Question

Suppose the market demand can be separated into two distinct markets, where p₁=80-5y₁,p₂=180-2y₂, and the common cost function is C=50+40(y₁+y₂).

a. Determine the equilibrium prices and quantities in each market and the overall profits that result from actions of a third degree price-discriminating monopoly.

b. Determine the price elasticity of demand in each market, evaluated at the equilibrium prices and quantities.

c. What is the relationship between the price elasticity of the demand in each market and the price prevailing in each market?

Answer 1

a)

Market 1:

p1 = 80 - 5y1

Total Revenue TR= p1*q1 = (80-5y1)*y1 = 80y1 - 5y1²

Marginal Revenue MR1 = dTR/dy1 = 80 - 10y1

C = 50+40(y1+y2)

The marginal cost of the derivative of total cost with respect to total output.

MC = 40

At equilibrium MR = MC

80 - 10y1 = 40

10y1 = 40

y1 = 4

p1 = 80-5y1 = 80-5*4 = 60

Market 2

p2 = 180 - 2y2

Total Revenue TR= p2*q2 = (180-2y2)*y2 = 180y2 - 2y2²

Marginal Revenue MR2 = dTR/dy2 = 180 - 4y2

MC = 40

At equilibrium MR = MC

180 - 4y2 = 40

4y2 = 140

y2 = 35

p2 = 180-2y2 = 180-2*35 = 110

Profit = p1*y1 + p2*y2 - TC = 4*60 + 35*110 - 50-40(4+35) = 240+3500 - 50 -40*39 = 3740 - 50 - 1560 = 2130

b)

Price elasticity of demand Ed= dQ/dP*P/Q

Market 1:

p1 = 80-5y1

y1 = 16 - 0.2p1

dy1/dp1 = -0.2

Ed = -0.2*60/4 = -3

Market 2:

p2 = 180-2y2

y2 = 90 - 0.5p2

dy2/dp2 = -0.5

Ed = -0.5*110/35 = -1.57

c)

We can see the market 2 has a lower elasticity of demand than Market 1, hence it has a higher price than market 1. Thus market having a lower price elasticity of demand will have a higher price.