Problem Set 2: Sketch of Solutions

Transcription

1 Problem Set : Sketch of Solutions Information Economics (Ec 55) George Georgiadis Problem. A principal employs an agent. Both parties are risk-neutral and have outside option 0. The agent chooses non-negative effort levels {a, a } to produce outputs q = a + e q = a + e where e, e N(0, s ) and e and e are independent. The agent s cost of effort is given by c (a, a ) = a +a + ka a, where k [, ]. The principal s profit is given by q + fq w (q, q ) where f > 0 and w denotes the wage paid to the agent.. Characterize the first-best outcome of this game.. Suppose that the principal offers a linear contract of the form w (q, q ) = a + b q + b q. Characterize the optimal linear contract. Now suppose that the principal cannot observe q (and hence the contract cannot depend on q ). 3. Characterize the optimal linear contract. 4. Compare the optimal contracts in parts and 3. Explain the intuition behind the differences. Solution of Problem : Part : In the first best outcome, the principal will pay a flat wage a to the agent to satisfy his (IR) constraints. The principal s expected profit given effort pair {a, a } is equal to E [q + fq a, a ] a = a + fa a Therefore, the first best outcome solves max a + fa a s.t. a a + a ka a 0

2 The agent s (IR) constraint will bind, and so a (a, a ) = a +a ka a. Therefore, the principal s problem can be re-written as ( ) ka a The first order conditions are max a,a a + fa a + a ka = a f ka = a This implies that Part : a fb = fk k a fb = f k k Because the agent is risk neutral, the optimal contract will yield the first best outcome. Therefore, it sufficies to determine the pair b, b such that the agent has incentives to exert the first best level of effort, and then choose the fix wage component a such that his (IR) constraint binds. Given an arbitrary contract w (q, q ) = a + b q + b q, the agent s expected utility is equal to E [a + b q + b q a, a ] c (a, a ) = a + b a + b a a + a ka a His first order conditions are b ka = a b ka = a Using the first order conditions from part, observe that the agent exerts the first best level of effort if b = and b = f. Then a = a fb + a fb + ka fb a fb Part 3: Given an arbitrary contract w (q ) = a + bq, the agent s expected utility is equal to E [a + bq a, a ] c (a, a ) = a + ba a + a ka a His first order conditions are b a ka = 0 ka a = 0

3 This implies that a (b) = a (b) = b k kb k The principal s problem can be written as max b a + fa a ba a s.t. a + ba + a a = b k a = kb k ka a 0 The agent s (IR) constraint will bind, and so we can re-write this problem as max b ( ) ( kf) b + 3k b k ( k ) The solution is given by b = ( k )( fk) + 3k. Part 4: Incentive goes down when task is more important: b f < 0. Incentive goes down when crowding out is more important: b k < 0. Problem. A risk-neutral principal employs two risk-averse agents. The agents (indexed by i {, }) choose nonnegative effort levels {a, a } to produce outputs q = a + # + d# q = a + # + d# where e, e N(0, s ) and e and e are independent. Given effort level a i and outputs {q, q }, agent i s utility is u (a i, q, q ) = e r[w i(q,q c ) a i ] where w i (q, q ) is the wage paid to agent i. The principal s profit is given by q + q w (q, q ) w (q, q ) Note that if d 6= 0, then their outputs are correlated, and hence the principal may find it optimal to reward each agent based on the other agent s output. 3

4 . Characterize the first best outcome of this game.. Suppose that the principal offers a linear contract to each agent of the form w (q, q ) = a + b q + g q and w (q, q ) = a + b q + g q. Characterize the optimal linear contract. 3. How does the optimal contract depend on d? Provide some intuition for the sign of b i and g i. 4. Compare the optimal contract from part to the first best outcome in part. What do you observe when d {0, }? Can you explain this intuitively? Solution of Problem : Part : The first best outcome is similar to the case with a single agent that we studied in class. In the first best outcome, each agent i receives a flat wage such that his (IR) constraint binds, and he is instructed to exert effort a fb i = c. Part : Given an arbitrary linear contract, each agent i chooses effort a i = b i c Recall that contracts are of the form s = a + b x + g x s = a + b x + g x Principal chooses g i to minimize agents exposure to risk, b i to optimally trade off risk and incentives, and a i to satisfy IR constraint. Using the same approach as we used in class to determine the optimal linear contract with a principal and a single agent, one can obtain the optimal bonus rates: b i = + d + d + rcs ( d ) and g i = d + d b i Parts 3 and 4: If d = 0, then we are back to the standard formula from the problem with a principal and a single agent that we analyzed in class. If d > 0, then g i < 0: Agent i is penalized if agent i performs better. b i increases in d and g i decreases in d. If d = {, }, then b i = so that a i = a fb i (and g i = d). Ea. agent s pay is independent of e i s (full insurance). 4

