You can find the consultant s raw data here:

Transcription

1 Problem Set 1 Econ 475 Spring 2014 Arik Levinson, Georgetown University 1 [Travel Cost] A US city with a vibrant tourist industry has an industrial accident (a spill ) The mayor wants to sue the company responsible and hires you, an economist, to estimate the lost consumer surplus to tourists You follow the suggestion of Hotelling in his letter to the National Park Service, and divide the region into 75 distinct zones On the basis of sample data, you estimate the number of visitors from each zone to your city with and without the spill, as well as the population of each zone, its average income and the average proportion of college graduates in each zone ANSWER: You can find the consultant s raw data here: wwwgeorgetownedu/faculty/aml6/econ475/ps1dataxls a Estimate the lost consumer surplus from the spill using a travel cost model Describe any assumptions you make along the way Note: There are a variety of ways to do this This answer sheet provides just one example Before getting into the specific calculations, let s first clear up some of the general concepts Consumer surplus can be measured as the area under the demand curve Thus finding the change in consumer surplus is simply a matter of estimating two separate demand curves, one before the spill and one after the spill, and subtracting the areas to find the lost consumer surplus Generically, a demand curve shows the relationship between price and quantity demanded This is tricky to estimate for goods like parks or tourist cities since these goods don t have prices The travel cost model allows us to back out a demand curve by finding the relationship between costs and quantity demanded Once we ve estimated a demand curve it is simple to find the area, ie the consumer surplus So here s a sketch of the strategy, motivated by Hotelling s letter: 1 Calculate the number of visitors per 1,000 people (v) in each zone before the spill 2 Calculate the travel cost (TC) for each zone (this is the same both before and after the spill) 3 Regress v on TC and any other relevant regional control variables to find the relationship between visits per 1,000 (not total visits!) and travel costs overall (this applies to all zones): Note: This is not the demand function This tells us how many people per 1,000 come from each zone for a given travel cost The demand function tells us the overall number of visitors for a given price They are similar concepts, but not the same Another Note: Excel can do simple regressions, but I recommend using Stata 4 For each individual zone, add some amount (F) to the travel cost and use the equation from part (3) to predict the visits per 1,000 people in each zone Multiply this by the population in each zone to get the total number of visits in each zone 5 Sum across all zones to get the total number of visits that correspond to a total travel cost of (TC+F) 6 Repeat steps (4) and (5) for many different values of (F) Plot the results with (F) on the vertical axis and total number of visits on the horizontal axis

2 7 The points that were graphed in step (6) map out a demand curve for visiting the City before the spill Approximate the area under the curve This is the total consumer surplus 8 Repeat steps (3)-(7) for visits after the Spill 9 Subtract the total surplus (from (8)) after the spill from total surplus (from (7)) before the spill The difference is the change in total surplus Before we begin, let s make several (additional) simplifying assumptions: First and foremost, let s assume we are in Hotelling s world: households are utility maximizers that care about visiting cities and other generic consumption (including leisure) and are subject to a total endowment budget constraint:, where p is the total cost of visiting the City The cost of travel is a flat rate For simplicity here, let s assume it is $10 per hour, including both out-of-pocket expenses and the opportunity costs of time Some of you might have gone further and assumed those opportunity costs are wages, measured as a proportion of household income, which differs by region In that case you have to worry about the income and substitution effects of changes in the wages Visitors to the City do not actually spend any time there, they simply travel to the City and travel home (ie time at site is zero) I also assume that road distances denotes round-trip travel time Step 1: First I imported the data in to STATA and calculated the visit rates before and after the spill insheet using "C:\Arik\Teaching\Ec475\Problem sets\ps1dtatxt" (7 vars, 75 obs) gen vnorate = vno/pop gen vsprate = vsp/pop sum vnor vspr Variable Obs Mean Std Dev Min Max vnorate vsprate Then I calculated travel cost as 10 times road distance gen travelcost = 10*rd Then I regressed the visit rate without the spill in travelcost, income, and college reg vnorate travelcost inc college Source SS df MS Number of obs = F( 3, 71) = Model Prob > F = Residual R-squared = Adj R-squared = 09570

3 Total Root MSE = vnorate Coef Std Err t P> t [95% Conf Interval] rd inc 196e e e e-07 college _cons Repeat with the spill reg vsprate travelcost inc college Source SS df MS Number of obs = F( 3, 71) = Model Prob > F = Residual R-squared = Adj R-squared = Total Root MSE = vsprate Coef Std Err t P> t [95% Conf Interval] rd inc 295e e e e-07 college _cons Next use the coefficients generated by the two regressions to predict the visit rate from each region under a variety of fees added to the travel cost (When the fee is zero, your prediction should be actual visitors from each region) Here s an extract of my calculations Visits When No Spill Fee Zone ,098 2,386 1, ,877 2,989 2,100 1, ,131 3,354 2,578 1,801 1, ,939 2,321 1,703 1, ,057 2,149 1, , , ,947 80,517 49,467 26,366 10,392 1,812

