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.
Regarding your answer to Q10: Do you think it would change matters if you found all 100 sibling pairs were IDENTICAL twins?
The sum of their utilities is maximized […] at […] approximately .724, which is the fraction of her income that Eve should be allowed to keep.
OK, I am happy that I got the tax rate right, but I don’t understand why the tax rate should be equal to the sum of their utilities at its maximum. Or did you mean “When the sum of their utilities is maximized, the fraction of her income that Eve should be allowed to keep is …”?
Merry Christmas to all (but you can interpret that as whatever seasonal greeting you prefer).
Snorri: That is indeed what I meant; I’m editing to make this clearer. Thanks for catching this.
RL:
Say the Smiths are not twins; the Joneses are fraternal twins; the Browns are identical twins.
A priori, the Browns should be most similar (same genes and same upbringing), the Joneses next (same upbringing, different genes) and the Smiths last (different genes, somewhat different upbringing).
Now I learn that in each pair, one went to college and one didn’t. For each pair, I upgrade my estimate of how different they are, by one amount for the Browns and Joneses, but by less for the Smiths, who were perhaps affected by a change in family circumstances.
So I stil think the Browns are more similar than the Joneses. The Smiths, depending on how much I’ve changed my estimates, could now rank anywhere on the list.
For #6, how do you know that Eve’s utility function isn’t
U= 10*(I^.5 – H) or U=.1*(I^.5 – H) ?
You can’t say “because that’s given in the problem” because, technically speaking, a person’s utility function is only unique up to an affine transformation (or really, any increasing transformation if there’s no uncertainty, and there really isn’t any in the problem above).
Also, if we’re going to go whole Rawlsian hog, then we should go back and *really* strengthen your language in “Of Jerks & Bullies.” Because after all, it’s just not very reasonable to say that behind the veil of ignorance/uncertainty/amnesia you *don’t* remember what your preferences are (or, to take Rawls seriously, how to calculate probabilities), but you *do* remember what country and year you were born in.
I would like to see an alternate question #9:
Suppose the government gives every citizen 15 cents for every hour worked, in addition to their usual wage. 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? Assume the minimum wage is lowered by 15 cents per hour to keep entry-level wages constant, and non-citizens were ineligible for the subsidy.
($.15 * 40 hours/week * 50 weeks/year = $300)
(Pigovian labor subsidy?)
Ryan: It doesn’t matter which affine transformation I apply to Eve’s utility function; she’ll still have the same preferences across wagers. She wants to maximize U(Eve’s income, Eve’s labor) + U(Adam’s income, Adam’s labor). If we apply an affine transformation to U, the solution to the maximization problem won’t change.
Michael: You are (more or less) asking for the effects of the Earned Income Tax Credit, no?
The EITC has a phase in/plateau/phase out shape for the total amount earned. Assuming Wikipedia is correct, for 1 person, no dependents:
Credit of 7.65% * x between $0 and $5970
Credit of $457 between $5970 and $7470
Credit of $457 – 7.65% * (x – $7,470) between $7470 and $13,440
Nothing above $13,440
Adding $0.15 per hour earned, regardless of base wage, gives everyone a tiny incentive to work a bit longer, thought it would be more of an incentive to those with lower wages. I’m not sure how close the effect would be to EITC, though, which is why I find it interesting (grin).
What if your result is that education has no effect on wages? Then it seems to me that you should be more confident of your results if you had a sample of twins.
The title of this post is “The Big Answers, Part I”; it should be “The Big Answers, Part II” :)
Roger Schlafly: I see your point, though I think there are a lot of ways you could argue this. One twin went to college and the other didn’t; therefore we know they’re very different; nevertheless they’re earning the same wage. So maybe that means education made up for the big difference.
Dr. Landsburg,
You’re right as long as I apply the same transformation to both utility functions (which is to say, as long as I assume they have the same utility function). But there’s no reason I have to do that; since each of those transformations describes the same preferences, I could use one utility function to describe her preferences and another to describe his. This would in fact change which distribution (first best or second best) maximizes the sum of utilities.
Ryan: I am not sure what experiment you’re envisioning. I am imagining that Eve knows her preferences but does not know whether she will have Eve’s consumption/labor or Adam’s consumption/labor. Likewise for Adam. We can apply any affine transformation you want to Eve’s utility function, and it won’t change her preferred tax rate. Likewise for Adam.
The given answer to Q10 is clever but it seems like it’s countered by an identical argument in the other direction.
You’re saying that when we find out the siblings are twins, their upbringings are now more similar than we previously thought, and since they still made opposite decisions about college, their genes must be more different than we had previously thought.
And therefore, (a) “Their genes are more different than we had thought, so in that respect the siblings are less similar than we had previously believed, so our results are less reliable.”
But why couldn’t you say equally in the other direction: (b) “Their upbringings are more similar than we had thought, so in that respect the siblings are more similar than we had previously believed, therefore our results are even more reliable.”
There could be a purely logical reason why (a) outweighs (b), but I’m not seeing it yet.
Bennett: You *can* say that. The question asked for arguments in both directions; here I gave an example of one important argument I’d hoped to see in any good answer. There are lots of other arguments that could appear in a good answer also.
Dr. Landsburg,
At first I wasn’t imagining any particular experiment — just trying to maximize the sum of two utility functions. But I would say that I’m imagining her being uncertain not only about which productivity she’s born with, but which utility function she’s born with as well. (If we don’t allow for this type of uncertainty, how do we allow for the case where they actually have different preferences as well — e.g., different labor vs. consumption tradeoffs or a different Euler equation?)
Ryan: If they have different utility functions, then I can’t see any justification for adding their utilities. First, as you point out, it would be meaningless to do this, since the functions are only defined up to an affine transformation. Second, even if it were meaningful, I can’t see what the justification would be.
What I tell my students is this: a) This whole procedure is predicated on the belief that, at least in the initial “behind the veil” state, everyone shares the same utility function, b) it is empirically the case that, at least to a very rough approximation, we do pretty much all seem to have pretty similar utility functions in the sense that we all seem to have coefficients of risk aversion that lie within a reasonably narrow range, and c) one might be able to use b) to get past the obstacle of a), but there are a lot of other obstacles, both practical and philosophical, along the way.
Dr. Landsburg,
That’s a fair argument, but it seems to me rather different (in that it is more reasonable!) than the standard approach in the optimal taxation literature. For instance, Saez (2001) specifically defines preferences over consumption and income, a bad, so that preferences are necessarily heterogenous (given heterogeneity in how easy it is to earn income).
It also seems to me that there really is a fair bit of heterogeneity of preferences, especially when we add a time element (discount rates vary quite a bit, I think), so I don’t see how to apply the “behind the veil” approach in actual taxation. It also seems to me that the veil motivates taxation, it must be that people know where and when they are born even behind the veil or the primary issue would be redistribution across borders and savings policy. But I may be taking up more than my fair share of time and space here.
Ryan: All of your points are excellent, and I mention them all in the classroom when I teach this stuff (except for the observed heterogeneity of discount rates, which I’ll make a point of adding to the lecture next time). Please do continue to take up time and space around here.