5 Problem 3. A (risk-neutral) municipal government considers funding an investment project put forward by an association (also risk-neutral). The cost of the project is known, but the government is unsure about its social value, and its assessment is at odds with that of the association. Specifically, if the project is of good quality, then its social value (net of the cost of the project) as assessed by the government is q G > 0, while the association would derive a private benefit v G > 0 from seeing it go through. If instead the project is of bad quality, then its net social value is q B < 0, but the association would derive a private benefit v B, higher than v G, if it went through. The association knows the quality of the project, while the government s (common-knowledge) belief is Pr(v B )=b. In the absence of information, the government is ready to fund the project, since we assume bq B +( b)q G > 0. However, since taxation is distortionary, the government has net value l > 0 for each unit of revenue raised from the association. However, the government would be unwilling to allow a bad-quality project to go through even if it were able to charge the association for its full private benefit; that is, we assume q B + lv B < 0. Assume the government has access to a (risk-neutral) expert who, when the project is of bad quality, manages to obtain an (unfalsifiable) proof of this fact with probability p, but observes nothing with probability ( p); nothing is also observed with probability when the project is of good quality. The expert starts with no financial resources and can therefore only be rewarded, not punished. The association is assumed to observe when the expert obtains a proof of bad quality, while the government has to be alerted by the expert.. Derive first the optimal scheme for the government when it cannot rely at all on the expert.. What is the optimal scheme when the government can rely on the expert and when collusion between the expert and the association is impossible because the expert is honest? 3. What is the optimal scheme when the government can rely on the expert but the expert is selfinterested and the association can promise the expert a side payment for not alerting the government when he obtains a proof of bad quality? Assume the collusion technology is such that, for every unit of money the association pays, the expert only collects an equivalent of k < units of money. 4. What is the optimal scheme when the government is unsure about the prospect for collusion, because it believes that with probability g the expert is honest and with probability g it is self-interested? Hint: The government imposes a tax T i when the association reports the project to be i {G, B}. In part, you are asked to characterize the optimal taxes {T G, T B }. Solution of Problem 3: Part : No Supervision Without supervision, the government faces a standard screening problem. If the government could observe the quality of the project, it would be willing to provide funding if the quality is good, but not if the quality is bad. In the absence of additional information, the government is assumed to be ready to fund the project. In order to extract information from the association, the government has two instruments at its disposal: the choice to fund or not and the tax imposed on the association. The assumption here is that the association is not penalized when a bad quality is announced. 5

6 Let us denote by T G and T B the taxes imposed when the agent announces that the quality is good and bad respectively. The government can choose to provide funding only if the agent announces the quality is good, or if the quality is good or bad. In order to induce the association to tell the truth in the first case, we must have the following incentive constraints which represent a tax under both the good and bad projects, but funding only for the good one: v G T G T B T B v B T G. These two conditions together contradict v B states, truth-telling requires v G. If the government decides to provide funding in both v G T G v G T B v B T B v B T G hence T B = T G. In the optimal contract the government will not be able to distinguish the quality of the project due pooling. Nevertheless, it will always provide funding and impose a tax T B = T G = v G. If it imposed T = v B, the association would only accept the project if the quality is bad. The expected value of the contract to the government is ( b)[q G + lv G ]+b[q B + lv G ]. The rent for the association is b[v B v G ]. Part : Honest Supervision If we assume that the expert has to be hired and paid before obtaining evidence, which he finds with probability bp (and in which case the project is not funded), then his added value is: b( p)(q B + lv G )+( b)(q G + lv G ) [bq B +( b)q G + lv G ]. If we assume that the honest expert demands a minimal expected remuneration z, the government will hire him as long as bp[q B + lv G ] z. Indeed, when the project is bad, with probability p the expert manages to obtain unfalsifiable proof of this fact. When the government is alerted about the bad quality, it will not provide funding for the project given that q B + lv G < q B + lv B < 0. Note that the association cannot be penalized when the project is found to be bad. With z low enough, the contract s expected value for the government is ( b)[q G + lv G ]+b( p)[q B + lv G ] z. 6

