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Econ 525 Farnham/Gugl ______________________________________________________________________________

Answer Key to Problem Set 2 Warm-up Exercise from ECON313 1. Poldi and Gerald are married. Poldi likes to light scented candles in their house while Gerald

hates the smell of scented candles. They both value money and their utility function over money (MP for Poldi’s money and MG for Gerald’s money) and smell of scented candles (S), measured in hours of the day in which the candles are lit, is given by UP(MP, S) = MPS for Poldi and UG (MS,S) = MG(24-S) for Gerald.

a) Is smell a positive or negative externality? Explain.

Smell is a negative externality because Gerald’s utility goes down with each unit of smell.

b) Is smell a consumption or production externality? Explain.

Smell is a consumption externality because it is created by the action of a consumer. This definition is based on who causes the externality not who is the “victim” of the externality. Since Poldi is a consumer and not a firm, this is a consumption externality.

Suppose both Gerald and Poldi own initially $100 each and both people are price takers in all markets (i.e. we have competitive markets for both goods). The price of one unit of money is, of course, equal to $1. c) If Gerald has the right to scent-free air and charges a price of q for every hour he lets

Poldi light her candles, how many hours will Poldi light her candles and how much money will she have to pay Gerald to compensate him for the smell? What are Gerald’s and Poldi’s utilities under this property right regime?

If Gerald has the right to scent-free air, his utility maximization problem is Max MG(24-S) s.t. MG = 100 + qS, because additionally to his $100 he is also receiving revenues from letting Poldi light her candles S hours during the day. Substituting for MG in his utility function we have MaxS (100 + qS) (24-S) and taking the derivative with respect to S and setting it equal to zero yields q(24-S)-(100 + qS) = 0 From this optimality condition we can make S explicit to obtain S = 12 –50/q. For Poldi: Max MPS s.t. MP + qS = 100, because she needs to decide how much of her $100 she wants to keep and how much of it she wants to pay to Gerald in order to acquire the right to light candles for S hours of the day. Substituting for MP in her utility function we have

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MaxS (100 - qS) S and taking the derivative with respect to S and setting it equal to zero yields -qS+(100 - qS) = 0 From this optimality condition we can make S explicit to obtain S = 50/q. Because both people are price takers, and with the assignment of the property right to Gerald we have created a market for scent, we find the competitive price for scent, q, by setting demand equal to supply. We can think of Gerald as the supplier of smell and Poldi as the demander of smell. That is, 12 –50/q = 50/q and therefore q* = 100/12 = 25/3 and S* = 6. Gerald lets Poldi light her candles for 6 hours and receives in return from her 6*25/3 = 50 dollars. This implies that Gerald’s utility is equal to 150*(24-6) = 2700 and Poldi’s utility is equal to 50*6 = 300 under this property right regime.

d) If Poldi has the right to light candles for 24 hours a day and charges a price of q for every hour she does not light her candles, how many hours will Poldi light her candles and how much money will Gerald have to pay her to compensate her for not being able to enjoy the scented candles the whole day? What are Gerald’s and Poldi’s utilities under this property right regime?

If Poldi has the right to scented air, her utility maximization problem is Max MPS s.t. MP = 100 +(24 – S)q, because additionally to her $100 she is also receiving revenues from letting Gerald have 24- S hours of scent-free air during the day. Substituting for MP in her utility function we have MaxS (100 +(24- S)q) S and taking the derivative with respect to S and setting it equal to zero yields -qS+(100 +(24- S)q) = 0 From this optimality condition we can make S explicit to obtain S = 12 +50/q. For Gerald: Max MG(24-S) s.t. MG + q(24-S) = 100, because he needs to decide how much of his $100 he wants to keep and how much of it he wants to pay to Poldi in order to acquire the right to scent-free air for (24-S) hours of the day. Substituting for MG in his utility function we have MaxS (100 – q(24-S)) (24-S) and taking the derivative with respect to S and setting it equal to zero yields q(24-S)-(100 - q(24-S)) = 0 From this optimality condition we can make S explicit to obtain S = 24 - 50/q. Because both people are price takers, and with the assignment of the property right to Poldi we have created a market for scent-free air, we find the competitive price for scent-free air, q, by setting demand equal to supply. We can think of Poldi as the supplier of scent-free air and Gerald as the demander of scent-free air. This means that the supply and demand of scent-free air (given by 24-S*) must be the same in the competitive equilibrium, and therefore also the amount of smell, S*. That is, 12 +50/q = 24 - 50/q and therefore q = 100/12 = 25/3 and S* = 18. Poldi forgoes to light her candles for 6 hours and receives in return from Gerald (24- 18)*25/3 = 50 dollars. This

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implies that Gerald’s utility is equal to 50*(24-18) = 300 and Poldi’s utility is equal to 150*18 = 2700 under this property right regime. We can see here that the optimal amount of smell in this economy depends on who is assigned the property right.

