Economics 812 Midterm

Prof. Bryan Caplan

Spring, 2004

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.  Suppose a million new immigrants enter the U.S., driving down the wages of low-skilled Americans by \$1/hour.

True, False, and Explain: According to Landsburg (Fair Play), this change usually increases Kaldor-Hicks efficiency, because the positive effect of population growth on innovation tends to outweigh the negative externality of falling wages.

FALSE.  While Landsburg does discuss the R&D externalities of population growth, he explains that falling wages are not an externality at all.  It is at worst a transfer from workers to employers, and if there is any demand elasticity the fall leads to additional surplus.

Several students also mentioned that immigration does not necessarily increase world R&D because one country's population must fall for another's to rise.  These answers received a couple extra points.

2.  Two types of agents consume guns (g) and butter (b).  The Type A agents have U=.5 ln g + .5 ln b, and initially own 1 unit of guns and 1 unit of butter.  The Type B agents have U=.3 ln g + .7 ln b, and initially own .5 units of guns and .5 units of butter.  10% of the agents are Type A; the other 90% are Type B.

True, False, and Explain: The price of guns will not equal the price of butter.

Plugging into the formula  from the notes:

.

3. True, False, and Explain: If everyone has lexicographic preferences for x over y, the conclusion of the First Welfare Theorem holds even though its assumptions do not.

TRUE.  With lexicographic preferences, no equilibrium price vector exists, since if py>0 everyone will want to sell all they have, and if py=0, everyone will want an infinite amount.  However, any distribution of resources will be Pareto efficient, because reallocating y or x between people always leaves one person better off and another person worse off.

4.  Consider the following 2-player game:

 Up Down Left 0,0 0,0 Right 0,0 0,0

True, False, and Explain:  This game has 4 PSNE but no MSNE.

FALSE.  There are 4 PSNE, but there is also an infinite number of MSNE.  Any probability mix leaves your opponent indifferent.  (Several students said that there was no reason to randomize, but that misses the whole point of MSNE.  If neither play can do strictly do better by changing, you have an equilibrium).

Questions 5 and 6 refer to the following information.

Suppose that two players repeatedly play the following game.

 Hawk Dove Hawk -10,-10 5,0 Dove 0,5 3,3

5.  True, False, and Explain: If the game is repeated infinitely, trigger strategies can sustain the socially optimal outcome as long as β>.6

FALSE.  The socially optimal outcome (social payoffs are maximized) occurs at Dove, Dove.  The trigger strategy would be: If you ever play Hawk, I will play Hawk forever afterwards to punish you, and you will play Dove.  It is therefore in a player's interest to play Dove if:

, which holds as long as b.4.

Many students assumed that the cheater would lose 10 every turn, but that would require the cheater to play Hawk, which he would definitely not do if he expected his opponent to play Hawk.  (Remember – hold all other player's behavior fixed and see if you can do better solely by changing your own behavior!)  Student who correctly reasoned from this assumption got 6 points.

A few other students assumed that the MSNE would be the trigger strategy, but punishing by playing Hawk forever is obviously a tougher punishment.  If you worked through this approach correctly, you also got 6 points.

6.  True, False, and Explain: If the game is repeated twice, β=1, and players get to flip a coin after turn 1, there is no NE where players play (Dove, Dove) in turn 1.

FALSE.  I propose the following NE.  We both play Dove in turn 1.  If we keep this agreement, then in turn 2 we play (Dove, Hawk) if the coin said Heads, and (Hawk, Dove) if the coin said Tails.  But if player 1 plays Hawk in turn 1, then in turn 2 we play (Dove, Hawk) regardless of the coin toss, and if player 2 plays Hawk in turn 1, then in turn 2 we play (Hawk, Dove) regardless of the coin toss.  The payoff from playing this strategy is 3+.5*5=5.5, which exceeds the payoff from cheating, which would be 5+0=5.

