Question

Once again, consider the Cobb-Douglas production function ? = ?? ?? ? . a) This time, derive the conditional input demands ? ∗ (?, ?, ?) and ? ∗ (?, ?, ?) and the associated long-run cost function ?(?, ?, ?) under the assumption that ? + ? = 1. b) Describe the average cost and marginal cost functions. How do they depend on output q and factor prices w and r? Explain. c) Continuing to assume ? + ? = 1, compare ?(?, ?, ?) and ?(?, ?,?̅, ?) from exercise 1.

Answer 1

a).

Consider the given problem here the production function is, => Q = A*L^a*K^b, where “a+b=1”.

=> Q = A*L^a*K^1-a, => MPL = A*K^(1-a)*a*L^a-1, => MPL = A*a*L^(a-1)*K^(1-a).

=> Q = A*L^a*K^1-a, => MPK = A*(1-a)*K^(1-a-1)*L^a, => MPK = A*(1-a)*K^(-a)*L^a.

=> MRTS = MPL / MPK = [A*a*L^(a-1)*K^(1-a)] / [A*(1-a)*K^(-a)*L^a].

=> MRTS = [a*L^(a-1)*K^(1-a)] / [(1-a)*K^(-a)*L^a] = (a/1-a)*[L^(a-1)*K^(1-a)] / [K^(-a)*L^a].

=> MRTS = (a/1-a)*(K/L).

Now, at the equilibrium MRTS must be equal to W/R.

=> MRTS = W/R, => (a/1-a)*(K/L) = W/R, => K = (W/R)*(1-a/a)*L….(1).

The production function is given by.

=> Q = A*L^a*K^1-a, => Q^(1/1-a) = A*L^(a/1-a)*K, => Q^(1/1-a) = A*L^(a/1-a)*[(W/R)*(1-a/a)*L].

=> Q^(1/1-a) = A*L^(1/1-a)*(W/R)*(1-a/a), => Q^(1/1-a) = A*(W/R)*(1-a/a)*L^(1/1-a).

=> Q = [A*(W/R)*(1-a/a)]^(1-a)*L, => L = [A*(W/R)*(1-a/a)]^(a-1)*Q.

=> L = A^(a-1)*(W/R)^(a-1)*(1-a/a)^(a-1)*Q.

Now, from the above condition (1), we have.

=> K = (W/R)*(1-a/a)*L = (W/R)*(1-a/a)*(A)^(a-1)*(W/R)^(a-1)*(1-a/a)^(a-1)*Q.

=> K = (A)^(a-1)*(W/R)^(a)*(1-a/a)^(a)*Q.

The above two equations are the conditional demand for labor and capital.

The LR cost function is given below.

=> C = W*L + R*K.

=> C = W*A^(a-1)*(W/R)^(a-1)*(1-a/a)^(a-1)*Q + R*A^(a-1)*(W/R)^a*(1-a/a)^a*Q.

=> C = W^a*R^(1-a)*A^(a-1)*(1-a/a)^(a-1)*Q + W^a*R^(1-a)*A^(a-1)*(1-a/a)^a*Q.

=> C(W, R, Q) = W^a*R^(1-a)*A^(a-1)*[(1-a/a)^(a-1) + (1-a/a)^a]*Q.

b).

The average and marginal cost functions are.

=> AC = C/Q = W^a*R^(1-a)*A^(a-1)*[(1-a/a)^(a-1) + (1-a/a)^a].

=> MC = dC/dQ = W^a*R^(1-a)*A^(a-1)*[(1-a/a)^(a-1) + (1-a/a)^a].

Here AC and MC are completely independent of “Q”, => as “Q” changes the AC and MC will not change.

Now, AC and MC are positively related to “W” and “R”, => dAC/dW, dAC/dR > 0, and dMC/dW, dMC/dR > 0, => as “W” or “R” increases the AC as well as MC both increases.

c).

Let’s assume the capital is fixed at “K bar”, => the production function.

=> Q = A*L^a*K^1-a, => (Q/A)*K^(a-1) = L^a, => L^a = (Q/A)*K^(a-1) , => L = (Q/A)^(1/a)*K^(a-1/a).

So, the associated cost function is.

=> C = W*L + R*K = R*K + W*(Q/A)^(1/a)*K^(a-1/a).

=> C(W, R, Q, K) = R*K + W*(Q/A)^(1/a)*K^(a-1/a).

=> MC = W*K^(a-1/a)*(1/A)^(1/a)*(1/a)*Q^(1/a-1).

If we compare C(W, R, Q) with C(W, R, Q, K), we got that for “Q=0”, => C(W, R, Q) = 0, but C(W, R, Q, K) is positive, => C(W, R, Q) < C(W, R, Q, K).

Now, the MC for C(W, R, Q) is constant and positive, on the other hand the MC for C(W, R, Q) is is positively related to “Q”, => as “Q” increases the MC also increases, => C(W, R, Q) < C(W, R, Q, K)..