ECO101 Lecture 06

Optimal Pollution is Not Zero

ECO101- Week 06 Environmental.pdf
ECO101 Textbook Quiz Chapter 17

Envionmentalists say we should have no pollution.
- That would be good if marginal costs didn't offset the benefits
$E = 100 - 2 A$
$C = 20 A + A^{2}$
$M C A = 20 + 2 A$
$B = 100 A - A^{2}$
$M B A = 100 - 2 A$
$A$ is abatement, something to reduce pollution
Ex:
- Cars
- Carpool
- Planning Ahead, do all pollution at one time
- Switch to e85
- Ecu tune
- Evs?
- The marginal cost of abatement gets more expensive.
- It's almost free to plan ahead, but very expensive to buy an ev and sell your current car.
- Marginal benefit of abatement slopes down because decreasing incremental benefits
- $A^{⋆}$ is optimal because it balances MC and MB
- Since $E$ depends on abatement. If there are $50$ units of abatement we will have no pollution.
- 3c
- 6b

Optimal Taxes (Pollution Fines) Should Reflect Damage Costs

1

$M C = 5 + Q$
$S M C = 10 + 2 Q$
$M B = 35 - 0.5 Q$
Pollution is a negative Externality
Private
- $M C = M B$
- $5 + Q = 35 - 0.5 Q$
- $Q = 20$
Social
- $S M C = M B$
- $10 + 2 Q = 35 - 0.5 Q$
- $2 Q = 25 - 0.5 Q$
- $2.5 Q = 25$
- $Q = 10$

    left=-0.5; right=40;
    top=60; bottom=-0.5;
    ---
    y=5+x
    y=20+x
    y=10+2x
    y=35-0.5x
    x=10|dashed
    x=20|dashed

$M B (10) = 30$
$M C (10) = 15$
Tax of $30 - 15 = 15$ dollars
Now when we tax $M C$
$20 + Q = 35 - 0.5 Q$
$Q = 10$
Now our private solution matches the social
Private agents opt for the socially optimal quantity due to the per unit tax.
Principal - Agent

2

(06 Environmental, p.2)
8. Robert's Roses is a plant nursery located in Darlington between a nuclear power plant and a steel mill. The total cost of producing roses is given by C = 1 + 0.1R2 + 0.5T2 – 0.004E where T is the tons of steel produced at the mill, E is the amount of electricity produced at the plant and R is the number of roses grown at the farm. Costs of production at the mill are given by C = T2 and at the power plant are given by C = 0.0025E2. In competitive markets, roses sell for $2.00, electricity sells for $0.02 and steel sells for $20. Calculate the system of Pigouvian per-unit taxes/subsidies that would result in the socially optimal levels of production. No diagram is required.

$p_{R} = 2$
$p_{S} = 20$
$p_{E} = 8$
$C_{R} = 0.1 r^{2} + 8 S - 12 E$
- Negative externality for rose producer
- Positive externality for the energy producer
- $M C_{R} = 0.2 R$
$C_{S} = S^{2}$
- $M S C_{S} = 2 S + 8 = P_{S}$
- $M S C_{S} = 2 S + 8 = 20$
- So $S^{⋆} =$
- Add 8 because that's the damage being done elsewhere.
$C_{E} = 2 E^{2}$
- $M S C_{E} = 4 E - 12 = P_{E} = 8$
- $E^{⋆} = 5$
- Sub 12 because that's the benefits elsewhere
Pollution fines need to reflect damages or benefits done elsewhere

