Final Exam (Solution) Economics 501b Microeconomic Theory

Transcription

1 Dirk Bergemann and Johannes Hoerner Department of Economics Yale Uniersity Final Exam (Solution) Economics 5b Microeconomic Theory May This is a closed-book exam. The exam lasts for 8 minutes. Please write clearly and legibly. Be especially careful in the de nition of the game, the payo function and the equilibrium notions. The allocated points are also a good indicator for your time budget. Please record the answers (; ), (; 4), and (5; 6) in separate bluebooks.

2 . () Recall the de nition of a signalling game. Nature draws player s type, a nite set, according to some distribution p on. Player knows, player does not. Player takes an action 4A, where A is a nite set of actions. Player obseres the realization a of and chooses then 4A, where A is a nite set of actions. This is the end of the game. Payo s are gien by u i (a; ), i = ;. (a) For this signalling game, de ne the notion of a weak perfect Bayesian equilibrium. [Solution] A weak PBE in the signaling game is an assessment f ; g such that () 8; (:j) arg max A u (a ; ; ) ; () 8a ; (:ja ) arg max (; Weak Consistency) (ja ) = and (triially) A X (ja ) u (a ; ; ) ; p () (a j) P p a j wheneer X p a j > ; (4) (ja ) is a probability distribution on for X p a j = ; (b) For this signalling game, de ne the notion of a sequential equilibrium. [Solution] A sequential equilibrium is an assessment f ; g that satis es () to (4) aboe (hence must be a PBE). In addition, it puts further restriction on (4) by requiring consistency. That is, 9 ( m ; m ) such that f ; g = lim m! (m ; m ) for some totally mixed strategy m (set of totally mixed strategies as de ned in notes) such that m for all m is deried from () aboe [notice since m, P p a j > for 8a A ]. (c) Proe, or proide a counterexample for, the following claim: in a signalling game, weak perfect Bayesian equilibria and sequential equilibria coincide. [Solution] Fudenberg and Tirole (99) shows that PBE and SE coincide in a multi-stage game of incomplete information with independent types if each player has at most two possible types or if there are just two periods [see 46 of their textbook]. We show here that the additional condition in (b) is always satis ed. Since SEs are PBEs. we just need to establish that PBEs are SEs for the signaling game. Take any PBE f ; g. The idea is to construct a totally mixed strategy pro le m that generates m!. Consider the following m : 8 >< m (a j) = >: m m m m m m mn() m m m (a j), if (a j) > and P p a j >, if (a j) = and P p a j >, if (ja ) > and P p a j = (ja )p(a ) p() where N () := # a A j (a j) = and P p a j > ; Otherwise (if (ja ) = and P p a j = ), m (a j) = m 6 4 X 8a A s.t.(ja )> & P p( ) (a j )= m m (ja ) p (a ) p () >=, >;

