Prof. Bryan Caplan

Spring, 2006

Part 1: True, False, and Explain

(10 points each - 2 for the right answer, and 8 for the explanation)

State whether each of the following nine propositions is true or false.  Using 2-3 sentences AND/OR equations, explain your answer.

1.  Situation A is a Kaldor-Hicks improvement over Situation B.  Situation A is Pareto efficient.

True, False, and Explain: Situation A must also be Kaldor-Hicks efficient.

FALSE.  Just because A is a Kaldor-Hicks improvement over B does not imply that there isn't another situation that would be a Kaldor-Hicks improvement over A.  The additional information that A is Pareto efficient is irrelevant, because virtually all situations are Pareto efficient.

2.  Suppose 50% of all agents are certain (p=1) that occupation of Iraq will reduce terrorism, and 50% of all agents are certain (p=1) that occupation of Iraq will increase terrorism.  A betting market exists where agents can bet on their beliefs.

True, False, and Explain: A general equilibrium can only exist if the prices of the two bets are equal.

FALSE.  Assuming agents cannot sell short, the price ratio will depend upon the amount that people are willing to bet, which in turn depends upon wealth.  If the people who believe that the invasion will reduce terrorism are willing on average to bet twice as much as the people who believe the opposite, for example, the betting market will clear when the "reduce" bet pays half as much as the "increase" bet.

In agents can sell short (or, equivalently, just issue notes offering to pay \$1 if X happens), there is no equilibrium.  Due to their certainty, both sides would happily offer offer an infinite amount of notes that pay off if the other side turns out to be right, so the price ratio would be undefined.

3. True, False, and Explain: According to Kreps, economic experiments confirm that general equilibrium analysis has little predictive value.

FALSE.  On p.198, Kreps says exactly the opposite: "The results obtained are usually striking in their support of Walrasian equilibrium."

Questions 4 and 5 refer to the following game:

 Left Right Left 2,1 0,0 Right 0,0 1,2

4.  This game has a MSNE with expected payoffs equal to (1.5, 1.5).

FALSE.  To make Player 2 indifferent, Player 1 plays L with probability σ:

1σ+0=0+2(1-σ), so σ=2/3

To make Player 1 indifferent, Player 2 plays L with probability φ:

2φ + 0=0 + (1-φ), so φ=1/3

To find expected pay-offs, simply plug the probabilities back into the above equalities: the expected payoffs in equilibrium are (2/3, 2/3).  Alternately, you could note that (L,L) happens 2/9 of the time, (R,R) happens 2/9 of the time, and something else happens 5/9 of the time, so expected payoffs for Player 1 are 2/9*2+2/9*1+5/9*0=2/3, and expected payoffs for Player 2 are 2/9*1+2/9*2+5/9*0=2/3.

5.  True, False, and Explain:  Played simultaneously, this game has two SGPPSNE.  Played sequentially (with Player 1 moving first), this game has two PSNE, but only one is SGP.

TRUE.  The simultaneous game only has one subgame, so both (L,L) and (R,R) are SGPNE.  The sequential game still has two PSNE: Player 1 will play L if he believes Player 2 will play L, and R if he believes Player 2 will play R.  However, once Player 1 plays L, Player 2 would definitely want to play L too, so by backwards induction, only (L,L) is SGP.

6.  Suppose there are two firms able to produce a good.  Firm #1 has TC=\$1000 + \$0Q;  Firm #2 has TC=\$0 + \$10Q.

True, False, and Explain: If demand goes up high enough, Firm #1 will drop out of the market and Firm #2 will set its price just below Firm #1's AC.

FALSE.  The opposite holds: Firm #2 has lower AC at low quantities, but Firm #1 has lower AC at high quantities.  If quantity>100, Firm #1 can charge just below \$10 and take the whole market at a profit.

7.  True, False, and Explain: If firms set prices (as opposed to quantities), firms would never want to split into additional firms, even if the game were infinitely repeated and β=1.

FALSE.  If β=1, Bertrand collusion is definitely sustainable, and if each firm gets an equal share of the monopoly profit, then splitting could increase your profits from a 1/N share of the monopoly profit to a 2/(N+1) share, which is greater for N>1.

Firms might also want to split if there are diseconomies of scale, though I gave less credit for this answer since it didn't use the information about repeated play and the discount rate.

