Differentiating permits allocation across areas∗
Jean-Philippe Nicolaı̈† Jorge Zamorano‡ §
This version: January 31, 2017
Abstract. This paper addresses the issue to differentiate permits allocation
across areas, which is linked to the risk of relocation. Two different areas
are considered, a domestic one in which the emissions are regulated and a
second one in which no environmental regulation is applied. The domestic
area is subdivided into the coastal area, located near to the border and the
inland area, far away from the border. The paper shows that the risk of
relocation is different across the two areas. Free allowances may be used
to reduce these risks and a possibility to minimize the cost of preventing
firms to relocate could be to differentiate permits across areas. However,
differentiating subsidies in a same sector is a politically sensitive initiative so
that the competition authorities may be reluctant to implement such policy.
Nevertheless, the paper shows that the inland firms would also benefit from
the subsidies given to the coastal firms.
Keywords. Relocation; Imperfect competition; Unilateral regulation.
JEL codes. L13, Q53, Q58.
1. Introduction
The negotiations during the COP21, which led to the conclusion of the Paris Agreement
– the most ambitious legal instrument adopted so far to fight global warming – also
highlighted the difficulties for countries to cooperate in reducing emissions. While some
States have already decided to implement (or intend to) pollution permit programs,
several major economies, among them the United-States, are still reluctant to adopt such
policies at the federal level. These unilateral policies may indeed increase the production
costs and push firms to relocate their activities in less environmentally-friendly countries,
which in turn would generate an increase in total emissions and also a deindustrialization.
∗
We thank Carmen Arguedas, Jean-Pierre Ponssard, François Salanié, participants at EAERE 2016,
FAERE 2016 and FSR 2016 conferences for their comments and useful references. Financial support
from Swiss Re Foundation is gratefully acknowledged.
†
ETH Zürich, Address: Chair of Integrative Risk Management and Economics, Zürichbergstrasse 18
8032 Zürich, Switzerland. E-mail: [email protected].
‡
Université Paris 1 Panthéon Sorbonne, Centre d’Économie de la Sorbonne, Address: 106-112 boulevard de l’Hôpital, 75647 Paris Cedex 13, France. E-mail: [email protected].
§
Universidad de Santiago de Chile, Departamento de Ingenierı́a Industrial, Address: Av. Ecuador
3769 Santiago, Chile. E-mail: [email protected].
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2
J.-P. Nicolaı̈ & J. Zamorano
It therefore transpires that a crucial question relating to the design of pollution permits
is: how to allocate permits to firms without causing massive relocation. This paper
proposes to take these relocation risks seriously and suggests a new way to allocate
permits to minimize the cost of limiting this risk.
Several authors have already shown that the relocation risk due to environmental
regulation is currently low. Branger et al (2015)[2] demonstrate that relocation due to
environmental motivations is very limited while Martin et al (2014)[6] found that free
allowances are sufficient to prevent firms from relocating. However, in order to meet the
European commitments of greenhouse gas (GHG) emissions reductions in the future, the
pollution caps need to be more stringent, which will necessarily increase permit prices.
Like this the relocation risk appears as a critical issue in the aftermath of COP21.
Pollution permits cover several sectors, which are ordinarily oligopolistic such as electricity, steel and cement. These two latter are also characterized by an exposition to international competition and high transportation costs. To be more specific, transportation
by land is very costly compared to maritime transportation in the cement sector. For
instance, the transport of one ton of cement from New-York (US) to Le Havre (France)
(5845 km) is as expensive as the transport of one ton of cement from Le Havre (France)
to Paris (France) (200 km).
Our analysis borrows from Ponssard and Walker (2008)[8], who first established that
the firms in areas that are far from the borders are less impacted by the implementation
of pollution permits than the firms located near to the borders. We consider two different
areas: a domestic area where the emissions are regulated and a foreign area where no
environmental regulation is applied. In addition, the domestic area is divided into two
smaller areas, a ”coastal area” which is located near the border and highly subject to
international competition, and an ”inland area” which is far from the border. Each firm
can sell in its own market or in other markets. Since the above-mentioned sectors are
oligopolistic, a competition ”à la Cournot” is assumed. The paper first considers that
the permit price is exogenous and that there is no demand in the foreign country.
The first contribution of this paper is to compare, in each of the two domestic areas,
the incentives for firms to relocate to the foreign country. When the two demands are
identical, the coastal firms have more incentives to relocate since they are more exposed
to international competition. Free allowances may be used to reduce these incentives and
a possibility to minimize the cost of preventing firms to relocate could be to differentiate
permits across areas. The paper demonstrates that the higher the difference between the
inland willingness to pay and the coastal willingness to pay, the higher the savings realized
by the regulator will be. However, differentiating subsidies in a same sector is however
a politically sensitive initiative so that the competition authorities may be reluctant to
implement such policy.
The second contribution of this paper is to show that the inland firms would also
benefit from the subsidies given to the coastal firms. In other words, we demonstrate
that the relocation of coastal firms to a foreign country reduces the profits of all domestic
firms. The relocation occurs if the effective marginal cost to produce in the coastal area
and to sell in one of the two domestic markets is higher than the effective marginal cost to
produce abroad and to sell in the same market. When coastal firms relocate in a foreign
Differentiating permits allocation
3
country, they can still sell in the inland area and they may do so since it is cheaper for
them to sell in the domestic market. Thus, the relocation makes a relocated coastal firm
more efficient and increases the competition in the inland market as well as in the coastal
market. Said differently, even firms that receive less or no free allowances will benefit
from such policy.
Relaxing the assumption of exogenous permit price, we show that the relocation of
coastal firms modifies the demand for permits in this sector. Since the pollution cap
is fixed, the relocation of coastal firms to a foreign country decreases the permit price,
which ultimately induces a profit increase. However, we demonstrate that the competition
effect prevails over the permit price decrease if the market for permits covers at least a
second sector, which is not exposed to international competition and whose market-size
is sufficiently high, such as the electricity sector. We show that the existence of demand
in the foreign country modifies the necessary condition to relocate. However, when the
demand in the foreign country is sufficiently high, the inland and coastal profits increase
with the relocation. Otherwise, the inland and coastal profits decrease with the relocation.
