Suppose that you are in the short-run and the production function for your firm

Question

Suppose that you are in the short-run and the production function for your firm is given as:

F(L,K)=K1/2L1/2

Also, in the short-run capital is fixed at K=16. You also know that the wage rate for workers is w=10, the price of capital r=20 and that the price for the product is P=100.

A) How many units of Labor should this firm hire in order to maximize profits/minimize costs in the short-run?

B) How many units of output does the firm produce in the short-run?

C) How much profits does this firm earn in the short-run?

Explanation / Answer

It is given that the short run Cobb Douglas Production function is:

F(L,K) = L1/2.K1/2 where L: Labour, K:Capital

Since it is the short run, capital is fixed at K=16. Additionally, price of the product if fixed at P=100

If w=10 is the wage rate of labour, and r=20 is the rental rate of capital then the total cost (TC) is:

TC = Fixed Cost (FC) + Variable Cost (VC) where----------1

FC = r*K because capital is fixed = 20*16 = 320 and VC = wL = 10L.

Since Q(output) = L1/2.K1/2,

L1/2 = Q/ K1/2

Or L= Q2/K

Or L = Q2/16---------2

Any Profit Function is defined as the difference between Total Revenue (TR) and Total Cost (TC) where TR = price * output.

Here, TR = P*Q or TR = 100Q.

Therefore, Profit = 100Y-FC-VC (From equation 1, replacing TC with FC and VC)

Or Profit = 100Q-320-10L-------3

Now, from equation 1, L = Q2/16

Therefore, Profit = 100Q-320-10(Q2/16)

Now, the first order condition (f.o.c) for profit maximization requires that the first order derivative is equal to zero , and the second order condition (s.o.c) for profit maximization requires that the second order derivative is negative. Thus,

From this example, f.o.c = 100-2*10Q/16 = 0

Or Q= 100*16/2*10

Or Q = 80.

Therefore L = Q2/16 = 400.

TC = 320 + 10*L = 320+4000 = 4320.

TR= 100*Q = 100*80=8000

Therefore Profit= 8000-4320= 3680.

A) Therefore, at the profit maximizing level, 400 units of labour are employed.

B) At the profit maximizing level, the firm produces Q=80 units of output.

C) The profit earned in short run at the profit maximizing level = 3680

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