Author: Zachary Bartsch

Using Economics to Save Presents from the Economists

Zachary Bartsch3 Comments

Economists like to hate on gift giving. Many of them consider purchasing a gift for another person as a futile attempt at imagining the preferences of another person. Given that you can’t perfectly know another person’s preferences, your gift selection will be sub-optimal. The argument goes that your friend or spouse or whomever would have been better off if you had given them money instead. Then they could have made the gift decision fully equipped with the information that is necessary to make them happiest.

There are some obvious things that are glossed over. Purchasing a good gift – or even writing a card – carries a big load of signaling value. People like to be liked and receiving a good gift signals that the giver cared enough to research appropriate gifts. Also, receiving money as a gift puts the onus of research and transaction costs on the receiver. If the recipient’s value of time is adequately high, then cash payments are even more resource destructive than giving a non-pecuniary gift. Especially if there is an expectation that the giver will later enquire about how the funds were used. At that point, the giver is saddling the recipient with all of the anxieties and costs of choosing a gift that makes another person happy.

But I want to talk about a non-obvious benefit of gift giving.

First, I want to talk about student loans (I promise, it’s relevant). Plenty of people argue that college students don’t understand debt and that they therefore don’t understand the future cost that they will bear by borrowing. When the lender is the department of education, there is no defaulting with the hope of bankruptcy. The debt will get repaid…. So far anyway.

If it’s true that students don’t understand debt, then we can appropriately construe future student loan payments as lump-sum costs. Of course there is deferment and forbearance – but put those to the side. The bottom line is that, almost regardless of a debtor’s activities, they must repay their debt. It doesn’t matter how the debtor earns or consumes, the debt must be paid. This fits the description of a lump-sum cost. Usually, things like lump-sum taxes are hypothetical and unpopular among the laity. But, if we accept that the decision-making-student has incomplete information in regard to the debt’s future payment implications, then the debt payments are exogenous and unavoidable from the future debtor’s perspective.

This is a good thing for the productivity of our economy. Because people are making tradeoffs between the two goods of leisure and consumption, a lump-sum tax causes individuals to work more than they would have worked otherwise. Lump-sum taxes don’t reduce the marginal benefit of working. Essentially, a debtor’s first several hours of work pay-off his debt first and then he gets to work for his own consumption.

Importantly, this ignores any human capital effects of the education. It doesn’t matter whether education actually makes people more productive. The seemingly exogenous debt payments cause debtors to work more and produce more for others. The RGDP per capita of our economy rises and we know that most of the benefits of work do not accrue to producers. Student debt, with the accompanying assumptions laid out above, therefore increases our incomes because it acts as a lump-sum tax.

Now it’s time to save presents from the economists.

As families get older and siblings drift apart, gift-giving begins to become less exciting. I’m tempted to say there is a natural process in which the first couple of adult-sibling Christmases include decent gifts. Then, the gifts become not-so-great as siblings become less familiar with each others’ preferences. Knowing this and still wanting to give a suitable gift, siblings may turn to gift cards. The less that a sibling knows the preferences of another, the more general the gift card.

If you’ve grown more distant from your brothers/sisters and you know that you’ll receive a gift, then it’ll probably be an Amazon, or Walmart, or some other gift card that permits spending on a broad variety of gifts. There comes a point when you’re spending $X on gift cards each year where $X = $x(n). That is, you’re spending some amount on each sibling for a total of $X each year. And for the sake of social cohesion and norms, all of your siblings are doing the same thing and spending the same amounts.

Importantly, you don’t control the social norms, nor your number of siblings. It might seem like you’re all just trading dollar bills at a unitary exchange rate, leaving no-one better or worse-off. But, trading cash is gauche. So, distant siblings trade broadly attractive gift cards in order to achieve that gift-like aura.

Social norms also say that gift giving is not a trade. If you don’t receive a gift, then you’re supposed to be ‘ok’ with that. So, each year you will spend $X on gift cards for your distant siblings and there is some probability that you get nothing in return. If you can’t control the number of siblings that you have and you can’t control whether you receive a gift card in return, then giving cash or cash-like gift cards to your siblings each year is a lot like a lump-sum cost. Socially – or maybe morally – you shouldn’t just ignore your siblings and it is incumbent upon you to give a gift.

