Question

2 .Suppose the production function of a firm is given by f (x1, x2) = 2x1 + 4x2

(a) Calculate the conditional demand functions of the firm assuming w1 = 2; w2 = 3, and y = 8

(b) Calculate the minimum cost of the firm to produce 8 units of the good when w1 = 2 and w2 = 3

Answer 1

y = f(x1, x2) = 2x1 + 4x2

Total cost = w1.x1 + w2.x2 = 2x1 + 3x2

(a) A linear production function means x1 and x2 are substitutes and isoquant is linear, touching both axes. Optimal solution lies on one of the corner points, signifying that either x1 or x2 are consumed.

y = 8 = 2x1 + 4x2

When x1 = 0, x2 = 8/4 = 2 and Total cost = 2 x 0 + 3 x 2 = 6

When x2 = 0, x1 = 8/2 = 4 and Total cost = 4 x 2 + 0 x 3 = 8

Since total cost is lower when x1= 0 and x2 = 2, this is the optimal bundle.

With given data,

Conditional demand for x1 = 0

Conditional demand for x2 = 2

(b) With given values and x1 = 0 & x2 = 2, total cost for 8 units is

Cost = 2 x 0 + 3 x 2 = 0 + 6 = 6