A restaurant owner has hired a manager to run the daily operations of her restau

Question

A restaurant owner has hired a manager to run the daily operations of her restaurant. Restaurant profits depend on both the manager’s unobservable effort level and random fluctuations in demand. The expected value of restaurant profits if the manager exerts effort, e, is given by e/2. Effort costs the manager 0.25e2. If the risk neutral manager’s salary is a percentage, c, of restaurant profits, how much effort should we expect the manager to exert? What is the optimal value of c from the owner’s perspective? How does the level of effort compare with the optimal level if the manager was also the owner of the restaurant?

Explanation / Answer

A restaurant owner has hired a manager to run the daily operations of her restaurant.

Restaurant profits depend on both the manager’s unobservable effort level and random fluctuations in demand.

i.e. p = f(e,d)

The expected value of restaurant profits if the manager exerts effort, e, is given by e/2.

i.e. p = e/2

The expected value of restaurant profits if the manager doe not exert effort is given by p = f(d), where d=demand fluctutation

Effort costs the manager : C(e) = 0.25e^2.

If the risk neutral manager’s salary is a percentage, c, of restaurant profits,he would exert effort to maximise his payoff:

i.e. he will max c*(e/2+f(d))/2 - C(e)

Differentiationg w.r.t. e, we get

c/4 - .5e = 0

which implies: c= 2e

the optimal value of c from the owner’s perspective will be, where his payoff is maximised

i.e. (1-c)*(e/2+f(d))/2

Differentiationg w.r.t. c, we get

(-)(e/2+f(d))/2=0

f(d) = -e/2

Solving the two equations simultaneously, would yield e=0

if the manager was also the owner of the restaurant, he would have maximised his payoff i.e. e/2 - 0.25e^2

Differentiationg w.r.t. e, we get

1/2 - 0.5e = 0

e = 1

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