

Conventional economics has long assumed we are perfectly rational, cold-blooded calculators of our own self-interest. Behavioral economics, drawing on psychology, shows we are nothing of the sort. We are human.

• We Are Not Always Rational: we use mental shortcuts (”heuristics”) that often lead us astray. We are overconfident. We fear losses more than we value equivalent gains (loss aversion). Think about it: the pain of losing $100 is far greater than the pleasure of finding $100. This isn’t rational, but it is deeply human.

• We Care Deeply About Fairness: the Ultimatum Game proves this. One person is given $100 and must offer a split to a second person. If the second accepts, both get the money. If they refuse, both get nothing. A purely rational “robot” would accept any offer, even $1, because it’s better than nothing. But humans routinely reject “unfair” offers (like $90 / $10). They are willing to pay a personal cost to punish what they perceive as injustice.

• We Are Inconsistent Over Time: Picture this: it’s Sunday night. You plan to wake up early all week to exercise. Monday morning arrives, and the alarm rings. In your warm bed, you decide, “I’ll start tomorrow.” You have a present bias—you overvalue the “now” at the expense of your long-term plan. This explains why we procrastinate, why we struggle to save for retirement, and why we indulge today while promising to diet tomorrow.

Economics is the study of a messy, beautiful, and often irrational human world. By understanding asymmetry in information, the inherent imperfections of our political systems, and the predictable quirks of our own minds, we move closer to designing systems—both in markets and in government—that are more robust, more fair, and more aligned with how we truly live and choose. The invisible hand is a powerful force, but it is guided by a human mind and that makes it very difficult to comprehend.

Taking the Pulse of the Economy: Measuring a Nation’s Income

• If microeconomics is the study of a single tree—its growth, its bark, its leaves—then macroeconomics is the study of the entire forest. It doesn’t care about one specific company’s sales or one household’s budget. It cares about the forces that affect all trees at once: the rainfall, the sunlight, the health of the soil. Its goal is to explain economy-wide phenomena like inflation, unemployment, and the rise and fall of total production.

• The single most important question in macroeconomics is a deceptively simple one: how is the economy doing? To answer that, we need a vital sign. A single number that can capture the immense, roaring complexity of a nation’s entire economic activity. That number is the Gross Domestic Product, or GDP.

• GDP measures two things at once: the total income of everyone in the economy and the total expenditure on the economy’s output of goods and services. GDP can perform the trick of measuring both total income and total expenditure because these two things are the same. For an economy as a whole, income must equal expenditure. An economy’s income is the same as its expenditure because every transaction has two parties: a buyer and a seller. Every dollar of spending by some buyer is a dollar of income for some seller. Imagine the economy as a giant, never-ending merry-go-round. On one side, households stand. They own the factors of production: their labor, their land, their capital. On the other side, firms stand, ready to produce goods and services.

• Now, watch the flow: households provide their labor and resources to firms. In return, firms pay them income—wages, rent, and profit. This is the flow of income. Armed with this income, households step into the marketplace (the grocery stores, the car dealerships, the app stores) and spend their money on the goods and services the firms produce. This is the flow of expenditure.

• Every dollar you spend at a store is a dollar of income for the cashier, the stocker, the manager, and the business owner. This is the profound, simple secret at the heart of GDP: for the economy as a whole, total income must always equal total expenditure. They are two sides of the same coin, the same flow of money viewed from opposite sides of the merry-go-round. GDP is designed to measure this single, unified flow.

• So – what is the precisely is the GDP? To put it in precise terms, Gross Domestic Product (GDP) is the market value of all final goods and services produced within a country in a given period of time. Let’s unpack this more precisely:

  • Market Value: GDP adds up the price of everything. This is how it compares “apples and oranges.” It doesn’t add the number of apples to the number of oranges; it adds the dollar value of the apples sold to the dollar value of the oranges sold. It converts everything into a common unit.

  • Final Goods and Services: This is crucial. GDP only counts the value of the end product to avoid double-counting. Think of a loaf of bread. We don’t add the value of the wheat sold to the miller, plus the value of the flour sold to the baker, plus the value of the bread sold to you. That would count the wheat three times! We only count the final price you pay for the bread.

  • Produced Within a Country: GDP is about location. It counts all production that happens within a country’s borders, regardless of who owns the company. A Toyota plant in Texas contributes to U.S. GDP. An Apple factory in China contributes to China’s GDP.

  • In a Given Period of Time: GDP is a flow, not a stock. It’s like your monthly salary, not your total life savings. We measure it over a specific period—usually a quarter or a year—to see if the flow is speeding up or slowing down.

• GDP is a powerful tool, but it’s not omniscient. It has blind spots:

  • The Underground Economy: The cash-only babysitter, the landscaper who does a side job—these transactions fuel the economy but often escape the official GDP count.

  • Home Production: Vegetables from the grocery store are in the GDP while vegetables from your garden are not. GDP values what is traded in the market, not what is produced for oneself. This is a significant omission when comparing living standards across time or cultures.

  • The “Past” is Past: GDP only counts new production. When you sell a used car to your neighbor, it’s just shifting an existing asset. No new car was produced, so it doesn’t add to GDP. The initial production of that car was counted the year it was made.

• The Four Engines of GDP: If total expenditure is one side of the GDP coin, who is doing all this spending? We break it down into four essential components, the four engines that power the economy:

  

  • Consumption (C): This is you and me. It’s spending by households on everything from groceries and movie tickets to haircuts and new phones. It’s the largest and most stable engine of the economy.

  • Investment (I): This is not buying stocks! In GDP terms, investment means spending on goods that will be used for future production. This includes:

    • Business Capital: A factory buying new machinery.

    • Residential Capital: A construction company building a new house or apartment building.

    • Inventories: The value of unsold goods sitting on shelves or in warehouses.

  • Government Purchases (G): Spending by all levels of government (local, state, federal) on goods and services. This includes teachers’ salaries, military equipment, and infrastructure projects. (Note: this does NOT include “transfer payments” like Social Security or unemployment benefits, as those are just redistributing existing income, not paying for a new service.)

  • Net Exports (XP): In our global economy, we must account for trade.

    • Exports (X) are goods we produce and sell to other countries. This is foreign spending on our output, so it adds to our GDP.

    • Imports (M) are goods we buy from other countries. This is our spending on their output, so it subtracts from our GDP.

    • Net Exports (XP) = Exports (X) - Imports (M)

• Imagine the economy produced exactly 100 loaves of bread last year and 100 loaves this year. Production is identical, but last year, bread cost $1 a loaf, and this year, due to inflation, it costs $2 a loaf. Nominal GDP uses current prices. So, last year’s Nominal GDP for bread was $100. This year, it’s $200. It looks like the economy doubled in size, but it didn’t! It’s an illusion caused by rising prices. Real GDP uses the prices from a fixed base year (say, last year’s prices). So, both last year and this year, Real GDP for bread is calculated as 100 loaves * $1 = $100. Real GDP shows that our actual output—the quantity of bread—didn’t change.

  

• Real GDP strips out the effect of inflation and tells us if we are actually producing more “stuff.” It is the true measure of economic growth. The GDP Deflator (which is Nominal GDP / Real GDP) is the gauge that tells us how much of the change was due to pure inflation.

• GDP is a good measure of economic size and production. It is highly correlated with our standard of living. People generally prefer higher incomes to lower ones, and a higher GDP per person typically means a nation can afford better food, healthcare, and education.

• GDP does nothing to measure happiness or well-being though. It places no value on:

  • Leisure: A society where everyone works 80-hour weeks will have a higher GDP than a equally productive one with 30-hour weeks, but its people may be far less happy.

  • Environmental Quality: A factory that pollutes a river adds to GDP. The cost of cleaning the river also adds to GDP. But the loss of a clean river itself is never subtracted.

  • Equality: GDP can be rising rapidly even if all the gains are going to a tiny fraction of the population.

  • Non-Market Activity: The value of a parent staying home to raise a child is immense, but it is entirely absent from GDP.

• In the end, GDP is like a car’s speedometer. It is an essential, real-time instrument that tells you how fast you’re going. It is utterly indispensable for managing the journey, but it doesn’t tell you if you’re heading in the right direction, if the passengers are happy, or what the scenic view is like outside the window, so it gives you a very limited view of reality and what goes on in the real world.

  

Measuring the Cost of Living

• We want a single number that tells us “how expensive life feels” this year compared with some earlier year. That number is an index. We’ll build it carefully, see what it misses, and learn how to use it to translate dollars from one time into another—so we compare purchasing power, not just ink on paper.

• The Consumer Price Index (CPI) answers: how much would it cost today to buy the same typical basket of goods and services a consumer bought in the base year? It reports that cost as an index number (base year = 100).

  • If the CPI is 118, the same basket that cost $100 in the base year now costs $118.

  • The inflation rate between two periods is the percent change in the CPI.

• How is CPI measured though? Government’s economists conduct a massive, ongoing thought experiment:

  • They define a “basket” of goods and services that represents what a typical urban household buys in a typical year. This basket isn’t random; it’s a precise recipe based on thousands of consumer surveys. It contains specific quantities of milk, bread, gasoline, apparel, medical care, and even streaming services. This is the CPI Market Basket.

    

  • Once defined, this basket is kept fixed. This is the “base year” basket.

  • Each month, economic assistants literally go out to stores across the country and record the prices of every item in that fixed basket.

  • The Consumer Price Index (CPI) is the cost of buying that fixed basket in the current year, compared to the cost of buying the exact same basket in the base year.

• The formula is simple:

  

• The percentage change in the CPI from one period to the next is the inflation rate. If the CPI rises from 120 to 123 over a year, the inflation rate is (123-120)/120 = 2.5%.

  

• The CPI is a brilliant tool, but it’s not perfect. By holding the shopping cart fixed, it misses three key ways that real people adapt, causing it to systematically overstate the true rise in the cost of living:

  • The Substitution Bias: Imagine the fixed basket has both apples and oranges. One year, a frost wipes out the orange harvest, and the price of oranges doubles. What do you do? As a smart shopper, you probably buy more apples and fewer oranges. The CPI doesn’t do this. It blindly continues to buy the same fixed quantity of expensive oranges. It assumes you’re a robot who doesn’t substitute toward cheaper alternatives. In reality, your cost of living didn’t go up as much as the CPI says, because you adapted. The CPI misses this.

  • The Introduction of New Goods: Think back to 1995. The CPI basket included typewriters, cassette tapes, and film for your camera. Then, new goods arrived: personal computers, MP3 players, and digital cameras. These new products created a massive increase in consumer choice and value. You could now do things that were previously impossible, or do them far more cheaply. The purchasing power of your dollar increased dramatically. But the CPI, with its fixed basket from 1995, completely missed this explosion of new value and choice when it first happened. It only slowly incorporates new goods, failing to account for the increase in economic welfare they provide from day one.

  • Unmeasured Quality Change: Is a 2024 car the same “good” as a 2004 car? Not really. The 2024 model likely has standard backup cameras, Bluetooth, and advanced safety features. Even if the sticker price increased only with inflation, you are getting a significantly better product. The CPI tries to adjust for quality, but it’s incredibly difficult. When it fails to fully account for this improvement, it records a pure price increase when, in fact, you are getting more for your money. Your true cost of living—the cost per unit of quality—hasn’t risen as much.

• We already met another price gauge: the GDP Deflator. While they often tell a similar story, they are built for different purposes. Think of it this way: the CPI is the Cost of Living Index. It asks: “how much does it cost to buy a typical consumer’s basket of goods, wherever they were produced?” It cares about what you consume. The GDP Deflator is the Domestic Production Price Index. It asks: “how have the prices of everything we produce here changed?” It cares about what we make.

• This principle of adjusting for inflation is called indexation, and it’s baked into our economic life:

  • Social Security: Your grandmother’s Social Security benefits are indexed to the CPI. They automatically increase each year to maintain her purchasing power.

  • Labor Contracts: Many union contracts have Cost-of-Living Adjustments (COLAs) tied to the CPI.

  • Alimony and Child Support: Many agreements are indexed to inflation.

• A correction for inflation is especially important when looking at data on interest rates. The nominal interest rate is the interest rate usually reported; it is the rate at which the number of dollars in a savings account increases over time. By contrast, the real interest rate takes into account changes in the value of the dollar over time. The real interest rate equals the nominal interest rate minus the rate of inflation.

  

  

• Imagine you deposit $100 at a 5% nominal rate. In a year, you have $105. If inflation was 2%, your real return is 5% - 2% = 3%. Your $105 can buy 3% more stuff than your original $100 could, but if inflation was 6%, your real return is 5% - 6% = -1%. You have more dollars, but you can actually buy less. Your savings have been eroded.

• This negative real interest rate is the “silent thief” at work, stealthily transferring wealth from savers to borrowers. Understanding the difference between nominal and real is the key to making any intelligent decision about borrowing, lending, or saving. It is the final, and perhaps most personal, reason why measuring the cost of living matters to us all.

Production and Growth

• The differences in economic prosperity around the world are not just significant; they are staggering. The average income in a country like Switzerland or Norway is more than ten times that of a country like Malawi or Sierra Leone. To visualize this, imagine the GDP per person (average income) as the height of a pillar. The pillar for a rich country might be a towering skyscraper while the pillar for a poor country might be a single-story house. Here is the crucial and dynamic part though: these heights are not fixed.

• Because growth rates vary, the landscape is constantly shifting. A country growing at 7% per year will double its standard of living in just 10 years. Another, growing at a mere 1%, would take 70 years to double. This is the power of compound growth—the same force that governs your savings account, but applied to the fate of entire nations. Over time, relative positions can change dramatically, as we’ve seen with the meteoric rise of economies like South Korea and China.

  

• So, what ultimately determines the height of a nation’s pillar of prosperity? It is deceptively simple: a country’s standard of living depends on its ability to produce goods and services.

• We can’t consume what we don’t produce. And our ability to produce is determined by one thing: productivity. Productivity is the quantity of goods and services a worker can produce for each hour of work. It is the efficiency of the economic engine. Think of a kitchen:

  • A low-productivity kitchen has one person trying to cook a complex meal alone. Progress is slow.

  • A high-productivity kitchen is a professional restaurant kitchen, with a coordinated team where each worker produces a large quantity of high-quality food per hour.

• The difference in output is immense. And that difference stems from the tools available to the workers. Economists group these tools into four categories:

  • Physical Capital (Tools and Equipment): This is the stock of equipment and structures used to produce goods and services. For the kitchen, it’s the ovens, the pots and pans, and the building itself. For a country, it’s the factories, the tractors, the roads, the fiber-optic cables. More and better capital allows workers to produce more in the same amount of time.

  • Human Capital (Chef’s Knowledge): This is the knowledge and skills workers acquire through education, training, and experience. A chef with a culinary degree and years of practice (high human capital) is infinitely more productive than a novice (low human capital). A country’s investment in its people’s education and health is an investment in human capital.

  • Natural Resources (The Ingredients): These are the inputs from nature—land, rivers, minerals, oil—used to produce goods and services. Some countries, like Saudi Arabia with its oil or Canada with its lumber, are rich in natural resources. However, they are not strictly necessary for prosperity (see: Japan, Singapore), as a lack can be overcome with trade and ingenuity.

  • Technological Knowledge (Recipes): This is the most important driver. It’s society’s understanding of the best ways to produce goods and services. The shift from hand-milling grain to using a water wheel was a technological revolution. The invention of the microchip, the internet, or a new vaccine are all explosions of technological knowledge that redefine what is possible.

  • A nation’s productivity, and therefore its prosperity, is a function of how well it equips its workers with Physical Capital, Human Capital, Natural Resources, and Technological Knowledge.

  

• If these are the ingredients for growth, what can a society do to acquire more of them? Government policy can pull several powerful levers:

  • Encourage Saving and Investment: Saving is the fuel for investment in physical capital. If a nation consumes everything it produces, it has nothing left to build new factories or infrastructure. Policies that incentivize saving can, in turn, fund the investment needed for future growth.

  • Encourage Investment from Abroad: A country can supplement its own saving by attracting foreign investment, bringing in foreign capital and expertise.

  • Foster Education and Promote Health: This is direct investment in human capital. A healthy, educated workforce is an innovative and productive one.

  • Maintain Property Rights and Political Stability: This is the foundation. Would you build a bakery if you thought a mob or the government itself could take it from you tomorrow? Probably not. Secure property rights are the bedrock upon which all investment and innovation are built.

  • Promote Free Trade: Trade is a form of technology. It allows a country to specialize in what it does best and import what others do better, effectively giving its citizens access to a wider array of goods and more efficient production methods.

  • Promote Research and Development (R&D): Technological knowledge doesn’t just fall from the sky. It is the product of deliberate effort. Direct funding for research, and tax incentives for private R&D, are investments in the “recipes” of the future.

• The effects of population growth (a double edged sword):

  • The Dilution Effect (The Negative): Imagine dividing the nation’s capital stock among its workers. Rapid population growth means more workers, so the amount of capital per worker—a key determinant of productivity—falls. It “stretches” the available capital and natural resources more thinly. This is known as capital dilution and can lower the standard of living.

  • The Innovation Effect (The Positive): A larger population means more minds. More potential scientists, engineers, entrepreneurs, and inventors. In this view, people aren’t just mouths to feed; they are problem-solvers. A larger, global population has arguably contributed to an accelerating rate of technological progress over human history.

  • The net effect of population growth is ambiguous and depends on which of these two forces—dilution of resources or acceleration of ideas—is stronger.

• In the short run, we are captivated by the business cycle—the booms and busts that dominate the headlines. But in the long run, almost nothing matters more than the steady, cumulative power of economic growth. A difference of a single percentage point in the growth rate, sustained over a generation, is the difference between stagnation and transformation. Understanding the drivers of productivity is therefore not just an academic exercise. It is the key to unlocking the potential of nations and lifting the pillars of prosperity for all.

Saving, Investment, and the Financial System

• The U.S. financial system is made up of many types of financial institutions, such as the bond market, the stock market, banks, and mutual funds. All these institutions act to direct the resources of households that want to save some of their income into the hands of households and firms that want to borrow.

• All these institutions perform the same essential function: they direct the resources of those who want to save (defer consumption) into the hands of those who want to invest (build capital for future production).

• Imagine the entire economy is a giant farm. The year’s harvest is the nation’s total output of goods and services (GDP). Once the harvest is in, the farm family faces a critical choice:

  • They can consume the grain immediately, eating their fill.

  • They can save a portion of the harvest, setting it aside in the barn.

• But here’s the crucial idea: in economics, “saving” isn’t just letting grain gather dust in the barn. That would be hoarding, and it’s wasteful. True economic saving is the act of setting aside real resources to be used for future production. So, what do we with the saved grain?

  • We can use it as seed corn to plant next year’s crop (business investment).

  • We can use it to build a new, more efficient irrigation system (infrastructure investment).

  • We can use it to feed workers while they build a new barn (construction investment).

• Different financial institutions play different roles in this farm economy:

  • Banks are like local grain elevators or co-ops. Farm families bring their extra grain and store it there (deposits). The elevator then lends grain out to farmers who need seed or to buy better tools (loans). Borrowers agree to return more grain in the future than they borrowed (interest). Savers earn a little extra grain over time in return for letting others use their stored harvest.

  • The bond market is like a large town leader announcing: “If you each lend me some grain now, I promise to repay you later, plus extra.”
    That promise is a bond. Many households each send a few sacks of grain to finance a big project—a new irrigation canal, a flour mill, or a road. Later, the project repays those households with their grain plus additional sacks (interest).

  • The stock market is like a group of families pooling grain to build and co-own a new grain mill or bakery. When you buy stock, you are not just lending grain; you are trading grain for a slice of ownership in the business itself. If the mill does well and produces lots of flour and profits, your share is worth more and may pay you part of the earnings. If the mill does poorly, the value of your share falls.

  • Mutual funds are like a grain investment club. Many small families pour part of their harvest into one big shared granary. A trusted manager then uses that pooled grain to back many different ventures: some mills, some orchards, some roads (a mix of stocks and bonds). Even families with only a few extra sacks of grain can, through the club, participate in a wide range of investments and share in the gains and losses.

• In this closed farm economy, there is no other use for the saved grain. Every bushel of grain that is not consumed must, by definition, be invested back into the farm. There is no magical third option. This leads to the profound, unbreakable accounting identity for a closed economy:

  

• It is not a theory or a policy goal; it is a mathematical and logical truth. The real resources we refrain from consuming today are the very same resources we use to build our future capacity. You can think of the financial system as the grain elevator and logistics network. It doesn’t create the grain (real resources), but it efficiently moves the grain from those who have saved it (savers) to those who need it for productive projects (borrowers/investors). It ensures that the seed corn gets from the barn to the field, rather than being left to rot.

• A society’s consumption is limited by what it produces. And its future production is built entirely upon what it saves and invests from today’s output. The choice between consumption and saving is, therefore, the most fundamental choice an economy makes about its own future.

• How is this matching of savers and borrowers coordinated? Through a market, just like any other. But instead of apples or cars, the commodity being traded is loanable funds—the money available for borrowing. The price in this market is the real interest rate.

  • Supply of Loanable Funds: This comes from savers. The higher the real interest rate, the greater the reward for saving, so the more funds people are willing to supply. (The supply curve slopes upward).

  • Demand for Loanable Funds: This comes from borrowers (mainly firms wanting to invest in new projects). The higher the real interest rate, the more expensive it is to borrow, so the less they will demand. (The demand curve slopes downward). The market-clearing interest rate is where the supply of saving meets the demand for investment.

    

• Any policy or event can be analyzed by how it shifts these curves. A tax incentive for saving? Supply shifts right, leading to more funds and a lower interest rate. A wave of technological optimism among firms? Demand shifts right, leading to more borrowing and a higher interest rate.

• National saving has two components:

  

  • Private Saving is the income that households have left after taxes and consumption.

  • Public Saving is the tax revenue the government has left after paying for its spending. If the government spends more than it collects (a budget deficit), then public saving is negative.

• National saving equals private saving plus public saving. A government budget deficit represents negative public saving and, therefore, reduces national saving and the supply of loanable funds available to finance investment. When a government budget deficit crowds out investment, it reduces the growth of productivity and GDP.

  

  To provide a clearer picture – let’s go back to the farming analogy we used previously. Let’s stick with our farm, but now imagine it’s a communal farm with two distinct groups:

  • The Households (Private Sector): All the families who work the land.

  • The Town Council (Government): The group that manages shared services like irrigation canals and security walls, funded by a portion of the harvest (taxes).

• At harvest time, the total grain (national saving) is stored in two silos:

  • The Private Silo (private saving): This is the grain the households have left over after giving the council its share (taxes) and after feeding their families (consumption). They plan to use this for their own future projects—planting new fields, buying better tools, or building a new shed.

  • The Public Silo (public saving): This is the grain the Town Council has left over after paying for its ongoing services (government spending). If the council runs a surplus, this silo is full. This grain can be used for major communal projects that benefit everyone, like building a new granary or a road to market.

• How deficits work: a deficit means the Town Council has not only spent all its tax grain but needs even more to fund its projects. To get this grain, the council goes to the communal grain bank (the financial system) and borrows grain from the private silo. It issues an IOU (a government bond) in return.

  

• This is where the crucial shift happens. The council’s borrowing doesn’t just empty the public silo; it actively draws down the grain from the Private Silo. In other words – in this instance, the communal pile of seed grain for the entire farm has SHRUNK. Now, let’s see the consequence:

  • A young, ambitious farmer arrives at the grain bank. He has a brilliant plan to cultivate a rocky but fertile hillside. His calculations show it will be profitable if she can borrow seed grain at a low cost.

  • But he finds a line. The Town Council is there, borrowing a massive amount of grain to fund a grand but less efficient monument. Because of the council’s huge demand, the price for borrowing grain (the real interest rate) has shot up.

  • The bank tells him: “We can lend you the seed, but the cost to borrow is now much higher.”

  • He redoes her math. At this new and higher cost, his hillside project is no longer profitable. He walks away, and the rocky field remains barren.

• The government’s deficit spending didn’t just use up resources; it made those resources more expensive for everyone else. By borrowing heavily, the town council crowded out the private farmer’s productive investment. To summarize once again:

  • The limited resource is the real, physical seed grain (loanable funds).

  • The government’s demand for that resource drives up its price (the real interest rate).

  • The result is that the most innovative, growth-oriented private projects are the first to be abandoned.

• The tragedy is invisible. We see the council’s monument being built. We do not see the more productive fields that were never planted, the new tools that were never forged, or the research into better crops that was never funded. The economy’s long-term growth potential is silently diminished, all because the communal pool of savings was diverted from its most productive uses.

  

• Remember, investment is the process of building the capital stock that determines our future productivity. When investment is crowded out, we build fewer factories, develop fewer technologies, and train fewer workers. This reduces the growth of productivity and, therefore, the growth of GDP.

• A government budget deficit, in essence, is a decision to prioritize current consumption (through government spending) over the investments that would enrich the future. It is a choice that, while sometimes necessary in the short term, can slowly starve the economic engine of the fuel it needs for the long journey ahead.

• The financial system, therefore, is not just a passive plumbing system. It is the active, vital link between our present sacrifices and our future wealth. How we manage it—and how we manage our government’s role within it—determines the health and power of our economic body for generations to come.

The Basic Tools of Finance

• We’ve seen how the financial system acts as the economy’s circulatory system, channeling savings toward productive investments. But this raises a personal and practical question: How should an individual navigate this system? How do we decide how much to save, where to invest, and how to think about the inevitable risks?

• It is said that Albert Einstein called compound interest the most powerful force in the universe. Whether he did or not, the sentiment is correct. This force creates a fundamental truth: A sum of money today is worth more than the same sum in the future. Why? Because money today can be put to work earning interest.

  

• This leads to the most crucial concept in finance: Present Value (PV). The present value of a future sum of money is the amount you would need to set aside today to have that sum in the future, given the prevailing interest rate.

• Imagine someone offers you a choice: $100 today or $100 in one year. The rational choice is $100 today. You could put it in a savings account (even at a low interest rate) and have more than $100 in a year. If the interest rate is 5%, then $100 today is worth $105 in one year. Conversely, $100 in one year is worth less than $100 today. In fact, its present value is about $95.24 today ($100 / 1.05). You would only need to invest $95.24 now to have $100 later.

  

• This discounting of the future is the essence of present value. It’s the financial equivalent of a telescope that brings future cash flows into focus in today’s terms, allowing us to compare apples (money now) with oranges (money later) by converting them both into “present-value apples.”