2. A soot-spewing factory that produces steel is next to a laundry. We will assume that the

factory faces a prevailing market price of P=$40. Its cost function is C=X2, where X is the steel output. The laundry produces clean wash, which it hangs out to dry. Suppose each unit of steel produced produces one unit of soot (S), that is, X = S. The soot from the steel factory smudges the wash, so that the laundry has to protect the laundry from the soot of the factory and this increases its costs of producing clean clothes. The cost function of the laundry is C = Y2 + ½ S, where Y is pounds of laundry washed. A pound of clean laundry sells for $10. Both firms face a competitive market.

a) What outputs X and Y would maximize the sum of the profits of these two firms? How

big is the joint profit?

Maximizing joint profits: Max{X,Y, S} 40 X +10 Y - X2 - Y2 - .5S but X=S Or equivalently Max{X,Y} 40 X +10 Y - X2 - Y2 - .5X Profit maximizing conditions: Price of output equals marginal cost of that output:

with respect to X: 40 =2X + .5 with respect to Y: 10 = 2Y

Hence X* = 39.5/2 = 19.75, Y* = 5. Total profit: 50+790 – (25 + 19.752 + .5*19.75) = 415.06.

b) If each firm individually maximizes its own profit, what will be the output of each firm?

How big is each firm’s profit? Steel Factory: Max{X} 40 X - X2 Profit maximizing conditions, Price of output equals marginal cost of that output: 40 = 2X Hence X = 20 Profit of steel factory: 800 – 400 = 400

Laundry: Max{Y} 10 Y - Y2 - .5S Profit maximizing conditions, Price of output equals marginal cost of that output: 10 = 2Y Y = 5. Profit of laundry: 50 – 25 - 10 = 15 Total profit: 400 + 15 = 415. This is less than in a).

c) What per-unit tax would we need to set on soot to obtain the outputs found in Part a) of this problem? What is the government’s revenue from this tax? What is the profit of each firm? Does this policy result in a Pareto improvement over the equilibrium outcome in b)? Explain carefully.

Steel Factory: Max{X,S} 40 X - X2 – t*S but S=X

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Or equivalently, max{X} 40 X - X2 – t*X Profit maximizing conditions, 40 =2X + t Comparing this with the optimality conditions from a) with respect to X where 40 =2X + .5, we must have t = .5. In this case X = 19.75 and therefore S = 19.75. Tax revenue is .5*19.75 = 9.875. The profit of the steel factory is: 790 – (19.752 + .5*19.75)= 390.06; the profit of the laundry is: 10*5 – 52 -.5* 19.75 = 15.125. This policy does not represent a Pareto improvement over the market equilibrium; the government and the laundry are better off, but the steel factory is worse off.

d) Draw the marginal benefit curve (marginal profit curve of steel factory) of soot and the marginal external cost curve of soot in a diagram with soot on the x-axis and $ on the y-axis. Indicate the socially optimal amount of soot and mark the government’s tax revenue from an optimal tax on soot.

In order to find the steel factory’s marginal profit as a function of soot write down the profit of the steel industry as a function of soot: Profit = 40 X - X2 but we also know that S=X so equivalently profit = 40 S - S2 Marginal profit as a function of soot is given by the derivative of the profit as a function of soot, that is MP = 40 –2S. In order to find the marginal external cost of soot, first write down the external cost of soot: e(S) = .5S. Then marginal external cost of soot is given by e’(S) = .5.

The socially optimal amount of soot is where the marginal benefit of soot to society (given by the marginal profit of the steel industry as a function of soot) equals the marginal external cost of soot. That is, 40 S - S2 = .5, and therefore S = 19.75. The Pigouvian tax (optimal tax on soot) is equal to the marginal external cost of soot at the socially optimal level of soot, that is t = . 5. Government revenues from this tax is equal to the area of the rectangular with height .5 and length 19.75. See lecture notes for graph.

e) Suppose the laundry has the right to clean air and is willing to let the steel factory pollute for a price of q per unit of soot. What is the equilibrium price of soot and how much revenue does the laundry get from selling its rights to clean air? What is the profit of each firm? Does this policy result in a Pareto improvement over the equilibrium outcome in b)? Explain carefully.

