SCORES Question Total Points Score Total Points = 100

Transcription

1 University of Toronto Department of Economics ECO Ajaz Hussain TEST 2 SOLUTIONS TIME: 1 HOUR AND 50 MINUTES YOU CANNOT LEAVE THE EXAM ROOM DURING THE LAST 10 MINUTES OF THE TEST REMAIN SEATED UNTIL ALL TESTS ARE COLLECTED, COUNTED AND THE PROCTOR ANNOUNCES THAT YOU CAN LEAVE THE ROOM IF YOU DETACH SHEETS IT S YOUR RESPONSIBILITY TO RE-STAPLE PAGES - GRADERS ARE NOT RESPONSIBLE FOR LOOSE SHEETS DO NOT HAVE A CELL PHONE ON YOUR DESK OR ON YOUR PERSON. ONLY AID ALLOWED: A CALCULATOR THERE ARE TWO WORKSHEETS AT THE END OF THE TEST GOOD LUCK! LAST NAME (AS IT APPEARS ON ROSI) FIRST NAME (AS IT APPEARS ON ROSI): MIDDLE NAME (AS IT APPEARS ON ROSI) STUDENT ID # (AS IT APPEARS ON ROSI) PLEASE CIRCLE THE SECTION IN WHICH YOU ARE OFFICIALLY REGISTERED (NOT NECESSARILY THE SECTION YOU ATTEND) MON 1-3 TUE TUE 2-4 WED 6-8 SIGNATURE: SCORES Question Total Points Score Total Points = 100 Page 1 of 35

2 Question 1 [5 Points] The following graphs are reproduced from Tella and MacCulloch s article Some Uses of Happiness Data in Economics. According to the graph on the left: despite income per capita rising, happiness levels are more or less the same According to the graph on the right: happiness levels rise with income per capita Longitudinal Survey of Americans 1975, 1980, 1985, 1990, 1995 Cross Section of Americans (1995) k 20k 30k 40k 50k Real Household Income/capita Please give a short explanation below for these seemingly contradictory conclusions. Answer: These are not contradictory statements: these graphs tell us that for Americans, like everyone else, it s not the absolute level of income that determines happiness but rather it s the relative income that determines happiness. If you are richer than everyone else, regardless of the dollar value of income, you re happier than the less fortunate citizens and denizens. Page 2 of 35

3 Question 2 [10 Points] (a) [5 Points] If a good has a downward sloping demand curve (i.e. it is ordinary good), then it may be a normal good or an inferior good -- true or false? Explain your answer, clearly stating any assumptions. Answer: True. We know that: We know that the substitution effect is always positive: For a good with a downward sloping demand curve (i.e. ) the price effect is positive (because as so that ) so that: What does this tell us about the income effect? Simply that the good may be a normal good or an inferior good. To see this, suppose the good is a normal good -- then the income effect is positive (because as so that ) so that: Thus, a good with a downward sloping curve may well be a normal good. But it can also be an inferior good provided the now negative income effect is smaller than the substitution effect: In sum: if the good has a downward sloping demand curve it may be a normal or an inferior good. Page 3 of 35

4 (b) [5 Points] A normal good must be an ordinary good (i.e. has a downward sloping demand) -- true or false? Explain your answer, clearly stating any assumptions. Answer: True. We know that: We know that the substitution effect is always positive: If the good is a normal good -- then the income effect is positive (because as so that ) so that: This implies that the price effect is definitely positive or that the good has a downward sloping demand curve (i.e. ) the price effect is positive (because as so that ): Page 4 of 35

5 Question 3 [20 points] Suppose the representative consumer has the following utility function over food (good 1) and everything else (good 2): Assume and that the consumption set is { }. (a) [5 points] Solve the consumer s UMP for the optimal amounts of goods 1 and 2 and the optimal level of utility and indicate whether goods 1 and 2 are ordinary goods, and whether goods 1 and 2 are normal/inferior goods. Assume all pecuniary parameters are and show all calculations. Answer: For simplicity, let s work with the log transformation: This is the log Cobb-Douglas utility function its indifference curves, like that of the Cobb-Douglas utility function -- don t touch either axis and if we assume that then the consumer has monotone preferences since for : With monotone preferences and the UMP becomes: The UMP is: The FOCs are: Page 5 of 35

6 Re-arrange equations ❶and ❷: Equating ❶and ❷: Multiplying both sides by -1 tells us that the optimal bundle is where curve = slope of the budget line: i.e. the slope of the indifference We can solve this equation simultaneously with equation : Now, from: Substitute in budget constraint: Page 6 of 35

