Optimal
Pollution
and
Tax
within
Profit-Maximizing
Labor-Managed
Cournot
Koji
Oligopolies
Okuguchi*
Abstract
Optimal
pollution
maximizing
Cournot
differentiation.
higher
than
managed
1.
oligopolies
the
marginal
with
for labor-managed
polluting
are identical,
value
of
and
lower
oligopoly
damage
is derived
the
the
firms
and
optimal
for profit-maximizing
the
value
profitproduct
tax rate is
damage
marginal
Cournot
without
pollution
environmental
than
and
for
labor-
of the envi-
oligopoly.
Introduction
A
labor-managed
surplus
per
Yugoslavia,
(see
has
rate
If all firms
cournot
ronmental
tax
unit labor.
and
Komiya
been
ducing
event
>&
firm
many
(1988)).
known
Labor-managed
Japanese
The
to behave
changes
(see
of an increase
is defined
firms
to be
a
firms
were
Ward
under
if the price
(1958)),
in the product
firm
that
price.
death while I was
few months
After my
some
maximizes
in
to be labor-managed
perfect
competition
is, its output
Labor-managed
it is pro-
decreases
Cournot
Yoshioka.
in
return to Japan, he sent me
time while he was
from
I heard of his
9
―
for a
on sabbatical leave.
there one of the papers cited here,
which was indispensable in writing this paper. I am
this kindness and for his long friendship with me.
―
the
oligop-
at Shiraz University in Iran. I was in Sydney
years ago at the same
its
the former
of the product
This paper is dedicated to the late Professor Moriyuki
sudden
which
prevalent
are considered
labor-managed
perversely
firm
very grateful to him
for
Optimal
oly
Pollution Tax
has
ior has
by
been
first formulated
been
analyzed
them,
these
Neary
works,
without
firms
the
(1980),
profit-maximizing
are
similar
analysis
duality
lemma
lution
perfectly
on
the
Okuguchi
the
rate
knowledge,
optimal
within
unique
then
Cournot
derive
prove
the
has
taxation
In
has
firms
(see
rate
has
efflu-
conducted
using
cost
firm's
within
harmful
a
Shephard's
function
Simpson's
that
and
analysis
emission
to
is
not
given
the
the
pollution
basis
tax rate
―
10
―
optimal
and
within
formulate
labor-
differentiation
pollution
of
tax
and
rate, there
this existence
that maximizes
and
the
oligopoly
2, I will
of pol-
firms
I will analyze
Cournot
product
the effects
for labor-managed
In Section
without
On
analyzed
this paper
oligopoly.
equilibrium.
the optimal
one
labor-managed
oligopoly
and
no
oligopoly.
managed
firms,
extended
pollution
Cournot
Cournot
has
where
firm's
The
Okuguchi
tax
firms'
duopoly
the
welfare
(1992),
(1995)
a
oligopoly
profit-maximizing
luting
Simpson
which
effluents.
pollution
between
(2003)
products
to mitigate
total social
that
all of
to its output.
Cournot
tax
optimal
Cournot
relationship
In
pollution
firms'
Ebert
assuming
outputs.
oligopoly
profit-maximizing
(1980)).
the
oligopoly,
Cournot
labor-managed
pollution
derived
to their
the best of my
and
Oates
as
on
the
its behav-
produce
policies
tax
competitive
and
others.
to
known
maximizes
and
among
governmental
pollution
which
Baumol
proportional
tax
of
for profit-maximizing
profit-maximizing
To
for
Cournot
demand.
necessarily
One
Oligopolies
Cournot
assumed
generally
Cournot
(1983),
(1993),
implicitly
effluents
rate
proportional
its factor
been
have
Waterson
to profit-maximizing
is to levy
(1994)
and
Okuguchi
tax
and
Yamazaki
ents
have
damage
investigated
and
and
and Profit-Maximizing
Hill
relation
harmful
pollution
Barnett
by
environment.
environmental
optimal
in
(1984)
emitting
damages
been
within Labor-Managed
with
exists
result, I
the net
pola
will
total so-
Optimal Pollution Tax
cial welfare.
equilibrium
oligopoly
tax
In
Section
for
a given
without
rate
duality
within Labor-Managedand
lemma
3, I will prove
pollution
product
maximizing
Okuguchi
firm, of which
unit
behaves
of a change
optimal
pollution
tax
rate
my
to the
analysis
Cournot
equilibrium
only
variable,
2.
one
The
to
there
firms
Cournot.
are
I use
n
firms
assumed
of the
z's labor
Xj : firm
f s output
h/ix,),
firm
have
reducing
using
already
I
Shephard's
a
of the surplus
per
Section
a
I
will
in re-
for the
involving
4 concludes.
to maximize
expectations
the
on
surplus
per
all its rivals'
unit
labor.
outputs
a la
notation.