Having to give away a lump-sum of money or money-like things no matter what else you do is a lump-sum cost. If people bear lump-sum costs, then they will work a little bit more and produce a little bit more for society. If gifts suboptimal but at least considered a ‘good’, then we’re better off: we work more to make others somewhat better off with resources that wouldn’t exist if we hadn’t chosen to give to others.

There are some caveats, of course. Economists are often not so popular at parties for a variety of reasons. One reason is that they flout social conventions. An economist might scoff at the social constraints as unbinding. Others would disagree. Another point of contention may be that an individual can choose to work no more, but to invest less instead. But this really just pushes the problem off until the individual has less income in the future and works more to compensate for it at a later time. A 3rd caveat is that we can choose the amount that we spend in others. But that just implies that at least part of the gift giving ritual isn’t a lump-sum cost. It does not imply that none of gifting giving is a lump sum cost.

Regardless, the social convention of giving gifts can provide for a Schelling point that makes us a more productive as a society. We spend on others, to a great degree beyond our individual control, in order to avoid severe social stigma. And, if we can’t control all of who counts as a worthy recipient of gifts, then we have a lump-sum cost to some degree. Giving gifts makes sense as a productive convention because it makes us a richer as part of a general equilibrium – if not a partial equilibrium. Merry Christmas.

Covid-19 & The Federal Reserve

Zachary BartschLeave a comment

I remember people talking about Covid-19 in January of 2020. There had been several epidemic scare-claims from major news outlets in the decade prior and those all turned out to be nothing. So, I was not excited about this one. By the end of the month, I saw people making substantiated claims and I started to suspect that my low-information heuristic might not perform well.

People are different. We have different degrees of excitability, different risk tolerances, and different biases. At the start of the pandemic, these differences were on full display between political figures and their parties, and among the state and municipal governments. There were a lot of divergent beliefs about the world. Depending on your news outlet of choice, you probably think that some politicians and bureaucrats acted with either malice or incompetence.

I think that the Federal Reserve did a fine job, however. What follows is an abridged timeline, graph by graph, of how and when the Fed managed monetary policy during the Covid-19 pandemic.

February, 2020: Financial Markets recognize a big problem

The S&P begins its rapid decent on February 20th and would ultimately lose a third of its value by March 23rd.  Financial markets are often easily scared, however. The primary tool that the Fed has is adjusting the number of reserves and the available money supply by purchasing various assets. The Fed didn’t begin buying extra assets of any kind until mid-March. There is a clear response by the 18th, though they may have started making a change by the 11th.  One might argue that they cut the federal funds rate as early as the 4th, but given that there was no change in their balance sheet, this was probably demand driven.

https://fred.stlouisfed.org/graph/?g=JYVL
https://fred.stlouisfed.org/graph/?g=JYVy

March, 2020: The Fed Accommodates quickly and substantially.

In the month following March 9th, the Fed increased M2 by 8.3%. By the week of March 21st, consumer sentiment and mobility was down and economic policy uncertainty began to rise substantially – people freaked out. Although the consumer sentiment weekly indicator was back within the range of normal by the end of April, EPU remained elevated through May of 2020. Additionally, although lending was only slightly down, bank reserves increased 71% from February to April. Much of that was due to Fed asset purchases. But there was also a healthy chunk that was due to consumer spending tanking by 20% over the same period.

https://fred.stlouisfed.org/graph/?g=JYWl
https://fred.stlouisfed.org/graph/?g=JYXj
https://fred.stlouisfed.org/graph/?g=JYYz

In the 18 months prior to 2020, M2 had grown at rate of about 0.5% per month. For the almost 18 months following the sudden 8.3% increase, the new growth rate of M2 almost doubled to about 1% per month. The Fed accommodated quite quickly in March.

April, 2020: People are awash with money

Falling consumption caused bank deposit balances to rise by 5.6% between March 11th and April 8th. The first round of stimulus checks were deposited during the weekend of April 11th. That contributed to bank deposits rising by another 6.7% by May 13th.

https://fred.stlouisfed.org/graph/?g=JYZs

By the end of March, three weeks after it began increasing M2, the Fed remembered that it really didn’t want another housing crisis. It didn’t want another round of fire sales, bank failures, disintermediation, collapsed lending, and debt deflation. It went from owning $0 in mortgage-backed securities (MBS) on March 25th to owning nearly $1.5 billion worth by the week of April 1st. Nobody’s talking about it, but the Fed kept buying MBS at a constant growth rate through 2021.

