Download Emission Permits Trading Across Imperfectly Competitive

Emission Permits Trading Across Imperfectly
Competitive Product Markets
Guy MEUNIER
CIRED-Larsen ceco
January 20, 2009
Abstract
The present paper analyses the efficiency of emission permits trading among several imperfectly competitive product markets. Even if
firms are price takers on permit markets the integration of permits
markets can decrease welfare because of imperfect competition on
product markets. With imperfect competition on product markets
the value of emissions should not be equalized between markets and if
an integrated permits market is implemented a corrective policy can
restore a second best optimum. The issue of permits market integration is then analyzed with uncertainty on markets conditions.
JEL Classification: D82, D43, L13,
Keywords: emissions permit market, demand uncertainty, market integration.
1
Introduction
With the implementation of the European Union emission trading scheme
(EUETS) and the Kyoto protocol emissions permits markets are used at
an unprecedented scale to regulate an externality. Main sectors concerned
by the EUETS are concentrated so they are often considered imperfectly
competitive. In particular, several geographically isolated and concentrated
electricity markets are concerned by emissions trading. The introduction of
an integrated permits market creates relationship between imperfectly competitive markets initially isolated. At the same time as numerous firms are
1
active on the emission permits market it can be assumed perfectly competitive. This paper deals with the efficiency of an integrated permits market
between two imperfectly competitive outputs markets. The two outputs considered are non-substitutable so markets are initially isolated. The aggregate
quantity of emissions is fixed and the issue is the allocation of this constraint
between outputs markets, that can be done either with two isolated emissions permits markets or an integrated one. Output markets considered can
be isolated in taste, space or time. Hence, the issue addressed concerns the
extent of a permits market along several dimensions.
The present work is related to the literature on environmental policy and
market power1 . Imperfect competition should be considered when designing
an environmental policy, as Buchanan (1969) stresses for an environmental
tax. An environmental tax should be lower than marginal damage when the
regulated market is a monopoly or a symmetric Cournot oligopoly (Barnett;
1980), because it encompasses a subsidy that corrects market power2 . Concerning the influence of market power on the efficiency of a permit market,
the literature can be crudely divided in three whether market power is introduced on the permits market, the good market or both simultaneously. This
paper belongs to the second strand.
Three previous articles have addressed the issue of imperfect competition in the goods market combined with perfect competition in the permit
market. Sartzetakis (1997, 2004) analyses the case of a duopoly exercising market power on an output market. He compares the efficiency of a
competitive emissions market to a command and control situation. In the
benchmark situation emissions of each firm are exogenously given. Emissions
trading modifies the allocation of emissions among firms and consequently
their production choices. In Sartzetakis (2004), the author shows that welfare can decrease when emissions trading is allowed between asymmetric firms
with different abatement and production technologies. The permit price that
clears the market is a weighted average of the value of emissions of firms under command and control, therefore, the inefficient firm cost is reduced while
the efficient one is increased when permits trade is introduced. The induced
reallocation of production on the output markets can offset the efficiency
1
See Requate (2005) for an extensive survey on environmental policy and imperfect
competition.
2
With asymmetric firms competing à la Cournot, the optimal tax can be higher than
environmental damage(Simpson; 1995) because of production allocation among firms effects.
2
gains from permits trading. Similarly the trading of permits between sectors
can worsen welfare by misallocating emissions. Hung and Sartzetakis (1998)
analyse the case of emissions trading between a monopolistic sector and a
competitive one. Market imperfection is transmitted from the monopoly to
the competitive sector via the emissions market: the monopoly consumes
fewer emissions than the optimum quantity and the competitive sector more.
In both situations, an integrated permits market can be strictly less efficient
than an optimal command and control.
I extend this literature by considering emissions permits trading between
two imperfectly competitive markets and explicitly analyzing a corrective
policy and the effect of asymmetric information on the attractiveness of markets integration. Actually, if an informed benevolent regulator can achieve
efficiency the introduction of market mechanisms can never outperform command and control. But, if asymmetric information is considered the regulator
may rely on markets to efficiently allocate emissions among agents or at least
to perform better than him. And, empirically, in the design of the EUAETS,
the allocation of emissions permits to producers of sectors concerned is done
on the basis of anticipated demand for outputs. These demands are unknown
by the regulator when setting the precise rules of the emissions markets. Furthermore, I address the issue of the effect of the misallocation on the choice
of emission cap.
I consider two imperfectly competitive markets and analyse how both
markets imperfections influence the allocation of emissions. It is shown that
the integration of permits markets can decrease welfare. I assume that the
regulator cannot directly correct output markets inefficiencies so a second
best allocation of emissions is introduced. The first best allocation is defined
in the context of perfectly competitive output markets while the second best
allocation is defined when output markets are imperfectly competitive. An
increase of emissions, besides decreasing costs of production, increases production. Because of market power, this increase of production is valuable.
Because of this effect, the second best optimal allocation of emissions cannot
be reached in general with an integrated market for emissions even if firms
are price takers on this market. The second best allocation can be sustained
with an integrated market for emissions if a subsidy or a tax on emissions of
one market is introduced. This subsidy is composed of the difference of price
cost margins weighted by the sensitivity of production on emissions. These
components reflects the value of an emission due to market power.
Next, I analyse the attractiveness of an integrated market under asym3
metric information or uncertainty. The regulator does not precisely know the
demand on output markets when deciding whether permits markets should
be integrated. The integration of markets has two contradictory effects: the
first one is the misallocation described above and the second one is the use
of information about markets conditions by firms. Without any additive
regulation the integration of markets is welfare enhancing if uncertainties
about the two markets are sufficiently negatively correlated, in that case the
second (positive) effect dominates. It stresses the value of markets as mean
to mobilize information, this positive effect of markets should be balanced
with imperfections when deciding whether or not to implement an integrated
market. Moreover, if accompanied by an ex-ante corrective subsidy an integrated market is always welfare enhancing because the subsidy can correct
the first (negative) effect. These results are closed to those obtained on decentralization in a context of asymmetric information and strategic behavior.
Concerning the design of the EUAETS, Malueg and Yates (2007) analyze the
decentralization process of the allocation of allowances between trading sectors and non trading one within each Member States. In their framework,
firms are not strategic but states are. Decentralization allows states to use
their private information about abatement costs but also to strategically allocates emissions rights, the trade-off between these two effects determines
whether centralization is preferred or not to decentralization.
