Environmental Policy and the Size Distribution
of Firms
Jessica Coria(a) and Efthymia Kyriakopoulou(a);(b)
(a)
(b)
University of Gothenburg
Beijer Institute of Ecological Economics
June 21, 2013
Abstract
In this paper we analyze the e¤ects of environmental policies on the
size distribution of …rms. We model a stationary industry where the observed size distribution is a solution to the pro…t maximization problem
of heterogeneous …rms. We compare the equilibrium size distribution under emission taxes, non-tradable and performance standards. Our results
indicate that each regulation a¤ects di¤erently …rms of heterogeneous size
and leads to di¤erent welfare levels.
1
Introduction
In recent years, several environmental regulations have been introduced to control emissions of local and global pollutants. These policies have a clear objective: to induce …rms to reduce emissions by investing in cleaner/energy saving technologies and promoting industrial turnover by modifying, among other
things, the possibility of entry of new …rms, exit of incumbent …rms, and the
relative competitive advantage of active …rms. Moreover, environmental regulations may also a¤ect the distribution of market shares and the related size
distribution of …rms if compliance changes the optimal plant size or the range
of optimal sizes. As pointed out by Evans (1986), the di¤erential e¤ect of regulation across …rm size is important for at least three reasons. First, when there
are scale economies in regulatory compliance, it might be optimal to exempt or
impose lighter regulatory burden on smaller …rms, or design regulations that are
neutral across …rm size to minimize the disproportionate impact of environmental regulatory requirements on small businesses ( See Brock and Evans 1985).
Second, society may have an interest in preserving small businesses because of
antitrust or other noneconomic reasons. Third, the incidence of regulatory costs
across …rm size may tell us something about the interest of certain groups of
businesses in supporting alternative regulatory policies.
1
While much of the existing literature assessing the impact of environmental
regulation on market structure tends to focus on entry, exit, and the number of
active …rms (See for instance, Katsoulacos and Xepapadeas 1995, 1996; Lahiri
and Ono 2007), the focus of this paper is on the e¤ects of the choice of policy
instruments on the size distribution of …rms and the dynamics of growth of
individual …rms over time. In this regard, as mentioned above, one could argue
that if there are some economies of scale in compliance, the minimum optimal
…rm size is larger under regulation. Thus, the market share of large …rms should
increase as small …rms withdraw from the industry or expand and become larger.
However, the e¤ects of environmental policy will depend on the choice of policy
instrument since di¤erent policy instruments redistribute intraindustry rents
di¤erently; if small …rms are treated less harshly, regulation will increase the
market share of small plants as large plants leave the industry or shrink in size.
In this paper we analyze and compare the resulting size distribution under three
policy instruments that di¤er in terms of their incidence across …rm size, namely
uniform emission standards, emission taxes and performance standards. Under
performance standards, the intensity of emissions is …xed by policy and neutral
across size. Instead, under emission standards, the regulatory goal is expressed
as an absolute emission limit, which should tend to favor smaller …rms as the
limit might not bind their emissions. Finally, emission taxes raise the per unit
cost of production for both small and large …rms in an industry. However, if
larger …rms use the input that originates emissions more intensively, an emission
tax should shift the supply curve of these …rms to a greater extent than it shifts
the supply curve of smaller …rms.
The size distribution of …rms has been extensively studied in the industrial
organization literature1 . Nevertheless, to the best of our knowledge, the dynamic e¤ects of the choice of environmental policy instruments shaping the size
distribution has not been studied previously. Perhaps the most closely related
work in the environmental economics literature is Pashigian (1984), and Sengupta (2010). Pashigian (1984) measures the e¤ects of environmental regulation
under the 1970s Clean Air Act on the size distribution of plants and the distribution of factor shares in United States over the period 1958-1977. He …nds
that environmental regulation had a signi…cant e¤ect on plant structure indicating the existence of signi…cant economies of scale on compliance; environmental
regulation favored large plants relative to small plants and raised capital intensity. Sengupta (2010) investigates the links between environmental regulation
and market structure by analyzing how regulation modi…es the optimal scale of
…rms and the dynamic path of the industry. Nevertheless, unlike our paper, she
1 Most
of the literature deals with the distributional properties of …rm size (see for instance,
Cabral and Mata 2003 and Angelini and Generale 2008). However, more recent research has
integrated the size distribution of …rms into standard economic theory. Attempts to explain
the size dynamics have investigated the e¤ects of bad productivity shocks (Hopenhayn 1992
and Ericson and Pakes 1995), learning (Jovanovic 1982), ine¢ ciencies in …nancial markets
(Cabral and Mata 2002 and Clementi and Hopenhayn 2006), the exogenous distribution of
managerial ability in the population (Lucas 1978 and Garicano and Rossi-Hansberg 2004) and
the e¢ cient accumulation and allocation of factors of production (Rossi-Hansberg and Wright
2007).
