API 111 midterm 2008 solutions

API 111 midterm 2008 solutions - Economics 2020a / HBS 4010...

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Economics 2020a / HBS 4010 / Kennedy School API-111 Microeconomic Theory Midterm Examination October 29, 2008 This exam contains three questions for a total of 80 points. Please complete all three questions. The exam is open book, so it is acceptable to use notes, books, calculators and laptop computers (assuming that the laptop computer is not connected to the internet). 1. (15 points) A consumer has Cobb-Douglas preferences corresponding to the (ordinal) utility function u(x, y) = x 1/2 y 1/2 , so that u(x, x) = x . Your friend says that he wishes to construct a different utility function v to represent these same preferences based on the rule that v ( x , 4 x ) = x in place of the rule u ( x , x ) = x . So for example, his utility representation satisfies v (1, 4) = 1; v (2, 8) = 2, and so on. (a) Graph two indifference curves labeled with corresponding utility values for his utility function. Note that his indifference curves are the same as those for u(x, y) = x 1/2 y 1/2 , with the only difference that the utility values associated with each indifference curve have been slightly changed. (Graph Omitted). (b) Find the utility function v resulting from his constructive method. By definition of v, v(x, 4x) = x. But u(x, 4x) = x 1/2 4x 1/2 = 2x. That is, for any given indifference curve, the utility value corresponding to function v is half the utility value corresponding to function u. That is, v(x, y) = x 1/2 y 1/2 / 2. An alternate approach would be to determine the indifference relationship between points of form (x, x), which determine the value of u, and points of form (x’, 4x’), which determine the value of v. Specifically, u(x, x) = x and u(x’, 4x’) = 2x’, so (x’, 4x’) ~ (x, x) if x = 2x’. That is (2x’, 2x’) ~ (x’, 4x’), so u(x’, 4x’) = u(2x’, 2x’) = 2x’. This implies in turn that v(x’, 4x’) = x’ = ½ u(x’, 4x’), so in general, v(x, y) = ½ u(x, y). (c) Identify a transformation function f so that v(x, y) = f(u(x, y)) . As shown in (b), v(x, y) = u(x, y) / 2, corresponding to transformation function f(u) = u / 2.
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2. (25 points) A consumer tells you that he is indifferent between lotteries L1 = ( p 1 = 0.2, p 2 = 0.5, p 3 = 0.3) L2 = ( q 1 = 0.4, q 2 = 0.2, q 3 = 0.4) where p j is the probability of outcome j in lottery L1, q j is the probability of outcome j in lottery L2, and where outcome 1 is strictly preferred to outcome 2 and outcome 2 is strictly preferred to outcome 3. Assume that preferences are consistent with expected utility maximization in parts (a) through
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API 111 midterm 2008 solutions - Economics 2020a / HBS 4010...

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