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# solutions - Problem Set 6 Suggested Solutions 1 First write...

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Problem Set 6 Suggested Solutions 1 First, write each firms profit as a function of q 1 and q. π 1 = R 1 ( q 1 , q 2 ) - C 1 ( q 1 , q 2 ) = (10 - Q ) q 1 - c 1 q 1 = (10 - [ q 1 + q 2 ]) q 1 - c 1 q 1 = 10 q 1 - q 2 1 - q 1 q 2 - c 1 q 1 π 2 = R 2 ( q 1 , q 2 ) - C 2 ( q 1 , q 2 ) = (10 - Q ) q 2 - c 2 q 2 = (10 - [ q 1 + q 2 ]) q 2 - c 2 q 2 = 10 q 2 - q 2 2 - q 1 q 2 - c 2 q 2 Since firm 1 will maximize profit by choosing q 1 , take the FOC of π 1 wrt q 1 . π 0 1 = 10 - 2 q 1 - q 2 - c 1 = 0 q 1 = 10 - q 2 - c 1 2 Since firm 2 will maximize profit by choosing q 2 , take the FOC of π 2 wrt q 2 . π 0 2 = 10 - 2 q 2 - q 1 - c 2 = 0 q 2 = 10 - q 1 - c 2 2 Combining these two equations gives q 1 = 10 - 10 - q 1 - c 2 2 - c 1 2 = 20 - 10+ q 1 + c 2 - 2 c 1 4 3 q 1 = 10+ c 2 - 2 c 1 q * 1 = 10+ c 2 - 2 c 1 3 q * 2 = 10 - 10+ c 2 - 2 c 1 3 - c 2 2 = 30 - 10 - c 2 +2 c 1 - 3 c 2 6 6 q 2 = 20 + 2 c 1 - 4 c 2 q * 2 = 10+ c 1 - 2 c 2 3 Substituting these quantities into the demand curve gives the price: P = 10 - Q = 10 - ( q 1 + q 2 ) = 10 - 10+ c 2 - 2 c 1 3 + 10+ c 1 - 2 c 2 3 = 10 - 20 - c 2 - c 1 3 = 10+ c 2 + c 1 3 Substituting these back into the profit functions gives: π * 1 = Pq 1 - c 1 q 1 = 10+ c 2 + c 1 3 * 10+ c 2 - 2 c 1 3 - c 1 10+ c 2 - 2 c

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solutions - Problem Set 6 Suggested Solutions 1 First write...

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