Public Bads, Growth, and Welfare
January 22, 2004
by
Toshihiro Ihori1 (University of Tokyo)
Hirofumi Shibata2 (Kanto Gakuen University)
Abstract
It is well recognized that considering the disexternalities of pubic bads such as
pollutants, economic growth does not necessarily enhance welfare. As an extension of
Shibata (2002)’s neutrality result, we shall show that when some agents are poor and
they face corner solutions, economic growth may produce undesirable outcome of
immiserizing growth. An increase in wealth differentials may improve the quality of
environment, which could benefit even the poor agent. Economic growth in the poor
country would hurt the rich country. Economic growth in the rich country might hurt
the poor country if it deteriorates environment much.
Key words: public bad, growth, welfare
JEL classification numbers: H41, F13, D62
1. Department of Economics, University of Tokyo, Hongo, Tokyo 113-0033, Japan
(phone) 03-5841-5502, (fax) 03-5841-5521 E-mail: ihori@e.u-tokyo.ac.jp
2. Kanto Gakuen University, Fujiakucho, Ohata, 373-8515, Japan
(phone) 0276-32-7800, (fax) 0276-32-3382 Email: hshibata@kanto-gakuen.ac.jp
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1. Introduction
It is well recognized that considering the disexternalities of pubic bads such as
pollutants, economic growth does not necessarily enhance welfare. The conventional
wisdom is that although growth may raise the amount of public bads such as
pollutants, the income effect of growth would be so beneficial that it would raise the
overall welfare. Hence, many developing countries are willing to have high growth
even if it may raise the amount of pollutants. Some economists, many of whom are
associated with international organizations, also advocate an international lump-sum
income transfer to the developing countries to help them to introduce pollution
abatement activities to improve the global environment.
It is thus important to
investigate the welfare implications of growth, which may deteriorate the quality of
environment.
However, there have been few papers to explore this possibility
analytically.
Although the paradoxical result of immiserizing growth has been
investigated, immiserizing growth due to pollutants is not fully addressed.
Shibata (2002) recently claimed that once we explicitly incorporate a public
bad of damaging welfare, economic growth cannot enhance welfare. Namely, the Nash
equilibrium quantity of a negative public good (a public bad) is independent of the
aggregate income of the set of contributors under some additional but plausible
assumptions
Since this neutrality theorem provides a useful starting point to analyze, we
first present this result in a concise way, following the idea of Shibata (2002). Then, it
is shown that if the neutrality result fails due to corner solutions and other reasons, we
might have some interesting results. Namely, when some agents are poor and they
face corner solutions, the larger the number of the poor in the community and the
larger the divergence of technology of abatement between the poor and rich agents, the
smaller would be the gap between the level of the Nash equilibrium public good and
that of the optimal level of public good.
In this sense, an increase in wealth
differentials may improve the quality of environment, which could benefit even the
poor agents. Furthermore, economic growth may produce undesirable outcome of
immiserizing growth. In particular, economic growth in the poor country would hurt
the rich country. On the other hand, economic growth in the rich country might hurt
the poor country if it deteriorates environment much.
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2. The Neutrality Theorem
As stressed by Shibata (2002), many public goods are a type of services that
lessens or reduces the amounts or magnitudes of the effects of undesirable public
outcomes
or
phenomena
(namely
public
bads).
With
respect
to
these
public-bad-reducing type public good, consumers’ real concern is not the quantity of the
public goods provided (for example, the quantities of police, fire stations, health
services) but the net magnitude of the public bad that remains in the community after
public-bad-reducing activities are undertaken. In other words, the magnitude that
consumers may actually concern is the undesirable conditions in the community that
the public goods attempt to reduce, such as the net state of the environmental pollution,
crime statistics, frequency of fires, the number of afflicted patients, etc.
In such a case it is important to formulate how the initial level of public bad is
accumulated.
If it were exogenously given and fixed, the conventional analytical
framework could be fine.
However, if it is endogenously accumulated within the
community and affected by economic activities, the conventional framework would not
be appropriate any more. We shall, therefore, analyze below consumers’ optimization
problem with respect to a public-bad-reducing type public good by explicitly considering
how the initial level of public bad is produced. Thus we incorporate a public bad of
damaging welfare as well as a public good of improving welfare. We take, as a concrete
example, pollution abatement activities.