The last issue addressed, that comes naturally to mind, is the effect of the
misallocation of the emission cap on the choice of this cap. More generally,
the issue is the way internal markets imperfections modify the choice of an
optimal macro variables. Intuitively, one may think that to relax the aggregate constraint would correct the misallocation and therefore, the optimal
cap should be higher with imperfections than without. The analysis reveals
that it is not the case, the relaxation of the emissions cap can either reinforce
or soften the misallocation so the sign of the change analyzed is no
I began to introduce the model (section) with general cost functions (section 2) and analyse the effect of market power within this framework (section
3). Then, I consider a more simple version with Leontief technologies in order
to analyze asymmetric information (section 4) and the choice of the emission
cap (section 5).
4
2
The model
2.1
Set up
I consider two polluting products markets indexed i = 1, 2, with inverse
demand function Pi (Qi ), i = 1, 2, where Qi is the aggregate quantity of good
i produced. Gross surplus from consumption of good i is Si (Qi ) with Si0 (Qi ) =
Pi . Inverse demand functions satisfy the following assumptions:
For each i = 1, 2 there is Qi > 0 such that:
• Pi is strictly decreasing and positive on 0, Qi
• Pi (Qi ) is null for Qi > Qi
• Pi is twice differentiable and satisfies: Pi0 + Qi Pi00 < 0 for Qi ∈ 0, Qi .
The last assumption, common in oligopoly literature (?) signifies that the
marginal revenue of a firm is decreasing with respect to the production of its
rival. It implies that quantities are strategic substitute and ensures existence
and uniqueness of Cournot equilibrium when firms have convex cost. On
market i = 1, 2, there are ni firms that produce the good. The production
of the good requires emissions, individual cost of production are Ci (qi , ei )
where qi and ei are quantities of output produced and emissions used by an
individual firm. The following assumptions are made on the cost function3 :
∀qi ≥ 0, ei ≥ 0:
• Cost are increasing and convex: ∂Ci /∂qi > 0, ∂ 2 Ci /∂qi2 > 0
• It is worth producing: Ci (0, ei ) = 0 and Pi (0) > ∂Ci /∂qi (0, ei )
• Cost and marginal cost are decreasing with respect to emissions: ∂Ci /∂ei <
0 and ∂ 2 Ci /∂qi ∂ei ≤ 0
• The effect of an increase of emissions is decreasing: ∂ 2 Ci /∂e2i ≥ 0
2
• And ∂ 2 Ci /∂qi2 .∂ 2 Ci /∂e2i − (∂ 2 Ci /∂qi ∂ei ) ≥ 0
3
I do not introduce upper bound on emissions to ensure that at equilibria considered
the emissions constraint is always binding.
5
Costs are increasing and convex with respect to output and decreasing
and convex with respect to emissions. Marginal costs are also decreasing with
respect to emissions. The last assumption ensures that the gross cost of a
firm is convex (cf appendix A) by limiting the effect of emissions on marginal
cost4 On each market i = 1, 2 an emission permits market is implemented
and the local price of emission is denoted σi , hence a firm on market i that
produces qi with ei emissions has a profit net of initial free allocations:
πi (qi , Qi , ei , σi ) = Pi (Qi )qi − Ci (qi , ei ) − σi ei
(1)
The aggregate quantity of emissions is e, the quantity of emissions on
market i = 1, 2 is Ei . Environmental damage is assumed separable and
depending only of the aggregate quantity of emissions. As this quantity is
fixed I do not explicitly introduce environmental damage. In sections 3 to 5
the issue addressed is the allocation of emissions among sectors and not the
choice of the aggregate constraint e that will be assumed fixed. In the last
section, I consider how the choice of this quantity is affected by local market
imperfections.
On each output market, all equilibria considered are symmetric so quantities of output and emissions are equally distributed among firms5 : individual
quantities are qi = Qi /ni and ei = Ei /ni . Welfare is the sum of surpluses net
of production costs on both polluting sectors:
X
W (Q1 , Q2 , E1 , E2 ) =
[S(Qi ) − ni Ci (Qi /ni , Ei /ni )]
(2)
i
Initial allocation to a market i=1,2 is denoted Êi , it is the quantity of emission
available to a market if there is no trade of permits across markets. I begin
to consider the first best optimum, i.e. the quantities and emissions on each
market chosen by a perfectly informed benevolent regulator able to enforce
4
These assumptions are satisfied for the two common specifications:
1. C(q, e) = c(e/q)q with c0 < 0, c00 > 0
2. C(q, e) = q + c(q − e) with c0 > 0, c00 > 0 for q > e.
The first specification represents the choice of a technology: a firm can lower its emission
rate (e/q) by increasing its marginal cost. And the second assumes separability between
production and abatement (q − e), so there is a technology to produce unitary abatement
or emissions permits, such as clean development mechanisms or carbon sequestration.
5
I do not consider that the regulator can discriminate among firms in a sector by
allocating different quantities of permits. Even if firms are symmetric this could increase
welfare as established by Amir and Nannerup (2005).
6
it in the next subsection. Then I introduce imperfect competition on output
markets and consider a second best optimum: the allocation of emissions
that maximizes welfare with market power exercice on output market.
2.2
Optimum and perfect competition
The objective of the regulator is to maximize welfare (2) subject to e ≥
E1 + E2 . First best quantities are denoted Q∗i , Ei∗ , i = 1, 2, they satisfy the
following first order conditions:
Si0 (Q∗i ) =
∂Ci ∗
(Qi /ni , Ei∗ /ni ) , i = 1, 2
∂qi
(3)
∂C1 ∗
∂C2 ∗
(Q1 /ni , E1∗ /n1 ) =
(Q2 /n2 , E2∗ /n2 )
(4)
∂e1
∂e2
If firms are price takers on both outputs and emissions markets, this optimum can be decentralized by an integrated emissions markets(Montgomery;
1972). In that case the permit price is σ = σ1 = σ2 , and on each markets
price taking behavior by firms ensures that (3) and (4) are satisfied. For
any initial distribution of emission permits among firms, emissions market
integration always improves welfare. If the regulator does not perfectly know
each market characteristics such as costs, demand and emission rates an integrated permit market is preferred to a tax or quotas to minimize the cost
to reach a given emission cap. Alternatively, if the regulator is informed he
can allocate emissions Êi = Ei∗ to each sector and not integrate emissions
markets, in that case both local permit prices would be equal.
