When I was a kid, my family didn’t get JELLO puddings – or any puddings for that matter. As an adult, I realized that a lot of those are just sugar, cornstarch, and stabilizers. So, they became less appetizing.
You’re going to laugh at me.
A few years ago my wife and I went to a nice little breakfast restaurant for brunch in old town Fredericksburg, VA. I got this coconut milk chia seed parfait. I was blown away. That seems silly to say, but it was really nice.
For years we spoke longingly of that chia parfait and we’d speculate about when we might go there again. It was one of those conversations that married people have.
“Hey, remember that really good thing?”
“Yeah, it was really good.”
“We should try that again sometime.”
Then one day, while visiting Virginia, we noticed that the restaurant had closed. It wasn’t surprising because the restaurant had only been ‘fine’, except for the healthy and delectable layered treat from years past.
Now we have a handful of kids and we try different things periodically to make the morning routines go more smoothly. Having a responsible treat to entice juveniles from their room isn’t the worst thing that we’ve tried.
My wife, in her laudable creativity, refined a new creation that’s inspired by our now frustrated longing for a nice chia parfait.
Below is a recipe for peanut butter chocolate chia seed pudding. Basically, you mix it the night before and stir it again in the morning and it’s ready to go. It’s a crowd pleaser.
The vast majority of business majors across the US are required to take two or more Economics courses. You can look across the spectrum. All of the top 20 business schools require two or more econ classes. In fact, Wharton is the top-ranked business school and their business program is actually an *economics* program. They don’t have finance/accounting/business degrees. Instead, they have an Economics degree with the various business concentrations. Again – the top business school in the country is an Economics program.
What about at the other end of the spectrum? I live in Florida. Every single Florida state school requires both Micro and Macroeconomics for business majors. These schools include everything from Florida State University to the local Florida state college down the road. I didn’t look at other state-run higher education systems in other states. There are a lot of states…
I teach at a private Catholic university. We’re listed in something called ‘The Newman Guide’ which recommends 17 Catholic schools. Many of these are liberal arts schools, but the list also includes Catholic University of America, which is an R1. Most of these schools also require two or more Economics classes in their Business major programs. The only exception is University of Dallas, which has Economics in the core curriculum.*
So, overwhelmingly undergraduate business programs across the country require two economics courses. But, why? The students are often not happy to be there, and I’ve even heard business professors demean the math as performatively rigorous and superfluous. They argue that plenty of people get rich or are otherwise successful without all of the quantitative skills that economics leverages.
I think that the fear of math is both a red herring and a scapegoat. Rather, Economics confronts students with the liberal arts – whether they like it or not. Be careful. Liberal Arts are not the same as Humanities. They include argumentation, the ability to write and communicate, clear and consistent logic, and, yes, even math. Accounting can tell you how to keep track of the money, but it doesn’t include a theory for when you should produce more or less in contrast to your competitors. Finance does better since it has the time value of money and ‘with vs without’ analysis. That’s closer to marginal thinking. But finance lacks a theory of markets outside of portfolio theory and arbitrage.**
I was listening to an episode of The Deduction, a podcast by the Tax Foundation. As if that first sentence isn’t evident enough, I was reminded of how confusing taxes are – period. Even experts disagree and see grey areas. As I was listening, I thought “man, they need a graph”. So, here we are.
Income Tax Vocabulary
The money that you are paid by your employer is your gross income. Not all of it is taxable. You can deduct money from your gross income to get your taxable income. Most people subtract the ‘standard deduction’ from their gross income, which is how I’ll proceed in this post. Since the standard deduction for 2026 is $16,100 for a single earner, that means that your taxable income is $16,100 less than your gross income. By following a formula, one can calculate the amount of money that they must pay the government. These payments can be all at once, throughout the year, or even directly from your paycheck. The total that’s due to the government by April 15 is called the total tax liability. Finally, the money that the government doesn’t take, and that you get to keep, is called your net income. It’s your income net of taxes.
If you’ve had a job, then you are probably most familiar with your gross income, what your employer pays you, and your net income, what you get to take home. The steps in between might include some hand-waving.
