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Working Paper Series
External Increasing Returns, Short-Lived
Agents and Long-Lived Waste
WP 91-02
A. John
Michigan State University
R. Pecchenino
Michigan State University
D. Schimmelpfennig
Michigan State University
S. Schreft
Federal Reserve Bank of Richmond
This paper can be downloaded without charge from:
http://www.richmondfed.org/publications/
Working Paper 91-2
EXTERNAL INCREASING RETURNS,
SHORT-LIVED AGENTS AND LONG-LIVED WASTE
A. John, R. Pecchenino, D. Schimmelpfennig,
and S, Schreft
EXTERNAL INCREASING RETURNS,
SHORT-LIVED AGENTS AND LONG-LIVED WASTE
A. John*, R. Pecchenino*, D. Schimmelpfennig* and S. Schreft**
*Department of Economics
Michigan State University
East Lansing, MI 48824
**Federal Reserve Bank of Richmond
Richmond, VA 23233
November 1990
We'would like to thank Donald Hester, Jeff Lacker, and seminar participants
at Michigan State University and the 1990 Western Economic Association
Meetings for useful comments. The views expressed in this paper are solely
those of the authors and do not necessarily reflect the views of the Federal
Reserve Bank of Richmond or the Federal Reserve System.
ABSTRACT
External Increasing Returns, Short-lived Agents and Long-lived Waste
Actions that affect environmental quality both influence and respond to
macroeconomic variables.
Further, many environmental and macroeconomic
consequences of current actions will have uncompensated effects that
outlive the actors.
This paper presents an overlapping-generations model of
environmental externalities and capital accumulation:
consumption of the
old generates long-lived garbage as a by-product, while young agents invest
in both capital and destruction of the existing garbage stock.
The model
-
also assumes external increasing returns:
increases in the capital stock
increase the future productivity of capital.
In the model, increases in the
natural rate of degradation do, and improvements in society's ability to
dispose of garbage may, encourage capital accumulation.
Multiple
Pareto-ranked equilibria can arise as a consequence of the interaction
between garbage and capital accumulation.
Underaccumulation of garbage,
analogous to dynamically inefficient overaccumulation of capital, can arise
in the model.
The family which takes its mauve and cerise, air-conditioned, power-steered,
and power-braked automobile out for a tour passes through cities that are
badly paved, made hideous by litter, blighted buildings, billboards, and
posts for wires that should long since have been put underground. They pass
on into a countryside that has been rendered largely invisible by commercial
art... They picnic on exquisitely packaged food from a portable icebox by a
polluted stream and go on to spend the night at a park which is a menace to
public health and morals. Just before dozing off on an air mattress,
beneath a nylon tent, amid the stench of decaying refuse, they may reflect
vaguely on the curious unevenness of their blessings. Is this, indeed, the
American genius?
J. K. Galbraith, The Affluent Society (1958)
What stands between us and a decent environment is not the curse of
industrialization, not an unbearable burden of cost, but just the need to
organize ourselves consciously to do some simple and knowable things.
Compared with the possibility of an active abatement policy, the policy of
stopping economic growth in order to stop pollution would be incredibly
inefficient. It would not actually accomplish much, because one really
wants to reduce the amount of, say, hydrocarbon emission to a third or a
half of what it is now. And what no-growth would accomplish, it would do by
cutting off your face to spite your nose.
R. M. Solow, "Is the End of the World at Hand?" (1973)
L
INTRODUCTION
There is currently a great deal of concern about the state of the
environment.
From the perspective of standard economic analysis, the
problem might seem evident, and the solution straightforward:
production
and consumption decisions generate external effects, implying a need for
Pigovian taxes or the creation of appropriate markets.
ignores two important aspects of environmental problems.
Such analysis
First, the
Actions that affect the environment
macroeconomic perspective is missing.
both influence and respond to macroeconomic variables.
Environmental policy
decisions frequently concern the intertemporal allocation of resources, and
so have implications for economic growth, capital accumulation, saving and
interest rates.
