Solutions to Exercises in Environmental economics. Spring 2015

Transcription

1 MÄLARDALEN UNIVERSITY School of Business, Society and Engineering Solutions to Exercises in Environmental economics Spring 2015 Innehåll 1 Welfare and markets Environmental valuation and cost benefit analysis Externalities and public goods Emission taxes markets and property rights Economic growth... 60

2 1 Welfare and markets 1.1 At perfect competition price will be equal to marginal costs and the supply curve equal to the marginal cost function. Since marginal cost is constant in this case the supply curve would be horizontal. Substitute mc as the price into the demand function: Q D = = 5 1 = 4 At perfect competition production will be 4 units and the price will be 2. To draw a figure we need the inverted demand function, i.e. the price as a function of the quantity: Q D = P 0.5 P = 5 Q D P = 10 2 Q D If marginal cost increases to 3 the supply curve will shift upwards: P D 3,5 4 5 Q The decrease in consumer surplus is equal to the marked area in the figure. In consists of a rectangle with the length 3.5 million and height 1 (the price increase of the units that are still consumed) and a triangle with the base 0.5 million and the height 1 (consumer surplus from those units no longer consumed) C = = = 3.75 Consumer surplus decreases with 3.75 million SEK.

3 Hamburgers 1.2 In this case both persons have identical preferences. The contract curve will then be a straight line from origin to origin. The indifference curves should be drawn so that MRS is equal for both persons as they pass the contract curve. Since we have decreasing marginal utility from one good the indifference curves should look like arcs. The area of Pareto improvements is limited by the indifference curves that pass through the initial allocation. After exchange we would expect to end up in an area of the box where Anna has a lot of cheese and Brewster has a lot of wine. Therefore the contract curve should be an arc passes through that area. The indifference curves should be drawn so that MRS is equal for both persons as they pass the contract curve. Since we have decreasing marginal utility from one good the indifference curves should look like arcs. The area of Pareto improvements is limited by the indifference curves that pass through the initial allocation A B Coca cola 1.4 a) When marginal cost of production increases the supply curve shift upwards to the left o market 2 since the supply curve is equal to the marginal cost curve under perfect competition. When the price of commodity 2 increase consumers will prefer commodity 1 so demand increase of commodity 1, demand curve shifts to the right. When demand of commodity 1increases, the price of commodity 1 increases. With a higher price of commodity 1 demand of commodity 2 increases, shifts the demand curve to the right. b) Producer surplus increase due to a higher price.

4 Kolstad 3:2 Majority vote: I everybody should pay an equal amount the cost per person would be: 60 % will vote no since they are willing to pay no more than$1 If we use majority vote there will be no clean up = 2 Pareto criterion We should not clean up since somebody is worse off. (Note also that if the Pareto criteria would have been used before the pollution was emitted we should not pollute either, since than someone was hurt from pollution. The Pareto criterion almost always supports the status quo.) The compensation principle - Kaldor Hicks If the 40 % that benefit benefits more than the losers loose we should clean up. Net benefit for the winners: (100 2) = Net cost for the losers: (2 1) = Total net benefit: = The winner win more than the losers loose, it would be possible for the winners to compensate the losers. Therefore the Kaldor Hicks criterion supports clean up even if no compensation is paid. (Another way to conclude this is that the total net benefit is positive. When we do cost-benefit analysis we uses the Kaldor Hicks criterion.)

5 Pindyck Rubinfeld 16:1 a) In equilibrium supply equals demand. Since supply is fixed we can substitute supply into the inverted demand equations: P G = P S P S = P G Which simplifies to: P G = P S P S = P G In a general equilibrium both these equations should be fulfilled. We have two equations and two unknown so it is possible to solve for the equilibrium prices. Substitute the price of gold into the equation for the price of silver gives: P S = ( P S ) P S = P S 0.75P S = 750 P S = 750 0,75 = Substitute this in the equation for the price of gold P G = = 1400 b) P G = P S P S = P G P G = P S P S = P G P S = ( P S ) P S = P S 0.75P S = P S = 7.5 0,75 = 950 P G = = 1300 The discovery of gold decreases the gold price but also the price of silver.

6 Pindyck Rubinfeld 16:2 A) If consumers are worried about the quality of the chicken, then they may choose to consume pork instead. This will shift the demand curve for chicken down and to the left and the demand curve for pork up and to the right. Feedback effects will partially offset the shift in the chicken demand curve, because some people may switch back to chicken when the price of pork rises. This will shift the demand curve for chicken back to the right by some amount, raising the price of chicken and shifting the demand curve for pork a bit further to the right. Overall, we would expect the price of chicken to drop, but not by as much as if there were no feedback effects. The price of pork will increase, perhaps by more than the increase in the absence of feedback effects. The other possibility is that the outbreaks of disease cause a reduction in supply of chicken. This would increase the price of chicken, which would then lead to an increase in demand for pork. The price of pork would increase in response, which would boost the demand for chicken by some (probably small) amount, raising the price of chicken again. This would increase the demand for pork some more, and so on. Ultimately these effects will peter out, but the final increases in prices will be greater than if there had been no feedback effects. B) The increase in tax raises the price of airline tickets, making it more costly to fly. The resulting increase in ticket prices would reduce the quantity of airline tickets sold. This in turn would reduce the demand for hotel rooms by out-of-town visitors, causing the demand curve for hotel rooms to shift down and to the left and reducing the price of hotel rooms. For the feedback effects, the lower price for hotel rooms may encourage some consumers to travel more, in which case the demand for airline tickets would shift back up and to the right, partially offsetting the initial decline in quantity demanded. The increase in airline demand would increase the demand for hotel rooms, causing hotel room prices to increase somewhat. In the end, we would still expect reduced quantities of both airline ticket sales and hotel rooms, higher airline ticket prices (due to the tax) and reduced prices for hotel rooms