2
focuses on the case of a speci…c regulation, namely environmental taxes, comparing the equilibrium paths corresponding to di¤erent exogenous tax levels
and outlining conditions under which industries with higher taxes are associated with higher investment in compliance cost reduction and higher shake-out
of …rms over time.
To study the e¤ects of the choice of policy instruments on the size distribution of …rms, we follow the seminal model by Lucas (1978) where the underlying
distribution of …rm’s sizes in the industry is the result of the existence of a productive factor of heterogenous productivity.2 In Lucas’s model, such a factor is
the managerial technology, while in ours, it is the energy e¢ ciency of …rms. In
such a setting, we introduce di¤erent environmental policies and analyze the resulting size distributions, as well as the variations on size distribution that arise
as a result of investments that reduce the cost of compliance with environmental
regulations. We also compute the e¤ects on welfare and illustrate our results
through numerical examples. Hence, the paper is organized in …ve sections. The
next section presents the model and the underlying distribution of …rm’s sizes
in the absence of environmental policies. The third section analyzes how the
choice of a policy instrument modi…es the size distribution of …rms. The forth
section presents some numerical simulations and analyzes welfare implications.
The …nal section concludes.
2
The Model
We consider a perfectly competitive stationary industry consisting of a continuum of risk-neutral polluting …rms of mass 1. Firms produce a homogeneous
good using two inputs, energy (e) and labor (l). Firms di¤er in terms of the
parameter re‡ecting energy e¢ ciency, which is assumed to be uniformly distributed on the interval ;
Assuming a Cobb–Douglas technology, the production function of …rm i is
then characterized as:
q( i ; e; l) = [ i ei ] li
_ ;
> 0;
+
< 1;
(1)
where q is the amount of output produced by a …rm using e units of energy and l
units of labor, i is the energy e¢ ciency of …rm i, and is a general productivity
parameter. In the absence of environmental policy, …rm i maximizes net pro…ts
i through the choice of inputs:
max
e;l
i
= p [ i ei ] li
wli
zei
F:
(2)
Where w and z are the equilibrium wage rate and energy price, and p rep2 The heterogeneity of the available physical capital with respect to productivity and emission intensity is well established in the environmental literature. Changes in energy use and
emissions pro…les of an industry are the result of complex interrelationships among a multitude
of time-varying technological and economic drivers (see Ruth et al. 2004 for discussion).
3
resents the output price. The …rst order conditions (FOCs) are given by:
p
i
p
1
ei
li = z;
1
ei li
i
(3)
= w:
(4)
Dividing by parts and solving wrt l; we get:
z NR
e :
w i
liN R =
(5)
R NR
Substituting equation (5) in the FOC, we can solve for eN
; li as:
i
h
R
eN
= p
i
h
liN R = p
1
w
1
z
w (1
i1 1
)
[1
]
i
z
i
i1
i1
1
:
(6)
:
(7)
1
h
1
Let k1 = p
w
z [1 ]
. We assume that 2 + < 1 analyzing the case where the optimal use of energy is a concave function of the energy
2 +
@eN R
1
3 +2
@ 2 eN R
1
1
1
e¢ ciency (i.e., @ i = 1 k1 i1
> 0, and @ i2 = [2[1 + 1]k
<
i
]2
i
i
0):
Firm i would operate in this market as long as its pro…ts are larger than the
R
scrap value F , i.e., N
F .3 In the continuum of …rms, the minimum energy
i
e¢ ciency consistent with this satis…es the condition N R ( 0 ) = F , or:
p
Substiting
l0N R
and
NR
0
R
eN
0
2
6
=4
0 e0 l0
wl0
ze0 = F
(8)
in (8) yields to:
1
p
(1
F
h
z w
)
1
+1
i1
31
7
5
(9)
Thus, the energy e¢ ciencyh of the i…rms operating in the market is uniformly
R
R
distributed on the interval N
. Note that N
is an increasing funtion of
0 ;
0
the inputs’ prices and a decreasing function of the output price p. Moreover,
the existence of a …xed cost implies economies of scale as large …rms can spread
the …xed cost across more output units than small …rms.
We can compute aggregate emissions in the absence of environmental
regui
h
NR
NR
lation E
by integrating individual emissions in (6) over the range 0 ; ,
which leads to:
Z
h
i1 1
1
ENR =
p
w
z [1 ] i
d :
(10)
NR
0
3 In the current analysis, we assume that each …rm has one plant. In this way we avoid
comparing multi-plant …rms with small …rms which creates some problems, especially in the
case of emission standards where …rms with multiple units would be treated in the same way
as one-unit …rms.