Under these circumstances the laws of thermodynamics normally imply that
there can be no such thing as a non-polluting production. It is hence plausible to
assume, for simplicity, that individual i ’s production of wealth wi necessarily emits a
quantity of pollutants, denoted by zi . The initial level of pollutants zi is increasing
with his production of wealth, wi .
Obviously, in the complete absence of his
production, a producer’s emission is zero. We assume that the quantity of pollutants
emitted is linearly dependent on production of the private good.
Therefore, the
pollution emission equation is given as
(1)
wi = mi z i ,
where mi is the exogenously given agent-specific coefficient of pollutants and reflects
the degree of efficiency of production activities that may minimize the amount of
pollutants. Obviously, when the individual produces no good, his emission is zero.
Individual i may not recognize the disexternalities of his income-earning activity on
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environment given by (1) at his optimizing behavior. Note that mi is not necessarily
the same among agents: mi ≠ m j for some i,j.
−
Suppose there are n+1 agents. Let Z ≡
n +1 −
∑z
i =1
. The utility is dependent on private
i
consumption and the quality of environment. Individual i ’s optimization problem is
thus formulated as below
(2)
−
Max U i = U i ( yi , ai + a −i − Z )
yi ,a i
s.t.
y i = wi − k i ai
where k i is the person-specific average cost of the abatement for individual i, and y i
is private consumption. ai and a −i
are the quantity of pollutants abated by
individual i and by individuals other than i , respectively. It is not necessary to
assume that a reduction of production has the same reducing effect on pollutants as an
increase of the abatement. We could assume that k i ≠ mi and k i ≠ k j for some i,j.
Hence his effective budget constraint is rewritten as
(3)
y i + k i ( A − Z ) = wi + k i (a −i − Z )
where ai + a −i = A . Note that the right hand side of (3) means the effective income of
individual i, which is the sum of his actual income wi and net externality effects from
public goods by the other weighted by k i . Denoting the aggregate (net) quantity of
pollutants emitted by Z , we have
(4)
Z ≡ Z − A,
Notice that in the above optimization, the initial quantity of aggregate pollutants
emitted by the individuals other than i works as if it reduces individual i ’s income.
The optimizing behavior on vector
(Z , yi )
may be formulated in terms of the
compensated demand functions
(5)
Z = Φ i (U i )
(6)
y i = Γi (U i )
(i = 1, 2, L n + 1)
Summing both all individuals’ budget constraints and individuals’ compensated
)
demand functions, the Nash equilibrium level of pollution (Z ) is given as the solution
of n+1 equations
4
n +1
mΦ i (U i ) = ∑ Γi (U i )
(7)
i =1
which are the equilibrium conditions of the private good, if the following condition is
satisfied.
∑m z − ∑k a
(8)
i i
i
i
= m( Z − A) .
where m is an exogenously given (average) coefficient. If mi = k i = m for all
individuals, (8) holds.
However, this condition is not necessary.
Namely, even if
mi ≠ k i ≠ m for some individuals, (8) could hold. This is an extension of Shibata
(2002)’s formulation.
The equilibrium condition (7) indicates that the Nash equilibrium quantity of
)
pollutants remaining in the environment (public bad), Z , is independent of
∧
interpersonal distribution of wealth, w i = ( wi , w2 L wn +1 ) . Thus the well known
neutrality theorem holds here too. See Shibata (1971), Warr (1983) and Bergstrom et al
(1986) among others. But different from the conventional equilibrium conditions, the
present equilibrium conditions (7) imply that the Nash equilibrium level of pollution is
independent of the aggregate wealth of the contributors of the pollution as well. This
type of neutrality of a Nash equilibrium provision of a public bad with respect to
aggregate income is referred to here as the “Second Neutrality”, following Shibata
(2002).
An intuitive explanation for the difference between our finding and that of the
conventional one is as follows: An additional wealth equal to ∆w may appear to shift
the community’s aggregate budget line upward. But in our framework as ∆w of
wealth is produced, that production activities necessarily emit a quantity of pollutant
equal to ∆w / m . In other words, by the laws of thermodynamics if the initial level of
net pollutants is to be maintained, the aggregate abatement has to be increased as the
same amount of ∆w / m , and hence the quantity of consumption physically possible
remains the same as before. Consequently, an additional wealth of ∆w does not shift
the average budget line upward when (8) holds.
