Kennedy: Theory of Environmental Regulation (Draft 1)
9. EMISSIONS TRADING WHEN ABATEMENT COST IS
UNCERTAIN
Recall from Chapter 6 that an emissions trading program and a pollution tax are
essentially equivalent policies in a setting with full information. That equivalence breaks
down in the presence of uncertainty about marginal abatement costs. In this chapter we
examine the properties of an emissions trading program when marginal abatement costs
are unknown by the regulator, and then compare the performance of emissions trading
with a pollution tax. In the final section of the chapter we examine a hybrid policy that
combines elements of a tax and an emissions trading program.
9.1 THE OPTIMAL EMISSIONS TARGET
Recall from Chapter 8 that under a tax policy, the tax rate is chosen by the regulator but
aggregate emissions in response to that tax are uncertain. Under emissions trading, the
regulator sets aggregate emissions directly but the equilibrium permit price that arises in
response is uncertain. We will examine the properties of the permit price in Section 9.2
but our first task is to determine the optimal level of aggregate emissions (and hence, the
optimal supply of permits).
9.1-1 The Quadratic Model
We will again work with our simple quadratic model where
(9.1)
D( E ) =
δE 2
2
and abatement cost for source i is given by
(9.2)
ACi (e) =
(eˆi − ei ) 2
2γ i
Emissions trading leads to the equalization of MACs across sources, and we know from
section 5.3 in Chapter 5 that the resulting marginal aggregate abatement cost is
(9.3)
MAC ( E ) =
Eˆ − E
ϕ
where
1
Kennedy: Theory of Environmental Regulation (Draft 1)
n
(9.4)
Eˆ = ∑ eˆi
i =1
and
n
(9.5)
ϕ = ∑γ i
i =1
9.1-2 The Optimal Emissions Target in a World with Two States
We begin with the simplest setting, in which Ê is known and ϕ has just two possible
values: ϕ =ϕ 1 and ϕ =ϕ 2 < ϕ1 . The regulator assigns probabilities π and 1 − π to these
two states respectively.
The full-information optimal quantities in this setting are
(9.6)
Ei* =
Eˆ
1 + δϕ i
for i = 1 and i = 2 respectively. These are illustrated in Figure 9-1.
If the regulator sets emissions at E1* then this choice will be socially optimal if ϕ = ϕ1 but
it will be too low if ϕ = ϕ 2 , and the associated loss of social surplus is the area LSS 2 ( E1* )
in Figure 9-2. If instead the regulator sets emissions at E2* then this choice will be
socially optimal if ϕ = ϕ 2 is the true state of the world, but it will be too high if ϕ = ϕ1 ,
and the associated loss of social surplus is the area LSS1 ( E2* ) in Figure 9-3.
As with the tax policy, the goal of the regulator is to strike a balance between these two
extreme outcomes, in this case by setting emissions somewhere between E1* and E2* .
~
Suppose the regulator sets the quantity at some intermediate value E , as in Figure 9-4. If
~
~
ϕ = ϕ1 then E will be too high, and if ϕ = ϕ 2 then E will be too low. Note that the
direction of these errors is opposite to those under the tax where emissions are too low if
ϕ = ϕ1 , and too high if ϕ = ϕ 2 ; recall Figure 8.24 from Chapter 8.
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Kennedy: Theory of Environmental Regulation (Draft 1)
~
~
The loss of social surplus associated with the errors when E = E are LSS1 ( E ) and
~
LSS 2 ( E ) respectively, as depicted in Figure 9-4. This emissions target is optimal if and
~
only if E minimizes the expected loss from these errors. That is, the optimal target is
~
E = arg min πLSS1 ( E ) + (1 − π ) LSS 2 ( E )
(9.7)
E
Once set, this target can be implemented at least-cost via an emissions trading program.
9.1-3 The Optimal Target in a World with More than Two States
The planning problem in (9.7) can be easily extended to a setting with k > 2 states, in
which case it takes the form
(9.8)
~
E = arg min
E
k
∑π
j =1
j
LSS j ( E )
where π k is the subjective probability of state j. However, the graphical interpretation of
the problem becomes less useful when k > 2 so it is more instructive to set up the
planning problem as one that minimizes expected social cost, just as we did in Section
8.4-2 in Chapter 8 when deriving the optimal tax when there are more than two states.
