Environmental and Resource Economics 27: 1–19, 2004.
© 2004 Kluwer Academic Publishers. Printed in the Netherlands.
1
On the Efficiency of Competitive Markets for
Emission Permits
EFTICHIOS SOPHOCLES SARTZETAKIS
Department of Accounting and Finance, University of Macedonia, 156 Egnatia Str., Thessaloniki
54006, Greece (E-mail: [email protected])
Accepted 8 January 2003
Abstract. It is typical for economists and policy makers alike to presume that competitive markets
allocate emission permits efficiently. This paper demonstrates that competition in the emission
permits market cannot assure efficiency when the product market is oligopolistic. We provide the
conditions under which a bureaucratic mechanism is welfare superior to a tradeable emission permits
system. Price-taking behaviour in the permits market ensures transfer of licenses to the less efficient
in abatement firms, which then become more aggressive in the product market, acquiring additional
permits. As a result, the less efficient firms end up with a higher than the welfare maximizing share
of emission permits. If the less efficient in abatement firms are also less efficient in production,
competitive trading of permits may result in lower output and welfare.
Key words: competitive trading of emission permits, economic efficiency, oligopolistic product
market
JEL classification: Q20, Q28, D60, D62
I. Introduction
Economists typically take as given that competitive markets allocate licenses efficiently, that is, to the user with the highest value. It has been shown, however, that
competitive trading of licenses does not necessarily yield social welfare maximization, and that in some cases a bureaucratic mechanism could be welfare superior.1
Despite these results, the belief that competitive markets for licenses will guide
firms to welfare maximizing decisions remains widely held among economists.
This apparent confusion also exists in the literature related to the use of emission
permits. Malueg (1990) was the first to point out the possibility that an emission permit trading program may reduce social welfare when the product market
is not perfectly competitive. Although the general idea is in the right direction,
the analysis relies on a peculiar specification of emission permits regulation and
on extreme differences in firms’ abatement technologies. Using the same model
specifications Sartzetakis (1997a) has shown that when the permits regulation is
correctly specified and firms have positive abatement costs, that is, only interior
solutions are admitted, trading of emission permits welfare dominates regula-
2
EFTICHIOS SOPHOCLES SARTZETAKIS
tions that prohibit trading. However, as pointed out by Sartzetakis (1997a), these
results depend on “. . . the assumption of constant and equal between firms cost of
production” (p. 76).
The present paper extends the work of Sartzetakis (1997a) by considering the
case in which firms differ in both production and abatement technologies. Our
analysis focuses on the interaction between a competitive market for emission
permits and an oligopolistic product market. Following the literature, we assume
that oligopolistic firms participate in a thicker market for emission permits, and
thus, they behave as price takers in that market while acting strategically in their
product market. We find that aggregate output and social welfare could decrease
as a result of competitive trading of permits under certain conditions regarding
abatement and production technologies of the regulated firms. To the best of our
knowledge, there is no formal proof of these results. Our goal is to specify these
conditions, and more importantly to reveal the intuition underlining these results.
In order to derive meaningful general conditions for the welfare comparison of
policy instruments, we separate production from abatement technologies. Abatement costs depend on the level of production and the number of emission permits
available to firms, establishing the link between emission permits and output
decisions. We show that welfare comparison can be expressed in terms of firms’
differences in abatement and production technologies. If the more efficient in
abatement firms are less efficient in production,2 competitive trading of emission permits is welfare superior to a bureaucratic mechanism. If, however, firms’
technological efficiencies in production and abatement are positively correlated,3
prohibiting trade of emission permits could dominate competitive trading of emission permits. Under the latter assumption regarding firms’ technologies, another,
and perhaps more interesting, result is that competitive trading of emission permits
could yield a reduction in industry’s output relative to a bureaucratic mechanism.
Our results should not be interpreted as calls for limiting the interest over tradeable
emission permits programs or other licenses trading programs. They are, however,
reasons for considering careful design of these programs, taking into account
the technological structure of the regulated industries, in order to improve their
efficiency.
To understand the intuition behind these results we focus on the role of the
product market through which the gains of competitive trading of emission permits
are realized. Price-taking behaviour in the emission permit market guarantees that
emission permits flow from the low to the high marginal abatement cost firms.
Industry’s marginal cost of abatement decreases, which assures welfare maximization when the product market is perfectly competitive. However, when the
product market is oligopolistic, profits are not necessarily positively correlated with
either industry’s output or welfare. If rival firms behave a la Cournot, acquisition
of emission permits increases the aggressiveness of firms with the higher marginal
abatement costs, while the more efficient firms react by decreasing production
to preserve a high price. The reduction of the efficient firms’ output decreases
ON THE EFFICIENCY OF COMPETITIVE MARKETS
3
their marginal cost of abatement yielding a further release of emission permits
to the less efficient firms. This latter transfer of emission permits does not occur
in price-taking markets. Thus, in the final distribution of emission permits, under
Cournot behaviour in the product market, the less efficient firms hold a higher
than the welfare maximizing share of emission permits. When the differences in
marginal costs among firms are more prominent in production rather than abatement, industry’s output could decrease. Social welfare could decrease either when
industry’s output decreases, or when the excess transfer of emission permits effect
on profits outweighs the effect of increased output on consumer surplus.
The need for special specification of emission permits regulation when the
product market is not competitive, that is, in the presence of double distortion, has
been identified in the literature in a variety of situations. Hahn (1984) examined the
case in which a firm could exercise monopoly power in the permit market, Misiolek
and Elder (1998), von der Fehr (1994), and Sartzetakis (1997b) examined the case
of exclusionary manipulation of emission permits. Fershtman and de Zeeuw (1996)
and van Long and Soubeyran (1977) pointed to the possibility that trading of emissions permits may facilitate cooperation among oligopolists by allowing credible
commitments that bypass antitrust regulation. The present paper differs from the
above literature by assuming a competitive market for permits and non cooperative
behaviour among firms in the product market. Closely related to our work are the
papers by Malueg (1990) and Sartzetakis (1997a) mentioned above, and the paper
by van Long and Soubeyran (2000), though the latter is not concerned with the
comparison of emission permits to command and control.