7 Substitute this in: We now solve for. From the 1 st FOC: But: so that: We can always rescale the utility function such that Therefore, To check whether the goods are ordinary and or normal, we need to differentiate the demand functions with respect to own prices and income: Page 7 of 35

8 Therefore good 1 is an ordinary good. Therefore, good 1 is a normal good. Therefore good 2 is an ordinary good. Therefore, good 2 is a normal good. Page 8 of 35

9 (b) [5 points] Suppose the representative consumer s utility function is: Currently, the representative consumer s budget constraint is where is the net income (i.e. income tax has been deducted at source). Now, suppose the government gives this consumer an additional net income in cash. Calculate the change in the optimal amounts of goods 1 and 2 and the change in optimal utility due to this income in cash program. Show all calculations. Answer: In this question we have: and Thus: Now, the government gives the consumer an additional constraint becomes: Thus: net income in cash. Therefore, the consumer s budget Therefore change in : Therefore change in : Therefore change in : Page 9 of 35

10 q U = 1.91 U = 2.25 (P 1, P 2, Y) = (10, 20, 280) (P 1, P 2, Y) = (10, 20, 200) q 1 An income in cash boosts consumption of both goods and raises consumer welfare (utility) Page 10 of 35

11 (c) [5 points] Once again, consider the consumer in part (b) who had the utility function: From part (b) recall that the consumer s budget constraint is where is the net income (i.e. income tax has been deducted at source). Now suppose that instead of giving the consumer income in cash, the government gives the consumer income in kind as food stamps (i.e. the food stamps can only be used to buy food). Calculate the change in optimal amounts of goods 1 and 2 and the change in optimal utility due to this income in kind program. Show all calculations. Answer: Now the consumer receives income in kind: a income in kind in food stamps. This allows the consumer to buy more food (good 1) but not everything else (good 2). Previously, the consumer could buy at most: units of food; now she can buy units of food. Notice that with an income in kind subsidy, the consumer won t be able to consume the bundle ( bundles in the yellow triangle are off limits under the income in kind subsidy scheme: ) since the q U = 1.91 U = 2.25 (P 1, P 2, Y) = (10, 20, 280) (P 1, P 2, Y) = (10, 20, 200) q 1 The utility maximizing bundle, subject to the income in kind budget constraint is at the kink or the corner of the budget set: Page 11 of 35

12 q U = 2.25 U = (P 1, P 2, Y) = (10, 20, 280) U = 1.91 (P 1, P 2, Y) = (10, 20, 200) q 1 Therefore, optimal consumption for good 2: Optimal consumption of good 1 can be solved in the following way: The corner is at and the utility is: Therefore change in : Therefore change in : Therefore change in : Page 12 of 35

13 (d) [5 points] Given your answers to parts (b) and (c), does the consumer prefer receiving as income in cash or as income in kind? What about the government does it prefer giving the consumer as income in cash or as income in kind? Show all calculations. Answer: Since increase in utility under income in cash is greater than utility under income in kind, the consumer prefers as income in cash. Since the government s costs for both the programs are the same, the government also would prefer the income in cash program since it s more popular. Page 13 of 35

14 Question 4 [20 points] Consider the two different consumers in part (a) and part (b) below with utility functions defined over { } Suppose each consumer s budget constraint is initially and then changes to. For each consumer below calculate the price effect, substitution effect and income effect of good 1. Show the effects graphically and show all calculations. (a) [10 points] Answer: The budget constraint is given by: Utility maximization condition for the above min utility function: Substituting this expression in the budget constraint yields: Price of good 1 changes to: Optimal consumption of good 1 after the price change can be computed from the following: To compute the substitution effect and income effect, we need to draw a budget line parallel to the new budget line that goes through the initial consumption point. Income corresponding to this parallel budget line can be computed as: Page 14 of 35

15 can be computed from the following: Price effect, substitution effect and income effect are calculated as follows: Price effect: Substitution effect: Income effect: Note that: Which implies: Substitution effect + Income effect =Price effect Page 15 of 35

16 q 2 C A=B (P 1, P 2, Y) = (10, 40, 200) (P 1, P 2, Y) = (5, 40, 200) (P 1, P 2, Y ) = (5, 40, ) q 1 Page 16 of 35

17 (b) [10 points] Answer: For simplicity, let s work with the log transformation (scale up the utility function by a power of 27): From Q3, part (a), we know that the solutions to this UMP will be of the following form: After the price change, To compute the substitution effect and income effect, we need to draw a budget line parallel to the new budget line that goes through the initial consumption point. Income corresponding to this parallel budget line can be computed as: can be computed from the following: Price effect, substitution effect and income effect are calculated as follows: Price effect: Substitution effect: Income effect: Note that: Which implies: Substitution effect + Income effect =Price effect Page 17 of 35

