Two weeks ago, I posted the first half of the honors exam that I administered last spring at Oberlin college. I am following up today with the second half. Once again, I’ve translated some of the questions from economese to English, but am fairly confident that nothing significant has been lost in the translation. This starts with Question 6:
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.)
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.
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?
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.
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.
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.
Question 9 – Money from a helicopter scenario
Short run:
output increases. The money will be spent before prices inflate back to parity with pre-check value. The increased demand for goods will be met by increased production of them. Decreased interest rates allow more to be borrowed for business expansion resulting in increased productivity.
Employment. Employment rises as a result of the increased demand for goods.
Interest rate. Interest rates drop.
Trade balance improves. Increasing the money supply lowers the value of the dollar which makes domestic exports more attractive. Less valuable currency likewise makes foreign imports less attractive.
Okay I’m just rehashing as much as I can remember of the ISLM model and I think my portrayal here is about as right as the model is. In reality, in this economic climate, human behavior isn’t reliably predicted. Off the cuff I think most of the increased cash would go into the bank.
Question 8: I draw inspiration (loosely) from the puzzle of the islanders who don’t know their eye colors.
Let’s start from Duke 5. If the King has no support when he arrives, then he will kill the King.
Duke 4 knows this, so he will support the King on day 4 (assuming that his visit is not canceled).
Duke 3 knows this, so he will kill the King on day 3, unless the King already has support.
Duke 2 knows this, so he will support the King on day 2 if necessary.
Duke 1 knows this, so he will kill the King on day 1.
The result is that Duke 1 becomes King on day 1, gains the support of Duke 2 on day 2, and all other Dukes cancel their visits.
Question 7 is pure game theory, but we need some more
information.
1. Where does the money come from? I.e.: is theis a zero-sum
game? If the players each throw their money into a pot and play
until the pot is empty, then the only winning strategy is to
maximize your own gains at the expense of the other player. If
the money is provided by a third party, the winning strategy
would be to maximize the returns over the long run. That
circumstance is easy: they each declare red without even looking
at their coins – $15 expected return per round.
2. On a practical basis, they can’t announce their colors
simultaneously, as this would keep each from hearing the color of
the other. Are we supposed to ignore this and assume there is no
information about each player’s color announcement before each
announces? Or, is when each announces their choice a part of the
strategy?
Ron:
1) The money comes from a third party.
2) They announce their colors simultaneously, outside each others’ hearing distance, but within hearing distance of the referee who makes the payoffs.
3) Both declaring red without looking at their coins is indeed one possible equilbrium outcome. It is not the only one.
Dr. Landsburg,
#6 is a number? Surely it’s a function. Even with additive utilities, it’s still a function of a weighting factor. (Also, given how much information we have, why are we using one linear income tax and not lump sum taxes indexed to the agent? Just because Frank Ramsey started with that arbitrary assumption doesn’t mean you have to.)
Question 10:
I note that the question does not say that the twins are identical.
The three factors that I can think of are:
a. Twins are raised in an environment more similar than that of siblings generally; that eliminates some confounding variables. (Even more confounding variables are eliminated if the twins are identical.)
b. If one of two twins goes to college and the other does not, then the one who goes to college must have been more talented (or perhaps less eager to start making money right away). (People who have identical twins in their family tell me that identical twins differ in personality much more than one might think, so this applies even to identical twins.)
c. Results obtained from people raised as twins, who nonetheless are sufficiently different so that only one of them goes to college, do not necessarily apply to the general population. For instance, sibling rivalry might stimulate both of them to work harder.
Simon,
I believe you’ve solved problem 8 correctly in terms of logic.
However, in this case, we’re not given the provison that the dukes
are logicians. What happens depends on how smart the dukes are and
how smart they believe the other dukes are.
If the primary priority is to stay alive, the first duke will
support the king. This guarantees that he will stay alive even
if duke number two fails to figure out the optimum strategy.
Thus, the king on May 6 is the original king.
Question 9:
Macroeconomics makes me uneasy, but I am keen to try my hand at this question.
Let’s distinguish 3 “pure” cases, in which the government finances the $300 checks entirely by a single method: [a] printing money (figuratively speaking); [b] cutting spending; [c] borrowing.
Case [a]: increasing the money supply will cause inflation. If the checks sent to the government are unexpected, and the recipients are not perfectly rational, then the checks will initially stimulate spending, and therefore increase output and employment in the short term. The trade balance will worsen in the short run, since some of the extra money will be used to buy from abroad.
