1. Suppose the principal is the buyer of a good and the agent is the seller of the good.  Suppose the buyer’s utility function is given by q1/3 - t and the seller’s utility function is given by t - θq4/3, where q > 0 is the amount of good received from the seller and t > 0 is the payment made to the seller.  Suppose that the buyer does not observe the seller’s cost parameter θ, but it is common knowledge that this parameter can be either θ = 1/4 with ν = 2/3 probability or θ¯ = 5/6 with 1/3 probability. Suppose also that the buyer offers a menu of contracts after the seller observes his cost parameter θ .

(a) Write down the buyer’s second-best optimisation problem by clearly stating the buyer’s objective function and the seller’s con- straints.

(b) Simplify this  problem  by  identifying  binding and  non-binding constraints, and making the necessary substitutions.

(c) Find the second-best optimal contracts (either as quantity-transfer pairs or quantity-information rent pairs) offered for each type of  seller.

(d) Find the expected utility of the buyer with the second-best op- timal contracts. Find the seller’s information rent for each type.

(e) Can the shutdown policy ever become optimal in this setting? Why or why not? Explain.

2. Suppose the principal is a monopolist and the agent is a consumer. When the quantity produced is q  > 0 and the transfer made is t > 0, suppose the consumer’s utility is given by θq1/2 - t and the monopoly’s profit is given by t - kq, where k = 1.  Suppose the monopolist does not observe the consumer’s taste parameter θ, but it is common knowledge that this parameter can be either θ¯ = 2 with ν = 2/5 probability or θ = 1 with 3/5 probability.  Suppose the monopolist offers a menu of contracts before the consumer observes his taste parameter.

(a) Write down the monopolist’s second-best optimisation problem by clearly stating the monopolist’s objective function and the consumer’s constraints.

(b) Simplify this  problem  by  identifying  binding and  non-binding constraints, and making the necessary substitutions.

(c) Find a menu of contracts (either as quantity-transfer pairs or quantity-information rent pairs) which solves the monopolist’s problem.

(d) Find the expected profit of the monopoly at the second-best menu of contracts. Identify each type of consumer’s information rent at these contracts.

(e) Suppose that the cost parameter increases from k = 1 to k = 2. How does this affect each type of consumer’s information rent? Do you think this is intuitive? Why or why not? Explain.

3. Suppose that a principal wishes to procure a public good x that affects the welfare of two individuals, i = 1, 2.  Suppose that both individuals have quasi-linear preferences over public good x e [0, 1] and  monetary transfers m  =  (m1 , m2 ).   Suppose  individual  1’s preferences are represented by the utility function v1 (x, m1 , θ 1 ) = θ 1x1/2 - x + m1 , while individual 2’s preferences are represented by v2 (x, m2 , θ2 ) = θ2x1/2  - x + m2 .  Suppose that individual 1’s realised type (i.e., his taste parameter θ1 ) is his private information, while it is common knowledge that θ 1 = 0 with probability 1/3 and θ 1   =  1 with probability 2/3.   On the other hand, it is common knowledge that θ2 =  for sure.

(a) Identify all possible states (i.e., type profiles) and find the first- best decision rule.

(b) Find the pivotal mechanism transfer rule and verify that budget balancedness is not satisfied.

(c) Find the expected externality transfer rule and show that budget balancedness is satisfied.

(d) Verify that the expected externality mechanism you found above is incentive compatible.

(e) Verify that the expected externality mechanism you found above is individually rational.

4. Suppose a risk-neutral landlord (i.e., the principal) wants to hire a risk-neutral tenant (i.e., the agent) to produce cherry.  The yield of cherry production is stochastic and it can be either q¯ =  100 tons or q = 25 tons.  When the agent does not exert effort, then with probability π0  = 1/5 the yield will be q¯ and with probability 4/5 it will be q .  When the agent exerts effort, then π 1  = 3/5 is the probability that the yield will be q¯ and 2/5 is the probability that the yield will be q .  Exerting effort is costly to the tenant and suppose this cost is equal to ψ = 1. Suppose the landlord receives a payoff from the production of cherry according to the payoff function S(q) = q1/2 .   Finally, suppose the contract consists of payments {(, t)} to the tenant, depending upon whether the yield is high or not.

(a) Suppose that the landlord can observe whether the tenant ex- erts effort or not.  Set the optimisation problem and find the optimal transfers when the landlord induces effort.  Show that the landlord prefers inducing effort over not inducing effort.

(b) Now suppose that the landlord cannot observe whether the ten-

ant exerts effort or not. Set the optimisation problem and find the optimal transfers when the landlord induces effort. What is the cost of inducing effort?

(c) Suppose again the landlord cannot observe whether the tenant exerts effort. Now suppose also that the landlord cannot impose a transfer less than -l, where l  =  1.   Set the optimisation problem and find the optimal transfers when the landlord induces effort. What is the expected limited liability rent of the tenant?

(d) Suppose once again the landlord cannot observe whether the tenant exerts effort. Now suppose also that the landlord cannot impose a transfer less than -l, where l = 0 this time. Set the optimisation problem and find the optimal transfers when the landlord induces effort.  What is the expected limited liability rent of the tenant?

(e) Suppose now that the landlord decides to make the tenant share- holder of the profits by using a simple linear sharing rule instead of paying him using transfers. Set the optimisation problem and find the optimal linear sharing rule.

5. Assume that a risk-averse entrepreneur (i.e., the agent) wants to start a project that requires an initial investment worth an amount L = 5. The entrepreneur has no cash of his own and thus must raise money from a risk-neutral lender (i.e., the principal).  The return on the project is random and equal to   =  12 with probability πe  and V = 2 with probability 1 - πe , where e e {0, 1} denotes the effort level of the entrepreneur.   In particular,  let π0   =  1/3 and π 1   =  2/3 and suppose that exerting effort is costly to the entrepreneur which is equal to ψ = 1.  Suppose the entrepreneur receives utility from any transfer t according to the function u(t) = t1/2 . Suppose the lender does not observe whether the entrepreneur exerts effort or not. Finally, suppose the financial contract consists of repayments {( , Z)} to the lender, depending upon whether the project is successful or not. (Note that u-1 (x) = x2 .)