The remainder of the paper is structured as follows. Section 2 introduces the modeling
assumptions and analyzes the incentives to relocate. Section 3 describes the economic
effects of relocation on profits. Section 4 discusses the robustness of the results. Section
5 concludes.
2. The model
2.1. The set-up
Geography. Assume two geographical areas, the domestic area and the foreign one,
respectively denoted by H and F . The domestic area is subdivided into the coastal area,
located near to the border and the inland area, far away from the border. Transportation
from one area to another is expensive and represented by a unit cost. Let τ1 be the
unit transportation cost between the inland area and the coastal area and τ2 between the
coastal area and the foreign one. The transportation cost between the inland area and
the foreign area is equal to τ1 + τ2 .
Sector. The paper focuses on a single sector economy, producing an homogenous good.
Technology is assumed to be polluting. This representative sector corresponds to the
steel and cement sectors. Let ni and nc be the number of firms in the inland area and the
coastal areas, respectively. Let nf be the number of foreign firms. Let cj be the constant
marginal cost of production for each firm located in j = i, c, f . We also consider the same
marginal cost of production for inland and coastal firms.
A firm is located in a single area, but can sell in several markets. Let qjk be the
quantity of a firm located in area j that is sent to market k. Let qj be the total quantity
produced by the firm located in area j and Qj be the total quantity produced in area
j.1 Since the above-mentioned sectors are oligopolistic, a competition ”à la Cournot” is
assumed.
1
qj =
P
k
k qj ,
Qj =
Pnj
1
qj ∀ j, k = i, c, f and QjT =
P Pnj
k
1
qkj ∀ j, k = i, c.
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J.-P. Nicolaı̈ & J. Zamorano
Demand functions. We assume that there is no demand for the good in the foreign
country. Later, in Section 4.4, we relax this assumption. The demand function in each
domestic area is given by QjT = Mj (aj − Pj ), where j = c, i. The interpretation is the
following. In market j, there are Mj consumers, whose maximal willingness to pay is aj
and they are ready to purchase several units of goods.
Figure 1 summarizes the structure of our model.
Figure 1: Summary of the Model Structure.
Regulation. In order to cut down the domestic pollution, the regulator implements
a market for permits covering the inland and the coastal markets. Firms should own
pollution permits to pollute. Let σ and µj be respectively the permit price and the
emission intensity in a market j. Suppose firms do not have access to any abatement
technology to reduce emissions. The permit price is first assumed to be exogenous.
Assumption 1. The emission intensity in the inland area is the same as the emission
intensity in the coastal area; µi = µc .
2.2. Relocation
We assume that firms have the choice between remaining in the domestic country or
relocating. They do not decide to close. Firms can relocate their production capacities
to the foreign country. First, we compare the incentives to relocate among the coastal
and inland firms.
A coastal firm has incentives to relocate its production, if, and only if, the profit
realized in the foreign area is higher than the current profit in the coastal area minus
the relocation fixed cost. Moreover, the firm takes into account that relocation modifies
market structures. Said differently, the firm relocates if:
(2.1)
πf (ni , nc − 1, nf + 1, σ) − CR > πc (ni , nc , nf , σ),
Differentiating permits allocation
5
where CR is the fixed cost of relocation, πc (ni , nc , nf , σ) is the profit of a coastal firm and
πf (ni , nc − 1, nf + 1, σ) is the profit of a foreign firm after the relocation takes place.
Relocation is also possible for inland firms as well and occurs if:
(2.2)
πf (ni − 1, nc , nf + 1, σ) − CR > πi (ni , nc , nf , σ),
where πf (ni − 1, nc , nf + 1, σ) is the profit of a foreign firm after an inland firm relocates.
The fixed cost of relocation for an inland firm is assumed to be the same as the one for
a coastal firm.
We compare now the incentives to relocate for firms in each of both domestic areas.
Incentives to relocate for inland and coastal firms. We analyze in which
area (inland or coastal) the incentives to relocate are the highest. This issue is crucial to
determine which firms face the higher relocation risk. The following equation illustrates
the conditions under which the coastal firms have higher incentives than the inland ones:
(2.3)
πf (ni , nc − 1, nf + 1, σ) − πc (ni , nc , nf , σ) > πf (ni − 1, nc , nf + 1, σ) − πi (ni , nc , nf , σ),
By developing equation (2.3), the following proposition is deduced.
Proposition 1. The coastal firms have more incentives to relocate than inland firms if
and only if
2 Mi ai − Mc ac + Mi + Mc τ1 (nc + nf − ni ) + 2Mc τ1 +
2 Mi − Mc (nf + 1)(cf + τ2 − cc − µc σ) − (cc + µc σ) > 0.
(2.4)
When the two demands are identical (Mi = Mc and ai = ac ) and ni < nc + nf + 1, the
coastal firms have more incentives to relocate since they are more exposed to international
competition. When the market in the inland area is sufficiently competitive (ni > nc +
nf + 1), the profits are weak and then the incentives for inland firms are higher. For
instance, inland firms have more incentives to relocate than coastal firms, when both
the market-size is sufficiently high and the willingness to pay in the inland market is
sufficiently weak, so that Mi ai < Mc ac .
Since in most cases, the incentives to relocate are higher for the coastal firms, we
assume for the remaining that coastal firms are more willing to relocate. Later, in Section
4.5, we focus on the case in which the inland firms have higher incentives to relocate than
the coastal ones.
Free allocation and incentives to relocate. The regulator may prevent relocation by granting to firms free allowances. First, assume differentiation is forbidden and
let be the number of free allowances given to each firm. The level of free allowances
which prevents coastal firms from relocating, is given by:
πf (ni , nc − 1, nf + 1, σ) − CR = πc (ni , nc , nf , σ) + σ,
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J.-P. Nicolaı̈ & J. Zamorano
Without differentiation, each inland firm receives σ.