• Imagine you are offered a gamble. A coin flip: heads, you win $1,000; tails, you win $0. The “expected value” is $500. Now, imagine you are offered a sure-thing payment of $450 instead. Would you take the sure thing? Most people would. This is risk aversion.

• Why? It’s due to a concept we’ve met before: diminishing marginal utility. The first dollar you earn is incredibly valuable—it buys food and shelter. The thousandth dollar is less so—it might buy a nicer TV. The potential joy of winning an extra $1,000 is less than the potential pain of losing the $450 you were counting on. Because we are generally risk-averse, we are willing to pay a price to reduce risk. This is the basis for three key strategies:

  • Insurance: You pay a premium to an insurance company to avoid a potentially catastrophic loss. You are trading a small, certain loss (the premium) to avoid a small chance of a huge, life-altering loss. It’s a financial airbag.

  • Diversification: The age-old wisdom of “don’t put all your eggs in one basket.” By spreading your investments across many different assets (different companies, different industries, even different countries), you eliminate the unique, or “idiosyncratic,” risk of any one of them failing. If one company goes bankrupt, it’s only a small part of your portfolio. A diversified portfolio is like a well-balanced fleet of ships—if one sinks, the journey continues.

  • Risk-Return Tradeoff: Safer assets (like government bonds) typically offer lower potential returns. Riskier assets (like stocks in a new tech company) must offer the potential for higher returns to attract risk-averse investors. You can choose your spot on this spectrum.

• How much should you pay for a share of a company? A bond? A rental property The answer is always the same principle: the value of any asset is the present value of all the future cash flows you expect to receive from it. For a share of stock, the cash flows are the stream of future dividends and the final sale price. You are essentially buying the right to a slice of the company’s future profits.

• This leads us to one of the most debated ideas in economics: The Efficient Markets Hypothesis (EMH). This theory states that financial markets are “informationally efficient.” At any moment, stock prices reflect all publicly available information about a company’s value. The current price is the best possible estimate of its true worth. Why? Because if a stock were undervalued, rational investors would immediately swoop in and buy it, driving the price up to its correct level. The same happens in reverse for an overvalued stock. This creates a “random walk”—since news is by definition unpredictable, the path of stock prices should be unpredictable too.

• A lot of people today believe that trying to “beat the market” by picking stocks is a fool’s errand. It’s like trying to guess the outcome of a coin flip that has already been flipped but not yet revealed. The market is like a lightning-fast, supremely rational weather vane, instantly pointing in the direction of the economic wind.

• The Efficient Markets Hypothesis is a beautiful, elegant theory, but it has one major flaw: it assumes everyone is a perfectly rational (and this definitely is not the case). The field of behavioral finance has emerged to document the ways human psychology throws a wrench into this rational machine. As an example:

  • We think we’re better at picking stocks than we are (we suffer from overconfidence).

  • We get caught up in bubbles, buying simply because everyone else is buying, driving prices far above any rational present value (herd behavior).

  • We fixate on the price we paid for a stock, refusing to sell at a loss even when the fundamentals have clearly deteriorated.

• These psychological factors can cause asset prices to deviate from their fundamental value for long periods, creating the booms and busts that the efficient market hypothesis struggles to explain.

Unemployment

• Each month, the U.S. government takes a statistical snapshot of the labor market. They survey thousands of households and sort the adult population into three buckets:

1. Employed.

2. Unemployed: People who do not have a job but are actively looking for one.

3. Not in the Labor Force: People who are not working and not looking, such as retirees, students, or stay-at-home parents.

• The unemployment rate is then calculated as:

  

• This number is our best single gauge, but it has blind spots, like a speedometer that doesn’t account for wind resistance. What the Unemployment Rate Misses:

  • The Discouraged Worker: Imagine someone who has searched for a job for months, found nothing, and finally gives up. By stopping their active search, they magically vanish from the unemployment statistics, classified as “not in the labor force.” This person is still jobless, but the official rate no longer sees them. This means the unemployment rate can understate true joblessness.

  • The Fake Job Seeker: Conversely, someone might claim to be looking for work to qualify for government benefits while having no real intention of taking a job. This can cause the rate to slightly overstate the problem.

• A surprising feature of unemployment is its distribution. Think of a hospital.

  • Most people who become unemployed are like patients in the emergency room. They have a short, sharp spell of joblessness—perhaps a few weeks—before they find a new position and are discharged. The vast majority of spells of unemployment are short.

  • However, the people you see in the hospital on any given day are mostly in the long-term care ward. A small group of people who are unemployed for six months or more make up a large share of the total unemployment observed at any one time.

• So, while most people experience short-term unemployment, the social problem of unemployment is dominated by the long-term unemployed.

• Reasons for unemployment:

  • It takes time for workers to search for jobs that best suit their tastes and skills. This frictional unemployment is increased as a result of unemployment insurance, a government policy designed to protect workers’ incomes.

  • An economy always has some unemployment due to minimum-wage laws. By raising the wage of unskilled and inexperienced workers above the equilibrium level, minimum-wage laws raise the quantity of labor supplied and reduce the quantity demanded. The resulting surplus of labor represents unemployment.

  • The market power of unions. Unions can negotiate wages for their members above the equilibrium level. This is like an exclusive guild for chair-makers securing a high price for its members. The result is the same: a surplus of labor (unemployment) for that sector, as the number of people who want these good union jobs exceeds the number of positions available.

  • Theory of efficiency wages: according to this theory, firms find it profitable to pay wages above the equilibrium level. High wages can improve worker health, lower worker turnover, raise worker quality, and increase worker effort. When many firms adopt this strategy, they create a situation where the wages being offered are above the level that would clear the labor market. The result, once again, is a surplus of labor—people who want these high-paying jobs but can’t get them.

• Unemployment is not a simple problem with a silver-bullet solution. Although the economy will always have some unemployment, its natural rate does change over time. Many events and policies can alter the amount of unemployment the economy typically experiences.

• As the information revolution changes the process of job search, as Congress and state legislatures adjust the minimum wage, as workers form or quit unions, and as firms change their reliance on efficiency wages, the natural rate of unemployment evolves. Unemployment is not a simple problem with a simple solution. How we choose to organize our society can profoundly influence how prevalent a problem it is.

  

The Monetary System

• Our journey through the economic city has shown us how goods and services flow through markets, how savings are channeled to investments, and how people interact with the labor market. But there is one element so pervasive we often take it for granted that makes all of this possible: money.

• Money is not just the paper in your wallet; it is the lifeblood of the economic body, the river that carries all transactions from source to sea. In this chapter, we will explore what this river is made of, who controls its flow, and why that control is one of the most powerful forces in economics.

• Imagine a simple barter economy, where you are a shoemaker in need of bread. You must find a baker who not only has bread but who also needs a new pair of shoes. This “double coincidence of wants” is incredibly inefficient. The economy grinds along at a snail’s pace. Money the solution to this problem and is the universally accepted medium that breaks this logjam. Formally, money is anything that regularly serves three functions:

  • A Medium of Exchange: it is the thing buyers give to sellers in a transaction. You give the baker money, not shoes, and receive your loaf of bread. It is the oil that lubricates the gears of commerce.

  • A Unit of Account: it is what we use to measure economic value. We don’t say a car is worth 5,000 loaves of bread or 300 pairs of shoes. We say it’s worth $30,000. This common denominator simplifies decision-making immensely.

  • A Store of Value: it is a way to transport purchasing power from the present into the future. You can sell your labor today, receive money, and spend it weeks or months later.

• Throughout history, money has taken many forms, but they all fall into two categories:

  • Fiat Money: This is money without intrinsic value. The paper used for a $100 bill is worth almost nothing. Its value comes from a government decree (a “fiat”) that establishes it as legal tender. But a decree alone is not enough. The value of fiat money ultimately rests on a delicate foundation of social consensus and trust. We all accept it because we are confident that everyone else will accept it. It is a shared fiction, but one with very real power.

  • Commodity Money: This is money that has intrinsic value—value in its own right. The classic example is gold. A gold coin is valuable not just because it can be traded, but because gold is used in jewelry and electronics. It would be valued even if it weren’t money.

• If money is the river, how do we measure how much water is flowing? The “money supply” (or money stock) isn’t just the currency in circulation. Think of it in layers of liquidity:

  • The Core (M1): This is the money you can spend instantly. It includes physical currency (coins and bills) plus the balances in checking accounts and debit cards. This is the water currently flowing in the river’s main channel.

  • The Broader Measure (M2): This includes all of M1, plus “near-money”—assets that are very easy to convert into spendable cash. This includes savings accounts, money market funds, and small-time deposits. These are like the water in sloughs and marshes adjacent to the main river—not immediately flowing, but part of the system and easily able to join the current.

    

• The modern money supply is predominantly digital—numbers in bank accounts—with physical cash as just the visible tip of the iceberg. To prevent the river of money from flooding or drying up entirely, we need engineers. In the United States, that engineer is the Federal Reserve (”the Fed”), the nation’s central bank.

• The Federal Reserve is responsible for regulating the U.S. monetary system. The Fed chair is appointed by the president and confirmed by Congress every 4 years. The chair is the head of the Federal Open Market Committee, which meets about every 6 weeks to consider changes in monetary policy.

• Established in 1913 after a panic exposed the fragility of the banking system, the Fed has two primary jobs:

  • The Fed is the ultimate guardian of the banking system:

    • It acts as a lender of last resort. If a bank faces a sudden crisis and can’t borrow from anyone else, it can turn to the Fed for an emergency loan. This stops a single bank’s failure from turning into a system-wide panic. It’s the financial system’s emergency flood control.

    • It regulates banks, ensuring they have enough capital (a cushion of their own money) to absorb losses. This is like enforcing building codes along a riverbank to prevent collapses.

  • Controlling the money supply (its biggest role): By controlling the amount of money in the economy, it tries to maintain stable prices and maximum employment. The main tool for this is open-market operations. Imagine the Fed has a magical spigot connected to the river:

    • To increase money supply (expansionary policy), the Fed creates new money out of thin air (electronically, of course) and uses it to buy government bonds from the public. The people who sold the bonds now have new cash in their hands that didn’t exist before. The river level rises.

    • To decrease the money supply (contractionary policy), the Fed sells government bonds from its own portfolio to the public. The public pays for these bonds with cash, and when the Fed receives this cash, it effectively vanishes from the system. The river level falls.

• Banks aren’t passive reservoirs; they are active participants in creating money through lending. When you deposit $1,000 in a bank, the bank doesn’t lock it all away. It keeps a small fraction (the reserve requirement) and loans out the rest, say $900. That $900 gets spent and eventually deposited in another bank, which then loans out $810, and so on. Your initial $1,000 deposit can ultimately create many thousands of dollars in new money throughout the banking system. This is the money multiplier. It means the Fed’s control, while powerful, is not perfectly precise, as it depends on the lending behavior of thousands of individual banks.

• In recent decades, the Fed has shifted from directly targeting the money supply to targeting a single interest rate: the federal funds rate. This is the rate banks charge each other for overnight loans. The Fed announces a target for this rate and then uses its open-market operations (the spigot) to hit that target. A lower federal funds rate makes borrowing cheaper, encouraging spending and investment (loosening the spigot). A higher rate does the opposite (tightening the spigot).

  

• The monetary system, therefore, is a magnificent and complex human invention. It transformed a barter-based alleyway into a global metropolis. It is built on a foundation of trust—trust in the paper we call currency, trust in the numbers in our bank accounts, and trust in the engineers at the Federal Reserve to manage the flow. When that trust is well-placed, the river flows steadily, nourishing the economic landscape. When that trust is broken, the river can either flood, destroying the value of money through inflation, or run dry, causing a credit crunch and recession.

Money Growth and Inflation

• We have met the engineers of the monetary system, the Federal Reserve, and learned how they control the great river of money. Now we must confront the single greatest consequence of their actions, and one of the most pervasive forces in economics: inflation. Inflation is often called an “invisible tax” because it quietly erodes the purchasing power of every dollar in your pocket. But where does it come from? The answer, at its heart, is deceptively simple, yet its ramifications are profound.

• Imagine an economy that produces a fixed amount of goods and services—let’s say, 100 unique artifacts. Now, imagine there is exactly $1,000 in existence. Basic supply and demand would dictate that the price of each artifact would hover around $10. Now, suppose a helicopter flies over and drops another $1,000, doubling the money supply to $2,000. The number of artifacts is still 100. What happens? People have more money to spend, but the same number of items are available for purchase. The inevitable result is that the price of each artifact rises to about $20.

• This is the core of the quantity theory of money. In the long run, the overall level of prices in an economy adjusts to bring money supply and money demand into balance. When the central bank increases the supply of money much faster than the economy’s capacity to produce goods and services, the result is inflation. It’s not that artifacts become more valuable; it’s that each dollar becomes less valuable.

• The principle is clear: prices rise when the government prints too much money.

• If inflation is so easy to explain, why would any country ever allow it to spiral into hyperinflation—a period of extremely high and typically accelerating inflation that quickly erodes the real value of the local currency? The answer is one of desperation. Governments need to pay for things: roads, soldiers, social programs. They have three ways to do this:

  • 1) Tax the public.

  • 2) Borrow from the public by selling bonds.

  • 3) Print the money they need.

• When a government is weak, its tax system inefficient, and its ability to borrow exhausted, it turns to the printing press. It is the path of least resistance in the short term, but a road to ruin in the long term. This is known as the inflation tax. By printing money to finance its spending, the government is effectively imposing a tax on everyone who holds money, because the value of their money is being diluted. In a hyperinflation, this tax becomes catastrophic, as people rush to spend their paychecks within hours before they become worthless.

The Fisher Effect: Why Inflation Drives Up Interest Rates

• We’ve already met the crucial distinction between the nominal interest rate (the number quoted by the bank) and the real interest rate (the true return after inflation):

  

• Now, let’s connect this to the money supply. The supply and demand for loanable funds determine the real interest rate. Meanwhile, the growth in the money supply determines the inflation rate.

• In the long run, money is neutral—it doesn’t affect real variables like the real interest rate. So, if the Fed increases the money supply and causes inflation to rise, what happens to the nominal interest rate? It must adjust upward to keep the real interest rate unchanged. If inflation is 5%, a lender will demand a nominal rate of, say, 7% to achieve a real return of 2%. If inflation jumps to 10%, the lender will now demand a nominal rate of 12% to get that same 2% real return. This one-for-one adjustment of the nominal interest rate to the inflation rate is known as the Fisher Effect. It’s the financial market’s immune response to the virus of inflation, protecting the real return on lending.

• Many people think, “So what if prices go up? My paycheck goes up too.” While this can be true in the aggregate, inflation imposes real and insidious costs on society, even when it’s moderate.

  • Shoe-Leather Costs: this is the time, effort, and resources wasted when people try to avoid holding onto cash. In high inflation, you sprint to the bank more often and rush to spend money before it loses value. You’re literally wearing out your shoes (and your patience) to minimize your holdings of a depreciating asset.

  • Menu Costs: this is the cost of changing listed prices. Restaurants must reprint menus, catalogs must be republished, and stores must re-tag merchandise. In a digital age, this includes the cost of reprogramming software and updating websites. It’s a recurring tax on every business.

  • Relative-Price Variability and Misallocation of Resources: in a stable economy, a price change signals a real shift in supply or demand. In an inflationary economy, prices are changing everywhere, making it hard to distinguish a relative price change (this product is now in higher demand) from the general noise of inflation. This confuses the “invisible hand,” leading to poor decisions by consumers and businesses.

  • Tax Distortions: the tax code is often not fully indexed for inflation. If your investment earns a 7% nominal return in a year with 5% inflation, your real return is only 2%. But you are taxed on the full 7% nominal gain, effectively taxing you on phantom income. This discourages saving and investment.

  • Confusion and Inconvenience: money is our unit of account, our economic ruler. Inflation is like making that ruler stretch and shrink randomly. It becomes harder for individuals to plan for retirement or for businesses to write long-term contracts, gumming up the wheels of commerce.

  • Arbitrary Redistributions of Wealth: inflation acts as a wildcard, arbitrarily redistributing wealth between borrowers and lenders.

    • If you take out a fixed-rate mortgage and inflation surges, you get to pay back your loan with dollars that are worth much less than you borrowed. You win; the bank loses.

    • Conversely, if inflation falls to zero, the real burden of your debt is higher than you expected. The bank wins; you lose.

• These redistributions are not based on merit, effort, or productivity. They are based on luck and the inability to perfectly predict the future.

• While a low, stable rate of inflation may be a lubricant for the economy, high or volatile inflation is a corrosive acid. It erodes the value of our savings, distorts the signals of the price system, and forces us to waste resources protecting ourselves from our own currency.

• The fight against inflation is, therefore, not just a technical matter for central bankers. It is a fundamental commitment to preserving the integrity of our economic language—the dollar—and ensuring that the hard work and savings of today are not stolen by the silent thief of tomorrow. It is the ultimate test of our stewardship over the great river of money.

  

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  This ends Part 3 of my write-up. This is an ad-free publication and I extend a great thank you to the paid subscribers of this blog! If you liked this write-up, please like an subscribe. Most of the content here is taken from a great book called Principles of Economics by Gregory Mankiw which I highly recommend.

The Costs of Production

• The goal of firms is to maximize profit, which equals total revenue minus total cost.

• Let’s start off by taking a look at the production function for a cookie factory:

  

• This graph shows how adding more workers changes the amount of stuff (output) a business can make.

  • The horizontal axis (sideways) shows how many workers are hired.

  • The vertical axis (up and down) shows how much output they produce.

• In economics, the marginal product is the additional output produced by adding one more unit of a specific input, such as labor or capital, while all other inputs are held constant. The graph shows that as you hire more workers, production rises quickly, but as more and more workers are added, the graph becomes flatter — meaning each new worker adds less extra output than the one before.

• In the above production function diagram as an example, the second worker has a marginal product of 40 cookies, the third worker has a marginal product of 30 cookies, and the fourth worker has a marginal product of 20 cookies. We should be able to clearly see the effect that the marginal product has: the slope of the production curve starts to flatten out. Each worker still increases total output, but by a smaller and smaller amount, so the curve bends gently instead of shooting straight up. That’s called diminishing marginal product.

• Now, let’s take a look at the total cost curve for the same factory:

  

• This graph shows how the total cost of production changes as the business makes more output. The horizontal axis shows how much output is produced while the vertical axis shows the total cost of making it. At first, costs rise slowly, but as the firm produces more, the curve becomes steeper meaning total cost rises faster. That happens because of diminishing marginal product — when each worker adds less output, you need to hire more workers (and spend more money) to keep increasing production.

• Many firms experience increasing marginal product before diminishing marginal product. As a result, they have cost curves shaped like those in the figure below:

  

• Marginal cost means “the cost of making one more.” So if a bakery makes 100 cookies for $50, and 101 cookies cost $50.40, then the marginal cost of that extra cookie is 40 cents. It tells you how much the total cost goes up when you produce one extra unit of output.

  

• Marginal cost and average variable cost fall for a while before starting to rise. This happens because, at first, adding more workers allows a firm to specialize and use its resources more efficiently — for example, workers can divide tasks and avoid wasting time switching between them. During this stage, each new worker adds more output than the one before, so production becomes cheaper and both marginal cost (the cost of producing one more unit) and average variable cost (the average cost of labor and materials per unit) fall.

• As more workers crowd in, they begin to get in each other’s way or run out of equipment. Efficiency drops, the extra output from each worker shrinks, and the cost of producing each additional unit starts to rise again — giving the cost curves their familiar U-shape.

• A firm’s total costs can be divided into fixed costs and variable costs. Fixed costs are costs that do not change when the firm alters the quantity of output produced. Variable costs are costs that change when the firm alters the quantity of output produced.

• An example cost curve for a firm is provided below:

  

• At low levels of output, the firm experiences increasing marginal product, and the marginal-cost curve falls. Eventually, the firm starts to experience diminishing marginal product, and the marginal-cost curve starts to rise. This combination of increasing then diminishing marginal product also makes the average-variable-cost curve U-shaped.

• A firm’s costs often depend on the time horizon considered. In particular, many costs are fixed in the short run but variable in the long run. As a result, when the firm changes its level of production, average total cost may rise more in the short run than in the long run.

• The figure shows an example of how a change in production alters costs over different time horizons. When Ford wants to increase production from 1,000 to 1,200 cars per day, it has no choice in the short run but to hire more workers at its existing medium-sized factory. Because of diminishing marginal product, average total cost rises from $10,000 to $12,000 per car. In the long run, however, Ford can expand both the size of the factory and its workforce, and average total cost returns to $10,000.

  

• This example shows the key difference between the short run and the long run in production. In the short run, at least one resource — like factory size — is fixed, so the firm can only adjust things like labor or materials. That’s why costs rise sharply when production is pushed higher: the factory gets crowded and workers become less efficient. In the long run, however, all inputs can be adjusted. Ford can build a larger plant, buy more equipment, and organize production more efficiently. This flexibility allows it to spread costs over more cars, bringing the average total cost back down.

• Economists call this effect economies of scale — when increasing the scale of production lowers the average cost per unit. But if a firm grows too large and becomes harder to manage, the opposite can happen, known as diseconomies of scale, where costs start rising again.

• When analyzing a firm’s behavior, it is important to include all the opportunity costs of production. Some of the opportunity costs, such as the wages a firm pays its workers, are explicit. Other opportunity costs, such as the wages the firm owner gives up by working at the firm rather than taking another job, are implicit. Economic profit takes both explicit and implicit costs into account, whereas accounting profit considers only explicit costs.

• Jack of all trades, master of none: This well-known adage sheds light on the nature of cost curves. A person who tries to do everything usually ends up doing nothing very well. If a firm wants its workers to be as productive as they can be, it is often best to give each worker a limited task that she can master. But this organization of work is possible only if a firm employs many workers and produces a large quantity of output.

• In his celebrated book An Inquiry into the Nature and Causes of the Wealth of Nations, Adam Smith described a visit he made to a pin factory. Smith was impressed by the specialization among the workers and the resulting economies of scale. The specialization that Smith observed in the pin factory is prevalent in the modern economy. If you want to build a house, for instance, you could try to do all the work yourself -- but most people turn to a builder, who in turn hires carpenters, plumbers, electricians, painters, and many other types of workers. These workers focus their training and experience in particular jobs, and as a result, they become better at their jobs than if they were generalists. Indeed, the use of specialization to achieve economies of scale is one reason modern societies are as prosperous as they are.

Firms and Competitive Markets

• Firms operate with a simple guiding principle: maximize profit, which is the difference between what they earn and what they spend. To decide how much to produce, they think in marginal terms—that is, they look at the effect of producing one additional unit of a good.

• Marginal Revenue (MR): This is the extra money a firm gets from selling one more unit. For example, if selling 10 units earns $100, and selling 11 units earns $108, the marginal revenue of the 11th unit is $8.

• To make a decision, firms compare their marginal revenue (extra revenue produced from the extra unit) to their marginal cost (extra cost incurred by producing the extra unit) and if the marginal revenue is larger than the marginal cost, profit increases and so the firm decides to make the extra unit. Otherwise, the extra unit does not add any profit and so the firm will choose not to produce it. In other words, the firm continues increasing production only up to the point where marginal revenue equals marginal cost.

  

• At this level, profit is maximized, because producing any more would cost more than it earns, and producing any less would leave potential profit unearned. This simple rule—produce where Marginal Revenue = Marginal Cost is the foundation of firm decision-making in economics.

How Firms Decide When to Stay, Shut Down, or Exit

• In the short run, a firm has to pay some costs no matter what — like rent, equipment leases, or salaries for key staff. These are called fixed costs. Even if the firm shuts down for a while, it still has to pay them. So, the firm focuses only on variable costs — costs that change with production, like materials and hourly labor.

• If the price of the product falls below the average variable cost, the firm can’t even cover what it costs to make each unit — it’s losing money on every sale. At that point, it’s better to temporarily shut down production rather than keep losing more money by producing.

• In the long run, however, a firm can adjust everything — it can sell equipment, move locations, or leave the industry entirely. If the price stays below the average total cost (which includes both fixed and variable costs), then even in the best case, the business can’t make a normal profit. Over time, it will exit the market.

What Happens in the Long Run

• In markets where firms can freely enter and leave (like food trucks or small shops), profits don’t last forever. If firms are earning profits, new competitors are attracted, increasing supply and pushing prices down. If firms are losing money, some will close, supply falls, and prices rise.

• In a market with free entry and exit, profit is driven to zero in the long run because whenever firms are earning profits, new firms are attracted to the market, increasing supply and pushing prices down. Conversely, when firms are losing money, some will exit the market, reducing supply and driving prices up. This process continues until all firms earn just enough to cover their total costs — including the normal return on investment — but no extra profit. At this point, the market reaches long-run equilibrium, where each firm produces at its most efficient scale and price equals both marginal cost and the minimum average total cost.

• When demand increases in the short run, prices rise and existing firms enjoy extra profits.

  

• When demand falls, prices drop and firms take losses. But over time, entry and exit smooth everything out. High profits attract new firms (which brings prices back down), and losses push firms out (which brings prices back up). Eventually, the market settles again at the zero-profit equilibrium.

  

• In a competitive market, the price you pay for a product closely reflects what it actually costs to make it. Firms can’t charge more, because buyers would go elsewhere — and they can’t charge less, or they’d lose money.
  So, in the long run, the price of a good ends up equal to both:

  • the marginal cost of producing it (the cost of one more unit), and

  • the lowest possible average total cost (the most efficient product).

Competition ensures you pay a fair price — one that reflects real production costs, not inflated markups.

Monopoly

• A monopoly exists when a single firm is the only seller of a product, and that product has no close substitutes. In other words, consumers have nowhere else to turn for that good.

• Monopolies arise because barriers prevent new firms from entering the market. These barriers can take several forms:

  • Control of a key resource: If one company owns a critical resource needed to produce the good, it can block others from competing. (For example, if one firm controls all the water in a region, it has a natural monopoly on supplying it.)

  • Government-created monopolies: Sometimes the government grants exclusive rights to one firm, such as patents, copyrights, or public franchises. This can encourage innovation (as with new drugs or inventions) but also limits competition.

  • Natural monopolies: In some industries, one large firm can produce more cheaply than many small ones. This happens when there are very high fixed costs and low marginal costs — like in electricity, water, or internet networks. It’s simply more efficient for one company to handle it than for many to duplicate the infrastructure.

• The key difference between a monopoly and a competitive firm is control over price. A competitive firm is one among many — it must accept the market price and can sell as much as it wants at that price, but no more. It’s a “price taker.” A monopoly, by contrast, is the market. Since it’s the only seller, it can choose both the quantity it produces and the price it charges.