Social cost is the sum of aggregate abatement cost and damage, so the generalized
planning problem is
(9.9)
⎡ ( Eˆ − E ) 2 δE 2 ⎤
min E ⎢
+
⎥
E
ϕ
2
2 ⎦
⎣
The first-order condition for this problem can be expressed instructively as
(9.10)
E[ MAC ( E )] = MD( E )
where
(9.11)
MD( E ) = δE
and
(9.12)
E[ MAC ( E )] = ν ( Eˆ − E )
and where
3
Kennedy: Theory of Environmental Regulation (Draft 1)
(9.13)
⎡1⎤
ν = E⎢ ⎥
⎣ϕ ⎦
is the mean of 1 / ϕ .
Solving the first-order condition in (9.10) yields the optimal level of emissions:
(9.14)
~ νEˆ
E=
ν +δ
See Figure 9-5 for the graphical representation in the two-state world.
It is important to note that this optimal level of emissions is not simply equal to the
probability-weighted average of the full-information optimal quantities. To see this, first
define a new parameter
(9.15)
θi =
1
ϕi
Then from (9.6) above, we can express the full-information optimal emissions in state i as
(9.16)
Ei* =
θ i Eˆ
θi + δ
Since ν is the expected value of θ , and since Ei* is strictly concave in θ i , it follows
~
from Jensen’s inequality that E > E[ E * ] . This relationship is illustrated in Figure 9-6
for a setting with two states.
~
To understand why E and E[ E * ] differ in this way, consider again the setting with just
two states. The MAC schedules in states 1 and 2 diverge as emissions fall, so the error
associated with emissions being too low is more costly than the error associated with
~
emissions being too high. The optimal quantity reflects this asymmetry, and so E is
closer to E2* than a simple probability-weighted averaging of E1* and E2* would dictate.
The same logic extends to a setting with more than two states.
4
Kennedy: Theory of Environmental Regulation (Draft 1)
Note the contrast between this relationship and that between the optimal tax and the
probability-weighted average of the full-information tax rates, as described in Figure 8-
29 from Chapter 8.
Expected Social Cost at the Optimum
~
Expected social cost under the optimal quantity policy is found by setting E = E in the
~
minimand from (9.9). Importantly, note that E is not a random variable – it is set by the
regulator – so the expectation in (9.9) is simply taken over 1 / ϕ , yielding the following
expression for expected cost:
(9.17)
νδEˆ 2
~
E[C ( E )] =
2(ν + δ )
This expected cost is increasing in the variance of ϕ . This property can be demonstrated
most clearly in the two-state setting, by examining the effect of a mean-preserving
spread. Specifically, let π =
1
2
, ϕ1 = μ + ε and ϕ 2 = μ − ε . Then the variance of ϕ is
σ 2 = ε 2 , and the mean of 1 / ϕ reduces to
(9.18)
ν=
μ
μ −σ 2
2
Making this substitution in (9.17) yields
(9.19)
~
E[C ( E )] =
μδEˆ 2
2( μ + δ ( μ 2 − σ 2 ))
~
This is clearly increasing in σ 2 . Note too that if σ 2 = 0 then E[C ( E )] reduces to cost
under full information.
9.1-4 Comparative Performance
Now that we have characterized the optimal emissions target under a quantity policy, we
can compare its performance with the optimal tax policy from Chapter 8. The basis for
that comparison is expected social cost at the optimum.
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Kennedy: Theory of Environmental Regulation (Draft 1)
Expected social cost under the quantity policy is given by (9.17). Expected social cost
under the optimal tax policy is given by (8.49) in Chapter 8, reproduced here as
(9.20)
E[C (~
t )] =
( μ + δσ 2 )δEˆ 2
2( μ + δμ 2 + δσ 2 )
where μ is the mean of ϕ , and σ 2 is its variance.
~
t )] and solving for δ , it is straightforward to show that the
By setting E[C ( E )] = E[C (~
optimal quantity policy outperforms the optimal tax policy if and only if δ > δ , where
(9.21)
δ =
μ ( μν − 1)
σ2
Since μ is the mean of ϕ while ν is the mean of 1 / ϕ , we know that μν > 1 when
σ 2 > 0 , so δ > 0 in that case. If there is no uncertainty, then μν = 1 and δ = 0 .