However, if differentiation is allowed, we determine c and i respectively the number
of allowances given to coastal and inland firms such that they have no strict incentives
to relocate. The values of c and i are determined such that:
(2.5)
(2.6)
πf (ni , nc − 1, nf + 1, σ) − CR = πc (ni , nc , nf , σ) + c σ,
πf (ni − 1, nc , nf + 1, σ) − CR = πi (ni , nc , nf , σ) + i σ.
Obviously, = c and > i . Differentiating permits allocation across areas reduces
the total cost by:
(2.7)
(2.8)
∆ =ni (c − i )σ,
τ1 ni σ(ni + nc + nf )
∆=
×
(ni + nc + nf + 1)2
h
2 Mi ai − Mc ac + Mi + Mc τ1 (nc + nf − ni ) + 2Mc τ1 +
i
2 Mi − Mc (nf + 1)(cf + τ2 − cc − µc σ) − (cc + µc σ) .
From equation (2.8), the following corollary is deduced.
Corollary 1. The higher the inland willingness to pay, the higher the savings due to
differentiation will be. The higher the coastal willingness to pay, the weaker the savings
due to differentiation will be. The effect of both market sizes on the savings is ambiguous.
Proof. Proof in Appendix B.
Corollary 1 shows that differentiation is particularly useful to reduce the transfer
between the regulator and the firms when the differences between the inland willingness
to pay and the coastal willingness to pay is high.
3. The effects of relocation on domestic profits
Differentiating subsidies in a same sector is a politically sensitive initiative so that the
competition authorities may be reluctant to implement such policy. We demonstrate
that, at the opposite, granting more free allowances to some firms is also beneficial for
the firms which receive less free allowances. Thus, we discuss the effect on the domestic
firms’ profits of coastal firms’ relocation.
Conditions for coastal firms’ relocation. As we have defined previously, a
coastal firm relocates if:
(3.1)
πf (ni , nc − 1, nf + 1, σ) − CR > πc (ni , nc , nf , σ),
This condition may be rewritten as:
cf + τ2 − cc − µc σ
(ni + nc + nf ) Mi qfi (ni , nc − 1, nf + 1, σ) + qci (ni , nc , nf , σ)
ni + nc + nf + 1
c
i
(3.2)
+Mc qf (ni , nc − 1, nf + 1, σ) + qc (ni , nc , nf , σ) > CR ,
−
Differentiating permits allocation
7
From equation (3.2), the following lemma is deduced.
Lemma 1. The relocation occurs if the effective marginal cost to produce in the coastal
area and to sell in one of the two domestic markets is higher than the effective marginal
cost to produce abroad and to sell in the same market.
Proof. Relocation cannot occur if the left side of the equation (3.2) is strictly negative.
Since the quantities produced are always positive, we deduce that it is equivalent to
(3.3)
cf + τ2 − cc − µc σ < 0.
Said differently, if a coastal firm relocates to a foreign country, the permit price is
c +τ −c
then higher than σ̄ = f µ2c c . Lemma 1 shows that for low values of the permit price,
relocation does not occur. From this lemma we deduce that, for the next phases of the
EU-ETS, relocation will occur if the permit price will be higher than this threshold.
Domestic profits. Let us start by analyzing the inland profits, when l coastal firms
relocate their production capacities.
Relocation modifies the market-structure and we know that if relocation occurs then
cc + µc σ > cf + τ2 . We can then easily establish the following proposition.
Proposition 2. The profits of the firms in the inland area decrease with the relocation
of firms.
Proof.
cf + τ 2 − cc − µ c σ i
∂πi (ni , nc − l, nf + l, σ)
=2
qi (ni , nc − l, nf + l, σ) + qic (ni , nc − l, nf + l, σ) .
∂l
ni + nc + nf + 1
The intuition is the following. The relocation occurs if the effective marginal cost
to produce in the coastal area and to sell in one of the two domestic markets is higher
than the effective marginal cost to produce abroad and to sell in the same market. When
coastal firms relocate in a foreign country, they still can sell in the inland area, and it is
then cheaper for them to sell in the inland market. Said differently, the relocation makes
a relocated coastal firm more efficient and increases the competition in the inland market.
Second, we analyze the effect of relocation on coastal firms’ profits. As previously we
can easily establish the following proposition.
Proposition 3. The profits of the remaining firms in the coastal area decrease with the
relocation of firms.
Proof.
cf + τ2 − cc − µc σ i
∂πc (ni , nc − l, nf + l, σ)
qc (ni , nc −l, nf +l, σ)+qcc (ni , nc −l, nf +l, σ) .
=2
∂l
ni + nc + nf + 1
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J.-P. Nicolaı̈ & J. Zamorano
To conclude, these two propositions show that relocation has a detrimental effect on
domestic profits.
The other redistributive implications. Let us discuss now the other redistributive
implications of preventing firms to relocate. We analyze the effects on the consumer
surplus in the two areas, the regulator’s revenue and the environmental damage. Let
assume that the damage function is given by D(ET ), where ET = Ei + Ec + Ef is the
total amount of emissions and D(·) represents the damage function, which is increasing
with emissions.
The following proposition presents the various redistributive implications of increasing
the number of firms which relocate.
Proposition 4. The consumer surplus in both domestic areas and the environmental
damage increase with the relocation of firms.
Proof. Proof in Appendix C.
Relocation is detrimental for the environment while it is beneficial for the consumers.
Indeed, relocation induces an increase in the foreign emissions. Since the firms which
have relocated are more efficient than before relocation, they produce more. Moreover,
the products prices decrease with relocation. The regulator’s revenue is constant since
the permit price is exogenous.
4. Discussion and Robustness
We study the robustness of our results.
4.1. Endogenous permits price
Until now we have assumed that the permit price is exogenous. We relax here this
assumption. Assume E to be the pollution cap and the market for permits to be competitive. The aggregated demand for permits is equal to the total amount of permits
that firms need to cover their emissions, µi Qi (σ, l) + µc Qc (σ, l). The market clears when
demand equals supply, or:
(4.1)
µi Qi (σ, l) + µc Qc (σ, l) = E.