/'s inverse
gi(x, ), firm
z's surplus
kj : firm
f s fixed
:product
that
simplify
problem
function
the
Oligopoly
aiming
s, : firm
p
and
properties
production
functions,
i. e. factor
mand
e, =
in
that the
oligopoly
existence
for
one
will show
damage.
the
pollution
mentioned,
asymmetric
problem
output.
Cournot
Cournot
profit-maximizing
environmental
3 by
to form
/, : firm
/, =
oligopoly
fixed-point
the following
have
Cournot
Cournot
be
without
I
a
unique
the optimal
for labor-managed
and
a
derive
is maximization
from
that is, industry
Labor-managed
Let
2
As
Oligopolies
profit-maximizing
this paper
value
in Sections
and
Cournot
of a
prtce. In
Cournot
marginal
for
welfare,
objective
in the product
existence
rate
(2003).
differently
for profit-maximizing
lation
tax
total social
labor-managed
event
the
differentiation
the
as in
of labor,
Profit-Maximizing
/'s emission
rate of harmful
per unit labor
cost
price
―
11
―
effluent
de-
Optimal
Pollution Tax within Labor-Managed
p
=f(Xxi),
inverse
w
: competitive
t :pollution
By
wage
unit effluent
=
loCj , industry
output
L
=
￡/,-, industry
factor demand
E
=
￡<?;, total emission
D
=
D(E),
the surplus
xt f&Xj)
Si
of the environmental
per
-
unit labor
wht
=
functions
&
h-t and
damage
for firm
)-tgj
rr
/,
Oligopolies
rate
tax rate per
value
Cournot
function
X
definition,
The
demand
and Profit-Maximizing
i is
(Xj )-ki
^
(1)
g,
are
assumed
h,' >
0,
h," ^
to
have
the
following
general
properties.
Assumption
1 : /' <
0,
0,
g! >
0,
g!f >
i
Inequality
firms
h"
behave
addition,
^
0 implies
under
Cournot
the maximum
maximization
labor's
expectations
is interior, firm
is given
on
=1,2,
marginal
their rivals'
/'s first order
･･･,≪.
product.
outputs
and
condition
(2)
If
all
if, in
for profit
by
ds'
A
dX;
h?(Xi)
where
A
=
non-increasing
0,
_
Q
'
{f&X;)
+ x, f'(Lxj)
- {x, f&c,)
- wh,
-
wh',(xi) - tgfc)}h,
(x,) -
tgi tot) - k, }h,%)
/ =
―
12
―
(x,)
l,2,---,≪.
(3)
Optimal
Let
Pollution Tax within Labor-Managedand
the numerator
in (3)
fied in light of v, =
^
be v,. Then
Profit-Maximizing
the second
order
Cournot
Oligopolies
condition
is simpli-
0 as
^<°<
i=1'2-'*'
=
dx,
(4)
h,(Xi)
where
^L=(2f+xir-WhlH-tgfjhi
-kh＼
dx,
^
J
(xj
-
I now
introduce
Assumption
This
linear,
not
2 : f
―'- <
dx,
Using
has
to
0 holds
the
<
-th,
been
i
definition
It holds
v, (*,,X,f)
=
in
the
Zx,
also if the
which
is an implicit
literature
/. e.
function
(5)
among
―
degree
function
second
order
of h"
^
0, g" ^
the first order
is
of its convexity
0. The
2 in view
/ =
demand
profit-max-
surplus
st >
, I rewrite
on
viable,
0
condition
per
is
unit
condition
and
S; ^
0.
(3) as
hKx^-tg'iixMbi)
- {x, f(X)-whl(x,)-tg,0,
1,2, ･ ･ ･ ,n.
is to be
= {f(X)+xif'(X)-wi
=
=
1,2, ■■･, n.
firm
Assumption
X
i
It is satisfied if the inverse
nonnegative,
under
=
used
If a labor-managed
be
)/*/',
assumption.
widely
oligopoly.
