Note Regarding Endogenous Emissions Rates in
Prices vs. Quantities
Erin T. Mansur
July 3, 2006
Abstract
This is a note on my paper titled “Prices vs. Quantities: Environmental Regulation
and Imperfect Competition” (located at
http://www.som.yale.edu/faculty/etm7/papers/mansur_envreg_note.pdf).
In this note, I discuss the robustness of the paper’s results to the assumption of
fixed emissions rates.
1
Model’s Set Up
Assume L independent product markets emit pollution into a common “airshed.” For example, suppose there are many independent industrial and electricity markets in the Northeast
that pollute nitrogen oxides, which cause smog throughout the region. Without loss of generality, consider product market l as the market of interest, which consists of N firms. Firm
i ∈ {1, .., N } produces qi and, if it does not control emissions, pollutes at rate ri . Abating
with technology that has an abatement rate ai will result in emissions ei = qi ri − qi ai . The
other L − 1 product markets, noted h ∈ {1, ..., l − 1, l + 1, ..., L}, have a total of M − N firms.
Let ϕjh = 1 indicate that firm j ∈ {N + 1, .., M} is located in product market h, otherwise
ϕjh = 0. Throughout the model, I assume firm j takes prices as given and pollutes.
1.1
Social Planner’s Problem
Social welfare (W ) depends on the (inverse) demand functions for product market l (P ) and
for each of the other markets (ω h ). Welfare also depends on production costs (ci ), abatement
technology costs (Ki ), and pollution damages (D). A social planner would choose qi and ai
1
for each i ∈ {1, .., M } to maximize W :
max
{q1 ,...,qM },
{a1 ,...,aM }
W =
Z
PN
i=1 qi
P (θ)dθ +
0
−
Z
L
X
h=1,h6=l
M
X
i=1
ci (qi ) −
M
X
i=1
PM
j=N +1 qj ·ϕjh
ω j (γ)dγ
(1)
0
M
X
Ki (qi , ai ) − D( {qi ri − qi ai }).
i=1
Assume the following conditions hold: P 0 < 0, ω 0h < 0, c0i > 0, c00i > 0, D0 > 0, and D00 > 0.
In addition, the abatement technology costs are assumed to meet these conditions: Kiq >
0, Kiqq > 0, Kia > 0, and Kiaa > 0, where Kij ≡ ∂Ki /∂j. To ensure a firm’s emissions increase
in qi , a sufficient condition that I assume holds is:
Kiqa (qi , ai ) · qi ≥ Kia (qi , ai ).
(2)
For example, if Ki (qi , ai ) = ki (ai ) · qi , then Kiqa (qi , ai ) · qi = Kia (qi , ai ) = ki0 (ai ) · qi .
Focusing on each firm i ∈ {1, .., N } in product market l, the resulting first order conditions
(assuming interior solutions, i.e., qi > 0, ai > 0, ai < ri ) are:
N
M
X
X
q
0
0
qi ) − ci (qi ) − Ki − D (
ei ) · (ri − ai ) = 0
(qi ) : P (
i=1
(ai ) :
−Kia
(3)
i=1
M
X
+D(
ei ) · qi = 0.
0
i=1
denoted
This implies the optimal choice for each firm (qi∗ , a∗i ). At the social optimum,
P
∗
∗
, a∗1 , ..., a∗M ), the marginal damages of emissions, D0∗ ≡ D0 ( M
e
W ∗ = W (q1∗ , ..., qM
i=1 i ), equal
the marginal costs of abatement both with respect to both qi and ai :
P
q ∗ ∗
∗
0 ∗
P( N
Kia (qi∗ , a∗i )
0∗
i=1 q i ) − ci (qi ) − Ki (qi , ai )
=
.
(4)
D =
ri − a∗i
qi∗
Firms in the other product markets have similar first order conditions with ωh in place of P .
1.2
Competitive Firm’s Incentives
In markets, however, firm i determines qi and ai and will internalize pollution damages only
when regulated, for example by pollution taxes or tradeable permit systems. If costs were
known with certainty, the social planner could use either market-based instrument to achieve
W ∗ . However, for reasons explained below, I assume that production costs and abatement
technology costs are uncertain to the planner.