7 Pindyck Rubinfeld 16:3 Given that MRS Bob MRS Jane, the current allocation of resources is inefficient. Jane and Bob could trade to make one of them better off without making the other worse off. Although we do not know the exact shape of Jane and Bob s indifference curves, we do know the slope of both indifference curves at the current allocation, because we know that MRS Jane = 4 and MRS Bob = 2. At the current allocation (point A in the diagram), Jane is willing to trade 4 sandwiches for 1 drink, or she will give up 1 drink in exchange for 4 sandwiches. Bob is willing to trade 2 sandwiches for 1 drink, or he will give up 1 drink in exchange for 2 sandwiches. Jane will give 4 sandwiches for 1 drink while Bob is willing to accept only 2 sandwiches in exchange for 1 drink. Any exchange that leaves both parties inside the lens-shaped area between points A and B will make both better off. For example, if Jane gives Bob 3 sandwiches for 1 drink, they will be at point C. Jane is better off because she was willing to give up 4 sandwiches but only had to give up 3. Bob is better off because he was willing to accept 2 sandwiches and actually received 3. Jane ends up with 4 drinks and 6 sandwiches and Bob ends up with 7 drinks and 7 sandwiches, and both are better off than at point A.

8 2 Environmental valuation and cost benefit analysis 2.1 We can use number of trips per person in the three areas and the travel cost to estimate a demand curve for the national park. Travel cost Visits per year Consumer surplus for one individual are equal to the area above the travel cost but below the demand curve: City Consumer surplus per person Population Total consumer surplus A-city = 40 2 B-city = C-city = 10 2 Total The total value according to this analysis is b) The travel cost method does only measure use values. If there are non use values of the national park we have made an underestimation. We assume that all people have the same preferences. If preferences between the population in a city and c city differ we get problems since the demand curve would not be correct in that case. We do not know whether this gives an over or an underestimation of the true value. The travel cost here is probably the price of bus tickets or something like that. Maybe we should also include the value of time as a travel cost? If we believe that such a measure of travel cost would be more relevant this valuation would be an underestimation.

9 2.2 According to the regression equation an increase in emission reduces the price with 100 dollar per μg m 3. The expected reduction in the price is thus = 1000 If the concentration is reduced with 30 we would expect house prices to increases with = Since there are houses in this area total house prices would increase with = 15 million dollars. If other people than the house owners benefits from the decrease in pollution the 15 million dollars will be an underestimation. (Moreover if house owners are hurt more than what they take into account when thinking about how much to pay for a specific house the regression equation will not reflect their true valuation. This valuation method assumes that everyone is perfectly informed and rational when acting on the market, an assumption that may not always be fulfilled.) 2.3 We can use number of trips per person in the three areas and the travel cost to estimate a demand curve for the beach Trips/year Consumer surplus for one individual are equal to the area above the travel cost but below the demand curve: City Consumer surplus per person Population Total consumer surplus Area 1 (80 4)/2 = Area 2 (60 3)/2 = Area 3 (40 2)/2 = Total The total value according to this analysis is 6.1 million.

10 b) The travel cost method does only measure use values. If there are non use values of the beach we have made an underestimation. We assume that all people have the same preferences. If preferences between the population in area 1 and area 3 differ we get problems since the demand curve would not be correct in that case. We do not know whether this gives an over or an underestimation of the true value. Kolstad 6:1 a) The net present value of the net benefit is about 25 million dollar, depending on what assumptions you make, this is a profitable investment b) The net present value of the net benefit is about -50 million dollar, depending on what assumptions you make, this is not a profitable investment c) About 4,5% Kolstad 6:2 a) b) c) , 100 = 72,6 investment is not profitable = investment is profitable, ,0 100 = is the maximum amount we will be willing to pay today to avoid this damage in the future.

11 Kolstad 7:4 a) Since Jose cannot buy air quality, the air is there for free, he cannot use his income for anything else than housing. Thus he consumes 5 units of housing. This is his budget constraint. With 5 units of housing and an air quality of 2 his utility becomes: U = 2 5 = 10 We get the indifference curve by calculating different combination that give the utility level of 10 housing Air quality calculation 2 5,00 10/2 3 3,33 10/3 4 2,50 10/4 5 2,00 10/5 6 1,67 10/6 7 1,43 10/7 8 1,25 10/8 9 1,11 10/9 10 1,00 10/10 H A

12 b) Increasing air quality to 4 keeping his income constant gives a utility level of: U = 4 5 = 20 We get the indifference curve by calculating different combination that give the utility level of 20 housing Air quality calculation 2 10,00 20/2 3 6,67 20/3 4 5,00 20/4 5 4,00 20/5 6 3,33 20/6 7 2,86 20/7 8 2,50 20/8 9 2,22 20/9 10 2,00 20/10 H 10 d a c b 10 A c) If air quality is improved from 2 to 4 Jose may reduce housing with 2,5 units and still get the same level of utility, reaching the same indifference curve. (Move from point a to point b) Therefore she will be willing to pay a maximum of 2,5 2 = 5 dollar. (How much wold she be willing to accept as a compensation if air quality is not improved? If she is compensated by another 5 units of housing she will reach the same indifference curve as she reaches by keeping 5 units of housing but get an increase in air quality to 4. Thus the lowest amount she is willing to accept to lose the improvement in air quality is 5 2 = 10. What is the value of improved air quality?)

