Eco 300 Intermediate Micro

Transcription

1 Eco 300 Intermediate Micro Instructor: Amalia Jerison Office Hours: T 12:00-1:00, Th 12:00-1:00, and by appointment BA 127A, aj4575@albany.edu A. Jerison (BA 127A) Eco 300 Spring / 27

2 Review of Chapters 1-4 and uncertainty 1. What is the marginal rate of substitution of one good for another? 2. Why might a person s marginal rate of substitution of food for clothes decrease as consumption of food increases? 3. Under what conditions must the marginal rate of substitution equal the negative of the slope of the budget line at the utility-maximizing bundle? Under what conditions can the marginal rate of substitution be different from the negative of the slope of the budget line? A. Jerison (BA 127A) Eco 300 Spring / 27

3 1. The marginal rate of substitution of food for books is the (maximum) quantity of books that the consumer is willing to give up for a given increase in the quantity of food. If the consumer s indifference curves are differentiable, the MRS is the negative of the slope of the indifference curve (evaluated at a particular market bundle). A. Jerison (BA 127A) Eco 300 Spring / 27

4 2. Even though we assume nonsatiation (the person always prefers more of a good to less, with amount of the other good fixed), the amount of clothes a person is willing to give up for an additional unit of food is likely to decrease as the person gets more food. This is because usually people derive most additional satisfaction from a good when they have less of it. The difference between having no food and a little food is much more important than the difference between having a huge amount of food and even more food. A. Jerison (BA 127A) Eco 300 Spring / 27

5 However, there can be cases where this is not true. Consider a good for which a certain amount is required in order to derive any utility from it. For example, suppose the good is time spent watching a movie. Suppose that the movie lasts 90 minutes and is very suspenseful until the end. So the additional benefit from watching the last 5 minutes of the movie is greater than the additional benefit of watching the first 5 minutes of the movie. Then the person s indifference curves over time spent watching the movie and time spent on other activities would be concave (the more they watch it, the more they want to watch it). A. Jerison (BA 127A) Eco 300 Spring / 27

6 3. If the consumer is consuming some of both goods at the utility-maximizing bundle, then the MRS (if it exists) must equal the negative of the slope of the budget line. This is an interior solution. But if the consumer is not consuming any of one good at the utility-maximizing bundle, then it is not necessary for the slope of the MRS to equal the negative of the slope of the budget line. A. Jerison (BA 127A) Eco 300 Spring / 27

7 4. Give an example of two goods that could be perfect substitutes, and an example of two goods that could be perfect complements. 5. Can there be only one utility function corresponding to a particular indifference curve? A. Jerison (BA 127A) Eco 300 Spring / 27

8 4. Two goods are perfect substitutes to a person if the MRS is the same everywhere. So for instance, if a person is willing to give up 5 cars for 1 house no matter how many cars and how many houses the person has, then houses and cars are perfect substitutes for that person. A more realistic example would be two brands of the same thing. Indifference curves are straight (downward-sloping) lines. Perfect complements are such that you need a certain additional amount of one good to be able to get any additional utility from the other good. For instance, tires on a car. A. Jerison (BA 127A) Eco 300 Spring / 27

9 5. A given indifference curve corresponds to infinitely many utility functions. If a utility function u(x, Y ) corresponds to an indifference curve I, then the utility function a u(x, Y ) for any positive constant a also corresponds to I. More generally if g is a function on the real numbers such that g (z) > 0 for all z, then g(u(x, Y )) corresponds to I if u(x, Y ) corresponds to I. A. Jerison (BA 127A) Eco 300 Spring / 27

10 6. How do you represent a change in income with prices held fixed? 7. How do you represent a change in price (with income held fixed)? A. Jerison (BA 127A) Eco 300 Spring / 27

11 6. A change in income can be shown on the graph as a parallel shift in the budget line. Since the equation of the budget line is C = (I/P C ) (P F /P C )F, the slope does not change when prices do not change. Only the intercepts change. 7. A change in the price of one good with price of the other good and income held constant can be represented by a pivoting of the budget line around the intercept of the good whose price didn t change. A. Jerison (BA 127A) Eco 300 Spring / 27

12 8. Revealed preference One can determine which of two bundles a consumer prefers, even if neither one has more of both goods than the other, by changing budget lines. Suppose a consumer chose bundle A when the budget line was l 1. Bundle B also lies of the budget line l 1. Then the prices and income change to give the consumer budget line l 2, which passes through B and D. Suppose the consumer now chooses B. We can conclude that the consumer prefers A to D. Since the consumer chose A when B was affordable, he must prefer A to B. Since he chose B when D was available, he must prefer B to D. So by transitivity of preferences, he prefers A to D. A. Jerison (BA 127A) Eco 300 Spring / 27

13 An example where revealed preference is used: Question 4. on the old exam. A consumer purchased the market basket (x, y) = (2, 4) in 1990 and (4, 1) in The prices of good X and Y were both $2 in 1990 but in 1991 the price of good Y rose to $3 per unit while the price of X was unchanged. a) Compute the consumer s Laspeyres price index for 1991 with base year The Laspeyres index is = 16/ A. Jerison (BA 127A) Eco 300 Spring / 27

