Question

consider an industry that uses capital, K and labor, L to produce output, X, according to a Cobb-Douglas production function: x=K L where 0 <X<1 is the share parameter for capital and 0<B,1 is the share parameter for labor. Denote the rental price of capital by r and the wage by w.

Determine the capital to labor ratio (K/L) wh9ich minimizes the cost of producing a fixed amount of output, X. Under what conditions does optimal ratio of capital to labor depend onX? Explain.

Suppose B=1-x. Compute the firm's marginal cost of production.

Answer 1

Solution:

Given a Cobb-Douglas production function: X = K^A*L^B; 0 < A, B <1

Rental price of capital = r, wage rate = w

Finding the cost minimizing capital to labor ratio:

With given rental prices of capital and labor, the cost function for this industry, TC = w*L + r*K

The cost is minimized at the input combination (L*, K*) where this cost function is tangent to the production function. This further means that slope of the two should equate at the optimum.

Slope of the cost function = w/r (fixed since cost curve is a straight line in input mix plane)

Slope of production function is also referred to as the marginal rate of technical substitution, MRTS

MRTS = Marginal product of labor (MPL)/Marginal product of capital (MPK)

MPL = $\partial X/\partial L$ = B*K^A*L^B-1

MPK = $\partial X/\partial K$ = A*K^A-1*L^B

Thus, MRTS = (B*K^A*L^B-1)/(A*K^A-1*L^B) = (B/A)*(K/L)

So, at the optimum, MRTS = slope of cost function

(B/A)*(K*/L*) = w/r

K*/L* = (A/B)*(w/r)

By substitution, we find the minimum total cost of producing X units of output.

With B = 1 - A, this ratio becomes: K*/L* = (A/(1-A))*(w/r)

Or K* = (A/(1-A))*(w/r)*L*

Substituting this in the production function, we get:

X = [(A/(1-A))*(w/r)*L]^A*L^1-A = [(A/(1-A))*(w/r)]^A*L

So, L = X*[(A/(1-A))*(w/r)]^-A and thus, K = (A/(1-A))*(w/r)*(X*[(A/(1-A))*(w/r)]^-A) = X*[(A/(1-A))*(w/r)]^1-A

So, Total cost as a function of output X, TC(X) = w*X*[(A/(1-A))*(w/r)]^-A + r*X*[(A/(1-A))*(w/r)]^1-A

TC(X) = X*[(A/(1-A))*(w/r)]^-A [w + r*(A/(1-A))*(w/r)]

TC(X) = X*[(A/(1-A))*(w/r)]^-A*(w/(1-A))

TC(X) = X*(w/(1-A))^1-A*(r/A)^A

So, marginal cost of production, MC(X) = $\partial TC(X)/\partial X$ = (w/(1-A))^1-A*(r/A)^A