1. Trading and Efficiency
© 1998, 2006,2017 Peter Berck

2. Topics Marginal Cost of abatement and Bid for Permits
Efficient Allocation of Emissions among firms
Cap and Trade Program
Coase Theorem

3. Reminder…
A Pareto improving exchange is one that makes at least one party better off and no party worse off.
Market transactions (without externalities) between willing buyers and sellers are pareto improving.
A redistributive tax is not. (Makes Larry Ellison worse off and John on the corner better off.)

4. MCA
Let
E be current emissions
E0 be initial emissions
E0-E is abatement
C(Q,E)
Costs go down when emissions go up.
Reducing emissions from E to E-1 is abating emissions by one unit.
MCA = C(Q,E-1) – C(Q,E)
Called Marginal Cost of Abatement

5. Emissions and Abatement Costs
E is emissions, A is abatement, E0 initial emissions E = E0 – A and A = E0 - E
C(Q,E) = C( Q, (E0-A) )
Increase A by 1 and find difference in cost
MCA = C( Q, E0-(A+1) ) – C(Q,E0 - A)
= C(Q, E – 1) - C(Q,E0 - A) > 0
MCE = C(Q, E+1) – C(Q,E) < 0
If approximation in units of 1 is good (small) enough to define marginal then, MCA = - MCE

6. Example C(Q, E) = 3 + 4Q -2E C(Q,A) = 3 + 4Q – 2( E0-A)
MCA = 2
MCE = -2

7. Abating 4 Units of Clean Air
Technique (20,50) costs 180 and is least
cost way to make Q* using 20 units of air
P other stuff = 2.

8. MCA
MCA = ( )/4

9. Change in output from q to q’
Let del = q’ –q be the change in output.
C(q + del) – C(q)
= C(q) + MC * del
If output goes up by a small amount, del, then cost goes up by mc times del. It goes up by mc per additional unit times del additional units.
This result brought to you by the calculus.

10. How much to pay for added E
Firm has rights to emit E and is offered rights to emit E+1.
Would pay added profits for added E
This is the bid for E
In a market those with highest bid get the E

11. Firm’s Bid for Clean Air Services
Profits as a function of effluent standard (E)
p(P,E) = P Q(.) - C(Q(.),E)
where Q(P,E) is the supply function
Firms will bid the amount of additional profits they could make from one additional unit of air (E+1) to receive one additional unit of air.
Bid(E) = p(P,E+1)-p(P,E)

12. Q is a function of E P = MC(Q,E) Supply function is Q(P,E)
Increase in P
Increase in E
MC(Q,E)
MC(Q,E+1) P
Q(E)
Q(E+1)

13. E goes up and …
Firm makes the output differently. Uses more E and less of other things per unit of output.
Isoquant diagram, moving to lower E.
This reduces cost and helps profits.
Because firm has lower cost, it likely has lower MC, and produces more.
Does this help profits?

14. Does making a little more q help profits?
Remember, p = mc(q).
So what happens when q goes up a little?
Gain p.
Lose mc
Net change: NOTHING.

15. Bid in terms of Cost Bid(E) = p(P,E+1)-p(P,E)
= P(Q(P,E+1) –Q(P,E))
+ C(Q(P,E), E) - C(Q(P,E+1), E+1)
Approximately* the same as Bid(E) =
= P( Q(P,E+1) –Q(P,E) )
price x change in quantity
+C(Q(P,E), E) – C(Q(P,E+1), E)
- mc x change in quantity
+ C(Q(P,E), E) - C(Q(P,E), E+1)
-MCE = MCA
This is Taylors theorem again. We can approximate a function by its original value and its partial derivatives. So d profit /dE = (p – mc) dq/dE - mca

16. Simplifying C(Q(P,E), E) - C(Q(P,E), E+1)
Is Marginal cost of abatement
Negative of the change in cost from adding a unit of emission
+C(Q(P,E), E) – C(Q(P,E+1), E)
MC is (change in cost from one more unit of Q)
MC * (Q(P,E)- Q(P,E+1)) is change in cost from
(Q(P,E)- Q(P,E+1)) of Q. Notice it is a negative number

