Economics 812 Midterm
Prof. Bryan Caplan
Spring, 2005
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 23 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, Paretoinferior 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 Paretoinferior 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 2player 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 HawkDove
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 (HawkDove)
has two equilibria. So you can punish
deviation in turn 1 in a selfenforcing way even in the last turn.
7. True, False,
and Explain: If the players play the
HawkDove 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
"cheaptalk" can affect games' outcomes.
FALSE. While Kreps says that "[I]t is typical
in the analysis of noncooperative 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 Paretoinferior 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 46 sentences AND/OR equations, answer all three of the following questions.
1. Suppose:
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 rentseeking. 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:
_{}
_{}
_{}