EC537 Microeconomic Theory for Research Students, Part II: Lecture 2 - PowerPoint PPT Presentation

EC537 Microeconomic Theory for Research Students, Part II: Lecture 2 Leonardo Felli CLM.G.4 15 November 2011

Moral Hazard: Consider the contractual relationship between two agents (agent 1 and 2) summarized in the following problem generated by the take-it-or-leave-it offer that agent 1 makes to agent 2: N � max p i v ( e , y i , w i ) e , w i i =1 (1) N � p i u ( e , w i ) ≥ U s.t. i =1 where: v ( · , · , · ) is agent 1’s utility function; u ( · , · ) is agent 2’s utility function; U is the reservation utility of agent 2. Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 2 15 November 2011 2 / 86

Moreover: e can be interpreted as agent 2’s effort or investment ; e enhances the random variable y interpreted as expected profit or expected outcome ; w i is a transfer contingent on y i from agent 1 to agent 2; θ is the state of nature and p i is the probability of state θ i . Assume that: e is chosen by agent 2 before the state of nature θ is realized; e is only observed by agent 2. It is his private information . Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 2 15 November 2011 3 / 86

Label: agent 2, who exerts effort, as the agent A ; agent 1, who benefits from the effort, as the principal P . Assume that: y is verifiable information (observable to all agents involved in the contract court included). Moreover, it is critical for the problem to be interesting that: y is not in a one-to-one relation with the effort e . In other case, the contracting problem will result to a highly simplified version of an optimal risk-sharing problem. Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 2 15 November 2011 4 / 86

First Best: Assume first that the effort chosen by the agent e is verifiable . Then w can be a function of e . Let e ∗ be the optimal effort from the principal’s view point. The Principal’s optimal contract then will specify: a state contingent payment { w 1 ( e ) , . . . , w N ( e ) } to the agent that is individually rational if and only if e = e ∗ : N � p i u ( e ∗ , w i ( e ∗ )) = U i =1 a state contingent payment { w 1 ( e ) , . . . , w N ( e ) } to the agent that is not individually rational (a punishment) otherwise. Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 2 15 November 2011 5 / 86

This can of course also be achieved with a y ∗ that corresponds to e ∗ in a one-to-one fashion. Therefore the problem is interesting only when output is a noisy signal of effort : y = f ( e , θ ) . The principal is thus restricted to offer the contract w ( f ( e , θ )) and e is chosen by the agent so as to maximize his expected utility: N � e ∈ arg max ˆ p i u ( e , w ( f ( e , θ i ))) . e i =1 The contract w ( f ( e , θ )) must be such that it is in the agent’s best interest to choose the “right” (desired) level of effort ˆ e . Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 2 15 November 2011 6 / 86

The principal’s problem is: N � max p i v (ˆ e , f (ˆ e , θ i ) , w ( f (ˆ e , θ i ))) ˆ e , w ( f ( e ,θ )) i =1 N � e , θ i ))) ≥ U s.t. p i u (ˆ e , w ( f (ˆ i =1 N � ˆ e ∈ arg max p i u ( e , w ( f ( e , θ i ))) e i =1 The latter constraint is known as the agent’s incentive compatibility constraint. The former constraint is the agent’s individual rationality constraint. Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 2 15 November 2011 7 / 86

The solution to the principal’s problem with both constraints is in general not a trivial matter . Key tradeoff: the one between insurance and incentives . Recall that without moral hazard the choice of e achieves ex-post allocative efficiency, while the choice of w achieves optimal risk sharing. This cannot be done in the presence of moral hazard. Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 2 15 November 2011 8 / 86

Second Best: risk neutral principal Assume that the principal is risk neutral: v ( e , f ( e , θ ) , w ) = f ( e , θ ) − w Optimal risk-sharing is achieved by giving the agent full insurance: ∀ i ∈ { 1 , . . . , N } . w ( f ( e , θ i )) = τ, The optimal choice of e will then be: ˆ e ∈ arg max e u ( e , τ ) . Clearly the agent will choose the same effort level independently of the contract ( τ is independent of e ). This means that any conflict of interest between the principal and the agent will not be ameliorated by the contract. Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 2 15 November 2011 9 / 86

Special case: effort as pure cost: Assume that e is a pure cost for the agent : u ′ > 0 , c ′ > 0 , c ′′ > 0 u ( e , w ) = U ( w ) − c ( e ) , Then the agent will minimize effort. If e ≥ 0 then: ˆ e = 0. This differs from the effort level ¯ e that the principal desires: � N p i f ( e , θ i ) − c ( e ) max e i =1 or N � ∂ f (¯ e , θ i ) = c ′ (¯ p i e ) ∂ e i =1 Optimal risk sharing implies no incentives . Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 2 15 November 2011 10 / 86

