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Lecture

# Homework 1 - Solutions.pdf

7 Pages
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Department
Economics
Course
ECO204Y5
Professor
Kathleen Wong
Semester
Fall

Description
ECO204: Homework 1 Solutions 1. True/False/Uncertain a. False. When income increases from I 1o I2, consumption of good Y increases (normal good), but consumption of good X decreases as incomes rise. This indicates that good X is viewed as an inferior good by this individual. Therefore, the statement that both X and Y are normal goods is false. In order for this to be so, we should observe a higher level of consumption for both goods as income increases. b. False. If Karen has convex preferences (we know this because the problem states that strawberries and apples are not perfect substitutes or perfect complements), her willingness to trade apples for strawberries (or in other words, her MRS) is not going to be constant. In particular, since her utility-maximizing demand for strawberries and apples will change with the prices, her MRS in the summer will be different from her MRS in the winter. c. True. Since demand is inelastic, the percentage decrease in Q when prices increase will be less than the percentage increase in P. So the overall decrease in Q will not be enough to offset higher prices. As a result, total expenditures on inelastic goods (Q x P) will increase when P increases. d. False. If Mike is maximizing utility, his indifference curve will be tangent to the budget line. Or, MU P PP the slopes of the two will be equal to each other:  . Given the information from the MPB PB MU 4 2 P 3 MU P problem, we can calculate that P   and P  , so we know that P  P . In order MPB 6 3 PB 5 MP B PB to move towards the bundle that guarantees MU P  PP , Mike should consume more pizza and MP B PB less burgers. If his current consumption bundle is at point A, the slope of his indifference curve MU P  is greater than the slope of the budget lne P P . He can gain a higher level of utility by MP B PB  consuming more pizza and less burger until he reaches point B, where the slope of his indifference curve and the budget line are tangent to each other. Another way to look at this is to apply the equal marginal principle, which is essentially a calculation of how much utility is provided by each dollar spent on each of the goods. Currently, utility/price for pizza = 4/3 = 1.33 and utility/price = 6/5 = 1.20. Since Mike gets more utility from spending \$1 on pizza, he should consume more pizza. Eventually, the utility/price from consuming pizza will fall until it equals the utility/price from consuming burgers. See graph on the next page. e. The statement is uncertain. Without a functional form for the utility function, we don’t know what Henrik’s preferences are. If we assume X and Y are perfect substitutes or that Henrik has extreme preferences (favoring one and completely hating the other), the statement would be false. The utility-maximizing bundle for both of these cases would result in a corner solution, meaning that one good would not be consumed at all. Thus, although the corner solution would be affordable (lying on the budget constraint), the utility-maximizing bundle would only contain positive values of one good, and zero units of the other. However, if the two goods are not perfect substitutes and we assume Henrik doesn’t have extreme preferences for the two goods, the utility-maximizing bundles would indeed lie on the budget constraint and contain positive values of both good X and good Y. The statement in this case would be true. B At A, the MRS >P-PB/P A B IC2 IC1 P 2. Income and Substitution Effects I = 24, P = 1, P = 2, U(C,D) = CD C D L  0 D P 0 D  C C P C L C  0 C PD 0  D PD L  0 I  PCC  D D 0 P C  PDD  I  Setting λs equal to each other, solve for C or D: D  C P P C D PCC D  P D PDD C  PC Substitute C or D into the budget constraint: PCC P D  I P PDD  P D I C P  D  C  PDD  PDD  I 2PDD  I D  I 2P D PCC P D  I PCC PCC P D P I D PDD  CC  I 2PCC  I C  I 2P C b. IfCP = \$1 andDP = \$2: C = \$24/2(\$1) = 12, D = \$24/2(\$2) = 6. Initial consumption bundle = 12C, 6D c. IfCP= \$1 and D = \$3: C = 12, D = \$24/2(\$3) = 4. Final consumption bundle: 12C, 4D. d. Initial utility: U(12, 6) = 12(6) = 72 Incorporating new prices:U C  PC  D  1  C  3D MU D PD C 3 Substituting C = 3D into U(C,D) = CD = 72 (3D)D = 72 3D = 72 2 D = 24 D = 2√6 ≈ 4.9 Substitute the value of D into the relationship C = 3D: C = 3(4.9) = 14.7 Decomposition bundle: 14.7C, 4.9D e. Substitution effect: 4.9D – 6D = -1.1D f. Income effect: 4D – 4.9D = -0.9D g. Total effect: SE + IE = -1.1 + (-0.9) = 2D When the price of donuts increased, Charlie decreased his consumption of donuts by 2 (to verify, compare answers for 2b and 2c). 3. Q(P) = 8 – 2P a. Q + 2P = 8 2P = 8 – Q P = 4 – 0.5Q b. on the next page. c. at P = \$1, Q = 8 – 2P = 8 – 2(\$1) = 8 – 2
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