Problem Set 8 Topi BI: Externalities 1. Suppose that a polluting firm s private osts are given by TC(x) = 4x + (1/100)x 2. Eah unit of output the firm produes results in external osts (pollution osts) equal to $2. The firms is a prie-taker, and an sell its output for $12 per unit. a) What is the profit-maximizing level of output? The firms profits are given by: Π = TR TC = 12x 4x (1/100)x 2. Profits are maximized where: dπ/dx = 12 4 (1/50)x = 0 x = 400. $800. b) What will the firms profits equal given the level of output in a)? Π = 12x 4x (1/100)x 2 = 12(400) 4(400) (1/100)(400) 2 = $1600. ) What do (total) external osts equal, given the level of output in a). d) Draw a marginal ost/marginal benefit diagram that illustrates your answers for part a) through ). $ MSC MPC 12 MB = p Π = a + b a External Costs = b + b 6 4 400 x 1
e) What is the soial surplus maximizing level of output? SS max is where MB = MSC 12 = 6 + (1/50)x x = 300. f) What will the firms profits equal given the level of output in e) (assume that the when the firm produes this level of output it an ontinue to sell eah unit produed for $8)? By how muh do profits hange, relative to what they were in b)? Π = 12x 4x (1/100)x 2 = 12(300) 4(300) (1/100)(300) 2 = $1500, a derease of $100. $600. g) What do (total) external osts equal, given the level of output in e). By how muh do external osts hange, relative to what they were in )? h) Show on your diagram from d) that the hange in external osts is greater than the hange in profits. $ MSC MPC 12 a MB = p Π by d External Costs = + d b 6 4 300 400 i) The government wishes to orret the market failure resulting from this externality, and is onsidering a number of different poliy instruments to do so. The first is a per unit tax of amount $t on the firm s output. The firms profits are given by: Π = TR TC tax osts = 12x 4x (1/100)x 2 - tx x 2
j) Write down the firm s profit funtion, given this tax (hint: profits will be a funtion of x and t now). Profits are maximized where: dπ/dx = 12 4 (1/50)x t = 0 x = 400 50t. Note that the tax inreases the firm s MPC by the amount t. k) Derive an expression that defines the profit-maximizing level of output for the firm, as a funtion of the tax t. Profits are maximized where: dπ/dx = 12 4 (1/50)x t = 0 x = 400 50t. Note that the tax inreases the firm s MPC by the amount t. l) What must the tax equal if it indues the firm to produe the soial surplus maximizing level of output? If t = 2, the firm will hoose x = 300, the SS maxing level of output. That is, the per unit tax must equal the MEC, whih is $2. Thus the tax ensures that the firm s new MPC oinides with MSC. m) By how muh will the firm s profits hange as a result of the tax (you should use your answer to part b) as the baseline for omparison)? Π = 12x 4x (1/100)x 2 tx = 12(300) 4(300) (1/100)(300) 2 2(300) = $900, a derease of $700. n) By how muh will external osts hange as a result of the tax (you should use your answer to part ) as the baseline for omparison)? External osts are $600, a derease of $200. o) How muh revenue does the government raise from the tax? The government raises tx = $600. 3
p) Draw a diagram illustrating the effet of the tax on the level of output, the firm s profits, external osts, and government revenue. t MPC + t = MSC MPC 12 a g d MB = p e b 6 4 The firm s TR = a + g + e + b + f The firms TC = f. The firm s tax osts = a + g So the firm s Π = e + b Before the tax, the firm s Π = a + g + d + e + b, so the tax dereases Π by a + g + d. External osts = b + g. Before the tax, external osts = b + g + d +. So the tax dereases external osts by d +. The government raises tx in revenue = areas a + g. f 300 q) Does this poliy represent a Pareto improvement? How about a potential Pareto improvement? The tax dereases Π, so is not a Pareto improvement. However, the derease in Π (a + g + d) is more than offset by the gains, in terms of external osts (d + ) and government revenue (a + g). The overall (net) gain of the tax is represented by the area, whih was the deadweight loss due to the externality, at the original equilibrium. Beause there is an overall gain, there is a potential for Pareto improvements. That is, this hange represents a potential Pareto improvement (PPI; see readings for Topi AI for an explanation of the onept of a PPI). 400 x 4
