EconS Microeconomic Theory II Midterm Exam #2 - Answer Key

Transcription

1 EconS 50 - Microeconomic Theory II Midterm Exam # - Answer Key 1. Revenue comparison in two auction formats. Consider a sealed-bid auction with bidders. Every bidder i privately observes his valuation v i for the object, drawn from uniform distribution U[0; 1], which is common knowledge among players. Assume that all bidders are risk neutral. (a) Find equilibrium bidding functions if the auction is: (1) rst-price auction, and () second-price auction. First-price auction. From previous exercises, we know that the equilibrium bidding function in this auction with bidders and with uniformly distributed valuations is b i (v i ) = 1 v i for every bidder i. For instance, in the case of = bidders, this bidding function becomes b i (v i ) = v i. If bidder j is the individual with the highest valuation for the object, he submits the highest bid (since bids are increasing in valuations) and pays a share 1 of his true valuation (this is what we normally denote as bid shading ). Second-price auction. From previous exercises, we also know that bidding according to his valuation is a weakly dominant strategy in the second-price auction, and also the Bayesian ash equilibrium of the game, that is, b(v i ) = v i for every bidder i. Therefore, if bidder i has the highest valuation for the object, that is v i > max j6=i v j, he submits the highest bid, winning the auction, and paying a price that coincides with the bid of the second highest bidder, b(v k ) = v k where v k = max j6=i v j denotes the highest valuation among all bidder i s rivals. (b) Evaluate the seller s ex-post revenue in each auction format (which is a function of the realizations of players valuations). Does the revenue equivalence theorem hold? Revenue from rst-price auction. From part (a), we know that b i (v i ) = v i, implying that the expected revenue of the seller in this auction format is 1 max fv 1; v ; :::; v g since the winning bid (and thus the price that the seller receives) is of the highest bidder s valuation. Revenue from second-price auction. Recall from part (a) that the individual i with the highest valuation submits the highest bid, b(v i ) = v i, winning the object, but he only pays a price equal to the bid of the individual with the second-highest valuation, b(v j ) = v j, implying that the seller s expected revenue is also v j. 1

2 Revenue comparison. Depending on the speci c realization of players valuations, the actual revenue that the seller receives can be di erent across auction formats. However, this is not contradictory with the revenue equivalence theorem. This theorem only tells us that, in expectation (that is, before valuations are drawn from the uniform distribution), revenues should coincide across these auction formats. (c) Find the seller s ex-ante (or expected) revenue in each auction format (which is not a function of the realization of players valuations but, instead, of the expected value of these valuations). Does the revenue equivalence theorem hold now? The ex-post pro t of rst-price auction for the winning bidder is i (v i ) = v 1 i {z } probability of winning = 1 v i 1 v i {z } value of the bid such that the ex-ante pro t of rst-price auction for all bidders is E [ (v)] = = X E [ i (v i )] i=1 Z 1 X i=1 0 = 1 1 v i f (v i ) dv i X i=1 = 1 = Z 1 0 v i dv i v+1 The ex-post pro t of second-price auction for the winning bidder is i (v i ) = v 1 i {z } v {z} j probability of winning value of the second-highest bid = v 1 i = 1 v i 1 v i where the second line originates from the fact of uniform distribution that sets v j to be 1 below v i, which establishes the equivalence between the ex-post pro t of rst-price and second-price auctions. Applying the same calculations as above, we can show that the ex-ante pro t of second-price auction is also

3 . Selten s horse. Consider the Selten s Horse game depicted in Figure 1. Player 1 is the rst mover in the game, choosing between C and D. If he chooses C, player is called on to move between C 0 and D 0. If player selects C 0 the game is over. If player 1 chooses D or player chooses D 0, then player is called on to move without being informed whether player 1 chose D before him or whether it was player who chose D 0. Player can choose between L and R, and then the game ends. Figure 1. Selten s horse. (a) De ne the strategy spaces for each player. Then nd all pure strategy ash equilibria (pse) of the game. [Hint: This is a three-player game, so you can consider that player 1 chooses rows, player columns, and player chooses matrices.] The strategy spaces of the players are as follows: S 1 = fc; Dg S = fc 0 ; D 0 g S = fl; Rg In Figure, we represent the strategies and payo s of the three players in the following normal form representation of the game, where Player 1 chooses between the rows, Player chooses between the columns, and Player chooses between the matrixes.

