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# PS7 - ECO 310 Fall 2008 Microeconomic Theory A Mathematical...

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ECO 310 - Fall 2008 Microeconomic Theory - A Mathematical Approach Problem Set 7 - Answer Key Question 1: (a) There were two hard parts to this question: nasty algebra, and the intuition. Firm 1°s pro±t is ° 1 = ( p ° c ) q 1 = f [ q 1 + q 2 ] ° 3 = 2 ° c g q 1 : So its Cournot FONC (taking q 2 as given) is @ ° 1 [email protected] 1 = [ q 1 + q 2 ] ° 3 = 2 ° c ° 3 2 [ q 1 + q 2 ] ° 5 = 2 q 1 = 0 Similarly for ±rm 2, @ ° 2 [email protected] 2 = [ q 1 + q 2 ] ° 3 = 2 ° c ° 3 2 [ q 1 + q 2 ] ° 5 = 2 q 2 = 0 Subtracting the two FONCs, we have q 1 = q 2 . This is very useful, and something we might have been able to guess since the two ±rms are symmetric. Write q 1 + q 2 = Q , the industry output, so q 1 = q 2 = Q= 2 . Using this in either FONC, 0 = Q ° 3 = 2 ° c ° 3 2 Q ° 5 = 2 X 2 = 1 4 Q ° 3 = 2 ° c : So Q ° 3 = 2 = 4 c , that is, p = 4 c , and Q = (4 c ) ° 2 = 3 with each ±rm producing half of this. The total pro±t is ° = ° 1 + ° 2 = ( p ° c ) X = (3 c ) (4 c ) ° 2 = 3 = 3 ± 4 ° 2 = 3 c 1 = 3 = 1 : 1906 c 1 = 3 : (b) With ° / c 1 = 3 , total pro±t increases as c increases. What is going on? The economic explanation is the game-theoretic interaction in the duopoly. In Cournot equilibrium, the ±rms are producing too much from the point of view of their joint pro±t-maximizing interest. It would be better for them to produce less and drive the price up. But under Cournot behavior they cannot credibly promise each other to keep outputs low and prices high. When the cost increases, they each produce less, which moves them in the right direction. This raise prices so much (since both are doing it) that pro±ts increase despite the increase in costs and decrease in quantity produced. Of course it wouldn°t pay either ±rm to increase its own cost unilaterally; they have to devise a way whereby both would commit to raising cost. For example, ±rms in such a situation would actually like the government to impose costly regulation on the industry. A recent case of British Airways and Virgin ²ying into Heathrow provides an example of this: the ±rms colluded by raising surcharges together, so that these were almost half of the price of a ticket. Some of these surcharges were costs imposed by Heathrow, which the ±rms may have liked. In fact, they seem to have liked the surcharges so much that they made up their

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PS7 - ECO 310 Fall 2008 Microeconomic Theory A Mathematical...

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