Unformatted text preview: Economics 3010
Fall 2009 Professor Daniel Benjamin
Cornell University Problem Set 9 This problem set is due in lecture on Friday, December 4. Please collect your answers into a stapled packet, and write your own name and your TA’s name on the front of each packet, as well as your netID and the time of the section you usually attend. 0. (Aplia) Please do the assigned problems on the Aplia website. These problems are due at 11:45pm on Thursday, November 3. 1. (Behavioral Economics: An economic model of procrastination) Consider a student who has a homework assignment due in two days. The student must decide whether to do the assignment today, tomorrow, or in two days (on the day it is due). Because she will have less flexibility later and less opportunity to talk with TAs and other students, it becomes more costly to do the assignment the more she delays. In particular, the cost of doing the assignment today is 1 util, the cost of doing the assignment tomorrow is 3/2 utils, and the cost of doing the assignment in two days is 9/4 utils. If she has not done the assignment by today or tomorrow, she must do it in two days (and incur the cost of 9/4 utils at that point). Just to be completely clear: • If she does the assignment in period 0, then uO = ‐1, u1 = 0, u2 = 0. • If instead she does the assignment in period 1: uO = 0, u1 = ‐3/2, u2 = 0. • Finally, if she does the assignment in period 2: uO = 0, u1 = 0, u2 = ‐9/4. (a) Suppose the student is very patient and so does not discount future (instantaneous) utility flows relative to current (instantaneous) utility flows. That is, her overall utility today (period 0) is UO = uO + u1 + u2, (1) where uO is instantaneous utility today, u1 is instantaneous utility tomorrow, and u2 is instantaneous utility in two days. If she does not do the assignment today, then tomorrow she will be making her decision using overall function (2) U1 = u1 + u2. Using equation (2), explain why if she has not done the assignment by tomorrow, she will choose to do it tomorrow rather than wait until the day it is due. Hence her decision today is whether to do it today or wait until tomorrow, at which point she will decide to do it right away. Using equation (1), explain why she will end up doing the assignment today. (b) Now suppose that instead of being perfectly patient, the student has a preference for indulging immediate gratification; she overweights the present relative to all future dates by a factor of two. That is, her overall utility today (period 0) is 1 UO = uO + ½ u1 + ½ u2, (3) and her overall utility tomorrow (period 1) is U1 = u1 + ½ u2. (4) (Economists say that an individual with this kind of discounting structure has a self‐control problem – or has time‐inconsistent preferences – because, as you will see, such an individual has preferences that are internally conflicting, depending on when you ask her.) Using equation (4), explain why if she has not done the assignment by tomorrow, she will choose to do it in two days rather than doing it tomorrow. Using equation (3), explain why from today’s point of view, she would prefer to do the assignment tomorrow instead of today, but if she had to choose between doing it today or doing it in two days, she would prefer to do it today. (c) Suppose that the student is sophisticated about her self‐control problem; that is, she is aware that her preferences today are equation (3) and her preferences tomorrow are equation (4). Explain why in that case, she will end up doing the assignment today. (Hint: What does she expect will happen if she does not do the assignment today?) (d) Now suppose that the student is naive about her self‐control problem; she does not realize that her preferences are internally conflicting. That is, even though her actual preferences are given by equation (3) and equation (4), when she’s making her decisions today, she believes that her preferences tomorrow will be U1 = ½ u1 + ½ u2, which is consistent with what her preferences are today. Explain why today she (incorrectly) believes that she if she postpones doing the assignment, she will do it tomorrow. What will she actually do tomorrow if she has not done the assignment by then? Explain why a student who is naïve about her self‐control problem will end up doing the assignment the day it is due, even though from today’s point of view (as well as from tomorrow’s and the next day’s point of view!) she would have preferred to do it today. (e) Self‐control problems may also explain why people borrow so much on credit cards (at very high interest rates), smoke and drink too much, and have difficulty dieting and exercising. However, the welfare‐maximizing policy response typically depends on whether individuals are sophisticated or naïve about their self‐control problems. Hence economists are very interested in understanding to what extent individuals are sophisticated or naïve. In a recent paper, economists at UC Berkeley found that members of a gym expected to use the gym 9.5 times per month, and so they paid a monthly membership fee of $70. However, these individuals ended up actually going to the gym an average of 4.2 times per month. As a result, they paid an average price of $70 / 4.2 ≈ $16.67 per visit, even though they could have paid $10 per visit on a pay‐per‐visit basis. Explain why these facts are consistent with gym‐goers having a self‐control problem getting themselves to go the gym and being naïve about that self‐control problem. 2 [ This evidence is from Della Vigna, Stefano, and Ulrike Malmendier (2006), “Paying Not to Go to the Gym,” American Economic Review, 96, 694‐719. If you’re interested, you can look at the paper: http://elsa.berkeley.edu/~sdellavi/wp/gymempAER.pdf ] 2. (Externalities: The steel mill and the paper mill) Consider two firms, a steel mill and a paper mill. Both firms are “small” relative to their industry and so take their respective output prices, ps and pp, as given. The firms’ profits per day are given by: πs = ps ys – cs(ys) πp = pp yp – cp(yp), where ys is the number of tons of steel produced daily by the steel mill, yp is the number of tons of paper produced daily by the paper mill, and cs(ys) = ys2 and cp(yp) = yp2 / 2 are the firms’ cost functions. (Of course, each cost function could be derived from the firm’s production function via the firm’s cost‐