3 By construction m (a j) > 8m; 8a A, and P 8a A m (a j) =. So m. Moreoer, m (a j)! (a ; ) as m!. Now for m, simply set it to play the strategies speci ed in with probability m, and to play eery strategy in A with probability m #A. Then clearly m! as m!. For belief m, only m is releant. It is easy to check that by construction for 8a A and, m p () m (a j) (ja ) = P p m a j! (ja ).. () Consider a nite normal form game with complete information. (a) De ne the notion of a strictly dominated strategy, rst a strictly dominated pure strategy, second a strictly dominated mixed strategy. [Solution] A pure strategy a i A i is strictly dominated if there exists some i A i such that 8 i j6=i A j, we must hae u i ( i; i ) > u i (a i ; i ). Equialently, the aboe can be written as the following: 8a i j6=i A j, we must hae u i ( i; s i ) > u i (a i ; s i ). The de nition for a strictly dominated mixed strategy simply replaces a i with i A i in the de nition proided aboe. (b) Then proe or disproe: a mixed strategy i is strictly dominated if and only if it assigns positie probability to a pure action a i that is itself strictly dominated. [Solution]. The forward direction (only if part) is false. Consider the following game, in which we only specify the row player s payo s: U M 4 D 4 The mixed strategy M + D is strictly dominated by the pure strategy U. Yet, neither M nor D is is strictly dominated by any strategy.. The backward direction (if part) is true. Suppose a mixed strategy i assigns positie probability p to a pure action a i that is strictly dominated by some i. Then construct a new mixed strategy ~ i by replacing action a i from i with i played with probability p. Since p > and i strictly dominates a i, then by construction ~ i strictly dominates i.. (4) Consider the following game between two sellers and a continuum of identical buyers. The sellers produce a homogenous good that the buyers alue at : The sellers contact the buyers by making simultaneously a price o er of for each rm i f; g: The buyers obsere the o ers and buys from the seller with the lower price if that price does not exceed : If the two prices are equal, then the market is equally split between the buyers. (a) Find the pure strategy Nash equilibria of this pricing game. [Solution] Let (p ; p ) be a pure strategy NE. Suppose p < p. If p <, seller can increase p to p + " to improe his payo while still winning the object. Hence, we consider p. In this case, seller makes a payo of. Howeer, by decreasing his price to ", seller wins the object and makes a payo of " >. Hence, with p < p, (p ; p ) cannot be a pure strategy NE Similarly, p > p will not gie an NE by symmetry. Hence, we just need to consider p = p. If p = p (; ), then both sellers receie a payo of : anyone of them can improe by setting p = to win the object and gain a payo of >..

4 If p = p (; ], then any seller can improe by decreasing his price just a little bit to gain the entire market, which improes his payo. Hence, we are left with p = p =, which is indeed the unique pure strategy NE. Gien p j =, seller i cannot improe his payo since setting = gies a payo of whereas setting > results in his losing the market and gaining the same payo of. In conclusion, the (unique) pure strategy NE is p = p =. (b) Suppose next that it costs c < to make the price o er. The buyers can purchase only from a rm that has made an o er. Again, the buyers buy from the rm with the lowest o er and if the prices are equal, then the market is split between the rms. Are there pure strategy Nash equilibria in the game? [Solution] No pure strategy NE. Suppose there exists a pure strategy NE. If the pure strategy NE inoles one seller (say seller i) opting out, then the other seller (seller j) should opt in and set the price p j = to gain the highest payo of c >. But gien that, seller i s opting out is not a BR since by opting in and setting price = ", seller i can gain the entire market and make a pro t of " c >. Hence, any pure strategy NE must inole both sellers opting in. Now suppose the pure strategy NE inoles both sellers opting in with prices p and p. If p < p, then seller loses for sure with payo c, and he is strictly better o opting out. Hence p 6= p cannot be a pure strategy equilibrium. Hence, we just need to consider p = p = p. If p > or if p < c, then both sellers strictly make losses; each can deiate to opt out to improe his payo strictly. Hence, we are left with p [c; ]. If c >, then we are done showing that there is no pure strategy equilibrium. So suppose c. Then p = p = p still cannot constitute an equilibrium because either seller can simply decrease price a little to p " to gain the whole market with a payo of p " c > p c. The argument aboe exhausts all possible pure strategies equilibrium candidates. We thus conclude that there is no pure strategy NE of this game. (c) Suppose now that there is no cost of making the price o er, but rm i incurs a priately obsered transportation cost of d i that is uniformly distributed on [; ]: Hence rm i makes a pro t of d i on the units that it sells. Consider again the game where the rms announce prices simultaneously. i. What is the releant solution concept for such a game? De ne the strategy and the equilibrium for this game. [Solution] The releant solution concept is Bayesian Nash Equilibrium (BNE). Seller i s (pure) strategy is a mapping : [; ] to [; ), for i = ;. The strategy pro le (p ; p ) is a BNE if 8i f; g, j f; g ni, E dj ui p i (d i ) ; p j (d j ) ; d i Edj ui p i (d i ) ; p j (d j ) ; d i ; 8di [; ] ; 8p i : [; ]! [; ). ii. Find the equilibrium of this game. [Solution] We look for a symmetric equilibrium with a linear form. Suppose seller j plays p j (d j) = d j + with 6= :Then, gien d i, seller i chooses p i (d i) to sole max E dj ui ; p j (d j ) ; d i = max E dj ( d i ) I < p j (d j ) + d i I = p j (d j ) = max E dj ( d i ) I f < d j + g + p i d i I f = d j + g = max Z = max ( d i ) ( d i ) dd j + (since = d j + occurs with probability ) 4.

