Transcript Some additional notes on transfer coefficients and spatial

Some additional notes on transfer
coefficients and spatial
considerations
Because almost everyone was
confused last time!
Notation
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Ei = Uncontrolled emissions from firm i.
ei = Post-control emissions from firm i.
Ai = Ei-ei = Abatement from firm i.
MCi(Ai) = Marginal cost of abatement (typically an
increasing function)
• MD(p) = Marginal damage from pollution (typically
an increasing function)
• p = p(e1,e2,…) = pollution as function of emissions
• dp/dei = ai = transfer coefficient = proportion of
emissions that cause damage.
Optimal amount of pollution
• 1 polluter: MC(A) = 2.5*A
• Marginal damage: MD = 10 (every unit of
pollution causes $10 in damage, so marginal
benefit of abatement is $10).
• Determine optimal amount of Abatement.
• Equivalently, we can determine optimal
amount of pollution.
Optimal Abatement
$
MC = MB
2.5A = 10
A*=4
MC of abatement
10
MB of abatement
A*=4
Abatement
Optimal emissions
$
Recall MC(A) = MC(E-e)=2.5(E-e)
If E=10, MB(e) = 25-2.5e
25
MC = MB
25-2.5e = 10
e* = 6
MC of emissions
10
6
MB of emissions
10
Emissions
This makes sense:
A=E-e, 4=10-6
Add space
• Pollution control cost: c(E-e)
• Pollution damage cost: D(p(e))
• Total cost to society of emissions e:
C(E-e) + D(p(e))
• Regulator’s objective:
Mine{C(E-e) + D(p(e))}
Solving this problem
• Take derivative, set = 0.
• dTC/de = -c’ + dD/dp * dp/de = 0
Marginal cost
of abatement
Marginal damage
from pollution
Effect of 1 unit of
emissions on pollution
(transfer coefficient, a)
Optimal amount of emissions must satisfy:
MC/a = MD
Add multiple polluters and space
• Regulator solves:
Min{ei}S ci(Ei-ei) + D(p(e1,e2,…,eN))
dTC/dei = -ci’ + dD/dp * dp/dei
Rule for any number of polluters:
MCi/ai = MD….so MCi/ai = MCj/aj
Example with 3 firms
• All MC the same: MC(Ai) = 2.5Ai
• Marginal Damage = 10 (so marginal benefit
of abatement = 10).
• a1 = 1.0, 2.5A1 = 10, A1* = 4.0
• a2 = .01, 2.5A2/.01 = 10, A2* = .04
• a3 = 10, 2.5A3/10 = 10, A3* = 40
• Firm with greatest influence must abate the
most (firm 3)…they will have high marginal
abatement cost relative to other two firms.