The social planner may establish a tradeable permit system that requires all M firms
to procure permits equal to the amount of pollution emitted. For each firm i, the planner
2
allocates permits (ei ) that trade at the market determined permit price τ = τ (e1 , .., eM ). I
assume that regulators set a non-compliance fine great
so that all firms comply with
PM enough
PM
the regulation. Therefore, the permit cap binds: i=1 ei = i=1 ei . In competitive market
l, firm i will maximize profits (πi ) with respect to qi and ai :
max π i = P qi − ci (qi ) − Ki (qi , ai ) − τ · (qi ri − qi ai − ei ).
qi ,ai
(5)
Firms that take the product and permit prices as given (P = P , τ = τ ) have these first order
conditions:
(qi ) : P − c0i − Kiq − τ · (ri − ai ) = 0
(ai ) : −Kia + τ qi = 0.
(6)
At τ = D0∗ (which I denote τ ∗ ), firms optimize by choosing qi∗ , a∗i . I assume the planner
sets the permit cap optimally assuming a competitive product market. Alternatively, the
planner may levy a Pigouvian tax (τ 0 ) also set equal to D0∗ that does not vary with market
conditions. Firms’ first order conditions will be as in (6) with τ = τ 0 .
1.3
Strategic Firm’s Incentives
Suppose that product market l is initially competitive and that regulators optimally set either
the permit cap or tax. The policy experiment that I focus on is the introduction of strategic
behavior into market l. While I focus on restructuring of wholesale electricity markets,
this may apply to mergers or the deregulation of other industries. The introduction, and
exercise, of market power in product markets can affect firms willingness to abate pollution.
In turn, this shock may affect policy makers’ optimal choice of environmental regulation.
The preferred regulation would minimize total welfare losses. In addition to the losses
corresponding to the pollution externality, regulators must consider the losses in the product
market.
To determine these effects, I begin by examining strategic firms’ incentives. A strategic
firm may be capable of setting the product price, P = Pi (qi ), and the permit price, τ = τ i (ei ).
Thus, the first order conditions are:
(qi ) : P + Pi0 qi − c0i − Kiq − τ · (ri − ai ) − τ 0i · (qi ri − qi ai − ei ) · (ri − ai ) = 0
(ai ) : −Kia + τ qi + τ 0i · (qi ri − qi ai − ei ) · qi = 0.
(7)
ai ). As in Hahn (1984),
I denote e
τ as the permit price given the firm’s optimal choices (e
qi , e
firms may have incentives to affect permit prices in order to minimize regulation costs. Hahn
shows that a firm will exercise monopoly or monopsony power depending on whether its sells
or buys permits, on net. However, if the allocation of permits across firms is such that
no trading would occur, (i.e., ei = qei ri − qeie
ai ), then firms will not profit from changing τ .
3
To simplify the problem, I assume that permits are allocated in this manner.1 Note that
assuming firms would not profit from trade alleviates the concern of transaction costs in
permit markets as noted by Stavins (1995). The relevant portion of the first order conditions
is thus:
(qi ) : P + Pi0 qi − c0i − Kiq − τ · (ri − ai ) = 0
(ai ) : −Kia + τ qi = 0.
(8)
Under a tax, these first order conditions would also hold.
1.4
Model of Market Structure
I model the structure of product market l as one dominated by a single firm i = 1 that
faces a competitive fringe f , which is the aggregation of price-taking firms 2, ..., N . As with
derived demand in wholesale electricity markets, I model demand as perfectly inelastic (q).
Therefore, the dominant firm’s residual demand (P1 ) is determined by the marginal cost of
the fringe. If the fringe’s emissions are also regulated then, from (6), this implies:
P1 = c0f (q − q1 ) + Kfq (q − q1 ) + τ · (rf − af ).
(9)
If the dominant firm recognizes that P1 is a function of τ , then—despite allocating permits
to avoid Hahn’s concern—the firm may have incentives yet to affect the permit price in order
to raise its rivals’ costs (RRC) (Misiolek and Elder, 1989; and Sartzetakis, 1997). Note that
RRC does not apply to taxes. To simplify the problem, I assume the dominant firm will not
be capable of raising its rivals’ costs through the permit market. There are two cases when
this may arise that I will refer to as Case A and Case B.
In Case A, I assume that the fringe does not pollute but that the dominant firm does
pollute and is regulated. In the case of electricity production, for example, this could be a
fringe that consists of hydroelectric, nuclear, and renewable power plants. Here, the dominant
firm cannot raise its rivals’ costs and P1 = c0f (q − q1 ). Similarly, RRC will irrelevant if all
firms pollute, but the pollution regulation only affects the dominant firm. For example, their
may be a local regulation that regulates Pennsylvania but not Maryland (as was the case of
the OTC regulation in 1999). Or, as during Phase I of Title IV of the 1990 Clean Air Act
Amendments, only the SO2 emissions from “Table A units” had to comply with the permit
system. Here, the dominant firm’s first order condition with respect to q1 can be written:
c0f (q − q1 ) − c00f (q − q1 ) · q1 − c01 − K1q − τ · (r1 − a1 ) = 0.
