Problem%20Set%209%20Solutions

Problem%20Set%209%20Solutions - Economics 3010 Fall 2009...

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Unformatted text preview: Economics 3010 Fall 2009 Professor Daniel Benjamin Cornell University Problem Set 9 1. (Intertemporal Choice: 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 (1) UO = uO + u1 + u2, 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. If the person does not do the assignment today (in period zero) then her period‐one self has utility given by U1 = u1 + u2. The choice set she maximizes her function over is binary: do the assignment in period one or do it in period two. If she does it in period one she receives ‐3/2 utils. And if she does it in period two she receives ‐9/4 utils. Period one self therefore chooses to do the assignment in period one. Today’s self (period zero) now faces the decision to do it now or put it off. She knows that if she puts it off until the future it will be done in period one. The choice in period zero is therefore between completing the assignment in period zero or in period one. She gets a utility of ‐1 by doing it today in period zero. If she waits she gets ‐3/2. The assignment is completed today. 1 (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 one‐ half. That is, her overall utility today (period 0) is 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. If the person doesn’t do the assignment in period zero she has the utility function given in equation four. Period one self seeks to maximize this function over the binary set consisting of {do it in period one, (or) do it in period two}. The utility from waiting until period two is ½*(‐9/4) = ‐9/8 and the utility from doing it in period one is ‐3/2. Period one self therefore waits until period two. Note: the choice that maximizes period zero’s utility is period one completion. Doing it in period one gives utility of ½*(‐3/2) = ‐3/4, which is higher than the utility of period zero completion (‐1) and period two completion (‐9/8). But if the choice were between doing it in period zero or doing it in period two she would choose period zero. For period zero utility is ‐1 and period two utility is ‐9/8. (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?) A sophisticated person backward inducts (i.e., solves backward) the solution to the problem. She begins by asking herself the following question: What happens if I wait? If the woman waits and does not do the assignment in period zero she has the utility function given in equation four. Period one self seeks to maximize this function over the binary set consisting of (1) do it in period one or (2) do it in period two. The utility from waiting until period two is ½*(‐9/4) = ‐ 9/8. The utility from doing it in period one is ‐3/2. Period one self therefore does it in period two. Today’s self now faces a choice between doing it now or later. If she puts the project off until the future it won’t get done until period two because period one self chooses to wait. Today’s decision is therefore between doing it today in period zero or in two days during period two. Completion in period two gives a utility of 1/2*(‐9/4) = ‐9/8. Finishing up today gives a happiness of ‐1. Today’s self thus completes the project in period zero (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 2 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 endup 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. If the girl is naïve then she believes period one’s preferences are U1 = ½ u1 + ½ u2. She concludes that the project will be done in period one if she postpones the assignment because period one completion maximizes her perceived set of period one preferences. She thinks she’ll get utility of ½*(‐ 3/2) = ‐3/4, which is higher than ½*(9/4) = ‐9/8. If tomorrow rolls around and she hasn’t done the assignment, she will wait until period two because the preferences in period one are actually U1 = u1 + ½ u2. As shown in the previous parts, the optimal choice is to wait until period two. In the end the naïve girl does the assignment on the day it’s due. She puts it off in the first period because she incorrectly believes her period one self will have time consistent preferences given by U1 = ½ u1 + ½ u2. But when period one comes her actual preferences are U1 = u1 + ½ u2. So she waits. The end result is that the assignment is done in period two. From period zero perspective this gives us utility of ½*(‐9/4) = ‐9/8, which is worse than just doing it immediately (a utility of ‐1). (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 whether 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. [ 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 ] Going to the gym requires an immediate effort cost and provides a distant reward (that six pack of abs you’ve always dreamed of). When both the cost of going to the gym and the benefits of improved 3 health are in the future, the gym members think that going to the gym is a good idea. But when the time comes to actually go, the gym goers are experiencing self‐control problems because they stay on their couch to avoid incurring the immediate cost of walking to the gym. They overvalue the cost relative to the benefit as much at any particular point in time because it is in the present. The people are naïve because they think they’ll go more than they actually do. When they sign up for the monthly plan they mistakenly believe that it will be cost‐effective because they think they’ll go to gym very often. But when the future actually arrives, they postpone their visit to the gym. 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. Steel Mill: The first step is to setup the profit maximization problem with ys as the choice variable. max psys – cs(ys) = max 6ys – ys2 The first order condition with respect to ys becomes: 6 ‐ 2ys = 0 2ys = 6 ys = 3 4 Paper Mill: The first step is to setup the profit maximization problem with as the yp choice variable. max pp yp – cp(yp) = max 4yp ‐ yp2 / 2 The first order condition with respect to yp becomes: 4 ‐ yp = 0 yp = 4 (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? The first step is to setup the social profit maximization problem with ys and yp as the choice variables. max ps ys – cs(ys) + pp yp – cp(yp) – csocial(ys, yp) = max 6ys – ys2 + 4yp ‐ yp2 / 2 ‐ (ys2 + yp2) / 2 The first order conditions become: (1) 6 ‐ 2ys ‐ ys = 0 (2) 4 – yp yp = 0 From (1) we obtain: 6 = 3ys ys = 2 From (2) we obtain: 4 = 2yp yp = 2 When firms do not take into account the social cost, the social profit is πsocial (ys , yp ) = πsocial (3,4). 