Thanks a lot Lafffont&Martimort, The Theory of Incentives, Chp4

4. Consider the following moral hazard problem with limited liability. Suppose a risk- nentml restaurant owner wants to hire a risk-neutral waitress to serve the customers in his restaurant. The number of customers served is stochastic; it can be either (j = 100 people or q = 50 people. When the waitress exerts effort, then 1T1 = 3/4 is the probability that the number will be it} and 1/4 is the probability that the number will be q. When the waitress does not exert effort, then no : 1/4 is the probability that thenumber will be (j and 3/4 is the probability that the number will be q. Exerting effort is costly to the individual and suppose this cost is equal to 1/) = 2. Sppose the owner receives a payoff from serving of the customers by the payoff function 3(q) = 43?. Suppose the contract the owner proposes consists of payments {(5.1)} to the individual1 depending on whether the serving number is high or not. (a) (b) Suppose that the owner can observe if the waitress exerts effort or not. Set the optimization problem and nd the optimal transfers when the owner induces effort. Show that the owner prefers inducing effort over not. inducing effort. Now suppose that the owner cannot observe if the waitress exerts effort or not. Set the optimization problem and nd the optimal transfers when the owner induces effort. What is the cost. of inducing effort? Suppose again the owner cannot observe if the waitress exerts effort. Now suppose also that the landlord cannot impose a transfer less than l, where l = 2. Set the optimization problem and nd the optimal transfers when the owner induces effort. What is the expected limited liability rent. of the waitress? Suppose once again the owner cannot observe if the waitress exerts effort. Now suppose also that the owner cannot impose a transfer less than l, where l = 0 this time. Set the optimization problem and nd the optimal transfers when the owner induces effort. What is the expected limited liability rent of the waitress? (0) Now suppose that the owner makes the waitress a shareholder of the prots by using a simple linear sharing rule. Set the optimization problem and nd the optimal linear sharing rule