4
Let h =
1
1
> 1. The solution to equation (10) can be represented as:
ENR =
k1
h
h
h
NR
0
ih
:
(11)
Finally, substituting equations (5) and (6) in (1), the output level of …rm i
is given by:
h
qiN R = p
+
w
i1
z
1
1
(12)
i
NR
To compute
total
h
i ouput with no regulation Q , we integrate (12) over the
NR
interval 0 ; , which leads to:
QN R =
z NR
E
p
Thus, the average emission’s intensity in the industry in the absence of
p
EN R
environmental regulations corresponds to Q
N R = z : Note that it is a decreasing
function of the price of energy and an increasing function of the share of energy
in the production process.
3
Environmental Regulation
Let us analyze now the e¤ects of environmental policies on the size distribution
in equilibrium. We assume that each unit of energy used as an input in the production process generates one unit of emissions. Moreover, we assume that given
the initial distribution of …rms’ size, the regulatory goal is to limit aggregate
emissions at some exogenously given level E by means of one of the following
three regulatory instruments: - a per-unit emission tax , - a uniform nontradable emission quota e, and - a uniform performance standard that de…nes
the maximum intensity of emissions. Finally, we assume that the stringency of
each policy remains unchanged regardless of the e¤ects of the instruments on
the initial distribution of …rms’size.
3.1
Environmental taxes
In case of emissions taxes, …rm i maximizes its pro…ts:
max
ei ;li
i
= p [ i ei ] li
wli
[z + ] ei
where is the per unit tax of emissions.
The FOCs are:
1
p
li = z + ;
i ei
p
i
ei li
5
1
= w:
F;
(13)
(14)
Dividing by parts, we get:
z+ T
e :
w i
liT =
(15)
Substituting equation (15) in the FOC, we can solve for eTi ; liT as:
h
eTi = p
Let k2 = k1
h
h
liT = p
z
z+
1
w
1
[z + ]
(1
w
)
[1
i1
]
i
(z + )
i1
i
ih
1
:
(16)
:
(17)
1
< k1 . As in the previous case, the optimal use of energy
h
ih N R
@eT
@ei
is a concave function of the energy e¢ ciency (i.e., @ i = z+z
@ i > 0, and
i
h
i
h
R
R
@ 2 eN
@ 2 eN
i
i
= z+z
< 0):
@ 2i
@ 2i
Individual output corresponds to:
h
qiT = p
+
w
[z + ]
i1
1
1
(18)
i
The cuto¤ value of the energy e¢ ciency in the case of taxes
condition T0 ( T0 ) = F , or:
h i
T
p
eT
lT = F + wl + [z + ] eT
0
T
0
satis…es the
(19)
Substituting liT and eTi in (19) yields to:
T
0
2
6
=4
F
(1
p
1
)
h
(z + ) w
1
+1
31
7
5
i1
(20)
By simple inspection of equations (9) and (20), it is easy to see that _ > 0,
R
> N
0 . As in the previous case, we can compute aggregate emissions and
T
T
output under taxes
(E
h
i ; Q ) by integrating individual emissions and output
over the range T0 ; , which leads to:
T
0
E T = k2
Z
1
T
0
i
QT =
d =
k2
h
[z + ] E T
p
h
h
T
0
ih
:
(21)
T
p
z+
where the average emissions’intensity in the industry corresponds to EQT =
. It is clear that with regards to the situation with no regulation, the
6
average emissions’intensity of the industry is decreased under taxes. Moreover,
h
ih
z
R
R
eTi = eN
< eN
_i. Hence, comparing equations (9) and (20), it
i
i
z+
is straightforward that with regards to the situation with no regulation, the
average energy e¢ ciency of the industry is increased under taxes since T0 >
NR
0 : Before the imposition of the regulation, …rms whose energy e¢ ciency was
lower than T0 earned positive pro…ts, but they did not take the social externality
cost into consideration. The tax on emissions corrects the divergence between
private and
by forcing …rms whose energy e¢ ciency belongs to
h social incentives
i
NR
T
the range 0 ; 0 out of business.
