468_ps3_2008_sol

468_ps3_2008_sol - ECON 468: Industrial Organization...

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Unformatted text preview: ECON 468: Industrial Organization Problem Set 3 April 24, 2008 1. Consider linear-city market in which 2 firms are located at the opposite end of the line: Figure 1: Linear city with two stores G 1 and G 2 G 1 = 0 x G 2 = 1 We know that in this market the demand functions at each store are given by: D 1 ( p 1 ,p 2 ) = p 2- p 1 2 t + t 2 D 2 ( p 1 ,p 2 ) = p 1- p 2 2 t + t 2 Consider the following two-period game: In period 1, Firm 1 can reduce its marginal cost by investing in research and development. In period 2, both firms compete in prices (i.e. Bertrand game with product differentita- tion). The cost function of the two firms is given by: C 1 ( q ) = ( c- x ) q C 2 ( q ) = cq where x is the result of the investment made in period one. The cost of reducing marginal cost by x is given by: g ( x ) = 1 2 x . (a) Given an arbibtrary value for x , what are the equilibrium profits of both firms in the second stage of the game? 1 The two profit functions are given by: 1 ( x ) = (3 t 2 + c- ( c- x )) 2 18 t 2 ( x ) = (3 t 2 + ( c- x )- c ) 2 18 t (b) Draw the reaction functions of both firms for two values of x . In which direction are equilibrium prices changing when we increase the amount of investment x ? What about profits? The reaction functions are given by: R 1 ( p 2 ) = t 2 + c- x 2 + p 2 2 R 2 ( p 1 ) = t 2 + c 2 + p 1 2 If we evaluate R 1 ( p 2 ) at x < x , the best-response function shift down as we go from x to x . Therefore the equilibrium price is decreasing in the level of investment. Similarly,....
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This note was uploaded on 08/08/2008 for the course ECON 468 taught by Professor Houde during the Spring '08 term at Wisconsin.

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468_ps3_2008_sol - ECON 468: Industrial Organization...

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