Next the government is onsidering a subsidy that it will pay the firm if the firm redues its output, relative to the level in a). That is, for every unit of x that that the firm redues output by, it will be paid an amount $s by the government. r) Write down the firm s profit funtion, given this subsidy (hint: profits will be a funtion of x and s now). The firms profits are given by: Π = TR TC + subsidy payments. Note that the firm is paid s per unit for eah unit is redues output by, below the original equilibrium level of 400. This means that the subsidy payment = s(400 x). So we an write the firm s profits as: Π = 12x 4x (1/100)x 2 + s(400 x) s) Derive an expression that defines the profit-maximizing level of output for the firm, as a funtion of the subsidy s. Profits are maximized where: dπ/dx = 12 4 (1/50)x s = 0 x = 400 50s. Note that the subsidy inreases the firm s MPC by the amount s, in the region where it is paid. That is, in the region where x < 400. This seems ounterintuitive (why would a subsidy inrease the firm s MC?), but isn t really one you think it through. Suppose that the firm is produing (say) 200 units of output (I ve just piked an number that is in the region of output where the subsidy applies). Think about the ost to it of produing an extra unit of output. It has to pay its osts (given by the regular MPC urve) but there is an additional ost, in that if the firm produes the 201 st unit of output it forgoes the subsidy that the government would have paid it, had it not produed that 201 st unit. There is an extra opportunity ost to prodution now. t) What must the subsidy equal if it indues the firm to produe the soial surplus maximizing level of output? If s = 2, the firm will hoose x = 300, the effiient level of output. That is, the per unit subsidy must equal the MEC, whih is $2. Thus the subsidy ensures that the firm s new MPC oinides with MSC, very muh like the tax. u) By how muh will the firm s profits hange as a result of the subsidy (you should use your answer to part b) as the baseline for omparison)? Π = 12x 4x (1/100)x 2 + s(400 x) = 12(300) 4(300) (1/100)(300) 2 + 2(100) = $1700, an inrease of $100. 5
v) By how muh will external osts hange as a result of the subsidy (you should use your answer to part ) as the baseline for omparison)? External osts are $600, a derease of $200. w) How muh does this subsidy ost the government? This osts the government s(400 x) = $200. x) Draw a diagram illustrating the effet of the tax on the level of output, the firm s profits, external osts, and government revenue. MPC 12 a d MB = p s b 6 f 4 300 400 x The firm s TR = a + b + f The firms TC = f. The firm s subsidy payments = + d. 1 So the firm s Π = a + b + + d Before the subsidy, the firm s Π = a + b + d +, so the subsidy inreases Π by. External osts = b. Before the tax, external osts = b + d +. So the tax dereases external osts by d +. The government pays s(400 x) = areas + d. y) Does this poliy represent a Pareto improvement? How about a potential Pareto improvement? 1 This follows, as the subsidy is equal to the vertial distane between the old and new MPC urves, and the subsidy is being paid for the units not produed between 300 and 400. 6
The subsidy osts the tax-payer money, so is not a Pareto improvement. However, this ost ( + d) more than offset by the gains, in terms of external osts ( + d) and the firm s profits (). The overall; (net) gain of the subsidy is represented by the area, whih was the deadweight loss due to the externality, at the original equilibrium. Beause there is an overall gain, it is a potential Pareto improvement. 2. Suppose that two polluting firms are able to abate (redue) pollution using a variety of different methods, not simply just by reduing output (for instane, firms an substitute less polluting inputs for more polluting inputs, install pollution ontrol equipment, hanges prodution proesses, et.). All these methods of abatement are ostly. Firm 1 s marginal abatement ost (MAC) measured in dollars is MAC 1 = 2A 1 and Firm 2 s is MAC 2 = 6A 2, where A measures the number of unit of pollution abated (that is, if A 2 = 20, for instane, this would mean that firm 2 had redued its emissions by 20 units). a) If the marginal private benefit of abatement is zero for eah of these firms, how muh abatement will they hoose to undertake (assuming that they are profit maximizes). If the MB of abatement is zero, no firm will undertake abatement, given that it is ostly. A 1 =A 2 =0. Now suppose that the government wish to redue aggregate emissions by 120 units. b) How many units should eah firm abate if we wish to produe this abatement effiiently? Hint: reall the basi ideas we learned about produtive effiieny in Topi AI part (ii). We know