13 Kolstad 9:6 Zone Distance Population Number of visitors Travel cost Total cost Visits per capita total Zone Distance Population Travel cost Total cost Visits per capita Number of visits total According to this analysis the number of visitors would decrease from to

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15 Consumption 3 Externalities and public goods 3.1 Air pollution 3.2 a) Take the derivative of total external cost: MC E = CE Q = 0,2 Q b) Supply equal demand: P = 10 P 240 = 12 P P = 40 = 20 Q S = = 200 Q D = = 200 We produce 200 at a price of 20 c) In optimum price is equal to social marginal cost. Private marginal cost comes from the supply function (replace P with MC): Q S = 10 MC P MC P = 0,1 Q Social marginal cost is the sum of private marginal cost and marginal external cost: MC S = MC E + MC P = 0,2 Q + 0,1 Q We may substitute social marginal cost as the price into the demand function to get the optimal quantity. Q = (0,2 Q + 0,1 Q) Q = 240 0,4 Q 0,2 Q 1,6 Q = 240 Q = 40,6 = 150

16 d) To achieve the social optimal production level the tax should be equal to marginal external cost at the optimum level of production. MC E (150) = 0,2 150 = 30. The optimal tax level is 30 e) To draw a figure we may derive the inverted demand and supply functions: Q D = P P = Q D Q S = 10 P P = 0.1Q S Supply function with the tax: P = Q S Price at the optimal output: P = = 45 (or = = 45 ) Price less tax at optimum: P = = 15 P 100 D S with tax S Decrease in consumer surplus: (45 20) + 50 (45 0) = = Decrease in producer surplus: 150 (20 15) + 50 ( 0 5) = = 875 Tax revenue: = External cost before tax: C E = = = External cost before tax: C E = = = Decrease in external cost: = Net benefit for society: = Society will have a net benefit of Q

17 3.3 a) Supply from the marginal cost function P = Q S Q S = P 200 Supply equal demand: 240 0,1 P = P = 1,1 P P = 440, = 400 Price will be 400, at that price demand is: Q D = 240 0,1 400 = = airplane tickets will be sold b) In optimum demand should be equal to social marginal cost: MC S = MC E + MC P = Q = Q We may substitute social marginal cost as the price into the demand function to get the optimal quantity: Q = 240 0,1 (300 + Q) Q = ,1 Q 1,1 Q = 210 Q = 0, = airplane tickets would be optimal. The optimal price is thus: Q D = 240 0,1 P 10Q D = 2400 P P = Q D P = = 490 c) 100. Equal to the marginal external costs

18 d) To draw a figure we need the inverted demand and supply functions. P = Q S P = Q D P D 1000 S with tax S Q Decrease in consumer surplus: 191 ( ) + ( 00 9 ) ( ) = = Decrease in producer surplus: 191 ( ) + ( 00 9 ) ( ) = = Tax revenue: = Decrease in external cost: = 900 Net benefit for society: = 450 Society will have a net benefit of 450.

19 3.4 a) Q = = 40 P = 20 Under perfect competition production is 40 and the price 20. b) $5 Equal to the marginal external costs c) Now MC = and thus p = 25 Q = = 30 P = 25 To draw a figure we need the inverted demand function: Q = 80 2 P => P = 40 0,5Q P M C MC 10 D Q Decrease in consumer surplus: = 175 Tax revenue: 5 30 = 150 External cost decreases by 5 10 = 50 Net benefit for society: = 25 Society will have a net benefit of 25.

20 P M C MC 10 MR D Q d) The monopolist set MR = MC MR = 40 Q (Since inverted demand is:p = Q ) MR = MC = 40 Q = 20 Q = 20 Price is from the demand curve => P = 30 e) MR = MC = 40 Q = 25 Q = 15 Price is from the demand curve => P = 40 0,5 15 = 32.5 Decrease in consumer surplus: = Tax revenue: 5 15 = 75 External cost decreases by 5 5 = 25 Change in profits:15 ( ) 20 (30 20) = = 87.5 Net benefit for society: = Society will have a loss of Produced quantity was below optimal production level and the tax reduced it even further.

21 3.5 To draw a figure we need the inverted demand function Q = P 2 P = 100 Q P = 50 0,5 Q Marginal social cost is equal to the sum of marginal private cost and marginal external cost, in this case 30. P M C MC D Q a) Q = = 60 Under perfect competition production is 60 and the price 20. b) Equal to the marginal external costs, that is 10. c) With a tax of 10 private marginal cost would be 30. Q = = 40 With a tax of 10 the production would be 40 and the price 30. Change in consumer surplus: = = 500 Increase in tax revenue: = 400 (tax rate times the quantity) External cost decreases with = 200 (cost per unit times the decrease in output) Net benefit of society: = 100 d) The monopolist set MR = MC when choosing how much to produce: P = 50 0,5 Q => MR = 50 Q MR = MC => 50 Q = 20 Q = 30 P = 50 0,5 30 = 35

22 P M C MC D Q The monopoly firm will produce 30 units to the price of 35. (In this case we are closer to the optimal quantity under monopoly than under perfect competing) e) With a tax of 10 private marginal cost is equal to the social marginal cost of 30 MR = MC => 50 Q = 30 Q = 20 P = 50 0,5 20 = 40 Decrease in consumer surplus: = 125 Tax revenue: = 200 External cost decreases by = 100 Change in profits:20 (40 30) 30 (35 20) = = 250 Net benefit for society: = 75 Society will have a loss of 75. Produced quantity was below optimal production level and the tax reduced it even further.

