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AMP2S.TEX Problem Set 2 29 November 2007 2 of 4 (i) Write down IC constraint and show, by deriving FOC and SOCs, that it is equivalent to LIC and monotonicity of q(·). (ii) Write down P constraint and show that it has to be binding for the lowest type and it is irrelevant for any higher type. (iii) Derive the formula for p(·) from LIC and put it into the objective function of the monopolist. (iv) Find optimal q(·) by ignoring monotonicity constraint. (c) For the optimal contract ( q SB (·), p SB (·) ) (i) Check if the optimal profile q SB (·) is increasing in θ. (ii) Check if information rent is increasing with θ (iii) Check if ‘no distortion at the top’ rule is preserved at the optimum. 2.3. (5%) Consider the moral hazard model with two outcomes and two effort levels discussed during the lecture and do one modification: the Agent is risk neutral. All other assumptions are preserved. (a) Characterize all optimal contracts for the P if he wants the A to choose high effort level. Assume additionally that high effort level produces more social surplus than the low effort level. (b) Show that it is the best for the P if the A chooses high effort level. (c) Show that there exists the optimal contract which involves the P getting a fixed rent from the A, who becomes then a residual claimant, bearing all the risk and taking all the profits and losses. 2.4. (7%) Consider a principal-agent problem with three exogenous states of nature, θ 1 , θ 2 , and θ 3 ; two effort levels, e L and e H ; and two output levels, distributed as follows as a function of the state of nature and the effort level: State of nature θ 1 θ 2 θ 3 Probability 0.25 0.5 0.25 Output under e H 18 18 1 Output under e L 18 1 1 The state of nature is not verifiable (it cannot be contracted on). The principal is risk neutral. The agent’s utility when choosing effort e and receiving monetary compensation w is √ w − g(e), where g(e L ) = 0 and g(e H ) = 0.1. The agent’s reservation utility is 0.1. (a) Derive the optimal contract assuming that the agent’s action is (observable and) verifiable. (b) Derive the optimal contract assuming that the agent’s action is not verifiable. Advanced Microeconomics I, Part 2
AMP2S.TEX Problem Set 2 29 November 2007 3 of 4 (c) Assume the principal can buy, for a price of 0.1, an information system that allows the parties to verify whether or not state of nature θ 3 happened. Will the principal buy this information system? 2.5. (3%) During the lecture we showed that in moral hazard model with two effort levels and continuum of outcomes if both parties are risk neutral, selling the project to the A (making him residual claimant) is optimal for the P. Show that such a contract is also optimal when the A is risk neutral and the P is risk-averse. 2.6. (5%) Consider the following variant of Clarke-Groves mechanism discussed in the class: • Every agent reports a valuation ˆθ i . • The bridge is built if and only ∑ I i=1 ˆθ i ≥ 0. • Each agent receives a payment equal to the sum of other bids if the bridge is built, i.e, T i = − ∑ j≠i ˆθ j . Hence, if ∑ j≠i ˆθ j > 0 agent i receives payment; otherwise she has to pay. • If bridge is not built then all T i = 0. (a) Show that the truth telling is a dominant strategy for every agent, hence the mechanism leads to efficient outcome. (b) Is this mechanism balanced? Since the above mechanism is not balanced, to improve it we may to introduce “tax” on agents to assure balanced budget. The basic idea is to require each agent i to make an additional payment that depends only on what the other agents do without affecting any of i’s incentives. Let ˆθ −i be a I − 1-dimensional vector of reports omitting the report of agent i and let h i (ˆθ −i ) be an extra payment made by the agent i always, i.e., independently if the bridge is built or not. Hence, we have, compared to the previous problem: T i = − ∑ ˆθ j≠i j + h i (ˆθ −i ) if bridge is built; T i = h i (ˆθ −i ) if bridge is not built. (c) Show that the truth telling is again a dominant strategy for every agent. (d) Assume that h i (ˆθ −i ) = ∑ j≠i ˆθ j if ∑ j≠i ˆθ j ≥ 0 and h i (ˆθ −i ) = 0 if ∑ j≠i ˆθ j < 0. Show that under this scheme every agent may be taxed but never subsidized. Is this new scheme balanced? 2.7. (3%) Show that in the model of ‘cheap talk’ game analyzed during the lecture, the equilibrium with two different messages is better for both sender and receiver than the ‘babbling’ equilibrium. 2.8. (6%) Consider a variant of the cheap talk game analyzed in the class in which payoff functions are: for Receiver −|y − t|; for Sender −|y − (t + b)|. Find the 2-report and 3-report Perfect Bayesian Equilibrium for this game. How they differ from analogous equilibria from the original model? Advanced Microeconomics I, Part 2
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