Second, the externalities are inter- as well as
1
intragenerational:
yet unborn.
actions taken today affect the welfare of generations as
Such external effects are intrinsically hard to internalize;
their existence almost surely alters the set of desirable policies.
This paper considers how agents make waste production and disposal
decisions in a setting where these have long-lasting effects.
Waste in this
model represents the long-lived, noxious by-products of consumption, be they
groundwater pollution from solid-waste landfills, toxic waste dumps, spent
nuclear fuel rods, medical debris on New Jersey beaches, or whatever.
The
model is formulated in terms of the accumulation of a stock of waste, but
this stock could equally represent some more general scalar index of
environmental quality.
Consumption activities generating waste as a
by-product are utility-enhancing, but the waste produced imposes a negative
externality on future generations that must either pay the costs of waste
disposal or accept a lower standard of living.
The analysis draws on a number of important and diverse economic
literatures.
The model utilizes the overlapping-generations framework of
Allais (1947) and Samuelson (1958), as extended by Diamond (1965) to
include capital accumulation, since this demographic structure lends itself
to analysis of situations where agents' actions have consequences beyond
their own lifetimes.
As Diamond (1965) embeds the features of Solow's
(1956) neoclassical growth model in an overlapping-generations framework, it
provides the natural starting point for analysis of the interaction between
capital accumulation and intergenerational environmental diseconomies.
2
A number of significant recent contributions to growth theory emphasize
the role of external increasing returns in the growth process; see Romer
(1990).
Romer (1986), Stokey (1988), Lucas (1988), and Schmitz (1989) all
study external increasing returns in infinitely-lived agent models, while
Freeman and Polasky (1989) and Mourmouras (1989) do so in an
overlapping-generations framework.
Such "new growth theory" models with
overlapping generations exhibit positive external effects across
generations:
actions.
agents alive today cannot capture all the benefits of their
External increasing returns are a feature of this model also,
permitting analysis of both positive and negative intergenerational
spillovers.
1
This work on growth theory is related in turn to the literature on
coordination failure in macroeconomics (see, for example, Cooper and John
(1988)).
That literature considers circumstances in which rational agents,
acting in their own self-interest, may achieve an inefficient equilibrium;
coordinated behavior may be required to achieve a Pareto-superior outcome.
Such coordination failures can arise in the model here.
Like Azariadis and
Drazen (1990), Murphy, Shleifer and Vishny (1989), and Weil (1989)‘ among
others, this paper provides a link between these two strands of the
literature.
Environmental questions have been analyzed extensively in the
environmental and natural resource literatures
(see, for example, Raumol and
1
Chatterjee and Cooper (1989). Pagan0 (1989) and John (1988) also examine
overlapping-generations models with externalities, but intergenerational
externalities are not their principal focus.
3
Oates (1988), Conrad and Clark (1987), Dasgupta and Heal (1979) and Neher
(1990)).
The externalities considered in this literature are often divided
into three categories:
common-property problems (Dasgupta (1982) and
Weitzman (1974a)); the upstream-firm problem (Loehman and Whinston (1970))
and pollution problems (Kneese and Maler (1973), Plourde (1972) and Weitzman
(1974b)).
Researchers have investigated mechanisms under which a
decentralized economy might successfully internalize environmental
externalities.
Such mechanisms include markets for effluents, pollution
licenses and Pigovian taxes.
On the basis of such analysis, policy
prescriptions have been offered and in some cases successfully implemented
(Hahn (1989)).
But by assuming that the life span of individuals and the
life span of the economy are the same (possibly infinite), researchers in
environmental economics have for the most part restricted themselves formally
to the analysis of intragenerational conflict.
While our analysis also
investigates policies that internalize the externalities present in the
model, the design of such policies is complicated by our explicitly
intergenerational focus.
Questions of intergenerational equity have been extensively discussed
in the exhaustible-resource literature (see, for example, Solow (1974,
1986)), but not, for the most part, in models of pollution.
It is
particularly striking that there is almost no use of an
overlapping-generations framework in either the exhaustible-resource or
pollution literatures, despite the appeal of this demographic structure for
explicit analysis of intergenerational issues.
