A firm's revenue R is stochastically related to the effort exerted by its employee. Effort is a continuous variable. The employee can choose any level of effort e E [0, ). The choice of effort affects revenue so that: E(R|e) = e and Var(R|e) = 1 where E(R|e) and V ar(R|e) denote the expected value and variance, respectively, of rev- enue when the employee exerts effort level e. The employer cannot observe the level of effort exerted by the employee. The employer wants to design a wage contract w based on the revenue and considers only contracts of the form: w=α+βR, and so the employee is guaranteed a payment a and then a bonus payment ßR which de- pends on revenue. The employee is a risk-averse expected utility maximiser. A contract w gives expected utility: Eu(w\e) = E(w\e) – eV ar(w\e) – c(e) %3D where E(wle) and Var(w|e) denote the expected value and variance of the contract, re- spectively, conditional on effort e, p is a parameter of risk aversion, and c(e) denotes the disutility of effort. For this employee, c(e) = ;e². If the employee rejects the contract, they receive reservation utility of zero. Explain how the employer can implement a level of effort ē. Show that the optimal contract has the property that the bonus payment decreases with the level of risk aversion. Note that for a random variable X and constants a and b we have ,Var(aX) = a²V ar(X), V ar(X + b) = V ar(X), E(aX) = aE(X) and E(X + b) = E(X) + b.
A firm's revenue R is stochastically related to the effort exerted by its employee. Effort is a continuous variable. The employee can choose any level of effort e E [0, ). The choice of effort affects revenue so that: E(R|e) = e and Var(R|e) = 1 where E(R|e) and V ar(R|e) denote the expected value and variance, respectively, of rev- enue when the employee exerts effort level e. The employer cannot observe the level of effort exerted by the employee. The employer wants to design a wage contract w based on the revenue and considers only contracts of the form: w=α+βR, and so the employee is guaranteed a payment a and then a bonus payment ßR which de- pends on revenue. The employee is a risk-averse expected utility maximiser. A contract w gives expected utility: Eu(w\e) = E(w\e) – eV ar(w\e) – c(e) %3D where E(wle) and Var(w|e) denote the expected value and variance of the contract, re- spectively, conditional on effort e, p is a parameter of risk aversion, and c(e) denotes the disutility of effort. For this employee, c(e) = ;e². If the employee rejects the contract, they receive reservation utility of zero. Explain how the employer can implement a level of effort ē. Show that the optimal contract has the property that the bonus payment decreases with the level of risk aversion. Note that for a random variable X and constants a and b we have ,Var(aX) = a²V ar(X), V ar(X + b) = V ar(X), E(aX) = aE(X) and E(X + b) = E(X) + b.
Chapter2: Mathematics For Microeconomics
Section: Chapter Questions
Problem 2.15P
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