Question

A firm’s short-run revenue is given by R= 10e- e2, where eis the level of effort by a typical worker(all workers are assumed to be identical). A worker chooses his level of effort to maximize wage less effortw- e(the per-unit cost of effort is assumed to be1). Determine the level of effort and the level of profit(revenue less wage paid) for each of the following wage arrangements. Explain why these different principal-agent relationships generate different outcomes.

a. w= 2 for; otherwise, w= 0.

b. w= R/2.

c. w= R- 12.5.

Short answer

1. There is no incentive for the worker to provide efforts beyond 1. The profits will be $7 in this case. Here, the agent does not have any incentive to put in the extra effort.
2. The wage and revenue of the firm, in this case, will be equal. Therefore, agents here are putting extra effort since wages reflect the revenue of the firm.
3. Again, in this case, we will see that the worker's wages match directly with revenue, depicting efforts on the agent's part for the principal.

Step-by-step solution

Explanation of part (a)

In the case of $w=2,e⩾1,$otherwise w = 0. There will be no incentive for the worker to put an effort beyond 1, as the wage received by the worker will be 2 if the worker provides one unit of effort. However, the wage will not increase if the worker provides any extra effort.

The profit for the firm will be equal to revenue - wages paid to the worker;

$\text{profits = (10)(1) - 12 - 2 = \$ 7.}$

The principal-agent problem will have this outcome since workers do not have much incentive to increase effort to earn more.

Explanation of part (b)

In this case,$w=\frac{R}{2}$. The worker will maximize the wages where the efforts put into obtaining those wages are equal. That is, the worker will maximize wage at,

$$\begin{gathered} \text{w - e =}\frac{{\text{10e -e}}^{\text{2}}}{\text{2}}\text{a} \\ {\text{= 4e - 0.5e}}^{\text{2}} \end{gathered}$$

To find the efforts that the workers are willing to put, take the derivatives concerning the efforts and make it equal to zero. Therefore:

$$\begin{gathered} \frac{\partial 4e-0.5e^2}{\partial 2}=4-e=0 \\ e=4 \end{gathered}$$

The efforts that workers are willing to put in is 4. Therefore, the wages received by workers at this effort will be equal to:

$$\begin{gathered} \text{w =}\frac{\text{R}}{\text{2}} \\ \text{=}\frac{\text{10}\left(\text{4}\right){\text{-4}}^{\text{2}}}{\text{2}} \\ \text{= 12} \end{gathered}$$

The revenue made by the firm will be:

$$\begin{gathered} \text{profits = 10}\left(\text{4}\right){\text{-4}}^{\text{2}}\text{- 12} \\ \text{= 12} \end{gathered}$$

The principal and agent, in this case, will reflect that the agent is putting extra effort since the revenue of the firm reflects the efforts put by the worker.

Explanation of part (c) 

In the case of $w=R-12.5$, the worker again will maximize wages at the efforts that are required to obtain those wages.

$$\begin{gathered} \text{w - e = (10e - e2 ) - 12.50 - e} \\ \text{= 9e - e2 - 12.50} \end{gathered}$$

For finding efforts, take derivates with respect to e and make them equal to zero.

$$\begin{gathered} \frac{\partial \left(9e-e^2-12.50\right)}{\partial 2}=9-e^2=0 \\ e=4.5 \end{gathered}$$

The efforts put by the workers will be equal to 4.5.

The wages at this effort will be:

$$\begin{gathered} w=R-12.50 \\ =\left(\left(10\right)\left(4.5\right)-4.52\right)-12.50 \\ =12.5 \end{gathered}$$

The revenue of the firm will be:

$$\begin{gathered} profits=\left(\left(10\right)\left(4.5\right)-4.52\right)-12.25 \\ =\$12.50 \end{gathered}$$

The agent and principal relationship here also reflects that the more the agent's efforts, the more revenue the principal will make.