23 3.6 a) A Harry Potter movie: The movie in itself is non rival (you only need to shoot and mix the movie once) A DVD is rival in the sense that we must produce one DVD for each customer that wants to buy it. At the cinema there may be some rivalry it the cinema is sold out, ( congestion good) From a legal point of view we have excludability since you need to pay to purchase the DVD or go into the cinema. However from a practical point of view it is quite simple to download the movie from the internet, limiting the excludability. b) A hockey stick: This is a rival good, every player needs his own stick. This is excludable, you need to pay to take the stick out of the shop. c) Watching a football game live Non rival as long as there are a lot of empty seats, rival when it is close to sold out, (congestion good.) Excludable since you need to pay to get into the arena. d) Cloudberries in the forest: These are rival, if I eat a couple of cloudberries you cannot eat the same berries. In Sweden non excludable since everyone has the right to pick berries everywhere, in a lot of other countries berries are excludable. e) Travelling by train from Örebro to Motala Non rival if there are a lot of empty seats, rival when the train is close to sold out, (congestion good) Excludable since you need to pay the ticket f) A swimsuit This is a rival good, it is difficult for two persons to swim in the same swimsuit at the same time. This is excludable, you need to pay to take the swimsuit out of the shop. g) A lamppost on the square Non rival as long as there are no congestion on the square, everone there wil benefit from the light of the same lamppost. Non excludable, everyone has the right to be on the square without paying. h) Fish in the Atlantic Sea Rival, If a catch a fish you cannot catch the same fish If we have no policy to regulate the fishery the fish is non excludable, with individual transferrable quotas the fish will be excludable, you need to purchase a quota to get the right of fishing.

24 3.7 a) Number of cars Travel time (minutes) Value per car Total value of the road b) 350 car owners will go by car. (If they are fewer, travel time is faster than the train and more people will choose to go by car. If they are more than 350, travel time is longer than by train and some of them will choose the train instead) Travel time will be 29 minutes and the value of the trip is equal to zero since you could as well go by train. c) 300 car owners will go by car. (If they were fewer the value of the car trip would be larger than the fee and more car owners will choose to go by car. If they are more than 350, the value if a trip is lower than the fee and some of them will choose the train instead). The total value of the road for society is For the car owners the value is zero since the fee they pay is equal to the value of the trip, but the government gets a revenue of at no cost to the car owners. (Those that go by train have a cost of increased congestion on the train) d) The value of the road is maximized with 250 cars on the road; the optimal fee is thus 50. e) Those that still go by car are indifferent; the value of the trip has increased as much as the fee. Those that went by car before and now chooses the train is indifferent since there value of going by car was 0 before. Those that went by train before gets a cost in increased congestion on the train.

25 f) All car owners benefit from the decrease in yearly tax payment. As in the previous exercise they are indifferent from the fee so this is a pure benefit. Those that went by train before gets a cost in increased congestion on the train. g) When you choose between going by train and going by car the time saving from car trips is reduced. The table below gives the valuation of a car trip relative a train trip when the train is faster. Number of cars Travel time (minutes) Value per car Total value of the road The number of cars will be 250. To value the benefit from the combined reform of those that still choose to go by car, we need the valuation of the car trip of 19 minutes compared with a car trip of 29 minutes. The car owners save 10 minutes at a value of 50. But they pay a fee of 25 Their net benefit is thus 25. Since they are 250 the total benefit to all that go by car is = Those that choses to go by train save 5 minutes that they value to 25. They get a cost however from increased congestion if not some of the money is used for longer trains.

26 h) We need to fill out the new table: Number of cars Travel time (minutes) Value per car Total value of the road people will choose to go by car. If you choose to go by car you save 5 minutes which you value to 25. But you get a cost of 25 so those that go by car is indifferent from the reform. Those that still go by train benefits from a reduction in congestion. g) If the number of inhabitants is just 350 travel time will be reduced to 19 minutes and the car owners will benefit 25. If not the wider road make this suburb more popular and attracts more inhabitants. This illustrates one important point on controversy in traffic planning. Do new and wider roads reduce congestion or increase traffic?

27 3.8 a) For a rival good we derive aggregate demand by adding up the quantities sold at every price level. To do so we must substitute for Q: D : P = 6 Q => Q = 6 P D : P = 10 0,5 Q => 0,5 Q = 10 P Q = 20 2 P Q = 6 P P Q = 26 3 P 3 P = 26 Q P = 6 Q 3 3 This is only true in the interval: 0 < P < 6 At prices above 6, p > 6 aggregated demand is: P = 10 0,5 Q Since only individual 2 buys anything at these prices. Optimal quantity is given from the cross section of aggregate demand and marginal cost. 2,5 Q = 6 Q 3 3 2,83 Q = 6 3 Q = 6 = 3,06 3,83 => P = 2,5 3,06 = 7,65 This is not in the interval 0 < P < 6. Thus we have used the wrong demand function. We must use the demand function for prices above 6: 10 0,5 Q = 2,5 Q 10 = 3 Q Q = 0 = 3,3 3 => P = 2,5 3,3 = 8,25 Optimal output is 3.3 and optimal price is 8,25. This is also the outcome of a competitive unregulated market. 10 D D 10 20