Marginal Tax Rates
One of the most confusing pieces of the income tax code is marginal income taxes. Below are the brackets for 2026.
Marginal Tax rates work like this: Every dollar that you earn faces a tax rate. If your taxable income would be below zero, then you pay zero in taxes. But if your taxable income is $5k, then it gets taxed at a rate of 10%. That part should be pretty straightforward. But what if your taxable income is $15k? According to the table, you face a tax rate of 10% for dollars earned up to $12,400. That would be a tax liability of $1,240. But the remainder of your $15k in taxable income exists in the next tax bracket. That portion of your taxable income faces a tax rate of 12%. Sticking with the example, $2,600 is in the 12% tax bracket, so the tax liability for that portion of your taxable income is $312 (=$2.6k*0.12). Therefore, your total tax liability would be the sum of your tax liabilities across all applicable tax brackets: $1,552 (=$1,240+$312).
There are some features of marginal tax rates that are worth mentioning. Since the tax rates on the lower taxable income brackets don’t change, earning more gross income never reduces your net income unless the tax rate exceeds 100% (which it doesn’t here). So, when someone says that their taxable income is in the 35% tax rate bracket, they probably just mean that their last dollar earned is there. They’re only paying 35% on the taxable income that’s above $256,225. They’re not paying 35% of all earned dollars to the Internal Revenue Service (IRS).
Below is a graph that details the different marginal tax rates with shaded areas. The blue line is the average tax rate. It’s calculated by dividing the tax liability by the gross income. Even though one might earn an income that’s greater than $257k where the marginal tax rate is 35% or greater, the average tax rate remains lower, topping out at about 30% in this figure. The average tax rate is lower than an earner’s top marginal tax rate because the income in those lower brackets never disappears or get taxed at a higher rate.
My children are getting more capable. They get more responsibility that comes with the independence that capability implies. Specifically, when getting ready in the morning they like to leave so that they arrive at school just barely on time. Except, when something comes up, they are rushed, flustered, short-tempered, and tardy. They lament that “if only the unforeseeable event X hadn’t happened, I would have been on time”.
It doesn’t matter what X is. Maybe they forgot to pack a lunch, or set out their clothes, or they have a flat tire on their bikes, or… whatever. The specific time-consuming event is unforeseeable. But, that *any* time-consuming event will occur is very foreseeable. What’s a Bayesian to do?
Before we even start the analysis, let’s acknowledge that being perfectly on time for some event usually involves stress and a lack of preparedness. Yes, you were ‘on time’, but given the probability of heavier traffic, difficulty finding a parking spot, or whatever, we know that tardiness is just one unforeseen event away.
Individual Punctuality
How long does it take to get somewhere? It takes both travel time and time preparing to depart. Let’s just generally call this ‘preparation’ time. Let’s assume that you complete everything that you would complete. That means that you aren’t forgoing a shower or breakfast or whatever lower priority you might choose to forgo to arrive at some obligation punctually.
Random events can occur either as you travel to work or as you prepare to depart, but let’s place the random travel events to the side and focus on what one can do to get out of the house ‘on time’. In my personal case, my children have a 30min interval during which they can arrive at school. They almost never arrive in the first 15min of that interval. That’s more of a policy choice than an accident. They don’t want to sit in a cold gymnasium for 20min if it’s avoidable. So, their planned arrival time has an effective 15min window.
Here is the problem. A time-consuming random event, X, is a right-skewed random variable. Discretely, the modal day includes X=0min. Though the most common delays are greater than 0min. See the distribution below. A 0min random event occurs 35% of the time. But, a time-consuming event happens 65% of the time. So, if you try to arrive exactly on time to your obligation, then you will be punctual 35% of the time and you will be tardy 65% of the time. That’s not a good look and not a good reputation to build – and that’s apart from building a habit of imprudence and the material consequence of not being ready for the task at hand.