5 6ys – ys2 + 4yp ‐ yp2 / 2 ‐ (ys2 + yp2) / 2 = 18 – 9 + 16 – 8 – 25/2 = 4.5 Social profit at the optimal amount is given by πsocial (ys , yp ) = πsocial (2,2). 6ys – ys2 + 4yp ‐ yp2 / 2 ‐ (ys2 + yp2) / 2 = 12 – 4 + 8 – 2 – 4 = 10. (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 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? The steel mill ends up producing 0.56569*3 = 1.69707 The paper mill ends up producing 0.56569*4 = 2.26276 The total amount of social profit is given by πsocial (ys , yp ) = πsocial (1.69707,2.26276). 6ys – ys2 + 4yp ‐ yp2 / 2 ‐ (ys2 + yp2) / 2 = 10.18242 – 2.88 + 9.05104 – 2.56 – 4 = 9.79346 (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. The Steel Mill’s profit maximization problem now becomes: max psys – cs(ys) = max 6ys – ys2 ‐ 2ys The first order condition becomes: 6 ‐ 2ys – 2 = 0 2ys = 4 ys = 2 The Paper Mill’s profit maximization problem now becomes: max pp yp – cp(yp) = max 4yp ‐ yp2 / 2 ‐ 2yp 6 The first order condition becomes: 4 ‐ yp – 2 = 0 yp = 2 Under the Pigouvian tax we found that both firms produced 2 units. The taxes paid by each firm in this case is 4, making for a total of 8 dollars in taxes paid. The government revenue here is also 8. Because the taxes paid and government revenue cancel each other out, we have total social profit under the Pigouvian tax being equal to: 6ys – ys2 + 4yp ‐ yp2 / 2 ‐ (ys2 + yp2) / 2 = 12 – 4 + 8 – 2 – 4 = 10 (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. 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. 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? We begin by setting up the maximization problem for the Steel Mill. max psys – cs(ys) = max 6ys – ys2 ‐ qys The first order condition becomes: 6 ‐ 2ys – q = 0 2ys = 6 – q ys = (6 – q) / 2 Let’s now do the Paper Mill. max pp yp – cp(yp) = max 4yp ‐ yp2 / 2 ‐ q yp 7 The first order condition becomes: 4 ‐ yp – q = 0 yp = 4 – q Total Demand is given by (6 – q) / 2 + 4 – q. Setting this equal to the total supply of 4 we get: (6 – q) / 2 + 4 – q = 4 q = 2 Plugging 4 back into the demand equation, we find that the Steel Mill demands (6‐2)/2 = 2 and the Paper Mill demands 4 – 2 = 2 in equilibrium. Notice that the cost of the permits is 4 for each firm because each factory buys 2 permits at a price of 2 dollars. This gives us a total cost of 8 dollars which is equal to the government revenue. The last two terms in the social profit function cancel out, leaving us with a total social profit given by: 6ys – ys2 + 4yp ‐ yp2 / 2 ‐ (ys2 + yp2) / 2 = 12 – 4 + 8 – 2 – 4 = 10 (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? The Pigouvian tax and the cap‐and‐trade system alter the firms’ maximization conditions in a way that makes the socially profitable quantity also the individual profit maximizing quantity (i.e., they cause the firms to “internalize the externality”). It is advantageous to use these methods of regulation because they raise government revenue and give firms a further incentive to find innovative ways of reducing pollution. 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 excludable? Is it a public good? Explain. 8 A good is a rival good if one person’s use of it diminishes another’s enjoyment. An apple would be an example of a good with complete rivalry in consumption. When I eat it, you cannot. In the case of a very large pool, there is no rivalry in consumption. A large pool can accommodate many swimmers. You and I can both consume the pool at the same time. A good is excludable if people can be prevented from using it. The pool in Pleasantville could be made excludable if the town decides it wants to charge an admissions fee or require the purchase of an annual pass. If Pleasantville chooses to charge an entrance fee, then the pool would not be a public good. Rivalry in consumption and excludability would make the pool a club good. Pleasantville could also decide to make the pool free. This would make it non‐rival and non‐ excludable. In this case, it would be a public good. (b) Would building the pool be an efficient use of resources? Building the pool would be an efficient use of resources if the total daily benefits exceed the total daily costs. Pleasantville residents receive a total daily benefit of $6000 for the pool (1000x3 + 1000x2 + 1000x1). The total cost is given at $5000. Because $6000 is greater than $5000, the pool should be built. (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? If Pleasantville charges $3 for pool entry, 1000 people will go swimming and the city will get $3000 in revenue. Clearly this is not enough to cover costs. If Pleasantville charges $2 per day, 2000 people will come, but the city will only get $4000 in revenue. At a price of $1, all potential pool users go to the pool, and the city gets $3000 in revenue. And with no fee, all demanders come, but the city gets no revenue. As we can see, no price will cover the cost of the pool. Moreover, the efficient allocation is that everyone consumes the pool because each type of pool user gets positive benefit from the pool, while the marginal cost of using the pool is zero. This efficient allocation is resources is only achieved when the price is $0 or $1. (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. Yes, if it is possible for the town to price discriminate. For example, suppose that the families that are willing to pay $2 or $3 are all families with children, and the families that are only willing to pay $1 are families without children. Then a price of $2 for families with children and $1 for families without children would generate revenue = $2 x 1,000 + $1 x 1,000 = $5,000, which is just enough to cover the cost of the pool. Moreover, with this pricing scheme, all potential pool users will in fact use the pool, so the allocation of resources will be efficient. 9 ...
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This note was uploaded on 12/28/2011 for the course ECON 3010 at Cornell.

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