1
(10)
These conditions would also arise if strategic firms could exercise market power in the product market
but not in the permit market. If many permit market firms incur similar “shocks” to incentives in the
product market, then the permit price could be affected by the exercise of market power in the product
markets. For example, PPL may be a dominant firm in the PJM electricity market and Sithe in the New
England market, but they take prices as given in the SO2 tradeable permits market. Both electricity markets
restructure and allow the firms to behave strategically, they will reduce output and could, collectively, move
the permit price.
4
As cf is convex, the dominant firm will reduce output relative to its behavior in a competitive
market: qe1 < q1∗ .2 Furthermore, it will reduce output from its most expensive methods of
production (e.g., relatively inefficient power plants).
In Case B, I assume the fringe pollutes, and is regulated, but the dominant firm does not
pollute. Again, in this case, RRC will be irrelevant. This will also be true if the dominate
firm does pollute but is not regulated by the same market as is the fringe. The first order
condition with respect to q1 is:
c0f (q − q1 ) + Kfq (q − q1 ) + τ (rf − af ) − q1 · c00f (q − q1 ) − q1 · Kfqq (q − q1 ) − c01 = 0.
(11)
Before examining the environmental and policy implications, I review the key assumptions:
• A social planner determines environmental regulation assuming competitive product
markets;
• Permits are allocated across firms such that, when firms behave strategically, no trade
is required: ei = qei ri − qeie
ai ;
• Only one firm in a product market has strategic incentives (versus an oligopoly model);
• Demand in the product market is perfectly inelastic (q); and
• The dominant firm does not consider raising rivals’ costs: either it pollutes and is
regulated; or the fringe pollutes and is regulated (but not both).
These assumptions allow for a concise description of point of this exercise. Relaxing these
assumptions would complicate the model but not change the underlying intuition.
2
Environmental Implications
Given the assumption of perfectly inelastic demand in the product market, the overall change
in production must sum to zero in equilibrium. Therefore, the reduction in output resulting
from the exercise of market power from the dominant firm (q1∗ −e
q1 ) exactly equals (in absolute
value) the increase in output from the fringe (e
qf − qf∗ ). If abatement technology is fixed in
the short run (or chosen independently of qi ), then the emissions implications of introducing
strategic behavior will be, at first, solely a function of the relative emission rates of the
dominant firm and the fringe.
However, if abatement technology may also be adjusted, then e
a may differ substantially
from a∗ . The change in total emissions from the exercise of market power will be:
2
ef − e∗ 1 − e∗f = qe1 (r1 − e
a1 ) + qef (rf − e
af ) − q1∗ (r1 − a∗1 ) − qf∗ (rf − a∗f )
ee1 + e
Note that cf is the horizontal sum of convex functions ci , i ∈ {2, .., N }.
5
(12)
In Case A, where the dominant firm pollutes (r1 − a1 > 0) and the fringe does not (rf = 0),
then the change in emissions from product market l (i.e., the “local” pollution) will be:
qe1 (r1 − e
a1 ) − q1∗ (r1 − a∗1 ). Under reasonable assumptions of abatement cost functions, the
exercise of market power by a dirty firm will lead to less local pollution (see Appendix
A). Figure 1 of the paper presents the “market” for emissions. The marginal abatement
cost slopes downwards as these costs fall when firms can pollute more, while the marginal
damages increase in emissions. As previously noted the regulator’s choice of the permit cap
and taxes are set under competitive product markets such that the optimality condition (4)
is met. Comparing the first order conditions under competition (6) with those for a strategic
firm (8), one can see that—for a given tax or permit price—less local pollution occurs when
the dominant firm exercises market power. I denote the new marginal abatement cost with
a dotted line. This curve includes the feedback effect that permits have on the product
market: firms will alter production decisions when marginal costs change, including permit
prices. For the permit market to be in equilibrium, the reduced demand for permits will
result in permit prices falling.