Steel Factory: Max{X,S} 40 X - X2 – q*S but S=X and so can write profit in terms of S: 40 S - S2 – q*S Set marginal benefit of soot equal to its marginal cost. Marginal benefit of soot to steel factory is how much marginal profit is created: 40 –2S, marginal cost of soot is equal to q. Thus optimal amount of soot is found by setting marginal benefit equal to marginal cost: 40 – 2S = q. The firm’s inverse demand for pollution rights is q = 40 –2S.

Laundry: Max{Y,S} 10Y – Y2 - .5*S + q*S

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Profit maximizing with respect to Y: 10 =2Y, so Y = 5. with respect to S, marginal cost of soot given by.5 needs to equal marginal benefit of soot given by q (now that steel factory has to pay $q to laundry for each unit of soot), so .5 = q. The firm’s inverse supply of pollution rights to the steel factory is therefore q = .5.

Setting inverse demand equal to inverse supply, 40 –2S = .5, we find S = 19.75. The laundry gets .5*19.75 = 9.875 from the steel factory. The profit of the steel factory is: 790 – (19.752 + .5*19.75)= 390.06; the total profit of the laundry (profit from laundering plus revenues from selling rights to pollute to steel factory) is: 15.125 + 9.875 = 25. This policy does not represent a Pareto improvement over the market equilibrium; the laundry is better off, but the steel factory is worse off.

f) Suppose the steel factory has the right to pollute the air up to S’ = 20. The steel factory is willing to cut down its pollution for a price of q per unit of soot abated. What is the equilibrium price of soot abated and how much revenue does the steel factory get from selling its rights to pollute? What is the profit of each firm? Does this policy result in a Pareto improvement over the equilibrium outcome in b)? Explain carefully.

Steel Factory: Max{X,S} 40 X - X2 + q*(20 – S) but S=X Or equivalently, max{S} 40 S - S2 + q*(20 – S) Profit maximizing condition: marginal benefit of producing one more unit of soot needs to equal marginal cost. Marginal benefit given by marginal profit, marginal cost given by lost opportunity to collect q from laundry for reducing soot. Thus, 40 –2S = q. Laundry: Max{Y, S} 10Y – Y2 - .5*S - q*(20 – S) Profit maximizing condition: marginal benefit of producing one more unit of laundry needs to equal marginal cost. With respect to Y: Marginal benefit given by marginal revenue of Y, marginal cost given by marginal production cost of Y. Thus 10 =2Y and Y=5. With respect to S: Marginal benefit given by saving $q that would otherwise have to be paid to steel factory and marginal cost given by increase in cost of producing laundry due to increase in soot: q = .5 From q = 40 –2S and q = .5, we find S = 19.75. The steel factory gets .5*.25 = .125 from the laundry. Total profit of the steel factory (profit from producing steel plus revenue from abating pollution) is: 790 – 19.752 + .125= 400.06; profit of the laundry is: 15.125 - .125 = 15. This policy does represent a Pareto improvement over the market equilibrium; the laundry is as well off as before, but the steel factory is better off. In all the solutions to overcome the externality problem, we see that the efficient amount of pollution is achieved. We also see that depending on the solution (i.e. who owns the rights) we have different distributional effects.

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3. Two firms, firm A and B, in a community pollute the environment. The government has

decided that 12 units of pollution must be abated and that each firm must cut pollution by 6 units. The total cost of pollution abatement is TCA =(1/6) qA

2 for firm A, and TCB = (1/3) qB

2 for firm B, where qA is the quantity of abatement for firm A and qB is the quantity of abatement for firm B. a) Is this solution cost efficient? Explain why or why not.

