Question

11.. Suppose we are given a specific cost function c(w, y) = yw1aW2-a How can we solve for its associated technology?

Answer 1

Shepherd lemma gives us the production function derived from cost function. Shepherd lemma states that the partial derivative of the cost function with respect to input prices is the firm's conditional input demand.

Thus,

$x_1 = \frac{\partial c(w,y)}{\partial w_1} = ya w_1^{a - 1} w_2^{ 1-a } = ya(\frac{w_2}{w_1})^{1-a}$

$x_2 = \frac{\partial c(w,y)}{\partial w_2} = ya w_1^{a} w_2^{ -a } = ya(\frac{w_2}{w_1})^{-a}$

Dividing both, we get:

$\frac{x_1}{x_2} = \frac{w_2}{w_1}$

Substituting this is demand for x2, we get:

$x_2 = ya(\frac{x_1}{x_2})^{-a} \Rightarrow y = \frac{1}{a}x_1^ax_2^{1-a}$

Hence the firm has Cobb Douglas technology. (and so constant returns to scale)