produtive effiieny is ahieved if MAC 1 = MAC 2 and A 1 + A 2 =120. MAC 1 = MAC 2 2A 1 = 6A 2, or A 1 = 3A 2. A 1 + A 2 =120 A 1 = 120 - A 2. So we have A 1 = 3A 2 and A 1 = 120 - A 2, whih we an use to solve for A 2 = 30 and A 1 = 90. Produtive effiieny would have firm 1 do three times as muh abatement as firm 2, given the target. ) Explain how an emissions tax will reate a private benefit of abatement for eah of these firms, and hene will indue abatement. Beause the firms pay nothing to pollute, there is no benefit from not polluting (abating). A tax per unit of emissions reates a private benefit to abate as eah unit abated will save the firm tax revenue. Given a tax rate of $t per unit of emissions, the MPB of abatement will equal $t, and eah firm hoose the abatement level where $t = MAC. d) At what level will the emissions tax need to be set, in order to indue 120 units of aggregate abatement. 7
If $t = 180 here, firm 1 will hoose A 1 = 90 and firm 2 will hoose A 2 = 30, so aggregate abatement will be 120. e) True or False? Emissions taxes result in produtive effiieny, but not neessarily alloative effiieny. Explain. (Hint: What further information do we need in order to determine the alloatively effiient quantity of abatement?). Beause the firms fae the same prie per unit of pollution, we end up with produtive effiieny in terms of their abatement hoies. In order to get alloative effiieny, we need to know something about the MSB of abatement, sine alloative effiieny will be where MSB = (equalized) MACs. With no information about MSB of abatement, then any emissions tax will ensure that the abatement that ours does so at least osts, but we will be unable to asertain whether it is the right level of abatement. 3. Suppose an apple orhard is loated right next to an apiary. 2 The presene of the bees in the apiary (speifially, their pollination ativities) benefits the apple produer in the following way. The more honey there is produed in the apiary, the more bees there are around, and the lower the osts of produing apples. Speifially, suppose that the ost funtion for the apple produer is: TC(A) = A 2 /50 + 3A + 100 H. a) Does inreased honey prodution derease the marginal osts of produing apples? If not, whih omponent of osts does honey prodution derease? We an find MC(A) by taking the derivative of TC(A). This gives us MC(A) = A/25 + 3. H does not appear in this expression, so inreased honey prodution must lower the fixed osts of apple prodution (note that this is not neessarily realisti: it just makes the question a little easier.). The osts of honey prodution are TC(H) = H 2 /40 + 4H. Apples and honey both sell in ompetitive markets at pries p A = $5 and p H = $7. b) If the honey produer maximizes its profits, taking as given the prie of honey, how muh honey will it produe? Draw a diagram illustrating this prodution hoie, as well as the area that represents profits to the honey produer. How muh profit does the honey produer make? 2 An apiary is just the name for a olony of bees. 8
Π(H) = 7H (H 2 /40 + 4H) Π (H) = 7 (H/20 + 4) = 0 H* = 60. Note that 7 is the MB of produing honey and (H/20 + 4) is the MC of produing honey. Π(H) = $90, given H = 50. ) If the apple produer maximizes its profits, taking as given the prie of apples, how many apples will it produe? Draw a diagram illustrating this prodution hoie, as well as the area that represents produer surplus to the apple produer. Why does produer surplus not equal profits for the apple produer, while produer surplus does equal profits for the honey produer? What do profits equal for the apple produer? Π(A) = 5A (A 2 /50 + 3A + 100 H) Π (A) = 5 (A/25 + 3) = 0 A* = 50. Note that 5 is the MB to the apple produing from produing apples and (A/25 + 3) is the MC of produing honey. Π(A) = $10, given A = 50. d) Now suppose that the apple produer and the honey produer merge their businesses. Write down the joint profit funtion from apple and honey prodution. Π T = Π(H) + Π(A) = 7H (H 2 /40 + 4H) + 5A (A 2 /50 + 3A + 100 H) e) What levels of apple and honey prodution maximize joint profits? Explain why this is different from the answers you found for parts (b) and ()? Π/ H = 7 (H/20 + 4) + 1 = 0 H* = 80. Π/ A = 5 (A/25 + 3) = 0 A* = 50. Now the MB of produing honey is equal to the $7 that honey sells for, PLUS (previously external) benefit that eah unit of honey produed reates in terms of reduing the osts of produing apples. If you look at the apple ost funtion, you an see that this benefit is equal to $1 per unit of honey. So MB(H) in the merged firm is $8. As a result, the merged firm produes more honey than the honey produer would independently. Nothing has hanged in terms of apple prodution, sine the MB and MC of apple prodution is the same in the merged firm as in the independently operating firm. f) How muh profit does this merged firm make, relative to the total profit earned by the apple and honey produers when they operated independently? Why have profits inreased? Given H = 80 and A = 50, Π T = $100. Aggregate profits have inreased by $10. 9
