Question

A firm’s production function is given by q = 40 ln(EW + EB + 1) where EW and EB are the number of whites and blacks employed by the firm, respectively. From this it can be shown that the marginal product of labor is MPE = 40 / (EW + EB + 1). Suppose the market wage for blacks is $50, the market wage for whites is $100, and the price of each unit of output is $20.

(a) How many workers of each race would a non-discriminating firm hire? How much profit is earned if there are no other costs?

(b) How many workers of each race would a firm with a discrimination coefficient of 0.6 against black workers hire? How much profit is earned if there are no other costs?

(c) How many workers of each race would a firm with a discrimination coefficient of 1.2 against black workers hire? How much profit is earned if there are no other costs?

Answer 1

a) The number of workers a non discriminating firm will hire will be uptil the marginal revenue product of the labor will be equal to their wage (i.e. The marginal revenue earned by that worker will be equal to their wage for the profit to be maximum, if the marginal revenue product was higher. the firm will keep increasing the number of that workers till it becomes equal to the wage)

Marginal product of labor (Additional number of units per labor)= MPE = 40/(EW+EB+1)
Marginal revenue product of labor = price* MPE = 800/(EW+EB+1)

Now for the black workers, this is equal to their wage, i.e.

50 = 800/(EW+EB+1)

EW+EB+1 = 16 , we have EB= 15-EW

Now for White workers, we have

100= 800/(EW+EB+1)

EW+EB+1= 8

EW= 7-EB

We cannot find EW and EB from these set of equations,
But since the EB and EW appear in addition, we can say that the level of production is only dependent on total number of workers, and a non discriminating firm will only hire the one with the lower wage i.e. It will only hire Black workers in this case since it will always have the same Marginal revenue product at each number of total workers and also a lower wage

With that in mind, we have

50 = 800/(EB+1) (from the first equation for blacks)

EB+1 = 16 , we have EB= 15
Thus the firm wil hire 15 black workers

The total quantity produced will be q= 40ln(16)= 110.9035 and the total revenue will be 20q= 2218.071

Total wages = 50*15= 750
Therefore the profits earned if there are no other costs= 2218.071-750= 1468.071

b) If the discrimination coefficient is d, then a firm with discrimination coefficient d against black workers will act like paying them a wage of wb will cost them wb(1+d) (For example it will still be paying the blacks 50$ wage in this case, but it will hire less as it will act as if the cost they are having to bear by hiring black workers is more than the wage that is 50(1+0.6)= 80$)

Since it is still less than the wage for whites which is 100$, the firm will again hire only black workers as it still costs them less and has the same marginal revenue product as the white workers.

80 = 800/(EB+1)
EB=9

Therefore EB=9

Thus the firm will only hire 9 black workers.

Total revenue= pq = 20q = 20*40ln(9)= 1757.7797

Total wages = 50*9=450

Total profit (In money terms) = 1757.7797-450=1307.7797.

c) Similarly, if the discrimination coefficient is now 1.2, the firm will act as if paying a black worker 50$ is costing them 50*(1+1.2) = 110$ for every worker

Thus it will now only hire white worker for 100$ as that would apparently cost less than hiring a black worker for 50$
It will hire white workers until their wage equals their Marginal revenue product i.e.
100 = 800/(EW+1)

EW+1= 8

EW=7

Thus the firm will hire 7 white workers

Total quantity produced = 40*ln(8)=83.177

Total revenue= 20*83.177=1663.5532

Total wages= 50*7=350

So total profit = 1313.5532

Hope it helps. Do ask for any clarifications if required.