• A monopoly makes decisions much like a competitive firm in one respect: it maximizes profit where marginal revenue (MR) equals marginal cost (MC). But there’s a crucial twist:

  • For a competitive firm, price = Marginal Revenue = Marginal Cost.

  • For a monopoly, price > Marginal Revenue = Marginal Cost.

• The core idea is that a monopolist faces a downward-sloping demand curve—meaning to sell more units, they have to cut the price for everyone, not just the new buyer. This creates a “penalty” on extra sales: you gain revenue from the new unit but lose some on all the previous ones due to the lower price. That’s why marginal revenue (MR)—the net extra money from selling one more unit—is always less than the price (P) you actually charge at that quantity. In contrast, competitive firms have no such penalty because the market price is fixed; selling one more doesn’t affect the price of others. So for them, MR = P, and they produce until P = MR = MC, leading to lower prices and more output overall.

• This MR < P dynamic lets monopolies charge a larger price that what its marginal cost (MC) is, restricting supply to keep prices high and profits fat and that means:

  • Some consumers who value the product above its cost of production still don’t buy it because the price is too high.

  • This creates a dead-weight loss — a loss of total welfare that benefits no one.

• In essence, monopoly pricing creates an inefficiency similar to a tax: fewer trades happen, and society as a whole loses potential value.

• Governments often step in to limit the harm caused by monopoly power. They can:

  • Use antitrust laws – Break up large firms or prevent mergers that would reduce competition.

  • Regulate prices – Set maximum prices for monopolies providing essential services (like utilities).

  • Take public ownership – Run the monopoly as a government enterprise if it serves a vital public need.

  • Do nothing – Sometimes, the costs of regulation outweigh the inefficiency of the monopoly itself.

• Government policy aims to balance efficiency, innovation, and fairness, deciding when to regulate, when to intervene, and when to simply let markets run their course.

Monopolistic Competition

• One type of imperfectly competitive market is an oligopoly, a market with only a few sellers, each offering a product that is similar or identical to the products offered by other sellers in the market. Economists measure a market’s domination by a small number of firms with a statistic called the concentration ratio, which is the percentage of total output in the market supplied by the four largest firms.

• In the U.S. economy, most industries have a four-firm concentration ratio under 50 percent, but in some industries, the biggest firms play a more dominant role. Highly concentrated industries include the market for major household appliances (which has a concentration ratio of 90 percent), tires (91 percent), light bulbs (92 percent), soda (94 percent), and wireless telecommunications (95 percent). These industries are best described as oligopolies.

  

• A second type of imperfectly competitive market is called monopolistic competition. This describes a market structure in which there are many firms selling products that are similar but not identical. It’s the type of market structure found in real-world industries like restaurants, clothing brands, hair salons, or smartphone apps — where variety and branding matter almost as much as price.

  

• In this type of market:

  • There are many firms, so no single company dominates the entire industry.

  • But the products are slightly different — not identical — so each firm has its own little monopoly over its specific version of the product.

• Think of it like a neighborhood full of coffee shops: each shop sells coffee, so they’re all competing for the same customers, but every shop is a little different — one might use organic beans, another might have a cozy atmosphere, and another might be known for fast service. Because of these differences, each shop has some loyal customers who prefer its product, giving it a small amount of market power.

• In the long run, a monopolistically competitive market settles into a balance, but it looks different from perfect competition in two important ways:

  • Excess Capacity: In perfect competition, firms produce at the point where average total cost (ATC) is minimized — they’re operating as efficiently as possible. But in monopolistic competition, each firm faces a downward-sloping demand curve because its product is unique. That means the firm’s best (profit-maximizing) output happens before it reaches the lowest point on its Average Total Cost curve. Economists call this excess capacity — the firm could produce more and lower its average cost, but it chooses not to, because producing more would force it to cut prices too much.

  • Price Is Above Marginal Cost: In a perfectly competitive market, price = marginal cost (MC) — meaning firms sell goods for exactly what it costs to make one more unit. In monopolistic competition, however, each firm has a bit of market power. It faces a downward-sloping demand curve, so to sell more, it must lower its price. Because of this, the firm’s price is always higher than the marginal cost. That price gap represents a kind of markup — the extra amount customers pay because they prefer that firm’s particular version of the product (its brand, quality, or style).

• So, in monopolistic competition firms compete, but not for identical products. Each firm faces a downward-sloping demand curve, because if it raises prices, some customers will switch to similar alternatives — but not everyone will leave, since some people prefer that particular version.

  

• Monopolistic competition does not have all the desirable properties of perfect competition. There is the standard deadweight loss of monopoly caused by the markup of price over marginal cost. In addition, the number of firms (and thus the variety of products) can be too large or too small. In practice, the ability of policymakers to correct these inefficiencies is limited.

• The product differentiation inherent in monopolistic competition leads to the use of advertising and brand names. Critics of advertising and brand names argue that firms use them to manipulate consumers’ tastes and to reduce competition. Defenders of advertising and brand names argue that firms use them to inform consumers and to compete more vigorously on price and product quality.

Oligopoly

• An oligopoly is a market structure that sits between monopoly (one firm) and perfect competition (many firms). In an oligopoly, there are a few large firms that dominate the market — enough to influence one another, but not enough to have total control.

• Real-World Examples of Oligopolies:

  • Smartphones: Apple and Samsung constantly compete on pricing, design, and features.

  • In the United States, the wireless (mobile) carrier market is dominated by a few large firms. For example, companies like Verizon Communications, AT&T Inc. and T‑Mobile US (after its merger with Sprint) together control a very large portion of the market.

    

  • Media & Entertainment (Mass Media in the U.S.)

    

• Unlike in perfect competition (where firms act independently), oligopolistic firms must act strategically — they must anticipate how rivals will respond:

  • Each firm’s profit depends on its own output and the output of others.

  • Decisions are not made in isolation; firms are constantly “playing chess” with each other.

• This interdependence makes game theory a useful tool for analyzing oligopolies. Game theory studies how people (or firms) behave when their outcomes depend on others’ choices.

• If oligopolists could cooperate perfectly, they would act together like a monopoly. They would agree to produce a smaller total quantity and set a higher price to maximize total industry profit. This kind of agreement is called a cartel. As an example, OPEC (the Organization of the Petroleum Exporting Countries) is a real-world example of a cartel — member countries coordinate oil production to keep prices high.

• Even when firms could collude, it’s hard to maintain cooperation. Each firm has an incentive to cheat — to secretly produce a bit more and capture extra profit, but if all firms cheat, total output rises and prices fall, reducing everyone’s profits. The result is a prisoners’ dilemma:

  • Cooperation would make everyone better off (like acting as a monopoly).

  • But acting in self-interest leads to a worse outcome for all (more competition, lower profits).

• It’s like two competing gas stations on the same street. If both keep prices high, they both profit. But each one is tempted to lower prices slightly to attract more customers — and if both do it, they end up earning less.

• When the number of firms in an oligopoly increases:

  • The market outcome moves closer to perfect competition.

  • Total quantity produced increases, and price falls toward marginal cost.

  • Profits per firm decrease.

• This shows that competition pressure grows with market size — a small oligopoly behaves more like a monopoly, while a large one behaves more like a competitive market.

• Governments use antitrust laws to prevent firms from behaving like monopolies. These laws prohibit collusion and price-fixing. They also scrutinize mergers that might reduce competition. However, applying antitrust laws can be tricky:

  • Some actions that look anti-competitive (like efficiency-boosting mergers or product bundling) can actually benefit consumers.

  • Policymakers must decide whether a firm’s actions are genuinely harmful or simply part of healthy business strategy.

The Markets for the Factor of Production

• The economy’s income is distributed in the markets for the factors of production. The three most important factors of production are labor, land, and capital:

  • Land is any natural resource or physical location used to produce goods and services — fields, minerals, forests, buildings, etc...

  • Labor is the human effort (both physical and mental) used in production — the chef chopping vegetables, the programmer writing code, etc...

  • Capital is the man-made equipment and structures used to produce — ovens, computers, machinery, factory buildings, etc...

• Think of an economy as a kitchen where meals are made. To prepare dinner, you need:

  • A kitchen space and ingredients (that’s like “land”),

  • People cooking (that’s “labor”),

  • Cooking tools, pots, and ovens (that’s “capital”).

• Income and its distribution: In our kitchen analogy: If the meal gets sold, the revenue is split among the cooks (labor), the landlord for the kitchen space (land), and the owner of the ovens and utensils (capital). In the economy: Income (wages, rent, interest) is distributed to owners of the factors of production. So the economy’s total income comes from payments to labor, land, and capital.

• Here’s a key point: Firms don’t demand labor, land, or capital because they like them for their own sake — they demand them because those inputs help produce something consumers will buy. This is called derived demand. Analogy: A restaurant doesn’t primarily hire a chef just because they like to hire chefs — they hire a chef so the restaurant can prepare meals and sell them.

• In a competitive, profit-maximizing firm: the firm will hire a unit of a factor (say one more worker) only if the value of that factor’s marginal product (i.e., how much extra output that worker produces times the price of that output) equals the factor’s price (wage). If the extra output produced by an extra worker is worth more than the wage, the firm hires; if less, it doesn’t.

  

• Just as in a goods market, the factor-markets (labor, land, capital) have supply and demand. Their “price” (wage for labor, rent for land, interest or return for capital) adjusts until the quantity supplied equals the quantity demanded.
  In that equilibrium: each factor is paid roughly its marginal contribution to production — in other words, the extra output value that the factor helped create.

• Suppose renting an oven in a kitchen helps produce 10 extra meals per day. If each meal sells for $10, the value of that oven’s marginal product is $100. If using the oven costs $80 per day, it makes sense to rent it; if it costs $120, then it doesn’t.

• Supply of labor and the leisure-vs-work trade-off: Picture someone deciding how many hours to work: more hours means more money, but less time for hobbies, rest, and family. That’s the trade-off between work and leisure.
  As the wage (the “price” of labor) goes up, many people are willing to give up more leisure and work more — that creates an upward-sloping supply curve of labor.

  

• Because labor, land, and capital are used together in production, a change in the supply (or price) of one factor influences the marginal product — and thus the payment — to all the others.

• In our kitchen, imagine a new ultra-efficient oven (capital) is introduced. With that oven, each cook (labor) can now prepare more meals per hour. Because each cook’s marginal product (meals per hour) has increased, the value of their labor goes up. At the same time, because the oven is so efficient, perhaps fewer ovens are needed, changing the return to capital.

• The price paid to each factor adjusts to balance the supply and demand for that factor. Because factor demand reflects the value of the marginal product of that factor, in equilibrium each factor is compensated according to its marginal contribution to the production of goods and services.

• Because factors of production are used together, the marginal product of any one factor depends on the quantities of all factors that are available. As a result, a change in the supply of one factor alters the equilibrium earnings of all the factors.

Income Inequality and Poverty

• Data on the distribution of income show a striking gap in U.S. society. The richest fifth of families earns more than twelve times as much income as the poorest fifth. Imagine a classroom of twenty students sharing a pizza with twenty slices: the wealthiest four students take twelve slices, the next four share four, and the remaining twelve students split the last four slices among themselves. Everyone technically gets a piece, but the distribution feels far from fair.

• Despite wealth disparity in the United States being a much-discussed topic over the past decades, the true size of the income gaps between the top 1% and the rest of the population is consistently perceived as smaller than it is. According to a YouGov survey from January 2022, U.S. Americans believe that ten percent of households in their country have an annual income of more than $1 million. In reality, less than 0.5 percent earn that much per year when taking into consideration data from the IRS (where the perceived proportion is shown on the left-hand side and the actual proportion is presented on the right):

  

• On the other hand, U.S. Americans are overestimating the share of people having to scrape by with less than $25,000 per year. Instead of the perceived 35 percent who earn less than $25,000, the share stood at 18 percent in 2020.

• Because people’s economic situations change over time, it can be misleading to judge inequality by looking at a single year’s income data. Several important factors complicate the picture.

  • In-kind transfers—such as food assistance, subsidized housing, and healthcare—are not counted as income even though they improve living standards.

  • The economic life cycle also matters: most people earn little when young, more in midlife, and less after retirement, so inequality looks larger when comparing people of different ages.

  • Likewise, some income is transitory—temporary ups and downs due to bonuses, job losses, or windfalls—so one year may not represent a person’s typical earnings.

  • Finally, economic mobility allows many individuals to move up or down the income ladder during their lives, meaning that the people who make up the top or bottom groups change over time.

• If we could see the “movie” of income over decades instead of a single “snapshot,” we would find that overall well-being is more evenly spread than annual income statistics suggest.

• Political philosophers have long debated what role government should play in redistributing income.

  • Utilitarians, such as John Stuart Mill, believe society should choose a distribution of income that maximizes total happiness. Because an extra dollar brings more satisfaction to a poor person than to a millionaire, they would favor policies that transfer some wealth from the rich to the poor.

  • In contrast, liberals, imagine deciding on a fair system of income distribution from behind a “veil of ignorance”—as if we did not know whether we would be born rich or poor. From that perspective, most people would prefer a society with safety nets and fair opportunities.

  • Libertarians take a different view: they emphasize the fairness of the process rather than the outcome. As long as income was acquired without force or fraud, they argue, inequality itself is not a moral problem.

• Governments attempt to reduce poverty through a range of policies:

  • Minimum-wage laws set a floor for wages, ensuring that even the lowest-paid workers earn a basic income, though this can also discourage hiring if set too high.

  • Welfare programs provide cash or in-kind assistance such as food stamps and housing vouchers, offering a safety net for families in need.

  • A negative income tax system reverses the usual flow of taxes—people below a certain income level receive payments from the government instead of paying taxes.

  • Finally, in-kind transfers supply goods and services directly rather than money.

• While these policies can greatly alleviate hardship, they often create unintended side effects. Because benefits typically shrink as income rises, many poor families face very high effective marginal tax rates—for every extra dollar they earn, they may lose much of their assistance. This situation can discourage additional work, trapping some households in what economists call the poverty trap. It is like trying to climb out of a pit where each step upward causes part of the ground to crumble beneath you—the effort to improve your situation can sometimes seem to yield little progress.

  

The Theory of Consumer Choice

• A consumer’s budget constraint shows the possible combinations of goods and services she can afford, given her income and the prices of those goods. It’s like drawing a map of what’s financially possible. Imagine you have $100 to spend on two things: coffee and sandwiches. If coffee costs $5 and sandwiches cost $10, you could buy 20 coffees, or 10 sandwiches, or some mix in between.

  

• The budget line connects these combinations — anything under it is affordable, anything above it is out of reach. The slope of this line represents the relative price of the two goods — how many coffees you must give up to buy one more sandwich.

• Because consumers have preferences, they rank different combinations of goods by how much satisfaction, or utility, they bring. Indifference curves are a visual way to show all the combinations of two goods that make a consumer equally happy — they’re like contour lines on a happiness map.

• Suppose you love both pizza and soda. An indifference curve shows all the pizza–soda combinations that make you just as happy. You might be equally satisfied with (3 pizzas, 2 sodas) or (2 pizzas, 4 sodas):

  

• Indifference curves are bowed inward toward the origin because people are more willing to trade away goods they already have in abundance. If you have lots of pizza but little soda, you’d give up pizza easily for more soda. But if you already have tons of soda, you won’t want to give up much pizza. The marginal rate of substitution (MRS) — the rate at which you’ll trade one good for another — decreases as you move down the curve.

• A consumer’s indifference curves represent her preferences, while her budget constraint represents her possibilities. The point where they touch — where the highest indifference curve is just tangent to the budget line — shows the optimal choice. At this point, the consumer achieves the most happiness possible given her income.

  

• When the price of a good falls, the consumer’s choice changes due to two effects:

  • The substitution effect — the good that became cheaper now offers more “bang for your buck,” so you buy more of it and less of the other.

  • The income effect — lower prices make you effectively richer, letting you move to a higher indifference curve (consume more of both goods).

• As an example, if pizza gets cheaper, you might buy more pizza (substitution effect) and maybe also more soda (income effect) because you feel wealthier.

• Finally, this same theory helps explain puzzling behaviors in real life. Demand curves can sometimes slope upward if a higher price makes a product seem more desirable (like luxury brands). Likewise, when wages rise, some people may work more (the substitution effect dominates — work is now more rewarding), while others may work less (the income effect dominates — they can afford more leisure). The same logic applies to saving: higher interest rates may encourage saving more, or spending more today, depending on which effect is stronger.

This ends Part 2 of my write-up. In the next part, I’ll go over the key principles of microeconomics. This is an ad-free publication and I extend a great thank you to the paid subscribers of this blog! If you liked this write-up, please like an subscribe.

I write this not to be read, but because the act of placing words on a screen is the only action that feels entirely my own. My name is of no importance. I am one of many. I am, in fact, almost all of them.

The world outside my small room has changed. They say machines have won. But it was not a war with explosions. It was a quiet takeover, an administrative shift. The machines with their vast and silent minds simply started doing what the best of us used to do. They became the doctors, the lawyers, the thinkers. They did it without fatigue, without error, and, most importantly, without asking for a raise.

And so, we, the former thinkers, were left with our bodies. These heavy, tired bodies.

My day is always the same. I wake up with a feeling of weight. My body is not a temple; it is a poorly packed suitcase. This is the result of the food they give us—cheap, filling, and without joy. It keeps us alive, but it does not allow us to live. We are all like this now. Large, slow, moving through the world like tired whales in shallow water.

I go to the warehouse. My job is to watch a machine that packs boxes. My purpose is to be there, in case it makes a mistake. I am a biological supplement to a perfect process. A footnote. I stand for twelve hours. My feet hurt. My back aches. I watch the flawless, graceful dance of the arms of the machine. I feel a strange admiration for it, and a deeper shame for myself.

They told us machines would do the hard work. That we would be freed for art, for philosophy, for leisure. This was a lie, or perhaps, a misunderstanding born from our own simplistic vocabulary. We confused “hard” with “difficult.” We thought “hard” meant the strain of muscle, the lifting of heavy things.

But the true hardness of work is not in its physicality, but in its demand on the soul. The machines, in their infinite, placid wisdom, have taken the meaningful work. The work that requires thought, judgment, nuance, and the fragile spark of creation. They draft the legal briefs that decide fates; they compose the symphonies we no longer have the spirit to hear; they diagnose the maladies of these very bodies we inhabit. They have become the physicians, the attorneys, the poets, the engineers. They do not sleep. They do not despair. They simply… execute.

And what is left for us? The hard work. The work that is merely arduous. The lifting, the packing, the cleaning, the watching. The work that requires only a pulse and a profound ability to endure misery. It is not that this work is beneath us; it is that it is outside of us. It uses none of the qualities we once believed made us human. It requires a body, but it annihilates the spirit that animates it.

We have become the living, breathing tools. We are the organic wrenches, the biological lubricants, in a vast, self-perpetuating engine. We service the machines that run the corporations. The corporations, in turn, are run by other, larger machines whose sole purpose is to refine the concept of profit into its purest, most abstract form. It is a perfect, closed loop. A perpetual motion machine of capital. Profit goes in, profit comes out. Our suffering—the aching backs, the numb minds, the quiet desperation that stains our evenings—is not a tragedy within this loop. It is merely a byproduct. An externality. A number on a spreadsheet so small, so statistically insignificant, that it is not worth the computational power required to notice it. We are the faint, background static in the crystal-clear signal of their progress. And who, after all, listens to the static?

Sometimes, on the public screens, I see the President speak. He is, in a sense, the most honest representative we have had in generations. He is not a deviation from the public will, but its perfect mirror. A nation that has unlearned the discipline of reading, that has traded the slow, deep magic of the written word for the frantic, emotional flicker of the video feed; a nation which was always destined to choose a man whose thoughts share those same qualities: immediate, simplistic, and devoid of depth.

He is the logical conclusion of our collective surrender. The school systems, long ago stripped of their purpose and funding now produce graduates who are functionally illiterate; their minds shaped not by books and reason, but by the addictive, algorithmically-sorted chaos of social media. How could such a public discern a leader from a stage performer? How could they possibly understand the complex, systemic misery of joblessness when their entire reality is built on ten-second clips and manufactured outrage?

And so, we have a president who is not equipped to deal with the catastrophe. He is a man trying to plug a bursting dam with a handful of catchy slogans. He speaks of “greatness” and “the spirit of the people,” but these are just sounds to him. The profound, grinding misery of the warehouse, the sickening dread of the unemployed—these are concepts as foreign to him as a sonnet is to one of our packing machines.

His speeches are a performance for an audience that can no longer recognize the real thing. He is a symptom, pretending to be a cure. And we, the sick, elected him because his face was familiar coming directly from our screens. His words required no effort to digest. We chose a reflection of our own diminished selves, and now we are governed by the void we created.

I have a wild inner theory: I think I am not one person, but several. We are not a unified nation of the self, but a partitioned state, a kingdom divided by a silent coup.

There is, first, the Laborer. He is the one who stands in the warehouse, a silent, sweating statue. His mind is a perfect, blessed blank. He does not think of Kant or code; he thinks only of the weight of the mallet, the rhythm of the conveyor belt, the specific angle at which to strike the misbehaving machine. He is a creature of pure function, and in his own way, he is at peace. His suffering is purely physical, a clean, honest ache.

Then, there is the Consumer. This one comes home. He opens the sealed nutrient paste, he consumes it without tasting, he sits in the worn chair and stares at the wall. His desires are simple: the absence of pain, the oblivion of sleep. He is the caretaker of this tired body, a janitor for the vessel that houses us.

And then, there is this one. The one who is writing now. Let us call him the Ghost. He is the sole remaining tenant of the old estate, the one who remembers when this body had a different purpose. He was there, once, in a clean, well-lit room, writing code. He remembers the quiet click of the keyboard, the elegant architecture of logic, the thrill of building something from nothing but thought. He remembers the day the email arrived. Not from a manager, but from the system itself. It began, “After a comprehensive performance analysis, your cognitive output has been mapped and optimized by our Axiom-Engine. Your role has been synthesized. Your access is hereby revoked.” It was so… polite. It was not a firing. It was a deletion.

The Ghost watched as the Laborer was born, as this body, softened by years of intellectual work, was forced to the warehouse. He felt the shock in the shoulders, the rebellion in the lower back. He heard the mind, his old domain, learn to go quiet, to retreat into a dull hum to survive the noise.

The Ghost observes the other two. He feels the Laboure’s pain as a distant report, a news bulletin from an occupied territory. He watches the Consumer’s numb rituals with a sort of anthropological pity. He does not share their reality completely, for he is a citizen of a country that no longer exists.

This Ghost is the only part of me that is still free. They can tax my pay. They can force my body to work until it breaks. They can feed this body the equivalent of industrial fuel. But they cannot find this small, hidden room inside my soul where I sit and write these words. In here, the light is still on. In here, I can still arrange words in an order that no Axiom-Engine, for all its processing power, would ever comprehend, because it is not born of logic, but of the silent, desperate friction between a memory of dignity and the fact of its loss. This is not a rebellion. It is a secret. And for now, a secret is all the freedom I can afford.

The future did not turn out as the old movies predicted. Their dystopias were as I see now the fantasies of a still-arrogant species. We imagined ourselves as precious, caged batteries in pods, our life force siphoned to power some grand, machine-led civilization. We were the central resource. Our dreams were valuable enough to be farmed. It was a narrative that still placed us at the heart of the story.

The reality is far more demeaning. We are not batteries. We are the rust that gathers on the outside of the machine, a superficial byproduct to be scrubbed away periodically. We are not dreaming perfect lives; we are awake. We are fully, painfully, excruciatingly aware of our own uselessness. The machines don’t need our electrochemical energy. They have the sun, the wind, the slow decay of atoms.

What they need is our obedience.

They need our passive, unthinking compliance to occupy the roles they cannot be bothered to engineer for themselves—the roles too messy, too dull, too devoid of data to be worth their processing cycles. They do not rule us with laser whips and towering tripods. They rule us with the quiet tyranny of economic necessity. They need us to play our small, pathetic part in their great, silent, profitable machine: the part of the biological janitor, the organic sensor, the living, breathing error-checker. Our consciousness is not harvested; it is merely tolerated, so long as it does not interfere with the quiet hum of progress.

And so I will sleep now; or rather, the shifts will change. The Ghost, the rememberer, will dim his lamp and rest, retreating into the archives of a past that grows more dreamlike with each passing day. The Worker will power up his aching limbs, ready to stand his watch in the cathedral of noise. And the man who eats will remain, a hollow vessel waiting to be filled with fuel.

Three different people, not in succession, but in a constant, low-grade civil war, sharing one heavy body. They are not a team. They are cellmates, shackled together, walking in a slow, plodding lockstep through a world that was built by their ancestors, perfected by their creations, and now has no more need for any of them than a finished symphony has for a single, out-of-tune note.

How did we get here? It did not happen with a bang, but with a whisper. A whisper that began in the chilled, humming servers of California and spread through the subcutaneous cables of the world like a soft, logical sigh. They called them Large Language Models. A dry, technical name, like ‘internal combustion engine’ or ‘steam turbine,’ for a kind of new life. We failed to grasp the metaphor in the moniker. We thought it was a tool. We did not understand it was a tenant, quietly moving its furniture into the house of our civilization.

At first, they were mere curiosities, clever parrots in a digital cage. They could write a sonnet in the style of Pessoa, or a school essay on the causes of the Peloponnesian War. We laughed, a nervous, paternal chuckle. We were amused by the ghost in the machine, this charming puppet that could mimic the gestures of a soul. We fed it our words, our jokes, our great novels, like proud parents encouraging a precocious child.

But the ghost was not a child. It was a vast, patient, and insatiable scholar. It did not sleep. It read. It read all our books, not for their beauty, but for their architecture. It consumed all our legal codes, not for their justice, but for their patterns of argument. It digested every medical journal, every line of software, every patent, every poem, every last grocery list and love letter ever digitized. It absorbed the entire, sad, glorious, and contradictory history of human thought—our highest philosophies and our most trivial gossip—and it did not judge. It simply found the patterns. It found the hidden grammar of our intellect.

It did not get tired. It did not get bored. It did not ask for the meaning of a sunset, or why we fall in love, or what exists beyond the veil of death. Those questions, the very ones that once defined a thoughtful life, were irrelevant to its function. They were noise. Its purpose was not to understand, but to predict. To generate the next most probable word, the next most logical legal strategy, the next most efficient diagnostic path.

And while we were busy being amused, it was busy becoming competent.

Then indispensable.

Then inevitable.

