A firm is jointly owned by Juan and Roda. The firm’s production function requires two inputs: effort by Juan, denoted by x, and effort by Roda, denoted by y. Effort is only observable by the person who exerts it. The cost to Juan of a unit of his effort is c j = 2 and the cost to Roda for a unit of her effort is cr = 2. The price received for the goods is p = 2. The production of the firm is given by Q = 10(ln(x + 1) + ln(y + 1)). Assume that both Juan and Roda are risk-neutral rational agents. a) What are the socially optimal amounts of effort x* and y*? What is the total surplus in that case? (Hint: Solve the problem of a social planner that cares equally for Juan and Roda.) b) Suppose that Juan and Roda have a contract that specifies that Juan pays a fixed amount w = 15 to Roda and that Juan gets to keep and sell all the output. What is the total surplus now? How much of that surplus goes to Juan? To Roda? c) Now suppose that the contract between Juan and Roda specifies that the total output is split evenly between them. What are the levels of x and y that Juan and Roda will choose in this case? Does this contract lead to the socially efficient outcome? d) Can you propose another contract where the total production is the same as in the socially optimal case? Briefly explain what this contract should look like or explain why such a contract is impossible.
A firm is jointly owned by Juan and Roda. The firm’s production function requires two inputs: effort by Juan, denoted by x, and effort by Roda, denoted by y. Effort is only observable by the person who exerts it. The cost to Juan of a unit of his effort is c j = 2 and the cost to Roda for a unit of her effort is cr = 2. The price received for the goods is p = 2. The production of the firm is given by Q = 10(ln(x + 1) + ln(y + 1)). Assume that both Juan and Roda are risk-neutral rational agents.
a) What are the socially optimal amounts of effort x* and y*? What is the total surplus in that case? (Hint: Solve the problem of a social planner that cares equally for Juan and Roda.)
b) Suppose that Juan and Roda have a contract that specifies that Juan pays a fixed amount w = 15 to Roda and that Juan gets to keep and sell all the output. What is the total surplus now? How much of that surplus goes to Juan? To Roda?
c) Now suppose that the contract between Juan and Roda specifies that the total output is split evenly between them. What are the levels of x and y that Juan and Roda will choose in this case? Does this contract lead to the socially efficient outcome?
d) Can you propose another contract where the total production is the same as in the socially optimal case? Briefly explain what this contract should look like or explain why such a contract is impossible.
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