Answer to Question #161296 in Microeconomics for Merki Kaleb

Question

Suppose that the firm has two possible activities to produce output. Activity a uses a1 units of good 1 and a₂ units of good 2 to produce 1 unit of output. Activity b uses b_l units of good 1 and b2 units of good2 to produce 1 unit of output. Factors can only be used in these fixed proportions. If the factor prices are (w_l, w₂), what are the demands for the two factors? What is the cost function for this technology? For what factor prices is the cost function not differentiable?

Answer 1

$$Solution$$

Here we derive the cost function with the help of the Cobb-Douglas production functions. Say, given $\alpha+\beta=1$ corresponds to constant returns to scale, $\alpha+\beta <1$ , showing decreasing returns and $\alpha+\beta>1$, corresponds to increasing returns.

Now, suppose that $c(w,q)$ is the cost function, then

$a_1(w_1,w_2,q)=q^\frac{1}{(\alpha+\beta)}(\frac{\alpha\ w_2}{\beta\ w_1})\\ a_2(w_1,w_2,q)=q^\frac{1}{\alpha+\beta}(\frac{\beta\ w_1}{\alpha\ w_2})\frac{\alpha}{(\alpha+\beta)}\\ c(w_1,w_2,q)=q\frac{1}{(\alpha+\beta)}[\frac{a}{\beta})\frac{\alpha}{(\alpha+\beta)}+(\frac{\alpha}{\beta})-\frac{\alpha}{\alpha+\beta}]w_1\frac{\alpha}{(\alpha+\beta)}w_2\frac{\beta}{\alpha+\beta}$ is thus the cost function for this technology.

While $\theta\varnothing(w_1,w_2)$ is the factor price for which the cost function is differentiable.