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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 Solutions 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. Answer: The question gives two key facts: one, that MRS is increasing and second that preferences are convex, i.e. consumers like variety. You already know that a consumer with an increasing MRS and good goods has the indifference curve: Indifference Curves with Increasing MRS 1 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain If preferences are convex, then the combination of two bundles A and B which both lie on the indifference curve are preferred to either A or B. In this case, the only way the bundle C (a combination of A and B) is preferred to either A or B is if Q1 and Q2 are "bad" goods, so that preferences are increasing towards the origin as depicted below. Bundle C a Combination of Bundles A & B Is Preferred to Bundles A or B In sum: Q1 and Q2 are "bad" goods. By the way, you can also draw other types of indifference curves: for example, there can be an upward sloping indifference curve. Draw one and ask yourself what kind of goods will generate such indifference curves? 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. Answer: Your utility is: U = Q1 Q2 U = Q11/4 Q23/4 1 = 11/4 Q23/4 Solving for Q2 yields Q2 = 1. Thus y = (1,1). 2 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain (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 . Answer: Because you remain on the same indifference curve, your utility continues to be 1. With Q1 = 2, the utility function is: U = Q1 Q2 U = Q11/4 Q23/4 1 = 21/4 Q23/4 1/21/4 = Q23/4 Q23/4 = 1/21/4 Q2= {1/21/4 }4/3 Q2= 1/2(1/4)(4/3) Q2= 1/21/3 Q2= 0.79 Observe how this answer makes sense: if you consume more of good 1 then, to stay on the same indifference curve, you must consume less of good 2.Thus, z = (2, 0.79). (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. Answer: A 60:40 combination of the bundles y and z is: (Combination Q1, Combination Q2)= (0.6(1) + 0.4(2) , 0.6(1) + 0.4(0.79)) (Combination Q1, Combination Q2)= (1.4, 0.916) This is depicted below: 3 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain You should pay careful attention to this graph since we'll be graphing convex combinations repeatedly. We had to graph a 60:40 combination of y and z. The "x" coordinate of the combination bundle is the 60:40 combination of the "x" coordinates of y and z: 0.6(1) + 0.4(2) = 1.4. But even though the combination bundle is a 60:40 combination of y and z, when you draw the graph, you "cut" the line in a 40:60 ratio. This is important to remember: cut the line in the opposing ratios. Here is why: the 60:40 combination of 1 and 2 is 1.4. Now, the total distance between 1 and 2 is 2 1 = 1. Imagine the "scale" starts at 1: the distance between 1.4 and 1 is 1.4 1 = 0.4 which is 0.4/1 = 40% of the distance between 1 and 2. Similarly, the distance between 1.4 and 2 is 2 1.4 = 0.6, which is 60% of the distance between 1 and 2. You should verify this for the y coordinates too. (d) Choose a bundle x such that y and z are at least as good as x. Show that the combination of y and z is at least as good x and thus that the CobbDouglas utility function represents convex preferences. Answer: In this case, any bundle x below the indifference curve will have y and z be least as good as x. Let's choose x = 0 which gives you utility 0. Clearly, the utility of the combination of y and z is greater than x and for that matter y and z: U = Q1 Q2 4 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain U = Q11/4 Q23/4 U = (1.4)1/4 (0.96)3/4 U = 1.04 This shows that a combination of y and z yields more utility than either y and z so that the consumer has a "taste for variety". 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 5 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain 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) Complements: lim 0 of U = [ Q1 ( 1)/ + Q2( 1)/ ] /( 1) (a) Assume 0 and derive the MRS of the CES utility function. Answer: The CES utility function is: U = [ Q1 ( 1)/ + Q2( 1)/ ] /( 1). To get the MRS we need dU/dQ1 and dU/dQ2: dU/dQ1 = {/( 1)} [ Q1 ( 1)/ + Q2( 1)/ ] {/( 1) 1} {( 1)/} Q1{( 1)/ 1} dU/dQ1 = [ Q1 ( 1)/ + Q2( 1)/ ] 1/( 1) Q1 1/ Similarly: dU/dQ2 = {/( 1)} [ Q1 ( 1)/ + Q2( 1)/ ] {/( 1) 1} {( 1)/} Q2{( 1)/ 1} dU/dQ2 = [ Q1 ( 1)/ + Q2( 1)/ ] 1/( 1) Q2 1/ Hence: MRS = (dU/dQ1)/(dU/dQ2) = Q1 1/ / Q2 1/ MRS = Q2 1/ / Q1 1/ = (/) (Q2/Q1) 1/ 6 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain (b) Compute the MRS of U = Q1 Q2. Answer: From U = Q1 Q2: dQ/dQ1 = Q11 Q2 dU/dQ2 = Q1 Q2 1 The MRS is MU1/MU2. Using the expressions above: MRS = (/)(Q2/Q1) (c) Substitute = 1 in the MRS of the CES utility function: do you get the MRS of U = Q1 Q2? Hint: obviously Answer: The MRS of the CES utility function is: MRS = (/) (Q2/Q1) 1/ If = 1 we get MRS = (/)(Q2/Q1) thus confirming the result that: lim 1 of U = [ Q1 ( 1)/ + Q2 ( 1)/ ] /( 1) will give the indifference curves of U = Q1 Q2. (d) Compute the MRS of U = Q1 + Q2. Answer: Now: dQ/dQ1 = dU/dQ2 = Thus: 7 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain MRS = MU1/MU2 = / (e) Substitute = in the MRS of the CES utility function: do you get MRS of U = Q1 + Q2? Answer: The MRS of the CES utility function is: MRS = (/) (Q2/Q1) 1/ As observe that 1/ 0 so that: MRS = (/)(Q2/Q1)0 = / thus confirming the result that: lim of U = [ Q1 ( 1)/ + Q2 ( 1)/ ] /( 1) will give the indifference curves of U = Q1 + Q2. CES utility function is groovy. 8 ...
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