The presentation of these results within a general model is undertaken in
Section III, following the specification of the model in Section II. In order to illustrate the mechanics of the trading of emission permits we proceed in Section IV
to employ a simple example that allows the derivation of closed form solutions
to be derived. We also present the results of some simulations providing a visual
illustration as well as some idea of the relative magnitude of the effects of emission
permit trading. Section V concludes the paper.
II. The Model
Assume a two sector economy; the first is a competitive numeraire sector; the other
is a homogeneous duopoly. Production generates a negative externality, emission E
of a pollutant. Consumer preferences are given by a utility function that is separable
in the numeraire good and in emission damages denoted by V(E),
U = u q 1 , q 2 + m − V (E)
(1)
where m is expenditure on a competitively supplied numeraire good and qi is
the output of firm i in the oligopolistic sector.4 To avoid unnecessary repetitions,
hereafter, i, j = 1, 2 and i = j.
4
EFTICHIOS SOPHOCLES SARTZETAKIS
Each duopolist has cost of production C i (q i ), with strictly positive marginal
cost of production, ci > 0. The amount of pollution emitted by each firm ei , is an
increasing function of its output level, ei = Hi (qi ), with Hi (0) = 0, hi = ∂H i /∂q i >
0 and hii = ∂hi /∂q i ≥ 0. Hereafter, subscripts denote partial derivatives. Firms
can reduce emissions by either reducing output or controlling emissions per unit of
output, that is, by engaging in abatement. Total abatement is defined as Z i = q i zi ,
where zi is firm i’s abatement per unit of output. Cost of abatement Ai (q i , zi ) is
strictly increasing in both its elements, that is, Aiq > 0 and Aiz > 0.
Let us consider two types of regulatory intervention that could be used by policy
makers to limit firms’ emissions. The first is a bureaucratic mechanism in the form
of licensing firm-specific emission quota, and the other is a tradeable emission
permits system. It is reasonable to exclude the possibility that the welfare maximizing distribution of emission permits can be implemented, because of information
and/or political constraints. Denoting by ēi firm
2 i’s emission quota, the aggregate
emission quota for the oligopolistic sector is i=1 ēi = ē.
Let ∂u/∂q i = p i (q 1 , q 2 ) be firm i’s inverse demand, and R i (q 1 , q 2 ) = pq i the
total revenue. We assume that q1 and q2 are strategic substitutes, that is, Rji < 0
and Riji < 0. Each firm in the oligopolistic sector sets output on the assumption
that the other firm holds its output fixed, arriving at the Cournot-Nash equilibrium.
In the case of emission permits trading we assume that although firms interact strategically in the output market, they are price takers in the emission permits market.
Thus, we rule out strategic interaction between the output and the emission permits
market. Output and abatement per unit of output are determined simultaneously.
III.
A. Non-tradeable Emission Quota
Under the bureaucratic mechanism, the regulator allocates ēi non-tradeable emission quota to firm i. Each permit allows the emission of one unit of pollutant. Firm
i chooses output and abatement per unit of output in order to maximize profits
subject to the regulatory constraint on emissions H i (q i ) − q i zi = ēi ,
(2)
max π i q i , q j , zi , ēi = R i q i , q j − C i q i − Ai q i , zi
i
i
q ,z
subject to H i q i − q i zi = ēi .
The first order conditions yield firm i’s output reaction function,5
πii = Rii − ci − Aiq − Aiz hi − zi /q i = 0.
(3)
where Aiq = f i (q i , ēi ), with fqi > 0, fēi < 0 and Aiz = g i (q i , ēi ), with qqi > 0,
gēi < 0 are the marginal costs of abatement with respect to output and abatement
per unit of output respectively. zi (q i , ēi ) is abatement per unit of output expressed
as a function of firm i’s output and emission allowance derived from the regulatory
ON THE EFFICIENCY OF COMPETITIVE MARKETS
5
constraint. The solution of the first order conditions yields firm i’s Coumot-Nash
equilibrium output as a function of firms’ production and abatement technologies
as well as their emission quota,
q i = ϕ i (ē1 , ē2 ) .
(4)
Assume that there is a transfer of emission quota from firm 1 to firm 2. We are
interested in deriving the output effect of this reallocation. Total differentiation of
the first order conditions, equation (3), with respect to output q1 , q2 , and emission
quota ē1 , ē2 , yields,
1
1
2
2
dq 1 + π12
dq 2 = − 1 d ē, π21
dq 1 + π22
dq 2 = 2 d ē;
π11
(5)
where d ē = d ē2 = −d ē1 , reflects the assumption of a constant aggregate emission
quota. i = fēi + gēi (hi − zi )/q i − g i (zēi /q i ) is the change in firm i’s marginal
cost of abatement6 caused by a change in its emission quota. Since fēi < 0, gēi <
0, zēi < 0 and assuming that fēi + gēi (hi − zi )/q i < g i zēi /q i then, i < 0, that
is, an increase in firm i’s emission quota yields a decrease in its marginal cost of
abatement. The solution of the two equations in (5) yields,
1
2
π 2 1 + π12
π 1 2 + π21
dq 1
dq 2
2
1
= − 22
, ϕē2 ≡
= − 11
,
(6)
d ē
d ē
Since πiii < 0, πiji = Riji < 0, and i < 0 the numerators of both expressions in
1 2
1 2
π22 − π12
π21 > 0,7 it follows that, ϕē1 < 0
(6) are positive, and given that ≡ π11
2
and ϕē > 0. Thus, if the allocation of emission quota is slanted in favour of firm 2,
its output increases, while that of firm 1’s decreases.