28 b) When deriving an aggregated demand curve for a non-rival good we add the amount that each consumer is willing to pay for each unit (since everyone can use the same unit) P = 6 Q ,5 Q P = 16 1,5 Q This is only true in the interval 0 < Q < 6 For Q > 6 aggregate demand is equal to: P = 10 0,5 Q Since only individual 2 has a positive MWTP at that quantity. Optimal quantity is given from the cross section of aggregate demand and marginal cost. P = 16 1,5 Q P = 2,5 Q 16 1,5 Q = 2,5 Q 16 = 4 Q Q = 4 (this is in the interval 0 < Q < 6, if it had been outside that interval we would have needed to do the calculation from the other demand curve) MWTP is: MWTP = 16 1,5 4 = 10 (This is however not the optimal price since no one would consume at that price. The price must be lower than 2 so that both consumers chooses to consume the products, in that case cost of production must be covered by taxes) 10 D D 10 20

29 3.9 a) A public goods is non rival and non excludable. With non excludable means that we cannot exclude anybody that doesn t pay from consuming the good. With non rival means that my consumption of the good does not decreases your possibility to consume the same good. Therefore we do not need to produce one unit for each consumer. TV programs on svt may be an example since it is difficult to exclude anyone who do not pay the license to watch them and my consumption of a TV show doesn t have any impact on your possibility to watch the same TV show. Other examples may be the defence and roads that are not crowded. b) For quantities between 0 and 3 both consumers have a willingness to pay. In that interval we sum the MWTP for both consumers: P = 6 2q + 6 q = 12 3q for 0 < q < 3 For quantities above 3 only consumer 2 has a willingness to pay. p = 6 q for 3 < q < x c) Optimal quantity is given from the cross section of aggregate demand and marginal cost. 12 3q = 4 + q 8 = 4q q = 2 Important to note; the efficient price is not equal to 6. At a price of 6 nobody would by it. The efficient price of a non-rival good is zero so that everybody that has a positive willingness to pay chooses to consume the good.

30 Kolstad 4:1 If we should reduce emissions in a cost effective way those firms that have the lowest cost of abatement should abate more than those that have high cost of abatement. Kolstad 4:2 a) To draw a figure we need to invert the demand equation: Q = 50 P P = 50 Q P MC P D Q At perfect competition 40 units will be produced and sold at a price of 10. b) P MC P D Total revenue for the monopoly firm is: 50 Q

31 R = P Q = (50 Q) Q = 50Q Q Marginal revenue becomes: R Q = 50 2 Q MR = MC gives; 50 2 Q = 10 2 Q = Q = 40 = 20 The monopolist will produce 20 units at the price:p = = 30 c) In perfect competition (question a) the consumer surplus is a triangle with the base 40 and the height = 800 Pollution damage is = 600 Net surplus for the society is: 200 In monopoly (question b) the consumer surplus is a triangle with the base 20 and the height = 200 Producer surplus is = 400 Pollution damage is = 300 Net surplus for the society is: 300. Kolstad 5:1 a) TC(d) = d + d 2 = d + d b) TC(d) TC(d) d d 3 = d 3 = d 3 = 2 3 d = 2 = 1 2 d 3 = 0 = = 1,2599 TC(1) = 1 + = 2 TC(1,26) = 1,26 +, 6 2 = 1,89 TC(1,5) = 1,5 +,5 2 = 1,94 c) In this case the Fritz only cost is the transportation cost which is lower the closer he is to the airport. According to the question he can live no closer than 0,1. Damage cost and the compensation will be: d 2 = 0, 2 = 0,0 = 100

32 Kolstad 5:2 Q H = 1 P H => P H = 1 Q H Q M = 2 2 P => P M = 1 0,5 Q M 1 1 Aggregated willingness to pay is the sum of the willingness to pay of each individual. (Sum the demand curves vertically) P = 1 Q H + 1 0,5 Q M = 2 1,5 Q At optimum marginal cost in production should be equal to aggregated willingness to pay.. Q = 2 1,5 Q 2,5 Q = 2 Q =,5 = 0,8

33 Kolstad 5:4 a) 2 p + 6 p = 8 p b) MD MS MD R 10 MD W 1 p c) The cheese factory will increase emission until marginal saving of emission is zero: M = 20 2p = 0 20 = 2p P = 10 Optimal level of emissions is where marginal saving equals marginal damage: MS = MD => 8 p = 20 2 p 10 p = 20 p = 2 d) Without abatement the level of emission would be p = 10. Total emission is equal to 10 less the amount of abatement: p = 10 a Substitute this into the MD equation: MD = 8 p = 8 (10 a) = 80 8 a Marginal damage is equal to the marginal willingness to pay for abatement so MWTP(a) = 80 8a e) Substitute p = 10 a into the MS equation: M = 20 2 p = 20 2 (10 a) = a = 2 a Marginal saving in the firm is equal to the marginal cost of abatement so MC(a) = 2a

