Question

1. Fred and Barney are fishermen who operate fishing ships out of Nantucket. They own similar ships that can each carry a day's catch of 800 pounds of cod. Marginal costs of fishing are constant at $4 per pound (for simplicity, assume they have no fixed - or sunk -- costs). For the current discussion, assume that Fred and Barney are the only fishermen and that the aggregate demand for Nantucket cod is characterized by the demand function Q 1600 100P, where P is the price per pound of cod and Q is in pounds of cod. This corresponds to an inverse demand function of P 16 .01Q, where Q = q qp Finally, the fish market operates in the following manner: fishermen go out in the morning and catch fish, then they return to the dock and drop the catch on the dock. Customers gather around and a market- clearing price is announced, at which point sales commence (a) What is Fred's residual (inverse) demand curve if his catch size is q and Barney's is q? (b) What is Fred's profit as a function of his catch q, and Barney's catch qip? (c) If Fred expects Barney to catch qa, what is Fred's best response function BR,(qB)? (d) What is Barney's residual (inverse) demand curve if his catch size is qg and Fred's is q? (e) What is Barney's profit as a function of his catch qa and Fred's catch q,? (f) If Barney expects Fred to catch q, what is Barney's beest response function BR(q,)? (g) Using the fact that each fisherman's equilibrium catch is a best response to the other's, solve for the equilibrium catch sizes qand q (h) What will be the resulting price? (i) Show that each fisherman's Cournot duopoly profits will be $1600.

Answer 1

1. P = 16 – 0.1Q

a)

P = 16 – 0.1(qF +qB)

Residual demand for Fred:

P = (16 – 0.1qF) – 0.1qB

(b)

Fred’s Profit:

πF = P*qF – MC*qF

πF = ((16 – 0.1qF) – 0.1qB)qF – 4qF

πF = 12qF – 0.1qF^2 – 0.1qBqF

(c) Best response function can be found out by differentiating πF wrt qF:

d πF/dqF = 12 – 0.1*2qF – 0.1qB

Equating to zero:

12 – 0.2qF – 0.1qB = 0

qF = (12 – 0.1qB)/0.2

qF = 60 – 0.5qB

(d) Barry’s Residual demand curve:

P = 16 – 0.1(qF +qB)

Residual demand for Fred:

P = (16 – 0.1qB) – 0.1qF

(e)

Fred’s Profit:

πF = P*qB – MC*qB

πF = ((16 – 0.1qB) – 0.1qF)qB – 4qF

πF = 12qB – 0.1qB^2 – 0.1qBqF

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