Question

There are 2 firms in a market producing differentiated products. The firms both have MC that is equal to 0

Firm 1 demand is

q1(p1,p2) = 6-2p1 + p2

Firm 2 demand is

q2(p1,p2) = 6-2p2 + p1

1. Firms compete in quantities- Cournot Competition. What are the inverse demand functions for firm 1 and 2?

2. Find and graph each firm’s best response functions. The quantities are strategic substitutes or complements?

3. Find the Nash equilibrium prices and quantities when both firms use Cournot. What are the firm’s profits?

Answer 1

Page numbers are given at the top of the page, follow them.

Vi= 6-2P,+ PR. 1 9:56=2P2 +P, >> 28,-P,= 6-9; 230 2P, -P,36-93 2) 3P, -2,36---01:. 4px - 21,5. 12. = 24-- Sebring the above two equations use get'; 28, - P2 = 6-q -26, +4P2 = 12 - 29, 3P, = 18-9, -29, >> P = 6- : Setting the value of Pe in equation we get; . 2P; - 6 + 1 2 2 + 2 2 2 2 = 6-9, .. 2P, = 612 - v, - § P,= 6 - 2 2 2 4 - 12 V Thus, the inverse demand function for the two forume are ; P,= 6 - 2/3 - 1 V 27 For Firm 1. P2 = 6 - 1 3 - 2 937 For Form - 2.

and TC = 0; an The profit function of at firm is given by; T Total Revenue - Total Cast | A: FR - TC , T = PxQ - o "TR = PxQ7 for form 1.- T, = P, xq, ),69,- 9,*- , % Differentisting it, with respect to V O ») 6-49,-=0. : >>26- 5 9,5.4.5-2012; This is the reaction er best quasfonce function Again for Firm 2:- . .. for firm. 1. : To = 6 4 2 - 1 / 2 V, V, - 2 / 32 V Differentiating with respect to of: 21:0 »> 6 - $ 0,- 49,=0 >> * * = 6-5 %, » V₂ = 4.5 - 1 ; This is the best for firm: - 2. response function

from the demand function ginene in the quaetion we can identify the nature of the goods. when P increases there is an increase in v Thúe implies that the two quantities are eubetitute to each other. 4.5 * lo Reebonce function of Finm - 1 Response Function of k firm - 2 . o. 4.5 is q.. 3) When the two firme compete in quantities we. have to solve the two respons function in order to find the equilibrium quantity ! We have; 9, = 4.5 - 1 1/2 and Va 45-129, -- Setting value of any in equation in ©; 9,=4.5(4:5 - 6) s> V, 54,5 - 1:125 + 0.06959, 3.375 0.9375 - 3-6 units

Setting 9, - 3.6 bine equation © ; . 9,= 4.5 - .0, >>V=4.5 - 5* 3.6 >> 9: 3.6 The equlibrium quantity for the two firms are; 9,9= 3.6 and q = 3.6 Setting the ralue of V, and of in the equations given below; P, = 6 - 2 399,- 1 3 / >> P,= 6- 3:6 - * 3:6 >> P, = 24 Again; P2 = 6 - 1 - 390,- 3 2 2 2 >> P = 6- $ x 3.6 - * 3.6 a) Pg = 214. The equilibrium price for the two firms are; 2,9=2.4 and Po' = 2:4.

UL Profit of firm 1: T;= P, xq, = 3.6* 2:4 = 8.64. Profit of firem 2 - T = P₂ X %= 8.68 2:4=8.64.