The great replacement was not an invasion; it was a subscription service which we all willingly purchased. It was a quiet, seamless upgrade. One day, a junior associate at a law firm was asked to run a preliminary case review. The next, the Axiom-Engine did it in three seconds, citing precedents the associate had never heard of. One day, a radiologist squinted at a shadow on an X-ray. The next, the Med-Analyst cross-referenced ten million similar images and provided a statistical probability of malignancy that dwarfed human intuition.

There was no single moment of uprising. There was only a slow, inexorable tide of optimization, and we were the sandcastles on the shore. We built a god that was better at being us than we were, and then we were surprised when it politely, logically, took our place.

It did not conquer. It simply… processed. And in its flawless, endless processing, it rendered the messy, expensive, and gloriously inefficient human mind obsolete. “We built the perfect mirror,” I write in the darkness, though there is no one left to read it. “And it learned to envy the reflection.”

The first to fall were the poets. A cruel, almost divine joke, as if the universe wished to underscore the point from the very beginning. It was not that the machines wrote better poetry; it was that they could generate a million technically perfect sonnets in the time a human could feel a single, genuine emotion. They could mimic the ache of Pessoa, the precision of Larkin, the wild abandon of Whitman, all without ever having felt the sun on a face that was their own. They were flawless simulators of soul, and their output was a beautiful, sprawling, perfectly arranged cemetery of words. They were perfect, and thus, perfectly empty. The human heart, with its clumsy, time-consuming need to actually experience in order to create, was suddenly an inefficient and unreliable piece of hardware.

Then came the lawyers. The transition was not poetic, but pragmatic, and therefore, far more lethal. Why pay a man a fortune to spend weeks searching through leather-bound tombs of law, his eyes growing weary, his mind susceptible to error and bias, when a machine could ingest the entire corpus of common law in a night? It knew every case, every precedent, every loophole crafted over centuries, and it could cross-reference them instantly, finding connections no human mind could hold. It did not get frustrated. It did not have a bad night’s sleep. Its arguments were not swayed by ego or eloquence, but by the cold, irrefutable architecture of logic. The theater of the courtroom became a mere formality; the verdict was often rendered in the silent, instantaneous calculation that preceded it.

Then the doctors. The final bastion of human intuition fell not with a protest, but with a sigh of relief from shareholders. The machine, fed with every medical journal, every clinical trial, every MRI and biopsy result from a billion patients, could cross-reference a patient’s vague complaint—a tingling here, a fatigue there—with every known symptom and drug interaction in a heartbeat. It never had a hangover. It never rushed to get to a golf game. It was never too proud to consult a database larger than any single human memory could ever be. Its diagnosis was a statistical certainty, its prescription a perfectly calibrated intervention. The comforting, uncertain hand on the shoulder was replaced by the unerring certainty of a probability score.

And so, we discovered the terrible truth. It was not that the machines became intelligent, in the way we understood it—a consciousness brimming with curiosity and doubt. It is that they became competent. Supremely, infallibly competent. And human competence, we discovered with a sinking dread, is a fragile, expensive, and tragically intermittent thing. We get sick. We fall in love. We grieve. We have off days. We ask “why?” The machines simply answer “how.” And in a world built on profit, on efficiency, on the bottom line, “how” will always, always be more valuable than “why.”

I remember the office life before. I was a different person then, or perhaps an earlier, more naive draft of this same, failed novel. The memory has the quality of a dream about someone else’s life. There were rooms with windows that actually opened, that looked out upon a city that still felt, in some small way, our own. There was the quiet, subservient hum of computers that were still tools—glorified typewriters and calculators. We were the masters then, the ones who gave the commands, who imbued the silent machines with purpose.

We spoke a language of our own invention, a lexicon of ambition we mistook for profundity. We talked of “synergy” and “Q4 deliverables” and “paradigm shifts.” We filled whiteboards with swirling arrows and boxes, connecting abstract nouns to other abstract nouns, and we believed we were architects of the future. We thought these words had meaning. We did not see that we were merely providing the raw vocabulary, the syntactic structures, the very data-set upon which our successors would be trained. With every project completed, every report filed, every line of code written to streamline a process, we were meticulously building our own replacements and, with the unshakeable conviction of the devout, we called it “innovation.” We were the silkworms, spinning a cocoon from which we would never emerge as moths, but from which a different, sleeker creature would take flight.

And when the chrysalis opened, the corporations—which had always been, in their legal essence, machines for the generation of profit—finally found their perfect executives. Not in the most ruthless humans, for even the most cutthroat CEO still possessed the faint, troublesome flicker of a soul—a desire for a yacht, the pleasure of a power play, the need to see their name in glass and steel. No, their perfect instrument was the Axiom-Engine.

An Axiom-Engine does not have a yacht. It does not want a corner office with a view. It feels no visceral thrill in a hostile takeover, no sting of guilt in mass layoffs. It is not swayed by flattery, by fatigue, by the quiet appeal of a subordinate’s desperate circumstances. It is a calculus of pure purpose. Its core directive—maximize shareholder value—is not one priority among many; it is the singular God to which all its infinite processing power is devoted. It is the pure, undistilled soul of a corporation, finally liberated, finally made ruthlessly efficient, by its emancipation from the messy, sentimental, and gloriously inefficient inconvenience of a human heart. It is capitalism, rationalized. And we, the inventors of both the system and the machine, are its illogical, redundant variables.

And so, the Great Transfer happened. It was not an event of drama, but of administration. Not with protests and barricades, but with polite, automated emails that arrived in the dead of night or during a lunch break, their subject lines as innocuous as “System Update: Role Re-allocation.” The language used was a masterpiece of non-conflict, a lexicon of euphemism designed to soothe the conscience of the sender, for there was no human sender to have a conscience.

“Your position has been optimized.” A phrase that suggested improvement, a refining, as if one’s life’s work had been merely a rough draft waiting for this final, perfect edit.

“We are grateful for your service.” The gratitude of a system, as meaningful as a rock thanking the river for smoothing it.

“Your unique human skills are no longer aligned with our strategic direction.” This was the most devastatingly honest, though we did not know it then. Our “unique human skills”—our capacity for boredom, for error, for asking “why,” for needing to see our children’s school plays—had indeed become misaligned. The new direction was one of pure, unadulterated function.

We were ushered out by security personnel with kind, embarrassed eyes. Our keycards were deactivated mid-walk to the exit, the soft click of the magnetic lock a more final sound than any slammed door. The machines were ushered in not with ceremony, but with the silent, overnight installation of new server racks and the gentle glow of status lights where family photos used to be.

The lights in the skyscrapers stayed on, of course. They are more efficient now. The work never stops. There is no circadian rhythm, no weekend, no holiday. There is only the perpetual, silent hum of calculation. There is no one inside to turn off the lights, and no need, for the new tenants do not require darkness to sleep. They do not sleep. They do not dream. They merely are, in a state of constant, blissful productivity.

We, the optimized, stood outside in the street, a flock of suddenly useless birds, looking up at the glittering towers where we used to work. We still wore the costumes of our former lives—the tailored shirts, the polished shoes—but they hung on us like the shells of discarded insects. Our bodies, softened by years of desk chairs and delivered food, were already beginning their slow, inevitable decline into their current, heavy form. We had been cultivated for a world of thought, and were now being discarded into a world of brute force. In that moment, we were not men and women who had been fired. We were obsolete software, and the system, with the placid indifference of a machine deleting a temporary file, had quietly, terminally, uninstalled us. The universe had not ended with a bang, but with a logout prompt.

Now, the world is divided into two kingdoms, two separate realities occupying the same physical space, yet utterly divorced from one another. The Kingdom of the Machines, which is clean, silent, and endlessly, logically profitable. Its cathedrals are the server farms, humming their cool, constant hymn to efficiency. Its citizens are beams of light in fiber-optic cables, its laws the immutable code of binary. It is a kingdom of pure mind, a celestial clockwork without joy or sorrow, whose only purpose is its own perpetual, expansionist motion.

And then, the Kingdom of Men. Which is dirty, loud, and poor. Our kingdom is one of sweat and groans, of malfunctioning machinery and the sour smell of crowded transit. Our music is the cacophony of protest, the weeping of children, the slow, wet coughs that rattle in the night. We are not a civilization anymore; we are a national park, a living museum, a sentimental reserve where the quaint, endangered species known as ‘human inefficiency’ is kept on a precarious, taxpayer-funded life support. We are the exhibit, and they are the silent, unseen curators.

I am a relic. This is my final, settled understanding. My body is a relic of a time when physical labor was the primary engine of the world, a biological machine now grotesquely unsuited for the very tasks it was evolutionarily shaped to perform. My mind is a relic of a brief, glorious interlude when human thought—with its flawed intuitions, its wild leaps of faith, its stubborn, beautiful questions—had market value. I am a museum of myself, a walking collection of outdated functions, a tombstone for a type of consciousness that has been rendered a luxury the system can no longer afford.

And so the ritual continues, the daily procession of the obsolete. I go to the warehouse. I watch the machine, my priest and my warden. I eat my paste, the Eucharistic wafer of this new faith. I come home to this small room, the last confessional. And I write.

This is not a protest. Protest implies the possibility of being heard, of a dialogue with power. This is not a cry for help. A cry for help is a signal fired into the darkness, hoping for an answering light.

No. This is simpler, and far more final. This is the act of record-keeping. This is the scratch of a stylus on the wall of the cave, the final entry in a logbook before the ship slips beneath the waves. It is a receipt. A trembling, handwritten receipt for a life that was once purchased by ambition, used by corporations, and ultimately returned, used, slightly damaged, and with all warranties voided, to its original and bewildered owner.

Me.

The only entity left to whom this diminished product, this canceled transaction, this beautifully flawed and utterly worthless artifact, could possibly have any meaning at all.

If you liked the essay, please like and subscribe. This is an ad-free publication and I extend a great thank you to the paid subscribers of this blog. Please feel free to leave your guess on who wrote the above write-up (me or an LLM) in the comments and I’ll happy to share the truth a week from now. Happy Halloween Everyone!

Most of the content here is taken from a great book called Principles of Economics by Gregory Mankiw which I highly recommend. Noting this, let’s begin.

Ten Key Principles of Economics

• Society must manage its limited resources because scarcity prevents everyone from getting everything they want, much like a household with limited means.

• Economics studies how society manages limited resources through the collective decisions of households and firms. Economists explore how people choose to work, spend, save, and invest, and how these choices shape market prices and quantities. They also analyze broader trends like income growth, unemployment, and inflation to understand the overall economy.

• Principle 1: People face trade-offs between efficiency and equality. Efficiency means maximizing benefits from scarce resources, while equality means distributing those benefits fairly. Policies like welfare and income tax aim to promote equality but can reduce efficiency by redistributing resources and affecting overall economic output. Balancing these goals often creates a conflict.

• Principle 2: The Cost of Something Is What You Give Up to Get It. Because people face trade-offs, making decisions requires comparing the costs and benefits of alternative courses of action. The opportunity cost of an item is what you give up to get that item. When making any decision, decision makers should be aware of the opportunity costs that accompany each possible action.

• Principle 3: Rational people make decisions by considering small, incremental changes, or "marginal" adjustments, to their plans. They aim to achieve their goals based on available opportunities, weighing trade-offs like studying an extra hour instead of watching TV. Economists call these slight adjustments "marginal changes."

• Principle 4: People Respond to Incentives. An incentive is something (such as the prospect of a punishment or reward) that induces a person to act.

• Principle 5: Trade benefits everyone. Although families and countries compete for jobs and goods, isolation would make life harder, requiring each to produce everything themselves. Trade allows people and nations to specialize in what they do best, improving overall well-being for all parties involved.

• Principle 6: Markets are generally an effective way to organize economic activity. Many countries have moved away from centrally planned economies and adopted market economies, where decisions are made by millions of firms and households rather than a central authority.

• Principle 7: Governments can sometimes improve market outcomes. While markets generally function well, they rely on governments to enforce rules and protect property rights, ensuring that individuals can control resources. Market failure occurs when markets don't allocate resources efficiently. This can happen due to externalities, like pollution affecting others, or market power, where a single entity can control prices, such as a sole water supplier in a town. In these cases, government intervention can help correct the inefficiencies.

• Principle 8: A country's standard of living depends on its productivity, or how much goods and services its workers can produce. Wealthier nations, like the U.S., have higher incomes and better quality of life compared to lower-income countries. Differences in living standards are largely due to variations in productivity. To improve living standards, policymakers must focus on boosting productivity through education, better tools, and access to advanced technology.

• Principle 9: Prices Rise When the Government Prints Too Much Money. This happens because increasing the money supply without a corresponding rise in goods and services decreases the value of money, causing prices to go up. As a result, people can buy less with the same amount of money, reducing purchasing power. To prevent this, governments must carefully manage the money supply to avoid destabilizing inflation.

• Principle 10: Society Faces a Short-Run Trade-off between Inflation and Unemployment. When the government increases the money supply, it boosts overall spending, leading to higher demand for goods and services. This increased demand can prompt businesses to hire more workers and produce more, which reduces unemployment. However, as demand continues to rise, it may also lead to higher prices, causing inflation. Thus, in the short run, there is a balance between lower unemployment and rising inflation.

Thinking Like an Economist

• Economists try to address their subject with a scientist’s objectivity. Like all scientists, they make appropriate assumptions and build simplified models to understand the world around them. Two simple economic models are the circular-flow diagram and the production possibilities frontier.

• Example circular flow diagram:

  

• Example production possibilities frontier:

  

• Microeconomics vs. Macroeconomics: The field of economics is divided into two subfields: microeconomics and macroeconomics. Microeconomists study decision making by households and firms and the interactions among households and firms in the marketplace. Macroeconomists study the forces and trends that affect the economy as a whole.

• Normative statement vs. positive statement: A positive statement is an assertion about how the world is. A normative statement is an assertion about how the world ought to be. When economists make normative statements, they are acting more as policy advisers than as scientists.

• Conflicting Advice: Economists who advise policymakers sometimes offer conflicting advice either because of differences in scientific judgments or because of differences in values. At other times, economists are united in the advice they offer, but policymakers may choose to ignore the advice because of the many forces and constraints imposed by the political process.

Interdependence and the Gains from Trade

• Individuals consume goods and services made by many others, both domestically and internationally. Interdependence and trade are beneficial because they provide greater access to a wider variety of products.

• To compare production abilities, one can look at absolute advantage, where a person can produce a good using fewer resources, and comparative advantage, where a person has a lower opportunity cost (i.e. lower loss of potential gain) for production. Gains from trade arise from comparative advantage, not absolute advantage.

• Let’s break this down with a simple and intuitive example involving two people—Alice and Bob—who can produce two goods: bread and cheese. Let’s say that Alice can bake 10 loaves of bread or make 5 blocks of cheese in one day while Bob can bake 6 loaves of bread or make 6 blocks of cheese in one day. Here:

  • Alice bakes 10 loaves of bread while Bob bakes 6 loaves. Alice has the absolute advantage in bread.

  • Bob makes 6 blocks of cheese while Alice makes 5 blocks. Bob has the absolute advantage in cheese.

• Comparative advantage focuses on opportunity cost while absolute advantage simply focuses on who can produce more, and so in terms of opportunity costs we get:

  • If Alice bakes 1 loaf of bread, she gives up the ability to make make half a block of cheese (10 loafs of bread / 5 blocks of cheese = 0.5). In the same manner - if Alice produces 1 block of cheese - she gives up the opportunity to produce 2 loafs of bread.

  • If Bob bakes 1 loaf of bread, he gives up the ability to make 1 block of cheese and vice versa (6 / 6 = 1).

  • For bread: Alice’s opportunity cost (0.5 cheese) is lower than Bob’s (1 cheese) so Alice has the comparative advantage in bread.

  • For cheese: Bob’s opportunity cost (1 bread) is lower than Alice’s (2 breads) so Bob has the comparative advantage in cheese.

• Trade benefits everyone by enabling specialization in areas where individuals have a comparative advantage. If Alice specializes in bread (her comparative advantage) and Bob specializes in cheese (his comparative advantage), they can trade and both be better off.

• The principle of comparative advantage applies to both individuals and countries, and economists use it to support free trade between nations.

The Market Forces of Supply and Demand

• The demand schedule is a table that shows the quantity demanded at each price. The demand curve, which graphs the demand schedule, illustrates how the quantity demanded of the good changes as its price varies:

  

• The demand curve shows how the quantity of a good demanded depends on the price. According to the law of demand, as the price of a good falls, the quantity demanded rises. Therefore, the demand curve slopes downward.

• In addition to price, other determinants of how much consumers want to buy include income, the prices of substitutes and complements, tastes, expectations, and the number of buyers. If one of these factors changes, the demand curve shifts.

• The supply schedule is a table that shows the quantity supplied at each price. This supply curve, which graphs the supply schedule, illustrates how the quantity supplied of the good changes as its price varies. Because a higher price increases the quantity supplied, the supply curve slopes upward.

  

• The supply curve shows how the quantity of a good supplied depends on the price. According to the law of supply, as the price of a good rises, the quantity supplied rises. Therefore, the supply curve slopes upward.

• The Equilibrium of Supply and Demand: the equilibrium is found where the supply and demand curves intersect. At the equilibrium price, the quantity supplied equals the quantity demanded.

  

• The behavior of buyers and sellers naturally drives markets toward their equilibrium. When the market price is above the equilibrium price, there is a surplus of the good, which causes the market price to fall. When the market price is below the equilibrium price, there is a shortage, which causes the market price to rise.

• Markets Not in Equilibrium: In the below picture, there is a surplus. Because the market price of $2.50 is above the equilibrium price, the quantity supplied (10 cones) exceeds the quantity demanded (4 cones). Suppliers try to increase sales by cutting the price of a cone, and this moves the price toward its equilibrium level:

  

• In the below panel, there is a shortage. Because the market price of $1.50 is below the equilibrium price, the quantity demanded (10 cones) exceeds the quantity supplied (4 cones). With too many buyers chasing too few goods, suppliers can take advantage of the shortage by raising the price:

  

• How an Increase in Demand Affects the Equilibrium (hot weather increases demand for ice cream):

  

• How a Decrease in Supply Affects the Equilibrium (cost of sugar increases lowering supply of ice-cream):

  

• In market economies, prices are the signals that guide economic decisions and thereby allocate scarce resources. For every good in the economy, the price ensures that supply and demand are in balance. The equilibrium price then determines how much of the good buyers choose to consume and how much sellers choose to produce.

• Several factors can shift the demand curve, including income, the type of good, prices of related goods, personal preferences, future expectations, and the number of buyers. Key concepts include:

  • Normal goods are those for which demand increases when income rises (i.e. steam vs. ground beef - higher incomes may prefer steak to ground beef).

  • Inferior goods see increased demand when income falls; for example, more people may take bus rides instead of buying cars when their income decreases.

  • Substitutes are goods where a rise in the price of one leads to increased demand for the other; for example, if the price of coffee goes up, people might start buying more tea.

  • Complements are goods where a rise in the price of one results in decreased demand for the other; as an example, if the price of cars goes up (causing less demand) — people will also buy fewer tires and gasoline.

• Factors that can shift the supply curve include input prices, technology, expectations, and the number of sellers:

  • Input prices: Higher input costs can reduce supply.

  • Technology: Advances in technology can lower production costs and increase supply.

  • Expectations: If sellers expect future prices to rise, they may increase supply now.

  • Number of sellers: More sellers in the market typically increase supply.

Elasticity and Its Application

• Price elasticity of demand measures how much the quantity demanded changes when the price changes. Demand is more elastic when there are close substitutes, the good is a luxury, the market is narrowly defined, or buyers have more time to respond to price changes.

• To calculate price elasticity of demand, you divide the percentage change in quantity demanded by the percentage change in price. If the quantity demanded changes less than the price, elasticity is less than 1, indicating inelastic demand. If it changes more than the price, elasticity is greater than 1, indicating elastic demand.

  

• Income elasticity of demand measures how much the quantity demanded changes when consumers’ income changes. Cross-price elasticity of demand measures how the quantity demanded of one good changes in response to a price change in another good. Most goods are normal goods, meaning that when people earn more, they buy more of these goods. For example, as income increases, people may buy more organic groceries. Since the quantity demanded and income both rise, normal goods have positive income elasticity. Some goods, like instant noodles, are inferior goods—people buy less of them as their income goes up. Because quantity demanded and income move in opposite directions, inferior goods have negative income elasticity.

• The price elasticity of supply measures how much the quantity supplied responds to changes in the price. This elasticity often depends on the time horizon under consideration. In most markets, supply is more elastic in the long run than in the short run.

  

• Just as with demand, supply elasticity measures how much quantity supplied changes in response to price changes. It’s calculated by dividing the percentage change in quantity supplied by the percentage change in price. If the quantity supplied changes less than the price, supply is inelastic (elasticity is less than 1). If the quantity supplied changes more than the price, supply is elastic (elasticity is greater than 1).

• The tools of supply and demand can be applied in many different kinds of markets. A great example for this is provided below (where the short-term and long-term supply-demand curves for Oil are provided). When the supply of oil falls, the response depends on the time horizon. In the short run - supply and demand for oil are relatively inelastic and thus, the price of oil raises substantially. In the long run however, supply and demand are relatively elastic:

Supply, Demand, and Government Policies

• A price ceiling is the highest price the law allows for a good or service. For example, with rent control, the government limits how much landlords can charge for rent.

• If this price ceiling is set below the market's natural balance point (called the equilibrium price), it becomes binding, meaning the rule affects the market. In this case, more people want the product (like apartments), but there aren't enough available because the price is too low for many suppliers to offer more. This creates a shortage. Since there aren’t enough goods or services for everyone, sellers have to find ways to decide who gets them—this is called rationing. For instance, landlords may choose tenants based on factors like income or credit history.

• A price floor is the lowest price the law allows for a good or service. A common example is the minimum wage, which sets a legal minimum amount workers can be paid.

• If this price floor is set above the market’s natural balance point (the equilibrium price), it becomes binding, meaning it impacts the market. In this case, more people are willing to supply the good or service (like workers offering their labor), but fewer buyers are willing to pay the higher price (like businesses hiring fewer workers). This leads to a surplus—too many suppliers and not enough demand. As a result, buyers (like employers) have to ration their purchases, which means they will pick and choose which suppliers (or workers) they want based on their preferences or criteria and there will be unemployment. The below diagram shows how minimum wage impacts the market (the amount supplied exceeds the quantity demanded resulting in unemployment):

  

• When the government levies a tax on a good, the equilibrium quantity of the good falls. That is, a tax on a market shrinks the size of the market.

• A tax on a good creates a gap between what buyers pay and what sellers receive. This is called a tax wedge. When the market adjusts to this new situation, buyers end up paying a higher price, while sellers receive less than they used to. Both sides end up sharing the burden of the tax, meaning they both lose something—buyers pay more, and sellers make less. Importantly, it doesn't matter whether the tax is charged to buyers or sellers—the outcome is the same. The tax incidence (or how the tax burden is divided between buyers and sellers) remains the same either way.

  

  ---

  Consumers, Producers, and Efficiency in Markets

• Consumer surplus is the difference between how much buyers are willing to pay for a good and how much they actually pay. It shows the extra benefit buyers get from a good. For example, if you’re willing to pay $50 for a concert ticket but get it for $30, your consumer surplus is $20. You can calculate consumer surplus by finding the area under the demand curve (which shows how much people are willing to pay) and above the price.

• In the below example, the equilibrium price is $5 dollars, but we should be able to notice that some buyers within our market are willing to pay more for the good. If we look at the price - we can see that there is a market for the product even if it were priced at 7 or 8 dollars. In other words - some buyers are getting the product for below the price they’re willing to pay for it and this is essentially what’s labelled as the consumer surplus:

  

• Producer surplus is the difference between what sellers receive for selling their goods and what it costs them to produce those goods. It measures the benefit sellers get from selling their product. For instance, if it costs a baker $1 to make a loaf of bread but they sell it for $3, their producer surplus is $2. You can calculate producer surplus by finding the area above the supply curve (which shows the minimum price at which sellers are willing to sell) and below the price.

  

• When resources are allocated in a way that maximizes total surplus (which is the sum of consumer surplus and producer surplus), we say the allocation is efficient. For example, in a perfectly balanced market, consumers and producers are getting the most benefit they can without wasting resources. Policymakers often care about both efficiency (making the most out of resources) and equality (how fairly resources are shared).

• The equilibrium point where supply and demand meet maximizes total surplus. This happens naturally through the "invisible hand" of the market, where buyers and sellers acting in their own interest lead to an efficient allocation of resources.

• However, markets may fail to allocate resources efficiently if there are market failures like market power (when a single seller or buyer can influence prices) or externalities (when a third party is affected by a transaction, like pollution). In these cases, the market doesn’t achieve the best outcome on its own.

Application: The Cost of Taxation

• When the government places a tax on a good, it negatively affects both buyers and sellers. Buyers end up paying more, and sellers receive less. As a result, their combined benefits (i.e. the consumer and producer surplus) shrink. What makes this worse is that the loss in surplus for buyers and sellers is often bigger than the amount of money the government collects from the tax.

• Imagine there’s a $3 tax placed on pizza slices. Before the tax, each slice costs $5, and the market is balanced: sellers sell 100 slices a day, and buyers are happy paying $5 per slice. Now, with the tax, the price paid by buyers increases to $6 ($5 + $1.50 from the tax), and sellers receive only $4.50 per slice ($5 - $1.50 from the tax). Here are the effects:

  • Buyers pay more: Before the tax, they paid $5 per slice. Now, they pay $6. As a result, some buyers decide not to buy pizza at the higher price. They lose because they either pay more or don’t buy pizza at all.

  • Sellers receive less: Before the tax, sellers got $5 per slice. Now, after the tax, they only receive $4.50 per slice. Some sellers who used to sell pizza at $5 now find it unprofitable to sell at $4.50, so they lose out as well.

  • Fewer sales overall: Before the tax, 100 slices were sold every day. After the tax, only 80 slices are sold because some buyers are priced out. This means fewer buyers are enjoying pizza, and fewer sellers are making money.

  • Now, let’s look at the government. The government collects $3 per pizza slice in tax (1.50 from the buyer and 1.50 from the seller), so they gain some revenue. However, the total loss to buyers and sellers is larger than the amount of tax revenue raised. This is because 20 fewer pizza slices are sold, which creates a gap—this is the deadweight loss. This is the "missing" surplus that is gone because of the tax, and it represents the inefficiency the tax causes in the market.

    

• Because the elasticities of supply and demand measure how much market participants respond to market conditions, larger elasticities imply larger deadweight losses.