34 MWTP 10 MC 2 8 a f) Optimal level of abatement is 8. If we abate 8 we will emit 2. We get the same answer regardless of whether we analyze emission or abatement, the problems are equivalent. Pindyck & Rubinfeld 18:2 Computer software is an example of a public good. Since it can be costlessly copied, the marginal cost of providing software to an additional user is near zero. (The fixed costs of creating software are high, but the variable costs are low.) Therefore, software is nonrival. We do not need to develop new software for each user. Furthermore, it is expensive to exclude consumers from copying and using software because copy protection schemes are available only at high cost or high inconvenience to users. Therefore, software is by and large nonexclusive. As both nonrival and substantially nonexclusive, computer software suffers the problems of public goods provision. The optimal price is zero since benefits from the software is larger the more that uses it. But with a price of zero there will be no economic incentives to develop new software. Basically we have two different kinds of software. Some software packages are licensed under the copyright law and these are illegal to use if you do not have a license. This is for example the case with Windows operating systems and Microsoft office. Other software packages are licensed under the GNU general public license. The GNU software is free to use and it is illegal to sell software developed from GNU software. Examples here are Linux operating systems and open office. Some companies work with both free and copyrighted software. Adobe reader and adobe flash player are free software. This creates a larger use of the adobe file formats and increases the demand for the copyrighted software packages to produce these kinds of files. The advantage with copyrighted software is that the income from selling them creates an incentive for developers to create new software. The drawback is that the code must be protected so that the users can not develop own versions for his own purposes. The advantage of GNU software is the code is open source so that everyone with enough knowledge can create own version according to her own purposes. It fulfills the marginal cost pricing principle since marginal cost of another copy of software is zero. The drawback is that developers must find other sources of revenue, for example paid support services.

35 Pindyck & Rubinfeld 18:3 a) To find the socially efficient level of emissions abatement, set marginal benefit equal to marginal cost and solve for A: A = A 300 = 25A A = 12 b) Substitute A = 12 into the marginal benefit and marginal cost functions to find the marginal benefit and cost MB = = 260 MC = = 260 c) Net social benefit is the area under the marginal benefit curve minus the area under the marginal cost curve. At the socially efficient level of abatement this is equal to area a + b + c + d in the figure below, or 0.5 ( ) 12 = $1800 million If you abate one million tons too many, the net social benefit is area a + b + c + d e, or ( ) = = $ If you abate 1 million too few tons, then the net social benefit is area a + b or 0.5( ) 11 + ( ) ( ) 11 = $ In either case, then, net social benefit falls by = $12.5 million. d) It is socially efficient to set marginal benefit equal to marginal cost rather than total benefit equal to total cost because we want to maximize net benefits, which are total benefits minus total cost. Maximizing total benefits minus total cost means that at the margin, the last unit abated will have an equal cost and benefit. Choosing the point where total benefits are equal to total cost would mean that net benefits equal zero, and would result in too much abatement. This would be analogous to choosing to produce where total revenue was equal to total cost. If total revenue is always equal to total cost by choice, then there will never be any profit. In the case of abatement, the more we abate, the costlier it is. Given that funds tend to be scarce, dollars should be allocated to abatement only so long as the benefit of the last unit of abatement is greater than or equal to the cost of the last unit of abatement.

36 Pindyck & Rubinfeld 18:6 a) The equilibrium price and output would be where quantity demanded is equal to quantity supplied: P = P 4 000P = P = $30 per 100 pounds lots, and Q = lots b) To find the socially efficient solution, we need to consider the external costs, as given by: MEC = Q S, as well as the private costs. Private marginal cost is achieved from the supply function by solving for P and replacing it with MC Q S = P P = Q S 20 MC = Q S 20 Therefore, M C = MC + MEC = Q s Q s M C = Q s 20 Solve the demand curve for price: P = Q. This is the marginal benefit curve. Setting marginal social cost equal to marginal benefit, Q 20 = Q Q = lots P = $48.75 per 100 pounds lots c) The equilibrium quantity declined and the equilibrium price rose in part b because the external costs were considered. Ignoring the external costs of paper production results in too much paper being produced and sold at too low a price.

37 Pindyck & Rubinfeld 18:7 a) Set demand equal to supply to find the competitive equilibrium. To do this, set price equal to marginal cost (which is the industry supply function): 100 Q = 10 + Q 90 = 2Q Q = 45 P = = $55 b) First, calculate the marginal social cost (MSC), which is equal to the marginal external cost plus the private marginal cost. Next, set MSC equal to the market demand function to solve for price and quantity: M C = MC + MEC = 10 + Q + Q = Q MSC = MSB: 100 Q = Q 90 = 3Q Q = 30 P = = $70 When all costs are included, the quantity produced will fall and the price will rise. c) The tax should be equal to the marginal external cost at the efficient level of output, ie t = MEC(30) = 30 The marginal cost function now becomes: MC = 40 + Q Market equilibrium: 100 Q = 40 + Q 60 = 2Q Q = 30 P = = $70 d) First we must determine marginal revenue. TR = Q(100 Q) = 100Q Q MR = TR Q = 100 2Q MR = MC gives: 100 2Q = 10 + Q 90 = 3Q Q = 30 P = = $70 So the monopolist produce at the optimal level e) The tax should be zero since we already have the optimal level of output. f) In this case it is actually the monopolist that yields the higher level of social welfare compared to the competitive market, because the monopolist s profit maximizing price and quantity are the same as the socially efficient solution. Since a monopolist produces less output and sets a higher price than the competitive equilibrium, it may end up producing closer to the social equilibrium when a negative externality is present

38 Pindyck & Rubinfeld 18:10a The efficient number of hours is the amount T such that the sum of the marginal benefits is equal to marginal cost. The demand curves represent the marginal benefits (i.e., willingness to pay) for each group. Therefore, add the demand curves vertically to determine the sum of all marginal benefits: M B = W + W + W 3 = 200 T T T = 760 5T Setting this equal to MC, 760 5T = 200 T = 112