Why does a high value of δ reduce the relative performance of the tax policy? The
reason relates directly to our discussion in Chapter 8 (section 8.4-2) of losses from the tax
policy under low- and high-damage scenarios, as summarized in Figures 8-33 and 8-34.
Recall from that discussion that a high damage parameter positions the tax-policy
problem in a region where uncertainty is effectively greater, because the gap between the
two MAC schedules is greater. This is also true of the quantity policy but there is a key
difference between the two policies in terms of their exposure to the uncertainty, in the
following sense.
Under the quantity policy, the quantity of emissions actually emitted is independent of
the true MAC; the quantity emitted is determined solely by the beliefs of the regulator.
In contrast, actual emissions under the tax policy are determined partly by the tax (which
is determined solely by the beliefs of the regulator) and partly by the true MAC, since
sources respond to the tax based on their true MACs. This is a vital difference between
the policies, and one that does not arise in a simple world with full information.
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Kennedy: Theory of Environmental Regulation (Draft 1)
One might be inclined to think that the tax policy’s capacity to allow the truth to
influence actual emissions is always a good thing. Not so; the endogenous quantity –
reflecting the response to the tax by sources – could be very different from what the
regulator expects, and hence very wrong. The cost of being wrong is highest when δ is
high (as indicated in Figures 8-33 and 8-34), so the tax policy is outperformed by the
quantity policy under those conditions. Conversely, if δ is low then the expected benefit
of allowing the true MAC to determine emissions outweighs the expected cost of being
wrong, and so the tax is a better policy under those conditions.
The Role of Variance *
Note from (9.21) that the performance threshold between the two policies depends on σ 2
but that relationship is less clear-cut than a cursory inspection of (9.21) might suggest. In
particular, δ is not necessarily decreasing in σ 2 because the product μν is itself
increasing in σ 2 . Recall that μ is the mean of ϕ while ν is the mean of 1 / ϕ , and hence
their product must rise when the variance of the random variable rises. The rate at which
it does so depends on the particular distribution specified for ϕ , so the relationship
between δ and variance also depends on that distribution.
To further understand this nuanced role of variance it is helpful to return to our setting in
which ϕ has only two possible values, ϕ1 and ϕ 2 with associated probabilities π and
1 − π respectively. In that simple setting, the variance of ϕ can be expressed as
(9.22)
σ 2 = π (1 − π )(ϕ1 − ϕ 2 ) 2
Thus, the variance of ϕ is increasing in the gap between ϕ1 and ϕ 2 , but it is nonmonotonic in π , and reaches a maximum at π =
(9.23)
2
σ max
=
1
2
where
(ϕ1 − ϕ 2 ) 2
4
In this two-state setting, it is straightforward to show that the critical performance
threshold reduces to
(9.24)
δ =
μ
ϕ1ϕ 2
7
Kennedy: Theory of Environmental Regulation (Draft 1)
We know that the tax policy is best when δ < δ , so we can use (9.24) to re-express that
condition in terms of π , yielding
(9.25)
π >π ≡
ϕ 2 (δϕ1 − 1)
ϕH −ϕL
Thus, the tax policy outperforms the quantity policy when π is large. Why?
t )] and δ
We know from our discussion in Chapter 8 on the relationship between E[C (~
(recall Figures 8-33 and 8-34 again) that the scope for a large error depends on the size
of the tax, because MAC1 ( E ) and MAC2 ( E ) diverge as emissions fall. Since the optimal
tax is low when π is high, and high when π is low, it follows that the scope for a large
error under the tax is lowest when π is relatively high. Hence, the tax policy outperforms
the fixed-quantity policy when π is high. Of course, if π = 1 or π = 0 then both policies
perform equally well (because we are then in a world with no uncertainty).
The relative performance of the two policies is illustrated in Figure 9-7, which plots
~
E[C (~
t )] − E[C ( E )] against π for the case where
(9.26)
ϕ2 = ϕ2 ≡
ϕ1
2δϕ1 − 1
At this value of ϕ 2 , π = 12 . Thus, in this case the two policies perform equally well when
σ 2 = 0 and when σ 2 is at its maximum (where π = 12 ). This fact is highlighted in
Figure 9-7 which also plots σ 2 (measured on the RHS axis) against π . The maximum
value of variance in this case can be shown to be
(9.27)
2
σ max
ϕ2 = ϕ2
= ϕ 22 (1 − δϕ1 )
We will refer to this case as the “medium-variance scenario”.