3
Imperfect competition
Imperfect competition on output markets is introduced: firms strategically
choose output quantities but are price takers on permits markets. I first
describe the equilibrium with two isolated emissions permits market then I
consider a second best allocation of emissions among markets with imperfect
competition. It is then shown that an integrated market of emissions permits
allocates emissions differently and can therefore decrease welfare. The analysis is carried first in the general framework introduced, then some insights
are derived from quadratic specifications.
7
3.1
General framework
On output markets firms compete à la Cournot by strategically choosing
output quantities whereas they are price takers on the emissions permits
markets. Assumptions on price and cost functions ensure existence of a
unique symmetric Cournot equilibrium for any σi (cf appendix A) on each
market. At this equilibrium all firms produce the same quantity of output
and consume the same quantity of emissions. The permit price clears the
local market for emissions.
In order to consider the effect of a change of the quantity of emissions,
productions are written as function of emissions. Given individual emissions
ei for each firm on market i = 1, 2, each firm maximizes its profit ( 1)
when choosing its production. At equilibrium individual production on each
market is denoted qiC (ei ) for i = 1, 2. These quantities satisfy the following
first order conditions:
∂Ci C
(q , ei ), i = 1, 2,
(5)
Pi + Pi0 qiC (ei ) =
∂q i
C
and the aggregate production is QC
i (Ei ) = ni qi (Ei /ni ). For a local permit
price σi the equilibrium individual demand of emissions of a firm of market
i is eC
i (σi ) that satisfies:
∂Ci C C C
(q (e ), ei ), i = 1, 2,
(6)
∂ei i i
the aggregate demand of firms of market i is EiC = ni eC
i . As firms are price
takers on the permits market, the initial distribution of permits between
firms does not influence the market outcome. So, it is equivalent to consider
that the regulator gives an allocation Êi /ni to each firm or an aggregate
amount of Êi to all firms on market i which is allocated between firms by
the emission permits market. In the latter case, the price σi clears the local
emissions market so that Êi = EiC (σi ).
With two isolated permits market, a regulator that can only set the quantity of emissions Êi but not production should consider the effect of the former on the latter. Such an allocation of emissions differs from the first best
described in section 2.
σi = −
Definition 1 A second best allocation (E1∗∗ , E2∗∗ ) of permits given market
power exercise is an allocation that solves:
C
max W (QC
1 (E1 ), Q2 (E2 ), E1 , E2 ) subject to E1 + E2 ≤ e
E1 ,E2
8
C
I assume that W (QC
1 (E1 ), Q2 (E2 ), E1 , E2 ) is twice differentiable and concave with respect to E1 and E2 so that there is a unique second best allocation
(E1∗∗ , E2∗∗ ) with E2∗∗ = e − E1∗∗ . On each market i = 1, 2 an additive emission
increases net surplus of:
∂Ci ∂qiC
∂Ci
Pi −
−
∂qi ∂ei
∂ei
The second term is the direct increase of surplus related to the decrease
of production costs, the first term is a corrective term related to market
power. This indirect effect is composed of two factors: the price-marginal
cost difference and the sensitivity of production to emissions. Because of
market power, and demand elasticity, this corrective term is strictly positive.
At the second best optimum, marginal net surpluses are equalized among
markets so the optimal allocation of emissions given the exercise of market
power on output markets satisfies the following first order condition:
∂C1
∂C2 ∂q2C
∂C2
∂C1 ∂q1C
−
= P2 −
−
(7)
P1 −
∂q1 ∂e1
∂e1
∂q2 ∂e2
∂e2
∗∗
Quantities produced at the second best allocation are denoted Q∗∗
1 and Q2
they satisfy:
C
∗∗
Q∗∗
i = Qi (Ei ) for i = 1, 2
The two corrective terms explain that allocations of emissions with an integrated market for emissions permits can be different than the second best
optimal one. Let define the difference of corrective terms:
s∗∗ = −P10
C
Q∗∗
Q∗∗ ∂q2C
1 ∂q1
+ P20 2
n1 ∂e1
n2 ∂e2
(8)
With an integrated market for emissions permits local permits prices are
equalized: σ1 = σ2 and the equilibrium price σ C clears the permits market:
e = E1C (σ C ) + E2C (σ C ). At this equilibrium:
−
∂C2 C C C
∂C1 C C C
(q1 (e1 ), e1 ) = σ C = −
(q (e ), e2 ),
∂e1
∂e2 2 2
the marginal values of emissions for each firms are equalized across output
markets. Therefore, given equilibrium productions the allocation of emissions
is optimal but if one considers the effect of emissions on production it is not
9
(in general) because of corrective terms. Even if firms are price takers on the
emissions permits market, the integration of emissions permits markets does
not increase welfare in general.
Proposition 1 If s∗∗ 6= 0, welfare is strictly lower with an integrated market
for emissions than with two isolated markets with initial allocations Êi =
Ei∗∗ , i = 1, 2.
The benefit of integration of permits markets depends of initial allocations Êi , if they are (E1∗∗ , E2∗∗ ) the integration of permits market decreases
welfare. But if the initial allocation departs from this second best, the welfare effect of markets integration is ambiguous. Welfare might be increased
by markets integration if Ê1 is too low or too high. The issue of emissions
permits markets integration boils down to the analysis of the choice of initial allocations. These initial allocations can be suboptimal if the regulator
lacks some information about market conditions when setting these. Before
analyzing this situation, I first analyze how the regulator can correct the
integrated permits market in order to reach the second best optimum.
Instead of allocating emissions to each sector and keep markets isolated,
the regulator can use a price instrument to correct the integrated market and
implement the second best optimum.
Corollary 1 The second best optimum can be established with an integrated
market for emissions permits and a subsidy s∗∗ of market 1 emissions.
The proof is in appendix B. The subsidy s∗∗ reflects the value of emissions that is not considered by firms while choosing their emissions with an
integrated market. It can be either positive or negative, it is positive if market 1 is the underemitting market. This subsidy is an indirect way to correct
market power, less efficient than a direct subsidy of production. More generally: strategic firms underproduce given the quantities of inputs they use
and, if the regulator cannot directly correct market power by subsidizing production, he can do it indirectly by subsidizing inputs and ‘distorting’ inputs
markets. Here, emissions are the subsidized input and a local subsidy would
C
be −Pi0 Q∗∗
i /ni ∂qi /∂ei on market i = 1, 2. The regulator can either set such
a subsidy on each market or only a subsidy s∗∗ on market 1, because the only
relevant variable is the difference between the two subsidies6 .
