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Unformatted text preview: Ecn 100  Intermediate Microeconomic Theory University of California  Davis October 25, 2010 John Parman Problem Set 3  Solutions This problem set will be graded and is due by 5pm on Wednesday, November 3rd in my mailbox in the economics department. You may turn in problem sets early by putting them in my mailbox in the economics department or by dropping them off in lecture. No late problem sets will be accepted. You are welcome to work in groups. If working in a group, everyone in the group must still submit an individual problem set. 1. Deriving Market Demand There are four individuals, Aaron, Bob, Carl and David, who buy widgets. Their individual demands are given by the following inverse demand functions: P ( x A ) = 10 x A (1) P ( x B ) = 20 2 x B (2) P ( x C ) = 20 x C (3) P ( x D ) = 15 2 x D (4) (a) Derive an expression for the market demand for widgets (you can assume that Aaron, Bob, Carl and David are the only people buying widgets). Graph this market demand with price on the vertical axis and widgets on the horizontal axis. To get the market demand, we need to sum the demand curves but we have to be careful about which consumers are demanding positive quantities of the good. From the inverse demand equations, we can see that above a price of 20, nobody has positive demand, between prices of 15 and 20 only Bob and Carl have positive demands, between prices of 10 and 15 Bob, Carl and David have positive demands and below prices of 10 everyone has positive demand. We can calculate market demand for each of these ranges of prices by adding up the relevant demand equations (not the inverse demand equations): D ( p ) = 0 for p > 20 D ( p ) = x B + x C = (10 1 2 p ) + (20 p ) = 30 3 2 p for 15 < p ≤ 20 D ( p ) = x B + x C + x D = 30 3 2 p +(7 . 5 1 2 p ) = 37 . 5 2 p for 10 < p ≤ 15 D ( p ) = x B + x C + x D + x A = 37 . 5 2 p + (10 p ) = 47 . 5 3 p for p ≤ 10 2 Problem Set 3  Solutions (b) Along which segment of the market demand curve is demand most elastic? Along which segment is demand most inelastic? Now that we know the demand function for each segment, we can find the elasticity as p q dq dp . For each segment, dq dp is just the slope of the de mand curve. Elasticity will vary along each segment because p and q are changing. What we can do is just plug in the values of p and q at the endpoints of each segment to find out what the range of elasticity values are along a segment: = p q dq dp∞ ≤ ≤  3 for 15 < p ≤ 20 4 ≤ ≤  1 . 14 for 10 < p ≤ 15 1 . 71 ≤ ≤ 0 for p ≤ 10 There is some overlap in the elasticity ranges due to the kinks in the demand curve, but we can see that the segment of the demand curve to the left is the most elastic while the segment all of the way to the right is the most inelastic. If a producer wanted to maximize total revenue, he would operate on this rightmost segment of the demand curve where he can set the price such that elasticity is equal to 1....
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 Winter '08
 PARMAN
 Economics, Supply And Demand, tax revenues

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