Question

Consider a weapons producer that is selling guns to two countries that are at war with one another. Guns can be produced at a constant marginal cost of $10 per gun. The demand for guns in each of the countries is given by:

p = 50 - 0.5Q (Country A)

p = 20 - 0.25Q (Country B)

a. If the weapons producer can charge different prices to each country, what price and quantity will it sell to each?

b. If the weapons producer cannot price discriminate, what price and quantity of guns will ist sell to each country?

c. Is the deadweight loss higher under price discrimination or a single-price? Show mathematically.

Answer 1

a) For country A, p=50-0.5Q is the inverse demand curve (Q=100-2p is the demand curve)

Then, marginal revenue for A, MRA = 50-Q

Now, for equilibrium in country A,

MRA = MC

or, 50-Q=10

or, Q=40 units is the equilibrium quantity for A

and p=50-(0.5*40) = $30 is the equilibrium price for A.

Similarly, for country B, p=20-0.25Q is the inverse demand curve (Q=80-4P is the demand curve)

Then, marginal revenue for B, MRB = 20-0.5Q

Now, for equilibrium in country B,

MRB = MC

or, 20-0.5Q=10

or, 0.5Q=10

or, Q = 20 units is the equilibrium quantity for B

and p=20-(0.25*20) = $15 is the equilibrium price for B.

b) If the producer cannot price-discriminate,

Q=QA+QB

or, Q = 100-2p+80-4p

or, Q = 180-6p

or, p = 30-(Q/6)

Then, marginal revenue MR = 30-(Q/3)

Now, for equilibrium, MR=MC

or, 30-(Q/3)=10

or, Q/3 = 20

or, Q = 60 units is the equilibium quantity when there is no price discrimination

and p = 30-(60/6) = $20 is the equilibium price when there is no price discrimination

c) Incase of price discrimination, in market A, dead-weight loss = 1/2(P-MC)*(QMC-Q) = 1/2*(30-10)*(80-40) = $400

Whereas, in market B, dead-weight loss = 1/2(P-MC)*(QMC-Q) = 1/2*(15-10)*(40-20) = $50

Thus, total dead-weight loss = $400+$50 = $450

But, in a single-priced monopoly, dead-weight loss = 1/2(P-MC)*(QMC-Q) = 1/2*(20-10)*(120-60) = $300

Thus, dead-weight loss is higher in case of price discrimation than for single-price monopoly.