This solution is not cost efficient because the marginal cost of abatement is not the same for both firms: MCA = (1/3)*qA and hence MCA(6) = 6/3 =2. MCB = (2/3)*qB and hence MCB(6) = 12/3 =4. If it is not the same, then we can show that we can save costs by having the firm with lower marginal cost of abatement abate one more unit and the firm with higher marginal cost of abatement abate one less unit, but still abate the same amount of pollution. Total cost of abating with each firm abating 6 units of pollution: 36/6 +36/3 = 18. Now let firm A abate one more unit and let firm B abate one less unit, then TCA = 49/6, TCB = 25/3, total cost of abating: 99/6 = 16.5. Clearly, there are cost savings if firm A abates 7 instead of 6 units and firm B abates 5 units instead of 6.

b) If the solution is not cost efficient, how much pollution should each firm abate at the cost

efficient outcome?

Both firms should abate a quantity that makes their marginal cost of abating equal. We also need to ensure that 12 units of pollution are abated. MCA = (1/3)*qA and MCB = (2/3)*qB ; qA + qB = 12. This means (1/3)*qA = (2/3)*qB and therefore qA = 2*qB and substituting into qA + qB = 12, we have 2qB + qB = 12, and therefore qB = 12/3 = 4. The optimal amount for firm A is qA = 2*qB = 8. Total cost of abating 64/6 + 16/3 = 96/6 = 16. Cost savings compared to a): 2 dollars.

c) Suppose the government tells firms to cut pollution by 6 units or trade with another firm so that in total pollution is cut by the desired amount. If there is a competitive market for pollution abatement, what would be the price for each unit a firm abates beyond its 6th unit on another firm’s behalf?

Let the price of abating on another firm’s behalf be denoted by p and the amount abated on another firm’s behalf be denoted by X. Since firm A is able to abate 6 units of pollution at a lower marginal cost than firm B, firm A will be the supplier of abatement and firm B will be the demander for abatement. This means that firm A’s marginal benefit of abating on firm B’s behalf needs to be equal to the cost of the last unit of abatement for firm A, i.e MCA(6+X) = p. For firm B the benefit of the last unit not abated by firm B needs to be equal to the cost it has to pay firm A for having firm A abate this unit on firm B’s behalf, i.e. MCB(6-X) = p. Thus we find the equilibrium amount of trade by setting MCA(6+X)= MCB(6-X). Plugging in the functional forms, we have (6+X)/3 = 2*(6-X)/3. Solving for X, we find X* = 2. The equilibrium price p* is equal to p* = MCA(6+X*) = MCB(6-X*) = 8/3. This means that firm A receives (8/3)*2 from firm B for abating 2 units on its behalf. It gains (8/3)*2 – [64/6 –36/6] = 16/3 – 14/3 = 2/3. Firm B saves [36/3 – 16/3] - (8/3)*2 = 4/3. Overall we create 2 more dollars of social welfare compared to a).

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Generalizing Results from Exercises 1 and 3. Note that we expect you to be comfortable with this level of formal analysis on the exam. 4. Poldi and Gerald are married. Poldi likes to light scented candles in their house while Gerald

hates the smell of scented candles. They both value money and their utility function over money (MP for Poldi’s money and MG for Gerald’s money) and smell of scented candles (S), measured in hours of the day in which the candles are lit, is given by the increasing and strictly quasiconcave utility function UP(MP, S) for Poldi and UG(MG, 24-S) for Gerald.

a) Is smell a positive or negative externality? Explain.

It is a negative externality because Gerald’s utility decreases with smell.

b) Is smell a consumption or production externality? Explain. It is a consumption externality, because it originates from an action of a consumer.

c) Give the conditions for the Pareto efficiency.

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maxS,MG ,λUG (MG,24 − S) + λ((UP (IG + IP − MG ,S) −UP )

FOCs :∂UG

∂MG

− λ∂UP

∂MP

= 0

−∂UG

∂(24 − S)+ λ ∂UP

∂S= 0

Hence ∂UG /∂MG

∂UG /∂(24 − S)=∂UP /∂MP

∂UP /∂S

Suppose Gerald initially owns IG income, and Poldi initially owns IP, and both people are price takers in all markets. The price of one unit of money is, of course, equal to $1. d) Show that assigning the right to scent-free air to Gerald is efficient. (Assume he

charges a price of q for every hour he lets Poldi light her candles.)