An increase in one firm’s emission quota, lowers its required abatement per
unit of output, which lowers its marginal cost, shifts its reaction function outward,
and increases its output and market share, holding the other firm’s emission quota
constant.8 Since the aggregate level of emission quota is assumed fixed, any
increase in one firm’s emission quota implies an equal reduction in the other firm’s
quota. Thus, if there is a transfer of emission quota from one to the other firm, the
increase in output and market share of the firm whose emission quota increases, is
augmented. The market share of the firm that receives the emission quota transfer
increases more when firms act as Coumot rivals compared to the case that firms
are price takers. The firm, whose share of emission quota decreases, reacts to its
rival’s increased output by reducing its own output, allowing its rival to become
more aggressive. Figure 1 illustrates the above discussion. The equilibrium moves
from point A to B as firm 2’s emission quota increases while firm 1’s decreases.
Thus, under the bureaucratic mechanism market shares depend on the allocation of
emission quota, given firms’ relative efficiency in production and abatement.
The aggregate output effect of the reallocation of emission quota depends on
the production and abatement cost structure of the industry. Summing up the two
expressions in (6) yields the aggregate output effect,
1
2 2
1
1
2
π11 − π12
− π22 − π21
1
2
(7)
Qē ≡ ϕē + ϕē =
ϕē1 ≡
6
EFTICHIOS SOPHOCLES SARTZETAKIS
Figure 1. Market share effects of emission quota reallocation.
Using (3) note that, πiii = 2p + q 1 p − cii − X i and πiji = p + q i p , where
X i is the change in firm i’s marginal cost of abatement caused by a change in its
output.9 Thus, πiii − πiji = p − cii − X i . The necessary and sufficient conditions
for πiii − πiji ≤ 0 is that the slope of the marginal production and abatement costs
are non-negative, cii ≥ 0 and X i ≥ 0. Assuming these conditions hold, both terms
on the numerator of (7) are positive and the sign of the aggregate output effect
depends on the magnitude of these terms. Without specifying firms’ technological
differences, the aggregate output effect is ambiguous.10
B. Competitive Trading of Emission Quota
Under the tradeable emission permits system the regulator issues permits, each
permit allowing emissions of one unit of pollutant, and distributes them free of
charge to the polluting sources. A total of ē emission permits are distributed to the
firms in the differentiated sector, each receiving ēi .11 After the initial distribution,
firms are free to trade emission permits. The regulatory constraint requires that firm
i’s emissions should not exceed its permits holdings after trading, that is, edi + ēi =
H i (q i ) − q i zi , where edi is firm i’s net demand for permits. The Coumot-Nash
equilibrium with permits trading is derived from maximization of the following
profit function,
(8)
max π i q 1 , q 2 , zi = R i q 1 , q 2 − C i q i − Ai q i , zi − P ε edi ,
q i ,zi
ON THE EFFICIENCY OF COMPETITIVE MARKETS
7
where P ε is the competitive emission permits price. The first order conditions of
the above maximization problem are the same in form with the first order conditions leading to (3). Thus, firm i’s equilibrium output resulting from the
solution of
the first-order conditions and the permits market clearing condition 2i=1 edi = 0,
are related in the same way as before to ēi , that is, q i = φ i (ēi , ēj ).
At the equilibrium, both firms’ marginal costs of abatement are set equal to
the permits price and thus, industry’s abatement costs are minimized. Competitive
trading of emission permits achieves the cost-effective distribution of abatement
efforts. On the contrary, at the non-trading equilibrium, the shadow value of emission permits is firm specific and thus, there is dispersion in firm’s evaluation of the
marginal emission permit.
Hereafter we assume that firm 2 is less efficient in abatement than firm 1. Therefore, firm 2 is a net buyer of emission permits and engages in less abatement effort
relative to the non-trading equilibrium. At the trading equilibrium, firms’ market
shares depend solely on production cost differences because marginal costs of
abatement are equalized. Recall that under the bureaucratic mechanism the initial
allocation of emission permits played a role in determining market shares. When
trading is allowed, the initial allocation of permits affects firms’ profits but not
market shares, that is, it only has distributional consequences.
The change in firms’ emission permit holdings after trading is determined endogenously and thus, simple comparative static analysis cannot be used to compare
the bureaucratic mechanism with trading of emission permits. The equilibria of
the two models are compared using mean value theorem analysis. For the relevant
version of the mean value theorem and the proofs of propositions see Appendix 2.
Proposition 1 compares firms’ market shares and industry’s output under the two
regulatory regimes.
Proposition 1.
i. Trading of emission permits results in output reallocation towards the less
efficient in abatement firm.
ii. Assuming that, p < 0, cii ≥ 0, X i ≥ 0 and i < 0, a necessary condition
for industry’s output to decrease as a result of competitive permits trading is
2|
≤
(p − ci1 − X 1 ) 2 − (p − ci2 − X 1 ) 1 < 0. A sufficient condition is |
| 1 |
(ci2 +X2 ) .
(ci1 +X1 )
Part i of Proposition 1 follows directly from equation (6). Since trading results
in a transfer of permits towards the less efficient in abatement firm, its market share
increases. Part ii of Proposition 1 states that there are situations in which industry’s
output decreases as a result of permits trading, since there are no conditions to
prevent (p − ci1 − X 1 ) 2 < (p − ci2 − X 1 ) 1 from arising. To understand
when these counterintuitive situations are likely to arise we resort to the sufficient
condition presented Proposition 1. ii. Recall that permits trading reallocates both
8
EFTICHIOS SOPHOCLES SARTZETAKIS
emission permits and output between firms. The above-stated sufficient condition
requires that the ratio of the output reallocation effect on firms’ marginal costs
(RHS of the condition) should be higher than the ratio of the emission permit
reallocation effect on firms’ marginal costs (LHS of the condition). In other words,
the increase in industry’s total costs due to firm 2’s gains in market share, should
dominate the reductions in industry’s abatement costs due to reallocation of abatement efforts from firm 2 to firm 1. In order for industry’s output to decrease as a
result of permits trading, the adverse effect from the resulting output reallocation,
when present, should outweigh the positive effect of emission permits trading on
abatement costs.