g) Note that the inrease in profits with the merged firms indiates that there exist potential Pareto improvements where the firms are not merged. Assuming one again that the firms operate independently, suggest a lump sum payment that the apple produer would be willing to offer the honey produer, suh that the honey produer will happily inrease output from 60 to 80. We know that when H = 60, Π(A) = $10. If H = 80, Π(A) = $30. So the apple produer is willing to pay the honey produer up to $20 for the inrease in H. We know that when H = 60, Π(H) = $90. If H = 80, Π(H) = $80. So the honey produer needs to be paid at least $10 by the honey produer in order to inrease A from 60 to 80. Any payment stritly greater than $10 and less than $20 would represent a Pareto improvement. 4. Suppose that your neighbor is planning a party the night before your first midterm for. If the party goes ahead, you will be unable to study at home, and instead will need to go to the library. You would rather study at home, and having to go to the library imposes a ost on you that you have estimated to be equal to $60 (think of this as the hassle osts of having to leave your home). Your neighbor has estimated that the value of the party to her is $100 (think of this as a dollar value of the utility she gets from holding the party). Assume that no-one else gets any benefit from the party, and that noone else has osts imposed upon them by the party. a) From the viewpoint of eonomi effiieny, should the party go ahead or not? Explain? The benefit of the party is $100, while its ost is just $60. The part has positive net benefits, and it is therefore effiient for the party to take plae. Suppose that you have the legal right to prevent your neighbor from holding the party, but that you are allowed to negotiate with your neighbor over whether the party will go ahead (that is, your neighbor an pay you to let her hold the party). b) Suggest a likely outome of these negotiations. In partiular, will the party take plae? If so, suggest a dollar amount for the payment that your neighbor would pay you (more than one dollar amount may work). Your neighbor is willing to pay up to $100 in order for the party to take plae, while you need to be paid at least $60 in order to be fully ompensated for the party. Any payment between $60 and $100 would therefore make both you and your neighbor stritly better off than if you simply asserted your right to prevent the party. So 10
after negotiations, we would expet to see the party take plae, whih we know from a) is the effiient outome. Now suppose that your neighbor has the legal right to hold the party, no matter what the effet on you, but that again you are allowed to negotiate with your neighbor over whether the party will go ahead (that is, you an pay your neighbor not to hold the party). ) Suggest a likely outome of these negotiations. In partiular, will the party take plae? If so, suggest a dollar amount for the payment that you would make to your neighbor to prevent the party. Your neighbor needs to be ompensated by at least $100 in order for her to agree not to hold the party, while you are only willing to pay $60 in order to prevent the party. So you will not be able to pay your neighbor not to hold the party. So (again) after negotiations, we would expet to see the party take plae, whih we know from a) is the effiient outome. d) Do we get the effiient outome, no matter who has the property rights in this ase (that is, no matter whether your neighbor has the right to hold the party, or whether you have the right to prevent the party)? Yes, the assignment of property rights only matters for who pays and who gets paid, but not in terms of the outome, whih is effiient in eah ase. e) Now suppose that instead of valuing the party at $100, your neighbor values it at just $50. How would your answers to parts b) and ) hange in this ase? Now it is not effiient for the party to take plae. If you have the right to prevent the party, your neighbor is