Microeconomic Theory II Preliminary Examination Solutions Exam date: June 5, 07. (40 points) Consider a Cournot duopoly. The market price is given by q q, where q and q are the quantities of output produced by the two firms. There are no costs. (a) Find the (Nash) equilibrium quantities of output. Solution: Firm i s profits are ( q q )q i, for first-order conditions of q q = 0 q q = 0, which solve for q = q = /. (b) Suppose that firm s owner first hires a manager, after which the manager of firm and owner of firm two simultaneously choose outputs q and q. The manager of firm is paid κπ (q,q ) + λq B, where q is the quantity chosen by the manager, π (q,q ) is the profit earned in the duopoly game (given outputs q, and q ), and κ, λ, and B are nonnegative constants chosen by the owner of firm. The outside option for the manager is 0. Assume that firm observes the values of κ, λ, and B before the two firms simultaneously choose their outputs. What is the subgame perfect equilibrium of this game. Compare the result to the outcome of the Stackelberg model (without managers). [5 points] Solution: Note first that there is a distinct subgame corresponding to each choice of (κ,λ, B) by firm. In a subgame perfect equilibrium, we first solve for a Nash equilibrium (which turns out to be unique) of the subgame reached by (κ,λ, B). The manager of firm chooses q to maximize First-order conditions are κ( q q )q + λq B. κ( q q ) + λ = 0 and q q = 0, which can be solved for best response functions q = + (λ/κ) q q = q. and
The values of κ and λ thus shift firm manager s best-response function. The solution is q (λ/κ) + (κ,λ) = q (λ/κ) (κ,λ) =. If λ/κ <, we have found the unique equilibrium of the subgame. If λ/κ, prices are zero, and firm optimally chooses to stay out of the market. It remains to determine the optimal values of (κ,λ, B). Note that we can restrict attention to λ/κ <, since if λ/κ, price is zero and firm makes a loss. Firm will choose κ, λ, and B to maximize its profits of ( κ)π (q (κ,λ),q (κ,λ)) λq (κ,λ) + B. Firm wants to minimze B consistent with the manager accepting the contract and so the choice leaves the manager indifferent between accepting the contract and taking the outside option: B = κπ (q (κ,λ),q (κ,λ)) + λq (κ,λ), and so the firm chooses κ and λ to maximize π (q (κ,λ),q (κ,λ)). In other words, firm chooses κ and λ to achieve the Stackelberg outcome with firm as leader, yielding λ/κ = /4, q = /, q = /4. Note that only the ratio λ/κ is determined. By encouraging its manager to choose an output larger than would maximize profits (since λ > 0), firm is encouraging firm to be less aggressive (which is what a Stackelberg leader does by committing to a quantity larger than the best reply to the follower s chosen quantity). (c) Now suppose that both owners hire managers, simultaneously making public the terms ((κ,λ, B ) and (κ,λ, B )) of the managers contracts, after which the managers simultaneously choose outputs. Solve for the equilibrium. [0 points] Solution: Now there is a distinct subgame corresponding to each choice of (κ,λ, B ) by firm and (κ,λ, B ) by firm. Set λ i := λ i /κ i. The best-response functions for the two firms are now q = + λ q q = + λ q,
which can be solved for q = + λ λ q = + λ λ As before B i is chosen to leave the managers indifferent between accepting the contracts, and the firm i chooses λ i to maximize its profits. Firm i s profit is + λ i + λ giving a first-order condition that reduces to giving λ = λ = /5. + λ i λ + 4 λ i + λ = 0,, (d) How do firms outputs and profits in the previous part compare to the those of the Nash equilibrium without managers? Explain your answer. Notice that your answer to the previous two parts should allow you to make these comparisons without calculating equilibrium outputs and profits. Solution: In equilibrium, both firms subsidize output. Equilibrium output must then be higher, and hence profits lower, than in a Nash equilibrium without subsidies. The firms are effectively playing a Prisoners Dilemma. (e) Suppose that a law is proposed making it illegal to disclose the compensation contracts of managers of firms. Given that the owners always have the option of not disclosing such contracts, why would such a law have any effect? Would you expect the owners of the two firms to support this law? Solution: Either firm would prefer to be the only firm offering an observable contract, but both fare worse when both do so. Nonetheless, in the absence of the law, it is not an equilibrium to have only one firm disclosing the contract. The firms would support such a law.. This question concerns an asynchronous version of an infinitely repeated game, with stage game given by A B A, 0, 0 B 0, 0,.
