Which Effort to Implement

Let $p_1 = 0.8$ , $p_0 = 0.2$ , $x_H = 2000$ , $x_L = 0$ , $\bar u = 20$ , $\alpha = 1/2$ , $c = 10$ .

Compute the principal's profit from implementing high effort and from settling for low effort, and determine which to choose.

\Pi_1 = 0.8 \times 2000 - C_1, \qquad \Pi_0 = 0.2 \times 2000 - \bar u^2

c = 10

\alpha = 1/2

— the current point lies in the "implement high effort" region.

Show proof

Implementing high effort gives expected output $0.8 \times 2000 = 1600$ , and the expected wage from the previous problem is $C_1 = 944.44$ . Hence $\Pi_1 = 1600 - 944.44 = 655.56$ .
Settling for low effort needs only the flat wage $\bar u^{1/\alpha} = 20^2 = 400$ , so $\Pi_0 = 0.2 \times 2000 - 400 = 0$ .
Since $\Pi_1 = 655.56 > 0 = \Pi_0$ , the principal implements high effort. The expected-output gain from effort, $(0.8-0.2) \times 2000 = 1200$ , comfortably exceeds the extra incentive cost $944.44 - 400 = 544.44$ .

With $\alpha = 1/2$ , find the threshold effort cost $c^*$ at which the principal is indifferent between implementing high effort and settling for low effort.

Use the signs at $c = 18$ and $c = 19$ to show $c^*$ lies between them.

\Pi_1(c) - \Pi_0 = 1200 - \left(C_1(c) - 400\right), \qquad c^* \approx 18.13

c = 18

\alpha = 1/2

— just left of the crossing, still in the "implement high effort" region.

Show proof

$\Pi_1(c) - \Pi_0 = 1200 - (C_1(c) - 400)$ . Since $C_1(c)$ is strictly increasing in $c$ (a higher effort cost widens the utility spread $u_H - u_L = c/0.6$ and raises the risk premium), the gap is strictly decreasing and the crossing is unique.
At $c = 18$ , $u_L = 14$ and $u_H = 44$ , so $C_1 = 0.8 \times 44^2 + 0.2 \times 14^2 = 1588$ and $\Pi_1 - \Pi_0 = 1600 - 1588 = 12 > 0$ .
At $c = 19$ , $u_L = 13.667$ and $u_H = 45.333$ , so $C_1 \approx 1681.44$ and $\Pi_1 - \Pi_0 = 1600 - 1681.44 = -81.44 < 0$ . The sign flips, so $c^* \in (18, 19)$ ; numerically $c^* \approx 18.13$ .

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