6
So, there is an infinity of corrective policies that could be implemented. The only
relevant feature of the corrective policy is that prices of emissions faced by firms on market
1 and on market 2 are different, and the difference should be equal to s∗∗ .
10
The subsidized market is the underemitting one, and it depends of the
extend of market power and the relation between production and emissions.
The influence of several parameters is investigated in a quadratic setting
developed in the next section.
3.2
Quadratic specifications
To get a more precise picture of the influence of parameters, a quadratic
specification is used. In this section, I content myself with determining which
sector underemits with an integrated market for emissions permits and should
therefore be subsidized. In the next section the description of the quadratic
framework is deepened to understand the influence of uncertainty.
It is assumed that on each market i = 1, 2: the marginal production cost
(without emissions and abatement) is null and emissions rates are constant
and set at 1 and there is a separable abatement technology:
On each market i = 1, 2:
• Consumers surplus on market is:
Si (Qi ) = (ai − 0.5bi Qi )Qi
(9)
• And cost of production:
Ci (qi , ei ) =
0 if qi < ei
0.5ci (qi − ei )2 otherwise
(10)
If the marginal cost of abatement is infinite, there is no opportunity to
reduce emissions but to reduce production and quantities of outputs and
emissions are equal.
With two isolated markets, quantities produced can be expressed as functions of local emissions. On market i = 1, 2, if production is strictly positive7
it satisfies the first order condition : ai − bi (ni + 1)qi = max {ci (qi − ei ), 0}
so aggregate production is:
C
QC
i (Ei ) = ni qi =
n i ai
ai
ni ai + ci Ei
if
≥ Ei ≥ −ni
(ni + 1)bi + ci (ni + 1)bi
ci
7
(11)
It is possible that the quantity of emissions of a sector is negative if this sector abates
more emissions and its production.
11
Some conditions (cf appendix C should be satisfied by parameters to
ensure that the emissions constraint is binding and that both sectors produce
a positive quantity with an integrated market. In this case, with an integrated
permits market, the allocation of emissions is such that:
C
C
c1 (q1C − eC
1 ) = c2 (q2 − e2 ),
marginal costs of production are equalized. Because of the separability
of abatement, the difference of abatement cost does not influence the market outcome8 . Thanks to this separability it is feasible to disentangle the
influence of parameters.
The marginal effect of individual emissions on individual production is:
∂qiC
ci
=
∂ei
(ni + 1)bi + ci
So it is increasing with respect to abatement cost ci and with respect to
market size 1/bi and decreasing with respect to the number of firms ni .
To determine which sector should be subsidized it is sufficient to compare
0 C
Pi qi ∂qiC /∂ei at the ‘uncorrected equilibrium’. As mentioned, marginal cost
are equalized so Pi0 qiC is determined by the aggregate abatement cost and
not by ci . So, if output markets are identical: P1 = P2 and n1 = n2 , the
less efficient firms with respect to abatement should be subsidized. At the
uncorrected equilibrium firms have similar marginal costs so they produce
similar quantities of outputs. To subsidize input of a sector decreases its
marginal cost and consequently increases production. So the market that
should be subsidized is the sector of which marginal cost is more sensitive to
an additive emission and, with quadratic specifications, the effect of emissions
on individual marginal cost is ci . So, s∗∗ > 0 if and only if c1 > c2 .
Similarly, it is worth considering the influence of the number of firms, if
price and costs are equal but n1 < n2 there are two effects that go in the
same directions: the price marginal cost margin is higher on market 1, and
the production on market 1 is more sensitive to emissions. Because of these
two effects sector 1 should be subsidized9 . And finally, the effect of market
sizes 1/bi can be investigated when a1 = a2 , c1 = c2 and n1 = n2 . At the
−1
8
The aggregate abatement cost is 0.5c(Q1 + Q2 − e)2 with c = (n1 /c1 + n2 /c2 ) . The
market outcome depends upon c and not the precise values c1 and c2 .
9
It is not a general result because with general price and cost functions the monotonicity
of the factor ∂qiC /∂ei with respect to ni is unclear.
12
uncorrected equilibrium, individual marginal costs are equal so −Pi0 qiC = bi qiC
are equal, and the biggest market should be subsidized because its production
increases more subsequently to an extra emission.
4
Imperfect information
As stated by proposition 1, the integration of emissions permits markets
can decrease welfare because of market imperfection on output markets. A
perfectly informed regulator is able to perform better than an integrated market by initially allocating the constraint among markets and keeping markets
isolated. But, if the regulator lacks information about market conditions he
might be unable to set optimal allocations and an integrated market can
improve efficiency despite imperfect competition.
4.1
Uncertainty
Uncertainty is added to the quadratic framework introduced in the previous
section. The approach is similar to the one developed by Weitzman (1974)
to compare price and quantity regulatory instruments. Costs are given by
(10), and gross consumers surplus on each market is random at the time
of allocation of permits: Si (Qi , θi ), where θi is a random parameter with
Eθi = 0, uncertainty is assumed additive:
Si (Qi , θi ) = (ai + θi )Qi −
bi 2
Q
2 i
And welfare in a particular state (θ1 , θ2 ) is :
X
W (Q1 , Q2 , E1 , E2 , θ1 , θ2 ) =
Si (Qi , θi ) − ni Ci (Qi /ni , Ei /ni )
(12)
(13)
i=1,2
Parameters θi can be interpreted in several ways. First, with additive
uncertainty and quadratic specifications they can encompass uncertainties
about marginal consumers surpluses and marginal costs. Second, the relevant feature of the model is that: when firms decide how much to produce
they have more information than the regulator when he designs the policy.
So, parameters θi represent either asymmetric information between the regulator and firms, or, uncertainty about future markets conditions due to the
13
time lag between the design of the emissions permits markets and markets interactions. Both are relevant and simultaneously at stake for the EU ETS: it
is reasonable to assume that firms have superior knowledge of market conditions than the regulator, and future trends of sectors concerned are uncertain
when the regulator designs emissions markets.
