Many maximum problems have an associated dual minimum problem. An imperfectly co

Question

Many maximum problems have an associated dual minimum problem. An imperfectly competitive firm wishes to minimize the costs of producing a given level of output Qdegree. Output is produced according to the production function: Q = AK^1/2L^3/4. Determine mathematically the degree of homogeneity and returns to scale for the production function given above. Determine mathematically the amount of capital and labor that will minimize the costs of producing Qdegree. Provide second order conditions for the minimum. Show graphically. Solve mathematically for the dual maximization problem for the imperfectly competitive firm and demonstrate that the solutions from the two models are identical. Provide second order conditions for the maximum. Show graphically.

Explanation / Answer

Production function: Q = AK1/2L3/4

a) For determining return to sacle of the production function:

Let K = aK and L = aL where a is a positive integer.

thus new production function new Q = Q2 = Aa1/2K1/2a3/4L3/4 = Aa1.25K1/2L3/4 = a1.25Q

Here New Q is more than a*old Q

as a1.25Q > aQ

Thus the return to scale is increasing return to scale.

And thus the degree of homogeinity is 1.25

b) Fo the minimum cost of capital and labor lets differentiate the production function as follows:

dQ/dL = 3/4AK1/2L-1/4 and dQ/dK = 1/2AK-1/2L3/4

Let w be wage rate of Labor and r be rental rate of capital

thus at minimum cost dQ/dL / dQ/dK = w / r

thus 3/4AK1/2L-1/4 / 1/2AK-1/2L3/4 = w /r

3/2 K/L = w/r

L = 3/2Kr/w is the optimum allocation of labor

and K = 2/3Lw/r is the optimum alloction of capital

verifying through second order differentiation as follows:

d2Q/dL2 = -3/16AK1/2L-5/4 and d2Q/dK2 = -1/4AK-3/2L3/4

As the above second order differntials are negative in terms. Verified.

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