Merry Christmas. As my gift to you, I present the long overdue answers to the remaining problems from my Oberlin honors exam. The original questions are here and here; the first round of answers is here.
Question 6. When Eve works, she produces exactly one apple per hour. Adam is completely unproductive and can produce nothing at all. Eve’s income is taxed at a flat percentage rate, with the proceeds delivered to Adam. What determines the optimal tax rate? What does “optimal” mean here, and what philosophical justification would many economists give for adopting this tax rate?
To make the problem concrete, you can assume that both Adam and Eve, if it were both possible and necessary, would be willing to work up to 1 hour for 1 apple, up to 2 hours for 4 apples, up to 3 hours for 9 apples, and up to x hours for x2 apples. Now what is the optimal tax rate? (Your answer should be a number.)
Answer. In preparing the answer to this question, I realized that my translation from the economese on the original exam to the English above sacrificed some critical information. The original exam specified that Adam and Eve have identical utility functions I1/2-H, where I is income and H is hours worked. That’s more than you could figure out from the problem as stated here. Mea culpa.
That having been said, it’s not hard (with a little calculus) to check that when Eve is allowed to keep a fraction S of her income, she works for S/4 hours, and earns a utility of S/4. Adam receives the fraction (1-S) of Eve’s income, which, given how much she works, comes to (1-S)S/4, and earns a utility equal to the square root of that. The sum of their utilities is maximized (again, using a little calculus) when S = (1/2)+Sqrt(5)/10, or approximately .724, which is the fraction of her income that Eve should be allowed to keep.
But why ever would we want to maximize the sum of utilities? The argument goes like this:
- The “right” tax rate is the one that Adam and Eve would agree on in a state of amnesia where neither was sure who was who—so that they’d be forced to set aside narrowly defined self-interest.
- The choice of tax rate in that state of amnesia is tantamount to a choice among wagers. Any tax rate amounts to a wager with some payout if you turn out to be Adam and some other payout if you turn out to be Eve.
- The utility function is *defined* to be the function whose expected value people seek to maximize when they choose among wagers. Therefore we can use the utility function to predict the rate Adam and Eve would agree on.
I teach my students that this is a powerful argument and well worth understanding in detail, though there are other powerful arguments that lead to different conclusions. In any event, this is what I expect an economics student to understand as the “optimal” tax rate in a problem of this sort. If a student had used an alternative—and equally interesting—definition of “optimal”, that could have been the basis of a worthy alternative solution.
Question 7. Jack and Jill play a game. First, each flips a coin. After seeing their own coins (but not each others’), each player (separately) says either “Red” or “Black”. If they name opposite colors, then the Black-sayer gets $4 and the Red-sayer gets nothing. If both say Black, then they both get either $5 (if both flipped heads) or $10 (otherwise). If they both say Red, then they both get either nothing (if both flipped heads) or $20 (otherwise). Assume both players play optimally. If Jack flips heads, what is the probability that he says “Black”? What if Jack flips tails?
Edited to add (in response to a comment from Ron): Assume that neither Jack nor Jill says either Red or Black with probability zero.
Answer. I’ve answered this one here.
Question 8. The five Dukes of Earl are scheduled to arrive at the royal palace on each of the first five days of May. Duke One is scheduled to arrive on the first day of May, Duke Two on the second, etc. Each Duke, upon arrival, can either kill the king or support the king. If he kills the king, he takes the king’s place, becomes the new king, and awaits the next Duke’s arrival. If he supports the king, all subsequent Dukes cancel their visits. A Duke’s first priority is to remain alive, and his second priority is to become king. Who is king on May 6?
Answer. I’ve answered this one here.
Question 9. Suppose the government mails every taxpayer a check for $300. Under a variety of assumptions, discuss the short run and long run effects on a variety of economic variables such as output, employment, the interest rate and the trade balance.
Answer. There are all kinds of directions to go with this one, but a few key points that I’d hope to find in any good solution. First, the answer depends on whether those $300 checks make people feel richer. If so, you’d expect them to respond by spending more and working less. In other words, they’ll want to consume more stuff while producing less. Where does that extra stuff come from? Presumably from abroad, so net imports rise. At the same time, competition for a limited supply of goods pushes the interest rate upward.
From there, a good answer can go in any number of directions: How does the story change in a world of imperfect markets? In what ways does it depend on the nature of those imperfections? And under what circumstances should we buy the assumption that those checks (in tandem with the combination of future tax liabilities and cuts in government services that will be necessary to fund them) really do make people feel richer? In what circumstances do they make people feel poorer (in which case all the effects are reversed)? And so on.
Question 10. Suppose you want to study the effect of education on wages. You have wage data for 100 pairs of siblings, where one member of each pair attended college and one didn’t. Based on these data, you make some estimates. Now you learn that all 100 pairs of siblings are in fact twins. Does this increase or decrease your confidence in your results? Make some arguments in both directions.
Answer. There are several things to say here, but here’s one that I was hoping for: If John goes to college and his younger brother Joe doesn’t, it might be because John and Joe are very different people—or it might be because their parents went bankrupt in the interim. On the other hand, if John and Joe are twins, the interim bankruptcy is ruled out, so we can be sure that John and Joe were somehow different. Let me repeat that: The fact that they are twins should make us more certain that there’s something different about them, not less—because it rules out a lot of alternative explanations for why only one of them went to college. That intrinsic difference should make you more skeptical of your results.