Interest rates will rise for 3 reasons: [1] the extra spending will increase the price of consumption (thanks Steve for The Armchair Economist), and therefore raise real interest rates; [2] nominal interest rates will rise even more, because of inflation; [3] a sensible central bank will raise interest rates to prevent inflation (this action seems to be dismissed as irrelevant in The Armchair Economist, which is one of the things that makes me feel uneasy).
Case [b]: demand decreases in the public sector but increases in the private sector. The net effect on any economic variable will be zero, as far as I can see; except that, presumably, private consumers will spend more money abroad, and therefore the trade balance will get worse to some extent.
I note that a permanent tax cut, as distinct from an equal and one-off amount sent to every taxpayers, would have a more substantial effect, because it would increase incentives to work.
Case [c]: Ricardian equivalence kicks in to some extent: people use part of the checks to buy government bonds to pay for future tax increases (i.e. to be ready to pay off public debt). However, few people are perfectly rational, and some rational people will be planning to emigrate, so many people will spend some of the money sent to them. As a result, this case is similar to case [a].
Since I spent time going through the first set, I might as well do these…
6.) A few different cases:
a.) If you want to stay equitable, then you want to set the tax rate to 50%. This is hardly optimal as she will work 0 hours unless she can work a fraction. As a result, there’s no apple for either.
b.) If you want to maximize total happiness, defined by total consumption by the two, then tax at 0%. That way, Jill works for 1 hour and there’s 1 total apple. (none to Jack).
7.) I hope my expected value computation is correct.
a.) The probability that he says Black is 4/9.
b.) The probability that he says Black is 8/13.
8.) The same king as May 1st.
9.) Under the Ricardian Equivalence, people will not change their consumption pattern (although reality might suggest otherwise…).
a.) Output: No change if the money from the government is otherwise taken out of the investments into production.
b.) Employment: No change, no income effect since they will have to pay up later.
c.) Interest Rate: No change if the money from the government is otherwise already taken out from the inter-temporal market.
10.)
Increase – each pair of observation is homogeneous (assumed) with only difference being education. You don’t have to worry about factors such as age difference, years of experience vs years in school.
Decrease – Some dependent variables (if any besides the binary for education) might have weights/coefficients that shouldn’t be there otherwise. This will require the results to be completely re-computed.
I wonder how people will do if these questions are on their prelims…
Snorri Godhi was right up top about the Dukes. Duke #1 will be king after killing the king in the first chance. Dukes 2 and 4 will support whoever is king when they have their chances.
Mark,
For your 6a, is equity necessarily the same as equality of consumption?
For 6b, is happiness (or utility) identical to consumption? Of course, it’s possible that utility is linear in consumption, and it’s possible that it’s one-to-one for all people, and it’s possible that “total” happiness is determined by simply summing utilities across individuals. Are they true here? (It may be true that utility is linear in consumption, but here certainly labor matters too.)
The Duke question doesn’t satisfy me because there isn’t enough information on how the Dukes make their choices. However, if the King is so weak that he needs to be propped up by the Duke, or can be easily killed by him, it is logical for Duke 1 to take advantage of the Kings weakness and kill him right away.
Who is King on May 6th? Still Duke 1 – for the time being. If he executed the King, he’s doubtlessly prepared to kill other major players. Duke 2 would arrive at Regicide Castle and realize this immediately. Since his priority is staying alive, he would support Duke 1, at least for the amount of time it took to ally with other Dukes and raise a significant army.
Question 8:
Duke One will be king on May 6th as he is the only Duke with the opportunity to achieve both priorities: staying alive and becoming king. Duke One arrives on the first day and kills the king – staying alive and becoming king. If Duke Two arrives and kills Duke One, he knows he might be (and likely will be) killed by Duke Three and so on until Duke Five is king on May 6th. Given that Duke Two’s first priority is to stay alive he will support the new king which makes Duke One the king on May 6th.
Gotta go with Ron on Q.8. The only way Duke 1 can guarantee he survives (by the rules as stated) is to support the king. Even assuming that all the dukes are logicians, how can Duke 1 know Duke 2 hasn’t made a deal with Duke 3. If his top priority is to stay alive, he has to support the king. (And nothing in Gene Chandler’s lyrics suggests a Duke of Earl might be a logician).
@Daniel Lee: that logic works, if Duke One has equally prioritized staying alive and becoming king. However, staying alive is his first priority, and becoming king is secondary to that. If Duke One kills the King, he opens himself up to the risk of being killed by Duke Two. If Duke One supports the King, no other Dukes will arrive, and therefore Duke One will remain alive — his biggest priority.
The original king remains king on May 6th.