https://fred.stlouisfed.org/graph/?g=JYZK

May, 2020 – December, 2021: The Fed Prevents Last-Time’s Crisis

Jerome Powell presided over the shortest US recession ever on record. The Fed helped to successfully avoid a housing collapse, disintermediation, and debt deflation – by 2008 standards. The monthly supply of housing collapsed, but it had bottomed out by the end of the summer. By August of 2021, the supply of housing had entirely recovered. The average price of new house sales never fell. Prices in April of 2020 were typical of the year prior, then rose thereafter. A broader measure of success was that total loans did not fall sharply and are nearly back to their pre-pandemic volumes. After 2008, it took six years to again reach the prior peak. A broader measure still, total spending in the US economy is back to the level predicted by the pre-pandemic trend.

https://fred.stlouisfed.org/graph/?g=JZ1Y

The Fed can’t control long-run output. As I’ve written previously, insofar as aggregate demand management is concerned, we are perfectly on track. The problem in the US economy now is real output. The Fed avoided debt deflation, but it can’t control the real responses in production, supply chains, and labor markets that were disrupted by Covid-19 and the associated policy responses.

What was the cost of the Fed’s apparent success? Some have argued that the Fed has lost some of its political insulation and that it unnecessarily and imprudently over-reached into non-monetary areas. Maybe future Fed responses will depend on who is in office or will depend on which group of favored interests need help. Personally, I’m not so worried about political exposure. But I am quite worried about the Fed’s interventions in particular markets, such as MBS, and how/whether they will divest responsibly.

Of course, another cost of the Fed’s policies has been higher inflation. During the 17 months prior to the pandemic, inflation was 0.125% per month. During the pandemic recession, consumer prices dipped and inflation was moderate through November.  But, in the 16 months since April of 2020, consumer prices have grown at a rate of 0.393% per month – more than three times the previous rate. Some of that is catch-up after the brief fall in prices.

Although people are genuinely worried about inflation, they were also worried about if after the 2008 recession and it never came. This time, inflation is actually elevated. But people were complaining about inflation before it was ever perceptible. The compound annual rate of inflation rose to 7% in March of 2021. But it had been almost zero as recent as November, 2020. That March 2021 number is misleading. The actual change in prices from February to March was 0.567%. Something that was priced at $10 in February was then priced at $10.06 in March. Hardly noticeable, were it not for headlines and news feeds.

https://fred.stlouisfed.org/graph/?g=JZ0f

PSNE: No More, No Less

Zachary Bartsch3 Comments

Today marks the 27th anniversary of John Nash winning The Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel for his contributions to game theory.

Opinions on game theory differ. To most of the public, it’s probably behind a shroud of mystery. To another set of the specialists, it is a natural offshoot of economics. And, finally a 3rd non-exclusive set find it silly and largely useless for real-world applications.

Regardless of the camp to which you claim membership, the Pure Strategy Nash Equilibrium (PSNE) is often misunderstood by students. In short, the PSNE is the set of all player strategy combinations that would cause no player to want to engage in a different strategy. In lay terms, it’s the list of possible choices people can make and find no benefit to changing their mind.

In class, I emphasize to my students that a Nash Equilibrium assumes that a player can control only their own actions and not those of the other players. It takes the opposing player strategies as ‘given’.

This seems simple enough. But students often implicitly suppose that a PSNE does more legwork than it can do. Below is an example of an extensive form game that illustrates a common point of student confusion. There are 2 players who play sequentially. The meaning of the letters is unimportant. If it helps, imagine that you’re playing Mortal Kombat and that Player 1 can jump or crouch. Depending on which he chooses, Player 2 will choose uppercut, block, approach, or distance. Each of the numbers that are listed at the bottom reflect the payoffs for each player that occur with each strategy combination.

Again, a PSNE is any combination of player strategies from which no player wants to deviate, given the strategies of the other players.

Students will often proceed with the following logic:

  1. Player 2 would choose B over U because 3>2.
  2. Player 2 would choose A over D because 4>1.
  3. Player 1 is faced with earning 4 if he chooses J and 3 if he chooses C. So, the PSNE is that player 1 would choose J.
  4. Therefore, the PSNE set of strategies is (J,B).

While students are entirely reasonable in their thinking, what they are doing is not finding a PSNE. First of all, (J,B) doesn’t include all of the possible strategies – it omits the entire right side of the game. How can Player 1 know whether he should change his mind if he doesn’t know what Player 2 is doing? Bottom line: A PSNE requires that *all* strategy combinations are listed.

