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PS17_solutions

# PS17_solutions - Intermediate Microeconomics 2008 Problem...

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Unformatted text preview: Intermediate Microeconomics, 2008 Problem Set No. 17: Solutions Problems (13.16) See solution in back of Perloff. (13.34) a) Use the product-differentiated Bertrand model to analyze the prices the hospitals set before the merger. Find the Nash equilibrium prices of the procedure at the two hospitals. Demand at the two hospitals is given by: q H = 50- . 01 p H + . 005 p N q N = 500- . 01 p N + . 005 p H with marginal costs m = 2000 and fixed costs f = 20000 for each hospital. To find the Nash equilibrium, first we find the best response function of each hospital, then set these equal. Beginning with H ’s profit function: Π H = ( p H- m ) q H- f = ( p H- m )(50- . 01 p H + . 005 p N )- f = 50 p H- . 01 p 2 H + . 005 p N p H- 50 m + . 01 p H m- . 005 p N m- f Taking the derivative with respect to p H , setting equal to zero and solving for p H yields: ∂ Π H ∂p H = 50- . 02 p H + . 005 p N + . 01 m = 0 ⇔ . 02 p H = 50 + . 005 p N + . 01 m p H ( p N ,m ) = 2500 + . 25 p N + . 5 m p H ( p N , 2000) = 3500 + . 25 p N Proceeding in similar fashion, we find that N ’s best response function is: 1 p N ( p H ,m ) = 25000 + . 25 p H + . 5 m p N ( p H , 2000) = 26000 + . 25 p H Substituting N ’s best response function into H ’s best response function, we find: p H = 3500 + . 25(26000 + . 25 p H ) ⇔ . 9375 p H = 10 , 000 p * H = 10 , 666 . 67 Substituting p * H from above into the best response function for N gives: p * N = 26 , 000+ . 25(10 , 666 . 67) = 28666 . 67. Hence prices in the Nash equilibrium are ( p * H ,p * N ) = (10666 . 67 , 28666 . 67). b) After the merger, find the profit-maximizing monopoly prices of the procedure at each hospital. Include the effect of each hospital’s price on the profit of the other hospital. After the merger, the hospitals’ objective is to maximize joint profits. Therefore, we construct the joint profit function: Π = Π H + Π N = ( p H- m ) q H + ( p N- m ) q N- 2 f = ( p H- m )(50- . 01 p H + . 005 p N ) + ( p N- m )(500- . 01 p N + . 005 p H )- 2 f = 50 p H- . 01 p 2 H + . 005 p N p H- 50 m + . 01 p H m- . 005 p N m + 500 p N- . 01 p 2 N + . 005 p N p H- 500 m + . 01 p N m- . 005 p H m- 2 f = 50 p H- . 01 p 2 H + . 01 p N p H- 550 m + . 005 p H m + . 005 p N m + 500 p N- . 01 p 2 N- 2 f Now take partial derivatives with respect to p H and p N , set each equal to zero and solve for p H and p N (respectively) in order to choose the optimal price at each hospital: 2 ∂ Π ∂p H = 50- . 02 p H + . 01 p N + . 005 m = 0 ⇔ . 02 p H = 50 + . 01 p N + . 005 m p H ( p N ,m ) = 2500 + . 5 p N + . 25 m p H ( p N , 2000) = 3000 + . 5 p N ∂ Π ∂p N = 500- . 02 p N + . 01 p H + . 005 m = 0 ⇔ . 02 p N = 500 + . 01 p H + . 005 m p N ( p H ,m ) = 25000 + . 5 p H + . 25 m p N ( p H , 2000) = 25500 + . 5 p H Now substitute p N ( p H , 2000) into p H ( p N , 2000) and solve for p H : p H = 3000 + . 5(25500 + . 5 p H ) ⇔ . 75 p H = 15750 p * H = 21000 Substituting p * H from above into p N ( p H , 2000) gives...
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PS17_solutions - Intermediate Microeconomics 2008 Problem...

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