- k-,
0,
if it is concave.
too large.
labor
+ xt f"
has
Cournot
and
wh,
the following
assumption
imizing
-
(*,-) -
h, }h!{Xi)
1,2, ･･･,/!,
x-nX
13
―
and
(6)
t. Simple
calculations
yield
Optimal Pollution Tax
within Labor-Managed
and Profit-Maximizing
Cournot
Oligopolies
^
=
(h,
+ x, hfif
-tg,
-
-k,)h"
(wh!1
+
<0
tg")
/ =
-
(*,./ - Wh,
1,2,..-,w.
(7.1)
^
=
xlfhi
+ JC￨./^―*/^0,
- 0 according
<
i =
it
――'- >
^
as a.{ =
t
―'- =
g,
a,
and
tively. Note
and
/' <
function).
Now
/?,･ are
that h"
0,
^
I have
Otherwise,
solve (6)
x,
with
elasticities of
0 implies
dv
―oX
>
/z,and
that ――
X'
0 if /" ^
the sign
respect
of ―-
g,
with
h' ^
0.
0 (convex
(7.2)
1,2, ･■･,/!,
(7.3)
/5;,
i =
where
1,2,...,≫.
respect
Hence,
or linear
to X; , respec-
in view
inverse
of this
demand
is indeterminate.
=<p'(X,t),
to x, to get
i
=
l,2,--.,n.
(8)
where
d^
dX
dx,
dp'
^T
dt
= ^'
'' =
1.2.-.≪-
―
14
―
(9-2)
Optimal
The
Pollution
Tax
signs
of
and
(7.3).
(7.2)
dustry
the
is
i.e.
partial
an
the
solution
the
the
It has
f
is
dv―'dx
dip
―
dX
3
clear
<
of
>
this
a unique
4
in
Cournot
of
light
of
equilibrium
one
variable
in-
function
equation,
(10)
<￡ are
d<p
―
dx
the
0,
i
assumption
solution
concave
and
1. Let
t)
under
sufficiently
0
derivatives
: (p' (0,
that
also
>
variable
indeterminate
t, the
fixed-point
Oligopolies
in
general
=
1,2,
indeterminate.
I introri"^
assumption.
Assumption
It is
a
of
Cournot
=f(X,t),
partial
following
one
are
value
as
of
Profit-Maximizing
derivatives
arbitrary
identifiable
X=l^'(X,t)
where
Labor-Managedand
above
Given
output
<p(X,t),
within
^0.
if 0
to
<
―-
<
in
solution
has
dip
<. ―
dX
ensure
Hence,
unique
H
(10)
this
of
■■･, n.
1.
0
case
(10)
a
unique
d<p
―
dX
Inequality
for
all i. If/
(10)
d<p
if ―
solution
has
a
<
is
0
is
0.
true
convex,
unique
<
I
if
get
solution
if
be
X=X{t＼
(11)
where
dip
dt
dX
_
dT
(12)
77^f
dX
dip
If ―
dX
<
However,
>
n
0,
0
or
dip
―
dX
the
consequently
sign
<
1,
of
&P
―
dt
the
dip
―
>
0.
denominator
of
the
above
expression
f
is
positive.
f
dip
<
0
On
is
indeterminate.
the
―
other
15
hand,
―
If
dty
―
dt
a,
<
0
>
p,
if a,
for
<
all
/,
J3, for
all
■
/.
Optimal
The
Pollution Tax
case
within Labor-Managed
d(f
―
dX
where
<
0
dtp
― >
dt
and
and Profit-Maximizing
0
hold
simultaneously
Cournot
Oligopolies
is shown
in
the
figure.