8.  N players are deciding whether to contribute to a public good.  The public good is discrete: it is produced at the optimal level so long as 1 person contributes.  Contributing costs the individual who contributes C, and 0 otherwise.  If the public good is produced, everyone gets a benefit of B; otherwise they get a benefit of 0.  B>C.

True, False, and Explain:  There are two PSNE, only one of which is Kaldor-Hicks efficient.

FALSE.  There are N PSNE, and all of them are Kaldor-Hicks efficient.  If no one is contributing, then any one player would want to deviate to contribute; if more than one player is contributing, then any one player would want to deviate to not contribute.  The only stable equilibria are those where exactly one player contributes.  Since B>C, but 1 contribution is all that is necessary, each of these equilibria is Kaldor-Hicks efficient.

9.  In Leviathan, Hobbes argues that, in the absence of government, individuals always prefer war to peace, leading to a "war of all against all" equilibrium.

True, False, and Explain:  This precisely what the Hawk/Dove game predicts.

FALSE.  For PSNE, the Hawk-Dove game predicts that you will never see universal war; instead, you'll see one aggressor and one appeaser.  In the MSNE of the Hawk-Dove game, (War, War) can occur, but it is an unlucky coincidence, not a typical result.

(20 points each)

In 4-6 sentences AND/OR equations, answer all three of the following questions.

1.  Give an example of lexicographic preferences.  How can lexicographic preferences preclude the existence of a general equilibrium?  Is it possible for a general equilibrium to exist given the existence of lexicographic preferences?  Explain.

I lexicographically preference my sons' lives to money – I wouldn't part with them for any amount of money.  Lexicographic preferences can preclude the existence of general equilibrium because if everyone shares the same preferences, they won't sell the lexicographically preferred good to buy more of other goods at any p>0 – but if the price of other goods falls to 0, they want more of these other goods than exist.  There are many ways that general equilibrium could co-exist with lexicographic preferences: (1)  Some people have these preferences, others don't, and the former buy all of the lexicographically preferred good from the latter; (2) People have different lexicographic preferences; (3) People only have lexicographic preferences up to a certain quantity (e.g. everyone might have a lexicographic preference for one kidney, but not two).

2.  Coordination equilibria are often persistent, but they also change.  Give a real-world example.  Then use game theory to explain how this change happened, paying particularly close attention to the incentives of the "first-movers."

My example: It used to be almost impossible to publish survey research in economics, and as long as no one would publish it, no economist wanted to do it.  This is no longer true.  The reason is probably that there were some economists who were burning to do survey research, so they worked on it even when the chance of publication was low.  But once they published some work, this raised the payoff.  This led slightly less enthusiastic economists to try survey research, which raised the chance of publication, which led to more survey research...  Heterogeneous preferences were the lever that got the survey research ball rolling.

Another class of good answer focused on the importance of market leaders.  A market leader thinks that other people will change if he changes too, so he feels safer doing something different.  Continuing with my example, many of the economists who started doing survey research – like Alan Blinder – were already famous for other accomplishments.  So they could expect people to take them seriously even if they "broke the rules," which helped change the rules.

3.  Landsburg (The Armchair Economist) argues that laws restricting cosmetic surgery are an inefficient restriction on competition.  Using the game theory you have learned so far, present and defend what you see as the strongest possible counter-argument to Landsburg's claim.  (You don't have to agree with your argument; just present it as forcefully as possible).

You can't just say that cosmetic surgery is a "Prisoners' Dilemma" for women, because efficiency calculations require us to count the value to both women AND men.  (Analogously, you can't just say that price supports are efficient because they benefit sellers; you have to show that the gain to sellers outweighs the gain to buyers).  Similarly, you can't just say that more cosmetic surgery reduces the supply of non-cosmetic surgery: That's true for all goods.  Better answers:

Maybe men value relative beauty, not absolute beauty; they want to marry a woman who is better-looking than other women, but don't particularly care how good-looking she is in absolute terms.  If so, cosmetic surgery burns up resources without making the average woman OR the average man better off.

Maybe people directly disvalue the average level of cosmetic surgery in their society, on the grounds that it "commodifies" or "cheapens" life.  But since the average level of cosmetic surgery is non-excludable, no one factors it into his/her decisions, leading to too much cosmetic surgery.  This is a common objection to e.g. human cloning.