By taking the derivative of the previous equation, we can deduce the following lemma:
Lemma 2. The relocation of either coastal or inland firms leads to a decrease in the
equilibrium permit price:
(4.2)
∂σ ∗
< 0.
∂l
Proof. Proof in the Appendix D.
This lemma shows that relocation affects the equilibrium permit price. Indeed, the
relocation of firms reduces the demand for permits in this sector while the pollution cap
is fixed. To enter into details, relocation induces two effects on the demand for permits:
Differentiating permits allocation
9
first, the number of domestic firms is reduced and, second, the relocation induces an
increase of the competition and then a reduction of the production for the remaining
firms.
We determine the derivatives of the inland profits with respect to l taking into account
the effect of the equilibrium permit price.
∂πi (ni , nc − l, nf + l, σ)
i
c
=2 qi (ni , nc − l, nf + l, σ) + qi (ni , nc − l, nf + l, σ) ×
∂l
−(n + 1 + l) ∂σ c + τ − c − µ σ f
f
2
c
c
(4.3)
+
,
ni + nc + nf + 1 ∂l
ni + nc + nf + 1
And for the coastal profits, we get:
∂πc (ni , nc − l, nf + l, σ)
=2 qci (ni , nc − l, nf + l, σ) + qcc (ni , nc − l, nf + l, σ) ×
∂l
−(n + 1 + l) ∂σ c + τ − c − µ σ f
f
2
c
c
(4.4)
+
.
ni + nc + nf + 1 ∂l
ni + nc + nf + 1
In both cases, the variation of profits with respect to relocation is compound of two
effects: a positive one, corresponding to the permit price decrease, and a negative one,
dealing with the increase of competition.
By calculating equations (4.3) and (4.4), we can easily deduce the following proposition.
Proposition 5. Under endogenous permit price, the profits of the firms in the inland
and coastal areas increase with the relocation of coastal firms.
Proof. Proof in the Appendix E.
Proposition 5 states that the effect, due to the equilibrium permits price decrease,
always prevails over the competition effect.
Thus the domestic remaining firms’ profits increase with relocation.
4.2. Two sectors included in the market for permits
We assume now that two sectors are included in the same market for permits. The
first sector corresponds to the one that we analyzed previously. The second sector is
not exposed to international competition and corresponds, for instance, to the electric
sector, which is also included in the EU-ETS. This sector is denoted B. The production
technology in sector B is also polluting.
Assume that nb firms in sector B produce a homogenous good and they compete ӈ la
Cournot”. These firms are assumed to be symmetric and to have the same marginal cost
of production (cb ), as well as the same emission intensity (µb ). For the sake of clarity, we
consider a simpler structure than in sector A. Indeed, since sector B is not exposed to
international competition, we do not consider a subdivision of the domestic area. Such
an assumption does not alter the results. We also assume the demand function to be
Qb = Mb (ab − Pb ), where Mb is the market-size of sector B, ab the willingness to pay of
sector B, and Pb the product price of sector B.
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J.-P. Nicolaı̈ & J. Zamorano
In the market for permits, the aggregated demand for permits is equal to µb Qb (σab ) +
µi Qi (σab , l) + µc Qc (σab , l). As we have previously discussed, the relocation of firms affects
the equilibrium permit price. The following lemma compares the effect of relocation of
firms belonging to sector A on the permit price under the present case with the case
discussed in Section 4.1.
Lemma 3. The effect of the relocation on the equilibrium permit price is weaker in the
case of a market for permits covering an exposed sector and a not exposed sector than in
the case of a single exposed sector included in the market for permits;
∂σab
∂σ
(σ) <
(σ) < 0.
∂l
∂l
(4.5)
Proof. Proof in the Appendix F.
In proportion, the variation of the demand for permits is less important since the
second sector is not directly affected by the relocation.
As in Section 4.1, the effects of coastal relocation on the inland profits and the coastal
ones are respectively given by the equations (4.3) and (4.4). We can then establish the
following proposition.
Proposition 6. There exists a threshold M̃b > 0 such that if the market-size is higher
than this latter (Mb > M̃b ), the profits of the remaining domestic firms decrease with the
relocation of firms.
Proof. Proof in Appendix G.
As we have seen in Section 4.1, relocation induces a negative competition effect on the
remaining domestic firms’ profits, and also a decrease of the permit price, which leads to
a profit increase. Said differently, the higher the market-size of the non-exposed sector is,
the weaker the permit price decrease will be, the more domestic profits in sector A will
decrease.
4.3. General demand function
Until now we have assumed that demand is linear. We consider now a general demand
function and we analyze whether the relocation of coastal firms to a foreign country
reduces the profits of all domestic firms.
Consider in each area that the inverse demand function is twice differentiable, positive
or null, and strictly decreasing when positive, and P (0) > 0. Let ηj = Pj00 Qj /Pj0 be the
elasticity of the demand slope in market j.
As we have defined previously, a coastal firm relocates if:
(4.6)
πf (ni , nc − 1, nf + 1, σ) − CR > πc (ni , nc , nf , σ),
Each firm sells in the two markets: the inland one and the coastal one. We can then
rewrite this condition as:
(4.7)
πf c (ni , nc −1, nf +1, σ)+πf i (ni , nc −1, nf +1, σ)−CR > πcc (ni , nc , nf , σ)+πci (ni , nc , nf , σ),
Differentiating permits allocation
11
where πjk (ni , nc , nf , σ) is the profit realized by a firm located in area j and selling
in market k. We deduce from equation (4.7) that either πf c (ni , nc − 1, nf + 1, σ) >
πcc (ni , nc , nf , σ) or πf i (ni , nc − 1, nf + 1, σ) > πci (ni , nc , nf , σ), or both at the same time.
We show in Appendix H that if the demand functions are not too curved (ηj > −2
for j ∈ {i, c}), πf c (ni , nc − 1, nf + 1, σ) > πcc (ni , nc , nf , σ) implies that cc + µc σ >
cf + τ2 . We deduce then that cc + τ1 + µc σ > cf + τ1 + τ2 . We deduce finally that
πf i (ni , nc − 1, nf + 1, σ) > πci (ni , nc , nf , σ). The condition such that cc + µc σ > cf + τ2
induces that domestic profits decrease with relocation.