Although the market shares reallocation effect is well understood (see for
example Malueg (1990)), the possibility that industry’s output might decrease as
a result of emission permits trading has, to the best of our knowledge, never been
raised before. The literature unanimously assumes that emission permits trading
yields an increase in industry’s output as long as firms are price takers in the
emission permits market. This is attributed to the fact that the interaction between
production decisions and permits trading has not been fully developed in previous
works. Neglecting the effect that emission permits trading has on production costs,
and given that industry’s abatement costs decrease (for a given level of output),
one is led to assume that industry’s output necessarily increases. Within the model
employed in the present paper, additional conditions must be satisfied for the
output effect of emission permits trading to be positive. These conditions establish
the appropriate connections between the effects of emission permits trading on
production and abatement costs. In Section IV we construct examples in which
these conditions are violated resulting in output decrease.
Within the partial equilibrium framework of our analysis we examine the effect
of emission permits trading on welfare, defined as the sum of consumer and
producer surplus,12
2
2
W ei , ej = u q i , q j −
Ci qi −
Ai q i , zi ,
i=1
(9)
i=1
where q i = ϕ i (ei , ej ) is given in (4) and zi = zi (q i , ei ) is given in equation
(A.1.3) in Appendix 1. ei denotes firm i’s level of allowable emission, which under
the bureaucratic mechanism is fixed at ēi and under emission permits trading is
ēi + edi . Differentiating (9) with respect to the change in firms’ allowable emission
due to emission permits trading, yields,
∂W
= pQē −
Aiz zēi ,
ci + Aiq + Aiz zqi ϕēi −
∂e
i=1
i=1
2
2
(10)
where ϕēi denotes the total effect of emission permits trading on firm i’s output,
given in equation (6), Qē denotes the effect on industry’s output, which is given
ON THE EFFICIENCY OF COMPETITIVE MARKETS
9
in equation (7), and Aiq and Aiz are the marginal cost of abatement with respect to
output and abatement per unit of output. Proposition 2 compares welfare under the
two regulatory regimes.
Proposition 2.
The necessary and sufficient condition for emission permits
trading to yield lower
welfare relative to a bureaucratic mechanism is pQē < 2i=1 (ci + Aiq + Aiz zqi )ϕēi +
2
i i
i=1 Az zē .
As it is illustrated in the next Section, the condition in Proposition 2 holds in a
variety of situations. What is interesting here is to determine the conditions under
which welfare is more likely to be lower under permits trading. Consider the case in
which output increases as a result of emission permits trading, that is, |ϕē2 | > |ϕē1 |,
keeping in mind that since firm 2 is the less efficient in abatement firm, ϕē2 > 0
and ϕē1 < 0. The direct effect of emission permits trading on industry’s cost of
abatement
should be negative, meaning that total abatement costs should decrease,
that is, 2i=1 Aiz zēi < 0. Thus, welfare could be lower under permits trading only
if the less efficient in abatement firm is also less efficient in production, that is,
c2 + A2q + A2z zq2 > c1 + A1q + A1z zq1 . Welfare is more likely to decrease the greater
is the technological difference between the two firms.
The condition in Proposition 2 can also be written as 2i=1 (Aiq + Aiz zqi )ϕēi +
2
i i
1 1
2 2
i=1 Az zē > (p − c )ϕē + (p − c )ϕē . Thus, it is the effect of emission permits
trading on industry’s cost of abatement that determines the sign of the welfare
effect. If the output reallocation effect is large, the increase in industry’s output
is small while the increase in industry’s abatement costs could be substantial and,
therefore, the value of the right hand side of the inequality could exceed the value
of the left hand side. If the benefits from the more efficient allocation of abatement
efforts are mainly absorbed in redistributing output and profits to the less efficient
firm rather than increasing industry’s output, emission permits trading is clearly
welfare inferior to the bureaucratic mechanism.
The intuition underlying results of this nature is easier understood in the context
of the second best problem. On the one hand, oligopolistic behaviour in the
product market yields a higher market share for the less efficient firm relative
to the cost-minimizing allocation of production. On the other hand, bureaucratic
mechanisms of allocating emission quota yield inefficient distributions of abatement efforts, that is, the firm with the less efficient abatement technology engages
in more than the efficient abatement effort.13 Allowing firms to transfer emission quota through a competitive market, corrects the abatement misallocation
problem but could aggravate the production misallocation problem. Removing only
one distortion could adversely affect both output and welfare. Emission permits
trading aggravates the output misallocation problem when firms’ technological
efficiency in production and abatement are positively correlated. Price-taking behaviour in the emission permits market assures that the less efficient in abatement
10
EFTICHIOS SOPHOCLES SARTZETAKIS
firm acquires emission permits, reducing industry’s marginal abatement cost (for
given output), but increasing industry’s production costs. If firms’ technological
differences are more prominent in production rather than abatement, industry’s
output may decrease. Furthermore, welfare may decrease even if industry’s output
increases as long as the increase in output is achieved by aggravating the output
missalocation problem.
IV. Example
In this Section, we examine a simple example of the economy described in Section
II with the following characteristics: utility is quadratic u(Q) = aQ − (1/2)bQ2 ,
yielding a linear demand function p = a − bQ; both firms have common, constant
rate of emission ρ, that is, ei = H i = ρq i ; constant marginal cost of production
ci ; and quadratic cost of abatement Ai = ei (Z i )2 , where Z i = zi q i . Under these
specifications firm i’s and industry’s output under the bureaucratic mechanism that
prohibits trading of emission quota are
i
i
i
j
j
i
j
j
a
−
2c
2a
−
c
+
c
−
ρ
2λ
−
λ
−
c
−
ρ
λ
+
λ
q ic =
, Qc =
, (11)
2b
3b
where the superscript c indicates equilibrium values under the bureaucratic mechanism and λi is firms i’s shadow value of emission quota, which at the equilibrium
equals firm i’s marginal cost of abatement.
Firm i’s and industry’s output under trading of emission permits is,
a − 2ci + cj − ρP ε
2a − ci + cj − 2ρP ε
, Qt =
,
(12)
3b
3b
where the superscript t indicates values at the emission permits trading equilibrium.
