Question

Two firms produce differentiated products with demand curves

p1 = a – q1 – bq2 and p2 = a – q2 – bq1. They both face constant average and marginal cost c and their profit functions are profit = (p1 – c)q1 and profit = (p2 — c)q2, respectively.

Solve the Bertrand game.

Hint: You need to solve the system of equations p1 = a − q1 − bq2 and p2 = a − q2 − bq1 for q1 and q2, which will allow you to write each firm’s profit as a function of prices and its marginal cost.

Answer 1

Page 1 (using ean:0 ; from 2 P₂ = a-qa-bq, %= a-bq, - p2 - 0 from 1 : Pi= a -9,- bqz > P, = 2 -9,- 6 (a-b9,-P2) * P, = Q -9,-'ab +629, +6P2 9,- 629,= a-ab + bp2 --P, » q = allo + 6P2-P, . (Itb)(+6) . 1-6² >> 2,5 6 + bo Put a, in ean. O 2,3 e forte + Bike Be

b=e- (et + b (tege) - face atob-ab - 1tb b(bp 1-62 Pinz atabat - b(02=P») – P2 235 - Corel) + ] 935 - ( 54 best 6-bit 2 bp₂-bpt l-b 1-62 - an In i -62 - . Poze b + bpi - Pz from egn. © & W 9, = + bonum P-P2 Profit of from 1: TG = (2-0)x9,= (1,-0) [146 + bietet Fec 20 9 (8-0[0-] fit ) = 0 TW-[-cubie te] =0 sonra 2-c-bp 176 407(1-6) » 2P,-C-6P2 = Q(1-6) 2p= all-b)+c+bP2 + por = a C1-b) +c+bP2 - : 2 Similarly, PBR = all-b)+C+ bp, 0

Solving W&@ simultaneously, Page 3 La featu]+ ct bo facebọcren)

P = Pa (1-6) + Qc + bal1-6) + bet be Page 4 too - → up, -b3p, = all-b) [a+b] + c(2+b] P. (22–62) - +b[ a[1-6] +c] = P, = (+6) [a (1-b)+c] +63(2-) = a(l-bte 2-b ? Naph Equilibrium I BERTRAND GAME = a (-6)+c 2-b

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