Conversely, in Case B, the dominant firm does not pollute and the fringe does. Figure
2 of the paper presents this case in the market for emissions. Given qe1 < q1∗ and perfectly
inelastic demand, qef > qf∗ . Therefore, for a given tax or permit price, more local pollution
will occur with the dominant firm exercises market power. Increased demand for permits
results in a higher permit price when the dominant firm exercises market power.
3
Implications for Policy Choice
Unlike product markets, incentive-based instruments—such as environmental taxes and tradeable permit systems—are regulatory constructs unlikely to respond optimally to shocks in
firms’ willingness-to-pay to pollute (i.e., marginal abatement costs). As shown in (4) and
figures 1 and 2, the optimal level of emissions is where marginal damages equal marginal
abatement costs. In the extreme cases, regulatory mechanisms can mimic society’s demand
for clean air: if the marginal damages are perfectly inelastic, then a permit system will
respond optimally; alternatively, taxes respond optimally when the marginal damages are
perfectly elastic. However, in general, when firms begin to exercise market power and distort pollution levels, the new optimal emissions level (where marginal damages equal the
new marginal abatement costs) will not be achieved by the previously determined regulation. Therefore, deadweight loss in the pollution market will result.
In Case A, when dirty firms begin to exercise market power, the local pollution decreases
and the permit cap is met with less abatement from the M −N firms in other product markets
(see figure 1). Given that firms are now exercising market power, at the initial permit cap,
the marginal damages exceed the new marginal abatement costs, implying that firms emit
more than the new optimal level. This new optimum has less pollution and lower permit
prices in comparison to the initial one.3 Alternatively, had regulators established an initially
3
If a permit market’s cap is initially equal to or greater than the optimal emissions level, then the
6
optimal tax instead, strategic behavior would result in less pollution as firms abate to the
point were marginal abatement costs equal the tax. At that point, the tax exceeds marginal
damages, implying emissions are less than socially optimal. The converse arguments can be
made in Case B, when the fringe pollutes, as is shown in figure 2. However, before policy
makers lower taxes, reduce the permit cap, or change the type of incentive-based instrument,
they may improve welfare by considering how permit prices affect welfare in the economy as
a whole.
Given this set up, I ask whether there is a policy implication of environmental instrument
choice between prices and quantity regulation. Specifically, I examine the welfare implications of setting a tax or a tradeable permit system under the first best, as in (4), given that
environmental regulators either may not consider the welfare implications in the product
market, or simply do not know whether there will be more or less pollution after a dominant firm exercises market power in the product market. As with Weitzman (1974), the
uncertainty of the marginal abatement costs imply that taxes and tradeable permits are not
equivalent but rather depend on the elasticities of marginal abatement costs and marginal
damages. The social planner will prefer a tax to a quota system when the slope of marginal
damages is less, in absolute terms, than that of marginal abatement costs. In contrast, the
quota system will be preferred if marginal damages are steeper. In the special case of equal
absolute elasticities, policy makers should be indifferent between the instruments; both cause
the same pollution-related welfare losses given a shock to marginal abatement costs, like the
introduction of strategic behavior.
Assume that, under perfect competition, regulators are indifferent between a tax and
tradeable permits that are set such that the tax (τ 0 ) equals the expected permit price
(τ ) where marginal damages and marginal abatement costs equal. This implies that the
slope of the marginal damages equals the absolute value of the slope of the marginal cost
of abatement with respect to both qi and ai (see Appendix B). Thus, any shock to costs
will result in welfare loss in the market for abatement that will have the same expectation
regardless of instrument choice. This is shown in both figures 1 and 2 by the shaded areas;
the gray area is the welfare loss associated with a permit cap while the black area is the
welfare loss associated with a tax.
The additional welfare implication that I want to address is: how does instrument choice
affect welfare in the product market? Relative to a tax, permit prices respond to production
decisions. In Case A, If the dirty dominant firm exercises market power, then permit prices
will fall. This will reduce the dominant firm’s production costs relative to a tax, and the
optimal qe1 for the strategic firm is greater than it would have been under a tax. Figure 3
depicts the welfare implications in q1 space. I plot residual demand (a function of the fringe’s
production costs), the marginal revenue of that function, and four cost curves: marginal private costs (c1 (q1 ) + K1 (q1 , a1 )), marginal social costs (marginal private costs plus D(q1 , a1 )),
marginal private costs plus the tax, and marginal private costs plus the permit price (recognizing the opportunity cost of polluting is that price). Holding abatement technology fixed,
introduction of anticompetitive behavior results in even greater welfare losses in comparison to an initially
optimized cap.
7
as permit prices fall relative to the tax, the strategic firm produces more (see Appendix C).