Aggregation of Multiple Sources

$C_{1} = A_{1}^{2} + 8$
- $M C_{1} = 2 A_{1}$
- $A_{1} = \frac{1}{2} M C$
$C_{2} = 2 A_{2}^{2} + 20$
- $M C_{2} = 4 A^{2}$
- $A_{2} = \frac{1}{4} M C$
$B = 28 A - 0.5 A^{2} + 20$
- $M B = 28 - A$
- $A = \frac{3}{4} M C$
- $M C = \frac{4}{3} A$
Aggregate the amount of abatement
Aggregate the marginal cost curves
- $M C = M B$
- $\frac{4}{3} A = 28 - A$
- $\frac{7}{3} A = 28$
- So $A^{⋆} = 12$
- Optimal amount of pollution is at abatement of $A^{⋆}$
Price pollution at true cost
- $M C_{1} = 2 A$
  - Low marginal cost of abatement
- $M C_{2} = 4 A_{2}$
  - High cost
- So it makes sense that $A_{2} = 2 A_{1}$
- Firm 1 can pollute twice as much as firm 2
- If $A_{1}$ and $A_{2}$ add to $12$
- $A_{1} + A_{2} = 12$
- $A_{1} + 2 A_{1} = 12$
- $3 A_{1} = 12$
- $A_{1} = 4$
  - Only 4 units of abatement
- $A_{2} = 12 - A_{1}$
- $A_{2} = 8$
  - Only 8 units of abatement
- $M C_{1} = 2 A_{1} = $ 8$
- $M C_{2} = 4 A_{2} = $ 8$
- A firm that abates, should get $$ 8$ of reward
Example
- $M B = 12$
  - Why
  - They incur costs of $$ 12$ per bag to collect and dump garbage
- Two types of households
- Single person household
- Family household
  - $n_{S} = 1000$
  - $C_{S} = 2 r_{s}^{2}$
    - Explain
    - $M C_{S} = 4 r_{s}$
    - $r$ is number of bags of garbage reduced
  - $n_{F} = 2000$
  - $C_{F} = 44 + r_{F}^{2}$
    - $M C_{F} = 2 r_{F}$
- Economies of scale to reduce pollution
  - Cheaper to a group to reduce pollution than a single
- Aggregate to calculate optimal level of reduction $r^{⋆}$
- Invert marginal cost curve
- $r_{S} = \frac{1}{4} M C$
- $r_{F} = \frac{1}{2} M C$
- $r = 1000 \frac{1}{4} M C + 2000 \frac{1}{2} M C$
- $r = 1250 M C$
- So $M C = \frac{1}{1250} r$
- $M B = 12$
- Every bag reduced, the city saves $12$ dollars
- $M C = M B$
- $12 = \frac{1}{1250} r$
- $15000 = r^{⋆}$
- Plan 1:
  - $n = 3000$
  - we need to reduce $15000$ bags
  - What if each household reduces by $5$ ?
  - Not feasible, because marginal cost for single household is very expensive.
  - See that $M C_{S} = 20 > M C_{F} = 10$
  - SIngles pay double than families
  - If marginal costs are different, then total costs are not at a minimum
  - $C_{S} = 2 (5)^{2} = 50$
  - $50 \cdot n_{S} = 5000$
  - $C_{F} = (44 + 25)$
  - $(44 + 25) \cdot n_{f} = 138000$
- Plan 2:
  - $M C_{S} = M C_{F}$
  - $4 r_{S} = 2 r_{F}$
  - $r_{F} = 2 r_{S}$
  - Total
  - $M C = 1000 r_{S} + 2000 r_{F} = 15000$
  - $5000 r_{S} = 15000$
  - $r_{S} = 3$
  - So single should reduce by $6$
  - Family reduce by $3$
  - $1000 (3) + 2000 (6) = 15000$
- How to achieve plan 2.
  - Quotas are bad because we need to monitor
    - Monitoring costs would exceed costs of the $$ 12$ of damages
  - $p = 12 = M B$
    - Charge people $$ 12$ to eliminate a bag
  - $M C_{S} = 4 r_{s} = $ 12 \to r_{s} = 3$
  - $M C_{F} = 2 r_{f} = $ 12 \to r_{f} = 6$
  - Same efficient quantity, by pricing pollution at its true cost
  - $C_{F} = 44 + r_{F}^{2} =$
  - $C_{S} = 2 r_{S}^{2} =$
    - Lower for both than plan 1
- You need to pay for an extra bag of waste in mississauga now.
- When you charge to get rid of garbage, people will find other cheaper ways to get rid of it.
- Use public garbage cans