14 b) Is it possible to tell for sure that the consumer was better off in 1990? The bundle (4, 1) was affordable to the consumer in 1990: With income of 12 in 1990, = 10 < 12, so the bundle could have been chosen in However, the consumer chose (2, 4) instead. Assuming strictly convex preferences, so that there is always exactly one most-preferred bundle on any budget line, this tells us that the consumer prefers (2, 4) to (4, 1). Therefore the consumer was better off in A. Jerison (BA 127A) Eco 300 Spring / 27

15 c) Suppose instead that the prices of X and Y were $3 and $2 in Is it possible that the consumer had the same standard preferences in both years and chose the market basket (2, 4) in 1990 and (4, 1) in 1991? Show that your answers are correct. We will show that (2, 4) was affordable in 1991 (but was not chosen) and (4, 1) was affordable in 1990 (but was not chosen): Income in 1990 was = 12. The money needed to buy (4, 1) in 1990 prices was = 10 < 12. So (4, 1) could have been chosen in 1990; instead (2, 4) was chosen, so (2, 4) must be preferred. A. Jerison (BA 127A) Eco 300 Spring / 27

16 However, the money needed to buy (2, 4) in 1991 prices was = 14. The consumer s income in 1991 was = 14. So (2, 4) could have been afforded in 1991; instead (4, 1) was chosen, so (4, 1) must be preferred. If the consumer s preferences did not change over the period, then there is a contradiction. Either (2, 4) is preferred to (4, 1) or (4, 1) is preferred to (2, 4), but not both. A. Jerison (BA 127A) Eco 300 Spring / 27

17 9. How would one find (theoretically) the ideal cost-of-living index? 10. How do you get an individual s demand curve for food from her price-consumption curve? A. Jerison (BA 127A) Eco 300 Spring / 27

18 9. We need to find how much income is required to achieve the base year s indifference curve with current prices. Graphically, shift the new budget line until it is tangent to the original indifference curve. The income needed to achieve that point with the current prices, divided by the base year s income, is the ideal cost-of-living index. A. Jerison (BA 127A) Eco 300 Spring / 27

19 10. Change the price of food, keeping price of clothes and income constant, and see how the amount chosen of food changes. To each price corresponds a quantity; graph price against quantity to get the demand curve. A. Jerison (BA 127A) Eco 300 Spring / 27

20 11. What does it mean if the quantity of clothes increases when the price of food increases? What does it mean if the quantity of clothes decreases when price of food increases? 12. Is it possible for the quantity of food to decrease when the price of food decreases? Is it possible for the quantities of both food and clothes to decrease when the price of food decreases? A. Jerison (BA 127A) Eco 300 Spring / 27

21 11. If quantity of clothes chosen increases when the price of food increases, then clothes are a substitute for food. If the quantity of clothes chosen decreases when the price of food increases, then clothes are a complement of food. 12. It is possible (the good is called a Giffen good). But it can t happen over the entire range of prices. It is not possible for the quantities of both food and clothes to decrease when the price of food decreases. This is because when the price of food decreases, everything that was affordable before is still affordable. So the person could just buy the original bundle, which must be better than a bundle containing less of both goods. A. Jerison (BA 127A) Eco 300 Spring / 27

22 13. What property of indifference curves ensures that the substitution effect will cause a decrease in amount of food consumed when price of food increases? 14. Use a single graph with indifference curves and budget sets to show that for a consumer with standard preferences it is possible that good Y is a gross substitute for good X but good X is a gross complement of good Y. A. Jerison (BA 127A) Eco 300 Spring / 27

23 13. It is the convexity of indifference curves. When the budget line becomes steeper, the point of tangency on the original indifference curve must lie to the left of the original bundle. If the indifference curves are not convex, the substitution effect will not necessarily cause a decrease in food consumption when price of food increases. A. Jerison (BA 127A) Eco 300 Spring / 27

24 Y X A. Jerison (BA 127A) Eco 300 Spring / 27

25 15. A particular competitive consumer spends 10% of his income on good X no matter what the prices of all goods are and no matter what his income is. Show all your work in answering the following problems. a) Find a formula for the quantity of good X that the consumer buys depending on the consumer s income and the price of good X. b) Use the formula in part a) to calculate the consumer s elasticities of demand for good X with respect to income and with respect to the price of good X and the price of any other good, Y. A. Jerison (BA 127A) Eco 300 Spring / 27

26 15. a. P X X/I = 0.1, no matter what P X and I are. So X = 0.1I/P X. b. Income elasticity of demand for good X is (dx/di)(i/x) = (0.1/P X )(I/X) = 0.1I/(P X X) = 0.1/0.1 = 1. So a given percentage change in income leads to the same percentage change in consumption of good X. This would have been true if the fraction of income spent on good X had been any other number. A. Jerison (BA 127A) Eco 300 Spring / 27

27 Price elasticity of demand for good X is (dx/dp X )(P X /X) = ( 0.1I/(P X ) 2 )(P X /X) = 0.1I/(P X X) = 1. Since I is exogenous, the expression for X does not depend on P Y. So dx/dp Y = 0, and the cross-price elasticity is zero (X and Y are independent). A. Jerison (BA 127A) Eco 300 Spring / 27