Question

can someone help on setting and solving for a-c

Problem Set 2 ENV ECON 147 DUE: Feb 26 11:59 PM There are four questions on this assignment. The first two questions are typical of the kind of questions I might ask on an exam. The last two question are more difficult. Feel free to come to office hours for help. Question 1: Bertrand Competition in theory) Two firms produce heterogeneous goods and compete in prious. The demand and cost functions for each firm are as follows: 4 12 - 2p, +2 C:(91) = 91 +92 92 Clea) = 2-2p+P =99z+ (a) Calculate each firms reaction (or strategy) function. (b) Given a = 1, calculate the equilibrium prices. (c) Suppose o rises. Graphically show the effect on the firms reaction functions and the equilibrium prices.

Answer 1

The reaction function is calculated by differentiating the respective profit functions with respect the corresponding price and then putting the first order Condition to zero. As per the definition of reaction function, reaction function of firm 1 is obtained with p1 a function of p2 and vice-versa.

qu = 2-2 Pi + P2 4691), at 92 a) M = P,91 - 4191) = 21 - 21, ²+ Papi - 9,-8,2 : 221 - 28,2+ PP, - 2 + 2p/ - P2 - Ca- & pit Pa)? = up - 2012 + P2P1 - 2 - P2-ly-api dm dpi - 4 - 4pi + Pz - 2(2- 2017 P2) (-2) -0 =0 = 0 4- 4Pc + Pzt 4 (2 - Q2 + P2] 4- 4 Pi + D2 + 8-8 PB + 4P2 122-1281 + $P2_=0 1 24 522 T2 = = = P292 - 6 192) 282 - 2P2²+ Pipz - (1927 922) 2P2 - 2P2²+ PIP2 - 212-2Pz +P) - 12-2/2 + P.) 2 2. 4P2 + Pi 22 - 2 (a- 2 Pz+A) (-2) = 0 DIT2, dP2 &+26+ Pi - 4P2 + 412-2P2 +P) - 0 27 2m + Po - 4P2+ 8 – 8 P2 + up, = 0 10+2 at Spi-12P2 so I lotza +5P1 = P2 12

b) If x= 1, form 2. Reaction function of P2 = 12 + SPL 12 two reaction funcions 75 (62+591) : 1241 Solwing the simultaneously ! 19 144 + 60 + 25p1 : 14481 204 = 144-25P1 P = 204 - 12 119 7 → 12 = +5 (12) - 1521

C) Reaction function of firm 1 is independent of value of alpha.

For firm 2 , as value of alpha will increase,the reaction function of firm 2 will shift outward and the value of p2 as well as p1 will also increase. Graphically :

12+542 la Pi lot 24 & 5P1 12 = P2 plo 1-1215 P2 10 42/5 -2.4 *=2 28 lo 1 -1415 P2 %10 1.12 Fami - f Reaction furetion af Figara (2:1) 3 -18:25 2 - 5 1 -0.5 0.5 l so .