Microeconomic Theory III Final Exam March 18, 2010 (80 Minutes)

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1 4. Microeconomic Theory III Final Exam March 8, (8 Minutes). ( points) This question assesses your understanding of expected utility theory. (a) In the following pair of games, check whether the players preferences over lotteries on the strategy profiles are identical (i.e. row player s preferences on the left to the row player s preferences on the right and column player s preferences on the left to the column player s preferences on the right). L M R L M R a,-, -,7 a,- 5, -, b,,4,4 b 5,,, c -,,7 -,-5 c -, 5,,- nswer: For the preferences to be the same, it must be that v i = a i u i +b i for some a i and b i where a i is positive and u i and v i are the utility functions of i in the left and the right game, respectively. Since u (b, M) =and v (b, M) =,we must have b =(Row player is and the column player is.) Since u (c, M) =and v (c, M) =5,we then have a =.But v (a, L) =6 = += u (a, L)+. Therefore, Player s preferences are different in two games. On the other hand, one can easily see that u =v +, showing that the preferences of Player are identical in the two games. (b) Under Postulates P-5 of Savage, let D,D,...,D n be disjoint non-null events such that D D D n,where º and are the at least as likely as and as likely as relations between events, derived from betting preferences as in the class. Given any subsets N and N of {,,...,n}, show that D N N i º D i. nswer: Under P-5, º is a qualitative probability, as you have shown in your homework. In particular, for any disjoint events B,C,D where C is not null, C  () B C º B D Cº D. () S S I will show that if S N = N S,then D i D i. This also implies that if N > N, then D i  S D i.[proof: S Take N to be a subset of N with N = N. Take also B = D i, C = \N D i, which is not null, and D =. By (), D i = B C B D = B D i, where the last indifference is by thefactiam aboutto show.] Toshow thisfact, use mathematical induction on N. If N = N =,both sides are empty sets,and

2 indifference is true. Suppose that the indifference holds for N = N = m, and consider N, N with N = N = m +.Let M = N\{i } and M = N \{j } for some i N, j N. Clearly, M = M = m, and by inductive hypothesis, D i D i. () i M i M Then, Ã! Ã! Ã! [ D i = D i D i D i D j D i D j = D i, i M i M i M where the first indifference is by () and the assumption that D i D j, and the second indifference is by () and ().. ( points) Consider an expected profit maximizing monopolist who faces an uncertain demand. He supplies q units of goods at zero cost and sells it at price θ q, where θ is unknown. [The price and the supply level can be negative.] (a) ssuming that θ N (y, σ ), compute the monopolist s optimal supply q and his expected profit under the optimal supply. nswer: The expected payoff of the monopolist is Hence, the optimal q is Thus, the payoff under optimal q is E [u] =q (E [θ] q) =q (y q). q = y/. U (y) =y /4. (b) Suppose that, through market research, the monopolist can learn about θ. In particular, by investing c, he can learn the value of a random variable Y before choosing his supply q, such that θ = + Y, N (, c) and Y N (,c). How much should the monopolist invest? [Note that the utility function of the monopolist is (θ q) q c.] nswer: Conditional on Y, θ is distributed with N (Y, c). Hence, by part (a), conditional on Y, the expected payoff of the monopolist under optimal supply is Y /4. Therefore, the expected utility from c is Y V (c) =E /4 c = E Y /4 c = c/4 c, where the first equality is by the law of iterated expectations and the last equality is by the fact that Y N (,c). Therefore, the optimal c is c =/8.

3 . (4 points) Consider the reduced normal form of the following game, in which the strategy set of Player is {,, B}, so that the equivalent strategies and B are represented by a single strategy. I B a b a b (a) Compute the set of rationalizable strategies. (Show your result.) nswer: The reduced game in normal form is B,,,,,, Note that strictly dominates B. Hence, B is eliminated. The remaining game is,,,, Nothing is eliminated in the new game ( is est reply to b; is est reply to a; a is est reply to and b is est reply to ). Therefore, S = {, } {a, b}. (b) Compute the set correlated equilibria. (Show your result.) nswer: Recall that a correlated equilibrium would put probability on nonrationalizable strategy profiles. Hence, a correlated equilibrium is p = (p,p,p,p 4 ) with p + p + p + p 4 =where the probabilities are as in the table p p p p 4 Note that it must be that p 4 =because if p 4 >, Player would strictly prefer to play a when she is asked to play b. Wethushave p p p

4 Since a weakly dominates b (as B has zero probability), she does not deviate when she is asked to play a. Likewise, when Player is asked to play, he knows that she plays a, and he does not deviate. When player is asked to play, he assigns probability p / (p + p ) on a and p / (p + p ) on b. Hence, the payoff from is and thepayoff from is p / (p + p ). Thus, he does not deviate iff p p. Therefore, the set of correlated equilibria is {p p (, a) p (, a),p (, b) =p (B, a) =p (B, b) =}. (c) Suppose that in addition to the type with the payoff function above, with probability., Player has a "crazy" type who gets if he plays and otherwise. Compute the set of all sequential equilibria. nswer: The only sequentially rational plan for the crazy type is I. Hence, the information set of Player is reached. Moreover, for the normal type of player, IB is not sequentially rational under any belief by part (a). Hence, at her information set, Player assigns probability on the event that player will play in the subgame. [Proof: she can assign positive probability only on the strategies in {, B, I} for the normal type. Conditional on I, either she assigns zero probability on normal type, concluding that she faces the crazy type and he will play with probability, or she assigns probability on that the normal type plays, in which case both types play.] Therefore, by sequential rationality, in any sequential equilibrium, she must play a with probability. Of course, this implies that the normal type plays I with probability in any sequential equilibrium. Therefore, the unique sequential equilibrium is s (normal) = I s (crazy) = I s = Pr (normal I) =.9. 4

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