Q2. The manager of a national retailing outlet recently hired an economist to es
ID: 1103166 • Letter: Q
Question
Q2. The manager of a national retailing outlet recently hired an economist to estimate the firm's production function. Based on the economist's report, the manager now knows that the firm's production function is given by Q=K^(12) L^(12) and that capital is fixed at 1 unit.
a) Calculate the average product of labor when 9 units of labor are utilized.
b) Calculate the marginal product of labor when 9 units of labor are utilized.
c) Suppose the firm can hire labor at a wage of $10 per hour and output can be sold at a price of $100 per unit. Determine the profit-maximizing levels of labor and output.
d) What is the maximum price of capital at which the firm will still make nonnegative profits?
Explanation / Answer
Q=K^(12) L^(12)
Where K = 1, then Q = L^(12)
a. APL = Q/L = L^(12)/L =1/L^(12)
APL when (L = 9) = 1/9^0.5 = 1/3
b.
MPL = dQ/dL = (½) * 1/L^(12)
MPL when (L = 9) = 0.5*(1/9^0.5) = 0.5*1/3 = 0.5/3 = 5/30 = 1/6
c.
Let us define the profit function –
= P*Q – W*L, where Q = L^(12), P = $100, W = $10
Or, = 100*Q – 10*L
= 100* L^(12) – 10 *L
Let us write the first order condition –
d/dL = 100/2L^(12) – 10 = 0
10/2L^(12) – 1 = 0
5/L^(12) – 1 = 0
5/L^(12) = 1
L^(12) = 5
L = 25 and Q = 25^0.5 = 5 are profit-maximizing levels of labor and output.
d.
Firm’s profit, = 100* L^(12) – 10 *L = 100* 25^(12) – 10 *25 = 250
Note that the firm employs only 1 unit of capital, then the maximum price at which it will make non-negative profit will $250 per unit.
When price of capital = $250, firm’s profit = 250 – 250 = 0
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