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econ105_10s_ps5sol0 - Problem Set 5 Solutions ECON105...

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Problem Set 5 Solutions ECON105 Industrial Organization and Firm Strategy Professor Michael Noel University of California San Diego 1. Two firms, located at either end of a linear city, compete in prices. Each firm has constant marginal cost equal to c. Each consumer i has unit demand, and receives utility of u ij = v - p j - tz ij if she buys from firm j located a distance of z ij away at price pj. She receives utility equal to zero if she does not buy. a. If not the whole market is served in equilibrium, find the equilibrium price and profits for each firm. Do profits rise or fall with an increase in t? If not the whole market is served, then there must be people (obviously in the middle of the line) who don’t buy from either firm because it is too far for the m to travel. Also, there is exactly one person (say at position X) who is indifferent between buying from firm 1 and not buying at all, and similarly one person (at position Y) must be indifferent between buying from firm 2 and not buying at all. The indifference of person X is mathematically described by the following equation: v p 1 tX=0, where the left-hand side is the utility from buying from firm 1 and the right-hand side is the utility of buying nothing assumed to be zero. We can modify this equation to obtain an expression of where exactly X would be on the interval as a function of the p 1 : X=(v- p 1 )/t . Using the same intuition for the person at position Y, we know that 0 Y) - t ( 1 - p - v 2 which implies that . t v - p 1 Y 2 Knowing where on the line points X and Y are located as functions of firm prices, it is a simple matter to calculate the demand curve of each firm: q 1 (p 1 )=X(p 1 )=(v- p 1 )/t and q 2 (p 2 )=1-Y(p 2 )=(v- p 2 )/t. Obviously, this is a symmetric problem (same demands, same costs, and simultaneous play) so we can do the profit maximization of either firm: 2 1 2 / ) ( 0 2 : * ) ( ) ( m ax p p c v p c p v FOC t p v c t p v p p cq p q p i i i i i i i i i i i p i 2 1 2 2 2 2 2 1 4 ) ( 4 2 2 2 ) ( 4 ) ) ( ( 2 2 / ) ( t c v t c cv c v t c v c t c v c v q q t c v t c v v t p v q i i i
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As long as the parameters v, c, and t are such that 1 2 1 t c v q q , the above equilibrium will exist because our initial assumption (that some don’t buy) will not be violated. 0
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