2. Suppose that the principal is a consumer and the agent is a producer of a...

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2. Suppose that the principal is a consumer and the agent is a producer of a given product. Suppose, when the quantity produced is q 0 and the transfer made is t 0, the monopoly's profit is t - Oq and the consumer's surplus is kq 1/2 - t, where k = 1. Suppose the consumer does not observe the monopoly's cost parameter 0, but it is common knowledge that this parameter can be either = 1 with 0 = probability v = 2/3 or 2 with probability 1/3. Suppose the consumer offers a menu of contracts before the monopoly observes its type. (a) Write down the consumer's optimization problem by clearly stat- ing the objective function and the constraints. (b) Simplify this problem by noting binding and non-binding con- straints, and making necessary substitutions. (c) Solve the simplified problem; that is, find an optimal contract as quantity-transfer pairs offered for each type of the monopoly. (d) Find the expected surplus of the consumer. Identify each type of monopoly's information rent. (e) Suppose that the taste parameter of the consumers increases from k = 1 to k = 2. How does this change affect the informa- tion rent of each type of the producer? Do you think the effect you found is intuitive? Why or why not? Explain. 4. Suppose a risk-neutral investor (i.e., the principal) wants to hire a risk-neutral broker (i.e., the agent) to manage his portfolio. The outcome of his portfolio is stochastic and can be either q = 8 bitcoin or q = 1 bitcoin. If the broker does not exert effort, then with probability To = 1/4 the outcome will be and with probability 3/4 it will be q. If the broker exerts effort, then with probability T = 1/2 the outcome will be and with probability 1/2 it will be q. Exerting effort is costly to the broker and suppose this cost is equal to 2 0.5. Suppose the investor receives a profit from the outcome of the portfolio according to the payoff function S(q) 9/2 while the broker receives a payoff of t from any payment t made to him. Finally, suppose the contract consists of payments {(t,t)} to the broker, depending upon whether the outcome is high or not. = = (a) Suppose that the investor can observe whether the broker exerts effort or not. Set the optimization problem and find the opti- mal transfers when the investor induces effort. Show that the investor prefers inducing effort over not inducing effort. (b) Now suppose that the investor cannot observe whether the bro- ker exerts effort or not. Set the optimization problem and find the optimal transfers when the investor induces effort. What is the cost of inducing effort? = (c) Suppose again the investor cannot observe whether the broker exerts effort. Now suppose also that the investor cannot impose a transfer less than -1, where l 0.75. Set the optimization problem and find the optimal transfers when the investor induces effort. What is the expected limited liability rent of the broker? (d) Suppose once again the investor cannot observe whether the broker exerts effort. Now suppose also that the investor cannot impose a transfer less than -1, where = 0.25 this time. Set the optimization problem and find the optimal transfers when the investor induces effort. What is the expected limited liability rent of the broker? (e) Suppose now that the investor decides to make the broker share- holder of the profits by using a simple linear sharing rule instead of paying her using transfers. Set the optimization problem and find the optimal linear sharing rule. 2. Suppose that the principal is a consumer and the agent is a producer of a given product. Suppose, when the quantity produced is q 0 and the transfer made is t 0, the monopoly's profit is t - Oq and the consumer's surplus is kq 1/2 - t, where k = 1. Suppose the consumer does not observe the monopoly's cost parameter 0, but it is common knowledge that this parameter can be either = 1 with 0 = probability v = 2/3 or 2 with probability 1/3. Suppose the consumer offers a menu of contracts before the monopoly observes its type. (a) Write down the consumer's optimization problem by clearly stat- ing the objective function and the constraints. (b) Simplify this problem by noting binding and non-binding con- straints, and making necessary substitutions. (c) Solve the simplified problem; that is, find an optimal contract as quantity-transfer pairs offered for each type of the monopoly. (d) Find the expected surplus of the consumer. Identify each type of monopoly's information rent. (e) Suppose that the taste parameter of the consumers increases from k = 1 to k = 2. How does this change affect the informa- tion rent of each type of the producer? Do you think the effect you found is intuitive? Why or why not? Explain. 4. Suppose a risk-neutral investor (i.e., the principal) wants to hire a risk-neutral broker (i.e., the agent) to manage his portfolio. The outcome of his portfolio is stochastic and can be either q = 8 bitcoin or q = 1 bitcoin. If the broker does not exert effort, then with probability To = 1/4 the outcome will be and with probability 3/4 it will be q. If the broker exerts effort, then with probability T = 1/2 the outcome will be and with probability 1/2 it will be q. Exerting effort is costly to the broker and suppose this cost is equal to 2 0.5. Suppose the investor receives a profit from the outcome of the portfolio according to the payoff function S(q) 9/2 while the broker receives a payoff of t from any payment t made to him. Finally, suppose the contract consists of payments {(t,t)} to the broker, depending upon whether the outcome is high or not. = = (a) Suppose that the investor can observe whether the broker exerts effort or not. Set the optimization problem and find the opti- mal transfers when the investor induces effort. Show that the investor prefers inducing effort over not inducing effort. (b) Now suppose that the investor cannot observe whether the bro- ker exerts effort or not. Set the optimization problem and find the optimal transfers when the investor induces effort. What is the cost of inducing effort? = (c) Suppose again the investor cannot observe whether the broker exerts effort. Now suppose also that the investor cannot impose a transfer less than -1, where l 0.75. Set the optimization problem and find the optimal transfers when the investor induces effort. What is the expected limited liability rent of the broker? (d) Suppose once again the investor cannot observe whether the broker exerts effort. Now suppose also that the investor cannot impose a transfer less than -1, where = 0.25 this time. Set the optimization problem and find the optimal transfers when the investor induces effort. What is the expected limited liability rent of the broker? (e) Suppose now that the investor decides to make the broker share- holder of the profits by using a simple linear sharing rule instead of paying her using transfers. Set the optimization problem and find the optimal linear sharing rule.