39 4 Emission taxes 4.1 P MC A MC A A a) The total cost of abatement can be calculated as triangels. The base is the quantity of abatement and the height is the marginal cost of abatement for the last abated unit. Total cost of abatement firm 1: 5 75 = 937,5 Total cost of abatement firm 2: 5 5 = 1562,5 Total cost of abatement 937, ,5 = 2500 b) For cost-efficient abatement, marginal cost of abatement should be equal in both firms: MC (A ) = MC (A ) => 3A = 5A We need another equation to solve this since we have two unknown variables. But we also know that the total level of emission should be 50 A + A = 50 Solving the first equation for A 1 A = 5 3 A And substitute into equation 2 5 A + A = A = A = 3 50 = 18, To calculate A1 insert 18,75 in the equation for A 1 A = 5 18,75 = 31,25 3 At these levels of abatement marginal cost of abatement in each firm becomes: MC (31,25) = 3 31,25 = 93,75 MC (18,75) = 5 18,75 = 93,75

40 P MC A MC A Total cost of abatement firm 1: 3, 5 93,75 = 1 464,84 50 A Total cost of abatement firm 12: 8,75 93,75 = 878,91 Total cost of abatement 1 464, ,91 = Total cost of abatement is reduced compared to exercise a since the abatement that firm 2 not need to do is more expensive than the extra abatement in firm 1. But if we increase abatement even more in firm 1 these extra units would be more expensive than the saving in firm 2. Therefore this is the cost minimizing distribution of abatement. c) We need a tax of for the firms to abate exact as much as in the cost effective solution. The firms will always abate until marginal cost of abatement is equal to the tax. Therefor marginal cost of abatement will always be the same in all firms if all firms face the same tax rate. If the marginal cost of abatement is the same in all firms we cannot reduce abatement cost by redistribution of the abatement between firms. Thus an emission tax will always give a cost effective abatement.

41 4.2 a) e The figure above illustrates the marginal saving of emission in agriculture. This is also the marginal cost of abatement. (If you emit you save the cost of abatement) If we reduce emission from 20 to 15 the total cost of abatement would be equal to the sum of the marginal cost of abatement for each unit, the marked area in the figure. Marginal cost of abatement for the 15 emitted unit is: M a (15) = 10 0,5 15 = 2,5 The area of the triangle is: 5,5 = 6,25 In the same way total cost of abatement in waste water treatment is: MC W (15) = = 20 The area of the triangle is: 5 0 = 50 Total cost of abatement in both sources with a 25 % reduction of emissions: ,25 = 56,25

42 b) To minimize total cost of abatement marginal cost of abatement/marginal saving from emission should be the same in all sources M a (e a ) = M w (e w ) = 10 0,5e a = 80 4e w We need another equation to solve this since we have two unknown variables. But we also know that the total level of emission should be 30 e a + e w = 30 We may write emission from waste water treatments as: e w = 30 e a Substitute this into the first equation gives: 10 0,5e a = 80 4(30 e a ) 10 0,5e a = e a 50 = 4,5e a e a = 50 4,5 = 11,11 e w = 30 11,11 = 18,89 At these emission levels marginal saving of emission is equal to: M a (11,11) = 10 0,5 11,11 = 4,44 M w (18,89) = ,89 = 4,44 In agriculture they must abate = 8.89 and total cost of abatement is: 8,89 4,44 = 19,6 In waste water treatment they must abate 20 18,89 = 1,11 and total cost of abatement is:, 4,44 = 2,42 Total cost of abatement at both sources is 22. By replacing expensive abatement at the waste water treatment with cheaper abatement in agriculture we have reduced total cost of abatement in society. An emission tax of 4.44 would give this emission levels since both sources abate until marginal cost of abatement is equal to the tax.

43 4.3 a) Marginal cost of abatement for the first abated unit, that is marginal saving for the last emitted unit, is M a (10) = = 0 M b (5) = = 0 Marginal cost for the third abated unit in firm a (marginal saving for the 7 th emitted unit) is: M a (7) = 10 7 = 3 Marginal cost for the third abated unit in firm b (marginal saving for the second emitted unit) is:m b (2) = = 6 Total cost of abatement at factory a is: 3 3 = 4,5 Total cost of abatement at factory b is: 3 6 = 9 Total cost of abatement for society is = 13.5 b) We may derive an aggregated marginal saving curve for both factories. M a (e a ) = 10 e a => e a = 10 M a M b (e b ) = 10 2 e b => e b = 5 0,5M b Under an assumption of cost efficient abatement M a (e a ) = M b (e b ) and could be replaced by MS. e = e a + e b = 10 M + 5 0,5M e = 15 1,5M M = 5 e,5,5 M = 10 3 e Marginal cost of abatement at a total emission level of 9 is equal to:: M (9) = = 10 6 = 4 We need a tax of 4 to get a total emission level of 9. Firms will abate until MS equals the tax level: Emission at factory a becomes: 4 = 10 e a e a = 10 4 = 6 (abatement is 10 6 = 4) Emission at factory b becomes: 4 = 10 2e b e b = 5 2 = 3 (abatement is 5 3 = 2) Total cost of abatement at factory a: 4 4 = 8 (A triangle with the base equal to total amount of abatement and the height equal MS at the emission level of 6) Total cost of abatement at factory b: 4 = 4 (A triangle with the base equal to total amount of abatement and the height equal MS at the emission level of 3) Total cost of abatement for society is = 12

