ECON 4910 Spring 2007
Environmental Economics
Lecture 10, Chapter 10
Lecturer: Finn R. Førsund
Unknown control costs
1
The regulator’s problems


Purification cost functions unknown to the
regulator
What to do?




Form the expected cost function
Ask the polluters about their cost functions
Assumption: only one polluter
Decision of the regulator.

What policy instrument to choose
Unknown control costs
2
The regulator’s problem with uncertainty
of the cost function

The social problem
Min e {E{c(e)}  D(e)}

The assumptions




One polluter
Only purification costs, output of polluter fixed
Damage function known
Standard curvature of functions
Unknown control costs
3
The solution to the regulator’s problem

First-order condition
Min e {E{c(e)}  D(e)} 
E{c '(e)}  D '(e)  0

Must have information to form the expected
marginal cost function
Unknown control costs
4
Illustration of social optimum
Probability distribution
D’(e)
-E{c’(e)}
-E{c’(e)} =
D’(e)
e
e*
Unknown control costs
5
The social solution with a specific
distribution

Simplifying assumption about the distribution
function of the cost function




Only two types; high cost, cH(e) and low cost, cL(e)
Probability 0.5 for each type
Only assuming a single firm
The expected marginal cost
E{c(e)}  0.5[cH (e)  cL (e)]

The social solution
0.5[cH (e)  cL (e)]  D(e)  e  e*
Unknown control costs
6
Choice of policy instrument

Emission tax
t  D(e* )

The adjustment of the firm
Min e {c j (e)  te}  cj (e)  t , j  H , L

The firm knows its own type
Unknown control costs
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Illustration of the simple distribution case
-E{c’(e)}
D’(e)
tH
Social loss if H
t
tL
-cH’
Social loss if L
-cL’
eL(t)
eL
e
e*
eH eH(t)
Unknown control costs
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Regulator asks about cost type

Assumptions




Emission tax is used as instrument
Regulator does not control if the information is
correct, i.e. regulator does not measure actual
emission
No ethical problems of the firm lying
It will always minimise costs of the firm if it
declares to be of the low cost type
Unknown control costs
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Illustration of firm giving cost
information
D’(e)
-cH’
Social loss if H
and lying
tH
tL
-cL’
eL(H)
eL*
e
eH*
Unknown control costs
eH(L)
10
Problems with the Kolstad model


Nothing said about whether emission are
measurable
Assume that emissions are measurable:

When regulator is given the information about the
cost function, then the regulator also knows the
marginal cost function, and optimal emissions eL*,
eH* can be calculated obeying
Min e j {c j (e j )  D(e j )}  cj (e j )  D(e j ) 
e j  e*j , j  L, H
Unknown control costs
11
Problems with the Kolstad model, cont.


When the firm reports its cost function then
the Regulator knows the correct emission ej*
and comparing this level and the choice of
emission will tell the Regulator whether the
firm is lying or not, thus inducing truth-telling.
The possibility to observe emission, but not
the cost function, is quite realistic. If
emissions cannot be observed, then the firm
may cheat about reported emission as well.
Unknown control costs
12
Kolstad: Introducing an incentive to tell
the truth


A high-cost firm can be compensated for
telling the truth
A reward for telling that the firm is high-costtype


How to minimise the reward, R, and still make the
firm willing to reveal that it is of the high-cost type
If high-cost is true then
cH (e )  t e  R  cH (eH ( L))  t L eH ( L)
*
H
*
H H
Unknown control costs
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Incentive to tell the truth, cont.

Setting the reward R such that it does not pay
to lie when the true type is low-cost
cL (eL* )  t L eL*  cL (eL ( H ))  t H eL ( H )  R

The range for R
cH (eH* )  t H eH*  cH (eH ( L))  t L eH ( L)  R  cL (eL ( H ))  t H eL ( H )  cL (eL* )  t LeL*

Left-hand side positive, total cost telling the truth
higher than cost when lying
Unknown control costs
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The Kolstad analysis is not correct

It is easy to demonstrate using diagrams that
there is no amount of payment R that
satisfies the equation above
Unknown control costs
15
Illustration of giving an incentive to the
high-cost firm to tell the truth
Difference in cost if H
between lying and truth
D’(e)
-cH’
tH
tL
-cL’
eL(H)
eL*
e
eH*
Unknown control costs
eH(L)
16
Illustration of giving an incentive to the
low-cost firm to tell the truth
Difference in cost if L
between truth and lying
D’(e)
-cH’
tH
tL
-cL’
eL(H)
eL*
e
eH*
Unknown control costs
eH(L)
17
An alternative model for asymmetric
information




