Question

Suppose that Jennifer produces good y by using input xi and x2. The production function which Jennifer faces is: y = x} + x3 The cost for every unit of xı is wi and the cost for every unit of x2 is w2. There is a fixed cost component F, which also forms a part of her total cost. (a) Write down the cost minimization problem. Solve this problem and express X1/X2 as a function of w2/w1.

Answer 1

(a)

Let output price be p. Then

Revenue (R) = py = p.(x1^1/2 + x2^1/2)

Cost (C) = F + w1.x1 + w2.x2

Cost minimization problem is:

Minimize C = F + w1.x1 + w2.x2

Subject to: R = p.(x1^1/2 + x2^1/2)

w1 > 0, w2 > 0, x1 > 0, x2 > 0, F > 0

Cost is minimized when MP1/MP2 = w1/w2

MP1 = $\dpi{80} \partial$ y/$\dpi{80} \partial$x1 = (1/2) / (x1^1/2)

MP2 = $\dpi{80} \partial$ y/$\dpi{80} \partial$x2 = (1/2) / (x2^1/2)

MP1/MP2 = [(1/2) / (x1^1/2)] / [(1/2) / (x2^1/2)] = (x2/x1)^1/2 = w1/w2

Squaring,

x2/x1 = (w1/w2)²

So,

x1/x2 = (w2/w1)²