Suppose a firm has a production function given by Q = 2Ö2LÖK.If the rental rate of capital is $100 per unit,the wage rate is $1,400 per week and the firm initially has 25 units of capital,what is the firm's short-run cost function? A) C(Q) = 2500 + 7Q² B) C(L) = 2500 + 1400L C) C(Q) = 0.25 + 7Q² D) C(L) = 0.25 + 1400L

Question

Quizplus · Accepted Answer

In short-run, we only consider the variable cost, which is the cost of labor. 
The firm's production function is Q = 2√(2LK)
Let's rearrange this to get labor demand: L = Q^2 / (8K)
The cost of labor is given by the wage rate multiplied by the amount of labor, so: C(L) = 1400L
Substitute the labor demand equation into the cost of labor to get: 
C(L) = 1400(Q^2 / (8K))
Simplify to get: C(Q) = 0.25Q^2 / K + 175Q
However, we need to find the cost function in terms of capital, not output. 
We know the rental cost of capital is $100 per unit and the firm has 25 units of capital initially, so the cost of capital is: 
C(K) = 100 * 25 = $2500
Therefore, the short-run cost function is: C(Q) = 0.25Q^2 / K + 175Q + $2500
which corresponds to choice B.

Suppose a Firm Has a Production Function Given by Q