18 q 2 A (P 1, P 2, Y) = (10, 40, 200) B C (P 1, P 2, Y) = (5, 40, 200) (P 1, P 2, Y ) = (5, 40, ) q 1 Page 18 of 35

19 Question 5 [20 points] (a) [10 points] You re interested in the impact on consumer welfare from a proposed excise tax on toothpaste. You do not have the representative consumer s utility function over toothpaste and everything else, but you do have the demand curve equation for toothpaste: Given that the representative toothpaste consumer buys tubes of toothpaste a year on average, calculate the impact on consumer welfare due to a tax on toothpaste (i.e. calculate the change in utility due to the excise tax on toothpaste). State all necessary assumptions and show all calculations. Answer: The demand curve for toothpaste is given by: To calculate the impact of a tax on toothpaste on consumer welfare based on such limited information, we have to make the following assumptions: Good 2 is the base good i.e. Interior solution Page 19 of 35

20 Initially Therefore, Initially, ] ] Then, a tax on shampoo is imposed. Therefore, Page 20 of 35

21 Therefore, Now, ] ] Change in consumer welfare ( ) can be computed as: Therefore, change in consumer welfare: Page 21 of 35

22 (b) [10 points] Use the demand curve equation for toothpaste in part (a) to calculate the representative toothpaste user s utility function over toothpaste and everything else. Use this utility function to compute the impact on utility from a 5% excise tax on toothpaste. Assume the representative toothpaste user has. State all necessary assumptions and show all calculations. Answer: We assume, and interior solution. This implies: At the interior solution, Recall that. Therefore, Integrating both sides with respect to yields: Where is a constant. Given this result, and ignoring the constant (since is simply a monotonic transformation of ), we get the following utility function: In order to calculate the impact of a tax on toothpaste on consumer welfare using the utility function, we need to set up the Lagrangian: Page 22 of 35

23 Using the envelope theorem: We can easily solve for the from the UMP. Taking first order condition with respect to, we obtain: Recall that Therefore, As a result, change in consumer welfare (utility ) is: Note that this result is very close but not exactly the same as the one we got in part (a). Page 23 of 35

24 Question 6 [5 Points] Please answer the question following this article from the New York Times (please focus on the bolded sentences below): November 7, 2011 Soda Bans in Schools Have Limited Impact By ANAHAD O'CONNOR State laws that ban soda in schools but not other sweetened beverages have virtually no impact on the amount of sugary drinks middle school students buy and consume at school, a new study shows. The study, which looked at thousands of public school students across 40 states, found that removing soda from cafeterias and school vending machines only prompted students to buy sports drinks, sweetened fruit drinks and other sugar-laden beverages instead. In states that banned only soda, students bought and consumed sugary drinks just as frequently at school as their peers in states where there were no bans at all. The study is among the first to directly examine the extent to which state policies on soda in schools influence students behavior. With obesity on the rise and teenagers getting about 15 percent of their daily calories from beverages, health groups like the Institute of Medicine have pushed for the removal of all sweetened beverages from schools, and some states have put in place all-out bans on sweetened drinks. California, for example, became the first state to ban the sale of soft drinks in grade schools, in 2003, and one city, Boston, moved earlier this year to forbid the sale and promotion of sugar-sweetened beverages and sodas on all city property. But the sale of soft drinks in schools has become a lucrative revenue source for many school districts, and a number of states have been reluctant to eliminate them from schools. Some states have instituted only partial bans that remove sodas from schools but not Snapple, Gatorade and other sugary drinks. In the latest study, published this week in The Archives of Pediatrics and Adolescent Medicine, researchers looked at soda and sweetened beverage consumption among roughly 7,000 fifth and eighth grade students across 40 states. The students were followed between 2004 and 2007, a time when many states and school districts were beginning to enact bans. The researchers found that the proportion of students who purchased sugary drinks at schools in states that removed only soda 28.9 percent was similar to the proportion in states that had no bans against any sugary drinks at all 26 percent. Both groups also reported having identical levels of access to sugary beverages at school: about 67 percent in each category. I think definitely the biggest message is that laws need to be comprehensive to have any positive effect at all, said Daniel R. Taber, an author of the study and postdoctoral research associate at the University of Illinois at Chicago. The most unequivocal finding was that laws that focus on soda are just not getting it done. If you really want to create a healthier school environment, you need more comprehensive laws. But the study, which was funded by the National Institutes of Health and the Robert Wood Johnson Foundation, had another surprising finding. In states with policies banning all sugary drinks in schools, students had less access to them on campus, but their overall consumption of the drinks did not fall, suggesting they were getting the drinks elsewhere. Dr. Taber said that finding indicated that removing sugary drinks from school campuses, while effective, could only go so far. He said more initiatives that target sweetened beverage consumption outside of schools like beverage taxes and regulations on their marketing to children were needed. The laws did exactly what they were designed to do, he said. They were designed to reduce kids access to sweetened beverages in schools, but you can t expect schools to do it all on their own. Dr. Taber said it made sense for some school districts to focus on soda at first, since it accounts for about two thirds of the calories teenagers get from sugary drinks. But that could also give some students the wrong idea about which drinks are best. Page 24 of 35

