1. On slide 6 of the lecture slides titled "product differentiation" two equations define equilibrium prices in a Hotelling model with linear transportation costs. The equations are: a- b Pi = c+t(1 -a- b)(1+ 3 b - a P2 = c+t(1-a- b)(1+ 3 Derive these equations.Analysis Each pair (a, b) defines a separate subgame. We have previously solved for equilibrium conditional upon fixed (a, b). This is the equilibrium that will prevail in each subgame. From our previous analysis, we know that, for any given vector (P1, p2, a, b), firm 1's profits are given by: TT' (P1, P2, a, b) = (p1 - c)0(P1, p2, a, b, t) where 0(P1, P2, a, b, t) = P1 - P2 1ta- b + 2t(1 - a - b) 2Analysis Moreover, we know that, in equilibrium: pi= ctt(1 -a - b) 1+ a - b 3 P2=ct t(1 -a -b) (1+ b - a 3 Now consider firm 1's optimal choice of a in stage 1. Note that: dpz and + da da Op1 da Op2 da Where the envelope theorem tells us that ap =0 when evaluated at py. One can evaluate the last two terms using the formulas reproduced above