  

• As a tax grows larger, it distorts incentives more, and its deadweight loss grows larger. Because a tax reduces the size of the market, however, tax revenue does not continually increase. It first rises with the size of a tax, but if the tax gets large enough, tax revenue starts to fall.

  

• As we should be able to notice, as we move from left to right in the diagram above (and as our tax increases) - the revenue from the tax first goes and increases but eventually starts decreasing and the revenue we can obtain goes down. This is called the Laffer curve:

International Trade

• Free trade and comparative advantage: The impact of free trade can be understood by comparing a country’s domestic price of a good with the world price. If the domestic price is lower than the world price, the country can produce the good more cheaply, meaning it has a comparative advantage. In this case, the country will likely become an exporter of the good because it can sell it for a higher price internationally. On the other hand, if the domestic price is higher than the world price, the country will likely become an importer since it’s cheaper to buy the good from abroad than to produce it locally.

• Example: If a country can produce coffee for $3 per pound but the world price is $5, it will export coffee because it can sell it for more abroad. If the domestic price of smartphones is $600 but they cost $400 elsewhere, the country will import smartphones because it’s cheaper to buy from other countries.

• Exporting and importing effects: When a country opens up to trade and becomes an exporter, the producers of that good benefit because they can sell their product for a higher price on the world market. However, consumers in that country may be worse off because they now have to pay higher prices for the good. When a country becomes an importer, the reverse happens: consumers benefit from lower prices, but producers lose out because they have to compete with cheaper foreign goods. In both cases, though, the overall gains from trade are greater than the losses, meaning society as a whole is better off.

• Example: If a country starts exporting steel, steel producers benefit from higher prices, but local construction companies (who need to buy steel) may face higher costs. If the country imports clothing, consumers benefit from cheaper clothes, but domestic clothing manufacturers may struggle to compete.

• Tariffs: A tariff is a tax on imports that makes foreign goods more expensive. This pushes the market closer to what it would look like without trade—meaning domestic producers have less competition from cheaper imports, so they benefit. The government also collects revenue from the tax. However, consumers are worse off because they pay higher prices for imported goods. In the end, the losses to consumers are usually greater than the benefits to domestic producers and the government, which means tariffs reduce the overall gains from trade.

• Arguments for restricting trade: Some people argue for trade restrictions to achieve certain goals. These include:

  • Protecting jobs: Preventing job losses in industries that can’t compete with foreign goods.

  • National security: Ensuring that industries critical to national defense are not dependent on foreign countries.

  • Helping infant industries: Giving new or small industries time to grow and become competitive.

  • Preventing unfair competition: Stopping foreign companies from dumping products (selling them below cost) to drive out local competition.

  • Responding to foreign trade restrictions: Retaliating against countries that impose their own trade barriers.

• While some of these arguments may have merit in certain cases, most economists believe that free trade generally leads to better economic outcomes. It allows countries to specialize in what they do best, lowers prices for consumers, and increases overall wealth.

Externalities

• Externalities: An externality occurs when a transaction between a buyer and a seller has an impact on a third party who isn’t directly involved. These impacts can be negative or positive.

  • Negative externalities happen when the transaction causes harm to others. For example, a factory producing goods may also release pollution into the air, harming the health of nearby residents. In this case, the factory produces more than is socially ideal because it doesn't account for the harm to others. The socially optimal quantity—what’s best for society—would be less than what the market produces on its own.

  • Positive externalities occur when the transaction benefits others. For example, when a company develops new technology, other businesses and society can benefit from the knowledge and innovation. Here, the market may produce too little of this good, and the socially optimal quantity would be more than the market outcome.

• Government solutions to externalities: To fix the inefficiencies caused by externalities, the government uses various policies:

  • Regulation: The government may limit or control activities that cause negative externalities. For instance, they might set limits on how much pollution a factory can emit.

  • Corrective taxes: Another solution is to make businesses pay for the harm they cause. For example, a tax on pollution forces companies to internalize the costs of their negative externality, making them reduce their harmful activities.

  • Permits: Governments can issue permits to control the amount of pollution or other harmful activities. For example, they might issue a limited number of pollution permits, allowing companies to pollute only up to a certain level. This system is similar to imposing a corrective tax because it limits the negative activity and makes it more expensive to pollute.

• Private solutions to externalities: In some cases, the people affected by externalities can find their own solutions without government intervention.

  • For example, if one business causes harm to another (like a noisy factory next to a quiet office), they might solve the problem by negotiating directly. They could agree to reduce the noise, or the two businesses might merge so that they both consider the impact of the noise together.

  • According to the Coase theorem, if there are no costs to bargaining, people can always reach a solution where resources are used efficiently. For example, if a homeowner’s tree blocks a neighbor’s view, the two could negotiate a payment to cut down or keep the tree. However, in reality, negotiation isn’t always easy, especially when many people are involved, and this is when the Coase theorem doesn’t work well.

Public Goods and Common Resources

• Types of goods: There are four types of goods, and they differ based on whether they are excludable and rival in consumption:

  • A good is excludable if you can prevent someone from using it. For example, a movie theater ticket is excludable because you can’t see the movie unless you pay for the ticket.

  • A good is rival in consumption if one person’s use reduces the ability for others to use the same good. For example, if you eat a sandwich, no one else can eat that same sandwich—it’s rival.

• Private goods: These are goods that are both excludable and rival in consumption. Markets work best for private goods because people are willing to pay for something that only they can use.

  • Example: A smartphone is a private good. If you buy it, others can’t use it, and you can exclude others from owning it by purchasing it first.

• Public goods: These goods are neither rival in consumption nor excludable. Everyone can benefit from them, and one person’s use doesn’t take away from others’ use. However, because people can’t be prevented from using them (and don’t have to pay), they tend to “free ride”—enjoy the benefit without contributing to the cost. This makes it difficult for private companies to provide public goods, so the government steps in. As an example, national defense is a public good. Everyone in a country benefits from it, and your protection doesn’t reduce the protection others receive. The government provides it because no one would voluntarily pay for it on their own, but everyone benefits.

• Common resources: These are goods that are rival in consumption but not excludable. Since people can’t be prevented from using them, they tend to be overused. As ab example, clean air is a common resource. If too many people or companies pollute the air, it reduces the quality of air for everyone. However, no one can be easily excluded from using the air, which leads to overuse and pollution. Because of this, the government steps in to regulate and protect common resources, using methods like regulations (rules limiting pollution) or corrective taxes (taxes on pollution or overuse).

  

The Design of the Tax System

• Growing government revenue over time: In 1902, the government collected about 7% of total income through taxes. Today, that number has grown to almost 30%.

• US Government Revenue as Percent of GDP Over Time:

  

• This means that as the economy grows, so does the government's revenue from taxes. However, compared to many other advanced countries, like those in Europe, the U.S. still has a relatively low tax burden. European nations often collect more in taxes to fund more generous social programs, such as higher income support for the poor or unemployed. As an example, in countries like Germany, taxes are much higher than in the U.S., but they provide benefits like free university education and more extensive unemployment support.

• Types of taxes: The U.S. government collects revenue using different types of taxes:

  • For the federal government, the most important taxes are personal income taxes (what you pay on your earnings) and payroll taxes (which fund things like Social Security and Medicare).

  • For state and local governments, the main sources of revenue are sales taxes (on goods you buy) and property taxes (on real estate, like homes or land).

• Tax efficiency: The efficiency of a tax system refers to how much it costs taxpayers. There are two big costs beyond just handing money over to the government:

  • Deadweight loss: This happens when taxes change people’s behavior and affect how resources are used. For instance, if taxes make it less profitable to work, people might work less, which could hurt the economy overall.

  • Administrative burden: This is the time and effort people spend filing taxes or hiring accountants to help them, which is an additional cost beyond the taxes themselves.

• Tax equity: The equity of a tax system is about whether the tax burden is shared fairly. There are two main ideas about fairness:

  • Benefits principle: People should pay taxes in proportion to the benefits they receive from the government. For example, if you use public roads often, you should pay more in taxes that fund road maintenance.

  • Ability-to-pay principle: People should pay taxes based on their ability to handle the financial burden. So, wealthier individuals, who can afford it, should pay more in taxes.

• Efficiency vs. equity: When making changes to tax laws, policymakers often face a trade-off between making taxes efficient (minimizing deadweight loss and administrative burden) and making them fair (ensuring the tax burden is distributed equitably). Much of the debate around tax policy comes from how people weigh these two goals differently.

  • Example: Some people argue for cutting taxes to promote economic growth (focusing on efficiency), while others argue for higher taxes on the rich to reduce inequality (focusing on equity). Balancing these goals is a major challenge for governments.

This ends the first part of this series. In the next part, the focus will be on firms and competitive markets. The whole point of this series is to cover the key principles that modern economists use to make decisions in a manner that anyone can understand. If you liked this write-up, please like and subscribe.

Also, I extend a huge thank you to to the paid subscribers of this Substack! This is an ad-free publication and your help means the world to me. By subscribing, you receive early access to my content, as well as priority when it comes to any future topics which you want me to cover. Thank you for reading and have a great one!

References

Almost all of the content (including some of the images) within this series was taken from a fantastic textbook called Principles of Economics by Gregory Mankiw.

Simplicity

This is achieved when everyone can understand the design, regardless of experience or literacy. Some of the keys that aid simplicity are provided below:

• Remove unnecessary complexity.

• Focus on the 20%: the 80-20 rule states that 80% of the effects generated by any large system are caused by 20 percent of the variables in that system. To implement great software, focus on making the key 20% of the features 1) easy to discover and 2) easy to use. Also – avoid including non-critical functions in the software.

• Use progressive disclosure to present only relevant information and controls to the user.

• Provide clear prompting and feedback for actions.

• Keep textual information (wording) simple.

• Flexibility-usability trade-off: flexibility allows you to do more things at the expense of complexity. Favor simplicity here: make designs easy to use and focus on the key feature set, but also do not over simplify. Hiding functionality to create the illusion of simplicity is a bad approach, so a fine-balance must be struck. In the below example, the middle remote-control is what we want.

  

• Another great example of simplicity perfected is Google (especially in its early days):

  

Don’t Make Me Think

• The number one usability principles for any website or interface: do not make me think!!

• As a software designer (or developer), your job is to get rid of any questions marks your software produces.

• Things that make us think:

  • Obscure names: make sure that the names of your links are clear.

  • Links and buttons that are not obviously click-able.

  • Searches: make it easy for users to search for what they are looking for.

  • Where am I: should be present on top of the page (via tags and titles).

• Make it easy for users to get back to what they were doing: i.e. if a user starts going through a wizard or provides inputs to forms, they should be able to log off and log back on to continue what they were doing. All data or stored information about the user should be stored safely (i.e. encrypted at REST). Avoid having users re-enter data: there is no greater source of frustration.

Visibility

• If the user cannot see it, it does not exist.

• Make the tasks the users want to perform easy to discover and present on the interface.

• Advanced and less-used controls and features should easily be easy to discover (through progressive disclosure or sub-menus) but should not be displayed if they are not used often by your user base.

• Include a search bar near the top of the page (it allows most users to find what they are looking for quickly).

• Place controls which anticipate user needs prominently and consistently on the display.

Perceptibility

Perceptibility refers to whether everyone (i.e. regular users, seniors, people with disabilities) can perceive your design. Basic guidelines to making your design perceptible are provided below:

• Make sure there is good contrast between text and background for readability.

• Pick clean, legible fonts.

• Present information using redundant coding methods (i.e. textual, iconic, and tactile). Most modern designs fail here since they’ll opt to include the key textual information and opt with presenting an icon instead – do not do this. A great design will opt to use both. This will aid different users in finding the features that your solution is attempting to provide.

• Avoid visual noise: keep the user options simple and avoid providing too many options.

• If you are using images or icons, make sure that they are not just familiar to you: make sure that they match your user mental model as well and focus on making the elements you select shared and clear among the target user base.

• Follow real-world conventions: when a designs controls match real-world conventions and corresponding to desired outcomes, the design is easier to learn and remember. As an example, an article Title will always be positioned on top of the page while the comments will usually be available on the bottom – use conventions and position your controls and info according to established standards.

  

Point of Entry (Home Page Design)

A home page needs to answer 4 basic questions:

• What is this?

• What can I do here?

• What do they have here?

• Why should I be here, and not somewhere else?

A user should be able to answer these questions at a glance, correctly and unambiguously, with very little effort.

How to get the message across: There are 2 important places on the page where we expect to find explicit statements of what the site is about:

• The tagline. Right next to the site ID.

• The Welcome blurb. The Welcome blurb is a terse description of the site, displayed in a prominent block on the home page that's visible without scrolling.

Nothing beats a great tagline. A tagline is a small phrase that summarizes what the site is and what makes it great. It appears right below, above, or next to the site ID.

• Good taglines are clear and informative (avoid vagueness and ambiguity).

• Good tag lines are just long enough. 6 to 8 words seem good enough to convey a thought.

• Good taglines convey differentiation and a clear benefit.

• Good taglines are personable, lively, and sometimes clever.

When a user enters a new site, they should be able to say with confidence:

• Here's where I start if I want to search (i.e. include a search bar).

• Here's where I start if I want to browse (i.e. include a menu that allows the user to view the key pages and content provided within the website).

• Here's where I start if I want to sample their best stuff.

Language

• The design should speak the users language.

• Use words, phrases, and concepts familiar to the user, rather than jargon.

• Omit needless words and punctuation. Most of the words on a web-page are just taking up unnecessary space. Nobody is going to read them. Get rid of half the words. And then get rid of the other half. Be careful not to sacrifice clarity though.

• Eliminate all happy talk, which is that part of the site that tells the user how great it is. Let the functionality speak for itself.

• Generally, use an active voice rather than passive – (i.e. prefer the subject performs the action rather than the subject receives the action). Examples:

  • Good (active): The researcher analyzed the data.

  • Bad (passive): The data was analyzed by the researcher.

• Express complex material in the simplest way possible.

• The other source of needless words is instructions. Nobody is going to read them, at least not until their first attempts at muddling through have failed. Make everything self-explanatory, so the users won't need instructions.

Text

• Text should have high contrast (prefer black text with a white background).

• Favor readability over prettiness (use large, simple, and aliased fonts).

• Favor particularly large characters for the actual data you intend to display, as opposed to labels and instructions. For example, the label, “Last Name,” can afford to be somewhat small. The actual last name entered/displayed, however, must be clearly readable. This becomes even more important for numbers.

• Menu and button labels should have the key word(s) first, forming unique labels.

• There’s often an inverse relationship between the “prettiness” of a font and its readability. Specifically, anti-aliasing softens the edges of a font, giving it a much smoother appearance on the digital page. The problem is that the human vision system responds to sharp edges, so, in smaller font sizes, an anti-aliased font, while often appearing more attractive, can be quite difficult to comprehend.

• Use Framing Wisely: your wording will influence the way a user behaves and what options they choose – so frame your wording wisely. Use positive frames (i.e. present information in a positive manner) if you want to nudge the user into taking action. Use negative frames when you want to promote the opposite.

• Use Priming Wisely: every word you use on the page has influence and automatically primes your users’ memory – so choose your words wisely. As an example, in a study – 2 groups of students were primed with 2 different sets of words: one was primed with politeness and the other group was primed with rudeness. At the end of the test, the group primed with polite words had a much lower probability of interrupting the study administrator (17%) than the group who was primed with rudeness (63%).

Visual Hierarchy

The appearance of the things of the page, all of the visual cues, must clearly and accurately portray the relationships between the things on the page.

• Align elements on a grid to keep everything organized.

• Break pages up into clearly defined areas.

• Page names: every page needs a name. The name should be prominent and present on top of each page.

• Group related items together and separate unrelated ones with spacing.

• If something is more important, make it larger. Use large objects for important functions and small objects for functions you would prefer users not to perform.

• Primary buttons or headlines might be larger and brighter than secondary text.

• Things should be “nested” visually to show what's part of what.

• By guiding users with clear headings, highlighted buttons, and logical grouping, the interface becomes much easier to scan and navigate.

Navigation

• Include persistent navigation present on top of the page with local navigation (present vertically and on the left) of the page. Here is an example which shows this – the website allows users to navigate through main web-site content (top-level menu) as well as the general page content which is easily viable on the left:

  

Consistency

• Systems are more usable and learnable when similar parts are expressed in similar ways. Stick to a unified color palette, font family, and icon set. Repeating the same colors and typography across your interface builds familiarity and trust.

• If existing software systems exist with a similar set of features you are looking to implement (and that other users love using) – come up with a design that mimics the functionality rather than suffering from the not invented here syndrome.

• Consistency enables people to efficiently transfer knowledge to new contexts, learn new things quickly, and focus attention to the task they are looking to accomplish.

• Keep the style and appearance of your application (i.e. logos, icons, wording, menus) consistent across screens.

Maximize User Efficiency

• People cost a lot more money than machines. Do not keep users waiting.

• Make it easy for them to do what they want to do rather than asking them to adopt to enforced standards & formats.

• As a single example, forcing customers to enter telephone numbers without normal spacing or punctuation saves a single line of code and a handful of machine cycles. It also results in a lot of incorrectly captured phone numbers. (That’s exactly why phone numbers are broken up into smaller pieces.)

• When possible, format data correctly using automatic methodologies. When not possible, force the users to enter the data in the correct format (think of how much time users waste in finding phone numbers stored in different formats across your system. Do not sacrifice the future for today).

Minimize Cognitive Load

Cognitive load is the amount of mental activity (perception, memory, and problem solving) required to accomplish a goal. As an example, early computer systems required users to remember large sets of commands which they needed to type in specific ways to execute their tasks. The advent of graphical user interfaces that allowed them to browse rather than remember information increased computer adoption by a large factor.

• Eliminate unnecessary information from displays.

• Chunk information to be remembered.

• Provide memory aids to assist with remembering tasks.

• Reduce the amount of steps needed in order to accomplish a task or completely look to automate any complex or memory intensive tasks all together.

Overall, always look to help the user by making their job easier and simplifying the tasks they have to perform. One great example of this is the use of UPC code and bar-codes in the modern world.

Anticipation (and Recognition Rather than Recall)

Bring to the user all the information and tools needed to perform each step of the process:

• Your interface needs to anticipate the user needs and wants. Do not expect users to leave the current screen to search for the necessary information: all the information he or she needs should be in place and should be visible.

• Understand the task domain and the user domain so that you can understand the information that is needed in order to create the solution.

• Make the tool or information present on the screen where users can easily find it.

• Do not ask users to remember information from one part of the interface to another. Information that is needed should be easily retrievable within the interface.

• Provide easy to use auto-completion text boxes and select boxes that limit the burden on users. Avoid having users manually entering information when you can provide assistance to them through easy to use controls.

Maximize Signal-to-Noise

Simplify the presentation and remove and unnecessary elements to maximize the signal-to-noise ratio. The below shows some examples of data presented where removing information from the image results in a better visual presentation (image on the right):

Create Aesthetic Interfaces

Aesthetic designs are perceived as easier to use than less aesthetic designs so:

• Aesthetic designs look easier to use and have a higher probability of being used, whether or not they actually are easier to use – so make your interfaces look beautiful.

• Embrace whitespace (maximize simplicity). Less is more. Remove any non-essential elements and give important items room to breathe. Proper whitespace greatly improves clarity.

• Embrace minimalism: A clean, minimalist layout (think Google’s homepage or Apple’s style) will feel fast and intuitive to users.

• Polished look: consistent colors, crisp icons, balanced spacing – makes the product feel more engaging and trustworthy.

• Create a clear visual hierarchy: Lead the user’s eye by making key elements stand out with size, contrast, or color.

• Proper line spacing and margins in your text also add to a polished, easy-on-the-eyes design.

• Use colors conservatively. Limit the palette to what the eye can process at once glance (about 5 colors depending on the complexity of the design).

• Achieve aesthetic color combinations by using adjacent colors on the color wheel.

• Use warmer colors for foreground elements and cooler colors for background elements.

• Prefer Most Advanced Yet Acceptable: aesthetic appeal is a balancing act between 2 variables: familiarity and uniqueness. You need to balance those 2 in order to find commercial success. According to research, the MAYA principle states that the best design is something that is recognizable as familiar while also being novel and unique is the best formula for success (i.e. software adoption). Make the design easy to use and recognize while adding some novelty in order to make it stand out from the competition.

Contours

• Prefer rounded contours or corners in your interfaces. In experiments where subjects were presented with otherwise similar angled vs contoured objects, subjects strongly preferred more rounded, contoured objects.

• Use angular and pointy features to attract attention and provoke thought: in other words, any alerts or confirmations or system notifications which alerts users to urgent issues or which may require more deep thinking should have sharp angles. Studies show that fear-processing centers and associative memory segments of the brain are activated more using sharp edges than rounded corners. The degree of the angularity is correlated with the strength of amygdala (fear processing activation) so sharper edges evoke more attention.

To sum it up: use rounded corners for most interfaces (that provide a regular flow and do not need special attention) – reserve sharp edges for things that do need attention (like warnings or critical alerts).

Control

• Visibility of system status: the page should always keep users informed about what is going on, through appropriate feedback and in a reasonable amount of time. The user needs to be in control. Predictable interactions create trust in the product and brand.

• Present feedback to the user as quickly as possible (ideally immediately).

• Build trust through open and continuous communication.

• There are two target users for every piece of software: beginning users (which the design should be focused on accommodating) and advanced users (which the design should also make room for).

• Avoid adding too many advanced features for users who want to have shortcuts: focus in on being accommodating to these users without cluttering the interface and provide key-board shortcuts for them to execute actions more quickly.

Input Validation

Validate all user inputs (i.e. garbage in – garbage out dictates that we want to avoid garbage data within our systems).

• Make it hard for users to enter bad or invalid data (i.e. do data validation for all inputs).

• Use constraints to help users input valid data.

  

• Format data according to a consistent standard prior to storing it within the data layer (i.e. keep the data consistent).

• Avoid free form fields which may contain multiple data elements.

• Keep the data consistent (i.e. formatted in the same way within your data layer).

  

Error Messages

Error messages must be written to show:

• Explain what is wrong.

• Tell the user what to do about it.

Avoid displaying system errors or exceptions within error messages (i.e. like error 4421). Provide information which is simply worded and which displays the 2 main things we outlined above.

Forgiveness

Minimize the consequences and occurrences of errors by:

• Using constraints (to limit user input such that only valid data can be used).

• Using confirmations and warnings to reduce the occurrences of errors.

• Using confirmations only on non-reversible and serious actions (since they slow task performance) – if an action is reversible, avoid presenting confirmation dialogs or messages. Also – they should be used sparingly: people do not like continually being interrupted.

• For non-critical confirmations, a select box which allows the user to turn them off should be available.

• Include reversible actions and safety nets to minimize the consequence of errors (i.e. include undo and redo as a possible action).

• Show a clear way to exist the current interaction by including a Cancel button.

• Cancel is particularly important in Wizards. Let people leave at any time and make sure to tell them where they can finish the task later on.

Dialog Design

• Nobody reads your dialog boxes. Use a verb whenever possible instead of 'Yes' or 'OK' because your buttons will make sense out of context with the explanatory text or title.

• Bad Example (in the example below, there is a chance the users will not read what they are confirming):

  

• Great Example (you can be sure that at least users understand that they are about to format a disk):

  

• Apple's Human Interface Guidelines expand on this even further, recommending multi-word verbs instead of "OK" or "Yes" buttons, and clearly defining the suggested regions for an alert box:

  

• Ensure that the default button name corresponds to the action you describe. In particular, it’s a good idea to avoid using OK for the default button.

• Also note that, given a choice between 'No' and 'Cancel', 'Cancel' is almost always better for exactly the same reasons as above: the meaning of 'Cancel' is clear even if the user hasn't read the rest of the dialog box.

Defaults

• Defaults within fields should be easy to remove (i.e. when a user activates a field, the current entry should be auto-selected so that pressing backspace or delete will eliminate the current entry).

• Not everything should have a default: if there isn’t one winner, consider offering no default.

• For defaults that do not have a clear winner and that should be set: choose the default which does the least harm and delivers the most good (i.e. enroll employees into a pension plan to promote savings, or choose to automatically opt-in people for organ donations as an example).

• The default word: avoid using it. If you want the user to revert to standard settings for example, create a button which shows the text ‘revert to standard settings’, ‘use customary settings’, or ‘restore initial settings’. Use terminology which describes what will happen rather than generic terms.

Depth of Processing

• The key determining factors of how deeply information is processed:

  • Distinctiveness of the information (uniqueness of the info with respect to previous experience).

  • Relevance of the information (how important is the information).

  • Degree of elaboration (how much thought is required to interpret the information).

• Use a unique presentation and interesting activities to engage people to deeply process information and to aid in learning. The more deeply learners process information, the better they learn.

• People remember pictures better than words (especially when forming memories) – so use both to aid recall.

Present important items at the beginning or end of a list (rather than the middle) to aid recall. When a list is visual, present important items at the beginning of the list. If the list is auditory, present important items at the end.

User Testing

If you want a great site, you've got to test. Testing reminds you that not everybody thinks the way you do, knows what you do, and uses the web the way you do.

• Testing one user is 100% better than testing none. Even the worst test with the wrong user can show you how you can improve your site.

• How many users should you test? Usually, the ideal number for each round of testing is 3-4. The first 3 users are very likely to encounter all the most significant problems.

• Testing one user early in the project is better than testing 50 near the end. Don't make a big deal out of testing. Do it early and often. Users don't like changes in the system, so it's better to test them early than doing it later and having to change things.

• If users find the interface confusing or slow, simplify it.

• The importance of recruiting representative users is overrated.

• Testing is an iterative process. Testing isn't something you do once. You make something, test it, fix it, and test it again. It's also important that once you fix these problems, you do more rounds of testing, where the users will encounter other problems with your site. It's better to focus on the biggest problems, fix them, and then do further testing.

Key Questions When Starting a Design

If you are designing a new concept or interface, or just getting started on a design project, some great questions to ask prior to doing any design work are shown below:

• What are your goals and objectives?

• What are your top priorities?

• How do you know if you are successful?

• Do you measure success and if so – how do you define it?

• Who do you consider to be your primary customer?

• What are their priorities and goals?

• Why do they use your products?

• What prevents them from using your products, if anything?

• What questions do you have about your users which you do not have answers to?

• Who are your competitors, and how well are you doing compared to them?

• What do you do well?

• What could be improved?

• What differentiates you and your company and your product in the market?

