Question

Problem # The agent selects his effort e from the interval [0,1]. Two outcomes are possible for the principal - the good one and the bad one; Pr(good) e. The principal's revenue is: R(good) 4, R(bad) 3; while the agent's utility is given by U(w,e) w e2. The 'outside option' is Uo 1. Propose the optimal remuneration scheme w for the case when: 5 a) e is observable b) e is not observable c) Repeat point a) with the utility function given by U(w,e)-e (use a calculator or provide an approximated solution) d) Repeat point b) with the utility function given by U(w,e) - w- e (do not solve for numbers, describe the principal's optimization problem and the interval within which the optimal effort level will be found)

Answer 1

1. When e is observable:

By e is observable would mean, that economic profit or benefit would be counted. That means, we will be more concerned about the Revenue when it’s good.

If Pr(good) = e, Then Pr(bad) would be = 1-e.

Since the event has only two outcomes, so Pr(good)=e=0.5

The principal’s revenue at R(good) = 4

At interval 0 the utility function for agent will be U₀= 1

At interval 1 the agent Utility function at good outcome would be stated as below:

U(w,e) = w-e²

U(w,e)= w-0.5²

U(w,e)= w-0.25

Maximum revenue earned by Principal at R(good) level = 4

Optimum level of remuneration would be 4-0.25 = 3.75 (because Principal’s payment for remuneration to agent would be his revenue less the agent’s utility)

2. When e is not observable:

By e is not observable would mean, that economic profit and loss both would be counted. That means, we need to ascertain the optimal solution after taking into consideration both the good and the bad probabilities.

We have already calculated the remuneration for agent at good level in a) i.e. 3.75.

Let’s calculate when revenue is bad.

Since the outcome for bad is equal. The Utility function for agent at bad level would be same.

U(w,e)= w-0.25

Maximum revenue earned by Principal at R(bad) level = 3

Optimum level of remuneration would be 3-0.25 = 2.75 (because Principal’s payment for remuneration to agent would be his revenue less the agent’s utility)

So, when e is not observable, there are two optimum remunerations for agent.

At Good level = 3.75

At Bad level = 2.75

3. Repeating point a utility function U(w,e) = w^1/2 – e²

U(w,e) = (3.75)^1/2- 0.5²

U(w,e) = 1.93 – 0.25

U(w,e) = 1.68

4. Repeating point b)

At interval 0, the agent’s utility is U₀ = 1.

So, the agent’s optimum remuneration wouldn’t be anything, because Principal doesn’t earn any revenue at agent’s no effort.

At interval 1, where there will be two outcomes good and bad. The probabilities of these two outcomes are equal. That means the agent’s utility function will remain same, but the optimum remuneration to agent will differ in the two circumstances.

At Good level of business the agent’s optimum revenue = w^1/2- e²

= (4)^1/2 – e²

At Bad level of business the agent’s optimum revenue = (3)^1/2- e²

The optimum solution would be achieved only at interval 1, because Principal doesn’t earn any revenue at interval 0. He will be at loss , if agent is paid at no effort.