Seade (1985)[10] shows that, in a Cournot framework under η < −2, an increase in
the constant marginal cost of all firms can increase all profits. Thus, in such a case
implementing an environmental regulation may increase all the firms’ profits. If the
demand slope is sufficiently elastic, an increase in the marginal cost induces an increase
in the mark-up, which prevails over the decrease in production. Note that the elasticity
of the slope of demand is constant with an iso-elastic demand function and equal to the
inverse of the demand elasticity. With an iso-elastic demand function, profits increase
when demand elasticity is weak and decrease otherwise. Cases in which profits increase
with a shock correspond to a pass-through rate that is greater than 100%. We do not
consider this pathological case in this paper.
As in Section 3, when the demand functions are not too curved the relocation occurs
if the effective marginal cost to produce in the coastal area and to sell in one of the two
domestic markets is higher than the effective marginal cost to produce abroad and to
sell in one of the two domestic markets. Moreover, the relocation still makes a relocated
coastal firm more efficient and increases the competition in the inland and coastal markets.
To conclude, the relocation of coastal firms to a foreign country reduces the profits of all
domestic firms, when the demand function is not too curved.
4.4. Demand in the foreign country
Until now we have assumed that there is no demand in the foreign area.
Let Pf (Qf ) be the demand in the foreign country and let πf f (ni , nc , nf ) be the equilibrium profits realized by a representative foreign firm selling in the foreign market. Let
nf the number of foreign firms. We consider that domestic firms do not sell in the foreign
market.
Now, focusing on the coastal firms’ relocation, we observe that the existence of demand
in the foreign country changes the necessary condition to relocate. Each domestic firm
sells in the two markets: the inland one and the coastal one. The foreign firms sell in the
three markets. The condition under which relocation occurs now is given by:
πf c (ni , nc − 1, nf + 1, σ) + πf i (ni , nc − 1, nf + 1, σ) + πf f (ni , nc − 1, nf + 1) − CR >
(4.8)
πcc (ni , nc , nf , σ) + πci (ni , nc , nf , σ),
In the same vain as Lemma 1, we deduce that, when πf f (ni , nc − 1, nf + 1) < Cr ,
cc + µc σ > cf + τ2 . When πf f (ni , nc − 1, nf + 1) > Cr , cc + µc σ may be higher or lower
than cf + τ2 .
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J.-P. Nicolaı̈ & J. Zamorano
Since only the foreign firms sell in the foreign market, the conditions under which
the relocation generates a decrease in profits are the same as in Section 3. Indeed, the
presence of consumers in the foreign area does not modify the competition effect.
The following proposition may be deduced.
Proposition 7. When πf f (ni , nc − 1, nf + 1) > Cr , the inland and coastal profits may
increase or decrease with the relocation. When πf f (ni , nc − 1, nf + 1) < Cr , the inland
and coastal profits decrease with the relocation.
When the demand in the foreign country is sufficiently high, the relocation may occur
even when the permit price is lower than σ̄. Said differently, in such a case, the remaining
firms’ profits increase with relocation. Indeed, the foreign firm will be less efficient than
the coastal one. Thus relocation reduces competition. However, when the demand in the
foreign country is sufficiently weak, the relocation occurs only if the permit price is higher
than σ̄. Relocation induces then an increase of competition and a decrease in domestic
profits.
4.5. Inland firms relocation
Until now, we have considered that coastal firms have more incentives to relocate their
production in the foreign area. We relax this assumption.
As we have defined previously, an inland firm relocates if:
(4.9)
πf (ni − 1, nc , nf + 1, σ) − CR > πi (ni , nc , nf , σ),
Each firm sells in the two markets: the inland one and the coastal one. We can then
rewrite this condition as:
(4.10)
πf c (ni −1, nc , nf +1, σ)+πf i (ni −1, nc , nf +1, σ)−CR > πic (ni , nc , nf , σ)+πii (ni , nc , nf , σ),
Assume πf c (ni − 1, nc , nf + 1, σ) > πic (ni , nc , nf , σ). Since firms compete ”à la
Cournot” and demand is linear, it implies that ci + τ1 + µi σ > cf + τ2 . Said differently, we
get ci + µi σ > cf + τ2 − τ1 . Assume now that πf i (ni − 1, nc , nf + 1, σ) > πii (ni , nc , nf , σ).
It implies that ci + µi σ > cf + τ1 + τ2 . However, cf + τ1 + τ2 > cf + τ2 − τ1 . Thus, if
inland relocation occurs ci + µi σ > cf + τ2 − τ1 . Said differently, the effective marginal
cost to produce in the inland area and to sell the coastal area is higher than the effective
marginal cost to produce abroad and to sell in the coastal area.
We deduce that when inland firms relocate in a foreign country, they still can sell
in the coastal area, and it is then cheaper for them to sell in the coastal market. In
other words, the relocation makes an inland firm more efficient in the coastal market and
increases the competition in the coastal market. Nevertheless, the relocated firm may be
less efficient in the inland market.
5. Conclusion
This paper justifies the protection of coastal firms and shows that the incentives to
relocate are different across areas. This effect justifies to differentiate firms across areas.
Differentiating permits allocation
13
However, differentiation is ordinarily forbidden since it may alter competition. This paper
demonstrates that, at the opposite, differentiation may be beneficial for the firms which
receive less or no free allowances. Indeed, the relocation of coastal firms to the foreign
country reduces the profits of domestic firms when the demand in the foreign country is
small and when the market for permits covers several sectors, whose some are not exposed
to international competition such as the electricity sector. Thus, the relocation makes a
coastal firm more efficient and increases the competition in the inland market.
One of the main ingredients of this paper is the transportation cost. However, this
latter may differ drastically across sectors and in function of the way to transport goods.
These differences go towards different treatments across the cement sector and the steel
sector, which is allowed in the EU, as it is done for the third phase of EU-ETS (different
treatment for the electricity sector as for the steel and cement sectors).