The mistaken student says ‘Fine’ and writes that the PSNE strategies are (J, BA) and that the payoff is (4,3)*.  And it is true that they have found a PSNE. When asked why, they’ll often reiterate their logic that I enumerate above. But, their answer is woefully incomplete. In the logic above, they only identify what Player 2 would choose on the right side of the tree when Player 1 chose C. They entirely neglected whether Player 2 would be willing to choose A or D when Player 1 chooses J. Yes, it is true that neither Player 1 nor Player 2 wants to deviate from (J, BA). But it is also true that neither player wants to deviate from (J, BD). In either case the payoff is (4, 3).

This is where students get upset. “Why would Player 2 be willing to choose D?! That’s irrational. They’d never do that!” But the student is mistaken. Player 2 is willing to choose D – just not when Player 1 chooses C. In other words, Player 2 is indifferent to A or D so long as Player 1 chooses J. In order for each player to decide whether they’d want to deviate strategies given what the other player is doing, we need to identify what the other player is doing! The bottom line: A PSNE requires that neither player wants to deviate given what the other player is doing –  Not what the other player would do if one did choose to deviate.

What about when Player 1 chooses C? Then, Player 2 would choose A because 4 is a better payoff than 1. Player 2 doesn’t care whether he chooses U or B because (C, UA) and (C, BA) both provide him the same payoff of 4. We might be tempted to believe that both are PSNE. But they’re not! It’s correct that Player 2 wouldn’t deviate from (C, BA) to become better off. But we must also consider Player 1. Given (C, UA), Player 1 won’t switch to J because his payoff would be 1 rather than 3.  Given (C, BA), Player 1 would absolutely deviate from C to J in order to earn 4 rather than 3. So, (C, UA) is a PSNE and (C, BA) is not. The bottom line: Both players must have no incentive to deviate strategies in a PSNE.

There are reasons that game theory as a discipline developed beyond the idea of Nash Equilibria and Pure Strategy Nash Equilibria. Simple PSNE identify possible equilibria, but don’t narrow it down from there. PSNE are strong in that they identify the possible equilibria and firmly exclude several other possible strategy combinations and outcomes. But PSNE are weak insofar as they identify equilibria that may not be particularly likely or believable. With PSNE alone, we are left with an uneasy feeling that we are identifying too many possible strategies that we don’t quite think are relevant to real life.

These features motivated the later development of Subgame Perfect Nash Equilibria (SGPNE). Students have a good intuition that something feels not quite right about PSNE. Students anticipate SGPNE as a concept that they think is better at predicting reality. But, in so doing, they try to mistakenly attribute too much to PSNE. They want it to tell them which strategies the players would choose. They’re frustrated that it only tells them when players won’t change their mind.

Regardless of whether you get frustrated by game theory, be sure to have a drink and make toast to John Nash.

*Below is the normal form for anyone who is interested.

Forgiveness is Underprovided

Zachary BartschLeave a comment

Forgiveness is Important

Whether one might socially offend us or whether one commits a crime, we face a fundamental tension between punishment and forgiveness. Punishment is important because it acts as a deterrent to the initial offense or to subsequent offenses. But punishment is also costly. Severing social or commercial ties reduces the number of possible mutually beneficial transactions. We lose economies of scale and lose gains from trade when we exclude someone from the market. Forgiveness is important because it permits those who previously had conflict to acknowledge the sunk cost of the offense and proceed with future opportunities for trade. However, an excess of forgiveness risks failure to deter destructive behaviors.

In the US, we enjoy a state that can prosecute alleged offenders and enforce punishments regardless of the economic status of the offended. While not perfect, the state incurs great cost by being the advocate of those who could not enforce great retributive punishment by their own means. A victim may choose to press charges against an offender, or the state can press charges despite a permissive victim.

In fact, our system of prosecution is somewhat asymmetrical. The state can press charges against a suspect, regardless of the victim’s wishes. While a victim can’t compel an unwilling state to press charges, say if the evidence is scant, an individual can engage in litigation against the accused.

Most of the possible combinations of victim and state strategies result in some kind of prosecution of the alleged offender. Except for litigation, our punishments in the US tend not to be remunerative – the victim isn’t compensated for the evils of the offender. ‘Justice’ is often construed as a type of compensation, however.

Herein lies a problem.