4 $Fee Visits When Spill Fee Zone ,398 1,712 1, ,002 2,145 1, ,331 2,583 1,834 1, ,279 1,684 1, ,202 1, ,188 50,582 27,406 11,316 2,415 - The bottom row of each chart plots the demand curve for site visits $120 Travel Cost Demand for Visits $100 $80 $60 $40 $20 No Spill Spill $0-50, , ,000 Visits Calculate the consumer surplus under each scenario The difference is the cost of the spill CS with no spill: $2,276,211 CS with spill: $1,323,136 Cost of spill: $ 953,075 The spill reduced CS by about 40 percent Note that different assumptions about travel costs will yield different costs, but approximately the same share of CS

5 b List some objections that defense council for the company might have to your estimate There are many possible criticisms of this technique Here are only a few examples: Only two data points (one before and one after) are not sufficient to establish a credible estimate of the number of visitors We don t know anything about why people are visiting the city If they are traveling to a nearby attraction and just stopping at the city, then the value is grossly overestimated (For example, if you were to do a travel cost model for a hot dog stand on the national mall, you might find that it is worth billions of dollars) People may be substituting for nearby cities, in which case damages may be overestimated (for example, if the state is suing for lost tax revenue, but tourists are visiting a different city in the same state, then there is no lost revenue) This estimate of damages doesn t take into account any spending that occurred during the clean-up process, such as hiring local labor 2 [Refresher on public goods] (a) To aggregate the demand curves vertically, first express the individual demand functions in terms of P; this is the "inverse demand function": P = q Now multiply the right hand side by 1000, the number of people who benefit: P = 100,000-2,500 q This is the aggregate marginal benefit curve for the public good The efficient level of the public good that should be provided can now be found by equating MC = P 25,000 = 100,000-2,500 q q = 30 miles (b) Total benefits are equal to the area under the demand curve up to q = 30 miles TB = (30) (25,000) + (05) (30) (100,000-25,000) = 1,875,000 Total costs equal the area under the MC curve up to q = 30 miles: TC = (30) (25,000) = 750,000 Net Benefits = TB TC = 1,125,000 (c) 32 miles (d) 10 miles

6 3 [Refresher on externalities] (a) In equilibrium, the cost of the two options will be the same, so f(n* ) = x + t/w (1) Differentiate with respect to w to see that As the wage increases, fewer people will drive Why? The two equal costs are wf(n* ) =w x + t That implies that x<f(n*) in equilibrium And the marginal increase in the cost of driving for an increase in the wage is f(n*)>x, so using a car becomes less attractive (b) Choose n to minimize total costs Solve: ( ) ( )( ) This gives the FOC ( ) ( ) (2) Compare this to equation (1) above Given our assumptions about f(), we know that (c) Differentiate (2) with respect to w to get As the wage increases, the socially optimal number of drivers also decreases 4 Kolstad problem #5, page 111 a Finding aggregate marginal damage for the public bad: Workers Marginal Damage (MD) = 2p Retirees MD = 6p Since pollution is a public bad, namely it is non-rival, we will add the MD curves vertically (both workers and retirees are harmed simultaneously from the same unit of pollution, so the damage to society is the sum of the damages to each) Society s MD = Worker MD + Retiree MD = 2p + 6p = 8p b Graph the marginal savings and aggregate marginal damage: $ 20 Society s Marginal Damage Factory s Marginal Savings 10 Pollution

7 c Find equilibrium pollution without regulation and society s optimal level of pollution: Without regulation or bargaining, the factory will maximize savings from polluting It does this by setting the marginal savings (ie the first derivative of savings) equal to zero: 20-2p = 0 P * = 10 units The optimal pollution for society occurs when the marginal damage to consumers (workers and retirees) is equal to the marginal savings to the factory Setting these equal we find: 20-2p = 8p 10p = 20 P S = 2 units d Find marginal willingness to pay for abatement for each consumer and society: First observe that pollution and abatement are closely related and opposite If there was zero abatement (the starting point for this question) there would be 10 units of pollution If we measure abatement (A) in the same units that we use to measure pollution, we have the relationship: P = 10 A At this point we also make a semantic leap: since pollution was a bad, increasing pollution was hurting society, hence the marginal damage Now, an increase in abatement is good (it causes less damage), thus we change the language to marginal benefit It s just nomenclature; nothing has changed in the math Plugging this into the marginal damage function for workers and retirees we get: Workers: marginal damage = 2P marginal benefit = 2(10 A) = 20 2A Retirees: marginal damage = 6P marginal benefit = 6(10 A) = 60 6A Society: marginal damage = 8p marginal benefit = 8(10 A) = 80 8A (Evaluating these at A=0 obviously yields $20, $60, and $80 respectively) e Find the firm s marginal cost of abatement and the optimal level of abatement: Making an equivalent semantic leap (from marginal savings to marginal cost ), we apply the same abatement-pollution relationship to the firm: marginal savings = 20 2P marginal cost = 20 2(10 A) = 2A (Evaluating this at A=0 obviously yields $0) Again, to find society s optimal level of abatement we set marginal benefit equal to marginal cost 2A = 80 8A A S = 8 units Reality check: starting from the un-regulated level of pollution (10 units), 8 units of abatement leaves 2 units of pollution, the answer from part (c) f Are optimal provisions of public goods and bads equivalent? Yes, it is exactly equivalent to solve for the social optimum using abatement or pollution They are opposite sides of the same coin (at their core, they both tell us the same information)