Nevertheless, an important difficulty that the regulator may face is to define the
different areas. An empirical analysis to determine in Europe the different incentives to
relocate and the relevant areas, would be a promising project.
References
[1] Branger, F. and Quirion, P. (2014), Climate Policy and the Carbon Haven Effect.
WIRES Climate Change 5(1): 53-71.
[2] Branger, F., P. Quirion and Chevallier, J. (2015), Carbon leakage and competitiveness of cement and steel industries under the EU ETS : much ado about nothing,
Accepted in The Energy Journal, CIRED Working Paper 2013-53.
[3] Demailly, D. and Quirion, P. (2006), CO2 abatement, competitiveness and leakage
in the European Cement Industry under the EU ETS: grandfathering versus outputbased allocation. Climate Policy Vol. 6, p. 93-113.
[4] Demailly, D. and Quirion, P. (2008), Concilier compétitivité industrielle et politique
climatique : faut-il distribuer les quotas de CO2 en fonction de la production ou
bien les ajuster aux frontières ?, La revue économique vol. 59(3):497-504.
[5] Ellerman, A.D., Convery, F. and de Perthuis, C. (2010), Pricing Carbon: The European Union Emissions Trading Scheme, Cambridge: Cambridge University Press.
[6] Martin, R., Muûls, M., De Preux, L. B., and Wagner, U. J. (2014), Industry Compensation under Relocation Risk: A Firm-Level Analysis of the EU Emissions Trading
Scheme.” American Economic Review, Vol. 104(8), p. 2482-2508.
[7] Nicolaı̈, J-P., Péchoux, I., Ponssard, J-P. and Pouyet, J. (2010), Environmental
Policy and border adjustment with imperfect competition, Revue économique vol.
61(1), p. 49-68.
[8] Ponssard, J-P. and Walker, N. (2008), EU Emissions Trading and the cement sector:
a spatial competition analysis, Climate Policy Vol 8,p. 467-493.
[9] Quirion, P., Meunier, G. and Ponssard. J-P. (2014), Carbon leakage and capacitybased allocations: Is the EU right?, Journal of Environmental Economics and Management Vol. 68(2), P. 262-279.
14
J.-P. Nicolaı̈ & J. Zamorano
[10] Seade, J. (1985), Profitable Cost Increases and the Shifting of Taxation, unpublished
University of Warwick Economic Research Paper.
Appendix
A. Firms’ quantities
A.1. Firms’ quantities under pollution permits without relocation
In domestic sectors, the profit of a representative inland firm, may be written as follows:
(A.1)
πi = Pi (QiT )qii + Pc (QcT )qic − (ci + µi σ)(qii + qic ) − τ1 qic ,
of a coastal firm:
(A.2)
πc = Pi (QiT )qci + Pc (QcT )qcc − (cc + µc σ)(qcc + qci ) − τ1 qci ,
And, of a foreign firm,
(A.3)
πf = Pi (QiT )qfi + Pc (QcT )qfc − cf (qfi + qfc ) − (τ1 + τ2 )qfi − τ2 qfc .
Firms maximize profits by choosing productions. Easily we obtain the first-order conditions and then we calculate the equilibrium quantities qjk (ni , nc , nf , σ), produced by each
firm for its local market as well as the productions allocated to other markets, i.e. the
exported quantities.
(A.4)
qii (·) =
Mi (ai −(nc +nf +1)(ci +σµi )+nc (cc +σµc )+nf cf +(nc +nf )τ1 +nf τ2 )
,
(ni +nc +nf +1)
(A.5)
qic (·) =
Mc (ac −(nc +nf +1)(ci +σµi )+nc (cc +σµc )+nf cf −(nc +nf +1)τ1 +nf τ2 )
,
(ni +nc +nf +1)
(A.6)
qci (·) =
Mi (ai +ni (ci +σµi )−(ni +nf +1)(cc +σµc )+nf cf −(ni +1)τ1 +nf τ2 )
,
(ni +nc +nf +1)
(A.7)
qcc (·) =
Mc (ac +ni (ci +σµi )−(ni +nf +1)(cc +σµc )+nf cf +ni τ1 +nf τ2 )
,
(ni +nc +nf +1)
(A.8)
qfi (·) =
Mi (ai +ni (ci +σµi )+nc (cc +σµc )−(ni +nc +1)cf −(ni +1)τ1 −(ni +nc +1)τ2 )
,
(ni +nc +nf +1)
(A.9)
qfc (·) =
Mc (ac +ni (ci +σµi )+nc (cc +σµc )−(ni +nc +1)cf +ni τ1 −(ni +nc +1)τ2 )
.
(ni +nc +nf +1)
B. Proof of Corollary 1
The derivatives of the savings with respect to the willingness to pay are equal to:
(B.1)
(B.2)
∂∆ 2Mi τ1 ni σ(ni + nc + nf )
=
> 0,
∂ai
(ni + nc + nf + 1)2
∂∆ −2Mc τ1 ni σ(ni + nc + nf )
=
< 0.
∂ac
(ni + nc + nf + 1)2
Then, the savings due to differentiation are increasing with respect to the willingness to
pay in the inland area, and they are decreasing with respect to the willingness to pay in
the coastal area.
15
Differentiating permits allocation
Calculating the derivative of the savings with respect to the two market-sizes, we get:
(B.3)
∂∆ τ1 ni σ(ni + nc + nf ) =
2a
+
τ
(n
+
n
−
n
)
+
2
(n
+
1)(c
+
τ
−
c
−
µ
σ)
−
(c
+
µ
σ)
,
i
1
c
f
i
f
2
c
c
c
c
| f
{z
}
∂Mi (ni + nc + nf + 1)2
<0
(B.4)
∂∆ τ1 ni (ni + nc + nf )σ =
−
2a
+
τ
(n
+
n
−
n
)
−
2
(n
+
1)(c
+
τ
−
c
−
µ
σ)
−
(c
+
µ
σ)
c
1
c
f
i
f
2
c
c
c
c
{z
}
| f
∂Mc (ni + nc + nf + 1)2
<0
C. Proof of Proposition 4
We calculate the consumer surplus in the domestic sectors and we derivate with respect
to l, we get:
(C.1)
∂CSi
cf + τ2 − cc − µσ i
=−
Mi ni qi (ni , nc − l, nf + l, σ) + (nc − l)qci (ni , nc − l, nf + l, σ)+
∂l
ni + nc + nf + 1
(nf + l)qfi (ni , nc − l, nf + l, σ) ,
(C.2)
∂CSc
cf + τ2 − cc − µσ c
=−
Mc ni qi (ni , nc − l, nf + l, σ) + (nc − l)qcc (ni , nc − l, nf + l, σ)+
∂l
ni + nc + nf + 1
(nf + l)qfc (ni , nc − l, nf + l, σ) .