Someone with just enough insight to be dangerous might say ‘Ah! Instead, leave with enough time to accommodate the expected unforeseen event’. Mathematically, that’s the weighted average. In this case, that’s six minutes. So, if you plan to arrive 6min early, then you will be punctual – on average. But even that’s not really what we’re after. We’d like to be on time for a preponderance of the days. Building in a 6-minute buffer does two things. 1) Every time that there is a 0min or 5min unforeseen event, you get to your destination 6min or 1min early. That’s good for your nerves, performance, and reputation. But, that also means that you’re late whenever there is a 10min, 15min, or 20min unforeseen event – and those occur 35% of the time!
My university recently hosted a guest speaker. Among their content, they included some nominal macroeconomic values from pre-2020, back in the era when inflation was very low. That roughly includes the years 2012-2019. Truly, inflation stayed below 2% through February of 2021, but I think that we can all agree that the economy was different in a few ways beginning in 2020.
I asked the speaker why not express the nominal values in real terms. They were emphatic that the low rates of inflation at the time implied that the signal-to-noise ratio was too low. Therefore, the ‘real’ inflation adjusted values would not be more precise because excessive noise would be introduced into the series during a period when not much deflating was necessary in the first place.
1) It may be information is incomplete and becomes more complete as time passes. This sort of noise only exists in the short-run and is resolved as more information becomes available later in time. Revisions tend to happen each month for prior months, as well as each year for prior years. There are also big revisions after methodological, consumption weight, and data source changes.
2) Another type of noise is due to incomplete information that is never resolved. After all, the government statisticians can’t see literally all of the transactions. Those unobserved transactions will never make it into the official inflation measures and we’ll never get a perfect picture.
3) Methodological artifacts may also include known biases. This type of noise doesn’t get corrected except after major changes to the series. If those changes never happen, then we just sort of live with imprecision. Luckily, so long as the bias is consistent, then percent change in the price indices will approximate the underlying true levels. However, if there are non-random biases in the percent change, then it can cause some trouble.
One way to get an idea for the amount of noise in the data is to observe the magnitude of revisions. Of course, this only helps us with the first type of noise above that eventually gets resolved with more information. It’s much harder to get a handle on the imprecision that is not identifiable. The Philadelphia Federal Reserve Bank provides an easy-to-use database that puts all of the archival and revised numbers for many macro series in a single place: the Real-Time Data Set (RTDS). It includes every historical PCE price index value for each publication month. Let’s limit our sample to the 21st century.
Economists talk about the ‘Covid Shock’ in 2020 because it was a mostly unpredictable event that had big, measurable effects. People spent a lot less time being in close quarters. Especially hit hard were movie theaters, and other events spaces.
In the several years prior to Covid, “Recreational Service” industry sales had been chugging along, growing at healthy annual rate of 3.4% (inflation adjusted). This category of services includes clubs, sports centers, theaters, and museums. In the blink of an eye, the covid shock drastically reduced spending in that category by more than 60%. See the graph below.
Unfortunately, we don’t have disaggregated series for the components of “Recreational Services”. But we do know that movie theaters were already well past their hay-day. Theaters had been closing and consolidating for more than a decade and ticket sales were down. Many give credit to the popularity of streaming video services and other digital media alternatives. Covid added insult to injury.
Now, going to a movie theater is exceptional. As a teenager in the early naughts, I’d go to the theater easily half a dozen times per year. Now, I don’t think that I’ve gone six times in the last five years. Real growth in the entire recreational service category has grown annually by an anemic 1.8% since 2019. It’s not dead, but that’s also the total industry. I’ve heard the news stories of sports events making a big comeback. I’ve not heard anything like that for movie theaters.
I went to the movies recently and it is not what you remember.
If you listen to or read major economists discussing what they think are big-picture problems, then their list usually includes three topics: Fertility, Culture, & the Fiscal Health. On the wonkier side, you’ll also hear that housing scarcity and affordability is a problem, but let’s stick with the first three.
Fertility
People are deciding to have fewer children for a variety of reasons. In no particular order, the reasons include greater access to financial institutions, more popular female education, higher female wages, lower infant mortality, and falling religiosity. Some also speculate that housing affordability, safety regulations, and social safety nets contribute too.
