Question

Two firms compete in a market with demand given by D(p) = 100 − p, where p is denoted in cents (p=100 is 1 dollar). Firms can only charge prices in whole cents – i.e. p can only take integer values, and not values like 1.5. Marginal costs for each firm are given by MC=10. Firms compete by simultaneously choosing prices. When prices are equal, each firm gets one half of total demand.

b. Find all the Nash equilibria of this duopoly game. Show that they are Nash equilibria by checking each firm’s best response. Calculate each firm’s profits under any equilibria.

Answer 1

1. Macket demand, DCPI = 100-p. (po 100 - DEPI Me=to (givens TC-- 104, Cassung no fixed cost) firms best response function is: TIG Y, (100-y, -Y2) - 104,! y = quantity of outpect produced by from s. toaga magia dyry, olay, Maximusing this profit dal - 0 dy, 100 - 29, -Y2 ~ 1050 29 = 90-Y Tyi - (90 - 42

ir ao yir biLY) o us y Similarly we find for firm 2. - Y2 (100-41-42) - 1042 max t = 100-14, - 24 – 1000 Y 2 = (90-4) 2 Oncombung? grapes 9 45 us Y = b2ly ) 90 Y 45 490 We need to find a pair (41142) of outputs mith the property that ? 8 Yobily) and Y2 = bg lyng That is, Y,= (90-422 and Y2 = 120-y) T2 ли Але ) 9,- 8o- (90-yil 2 TITT