Supply & Demand, with Tables?

When I was a graduate student, I paid for my tuition by tutoring for the university athletics department. I tutored stat, math, micro, macro, excel, and finance. I tutored the same students each week, so I got to know them pretty well over the course of the semester. I also got to know their strengths and weaknesses. It was at this time that I realized most quantitative or even analytical ideas could be described in 4 potentially equivalent ways:

  1. Mathematically
  2. Using logic in English
  3. Graphically
  4. With a Table

In this post I want to share the Supply & Demand cheat-sheet that I use to help my students learn about the effects of supply and demand.

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Cheers to Sumproduct!

I teach macroeconomics, finance, and other things.

Often, I use Excel to complete repetitive calculations for my students. The version that I show them is different from the version that I use. They see a lot more mathematical steps displayed in different cells, usually with a label describing what it is. But when I create an answer calculator or work on my own, I usually try to be as concise as possible, squeezing what I can into a single cell or many fewer cells. That’s what brings me to to the sumproduct excel function that I recently learned. It’s super useful I’ll illustrate it with two examples.

Example 1) NGDP

One way to calculate NGDP is to sum all of the expenditures on the different products during a time period. The expenditures on a good is simply the price of the good times the quantity that was purchased during the time period. The below image illustrates an example with the values on the left, and the equations that I used on the right. That’s the student version. There is an equation for each good which calculates the total expenditure on the individual goods. Then, there is a final equation which sums the spending to get total expenditures, or NGDP.

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In Praise of the FRED Excel Add-in

Sometimes, large entities have enough money to throw at a problem that by sheer magnitude they produce something great (albeit at too high a cost). The iPhone app from the FRED is not that thing. But the Excel add-in is something that every macroeconomics professor should consider adding to their toolkit.

Personally, I include links to FRED content in the lecture notes that I provide to students. But FRED makes it easy to do so much more. They now have an add-in that makes accessing data *much* faster. With it, professors can demonstrate in excel their transformations that students can easily replicate. The advantage is that students can learn to access and transform their own data rather than relying on links that I provide them.

The tool is easy enough to find – FRED wants you to use it. After that, the installation is largely automatic.

Installed in excel you will see the below new ribbon option. It’s very user friendly.

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Teaching Price Controls (Poorly)

Economics textbooks differ in their treatment of price controls. None of them does a great job, in my opinion. The reason is mostly due to the purpose of textbooks. Despite what you might suspect, most undergraduate textbooks are not used primarily to give students an understanding of the world. They are often used as a bound list of things to know and to create easy test questions. If a textbook has to change the assumptions of a model too much from what the balance of the chapter assumes, then the book fails to make clear what students are supposed to know for the test.

I think that this is the most charitable reason for books’ poor treatment of price controls – even graduate level books. The less charitable reasons include sloppy exposition due to author ignorance or an over-reliance on math. I honestly would have trouble believing these less charitable reasons.

I picked up 5 microeconomics text books and the below graph is typical of how they treat a price ceiling.

The books say that the price ceiling is perfectly enforced. They identify producer surplus (PS) as area C and consumer surplus (CS) as areas A & B. There are very good reasons to differ with these welfare conclusions.

Problem #1

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PSNE: No More, No Less

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.

You Current Grade: It’s Complicated

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.

Reflections on Teaching in Fall 2020

As the Fall semester comes to close on college campuses, it’s a good time to reflect on and assess how the past semester went. Many universities went to almost exclusively virtual learning, but other schools tried to make Fall 2020 as normal as possible given the circumstances of the COVID-19 pandemic.

My school, the University of Central Arkansas, chose the route of trying to have things as normal as possible — by which I mean students live on campus, classes are mostly in-person — while still accommodating students and faculty that preferred a more physically distant atmosphere. For example, UCA increased the number of fully online courses available, roughly trying to meet faculty and student demand. I normally teaching one online course per semester anyway, and I continued that this semester. Other faculty had more online classes than usual, or moved their class to be partially online.

So what was my experience?

First, the students, the most important part of the teaching process. Overall, I would say my students did very well. At least in the classroom, they complied with all the rules the University set forth: wearing masks, physical distancing in classrooms (seen in the image below), even the one-way entrances and exits to the building. There were only 3-4 times I can recall this semester when a student entered my classroom without a mask, and they immediately asked me for one upon realizing their mistake (I kept a pack of surgical masks with me).

My classroom at the University of Central Arkansas, with chairs blocked off for physical distancing.

As far as academic performance of students, I was very pleased with the students. For those students that were able to stick with the class and keep up, which was most students, they perform as well or better than previous semesters. Some students, due to personal circumstances, had trouble keeping up. I tried as much as possible to accommodate students in these situations, by being flexible with deadlines, offering additional resources, and generally just trying to listen to them and empathize. It was hard for everyone.

On my end, I tried to make the teaching atmosphere of the classroom as normal as possible. I usually do have some interactive aspects of the classroom, where students work in small groups, talk to their neighbors, etc. Most of those activities didn’t happen, unfortunately. But otherwise, the classroom atmosphere operated as usual.

As my students did, I also wore a mask in the classroom while I lectured. For students that had to miss class due to quarantine, isolation, or other reasons, we were asked to record every lecture and have an option for students to watch the lecture virtually if needed. Making sure that the video was properly recording and the I had set up the Zoom link for students that needed to be remote added an extra element to think about at the beginning of each class, but it was the kind of thing that once you get used to it, it just became normal.

I will say that I often felt very exhausted after teaching each day. The mental load of making sure everything was working right in the classroom, combined with the constant sense of doom in the world around us, made this a challenging semester mentally. I’m sure this was even more true for some of my students. But, we made it.

