Question

A firm has the cost function That is, it has a fixed production capacity i, below which marginal cost is constant, at e (a) Sketch the firm's marginal- and average-cost function. (b) Solve for its profit-maximizing output if it sells in a perfectly competitive market. (e) Describe the solution possibilities for output if the firm is a profit- maximizing monopoly with linear demand (d) Identify the "shadow price of capacity in each of cases (b) and (c).

Answer 1

(a) The graphs is as below.

MCAO -bar

The MC is c, and the AC would be $AC = \frac{C}{x} = c$ .

(b) The marginal cost is constant at c. The perfectly competitive equilibrium would be where MC is equal to MR=P, and since MC=c, the profit maximizing output would be where . If the price is lower than c, the firm produces 0 output, and if the price is equal to or greater than c, the firm produces output $\bar x$ . Hence, the profit maximizing output would be
r={1 for c P
.

(c) In case of a linear inverse demand , the total revenue would be and the marginal revenue would be $MR = \frac{\partial }{\partial x}(TR) = a - 2bx$ . The profit maximizing output would be where or or $x = \frac{a - c}{2b}$ for $\frac{a - c}{2b} \leq \bar x$ . The price would be $P = a - b(\frac{a - c}{2b})$ or $P = \frac{2a - a + c}{2}$ or $P = \frac{a + c}{2}$ .

(d) The shadow price of a product is what an individual is willing to sacrifice to get that product, or the price of a product for which the firm is willing to produce. In part b, the shadow price of capacity is at least c, since below c the output is zero. In part c, the shadow price of capacity is $\frac{a + c}{2}$ for the condition $\frac{a - c}{2b} \leq \bar x$ .