5 FOC yields, p i (d i ) = p i (d i) ( + ) d i + + ( + ). ( (d i ) d i ) = Matching this to p i (d i) = d i +, we get = ( + ) 6=, and + ( + ) = equialent to = + and = +. Hence, a BNE of the game is for each seller i to bid p i (d i) = + d i + +. Comment: Notice that as!, p i (d i)! for i = ; as in Question (a). 4. (4) Consider the following Bayesian game. Player is interested in selling his watch to player. The cost of selling the watch is zero and his utility is f he sells the watch at price p. Player has alue for the watch and would obtain utility f he buys the watch for price p. Only player knows the alue of. It is common-knowledge that player beliees that is uniformly distributed on the interal [; ]. They will bargain oer the price through a mediator. Each player i will submit a number b i in a sealed enelope to the mediator. The mediator unseals the enelopes and implements a trade according to the following rules. If b > b then there will be no sale. If b b then the watch will be sold to bidder at price p = b. (a) What are the sets of pure strategies in this Bayesian game for players and? [Solution] Player s set of pure strategy is the interal [; ). Player s set of pure strategy is all the functions that map the interal [; ] to [; ). (b) For what set of alues b does there exists a pure-strategy Bayesian Nash equilibrium in which player submits b? Explain carefully how you reach your conclusion. [Solution] In a pure strategy BNE, if player chooses b, then gien, player s pure strategy best response b () must sole max b ( b ) I fb b g. Thereofre b must necessarily satisfy the following: 8 < b () = : To make b an optimal choice, it must sole max b b if > b ; < b i if < b ; [; ) if = b. E [b I fb b ()g] = max b 9 = ; (). b I fb b ()g d (). (b ; b (:)) is a pure stratege BNE i it satis es both () and (). (i) It is clear that any b [; ) with b () = is a pure strategy BNE, since they satisfy () and the expression in () is regardless of what b is. (ii) Consider b (; ). Examine b () = b if b ; if < b.. 5

6 Obiously () is satis es. We need to show b soles (). Now If player deiates to b > b, then b I fb b ()g d = b b d = ( b ) b >. b I fb b ()g d = since b () b 8 [; ], so b > b (), 8 [; ]. On the other hand, if player deiates to < b < b, Finally, if player deiates to b =, b I fb b ()g d = b ( b ) < b ( b ). b I fb b ()g d =. Hence, we e demonstrated that b = b indeed maximizes R b I fb b ()g d. (iii) Lastly, it is clear that b = and b () = ; 8 [; ] constitute a pure strategy BNE. Obiously () is satis ed. Gien b () = ; 8 [; ], R b I fb b ()g = for all b [; ). So () is triially satis ed. Hence, any b [; ) can be part of a pure strategy BNE. (c) Gie an example of a strategy for player which is weakly dominated. (It makes no di erence here or in the next part whether you use the ex ante or interim ersions of weak dominance.) [Solution] Consider b () = > 8. This strategy gies player an ante payo of ( b ) I fb b () = g d: We show that b () = weakly dominates it: for all b [; ), = = ( b ) I fb b () = g d ( b ) [I fb g I fb g] d ( b ) I fb b () = g d ( b ) I f < b g d, since b for all b (; ]. We just need to establish strict inequality ( b ) I fb b () = g d > for some b. Consider b = :5: Then ( :5) I f:5 b () = g d = ( b ) I fb b () = g d, whereas ( :5) I f:5 g d = ( :5) d = :5 = <. 6