Next consider a high-variance scenario, where ϕ 2 < ϕ 2 . This case is illustrated in Figure
9-8. Note that π <
1
2
in this case; the tax policy outperforms the quantity policy over a
larger range of π .
8
Kennedy: Theory of Environmental Regulation (Draft 1)
In contrast, Figure 9-9 depicts a low-variance case where ϕ 2 > ϕ 2 . In this case, π > 12 ;
the tax policy outperforms the quantity policy over a smaller range of π .
These results tell us that variance is an important factor in the relative performance of the
two policies but that the relationship is not a simple monotonic one. In particular, the
skewness of the distribution (determined in the two-state world by the size of π relative
to
1
2
) is also a critical factor.
9.2 PERMIT PRICES
Under a tax policy, the regulator sets the price of emissions but the emissions that arise is
in response to that price is a random variable. In contrast, under emissions trading the
regulator sets the quantity of emissions but the price of emissions is a random variable.
We now want to examine the properties of that random variable.
~
The regulator sets the quantity of emissions at E , as determined by (9.10) above. All
sources equate their own MAC to the permit price, and consequently, marginal aggregate
abatement cost is equated to the permit price. Thus,
~
~
pi = MACi ( E )
(9.28)
in state i. Figure 9-10 illustrates this outcome for a two-state world.
It follows that the expected value of the equilibrium price is
~
E[ ~
p ] = E[ MAC ( E )]
(9.29)
~
~
~
Moreover, since E has been set to ensure that E[ MAC ( E )] = MD( E ) , it follows that
(9.30)
~
E[ ~
p ] = MD( E )]
Figure 9-11 illustrates this outcome for a two-state world.
9.3 PERMIT PRICES IN THE QUADRATIC MODEL
Let us now examine the properties of the permit price in the context of our simple
quadratic model. We know from (9.14) above that
9
Kennedy: Theory of Environmental Regulation (Draft 1)
(9.31)
~ νEˆ
E=
ν +δ
where ν is the mean of 1 / ϕ . Thus, from (9.28) we know that the equilibrium price of
permits in state i is
(9.32)
~
Eˆ − E
~
pi =
=
ϕi
δEˆ
(ν + δ )ϕ i
In the special case of just two states,
(9.33)
ν=
π 1− π
+
ϕ1 ϕ 2
so we have
(9.34)
~
pi =
δEˆ ϕ j
πϕ 2 + (1 − π )ϕ1 + δϕ1ϕ 2
where i ≠ j . Figure 9-12 depicts the properties of these prices in the two-state world, in
terms of their relationships to π . To understand these relationships consider Figure 9-13.
It illustrates the impact of an increase in π : E[MAC(E)] pivots closer to MAC1 ( E ) . This
~
causes a reduction in E , and this in turn drives up the equilibrium price in both states.
In the limit as π → 1 we move towards a world with full information where the true state
is state 1, and so ~
p1 → t1* . That is, the permit price converges to the full-information
Pigouvian tax rate. Conversely, as π → 0 we move towards a world with full
information where the true state is state 2, and so ~
p2 → t 2* .
Importantly, note from Figures 9-12 and 9-13 that ~
p 2 rises at a faster rate than ~
p1 does
as π rises; that is, the two prices diverge as π rises. Why?
~
The divergence of MAC1 ( E ) and MAC2 ( E ) as E falls means that as π rises (and E
falls), ~
p1 and ~
p 2 also diverge. This effect is highlighted in Figure 9-14, which focuses
~
on the relationship between π , E and the permit prices in the two states of nature.
10
Kennedy: Theory of Environmental Regulation (Draft 1)
~
Figure 9-14 illustrates that as π rises from a low value π L to a higher value π H , E
falls, and the gap between ~
p1 and ~
p 2 rises. Note the importance of the divergence of the
two MAC schedules in this result.