It looks like most people are assuming facts not in evidence for question 8. The key for each Duke is knowing their position in the line. But the problem, as stated, does not indicate that the Dukes know where they are in the line. Duke 1 doesn’t know that he is the first Duke, and doesn’t know how many more Dukes will follow.
That being the case, the optimal strategy is for Duke 1 to support the original King.
I think you people are way to into trying to think this duke thing out and have made a major error along the way. Reread the last part of the question, if the dukes priority is to live he will support the king and all the other duke stay home. I think if this question was asked of a group of children they would have gotten that part and not blathered on about the thought process of the other dukes who they can’t know because they are not real and are actually inconsequential because the question itself doesn’t give the information so you are assuming it and we all know what assuming does. LOL
Any way we could get the economese for these questions as well?
Assuming that all 5 Dukes would make the same decision in the same circumstance, there are only 2 possible answers: either the original king remains king OR Duke Five is the king.
If Duke One kills the king then we are left with the exact same problem, only with 4 Dukes. Duke Two is in the exact same position that Duke One was in yesterday, and will make the same decision Duke One made.
So there is no reason to for us to think that if Duke One kills the king Duke Two will supports Duke One (though Duke One may wish it were the case).
Inductively, as long as the number of Dukes is greater than 1, the problem is the same.
To come to a different conclusion, we need more information about the Duke’s individual motivations and alliances.
To know which of the two possible inductive scenarios hold, we need to make a judgement about their priorities.
I take “First priority is to remain alive” to mean that they will not risk their lives at all to become king, in which case the original king remains king.
Or, to argue the opposite, Duke 1 will remain king because Duke 1 always kills the king if there are an odd number of Dukes, and supports the king if there is an even number.
If there is 1 Duke, then he kills the king and remains alive.
If there are 2 Dukes, then Duke 1 supports the king since, if he kills them, there will be only 1 Duke and we know what happens then.
If there are 3 Dukes, then Duke 1 kills the king, knowing that Duke 2 cannot kill the king for fear of Duke 1.
If there are 4 Dukes, then we must support the king for fear of Duke 3.
Etc.
If I provide the correct answers, do I also get an honors degree in economics from Oberlin?
6
You say they would both be able and willing to work, and yet adam is completely unproductive. You give figures that look like an equation is possible, but there is no equation. How much does adam love eve. The answer is between %30 and 40% – depending on their relationship, and <%30 if she really hates him – which she should because he is completely unproductive. Maybe he is good in bed. Call it %35
A similar problem to the duke problem is one involving pirates: There are 100 pirates dividing 50 gold pieces (no fractions of a piece is allowed). The first ranking pirate suggests a division which all pirates vote on, if at least half the pirates (including the dividor) agree on the division the split is made, otherwise he is killed and the next pirate makes the deal. The pirates are perfect logicians and perfectly rational. They value staying alive above all else, if they stay alive either way they will pick the option which gets them the most money, and all else equal they will pick the option that kills the most people. (As an example take three pirates dividing the coins, the second to last pirate cannot die [if he gives the last pirate nothing the vote is tied 1-1 and he gets all the money] therefore even if the third to last pirate gives him all the money he would vote against him because that results in less bloodshed). How is the money divided in the end?
7
Sorry. Question makes no sense
Duke1’s dying scream of “B-b-but that’s not logicaaaaaal!” echoes around the castle as grinning, swivel-eyed Duke2 guts him like a halibut.
8
The king will still be king
Crucially,
They will either spend it, or pay down debt, or if they are not in debt then they will save it. In all of these cases it is better than if it were given to Goldman Sachs. Oops!
10
Now that you know that they are twins you know that the one that they are equally able (I assume that is what you are getting at by twins) and you also know that the one that went to college is massively in debt. She should be able to get a better job, but she has to service $150,000 in loans. Her college free, debt free twin is probably better off, because there are no jobs that will enable her to pay off her debt – they are in luck however. As twins they can do porn – and they will make shit loads. Ask the Milton Twins
Disclaimer: Never opened an econ book.
1. Not enough info. Implied value is not true value. eg. When peanuts are cheap she MAY be drinking as much rootbeer as she can handle. If her income raises her rootbeer will still be at her consumption peak.
2. maths- those solutions depend on rationality. is he perfectly rational?
3. False. he should be shot anyway, monopolists who maximize income are scum. he sounds like a slum lord.
4. Store wins, customer is even or loses. Total efficiency declines versus cash due to costs of card production and marketing.