7 (d) What is the set of Bayesian Nash equilibria in which player does not use a weakly dominated strategy? Explain carefully how you reach your conclusion. [Solution] We consider the interim ersion of weak dominance. Gien [; ], player, choosing b (), receies a payo of ( b ) I fb b ()g. We show that b () = is a weakly dominant strategy. Examine the table below, which shows player s payo for di erent b alues gien b () < ; b () =, and b () > : payo to b () < b () = b () > b < b > or b > b > : b = b > b < or By obsering the table aboe, we can see that clearly b () = gies player weakly better payo than any b () 6= for all b [; ) (and strictly better payo for some alues of b ). Hence, we conclude that b () = is the unique weakly dominant strategy. So the set of BNE in which player does not use a weakly dominated strategy is simply the singleton fb () =, 8 [; ]g. 5. (4) Consumers buy insurance from a competitie insurance industry. Each consumer is an expected utility maximizer with utility function u (x) = p x for nal consumption x. Each consumer has initial wealth $. There are two types of consumers, high risk and low risk. High risk consumers hae a probability of losing $, while low risk consumers hae a probability of losing $. The insurance industry is perfectly competitie and risk-neutral, that is each insurance company is maximizing expected pro t but because of perfect competition each rm earns zero expected pro t from the insurance policy that it sells. (a) First, suppose that insurance companies can distinguish high and low risk consumers, i. What premiums do they charge to the two types of consumers in a free entry competitie equilibrium? Gie an argument supporting your claim. [Solution] A high risk consumer without insurance is facing a consumption plan: ; ; ; ; which reads "he gets $ with prob. and $ with prob.." A high risk consumer with an insurance contract (; ) is facing a consumption plan: ; ; + ; ; where is the premium he pays to the insurance company in return for which he will be paid + in the case of accident. Likewise, a low risk consumer without insurance is facing a consumption plan: ; ; ; : A low risk consumer with an insurance contract (; ) is facing a consumption plan: ; ; + ; : b > if b b (); otherwise :Notice that both cases are feasible since b = will always gie b >. b < if b b (); otherwise. Notice that both cases are feasible since for any b >, we can nd some b () > b. 7

8 Since insurance companies can di erentiate the two types of consumers, it can proide two di erent insurances to the two types of consumers (high/low insurance to the high/low risk type). For the high risk consumers only. Its expected pro t is H H. Since consumers are free to accept the serice, his expected utility should be no less than the leel if he opts out of the insurance: H + p q + H p p + t :6: Competitieness requires that the pro t should be equal to zero, so H H = ) H = H. Second, it also requires the rm to o er a contract that maximizes the consumer s expected utility. H ; H p p max + + ; s:t:zero pro t condition or H = H : ; The problem is then equialent to the FOC of which requires H max p p + +, H + + H = 6, H =, which implies full insurance: the consumer is guaranteed the same amount of consumption under di erent states. Hence, H = ; H = : q 4 Gien this, the expected utility for a high risk consumer is: t :65 > :6, the leel such a consumer receies if not buying insurance. Hence, the high risk consumer is better o through the purchase of such a contract. In a similar fashion, if the insurance companies face low risk consumers, they will set L = L, L = L and sole FOC yields Soling, we obtain L max p p + +. L + + L =. L = ; L =. This also assures full insurance in the consumption q 5 t 4:8, which is strictly higher than plan. The corresponding expected utility is gien by p + p t 4:4, the leel that the low-risk consumer receies without the insurance. ii. How much insurance do the two types of consumers buy in a free entry competitie equilibrium? Gie an argument supporting your claim. [Solution] As argued aboe, since each type of consumers is strictly better o with his corresponding insurance, all customers will demand insurances, with supply matched due to perfect entry competition. (b) Now suppose that insurance companies cannot distinguish high and low risk consumers. 8