4.2
Production
In all regulatory options considered the regulator cannot control production
but only emissions markets, so it is worth establishing reduce form of welfare
as function of emissions. For i = 1, 2 local production depends upon local
emissions and market conditions. If ai + θi is sufficiently high10 , production
is similar to (11) so:
QC
i (Ei , θi ) =
ni (ai + θi ) + ci Ei
(ni + 1)bi + ci
(14)
Given a fixed quantity of emissions, production is adapted to local market
condition θi . Thanks to the linearity of our framework adaptation to θi and
to emissions are not intertwined; the sensitivity of production to emissions
is independent of random market condition and denoted αi :
ci
∂QC
i
=
.
αi =
∂Ei
(ni + 1)bi + ci
(15)
Local net surplus can be expressed in a reduced form as a function of emissions:
Wi (Ei , θi ) = (Ai (ai + θi ) − 0.5Bi Ei ) Ei + Ki ,
(16)
where coefficients are:
(ni + 2)bi + ci
,
(ni + 1)bi + ci
(ni + 1)2 bi + ni ci
= bi αi
,
ni ((ni + 1)bi + ci )
C
= ai − 0.5(bi + ci )QC
i (0, θi ) Qi (0, θi ).
Ai = αi
Bi
Ki
10
If the quantity of emissions is negative and lower than ni (ai + θi )/ci ) ≤ −Ei it is
not worth producing. And if the quantity of emissions is higher than the unregulated
production ni (ai + θi )/(ni + 1), firms produce this quantity.
14
And summing over both markets, a reduce form of aggregate welfare is (with
a slight abuse of notation):
W (E1 , θ1 , θ2 ) = W1 (E1 , θ1 ) + W2 (e − E1 , θ2 )
= [A1 (a1 + θ1 ) − A2 (a2 + θ2 ) + B2 e − 0.5(B1 + B2 )E1 ] E1
+ [K1 + K2 + (A2 (a2 + θ2 ) − 0.5B2 e) e]
Within the reduce form of welfare, uncertainties are still additive but weighted
by coefficients Ai . These coefficients together with Bi , encompass the relationship between emissions and productions.
Furthermore, on each market i = 1, 2, in any demand states θi , the deC
mand for emissions permits is EiC (σi , θi ) that satisfies: ci (QC
i − Ei ) = σi
so:
1
EiC = (αi ai − σi )
(17)
βi
where:
(ni + 1)bi
βi =
αi , i = 1, 2
(18)
ni
Some assumptions are required on parameters to ensure that in all configuration considered and in all states, the emission constraint is binding and
both goods are produced. Those conditions (listed in appendix C) consist
mainly in that ai , i = 1, 2 are sufficiently high and not too different, and the
support of θi , i = 1, 2 should be restricted.
4.3
Regulatory options
The regulator does not know the value of random parameters θi when deciding to allocate emissions among firms. He decides whether or not to implement an integrated market for emissions permits, if markets are integrated,
the initial allocation of permits does not influence the market outcome. I
analyze both cases whether the regulator set an ex-ante subsidy to partly
compensate market power or not.
So the three regulatory options are:
1. The regulator allocates emissions Êi to firms of market i = 1, 2 and
emissions permit markets are isolated so productions are QC
i (Êi ) and
expected welfare is:
Ŵ = Eθ1 θ2 W (Ê1 , θ1 , θ2 )
15
2. Emissions permits market are integrated and a subsidy s on market 1
emissions is fixed ex ante. Firms choose production and emission once
θ1 , θ2 are revealed, the permit price σ(θ1 , θ2 , s) clears the market for
emissions so that:
E1C (θ1 , σ − s) + E2C (θ2 , σ) = e
(19)
An expected welfare is denoted WI (s):
WI (s) = Eθ1 θ2 W (E1C , θ1 , θ2 )
3. Emissions permits market are integrated and no corrective policy is
implemented. So expected welfare is WI (0).
4.4
Comparison
In the benchmark situation the regulator allocates emissions Êi to firms of
market i = 1, 2. Firms may trade permits within each markets but not across
markets. The regulator maximizes expected welfare: EW (E1 , θ1 , θ2 ). Hence,
he fixes the allocation:
Ai ai − Aj aj + Bj e
Êi =
, i, j = 1, 2, j 6= i
(20)
Bi + Bj
With this allocation of emissions expected welfare is Ŵ . Emissions are not
influenced by random parameters but productions are. With an integrated
market for emissions permits, the allocation is conditional on random markets
conditions. There are potential gains from market integration that come from
the adaptation of emissions to these conditions. If an integrated market is
introduced, the initial allocation of permits does not influence the outcome
because firms are competitive on the permits market. In order to compare
both situations with and without a corrective subsidy on the permit market,
I determine a general expression for any subsidy s on market 1 emissions.
If a subsidy s is set on emissions on market 1, firms from market 1 face
the permit price σ − s and firms from market 2 the permit price σ. At
equilibrium, the permit price clears the permits market and equation (19) is
satisfied. So the equilibrium quantity of emissions is:
1
[α1 (a1 + θ1 ) + s − α2 (a2 + θ2 ) + β2 e]
E1C (θ1 , θ2 , s) =
β1 + β2
1
= E1 (s) +
(α1 θ1 − α2 θ2 )
(21)
β1 + β2
16
This expression consists of a fixed term and a random one. The latter
represents the adaptation of emissions allocations to revealed market conditions θi , i = 1, 2. The nice feature of the quadratic framework is that this
effects can be isolated: the subsidy only modifies the expected allocation of
emissions and not its random part. Hence, the variation of the allocation can
be isolated and injecting expressions into expected welfare allow to isolate
effects of both terms on welfare:
WI (s) = E W E1C , θ1 , θ2
(22)
= E W (E1 , θ1 , θ2 ) + I
2
= Ŵ − 0.5 (B1 + B2 ) Ê1 − E1 + I
Where I does not depend upon s and contains the effects of adaptation of
emissions to market conditions:
B1 + B2
α1 θ1 − α2 θ2
B1 + B2
α1 θ1 − A2 −
α2 θ2
(23)
I = E A1 −
β1 + β2
β1 + β2
β1 + β2
The issue of market integration boils down to the comparison of the ‘uncertainty’ effect and the misallocation related to market power. With a subsidy
the latter can be canceled.
