Question

Two firms compete by choosing price. Their demand functions are

Q_1=20-P_1+P_2 and Q_2=20+P_1-P_2

where P_1 and P_2 are the prices charged by each firm, respectively, and Q_1 and Q_2 are the resulting demands. Note that the demand for each good depends only on the difference in prices; if the two firms collude and set the same price, they could make that price as high as they wanted, and earn infinite profits. Marginal costs are zero.

a. Suppose the two firms set their prices at the same time. Find the resulting Nash equilibrium. What price will each firm charge, how much will it sell, and what will its profit be? (Hint: Maximize the profit of each firm with respect to its price.)

b. Suppose Firm 1 sets its price first and then Firm 2 sets its price. What price will each firm charge, how much will it sell, and what will its profit be?

c. Suppose you are one of these firms and that there are three ways you could play the game: (i) Both firms set price at the same time; (ii) You set price first; or (iii) Your competitor sets price first. If you could choose among these options, which would you prefer? Explain why.

Answer 1

Two firms compete by choosing price. Their demand functions are

Q_1=20-P_1+P_2 and Q_2=20+P_1-P_2

where P_1 and P_2 are the prices charged by each firm, respectively, and Q_1 and Q_2 are the resulting demands. Note that the demand for each good depends only on the difference in prices; if the two firms collude and set the same price, they could make that price as high as they wanted, and earn infinite profits. Marginal costs are zero.

Demand functions are :

Q1=20-p1 + pe
Q2= 20 - p2 + p1
MC = 0

a. the two firms set their prices at the same time. Find the resulting Nash equilibrium. What price will each firm charge, how much will it sell, and what will its profit be?

Bertrand price competition,
Firm 1 try to maximize profit , given p2
π1 = 20 p1 - p12 +p1p2
δπ2/δp1= 20 - 2p1+2p2=0
20+p2 = p1 (2)
P =20+p2 /2 = 10+1/2 p2
Similarly,from 2 wants to maximize profit ,given p
π2= 20p 2 - p22+ p1p2
δπ2/δp2= 20- 2p2 + p1 =0
( P2= 10+ 1/2 p1)
=: P2 = 10 + 1/2 ( 10+1/2 p2)
P2 - 1/ 4 p2 = 10+5 =15
P2 - 1/4 p2 = 10+5=15
P2 = 15× y/3

= 20
P1 = p2 = $ 20
Q1 = 20 - 20 +20

=20
Q2 =20
π1 = 20×20

= $ 400
π2 = $400

b. Suppose Firm 1 sets its price first and then Firm 2 sets its price. What price will each firm charge, how much will it sell, and what will its profit be?
:- Stackelberg price competition
We solve this by using backward induction, At stage 1 firm 2 try to maximize profit given P1
π2 = 20p2 - p22 + p1p2

δπ2/δp2 = 20 - 2p2 + p1 = 0

P2 = 10 + 1/2 p1

At stage 2, firm 1 observes p2 and then maximize profit

π1 = 20 p1 - p12 + p1p2

= 20 p1 - p12 + p1(10 + 1/2 p1)

= 20 p1 - p12 + 10 p1 + 1/2p12

= 30p1 - p12/2

δπ1/δp1 = 30- p1= 0

P1 = 30

P2 = 10 + 1/2(30)

= $25

Q1= 20-30+25

= 15

Q2 = 20+30-25

= 25

π1= 30(15)

= $450

π2 = 25(25)

=$625

There is a second mover advantage.

c. Suppose you are one of these firms and that there are three ways you could play the game: (i) Both firms set price at the same time; (ii) You set price first; or (iii) Your competitor sets price first. If you could choose among these options, which would you prefer? Explain why.

:-If I would be a firm , I will choose to set a price second or let my competitor sets price first, in by this I can maximize my profit($625) which is the that I can earn given this conditions.