3.2
Non-Tradable Uniform Emission Quotas
Under a non-tradable uniform emission quota, the government restricts the individual emissions generated during the production process to the level e. In
our setting, this restriction is equivalent to a restriction on the use of the energy
input. Thus, …rm i maximize pro…ts given by the constraint ei e; or:
max
ei ;li
i
= p [ i ei ] li
wli
zei
F
s:t: ei
e
(22)
If the quota is not binding, the choice of inputs proceeds as in the case
without regulation. Instead, if the quota is binding, the FOC wrt li is:
liS =
p
i
1
e
1
(23)
w
Since aggregate emissions under the emission tax and non-tradable quotas
are equivalent ex-ante, we can solve for the uniform
h
inon-tradable quota e by
T
integrating equation (16) over the distribution 0 ; , which leads to the following condition:
k2
Z
k2
Z
1
T
0
T
0
i
h 1
i
d =
=e
h
Z
ed ;
NR
0
NR
0
i
:
Therefore, the emission standard e can be represented as:
2
h ih 3
h
T
0
k2 6
7
e=
:
4
NR 5
h
0
(24)
Since the standard is uniform and …rms are heterogenous, we should expect
it to be binding only for some …rms. Taking the case without regulation as
7
R
the baseline, we should expect the standard to be binding if eN
e. By
i
comparing equations (6) and (24), it is easy to show that there is a critical
value b1 that de…nes whether the emission standard is binding. The critical
value b1 corresponds to:
2
b
z
z+
b =6
4h
1
2
h
6
4
h
T
0
ih 33 h 1 1
77
55
NR
0
Note that @@h1 < 0, which implies that the larger the share of energy on the
production process, the lower the critical value b1 . By analogy, the larger the
share of labor on the production process, the larger the critical value b1 . Moreover, b1 his positively
related to the length of the energy e¢ ciency distribution’s
i
NR
interval 0 ; , meaning that the more heterogeneous the …rms are in terms
of the energy e¢ ciency, the larger the critical value de…ning whether emission
standards are
h binding.
i
R b
If i 2 N
0 ; 1 , the emission standard is not binding and individual emissions and output are given by equations (6) and (12), respectively. Emission’s
p
intensity
h
i corresponds to z as in the case with no regulation. Instead, if i 2
b ; , the emission standard is binding and individual emissions are equal to
1
e.
h
i
Integrating individual emissions over the range b1 ;
leads to E S b[b ; ] =
1
h
i
b . Thus, aggregate emissions under the standard are equal to:
e
1
ES
ES
= k1
=
Z
b
1
NR
0
k1 bh
1
h
1
d +e
i
h
NR
0
ih
h
+e
b
h
b
Sustituting equation (24) in (25) yields to:
ES =
2
k1 6 bh
4 1
h
h
NR
0
ih
+
z
z+
h
2
6
4
h
1
h
T
0
i
1
;
i
ih 3
NR
0
(25)
7
5
h
i
3
b 7
1 5:
On the other hand, individual ouput under binding standards is equal to:
qiS = [ i e]
8
liS
substituting liS , we get:
h
qiS = p
to:
i11
w
1
e1
:
i
(26)
h
i
+
b
Substituting e in equation (26) and aggregating over the range b1 ;
QS b[b
1;
]=
2
h 1
zk1
z
p z+
h
h
6
4
T
0
ih 3 1
1
7
5
NR
0
1
1
1
+
yields
1
+
1
:
1
Thus, the emission intensity of the standard for the range where the standard
is binding is equal to
"
S
p
E
b[ b ; ] =
S
1
Q
z+
where
=
Note that
1.
1
h
+
2
(1
ES
b
QS [b1 ;
2
1
2
[1
]
#
):
]<
ET
QT
2
h
h
6
4
h
ih 3 h1 "
7
NR 5
T
0
[1
]
2
i
h
Finally, integrating individual output over the range
QS =
zk1
ph
bh
1
h
NR
0
ih
1
+h
1
h
[
+
+
Hence, the average emission’s intensity corresponds to:
2
ES
p6
6
=
6
QS
z 4
3.2.1
bh
h
1
bh
1
h
NR
0
ih
+h
h
NR
0
z
z+
ih
ih
+
1
h
h
z
z+
[
ih
T h
0
NR
0
]
h
1
Taxes vs. Quotas
[
i
b
1
1
]
NR
0 ;
T h
0
NR
0
h
]
1
1
+
;
h
b
1
b
1
i
<
leads to:
+
1
:
1
h
1
1
1
h
T h
0
NR
0
#
1
b
0
2
if and only if 1
b
i
b
1
1
i
1
3
+
b
From the comparison of equations (16) and (24), it can be shown that there
is a critical value b2 that de…nes whether the emission standard induces less
emissions than taxes. The critical value b2 corresponds to:
2 2
h
h
16
b =6
4 4
2
h
9
T
0
ih 33 h 1 1
NR
0
77
55
:
1
+
1
1
7
7
7:
5
h
i h1 1
It is easy to see that b1 = b2 z+z
< b2 . Hence, individual emissions
under no-regulation, emission taxes and the emission standard e can be compared as:
i
h
R
T
R
eTi = 0 < eN
< e if 2 N
0 ; 0
i
i
i
R
0 < eTi < eN
< e if 2 T0 ; b1
i
i
i
R
0 < eTi < e < eN
if 2 b1 ; b2
i
i
i
R
0 < e < eTi < eN
if 2 b2 ;
i
To compare aggregate emissions under taxation and under standards, we
compare emissions in the di¤erent intervals de…ned before. Let E T S to denote
the di¤erence in emissions under these policies, i.e., E T S = E T E S . We have:
R
T
1. If 2 [ N
o ; o ); …rms operate under emission standards, but not under
TS
taxation. E
in this case are given by:
E1T S
=
k1
Z
T
o
NR
o
h 1
d
i
k1
h
=
h
T
o
NR
o
h
:
2. If 2 [ To ; ^ 1 ]; the emission standard is not binding for individual …rms,
…rms generate the same amount of emissions as in the non-regulation case. As a
result, emissions under standars are higher than emissions under taxation. The
di¤erence between the two levels is given by:
#
"
#
"
Z ^1
h Z ^1
h
z
z
k1
h 1
h 1
TS
^
d
d =
1
E2 = k1
1
i
i
T
T
z+
h
z+
o
o
3. If 2 [ ^ 1 ; ^ 2 ]; the emission standards becomes binding for individual
…rms, but …rms still generate more emissions compared to …rms under taxation.