3. Endogenous
Endogenous production activities
We may incorporate endogenous labor supply or endogenous production
activity explicitly.
Suppose the utility function now includes leisure, xi as well.
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Then the optimization problem (2) may be rewritten as
(9)
Max U i = U i ( yi , ai + a −i − Z , xi )
yi , ai , xi
s.t.
y i + wi xi = wi H i − k i ai
where wi now denotes the wage rate and H i denotes the initial labor endowment,
respectively. The more people work, the more people pollute. This is a simple way to
incorporate pollutants caused by endogenous production activities. Hence, in place of
(1) we have as the pollution emission equation
(10)
wi ( H i − xi ) = mi z i
From the utility maximizing behavior subject to the above budget constraint,
we now have as the similar compensated demand functions with respect to yi and Z
as (5) and (6).
(5)’
Z = Φ *i (U i )
(6)’
yi = Γi* (U i )
Substituting the compensated demand equations (5)’ and (6)’ into the above budget
constraint (10) and adding up, we have as the equilibrium conditions of the private
good in place of (7) if (8) holds.
(11)
n +1
∑Γ
i =1
*
i
(U i ) = mΦ *i (U i )
where wi H i does not appear. Thus, the real equilibrium does not depend on initial
endowment, wi H i . We have the second neutrality theorem in this case as well.
4. Growth and Welfare
4.1 Wealth differentials
Some economists, many of whom are associated with international
organizations, advocate an international lump-sum income transfer to the developing
countries to help them to introduce pollution abatement activities (i.e. emission taxes)
to improve the global environment. But as implied by the conventional neutrality
theorem, the effect of lump-sum income transfer on the recipient’s emission is zero at
best. Because a country’s aggregate income has a neutral effect in determining the
Nash equilibrium pollution level when all consumers are rich enough to emit the
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equilibrium level of pollution.
In the preceding discussions we assumed that all agents attain their
respective internal solutions at the Nash equilibrium and, consequently, the set of
contributors to the abatement activities is fixed. Hence, wealth differentials with the
same aggregate wealth do not matter at all. However, it could happen that some
agents in the community with small wealth choose not to contribute to the abatement
because the Nash equilibrium level of abatement by others ( a −i ) is larger than their
individually desirable level of total abatement (A). They do not contribute to the
abatement at all. Also it could happen that condition (8) does not hold. In such cases
changes in wealth differentials due to economic growth or redistribution would have
real effects.
Suppose the world consists of two types of n+1 countries, the identical poor
countries 1, 2,..., n and the rich country n+1, w1 = ... = wn = wL < wn +1 = wH . Subscript
L and H denotes poor and rich, respectively. Country L does not contribute at the
corner solution: a L = 0 First of all, suppose income is redistributed from country L to
country H, so that wL declines and wH rises but nwL + wH does not change, The
initial pollutants Z does not change either. As the effective income of country H,
wH − k H Z , increases, from the income effect y H and A − Z increase, and hence
U H and a H also increase. Since country L faces the corner solution, a decrease in
wL results in the same decrease in y L without reducing a L . U L will decrease.
Therefore, an increase in wealth differentials would normally benefit the rich country
and hurt the poor country, and the overall quality of environment will be improved. At
the same time, an increase in a H
means an improvement of the quality of
environment, A − Z , which will benefit country L to some extent. If the abatement
activity of country H is efficient ( k H < k L ) or the number of poor countries (n) is large,,
the poor country could also gain.
Analytically, from the utility function (2) as to the poor country
(12)
dU L = U y dwL + U A d ( A − Z )
Since d ( A − Z ) = da H = d ( wH − y H ) / k H and ndwL = −dwH , (12) may be rewritten
as
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(12)’
dU L = −(
Uy
n
−
UA
U
)dwH − A dy H
kH
kH
At the corner solution, we know U y −
necessarily imply
Uy
n
−
UA
>0.
kH
UA
>0 for the poor. However, it does not
kL
If k H is low or n is high,
Uy
n
−
UA
could be
kH
negative and this term may dominate the second term, so that the sign of (12)’ could
also be positive. In other words, dU L > 0 if and only if
(13)
k H U A dwH − dy H
<
n Uy
dwH
Condition (13) could hold even if k H = k L . When the number of poor countries is large
and/or the rich has more efficient technology of abatement than the poor, a transfer
from the poor to the rich enhances the quality of environment and may benefit the
poor.