Proposition 2 With asymmetric information
i With a corrective subsidy expected welfare is greater with market integration if and only if I ≥ 0,
ii without corrective subsidy, expected welfare is greater with market integration if and only if:
2
(24)
(B1 + B2 ) Ê1 − E1 ≤ 2I
This proposition stresses the two effects at stake when evaluating market
integration. On one hand there is a welfare loss related to market imperfection, but, on the other hand there is an eventual welfare gain related to
uncertainty. In a context of asymmetric information, the potential gain from
market integration is related to the mobilization of private information of
firms when allocating the emission cap. With an integrated the allocation is
contingent on market conditions, and this contingency can enhance welfare.
So, the main question is the sign of the uncertainty effect I. Without further
assumptions on parameters values and distributions, the sign of this term
cannot be set.
17
Without abatement
A special case worth considering is the case without separable abatement:
c1 = c2 = +∞. The quantity produced and the quantity of local emissions
are identical and at equilibrium:
n2 + 1
1
C
C
(a1 − a2 ) +
b2 e + s + (θ1 − θ2 )
Q1 = E1 =
(25)
β1 + β2
n2
θ1 − θ2
= E1C +
β1 + β2
where βi = (ni + i) bi /ni can be interpreted as a measure of aggregate
market power. The random term represents the adaptation of productions
to revealed market conditions θi , i = 1, 2.
Corollary 2 With asymmetric information and no separable abatement,
(i) Without any subsidy an integrated market improves expected welfare if
and only if:
var(θ1 − θ2 ) >
[b1 (a1 − a2 + b2 e) /n1 − b2 (a2 − a1 + b1 e) /n2 ]2
[(1 + 2n1 ) b1 + (1 + 2n2 ) b2 ] (b1 + b2 )
(ii) With an optimal ex ante subsidy, markets integration always increases
expected welfare.
∗
The optimal subsidy is such that QC
1 (s ) = Q̂1 and the welfare gain is :
var(θ1 − θ2 )
1
B − (b1 + b2 )
B2
2
Calculations are in appendix D. The choice to integrate market is based
on the comparison of the two effects mentioned. Without separable abatement, the uncertainty term is always positive and proportional to the variance
of the difference of marginal consumers surplus. Once the emission cap is
fixed, the determining parameter for the allocation of this cap is not absolute
marginal consumers surplus but relative one. The gain from market integration is related to the adaptation of productions to markets conditions in the
right direction. Both optimal allocation and the integrated market one are
18
proportional to the difference θ1 − θ2 . It explains that welfare gain is increasing with respect to the variance of relative marginal surplus. This variance is
var(θ1 )+var(θ2 )−2cov(θ1 , θ2 ), it is decreasing with respect to the correlation
of random parameters, so most gains from market integration are obtained
if markets are negatively correlated. This positive effect emphasizes the role
of markets to aggregate information and coordinate decentralized decisions.
The negative term due to market power is related to concentration on each
market. Some comparative static can be done. First, it should be noticed
that with constant production cost an emission rates, first best and second
best allocations in any state (θ1 , θ2 ) coincide. Compared to the optimal
allocation, firms of a market produce too much and firms of the other not
enough. Market power rather than decreasing production misallocates the
constraint. Second, it appears from the expression of production (25) that
if firms of market 1 underproduce on average they always do. Therefore, an
increase of the number of firms on the underemiting market increases welfare
in all demand states and subsequently the appeal of market integration. An
increase of the number of firms in the overemitting market has the opposite
effect: it worsen the misallocation in all demand states and subsequently the
attractiveness of market integration. If an ex ante subsidy is set to correct
the effect of market power, an increase of the number of firms in any market
increases welfare.
With no opportunity of external abatement, the uncertainty effect is positive whatever the distribution of demand states because the optimal production on market 1 and the production with an integrated permits market
are both linearly related to the difference θ1 − θ2 . When markets conditions vary, the integrated market quantity eC
1 goes in the right direction: it
increases when θ1 − θ2 does.
With abatement
With separable abatement, the comparison of regulatory options is described
by expressions (22) and (23). First, Ai > αi and Bi > βi so Ai − (B1 +
B2 )αi /(β1 + β2 ) > 0. If uncertainties are independently distributed, the
uncertainty effect is always positive because if consumers surplus on one
market increases, the quantity of emissions permits consume by this market
increases. In that case, introduction of abatement does not qualitatively
modify the comparison because the information processing rule of market is
still positive and can justify the integration of permits market.
19
The situation is more interesting when correlation are considered. If
market conditions simultaneously varies, i.e. are correlated, and the market
moves in the opposite direction direction there is a loss. It can be the case
if the optimal contingent allocation and the integrated market one are not
proportionally related i.e.
A2 B1 + B2
A1 B1 + B2
−
6=
−
α1
β1 + β2
α2
β1 + β2
, simplications of this expression, it is equivalent to:
(n2 − n1 )b1 b2 6= b2 c1 − b1 c2
Corollary 3 If θ1 = xθ2 with x ∈ R:
If (n2 − n1 )b1 b2 > b2 c1 − b1 c2 , there is u > 1 such that:
I<0⇔
α2
α2
<x<u
α1
α1
When abatement is introduced the positive attribute of integrated market
vanishes for some distributions of random variables. A particular case is
exhibited in corollary (3). Abatement creates a further difference between
firms actions and public interest. With abatement firms have two ways to
reduce their emissions and the arbitrage they do
5
The choice of the emission cap
In previous sections the emission cap was assumed fixed and the issue addressed was how market power on output markets misallocates a fixed emission cap. A corrective policy was determined but it is not likely that such a
policy will be put in place in the EUETS in order to limit possible regulatory capture or strategic interventionism by European states. Therefore, a
natural question that arises is the effect of such misallocation on the optimal emission cap. Even if the political process that fixes emissions cap can
hardly be seen as a benevolent regulator maximizing welfare, such a question
is relevant to understand how internal imperfections influence aggregate environmental policies. The issue can be generalized: it is to understand how
internal imperfections influence the choice of an aggregate constraint. And
20
far from providing any general answer to this question, I establish here with
a relatively simple example that it can be in either ways. Internal misallocations of a constraint can imply that the optimal, second best, aggregate
quantity is either higher or lower than the first best one (without internal
imperfections).
In the general case (i.e. with general cost function) three situations should
be distinguished: with and without market power and in the former case
with and without the corrective policy. As I consider here the linear framework without abatement, the competitive case (first best) coincides with the
case of corrected market power (second best). As environmental damage is
assumed separable and convex, it is sufficient to analyze the derivative of
welfare (2) with respect to e with and without the exercise of market power.