z
z+
E3T S = k1
E3T S =
k1
h
z
z+
h
0
Z
h
h
B
B
@
^
^
2
Z
h 1
d
i
1
^
2
ed ;
^
1
h
T
o
NR
o
h
^
^
2
1
i
^
h
2
^
h
1
1
C
C:
A
4. Finally, if 2 [ ^ 2 ; ]; emissions under taxation are higher than emissions
under standards, and the di¤erence is given by:
E4T S
=
Z
ed
^
k1
2
10
z
z+
h
Z
^
h 1
d
i
2
:
h
T
o
h
:
E4T S =
k1
h
h
z
z+
0
h
B
B
@
1
h
T
o
h
NR
o
^
2
i
h
^
C
C:
A
h
2
Aggregate emissions under taxes are lower than under standards if:
E1T S +
E2T S +
E3T S +
E4T S < 0:
After some manipulation the following proposition regarding aggregate emissions can be derived,
Proposition 1 There is a critical ratio z+z that de…nes whether aggregate
emissions under emission taxes are higher or lower than under emission standards. If z+z > ( z+z ) , aggregate emissions under taxes are higher than under
emission taxes. If z+z < ( z+z ) the reverse holds.
The critical ratio ( z+z ) corresponds to:
(
2
6
z
) =6
4
z+
h
h
^
h
NR
o
1
NR
o
i
+(
i
^
NR h
o )
h
T
o
1
h
^
1
i
h
T
o
h
h
NR
o
3 h1
7
i7
5 :
Note that the critical value depends strongly on the threshold ^ 1 , (i.e., the
sooner the standard is binding, the smaller the critical value. At the limit, if
^ ! N R , ( z ) ! 1:
1
o
z+
3.3
Performance Standard
Under a performance standard, the average emissions intensity is …xed by policy.
As in the case of the emission standard, we assume that aggregate emissions
under the performance standard and the emission tax are equivalent ex-ante.
Therefore, the performance standard is equal to the average emission’s intensity
p
under taxes and corresponds to z+
:Thus, …rm i maximizes
maxp ( i ei ) li
ei ;li
wli
zei
F s.t.
ei
qi
p
z+
() ei
p
qi :
z+
Note that since the average intensity with no regulation is larger than under the
tax (and hence, under the performance standard), the constraint is binding _
i, and thus individual emissions are equal to:
ei =
p
z+
( i ei ) li
or
11
S
eP
=
i
"
p
i li
z+
#1 1
;
(27)
the pro…t maximization problem becomes:
S
maxp ( i eP
i ) li
S
zeP
i :
wli
li
Solving the FOC wrt li we get:
1
1
p
i li
z+
liP S =
"
(1
1
)
w
p
1
1
]+ ]
i
(z + ) w1
)
1
p
z+
1
[z [1
(1
z
(1
i li
)
=0
1
#1
:
(28)
Substituting (28) into (27), we get:
S
eP
=
i
Note that
S
eP
i
eT
i
eTi _i since
h
=
h
"
1
p
z[1
]+
[z+ ][1
]
]+ ]
[z + ] [1
z[1
]+
[z+ ][1
]
i
[z [1
i1
] w
i
#1
1
:
(29)
S
; which means that eP
= eTi
i
> 1:
h
z[1
]+
[z+ ][1
]
i1
S
More explicitly, we show that eP
> eTi
i
"
p
1
[z [1
[z + ] [1
]+ ]
] w
[z [1
#1
]+ ]
[z + ] [1
[z [1
[1
z+
i
1
>
>
]
]+ ]
]
z >z+
"
1
p
1
[z + ]
1
1
[z + ]
i
w
#1
1
)
)
> [z + ] )
z
)
0>
which is true.
