outputs, y the output. (monthly revenue) a the input (effort). (work hard and smart) E the random shock that is beyond the agent's control, to capture all the other things that may affect the output. (covid, weather) Is there anything left out from this production? The Principal's Payoff Function The principal's payoff is the profit Since y is Stochastic due to the random Shock E, the objective function of the principal (who is assumed to be risk neutral) is to maximize the expected profit so get where the notation E(x) denotes the expected value of the random variable x. Example: Optimal Payoffs Example: Principal's Optimal Choice The agent's expected utility is still uo. Then together With the principal's FOC we get where Her expected wage is equal to her cost of effort 0.5 plus the reservation utility uO : And thus Ew*-o.5 Then the principal's optimal expected profit is -0.51 -08. -u, = 1 0.5 uo=O.5 u. s* The optimal incentive Contract is That is, 0.5-uo. This means that for O, it must be s O. u, +0.5. (1) Suppose the production function is instead y=ga+E, where g > O indicates the productivity of a person with ability g. Solve the basic principal-agent model where the cost function is c(a) g (2) Suppose the cost function is 2g , while the production function is still y=a+E. Solve the basic principal-agent model. Incentive Contracts Production: Food Court Shop The incentive contract links the agent's The technology Of production links the inputs to performance With her compensation/reward. The simplest one is the linear contract. w the total compensation (the wage). e fixed base wage. Y the performance (which is usually positively correlated with the output y=a+E). the bonus rate it measures how strong the incentive is; When O, then W=s) Example: Agent's Optimal Choice Suppose the cost function is c(a) = 0.5a2 Then a. The agent's FOC, bc'(a*) O, becomes