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.
a) What prices will our monopolist charge in the two separate markets? (6 m)
b) Calculate the price elasticity of demand in each market and explain the intuition behind the
relationship between the prices and elasticities in these two separate markets. (4 m

Answer 1

Profit maximization for monopoly

a) $MC=MR$

$P=100-q$

$TR=(100-q)q$

$=100q-q^2$

$MR=$ $dTR \over dQ$ $= 100-2q$

$MR=MC$

$100-2q=$ $2\over 3$

$=49$ $2\over 3$

$MR=$ $dTR\over dQ$

$=(160-2q)q=160q-2q^2$

$MR=160-4q$

$MR=MC$

$160-4q$ $=$ $2\over 3$

$=39$ $5\over 6$

$P=100-q$

$100-49$ $2\over 3$ $=50$ $1\over 3$

$P=160-2$ $($ $39$ $5\over 6$ $)$

$=80$ $1\over 3$

b) Price elasticity market A for inverse demand.

Elasticity$=$ ${P \over Q}* {1\over {dP(Q) \over dQ}}$

$P(Q)=100q-q^2$

$dP(Q) \over dQ$ $=100-2q$

$=100-2*49{2\over 3}$

$=$ $2\over 3$

$=$ $3\over 2$ $*$ $50{1\over 3}\over 49{2\over 3}$

$1.5*$ $50.333\over 49.67$ $=1.52$

Market B

Elasticity$=$ $1\over {dP(Q) \over dQ}$ $*$ $P\over Q$

$P(Q)=(160-2q)q$

$=160q-2q^2$

$dPQ\over dQ$ $=160-4q$

$160-4(39$ $3\over6$ $)$ $=$ $2\over 3$

$3\over 2$ $=$ $80.333 \over 39.833$

$=3.025$