Now consider the expected value of the permit price:
(9.35)
E[ ~
p] =
νδEˆ
ν +δ
In the special case of just two states, this expected price is
(9.36)
E[ ~
p] =
δEˆ (πϕ 2 + (1 − π )ϕ1 )
πϕ 2 + (1 − π )ϕ1 + δϕ1ϕ 2
Note that E[ ~
p ] is decreasing in π , as depicted in Figure 9-15.
How can the expected price be decreasing in π when both possible prices are increasing
in π , as illustrated in Figure 9-12. The algebra is straightforward: a higher value of π
puts more weight on ~
p1 and less weight on ~
p 2 in the expected value, and since ~
p1 < ~
p2 ,
the expected value must fall.
In terms of the economics, a higher value of π means a lower optimal level of emissions
– reflecting the higher probability that abatement cost is low – and this means that
~
marginal damage at the optimum is lower. Since E[ ~
p ] = MD( E ) , it follows that E[ ~
p ] is
decreasing in π . We can also see this property of E[ ~
p ] from Figure 9-16, in which
E[MAC(E)] pivots towards MAC1 ( E ) as π rises.
Note also that E[ ~
p ] falls at an increasing rate as π rises; see Figure 9-15. This reflects
the fact that the difference between ~
p 2 and ~
p1 rises as π rises. This property of E[ ~
p]
means that
(9.37)
E[ ~
p ] > E[t * ] = πt1* + (1 − π )t 2*
as illustrated in Figure 9-17, where E[t * ] declines linearly as π rises.
11
Kennedy: Theory of Environmental Regulation (Draft 1)
Now let us compare the expected permit price with the optimal tax rate from Chapter 8.4,
given by
(9.38)
~
t =
μδEˆ
μ + δ (μ 2 + σ 2 )
p ] and ~
t yields
Taking the difference between E[ ~
(9.39)
⎛
⎞ 2
δEˆ
⎟ νμ + νσ 2 − μ
E[ ~
p] − ~
t = ⎜⎜
2
2
⎟
⎝ (ν + δ )(δμ + δσ + μ ) ⎠
(
)
The first bracketed term is clearly positive. The second term is also positive when σ 2 > 0
since in that case νμ > 1 , which in turn means that νμ 2 > μ . Thus, E[ ~
p] > ~
t when
σ 2 > 0 . When combined with our other results from Chapter 8, we can conclude that
(9.40)
p ] > E[t * ] > ~
t
E[ ~
when σ 2 > 0 .
These relationships are depicted for the two-state world in Figure 9-18, where all
variables are plotted against π . Ultimately, these relationships reflect the divergence of
the MAC schedules as aggregate emissions fall.
It is important to remember that ~
t is not a random variable – it is chosen by the regulator
– but the permit price is a random variable. The permit price is almost surely never
p ] ; in the two-state world, it is equal to either ~
actually equal to E[ ~
p or ~
p .
1
2
Moreover, recall that the gap between ~
p 2 and ~
p1 in the two-state world rises as π rises,
reflecting the fact that the gap between MAC2 ( E ) and MAC1 ( E ) rises as optimal
emissions fall in response to a rising π ; recall Figure 9-14. This rising gap between ~
p2
and ~
p1 is a useful gauge of the rising risk associated with a quantity-based policy as π
rises. Recall from section 9.1-4 above that if π is high enough then the quantity-based
policy is outperformed by a tax policy for precisely that reason. In particular, we know
from (9.25) that if π > π then a tax is the best policy. Figure 9-19 depicts this critical
12
Kennedy: Theory of Environmental Regulation (Draft 1)
threshold for π alongside the permit prices and the tax. This figure provides a useful
summary of the key points of comparison between the tax and quantity policies.
9.4 EMISSIONS TRADING WITH A “SAFETY VALVE”
Our analysis to date has shown that a quantity-based policy with emissions trading is
better than a tax policy if there is a relatively low probability that MAC (E ) is low.
Conversely, a tax policy is best if there is a high probability that MAC (E ) is low.
This raises a natural question: can we use a combination of the two policies in a way that
is better than using either policy on its own? In this section we investigate a combined
policy that uses a quantity-based policy with a “safety valve”. We will examine this
policy in the context of our simple model with just two possible states.
