Question

1.     Assume the following for price discriminating monopolist aimed at maximizing profit. Total demand for the product of the monopolist is Q = 50-5P (P = 100-2Q)

·        Demand in Market one is Q1= 32-0.4P1(P1= 80-2.5Q1)

·        Demand in Market two is Q2= 18-0.1P2(P2= 180-10Q2)

·        Cost function is C = 50+40Q (where Q = Q1+ Q2)

a.      Find equilibrium quantities (Q1and Q2),

b.     Find equilibrium prices (P1and P2),

c.      Calculate profit (π),

d.     Calculate and Interpret elasticities (ε1 and ε2)

Answer 1

a.

Total revenue in market 1

$= 80Q_1 - 2.5 Q_1^2$

Marginal revenue in market 1

$=\frac{ ∆TR}{ ∆ Q_1}\\ = 80 - 5 Q_1 \\ MC = 40 \\ 80 - 5 Q_1 = 40 \\ 40 = 5 Q_1 \\ Q_1 = 8$

Total revenue 2

$= P × Q = 180Q_2 - 10 Q_2^2$

Marginal revenue 2

 $= 180 - 20 Q_2 \\ MC = 40 \\ Equating \space MR = MC \\ 180 - 20 Q_2 = 40 \\ 140 = 20 Q2 \\ Q_2 = 7 \\$

b.

$P_1 = 80 - 2.5 (8) = 60\\ P_2 = 180-10Q_2 = 110$

c.

$Profit = TR_ 1 + TR_ 2 - TC \\ = 8(60) + 7(110) - 50 - 40(15) \\ = 480 + 770 - 50 - 600\\ = 600$

d.

$Q_1= 32-0.4P_1 \\ \frac{∆Q_1}{∆P_1 }= - 0.4 \\ e = \frac{∆Q_1 }{ ∆ P_1} \times \frac{ P _1 }{Q _1}\\ e = -0.4 \times (\frac{60 }{ 8}) = - 3\\ Q_2= 18-0.1P_2 \\ \frac{∆Q2}{ ∆ P_2 }= -0.1 \\ e =\frac{ ∆Q2 }{∆ P_2 }\times \frac{P_2 }{Q _2}= -0.1(\frac{110}{7}) = -1.57$

Good 2 is less price elastic. 

A higher price is charged to the low elasticity segment and a lower price is charged to the high elasticity segment.