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Exercise 3. Consider a portfolio choice in a two-period economy with one risky and one riskless assets. The safe asset gives back the initial investment (it yields one unit of the good at date I for each unit invested at date O). The risky asset returns Ry = 1.15 with probability a = _ and RL = 0.8 with probability (1 - x) per unit invested. Consider a consumer who owns initial wealth Wo = 100 with preferences represented by log(C). 1. Denote the fraction of wealth invested in the risky asset by 0. Write down the contingent consump- tions in the high and low state, respectively Can and Cr- 2. Write down the mathematical relation between consumptions Cu and CL, and represent it in a cartesian space (Cu, CL). Compute the slope of the frontier of the set of feasible allocations for the consumer. Explain. 3. Write down the agent marimization problem. Identify the indifference curve and compute the MRS (Marginal Rate of Substitution) 4. Find the optimal portfolio choice 0* .4 Indifference Curves (18 points) In each of the following examples, the consumer consumes only two goods, a and y. Based on the information given in each statement, sketch a plausible set of indifference curves (draw at least two curves on a set of labeled axes and indicate the direction of higher utility). Then, write down a possible form of the utility function u(x, y) that is consistent with your graph. 1. (6 points) Alan likes wearing both right shoes (r) and left shoes (y). He always needs to wear them as a pair, having a right shoe is useless without the left one and viceversa.1 Positive vs. Normative Statements (16 points) Identify whether each of the following statements is positive or normative. Briefly justify your answer. 1. (4 points) The government has a duty to provide basic healthcare and education to every citizen. 2. (4 points) The cost of health insurance is too high. 3. (4 points) The median earnings of a full-time worker with a college degree are almost twice as high as those of a high-school graduate with no college education. 4. (4 points) The current unemployment rate is 3.9%, the lowest it has been since December 2000.Short Answer Problems FIGURE 8.9 1. Why do indifference curves have negative slopes? Short Answer Problem 4 2. Jan and Dan eat bread and peanut butter and have the same income. Because they face the same prices, they have identical budget lines. Currently, Jan and 5 Dan consume the same quantities of bread and Lizards (number per year) peanut butter; they have the same best affordable 4 consumption point. Jan views bread and peanut butter as close (though not perfect) substitutes, 3 whereas Dan considers bread and peanut butter to be quite (but not perfectly) complementary. On the 2 same diagram, draw a budget line and representa- tive indifference curves for Jan and Dan. (Measure 1 the quantity of bread on the horizontal axis.) FIGURE 8. 8 Short Answer Problem 3 Snakes (number per year) 4. Figure 8.9 again shows Igor's indifference curves. a. Snakes cost $20, lizards cost $20, and Igor's income is $60. Draw Igor's budget line in Fig- Lizards [number per year) ure 8.9. How many snakes does Igor buy? How many lizards? b. For his superior work in finding brains, Igor's master gives him a raise to $120. In Figure 8.9 draw Igor's new budget line. After the raise, how many snakes does Igor buy? How many lizards? For Igor, are snakes and/or lizards a normal good? 5. Ms. Muffet consumes curds and whey. The initial 3 price of curds is $1 per unit, and the price of whey Snakes (number per year) is $1.50 per unit. Ms. Muffet's income is $12. 1. What is the relative price of curds? 3. Three of Igor's indifference curves are illustrated b. Derive Ms. Muffet's budget equation and draw Figure 8.8. her budget line on a graph. ( Measure curds on a. Suppose that the price of a snake is $10, the the horizontal axis.) price of a lizard is $20, and Igor has $60 to c. On your graph, draw an indifference curve so spend on snakes and lizards. Carefully draw his that the best affordable point corresponds to 6 budget line in Figure 8.8. How many snakes units of curds and 4 units of whey. does Igor buy? How many lizards? d. What is the marginal rate of substitution of b. Suppose that the price of a snake spirals to $20 curds for whey at this point? while the price of a lizard does not change. If c. Show that any other point on the budget line is Igor's income stays at $60, draw his new budget inferior. line in Figure 8.8. Now how many snakes does gor buy? How many lizards' Are snakes and liz- . For the initial situation described in problem 5, ands substitutes or complements? suppose that Ms. Muffet's income now increases. c. When is Igor better off: before or after snakes go a. Illustrate graphically how the consumption of up in price? How can you tell? curds and whey are affected if both goods are10. Diminishing marginal rate of substitution means 13. Which of the following statements about Figure 8.6 that is correct? a. the budget line has a negative slope. a. Point a is preferred to point a, but a is not b. the budget line does not shift when people's affordable. preferences change. . The consumer is indifferent between points d indifference curves might have a positive slope. and c, but e is more affordable. d. indifference curves will be concave. Point & is preferred to point d , but b is not affordable. 11. If two goods are perfect substitutes, their d. Both points a and d cost the same, but a is pre- a. indifference curves are positively sloped straight ferred to d. lines. b. indifference curves are negatively sloped straight 14. What is the best affordable point of consumption? lines. a. a c. indifference curves are L-shaped. b. b d. marginal rate of substitution is infinity. d d 12. If the indifference curves between two goods are L- shaped, the goods are 15. A consumer is in equilibrium when the consump- a. complementary goods. tion point is on b. substitute goods. a. the budget line. c. normal goods. b. an indifference curve. d. inferior goods. c. the highest indifference curve that just touches the budget line. Predicting Consumer Behavior d. None of the above. Use Figure 8.6 for the next two questions. FIGURE 8.6 16. Which of the following is true when the consumer is Multiple Choice Questions 13 and 14 at the best affordable point? a. The point is on the budget line and highest at- cainable indifference curve. b. The slope of the budget line equals the slope of the indifference curve. c. The MRS equals the relative price. d. All of the above are true at the best affordable Hot dogs (quantity per month) point. d 17. Which of the following statements is true? a. The law of diminishing marginal rate of substi cution means that indifference curves are convex b (bowed out). b. A demand curve can be derived from the indif- ference curve/budget line analysis. Hamburgers (quantity per month] c. Demand curves and indifference curves measure the same things. d. Demand curves and indifference curves have negative slopes for the same reason