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Free Money, Courtesy of Credit Cards

Zachary Bartsch1 Comment

In grad school, I learned about the overlapping-generations model. The idea is that we simplify people down to the fundamental parts of their life-cycle. Each person lives for 2 periods. In the first period, they can produce only. In the second period, they can consume only. A popular conclusion of the model pertains to old-age benefit programs such as Social Security.

The first beneficiaries receive a gift that is free to them, then each subsequent generation accepts the debt, pays it off, and then passes on new debt to the proceeding generation. In this manner, the program benefit of the current generation is limited by the income of the following generation. Therefore, every single generation can consume as if they lived a generation later – and a generation richer – in time. That’s exciting.

But this model is not unique to governments. With a little bit of finance, we can model every person as their own self-encapsulated overlapping-generations model – with two similarly exciting conclusions. Let’s consider a person who has monthly consumption expenditures of $1k per month and let’s assume a discount rate of half a percent per month.

Life is pretty good for this person. They earn income each month and they spend $1k of it during the same period. Now let’s give the person a credit card. It doesn’t matter what the interest rate is – they’re going to pay it off each subsequent month. Now let’s see what’s possible.

What’s going on here? The difference in the consumption pattern is that the first month with a credit card can enjoy twice the consumption. How’s that? $1k of that January consumption is just the typical monthly spending. The other $1k is running up a month’s worth of spending on the credit card. So long one pays-off the card in the following month, there are no interest charges. But wait – if one pays-off the credit card in February, then how does one consume in February? By borrowing from March’s income, of course! And so the pattern repeats ad-infinitum. With a credit card one can borrow against next month’s spending. You too can borrow from your future self. And your future self won’t mind because they’ll do the same thing.

Conclusion #1: Having a credit card entitles you to one free month of double consumption.

The above example includes identical income over time. But, what if your income grows? Let’s assume that your income and commensurate consumption grow at a rate of one quarter percent per month. Our consumption without a credit card is tabulated below.

Obviously, having income and consumption that grow is more enjoyable than ones that are constant each period. Now let’s observe below what happens when we again introduce a credit card that one pays-off each month.

What’s going on here? Just as happened previously with a credit card, one can enjoy an extra boost to consumption in the first period. But what does growing income do for us besides greater complication? Just as previously, one can pay their debt each period and consume by borrowing against the next month’s income. But with growing income, having a credit card means that one can enjoy the next month’s level of consumption today. That is, next month’s higher consumption is shifted sooner in time by one month. Notice that, with growing income, consumption for July without a credit card ($1,018) is the same as the consumption in June with a credit card. Even without the first-month-gift, credit cards increase the present value of one’s consumption by making next month’s greater income available today – and the same is true for every single month.

Conclusion #2: Having a credit card today entitles you to next month’s greater income.

How big a deal is this? Obviously, it will differ with the discount rate and the rate of income growth. Using the numbers above, having a credit card permits one to consume with a present value that is 10.5% higher. Let that sink in. People who have access to credit consume as if they are 10.5% percent richer. Access to credit can make the difference between a pleasant Christmas, having quality internet, paying for car repairs, and so on. Being poorer is one thing. Being poorer and lacking access to credit is like taking an instant haircut to one’s quality of life. On the flip side, people can be made better-off without additional improvements to their productivity. Increasing access to credit may be a less costly improvement to the value lifetime consumption than many of the other less politically feasible improvements to labor productivity.

Compulsory Schooling by Gender & Age

Zachary Bartsch1 Comment

This weekend I’ll be at the Southern Economic Association Conference in Houston Texas. I’m organizing and chairing a session called Education Policy Impacts by Sex (you should come by and see me if you will be there too!).

Personally, I will be presenting on the impact of compulsory school attendance laws on attendance. Today I just want to share and discuss a single graph that’s not my presentation.

Prior to my research, there was already a canon of existing literature on compulsory attendance legislation (CSL) and I’ve previously written on this blog about it (attendance, CSL, and differences by sex). However, the literature had some limitations. Authors examined smaller samples, ignored gender, or ignored different effects by age.

I examine full-count IPUMS data from the 1850-1910 US censuses of whites in order to investigate the so-far-omitted margins mentioned above. Here are some conclusions:

Prior to CSL:

  • Males and females attended school at similar rates until the age of 14.
  • After 14, women stopped attending school as much as men.
    • By the age of 18, the attendance gender gap was 10 percentage points.