And the derivative of the environmental damage:
(C.3)
c + τ − c − µ σ ∂D
f
2
c
c
= −δµc (Mi + Mc )
.
∂l
ni + nc + nf + 1
From Lemma 1, we obtain that
∂CSi ∂CSc
, ∂l
∂l
and
∂D
∂l
are positives.
D. Proof of Lemma 2
From the equilibrium in the market for permits considering the relocation of coastal firms,
we know that:
(D.1)
µi Qi (σ, l) + µc Qc (σ, l) = E.
Calculating and rearranging the terms, we obtain:
∂Q (σ, l)
∂Qc (σ, l) ∂σ
i
(Mi + Mc ) µi
+ µc
=
∂σ
∂σ
|
{z
} ∂l
<0
(D.2)
2(Mi + Mc )(µi ni + µc (nc − l))(cf + τ2 − cc − µc σ)
i
c
−
+ µc qc (σ, l) + qc (σ, l) .
ni + nc + nf + 1
|
{z
}
>0
16
J.-P. Nicolaı̈ & J. Zamorano
i (σ,l)
On the left hand side, the first term in brackets is always negative since ∂Q∂σ
<0
∂Qc (σ,l)
and ∂σ < 0. Moreover, from Lemma 1, we know that cc +µc σ > cf +τ2 , i.e. the right<0
hand side of the equation above is positive. Therefore, we easily conclude that ∂σ
∂l
in equilibrium. The relocation of coastal firms has a negative effect over the endogenous
permit price. Similarly we can conclude when we consider the relocation of inland firms.
E. Proof of Proposition 5
Now, we analyze the effect of relocation on domestic profits. The profit of an inland firm
is given by:
2 1 2
1 i
c
q (ni , nc −l, nf +l, σ) +
q (ni , nc −l, nf +l, σ) ,
(E.1) πi (ni , nc −l, nf +l, σ) =
Mi i
Mc i
and for a coastal firm, we get:
(E.2) πc (ni , nc −l, nf +l, σ) =
2 1 2
1 i
qc (ni , nc −l, nf +l, σ) +
qcc∗ (ni , nc −l, nf +l, σ) .
Mi
Mc
Since we analyze the effects of relocation on profits, we calculate the derivative of
profits due to the relocation of coastal firms. Then, for an inland firm, we get:
c
i
qi (ni , nc − l, nf + l, σ) + qi (ni , nc − l, nf + l, σ)
∂πi (ni , nc − l, nf + l, σ)
=2
×
∂l
(Mi + Mc )(ni + nc − l)
qci (ni , nc − l, nf + l, σ) + qcc (ni , nc − l, nf + l, σ) ,
(E.3)
and for a coastal firm, we get:
(E.4)
∂πc (ni , nc − l, nf + l, σ)
=2
∂l
2
qci (ni , nc − l, nf + l, σ) + qcc (ni , nc − l, nf + l, σ)
(Mi + Mc )(ni + nc − l)
.
From equations (E.3) and (E.4), we can conclude that the relocation of coastal firms
increases the profit of the inland and coastal firms ( ∂π∂li (·) > 0 and ∂π∂lc (·) > 0).
F. Proof of Lemma 3
The profit of a representative firm in sector B, may be written as follows:
(F.1)
πb = Pb (Qb )qb − (cb + µb σab )qb .
Where Qb = Mb ab − Pb is the demand in the sector B and firms maximize their
profits by choosing productions. Thus, we obtain the first-order conditions and then we
calculate the equilibrium quantities.
b σab
(F.2)
qb (σab ) = Mb ab −cnbb−µ
.
+1
Differentiating permits allocation
17
And the equilibrium profit is given by:
πb =
(F.3)
1
Mb
2
qb (σab ) .
First, we start analyzing the sign of the derivative of the permit price with respect to
the relocation. From the equilibrium in the market for permits, we know that:
(F.4)
µb Qb (σab ) + µi Qi (σab , l) + µc Qc (σab , l) = E.
Since we analyze the effect of relocation on the permit price, we study the derivative of
equation (F.4) with respect to l, and we obtain:
h
(F.5)
−
|
µb
|
∂Qi (σab , l)
∂Qc (σab , l) i ∂σab
∂Qb (σab )
+ µi
+ µc
=
∂l
∂σab
∂σab
∂σab
{z
}
<0
2(µi ni + µc (nc − l))(cf + τ2 − cc − µc σab )
+ µc qci (σab , l) + qcc (σab , l) .
ni + nc + nf + 1
{z
}
>0
Focusing on the left hand side, the first term in brackets is always negative since
c (σab ,l)
i (σab ,l)
< 0, ∂Q∂σ
< 0 and ∂Q∂σ
< 0. Thus, by Lemma 1, we know that cc +µc σab >
ab
ab
cf + τ2 , i.e. the right-hand side of the equation above is positive. Therefore, we easily
conclude that ∂σ∂lab < 0 in equilibrium.
Second, focusing now on the derivative of the permit price with respect to the relocation, we observe that it is a linear function depending on σ.
From equation (F.5) we calculate ∂σ∂lab (σ) and we obtain that:
(F.6)
h
i ∂σ
∂σab
(Mi + Mc )(nb + 1)(nf + l + 1)(ni + nc − l)µ2c
(σ) =
(σ).
