eco204_summer_2009_practice_problem_3

Eco204_summer_2009_p - University of Toronto Department of Economics ECO 204 Summer 2009 S Ajaz Hussain ECO 204 Summer 2009 S Ajaz Hussain Practice

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Unformatted text preview: University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain ECO 204 Summer 2009 S. Ajaz Hussain Practice Problems 3 Please help improve the course by sending me an email about typos or suggestions for improvements Note: Please don't memorize these solutions in the expectation that similar questions will appear on tests and exams. Instead, try to understand how to derive the answer as you'll be tested on techniques and applications, not on memorization. Moreover, tests and exams will cover topics and techniques that may not be in these practice problems. You are urged to go over all lectures, class notes and HWs thoroughly. Question 1 (20082009 Test 1 Question) In a marketing survey, consumers indicate they have an increasing marginal rate of substitution between Q1 and Q2. The survey also indicates that these consumers have convex preferences over Q1 and Q2. What can be said about the nature of Q1 and Q2? Give a brief explanation using a diagram to illustrate your answer. Question 2 In this question, you will practice checking for convex preferences via a mathematical approach. You know that the CobbDouglas utility function U = Q1 Q2 represents convex preferences. It may be helpful to review the definition of convex preferences again. Suppose = and = . (a) Suppose you're currently consuming a unit of good 1. If your utility is 1, how many units of good 2 are you consuming? Label this bundle y. (b) Suppose you increase your consumption of good 1 by a unit. How much of good 2 must you consume to remain on the same indifference curve? Label this bundle . (c) Suppose you combine bundles y and z in a 60:40 ratio. What is the combination bundle? Graph the combination, y and z bundles below. (d) Choose a bundle x such that y and z are at least as good as x. Show that the combination of 1 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain y and z is at least as good x and thus that the CobbDouglas utility function represents convex preferences. Question 3 In this question you will practice another common utility function in economics and finance: the CES utility function. The nice thing about it is that it encompasses other common utility functions such as the CobbDouglas, perfect substitutes and complements as special cases. Consider a consumer with a consumption set over two goods Q1 and Q2. The CES utility function is: U = [ Q1 ( 1)/ + Q1 ( 1)/ ] /( 1) where , and are constants. Often as in the lectures the term ( 1)/ is expressed in terms of the parameter ("row") where: = ( 1)/ so that the CES utility function becomes: U = [ Q1 + Q1 ] 1/ I am going to work with U = [ Q1 ( 1)/ + Q1 ( 1)/ ] /( 1) but be forewarned that you may see the CES utility expressed differently in the future. In this question you will show that the utility functions: Imperfect Substitutes: U = Q1 Q2 Perfect Substitutes: U = Q1 + Q2 are "special cases" of the CES utility function by showing that as varies, the MRS of the indifference curves of the CES utility function are approximately equal to the MRS of the indifference curves of the imperfect substitutes and perfect substitutes utility functions. You are not responsible for proving the result for complements utility function, although it would be a nice exercise applying L'Hospital's rule. Imperfect Substitutes: lim 1 of U = [ Q1 ( 1)/ + Q2 ( 1)/ ] /( 1) Perfect Substitutes: lim of U = [ Q1 ( 1)/ + Q2( 1)/ ] /( 1) 2 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain Complements: lim 0 of U = [ Q1 ( 1)/ + Q2( 1)/ ] /( 1) (a) Assume 0 and derive the MRS of the CES utility function. (b) Compute the MRS of U = Q1 Q2. (c) Substitute = 1 in the MRS of the CES utility function: do you get the MRS of U = Q1 Q2? Hint: obviously (d) Compute the MRS of U = Q1 + Q2. (e) Substitute = in the MRS of the CES utility function: do you get MRS of U = Q1 + Q2? 3 ...
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This note was uploaded on 05/02/2011 for the course ECO 204 taught by Professor Hussein during the Fall '08 term at University of Toronto- Toronto.

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