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maxS,MG ,φUG (MG,24 − S) + φ(IG + qS − MG )

FOCs :∂UG

∂MG

−φ = 0

−∂UG

∂(24 − S)+ φq = 0

Hence ∂UG /∂MG

∂UG /∂(24 − S)=1q

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maxS,M P ,ξUP (MP ,S) + ξ(IP − qS − MG )

FOCs :∂UP

∂MP

− ξ = 0

∂UP

∂S− ξq = 0

Hence∂Up /∂MP

∂UP /∂S=1q

Both are price takers and hence at the equilibrium price q, we have

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∂UG /∂MG

∂UG /∂(24 − S)=∂UP /∂MP

∂UP /∂S

e) Show that assigning the right to light candles for 24 hours a day to Poldi is efficient.

(Assume that she charges a price of q for every hour she does not light her candles.)

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max(24−S ),MG ,φUG (MG,24 − S) + φ(IG − q(24 − S) − MG )

FOCs :∂UG

∂MG

−φ = 0

∂UG

∂(24 − S)−φq = 0

Hence ∂UG /∂MG

∂UG /∂(24 − S)=1q

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maxS,M P ,ξUP (MP ,S) + ξ(IP + q(24 − S) − MG )

FOCs :∂UP

∂MP

− ξ = 0

∂UP

∂S− ξq = 0

Hence∂Up /∂MP

∂UP /∂S=1q

Both are price takers and hence at the equilibrium price q we have

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∂UG /∂MG

∂UG /∂(24 − S)=∂UP /∂MP

∂UP /∂S

f) Can you tell under which conditions the level of scent will be the same in d) and e)? Note that we give more money to one person than the other in the two alternative assignments of property rights. If Gerald has the right to scent-free air in d) he will spend IG in addition to his revenues from selling pollution permits to Poldi. Poldi, on the other hand will have less than IP to her disposal as she pays for the pollution permits. If Poldi has the right to scented air in e) she will spend IP in addition to her revenues from selling scent-free hours to Gerald. Gerald, on the other hand will have less than IG to his disposal as he pays for the scent-free hours. As long as the marginal rates of substitution are functions of disposable income (Mi) and scent, keeping the efficient level of scent the same under both property right regimes is impossible. Thus, as long as the MRS for each person depends on the amount of money a person has and the level of smell, the efficient level of scent will be different under different property right regimes. Only if the MRS are independent of disposable income (Mi) can the values of the MRS that characterize the efficient level of scent be the same under both property right regimes and simultaneously involving the same amount of scent. Utility functions that satisfy this property must be quasi-linear and of the form UP(MP, S) = MP + vP(S) for Poldi and UG(MG, 24-S) = MG + vG(24-S) for Gerald, where vi( ) is a strictly concave and increasing function.

5. Two firms, firm A and B, in a community pollute the environment. The government has

decided that Q units of pollution must be abated and that each firm must cut pollution by Q/2 units. The total cost of pollution abatement is cA(qA) for firm A, and cB(qB) = 2cA(qB) for firm B, where qA is the quantity of abatement for firm A and qB is the quantity of abatement for firm B. d) Is this solution cost efficient? Explain why or why not.

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mincA (qA ) + 2cA (qB ) s.t. qA + qB = QminqA

cA (qA ) + 2c2(Q− qA )

FOC : cA' (qA ) = 2cA

' (Q− qA )but cA

' (Q /2) ≠ 2cA' (Q /2)

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The solution is not cost efficient because the marginal cost of abatement for each firm is different at Q/2, but for cost efficiency we need them to be the same.

e) If the solution is not cost efficient, give the condition of how much pollution should each firm abate at the cost efficient outcome.

See above. Each firm should abate until the marginal abatement cost of each firm is the same. Given the relationship between the cost functions of each firm this is achieved if firm A abates and amount such that its marginal abatement cost is twice the amount that it would be if it abated firm B’s optimal amount. Note that this does not imply that firm A should abate twice as much as firm B. For example, if cA(qA) = qA

3 then

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qA = 2Q /(1+ 2).

f) Suppose the government tells firms to cut pollution by Q/2 units or trade with another firm so that in total pollution is cut by the desired amount. If there is a competitive market for pollution abatement, what would be the price for each unit a firm abates beyond its (Q/2)th unit on another firm’s behalf?

