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ps3ans_Solutions - AEM 4240 Problem Set#3 Answer Key 1 Rad...

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AEM 4240 Problem Set #3 Answer Key 1. Rad and Bug (Bertrand competition with differentiated products) a) What is RAD's best response price as a function of the price that BUG charges? Writing out RAD’s profit function in terms of P R and P B , П R = P R *[1,000 - 10 P R - (10,000/P B )] - 10[1,000 - 10 P R - (10,000/P B )] = 1,000P R - 10 P R 2 -10,000 P R / P B – (10,000-100 P R - 100,000/P B ) and taking the derivative with respect to P R , setting that equal to zero, gives RAD’s best response function: П R / P R = 1,000 - 20 P R – ( 10,000 / P B -100) = 0 20 P R = 1,100 - 10,000 / P B P R = 55 – 500/P B . Likewise, BUG’s best response function is P B = 55 – 500/P R . b) Show both firms charging P R = P B = 43.51 is a Nash equilibrium. Plug 43.51 as the prices in both reaction functions. You will see that the two functions are equal to each other at that value and thus P R = P B = 43.51 is a Nash equilibrium. You do not need to solve to get 43.51. But, read on if you want to know how to derive the value of 43.51. Substituting BUG's best response function into RAD's yields We know by symmetry of the two best response functions that P B = P R, so we can simply use the same variable, P B , to represent both: P B = 55 – 500/P B Multiply through by P B : P B 2 = 55 P B – 500 And rearrange to get into standard quadratic form: P B 2 - 55 P B + 500 = 0 Then use the quadratic formula: P B = -b +/- (b 2 - 4ac), where in this case a = 1, b = -55, and c = 500, to get [55 + (3025 - 2000)]/2 43.51 [55 - (3025 - 2000)]/2 11.49 We can substitute these two possible solutions into the profit function and see that 43.51 gives a higher value. P B = 43.51, P R = 43.51. c) If BUG found a way to produce its product (exactly the same product) with a MC=0, would it change the demand that either firm faced? Would it change either firm's reaction function? Would it change the Nash equilibrium? It would not change the demand functions that either firm faced. It would change BUG's best response function, but it would not change RAD's best response function. BUG's best response to any given price of RAD would now be lower than before:
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П B = P B *[1,000 - 10 P B - (10,000/P R )] - 0 = 1,000 P B - 10 P B 2 - 10,000 P B / P R П B / P B = 1,000 - 20 P B - 10,000 / P R = 0 P B = 50-(500/ P R ). As a result, it would change the equilibrium price to a point at which both firms charge lower prices than in (a). Note that this is as far as the question requires you do go. However, if you wanted to derive the actual Nash values, the method to do so is below: P B = 50 – 500/[55-(500/P B )] [55-(500/P B )]*P B = 50*[55-(500/P B )] – 500 55P B – 500 = 2,750 – 25,000/P B – 500 multiply through by P B and rearrange to get the equation into standard quadratic form: 55P B 2 - 2,750P B + 25,000 = 0 P B = [55 + (2,750 2 - 25,000*55*4)]/110 38.06
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This homework help was uploaded on 02/20/2009 for the course AEM 4240 taught by Professor Blalock,g. during the Fall '07 term at Cornell.

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ps3ans_Solutions - AEM 4240 Problem Set#3 Answer Key 1 Rad...

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