The equilibrium price of emission permits lies between the two firms’ evaluation
of emission permits when permits are not tradeable, that is, λ1 < P ε < λ2 . The
permit price can be expressed as a weighted sum of firms’ shadow values of their
emission quota under the bureaucratic mechanism,
ej 3b + 2ei ρ 2 λi + ei 3b + 2ej ρ 2 λj
ε
(13)
P =
3b ei + ej + 4ei ej ρ 2
q it =
The relationship between firms’ evaluation of emission permits under the two
regulatory equilibria facilitates the comparison of firms’ and industry’s output
under the two equilibria. From equations (11), (12) and (13) we derive,
ρ b ei + 2ej + 2ei ej ρ 2 λi − λj
it
ic
,
(14)
q −q =
b 3b ei + ej + 4ei ej ρ 2
Qt − Qc = ρ ei − ej λi − λj .
Trading of emission permits increases the output of the firm with the higher evaluation of its emission permits, that is, the higher marginal cost of abatement at the
ON THE EFFICIENCY OF COMPETITIVE MARKETS
11
bureaucratic equilibrium. Industry’s output increases as a result of permits trading
if ei > ej ⇒ λi − λj . Thus, allowing trading of emission permits yields an
increase in industry’s output if and only if the less efficient in abatement firm, that
is, firm 2 (e2 > e1 ), has the higher marginal cost of abatement at the bureaucratic
equilibrium, that is, λ2 < λ1 . The difference of firms’ shadow value of emission
permits is,
λi − λj =
(15)
2
2
i
j
ρ c − c + b ēi − ēj
3b ei − ej ρ Q̂ − ē − 3b ei + ej + 4ei ej ρ
,
K
where K = 3b2 +4ρ 2 ei ej ρ 2 +b(ei +ej ), and Q̂ denotes the Coumot-Nash output
in the absence of any restriction in emission. Industry’s emission quota cannot
exceed unregulated emission, that is, ρQ > ē. Thus, firm 2 has lower evaluation
of emission quota at the equilibrium, that is, λ2 < λ1 , if it is less efficient than firm
1 in production c2 < c1 , and receives a higher share of emission permits ē2 > ē1 .
Result 1 is a specialized version of Proposition l.
Result 1.
Assuming linear demand, constant marginal cost of production, constant emission
rate per unit of output and quadratic abatement costs, a bureaucratic mechanism
yields higher output iff ei > ej results in 3b2 (ei − ej )(ρ Q̂ − ē) < [3b(ei + ej ) +
4ei ej ρ 2 ][ρ(ci − cj ) + b(ēi − ēj )].
The possibility that the bureaucratic mechanism dominates competitive trading
of emission permits is higher, the less restrictive is the aggregate emission reduction
requirement, that is, the smaller the value of ρQ − e. It is also worth noticing that a
bureaucratic mechanism allocating emission quota in proportion to firms’ unregulated levels of emissions, that is ēi = ξρqt ,14 assures that ci > cj ⇒ ēi < ej .
Therefore, such an allocation of emission permits, makes the second term in the
numerator of (15) smaller and thus, it weakens the possibility that a positive correlation in firms’ abatement and production efficiency will result in the dominance
of the bureaucratic mechanism.
To provide a graphical illustration of Result 1 we present simulations of the
model. The initial values of parameters are chosen so as to satisfy all nonegativity
and other necessary conditions.15 Figure 2 illustrates the difference in industry’s
output between the two regulatory regimes (z axis) as a function of firms’ difference in abatement (denoted as e = e2 − e1 on the y axis) and production
technology (denoted as c = c2 −c1 on the x axis). To facilitate clear exposition of
the results, we focus on the region in which firm 2 is less efficient in both abatement
and production, that is, when production and abatement technologies are positively
correlated. At e = 0 and c = 0 firms have the same abatement and production
technologies, and as e and c increase, firm 1 becomes more efficient in both
abatement and production. To improve presentation, Figure 2 includes the plane
corresponding to zero difference in output, which is presented in solid black colour.
12
EFTICHIOS SOPHOCLES SARTZETAKIS
Figure 2. Difference in industry’s output.
Figure 2 illustrates that the difference in industry’s output is negative Qt < Qc for
a range of parameter values in the area in which firm 2’s inefficiency is much more
prominent in production rather than in abatement.
As noted above, the initial distribution of emission permits affects the range
of parameters within which the bureaucratic mechanism results in higher output.
In the case presented in Figure 2, the allocation of emission permits is inversely
related to firms’ marginal cost of production. When c is large, firm 2’s share of
emission quota is very small and its marginal abatement cost very high, and thus, its
output very low at the bureaucratic equilibrium. When trade of emission permits
is allowed, acquisition of additional emission permits leads to a rapid expansion
of firm 2’s output, which is faster than the reduction in firm 1’s output. Thus,
the possibility that Qt < Qc is reduced when emission permits are distributed
according to historical emission levels. On the contrary, if emission permits were
allocated equally between firms, the range of parameters for which the bureaucratic
regulation results in higher output increases. Thus, the initial distribution of emission permits determines the range of parameter values for which a bureaucratic
mechanism yields higher output than emission permits trading.16
We now proceed with the discussion of the welfare effect of emission permits
trading. Under this Section’s demand and technology specifications, the welfare
difference between the two regulatory regimes is,
(16)
W t − W c = (1/2) p t − p c Qt − Qc − ci q it − q ic −
2
it
A − Aic ,
cj q j t − q j c −
i=1
where pt and pc denote output prices under the two regulatory regimes. Welfare
increases as a result of emission permits trading when industry’s output increases
and the resulting increase in consumer surplus17 exceeds industry’s costs incurred
ON THE EFFICIENCY OF COMPETITIVE MARKETS
13
Figure 3. Welfare difference.
in achieving this increase. Substituting the equilibrium values in (16) and simplifying, the welfare difference can be written as,
(17)
W t − W c = λ2 (ω2 e + ω3 ) + λ (ω4 e − ω5 c) /ω1 ,
where λ = λi − λj and ωi , i = 1, 5 are all positive parameters.18 Not surprisingly,
the same parameters that determine the sign of industry’s output difference are
present in expression (17) albeit in one order higher. Result 2, which is a special
version of Proposition 2, summarizes the conditions resulting from expression (17).
