Question

A firm is selling a single product in 2 markets (San Antonio and Houston). Demand schedules for the two markets are:

Q_SA = 140 – 2P_SA or P_SA = 70 - .5 Q_SA

MR_SA = 70 - Q_SA And Q_H = 40 - .2P_H or P_H = 200 – 5Q_H and MR_H = 200 – 10Q_H

Total demand is: Q_SA + Q_H at equal prices, $\sum Q=120-2P+36.67-.3333P=156.67-2.33P$

P = 67.15 - .4286Q, and $\sum MR=67.15-.8572Q$

Total cost of supplying customers is TC = $650 + $20Q.

a.) How much should be sold in each market if the firm price discriminates?

b) What prices should be charged in each market if the firm price discriminates?

c.) What are the price elasticities of demand in each market if the firm price discriminates?

Answer 1

a)

P_SA = 70 - 0.5Q_SA

MR_SA = 70 - Q_SA

P_H = 200 - 5Q_H

MR_H = 200 - 10Q_H

Q = Q_SA + Q_H

TC = 650 + 20Q

MC = 20

Profit maximising condition

MR_SA = MR_H = MC  

MR_SA =  MC  

70 - Q_SA = 20  

70 - 20 = Q_SA  

Q_SA = 50

firm should sell 50 units in San Antonio market

MR_H = MC  

200 - 10Q_H = 20  

200 - 20 = 10Q_H

180 = 10Q_H

Q_H = 18

firm should sell 18 units in Houston market

b)

P_SA = 70 - 0.5Q_SA

P_SA = 70 - 0.5(50) = 70 - 25 = 45

So firm should charge 45 in Sen Antonio market

P_H = 200 - 5Q_H

P_H = 200 - 5(18) = 200 - 90 = 110

Therefore firm should charge 110 in Houston market

c)

P_SA = 70 - 0.5Q_SA

Q_SA = 140 - 2P_SA

Price elasticity of demand

Ed = (dQ/dP)(P/Q)  

dQ_SA /dP_SA = - 2  

Ed = (-2)(45)/50 = - 90/50 = - 1.8

|Ed| =  |-1.8| = 1.8  

Thus price elasticity of demand in San Antonio market is 2

P_H = 200 - 5Q_H

Q_H = 40 - 0.2P_H

dQ_H/dP_H = - 0.2  

Ed = (- 0.2)(110/18) = - 22/18 = - 1.22

|Ed| = |- 1.22 | = 1.22

Thus price elasticity of demand in Houston market is 1.22