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.  A special medical test always detects the presence of a disease if a person has it; however, 5% of perfectly healthy people will test positive as well (there is a 5% "false positive" rate). Suppose that .1% of people actually have the disease, and that members of the population are tested at random.

True, False, and Explain: The approximate conditional probability of having the disease given the fact that you test positive is 95%. (Hint: Remember Bayes Rule!)

 

FALSE.  Applying Bayes' Rule, the P(have disease| you tested positive)=

 

.

 

2.  Two types of agents consume guns (g) and butter (b).  The Type A agents have U=.5 g + .5 b.  The Type B agents have U=.3 g + .7 b.  All agents initially own 1 unit of guns and 1 unit of butter.  50% of the agents are type A; 50% are type B.

 

True, False, and Explain: General equilibrium does not exist because the agents' demand functions are discontinuous in price.

 

FALSE.  Since these are linear utility functions, they DO generate demand functions that are discontinuous in price.  (E.g., the Type A agents want only guns if p_g<p_b, and only butter if p_b<p_g).  However, this continuity is not a necessary condition for GE to exist (rather it is one item on a list of sufficient conditions).  In fact, GE exists in this economy if p_g=p_b.  Then the type Bs sell all of their guns, and the type As are not unwilling to sell all of their butter.  In equilibrium, then, the Type As have all the guns and the Type Bs have all of the butter.

 

Many students used the price formula ratio from the notes, but that only works for log utility!

 

3. True, False, and Explain: In a Coordination game, Pareto-inferior equilibria are not subgame perfect.

 

FALSE.  In the simultaneous Coordination game that we analyzed in class, there is only one subgame, so both of the equilibria are subgame perfect. 

 

It is however true that in a sequential version of the Coordination game, the Pareto-inferior equilibria are not subgame perfect.  I gave partial credit for students who answered TRUE on this ground.

 

4.  True, False, and Explain:  Landsburg's "Indifference Principle" (The Armchair Economist) is inconsistent with the concept of the MSNE.

 

FALSE.  MSNE is actually a special case of Landsburg's Indifference Principle.  The Indifference Principle states that in equilibrium, people will be indifferent between all of their choices.  If one were better than the other, everyone would do it.  And this is precisely what happens in a MSNE: ever agent is indifferent between every choice.

 

5.  Consider the following 2-player game:

 

	Up	Down
Left	10,0	10,0
Right	0,10	0,10

 

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

 

FALSE.  Since Left is strictly dominant for Player 1, and Up and Down have equal payoffs for Player 2, there are two PSNE: (Left, Up), and (Left, Down).  But there are also infinitely many MSNE, where Player 1 plays L with p=1, and Player 2 plays Up with any probability.

 

Questions 6 and 7 refer to the following information.

 

Suppose that two players play an Ultimatum game where Player 1 divides a payoff of 10 between himself and Player 2.  Then the players play the following Hawk-Dove game ONCE.  Players do not discount the future (β=1).

 

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

 

 

6.  True, False, and Explain: An even split (5/5) in the Ultimatum game is a focal point but cannot be a SGPNE.

 

FALSE.  It would be a SGPNE given the following strategies: If Player 1 offers an even split or better in turn 1, then in turn 2 they play (Hawk, Dove); otherwise, they play (Dove, Hawk).  If Player 1 follows this strategy, he gets 5+5=10.  If he deviates and keeps all 10, then he still is not strictly better off, because he gets 0 in turn 2.  Player 2 would not want to deviate either; refusing his offer in the Ultimatum game obviously makes him worse off, and deviating from his proposed strategy in turn 2 – holding constant Player 1's action – also makes him worse off.  If he plays Dove when he's expected to play Hawk, he loses 2; if he plays Hawk when he's expected to play Dove, he loses 10!

 

How is this possible?  It is possible because the last game (Hawk-Dove) has two equilibria.  So you can punish deviation in turn 1 in a self-enforcing way even in the last turn.

 

7.  True, False, and Explain: If the players play the Hawk-Dove game first, and the Ultimatum game second, exactly two SGPPSNE exist.

 

TRUE.  In the last turn, it is only subgame perfect for Player 1 to offer 0 and keep 10 for himself, and Player 2 to accept.  But there are two PSNE in the first turn: (Dove, Hawk) and (Hawk, Dove).  So for the whole game, there are two SGPPSNE: (Dove, Hawk, 0, Accept) and (Hawk, Dove, 0, Accept). 

 

8.  True, False, and Explain:  According to Kreps, game theory rules out the possibility that "cheap-talk" can affect games' outcomes.

 

FALSE.  While Kreps says that "[I]t is typical in the analysis of non-cooperative games to omit such communication from formal models," he immediately adds that "This doesn't mean that these possibilities won't affect the outcome of games; they do so in some important ways in various situations."  Most obviously: in Coordination games with Pareto-inferior equilibria.

 

9.  Suppose the demand curve for a contestable monopoly crosses the AC curve at more than one point.

 

True, False, and Explain:  There are multiple equilibria.

 

FALSE.  Under contestable monopoly, P=AC, but only the higher quantity/lower price intersection is an equilibrium.  At the higher intersection, even a monopolist that DID NOT face potential competition would want to cut price somewhat to earn additional profits.  If you do face potential competition, you would have to cut your price all the way down to the lowest intersection of demand and AC to prevent entry.

 

Part 2: Short Answer

(20 points each)

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

 

1.  Suppose:

 

• 40% of all agents (the Type As) in an economy have U=ln x + ln y, and the other 60% (the Type Bs) have U=3 ln x + ln y.
• All agents start with one unit of x and one unit of y. 

How will redistribution of x from As to Bs affect the general equilibrium ?  Write down the formula for , using  to indicate the quantity of x you let the Type As keep. (Hint: Remember hw#2, problem 4!)

 

In hw#2, problem 4,

 

The current problem is different in a few ways:

 

1.  The utility function of the Type Bs normalizes to U=.75 ln x + .25 ln y, not U=2/3 ln x + 1/3 ln y.

 

2.  There are 40% As and 60% Bs, not 50/50.

 

3.  As a corollary of 2, taking 1 unit of x from each A does NOT give 1 unit of x to each B!  Instead, if the As have  each, then the Bs have = each.  Five points off if you did not figure this out.

 

So the correct formula for

 

 

2.  What kind of a bargainer are you?  Why?  Explain your typical strategies in game theoretic terms.  Are there any focal points that you frequently rely on?

 

I am usually a soft bargainer, probably because I have greater disutility of conflict than most people.  I value my tranquility.  But it depends greatly on the bargaining partner.  I am an extremely soft bargainer with my kids, because I put a lot of weight on their welfare.  I am a hard bargainer with my parents, because I know they put a lot of weight on my welfare.  I generally am a hard bargainer with students, in part to maintain a reputation that discourages rent-seeking.  One focal point that I often rely on is accepting the other party's FIRST offer without negotiation as long as it's "reasonable."  This cuts down on transactions costs, and if people expect me to follow it, they make their first offer a reasonable one.

 

3.  Suppose an Incumbent infinitely repeats the following entry deterrence game.  Suppose that in order to have a reputation for Fighting, an Incumbent must be willing to accept the (In, Fight) result for one turn.  (After that turn, he earns the (Out, Fight) payoff forever).  Write down the inequality the Incumbent must satisfy to have this reputation.  Then solve for the critical value of b.   

 

The Incumbent can have this reputation as long as: