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July 1st, 2013, 04:03 AM  #1 
Newbie Joined: Jul 2013 Posts: 1 Thanks: 0  Hotelling Model
Hi, I'd like some help with the problem below. There are 2 companies A en B. Both have fixed locations 0 and 4. Consumers are uniformly distributed on the interval [0,4+$]. Every consumer has linieair transportcosts, and buys 1 unit of the product where it is the cheapest. I.E., if prices are pA and pB and the distance from the consumer to company A is equal to x and the distance to company B is y. Then the consumer buys at A when pA +x < pB + y. If the consumer is indifferent he chooses the company with chance 1/2. The companies have constant marginal cost equal to zero and prices pA and pB are strategic variables. The locations are fixed My question: Determine the BertrandNash equilibrium in this model and show that this equilibrium only exists when 36  12$  5$^2 >= 0 holds. (This is what I have come up with to solve) To answer the question I first write down the demand for the product for producer A and B: qA= { 4 + $ als pA<pB4 { 4 + 0,5$ als pA=pB4 { 2 + 0,5(pBpA) als pB4<pA<pB+4 { 0 als pA>=pB+4 qB= { 4 + $ als pB<pA4 { 4 + $ als pB=pA4 { 2 + 0,5(pBpA)+ $ als pA4<pB<pA+4 { 0 als pB>=pA+4 To find the BetrandNash equilibrium I maximalize both profitfunctions, and find the reactioncurves. piA=(2+0,5(pBpA))pA > via 1st derivative > pA=2+0,5pB Als for piB > pB=2+0,5pA+$ Using A in B gives the Bertrand Nash equilibrium: pA=4+2/3 $ pB=4+4/3 $ From this point on I don't know how to find the equation: 36  12$  5$^2 >= 0 ??? Many thanks in advance! 

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