Trading and Efficiency
© 1998, 2006 Peter Berck
Topics
• Marginal Cost of abatement and Bid for
Permits
• Efficient Allocation of Emissions among firms
• Cap and Trade Program
• Coase Theorem
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.)
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
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.
MCA
• MCA = (180-160)/4
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
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)
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)
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)
• true if “1” is small relative to Q
• Approximately the same as Bid(E) =
• = P(Q(P,E+1) –Q(P,E))
• + C(Q(P,E), E) - C(Q(P,E), E+1)
• +C(Q(P,E), E) – C(Q(P,E+1), E)
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
Putting it together
• Bid(E) =
• P(Q(P,E+1) –Q(P,E))
• + C(Q(P,E), E) - C(Q(P,E), E+1)
• +C(Q(P,E), E) – C(Q(P,E+1), E)
• Bid(E) =
• (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
Bid as a function of emissions
• Bid = change in total cost/change in pollute
Bid
tons of E emitted (units of air used up)
MCA as a function of Abatement
• Costs more to clean up a ton as more tons
cleaned up.
MCA
tons of abated
Trading lowers total cost
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)
How much Air to each?
• Bid is marginal cost emissions
• Area under bid is avoided costs
$10000
$7000
10
20
Tons emitted
Bids for two firms.
50/50 split
Bid 1
Bid 2
Tons emitted
How much Air to each?
• Intersection minimizes total cost
• Treating firms the same has losses
50/50 split
Bid 1
Bid 2
Tons emitted
Dead Weight Loss
50/50 split
Bid 1
Bid 2
tons
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
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
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.
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
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.
Optimal Pollution
8000
7000
6000
Dollars
5000
 Marginal Benefit
4000
3000
2000
Marginal Cost 
1000
0
0
1
2
3
4
5
6
Million Tons Abated
7
8
9
10
Figure 12.1. Marginal Cost and Marginal Benefit of SOx
Emissions Abatement
Coase
Zero Pollution
8000
7000
6000
Dollars
5000
4000
3000
2000
1000
0
0
1
2
3
4
5
6
Million Tons Abated
The shaded area is the deadweight
loss if all 9.1 million tons of S0x
are abated.
7
8
9
10
Zero Abatement
8000
7000
6000
Dollars
5000
4000
3000
2000
1000
0
0
1
2
3
4
5
6
Million Tons Abated
7
8
9
10
Figure 12.3. Deadweight Loss if Emissions are 9.1 Million
Tons
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.)
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.