ak3 - Homework #3 Answer Key due September 3 in class Econ...

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Homework #3 Answer Key due September 3 in class University of California, Los Angeles Each question is worth 5 points, for a total of 25 points. 1. Suppose there is a market made up of two types of individuals with the following demand curves. x 1 ( P ) = 13 2 P x 2 ( P ) = 5 P where P is the market price of good x and x i for i = 1 ; 2 is the quantity demanded of each type at every price. (a) If there is only one consumer of each type, ±nd the market demand. ANSWER: If P > 6 : 5 , market demand is zero; Q D ( P ) = 0 . If 5 < P < 6 : 5 , market demand consists only of type 1 ; Q D ( P ) = 13 2 P . If P < 5 , market demand consists of Q D ( P ) = x 1 ( P ) + x 2 ( P ) = 18 3 P . (b) Suppose there is a single, pro±t-maximizing producer (with no in²u- ence over the price) with a cost function C ( x ) = 1 6 x 2 + 25 where x is the quantity supplied. Find the ±rm³s supply function. ANSWER: We know the ±rm will produce where P = MC : P = MC ( x ) = 1 3 x This means his supply function is Q S ( P ) = 3 P (c) Given the supply function from part b , what is the market price P for good x and how much is sold? ANSWER: The market demand must equal market supply for the market to clear. Since we know market demand falls into three parts, we will check if P < 5 ±rst: Q D ( P ) = Q S ( P ) 18 3 P = 3 P P = 3 Since P < 5 , we know that we have found the market equilibrium. At P = 3 , Q S (3) = 3 (3) = 9 . 1
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(d) At that price, what is the producer surplus? What is the consumer surplus enjoyed by each type of consumer? ANSWER: Producer surplus is the amount below the market price Notice that both supply and demand for each type of consumer is a straight line. That means this amount is calculated as the area of a triangle. the P -intercept, we calculate inverse supply and demand: P = 1 3 Q P = 13 2 1 2 x 1 P = 5 x 2 Here, that area is PS = 1 2 (3 0) (3 (3)) = 27 2 Consumer surplus of each type consists of the area above the market price each consumer would have been willing to purchase. For the CS 1 = 1 2 13 2 3 ± (13 2 (3)) = 12 : 25 CS 2 = 1 2 (5 3) (5 3) = 2 (e) Suppose there are two consumers of type 1 and eight of type 2. Find the market demand, the market price if there is only one producer, and the quantity sold. ANSWER: If P > 6 : 5 , market demand is zero; Q D ( P ) = 0 . If 5 < P < 6 : 5 , market demand consists only of type 1 ; Q D ( P ) = 2 [13 2 P ] = 26 4 P . If P < 5 , market demand consists of Q D ( P ) = 2 [13 2 ( P )] + 8 [5 P ] = 66 12 P . The market supply function has not changed. Again, we know markets must clear, so we check to see if the market-clearing price is less than
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This note was uploaded on 09/23/2008 for the course ECON 11 taught by Professor Cunningham during the Summer '08 term at UCLA.

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ak3 - Homework #3 Answer Key due September 3 in class Econ...

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