The discount factor is given by δ (0, ), and time is indexed by t = 0,,,.... For t > 0, player (the row player) chooses in odd periods, and player (the column player) chooses in even periods. So, for t > 0, each player s action is fixed for two periods. Period 0 is different in that both players choose simultaneously in that period, and player gets to choose again in the next period, period. (a) Prove that the strategy profile in which each player always plays B is not a subgame perfect equilibrium for any δ. [0 points] Solution: Let (s, s ) be the strategy profile in which each player always plays B. In particular, this means each player plays B after every history. Consider player in period after player had chosen A in period 0. By following the strategy, player receives a payoff of δ, while by deviating to A this period (and so in period ), and then playing B thereafter, player has payoff (since player, by assumption, plays B thereafter), ( δ) + δ, and which is true for all δ (0, ). ( δ) + δ > δ 4δ + δ > 0 ( δ)( δ) > 0, (b) Prove that there exists a δ such that the infinite repetition of B B is a subgame perfect equilibrium outcome if δ < δ. What is the value of δ? [0 points] Solution: Consider the profile that specifies that both players choose B in period 0; if B was chosen the previous period by player i, then player chooses B this period; and if A was chosen the previous period by player i, then player chooses A this period. It is clearly optimal (for any δ) for player to play A if player i had played A in the previous period. Suppose player i had played B in the previous. By following the strategy, player i receives a payoff of, while by deviating to A this period (and so in the next period), and then following the strategy, player i has payoff (since player, by assumption, is also following the strategy) of δ, and δ δ < =: δ. A similar argument shows that player does not want to deviate in period 0 for δ <. Finally, player will not deviate in period 0, since B is a myopic best reply to B, and s choice in period 0 has no future payoff implications. 4
Questions and 4 concern the same setting, but are otherwise independent. In the common setting, two players ointly own an asset (with equal shares) and are bargaining to dissolve the oint ownership. Suppose v i is the private value that player i assigns to the good, and suppose v and v are independently drawn from the interval [0, ], according to the distributions F i, with densities f i, for i =,. Efficiency requires that player i receive the asset if v i > v. If the partnership is not dissolved, player i receives a payoff of v i.. The players bargain as follows: the two players simultaneously submit bids, with the higher bidder winning the obect (with ties resolved by a fair coin flip), and the winner pays the loser the winning bid. (a) What are the interim payoffs for each bidder? Solution: Since there is always a winner in this bargaining game, the partnership is always dissolved. Suppose bidder is playing σ. Then i s interim payoff from bidding b i with value v i can be written as either U i (v i,b i ;σ ) = or as σ (v )<b i (v i b i )d F (v ) + σ (v )>b i σ (v ) d F (v ) + v i Pr{σ (v ) = b i } U i (v i,b i ;σ ) = E [v i b i σ (v ) < b i ] Pr{σ (v ) < b i } + E σ (v ) σ (v ) > b i Pr{σ (v ) > b i } + v i Pr{σ (v ) = b i }. (b) Suppose (σ,σ ) is a Bayes-Nash equilibrium of the auction, and assume σ i is strictly increasing and differentiable function for i =,. Describe the pair of differential equations the strategies must satisfy. [0 points] Solution: Since σ is strictly increasing, and there are no atoms, Pr{σ (v ) = b i } = 0. Interim payoffs can then be written as U i (v i,b i ;σ ) = (v i b i )F (σ (b i )) + The first order condition (with respect to b i ) is 0 = F (σ Simplifying, v >σ (b i ) (b i )) + (v i b i )f (σ (b i ))(σ (σ (b i ))) σ (σ σ (v ) f (v )d v. (b i ))f (σ (b i ))(σ (σ (b i ))). 0 = F (σ (b i )) + v i b i σ (σ (b i )) f (σ (b i ))(σ (σ (b i ))). 5
