Problem 1: Assume a firm owner (she) is an expected utility maximizer and has vNM utility function u(x) = ln(x) where x is final wealth. She has initial wealth w and receives a price p = 10 per unit of output y. There is uncertainty about the production cost per unit of output: with probability 1/2 the unit cost is 6 and with probability 1/2 the unit cost is 12. Fixed costs are 0.
(a) How much output y does the firm owner choose to produce?
(b) How does the optimal production, determined in a), change if the initial wealth level increases? Relate this to the firm owner's risk attitude.
(c) Given the price, costs and probabilities as specified above, the firm owner is indifferent between scenario A and scenario B:
A: having initial wealth w = 80 and producing y = 10. B: having initial wealth w0 and producing y0 = 0.
What is the value of w0? Round your result to two decimal places.
Is w' larger or smaller than the expected final wealth in scenario A? Relate the answer to this question to the firm owner's risk attitude.
(d) Now suppose the firm owner has utility function u(x)= x2 instead of u(x)= ln(x). Price, costs and probabilities remain the same.
For the firm owner to be indifferent between scenario A and B, as specified in (c), what value does w0 have to take now? Round your result to two decimal places.
How does w0 now compare to the expected final wealth in scenario A? Relate the answer to the risk attitude implied by u(x)= x2.
Problem 2: A worker (he) produces a product for a principal (she) and receives a wage in return. The principal's revenue for the sale of the product is 10 if the product has no defect, otherwise her revenue is 0. The worker can choose between two different effort levels eH and eL. If he chooses eH, the product has a defect with probability 1 , if he chooses eL, the product has a defect with probability 3 . The defect is observable and hence the principal can condition the wage on whether the product has a defect or not. The principal and the worker both have an expected utility function. The principal is risk neutral and wants to maximize her expected profit. The worker is risk averse and his von Neumann-Morgenstern utility function over wage income w and effort e is given by
U (w, e) = 80 - 2/w - v(e)
for w ≥ 0 and e ∈ { eH, eL} where the cost of effort, v(e), is given by v(eH)= 50 and v(eL)= 0. The worker's reservation utility is U = 0. Let w1 be the wage paid in case there is no defect and let w2 be the wage paid in case of a defect. In the following analysis, you can restrict to strictly positive wages.
(a) Suppose effort is observable and verifiable.
i. What is the optimal wage schedule if the principal requires eL from the worker? State and solve the principal's optimization problem.
ii. What is the optimal wage schedule if the principal requires eH from the worker? State and solve the principal's optimization problem.
iii. Which effort level will the principal require from the worker?
(b) Now suppose effort is unobservable.
i. What is the optimal wage schedule if the principal wants the worker to choose eL? Explain.
ii. What is the optimal wage schedule if the principal wants the worker to choose eH? State and solve the principal's optimization problem via the Kuhn-Tucker method.
iii. Which one of the two wage schedules does the principal offer to the worker?
Problem 3: Ann wants to order pies from the local caterer (he). The caterer can be an efficient caterer with a costs of 5 per pie or inefficient caterer with a cost of 10 per pie. The utility of the caterer from a contract (n, p) to deliver n pies in exchange for a total payment of p then is
if he is efficient, and
Ue(n, p)= p - 5n
if he is inefficient.
Ui(n, p)= p - 10n
If the caterer delivers to Ann, then he cannot deliver to Mr. Smith and has to forgo 60 utility units. Thus the caterer's reservation utility is 60.
Ann's utility from ordering n pies for a total payment of p is Π(n, p)= 80√n - p.
(a) Suppose Ann can observe the caterer's type. How many pies does Ann order and for which total payment if 1) the caterer is efficient and 2) the caterer is inefficient?
For each case, state and solve Ann's optimization problem.
(b) Now suppose that Ann cannot observe the caterer's type but she knows that the caterer is efficient with probability 2/3.
i. Seeing the two contracts from a), the caterer assures Ann that he is inefficient. Should Ann believe the caterer?
ii. Ann decides to recalculate the contracts. Which contract menu does Ann offer to the caterer if Ann wants to contract with the caterer independent of his type? State and solve Ann's optimization problem via the Kuhn-Tucker method.
iii. Which (single) contract does Ann offer if she only wants to contract with the caterer if he is efficient? Explain.
iv. Is it better for Ann to contract with the caterer independent of his type or to only contract with the caterer if he is efficient?
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