What’s wrong with lower fertility? In an objective sense, there is nothing wrong. But, in the sense that people value similar things, we are in somewhat uncharted territory. Realized fertility is dropping across the globe. We know that economies of scale increase productivity and real wages. We also know that technological innovation comes from having more minds engaged with economic problems. It’s possible that labor productivity rises faster than the productivity that we lose with smaller scale, but it’s an open question. What happens to the liberal societies and polities when the liberals fail to persist? These are big geopolitical concerns.
Culture
People seem to be more fragmented religiously and culturally. Social scientists used to discuss Judeo-Christian norms more often. Sometimes you’d hear about English or Roman legal tradition or enlightenment values. But now, there seems to be very little in terms of common social cohesion. In the USA, the general common culture seems to be ‘smile and be nice’. That’s not the worst common rule, but it’s not enough to hang our hat on for a capable liberal state.
The lack of cultural cohesion isn’t my own particular concern – public intellectuals in economics and elsewhere feel like there is a problem. There is a mix of reasoning behind the concern. Some people are worried about transmitting values to the next generation, some are worried about how people behave when no one’s watching, and still others are worried about simply lacking a Schelling point that coordinates large scale economic cooperation.
Americans like their food. Holidays are often known by the dishes that we serve. Thanksgiving is a bit unique in that most of us converge on turkey, though diversity obviously exists. What about Easter? There’s not really the same focus on a single food like there is for Thanksgiving. My impression is that people eat daytime or lunch foods that include ham, lamb, or just about anything. My family tends to make tacos.
What am I saying?! We eat candy! Solid or hollow chocolate bunnies, jellybeans, peeps, and on and on. We fill Easter eggs and keep candy around the office. We literally have baskets full of candy.
A Chocolate Bunny? In this economy?
Have you seen the price of chocolate? Yeesh! The latest figures are from February and the prices for chocolate and cocoa bean products are down 11.7% year-over-year. That’s nice, you may think, our budgets can fit a bit more chocolate into our consumer – I mean Easter – baskets. Great news. The news seems a little less great when you realize that February’s price of chocolate was 90% higher than it was four years earlier in 2022. 90% higher is a lot like 100%, and 100% is double! In fact, the price had peaked at 142% higher by September of 2025, and now prices are quickly falling. See the chocolate-colored line in the graph below.
This post solves for the equilibrium quantity of production with quadratic total cost under Cournot and Stackelberg competition.
Say that there are two firms. They produce the exact same quality and type of goods and sell them at the same price. Let’s also assume that the market clears at one price. Finally, let’s assume increasing marginal costs.
Let’s say that they face the following demand curve:
The firms have a total cost of:
The marginal cost is the derivative with respect to the choice variable for each firm, or their respective quantities produced:
The total revenue is just the price times the quantity sold.
This is all standard fare for economic modeling. You’re free to make different assumptions. You can even adopt different slopes in the demand curve to reflect goods with different characteristics.
Cournot Competition
If you imagine a lengthy production process, or otherwise that they physically attend the same market, then it’s reasonable to assume that they don’t know one another’s choice of quantity produced.
We know how firms maximize profit: They produce the quantity at which the marginal revenue equals the marginal cost. But, what is marginal revenue? The derivative of total revenue with respect to the choice variable:
Now we can set the marginal revenue equal to marginal cost and solve for the optimal level of output:
Notice that the optimal level of output depends on the production decision of the other firm. These are called response functions. If we solve for the quantities at which they intersect, then we are solving for where both firms are producing the best response to one another. This is known as a Pure Strategy Nash Equilibrium (PSNE).
Luckily, in many applications, one or more of the above terms are zeros, which makes things much simpler.
The general process for solving for the Cournot equilibrium is:
If you’ve been on LinkedIn recently, then you may have seen the chatter about teaching your artificial intelligence to have various skills. I saw one post by a guy who claimed to have created several skills, each representing a tech billionaire.
At first, I thought “I am behind the 8-ball. What is this new thing?”. Obviously I know what the word “skill” is and how people use it, but I had not encountered its use in the context of AI having it. What does it mean for an AI to have a skill? I somewhat dreaded the the work of learning the new skill of teaching my AI skills.
Then I had lunch with a computer scientist and I learned that skills are nothing new.
Economist Writing Every Day 