Question

(Dominant Firm) Consider a market in the short-run. The market demand curve is given by D(p) = 2000 − 30p. There is a dominant firm in the market with cost function C(q) = 20q. There are also 10 price-taking fringe firms, each with cost function C(q) = 20q + ^q2 . Their marginal costs are therefore M C(q) = 20 + q.

A) Find the supply function for an individual fringe firm, q(p). (You do not need to define this piecewise, but be aware that the expression may be negative for some values of p.) Also, find the total supply curve for the fringe, Q^s(p).

B) Give an expression for the dominant firm’s residual demand curve, D^R(p).

C) What price does the dominant firm set, p^dom? How much does it produce, q^dom? How much is sold by the (entire) fringe, Q^ff ?

D) How much does each individual fringe firm produce, q^ff ? How much profit does each individual fringe firm make, π^ff ?

Answer 1

The market demand curve is given by Q = 2000 − 30P. The dominant firm has cost function C(q) = 20q. Its marginal cost is 20. There are also 10 price-taking fringe firms, each with cost function C(q) = 20q + q2 . Their marginal costs are therefore M C(q) = 20 + 2q.

A) The supply function for an individual fringe firm, q(p) is the rising marginal cost function. For one firm it is given by P = 20 + 2q or q = 0.5P - 10. Note that price must be higher than 20 because minimum AVC is 20. For 10 firms, the total supply curve for the fringe is 10q = Qs(p) = 5P - 100.

B) The expression for the dominant firm’s residual demand curve, DR(p) is given by

Qd = Market demand - fringe supply

= 2000 − 30P + 100 - 5P

= 2100 - 35P.

The inverse demand function is P = 60 - (1/35)Qd, Marginal revenue is MR = 60 - (2/35)Qd

C) The does the dominant firm set is, pdom

MR = MC

60 - (2/35)Qd = 20

Qdom = 700 units and Pdom = 60 - (1/35)*700 = $40

The (entire) fringe, is now selling Qff = 5*40 - 100 = 100 units

D) Now each individual  fringe firm produce, qff = 100/10 = 10 units. The profit does each individual fringe firm make, πff = 10*40 - ( 20*10 + 100) = $100.