A
■―
-...._
/
, <p(X,t2)
<p(X,tx)
/
^￡_
"'･-.//
＼
j＼ tincreases
I
I
＼
＼
＼
,
0
X(u)
Figure: Equilibrium
I am
now
Cournot
be maximized
with
of the firms'
from
by
firms'
to determine
oligopoly.
respect
tax
minus
effluents.
the optimal
The
total social
to the pollution
profits, the consumers'
pollution
X
industry output as an increasing function of t, where
in a position
labor-managed
X(u)
welfare
W,
tax rate, is defined
surplus
the value
pollution
of the
and
tax rate
which
for
is to
as the sum
the governmental
environmental
tj < t2 .
damaged
revenue
caused
Thus
fx
W
=
=
Ztt. +
I
/
Jo
f(l)di
/(7V7
-
-
wXhj
―
Xf(X)
+
tE
(Xj) - Ikj
-
16
―
-D(E)
D
fa
(Xj)),
(13)
Optimal
Pollution Tax within Labor-Managedand
where
and
X
and
x} s in the
are functions
expression
of t. Differentiating
dW
dt
above
dx
=
f(x)
J
wish,
dt
J
-(.-D'J^
where
I have
made
Profit-Maximizing
dxj
―-―
dt
are
evaluated
W
with
respect
D
Ze,.tdxj
――
Sj dt
Cournot
at the
Oligopolies
equilibrium
to t, I get
+ I^V-^/)^-.
use
of the first order
(14)
condition
(3)
as well
as the defi-
dW
nition
of st . The
optimal
pollution
tax rate must
satisfy
=
0. Hence
dt
l^hj-Xjf)
(15)
t =D'+
.
Note
that the sign
ever,
I
quently,
can
that
get t―D'in
In
all firms
order
not
of the
say
of the
anything
term
on
is positive
regarding
the right
the
hand
sign
side
for all j. How-
of
dx
――,
dt
of (15).
conse-
Hence,
general.
to get a clearer
/,. =hi
(Xi) =
=8i(Xi)
=
dx>
n ―dt
result, I consider
a symmetric
to be identical, i, e.
h(Xi)
=
g(Xi)
for all i. In this case, xt
dX
―
dt
s, h- ― Xjf
definite
second
are assumed
e>
expression
=
x
for all i and
>
according
as a,
―
17
=a-p
―
.
= (3i for all i.
case
in which
I
Optimal Pollution Tax
Hence,
regardless
ambiguously
within Labor-Managed
whether
a
>
positive, leading
and Profit-Maximizing
j3 or ex.<
ft, the second
Cournot
term
Oligopolies
in (15)
is un-
to
t>D'.
To
(16)
wit, if all firms
at a level
higher
a corollary
3.
the marginal
of this result, I can
labor-managed
mental
than
are identical,
monopoly
value
optimal
pollution
than
the
this section
maximizing
will use
Cournot
I
Cournot
as in Okuguchi
instead
(2003).
the production
of the
pollution
marginal
value
As
tax rate
for
of the environ-
Section
2
Therefore,
to
firm
derive
the
optimal
without
pollution
using
fi<0,
function
function
possible
lemma
2. However,
of/
1,2, ...,,,
I
(17)
h, (x{). Moreover,
= p(X)
confusion
instead
with
the
of p
I
will
=f(X)
producton
de-
as in
function.
as
-tgi(ft(lt))
= pCZfj(lj))fi(l,)-wli
-tg,
(/,≪,)),
i =
for profit
duality
in Section
i =
/, =
as p
i's profit ?r, is given
assumed
for profit-
function,
7T, =P(X)xi-wli
I have
tax rate
Shephard's
I will stick to the notaton
demand
demand
avoid
Oligopoly
f!>0,
factor
the inverse
will
oligopoly,
*,-=//('/).
tions
damage.
damage.
In
where
tax rate is set
of the environmental
assert that the optimal
is higher
Profit-Maximizing
note
the
away
maximization
fixed
cost. The
for firm
―
18
first and
i's profit under
―
l,2,---,n.
second
order
the Cournot
(18)
condibehav-
Optimal Pollution Tax
within Labor-Managedand
ioristic assumption
yield (19)
and
Profit-Maximizing
Coumot
Oligopolies
(20), respectively.
^L
=
p'(Xfj(ij))fi(ii)f!(ii)+P(Xfj(ij))fi'(ii)
-yv-tg'i(fi(i,))f!(h)
3 TV―r
=
(p
+fi
Pj)fi
=o,
+ (p
-tig-f+glf")
where
not
in (19),
I have
clear if (20)
resolve
P
)fi
away
I therefore
=
As
the
following
0,
g'/f!2 +
4 : p
+
x{ p1
>
0,
p' + x, p"
<
i
first inequality
implies
increase
in its output
revenue
is decreasing
holds
f"
― 0.
