Question

If we return to the haircut production function from Part C, question 5. ?(?, ?) = (50?^1/2)(?^1/2), where ? is capital and ? is labour. The marginal products are ??? = (50(1/2)?^1/2)(?^−1/2), and ??? = (50(1/2)?^−1/2)(?^1/2)

a. Assume that a haircut has the price, p, that the cost of purchasing labour is ?? and the cost of purchasing capital is ?.

We will also assume that in the short run that capital is held constant. What is the equation for a short run isoprofit line?

b. Suppose that the price of capital increases (and capital is held constant in the production function).

What happens to the firm’s profit?

What happens to the firm’s use of labour?

c. What is the condition for profit maximization in the short run?

d. Draw a graph showing the short run profit maximization with a few isoprofit lines and the production function.

e. Solve for the firm’s short run demand for labour with capital held fixed.

f. Solve for the firm’s short run supply of haircuts.

g. In the long run, when both inputs can vary, what is the optimal amount of each input?

h. What is the long run profit maximizing output, the number of haircuts in the long run?

Answer 1

Solution:

a. In short run, with fixed capital, say K*, so, the cost of hiring this amount of capital is rK*

With the cost of purchasing as wL, Total cost (TC) in short run becomes: wL + rK*

Total revenue, TR = p*Q

Iso-profit line is the line which has combinations of labor input and quantity/output which generate same level of profit along the entire line.

Profit, Z = total revenue - total cost

Z = pQ - wL - rK*

So, the above equation is for short run iso-profit line, generating $Z amount of profit. Rearranging the equation, we get:

Q = (Z + rK*)/p + (w/p)*L

b. Price of capital is r. If r increases, we ought to see that how firm's profit and labor use changes.

With Profit = pQ - wL - rK*

Taking the required partial derivative: = -K*, so as price of capital increases, profit decrease by $K*.

Remaining things equal, if r increases, with fixed K*, cost of purchasing capital has increased, thus in order to level the cost, labor used will decrease. Notice that this will also decrease the total revenue (since output depends on the amount of labor used), but decrease in cost will be more.

c. With production function: Q(K,L) = 50K*^1/2L^1/2, we can write

Profit, Z = pQ - wL - rK*

Solving the first order condition to maximize profit, by differentiating profit with respect to L and equating to 0: = 0

= p*(dQ/dL) - w = 0

p*MPL = w

Which is, p*(50)(1/2)K*^1/2L^-1/2 = w

e. Rearranging the above equation, we get short run demand of labor, L*

25p*K*^1/2/w = L^1/2

On squaring on both sides, we get

(25p/w)²*K* = L

f. With the above demand for labor, haircuts supplied:

Q* = 50K*^1/2((25p/w)²*K*)^1/2

Q* = (50*25p/w)K*

Q* = (1250p/w)K*

g. In the long run, when both inputs can vary, profit maximization occurs where: MPL/MPK = w/r

So, with MPL = 25(K/L)^1/2 and MPK = 25(L/K)^1/2

MPL/MPK = [25(K/L)^1/2]/[25(L/K)^1/2] = K/L

So, condition is: K/L = w/r or rK = wL

To solve further, we require a constraint, like the budget (that is cost which can be spent) or the quantity (which needs to be produced). But since for next part, we are asked the quantity, that is haircuts, we mainly require the former, that is constraint on cost that can be spent.