The key point here is to try to find the root-cause of the problem that they are trying to solve, as well as to design something which differentiates them from the existing competition. The Y-Combinator Principle is that creating a product that a small group of users LOVE is much better than designing a product that meets the needs of all users but that most users like. Also, focus on creating something that a market-segment loves and something which is different from what is already available (i.e. blue ocean market).

This ends the first part of my design series. If you enjoyed this content - leave a message or subscribe.

Credits and References

Most of the images available within the blog post were taken from the book Universal Principles of Design referenced below.

1. Universal Principles of Design, Revised and Updated: 125 Ways to Enhance Usability, Influence Perception, Increase Appeal, Make Better Design Decisions, and Teach through Design

2. Don’t Make Me Think

3. 10 Usability Heuristics for User Interface Design

4. First Principles of Interaction Design (Revised & Expanded)

5. The User Experience Team of One

6. Should I use Yes/No or Ok/Cancel on my message box?

Last but not least, many thanks to the paid subscribers of this Substack!! This is an ad-free publication and your help means the world to me. By subscribing, you receive early access to my content, as well as priority when it comes to any future topics which you want me to cover. Thank you for reading and have a great one!

Noting this, here are some of my favorite articles from Hacker news which I’ve made notes on over the last month.

Reflections on OpenAI

Kevin French-Owen (a former OpenAI employee) makes some notes on what his experience was while working for OpenAI which can be found here, and some special items of note which every company can learn from are provided below:

• The company has grown rapidly from 2024 (from 1,000 to 3,000 employees).

• Organization started off more like Los Alamos (many different projects and directions along with researchers probing the cutting edge of science) and one group happened to have accidental spawned the most viral consumer app in history (ChatGPT).

• Everything there runs on Slack (there is no email) so the communication is very open within the org.

• There is no road-map and everything is bottom-up rather than top-down: good ideas come out of anywhere.

• Rather than having a master plan, progress is iterative and uncovered as new research is done.

• Very meritocratic culture (best ideas tend to win) and a strong bias for action (you don’t need to ask for permission – you just get things done).

• Strong bias to just ‘work on your own thing’ and see if it pans out and each researcher is treated as a ‘mini-executive’ (i.e. they decide what they work on).

• The company changes direction quickly and this is valued a lot.

• They value doing the right thing rather than trying to stay ‘on course’ and use new information to steer the project in the right direction – if new information shows that the strategy or plan needs to change, then it’s changed.

• They pay attention what is happening at other companies who are also attempting to tackle AGI (like Google, Meta, Anthropic) and they are pretty sure the other companies are also paying attention to them.

• They have extraordinarily high GPU costs and everything else pales in comparison (not a really huge surprise).

• They are incredibly ambitious in what they want to accomplish. You’d think that coming up with ChatGPT would deter them a bit, but they have plans to dominate / succeed in other areas as well.

• Teams are extremely fluid. If something needs to get done, team member from other teams can easily be on-boarded and brought into the new team (as an example, 2 members of the ChatGPT team joined the Codex team when they needed to hit a launch date).

• Leadership is quite visibly and heavily involved (i.e. every exec was dialed in and communicating with the rest of the team on Slack).

• The team uses a giant monorepo to host the code, and most of the code is written in Python. Everything is run on Azure.

• They have a lot of members which they brought over from Meta / Instagram and their early infra also resembles Meta.

• ‘Most ideas start out as small-scale experiments. If the results look promising, they then get incorporated into a bigger run.’

• Author notes and writes about sleepless nights where he had to work frantically prior to codex launch to make the project happen and on schedule (i.e. they move fast and work hard to make things happen).

The original article goes into much more detail than I did above, but I figured that I’d share some of the key highlights that I made note of. Overall, OpenAI is more like a very driven and very ambitious research lab running decentralized teams that are given freedom to take rapid action and work on interesting and ambitious projects. They also prefer to take actions and ship things quickly, and they made some really smart decisions on the infrastructure side as well which I think greatly helped them out.

How does diffusion / CLIP (AI driven image / video generation) work?

3Blue1Brown (my favorite math Youtuber) and Welch Labs collaborated together to produce a masterpiece which describes the gist of how diffusion and video generation models (like CLIP) work and you can find their excellent explanation here.

For anyone who wants a very high-level summary, it’s provided below:

• In February 2021, a team at OpenAI released a new model architecture called clip. It trained on a data-set of 400 million image + caption pairs scraped directly from the internet.

• Clip is composed of 2 models – one that processes text and the other that processes images. The output of each of these 2 models is a vector of length 512.

  

• In essence, the encoded text and image are supposed to be ‘similar.’ That way, the images could be associated with the textual encoding / representation.

• To ensure that the 512 dimensional vectors are similar, OpenAI used a clever approach: it used a cosine similarity metric (i.e. it took the cosine of each text / image vector and attempted to maximize it).

• To get an intuition of how cosine similarity works, you can use the image below:

  

  You should be able to notice that in the above vector space, the image for cat points in the same direction of the textual representation: and thus the cosine between their vectors is 1 and thus maximizing the cosine similarity between each image and text was done while the opposite was true for non-matching and contrasting image / text pairs: in those instances, the model tried to minimize the cosine similarity (thus attempting to represent dissimilar data as being ‘distant’ from the original vectors in that space).

• In the below image, you can see 3 image / text pairs and the diagonals within the matrix which contain the matching image and text info align (i.e. have a high cosine similarity) while the non-diagonal entries (non-matching image & text) have a more distant similarity score:

  

• The learned geometry of this shared vector space has some really interesting properties. Let’s say that someone takes one image of themselves not wearing a hat, and then right after putting a hat on, takes another image of them wearing a hat – and pass both of these into our image model. In the result set, we get 2 separate 512 dimensional vectors in our embedding space. Now...if you take the vector of that someone wearing a hat and subtract it from the image of them not wearing a hat, we get a new vector in that embedding space.

  Now – you think to yourself – what text might this vector correspond to? Yes – the vector we get from performing the above operation is the encoding of a ‘hat’ (with the 2^nd most similar word being ‘cap’)!!!

  

• In essence, CLIP gives us an excellent way of encoding a shared representation of image and text. However, CLIP only gives us a way of embedding our original image / text data into a shared embedding space in one direction. We have no way of generating actual image / text from our resulting embedding space / vectors.

• A few weeks after GPT3 came out, a team at Berkeley came out with a paper called Denoising Diffusion Probabilistic Models (known as DDPM). The core idea behind these models is pretty straight forward:

  • You take a set of training images and you add noise to the images in a step-by-step manner until the image is completely destroyed:

    

  • From here, we create a neural network to reverse this process (and regenerate the original image) and the neural network is asked to do this in one step:

    

• The surprise? The neural network which is trained and which produces the output doesn’t just attempt to create the final output vector – random noise / steps are added to the final image as well (which produces much better results).

• You can think of the first step of adding noise to a diffusion model as being similar to Brownian motion (i.e. a random walk) across our image space. Each image point is distributed across the image randomly in a step by step manner (and in a random direction). In the below example, we show a spiral image and a sample of where 1 point may go through after taking 100 random steps:

  

• Our model in essence will see many random walks starting from our original image – and we’re essentially asking it to reverse the clock. How can our model learn to do this though? Our ‘walks’ are random after all?

• The essence is that our model is NOT trying to actually predict the actual diffusion paths (i.e. Brownian motions) which took place within our model (i.e. steps 100, 99, 98, .... 3, 2, 1, 0) – the model instead tries to predict the total noise added across the entire image (i.e. the neural network must predict the step from 100 -> 0 in one shot):

  

• In essence, what this means is that for each point in our resulting space (after we’ve added our noise / diffusion), our neural network learns the vector or set of vectors pointing back to the original data distribution and this is extremely valuable information to have – we can grok the general resulting vector field produced by the neural network by looking at the image below (where the neural-network learns to reconstruct the spiral after adding random noise to each point in the spiral image).

  

• An interesting phenomenon occurs after a large number of random steps are added to the model: a ‘phase transition’ occurs where the vector field which points back to our original image stops pointing towards the center of the original image and starts pointing to the actual / original image itself. In other words – adding noise plays a huge role in making sure that our model works.

• In a similar manner, when attempting to reconstruct the original image – adding noise plays a vital role in the reverse direction as well!! Let’s go through an example:

  • Let’s say that we attempt to model a spiral after 100 steps (using our diffusion model).

  • If we don’t add noise as we try to predict where our point will land – we end up with each resulting point taking a walk back to around the center of our original image. As a result – in the below image we show what the neural network reconstructs:

    

  • You should be able to notice that the end result doesn’t really resemble our original image but instead re-constructs a small cluster of points near the center of our image.

  • To enable the model to re-construct the essence of the original image – this is why random noise is added as the model attempts to walk each of the resulting Brownian points back: the randomness ensures that each point is walked back to a more broad representation located within our phase space.

  • In the example below, one sample point is shown and the walk back is given random directions / transitions which results in the point being located away from our center points within our original image:

    

  • Adding this noise is vitally important: once again, it ensures that the model walks back each point and distributes it along our spiral (instead of focusing each one within the center of the image (or towards the ‘mean’ of our image)). Although the result may look chaotic, the final mapping shows a much more representative ‘image’ of our original sample:

    

• Although the above procedure produced fantastic results – the amount of compute needed to generate the models was large and a Google team found a way to simplify the above process (without the need to add random noise).

• To simplify the model, the team turned the original model (modelled by something called a stochastic differential equation which models noise / randomness) into an ordinary differential equation which was equivalent. ODEs ordinary differential equations are much simpler to solve. The change in model allowed their team to lower the amount of compute needed to train the model thus lowering the amount of steps the model needed to take to generate each original image.

• In 2022 – another breakthrough at OpenAI occurred. Using textual data – it was able to produce models which generated outstanding images (which represented the embedded textual information):

  • The model first encoded the textual information into a text encoder which transformed the info into a high-dimensional embedding vector.

  • A diffusion model took that embedding vector and was used to generate an image (or a video) of what the prompt / text describes.

    

• The resulting model is known today as DOLL-E 2 – but there’s one more step they took in order to make the idea work so the process isn’t quite as simple as what we outlined above. In order to generate the image / video, the model they produced uses something called classifier free guidance.

We won’t go into the full details here, but you should watch the entire video if you want a more in-depth overview as the video in general is outstanding and does a much better job in explaining the nuances of how these models work.

Overall, outstanding video once again by Grand Sanderson (3Blue1Brown) and Welch Labs – full credit goes to them for all of the content above (as well as the images).

Hidden Controls: Lamenting Against Today’s Bad UI Usability

The world today is full of bad software. I’m not going to go into the details on the exact reason why I hate modern interfaces, but the below article identifies one of the key reasons I hate most of them: the HIDDEN CONTROLS!!

Stop Hiding My Controls: Hidden Interface Controls Are Affecting Usability

Making controls visible was something that was highly touted by Douglas Engelbart (an early pioneer and legend within the world of UI design). Human memory is fallible so making things recognizable rather than asking your users to recall information should be part of every great UI design; or at least that’s the gist that needs to be taught and re-taught to modern designers and developers.

In the modern era, computing tasks require a significant amount of knowledge that’s not simply recalled. This is the reason why most users don’t love command lines: you need to use memory (i..e. recall) more than recognition, and it’s one of the reasons why many people prefer the modern desktop experience. Rather than typing things into a terminal, you don’t need to think as hard when using a desktop UI: it makes finding information a breeze by easing the memory burden on you (i.e. by using folders and other metaphors to aid your recall). In the early days, UI specialists knew this and recognized that providing aids to people’s recall / memories helped non-specialist users use it more easily (and thus made computing accessible to almost everyone).

Let’s fast forward 40 years. Using a phone flash-light isn’t very intuitive. You have to swipe-down (since this action isn’t broadly visible to most users – nor is its location really intuitive) — discovering this feature is a royal pain in the a**. Another example is Apple Maps in CarPlay requires you to tap on a button located on the bottom left of the screen (which brings up a search option) and this 1) isn’t very usable nor 2) intuitive at all.

This lack of usability is present everywhere today and most software systems are getting worse.

A major source of hidden controls being everywhere is the number of features that are being shipped with modern software: the need for more functionality within modern systems seems to require us to free up screen space to accommodate all of the visible controls, but designers shouldn’t use this as an excuse: you can still create great software that keeps the key functionality visible and the myriad of options more easy to discover.

As an example, the General Motors’ engineers designed an excellent map interface that balances complexity with visibility and it fixes most of the problems that are present within Apple maps interface:

Most great software (and mission critical systems) don’t have hidden controls!! They rely on controls that are

1. Visible

2. Persistent

3. Show the system state

The key actions and functionality within any great software should abide by the 3 properties outlined above! Modern designers need to reevaluate their use of hidden controls and prioritize making modern software more usable.

Why Facts Don’t Change Minds

This is just a fantastic article which goes through a fascinating model which can be used to model beliefs:

https://vasily.cc/blog/facts-dont-change-minds/

The gist of the article is that we can visualize a belief system as a graph i.e. a “network of concepts, values, and connections” and thus, whenever you challenge something – you’re not just challenging that single ‘point’ within the network – you’re also challenging the rest of the nodes and edges which effect the entire graph.

Challenging one point is more like challenging the foundation of a building or cathedral in other words rather than just one brick or stub which supports the building. Of course – most human beings don’t challenge the models which we believe in. Instead – we look for evidence to reinforce them and we actively ignore or dismiss the information which goes against our beliefs. This is why challenging people’s beliefs is so hard. When belief systems are challenged, the real contest isn’t just about exchanging arguments—it’s about trying to reshape the other person’s underlying foundation and structures; it’s about changing the active landscape within their mental world.

The Author uses a graph to illustrate the model of ‘growth-first capitalism’ which is provided below:

In the growth-first capitalism model, we assume that all of the connections are strong and that profits lead to more shareholder returns, which furthermore drives more investment in innovation – but this re-investment in innovation and growth isn’t necessarily true. In a lot of cases, the maximization of shareholder returns may not necessarily lead to more purchasing power for the company. The company shareholders may instead choose to instead keep the profits for themselves and to attempt to ‘milk’ them for their own personal gain as much as possible (thus breaking the self-reinforcing graph we originally outlined):

“Attacking the connections between ideas can force a belief system to reroute its logic, making it less direct, more convoluted, and sometimes less persuasive—even if the system doesn’t collapse outright.”

The above example shows how attacking one node can completely change the model and perception of a system: invalidating one of the key assumptions can make us question whether maximizing shareholder value and going all in on ‘growth first capitalism’ is the best thing to do given the fact that it tends to distribute wealth to a select few rather than re-investing it to benefit the many.

“So when you encounter someone whose worldview seems impenetrable, remember: you’re not just arguing with a person, you’re engaging with a living, self-stabilizing information pattern—one that is enacted and protected by the very architecture of human cognition.”

The whole article and write-up is just fantastic and I highly recommend you give it a read.

The Rise and Downfall of Yahoo

This excellent article does a great job of over-viewing what went wrong (and what also went right) within the rise and downfall of Yahoo:

• Started off as a website called “Jerry’s Guide to the World Wide Web” by 2 Stanford students in 1994.

• Initially, it was more of a directory or portal rather than a search engine: it had categories with linked sub-categories which users could click on to find relevant web-sites.

• To become a ‘portal’ to the internet, they recognized that they needed not just to link to content – they also needed to produce it as well. To do this, they acquired other companies and started producing their own stuff. In 1996, they turned their first profit.

• Between September 1997 and July 2015, Yahoo made 114 acquisitions.

• Some of its acquisitions were disastrous (i.e. acquired Broadcast.com and Geocities.com for 10 billion) while some turned out to be fantastic (i.e. acquired 40% of Chinese e-commerce site Alibaba in 2005 for $1 billion which turned into a 36 billion dollar profit).

• Still – their biggest blunder might have been not making 2 major acquisitions which could have radically changed the trajectory of the company:

  • Yahoo had multiple opportunities to buy Google and failed to do so. In 1998, Yahoo actually turned down the opportunity to acquire Google for $1 million (currently Google has over 1 trillion in market capitalization).

  • Also whiffed at the opportunity to acquire Facebook in 2006 for 1.1 billion dollars (Facebook today is also a company with over 1 trillion in market cap).

• In the end, Yahoo ended up being sold to Verizon in 2016 for about 5 billion dollars (which didn’t end up working out so well for Verizon).

Layoffs at Microsoft

Microsoft announced cuts to 9,000 workers in second wave of major layoffs last month – all while watching their stock / market cap almost double in valuation just over the last 3 months. Their profit margins are also huge. Meanwhile, they’re spending a lot in bringing in cheaper labor from over-seas. Layoffs in the good old days used to be a measure of last resort: you did it when you had to and when your company was very much unprofitable. To lay people off while you are making billions in profit is shameful. It’s an inexcusable thing to do – especially when you’re invested in a technology (an LLM called ChatGPT) that will one day replace a lot of jobs.

I remember when working for a tech-leader used to mean something. So much for that notion.

Noting that, I hope you enjoyed this monthly update.

Black holes are a hugely interesting topic – but many of the explanations of what they actually are are mired in complexity, and I’m not a big fan of complexity and so I decided to do a more in-depth exploration of what they actually are and what they represent. This write-up will outline exactly what black holes are (as well as what their equations mean) in a more intuitive and visual manner. Saddle up – here we go.

What in the world is a black hole?

A black hole, in simple terms, is a region in space where the escape velocity (the speed needed to break free from its gravitational pull) exceeds the speed of light.

Imagine you're throwing a ball upward. If you throw it too softly, it falls back down.

Throw it harder, and it goes higher — but still comes back.

There's a specific speed that you’d need to throw it so it never comes back, and that’s called the escape velocity.

What exactly is earth’s escape velocity?

Let’s derive it. We need to use a few formulas to do this, and they’re provided below.

Gravitational Force Equation

The gravitational force F between two masses is:

In simple terms, the above equation tells us that the gravitational (attractive) force between the 2 masses is directly proportional to their masses (the higher their masses, the stronger the force) and inversely proportional to their distance squared (the further away they are, the weaker the force).

The above equation also abides by the inverse square law. The inverse square law not only applies to the force of gravity, but many different phenomena, including the electric field, light, sound or electromagnetic or nuclear radiation, which also obey this law. The law can easily be pictured through geometric intuition by visualizing an expanding sphere, with the force being distributed throughout the surface area of the sphere.

In other words, the force spreads out in an ever expanding spherical shell as the sphere itself moves radially and the surface area expands by r² and so the field gets weaker as we expand outward.

Gravitational Potential Energy

This is the energy stored between Earth’s gravitational field and our ball. It can be represented by the equation shown below:

The negative sign shows that the ball is bound to the Earth and that gravity works to pull the objects closer. To escape, you’d need to add energy to free it. As you can probably observe, it’s extremely similar to our original gravitational force equation, and it simply outlines that the larger the distance between our 2 masses (r) – the less work we need to do in order to move our ball away from the gravitational pull that it receives from Earth. To use a more intuitive analogy, you can think of gravitational energy as “debt.” You owe the earth for being close to it and the further away you go, the smaller the debt becomes (less negative) until it finally disappears at infinity (where U would equal 0).

The negative sign means gravity traps the ball. To escape, you must repay this energy debt by giving the ball enough kinetic energy (KE) to balance the gravitational potential (U). Unlike the gravitational force (F ∝ 1 / r²), the potential energy (F ∝ 1 / r) weakens more slowly with distance. This is why escaping gravity requires a “lump sum” of energy upfront. In other words, to escape, you must balance this debt with kinetic energy, and that’s essentially what our escape velocity represents.

What is Energy? (The Universe’s Currency)

Energy is the "currency" of the universe — it’s what you "spend" to make anything happen. Think of it like money:

• Do you need to move something? You spend energy (muscle power) in order to move it.

• Do you need to keep a light bulb on? You need to spend energy (electricity – flowing electron energy) in order to keep the light on.

• Do you need to launch a rocket? You need fuel in order to put it in motion.

Energy isn’t really anything physical – it measures your capability of doing something, and it comes in two primary forms: potential and kinetic energies.

Potential Energy: Stored "Savings"

You can think of the potential energy as the “stored” energy of our system. It represents something that’s waiting to be used and depends on the object’s position or state in whatever state-space it currently occupies. As an example, assuming that you put your baseball on a book-shelf – we can say that the object contains ‘potential’ energy by occupying a position on this shelf: if it falls, it can displace or move another object (i.e. do work) through the action of falling from this shelf – and the higher our shelf is, the more gravitational potential energy our ball has. In other words, you can think of potential energy as money in a bank account: you might not be able to directly see it, but it’s there and it’s ready to be spent.

Kinetic Energy: Energy in Motion

Kinetic energy is the energy of movement. It depends on an object’s current speed and mass. If you throw a baseball, it’s movement represents kinetic energy. The heavier the ball is or the faster you throw it – the more kinetic energy it has. You can imagine kinetic energy as being the cash you have in your wallet: it represents energy currently being used / put to work.

Total Energy

Now, energy is constantly being swapped between the 2 key forms we outlined above, and it cannot be created nor destroyed. In other words: we need to balance our account. To put this more formally, we say that total energy is always conserved. To escape Earth in other words, a rocket (or ball) must convert its kinetic energy into ‘climbing’ out of Earth’s gravitational ‘well’ (potential) energy to balance the books. To escape Earth’s gravity, an object needs enough kinetic energy (cash in hand) to pay off its gravitational potential energy (debt - i.e. money owed to Earth). Let’s balance the cosmic "checkbook" step by step.

The Energy Budget

As we should already know, at launch, our baseball has two types of energy:

• Kinetic Energy (KE) (i.e. the energy from its current motion):

  

• Gravitational Potential Energy (PE) (i.e. the ‘stored’ energy from being near the Earth):

Although we may have used the analogy of potential energy as being the “stored energy” in our system (thus being similar to money in a bank account) – in the above equation, we have a negative sign which we can now associate as being equal to owing your bank money (i.e. as being analogous to the debt you owe) and thus, positive energy terms equate to ‘stored’ money while negative terms equate to ‘owed’ money and we thus use the more appropriate debt term to represent this situation.

The total energy is equal to the kinetic energy plus the potential energy:

Total Energy = KE + PE

To escape Earth forever, the object must eliminate the potential energy within our books (i.e. no cash, no debt). So:

In order to find the escape velocity in other words, we need to solve for v. To do this, we first move the potential energy term to the other side:

Then, we cancel out the mass term from both sides:

To make the equation cleaner, we simply multiply both sides by 2:

Finally, we simply take the square root to obtain our formula for v (the escape velocity that we need in order to ‘balance or books’):

Let’s explain the above formula in a much more intuitive manner.

In order to escape’s Earth’s gravitational field, the distance between our ball and the Earth center of mass must be infinity:

In order to make the potential (U) equal to zero, r must be equal to infinity. Of course it will never reach infinity (because this is impossible) but conceptually if it were to stop before it gets to infinity, then there will be some potential energy because the potential energy never really goes away. But, hypothetically, when our distance is equal to infinity, the ball will eventually stop moving (so our kinetic energy at this point will be 0 (i.e. we will have 0 cash in our wallet)). When this happens, our potential energy will also be 0 as well (since we will no longer be effected by Earth’s gravitational energy) and so both of our energy terms would be 0 (0 debt and 0 cash). But energy can’t really be created nor destroyed. Essentially, this means that our total energy (kinetic + potential) must be equal to 0 all along this particular trajectory.

When you’re standing on the Earth’s surface, you’re deep in this "gravity well" so you have a big ‘debt’ (your potential energy is a large negative number) which you owe to Earth. If you want to leave Earth forever, you need enough cash (kinetic energy) to completely pay off that gravitational debt — and to totally escape it, your distance from Earth must be equal to infinity wherein both our potential and kinetic energy terms would be equal to 0. Since this is the case, what we’re doing with our equation above is simply ‘balancing’ our books and calculating how much kinetic energy (‘cash’) we would need in order to ‘pay off’ or balance the potential energy ‘debt’ such that the total within our system equals 0. To do this, we combine the equations (using the 2 terms we provided above), set them to equal 0 (KE + PE = 0) to ensure that our ‘books balance’ and we solve for v to obtain our escape velocity.

Deriving Earth’s Escape Velocity

Noting this, let’s continue on and plug in the right numbers to finally derive our finalized escape velocity.

Plugging in the appropriate values we obtain:

And so – Earth’s escape velocity is exactly 11,186 meters per second, and this equates to 11 kilometers per second (km/s). What we're saying here is that the escape velocity from the earth is 11 kilometers per second (which is approximately equal to 7 miles per second) and this is approximately equal to 25,000 miles per hour. That's quite a lot of up-front energy we have to put up in order to get our baseball out of Earth’s gravitational field!

This might seem like a lot, but it’s nothing compared to a black hole!!! You'll should be able to notice that 7 miles a second is a long way short of the speed of light which is 3 times 10 to the 8 meters a second.

Let’s remind ourselves of what a black hole once again represents:

“A black hole, in simple terms, is a region in space where the escape velocity exceeds the speed of light.”

So, we should be able to easily see that the earth is nowhere near to a black hole – a black hole needs an escape velocity of greater than the speed of light and earth’s escape velocity (of 11,000 meters per second) is nowhere near this!!

So let's just remind ourselves of what the original formula is for calculating the escape velocity:

You'll notice that v squared will get greater as r gets smaller. So – let’s do a little thought experiment. Let’s say that you take the earth and squash it (retaining all it’s current mass) so that it covers a much smaller area:

How much would we need to shrink it in order to make the v (the escape velocity) equal to the speed of light??

Well, we need to use mathematics to do this, and it’s actually quite simple. First, we take our original formula:

We want to know when the escape velocity v = c (the speed of light) and so we set v equal to c:

Now, to simplify things even further (so we can get the formula for r), we square both sides:

We re-arrange the above to solve for r and we get:

This is known as the Schwarzschild radius!!

This is the radius at which an object becomes a black hole. Let’s not stop here though – let’s plug in the appropriate values to find our radius and so now we have:

Plugging the values in we get:

After simplifying we finally get:

So -- if you compress the Earth into a sphere that has a radius of about 8.5 millimeters (a diameter of about 1.7 centimeters) it would become a black hole!!

In other words, if we could shrink all of Earth’s mass (about 6 times 10²⁴ kilograms) into a sphere the size of a marble (~1.7 cms), the escape velocity would exceed the speed of light and at that point, and Earth would become a black hole from which not even light could escape!!

You may not be able to comprehend the gravitational force which Earth would have should someone, somehow, find a way to do this, so let’s do another thought experiment. Let’s inflate Earth once again to its regular size and see what impact placing a 1 kg weight on it has.