It is obvious that firm A will be selling pollution permits to firm B. The price of the pollution permits, denoted by p, and the quantity of pollution permits, denoted by X, are equal to

Allocating Pollution Permits by Means of a Second-Price Auction

6. There are three firms in a community that pollute the environment. The government has decided that 18 units of pollution must be abated. The marginal cost of pollution abatement for each firm is given in the table below. Unit abated Firm A Firm B Firm C 1 1 2 3 2 2 4 6 3 3 6 9 4 5 8 12 5 7 10 15 6 10 12 18 7 11 14 21 8 12 16 24 9 16 18 27 10 25 20 30

a) The government mandates that each firm must cut pollution by 6 units. Is this solution cost efficient? Explain why or why not.

�

p = cA' (qA

* )where : cA

' (qA* ) = 2cA

' (Q− qA* )

X = qA* −Q /2

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This solution is not cost efficient. To see why, first look at the cost of each firm abating 6 units. Unit abated Firm A Firm B Firm C 1 1 2 3 2 2 4 6 3 3 6 9 4 5 8 12 5 7 10 15 6 10 12 18 Abatement cost per firm

28 42 63

Total cost of abating 18 units of pollution if every firms cuts pollution by 6 units = 28+42+63 = 133. The cost of abating the 6th unit of pollution is not the same for the three firms. It is cheaper for firm A ($10) than for Firm B ($12) and firm C (18). By reallocating units of abatement among firms we can save costs. Since it is cheaper for firm A to abate the 7th unit ($11) than it is for firm C to abate the 6th unit ($18), we can save costs if firm A would abate one more unit and firm C would abate one unit less. The last three units abated cost 10+12+18 =40 with 6 units abated by each firm. The cost of abating the last three units with firm A abating one more unit and firm C abating one unit less (of the last three units two are abated by firm A and one is abated by firm B) is 10+11+12=33. The same amount of pollution would be abated, but at lower cost and therefore each firm abating 6 units is not cost efficient.

b) If the solution in a) is not cost efficient, how much pollution should each firm produce at

the cost efficient outcome? Each firm should abate until the cost of abating the last unit of pollution is the same for all firms. As we have seen from the answer to a), if firm A abates one more unit and firm C abates one unit less, costs of abating decrease. We can make the same argument to have firm A abate the 8th unit and have firm C abate one more unit less (that is, firm C abates only 4 units). So if firm A abates 8 units, firm B abates 6 units and firm C abates 4 units, every firm abates up to the point where their MC of abating is the same. Total cost of abating is then Unit abated Firm A Firm B Firm C 1 1 2 3 2 2 4 6 3 3 6 9 4 5 8 12 5 7 10 6 10 12 7 11 8 12 Abatement cost per firm

51 42 30

Total cost of abating = 51 + 42 + 30 = 123.

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Suppose the government auctions pollution permits using a second price auction. (That is, the highest bidder receives the permit at a price equal to the second highest bid). Without any pollution permits, each firm must abate all of its pollution, that is 10 units. It is a well-established economic result, that in second price auctions, people are best off if they bid equal to their true willingness to pay if one object is auctioned. In our setup, more than one permit is auctioned and so firms can find it in their interest to bid less than their true willingness to pay.1 However, as the number of firms becomes very large, firms will again bid their true willingness to pay. Although we deal with three firms only, for simplicity, we’ll assume that they bid according to their true willingness to pay. That is, if for example, firm A must abate 10 units but now has an option to get a pollution permit, this permit is worth to firm A $25, because this is exactly how much firm A would save by having to abate one unit of pollution less. So firm A would bid $25 in order to get its first pollution permit. Note that sometimes two firms are the highest bidders. In this case only one firm can receive the permit. To break the tie, give the permit to the firm whose name appears first in the alphabet.

c) Write down the bids of each firm for each of the 12 pollution permits.

Permit Firm A’s bid Firm B’s bid Firm C’s bid Winner Price paid 1st 25 20 30 Firm C 25 2nd 25 20 27 Firm C 25 3rd 25 20 24 Firm A 24 4th 16 20 24 Firm C 20 5th 16 20 21 Firm C 20 6th 16 20 18 Firm B 18 7th 16 18 18 Firm B 16 8th 16 16 18 Firm C 16 9th 16 16 15 Firm A 15 10th 12 16 15 Firm B 15 11th 12 14 15 Firm C 14 12th 12 14 12 Firm B 12 Explanation of table entries: Without a permit, a firm needs to abate 10 units of pollution. With one permit it needs to abate 9 units of pollution. So how much is it worth to the firm to receive one permit? It’s equal to cost savings of not having to abate the 10th unit of pollution. Thus when the government auctions off the first permit, firms’ bids equal their MC of abating the 10th unit of pollution. Once a firm has purchased a permit, it values another permit equal to the cost savings of not having to abate the 9th unit of pollution, and so on and so forth. Note that sometimes two firms are the highest bidders. In this case only one firm can receive the permit and I have given the permit to the firm whose name appears first in the alphabet. If you

