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 = 530 - 5x and China's is P = 954 - 8y, where P is in millions of dollars. The marginal cost of missiles is MC = 2Q, 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.

Clean Handwriting

Answer 1

Firm will determine its profit maximizing quantity in both markets by equating Marginal revenue (MR) with MC.
MC = 2Q = 2x + 2y

India:
P = 530-5x
Total Revenue, TR = P*x = (530-5x)*x = 530x - 5x²
So, MR =

Now, MR = MC gives,
530 - 10x = 2x + 2y
So, 2y = 530 - 10x - 2x
So, 2y = 530 - 12x
So, y = (530/2) - (12/2)x
So, y = 265 - 6x

China:
P = 954 - 8y
TR = (954-8y)y = 954y - 8y²
MR =

Now, MR = MC gives,
954 - 16y = 2x + 2y
So, 2x = 954 - 16y - 2y = 954 - 18y = 954 - 18(265 - 6x) = 954 - 4770 + 108x
So, 108x - 2x = 106x = 3816
So, x = 3816/106 = 36

Thus, x = 36
y = 265 - 6x = 265 - 6(36) = 265 - 216 = 49
So, y = 49

Price in India: P = 530 - 5x = 530 - 5(36) = 530 - 180 = 350
Price in China: P = 954 - 8y = 954 - 8(49) = 954 - 392 = 562

b. India:
Price elasticity of demand, e1 =

dp/dx = -5 (From demand function)
So, dx/dp = 1/(-5) = -0.2
Thus, e1 =

China:
dp/dy = -8 (From demand function)
So, dy/dp = 1/(-8) = -1/8 = -0.125
So, e2 =

Thus, we can 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.