7 For the association, the rent is reduced to b( p)[v B v G ]. Part 3: Self-Interested Supervision First of all, the agent knows that the government always provides funding except if hard evidence is available. Therefore the association will promise a side payment x for not alerting the hard evidence as long as x apple v B. The expert collects only kx, so the government can convince the expert not to collude by offering w kx. By offering w = kv B the government can make the contract collusion-proof. Given that the expert is risk-neutral, the entire payment can be made when there is a report. The condition for using the self-interested expert is almost the same as for the honest expert, that is, z must be smaller than bp[q B + lv G ] but there is the added constraint to preclude collusion. Therefore, the government will only use the expert if bp[q B + lv G ] max {z, bpkv B }. Part 4: Mixed Supervision If z bpkv B, there is no additional cost to the government to induce the expert to honestly alert it about the bad quality, since it can make the full payment only when there is a report. So the expert again will be used if bp[q B + lv G ] z. If z < bpkv B the total amount that has to be paid is higher due to the potential incentive for collusion, and thus we need to compare a collusion-proof contract and a contract which is not collusion-proof. Here if the payment (which again can be paid in full in the state where there is a report) is smaller than kv B then the government gains only bpg[q B + lv G ] from the contract, since the expert has no incentive to alert the government when he is self-interested. Thus the government is comparing the total gain under the incentive scheme bp[q B + lv G ] bpkv B with the total gain under the no-incentive scheme bpg[q B + lv G ] z. It will choose the incentive scheme if bp[q B + lv G ] bpkv B max{ bpg[q B + lv G ] z,0} and the no incentive scheme if bpg[q B + lv G ] z max{ bp[q B + lv G ] bpkv B,0} and no expert if both are smaller than 0. Notice that only in this case (where it chooses not to incentivize) is the government better off by having the probability g of the expert being honest instead of just having a self-interested expert for sure. The incentive scheme works even if g = since the association knows the state when the expert obtains proof of bad quality. Problem 4. This problem involves a comparison of long-term vs. short-term wage contracts in an efficiency wage setting. Suppose workers and firms live two periods instead of one. Workers have utility y e per period, where y 0 is income and e {0, } is effort. Firms cannot directly observe effort, but if it is equal to zero, then they have a chance q of detecting this. This chance q is chosen by the firm at a cost c (q), where c 0 > 0, 7

8 c 00 > 0. The outside option of the worker is u each period; assume that for any contract the firm offers, the participation constraint of the worker binds.. A short term contract lasts one period. It consists of a wage paid in case shirking is not detected (0 is paid if it is) and a monitoring probability q s. Write the incentive compatibility constraint, argue that it binds, and then write an expression for the total monitoring costs incurred by a firm if it offers two consecutive short term contracts.. A long term contract with deferred compensation lasts two periods. In the first, if shirking is detected (with probability q ), 0 is paid and the contract is not renewed; the worker then seeks a short term contract in the labor market (assume he can always find one). If shirking is not detected, the worker is not paid anything, but the worker is allowed to continue with the firm. In the second period, if shirking is detected (probability q ), 0 is paid; if not, the deferred compensation W is paid. (a) Write the incentive compatibility condition that must hold in the second period, assuming the worker worked in the first period. (b) Do the same assuming the worker shirking in the first period but escaped detection. (c) Show that if q satisfies IC in case (i), it does so in case (ii) and write an expression for it in terms of u. (d) Write the incentive compatibility constraint for the first period, assuming that the worker will have the incentive to exert effort in the second period regardless of what he does in the first period; use this to derive an expression for q in terms of u. 3. Compare q and q. Interpret. 4. Write down expressions for the total monitoring cost (per worker) under short- and long-term contracts. Show that the long-term contract is less costly in terms of monitoring. Provide some intuition for this result. 5. How does the benefit of the long term contract (the cost of short term less that of long term) depend on u? Provide some intuition. 6. Suppose that the outside option u of the workers increases. What do you expect will happen to the prevalence of long term contracts? Solution of Problem 4: Part : The incentive compatibility condition is y ( q s )y or y and it binds q s because the principal wants to minimize the cost (y) as much as she can (up to y = ). Since the participation constraints bind and y = u +, q s = each period s =,. Therefore, total monitoring costs will be c(q s)=. Part : (a) +(W ) +( q )W (b) W ( q )W (c) Clearly, q which satisfies (a) satisfies (b) because we can cancel out in both sides. Equivalently, the first period effort cost is sunk at time the second period effort decision is made and so doesn t affect that decision. Now, the participation constraint in this case will be +(W ) =u or, substituting into the incentive constraint, u = +( q )W. Therefore, q = u+. 8

9 (d) Incentive compatibility condition will be +(W ) ( q )(W )+q u. By the participation constraint, +(W ) =u ) W = u + and u =( q )(u + )+q u ) q =. Part 3: As we see, q = > u+ = q in the long-term contract. If an agent is detected in the first period, she may get u in the second period. However, if an agent is detected in the second period, she gets nothing. Therefore, the opportunity cost of shirking in the second period is larger than that in the first period. It makes the principal save on detection costs in the second period. Part 4: Short-term total monitoring cost is c and long-term cost is c + c u+. Therefore, STMC LTMC = c c u+ 0 since c( ) is an increasing function. Part 5: The benefit of the long-term contract is c c u+. Since c( ) is increasing convex function, as u increases ( decreases), the difference becomessmaller. If B(u) =c c u+, then B 0 (u) = h i c 0 c0 u+ < 0, since c 0 c0 u+ > 0 when c( ) is convex. Like the interpretation in c), as u gets bigger, the opportunity cost of shirking in the second period gets bigger, so it s not necessary to monitor as much. Thus when u is high, saving on monitoring costs isn t worth as much. Part 6: Since B is a decreasing function of u, if the distribution of u increases in the first-order stochastic dominance sense, the average value of B will fall, so we should expect to see fewer long term contracts. Similar effects might be expected from a decrease in the cost of monitoring say from improvements in information technology. 9