Consider only two types of firms, high-cost H
and low cost L
Assume that a firm earns a profit π from its
economic activity independent of type
Emissions are measurable, but not cost
functions
The regulator offers contracts specifying
permitted emission and a type-specific
subsidy/tax - transfer
Unknown control costs
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The contracts for the two cost types for
emission quantity and tax

Contract for low- cost type


Contract for high-cost type


π –TH* - cH(eH*) = 0 → TH *= π - cH(eH*)
Low-cost type choosing high-cost contract


π –TL*- cL(eL*) = 0 → TL* = π - cL(eL*)
π –TH* - cL(eH*) = π –TH* - cL(eH*) + cH(eH*) cH(eH*) = {π –TH* - cH(eH*)} + cH(eH*) - cL(eH*)
If high-cost gross profit is ≥ 0, then it pays the
low-cost type to choose a high-type contract
Unknown control costs
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The contracts for the two cost types, cont.

Can it be profitable for the high-cost type to
take a low-cost type contract



π –TL* - cH(eL*) = π –TL* - cH(eL*) + cL(eL*) - cL(eL*)
= π – {TL* - cL(eL*)} - cH(eL*) + cL(eL*) =
π- π - cH(eL*) + cL(eL*) < 0
It cannot be profitable for a high-cost type to
take a low-cost type contract
The problem for the regulator is the
behaviour of the low-cost type taking the
high-cost type contract
Unknown control costs
20
Contract to avoid low-cost firm taking a
high-cost type contract

The contracts should fulfil two objectives


Ensure participation of the firm, i.e. the gross
profit must be non-negative
Give incentive to tell the truth about the cost type,
i.e. transfer according to type must induce truthtelling
Unknown control costs
21
Contracts

Participation:




UL = π -TL - cL(eL) ≥ 0
UH = π -TH - cH(eH) ≥ 0
The tax must not be so high that the firms
close down
The quantities to be permitted emitted in the
contracts are endogenous and still not
determined
Unknown control costs
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Incentives


Define gross profit (pure profit) when telling
the truth as Uj, j=L,H
Incentives to tell the truth



UL ≥ π -TH - cL(eH)
UH ≥ π -TL - cH(eL)
It should give higher gross profit to tell the
truth than to lie.
Unknown control costs
23
Combining participation and incentive


The high-cost type will not take a low-costtype of contract
The tax-quantity pair in the contract can
therefore be set such that gross profit is zero



UH = π -TH - cH(eH) = 0
The incentive-condition for a high-cost type
will not be a problem, i.e. fulfilled
The participation condition for low-cost type is
fulfilled when UH ≥ 0 implying UL > 0
Unknown control costs
24
The active participation- and incentive
conditions

The four original conditions are reduced to
two


UH = 0
UL = UH + cH(eH) - cL(eH) = cH(eH) - cL(eH)
The last condition was established with eH = eH*
above
Unknown control costs
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Determination of emission- and tax
quantities of the contracts

The objective function for the Regulator is



W = V + αU = T- D(e) + αU =
π
– c(e) – U – D(e) + αU = π – c(e)– D(e) + (1-α)U
Assume that low-cost type appears with
probability p and high-cost type with
probability (1-p)
The two possible objective functions are


WL = π – cL(eL)– D(eL) + (1-α)UL
WH = π – cH(eH)– D(eH) + (1-α)UH
Unknown control costs
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Determination of emission- and tax
quantities of the contracts, cont.

Maximising the expected value of the
objective function
E{W} = p(π – cL(eL) – D(eL) - (1-α)(cH(eH) - cL(eH) )
+ (1-p)(π – cH(eH) – D(eH) )
(setting UH = 0 )
Differentiating:

E{W }
 p (cL (eL )  D(eL ))  0
eL
E{W }
 p (1   )(cL (eH )  cH (eH ))  (1  p)( cH (eH )  D(eH ))  0
eH
Unknown control costs
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Determination of emission- and tax
quantities of the contracts, cont.

The low-cost contract for emission is set by
cL (eL )  D(eL )  0


This is the standard condition.
The condition for emission allowed on the
high-cost type contract is implying a higher
emission than eH*.
Unknown control costs
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The allowed emission for the high-cost
type

Rearranging the second condition for maximising
the objective function
p(1   )
D(eH )  cH (eH ) 
(cL (eH ))  cH (eH )) 
(1  p)
cH (eH )  cL (eH )  D(eL )

High-cost- emission set such that marginal damage
equals the sum of two terms, the marginal
purification cost and a term reflecting the negative
effect for the Regulator caused by the low-cost type
trying to take a high-cost-type contract
Unknown control costs
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