Permit Trading

Cap and Trade System
$e_{1} = 10 - a_{1}$
$c_{1} = 10 a_{1} + a_{1}^{2}$
$e_{2} = 10 - a_{2}$
$c_{2} = 20 a_{2} + 0.5 a_{2}^{2}$
$M C A_{1} = 10 + 2 a_{1}$
$M C A_{2} = 20 + a_{2}$
$e^{⋆} = 12$
So $a_{1} + a_{2} = 8$
- Why?
- If I need a total of $12$ emissions
- $e^{⋆} = e_{1} + e_{2} = 12$
Plan 1:
- $a_{1} = 4$
- $a_{2} = 4$
- $M C A_{1} = $ 18$
- $M C A_{2} = $ 24$
- Costs are not at a minimum
- $C_{1} = 10 (4) + 4^{2} = 56$
- $C_{2} = 20 (4) + 0.5 (4)^{2} = 88$
- So we get $144$
Plan 2:
- $M C A_{1} = M C A_{2}$
- $10 + 2 a_{1} = 20 + a_{2}$
- $a_{2} = 2 a_{1} - 10$
- $a_{1} + a_{2} = 8$
- Sub
- $a_{1} + 2 a_{1} - 10 = 8$
- $3 a_{1} = 18$
- $a_{1} = 6$
- So then $a_{2} = 2 (6) - 10 = 2$
- $M C A_{1} (6) = 22$
- $M C A_{2} (2) = 22$
- When marginal costs are equal, totals costs are minimized.
- Check:
  - $C_{1} = 10 (6) + 6^{2} = 96$
  - $C_{2} = 20 (2) + 0.5 (2)^{2} = 52$
  - $138$
  - These are lower costs than plan 1
3:
- Would firm $1$ agree?
  - They used to have less costs, but now has a lot more costs to abate.
  - However now we can sell the permits at $$ 22$ , we can sell the surplus abated emissions
  - So we can sell permits to make profits on these.
- Firm $2$ agrees?
  - They save money on abatement by just buying permits.
4:
- GreenPeace
- Environmental group
- They can raise money and buy permits to clean up the environment.
- If greenpeace bought $3$ permits, that's less pollution that firms can do, but they still make profits so they'll happily accept.
- $a_{2} = 2 a_{1} - 10$
  - Says that they're equal
- $a_{1} + a_{2} = 11$
  - Says that we need to equal $11$ because greenpeace just bought $3$ permits
- $a_{1} + 2 a_{1} - 10 = 11$
- $a_{1} + 2 a_{1} = 21$
- $3 a_{1} = 21$
- $a_{1} = 7$
- $7 + a_{2} = 11$
- $a_{2} = 4$
- These are the Equilibrium trades. Firm 1 abates 1 more.
- $M C A_{1} = 24$
- $M C A_{2} = 24$
- $p^{⋆} = 24$
- $C_{1} = 10 (7) + 7^{2} = 119$
- $C_{2} = 20 (3) + 0.5 (3)^{2} = 64.5$
- Increase in costs are more than compensated by the sales of permits.
Tradeable permits were introduced
Politics
- Us electrical industry
- Coal creates $S O_{2}$ which creates acid rain
- Polluters were being given the right to pollute
  - Bad for politics
- Auction
  - It would make sense for the EPA to bid for the right to pollute
Regional
- All the good quality coal is in the southwest USA
- Before Cap and Trade System, firms not in the southwest, had to ship coal from the southwest to them to meet the cap.
- Now they just buy permits and burn crappier coal
- Southwest sold permits and got cleaner.
- Non southwest bought permits, but got dirtier.
- Made the acid rain problem a bigger deal
China
- Environmental problem
- They need to clean up their environment, but not to cripple industries.
- They should implement a cost effective reduction strategy.
@J. Dales created this Cap and Trade System.
Coase Theorem, assign property rights, then bargain.