Kemp and Long (1980),
Mourmouras (1990) and Sandler (1982) are important exceptions:
Kemp and Long
and Mourmouras construct overlapping-generations models of natural resources;
-
Sandler analyzes the optimal provision and maintenance of club goods in a
finite-horizon economy.'
Intergenerational externalities are also a feature
of John, Pecchenino and Schreft's (1989) analysis of the stockpiling of
nuclear weapons.
In a sense, this paper revisits the "limits to growth" debate of the
1960s and 197Os, as perhaps exemplified by the Galbraith and Solow quotes at
the start of the paper.
The questions addressed in that literature are
again timely, since modern theory is providing new arguments for subsidizing
growth at a time when environmental concerns are gaining prominence in
political debate.
Environmental models with an explicit intergenerational
and macroeconomic focus are needed.
2
Schimmelpfennig (1990) studies the greenhouse effect in an economy with
renewable resources and an overlapping-generations structure. Howarth and
Norgaard (1990) use a three-period model with distinct generations to
consider the impact of property rights on intergenerational equity.
S
II
THE ENVIRONMENT
Preliminaries
Consider an infinite-horizon economy comprised of finitely-lived individuals
and perfectly competitive firms.
A new generation (called generation t)
is born at each date t - 1, 2,..., and lives for two periods.
Assume no
population growth and normalize the size of each generation to unity.
Young
agents have preferences defined over consumption in old age, c~+~, and the
stock of garbage when they consume, g
t+1'
These preferences are represented
by the utility function
%Cl)
- d(gt+l)*
Assume that U(e) is strictly concave, 4(e) is strictly convex, and both are
increasing and twice continuously differentiable.
Assume also that
lim U' (s) - =.3
c+o
Young agents are each endowed with one unit of labor which they supply
to firms inelastically.
They divide their wage, w
t*
between saving for old
age consumption, s
and investment in the destruction of garbage, dt. When
t'
old, agents supply their saving inelastically to firms and earn the gross
return (1 + r
t+1
).
The garbage stock evolves according to
gt+1 - (1 - b)g, + Bct - 7dt
where b is the natural biodegradation of garbage, b E [0, 11, PC, is the
augmentation of the garbage stock by the consumption of the old at t, fi> 0,
3
Note that we exclude intergenerational altruism.
6
and Tdt is the diminution of the stock of garbage by the destruction efforts
of the youig at t, 7 > 0.
To enjoy the fruits of their labor fully, agents may choose to clean up
their environment prior to consuming.
While its efforts reduce the existing
stock of waste for itself and for future generations, the present generation
is concerned only with its own welfare and ignores any benefits bestowed on
its progeny.
Consumption of those now alive, however, produces garbage
that
impinges on future generations' utility; this effect, too, is ignored by
those currently alive.
The firms are perfectly competitive, profit maximizers who produce
output using the production function Yt - a$(Kt JF(Kt,Nt).
The function
F() is a standard neoclassical constant-returns production function,
implying that output per worker can be written as y, - a$(K,-,)f(k,), where
kt is the capital-labor ratio, and where f'(m) > 0, f"(s) I 0 and
I;in&
f'(e) - 0.
The function $(Ktml) represents enhancements to productivity
from last period's capital ($'(-) > 0); it is thus a technological
externality.
Because Kt 1 is predetermined at time t, $(Kt 1) enters the
production technology as a constant from the perspective of current
producers.
Since the population is normalized to one, $(Ktol) can be
written as $(ktml). It is assumed that the capital stock depreciates at
rate 6, with depreciation occurring during the production process.
Finally,
a is a technology parameter.
The inclusion of last period's capital stock in the current production
technology is motivated by the recent literature on external increasing
7
returns in growth models, emphasized in particular by Romer (1986); see also
Weil (1989).
Romer's insight, drawing on Arrow's (1962) analysis of
learning-by-doing and Young's (1928) analysis of increasing returns, is that
production generates knowledge as a by-product.
