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

Third degree price discrimination involves monopolists having the ability to sell commodities at different prices in different markets.

Market A

P=100 – Q_A

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

TR_A = $(100 – Q_{A}) \times Q_{A}$

TR_A = $100 Q_{A} – Q_{A}^2$

MR_A =$\varDelta TR_{A} / \varDelta Q_{A} = \varDelta(100Q_{A} – Q_{A}^2)/ \varDelta Q_{A}$

MR_A = 100 – 2Q_A

Market B

P=160−2Q_B

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

TR_B = $(160−2Q_{b}) \times Q_{b}$

TR_B = $160Q_{b} −2Q_{b}^2$

MR_B =$\varDelta TR_{b}/ \varDelta Q_{b} = \varDelta (160Q_{b} −2Q_{b}^2)/ \varDelta Q_{b}$

MR_B = 160 – 4Q_B

Equilibrium: MR = MC

Market A

MR_A = 100 – 2Q_A = $\frac 23 Q_{A}$

$\frac 23$ Q_A + 2Q_A = 100

$\frac 83Q_{A}$ = 100

Q_A = $100 \times \frac 38$

Q_A = 37.5

P=100 – Q_A

P=100 – 37.5

P= 62.5

Total Revenue

TR_A = $100Q_{A} – Q_{A}^2$

TR_A = 100(37.5) – 37.5²

TR_A = 2,343.75

Market B

MR_B = 160 – 4Q_B =$\frac 23Q_{b}$

$\frac {14}{3}$ Q_B = 100

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

Q_B = 21.43

Price

P=$160−2Q_{b}$

P= $160 − 2 \times 21.43$

P = 160 – 42.86

P = 117.14

Total Revenue

TR_B = $160Q_{b} −2Q_{b}^2$

TR_B = 160(21.43) – 2(21.43²)

TR_B = 3,428.8 – 918.4898

TR_B = 2,510.31