Question

5. Let the firm's production function be given by y 1+2. Note that the inputs r1 and 2 are perfect substitutes in this production process. Suppose wi 2 and w2 1 (a) Derive the conditional factor/input demands and use them to find the long-run cost function for this firm. (b) For these factor prices, derive the firm's long-run supply curve. (c) For these factor prices graph the firm's long-run supply curve. (d) Suppose the price of the second input, w2, rises to 2 per unit. What is the long-run cost curve? Derive and graph the new supply curve.

Answer 1

a. The firm treats the two inputs as perfect substitutes and would only employ the cheaper one. Since the cost of input two is less than the cost of input one, the firm will only employ input two. $\dpi{80} \fn_phv \small x_1=0,x_2=y$
The firm's cost function is:
$\dpi{80} \fn_phv \small Cost=w_1x_1+w_2x_2=y$

b. The firm equates its marginal cost to the price set in the market by the industry:
$\dpi{80} \fn_phv \small MC=1=p$
The firm's supply is:
$\dpi{80} \fn_phv \small Supply=\begin{cases}\infty&p>1\\ [0,\infty)&p=1\\ 0&p<1 \end{cases}$

c. Plotting the firm's supply curve:

d. When price of input one rises to 2, the firm would be indifferent between employing input one or two. The long run cost curve is:
$\dpi{80} \fn_phv \small Cost=2y$
$\dpi{80} \fn_phv \small MC=2=p$
$\dpi{80} \fn_phv \small Supply=\begin{cases}\infty&p>2\\ [0,\infty)&p=2\\ 0&p<2 \end{cases}$