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Unformatted text preview: CHAPTER 4 APPENDIX DEMAND THEORY – A MATHEMATICAL TREATMENT EXERCISES 1. Which of the following utility functions are consistent with convex indifference curves and which are not? a. U(X, Y) = 2X + 5Y b. U(X, Y) = (XY) 0.5 c. U(X, Y) = Min(X, Y), where Min is the minimum of the two values of X and Y. Indifference maps for the three utility functions are presented in Figures 4A.1(a), 4A.1(b), and 4A.1(c). The first is a series of straight lines, the second is a series of hyperbolas and the third is a series of Lshaped curves. Only the second utility function has strictly convex indifference curves. To graph the indifference curves which represent the preferences given by U(X,Y) = 2X + 5Y, set utility equal to some level, U , and solve for Y to get X U Y 5 2 5 = . Since this is the equation for a straight line, the indifference curves are linear with intercept U 5 and slope 5 2 . The graph shows three indifference curves for three different values of U , where U < U 1 < U 2 . U U 1 U 2 U 2 U 1 2 U 2 2 U 5 U 1 5 U 2 5 Y X Figure 4A.1(a) To graph the indifference curves that represent the preferences given by U ( X , Y ) = ( XY ) 0.5 , set utility equal a given level U and solve for Y to get Y = U 2 X . By plugging in a few values for X and solving for Y, you will be able to graph the indifference curve for utility value U , which is illustrated in Figure 4A.1(b), along with the indifference curve for a larger utility value, U 1 . X Y U U 1 Figure 4A.1(b) To graph the indifference curves which represent the preferences given by U ( X , Y ) = Min ( X , Y ) , first note that utility functions of this form result in indifference curves that are Lshaped and represent a complementary relationship between X and Y. In this case, for any given level of utility U , the minimum value of X and Y will also be equal to U . If X increases but Y does not, utility will not change. If both X and Y change, then utility will change, and we will move to a different indifference curve. See the following table which illustrates how the utility value depends on the amounts of X and Y in the consumption bundle. X Y U 10 10 10 10 12 10 12 12 12 12 11 11 8 11 8 8 9 8 X Y U o U 1 U U 1 U U 1 Figure 4A.1(c) 2. Show that the two utility functions given below generate the identical demand functions for goods X and Y: a. U(X, Y) = log(X) + log(Y) b. U(X, Y) = (XY) 0.5 If two utility functions are equivalent, then the demand functions derived from them are identical. Two utility functions are equivalent if you can transform them are identical....
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This note was uploaded on 01/04/2012 for the course ECON 101 taught by Professor Pr.makushi during the Spring '11 term at San Diego Christian.
 Spring '11
 Pr.Makushi
 Utility

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