44 4.4 a) They will emit as long as the marginal saving of emission is positive. The emission levels are thus found by setting the MS functions equal to zero. M (x ) = x = 0 => x = 40 M (x ) = x = 0 => x = 30 b) At optimum: M (x ) = M (x ) = MD(x) x = x x = (x + x ) x = x 10 = 0.1 x x 100 = x + 6 x x = x (100 6 x ) = x x = x 2 x = 30 x = 15 x = x = 10 x = x + x = = 25 M (10) = = 7.5 M (15) = = 7.5 MD(25) = = 7.5 The optimal level of emission is equal to 25 units Firm 1 emits 10 units, i.e abates 30 (40-10) Firm 2 emits 15 units, i.e abates 15 (30-15) Alternative solution: Derive the aggregated MS function 10 M = 0.25x 40 4 M = x 30 2 M = x x = 70 6 M 6 M = 70 x M = x M = MD => x = x 0.266x = x = 25 MD(25) = = 7.5

45 c) The optimal level of an emission tax is equal to the marginal damage that is 7.5 (the marginal external cost) 20 M x 10 M x MD x M x d) 20 Total cost of abatement firm 1: = x Total cost of abatement firm 2: = total cost in both firms = e) Total abatement must be 45, therefore each firm abate 22.5 Since the emission tax minimizes cost of abatement for a specific total emission level this distribution must have higher cost of abatement. No calculations are required but just to check we can do it: Firm one emit = 17.5 Firm two emit = 7.5 M (17.5) = = M (7.5) = = Total cost of abatement firm 1: = 63.3 Total cost of abatement firm 2:.5. 5 = Total cost in both firms = 190 which is higher than

46 4.5 a)

47 b) c) An emission tax of 80 gives a reduction of 60 units since both firms set there marginal cost of abatement equal to the emission tax. An emission tax gives always cost efficient abatement since all firms set their marginal cost of abatement equal to the tax marginal cost of abatement must be the same in all firms.

48 M e MD e M e M e e a) Firm 1 would emit 1000 units and firm 2 would emit 1400 units b) 800 (200 in firm 1 and 600 in firm 2) c) 80 (equal to MD and MS) d) Both firms abate 800, total cost of abatement is in each firm, so for society as a whole total cost of abatement is equal to When subsidies are increased to ecological farming the supply of ecological milk increases. In the intersection between D 1 and S 2 we get a lower price of ecological milk. The lower price of ecological milk makes more consumers choose ecological milk instead of conventional milk, the demand of conventional milk shifts to the left from D 1 to D 2. The decreases in demand of conventional milk reduces the price of conventional milk. When the price of conventional milk is reduced some consumer would replace ecological milk with conventional milk and demand of ecological milk shifts down from D 1 to D 2.

49 Kolstad 12:4 a) M e 5 M e M e b) M e MD e 5 p M e MD e M e e e e

50 c) M (e ) = 5 e e = 5 M (e) M (e ) = 8 2 e 2 e = 8 M (e) e = 4 0,5 M (e) e = e + e = 5 M (e) + 4 0,5 M (e) = 9 1,5 M (e) 1,5 M (e) = 9 e M (e) = 6 e 3 MD(e) = MD (e) + MD (e) = e + e = 2 e Optimal level of emission is where MS = MD 6 3 e = 2 e 6 = 2,67 e e = 6 = 2,25,67 p = MD(e ) = 2 2,25 = 4,5 e = 5 p = 5 4,5 = 0,5 e = 4 0,5 p = 4 0,5 4,5 = 1,75 Kolstad 12:5 If we have a larger externality, i e if MC T shifts upward the optimal level of output would be smaller. So whether a monopolist produces more than, less than or exactly equal to the optimal level of output depend upon the size of the externality.

51 Kolstad 12:6 p = 60 TC F = Q F MC F = 4Q F TC G = Q G MC G = 2Q G a) Price equals marginal cost: 4Q F = 60 => Q F = 60 4 = 15 2Q G = 60 => Q G = 60 = 30 Fireyear produces 15 ton and Goodstone produces 30 ton. π F = = = 150 π G = = = 400 b) A pigovian tax should be equal to marginal damage cost in this case 12. Cost functions with the tax included: TC F = Q + 2Q F MC F = Q F TC G = Q + Q G Price equals marginal cost: Q F = 60 => Q F = 48 = 12 4 MC G = Q G Q G = 60 => Q G = 48 = 24 Fireyear produces 12 ton and Goodstone produces 24 ton. π F = = = 12 π G = = = 76 c) The subsidy should be 12 per unit of reduced emission. Cost functions with the subsidy included: TC F = Q F 12(15 Q) = Q + 2Q F TC G = Q G 12(30 Q) = Q + Q G MC F = Q F MC G = Q G Since marginal cost functions are the same as with the tax, Fireyear produces 12 ton and Goodstone produces 24 ton. π F = = = 168 π G = = = 436 d) With a pigovian tax fireyear makes losses in the short run. If they do not expect the price to increase they should shut down production. (Which reduces pollution even more) With the subsidy both firms will stay in the market. Moreover if the firms expect that a subsidy will be introduced they should increases output in order to make the subsidy being calculated from a higher level of pollution. If the tax/subsidy where to be introduced to all firms in the world and not only these two, the tax would probably lead to a reduction in supply and a price increase while the subsidy would lead to a increase in supply and decrease of the price. Thus only the tax gives the right prices in the consumer market.