The cuto¤ value of the energy e¢ ciency in the case of performance standards
PS
PS
S
satis…es the condition P
0
0 ( 0 ) = F , or:
h
i
PS
p
eP S
lP S = F + wlP S + zeP S :
(30)
0
S
Substituting liP S and eP
in (30) yields to:
i
12
>
PS
0
=
"
p
F1
[z + ] w [1
[z [1
]+ ]
1
#1
1
]
[1
1
[1
]]
:
(31)
Comparing the cutt-o¤ value in the case of PS with the one in the case of
T:
T
0
We can show that:
2
6
=4
1
F
h
1
p
PS
0
[z [1
]+ ]
1
] + [1
{z
]]
<
T
0
<
]]
1
]
[1
1
A
}
|
h
1
<h
1
[1
]]
1
[z [1
|
7
5
)
1
[1
[z + ] [1
]+ ]
(1
]
[1
[z [1
[z + ] w
i
2 1
1
[z + ] [1
1
31
1
1+
{z
}
B
(1
< [z [1
|
[z + ]
(1
2
)
[z + ]
i
2 1
i1
)
1
]+ ]
{z
C
[1
}|
[1
1
{z
]]
}
D
S
We can show that C < A and B < D, which implies that P
< T0 : Since taxes
0
impose stricter restrictions on …rms, making them reduce their individual emissions even more than in the case of performance standards, the marginal …rm
in the case of taxation should be more energy e¢ cient that the corresponding
one in the case of performance standards.
PS
Aggregate emissions under performance standard
are calculated by inh
iE
PS
tegrating individual emissions over the range 0 ; , which leads to:
EP S =
3.3.1
z [1
]+
[z + ] [1
]
Z
1
PS
0
eTi =
]+
k2 z [1
h [z + ] [1
]
1
h
h
PS
0
(32)
Performance Standards vs. Taxes and. Quotas
From the comparison of equations (16),(24) and (29), it can be shown that
there is a critical value b3 that de…nes whether performance standards induce
less emissions than emission standards. This critical value is equal to:
^ =
3
z+
z [1
]+
2 2
6 6
4h 4
13
h
h
T
0
ih 33 h 1 1
NR
0
77
55
:
ih
:
Moreover, ^ 3 = ^ 2
h
(z+ )(1
)
z[1
]+
i
S
< ^ 2 . Thus, as mentioned above, eP
>
i
eTi _ i:
R
Individual emissions under no-regulation eN
, emission taxes eTi , emission
i
PS
standard e and performance standard hei can now
i be ranked as:
R
S
< e if
= eTi = 0 < eN
eP
i
i
S
R
eTi = 0 < eP
< eN
<e
i
i
NR
2
if
S
R
0 < eTi < eP
< eN
<e
i
i
if
S
R
0 < eTi < eP
< e < eN
i
i
if
2
i0
2
i
;
PS
0
PS
0 ;
T
0
T b
0; 1
i
i
i
i
2 b1 ; b3
i
i
S
NR
b ;b
0 < eTi < e < eP
<
e
if
2
i
i
i 3 2i
T
PS
NR
0 < e < ei < ei < ei
if 2 b2 ;
From this ranking, the following propositions can be derived,
Proposition 2 Aggregate emissions under emission taxes are lower than under
performance standards.
The intuition behind this proposition is straightforward; with regards to
emission taxes, performance standards induce more emissions for all …rms, and
a lower cuto¤ value consistent with non-negative pro…ts.
Proposition 3 There is a critical ratio z+z that de…nes whether aggregate emissions under performance standards are higher or lower than under emission
standards. If z+z > ( z+z ) , aggregate emissions under performance standards
are higher than under emission standards. If z+z < ( z+z ) the reverse holds.
The critical ratio ( z+z )
(
z
)
z+
2 h
6
=6
4
(z[1
]+ )
(z+ )(1
)
corresponds to:
i1
1
h
( )h
PS h
0 )
(
z
1
1
i
h
z
( )h
1
1
(
h
( ^ 1 )h
i
(
NR h
0 )
T h
0)
We can show that ( z+z ) > ( z+z ) . This means that compared to taxes,
aggregate emissions under performance standards are more likely to exceed the
aggregate emissions under emission standards.