Result 2.
Assuming linear demand, constant marginal cost of production, constant emission
rate per unit of output and quadratic abatement costs, a bureaucratic mechanism
yields higher welfare iff λ[(ω2 λ + ω4 )e + ω3 λ − ω5 c] < 0.
Although the possibility that output and welfare decrease as a result of emission
permits trading arises in the same region, it is not at all necessary that welfare
decreases when output decreases. Figure 3 presents the simulations of the welfare
difference. Welfare increases when industry’s output decreases, while it decreases
for a range of parameters that yield an increase in industry’s output (compare
with the aid of Figure 4 that reports the
Figure 2 to 3).19 This result is explained
difference in industry’s abatement costs, 2i=1 (Ait − Aic ).
Industry’s abatement costs increase with industry’s output. However, the rate
of change in industry’s abatement costs is greater than the rate of change in
industry’s output around the curve tracing zero values in the respective surfaces.
As output starts increasing, industry’s abatement costs are rising fast enough to
accelerate the decrease in industry’s profits to a level higher than the rate of
increase in consumer surplus. After a point, the increase in industry’s abatement
costs smoothens, inducing a similar effect on the decrease in industry’s profits and
thus, welfare increases.
14
EFTICHIOS SOPHOCLES SARTZETAKIS
Figure 4. Difference in industry’s abatement costs.
VI. Conclusions
We have demonstrated that competitive markets cannot assure the efficient allocation of emission permits. We have proved that a bureaucratic mechanism could
yield higher output relative to competitive emission permits trading. We have also
showed that a bureaucratic mechanism can be welfare superior to competitive
trading of emission permits in a number of situations. In deriving these results
we focused on clarifying the intuition behind the result that, under oligopolistic
behaviour in the product market, the less efficient firms hold a higher than the
welfare maximizing share of emission permits.
Our results should be seen as cautioning policy makers not to presume that
competitive trading of permits assures social efficiency. Our results should not
be perceived as arguments against the use of tradeable emission permits. As the
simulations in Section IV indicate, allowing trade of emission permits generates, in
the majority of situations, large output and welfare gains. However, policy makers
should be aware of the potential problems so as to reserve some flexibility for
their response when welfare-decreasing situations arise. One possible direction
of policy makers’ response is to restrict the amount of trades based on firms’
production efficiency. Clearly, this solution requires a great deal of information
and could have a great cost in forgone welfare-maximizing trades in case of error.
That is why we believe that more effort, both at theoretical and empirical level,
should be devoted in evaluating modifications to the unrestricted trading, taking
into account the characteristics of the environment in which the market for quota
licenses operates. It is also important to note that the initial allocation of permits
plays an important role in determining the range of possible inefficiencies.
Although our analysis builds around tradeable emission permits, the general
formal model we use admits several interpretations, and thus, our results have
broader consequences for public policy since they apply to a wide range of govern-
ON THE EFFICIENCY OF COMPETITIVE MARKETS
15
ment licensing. Consider for example the case that a government uses quotas to
restrict imports of a factor used as an intermediate input in a domestic oligopolistic
industry. Import quotas are imposed to support domestic production of this factor
that is undertaken by subsidiaries of the final product oligopolists, which exhibit
technological differences. Competitive trading of import quotas will result in transferring quotas towards the less efficient firms. Assuming Cournot reactions in the
final product market, firms with higher opportunity cost of quotas will acquire more
than the welfare-maximizing number of quotas. If less efficient oligopolists are
associated with less efficient subsidiaries, the possibility arises that competitive
trading of import quotas is welfare inferior to a bureaucratic allocation of input
quotas.
Acknowledgements
I would like to thank Barbara Spencer, Peter Tsigaris, Gorgon Tarzwell, Thomas
Ross and Christos Constantatos for their comments and suggestions on earlier
versions of this paper. I would also like to thank two referees of this journal
for their valuable input on the final version of the paper. Most of this work was
completed while I was a faculty member at the University College of the Cariboo.
I gratefully acknowledge financial support of the Scholarly Activity Committee of
the University College of the Cariboo.
Notes
1. Borenstein (1988) demonstrates that lumpiness of operating licenses and the transfer of production to inefficient entrants due to product market imperfections inhibits the ability of competitive
markets to efficiently allocate operating licenses. Spencer (1997) finds that a bureaucratic scheme
that prohibits resale of quotas for imported capital equipment could dominate a tradeable quota
system by supporting an “infant” domestic capital equipment industry.
2. Throughout the paper we refer to firms as more efficient in production/abatement, to indicate
firms with relatively lower marginal production/abatement costs. When a more specific definition
is required there is a special note such as in the case of footnote 13.
3. That is, if the firm with the lower marginal production cost has also the lower marginal abatement
cost. If the age of capital is considered, it is reasonable to expect that production and abatement
technologies are positively correlated. Newly acquired capital is more productive, less polluting
and more likely to be associated with newer, more productive abatement equipment.
4. Thus, marginal utility of income is constant and equal to one.
5. Appendix 1 provides some details of the calculations leading to (3).
6. Marginal cost of abatement with respect to output and abatement per unit of output – the latter
weighted by the uncontrolled emission per unit of output – that is, i = ∂Aiq + Aiz (hi −
zi )/q i /∂e.
7. See Appendix 1 for a discussion regarding the meaning and importance of this condition.
8. Total differentiation of (3) with respect to q i yields the slope of firm i’s reaction function,
which is negative given the second order conditions and strategic substitutability, dq i /dq j =
i /π i < 0.
