Question

During the war, the same arms merchant often sells weapons to both sides of the conflict. In this situation, a different price could be offered to each side because there is no danger of resale. Suppose a US arms merchant has a monopoly of a special air-to-sea missiles and is willing to sell them to both India and China. India's demand for missiles is P = 64 - 4x and China's is P = 80 - 6y, where P is in millions of dollars. The marginal cost of missiles is MC = 4Q, where Q = x + y.

(a) How many missiles will be sold to each country and what price will be charged to each country? Show your work in details.

(b) Find out the price elasticity of demand for each country at that price and check that the country with less elastic demand is indeed paying a higher price.

Answer 1

Firm will determine its profit maximizing quantity in both markets by equating Marginal revenue (MR) with MC.
MC = 4Q = 4x + 4y

India:
P = 64 - 4x
Total Revenue, TR = P*x = (64-4x)*x = 64x - 4x²
So, MR =

Now, MR = MC gives,
64 - 8x = 4x + 4y
So, 4y = 64 - 8x - 4x
So, 4y = 64 - 12x
So, y = (64/4) - (12/4)x
So, y = 16 - 3x

China:
P = 80 - 6y
TR = (80-6y)*y = 80y - 6y²
MR =

Now, MR = MC gives,
80 - 12y = 4x + 4y
So, 4x = 80 - 12y - 4y = 80 - 16y = 80 - 16(16 - 3x) = 80 - 256 + 48x
So, 48x - 4x = 256 - 80
So, x = 4

Thus, x = 4
y = 16 - 3x = 16 - 3(4) = 4
So, y = 4

Price in India: P = 64 - 4x = 64 - 4(4) = 48
Price in China: P = 80 - 6y = 80 - 6(4) = 56

b. India:
Price elasticity of demand, e1 =

dp/dx = -4 (From demand function)
So, dx/dp = 1/(-4) = -0.25
Thus, e1 = -0.25(48/4) = -3

Similarly, for China:
dp/dy = -6 (From demand function)
So, dy/dp = 1/(-6) = -0.1667
So, e2 = -0.1667(56/4) = -2.334

Thus, we see that absolute value of e2 < absolute of e1 and price paid by China is greater so the country with less elastic demand is indeed paying a higher price.