Question

Consider a firm using quantities L1 and L2 of two kinds of labour as its only inputs in order to produce output Q=L1+L2. Thus, each unit of labour produces one unit of

output. Suppose that we also have two segmented labor markets, with the following inverse labor supply functions.

w1=α1+β1L1
w2=α2+β2L2

which shows the wage that must be paid to attract a given labor supply. Assume that the firm is competitive and take price of output P as given. (α1, α2, β1, β2 are positive.)

(a) Write the profit function π(L1,L2).

(b) Find the profit-maximizing (L1^*,L2^*).

(c) Find the maximized profit as a function of (P, α1, α2, β1, β2)

Answer 1

The production function is . The cost function in this case would be . The firm is competitive, and takes the price P as given. Hence, the revenue of the firm will be .

(a) The profit function can be founded as since profit is equal to revenue minus cost, we have as the profit function. Further, putting the values of C and Q, we have or . Putting the value of the inverse supply functions, we have or or .

(b) The profit maximizing inputs can be derived by maximizing the profit function. The FOC's are as below.

or , and for , we have or .

or , and for , we have or .

The SOC's are as below:

As , we have , ie , and as , we have , ie . Also, for , we have , and hence or or . Hence, there is a maximum at and .

(c) The maximized profit function, for , is hence or or or or .