Although production at any
time period is a constant-returns activity, the model exhibits increasing
returns from an intertemporal social perspective.
Since our focus is on external effects across generations, we wish to
abstract from the well-understood free-rider problems,within a generation.
We assume that each generation elects a government at the beginning of each
date for a one-period term.
This government behaves myopically, carrying
out policies made solely for the welfare of agents alive during its term in
office.
The government has the power to levy lump-sum and distorting taxes
and transfers.
Specifically, it levies lump-sum taxes on the young to
achieve the desired destruction of garbage.
That is, an agent's choice of
destruction can be interpreted as arising from the collective provision of
waste depletion, a public good.
This allocation could be achieved as a
Lindahl equilibrium.
In our analysis we also consider the effects of a distorting tax
(transfer) on the return to capital and a proportional (lump-sum) tax
(transfer) on wages accompanied by a lump-sum transfer of the proceeds to
the old agents.
These policies may be used to alter the rate of capital
accumulation and thus the augmentation of the stock of garbage.
A
myopic government, concerned only about the short-run effects of garbage
stock diminution, would never have an incentive to impose such taxes.
contrast, an infinitely-lived institution, set up with the goal of
8
By
-
implementing policies to improve the welfare of the current and all future
generations, may choose a policy that internalizes the' intergenerational
externalities.
VI.
The problem of such an institution is considered in Section
In the remainder of this section and in Sections III - V, only a
short-lived government is assumed to exist.
The representative agent's optimization problem
The representative agent takes as given the wage, w
t.’
the return on saving,
;t+l, and the stock of garbage at the beginning of period t, gt.
Thus, the
competitive lifetime choice problem of a representative agent is to choose
C t+l, dt
and st to maximize
(1)
- a-st+l)
u(ct+l)
subject to
W
t
-st+d
C
t+1
g
t+1
C
t+1'
(2)
t
- (1 + :t+l>st
-
(1
d
t'
- Wgt
+
PC,
(3)
(4)
- 7dt
St z 0.
After substituting equations (2)
(4) into (l), utility maximization
yields the following first-order condition:
U'(*)(l + Gt+r) - 74'(') - 0.
(5)
The individual chooses st to equate the marginal utility of consumption,
multiplied by the return to saving, to the marginal disutility of garbage,
multiplied by the return from destruction.
Equation (5) assumes an interior
solution with a positive level of destruction, but nothing in the model
9
precludes the possibility that agents might choose not to engage in any
destruction, and hence set d
t
- 0.
We discuss this case further below.
4
The representative firm's problem
The individual firm takes the wage and the rental rate on capital as given.
It hires labor until the marginal product of labor equals the wage,
aWt-l)
[f(kt)
- ktf’ (kt) 1 - wts
(6)
and hires capital until the marginal product of capital equals the rental
/
rate [rtl,
a$(kt-l)f' (kt) - rt.
(7)
Goods market clearing
For the goods market to clear the demand for goods must equal the supply
of goods.
S
t
Thus,
+dt+c
t
- wt + rtkt + (1 - 6)kt.
(8)
Combining equations (2), (3) and (8) yields
(9)
kt - y-1)
given that ;t+l - rt+l - 6:
capital.
the return on saving equals the net return on
Thus, the capital stock at t is fully determined by saving at
t - 1.
4
Note that the boundary conditions on the utility function ensure that agents
will always wish to choose a strictly positive level of consumption. In the
absence of transfers to the old, this is sufficient to guarantee that saving
will also be strictly positive.
10
IIT
THE STEADY STATE
Definition
A
competitive steady-state equilibrium for this economy is given by (c, d,
g, w, r, s, k) such that:
(a)
agents maximize (1) subject to (2) - (4), given w, r, 6, and g;
(b)
the factor markets clear:
(c)
w - a$(k)[f(k) - kf'(k)]
(10)
r - a$(k)f'(k);
(11)
the goods market clears:
w+(l+r-6)k-s+d+c;
(d)
the stock of garbage is constant:'
g
A
(12)
B
o-c-b
:, d.