After CSL

  • Male and female attendance increased from the ages of 6 to 14
  • Women began attending school more than prior to CSL until about age 18.
  • After the age of 18, women experienced no greater attendance than previously.
  • But, both sexes attended school less than prior to CSL for ages 5 and younger.
  • Men began attending school less after the age of 17.
  • CSL increased lifetime attendance for both males and females

Overall, examining the impact of CSL across many ages allows us to see when and not just whether people attended more school. Previous authors would say something like “CSL increased total years of school by about 5% on average”. For men, almost all of those gains were between the ages of 6 & 16. But women experienced greater attendance from ages 6 to 18.

Additionally, examining the data by age reveals that there was some intertemporal substitution. Once it became legally mandatory for children to attend school between the ages of 6 & 14, parents began sending their younger children to school at lower rates. Indeed – why invest in education for two or three early years of life if you’ll just have to send your children to school for another eight years anyway. Older boys dropped out of school at higher rates after CSL too. Essentially, the above figure became compressed horizontally. People ‘put in their time’, but then reduced investments at non-mandatory ages.

This reveals a shortcoming of the current literature, which focuses mostly on 14 year olds. By focusing on a popular age of attendance that was also compulsory, previous authors have missed the compensating fall in attendance at other ages. Granted, the life-time effect is still positive – but it’s attenuated by a richer picture. The picture reveals that individuals were not attending school by accident. Students or their parents had in mind an amount of educational investment for which they were aiming. When children were forced to attend school at particular ages, the attendance for other ages declined.

Buying in Bulk: Money Saver or Self Sabotage?

Zachary Bartsch1 Comment

Recently, I’ve been buying a lot more non-durable goods when they are on sale. Whereas previously I might have purchased the normal amount plus one or two units, now I’m buying like 3x or 4x the normal amount.

What initially led me here was the nagging thought that a 50%-off sale is a superb investment – especially if I was going to purchase a bunch eventually anyway. I like to think that I’m relatively dispassionate about investing and finances. But I realized that I wasn’t thinking that way about my groceries. The implication is that I’ve been living sub-optimally. And I can’t have that!

If someone told me that I could pay 50% more on my mortgage this month and get a full credit on my mortgage payment next month, then I would jump at the opportunity. That would be a 100% monthly return. Why not with groceries? Obviously, some groceries go bad. Produce will wilt, dairy will spoil, and the fridge space is limited. But what about non-perishables? This includes pantry items, toiletries, cleaning supplies, etc. 

Typically, there are two challenges for investing in inventory: 1) Will the discount now be adequate to compensate for the opportunity cost of resources over time? 2)  Is there are opportunity cost to the storage space?

For the moment, I will ignore challenge 2). On the relevant margins, my shelf will be full or empty. I’ve got excess capacity in my house that I can’t easily adjust it nor lend out. That leaves challenge 1) only.

First, the Too Simple Version.

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You Current Grade: It’s Complicated

Zachary BartschLeave a comment

By now, most US universities are 4-5 weeks away from the end of the fall semester. Whether it’s now, or just prior to the withdrawal deadline, student tend to demonstrate increased interest in their grade for their courses. They say that they want to know how they are doing. But they often prefer to know what grade they will earn at the conclusion of the course. The answer to the latter question could include all kinds of assumptions. But “What is my grade right now?” is a deceptively subtle question.

It seems direct. We could easily be curt and claim that it shouldn’t be complicated to tell a student what their grade is, and that it’s a failure of the teacher or of the education system writ large if it is complicated. While I entirely agree that a teacher should have an answer, it’s important to emphasize that “What is my grade right now?” is an ill-defined question. The problem is that a student can mean two different things when they ask about their grade.

Q1) What proportion of possible points have I earned so far?

Q2) What proportion of points will I have earned if my performance doesn’t change?

It’s important for teachers to ensure that their students understand which question is being answered.

First, I’ll illustrate when there is no distinction between the answers. Let’s say that there are two types of assignments: Exams, which are worth 75% of the course grade, and quizzes, which are worth 25%, of the course grade. So long as the two assignment types are identically distributed throughout the semester, Q1 & Q2 have the same answer. Below is a bar chart that illustrates a distribution of points over 4 weeks. The proportion of points for each assignment type is identically distributed over time (not necessarily uniformly distributed).