4
Numerical Example
In this section we present some numerical examples of the size distribution
induced by the di¤erent policies under analysis. For such a purpose, we provide
values for some of our key parameters and calculate the resulting choice of
inputs, pro…ts, and aggregate emissions and output.
14
i h
NR
o
i 3 h1
7
7 :
5
Table 1: Param eter Values
a
0:2
0:5
2
p
5
w
1
z
1:6
0:2
F
21
N
30
1
0
The production elasticity of emissions and labor is set at a = 0:2 and = 0:5
respectively. The genaral productivity parameter, ; is equal to 2: The price of
the output is set at p = 5; while the wages and the price of energy are also set
at w = 1 and z = 1:6 respectively. We set the emission tax = 0:2 and the
…xed entry cost of …rms F = 21: Finally, we assume that the initial number of
…rms is 30 (N ), and these …rms are uniformely distributed in the interval [0; 1] ;
which means that the upper bound of the distribution is given by = 1 and
the lower bound = 0:
Using the parameter values presented in Table 1, we study the size distribution of …rms before and after the implementation of the environmental
regulation. Table 2 summarizes the main results. If no policy is enforced, 7 out
of 30 …rms cannot operate due to the …xed entry cost which makes them non
pro…table. This means that …rms with low energy e¢ ciency [0; 0:267) will not
be active, even in case of no regulation.
Firms need to be more energy e¢ cient in order to survive if environmental
taxes are imposed. The cutt-o¤ value in this speci…c numerical example is 0.3.
The consideration of the social cost of emissions made …rms belonging to the
interval [0:267; 0:3) exit the market. The case of standards is di¤erent. For the
…rms lying on the left hand side of the distribution ([0:267; 0:43]), the emissions
standard seems not to be binding. Those …rms produce the same amount of
output and solve the same maximization problem as in the non regulation case.
The …rms that have to comply with the rule are those lying on the right hand
side of the distribution, i.e. [0:43; 1] in this particular numerical example. Those
…rms produce less than before, as it is illustrated in Figure 1. To make the
comparison with taxes, small …rms ([0:267; 0:57]) produce more output in the
case of the emission standard, while large …rms ([0:57; 1]) produce more in the
case where taxes are chosen as an instrument to reduce pollution. The point
0:57 is the one where the two curves (qT; qS) cross each other. In the same
graph, we can also observe the output distribution in the case of performance
standards (PS). It is very clear that output under trandable permits is higher
than output under taxation, which is easily explained if we take into account
that …rms pay taxes for the total of the emissions they generate, while in the
case of performance standards, they are restricted by the regulation, but there
is no extra, monetary cost that they have to pay in order to comply with this.
Using the speci…c parameter values assumed here, we can rank the four regimes
as follows, QN R > QP S > QT > QS : This ranking is in line with the predictions
of our theoretical model.
15
Figure 1: Distribution of Output across the Di¤erent Types of Firms (NR:
Non-regulation, T:taxes, S:standards, PS: performance standards.
Table 2: Static Case, Num erical Results
b
E
Q E=Q W I
1
0:267
355 567 0:63 368
0:3
284 511 0:56 361
0:267 0:43 265 502 0:52 349
0:3
297 535 0:56 363
0
Non Regulation
Taxes
Standards
Performance Standard
Table 2 also shows the total emission level in each case, while Figure 2
presents the emissions generated by each type of …rms. To start with the aggregate amount, the higher the output, the higher the total amount of emissions,
which means, E N R > E P S > E T > E S : So, under no regulation, we observe
an increasing amount of emissions, as we move to the right of the distribution,
which is highly connected to the higher output levels and the implied size of
the …rms. Taxes and PS result in lower levels of emissions (still higher in the
case of PS, where …rms do not pay a tax per unit of emissions they generate).
