Homework 2 Solution1 Econ 4001.03, Fall 2013 Prof. Lixin Ye Homework 2 Solution 1. True or False, and explain briefly. 1) The assumption that more is better implies that the indifference curves are upward sloping. False. If the indifference curves are upward sloping, the monotonicity assumption that more is better is violated. 2) Convexity of indifference curves implies that consumers are willing to give up more “y” to get an extra “x” the more “x” they have. False. Convexity of indifference curves implies that consumers are willing to give up more “y” to get an extra “x” the less “x” they have. 3) Consider the following three bundles. Bundle Good x Good y A 3 9 B 9 6 C 7 7 If Bundles A and B are on the same indifference curve, preferences satisfy all the usual assumptions introduced in the lecture, Bundle C is preferred to Bundle A. True. It can be checked that 1233C AB= +, thus C is a convexed combination of A and B. By the convexity of the preferences, C is preferred to A. 4) The utility functions 23(, )Uxy xy= and ( , ) 2log 3log 5Vxy x y=++represent the same preferences over goods xand y. True. Note that (, ) log (, ) 5Vxy Uxy= +, so Vis a positive monotone transformation of U. Alternatively we may calculate the marginal rate of substitutions under both utility functions. It can be verified that both functions give rise to the same marginal rate of substitution:,,23UVxy xyyMRS MRSx= =. 2. Consider two goods, A and B. For each of the following scenarios, develop the utility function (,)U AB that matches the given information.2 1) The consumer believes that goods A and B are perfect substitutes with one unit of A equivalent to two units of B. Since A and B are perfect substitutes, and one unit of A is equivalent to two units of B, we have2ABMU MU. The utility function can thus be represented by: (,) 2U AB A B= + (1.1) (To verify (1.1) is correct, we know 2AMU  is the slope with respect to A , and 1BMU  is the slope with respect to B. Thus 2ABMU MU holds. Also note that the representation of this preference is not unique, as any positive monotone transformation of (1.1) also represents the same preference. For example, (,)2(2 )4 2U AB A B A B   can be another utility function to represent the preference described in the question.) 2) The consumer believes that goods A and B are perfect complements and always uses two units of B for every unit of A. This means that for the most efficient use of A and B, two units of B should go with every unit of A. In other words, the most efficient bundles should satisfy equation2BA. Given this, the preference can be represented by the following utility function: (,) min(2,)U AB AB= (1.2) (Again the utility function is not unique, as any positive monotone transformation, say 1min(2 , ) min( , / 2)2AB AB=, also represents the same preference.) 3. Suppose a consumer has preferences over two goods that can be represented by the utility function 3UX Y= +. 1) What are XMUand YMU? Describe the relationship between good X and good Y. 1=XMUand 3YMU=. Goods Xand Yare perfect substitutes in a three-for-one ratio (3 units of Xis equivalent to 1 unit of Y). 2) Describe the shape of the indifference curves. The indifference curves are straight lines with a slope of -1/3.3 3) Describe the special properties of theYXMRS,. Because the rate at which the consumer will trade good Xfor good Ydoes not vary along the indifference curve (due to the fact that the two goods are perfect substitutes), the YXMRS,is constant. 4. Suppose a consumer has preferences over two goods that can be represented by the utility function min{ ,3 }U XY=. 1) Describe the relationship between good X and good Y. Goods Xand Yare perfect complements for each other in a fixed proportion. That is, the consumer wants to consume 3 units of Xwith each unit of Y. 2) Describe the shape of the indifference curves. The indifference curves are “L-shaped” with the “points” of the “L’s” occurring where /3YX= (which is the efficient line). 3) Describe the special properties of the YXMRS,. The marginal rate of substitution doesn’t have a “normal” meaning in the context of two goods that are perfect complements. The YXMRS,is undefined at the corner point of the L. 5. Consider the utility function (, ) 2Uxy x y= +. 1) Is the assumption that “more is better” satisfied for both goods? Yes. 1/xMU x= and 1yMU =. Since the marginal utility is greater than zero for both goods, increasing consumption of either good will increase total utility. 2) What is ,xyMRS for this utility function? ,1xxyyMUMRSMUx=−=− 3) Is the marginal rate of substitution diminishing, constant, or increasing in x as the consumer substitutes x for y along an indifference curve?4 ,| | 1/xyMRS x= is decreasing inx. 4) Will the indifference curve corresponding to this utility function be convex to the origin, concave to the origin, or straight lines? Explain. The indifference curves will be convex to the origin since the marginal rate of substitution is decreasing as the consumer substitutes x for y along an indifference curve. More specifically, that ,/ | 1/xyUUMRS dy dx x== = − is increasing in x implies that 22/| 0UUd y dx=>, i.e., the indifference curve is convex to the