Finally, how about the administration of my University. I’ll bite my tongue a little here: I am up for tenure this year! But really, I don’t have anything major to complain about. Guidance was communicated well, although sometimes big changes were rolled out a bit more quickly than the faculty liked. UCA provided isolation and quarantine dorms for students, though these never came close to capacity. Weekly updates on testing, cases, and related data were provided to everyone (and made publicly available, so I’m not revealing any secrets here).

Testing data for UCA students. This data excludes athletes, since they were required to get tested regularly, which could have skewed the data.

As you can see above, the general student body at UCA did report positive COVID cases every week. And some weeks the positive test rate was a little higher than I was comfortable with! But we never had a large spike in cases, and the University held firm to its commitment to offer in-person classes for everyone that wanted them, as long as the campus was generally safe.

All in all, I think it was the best semester we could have had under the circumstances. The only thing really weighing on my mind: we are going to do it all over in the Spring semester. And we’ll do it as well as we can.

Spontaneous Emergence of Property Rights in the Classroom

Last week I posted about Bart Wilson’s talk on his new book “The Property Species” and promised to share a class demonstration about the emergence of property rights in the classroom. But first let me tell you why I did this demonstration.

When I was a student I hated assignments that go through the motions of learning, but provide no learing. Building a paper maché volcano, while fun for some, teaches little about volcanic eruptions. Shaking and opening a soda bottle (pop?) is more instructive: it’s the fall in pressure as the bottle is opened that leads to the rapid release of the gas disolved in the liquid, the same thing happens to magma. And while being able to algebraically solve for the equilibrium price given supply and demand functions is a very necessary evil (to a point), it teaches little about the process of competition and price formation.

This is why I was reluctant to having my first Intro to Economics class write their own version of “I pencil”, quite a few years ago. Driving the point of how largely anonymous exchange and specialization, coordinated peacefully through property, prices, and profits and loss makes the modern world possible is very important. But how much can you really learn about this by watching and transcribing an episode of “How It’s Made”? For most students, not much at all. Partly in dread of reading and grading 80 versions of “I whiteboard marker”, or “I toothbrush”, and partly following my conscience I decided to throw in a twist.

The twist may seem evil and arbitrary at first. Students still had to choose a good and write their own version of “I _____” , but if two students wrote about the same good I would divide their grade by 2. If three students wrote about the same good I would divide their grade by 3 and so on. I did not give any additional prompts about how they should sort out potential conflicts or coordinate amongst themselves. These were just the rules of the assignment.

Without this seemingly arbitrary grading rule, goods to write about were not scarce. By changing the grading rules, goods to write about became scarce. While there are many more goods to write about than students, certain goods stand out in the mind, and extra effort must be devoted in thinking up a new good, and finding out if someone had already looked around their room and chosen the same good. Now students also had to coordinate amongst themselves or run the risk of a fairly severe penalty to their grade.

As expected, I have never had to enforce the the harsh grading penalties (anecdotal, I know). Students always find a way to coordinate and establish property rights over suddenly scarce goods. The point of the assignment was no longer about I pencil, but about the emergence of property rights and social coordination (and hopefully a little bit about I pencil as well). I didn’t act as a central authority that imposed and enforced property rights. I merely changed the incentives and constraints, hoping that the costs of coordinating and setting up agreements was smaller than the costs of not doing this.

When they turned in their assignment, we discussed how they had actually coordinated. Over the years I have seen multiple ingenious mechanisms. From class forums using the university platform, to a simple spreadsheet circulated amongst the students via email or WhatsApp. In the good old times before the pandemic they would sometimes meet after class and sort it out in person. Sometimes they created a common pool of goods and one of their classmates is chosen to distribute them among their peers. Leaders emerge to fill various roles from dispute resolution to registering claims. How this person is chosen also varies from class to class. Some students volunteer, others have it thrust upon themselves. The use of a homesteading rule is fairly common, first to choose gets the good in cases where there are multiple claims. In class we discuss why they use this rule, rather than last to choose gets the good, and the problems this alternative would entail.

I have only had one instance of a strong and contested dispute among “property owners”. That semester students had to not only write but present their work. Two groups (that semester “I _____” was a group assignment) wanted to do a good they thought would be amusing to present in class. I’ll leave it up to your imagination what good students in their late teens and early twenties might find to be amusing to present in class. The two groups of students underwent a rather complicated dispute resolution system with the rest of the class playing the role of arbiters of the multiple claims to the same good. Neither group wanted to budge, but one group ended up ceding the rights in the end.

What I like about this little classroom demonstration is that it makes it easier to teach the emergence of institutions as the products of human action but not human design. Order without design is a difficult concept to grasp, but maybe even more importantly it is a concept that is difficult to accept. But after this demonstration, not anymore, students experience the emergence of property rights. An added bonus is that in this case scarcity is clearly a product of the relation between their minds and how they relate to the world, not about objective quantities of goods.

Property rights emerge through their coordination but are not centrally imposed. They coordinate because a change in the environment turned a previously free good, the subject of their short “I ____” essay, into a scarce (economic) good. As you can probably tell Harold Demsetz is one of my favorite economists of all time. After the barrier of disbelief is breached, we can easily talk about the spontaneous emergence of money, cover a little about how property rights emerged in whaling on the high seas, and the spontaneous origin of law (very useful for future law students usually educated in the positivist tradition, as is the norm in Ecuador).

I later learned of the fish game (I am not an experimentalist). But, no disrespect intended, it seems a little contrived. I still like my assignment better. While the goldfish game teaches the tragedy of the commons, the “I _____” assignment teaches how the tragedy can be solved without a centralized authority by having students solve if for themselves and come to grips with the real limitations and problems they faced, albeit on a much smaller scale. I am still hoping for an experimentalist that thinks something serious can be made out of my little classroom demonstration.