On the contrary, the redistribution policy, which intends to reduce wealth
differentials between rich and poor countries will deteriorate environment and may
reduce welfare of the poor country as well as the rich country. We have explored the
possibility that an increase in wealth differentials may be desirable for improving the
quality of environment. In the standard framework of an interior solution, as shown in
Ihori (1996), an agent with high productivity does not necessarily enjoy high welfare.
Namely, if k L > k H and agent L has less efficient technology than agent H, then, both
agents can gain by transferring income from agent L to agent H. The above analytical
result has shown that we may derive the similar policy implication in the case of corner
solution under condition (13). The relative size of poor countries has qualitatively the
same effect as the relative efficiency of abatement technology in condition (13).
Itaya et. al (1997) showed that social welfare can be raised by creating
sufficient income inequality that only the rich provide the public goods. Under the
identical technology of proving the public good they considered the case where the poor
is just indifferent between contributing to the public good and not contributing and
hence the utilities are initially equal before the transfer. On the contrary, we have
considered the case where the poor chooses not to contribute and the utilities are
initially divergent between the two types of agents. Then, we have shown that even the
poor could gain if condition (13) is satisfied. In this sense, our result is an extension of
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Itaya et. al (1997).
Consequently, the quantity of pollutants remaining in the environment after
the Nash equilibrium quantity of abatement was undertaken would become smaller
than the level achievable had all consumers been in the set of contributors with the
same aggregate wealth. This would benefit all agents. The larger the number of the
poor in the community, the larger would be the Nash equilibrium level of abatement.
In other words, the larger the wealth differentials, the smaller would be the gap
between the level of the Nash equilibrium abatement with corner solutions and that
with the Pareto optimum. Accordingly, the Nash equilibrium level of abatement
becomes larger than that would be attainable had all agents been in the contributor’s
set with the same level of aggregate wealth.
4.2 Immiserizing growth
We have so far shown that economic growth cannot enhance economic welfare,
as implied by the second neutrality theorem. When we consider the case of corner
solutions due to wealth differentials and/or do not require (8), this neutrality result
does not hold anymore.
The world-wide economic growth may reduce welfare by
increasing the overall level of emission of pollutants if economic growth makes some
poor countries who were previously not achieving interior solutions become richer. In
such a case, we would have an interesting result, worse than the second neutrality
result.
Suppose as in section 4.1 the world consists of two countries, the rich and the
poor, and economic growth occurs both for the rich and poor countries. For simplicity
we assume the two-country world; L=1 and H=2. First of all let us consider the case
where w2 only increases. It would raise Z . Then, if w2 − k 2 Z increases (decreases),
y 2 and A − Z increase (decrease), and hence U 2 increases (decreases).
On the
other hand, if w2 − k 2 Z does not change, y 2 and A − Z does not change, and hence
U 2 remains the same as before. In such a special case, as at the interior solution of
section 2, an increase in wealth would result in an increase in abatement by the same
amount, so that welfare will not change. If A − Z increases (decreases), it would
benefit (hurt) country 1.
Next, let us consider the case where w1 only increases. For the poor country
at the corner solution where the poor does not abate at all, a (marginal) increase in its
9
wealth will not induce an increase in abatement. Thus, y1 increases by the same
amount, and it will benefit country 1. However, the overall level of pollutants will
increase, hurting the rich country.
Since the world-wide economic growth means that both w1 and w2 increase,
from the above analytical results we may say that the rich country would normally lose
by the world-wide economic growth unless w2 − k 2 Z increases much. Growth will
benefit the poor to some extent since the poor can enjoy more private consumption.
However, if growth itself reduces the quality of environment much, the damaging effect
of economic growth on the environment may outweigh the positive income effect for the
poor country.
In such a case the poor country would also lose by the world-wide
economic growth.
Analytically, as to the poor country we have
d ( A − Z ) = da 2 − d ( w1 + w2 ) / k = −(dy 2 + dw1 ) / k
where we assume k1 = k 2 = m1 = m 2 = k
for simplicity.
Then, substituting this
equation into the equation of dU 1 , (12), we have
(14)
dU 1 = U y dw1 −
UA
U
U
(dy 2 + dw1 ) = (U y − A )dw1 − A dy 2
k
k
k
At the corner solution the first term of the right-hand side is positive, while the second
term is negative.