4 Figure. Selten s horse - Matrix representation. We next underline the best responses of the three players in Figure, and identify that (C; C 0 ; R), (D; C 0 ; L), and (D; D 0 ; L) are the pure strategy ash equilibria of this game. Figure. Selten s horse - Underlining best response payo s. (b) Argue that one of the three pses you found in part (a) is not sequentially rational. A short verbal explanation su ces. (D; C 0 ; L) is not sequentially rational. If Player 1 chooses D, then Player s belief is = 1, responding with L (see left-hand side at the bottom of the tree). Anticipating that Player choosing L, Player compares his payo from C 0, 1, against that from D 0 (which is followed by Player responding with L), 4, and thus chooses D 0. Therefore, Player choosing C 0 is not sequentially rational. (c) Show that strategy pro le fc; C 0 ; Rg can be sustained as a PBE of the game. (You don t need to show that this is actually the unique PBE we can sustain in this game.) Discuss that this strategy pro le is based on credible beliefs by player. We check the pooling strategy pro le, C; C 0, where Player 1 chooses C and Player selects C. As depicted in Figure 4, since player 1 chooses C (as illustrated by the blue horizontal arrow) and player chooses C 0 (as illustrated by the green horizontal arrow), messages D and D 0 are o -the-equilibrium path, leaving the 4

5 beliefs of Player unrestricted, that is, [0; 1]. In other words, Player s information set should never be reached in this strategy pro le. Figure 4. Pooling Strategy Pro le C; C 0 Therefore, if Player is ever called out to move, he compares the expected payo from responding with L and R, as follows: EU (L) = + 0 (1 ) = EU (R) = (1 ) = 1 Player then responds with L if > 1, which simpli es to > 1. Otherwise, he responds with R. This gives rise to two cases (one in which > 1, and Player responds with L; and another in which 1 and Player responds with R), which we separately analyze below. - Case 1, > 1. As depicted in Figure 5a, Player responds with L (as illustrated by the red arrows) since > 1. In this context, Player can improve his payo by deviating from C 0, which yields a payo of 1, to D 0, which yields a payo of 4. Therefore, the pooling strategy pro le C; C 0 cannot be supported as a PBE of this game when Player s beliefs satisfy > 1. 5

6 Figure 5a. Pooling Strategy Pro le C; C 0 when > 1. - Case, 1. As depicted in Figure 5b, Player responds with R (as illustrated by the red arrows) given that his beliefs are 1. In this context, Player does not deviate because his prescribed strategy, C 0, gives him a payo of 1, while deviating to D 0 would give him a payo of 0. Similarly, Player 1 does not deviate because his prescribed strategy, C, gives him a payo of 1, exceeds his payo from deviating to D, zero. Therefore, strategy pro le C; C 0 can be supported as a PBE of this game when Player s beliefs satisfy 1. Figure 5b. Pooling Strategy Pro le C; C 0 when 1.. [Principal-agent problem] Consider a situation where a principal has pro t function u p (e; w) = e where e denotes the e ort that agent (e.g., employee) exerts, which is transformed into pro ts at a rate > 1; and w represents the salary that principal pays the agent. 6 w

7 While the principal cannot observe the agent s type i ; where i = fl; Hg and L = 1 and H = ; he knows the frequency of worker with high-type, p; and low-type, 1 p: Utility function of each agent is u i (e; w) = w i e where i = fl; Hg: Intuitively, the agent s utility increases in the salary that he receives, but decreases in the e ort he exerts. Hence, the second term i e can be interpreted as the agent s cost of e ort, which is increasing and convex in e ort, and where the (absolute and marginal) cost of e ort is larger for the high type than for the low-type since H > L. The reservation utility of the agents is zero. (a) Symmetric Information: Find the contract(s) that will be o ered by the principal when he can observe the agent s type. Since the principal can observe each agent s type, he solves, for every worker type i ; max w i ;e i e i w i subject to w i i e i 0 (P C) As usual, the P.C. constraint must be binding, i.e, w i = i e i for all i = fl; Hg: Otherwise, the rm manager could still lower salaries and extract a larger surplus. We can thus substitute the binding PC constraint into the principal s objective function, to obtain the following unconstrained problem max e i e i i e i Taking the FOC with respect to e i ; yields i e i = 0 which, solving for e i ; helps us obtain the optimal e ort level under symmetric information e i = for all i = fl; Hg i That is, since L = 1 and H =, e ort levels are e L = and e H = 4 ; thus prescribing a higher e ort level for the worker with a low disutility from e ort, i.e., e L > e H : Salaries in this context become w i = i e i, which entail w H = = 4 8 and w L = = 4 and (b) Asymmetric Information: Find the contract(s) that the principal o ers when he cannot observe the types of each agent. Remark: For generality, I rst solve the incomplete information game without assuming speci c values for parameters p and, and then evaluate the results at the parameters given in the exercise p = 1 and =. 7