In this short run case, the only manner of abatement is to reduce output so all welfare effects are shown. The welfare loss from a dominant firm facing a tradeable permit is the area
between the residual demand curve and the marginal social costs from the point where the
dominant firm opts to produce (e
q1 (e
τ )) to the optimal point where the lines intersect q1∗ (the
hashed area). A tax would result in addition welfare loss (the gray area), as the dominant
firm would produce even less (e
q1 (τ 0 )).
In Case B, if the clean dominant firm exercises market power, then permit prices will rise
as the dirty fringe produces more. Relative to a tax, the fringe’s production costs increase.
This implies that the dominant firm’s marginal revenue increases so the optimal q1 for the
strategic firm is greater than it would have been otherwise (see Appendix C). The welfare
implications are shown in figure 4. As in the previous case, the strategic firm produces
where marginal revenue equals marginal cost. As this figure has four residual demand curves
(corresponding to the four cost curves in figure 3), I have opted not to draw all marginal
revenue curves. As can be seen, the residual demand with the permit price lies above that
with the tax. Therefore, for a given fringe emissions rate, the marginal revenue for the
residual demand with permits is greater than that with taxes, implying more welfare loss
under the tax (the same area shading is used as in the previous figure).4
In both cases, the dominant firm produces more and, therefore, reduces welfare loss.
Note that even when the marginal damages and marginal abatement costs have different
absolute elasticities, this finding suggests that more consideration should be given to permit
markets. Another important consideration is that transaction costs imply improved welfare
from using a tax instead of a tradeable permits system (Stavins, 1995). All else equal, I find
that given the presence of strategic firms in product markets, welfare losses will be reduced
if environmental policy makers opt for tradeable permit systems in comparison to pollution
taxes.
4
In the general case, the distortions of the abatement technology costs will also be taken into account
f (τ 0 ) and W
f (e
when comparing W ∗ with W
τ ).
8
Appendices
Appendix A
If e
a1 is sufficiently less than a∗1 for the dominant firm, the firm may actually increase emissions
when reducing output. To determine whether this is feasible, define b
a1 such that:5
qe1 r1 − qe1b
a1 = q1∗ r1 − q1∗ a∗1 .
This abatement rate is the break even rate where the emissions are as large as under a
competitive regime, given qe1 . Now, I ask: can the dominant firm increase profits by increasing
a1 above the rate b
a1 ? Returning to the modified first order condition of the dominant firm
with respect to a1 from (8),Pand recalling that the dominant firm
in
PM is the∗ only polluter
M
∗
∗ ∗
∗
∗ ∗
∗ ∗
product market l (implying i=1 {qi ri − qi ai } = q1 r1 − q1 a1 + i=N+1 {qi ri − qi ai }.) and
P
∗
∗ ∗
0∗
a ∗ ∗
∗
that the social planner sets τ = τ ( M
i=1 {qi r i −qi ai }) = D = Ki (qi , ai )/qi , ∀i ∈ {1, ..., M } :
M
X
∂π 1
a
|q =eq ,a =ba = −K1 (e
q1 , b
a1 ) + τ (e
q1 r1 − qe1b
a1 +
{qi∗ ri − qi∗ a∗i })e
q1
∂a1 1 1 1 1
i=N+1
q1 , b
a1 ) + τ (q1∗ r1 − q1∗ a∗1 +
= −K1a (e
= −K1a (e
q1 , b
a1 ) + K1a (q1∗ , a∗1 )
qe1
.
q1∗
M
X
(A1)
{qi∗ ri − qi∗ a∗i })e
q1
i=N+1
To determine the sign, I use a first order Taylor series expansion of ∂π 1 /∂a1 around
noting that K1a (q1∗ , a∗1 ) is a constant:
(q1∗ , a∗1 )
∗
∂π1
a ∗ ∗
a ∗ ∗ q1
∗
∗
|q =q ,a =a ≈ (−K1 (q1 , a1 ) + K1 (q1 , a1 ) ∗ )
(A2)
∂a1 1 1 1 1
q1
K a (q ∗ , a∗ )
+(−K1qa (q1∗ , a∗1 ) + 1 1∗ 1 ) · (q1 − q1∗ ) − K1aa (q1∗ , a∗1 ) · (a1 − a∗1 ).