Risk neutral agent: The special case with no tradeoff between incentives and insurance is when the agent is risk neutral . The solution is to sell the firm/activity to the agent . Let w ( f ( e , θ i )) = f ( e , θ i ) − κ Then ˆ e = ¯ e since: � N e ∈ arg max p i f ( e , θ i ) − κ − c ( e ) ˆ e i =1 While κ is independent of e and such that N � p i f (ˆ e , θ i ) − κ − c (ˆ e ) = U i =1 Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 2 15 November 2011 11 / 86

General Characterization: We provide a more detailed characterization of the moral hazard problem by considering its simplest general form. The principal hires the agent to perform a task. The agent chooses his effort intensity, e , which affects the outcome of the task, q . The principal only cares about the outcome, but effort is costly for the agent, hence the principal has to compensate the agent for incurring the cost of effort. Effort is observable only to the agent , hence the agent’s compensation has to be contingent on the outcome q : a noisy signal of effort. Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 2 15 November 2011 12 / 86

Assume that the outcome of the task can take only two values: q ∈ { 0 , 1 } . We assume that when q = 1 the task is successful and when q = 0 the task is a failure . The probability of success is: P { q = 1 | e } = e . The principal’s preferences are represented by: V ′ ( · ) > 0 , V ′′ ( · ) ≤ 0 V ( q − w ) , where w is the transfer to the agent. Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 2 15 November 2011 13 / 86

The agent’s preferences are represented by the utility function separable in income and effort: U ′ ( · ) > 0 , U ′′ ( · ) ≤ 0 U ( w ) − c ( e ) , where c ′ ( · ) > 0, c ′′ ( · ) ≥ 0. For convenience we take c ( e ) = e 2 / 2 and we normalize the agent’s outside option: U = 0 . Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 2 15 November 2011 14 / 86

First Best Contract: The first best contract can be contingent on e . It is obtained as the solution to the problem: e V (1 − w 1 ) + (1 − e ) V ( − w 0 ) max e , w i e U ( w 1 ) + (1 − e ) U ( w 0 ) ≥ e 2 s.t. 2 The optimal pair of transfers w ∗ 1 and w ∗ 0 are such that the following FOC (Borch optimal risk-sharing rule) are satisfied: V ′ (1 − w ∗ = V ′ ( − w ∗ 1 ) 0 ) U ′ ( w ∗ U ′ ( w ∗ 1 ) 0 ) Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 2 15 November 2011 15 / 86

These transfers are paid only if the effort level coincides with e ∗ satisfying: U ′ ( w ∗ 1 ) e ∗ = [ U ( w ∗ 1 ) − U ( w ∗ 1 ) [ V (1 − w ∗ 1 ) − V ( − w ∗ 0 )] + 0 )] V ′ (1 − w ∗ Finally the agent’s expected utility coincides with the outside option: e ∗ U ( w ∗ 1 ) + (1 − e ∗ ) U ( w ∗ 0 ) = ( e ∗ ) 2 / 2 Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 2 15 November 2011 16 / 86

If the principal is risk neutral: V ( x ) = x Then the conditions above become: w ∗ 1 = w ∗ 0 = w ∗ and e ∗ = U ′ ( w ∗ ) U ( w ∗ ) = ( e ∗ ) 2 / 2 , If the agent is risk neutral: U ( x ) = x Then the optimum entails: e ∗ = 1 . w ∗ 1 − w ∗ 0 = 1 , Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 2 15 November 2011 17 / 86

Second Best Contract: If e is not verifiable then for every w 1 and w 0 it is determined so that: e U ( w 1 ) + (1 − e ) U ( w 0 ) − e 2 / 2 max (2) e The second best contract can be contingent only on q . It is obtained as the solution to the problem: max ˆ e V (1 − w 1 ) + (1 − ˆ e ) V ( − w 0 ) ˆ e , w i e ) 2 / 2 s.t. e U ( w 1 ) + (1 − ˆ ˆ e ) U ( w 0 ) ≥ (ˆ e U ( w 1 ) + (1 − e ) U ( w 0 ) − e 2 / 2 ˆ e ∈ arg max e Leonardo Felli (LSE) EC537 Microeconomic Theory for Research Students, Part II: Lecture 2 15 November 2011 18 / 86