25 Soda is definitely the most popular choice among kids, he said. But there s a lot of misconceptions about which beverages are healthy. Many kids think beverages like Gatorade are a healthy alternative to soda. According to the article: In states with policies banning all sugary drinks in schools, students had less access to them on campus, but their overall consumption of the drinks did not fall, suggesting they were getting the drinks elsewhere. Please model this effect of banning soda at school graphically. Show the before and after soda ban budget constraints and indifference curves (choose the utility function wisely). Be sure to label the axes and give a brief explanation below. Answer: There are many ways to model the ban on sugary drinks in schools inducing students to buy sugary beverages elsewhere. One can model a ban as an infinitely high price. We want to show a case where school kids switch from consuming sugary drinks at school and outside school to now only consuming drinks outside schools, maintaining the same level of consumption. One way to do this is by the linear utility model, with, i.e. school kids perceive a sugary drink purchased at school (good 1) to be the same as a sugary drink purchased outside school (good 2). Before the ban, assume the price of sugary drinks in and outside schools is the same so that which means that before the ban, school kids could ve consumed any bundle on the budget constraint. After the ban, with the budget line becomes vertical so that the optimal choice is to purchase sugary drinks outside school: Other answers are possible (such as, an indifference that touches both axes and with decreasing. Page 25 of 35

26 Question 7 [10 Points] Consider a 2 period economy with a single good (say, corn). Each consumer receives real income (in units of corn) at the beginning of respectively. Denote the real interest rate by (assume a common at ). Suppose a consumer has the following utility function defined over the consumption set { }: Assume all pecuniary variables. (a) [5 points] Derive an expression for the of (Future Value of Total Lifetime Income), and the of (Present Value of Total Lifetime Income). State all assumptions and show all calculations. Answer: The intertemporal UMP is to choose consumption in each period subject to the constraint that income: consumption equals This UMP cannot be solved by calculus because the utility function is not differentiable everywhere. Instead, we exploit the fact that at the optimum: Solve this equation and the intertemporal budget constraint simultaneously. From above: Substitute in: [ ] Page 26 of 35

27 Substitute in: Thus: From these we can get the of and of by using direct calculation: the optimal utility is: The of : To compute of, express in terms of Page 27 of 35

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29 (b) [5 points] Under what conditions will the consumer save corn at? Characterize: The relationship between and The relationship between and Show all calculations. Answer To be a saver at, this consumer would have to consume less than her income: To characterize the relationship between vs., we need to first compute. Page 29 of 35

30 Now, we can find out how and are related: Since for the consumer to be a saver at. To characterize the relationship even better, we need to compute the second order derivative: Since for the consumer to be a saver at. This implies a convex relationship. Therefore, if the consumer is a saver in, then as increases, falls and falls at a decreasing rate. Now, To characterize the relationship between vs., we need to first compute. [ ] Now, we can find out how and are related: Page 30 of 35

31 [ ] [ ] [ ] Since (which implies ) for the consumer to be a saver at. To characterize the relationship even better, we need to compute the second order derivative: Since (which implies ) for the consumer to be a saver at. This implies a convex relationship. Therefore, if the consumer is a saver in, then as increases, falls and falls and falls at a decreasing rate. Page 31 of 35

32 Question 8 [10 Points] Consider a 3 period economy with a single good (say, corn). Each consumer receives real income (in units of corn) at the beginning of respectively. Denote the real interest rate by (assume a common at ). Suppose a consumer has the following utility function defined over the consumption set { }: That is, the consumer perceives consumption at and as complements, and consumption at versus literally as perfect substitutes. Under what conditions will and? State all assumptions and show all calculations. Assume all pecuniary variables. Answer: Note: This problem is similar to the one in consumer theory where a consumer perceives food and alcohol as complements, and wine and beer as perfect substitutes for alcohol. The agent s problem is: For simplicity, write this as: At the optimum it must be that: We can express the budget constraint in terms of (say) and only: { } { } Note that if then is rising in so that: { } { } Page 32 of 35

33 One can substitute these in the budget constraint to get the actual values for Thus, when: On the other hand, if then is decreasing in so that: One can substitute these in the budget constraint to get the actual values for Thus, when: Page 33 of 35

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