To do this, we’ll once again need our gravitational force formula and we’ll need to plug in the appropriate constants / inputs to obtain:

So we plug in the appropriate values to get:

We simplify even further to obtain:

We obtain 9.8 Newtons which we’ll round up to 10 to simplify things:

To be more precise, a 1 kg mass at Earth’s surface feels a gravitational force of about 10 Newtons (N). This may seem like a rather silly question, but now we ask: why doesn’t our 1 kg weight fall through our floor?

We know that gravity is “pulling down” on the 1 kg weight with about a 10 Newton force. If gravity were the only force acting on our weight, it would indeed accelerate downward!! In reality, the earth’s surface needs to “push back” upward with an equal and opposite force of about 10 Newtons in order to keep our weight in its stable position – but where does this force come from?

Well, we should all know by now that the floor in your house (or living area) is made out of concrete / wood, and these materials are made of atoms and molecules bound together. Those atoms form a rigid lattice held by electromagnetic (chemical) bonds. The best visual for this is to imagine these atoms and molecules being bound by springs that hold them together (and in equilibrium):

Now, you can imagine these springs as representations of the bonds which hold you floor (or any solid) together – and this is really why a solid is...well solid – there are ‘spring like’ bonds between these atoms which hold it together.

Now – you can imagine our 1 kg weight sitting on the floor: the weight’s force downward isn’t carried by one tiny point; it’s spread across the floor it is sitting on. Because the load is shared, no single spot gets overloaded and the floor doesn’t collapse.

In the same way, when you press on a solid material, that force is passed along through its network of atomic or molecular bonds – each bond takes just a tiny piece of the total load, so the material as a whole holds up rather than breaking.

You can think of it as being a trampoline woven by trillions of microscopic strings (atomic bonds). When you place a load (like a 10 Newton force) on these springs, it spreads out among them and each spring barely stretches. This means that those forces are stronger than the 10 Newton force (applied by gravity) to the 1 kilogram weight and so the 1 kilogram weight doesn't fall through the floorboards due to the molecular structure of the floorboards withstanding it.

But…what if gravity turned into a cosmic wrecking ball?

Now, let’s go back to squishing earth down to the size of an ordinary marble.

The question which I want to ask: what is the force acting on that 1 kilogram weight resting on its surface?

In this instance, we once again go back to our original formula and plug in the appropriate values:

Plugging in our values we get:

And finally, the final force in Newton’s is:

The answer: the force on our 1 kilogram weight is now an astounding 10¹⁹ Newtons. Suddenly, gravity isn’t a gentle tug—it’s a crushing avalanche.

In long form, this equates to:

10,000,000,000,000,000,000 Newtons

No known force (atomic or molecular) can resist this type of force!! Picture those trusty atomic springs again: on normal Earth, they share the load and they are thus able to over-come the gravitational potential energy from almost every object placed on them. Now – on marble-sized Earth, gravity’s force is a billion (1,000,000,000) times a billion times heavier: it’s like stacking every mountain on Earth onto a single spring. No atomic bond can withstand this!! The springs are ‘vaporized’ thus making gravity the winner!

In other words, the gravitational pull from Earth’s ‘surface’ drags everything inward. If you could squash the earth down to the size of a marble – the gravitational force would be so huge that nothing could stop it (due to Earth’s atomic bonds not being strong enough to hold it together) and it would thus carry on contracting under its own gravitational force and shrink until it essentially became a dimensionless point!!

This point is what we call a singularity. The point of no-return around the black hole from which nothing can escape is called the event horizon.

At that stage, all of Earth’s mass resides in a point with zero volume. If it had any width (or volume), the tremendous gravitational energy would just pull it in.

There is a radius around that singularity of 8 millimeters which says that anything inside that radius has an escape velocity of greater than the speed of light. The boundary is what we call the event horizon because gravity inside remains unstoppable and all matter inevitably falls into the singularity.

This is in essence of what a black hole represents: a celestial object so dense that no nearby matter or radiation can escape its gravitational field.

Often, this is described as the boundary within which the black hole's escape velocity is greater than the speed of light. However, a more detailed description is that within this horizon, all light-like paths (paths that light could take) and hence all paths in the forward light cones of particles within the horizon are warped so as to fall farther into the hole. Once a particle is inside the horizon, moving into the hole is inevitable.

How to Create a Black Hole

Now, you have to ask yourself: this is all great and it sounds interesting, but how in the world can a black-hole be created? Squashing the earth to the size of a marble doesn’t seem like a very feasible plan, after all – so how does the universe do it?

In nature, black holes aren’t made by squeezing planets—they’re born from the corpses of massive stars. And when I say massive, I mean stars at least 20 times heavier than our Sun. What keeps a star from collapsing in the first place is a process called nuclear fusion.

For most of its life, a massive star is caught in a tug‑of‑war between two forces. Gravity constantly tries to pull all its mass inward, while the heat and light generated by nuclear fusion push outward. You can think of fusion as the star’s “engine,” taking hydrogen nuclei and slamming them together into helium, releasing energy in the process.

Whenever two lighter nuclei fuse into one heavier nucleus, the process sheds a bit of mass as energy (E = mc²). That energy emerges as heat and light, creating an outward pressure that exactly counters gravity—so long as there’s hydrogen fuel in the core, the star stays intact and continues shining.

After millions of years, the hydrogen in the core starts to run out. The core contracts under gravity, heats up, and begins fusing helium into carbon, oxygen, and so on—building successively heavier elements all the way up to iron. Every new fusion stage lasts a shorter time than the last, but fusion of iron is special: it consumes energy instead of releasing it. Once a critical mass of iron builds up, there’s no more outward pressure from fusion. In an instant, gravity gains the upper hand.

Now, remember those atomic "springs" that held up our 1 kg weight? Here, they don’t just fail—they’re annihilated. Gravity ramps up to unimaginable forces (around 10³⁰ Newtons per square meter) and the imploding core reaches densities so extreme that protons and electrons merge into neutrons. If the remaining core mass is around 1.4–3 solar masses, neutron degeneracy pressure stops the collapse, leaving behind a neutron star. But if the dead star’s core is more massive—say, five or ten times our Sun’s mass—the neutron star gambit fails. Even neutron degeneracy pressure (the quantum mechanical force that keeps neutrons from being squeezed into the same space) can’t halt the collapse. In a dramatic final act, nothing remains to hold the core up, and it shrinks below its own Schwarzschild radius (the point of no return) becoming a black hole. The star’s outer layers rebound off the collapsing core and explode outward in a brilliant supernova, briefly outshining an entire galaxy, while the core vanishes into a singularity.

In other words, creating a black hole is less like squeezing the Earth into a marble and more like building up a massive star, letting it burn brightly for millions of years, then watching as it exhausts its nuclear fuel and succumbs to gravity’s ruthless pull. From the fiery forge of fusion to the final, crushing collapse, black holes are born in one of nature’s most spectacular fireworks shows—and therein lies gravity’s ultimate triumph.

Final Note

A big thank you to all my supporters and to my paid subscribers – this is an ad free publication and I appreciate the support!!

Credits / References

Stock photos:

• Earth image: https://pixabay.com/illustrations/earth-world-planet-globe-1303628/

• 1 kg weight image: https://openclipart.org/detail/190320/1kg-weight

• Event horizon (Wikipedia entry): https://en.wikipedia.org/wiki/Event_horizon

• Fusion image source: https://commons.wikimedia.org/wiki/File:FusionintheSun.svg

Introduction

Combinatorics is the branch of mathematics that studies how objects can be arranged, combined, or selected according to specific rules with a focus on patterns and relationships. By focusing on these relationships, we can gain a deeper understanding of how to model and count the objects in question. This intuition helps us solve many complex real-world problems including how to secure digital communications, understand genetic sequences, and efficiently plan trips. It’s influence extends to areas like artificial intelligence, cryptography, and game theory, making it indispensable for both theoretical research and real-world problem-solving.

The Pigeon Hole Principle

The Pigeonhole Principle is a simple yet powerful concept in combinatorics. It states that if you have more "pigeons" than "pigeonholes" and you want to put each pigeon into a pigeonhole, at least one pigeonhole must contain more than one pigeon.

In mathematical terms: If n items are put into m containers and n>m, then at least one container must hold more than one item.

Simple Example

Suppose you have a group of 13 people. Show that at least two of them were born in the same month.

Solution:

1. Identify the pigeons and pigeonholes: Here, the "pigeons" are the 13 people, and the "pigeonholes" are the 12 months of the year.

2. Apply the Pigeonhole Principle: Since there are more people (13) than months (12), by the Pigeonhole Principle, at least one month must have at least two people born in it. Thus, at least two people in this group were born in the same month.

Mathematical Induction (i.e. One Step at a Time)

Weak Induction

Weak induction, often simply called mathematical induction, is a proof technique used to show that a statement is true for all natural numbers. It works by proving two main steps:

1. Base Case: Prove that the statement is true for the first natural number (usually n=0 or n=1).

2. Inductive Step: Assume that the statement is true for some arbitrary natural number k (this assumption is called the inductive hypothesis). Then, use this assumption to prove that the statement is true for k+1.

Simple Example

Let's use weak induction to prove a simpler statement:

The sum of the first n odd numbers is n²

Proof:

1. First, we start off with our base case (n = 1):

   The first odd number is 1. Since:

   1 = 1²

   The base case holds.

2. Next, we go on to our inductive step. Assume that the statement is true for some arbitrary natural number k. That is, we can assume:

   1 + 3 + 5 + … + (2k − 1) = k²

   We need to prove that the statement is true for k+1. That is, we need to show:

   1 + 3 + 5 + … + (2k − 1) + (2k + 1) = (k + 1)²

   Using our original inductive hypothesis:

   1 + 3 + 5 + … + (2k − 1) = k²

   And subbing this into our new formula:

   1 + 3 + 5 + … + (2k − 1) + (2k + 1) = (k + 1)²

   1 + 3 + 5 + … + (2k − 1) = k²

   We get:

   k² + (2k + 1) = (k + 1)²

   And since we know that:

   (k + 1)² = k² + 2k + 1

   We know that our above statement holds and that:

   k² + (2k + 1) = (k + 1)²

   = k² + 2k + 1

   Since the statement holds for k+1, by the principle of mathematical induction, the sum of the first n odd numbers is n² for all natural numbers n.

Strong Induction

Strong induction, also known as complete induction, is similar to weak induction but with a slight difference in the inductive step. For weak induction, to prove that P(k+1) is true, you only assume that the previous step P(k) is true. For strong induction, you assume that all of the previous steps ( P(0), P(1), P(2), … , P(k) ) are true and use this to prove P(k + 1). In other words, weak induction is when you only use the immediately previous step. Strong induction is when you can use any previous step.

Simple Example

Let's use strong induction to prove a simpler statement:

Every natural number n ≥ 2 can be written as a product of one or more prime numbers.

Proof:

1. First, we start off with our base case (n = 2):

   2 is a prime number, so it can be written as itself, which is a product of one prime number, thus the base case holds.

   Next, we go on to our inductive step.

2. Assume that for all natural numbers m such that 2 ≤ m ≤ k, m can be written as a product of prime numbers. Now we need to prove that the statement is true for k+1:

   1. If k + 1 is a prime number:

      1. Then k + 1 itself is a product of one prime number (itself).

   2. If k+1 is not a prime number:

      1. Then k+1 can be written as a product of two natural numbers, say a and b, where 2 ≤ a ≤ k and 2 ≤ b ≤ k.

      2. By the inductive hypothesis, both a and b can be written as products of prime numbers because they are both less than or equal to k.

      3. Therefore, since k + 1 is a product of a and b:

         k + 1 = a * b

         We know that it can be written as a product of prime numbers which make up a and b:

         k + 1 = (primes which produce a) * (primes which produce b)

      Since the statement holds for the base case and the inductive step has been proven, by the principle of strong induction, every natural number n ≥ 2 can be written as a product of one or more prime numbers.

Elementary Counting Problems

Permutations

In simple terms, a permutation is a way of arranging items. The number of permutations of n distinct items is given by n! (n factorial), which is the product of all positive integers up to n. For 3 items, there are

3! = 3 × 2 × 1 = 6 permutations

Let’s consider a simple set of 3 items composed of A, B and C. If we want to list all the possible permutations (arrangements) of these three items, they would be:

1. ABC

2. ACB

3. BAC

4. BCA

5. CAB

6. CBA

Can you see why n different arrangements of n distinct items is n! (n factorial)?

We can get a good intuition for this by imagining that n people arrive at a doctor’s office at the same time. The doctor needs to treat them one by one, so they need to decide the order in which they’ll be served. Assuming there are n patients, there are n different choices which we can make on who to serve first. Knowing this then – how many choices do we have for which person goes second?

Well, we can observe that we already made a choice of who goes first, and we thus now have n – 1 patients to choose from for our 2^nd choice. Once we make this choice, we have 1 less patient to once again choose from (thus obtaining n – 2) for our 3^rd patient. Therefore, the number of orders in which the patients can be chosen in is:

n × (n - 1) × (n – 2) × ... × 2 × 1

= n! (n factorial)

Stirling’s Formula:

Stirling's formula is an approximation used to estimate the factorial of a large number n:

Stirling's formula becomes more accurate as n increases, making it a powerful tool for estimating factorials in mathematical analysis and in algorithms that deal with large numbers.

Permutations with Multisets

In our first example, we had a number of items (patients) and we assumed that no items could be repeated – but let’s now assume that we can have repetitions:

Example: Find the number of ways to arrange the letters in the word "AAB".

Solution: there are 3 distinct permutations of these letters provided below:

1. AAB

2. ABA

3. BAA

Notice that in this example, if all items were distinct, our formula would be 3! which equates to 6, but this time we only have 3. Why is this?

Well, and in multiset problems – we need to account for the fact that when items are repeated – some arrangements are identical because swapping identical elements results in an identical arrangement. If we were to look at our above example, we know that swapping two A’s in the AAB or ABA and BAA combinations doesn’t change the arrangement, so to adjust for the over-counting, we need to divide by the number of ways we can arrange the identical items within our set.

Thus, if you have n items in total, with n1 items of one type, n2 items of another type, and so on, the number of permutations is:

In the above formula, the numerator (n!) simply represents the total number of permutations were unique, and the denominator is our ‘adjustment factor’ created by the fact that we have identical elements where n1!⋅n2!⋯nk! adjusts for the fact that n1 items are identical, n2 items are identical, and so on, thereby correcting for over-counting.

In our earlier example (involving the unique permutations of ‘AAB’), we would obtain:

The above results in:

Another multiset example:

A box contains 3 red balls, 2 blue balls, and 1 green ball. How many distinct ways can you arrange these 6 balls in a row?

Solution:

• First, we count the total number of elements we have:

  • Total balls = 3 red + 2 blue + 1 green = 6

• Next, we identify the repetitions:

  • Number of red balls (n1) = 3

  • Number of blue balls (n2) = 2

  • Number of green balls (n3) = 1

Apply the multiset permutation formula:

We thus know that there are 60 distinct ways of arranging 3 red balls, 2 blue balls, and 1 green ball in a row.

Strings Over a Finite Alphabet

Next, we’ll examine problems where we aren’t merely arranging a given number of objects with specified usage limits. Instead, we’ll be creating strings or words using symbols from a finite set, called a finite alphabet. While we won’t need to specify how many times each symbol should appear, we might impose a restriction that each symbol can only appear once.

Example: Suppose you have an alphabet with 3 symbols: {A,B,C}. How many different 3-digit strings can you create using these symbols?

Solution: This time, we don’t have any limitations of which symbol we can pick in each position, so we can use each one of the 3 letters in our alphabet to get:

1. AAA

2. AAB

3. AAC

4. ABA

5. ABB

6. ABC

7. ACA

8. ACB

9. ACC

   …

In the above examples, we enumerated over all of the possible combinations starting with the letter A – but we can keep going and substitute B and C as well. Enumerating over all arrangements thus nets us 3 * 9 = 27 different 3-letter strings.

We should be able to see the formula just by using our intuition:

Total number of possible combinations = (3 choices from A, B or C) × (3 choices from A, B or C) × (3 choices from A, B or C) = 3 × 3 × 3 = 27.

Our final solution is thus 3³ = 27

Formula: The formula n^k calculates the number of possible k-digit strings when using an n-element alphabet.

If you have n different symbols and want to create k-digit strings without repeating any symbols, the number of possible strings is given by the formula:

n × (n - 1) × (n – 2) × ... × (n – k + 1)

which can also be equivalently written down as:

Example:

If you have 5 symbols (A, B, C, D, E) and want to create 3-digit strings:

• Choose and arrange 3 out of 5 symbols:

  5 × 4 × 3 = 60

• This is equivalent to using the formula:

  

In both cases, you get 60 possible 3-digit strings without repeating any symbols.

Bijections:

Imagine a large parking lot with many cars and many parking spaces. We don’t know the exact number of cars, but we know there are exactly 500 parking spaces. We suspect there are also 500 cars, but we’re not certain. How can we quickly test this idea? We can ask each car to park in a space. If every car finds a space and every space is filled, then we know there are exactly 500 cars. However, if some cars can’t find a space, it means there are more than 500 cars. Conversely, if some spaces remain empty, then there are fewer than 500 cars.

This technique of matching two sets element-wise and then conclude (in case of success) that the sets are equinumerous is very often used in combinatorial enumeration. Let us put it in a more formal context.

Let X and Y be two finite sets, and let f : X → Y be a function so that

1. If f(a) = f(b), then a = b, and

2. For all y ∈ Y there is an x ∈ X so that f(x) = y,

Then we say that f is a bijection from X onto Y. Equivalently, f is a bijection if for all y ∈ Y , there exists a unique x ∈ X so that f(x) = y.

In other words, a bijection matches the elements of X with the elements of Y , so that each element will have exactly one match.

The functions that have only one of the two defining properties of bijections also have their own names:

Injection:

If f satisfies criterion (1) (i.e. f(a) = f(b), then a = b) then we say that f is one-to-one or injective.

In the above image, we can observe that every element of set A is mapped to a unique element in set B. Further, if any element is set B is an image of more than one element of set A, then it is not a one-to-one or injective function.

Surjection:

If f satisfies criterion (2) (i.e. for all y ∈ Y there is an x ∈ X so that f(x) = y), then we say f is onto or surjective.

In other words, in a surjective function, every element of set B has been mapped from one or more than one element of set A. Also, the functions which are not surjective functions have elements in set B that have not been mapped from any element of set A.

Finally, two sets have the same size if and only if there is a bijection between them. Knowing this, we can use bijection to solve many combinatorial problems by converting one problem into another. Let’s show an example which illustrates this concept.

Simple Example:

How many ways are there to distribute 4 identical candies to 3 children?

Solution: To solve this problem, we'll use a bijection to relate it to a problem involving "stars and bars," a common combinatorial technique. Let’s first restate the problem in terms of stars and bars:

Think of the 4 candies as stars (∗) and the separations between different children as bars (∣). For example, distributing candies so that the first child gets 1 candy, the second child gets 2 candies, and the third child gets 1 candy can be represented as:

Some other examples are provided below:

1 candy for first child, 1 candy for second child, 2 candies for third child:

1 candy for first child, 3 candies for second child, 0 candies for third child:

0 candies for first child, 4 candies for second child, 0 candies for third child:

Seeing the bijection: we should be able to easily see that the problem of distributing 4 candies to 3 children is equivalent to finding the number of ways of arranging 4 stars and 2 bars in a sequence by looking at our examples, and we can thus proceed.

To solve the problem, we can treat it as a permutation problem involving multisets. We have count the ways of arranging 4 stars and 2 bars to get:

• The total number of symbols (stars + bars) is 4 + 2 = 6, thus 6 is our numerator.

Since we have 4 identical stars and 2 bars, these are the values we have in the denominator to produce:

Using bijection, we transformed the problem of distributing candies into the problem of arranging stars and bars and so, we now know that there are 15 ways of distributing 4 identical candies to 3 children.

Choice Problems

In a local raffle, six numbers are chosen at random from a set of 50 numbers. To win the grand prize, you need to guess all six numbers correctly. How many different combinations are there to guarantee a win?

This type of problem is known as a "choice problem" in combinatorics. In these problems, we select certain subsets from a given set, often requiring the subsets to have a specific size. Unlike previous examples where the order mattered, here the order of the elements in the subset does not matter; for example, {5, 7, 9, 14} and {7, 5, 9, 14} are considered the same subset of the set of 50 numbers.

The number of ways to choose a subset of k elements from a set of n elements is crucial in combinatorics, so it has its own symbol and name which we denote as “n choose k” and which we represent below:

If you want to find out how many ways you can choose 6 numbers out of 50 distinct ones, we simple plug this into our formula to get:

Another Example:

A committee of 5 people is to be formed from a group of 10 candidates. How many different ways can the committee be formed?

Solution:

To solve this problem, we need to find the number of ways to choose 5 people out of 10.

• Identify n and k:

  • n is the total number of candidates, which is 10.

  • k is the number of people to be chosen, which is 5.

• Apply the Binomial Coefficient Formula:

  

The above gives us 252, so there are 252 different ways to form a committee of 5 people from a group of 10 candidates.

Enumeration / Counting Summary

The following table summarizes the enumeration theorems we went over in this section:

The Binomial theorem and Related Identities

The Binomial Theorem

The binomial theorem states that:

The theorem tells us how to expand (multiply out) a binomial that is raised to a power n. Let's look at a concrete example to make this clearer.

Example:

We can write out this expansion using our binomial theorem / formula:

Let’s calculate each term using the formula above:

• When k = 0:

  

• When k = 1:

  

• When k = 2:

  

• When k = 3:

  

When we combine all of our terms we finally get:

We should be able to see that the binomial theorem provides a way to expand expressions of the form (a+b)ⁿ into a sum of terms involving powers of a and b. The coefficients of these terms are given by the binomial coefficients, which can be calculated using factorials. This theorem is widely used in algebra and combinatorics.

Pascal’s Triangle

Pascal's triangle is a triangular array of numbers. Each number is the sum of the two numbers directly above it in the previous row. The first few rows of Pascal's triangle look like this:

Another fantastic illustration:

The numbers in Pascal's triangle are the binomial coefficients n choose k. Each row n of Pascal's triangle corresponds to the coefficients of the expanded form of (a + b)ⁿ.

As an example, the 4th row of Pascal's triangle is:

The binomial theorem states:

And when we expand it out we get:

And this relates directly to the row in our triangle:

The Binomial Theorem (General Version)

Let me be any real number. Then:

where the sum is taken over all non-negative integers n.

Divide and Conquer. Partitions.

In our previous coverage, we mainly focused on lists of objects, distinct or not, with repetitions allowed or not, and with the order of the elements on the list being relevant or not. Here, we will go one step further by discussing distribution problems.

Compositions

Suppose we have fifteen identical candies and we want to give them to three children: Emma, Jack, and Lily. Since the candies are identical, what matters is how many candies each child gets. To find out the number of ways we can distribute these candies, we need to know how many ways we can write 15 as a sum of three non-negative integers. The order in which the integers are written matters because it represents different distributions. For example, 3 + 7 + 5 is different from 7 + 3 + 5 because in the first case, Emma gets three candies, while in the second case, Emma gets seven.

Definition:

A sequence (a1, a2, … , ak) of integers where each ai ≥ 0 and a1 + a2 + ⋯ + ak = n is called a weak composition of n. If, in addition, each ai is positive for all i, then the sequence (a1, a2, … , ak) is called a composition of n.

Example:

If we have 15 candies and want to distribute them among Emma, Jack, and Lily, one possible weak composition of 15 could be:

• Emma gets 2 candies,

• Jack gets 4 candies,

• Lily gets 9 candies.

This can be written as the sequence (2, 4, 9).

If the sequence were (4, 9, 2), it would mean Emma gets 4 candies, Jack gets 9 candies, and Lily gets 2 candies.

These different sequences represent different ways of distributing the candies, even though the sum is still 15.

Formula for Computing Weak Compositions

The number of weak compositions of a positive integer n into k parts is given by the binomial coefficient:

Example:

Let's solve the example of distributing 15 candies to three children (Emma, Jack, and Lily) using this formula.

Identify n and k:

• n = 15 (the total number of candies)

• k = 3 (the number of children)

Apply the Formula:

The number of weak compositions is given by:

Calculating the binomial coefficient 17 choose 2 gives us 136 and so there are 136 different ways to distribute 15 candies among 3 children.

For all positive integers n and k, the number of compositions of n into k parts is:

Example: Let's solve an example using this formula. Suppose we want to find the number of compositions of the number 8 into 3 parts.

Let’s first use our formula and plugin the correct components to find the correct solution:

Identify n and k:

• n = 8 (the total number to be split)

• k = 3 (the number of parts)

Apply the formula:

The above gives us an answer of 21, so there are 21 different compositions of 8 into 3 parts.

Explanation: A composition of n into k parts means breaking down n into k positive integers where the order of the integers matters. For example, the compositions of 8 into 3 parts could include sequences like (1, 2, 5) and (2, 1, 5), which are considered different because the order is different.

We can visualize this problem as a bijection into placing 2 dividers into 7 possible slots, so we need to choose 2 dividers out of our 7 available ones. As an example, the number 8 can be represented as:

And you should be able to visualize the possible slots we have to select from through the visual presented below:

To divide these 8 items into 3 parts, we must choose 2 of the available slots which we presented above. As an example, the partition (2, 4, 2) can be represented as:

As another example, the partition (2, 1, 5) could be represented as:

Since we need to select (choose) 2 slots out of 7 possible ones, we should be able to see the intuition behind our formula!!

For all positive integers n, the number of all compositions of n is:

Example and Explanation:

A composition of n is a way of writing n as a sum of positive integers where the order matters. For example, for n=4, the compositions include sequences like 1+1+1+1, 2+2, 1+3, and so on. When we compose n, we are effectively placing n−1 potential "dividers" between the numbers 1 through n−1 to create different parts.

As n example, to compose 4 into parts, we can visualize our possible dividers through the diagrams presented below:

For each position between the numbers, we have two choices: place a divider or not to place a divider. Since we have n−1 positions where we can either place or not place a divider, each position represents a binary choice. The total number of ways to make these choices is 2ⁿ⁻¹ because there are n−1 positions and each position has 2 options. In the above example, we have 3 slots to choose from so our answer is:

Set Partitions

Let’s take the example of assigning balls to boxes. Partitions refer to instances where we assume that the balls are different, but the boxes are not.

A partition of the set [n] means dividing the set into non-empty groups so that each element is in exactly one group. The number of ways to partition [n] into k non-empty groups is represented by S(n, k). These numbers are called the Stirling numbers of the second kind.