17. Putting it together Bid(E) = P(Q(P,E+1) –Q(P,E))
+C(Q(P,E), E) – C(Q(P,E+1), E)
+ C(Q(P,E), E) - C(Q(P,E), E+1)
(P – MC )(Q(P,E+1) –Q(P,E)) + MCA
= MCA
Firm’s will pay their marginal cost of abatement in order to get another unit of emissions

18. Further example Suppose mca = 3 A As A goes up, so does MCA E0 = 10
A = E0 – E = 10 - E
Bid = MCA(A) = MCA(E0-E) = 3E0 – 3E=30-3E
So A = 0, E = 10, bid = 0
A = 5, E = 5 bid = 15
A = 10, E = 0 bid = 30

19. MCA as a function of Abatement
Costs more to clean up a ton as more tons cleaned up.
MCA
tons of abated

20. Bid as a function of emissions
Bid = change in total cost/change in pollute
Bid
First tons are the most valuable. If had no emission rights would have to use all known clean up techniques, including very expensive ones.
tons of E emitted (units of air used up)

21. Trading lowers total cost

22. Fixed Amount of Air: Two Firms
Read firm 1 from left
Read firm 2 from right
Point splits total between the firms.
$/unit
Total Tons To be Emitted
Firm 1 emissions
Firm 2 emissions
tons of E emitted (units of air used up)

23. How much Air to each? Bid is marginal cost emissions
Area under bid is avoided costs
$10000
$7000
If the firm is allowed to emit 10 tons, it saves 10,000 dollars; 20 tons, 17,000. 20 10
Tons emitted

24. Bids for two firms.
50/50 split
Bid 1
Bid 2
Tons emitted

25. How much Air to each? Intersection minimizes total cost
Treating firms the same has losses
50/50 split
Bid 1
Bid 2
Firm 1 saves blue and firm 2 saves the green.
We can do better.
Tons emitted

26. Dead Weight Loss
50/50 split
Bid 1
Bid 2
tons

27. Losses
Older studies show that the total cost of achieving clean air is much higher with a uniform TBES than it would be with TBES set for each firm.
California appears as the exception
Spent much more on regulation than other states
Meredith Fowlie does this with Cars and Powerplants and finds big losses from making powerplants too clean and cars not clean enough

28. Cap and Trade
One way to get the firms to the intersection point is to give each of them a pollution allotment and let them trade.
So long as they don’t trade allotments that they don’t have (fraud) the total amount of pollution remains constant
Acid rains section of CAA is a Cap and Trade Program as is RECLAIM

29. Avoiding loss
Trading.
let plants within a firm
let firms within an airshed
Jointly meet standards
Firm with higher MCA at the standard buys right to pollute from firm with lower MCA
Both have more money
Pollution is same.

30. However
When two firms trade the spatial distribution of the pollution will be different.
Trading can be a mechanism to inflict pollution on the poor.
Trading can lead to pollution in places that are more sensitive--have more people and more health damage

31. Right Amount of Pollution
MCA and Marginal benefit of Abatement the same.
Or: Bid (firms marginal benefit from pollution) = Marginal damage from pollution
Same thing.

32. Optimal Pollution
Figure Marginal Cost and Marginal Benefit of SOx Emissions Abatement

33. Coase

34. Zero Pollution
The shaded area is the deadweight loss if all 9.1 million tons of S0x are abated.

35. Zero Abatement
Figure Deadweight Loss if Emissions are 9.1 Million Tons

36. Coase
Coase’s contribution was to recognize that trading is costly, sometimes prohibitively so.
We call the following the Coase theorem:
When trading costs are low enough, it does not matter which firms originally get the rights to pollute. The costs and pollution will be the same. But not the amounts of money the firms get (the ones with the permits get more money.)

37. Coase
But Coase’s real contribution was to say that without trading one could get either of the DWL figures.
Giving the rights to the polluter causes a lower DWL in the figures than giving the rights to the breathers.
Of course, rights allocations in between do better than either.