Question

Suppose the production function of a firm is given by f (x1; x2) = min{x1, x2}

(a) Calculate the conditional demand functions of the firm assuming w1 = 2; w2 = 4, and y = 8

(b) Calculate the minimum cost of the firm to produce 8 units of the good when w1 = 2 and w2 = 4:

Answer 1

y = f(x1, x2) = min{x1, x2}

Total cost (C) = w1.x1 + w2.x2

(a) This is a fixed proporiton production function where cost is minimized when x1 = x2.

q = 8 = min{x1, x2}

When x1 = x2,

min{x1, x1} = 8

x1 = 8 [Specific conditional demand for x1]

x2 = x1 = 8 [Specific conditional demand for x2]

Substituting in generalized total cost function,

C = w1.x1 + w2.x1 = x1.(w1 + w2)

x1 = C / (w1 + w2) [Generalized Conditional demand for x1]

Again, from generalized total cost function,

C = w1.x2 + w2.x2

C = x2.(w1 + w2)

x2 = C / (w1 + w2) [Generalized Conditional demand for x1]

(b) With given values, Minimum cost of producing 8 units is

C = 2 x 8 + 4 x 8 = 16 + 32 = 48