1 Rodriguez, Gustavo E. (2009) "Sequential Auctions with Multi-Unit Demands," The B.E. Journal of Theoretical Economics: Vol. 9 : Iss. 1 (Contributions), Article 45. DOI: 10.2202/1935-1704.1534 Available at: http://www.bepress.com/bejte/vol9/iss1/art45

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give the permit to the other firm, nothing changes in terms of overall costs of the firms. Here is what happens if you break the ties differently: Permit Firm A’s bid Firm B’s bid Firm C’s bid Winner Price paid 1st 25 20 30 Firm C 25 2nd 25 20 270 Firm C 25 3rd 25 20 24 Firm A 24 4th 16 20 24 Firm C 20 5th 16 20 21 Firm C 20 6th 16 20 18 Firm B 18 7th 16 18 18 Firm C 16 8th 16 18 15 Firm B 16 9th 16 16 15 Firm B 15 10th 16 14 15 Firm A 15 11th 12 14 15 Firm C 14 12th 12 14 12 Firm B 12

a) How many units does each firm abate and how many permits does each firm buy? Firm A buys 2 permits and abates 8 units of pollution, firm B buys 4 permits and abates 6 units of pollution, firm C buys 6 permits and abates 4 units of pollution.

b) What is government revenue from this auction? Government receives 25+25+24+20+20+18+16+16+15+15+14+12 = 220

c) What are the cost savings to society from this policy compared to a)? (Subtract government revenue from costs of firms and compare with costs of firms under a)!)

With permits: The cost of permits to firms is equal to what the government receives in revenues and hence the permit costs to firms and government revenues cancel out. This means, the social cost of abating with permits is equal to the cost of firms abating 18 units of pollution. Unit abated Firm A Firm B Firm C 1 1 2 3 2 2 4 6 3 3 6 9 4 5 8 12 5 7 10 6 10 12 7 11 8 12 Abatement cost per firm with permits

51 42 30

Social cost of abating 18 units of pollution with permits is equal to 51+42+30 = 123. This amount is

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$10 less than in a). It is also equal to the cost efficient amount found in b). Auctioning permits is cost efficient. 7) Angrist and Krueger (1991) use quarter of birth (QOB) as an instrument for years of education in their famous paper discussed in lecture. Criticism of this approach emerged quite quickly. An early example is Bound, Baker, and Jaeger (1995), who argue that both the relevance assumption and exclusion restriction assumption are probably violated when QOB is used as an instrument. For example, they cite evidence that rates of mental health problems vary with quarter of birth, that school performance varies by quarter of birth, and that children of higher income families are less likely to be born in the winter. More criticism of the QOB instrument can be found here. a) Supposing this criticism is correct, what problems are likely to arise from the use of QOB as an instrument for education? In order for QOB to be a valid instrument, it must 1) affect the endogenous variable for which we’re instrumenting (in this case, it must affect years of schooling); and 2) only affect the outcome variable (wages in this case) through its effect on the endogenous RHS variable (years of schooling). We sometimes call assumption 1 the “relevance” assumption, and assumption 2 the “exclusion restriction.” The problem in this case is twofold. QOB is only very weakly relevant to years of education. As Bound, Jaeger, and Baker note, the R-squared of a regression of years of education on QOB is between 0.0001 and 0.0002. In general, this means that standard errors of the IV estimates will be very large (because we’re only using a tiny fraction of the variation in x—that supplied by variation in the instrument—to determine the effect of x on y). Angrist and Krueger have a very large sample size which allows them to produce fairly precise estimates in spite of the weak relevance. So long as the exclusion restriction holds (that is, the QOB instrument is exogenous), then they can expect consistent estimates. But IV is biased in finite samples, and this bias is inversely proportional to the amount of variation in the endogenous variable, X, that is explained by the instruments, Z. So in cases where the instruments are “weak”—that is they have little explanatory power for X—we may see large finite sample bias, even when the exclusion restriction holds. The second problem with QOB as an instrument is that it likely violates the exclusion restriction. In other words, QOB may affect wages through channels other than QOB’s effect on years of education. Bound, Jaeger, and Baker cite various studies that find seasonality in birth rates, differences in mental health outcomes by birth month, and differences in IQ and school performance by season born. This suggests that QOB is probably at least weakly endogenous (i.e. correlated with the error term in the structural equation). This violation of the exclusion restriction not only contributes to bias of the IV estimator; it also causes inconsistency. And the contributions to bias and inconsistency will be largest in cases where QOB is a weak instrument.