At the equilibrium, b i = σ i (v i ), and so σ (σ (σ i (v i )))F (σ (σ i (v i ))) = v i σ i (v i ) σ (σ (σ i (v i ))) f (σ (σ i (v i ))). (c) Suppose v and v are uniformly and independently distributed on [0, ]. Describe the differential equation a symmetric increasing and differentiable equilibrium bidding strategy must satisfy, and solve it. [Hint: conecture a functional form.] [0 points] Solution: Substituting F i (v i ) = v i, f i = and σ = σ = σ yields, for all v [0, ], σ (v )v = (v σ(v ). Suppose σ(v ) = k v. Then, so that k =, that is k v = (v k v ), σ(v ) = v solves the differential equation. (d) Prove the strategy identified in part (c) is a symmetric equilibrium strategy. Solution: If player is following the strategy σ(v ) = v /, then for any b i [0,/], F ( σ (b i )) = b i, and so interim payoff for i is U i (v i,b i ; σ) = (v i b i )b i + b i v d v = (v i b i )b i + 9b i 6 which is a quadratic in b i, with b i having a negative coefficient. That is, U i is strictly concave in b i [0,/]. Since bidder never bids more than /, it is clear that a bid above / is not a profitable deviation. Hence, σ is a symmetric equilibrium strategy., 4. (a) Describe the class of Groves mechanisms to efficiently dissolve the partnership. What attractive property do Groves mechanisms have in terms of strategic behavior? Solution: Given reports of valuations (v, v ), a direct mechanism specifies ownership shares, with x i denoting i s share, and transfers t i to i. A mechanism (x, t ) is efficient if, v i > v, x i (v i, v ) = 0, v > v i. 6
The payoff to player i under the mechanism (x, t ) from the report ( ˆv, ˆv ) is x i ( ˆv i, ˆv )v i + t i ( ˆv i, ˆv ). Groves mechanisms are the class of mechanisms (x, t ) with efficient dissolution x and transfers t i (v i, v ) = x (v i, v )v + k i (v ). The attractive property is that truth telling is a dominant strategy equilibrium of the direct mechanism. (b) Prove that there is a Groves mechanism satisfying ex ante budget balance and interim individual rationality. [5 points] Solution: The mechanism satisfies ex ante budget balance if E t (v, v ) + t (v, v ) = 0. The payoff to player i under the mechanism (x, t ) from the report ( ˆv, ˆv ) is x i ( ˆv i, ˆv )v i + t i ( ˆv i, ˆv ). The mechanism satisfies individual rationality if, for all v i [ˉv, ˉv ], E v x i (v i, v )v i + t i (v i, v ) v i. A Groves mechanism satisfies individual rationality if i.e., E v x i (v i, v )v i + x (v i, v )v + k i (v ) E v x i (v i, v ) so individual rationality is satisfied if v i + x (v i, v )v + k i (v ) v i, 0, min E v x i (v i, v ) v vi i + x (v i, v )v E v k i (v ). We now evaluate the left side: From part 4(a), and ignoring ties (since these have zero probability) E v x i (v i, v ) v i + x (v i, v )v = v i Pr(v i > v ) + E v v v i v > v i Pr(v > v i ) = v i F (v i ) v i ( F (v i )) + E v v v > v i Pr(v > v i ) = v i (F (v i ) ) + v i v f (v ) d v. () 7
Differentiating with respect to v i gives F (v i ) + v i f (v i ) v i f (v i ) = F (v i ), and the function is minimized at v i solving F (v i ) = (we know the first order condition is sufficient for minimization because the second derivative is nonengative). Moreover, the value of () at v i is v i v f (v ) d v =: κ i > 0, so individual rationality is satisfied by any E v k i (v ) κ i. We can choose k i and k to satisfy individual rationality and ex ante budget balance if E x (v, v )v + x (v, v )v κ κ 0. This inequality is equivalent to E x (v, v )v + x (v, v )v κ + κ, () Assume F = F, so that v = v = v. (It is true in the asymmetric case as well, as long as the asymmetries are not severe, but the calculations are more involved.) See Figure. First note that the left side of () is simply E max{v, v }, and is the expectation is the integral over the square in Figure. The right side of () is κ + κ = v d F (v ) + = = v C D E B C D E B C D E v v d F (v ) v d F (v )d F (v ) + B C D max{v, v } d F (v )d F (v ) + max{v, v } d F (v )d F (v ) + But since Pr(A) = Pr(C D), we have verifying (). C D v d F (v )d F (v ) A v d F (v )d F (v ) C D C D min{v, v } d F (v )d F (v ) v d F (v )d F (v ). max{v, v } d F (v )d F (v ), 8
v B C D v A E v v Figure : Figure for Question 4(b). The vertically shaded region ( B C D) is the region of integration for κ, while the horizontally shaded region (C D E ) is the region of integration for κ. (c) Is there a Groves mechanism satisfying ex post budget balance? Provide a proof. [0 points] Solution: There is no Groves mechanism satisfying individual rationality and ex post budget balance: Ex post budget balance requires for all v and v, x (v, v )v + x (v, v )v + k (v ) + k (v ) = 0. (Note that there is no seller to break the budget. ) Choose any v < v < v. Then ex post budget balance implies and so that v + k (v ) + k (v ) = 0 v + k (v ) + k (v ) = 0, v = v + k (v ) k (v ), a contradiction (simply choose v v + k (v ) k (v )). 9