∂l
(Mi + Mc )(nb + 1)(nf + l + 1)(ni + nc − l)µ2c + nb Mb (ni + nc + nf + 1)µ2b ∂l
∂Qa (σab )
∂σab
Where ∂σ
(σ) corresponds to the variation of the permit price with respect to the
∂l
relocation, when the market for permits only covers the domestic sector A. Moreover, we
observe that the expression in parentheses is always positive and it is smaller than one.
Another important feature is that this expression is decreasing with the number of firms
in the sector B, i.e when nb is large enough, the value of the expression is close to zero.
Otherwise when there is no sector B (i.e. nb = 0) the derivatives of permit price with
respect to the relocation are equivalents.
(σ) < ∂σ∂lab (σ) < 0.
Then, from (F.6) we conclude that ∂σ
∂l
G. Proof of Proposition 6
We start presenting the derivative of the inland profit with respect to the relocation, and
it is given by:
∂πi (ni ,nc −l,nf +l,σab )
i
c
= 2 qi (ni , nc − l, nf + l, σab ) + qi (ni , nc − l, nf + l, σab ×
∂l
−(nf +l+1)µc ∂σab
cf +τ2 −cc −µc σab
(G.1)
+
,
ni +nc +nf +1 ∂l
ni +nc +nf +1
18
J.-P. Nicolaı̈ & J. Zamorano
and, for coastal firms:
∂πc (ni ,nc −l,nf +l,σab )
∂l
(G.2)
= 2 qci (ni , nc − l, nf + l, σab ) + qcc (ni , nc − l, nf + l, σab ×
−(nf +l+1)µc ∂σab
cf +τ2 −cc −µc σab
+ ni +nc +nf +1 .
ni +nc +nf +1 ∂l
The idea is to find, in function of the market size of the non-exposed sector, the
conditions under which the relocation may induce a decrease of inland and coastal profits.
Let us suppose now that the derivative of the inland and coastal profits with respect
to the relocation are nulls. We are considering that the competition effect is equal to
permit price decrease effect for a given M̃b .
Then, from both equations, (G.1) and (G.2), we obtain the following expression:
(G.3)
cf + τ2 − cc − µc σab
−(nf + l + 1)µc ∂σab
(M̃b ) +
= 0.
ni + nc + nf + 1 ∂l
ni + nc + nf + 1
Now, from equation (F.5) we obtain the value of ∂σ∂lab , which we express in function
of M̃b . Then, we replace it in equation (G.3) and we get:
(G.4)
c
i
2
(nb + 1)(nf + l + 1)µc qc (ni , nc − l, nf + l, σab ) + qc (ni , nc − l, nf + l, σab )
M̃b =
nb µ2b − cf − τ2 + cc + µc σab
Thus, from Lemma 1 we know that cc + µc σab > cf + τ2 . Hence, this threshold is
positive (M̃b > 0).
Moreover, from equation (F.6) we know that the ∂σ∂lab < 0 and it increases with sector
B’s market-size. Then, when the Mb > M̃b , the permit price decrease effect, which is
a positive effect, decreases and the competition effect prevails, we obtain that equations
(G.1) and (G.2) are negatives ( ∂π∂li (·) < 0 and ∂π∂lc (·) < 0).
H. General demand function
The goal of this Appendix is to demonstrate that a firm has never incentives to choose
the technology with the highest marginal cost of production if the demand functions are
not too curved (η = P 00 Q/P 0 > −2).
Assume two different kinds of firms 1 and 2. The marginal cost of production are
different and c1 > c2 . Let n1 and n2 be the number of firms respectively of kind 1 and
kind 2.
We consider another firm, called e characterized by a marginal cost denoted by ce . We
analyze whether it is profitable for the firm to increase the marginal cost of production.
The profit of this latter is given by:
(H.1)
πe = (P (qe + Qe ) − ce )qe
where Qi represents the production of all the firms except the one of firm i. The first-order
19
Differentiating permits allocation
condition is given by:
(H.2)
∂πe
= P 0 (qe + Qe )qe + P (qe + Qe ) − ce = 0
∂qe
We derive the profit according to the marginal cost
(H.3)
∂πe
= P 0 (qe + Qe )qe + P (qe + Qe ) − ce + P 0 qe Q0e − qe
∂ce
Unsing equation (H.2), we get:
(H.4)
∂πe
= (P 0 Q0e − 1)qe
∂ce
Since P 0 < 0, if we show that Q0e > 0, we demonstrate then that
∂πe
∂ce
< 0.
Let consider the first-order conditions of the other firms. The profit of a firm of kind
1 is given by:
(H.5)
π1 = (P (q1 + Q1 ) − c1 )q1
The first-order condition for a firm of kind 1 is given by:
(H.6)
∂π1
= P 0 (q1 + Q1 )q1 + P (q1 + Q1 ) − c1 = 0
∂q1
The profit of a firm of kind 2 is given by:
(H.7)
π2 = (P (q2 + Q2 ) − c2 )q2
The first-order condition for a firm of kind 2 is given by:
(H.8)
∂π2
= P 0 (q2 + Q2 )q2 + P (q2 + Q2 ) − c2 = 0
∂q2
If we sum all the first-order conditions except the one relative to firm e, we get:
(H.9)
(n1 + n2 )P + P 0 (qe + Qe )Qe = n1 c1 + n2 c2
From equation (H.9), we deduce that
(H.10)
Qe (qe ) = φ(qe (ce ))
By substituting Qe (qe ) by φ(q(ce )) in equation (H.9), we get:
(H.11)
(n1 + n2 )P (qe + φ(qe )) + P 0 (qe + φ(qe (ce )))φ(q(ce )) = n1 c1 + n2 c2
By derivating equation (H.11), we get:
(H.12)
φ0 = −
(n1 + n2 )P 0 + P 00 φ
(n1 + n2 + 1)P 0 + P 00 φ
20
J.-P. Nicolaı̈ & J. Zamorano
Since P 00 Q/P 0 is assumed to be higher than −2, we deduce that φ0 > 0 and consequently
e
<0
that ∂π
∂ce