Ten workers work jointly on a project.All 10 workers are equally skilled.The total value of the output produced is $60 times the sum of the number of hours worked by all 10 workers.Each worker's utility is equal to his income minus the square of the number of hours he works.Each worker is selfish.The employers have no way of keeping track of any individual's work effort, so they decide to let each person work as long as he wants to and they divide the total value of the output equally among the workers.How much income will each worker get?
A)$1,800
B)$180
C)$30
D)$195
E)None of the above.

Question

Quizplus · Accepted Answer

Let $x$ be the number of hours each worker works. Then, the total number of hours worked is $10x$.
The total value of the output produced is $60(10x)=600x$.
Each worker's income is $\frac{600x}{10}=60x$.
Each worker's utility is $60x-x^2$.
To maximize utility, we must find the value of $x$ that maximizes $60x-x^2$.
Completing the square, we get $-(x-30)^2+900$. Since $-(x-30)^2$ is always non-positive, the maximum value of $60x-x^2$ is $900$ when $x=30$.
Thus, each worker's income is $60(30)=180$, which is choice B.

Ten Workers Work Jointly on a Project