5. Depends on your world view. Even with “equal” tax burden.
6. You state Adam never works, then you state he would. Your use of Adam and eve also implies a relationship. Optimal tax depends on the intangibles Adam brings outside of production. Also, How many apples must be eaten to survive? If Eve values Adam for procreation and other intangibles then whatever tax keeps him happy. So 50% if she she does the bare minimum for survival. If not the optimum is 0%. Partial value would provide an answer between. Also does Eve enjoy picking apples?
7. If they both call black 25% of the trials would be $5 and 75% would be $10 they win $35 on average every 4 games.
If they both call red: 25% for $0 and 75% $20 they win $60 every 4 games
If they base their call of the result of the flip and Jack has heads: he may gain $4 at the expense of causing Jill to come off the Optimum money maker o $60 garanteed.
8. Given that top priority is to survive. All Dukes but the last to visit will support the king. The odd/even arguement only works if the Dukes are perfectly rational, you didnt state that they are.
9. lol- what happened with that?
10. good question. overall: less confidence, if he wasnt with it enough to do his research on the siblings first…Who is that negligent? Thats a big thing to miss. Would the age data have to be correlated to the overall economic situation at each stage of wage earning? You’d ahve to account for a cohort who goes into the economy from school during a bad job market, etc.
“If he supports the king, all subsequent Dukes cancel their visits.” Am I the only one who thinks this is the most important line in the riddle?
Duke 1, wanting to stay alive, supports the King and all subsequent Dukes cancel their visits. Thus the king remains the king.
@ Ryan Yin:
“Equitable Consumption” in that sense means Jack and Jill consume the same amount. Or equivalently, total production divide by 2.
From the question: “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?”
“Total Happiness” is a bit vague but it really assumes that if one is willing to work for up to x hours for x^2 apples, he must be “happy” to do that. Not making any assumption about utility as one will need to discuss about the existence of it. We can answer it simply by their willingness to work. This happiness is in turn Jack’s plus Jill’s.
Hope it makes more sense.
Mark
Hi,
Can we have it in original economese, a PDF like in the first part? Thanks.
Wanderer.
Mike and Wanderer: I’ve posted the economese at http://www.TheBigQuestions.com/part2.pdf .
Thanks Steve!
So for the dukes Question:
Who will be King on the 6th day. Not too sound daft but this sounds like one of those silly riddles.
Regardless of who kills who – the king will be the king on the 6th day because he’s the king now – not a duke.
>9. lol- what happened with that?
The Australian government sent almost all taxpayers AU$900 each this year. Other groups (some non-taxpayers) received other “stimulus package” payments. The instruction was to “don’t save it, spend it”.
One way or another, Australia technically avoided a recession this year. While this kind of strategy keeps voters happy, what do economists think? Is it a significant contributing factor to avoiding a recession?
Question 6:
Thank you for the economese, it makes question 6 much easier (for me). Just to be clear: the following is based entirely on the economese formulation.
Eve’s utility is
E = sqrt(I) – H
Given her rate of production and tax rate T, her income is
I = H.(1-T)
[where the dot indicates multiplication.]
Eve will work until her marginal utility is zero.
If I computed the zero crossing correctly, Eve will work H0 hours, where
H0 = (1-t)/4
At that point, the utilities for Eve and Adam are:
E = sqrt[H0.(1-T)] – H0 = (1-T)/4
A = sqrt(H0.T) = sqrt[T.(1-T)]/2
Maximizing total utility U = E+A subject to my doing the math properly, I obtain an optimal tax rate:
T0 = [5 – sqrt(5)]/10
or approximately 27.6%.
A remark: if economic growth is affected by the taxation rate, then Adam might be better off in the long term if he votes for a tax rate lower than the “optimal” rate.
Another remark: as I remember, happiness studies show that happiness increases with income only for those who work for their income: increasing welfare benefits does not increase the happiness of the unemployed (as long as they have enough to live, of course).
8:
Duke 1’s reasoning as I see it (assuming he knows about the other dukes’ preferences, how many of them there are, and is certain they will behave rationally):
[If I kill the king, the 2nd duke should reason:
(If I kill 1, 3 should reason:
,
therefore I, Duke 2, should not kill 1),
therefore I, Duke 1, should kill the king.]
So Duke 1 is king on May 6th, at least if he has the balls to trust the rationality of his fellow dukes.
In real life, however, no one touches the king.
My brilliant comment is awaiting moderation although it seemed moderate enough to me…
Is the economese for #6 the same as the question posted above? When I calculate the FOC for Eve from the utility functions, I have her choosing to work for W/4 hours, where W is her real (after tax) wage, which doesn’t seem to be the same as above.
Oh, wait — does the description above mean they’re indifferent between (zero apples, zero work) and (X^2 apples, X work)? Ok, got it.