What is the student’s grade at the end of week 2 if they have scored 90% on the exams and 70% on the quizzes? By the end of week 2, there have been 30 possible exam points and 10 possible quiz points. The student has earned 34 of the 40 possible points so far. The math for Q1 is:

(0.9)(30)+(0.7)(10) = 27+7=34

34/40 = 85%

And, if they continue to perform identically in each assignment category, then they can expect to earn an 85% in the class. The math for Q2 is:

(0.9)(75)+(0.7)(25) = 67.5+17.5 = 85%

Both Q1 and Q2 have the same answer. And, honestly, principles or introductory courses have formats that often lend themselves well to having assignments distributed similarly over time. My own Principles of Macroeconomics class matches up pretty well with the above math. Each week, there is a reading, a homework, and a quiz. By the time students complete the first exam, they’ve completed about one third of all points in each assignment category.

Higher level classes or classes with projects tend *not* to have identical point distributions across time among assignments. Maybe there are presentations, projects, or reports due throughout the semester or at the culmination of the course. For example, my Game Theory class has two midterm exams, but no final exam. It has homework in the first half of the semester, and term paper assignments in the latter half.

The bar chart below displays a point-split among the same quizzes and exams, but they now are differently distributed throughout the semester. Quiz points have been frontloaded.

What is the student’s grade at the end of week 2 if they have scored 90% on the exams and 70% on the quizzes? By the end of week 2, there have been 30 possible exam points and 15 possible quiz points. The student has earned 37.5 of the 45 possible points so far. The math for Q2 is:

(0.9)(30)+(0.7)(15) = 27+10.5=37.5

37.5/45 = 83.33%

And, if they continue to perform identically in each assignment category, then they can expect to earn an 85% in the class. The math for Q2 is:

(0.9)(75)+(0.7)(25) = 67.5+17.5 = 85%

All I did was frontload 5 percentage points for quizzes and now the answers to Q1 and Q2 differ by 1.66 percentage points. That may seem like small potatoes. But consider that a) many students and universities use and care about the +/- system of grades, and b) a grade difference of 1.66 points was caused by a mere change of 5-point change in the distribution. Bigger changes result in bigger differences. Frontloading the remaining 5 quiz points from the end of the semester would result in a Q1 score of 82% – yielding a 3 point difference between the two calculation methods.

The differences between Q1 & Q2 illustrated above are even more pronounced once you begin to include extra credit. One point of extra credit has a smaller effect on the answer to Q1 as more and more possible course points have been earned.

If students only care about their ultimate grade in the course, then they will always prefer to receive the answer to Q2. But, students may also want to know how effective their recent study habits have been so that they can re-evaluate them conditional on the knowledge of the assignment point distributions. Q2 requires more assumptions if an assignment type hasn’t even occurred yet. Students can ask “Have I given this course the appropriate amount of attention given the types of assignments that we’ve had?”.

For example, my Principles of Macroeconomics course has the first exam at week 5. Students should have an average score that is greater than 90% by the end of week 4 because the reading assignments are simple, the homeworks are lenient, and the quizzes permit practice attempts. Students who have an 80% by the end of week 4 are going to have a rougher time once they encounter an exam.

Reasonable people can disagree about which calculation is more useful. And more mathematically inclined students can calculate their own grades anyway. Therefore, after every exam, I send a mail-merge email to each of my students in order to update them about their grade. I give them the answer to both Q1 & Q2, and I illustrate the impact of several alternative scenarios for their future performance. If there is information that a student wants about their grade, then it’s in that email.

In conclusion, teachers should take great care in making student grades and progress reports clear. Students should take great care to understand what they are asking and and what the answer means. Grades can be very important for students who are close to the margin for scholarships, academic probation, or failure. While students may care too much about their grades, teachers should be sensitive to the fact that the care is real none the less. Teachers owe their students a firm and clear indicator of performance.

*There is another case in which Q1 & Q2 have the same answer. It’s when the student earns exactly the same grade in each assignment category, regardless of whether the category points are distributed identically across time.

Stocking Stuffers: First Mover Advantage & Nested Utility Functions

Zachary BartschLeave a comment

I have two gift recommendations for you this year. Typically, I purchase a lot of very practical items. My wife makes fun of me for requesting tools and hardware as gifts – but hopefully the following list will provide some crossover between practicality and good gift ideas.

Depending on your family’s traditions both of these gifts are stocking stuffers.