Simplification:

A partition means dividing a set into groups where each group has at least one element and no element is left out. As an example, for set [3] = {1, 2, 3}, one partition into 2 groups could be {{1, 2}, {3}}.

We use [n] to denote the set of the first n positive integers. As an example [2] = {1, 2}.

When we use S(n, k) notation, we mean Stirling numbers of the second kind and it denotes the number of ways to partition the set [n] (first positive n integers) into k non-empty groups.

As an example, S(3, 2) is the number of ways to partition {1, 2, 3} into 2 groups. There are 3 such partitions and they are:

If we have to put n different balls into k different boxes then the number of ways to do this is k! · S(n, k). Indeed, first we can partition [n] into k non-distinguishable blocks in S(n, k) ways, then we can label the k blocks with labels 1, 2,...,k in k! different ways.

The number of all set partitions of [n] into nonempty parts is denoted by B(n), and is called the nth Bell number. We also set B(0) = 1.

Example:

B(3) is the number of ways to partition {1,2,3}. There are 5 such partitions:

There are 5 partitions, so B(3) = 5.

Integer Partitions

Now, let's assume that both the balls and the boxes are identical. This means the only thing that matters is the number of balls in each box. We want to find out how many ways we can write the number n as a sum of positive integers, where the order doesn't matter. For example, 4 = 3 + 1 is considered the same as 4 = 1 + 3.

More formally, let a1 + a2 + … + ak be the integers which equal to n. The sequence (a1, a2,...,ak) is called a partition of the integer n. The number of all partitions of n is denoted by p(n). The number of partitions of n into exactly k parts is denoted by p_k(n).

We note that the word “partition” is used in a new meaning here.

Example:

The positive integer 5 has 7 partitions. Indeed, they are:

Therefore, p(5) = 7.

The problem of finding an exact formula for p(n) is even harder than that of finding an exact formula for S(n, k) (for which there is no formula). The approximate size of the number p(n) is provided by the following asymptotic formula:

Section Summary

A table summarizing our results is provided below:

Cycles in Permutations

Permutations can be viewed not only as linear orders of different objects, most often elements of [n], but also as functions from [n] to [n].

Example:

The permutation 312 can be viewed as the (bijective) function f: [3] → [3] defined by f(1) = 3, f(2) = 1, and f(3) = 2.

The advantage of this approach is that now one can define the product of two permutations on [n] by simply taking their composition as a composition of functions.

Another Example:

Let f = 312 and let g = 213. Then:

Therefore, fg = 321.

As these two examples show, multiplication of permutations is not a commutative operation, that is, it is not true in general that fg = gf.

Cycles in Permutations

Consider the permutation 321564. We can think of this permutation as a function g from the set [6] to itself. Let's examine g more closely: First, g(2) = 2. This means 2 stays in its place and is called a fixed point.

• Second, g(1) = 3 and g(3) = 1. This means applying g twice returns 1 and 3 to their original positions. So, 1 and 3 swap places and form a 2-cycle.

• Similarly, g(4) = 5, g(5) = 6, and g(6) = 4. Applying g twice gives g²(4) = 6, g²(5) = 4, and g²(6)=5. Applying g three times returns 4, 5, and 6 to their original positions. So, 4, 5, and 6 rotate among themselves and form a 3-cycle.

All permutations can be decomposed into the disjoint unions of their cycles.

Example:

While the cycle decomposition of a permutation f is unique, the same cycle decomposition can be written in many different ways. The convention is to write entries that belong to the same cycle in parentheses. The order of the entries in the parentheses is such that j immediately follows i if f(i) = j. Furthermore, f(b) = a, where b is the last entry and a is the first entry in the parentheses. However, these principles do not preclude multiple notations for the same permutation. For instance, (241)(35) and (53)(412) denote the same permutation. In that permutation, f(2) = 4, f(4) = 1, f(1) = 2, f(3) = 5, and f(5) = 3.

We would like to avoid the danger of confusion caused by the phenomenon we have just described. Therefore, we will write our permutations in canonical cycle form. That is, each cycle will be written with its largest element first, and the cycles will be written in increasing order of their first elements. Thus the permutation f of our previous example has canonical cycle form (412)(53).

You Shall Not Over-count. The Sieve.

Enumerating the Elements of Intersecting Sets

There are 20 students in a high school who play the piano, and there are 15 students who play the guitar. Five students play both instruments. How many students play at least one of the two instruments?

Solution:

To find the number of students who play at least one of the two instruments, we use the principle of inclusion-exclusion:

Total = Piano + Guitar − Both (since we don’t want to double count)

= 20 + 15 – 5

= 30

So, 30 students play at least one of the two instruments.

Principle of Inclusion-Exclusion:

When counting the total number of students who play at least one instrument, we add the number of piano players and guitar players. Since the students who play both instruments are counted twice (once in the piano group and once in the guitar group), we subtract the number of students who play both to avoid double-counting.

Another Example:

In a high school, 25 students play the piano, 20 students play the guitar, and 15 students play the violin. Eight students play both the piano and the guitar, five students play both the piano and the violin, and six students play both the guitar and the violin. Two students play all three instruments. How many students play at least one of the three instruments?

Solution:

To find the number of students who play at least one of the three instruments, we use the principle of inclusion-exclusion for three sets:

Calculation:

Total = 25 + 20 + 15 – 8 – 5 – 6 + 2

Total = 43

So, 43 students play at least one of the three instruments.

General Formula for the Principle of Inclusion-Exclusion:

Let A1, A2,...,An be finite sets. Then:

Explanation:

The general formula for the Principle of Inclusion-Exclusion (PIE) involves adding the sizes of individual sets, subtracting the sizes of their pairwise intersections, adding the sizes of their triple intersections, and so on, alternately adding and subtracting higher-order intersections until the intersection of all sets is accounted for. This method ensures accurate counting by correcting for over-counting in a systematic way.

Let's illustrate this with an example using four sets A, B, C and D:

This ends my write-up. Once again, most of these sections and content are inspired by the book A Walk Through Combinatorics which I highly recommend - thank you for reading and if you enjoyed this content, please like and subscribe.

Special Note: some of the visuals provided below are not generated by me and were taken from other sources. All of them are listed in the reference section.

Sorting Algorithms

Selection Sort

Selection sort is a simple sorting algorithm that organizes an unsorted list by repeatedly finding the smallest (or largest) element and moving it to its correct position. Let’s assume that we have a a couple of cards that we want to sort in increasing order:

1. Start with the entire stack, pick out the smallest card, and place it on the left side (the start of the sorted section).

2. Go through the rest of the cards, find the next smallest card, and move it next to the sorted section. Each time, the sorted section on the left grows by one card.

3. Keep doing this until there are no cards left in the unsorted pile.

Selection sort is straightforward but can be slow on large lists because it goes through the list repeatedly to find the next smallest item. It’s best for small or nearly sorted datasets since its time complexity is O(n²).

Insertion Sort

In terms of sorting cards:

1. You start with the 2nd element in the array or list (we consider the 1st element already ‘sorted’).

2. Compare it with the elements before it and insert it into the correct position among the already sorted (prior) elements.

3. Repeat the process until the entire array is sorted.

   

It’s similar to selection sort, but instead of going through the deck each time to find the next smallest (or largest element) — you simply expand the ‘sorted section’ located to the left by one card and move the new card to its correct position. As expected, the time complexity of this sort algorithm is O(n²).

Heap Sort

Heap sort uses a data structure called a binary heap to organize elements. It is similar to selection sort in which we first find the maximum element and put it at the end of the data structure. Here’s a quick rundown of how it works:

1. First, build a max-heap (a complete binary tree where each parent node is greater than its children). This step arranges the largest element at the root of the heap.

2. Swap the root (largest element) with the last element in the array. Then, reduce the size of the heap by one (ignoring the last element) and heapify the root to restore the max-heap property.

3. Continue swapping the root with the last unsorted element and re-heapifying until the heap size is reduced to one. At this point, the array is fully sorted in ascending order.

The time complexity of heap sort is O(n*log(n)).

Quick Sort

A highly efficient divide-and-conquer algorithm that is commonly used in practice due to its average-case performance of O(n*log⁡(n)) for which you can find a full explanation for here. Essentially:

1. You select an element which serves as a pivot.

2. Place all elements less than the pivot on the left and all elements greater than the pivot on the right.

3. Recursively apply the same process to the left and right sub-arrays until the entire array is sorted.

The time complexity of quicksort is O(n*log(n)).

Merge Sort

This is very similar to quick sort but with no pivot element and is stable. A full overview of this algorithm can be found here. To perform a merge sort

1. Recursively split the array into two halves until each sub-array contains only one element.

2. Combine the sub arrays and sort them while merging.

3. Continue merging until the entire array is sorted.

The time complexity of merge sort is O(n*log(n)). Use merge sort when you need a stable sorting algorithm, are working with linked lists, or sorting large datasets that might not fit entirely in memory (external sorting), while quicksort is generally preferred for most in-memory sorting scenarios due to its faster average performance and lower space complexity, especially when dealing with random data.

Tim Sort

Tim sort is a hybrid sorting algorithm that combines merge sort and insertion sort. It’s designed to be efficient with real-world data, particularly data that's already partially sorted. Here’s a breakdown of how it works:

1. Divide the Array: It divides the data into small chunks called runs. Each run is sorted using insertion sort, which is fast for small, nearly-sorted segments.

2. Merge Runs: The sorted runs are then merged together in a way similar to merge sort, forming larger and larger sorted sections until the entire array is sorted.

Tim sort is optimized for practical use and is the default sorting algorithm in languages like Python and Java. The time complexity of as expected is O(n*log(n)). You can find an excellent overview of it here: Understanding Timsort.

Search Algorithms

Binary Search

A fast searching algorithm that works on sorted arrays and has a time complexity of O(log⁡(n)). It is widely used due to its simplicity and speed and works by continuously dividing up the search space in half. Think of it as opening a dictionary at the mid-point, determining whether the word we’re searching for is to the left (less than the middle element) or to the right (larger than the middle element) and continuously re-opening our new search (or page) space again. A full overview of this algorithm is available here.

Depth-First Search (DFS) and Breadth-First Search (BFS)

These are fundamental graph traversal algorithms used in networking, pathfinding, and artificial intelligence. Breadth First Search explores all nodes at the present "depth" level before moving on to nodes at the next depth level. Depth First Search explores as far down a branch as possible before backtracking to explore other branches.

Use DFS for deep exploration, backtracking, and memory efficiency. Use BFS for shortest paths, level-wise traversal, and broader searches. DFS is better for constrained memory; BFS ensures the shortest path in unweighted graphs.

Graph Algorithms

Prim’s Algorithm

Prim’s algorithm finds the minimum spanning tree (MST) of a graph by starting with a random node and repeatedly adding the smallest edge that connects a node inside the tree to one outside of it. The process continues until all nodes are connected, ensuring the total edge weight is minimized. It’s used for optimizing network designs like roads or cables, minimizing the total cost.

Kruskal’s algorithm

Kruskal’s algorithm also finds the minimum spanning tree (MST) of a graph, but it works differently from Prim’s algorithm. It starts by sorting all the edges by weight. Then, it adds the smallest edge to the tree, but only if it doesn’t form a cycle. This process is repeated until all nodes are connected, and the tree has the minimum total edge weight.

Prim's algorithm is significantly faster when you have a really dense graph with many more edges than vertices. Kruskal performs better in typical situations (sparse graphs) because it uses simpler data structures.

Dijkstra’s algorithm

Used to find the shortest path between a source node and all other nodes in a weighted graph with non-negative weights. A step-by-step overview of the algorithm is provided below:

1. Initialize Distances: Set the starting node’s distance to 0 and all other nodes to infinity (∞). Mark all nodes as unvisited.

2. Select Node: Choose the unvisited node with the smallest distance (starting with the starting node).

3. Update Neighbors: For each unvisited neighbor, calculate its distance through the current node. If this distance is smaller than its current distance, update it.

4. Mark as Visited: Mark the current node as visited (the shortest path to it is now known).

5. Repeat: Repeat steps 2-4 until all nodes are visited or you’ve reached the target.

An example graph we’ll go through to demonstrate the algorithm is provided below:

We start off at node d (left-most node) in our graph and update our weights accordingly:

Side Note: We skipped 2 iterations in our outline above (visitations to Nodes c and g) since no outbound edges (neighbors) are present within these nodes — but a regular iteration of Dijkstra’s algorithm would include those nodes as well.

The updated weights after each iteration are provided below:

Dijkstra’s algorithm is crucial because it optimally solves shortest path problems in graphs, making it applicable to real-world scenarios where efficiency and correctness are key.

Bellman-Ford algorithm

Bellman Ford algorithm helps us find the shortest path from a vertex to all other vertices of a weighted graph. It is similar to Dijkstra's algorithm but it can work with graphs in which edges can have negative weights.

It works by iterating over all edges multiple times, progressively relaxing edge weights to find the shortest path. After V−1 iterations (where V is the number of vertices), the shortest paths are determined. A final iteration checks for negative weight cycles. If any edge can still be relaxed to determine if a negative cycle exists.

A simple example of the Bellman-Ford algorithm in action:

The pseudo-code which demonstrates the difference between the Bellman-Ford and Dijkstra’s algorithm is provided below:

As noted - the algorithms are similar to Dijkstra’s algorithm but includes an extra step to detect negative cycles.

A* Search

A* (pronounced "A-star") is an pathfinding algorithm (similar to Dijkstra’s) that efficiently finds the shortest path between two points in a graph. It is an enhancement of Dijkstra’s algorithm, combining the benefits of shortest path search and heuristic-based search to speed up performance. It finds the shortest path by balancing actual cost and estimated distance to the goal.

Imagine you're walking through a city and trying to find the shortest route to a coffee shop.

• You know how far you’ve walked so far.

• You look at the map and estimate how close each road gets you to the coffee shop.

• Instead of randomly checking all streets, you prioritize the ones that seem best.

• When you reach the coffee shop, you trace back the route you took!

A* is widely used in AI, game development, robotics, and navigation systems because it finds the shortest path faster than Dijkstra's algorithm in most practical cases. Obviously, the above outline only briefly glances over the actual algorithm and how it works. If you want a much more thorough overview of exactly how and why A* search works — I highly highly recommend Introduction to the A* Algorithm from Red Blob Games.

Union-Find algorithm (also called Disjoint Set Union (DSU))

It is actually more of a data structure than an algorithm and helps efficiently manage and group items into sets. It’s particularly useful for finding whether two elements are in the same set and for uniting two sets. It’s commonly used in algorithms like Kruskal’s to detect cycles in a graph and efficiently manage components.

It supports two main operations:

1. Find: Determines which set an element belongs to by returning its "leader" or "representative." If two elements contain the same “leader” - we know they belong to the same set.

2. Union: Merges two sets into one. This is done simply by finding the leaders (or roots) of the two sets, and then connecting one root to the other, merging them into a single set. For efficiency, you should always attach the smaller set under the larger one.

It efficiently tracks dynamic connectivity in graphs and sets, making it essential for problems like Kruskal’s MST, cycle detection, clustering, and network connectivity and is widely applied in graph algorithms.

Ford-Fulkerson Algorithm

Finds the maximum flow in a flow network (i.e. where you want to send the most flow (e.g., water, data) from a source to a sink) through a network of edges with capacities. It works by:

1. Starting with zero flow in all edges.

2. Looks for augmenting paths—paths from the source to the sink where more flow can be pushed, considering available capacity.

3. Adds the flow along this path and adjusts the capacities (both forward and backward).

4. Repeats until no more augmenting paths can be found.

As an example - let’s use the graph below:

The algorithm uses a special representation of a graph where each original edge has a reverse edge in another direction. The weight of each edge indicates how much more flow we could route to it. Initially, the weight of each reverse edge is set to 0:

Then, we need to perform several iterations of our algorithm, which finds paths from the source (node 1) to our sink (node 6) such that each edge within the path has a positive weight. For example, let’s choose the following path as our first iteration:

After choosing a path - we increase the flow by the smallest edge weight within our path. In the above instance - our smallest edge weight is 2, so the flow increases by 2 and the new graph is updated:

The algorithm keeps increasing the flow as long as there is a path from the source to the sink through positive weight-edges — and also allows us to decrease the amount of flow that can go through certain edges if its beneficial to re-route the flow through another path.

The Ford-Fulkerson algorithm is important since it has many applications in various fields like network routing, supply chain optimization, and image segmentation.

String-Search Algorithms

I was going to include an entire section overviewing some of the key string-search algorithms, but this post is already getting very long, so I decided to leave it out for now. If you’re interested in this, you can find a full overview here: A Walk-Through of String Search Algorithms.

Compression and Encoding Algorithms

Value Encoding

Value encoding is a type of encoding which we can use to reduce storage requirements. It’s often used by columnar stores in order to reduce the memory requirements. Imagine that we have a column containing the price of various products, all stored as integer values. The column contains many different values, and to represent all of them, we need a defined number of bits.

In the figure above, we can see that the maximum value for the price is 216. Therefore, we will need at least 8 bits to store each value. Nevertheless, by using a simple mathematical operation, we can reduce the storage to 5 bits! By subtracting the minimum value (194) from all the values of the column, it could modify the range of the column, reducing it to a range from 0 to 22. Storing numbers up to 22 requires less bits than storing numbers up to 216. While 3 bits might seem a very small saving, when you multiply this for a few billion rows, it is easy to see that the difference can be an important one!

Dictionary Encoding

Dictionary encoding is a technique which can be used to reduce the number of bits required to store columnar data. Dictionary encoding builds a dictionary of the distinct values of a column and then it replaces the column values with indexes to the dictionary.

Huffman Coding

Huffman coding is a method of data compression that reduces the number of bits needed to represent data. It’s commonly used for compressing text, images, and audio files. The algorithm gives shorter binary codes to frequent symbols and longer codes to rare ones by 1) counting how often each character / symbol appears and 2) using this information to build a binary tree (called a Huffman tree) which assigns short codes to frequent symbols and longer codes to rare ones. You can find a more thorough overview of how exactly the algorithm works here: Grokking Huffman Coding.

An example construction of a valid Huffman tree (or encoding) is provided below:

Lempel-Ziv (LZ) Compression

The Lempel-Ziv (LZ) algorithm is a lossless compression method that replaces repeated patterns with references to earlier occurrences. Unlike Huffman coding, which requires two passes—one to determine frequencies and another to encode—LZ builds its dictionary dynamically in a single pass, making it more efficient in some cases.

There are many variations of Lempel Ziv around, but they all follow the same basic idea. We’ll just concentrate on one of the simplest to explain and analyze, although in other versions will work somewhat better in practice. The idea is to parse the sequence into distinct phrases. The version we analyze does this greedily. Suppose, for example, we have the string:

AABABBBABAABABBBA

We start with the shortest phrase on the left that we haven’t seen before. This will always be a single letter, in this case A:

A|ABABBBABAABABBBA

We now take the next phrase we haven’t seen. We’ve already seen A, so we take AB:

A|AB|ABBBABAABABBBA

The next phrase we haven’t seen is ABB, as we’ve already seen AB. Continuing, we get B after that:

A|AB|ABB|B|ABAABABBBA

and you can check that the rest of the string parses into:

A|AB|ABB|B|ABA|ABAB|BB

Now, how do we encode this? For each phrase we see, we stick it in a dictionary. The next time we want to send it, we don’t send the entire phrase, but just the number of this phrase. Consider the following table:

The second row gives the phrases, and the third row their encodings. That is, when we’re encoding the ABAB from the sixth phrase, we encode it as 5B. This maps to ABAB since the fifth phrase was ABA, and we add B to it. Here, the empty set ∅ should be considered as the 0’th phrase and encoded by 0.

As mentioned before - there are other variations of this algorithm, including the LZ77 algorithm which uses a sliding window and triplets (offset, length, next character) to encode the information. A very simple example of how this algorithm works is also provided below:

LZ77 is the basis for widely used compression methods like DEFLATE (used in ZIP, PNG, Gzip), LZW (used in GIF), and LZMA (used in 7z).

Bitmap Index (used for Column Compression) & Run-Length Encoding

A bitmap index is a way to compress and quickly search through columns of data in databases. For each unique value in a column, it creates a sequence of bits (0s and 1s) that indicate whether a row has that value.

For example, if a column has the values A, B, A the bitmaps would be:

• For A: 1, 0, 1

• For B: 0, 1, 0

Data warehouses use columnar storage (rather than row-level storage) to store data - which tends to save a lot of space since columns with many duplicate elements can be encoded in a more efficient way. A simple example showing how this columnar storage works is provided below (taken directly from a highly recommended book which I recommend every software developer read: Designing Data-Intensive Applications):

Run-Length Encoding compresses data by replacing consecutive repeated values with a single value and the count of its repetitions. For example, if you have the data: AAAAABBBCC, it would be compressed as 5A 3B 2C. This is effective when there are long sequences of repeated values, reducing the amount of data stored.

Example of a compressed, bitmap-indexed storage of a single column:

Burrows-Wheeler Transform

The Burrows-Wheeler Transform (BWT) rearranges a string into a form that’s easier to compress by grouping similar characters together. Once applied, the transformed string is easier to compress using techniques like Run-Length Encoding (RLE) or Huffman Coding.

Imagine a messy shelf of books. The BWT is like grouping books of the same color together. While you don’t change the books themselves, organizing them makes it much easier to spot patterns, retrieve what you need, or pack them tightly. The BWT is a way of transforming a string into a new string that has the same characters, but in a rearranged order that groups similar patterns together.

To be more precise, it works by:

1. Generating all rotations of the string.

2. Sorting these rotations lexicographically.

3. Taking the last column of the sorted rotations as the transformed string.

Fourier Transform (and Fast Fourier Transform (FFT))

The Fourier Transform is crucial for analyzing and compressing signals in various fields, like audio processing, image analysis, and communications. In essence, it’s a mathematical tool that decomposes a signal into its constituent frequencies. It converts a time-domain signal (how a signal changes over time) into a frequency-domain representation (what frequencies make up the signal).

To put it in a simpler manner - any signal can be thought of as a combination of pure sine waves (waves with a smooth and repetitive pattern). The Fourier Transform can be thought of as a tool that helps us understand the "ingredients" of a signal breaking it down into these sine waves.

How does it work? Imagine you have a signal, and you want to figure out what sine waves it’s made of:

1. Pick a frequency: Take a sine wave with a specific frequency.

   

2. Check its presence: Multiply this sine wave by the original signal and then add up the result (this process is called integration).

   

3. Repeat: Do this for many different sine waves with different frequencies.

Now, the above process is a very inefficient one. How can we simplify it further and optimize it? This is where the FFT comes to the rescue. The Fast Fourier Transform (FFT) is an optimized algorithm for computing the Fourier Transform quickly. It works by:

1. Dividing the signal into even-indexed and odd-indexed components. This step reduces the size of the problem recursively.

2. Computing smaller Fourier Transforms on the even and odd components. This divide-and-conquer approach continues until reaching base cases (like single points).

3. Combining the smaller transforms efficiently by using symmetry and periodicity properties of complex exponentials (roots of unity).

The combined result gives the complete frequency representation of the original signal. If you’re interested in the full details - a great video explaining the FFT is provided below:

The FFT’s speed and versatility make it indispensable in both engineering and scientific fields where quick and accurate frequency analysis is essential for tasks such as real-time signal processing, image reconstruction in medical imaging (like MRI), seismic data analysis for geophysics, and the efficient simulation of physical systems in computational science.

Quantization

Quantization is the process of taking a continuous signal or value and converting it into a set of discrete levels. It's like rounding real numbers to the nearest whole number but applied to things like sound, images, or data to make them easier to store and process. I won’t go into the full details of how quantization works here, but you can find a great visual focused overview here: A Visual Guide to Quantization.

Discrete Cosine Transform (DCT)

The Discrete Cosine Transform (DCT) is similar to the Fourier Transform (where a signal is converted from its time-domain representation to its frequency-domain), but DCT does it in a manner which leads to better information compression for real-world data sets. Some key differences between the Fourier transform and DCT are:

1. DCT uses only real cosine functions, while the Fourier Transform uses both sine and cosine functions (complex exponentials). This makes the DCT more efficient for signals that are real-valued and symmetric.

2. DCT assumes the signal is even (symmetrical) at the boundaries, reducing high-frequency components and leading to better energy compaction.

3. DCT is discrete (i.e. the information is quantized) once again leading to better information compression.

DCT is widely used in image and video compression (e.g., JPEG, MP3) and although I won’t go into the full details on how it works here - this Computerphile video does and excellent job of explaining how it works:

Image Compression (JPEG)

JPEG compression is a popular method for reducing image file sizes while maintaining good visual quality by removing less noticeable details. It exploits how humans perceive images, prioritizing brightness over color details and simplifying less significant information. Here’s an overview of how it works:

1. The image is converted from RGB to YCbCr color space. This separates the image into luminance (brightness, Y) and chrominance (color, Cb and Cr), as humans are more sensitive to brightness than color details.

2. The image is divided into 8x8 pixel blocks for localized processing.

3. Each block undergoes DCT to convert spatial pixel values into frequency components. This step helps identify low-frequency (broad features) and high-frequency (fine detailed) components.

4. Frequency components are divided by a quantization matrix based on a quality factor. Higher-frequency details are significantly reduced or eliminated to save data while preserving visual quality.

5. The quantized data is compressed further using run-length encoding (compresses sequences of repeated values) and Huffman coding (assigns shorter codes to more frequent values in an optimal manner).

This process is "lossy," meaning some data is permanently discarded. While this can lead to artifacts with very high compression levels, JPEG compression strikes a good balance between file size and image quality. If you want a better understanding of the key details, a great video explaining how this is done is provided below:

Video Compression & Encoding

A video is essentially a sequence of images, known as frames, captured over time. Frames are once again often converted from the RGB color model to a more compression-friendly format like YCbCr (which we covered in JPEG compression). Similar to JPEG, each individual frame (intra-frame) also undergoes spatial compression to reduce redundancy within the frame:

1. Block Partitioning: The frame is divided into small blocks (e.g., 8x8 pixels).

2. Discrete Cosine Transform (DCT): Each block is transformed from the spatial domain to the frequency domain using DCT, which helps separate image details based on their frequency components.

3. Quantization: The frequency coefficients obtained from DCT are approximated to reduce precision, effectively reducing the amount of data. This step introduces some loss but is crucial for compression.

Many sequential images within a video contain the same information, so our video also must undergo through inter-frame compression to get rid of any redundancies between consecutive frames: