EC476 Contracts and Organizations, Part III: Lecture 2 Leonardo Felli 32L.G.06 19 January 2015
Moral Hazard: ◮ Consider the contractual relationship between two agents (a principal and an agent) ◮ The principal hires the agent to perform a task. ◮ The agent chooses his effort intensity, a , 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 , (it is the agent’s private information). Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 19 January 2015 2 / 48
General Setup ◮ 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 ′ ( · ) > 0 , p ′′ ( · ) < 0 . P { q = 1 | a } = p ( a ) , where p (0) = 0, lim a →∞ p ( a ) = 1, and p ′ (0) > 1. ◮ 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) EC476 Contracts and Organizations, Part III 19 January 2015 3 / 48
General Setup (2) ◮ The agent’s preferences are represented by the utility function separable in income and effort: U ( w ) − φ ( a ) , U ′ ( · ) > 0 , U ′′ ( · ) ≤ 0 where φ ′ ( · ) > 0, φ ′′ ( · ) ≥ 0. ◮ For convenience we take φ ( a ) = a and we normalize the agent’s outside option: U = 0 . Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 19 January 2015 4 / 48
General Setup (3) Assume that: ◮ a is chosen by the agent before uncertainty is realized; ◮ a is only observed by the agent. It is his private information . ◮ q is verifiable information (observable to all agents involved in the contract Court included). ◮ the transfer w can only be contingent on the verifiable information q . ◮ q is not in a one-to-one relation with the effort a . Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 19 January 2015 5 / 48
First Best Benchmark ◮ The contract theory literature defines the first best world as a world where there are no frictions. ◮ In the current setting this implies that the contract offered by the principal can be contingent on the effort a . ◮ In other words, the effort a is verifiable (observable to all agents involved in the contract Court included). Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 19 January 2015 6 / 48
First Best Contract ◮ The first best contract is obtained as the solution to the problem: max p ( a ) V (1 − w 1 ) + (1 − p ( a )) V ( − w 0 ) a , w i s.t. p ( a ) U ( w 1 ) + (1 − p ( a )) U ( w 0 ) ≥ a ◮ 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 ∗ 1 ) U ′ ( w ∗ 0 ) Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 19 January 2015 7 / 48
First Best Contract (2) ◮ These transfers are paid only if the effort level coincides with a ∗ that satisfies the following FOC: � V (1 − w ∗ 1 ) − V ( − w ∗ 0 ) + U ( w ∗ 1 ) − U ( w ∗ 0 ) � 1 p ′ ( a ∗ ) = V ′ (1 − w ∗ 1 ) U ′ ( w ∗ 1 ) U ′ ( w ∗ 1 ) ◮ Finally the agent’s expected utility coincides with the outside option: p ( a ∗ ) U ( w ∗ 1 ) + (1 − p ( a ∗ )) U ( w ∗ 0 ) = a ∗ Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 19 January 2015 8 / 48
First Best Contract – Risk Neutrality ◮ If the principal is risk neutral: V ( x ) = x ◮ Then the conditions above become: w ∗ 1 = w ∗ 0 = w ∗ and 1 U ( w ∗ ) = a ∗ , p ′ ( a ∗ ) = U ′ ( w ∗ ) ◮ If, instead, the agent is risk neutral: U ( x ) = x ◮ Then the optimum entails: w ∗ 1 − w ∗ p ′ ( a ∗ ) = 1 . 0 = 1 , Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 19 January 2015 9 / 48
Second Best Contract ◮ If a is not verifiable then, for every w 1 and w 0 , a is determined so that: max p ( a ) U ( w 1 ) + (1 − p ( a )) U ( w 0 ) − a (1) a ◮ The latter is the agent’s incentive problem. ◮ Only the agent controls a and hence incentives for the agent to choose the principal’s desired level of a have to be induced through the contingent trasfer w ( q ). ◮ In other words, the second best contract can be contingent only on q . Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 19 January 2015 10 / 48
