Question

question (e) and question (f)

6. Suppose you have a job analyzing a perfectly competitive market. The aggregate demand is (p) = 98 - and the cost function for the firms is C(q) = 772 +35. Suppose all firms use the same cost function. (a) Setup and solve the profit maximization problem over quantity. Write the quantity an individual firm will produce as a function of the sale price. (3 points) (b) Solve for the price, quantity, and profits for each individual shop and then also for aggregate quantity in equilibrium when the number of firms is fixed at N = 42. Show your work. (Remember: In short-run equilibrium, p = MC and Q=0°) (6 points)

Answer 1

a. A representative firm's profit maximizing problem is:
$\dpi{80} \fn_phv \small max \ pq_i-7q_i^2-35$
solving this problem by differentiating the objective function with respect to the quantity produced by the representative firm:
$\dpi{80} \fn_phv \small p-14q_i=0\rightarrow q_i=\frac{p}{14}$

b. For the given number of firms, total supply in the market is:
$\dpi{80} \fn_phv \small Q=42*\frac{p}{14}=3p$
Substituting this into the demand function:
$\dpi{80} \fn_phv \small 3p=98-\frac12p\rightarrow p=28$
Quantity produced by each firm at this price is:
$\dpi{80} \fn_phv \small q_i=\frac{28}{14}=2$

Profit of each firm is:
$\dpi{80} \fn_phv \small \pi_1=2*28-7*4-35=-7$

c. Firms shut down when they are not able to cover their fixed costs. Since profit is negative, firms are not able to cover their fixed costs of 35 and hence will shutdown.

d. In the short run competitive equilibrium, each firm solves its profit maximizing problem by equating its marginal cost to the price in the market. Firms can earn positive profits in the short run but earn zero profits in the long run as any positive profit in the short run induces entry of new firms till profit earned by each firm is zero.