Ht
is positive.
with
I rewrite
(h
that firm
,X,t)
/'s marginal
The
respect
second
one
to increase
the first order
it stands,
it is
assumption
to
(22)
with
/, =^
respect
(X,t),
g!f!' >
asserts
(19)
0,
l,2,.--,n.
with
(21)
respect
to
that the marginal
in its output.
condition
=p'(X)f(li)fi(li)
=
revenue
The
third one
=l,2,---,n.
(22)
as
+p(X)fi'(li)
-tg'i(fi(li))f!(li)=O,
Solving
(20)
this indeterminacy.
Assumption
The
1,2,-･■,/!,
maximum.
introduce
(19)
ft
i
corner
i,2,.･･,/!.
+p
<0,
assumed
holds.
+fi
i =
i
-w
to //,I get
i =
1,2,--.,≪,
(23)
where
X
dX
P' (/!'￡ +f!2)
+Pf>'
-
t(g;>f!2 + g!f)
i =
―
19
―
1,2, ･･･,≪.
'
(24.1)
Optimal
Pollution Tax within Labor-Managed
and Profit-Maximizing
&d>'
z'-'f^+g'f"
ft
P' (ff,
+/;2)
+Pf,"
-
t(g;'f;2 +
i
The
Cournot
a single
equilibrium
variable
X
Cournot Oligopolies
industry
function
output
ip(x,t),
=2fj(il>'(X,t))
=
g;f)
1,2, ･ ･･,/!.
is indentical
to the fixed
/. e. it is the solution
(24.2)
point
of
of
=^(X,t),
(25)
where
^
=
^X=
V,
Introduce
^
given
= If;^t
the following
Assumption
As
=^
X0X<O.
(26.1)
<0.
(26.2)
assumption.
5: tf {0, rj >
0,
y
is strictly decreasing
in
t, (25)
solution,
has
light of (26.2)
a
unique
x
=
and
1,2, ･ ･ ･, n.
ty{ 0,t)
which
>
0
under
this
assumption,
is strictly decreasing
in
t in
and
f
=
-^<0.
1 ― ipx
ax
On
the other
is ambiguous
hand,
(27)
the individual
firm's
output
response
to a change
in
t
as
^L-f^L.
dt
fi
(78)
(
}
dt'
f
it^x
+ ^O-
(29)
―
20
―
Optimal
Pollution Tax
The
total social welfare
W
=
5k,
=
Noting
agj(fj(lj))
that W
dX
taken
nate
in
optimal
in (31).
general
due
pollution
value
f-,=f
and
Hence,
dt
differentiating
the first order
the optimal
tax rate which
term
on
the right
hand
to the indeterminacy
may
be
lower
damage.
to derive
maximizes
x{ =x.
n
dt
21
of (32)
sign
than
If all firms
/, =/,
―
side
of the
higher,
for all i, I have
XdX
(19)
W
the
is
(32)
of the environmental
n
condition
fj Xj
dT
-^-.
tax rate
gj = g
dx
and
,dl,
account
of the second
ginal
of t in the equilibrium
dL
into
p
sign
(30)
z{P(x)f;-w-D'zg;f:}^r
t=D'+
The
by
to /, I have
=
last equality
Oligopolies
-D(I.gj(fj(lj))
is a function
dW
I have
Cournot
J?p(T)dT-wL-D(I.gi(fj(lj)).
respect
where
is defined
Profit-Maximizing
+J*p(T)dr-p(X)X
+
it with
within Labor-Managedand
―
Hence,
of
is in
determi-
―*―. Hence,
dt
or equal
to the
are symmetric,
the
mari.e.
Optimal Pollution Tax within Labor-Managed
and Profit-Maximizing
Cournot
Oligopolies
therefore,
dx
dl
_
It
dt
which
f
leads
to
t<D'.
This
inequality
gopoly.
optimal
to (16)
also
for labor-managed
for profit-maximizing
that if the product
tax rate is equal
market
to the
Cournot
oli-
monopoly.
is perfectly
marginal
Note
competitive,
value
of the
taxation
on
envi-
damage.