However, my comment about the solution being a function and not a single number stands, even restricting attention to “social welfare functions” that are strictly additive in utilities. After all, Eve’s preferences can be equally represented by U = 2I^.5 – 2H or (in a zero uncertainty world) U = e^(I^.5-H)
@ Mark Liu,
It’s fine to use a “equality of consumption” as a standard. I’m just saying that it’s better to use that term (or even just “equality”) rather than “equity”.
It’s also fine to take a stance that we infer utility from willingness to work. My point was that you can’t assume that preferences are purely defined over consumption, or that utility functions are linear in consumption (concavities mean you might want to redistribute). I agree with your question about the existence of utility functions, but I think that cuts the opposite direction — there are an infinite number of possible utility functions, not just one or zero.
#8 Duke #1 will be king,assuming he knows the rules of the game and he knows the number of duke in the kingdom. (Both rational assumptions.) He supports the King until the 6th, when no more dukes would be coming, because their invitations were cancelled. Then he kills him. It is never stated the other dukes would start coming after the first days of May.
Question 8:
1. Duke 1 knows that Duke 2 would kill him and then, as a monarch in the style of Julius Caesar or Caesar Augustus, rename months so as to eliminate the rest of the month of May, or create so many new months as to eliminate the dates May 3, 4 and 5. Thus Duke 1 would support the King.
2. Duke 4 decides the play chicken. Realizing that the situation is unstable so long as there are an odd number of dukes remaining, he kills the King and then, as monarch, establishes a new dukedom and names a new Duke 6. Duke 5, now realizing there’s a new Duke on the way to menace him, is confronted with a dilemma. He can do the same thing as Duke 4 – kill the king, create another duke – and play chicken with Duke 6. Or he can take the safe way out and support Duke 4.
3. Along the same lines, any monarch with the power to create new dukes might do so in abundance. Sure, ideally he’d ensure an even number. But in any event he’d create a circumstance in which the next duke in line would realize that, given the large number of potential claimants and the variations in tastes, preferences and analytical skills you find in any human population, the odds that SOMEBODY behind him would have lethal ambitious is great. Ironically, the less stable the King’s position is, the more likely that Duke 1 would have an interest in supporting the King.
Question 8 attracts most of the attention, and no wonder! a story of intrigue and murder is a lot more exciting than people living on welfare, games of chance for small stakes, macroeconomics, or statistics.
One could make a movie out of this. Duke 1 arrives on May 1, the King tells him that there is only one more Duke coming, but Duke 1 takes his chances and kills the King anyway. When Duke 2 arrives, the new King tells him that another Duke will arrive on the next day. What will Duke 2 do? in any case, imagine the surprise of whoever is around when Duke 3 arrives on May 3!
It’s a bit like Macbeth, only better.
How often do you derive a game theoretical benefit from the weakness of your own position?
1. Playing chicken, when you throw your steering wheel out of the car, you create a credible claim that you cannot be dissuaded for your current course of action, thereby forcing others to abandon efforts to discourage you. Burning bridges and ships, crossing Rubicons, etc., all create these same signals.
2. Insurance companies are willing to extend the benefits of group insurance policies not only to employees, but to their spouses, on the theory that employees are unlikely to acquire spouses on any basis that would correlate with abnormally high health costs. True, people might try to game this system – an employee might marry a someone with leukemia simply to provide health benefits to a friend. But the burdens of divorce are sufficiently great as to discourage this kind of gamesmanship. It is precisely the burdens of marriage that make it a good marker for insurance companies. Unmarried couples – including same sex couples – lack the burdens of a marital contract in most states, and therefore lack the markers insurance companies look for. The lack of burden creates a burden.
Other examples?
10: Fraternal or identical twins? It makes a BIG difference.
Thanks for posting the original version. I agree that it makes question 6 easier, since in the ‘translation’ it is not clear how one would add Adam and Eve’s utility functions together to create a social utility function.
Mike,
If it’s clear in the original how to add utilities together to get social utility, so much the worse for the original.
Poor Euthyphro. He should have just said, “gee Socrates, the Good is simply adding up utility functions. Duh. After all, there is just one unique utility function that describes a person’s preferences, and just one way to aggregate across utility functions.”
I’m slightly confused by question 6.
You state that Eve can produced exactly one apple per hour and also that she would be willing to work for up to x hours for x^2 apples.
Surely under both of these conditions she will only ever work 1 hour for 1 apple and a tax rate of anything greater than 0% will discourage her from working entirely since her post-tax income will be less than 1 apple per hour?
Al: I agree with you that under these circumstances she will never work more than one hour. It does not follow that she works zero.