not willing to pay you enough to gain your permission for the party (she s only willing to pay $0 now, but you need at least $60 in ompensation), so the party will not take plae (whih is effiient). Similarly, if she has the right to hold the party, then you are willing to pay her up to $60 to prevent the party, while she only needs $50 in ompensation. Again, the party will not take plae. Now suppose that you are not the only one adversely affeted by the party. In fat, assume that there are 100 apartments in your apartment omplex, and the party will impose some ost on eah resident of all apartments. f) Do you think we are likely to get the effiient outome in this ase? If so, why? If not, why not? As we inrease the number of affeted parties, the osts to negotiations inrease, whih makes it less likely that we will be able to negotiate the effiient outome. The idea that externality-type problems an be solved by individual negotiations is 11
knowns and the Coase theorem. Note that the Coase theorem works sine SS isn t maximized in the presene of externalities. The Coase thereom, however, begins to break down one we have positive transation osts. The Coase theorem offers little useful insight to most real word extenality problems with whih we are onerned. 5. Consumer A gets utility from igarettes, denoted. All other goods are onsidered as a omposite ommodity, whih we will all good y. The prie of good y is $1 (hene we an thinking about y as how muh the onsumer spends on all goods that are not igarettes). Cigarettes ost $0.25 eah, and onsumer A has $100 a week to spend on igarettes and all other goods. Her utility funtion is UA = (16.25 ) + y. a) Derive onsumer A s demand urve (marginal private benefit urve) for igarettes. Draw a diagram illustrating this demand urve. Given the prie of $0.25 per igarette, how many igarettes will she smoke eah week? MRS A = 16.25 2. Setting MRS A = p /p y 16.25 2 = 0.25 16 = 2 = 8. $ 16.25 5 MPB 8 8.125 MPC Consumer A is married to onsumer B, who does not smoke. In fat, he hates the fat that A smoke. His utility funtion desribing his preferenes is U B = - + y, where is the number of igarettes that A smokes and y is B s onsumption of all other goods. b) Draw a diagram of B s indifferene urves between and y. Hints: (i) think arefully about why typial indifferene urves slope downwards and (ii) ask yourself what would need to happen to the quantity of y if the quantity of inreases, in order to hold B s utility onstant. y M 10 U B 6 L 4 Bundles L and M both yield the same utility (U = 6) for onsumer B. Moving from L to M, by 4 units, whih, eteris paribus, makes B worse off. If however, he reeives four units more of y as by 4, this is suffiient to fully ompensate him for the in the good that is a bad. Hene ICs slope upwards when one good is a bad. Note arefully the diretion of U here. Straight up is good (more y, onstant), while straight right is bad (more, y onstant). 12
) If B is the only onsumer adversely affeted by A s smoking, what is the marginal external ost of A s smoking? On your diagram from part (a), draw the marginal soial ost urve assoiated with A s smoking? The ost to B per igarette smoked by A is 1 unit of good y. In other words, $1 per ig smoked. So MEC = $1.00 and MSC = MPC + MEC = $0.25 + $1.00 = $1.25. 1.25 0.25 $ 16.25 5 MPB T S Q R MSC MPC 7.5 8 8.125 d) Demonstrate the when A hooses her smoking level to maximize her utility, soial surplus is not maximized. What smoking level does maximize soial surplus? Given = 8, private surplus to A is equal to areas T+S+R. But soial surplus = T Q, sine EC = S+R+Q. Reduing by one half (from 8 to 7.5) would redue private surplus by area R ($0.25, or ¼ of a unit of good y), but would redue EC by R + Q ($0.50, or ½ of unit of good y). So soial surplus would inrease by area Q ($0.25, or ¼ of a unit of good y), if we ould redue from 8 to 7.5. At =7.5, SS is maximized. e) Devise a poliy that A and B an implement suh that A hooses the soial surplus maximizing level of smoking. Is your poliy a PPI or a PI? Explain. If B paid A (say) $0.40 in return for A reduing igarette onsumption by one half, this would be an atual PI. B paid just $0.40 for a half unit redution in, whih yields him $0.50 of extra benefits (redued external osts). That last half ig yielded private surplus of R to person A, where area R equals $0.25, but she was given more than this ($0.40) in ompensation by B. It s a win-win sine both are better off. Any payment greater than $0.25 and less than $0.50 would represent suh a PI. 13