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July 1st, 2013, 04:03 AM   #1
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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 Bertrand-Nash 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<pB-4
{ 4 + 0,5$ als pA=pB-4
{ 2 + 0,5(pB-pA) als pB-4<pA<pB+4
{ 0 als pA>=pB+4

qB=
{ 4 + $ als pB<pA-4
{ 4 + $ als pB=pA-4
{ 2 + 0,5(pB-pA)+ $ als pA-4<pB<pA+4
{ 0 als pB>=pA+4

To find the Betrand-Nash equilibrium I maximalize both profitfunctions, and find the reactioncurves.
piA=(2+0,5(pB-pA))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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