Conclusion
managed
has
Section
ginal
firm
as well
on
the
differentiation
and
polluting
taxation,
and
pollution
value
identical,
the
differentiation
for
and
general
be
demand
pollution
damage.
damage.
functions
tax
In
rate is higher
section
profit-maximizing
with
polluting
and
3, I have
Cournot
firms.
―
22
If firms
―
are
emission
than
the
derived
oligopoly
the
maxi-
firms,
or equal
If all firms
to
which
asymmetric
than
without
subject
tax rate
with
lower
are
labor-
oligopoly.
oligopoly
which
pollution
case
higher,
environmental
optimal
rate
In
Cournot
Cournot
firms
the optimal
welfare.
their labor
of the environmental
tax
derived
tax rate may
of the
that
with
pollution
as on labor-managed
labor-managed
sense
lution
research
formulated
the total social
optimal
no
2, I have
pollution
mizes
been
competitive
product
the
opposite
is true
shows
pollution
There
In
that (33)
that (32)
ronmental
4.
is entirely
Note
further
the
(33)
to the
the
mar-
symmetric
in
functions
are
marginal
value
the optimal
without
are asymmetric,
pol-
product
the
opti-
Optimal Pollution Tax
mal
pollution
value
of the
managed
tax rate
Cournot
pollution
mental
damage.
be
may
oligopoly.
to
lower
However,
than
asymmetric
and
the
than
the
if
the
case
of
value
the
labor-
identical,
the
of the environ-
optimal
pollution
oligopolies
to product
profit-maximizing
to the marginal
are
Cournot
responses
Oligopolies
of asymmetric
marginal
profit-maximizing
and
or equal
all firms
properties
asymmetric
labor-managed
Profit-Maximizing Cournot
as in
tax rate is lower
The
comparable
higher,
damage,
for labor-managed
competitive
be
environmental
optimal
rate
within Labor-Managedand
price
tax
may
change
of
firms.
References
Barnett, A. H. (1980), "The
nomic
Baumol,
Review
W.
J. and W.
ond ed. Cambridge
Bonin,
J. P. and
Managed
L. Putterman
(1987),
London
Different Abatement
Tax
(1983).
Ireland, N. J. and P. J. Law
Croom
R. (1988), "The
Economics
of Environmental
of Cooperation
Academic
Structure: The
Eco-
Policy, Sec-
and
the Labor-
Publishers.
Case
of Oligopoly
and
Archiv 49, 154-163.
"Labor-Managed
(1982),
Cournot
Economics
Oligopoly
and Industry
7, 43-51.
The Economics
of Labour-Managed
Structural and Behavioral
2", Tokyo
Journal
K.
Theory
Finanz
University Economic
Neary, H. M. (1984), "Labor-Managed
Managed
American
Enterprises,
Helm.
Firm, Partsland
Okuguchi,
The
and Market
Technology,",
and M.Waterson
Comment",
under Monopoly",
and N. Y., Harwood
Output", Journal of Comparative
Komiya,
Rate
University Press.
Economy,
London:
Tax
E. Oates (1980),
Ebert, U. (1992). "Pigouvian
Hill, M.
Pigouvian
70, 1034-1041.
(1993),
Cournot
Cournot
of Comparative
"Comparative
Characteristics of the Japanese
Review
Oligopoly
Economics
54, 2-16 and 54-66.
and Industry Output
Statics for Profit-Maximizing
Oligopolies", Managerial
and
: A
8, 322-327.
and
Decision Economics
Labor14, 433-
444.
Okuguchi,
Optimal
K. and
T. Yamazaki
Pigouvian
Tax
(1994),
within Cournot
"Ad
25-32.
―
23
Valorem
and
Specific Taxes,
Oligopoly", Keio Economic
―
and
Studies 31,
Optimal
Pollution Tax within Labor-Managed
Okuguchi,
K. (2003),
nomic
Simpson,
Pollution Tax
in Cournot
Cournot
Oligopoly",
Oligopolies
Keio
Eco-
Studies, forthcoming.
R. D (1995),
ronmental
Ward,
"Optimal
and Profit-Maximizing
13 (1958),
Review
"Optimal
and Resource
"The
Pollution Taxation in a Cournot
Economics
Duopoly",
Envi-
6, 359-369.
Firm in Illyria: Market
68, 566-589.
Z4
Syndicalism",
American
Economic