Homework 2 Solutions

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1 Homework 2 Solutions Econ 5 - Stanford Universit - Winter Quarter 215/16 Januar 22, 216 Eercise 1: Based on Varian, Microeconomics, 8e, Problem 4.1 A college football coach sas that given an two linemen, A and B, he alwas prefers the one who is bigger and faster. (a) Is this preference relation transitive? Wh or wh not? (b) Is this preference relation complete? Wh or wh not? (c) Is this preference relation monotonic? Wh or wh not? Answer: (a) Yes. A being bigger and faster than B and B being bigger and faster than C implies that A is bigger and faster than C. Therefore, A B,B C = A C (b) No. If A is bigger but B is faster, then the preference is not defined. (c) Yes. More of either qualit is better. Eercise 2: Utilit Functions (Lecture 5) For each of the following utilit functions, sketch the indifference curve u(, ) = 128, and write the epressions for the partial derivatives u(,) and u(,) and the marginal rate of substitution. (a) u(, ) = 2 3 (b) u(, ) =2 +4 (c) u(, ) =min{, 2} (d) u(, ) = + (e) u(, ) = +ln (f) u(, ) = (g) u(, ) = 1

2 Answer: (a) u =23, u =32 2, MRS = 2 3. (b) u u =2, =4, MRS = 1 2. (c) If <2, u(,) = 1, u(,) =, and MRS =. If>2, u(,) =, u(,) = 2, and MRS 2

3 =. If =2, the derivatives are undefined. (d) 5 5 u = 1 2, u = 1 2, MRS =. (e) u u =1, = 1, MRS =. (f) u u =2, =2, MRS =. 3

4 (g) u u =1, = 1, MRS = 1. Eercise 3: Preferences and Utilit (Lectures 4 and 5) In Lecture 4, we said that a preference relation was conve if it satisfied the following condition: if A and B are on the same indifference curve, and C is a conve combination of A and B (i.e., lies along a line connecting A and B), then C is preferred to both A and B. For each of the utilit functions from Eercise 2, determine whether the preferences represented b that utilit function is conve or not. In particular, for each function, choose points A, B, and C that fit the above definition, draw the relevant diagram showing those points, and show whether the utilit at point C is higher than, equal to, or less than the utilit at points A and B. Answer: (a) u(, ) = 2 3 As shown in the indifference curves below, the utilit at point C is higher than the utilit at A or B, as C is on an indifference curve with higher utilit. The preference relation is conve. (b) u(, ) =2 +4 As shown in the indifference curve below, the utilit at point C is the same as the utilit at A or B, as the three points are on the same indifference curve. The utilit function is not strictl conve (though it is weakl conve). In fact, it s a straight line and goods and are perfect substitutes. 4

5 (c) u(, ) =min(, 2) If we choose two points A and B on different sides of the gre line, the linear combination C has a higher utilit. However, if ou choose two points A and B on the same sides of the gre line, the linear combination C has the same utilit as A and B. Therefore, the preference as a whole is not strictl conve. (It is weakl conve.) (d) u(, ) = + As shown in the indifference curves below, the utilit at point C is higher than the utilit at A or B, as C is on an indifference curve with higher utilit. The utilit function is conve. 5

6 (e) u(, ) = +ln As shown in the indifference curves below, the utilit at point C is higher than the utilit at A or B, as C is on an indifference curve with higher utilit. The utilit function is conve A C B (f) u(, ) = As shown in the indifference curves below, the utilit at point C is lower than the utilit at A or B, as C is on an indifference curve with lower utilit. The utilit function is not conve, but concave. 6

7 (g) u(, ) = As shown in the indifference curve below, the utilit at point C is the same as the utilit at A or B, as the three points are on the same indifference curve because the indifference curves are lines. The utilit function is not strictl conve C B A

8 Eercise 4: Utilit Functions (Lecture 5) Which sets of the following utilit functions represent the same preferences? How do ou know? (a) u(, ) =3 +4 (b) u(, ) = (c) u(, ) =3ln +4ln (d) u(, ) = 3 4 (e) u(, ) = (f) u(, ) = 3 4 (g) u(, ) = (h) u(, ) = 1 3 ln ln Answer: Compute the MRS for each of the utilit functions. Utilit functions with the same MRS represent the same preferences. (a) MRS = 3 4 (b) MRS = (c) MRS = 3 4 (d) MRS = 3 4 (e) MRS = 4 3 (f) MRS = 3 4 (g) MRS = 3 4 (h) MRS = 4 3 So (a) and (g) represent the same preferences. Similarl, (c), (d), and (f) represent the same preferences. Finall, (e) and (h) represent the same preferences. No other utilit function represents the same preferences as (b). This problem could also be solved using monotonic transformations. If one utilit function is a monotonic transformation of another, then the two utilit functions represents the same preferences. Eercise 5: Wow. I feel completel satisfied! (Lectures 4 and 5) Suppose ou onl consume two goods: chocolate kisses () and peanuts (). Your marginal utilit of chocolate kisses is positive for the first twent kisses ou consume each da; beond the twentieth kiss, though, an additional kisses will make ou less and less happ. Similarl, ou enjo positive marginal utilit for the first ten ounces of peanuts ou consume each da, but negative marginal utilit for an peanuts beond that. (a) Write down a utilit function U(, ) that is consistent with these preferences; also sketch an indifference curve diagram, indicating with arrows the direction of higher utilit. (b) Are our preferences conve? Eplain intuitivel, making reference to the indifference curves ou drew in part (a). Answer: 8

9 (a) U(, ) = ( 2) 2 ( 1) 2. Other functions are possible but this is the easiest. An function that correctl follows the instructions will be accepted (b) Yes, these preferences are conve. Note that the indifference curves are circles and that the direction of increasing utilit is inwards. The conve combination of an two points on an indifference curve lie inside the circle, which is preferred. 9