9 i. What problem would arise if insurance tried to sell insurance to consumers as described in part (a)? [Solution] A high risk consumer would prefer L = ; L = : r + r + r r 5 4 = > : The expected pro t that a company earns from a high risk consumer who chooses a contract for a low risk consumer turns out to be negatie: + = < : In a similar fashion, we can show that the low-type will NOT deiate to choose the high insurance. If some or all high risk consumers choose the contract for the low risk consumers, the insurance company s expected pro t is actually negatie (since what the company can earn from a low risk consumer choosing a low insurance is zero) this will thus result in the insurance companies exiting the market. ii. Suppose there were a "separating equilibrium," where di erent types of consumers bought di erent contracts. Describe what the equilibrium must graphically and/or algebraically. (Hint: You do not hae sole for the equilibrium solution analytically. It su ces to describe the properties of the equilibrium allocation from the conditions of incentie compatibility and zero pro t/free entry condition of the insurance rms.) [Solution] Suppose an insurance company o ers two contract L ; L and H ; H. In a separating equilibrium, low-risk consumers must choose the former and high risk consumers must choose the latter. This translates to the following IC conditions: (IC H ) : H + p q + H (IC L ) : L + p q + L p L + q + L ; p H + q + H : Moreoer, indiidual rationality for each type requires: (IR H ) : H + p q + H p p + t :6; (IR L ) : L + p q + L p p + t 4:4: (i) We rst argue that due to the assumption of free entry/ competition the expected payo from EACH contract should be zero: i.e., we must hae H = H and L = L. (ii) We now argue that only (IC H ) matters while (IC L ) will be satis ed automatically. Verbally, this means that low risk consumers hae no incentie to pretend themseles to be of high risk. To see this, summing up the two self-selection conditions, (F) : p L q + L p H q + H : We can now show that (IC L ) will be satis ed by (F) : L + p q + L = L + p q + L + p L q + L (F) L + p + p H q + H q + L 9

10 = = p L + p H p L + q + L q q + L + + L q + H (iii) Gien this obseration, it inoles full insurance for high risk consumers since there is no reason to distort their choice of insurance as low risk consumers will not mimick high risk consumers. In this sense, H = ; H = is the equilibrium contract. (i) To pin down the contract for low risk, note that L ; L should be the solution to: This translates to max L + ( L ; L ) p q + L s:t: L = L (non-negatie pro t), (F) r 4 (IC H ) : L + p q + L : s.t. r 4 max L + p p + L L p L + p + L. The constraint must be binding for otherwise we are back to the situation in b(i). So we just need to sole r 4 = L + p p + L. The solution is L = 7 4p 7 = :9. This gies a payo of L + p p + L = 4:4. A key result here is the absence of full insurance for low risk consumers: L > + L. Low risk consumers hae to shoulder some degree of risk. (c) Suppose that a proportion of consumers were high risk. For which alues of would another rm be able to enter and earn positie pro ts selling to both high and low risk consumers relatie to the "separating equilibrium"? [Solution] Suppose that an entrant o ers a pooling contract (; ) which consumers of both types would like to accept. The probability of accident is + ( ) = ( + ). The probability of no accident is ( + ) =. The expected pro t from this contract is ( + ) + = [ ( + ) ] : We see the pro t is positie i ( + ) i + = + = +, decreasing in and increasing in. We want to nd the upper bound for the. In other words, we want to minimize or equialent maximize subject to the following: + +

11 (a) The high risk consumers would prefer this contract i : p p + + H + p q r 4 + H = ; and (b) The low risk consumers would prefer this contract i : p p + + L + p q + L = 4:4.