A certain city has two kinds of workers, alphas and betas.An alpha can produce $100 worth of output per day working for himself.If he works in the local factory, he produces $120 worth of output a day.A beta produces $60 worth of output per day working for himself, and he produces $80 worth of output per day if he works for the local factory.Workers either work for themselves or work in the factory.The factory owner can't tell alphas from betas.He pays a wage equal to the average product of his labor force and he has at least some alphas working for him.Workers are free to choose to work for themselves or the factory, depending on which offers more money.
A)At least 5/6 of the factory's employees must be alphas.
B)At least half of the factory's employees must be betas.
C)At least half of the factory's employees must be alphas.
D)None of the factory's employees can be betas.
E)No more than 5/6 of the betas can work in the factory.

Question

Quizplus · Accepted Answer

Let's assume that the factory owner has $x to pay to his employees. Then, the average output of the labor force is also $x (since the number of workers and the total output are equal in the factory).

Let's consider two cases: If the factory owner hires alphas only, then the total output will be $120 for each alpha he hires. Hence, he can hire x/120 alphas. On the other hand, if the factory owner hires betas only, then the total output will be $80 for each beta he hires. Hence, he can hire x/80 betas.

If the factory owner wants to maximize his profit, he should find the optimal combination of alphas and betas. Let's assume that he hires y alphas and z betas. Then, we have two equations:

y + z = total number of workers (which is also equal to x/120 + x/80)

100y + 60z = x (since he pays the average output to his employees)

Simplifying these equations, we get:

3y + 2z = x/40

5y = x/2

Therefore, we have:

y = x/10 and z = 2x/15

This means that the factory owner should hire 2x/15 betas and 8x/15 alphas to maximize his profit.

Now, let's prove that at least half of the factory's employees must be alphas:

Assume the opposite, i.e., more than half of the factory's employees are betas. Then, we have:

z > y

2x/15 > 8x/30

2x/15 > 4x/15

x > 0

This contradicts the assumption that the factory owner has some alphas working for him (which is given in the problem statement). Therefore, at least half of the factory's employees must be alphas.

(Note that the answer choices A, B, and D are false, and the answer choice E is not necessary to consider.)

A Certain City Has Two Kinds of Workers, Alphas and Betas.An