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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Transcribed Image Text: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, 0). The choice of effort affects revenue so that:
E(R|e) = e
and Var(R|e) = 1
where E(R|e) and Var(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 BR which de-
pends on revenue. The employee is a risk-averse expected utility maximiser. A contract w
gives expected utility:
1
Eu(w\e) = E(w\e) - jeV ar(w\e) – c(e)
pV.
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) =
e?. 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),
Var(X + b) = Var(X), E(aX) = aE(X) and E(X + b) = E(X)+ b.
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