In the case of perfect competition, the derivative of welfare (2) is Si0 (Q∗i ),
i=1,2 which is the value of the emission constraint and the price of emissions
permits. With Cournot competition on output markets the effect of the cap
on quantities produced
should be considered and the derivative of welfare is
0
C
0
C
C
S1 (Q1 ) − S2 (Q2 ) ∂Q1 /∂e + S20 . The effect of misallocation on the optimal
emissions cap is determined by the sign of the difference of both derivatives:
∂QC
1
C
∗
4 = P1 (QC
)
−
P
(Q
)
+ P2 (QC
2
1
2
2 ) − P2 (Q2 )
∂e
(26)
From this equation it appears that the answer is not obvious for two
contradictory effects are at stake. Let’s assume that market 1 is the underemitting market, in that case the first term is positive: an increase of the
emission cap has a positive effect by increasing the emissions of market 1,
but the second term is negative because market 2 is overemitting. Hence,
an increase of the emission cap partly corrects the misallocation but has a
lower direct effect on welfare. Whether one effect dominates the other is not
evident from this equation. It is unclear whether the misallocation implies a
more or less stringent environmental policy.
∗
With the linear
P2 (QC
2 ) − P2 (Q2 )
case: Pi = ai − bi Qi , the∗ difference
C
∗
C
is −b2 . Q2 − Q2 and this is equal to −b2 Q1 − Q1 . Similarly
the differ
ence P1 − P2 can be simply expressed as (b1 + b2 ) Q∗1 − QC
.
And
finally it
1
appears that the only relevant quantity is the sensitivity of productions to
the emissions cap and this sensitivity is determined by the relative extent of
market power on both markets.
Proposition 3 The optimal emission cap is higher with market power than
21
without if and only if:
(n1 − n2 ) Q∗1 − QC
1 ≥ 0
(27)
∂QC
Calculations are straightforward to establish that (b1 + b2 ) ∂e1 − b2 is
proportional to n1 − n2 . The difference of markets concentrations is sufficient
to compare both effects mentioned above. If market 1 is under emitting and
n1 ≥ n2 the corrective influence of the emission cap dominates and the
optimal emission cap is higher with market power than without, i.e. the
environmental policy is less stringent. But if the concentration on market 1
is higher than on market 2, the misallocation due to market power implies a
more severe environmental policy. Hence, with two similar markets (P1 (Q) =
P2 (Q)) the optimal emission cap is always increased by market imperfection.
6
Conclusion
In this paper I analyzed how imperfect competition on two output markets
influences the efficiency of an integrated market of emissions permits. If output markets are imperfectly competitive the integration of permits markets
can decrease welfare. Market power on output markets explains that the
allocation of the emissions constraint is suboptimal with an integrated permits market. This misallocation can be corrected by a subsidy on emissions
of one sector that encompasses the effect of emissions on production and
market power distorsion.
A perfectly informed regulator can do strictly better with two isolated
permits markets than an uncorrected integrated market. However, it is more
realistic to consider that there is asymmetric information between firms and
the regulator about market conditions. By introducing, in a quadratic specification of the model, additive uncertainty on demand conditions the issue
of permits market integration under asymmetric information has been analyzed. Additionaly to the misallocation previously stressed, an uncertainty
effect is introduced. An integrated markets for emissions has the additional
feature of adapting the allocation of the emissions cap to revealed market
conditions. The decision whether to integrate markets for emissions should
be based on the comparisons of the two effects. If there is no possibility
of abatement, the uncertainty effect is always positive and decreasing with
markets corelations. An integrated markets have
22
Many markets concerned by the regulation of emissions are concentrated
such as electricity markets, and more firms are present on the emissions
market than on output markets, hence, it is justified to analysis how local
imperfections interacted via emissions markets. The scope of an emission
market can therefore be limited in order to avoid imperfection ‘contagion’.
References
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23
Sartzetakis, E. (2004). On the Efficiency of Competitive Markets for Emission
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A
Existence of uniqueness of Cournot equilibrium
Let i ∈ {1, 2}, I establish that for any permit price σi there exists a unique
equilibrium which is symmetric on market i. To do so I consider the gross
cost function Γ (qi , σi ) = min{Ci (qi , ei ) − σi ei }.
ei
This function is convex with respect to qi by derivation:
∂Ci
∂ 2Γ
∂ 2 Ci
∂Γ
=
(qi , ei ) and
=
−
∂qi
∂qi
∂qi2
∂qi2
∂ 2 Ci
∂qi ∂ei
2 ∂ 2 Ci
∂e2
−1
≥0
Thanks to the assumption on the price function Pi0 + Pi00 Qi , ∀Qi there is
a unique Cournot equilibrium for all σi (Novshek; 1985).
Equilibrium quantities satisfy the two equations:
∂Ci
∂qi
∂Ci
=
∂ei
Pi + Pi0 qi =
σi
C
Hence, for any permit price the strategy qiC (eC
i ), ei (ni , σi ) is the unique
(symmetric) equilibrium of the game considered.
B
Proof of corollary 1
With a subsidy s on emissions of market 1 and an integrated market for
emissions with a price σ, the price of emissions faced by firms on market
24
1 is σ − s and by firm on market 2 σ, therefore, at equilibrium respective
C
emissions per firm are eC
1 (n1 , σ − s) and e2 (n2 , σ)and therefore:
∂C2
∂C1
=
+s
∂e1
∂e2
(28)
For a subsidy s∗ , emissions satisfy the first order condition of the second
best equation (7), so, thanks to the assumption of welfare concavity, it is the
second best allocation.