The combined policy works as follows. The regulator issues a quantity of permits E1*
such that
(9.41)
MAC ( E1* ) = MD( E1* )
as illustrated in Figure 9-20. If permits trade at p1* , as depicted in Figure 9-20, then the
regulator knows that the true state of nature is indeed state 1, and the social optimum has
been implemented. If instead permits trade at p2 ( E1* ) , as depicted in Figure 9-20, then
the regulator infers correctly that the true state of nature is state 2.1
If the regulator does observe p2 ( E1* ) then it could in principle issue more permits, raising
the total supply to E2* , thereby inducing a reduction in the trading price to p2* , as
illustrated in Figure 9-21.
The problem with this approach in practice is that many of the abatement actions taken in
response to the initial high permit price may be irreversible. This is especially true of
investments in new technology. The costs of these investments generally cannot be
1
Note the importance of a competitive permit market here: no source can be so large relative to the market
that it can manipulate the permit price and thereby send a false signal to the regulator.
13
Kennedy: Theory of Environmental Regulation (Draft 1)
recovered by “undoing” them. Thus, the regulator would like to prevent these suboptimal investments from happening before they are made.
A natural solution is to announce ahead of time that unlimited additional permits can be
purchased from the regulator at a price p2* should the demand arise (as it will if the true
state is state 2). This means that the trading price will never rise above p2* , and that no
sub-optimal investments will be made in response to an erroneously high price. This
commitment to sell additional permits at a fixed price works like a “safety valve”: if
pressure on the permit price pushes it towards a value higher than p2* , then the sale of
additional permits at price p2* immediately relieves that pressure.
This solution to the information problem works perfectly if there are only two possible
states of nature, since the regulator knows exactly how many permits to issue initially,
and at what price to set the safety valve. In a setting with more than two possible states of
nature, the problem cannot be solved quite so easily. In such a setting, it is generally not
possible to achieve the full-information outcome, and the optimal safety-valve policy
must be chosen under uncertainty about the outcome.
This optimization problem is complicated, and we will not investigate it here. However,
the simple two-state example serves to illustrate that a combined policy – emissions
trading coupled with a safety valve – can generally do better than either a fixed-quantity
policy or a tax policy alone.
14
Kennedy: Theory of Environmental Regulation (Draft 1)
$ per unit
MD(E )
MAC 2 ( E )
MAC1 ( E )
E1*
E2*
Ê
E
Figure 9-1
15
Kennedy: Theory of Environmental Regulation (Draft 1)
$ per unit
MD(E )
MAC2 ( E )
LSS 2 ( E1* )
MAC1 ( E )
E1*
E2*
Ê
E
Figure 9-2
16
Kennedy: Theory of Environmental Regulation (Draft 1)