(13)
steady-state equilibrium can be characterized by the first-order
condition (5) evaluated at the steady state, the market clearing conditions,
and the law of motion for the stock of garbage.
Equations (10) - (12) and
the agent's budget constraint can be combined to yield the following
equilibrium relationships:
s - k,
(14)
f' 04 ,
r(k) - alp(k)
(15)
y(k) - aWd
f(k) ,
(16)
w(k)
- r(k)k
(17)
- y(k)
(18)
c(k) - [l + r(k) - 6]k,
S
This is the steady-state condition for b > 0.
b- 0 below.
11
We consider the case where
d(k) - w(k) - k.
(19)
Substituting (17) - (19) into (13), we obtain
g(k) - (B/b)(c(k)) - (r/b)[wW
- kl.
It is convenient to write this as
(20)
[8(1-6) + ylk + p(k)
where
PW
- pr(k)k - v(k)
(21)
Equation (20) gives the steady-state level of waste as a function of
the capital stock.
Suppose, counterfactually, that young agents devoted
their entire wage to destruction.
Then labor's share of output would go
towards destruction and capital's share of output would go to consumption.
The net effect on garbage would be given by p(k) - @(k)k
young actually devote some of their wages to saving.
- v(k).
Now the
Each unit of saving
(capital) implicitly represents one less unit of destruction and (l-6) more
units of consumption when old; thus each unit of capital also contributes
(p(l-6) + 7) to the garbage stock.
The net addition to the garbage stock
each period therefore equals [p(l-6) + y]k + p(k), which must equal the
natural depletion of the garbage stock (bg) in steady-state equilibrium.
We can thus identify a number of ways in which changes in the capital
stock will affect the steady-state stock of garbage.
First, as just noted,
additional capital directly implies less destruction and more consumption, so
that if p(k) - 0, garbage increases linearly with the capital stock.
Second,
increases in the capital stock increase output, and thus imply higher
payments to capital and labor.
The effect of this change on the garbage
12
stock depends on whether or not the young's destruction exceeds the old's
addition to garbage at the margin; that is, it depends upon whether p(k) is
positive or negative.
'Ihe sign of p(k) depends on the relative shares of
capital and labor, the size of the consumption externality, and the
productivity of the destruction technology.
Third, changes in the capital
stock may change the stock of garbage by altering relative factor shares.
If, for example, a higher capital stock is associated with higher payments
to capital (so r(k)k/y(k) is increasing in k), then increases in the capital
stock will, through this channel, increase garbage.
A steady-state equilibrium for this economy is represented by the
steady-state garbage equation (20) and the first-order condition
U'(c(k))(l + r(k) - 6) - -d’(g)
(22)
- 0,
where c(k) and r(k) are as just defined.
An equilibrium for this model is
illustrated in Figure 1.
g
SSG
cc
x
I
k
Figure 1
This figure is drawn under the assumptions that steady-state garbage (SSG) is
increasing in k and that the first-order condition (FOC) defines an inverse
relationship between k and g.
We discuss these assumptions further below.
Before considering the comparative static properties of the model and the
possibility of multiple equilibria, we briefly turn to some special cases,
to which we will then refer throughout the paper.
13
zero destruction (d - 0)
Special Case I:
As noted in the discussion of the agent's problem, the equilibrium may entail
a corner solution where agents engage in zero destruction.
If young agents
are not engaging in destruction, then they are saving their entire wage
income.
k-
This implies in turn that, in steady state,
s - w(k) - a$(k)[f(k) - kf'(k)].
(23)
This equation implicitly defines the capital stock, kzd. Assuming
zero
destruction, the garbage stock is given by
t (c(k)).
(1
Recalling that g(k) - (b/b)c(k) - (r/b)d(k), it
g - gzdW
-
g(kz,) - gzd(kzd) .
Finally,
(24)
follows that
note that zero destruction will be an
equilibrium if and only if
U’(c(kzd))
(1 + r(kzd)
- 6) - ~4’ (gzd)
r 0.