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b) What if Angrist and Krueger had used a regression discontinuity (RD) approach, instead of the IV approach they actually used? Would this make you more confident in their estimates of the effect of education on the wage? What would have to be true about the bandwidth selected for such a study, given the concerns raised above? With a large enough sample, Angrist and Krueger could compare wages of people born on Dec 31 with those born on Jan 1 (or expand the bandwidth a bit and compare wages of people born Dec 28-31 with those born Jan 1-4). This would be an example of “fuzzy” regression discontinuity. Birth on Dec 31 does not deterministically mean you receive more education than someone born Jan 1. But due to legal institutions governing school starting age and age of legal dropping out, it can be expected to affect the years of education received. One of the criticisms of Angrist and Krueger is that QOB is an endogenous instrument for years of education. People have pointed out that someone born on March 15 may be different (on average) than someone born on October 15 along dimensions other than the amount of schooling they receive. If some of these other dimensions (mental health, school performance, parental income, etc.) matter for wage determination, then unobserved heterogeneity along these dimensions will lead to biased and inconsistent IV estimates of the effect of education on the wage. Since what Angrist and Krueger do is very similar to RD to begin with, we could reformulate their approach as an RD approach and think about reducing the bandwidth around the threshold from ~91 days on each side (i.e. 1 quarter) down to just a few days on each side. Arguably whatever differences, on average, exist between people born on March 15 and October 15, would disappear if we compared people born on Dec 31 and Jan 1. In such a short time frame, differences are likely to be completely random, and therefore the treatment and comparison groups should be, on average, identical to each other, except in assignment to treatment or control. The problem with this approach is likely to be one of sample size. Because the difference in birth date is likely to only have a very small effect on years of education, the standard errors are likely to be large. So this may not prove a viable fix to the criticism of Angrist and Krueger, in practice. c) Would taking such an RD approach eliminate the criticism that the Angrist and Krueger estimates are just providing a LATE? No, we’d still be just measuring the effect of a little more education among high-school dropouts. This is different from measuring the effect of a little more education for the average person, or someone elsewhere in the distribution of years of education. d) Is the “no bunching” assumption, required for identification in the RD approach, likely to be violated?

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Probably not. It’s arguably going to be difficult to precisely manipulate whether your child is born on Dec 31 or Jan 1. Certainly, no one would bother to do it for reasons of manipulating their kid’s school entrance age. After all, you can always hold your kid back, if you think they’re too young. By the way, there’s arguably good reason to do this, as noted in a famous paper (http://qje.oxfordjournals.org.ezproxy.library.uvic.ca/content/121/4/1437.full.pdf+html) by a former UVic Econ Honours student (Kelly Bedard). And the findings of this paper (that older kids within a grade have longlasting academic advantages over younger kids within a grade) further invalidate the exclusion restriction in Angrist and Krueger. After all, the Bedard results suggest that students born on Jan 1 are likely, on average, to outperform students born on Dec 31 in school. That means that even if we’re comparing students born Dec 31 with students born Jan 1, there may be systematic differences across groups along dimensions (other than years of education) that matter for wage determination. Back to the bunching question. One might want to investigate a possible story for why we might observe bunching. Tax rules, in many countries, are such that you can claim a child tax credit if you have a baby born within the tax calendar year. So by having your kid on Dec 31 rather than January 1, you may gain a couple thousand dollars. If such tax credits aren’t pro-rated (e.g. you only get 1/365th of the credit if you only have the child for one day in the tax year), then parents may have an incentive to schedule C-sections, induced labour, etc. in order to speed up birth at the end of the year. If parents who do this are systematically different from parents who don’t, in ways that may effect the child’s later earnings, this could be a problem. A quick search on the web produced the following paper (http://www.jstor.org/discover/10.1086/250054?uid=3737720&uid=2&uid=4&sid=21101634666203), which addresses this point. So maybe my “probably not” (above) was a bit hasty.