(15)
Since
k<
dU 1 < 0 if and only if
U A dw1 + dy 2
Uy
dw1
dw1 + dy 2
> 1 , condition (15) may hold even when the poor country is at the
dw1
corner solution k >
UA
. This condition means that m is relatively small and hence
Uy
growth deteriorates environment much. On the other hand, for the rich country we
have
(16)
dU 2 = U y dw2 −
Since k =
UA
U
U
U
(dy 2 + dw1 ) = (U y − A )dw2 − A dy 2 = − A dy 2
k
k
k
k
UA
for the rich country, the sign of (16) is always negative. Thus, under
Uy
condition (15) we have the seemingly paradoxical result of immiserizing growth.
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As Bhagwati (1958) showed, the paradoxical possibility of immiserizing
growth in the field of international trade requires that either growth be ultra-biased
against production of the importable or the foreign offer curve be inelastic.
Ihori
(1994) showed that immiserizing growth may occur due to non-cooperative private
funding of impure public goods. Cornes and Sandler (1996) pointed out that technical
progress in the production of the private good may not be beneficial and explored one
possibility of immiserizing growth in the presence of pure public goods. We have shown
that immiserizing growth may also occur even in the case of pure public goods once we
allow for wealth differentials and pollutants caused by production.
5. Conclusion
As Shibata (2002) pointed out, the Nash equilibrium quantity of a negative
public good (a public bad) is turned out to be independent not only of distribution of
income but also of the aggregate income of the set of contributors. He has referred to
the former independence as the first neutrality and the latter, the second neutrality. In
this paper we have clearly presented the second neutrality theorem using a simple
theoretical model with more general technology.
An increase in wealth differentials due to economic growth or international
redistribution may be desirable for improving the quality of environment. In particular,
when some agents are poor and they face corner solutions, the quantity of pollutants
remaining in the environment after the Nash equilibrium quantity of abatement was
undertaken would become smaller than the level achievable had all consumers been in
the set of contributors with the same aggregate wealth. Namely, when the number of
poor countries is large and/or the rich has more efficient technology of abatement than
the poor, a transfer from the poor to the rich enhances the quality of environment and
may benefit the poor.
We have then shown that in our theoretical framework economic growth does
not always enhance welfare. Also, in such a case the world-wide economic growth may
produce undesirable outcome of immiserizing growth. The rich country would normally
lose by the world-wide economic growth.
If growth itself reduces the quality of
environment much, the damaging effect of economic growth on the environment may
outweigh the positive income effect for the poor country.
country would also lose by the world-wide economic growth.
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In such a case the poor
References
Bergstrom, T. L. Blume, and H. Varian (1986) “On the Private Provision of Public
Goods,” Journal of Public Economics, 29, 25-89
Bhagwati, T.L., (1958), “Immiserizing Growth: A Geometrical Note”, Review of
Economic Studies 25, 201-205.
Cornes, R. and T. Sandler (1985), “Easy Riders, Joint Production and Public Goods,”
Economic Journal, 94, 580-598
Cornes, R. and T. Sandler (1996), The Theory of Externalities, Public Goods and Club
Goods, Cambridge UP.
Ihori, T., (1994), "Immiserizing Growth with Interregional Externalities of Public
Goods," Regional Science and Urban Economics, 24, 485-496.
Ihori, T., (1996), “International Public Goods and Contribution Productivity
Differentials”, Journal of Public Economics, 61, 567-585.
Itaya, J., D. de Meza, and G. D. Myles (1997), “In Praise of Inequality: Public Good
Provision and Income Distribution”, Economics Letters 57, 289-296.
Sandler, T. (1992) Collective Action: The Theory and Application, Ann Arbor University
of Michigan Press
Shibata, H. (1971) “A Bargaining Model of a Pure Theory of Public Expenditure,”
Journal of Political Economy, 1-28
Shibata, H., (2002), “Independence of the Nash Equilibrium of the Public Bads of the
Aggregate Income: The Second Neutrality Theorem”, manuscript presented as
Presidential Address at the 58th Congress of the International Institute of Public
held at Helsinki, August 27, 2002.
Warr, P. G. (1983) “The Private Provision of a Public Good is Independent of the
Distribution of Income,” Economic Letters, 13, 207-211.
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