q1
a1 :
Substituting q1 = qe1 and a1 = b
K a (q ∗ , a∗ )
(e
q1 − q1∗ ) · (r1 − a∗1 )
∂π 1
≈ (−K1qa (q1∗ , a∗1 ) + 1 1∗ 1 ) · (e
q1 − q1∗ ) − K1aa (q1∗ , a∗1 ) · (
(A3)
)
∂a1
q1
qe1
r1 − a∗1
K a (q∗ , a∗ )
= (q1∗ − qe1 ) · [K1qa (q1∗ , a∗1 ) − 1 1∗ 1 + K1aa (q1∗ , a∗1 )
]
q1
qe1
Noting q1∗ > qe1 , K1aa > 0, and the assumptions (2) and that solutions are interior solutions,
1
| q1 ,a1 =ba1 > 0, so the dominant
implies that profits are increasing in the abatement rate, ∂π
∂a1 q1 =e
firm’s optimal e
a1 must be greater than b
a1 . Therefore, emissions from the dominant firm are
reduced when the firm produces less.
5
I thank Barry Nalebuff for suggesting this approach.
9
Appendix B
I assume the slope of the marginal damages equals the absolute value of the slope of the
marginal cost of abatement with respect to both qi and ai for all permit market firms. In
both Case A and B, each firm j ∈ {N + 1, .., M } will have equal slopes implying:
M
X
ω 0h (qj∗ ) − c00j (qj∗ ) − Kjqq (qj∗ , a∗j )
Kjaa (qj∗ , a∗j )
∗
ei ) = |
|=
.
D (
(rj − a∗j )2
(qj∗ )2
i=1
00
(B1)
In Case A, the dominant firm in product market l has equal slopes implying:
M
X
−c00f (q − q1∗ ) − c001 (q1∗ ) − K1qq (q1∗ , a∗1 )
K1aa (q1∗ , a∗1 )
∗
D (
ei ) = |
|
=
.
∗ 2
∗ 2
(r
−
a
)
(q
)
1
1
1
i=1
00
(B2)
In Case B, each of the fringe firms i ∈ {2, .., N } in product market l will have equal slopes
implying:
M
X
−c00 (q∗ ) − Kiqq (qi∗ , a∗i )
Kiaa (qi∗ , a∗i )
00
e∗i ) = | i i
|
=
.
(B3)
D (
∗ 2
∗ 2
(r
−
a
)
(q
)
i
i
i
i=1
From these firms’ perspective, the product price is constant. Hence, the assumption that P
is perfectly inelastic does not imply that tradeable permits or pollution taxes will necessarily
be the preferable regulation.
Appendix C
Strict monotone comparative statics may be used to determine the sign of the impacts of
the permit price on strategic firms’ production decisions (Edlin and Shannon, 1998). If
abatement technology is fixed, then (in Case A) taking the partial derivative of the first
order condition (10) with respect to the permit price implies:
∂(c0f (q − q1 ) − c00f (q − q1 ) · q1 − c01 − K1q − τ (r1 − a1 ))
∂ 2π1
=
= a1 − r1 < 0.
∂q1 ∂τ
∂τ
(C1)
Therefore, as permit prices fall, the strategic firm produces more.
In Case B, I take the partial derivative of the first order condition (11) with respect to
the permit price:
∂(c0f (q − q1 ) + Kfq (q − q1 ) + τ (rf − af ) − c00f q1 − Kfqq q1 − c01 )
∂ 2 π1
=
= rf − af > 0. (C2)
∂q1 ∂τ
∂τ
As permit prices rise, the strategic firm produces more. In both cases, the dominant firm
produces more.
10
References
[1] Edlin, Aaron S. and Chris Shannon. 1998. “Strict Monotonicity in Comparative Statics,”
Journal of Economic Theory, 81(1): 201-219.
[2] Hahn, Robert W. 1984. “Market Power and Transferable Property Rights.” Quarterly
Journal of Economics, 99(4): 753-765.
[3] Misiolek, W. S. and H. W. Elder. 1989. “Exclusionary Manipulation of Markets for
Pollution Rights,” Journal of Environmental Economics and Management, 16(2): 15666.
[4] Sartzetakis, Eftichios Sophocles. 1997. “Raising Rivals’ Costs Strategies via Emission
Permits Markets,” Review of Industrial Organization, 12(5-6): 751-765.
[5] Stavins, Robert N. 1995. “Transaction Costs and tradeable Permits,” Journal of Environmental Economics and Management, 29: 133-147.
[6] Weitzman, Martin L. 1974. “Prices vs. Quantities,” Review of Economic Studies, 41:
477-491.
11