(25)
This equilibrium is illustrated in Figure 2, where equation (24) is graphed
as SSG
zd’
c
k
k
rd
Figure 2
We can divide k-g space into two regions depending upon whether or not
agents engage in positive destruction.
A member of generation t takes as
given the garbage stock at time t and the capital stock at t and t-l.
We
obtain the zero-destruction locus (ZDL) dividing the regions by setting
k
t-1
- kt and finding (kt, gt) pairs such that the first-order condition (5)
14
is satisfied with equality when dt - 0.
We illustrate positive- and
zero-destruction equilibria in Figures 3a and 3b.
k
6
k
zd
zd
Figure 3a
Figure 3b
6
Under the assumptions that d - 0 and k - ktel, it follows that
t
t
k
- w(kJ,
t+1
where w() is the steady-state function defined in equation (17).
r
- aWt)
f' (w(kt) > ;
- &Q
Therefore,
t+1
C
t+1
-
&ct)
- (1 + :(kt) - 6)w(kt).
Also,
gt+1 - (1 - Wgt
+ Bc(kt).
Using these definitions, the first-order condition will be satisfied with
equality at time t, assuming dt - 0, when
U' (~~+~)(l+ rt+l - 6) - ~4' (gt+l)- 0.
The ZDL As thug defined by
*
U'(cW)[c(W/wWl
- r4'((1 - Wgmt
+ Be(k)) - 0.
At kzd, ';(k)- r(k), z(k) - c(k) and w(k) - k.
Comparing this equation with
equation (22), it follows that the FOC will lie above the ZDL at kzd if and
only
if V/b)c(kzd) > gmL-
Since the left-hand side of this inequality is
simply gzd(kz,),we can observe either Figure 3a, where the SSGzd line lies
above the FOC, which in turn lies above the ZDL at ktd; or we can observe
Figure 3b, where the ordering is reversed. Note, finally, that the ZDL
can be shown to be downward sloping if (1 - o) > 0, if c'(kt) > 0, and if
there are no external increasing returns..
15
For a positive destruction equilibrium, it must be the case that the SSG
line intersects the FOG line above the ZDL.line, as in Figure 3a.7
In a
zero-destruction equilibrium, the SSGzd line must lie below the ZDL at kZd,
as illustrated in Figure 3b.
Special Case II:
zero garbage (p - 0)
Since an aim of this paper is to understand how an environmental externality
affects capital accumulation, it is useful to consider the no-garbage economy
as a reference point.
The economy in this case can be understood as a
special case of the no-destruction economy, since agents will obviously not
engage in destruction if there is nothing to destroy.
Agents thus save all
their wage income, and the equilibrium capital stock will, as above, be
implicitly defined by equation (23).
Note that this example is also the
special case of the Diamond (1965) model where agents do not consume in
youth.
zero degradation (b - 0)
Special Case III:
In practice, the natural rate of biodegradation depends in part on how
garbage is stored.
When biodegradable garbage is placed in the anaerobic
conditions of a landfill, the biodegradation rate has been found to be
extremely slow.
Hanson (1989) reports that, after 25 years in a landfill,
7
In addition, it must be the case that w(k) > k. Unlike the
zero-destruction locus, which summarizes agents' behavior, this is
essentially a technological restriction on k for destruction to be positive.
In terms of the figures, it implies that SSG must lie below SSGzd at a
positive-destruction equilibrium.
w(k) > k for k < kzd.
Figures 3a and 3b are drawn such that
16
-
waste retains its original weight, volume and shape.
8
Given this, it is
worth examining the limiting case where garbage does not degrade at all.
9
A steady-state equilibrium in the absence of degradation will be
characterized by garbage destruction by the young that exactly offsets the
addition to the garbage stock as a result of the consumption of the old.
That is, at a steady state with b - 0, it must be the case that
=b
(26)
p(k) - - IP(l-6) + rlk
In this case, the garbage equation serves to pin down the equilibrium capital
stock.
