Econ 200 Problem Set 3 Solution

Econ 200 Problem Set 3 Solution - Answer Key: Problem Set 3...

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Answer Key: Problem Set 3 February 8, 2005 Problem 1 Consider a market with inverse demand P =11 Q ,whe re Q is the sum of output produced by all f rms in the market. Firms have identical cost functions c ( q )=2+ q . Question a Suppose there is a single f rm in this market. What price and quantity will the f rm choose? What will be the consumer surplus? What will be the f rm’s pro f t? ANSWER: The f rm solves max q { (11 q ) q (2 + q ) } It’s easy to see the f.o.c. is 11 2 q 1=0 which gives q m =5 ,p m =6 π m =( 1 0 5)5 2=23 CS m · 5 · 1 2 =12 . 5 Question b Suppose there are n f rms in the market that compete by simultaneously choos- ing quantities. What quantities, q c ( n ) , will they choose in equilibrium? How do these quantities change with the number of f rms? What is the total quantity Q c ( n ) produced in equilibrium and how does it change with the number of f rms in the market? What is the pro f t π c ( n ) and how does it change with n ?Wha t is the consumer surplus and how does it change with n ? 1
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ANSWER: Firm i solves max q i { (11 q i Q i ) q i (2 + q i ) } The f.o.c.(w.r.t. q i )is 11 2 q i Q i 1=0 Symmetry implies Q i =( n 1) q i ,which reduces the f.o.c. to 10 ( n +1) q i = 0 . (Note that if you make this substitution before deriving the FOC, you will not get the right soultion; remember that f rm i does not get to choose the quantities of its competitors!) Therefore: q c i ( n )= 10 n +1
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Econ 200 Problem Set 3 Solution - Answer Key: Problem Set 3...

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