8 Under incomplete information, the principal maximizes his expected utility max p(e H w H ) + (1 p)(e L w L ) w L ;w H ;e L ;e H subject to the participation constraints for each agent w L e L 0 (P C L ) w H e H 0 (P C H ) and the incentive compatibility constraints for each agent w L e L w H e H (IC L ) From the above equations w H e H w L e L (IC H ) w L e L {z} From IC L w H e H > w H e H: This indicates that the P C H is binding. So we can now set up our Lagrangean as follows: L = p(e H w H ) + (1 p)(e L w L ) Taking the FOCs + 1 (w H e H) + (w L e L w H + e H) + (w H e H w L + e = p = 0 =) = p = (1 p) + = 0 =) = 1 p = p 4 1 e H + e H 4 e H = 0 =) e H H ( 1 L = (1 p) + 4 e L e L = 0 =) e L = (1 p) ( ) In addition, from the rst and second FOC, we obtain that p 1 = p 1, or 1 = 1, thus con rming that its associated constraint, P C H, binds, i.e., w H e H = 0. We can then use this result into the expression of IC H to obtain that w H e H {z } 0 w L e L which means that the high-type agent would receive a negative utility should he select the contract meant for the low-type. Hence, the IC H must slack (i.e., hold strictly) entailing that its associated Lagrange multiplier is nil, = 0. Using the second FOC, = 1 p, we then nd that = 1 p > 0. 8

9 We can now evaluate the e ort levels found in the last two FOCs at 1 = 1, = 0, and = 1 p, which yields e H = e L = p ( (1 p)) = p (1 + p) (1 p) (1 p) = and Let us now compare these e ort levels against those under complete information found in part (a), where we obtained that e H = for the high type and 4 e L = for the low type. Therefore, the e ort level of the high-type agent is p lower under incomplete than complete information since < simpli es (1+p) 4 to p 1, which holds by de nition; but the e ort level of the low-type agent coincides across information settings. In other words, there is no distortion for the low-type worker (the most e cient worker), while there is distortion in the e ort required from the high disutility of e ort worker. (In addition, note that while e H = p increases in p, it lies below e (1+p) L = for all values of p, including at p = 1.) Regarding salaries, we nd that w L = e H + e L = w H = e H = p (1 + p) p (1 + p) + 4, and In addition, note that when p = 1, our results converge to the nding under symmetric information of part (a), i.e., e ort levels become e H = 4 and e L =, and salaries are w L = + = 5 and w H =. 8 umerical example: Consider that p = 1 ; and = : Then the optimal e orts under symmetric information are e L = 16 and e H = 8 while under asymmetric information are e L = 16 and e H = 4 Similarly, salaries under symmetric information are w L = 56 and w H = 18 whereas under asymmetric information w L = 7 and w H = : (c) Find the information rents of the worker with low cost of e ort and of the worker with high cost of e ort. Interpret. 9

10 We can measure the information rents of each type of agent, by comparing his utility under asymmetric information (where it can be positive) and symmetric information (where it was zero). In particular, since P C H binds, the high-type does not retain information rents. However, the low-type agent obtains a utility level of u L = w L e L = e H + e L e L = e p H = {z } (1 + p) w L which is positive under all parameter values, and thus larger than his h zero utility level under symmetric information. His information rent is thus p, (1+p)i which is increasing in the probability of the agent being of high type, p. Intuitively, as the proportion of low-cost agents decreases, they require a higher salary for the principal to distinguish them from the high-cost agents. In the above numerical example, this information rent becomes = (1+ 1 ) 10