$ per unit
MD(E )
MAC2 ( E )
LSS1 ( E2* )
MAC1 ( E )
E1*
E2*
Ê
E
Figure 9-3
17
Kennedy: Theory of Environmental Regulation (Draft 1)
$ per unit
MD(E )
MAC2 ( E )
~
LSS 2 ( E )
~
LSS1 ( E )
MAC1 ( E )
E1*
~
*
E E2
Ê
E
Figure 9-4
18
Kennedy: Theory of Environmental Regulation (Draft 1)
$ per unit
MD(E )
MAC2 ( E )
E[ MAC ( E )]
MAC1 ( E )
E1*
~
*
E E2
Ê
E
Figure 9-5
19
Kennedy: Theory of Environmental Regulation (Draft 1)
tons
E2*
~
E
E[ E * ]
E1*
1
π
Figure 9-6
20
MEDIUM VARIANCE:
ϕ2 = ϕ2
$
VARIANCE
Kennedy: Theory of Environmental Regulation (Draft 1)
2
σ max
= ϕ 22 (1 − δϕ1 ) 2
σ2
~
~
E[C ( t )] − E[C ( E )]
TAX POLICY
IS BEST
π
0
QUANTITY POLICY
IS BEST
π =
1
2
π =0
π =1
Figure 9-7
21
HIGH VARIANCE:
ϕ2 < ϕ2
$
2
σ max
> ϕ 22 (1 − δϕ1 ) 2
VARIANCE
Kennedy: Theory of Environmental Regulation (Draft 1)
σ2
~
~
E[C ( t )] − E[C ( E )]
TAX POLICY
IS BEST
0
QUANTITY POLICY
IS BEST
π<
π =0
π
1
2
1
2
π =1
Figure 9-8
22
VARIANCE
Kennedy: Theory of Environmental Regulation (Draft 1)
LOW VARIANCE:
ϕ2 > ϕ2
$
2
σ max
< ϕ 22 (1 − δϕ1 ) 2
σ2
~
~
E[C ( t )] − E[C ( E )]
TAX POLICY
IS BEST
1
2
π
0
QUANTITY POLICY
IS BEST
π >
π =0
1
2
π =1
Figure 9-9
23
Kennedy: Theory of Environmental Regulation (Draft 1)
$ per unit
MD (E )
MAC2 ( E )
~
p2
E[ MAC ( E )]
~
p1
MAC1 ( E )
~
E
Ê
E
Figure 9-10
24
Kennedy: Theory of Environmental Regulation (Draft 1)
$ per unit
MD (E )
MAC2 ( E )
~
p2
E[ MAC ( E )]
E[ ~
p]
~
p1
MAC1 ( E )
~
E
Ê
E
Figure 9-11
25
Kennedy: Theory of Environmental Regulation (Draft 1)
$ per unit
δEˆ ϕ1
ϕ 2 (1 + δϕ1 )
~
p2
t2*
t1*
~
p1
δEˆ ϕ 2
ϕ1 (1 + δϕ 2 )
1
π
Figure 9-12
26
Kennedy: Theory of Environmental Regulation (Draft 1)
$ per unit
MD (E )
MAC2 ( E )
E[ MAC ( E )]
~
p2
~
p1
MAC1 ( E )
~
E
Ê
E
Figure 9-13
27
Kennedy: Theory of Environmental Regulation (Draft 1)
$ per unit
~
p2 (π H )
MAC 2 ( E )
~
p1 (π H )
~
p (π )
2
L
~
p1 (π L )
MAC1 ( E )
~
E (π H )
~
E (π L )
Ê
E
Figure 9-14
28
Kennedy: Theory of Environmental Regulation (Draft 1)
$ per unit
δEˆ ϕ1
ϕ 2 (1 + δϕ1 )
~
p2
t2*
E[ ~
p]
t1*
~
p1
δEˆ ϕ 2
ϕ1 (1 + δϕ 2 )
1
π
Figure 9-15
29
Kennedy: Theory of Environmental Regulation (Draft 1)
$ per unit
MD (E )
MAC2 ( E )
E[ MAC ( E )]
E[ ~
p]
MAC1 ( E )
~
E
Ê
E
Figure 9-16
30
Kennedy: Theory of Environmental Regulation (Draft 1)
$ per unit
δEˆ ϕ1
ϕ 2 (1 + δϕ1 )
~
p2
t2*
E[t * ]
E[ ~
p]
t1*
~
p1
δEˆ ϕ 2
ϕ1 (1 + δϕ 2 )
1
π
Figure 9-17
31
Kennedy: Theory of Environmental Regulation (Draft 1)
$ per unit
δEˆ ϕ1
ϕ 2 (1 + δϕ1 )
~
p2
t2*
~
t
E[t * ]
E[ ~
p]
t1*
~
p1
δEˆ ϕ 2
ϕ1 (1 + δϕ 2 )
1
π
Figure 9-18
32
Kennedy: Theory of Environmental Regulation (Draft 1)
$ per unit
δEˆ ϕ1
ϕ 2 (1 + δϕ1 )
EMISSIONS
TRADING POLICY
IS BEST
TAX POLICY
IS BEST
~
p2
t2*
E[ ~
p]
~
t
t1*
~
p1
δEˆ ϕ 2
ϕ1 (1 + δϕ 2 )
π
1
π
Figure 9-19
33
Kennedy: Theory of Environmental Regulation (Draft 1)
$ per unit
MD (E )
MAC2 ( E )
p2 ( E1* )
p1*
MAC1 ( E )
E1*
Ê
E
Figure 9-20
34
Kennedy: Theory of Environmental Regulation (Draft 1)
$ per unit
MD(E )
MAC2 ( E )
p2 ( E1* )
p2*
p1*
MAC1 ( E )
E1*
E2*
Ê
E
Figure 9-21
35