(0, 1) (1, 0) (3, 5) (4, 2) (3, 10) (4, 8) (8, 3) (16, 6)

Transcription

1 1. Consider a person whose preferences are represented by the utility function u(x, y) = xy. a. For each pair of bundles A and B, indicate whether A is preferred to B, B is preferred to A, or A is indifferent to B: A B (0, 1) (1, 0) (3, 5) (4, 2) (3, 10) (4, 8) (8, 3) (16, 6) b. For this person, and for the eight bundles in a., list the bundles in order of preference from most to least preferred. If two bundles are indifferent, put them side-by-side in your list. c. Consider any pair of bundles (x 0, y 0 ) and (x 1, y 1 ) between which this consumer is indifferent. Prove that with this utility function she will also be indifferent between the pair of bundles containing twice at least as much of each good as in the original pair of bundles. d. Can you suggest another utility function that will represent this same consumer s preferences? Briefly justify your suggestion. 2. A woman I know has funny tastes. If she has more Y than X, she will always trade 2Y for 1X and be indifferent. If she has more X than Y; however, every time she gives up 1Y she requires 2X to remain indifferent after exchange. a. Sketch this woman s indifference map. Y X = Y 45 o 0 X b. What is her MRS YX at any bundle (x 0, y 0 ) where x 0 < y 0? c. What is her MRS YX at any bundle (x 1, y 1 ) where x 1 > y 1? d. Do her preferences satisfy convexity? Why or why not? 1

2 3. Some indifference curves for a consumer whose preferences are complete and transitive are sketched in the following figure. The figure has been divided into quadrants labeled I IV relative to the point (x 0, y 0 ). The number to the right of each indifference curve are the numbers assigned to all bundles on the curve by the consumer s utility function. Notice that the direction of increasing preference is always toward the point (x 0, y 0 ). Y II X = Y III y 0 I For each of the quadrants I IV, answer each of the following questions: a. Is MU X positive, negative, or zero? x 0 IV X b. Is MU Y positive, negative, or zero? c. Do the preferences satisfy the assumption of monotonicity in this quadrant? Why or why not? d. Do the preferences satisfy the assumption of convexity in this quadrant? Why, or why not? 4. In your own words, explain what the MRS is. 5. Draw indifference map that represent the following preferences for hamburgers () and sodas (S). Indicate the direction in which utility is increasing, and put hamburger consumption on the vertical axis and soda consumption on the horizontal axis: a. Joe has convex preferences, but dislikes both hamburgers and sodas. 2 S

3 b. Bob loves hamburgers, and dislikes sodas. Assume he will drink a soda if it is served to him. c. Em loves hamburgers and sodas, but insists on drinking exactly one soda for every hamburger he eats. d. Bill likes hamburgers, but neither likes nor dislikes sodas. 3

4 e. Mary always gets twice as much utility from an additional hamburger than from an extra soda. 6. Write down the utility function associated with the following preferences for hamburgers () and Soda (S). a. Dan loves hamburgers and sodas, but insists on drinking exactly two sodas for every hamburger he eats. b. Scott loves hamburgers and sodas, but insists on drinking exactly two sodas for every three hamburgers he eats. c. Betty always gets 3 times as much utility from an additional hamburger than from an extra soda. 7. John claims that given the choice between 2 pizzas A and B, he always chooses the one with more pepperoni and more anchovies. Is this preference relation complete? Is it transitive? 8. Sarah has Cobb-Douglas preferences for 2 goods, pizza (P) and milk (M). Let M be the X-good (i.e., the good on the horizontal axis) and P be the Y good, i.t., the good on the vertical axis. U(P, M) = PM a. Find the marginal utility of P, the marginal utility of M, and the MRS for Sarah s utility function. b. Verify that these preferences are convex, by sketching a typical indifference curve (with consumption of P on the vertical axis and consumption of M on the horizontal axis.). Use the following steps. (i) Suppose Sarah consumes 9 pizzas and 4 milks. Calculate Sarah s utility. (ii) Fix the utility function at a certain level, k (k is the number you calculated in part (i)). PM =k (iii) Solve the above equation for P in terms of k and M. (iv) Pick a number of different consumption levels of M. Then use the expression you derived in part (ii) to find the values of P that keep Sarah indifferent between each of the new bundles and the bundle consumed in part (i). 4

5 (v) Plot the different pairs of P and M in a graph with P on the vertical axis, and trace out the indifference curve. P c. Using the different consumption bundles from part b., verify that as M increases (P should be decreasing), the absolute value of MRS diminishes. M Al has Cobb-Douglas utility function U ( F, = F C and John has Cobb-Douglas utility function U ( F, = FC. J A a. Calculate Al s and John s MRS s. What do you notice about them? b. Bill has the utility function U ( F, = F C. Calculate Bill s MRS. B c. What do you notice about those MRS s? Relate your answer to the indifference curves for each of the individuals and to each of the individual s underlying preferences. d. Suppose Al consumes 1 unit of C. Calculate the marginal utility function of F for Al. Does Al s utility function appear to obey increasing or decreasing marginal utility of F? e. Repeat part d. for Bill (Remember Bill s utility function is U ( F, = F C.) B f. ow do you make sense of your answers to parts 7c, 7d, and 7e? 10. Professor Keith always gives two midterms in his class. e only uses the higher of the two scores that a student gets on the midterms when he calculates the course grade. a. Nancy wants to maximize her grade in this course. Let x 1 be her score on the first midterm and x 2 be the scroe on the second midterm. Which combination of scores would Nancy prefer, x 1 = 20 and x 2 =70 or x 1 = 60, and x 2 = 60. 5

6 b. Plot an indifference curve showing all of the combinations of scores that Nancy likes exactly as much as x 1 = 20 and x 2 =70. Also plot a indifference curve showing the combinations that Nancy likes exactly as much as x 1 = 60, and x 2 = 60. x 2 x 1 c. Does Nancy have convex preferences over these combinations? Why, or why not? d. Nancy is also taking a course in economics from Professor Scott. e gives two midterms. Instead of discarding the lower grade, he discards the higher one. Let x 1 be her score on the first midterm and x 2 be her score on the second midterm. Which combination of scores would Nancy prefer,, x 1 = 20 and x 2 =70 or x 1 = 60, and x 2 = 60. e.. Plot an indifference curve showing all of the combinations of scores that Nancy likes exactly as much as x 1 = 20 and x 2 =70. Also plot a indifference curve showing the combinations that Nancy likes exactly as much as x 1 = 60, and x 2 = 60. Does Nancy have convex preferences over these combinations? Why, or why not? x 2 x 1 6