Ec5p032007 - Economics 5 Principles of Microeconomics Third...

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Economics 5 D. Richards Principles of Microeconomics Fall, 2007 Third Problem Set **Note: For some reason, my computer changed some of the ‘>’s to ‘ ’s and it wouldn’t let me change them back and I don’t know how to fix it** 1. The market for widgets has fifty (50) identical consumers. Each chooses that combination of widgets, X, and a composite commodity, Y , (representing an agglomeration of all other goods) that maximizes his or her budget subject to a budget constraint. The marginal utility of all other goods taken as a collection is constant. That is, within the range of changes considered here, the amount a consumer spends on these other commodities as a group does not affect the marginal utility of Y because so little is spent on any one commodity that the overall effect is insignificant. To make this concept operational, assume that the marginal utility of Y is fixed at 20. The same does not hold for the marginal utility for widgets or X. This is just one good—not a collection—and it is assumed to obey the natural assumption of diminishing marginal utility. Specifically, the marginal utility of X is assumed to obey the following equation: MU X = 400 – 20X. Each consumer has a budget of $10,000 (think of this as an amount per month) and the price index for Y is given and is equal to $10. a. Use the information you have and the equimarginal condition for utility maximization to determine how many widgets an individual widget consumer will buy at a price of: P X = $200; P X = $180; P X = $160; P X = $140; P X = $120; P X = $100; P X = $80; P X = $60; P X = $40; P X = $20; and P X = 0. b. Multiply the amount purchased by one widget consumer at each price by 50 (the total number of such consumers) to obtain the market demand at each of the prices noted in part (a). MUx/P=2>MU=2P>400-20X=2P Px X (a) 50X (b) 200 0 0 180 20 1000 160 40 2000 140 60 3000 120 80 4000 100 100 5000 80 120 6000 60 140 7000 40 160 8000 20 180 9000 0 200 10000 c. Show that the market (inverse) demand curve just obtained may be represented by the equation: P = 200 – Q/5. MU=2p, (X=Q/50), 2P=400-20Q/50>2P=400-2Q/5>P=200-Q/5 2. An industry is comprised of 60 firms each of which has the following Total Cost function: TC = 15 q + 7.5 q 2
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a. One way to define Marginal Cost is as the increase in Total Cost when output is increased by one unit. That is, one definition of Marginal Cost for the typical firm is as follows: MC A = 15( q + 1) + 7.5( q + 1) 2 – [15 q + 7.5 q 2 ] Show that this measure of Marginal Cost yields a value of MC A = 15 q + 22.5. MC A =15q+15+7.5q 2 +15q+7.5-15q-7.5q 2 15q+15q-15q+7.5+15 15q+22.5 b. A second way to define Marginal Cost is as the decrease in Total Cost when output is reduced by one unit. This alternative definition of MC thus is: MC B = 15 q + 7.5 q 2 – [15( q -1) + 7.5( q - 1) 2 ] Show that this measure of Marginal Cost yields a value of MC B = 15 q + 7.5 MC B =15q+7.5q2-15q+15-7.5 2 -15q-7.5 15q+7.5 c. Since the two measures of marginal cost are not identical, the best way to proceed is to take an average of MC A and MC B .
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