minimization problem.) The market price of a ton of steel is ps = $6, and the market price of a ton of paper is pp = $4. Unfortunately, producing steel and producing paper requires polluting a local river. An increasing number of tons of pollution is generated by production of each additional ton of steel and each additional ton of paper. The daily cost to society (e.g., need for more extensive water sanitation facilities, loss of public swimming areas, loss of wildlife) equals the total amount of pollution (in tons), which is: csocial(ys, yp) = (ys2 + yp2) / 2. (a) The firms choose their output levels to maximize profit. Since the firms only have to pay their private costs of production (and not the social costs), show that the steel mill produces 3 tons of steel, and the paper mill produces 4 tons of paper. (b) The socially optimal levels of output take into account the social costs. In economic analysis, we often imagine the problem faced by a so‐called social planner, a benevolent dictator whose goal is to maximize society’s well‐being. In this case, the social planner’s problem is to choose output levels ys and yp to maximize “social profit” – which is private profit (which is ultimately income of the firm’s shareholders, used by them to purchase consumption goods) less the social cost: πsocial = ps ys – cs(ys) + pp yp – cp(yp) – csocial(ys, yp). Show that the socially optimal levels of output are 2 tons of steel and 2 tons of paper per day. What is the total amount of social profit when firms maximize profit without taking into account the social cost? What is the total amount of social profit at the socially optimal levels of output? (c) The amount of pollution produced by the free market is (32 + 42) / 2 = 12.5 tons per day, while the level of pollution at socially optimal production levels is (22 + 22) / 2 = 4 tons per day. Suppose the Environmental Protection Agency (EPA), in order to get overall level of pollution to be 4 tons a day, sets a regulation requiring each firm to reduce its daily output by a factor of (4/12.5)1/2 = 0.56569. What will 3 be the steel mill’s level of output under that regulation? The paper mill? What is the total amount of social profit under that regulation? (d) An alternative to regulation is a Pigouvian tax. Show that a tax of $2 per ton of pollution causes each firm to choose the socially optimal amount of output. Now that the firms are paying a tax (which reduces their profit) and the government is collecting revenue (which could be distributed to individuals to increase consumption), the social profit is: πsocial = ps ys – cs(ys) + pp yp – cp(yp) – csocial(ys, yp) – (taxes paid) + (gov’t revenue) Show that the total amount of social profit under the Pigouvian tax is the same as the total amount of social profit when the social planner can choose the firms’ output levels. (e) Another alternative to regulation is a cap‐and‐trade system. Suppose that the government sells 4 pollution permits every day, each of which gives the right to emit one ton of pollution into the river for that day. After selling them, the government allows firms to trade these permits among themselves in a market. We will calculate the equilibrium price and allocation of these pollution permits. Show that if the price of a pollution permit is $q, then the steel firm’s quantity demanded of permits will be (6 – q) / 2, and the paper firm’s will be 4 – q. Set total quantity demanded equal to total quantity supplied to show that the equilibrium price will be $2. How many permits will the steel firm buy in equilibrium? The paper firm? Now social profit must take into account that firms pay an additional cost to purchase the permits and the government is collecting revenue from sale of the permits: πsocial = ps ys – cs(ys) + pp yp – cp(yp) – csocial(ys, yp) – (cost of permits) + (gov’t revenue) Show that the total amount of social profit under the cap‐and‐trade system is the same as the total amount of social profit when the social planner can choose the firms’ output levels. (f) Why do the Pigouvian tax and the cap‐and‐trade system lead to a higher total social profit than regulation? Relative to regulation, what are some other advantages of these solutions to the externality problem? 3. (Public Goods: An example of a club good) The town of Pleasantville is thinking of building a swimming pool. Building and operating the pool will cost the town $5,000 per day. There are three groups of potential pool users in Pleasantville: (1) 1,000 families who are each willing to pay $3 per day for the pool; (2) 1,000 families who are each willing to pay $2 per day for the pool; and (3) 1,000 families who are each willing to pay $1 per day for the pool. Suppose also that the intended pool is large enough so that whatever number of families come on any day will not affect what people are willing to pay for the pool. (a) Is the pool rival? Is it excludeable? Is it a public good? Explain. (b) Would building the pool be an efficient use of resources? 4 (c) Consider four possible prices for family admission to the pool: $3, $2, $1, $0. Which of these possible prices would cover the cost of the pool? Which would achieve an efficient allocation of resources? (d) Is there any pricing scheme for admission to this pool that would both cover the pool’s cost and achieve an efficient allocation of resources? Explain. 5 ...
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