−Rig
ii
9. X is defined in a similar way as i in footnote 5. That is, Xi = ∂Aiq + Aiz (hi − zi )/q i /∂q.
16
EFTICHIOS SOPHOCLES SARTZETAKIS
10. In the absence of technological differences, a reallocation of emission quota leaves aggregate
output unaffected, while increasing the market share of the firm whose emission quota increases.
11. For simplicity we assume that the initial emission permits allocation as well as the total amount
of allowable emission are the same as under the bureaucratic mechanism.
12. Assuming that the aggregate emission quota remains the same under both regulatory systems,
the emission damage is identical under both regulatory systems, and thus, we neglect the V(E)
term.
13. This result is independent of the initial allocation of emission quota.
14. Where ξ < 1, is the rate by which the government wishes to decrease emissions.
15. The initial values of the parameters used for the simulations are: a = 1,500; b = 0.20; c1 = 200;
e1 = 0.10; ρ = 0.60, and ξ = 0.40. Figures 2, 3 and 4 present simulations for the following range
of values: 200 ≤ c2 ≤ 500 and 0.10 ≤ e2 ≤ 0.70.
16. The point that the initial allocation of permit can affect the efficiency of emissions trading
markets when the product market is non competitive has been raised in a number of works,
and in particular in Hahn (1984).
17. The first term in (16) corresponds to the area under the linear demand pt (Qt −Qc )+(1/2)(pc −
pt )(Qt − Qc ).
18. The values are: ω1 = 36bei ej x 2 , ω2 = 3b[27b2 ej (ei + 2ej ) + 2ei ej ρ 2 [3b(3ei + 5ej ) −
2ei ej ρ 2 ]], ω3 = 18bej 2 [4ei 3 ρ 4 + 9b2 ej + 6b(ei + ej )ei ρ 2 ], ω4 = 12bei ej ρx(α − ci − ρλi ),
ω5 = 4ei ej ρx[2(2x + ei ej ρ 2 ) + 3bej ], and x = 3b(ei + ej ) + 4ei ej ρ 2 . Detailed calculations
leading to (17) are available from the author upon request.
19. Welfare increases as a result of emission permits trading, when the more efficient in abatement
firm is less efficient in production. This case is not shown in Figure 3.
20. Given that fqi > 0, gqi > 0 a necessary and sufficient condition for Xi ≥ 0 is that the sum of the
first two terms is greater than the absolute value of the last term, when this term is negative. For
example, this term is negative if emissions are assumed proportional to output, H i = q i . In
this case, and given the value of zi in (A.1.3), the term in brackets in the right hand side becomes
−2ē/q i .
21. For sufficient conditions for the existence, uniqueness and stability of Cournot equilibrium see
Friedman (1977), Nishimura and Friedman (1981) and Gaudet and Salant (1991).
22. We adopt the version of the theorem from Brander and Spencer (1983), p. 233.
23. Firm 2 buys emission permits for as long as the cost reduction generated by the additional
emission permits exceeds the permit price.
References
Borenstein, S. (1998), ‘On the Efficiency of Competitive Markets for Operating Licenses’, Quarterly
Journal of Economics 103, 357–385.
Brander, J. A. and B. J. Spencer (1983), ‘Strategic Commitment with R&D: The Symmetric Case’,
The Bell Journal of Economics 14, 225–235.
Fershtman, C. and A. de Zeeuw (1996), ‘Transferable Emission Permits in Oligopoly’, Manuscript.
The Netherlands: Center, Tilburg University.
Friedman, J. W. (1977), Oligopoly and the Theory of Games. Amsterdam: North Holland Publishing
Company.
Hahn, R. W. (1984), ‘Market Power and Transferable Property Rights’, Quarterly Journal of
Economics 99, 753–765.
Gaudet, G. and S. Salant (1991), ‘Uniqueness of Cournot Equilibrium: New Results from Old
Methods’, Review of Economic Studies 58, 399–404.
Long, Ngo van and A. Soubeyran (1997), ‘Cost Manipulation in Oligopoly: A Duality Approach’,
S.E.E.D.S. Discussion Paper 174, Southern European Economics Discussion Series.
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Long, Ngo van and A. Soubeyran (2000), ‘Permis de Pollution et Oligopole Asymétrique’, Economie
& Prevision numéro 143–144, 83–89 (avril–juin 2000/2–3).
Malueg, D. A. (1990), ‘Welfare Consequences of Emission Credit Trading Programs’, Journal of
Environmental Economics and Management 19, 66–77.
Misiolek, W. S. and H. A. Elder (1988), ‘Exclusionary Manipulation of Markets for Pollution Rights’,
Journal of Environmental Economics and Management 16, 156–166.
Nishimura, K. and J. W. Friedman (1981), ‘Existence of Nash Equilibrium in N Person Games
without Quasi-concavity’, International Economic Review 22, 637–648.
Sartzetakis, E. S. (1997a), ‘Tradeable Emission Permits Regulations in the Presence of Imperfectly
Competitive Product Markets: Welfare Implications’, Environmental and Resource Economics 9,
65–81.
Sartzetakis, E. S. (1997b), ‘Emission Permits Markets as Vehicles for Raising Rivals’ Cost
Strategies’, Review of Industrial Organization 12, 751–765.
Spencer, B. J. (1997), ‘Quota Licenses for Imported Capital Equipment: Could Bureaucrats Even Do
Better Than the Market?’, Journal of International Economics 43, 1–29.
Von der Fehr, N.-H. M. (1993), ‘Tradeable Emission Rights and Strategic Interactions’, Environmental and Resource Economics 3, 129–151.
Appendix 1
For positive values of output and abatement per unit of output, the first order conditions of
the constrained profit maximization problem in (2) are,
(A.1.1)
π i i ≡ Ri i q i , q j − ci q i − Aq i − λi hi + λi zi = 0,
−Az i + λi q i = 0 ⇒ λi = Az i /q i ,
(A.1.2)
along with the regulatory constraint that is binding assuming λi > 0. Where λi , the
Lagrange multiplier of the constrained maximization problem, should be seen as the
shadow value of firm i’s emission permits. The regulatory constraint is solved for firm
i’s abatement per unit of output zi as a function of q i and ēi ,
(A.1.3)
zi = H i q i − ēi /q i = zi q i , ēi .
