Principal-Agent Problems and Designing Contracts Under Asymmetric Information: Practice Problems and Solutions

Mathematical Economics Problems and Solutions - Section 12

See Mr. Stolyarov's complete list of Mathematical Economics Problems and Solutions.

Note: Here, I will present solve problems typical of those offered in a mathematical economics or advanced microeconomics course. Some of the problems were authored by Dr. Charles N. Steele and are reprinted with his generous permission. The solutions to the problems are my own work and not necessarily the only way to solve the problems.

Problem 1. Ludwig hires Frederic to work on a project that will yield $700 in revenue if it succeeds and $100 in revenue if it fails. Frederic's opportunity cost of working without substantial effort is $160. His additional cost of working hard is $50. If Frederic works without substantial effort, the probability that the project will succeed is 0.3. If he works hard, the probability that the project will succeed is 0.6. Design a contract whereby Ludwig will pay Frederic so that Frederic has an incentive to work hard.

Solution 1. Let x be the payment to Frederic in the event of the project's success.

Let y be the payment to Frederic in the event of the project's failure.

Let p = 0.6 be the probability of success if Frederic works hard.

Let q = 0.3 be the probability of success if Frederic does not work hard.

Let w = 160 be Frederic's base opportunity cost.

Let e = 50 be Frederic's additional cost of working hard.

Frederic will work hard if

(i): (p-q)(x-y) ≥ e

(ii): y + p(x-y) ≥ w + e

(i) means that the expected gain to the manager from the extra effort is at least as high as the cost to him of the extra effort he expends when working hard.

(ii) means that the expected earnings of the manager (y + p(x-y)) are greater than his opportunity cost w + e.

We set (p-q)(x-y) = e: (p-q)(x-y) = 0.3(x - y) = 50, so x - y = 166.66666667

We set y + p(x-y) = w + e. We substitute 166.66666667 for (x - y). Thus,

y + 0.6*166.66666667 = 160 + 50 and y = 110. Therefore, x = 110 + 166.66666667 = x = 276.6666666667.

So the optimal contract will be as follows.

Contract: Ludwig pays Frederic 276.6666666667 if the project succeeds and 110 if the project fails.