The Principal-Agent Problem

Transcription

1 The Principal-Agent Problem Class Notes A principal (she) hires an agent (he) or more than one agent for one perio. Agents effort levels provie a revenue to the principal, ho pays a age to each agent. We ant to fin the contractual arrangements that is best for the principal in terms of her profit. Suppose the principal hires one agent. The agent s effort, enote e, provies a revenue R(e) to the principal, ho pays a age W to the agent. The agent s cost of effort is given by the function c(e). So the principal s payoff hen the agent orks e is given by π p = R(e) W, an the agent s payoff is π a = W c(e). To be consistent ith reality, e ant the functions R( ) an c( ) to be increasing. We may also impose concavity on R an convexity on c, an that c(0) = 0. Suppose n agents are hire by the principal, here e i enotes Agent i s effort, i = 1,..., n. As before the cost of effort to Agent i is given by c(e i ) but the principal s revenue is no given by R(e 1,..., e n ). Let W i enote the age pai to Agent i. The principal s payoff in this case is π p = R(e 1,..., e n ) W i, an Agent i s payoff hen orking e i is π a = W i c(e i ). We first look at the case here agents can be perfectly monitore to then look at cases here agents cannot be perfectly monitore. 1

2 1 Agents can be Perfectly Monitore The principal in this case ill be able to pay each agent accoring to his effort level, i.e. an agent s overal age ill be given by W i = e i, is a age rate. Agent i s problem is then e i e i c(e i ). The solution to this problem, enote e i, si such that e i (e i c(e i )) = 0 = c (e i ), here c (e i ) = c(e i) e i. Since e i epens on the age rate, e ill rite e i = e i (). The principal s objective is to choose a age that solves R(e 1 (), e (),..., e n ()) e i (). The age rate that imizes the principal s payoff, enote, is then such that ( R(e 1( ), e ( ),..., e n ( )) = e i ( ) + e ) i( ) Example 1 Let n = 1, R(e) = e, c(e) = e, an suppose that the principal is perfectly able 100 to monitor the agent s effort. Then, given a age rate, the agent s effort, e, is such that = c (e ) = e 100 = e 50, hich gives us e = e() = 50. The principal s problem is then e() e() 50 50, an the age rate that imizes her payoff is such that = 0 = 1.

3 Hence for this problem, the agent s effort level is the overall payment to the agent is an the principal s payoff is The agent s payoff in this case is e( ) = 50 1 = 5, e( ) = 1 5 = 1.5 R(e( )) e( ) = = 1.5. e( ) c(e( )) = = 6.5. The Pareto Efficient Allocation Suppose that the principal orks for herself, an let e i enote the effort level she evotes to task i, say. Her problem is then e 1,...,e n R(e 1,..., e n ) c(e i ), an thus the effort levels that imize her payoff are such that e i R(e 1,..., e n) = c (e i ) for all i. This allocation is calle the Pareto efficient allocation, it is the allocation that imize the overall ealth (the sum of all payoffs). Example Let s fin the Pareto efficient allocation for the problem escribe in Example 1. Here e have R (e ) = c (e ) 1 = e 50 e = 50. 3

4 The firm s revenue is then 50, the cost of effort is = 5 an the firm s profit is 5, hich is the overall ealth, as it can be ivie beteen the principal an the agent. The overall ealth here is greater than in the previous example since then the sum of both payoffs as = A Revenue Sharing Plan Suppose that agents cannot be perfectly monitore an that the principal ants to motivate them by sharing her revenue ith them. That is Agent i s compensation consists of a share s i of the total firm s revenue, here (s 1, s,..., s n ) is such that s i 0 for all i an i s i 1. Agent i s problem is no e i s i R(e 1,..., e n ) c(e i ), an the Agent i s choice of effort, e i, is such that (s i R(e 1,..., e n ) c(e i )) = 0 s i r(e 1,..., e n ) = c (e i ). e i e i Example 3 Consier a situation as in Example 1 but instea of paying a age rate, the principal gives a fraction s if her revenue as a age to the agent. The agent s choice of effort, e, is such that sr (e ) = c (e ) s = e 50 e = 50s. The principal s payoff is simply the amount of revenue left, i.e. (1 s)r(e ). If s = 1, for instance, the agent s effort is 10 an the principal s revenue is 4 10 = Note that this is much loer than the principal s payoff hen she can perfectly monitor effort. For the agent to ork as much as in the perfect monitoring case, i.e. e = 5, the share of revenue that the principal must give to the agent is 1. 4

5 4 The Shirking Moel Suppose that the principal ants each of her agents to provie a level of effort equal to ê, here ê is some arbitrary number. The principal cannot monitor perfectly, but she can monitor. That is, each employee s effort may be monitore ith probability p. If an employee is caught orking less than hat is emane, ê, he ill be fire. The age pai by the principal is enote Ŵ. In this case, e assume that the agent has an outsie option, hich is a payment of W. More specifically, if an agent is fire by the principal, then he receives a payment of W. The agent s expecte payoff is then Ŵ c(e) if e ê, π a = pw + (1 p)ŵ c(e) if e < ê, From this payoff function, e can see that the agent ill either ork ê or e = 0. So in orer for the principal to be able to hire some employees, the age Ŵ has to be such that Ŵ c(ê) pw + (1 p)ŵ Ŵ W + c(ê) p. Note that the agent s expecte payoff hen shirking is pw assume that c(0) = 0. + (1 p)ŵ since e have Example 4 Consier a situation as in Example 1, an assume that W = 6.5, the agent s payoff effort can be perfectly monitore. If p = 1, that must Ŵ be to inuce the agent to ork ê = 5? Ŵ W + c(ê) p. = (5) /100.5 = $ Note that the principal has to pay a significant premium to obtain this effort level. 5 A Forcing Contract Suppose the principal tells her employees: Either revenue is at least R, in hich case you get pai a age W, or either revenue is belo R an you on t get pai at all. What is then the equilibrium? I ll leave this one to you. 5