The required abatement per unit of output is decreasing in the level of emission quota ēi
and is increasing in output q i , i.e. zē i < 0 and zq i > 0 since we have assumed that hi i ≥
0. Substituting the value of zi from (A.1.3) into Aq i and Az i we derive Aq i = f i (q i , ēi ),
with fq i > 0, fē i < 0 and Az i = g i (q i , ēi ), with gq i > 0, gē i < 0. Substituting into
(A.1.1) the value of λi from (A.1.2) and the values of Aq i and Az i as defined above, firm
i’s output reaction function is expressed as a function of firm i’s emission quota,
(A.1.4)
π i i = Ri i q 1 , q 2 − ci q i − f i q i , ēi −
hi q i − g i q i , ēi /q i = 0.
g i q i , ēi
The second order conditions with respect to output require, π i ii = Rii i − ci − Xi <
0, where Xi = fq i + gq i [(hi − zi )/q i ] + g i (hi i − zq i )q i − (hi − zi )/(q i )2 is the
18
EFTICHIOS SOPHOCLES SARTZETAKIS
change in firm i’s marginal cost of abatement caused by a change in its output, that is
∂Aq i + Az i [(hi − zi )/q i ]/∂q i . This marginal change is assumed nonnegative.20
We further assume that own effects of output on marginal profit exceed cross effects,
1 2
1 2
≡ π11
π22 − π12
π21 > 0.
(A.1.5)
This assumption is related to conditions for uniqueness of the equilibrium and to reaction function stability. To avoid complicating the presentation unnecessarily, we consider
situations in which the values of the parameters yield unique and stable equilibria.21
Appendix 2
Mean value theorem22
Let f (e) be a continuously differentiable real-valued function defined on a convex subset
of Euclidean n-space. Let ec and et be two vectors in this subset. Then, there exists a point
e∗ such that,
(A.2.1)
f ≡ f et − f ec = ∇f e∗ · et − ec ,
where ∇f (e∗ ) is the gradient of f evaluated at e∗ , and e∗ = ec + θ (et − ec ) for some
θ ∈ (0, 1). The two vectors we are interested in are firms’ holdings of emission quotas at
the bureaucratic and the emission permits trading equilibria, that is, ec = (ē1 , ē2 ) and et =
di
t
c
(ē1 + ed1 , ē2 + ēd2 ). The difference in emission
quota for firm i is ei = ei − ei = e .
Competition in the permits market requires 2i=1 edi = 0 and thus, ed2 = −ed1. It follows
that e = e2 = −e1 .
Recall that we have assumed that firm 2 is less efficient in abatement than firm 1. Thus,
for any allocation of emission permits before trading of emission permits is completed, the
change in cost and marginal cost of abatement due to the change in permits holdings is
stronger for firm 2 than firm 1, that is, ∀e∗ ∈ (ec , et ) we have that |A2z zē2 | > |A1z zē1 | and
| 2 | > | 1 |. Further, firm 2 will be a net buyer while firm 1 will be a net seller of emission
permits, that is, ed2 > 0 and ed1 < 0.23
Proof of Proposition 1.
(i) Applying (A.2.1) to q i = ϕ i (e1 , e2 ) yields
q i = ϕēi 1 e1 + ϕēi 2 e2 ,
(A.2.2)
where ϕēi i denotes the partial effect of the emission permits reallocation on firm i’s output,
with the total effect, ϕēi = −ϕēi 1 + ϕēi 2 given in (6). ϕēi is evaluated at a vector e∗ , where
e∗ ∈ (ec , et ). Since e = e2 = −e1 , equation (A.2.2) yields q i = ϕēi e. Recall
that firm 2 is assumed to be less efficient in abatement relative to firm 1, which implies
that ϕē2 > 0 and ϕē1 < 0. Therefore, firm 2’s market share increases while that of firm 1’s
decreases as a result of emission permits trading. (ii) Applying (A.2.1) to industry’s output Q = 2i=1 q i = 2i=1 ϕ i (ēi , ēj ) yields
Q = Qē1 e1 + Qē2 e2 ,
(A.2.3)
ON THE EFFICIENCY OF COMPETITIVE MARKETS
19
where Qēi denotes the partial effect of the emission permits reallocation on aggregate
output with the total effect Qē given in (7). Qē is evaluated at a vector e∗ , where e∗ ∈
(ec , et ). Since e = e2 = −e1 , equation (A.2.3) yields Q = Qē e. From equation
(7) we get that the necessary and sufficient condition for Q > 0 is, (p − ci 1 − X1 ) 2 −
(p − ci 2 − X1 ) 1 > 0. Given that when evaluated at e∗ we get | 2 | > | 1 |, then
p ( 2 − 1 ) > 0. Therefore, a sufficient condition for Q > 0 is −(ci 1 + X1 ) 2 +
(ci 2 + X1 ) 1 > 0. Taking into account the assumption regarding marginal cost changes,
2|
2
2
i +X )
≥ (c
that is, ci i + Xi ≥ 0 and i < 0, the sufficient condition can be written as |
| 1 |
(ci 1 +X 1 )
Q.E.D.
Proof of Proposition 2.
Applying (A.2.1) to welfare W (e1 , e2 ) yields
W = Wē1 e1 + Wē2 e2 ,
(A.2.4)
where Wēi denotes the partial effect of the emission permits reallocation on social welfare
with the total effect Wē given in (10). Wē is evaluated at a vector e∗ , where e∗ ∈ (ec , et ).
Since e = e2 = −e1 , equation (A.2.4) yields W = Wē e. Therefore the
necessary
and sufficient condition for W
< 0 is derived from equation (10), that is,
pQē < 2i=1 (ci + Aq i + Az i zq i )ϕēi + 2i=1 Az i zē i . Q.E.D.