final_sol_f03 - Name Davidson College Department of...

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Name Davidson College Mark C. Foley Department of Economics Aug - Dec 2003 Intermediate Microeconomic Theory Final Examination Selected Suggested Solutions Structure: There are 175 points on the exam. Problem 1 is worth 40 points. Problems 2 and 3 are worth 30 each. All other problems are worth 25 points each. Directions: You must show all your work to receive full credit. Any assumptions you make and intermediate steps should be clearly indicated. Do not simply write down a final answer to the problems without an explanation. This review is to be taken under the honor code. It is closed-book, closed-notes, untimed, and you may use a calculator. I do require that you take it in one sitting. Sign the honor pledge. Carpe diem. And have a safe, enjoyable, and productive break! Honor Pledge
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Section I: Problems Answer these 4 questions (No choice L ). Consider the following utility function: X XY Y X U + = ) , ( . (a) Are the preferences represented by this utility function homothetic? Why or why not? Homothetic utility functions have a marginal rate of substitution which depends only on the ratio of the quantities of the two goods, not the absolute total amounts. This is useful since indifference curves of higher utility are simply copies of those at lower utility levels, thus results obtained by focusing one (or a few) indifference curve will likely not differ greatly if a different indifference curve (or range of curves) was studied. Homothetic preferences have drawbacks however. The income-consumption curves are all straight lines through the origin. That is, when income is increased (or decreased) by s%, the optimal consumption bundle increases (or decreases) by s%. What is the MRS? X Y MU MU Y U X U MRS Y X 1 + = = = which is not a function of the ratio of X to Y ) , ( Y X U is not homothetic. (b) Using the Lagrangian method of optimization, derive the uncompensated (Marshallian) demand functions for X and Y. Let I represent the individual’s income, X P the price of good X, and Y P the price of good Y. Explain intuitively what the first order conditions mean. The first order conditions (F.O.C.) are necessary conditions for maximizing utility. They are not sufficient to ensure utility maximization. Satisfaction of the F.O.C.’s identifies a potential optimal consumption bundle. Intuitively, if one of the partial derivatives were not equal to zero (more or less than zero), then utility could still be increased by increasing (or decreasing) the amount of X and Y. [ ] Y P X P I X XY L Y X - - + + = λ ± 0 1 0 = - + = X P Y X L ² 0 0 = - = Y P X Y L ³ [ ] 0 0 = + - = Y P X P I L Y X Combining ± & ² yields, ) 1 ( 1 + = = + = Y P P X P X P Y X Y Y X Substituting into ³ and solving for Y yields: Y Y X Y X Y P P I Y I Y P P P Y P 2 ) 1 ( * - = = + +
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X Y Y Y Y X Y Y Y X Y X Y P P I X P P P I P P X P P I P P X Y P P X 2 2 ) 2 ( 1 2 ) 1 ( * + = + - = + - = + =
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This note was uploaded on 08/27/2011 for the course ECON 101 taught by Professor Hal during the Spring '11 term at Ewha Womans University.

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final_sol_f03 - Name Davidson College Department of...

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