Question

A monopolist faces two totally separated markets with inverse demand p=100 – qA and p=160−2qB respectively. The monopolist has no fixed costs and a marginal cost given by mc= 2 /3 q Find the profit maximizing total output and how much of it that is sold on market A and market B respectively if the monopoly uses third degree price discrimination.

Answer 1

Marginal Costs (MC) = $\frac {2}{3}q$

Market 1 Inverse demand fxn: P=100 – qA

Total Revenue (TR) = P*Q

TR = $(100 – Q) \times Q$

TR = $100Q – Q^2$

MR = $\varDelta TR / \varDelta Q = (100Q – Q^2)/ \varDelta Q$

MR = 100 – 2Q

Market 2 Inverse demand fxn: p=160−2qB

Total Revenue (TR) = $P\times Q$

TR = $(160 – 2Q) \times Q$

TR = $160Q – 2Q^2$

MR = $\varDelta TR / \varDelta Q = (160Q – 2Q^2)/ \varDelta Q$

MR = 160 – 4Q

Equilibrium

MR = MC

Market 1: $100 – 2Q = \frac{2}{3}Q$

$\frac{8}{3}Q = 100$

Q = $100\times \frac{3}{8}$

$Q_{1} = 37.5$

P = 100 – Q

P = 100 – 21.43

P = 78.57

TR = $100Q – Q^2$

TR = $100(37.5) – 37.5^2$

TR = $160(37.5) – 2(37.5^2)$

TR = 2,343.75

Market 2: 160 – 4Q = 2/3Q

14/3Q = 100

Q = $100\times \frac{3}{14}$

$Q_{2} = 21.43$

P = 160 – 2Q

P = $160 – 2\times21.43$

P = 160 – 42.86

P = 117.14

TR = $160Q – 2Q^2$

TR = $160(21.43) – 2(21.43^2)$

TR = 3,428.8 – 918.4898

TR = 2,510.31