C
Conditions with quadratic specifications
For the emissions constraint to be binding the following condition should be
satisfied:
n 1 a1
n 2 a2
+
≥e
(C1)
(n1 + 1)b1 (n2 + 1)b2
And, to ensure that both sectors produce a positive quantity with an
integrated market, the equilibrium permit price should be lower than ai
ni
n1 n2
(ai − aj ) ≤ e + aj
+
(C2)
(ni + 1)bi
c1
c2
D
Proof of proposition 2
With quadratic expressions welfare could be written (22) as in the main
text. When an integrated market is implemented equilibrium production on
C
market 1 is given by expression (refeq:prodcournot): QC
1 = Q1 +(θ1 − θ2 ) /B
and the expected production is:
n2 + 1
1
(a1 − a2 ) +
b2 e + s
(29)
B(n1 , n2 )
n2
Quadratic expressions and additive uncertainty on marginal surpluses
allow to separate effects of expected production and random term as in expression (refeq:welfarecournotquad). The effect of expected production is:
EW
C
QC
1 , Q2 , θ1 , θ2
2
b1 + b2 C
∗
= Ŵ −
Q1 − Q1
2
25
And the difference between Cournot production and optimal one is:
1
n2 + 1
1
∗
C
Q1 − Q1 =
(a1 − a2 ) +
b2 e −
[(a1 − a2 ) + b2 e]
B
n2
b1 + b2
1
n2 + 1
(a1 − a2 ) (b1 + b2 − B) + b2 e
(b1 + b2 ) − B
=
B. (b1 + b2 )
n2
1
b1
b2
1
1
=
(a2 − a1 )
+
+ b1 b2 e
−
B. (b1 + b2 )
n1 n2
n2 n1
Therefore, the difference between command and control and an integrated
permits market without subsidy is:
4(0) =
var(θ1 − θ2 )
1
(B − (b1 + b2 ))
2
B
2
b2
1
1
b1
1
+
−
− 2
(a2 − a1 )
+ b1 b2 e
2B (b1 + b2 )
n1 n2
n2 n1
And an integrated market increases welfare if and only if:
var(θ1 −θ2 ) ≥
1
[(a2 − a1 ) (n2 b1 + n1 b2 ) + b1 b2 e (n1 − n2 )]
((2 + n1 )b1 + (2 + n2 )b2 ) (b1 + b2 )
.
D.1
Uncertainty and separable abatement
To the above framework I add here the possibility for firms to abate emissions.
I only consider two local monopolies: n1 = n2 = 1. The aim is to show how
the introduction of abatement can modify results of section 4, and more
particularly the sign of the uncertainty effect. Consumers surpluses are still
quadratic in production. But the cost of production of firms is:
0 if qi < ei
(30)
Ci (qi , ei ) =
0.5ci (qi − ei )2 otherwise
For i = 1, 2, for any quantity ei , production on market i is:
Qi =
ai + θi + ci ei
2bi + ci
26
From this expression, it appears that even if emissions are fixed firms adapt
production to market conditions. It is possible to determine a reduce form
of net surplus on market i with respect to ei :
Wi (ei , θi ) = Si (Qi (ei , θi ), θi ) − Ci (Qi (ei , θi ), ei )
= (Ai (ai + θi ) − 0.5 Bi ei ) ei + Ki
(31)
(32)
where coefficients are:
3bi + ci
ci
(2bi + ci )2
bi c i
(4bi + ci )
=
(2bi + ci )2
ai + θi
bi + ci ai + θi
=
ai + θi −
2 2bi + ci 2bi + ci
Ai =
Bi
Ki
From these expressions of local gross surplus, welfare can be written as a
quadratic function of emissions e1 :
W (e1 , θ1 , θ2 ) = W1 (e1 , θ1 ) + W2 (e − e1 , θ2 )
= [(A1 (a1 + θ1 ) − A2 (a2 + θ2 ) + B2 e) − 0.5(B1 + B2 )e1 ] e1 + K
where K = K1 +K2 +(A2 (a2 + θ2 ) − 0.5B2 e) e. So, with separable abatement
welfare in anay state can still be written as a quadratic function of e1 with
an additive uncertainty. So the optimal allocation of emissions with isolated
market is:
eˆ1 = [A1 a1 − A2 a2 + B2 e] [B1 + B2 ]−1
The following step is to determine the quantity of emissions emitted by market 1 when a subsidy s is set with an integrated market for emissions. Equilibrium quantity satisfies:
c1 (Q1 − e1 ) = c2 (Q2 − (e − e1 )) + s
Injecting the expression of production into this equality gives the following
result:
−1
eC
1 (θ1 , s) = [α1 (a1 + θ1 ) − α2 (a2 − 2b2 e + θ2 )] β
1
= eC
(α1 θ1 − α2 θ2 )
1 (s) +
β
27
where coefficients are:
αi =
ci
, i = 1, 2
2bi + ci
2b2 c2
2b1 c1
+
2b1 + c1 2b2 + c2
The uncertainty effect can be isolated and expected welfare with a subsidy s can written as a sum of expected welfare with isolated markets, a
negative term related to the ‘misallocation’ phenomenon and a term related
to adaptation of the allocation to market conditions denoted I:
β=
2
B1 + B2 C
WI (s) = Ŵ −
eˆ1 − e1 + I
(33)
2
This expression is very similar to the expression (refeq:deltasurplus) in
the main text. The issue here is the sign of term I and whether it can be
negative. With the expression of coefficients Ai and eC
1 , the sign of I is the
sign if the expectation:
B1 + B2
B1 + B2
E A1 −
α1 θ1 − A2 −
α2 θ2 [α1 θ1 + α2 θ2 ]
2β
2β
And, for i = 1, 2, Ai > αi and B1 + B2 < 2β so Ai − (B1 + B2 )αi /(2β) > 0.
So, if variables are independently distributed I > 0.
Furthermore, Ai = (3bi +ci )/(2bi +ci )αi , so, if b1 c2 =b2 c1 then (3b1 + c1 ) / (2b1 + c1 ) =
(3b2 + c2 ) / (2b2 + c2 ), and I > 0. In both these cases, the situation is not
much different of the situation without abatement. The interesting case is
when random variables are correlated and
b1 c2 6= b2 c1
In that case it is feasible to exhibit situations where I < 0:
Lemma 1 If b1 c2 < b2 c1 , for all x such that:
α2
α2 2βA2 /α2 − (B1 + B2 )
<x<
α1
α1 2βA1 /α1 − (B1 + B2 )
The term I is strictly negative and the integration of permits market always
decreases welfare
28
The proof of this lemma is straightforward calculation:
Let denote:
2βA2 /α2 − (B1 + B2 )
u=
2βA1 /α1 − (B1 + B2 )
If b1 c2 < b2 c1 then u > 1 because A1 /α1 = 1 + 1/(2 + c1 /b1 ) < 1 + 1/(2 +
c2 /b2 ) = A2 /α2 .
The sign of I is the sign of
E [(α1 θ1 − uα2 θ2 )(α1 θ1 − α2 θ2 )]
And injecting in this expression θ1 = xθ2 :
signI = sign(x − uα2 /α1)(x − α2 /α1 )
Which is negative if and only if uα2 /α1 < x < α2 /α1.
29