Finally, the enforcement of emission standard leads to the same emission levels
as in the non regulation case for the …rms lying on the left hand side of the distribution, but there is a signi…cant decrease in the emissions generated by the
rest of the …rms. This decrease also a¤ects the emissions-output ratio, which
16
has been calculated in Table 2 for all the cases under study. More speci…cally,
S
T
TP
NR
in this table we can see that (E=Q) < (E=Q) = (E=Q)
< (E=Q)
:
The last column of Table 2 provides the values of a welfare indicator for each
policy instrument. This indicator is equal to aggregate pro…ts of active …rms in
the di¤erent policy senarios. Two factors determine the observed di¤erences in
the welfare values: (i) the cost of compliance with the di¤erent environmental
policies and (ii) the di¤erent number of …rms exiting the market after the implementation of each environmental policy. The formulas for each welfare indicator
(W I) are given below:
WI
W IT =
W IS =
Z
Z
T
0
^
1
NR
0
NR
=
Z
pqiT ( i )
NR
0
wliT ( i )
pqiN R ( i )
W IP S =
Z
pqiN R ( i )
PS
0
wliN R ( i )
(z + )eTi ( i )
wliN R ( i )
pqiP S ( i )
R
zeN
( i)
i
wliP S ( i )
R
zeN
( i)
i
F d
2R
T 4
0)
F d i +(
Z
F d +
S
zeP
i ( i)
^
pqiS ( i )
(33)
T
0
(
eTi ( i )d
T
0)
(34)
wliS ( i )
1
(35)
F d
(36)
Note that in equation (34) that presents the welfare indicator in the case of
environmental tax, there are two additive parts. The …rst part shows the aggregate pro…ts of active …rms after having paid the cost of taxation. The second
part shows that these money are returned to the …rms in the form of a lumpsum subsidy which is equal for every …rm. This is a only policy that implies
an actual, monetary cost for the …rms and this should be taken into account
by the social planner. Equations (35) shows the aggregate pro…ts in the case
of emissions standards. More speci…cally, the …rst part presents the derived
pro…ts when the standard is not binding, while the second integral term shows
the aggregate pro…ts of active …rms that …nd the imposed standard binding.
The results based on our numerical experiments provide the following ranking: W IN R > W IP S > W IT > W IS : This welfare ranking is in line with the
analysis and the comparison of the di¤erent policies above and can be used to
provide some policy implications for the environmental tools under study. In
this context, performance standards are not only considered to be the most
neutral policy among the three, but it is shown to lead to higher welfare levels. Environmental taxes lead to lower welfare levels compared to performance
standards, while emissions standards are proved to be the worst instrument in
terms of welfare. This could be explained by the fact that they impose strict
restrictions not only to emission levels but also to the level of the output. In
general, policies that favor signi…cantly either small or large …rms (e.g. emissions standars, environmental taxes) lead to lower welfare levels compared to
17
i
3
5
ze
F d
Figure 2: Distribution of Emissions across the Di¤erent Types of Firms (NR:
Non-regulation, T:taxes, S:standards, PS: performance standards.
policies that are cosidered to impose relatively the same restrictions to …rms of
di¤erent sizes (e.g. performance standards).
In order to evaluate the di¤erent kinds of environmental policy and and see
how they a¤ect …rms of di¤erent sizes, we calculate the absolute and the percentage change in the pro…ts shares of individual …rms after the implementation of
the policy instruments under study. As we can see in Figures 3 and 4, emission
standards are much more stringent for larger …rms, while they have a positive
e¤ect on small …rms which seem to increase their pro…ts share. Performance
standards are considered to be a neutral policy that imposes relatively more
constraints to smaller …rms which, due to their smaller output, have to face
stricter environmental constraints. This kind of policy is somehow equivalent to
a di¤erentiated emission standard, where the lower the output, the more restrictive is the emission standard. Under performance standards, small …rms lose
a small part of their pro…ts while larger …rms increase marginally their pro…ts
shares. Taxes clearly impose a higher cost to smaller …rms, which do not have
high surpluses, leading to the exiting of the …rms that were closer to the lower
boundary of the distribution. On the contrary, environmental taxes favor larger
…rms increasing their pro…ts shares.
18
Figure 3: Change in Pro…ts Shares of Individual Firms
Figure 4: % Change in Pro…ts Shares of Individual Firms.
19
5
Conclusions
In this paper we study the e¤ects of the choice of policy instruments on the size
distribution of …rms. We have shown that each regulation a¤ects di¤erently
…rms of heterogeneous size, favoring either small or large …rms. For instance,
compared to taxes, uniform emission standards are much more stringent for
larger …rms - that emit in absolute terms more than small …rms. Moreover,
the fact that a di¤erent number of …rms goes out of business under di¤erent
policy instruments provides some implications for their level of strigency. In the
current study, the implied heterogeneity among …rms di¤erentiates the e¤ects
of the alternative policies studied here.
To sum up, the internalization of the social cost, coming from the polluting
activity of …rms, leads to lower production levels for each "type" of …rm. More
speci…cally, emission taxes will a¤ect more small …rms with signi…cantly low
surpluses as the use of energy now becomes more expensive. The result here is
equivalent to an increase in the price of energy. Emission standards, as stated
above, a¤ect only the large …rms who have to comply with the new regulation.
Small …rms keep producing the same amount of output, using the same amount
of energy, since the constraint is not binding in their case. Finally, performance
standars favor marginally large …rms that produce high levels of output and do
not …nd the regulation so restrictive. However, compared to the other two policy
instruments, performance standards do not a¤ect pro…ts shares signi…cantly,
while at the same time, they lead to higher welfare levels.
6
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