1) Laurie Berkner CDs

Having children means that you hear opinions and preferences from more people. And children are sure to share those opinions. When you’re in the car, I recommend that you strike first with 2 different CDs (or mp3 albums) by Laurie Berkner. Laurie Berkner is a singer songwriter who creates outright good children’s music. She has variety and produces earworms that are not too bad to have around. The Ultimate Laurie Berkner Band Collection is a crowd-pleaser. If you’ve got a more intense personality and your children can handle it, then I strongly recommend The Dance Remixes. It rocks.

The idea here is game theoretical. Your children are going to find something that they like. A lot. Odds are good that waiting for them to encounter something won’t bode well for your happiness once they find it. Take the first-mover advantage and introduce them to Laurie Berkner. They’ll get hooked and you’ll be stuck listening to a lot of children’s music. But at least it will be good/tolerable that you also enjoy… Unlike some other alternatives…

2) Highly Specific Treats

We live in a rich society. Most of us walk the store aisles implicitly saying ‘no’ to the vast majority of goods. Even the ones that we like. Take the opportunity that the holiday season provides and say “yes” to getting some special treats. These treats fall into two categories: 1) “Nostalgic Treats” & 2) “I’ve never tried it”.

1) Sharable Nostalgic Treats

When I was about 4-5 years old, I remember getting great big bags of pretzels that were covered in a mustard powder (“mustard pretzels”). As it turns out, they are only a regionally available product and I never saw them again after my family moved from Tennessee. But 33 year old me thought “Surely, the internet has them”. And indeed they do! I made this purchase at a per-unit price that I would not typically indulge. However, I got to share the story and the experience with my family. It pleased me to share a deep memory with them and it pleased them to get a ‘special’ snack. For me, it was mustard pretzels. For my wife, it was a bulk pack of Heath and Skor bars.

2) I’ve never tried it

Separately, while watching Captain America and the Winter Soldier, it occurred to me that I had never knowingly had Turkish Delights. So, I found a variety pack of fancy ones. First, they’re delicious and you feel fancy while eating them. Second, this is 21st century America. What’s the point in saying that we’re rich if we’re not willing to act like it a little? Maybe it’s not Turkish Delights for you. Maybe it’s Pilipino rice candies or Mexican Tamarind candies. Make sure that you get a couple of new treats and share them with others. The purchases are much more worth the price when you consider the nested utility function among your loved ones.

The (Employment) Depressing Child Tax Credit

Zachary BartschLeave a comment

For those who didn’t know, as part of the American Rescue Plan, there were some changes made to the Child Tax Credit (CTC) for the tax year 2021.

  • First, the credit was expanded from $2k to $3,600 per child for children under 6 years of age (to $3k otherwise). It’s also fully refundable.
  • Second, half of the credit is being disbursed to tax-filers early: over the latter 6 months of 2021.

What does this mean?

For context, I have 3 children all under that age of 6. My total 2021 CTC is $10,800. Half of that is being distributed as monthly checks from July through December. For me, that’s a check for $900 per month that I had not anticipated. While it is true that I will see less of a remaining credit when I file my taxes by April of 2022, I strongly suspect that most similar households are somewhat short-sighted about these funds.

Depending on the number of children in a household, the monthly check from the IRS can be quite significant. From the parent point of view, there has been a lump-sum transfer. There is no endogenous response to obtain more children – there’s no time for that. The transfer also occurs regardless of any activities, economic or otherwise. In essence, tax-filers with children have experienced a positive income shock.

The big question is: What is the effect on employment?  

In one sense, the effect is ambiguous and depends on preferences: People can now afford more leisure and more consumption. How they engage in more of each is a matter of preference. But, given that both are goods, both will increase by some amount.

That’s my simple model. I hereby make multiple predictions:

  1. Parents with children will have consumed more in the 3rd and 4th quarters of 2021.  
  2. Parents with children will have lower employment growth for those quarters.
  3. The effects will be stronger for parents with children under 6 years of age.
  4. The employment rebound in the 1st quarter of 2022 will be stronger for these groups (and stronger than forecasted overall).
  5. Finally, while I’m feeling silly enough to make predictions publicly, I predict slower growth in consumer durable expenditures in 2022 Q1.

I looked at the BLS for data to corroborate my predictions. Excitingly, the Current Population Survey (CPS) does slice the data by sex, age, and age of own children (conveniently by younger than 6 and 6-17 years of age). This is where I post the great visual to demonstrate the veracity of my claims, right?

WOMP WOMP.

The relevant data is currently only available annually as recent as 2020…