Second Best Contract (2) ◮ The second best contract can be obtained as the solution to the problem: max p (ˆ a ) V (1 − w 1 ) + (1 − p (ˆ a )) V ( − w 0 ) ˆ a , w i s.t. p (ˆ a ) U ( w 1 ) + (1 − p (ˆ a )) U ( w 0 ) ≥ ˆ a ˆ a ∈ arg max p ( a ) U ( w 1 ) + (1 − p ( a )) U ( w 0 ) − a a ◮ The first constraint is known as the agent’s individual rationality constraint, ◮ The second constraint is known as the agent’s incentive compatibility constraint. Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 19 January 2015 11 / 48
Second Best Contract (3) ◮ The FOC of the incentive compatibility constraint are: p ′ (ˆ a ) [ U ( w 1 ) − U ( w 0 )] = 1 (2) ◮ A first observation: from this condition it is clear that full insurance: w 1 = w 0 leads to no incentives: p (0) = 0 ◮ Assumptions on p ( · ) imply that the solution to this condition is unique for any pair ( w 0 , w 1 ). ◮ We can replace the agent’s (IC) by the set of FOC in (2). ◮ In general replacing the (IC) constraint with the FOC of the agent’s effort choice problem is not a valid approach. Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 19 January 2015 12 / 48
Risk Neutral Agent ◮ Consider now the case in which the agent is risk neutral: U ( x ) = x we have seen that first best optimality requires p ′ ( a ∗ ) = 1 ◮ In this case the FOC of the (IC) constraint becomes: p ′ (ˆ a )( w 1 − w 0 ) = 1 ◮ Therefore setting w 1 − w 0 = 1 leads to the first best allocation: optimal risk sharing and optimal incentives . Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 19 January 2015 13 / 48
Risk Neutral Agent (2) ◮ The reason is that: ◮ optimal risk sharing requires that the agent bears all the risk in the environment, ◮ optimal incentives requires that the agent is residual claimant. ◮ This is achieved by selling the activity to the agent at a fix price − w 0 > 0 so that the risk averse principal receives full insurance. ◮ Notice that in this case we need the agent to have deep enough pockets: when the outcome is q = 0 the agent’s payoff is w 0 < 0. ◮ The agent must be willing to incur a loss with a strictly positive probability. Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 19 January 2015 14 / 48
Resource Constrained Agent ◮ It is often natural to assume that the agent has no resources to put in the activity. ◮ This implies a resource constraint : w i ≥ 0. ◮ In this case the problem becomes: max p (ˆ a ) V (1 − w 1 ) + (1 − p (ˆ a )) V ( − w 0 ) ˆ a , w i s.t. p (ˆ a ) w 1 + (1 − p (ˆ a )) w 0 ≥ ˆ a p ′ (ˆ a )( w 1 − w 0 ) = 1 w i ≥ 0 ∀ i ∈ { 0 , 1 } Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 19 January 2015 15 / 48
Resource Constrained Agent (2) ◮ In the situation in which the agent is resource constrained not all the risk can be transferred to the agent: the constraint w i ≥ 0 will be binding for the transfer w 0 : w 0 = 0 ◮ It is still possible to create first best incentives but for this purpose the agent’s needs to be rewarded. ◮ If w 1 − w 0 = 1 then the agent’s payoff is: p ( a ∗ ) − a ∗ > 0 since p ′ (0) > 1 and p ′ ( a ∗ ) = 1. ◮ In other words the (IR) constraint is not binding. This is not necessarily optimal for the principal. Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 19 January 2015 16 / 48
Resource Constrained Agent (3) ◮ In particular, if we assume that the principal is risk neutral as well: V ( x ) = x then the principal’s problem is: max p (ˆ a ) (1 − w 1 ) − (1 − p (ˆ a )) w 0 ˆ a , w i s.t. p (ˆ a ) w 1 + (1 − p (ˆ a )) w 0 ≥ ˆ a p ′ (ˆ a )( w 1 − w 0 ) = 1 w i ≥ 0 ∀ i ∈ { 0 , 1 } ◮ The solution implies that 1 w 0 = 0 , w 1 = p ′ (ˆ a ) Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 19 January 2015 17 / 48
Resource Constrained Agent (4) ◮ Moreover, ˆ a solves the constrained problem: max p (ˆ a ) (1 − w 1 ) ˆ a s.t. p ′ (ˆ a ) w 1 = 1 or a ) = 1 − p (ˆ a ) p ′′ (ˆ a ) p ′ (ˆ a )) 2 ( p ′ (ˆ ◮ Given that p ′′ ( · ) < 0 then we conclude: ˆ a < a ∗ . ◮ The resource constraint implies a second best level of effort . Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 19 January 2015 18 / 48
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