Incidentally, you could sustain cooperation even without a coin flip.  Just do the MSNE in turn two if both players played Dove in turn 1.  But if player 1 plays Hawk in turn 1, then in turn 2 we play (Dove, Hawk), and if player 2 plays Hawk in turn 1, then in turn 2 we play (Hawk, Dove).  Since both players play Hawk with p=1/6 in the MSNE, the expected MSNE payoff is once again:

1/36*-10+5/36*5+5/36*0+25/36*3=2.5.

7.  Suppose there is a Cournot industry with the demand function Q=a-bP, 8 firms, 0 MC, and no fixed cost.

True, False, and Explain:  A firm would want to split into two firms because it would earn 62% more profit as a result.

TRUE.  From the hw, we know that profits as a function of N equals .  So profits for one firm in a industry with 8 firms are , while profits for two firms in a industry with 9 firms are .  The latter is 62% greater than the former.

8.  True, False, and Explain:  Kreps argues that in the "noisy" real world, trigger strategies are not a practical way to sustain collusion.  Instead, colluding firms would only punish if prices were significantly below the agreed level.

FALSE.  Kreps specifically states that "you cannot go too far in lessening the severity of punishment, making it of shorter duration, say, or only triggering punishment when prices fall very far.  This would encourage each side to chisel somewhat..." (p.526)  Intuition: If no small deviations get punished, then everyone will always engage in a small deviation.

9.  Suppose firms in an industry have fixed costs and increasing marginal costs.  The industry demand curve lies strictly above firms' AC curves.

True, False, and Explain:  Bertrand competition will allow only one firm to survive in this market, implying no productive inefficiency but some allocative inefficiency.

FALSE.  With increasing MC, the AC curve eventually turns up – it will have a U-shape.  So if demand is high, multiple firms can definitely survive.  Ignoring divisibility problems, it is possible for multiple firms to survive in such an industry, setting P=AC=MC.

(20 points each)

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

1.  "The assumptions of the Arrow-Debreu model are sufficient but not necessary conditions for the efficiency of laissez-faire."

Discuss this statement, using two specific examples from game theory to illustrate your point.

Arrow-Debreu assumes that firms have no control over prices.  But in the Bertrand oligopoly game, firms set prices, but can still reach the first-best outcome where P=MC.  Similarly, in the Ultimatum game, there are no enforceable trades at all, and yet the game theoretic prediction (\$.01 for you, rest for me) is perfectly efficient.  In both cases, then, laissez-faire is fully efficient even though the A-D assumptions do not hold.

Other good examples some students used: repeated play (including reputation), coordination games.

2.  Use Tullock's insights on rent-seeking to analyze students' allocation of effort between studying and cheating.  What changes if students can bribe professors for better grades?

Tullock would predict that students would keep cheating until the rate of return from studying equals the rate of return from cheating.  In equilibrium, both methods of earning grades would have to be equally effective.  If students can bribe professors, this leads students to cut back on both studying and cheating in favor of cash bribes.  On the surface, this seems clearly more efficient.  But there is an externality on the value of the diploma.  More interestingly, if professors can extract bribes from their students, this will lead to additional competition for academic jobs (like getting a second Ph.D.), presumably with little social benefit.

3.  Suppose you have a 2-player version of the voluntary donation game from the notes, with one difference: Each agent cares somewhat about the other, so player one maximizes U1=c1D+.5*c2D, and player two maximizes U2=c2D+.5*c1D. Carefully set up each player's maximization problem.  Then solve for the symmetric Nash equilibrium.

Note that D=d1+d2=2-c1-c2.  So player 1 maximizes:

U1=c1(2-c1-c2)+.5*c2(2-c1-c2)=2c1-c12-1.5c1c2+c2-.5c22

And player 2 maximizes:  U2= 2c2-c22-1.5c1c2+c1-.5c12

Differentiating U1 wrt c1: 2-2c1-1.5c2=0

Differentiating U2 wrt c2: 2-2c2-1.5c1=0

Solving 2 equations in two unknowns implies c1=c2=4/7.  Each player keeps 4/7 for himself and donates 3/7 to the common fund, whereas in the simple model without altruism, each agent keeps 2/3 for himself and donates only 1/3.  Altruism partially solves the public goods problem.