Econ11Fall2007ProblemSet_4AnswersDue31Oct

Econ11Fall2007ProblemSet_4AnswersDue31Oct - A. Yezer THE...

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THE GEORGE WASHINGTON UNIVERSITY Department of Economics A. Yezer Answers to 4 th Problem Set in Econ 11-10 Fall 2007 1a. Assume that Jahangir currently consumes 10 apples per month and that his monthly marginal utility for apples is given by MU A = 20 - A, where A is the number of apples consumed per month and MU is marginal utility. Assume further that his marginal utility of grapes and bananas are given by MU G = 40 - 4G, and MU B = 60 - 6B, where G and B are in pounds per month. You are told that apples cost $1 each, grapes cost $2 per pound and bananas cost $3 per pound. If Jahangir currently consumes 10 apples per month, how many pounds of grapes and bananas does he consume per month? How would your answer change if Jahangir consumed 5 apples per month? In this problem you are to apply the golden rule for utility-maximization from consumer's demand theory which states that, for all goods consumed, the marginal utility of the last dollar spent on each good must be equal. Now the marginal utility of the last dollar spent on a good is the quotient of marginal utility divided by price. In this case, we know that MU A = 20 - A, and P A = 1, so MU A /P A = (20 - A)/1 = 20 - A, and we see that it is a simple decreasing function of A. If Abhijit consumes 10 apples per month, then MU A /P A = 20 - 10 = 10. We also know that MU B = 60 - 6B, and P B = 3, so MU B /P B = (60 - 6B)/3 = 20 - 2B. Furthermore, MU G = 40 - 4G and P G = 2, so MU G /P G = (40 - 4G)/2 = 20 - 2G. Given that Abhijit consumes 10 apples, we know that the marginal utility of the last dollar spent on apples is 10 as established above. Therefore, the marginal utility of the last dollar spent on bananas and grapes must also be 10 for him to maximize utility. How many B and G does it take to achieve this result? Setting 10 = 20 - 2G and find that G = (10 - 20)/-2 = -10/-2 = 5 pounds of grapes and setting 10 = 20 - 2B, find that B = 5 pounds of banannas is the combination of B and G that maximizes utility! 1b. Assume that biologists have created a new fruit, the lemopear, by crossing lemons with pears and that Jahangir 's MU for lemopears is given by MU L = 10 - L. This new creation has been priced at $5 each. How many lemopears will Jahangir consumer per month? Let's solve for the relation between the number of lemopears and marginal utility of the last dollar spent on L as follows: MU L /P L = (10 - L)/5 = 2 - 0.2L. But Jahangir already gets MU/P = 10 from apples. But even when L = 0 MU L /P L = 2, so there is no way that lemopears can generate sufficient marginal utility per dollar of expenditure for them to be attractive to Jahangi r . He maximizes utility by consuming 0 lemopears. This is called a "corner solution" and we all know lots of goods that we don't consume. For these goods, the marginal utility of the first dollar of expenditure is low enough so that it does not attract us compared to alternatives. Likely most of you would not be attracted by a fruit produced by crossing a lemon and a pear - a silly, proposition that will never be undertaken unless
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Econ11Fall2007ProblemSet_4AnswersDue31Oct - A. Yezer THE...

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