This equilibrium is illustrated in Figure 4.
k
b-0
k
Figure 4
8
William Rathje, an archeologist and organizer of a study of the waste
production of households (Le Projet du GarbBge), has excavated a number of
landfills, finding, among other things, forty-year-old readable newspapers
and hot dogs that are still recognizable after decades. See Rathje (1984)
for'a discussion of Le Projet du GarbAge, and Luoma (1990) for a more
general discussion of Rathje's work.
9
Recall also that the variable we call garbage might represent some more
general index of environmental quality. Under such an interpretation, it is
not obvious that a positive rate of degradation is an appropriate assumption.
17
Special Case IV:
Absence of External Increasing Returns (4(k) - 1, Vk)
At various points throughout the paper, we consider the role of external
increasing returns for the analysis.
For the present, we note simply that
we can eliminate the external increasing returns from the analysis by
setting $(k) everywhere equal to 1.
ia
z
COMPARATIVE STATICS
The comparative static results that follow are based on the equilibrium with
positive destruction, as illustrated in Figure 1.
Where applicable, we note
how our results. are altered for the special cases considered above.
1
Total
differentiation of equations (20) and (22) yields the following system:
p(l-6) + -y+ p'(k)
2
0
-
$J'(l-o)r
_izk
dr
dS
U’(l-a)
db
-
(27)
da
where r - U'() (1 - a)r'(k) - 0%
and Q - - U"()c/U'().l"
k2I
The determinant of the left-hand side matrix is
A - U'(a) (1 - o)r'(k) - a5
k2] - -~$j(8(1-6)
+ -r+ p'(k)).
The following conditions are together sufficient for A < 0:
(i)
(ii)
(1 - a)r'(k) c 0,
p'(k) 10.
The (1 - G) term in condition (i) indicates the response of saving to
changes in the interest rate.
If (1 - a) > 0, then the substitution
effect dominates and saving is increasing in the interest rate, whereas if
(1 - 4) < 0, the income effect dominates and saving is decreasing in the
10
The elasticity of substitution, u, is a function of k unless the utility
function exhibits constant relative risk aversion. For compactness of
notation, we suppress this dependence.
19
interest rate.
In the absence of external increasing returns, higher values
of the capital stock are associated with a lower rate of interest, so
If external increasing returns are present, r'(k) may be
r'(k) < 0.
positive.
To understand condition (ii), recall that
p(k)
7 y(k).
1
-
If factor shares are constant, then p(k) will be (perhaps inversely)
proportional to output (y(k)).11
implied by p'(k) > 0.
In this case, p(k) > 0 implies and is
This is true a fortiori if capital's share of output
increases as the capital stock increases.
Referring back to Figure 1, condition (i) implies that the first-order
condition is downward-sloping (since it implies that r < 0).
(ii) ensures that steady-state garbage is increasing in k.
and condition
The following
propositions set out the comparative static behavior of the model when
conditions (i) and (ii) hold.
Proposition 1:
12
Economies with better destruction technologies accumulate
less garbage in steady state, but may have more or less capital than
economies with worse technologies.
11
For example, if f(k) - kV, then p(k) - [(p + y)v - r]a$(k)kY.
12
*
Conditions (i) and (ii) are considered in greater detail in Section V,
where we discuss the possibility of multiple equilibria in the model. The
comparative static results derived below are easily reinterpreted if (i) or
(ii) does not hold but A is still negative. If A > 0, the equilibrium is not
locally stable.
20
_
An economy's destruction technology is better if 7 is higher.
Proof:
From
the system of equations characterizing the steady state,
II
p(l-6) + -y+ p'(k)
< 0;
Agents allocate their wages so that at the margin they are indifferent
between destroying garbage and consuming.
Thus, an economy with a more
productive destruction technology can devote fewer resources to destruction
to achieve a given garbage stock.
Agents in such economies thus have an
incentive to substitute consumption and saving for destruction, but saving
is relatively less effective and destruction relatively more effective for
increasing utility.
In terms of Figure 1, an increase in 7 shifts both
curves downward.
Changes in the productivity of the destruction technology are naturally
irrelevant in any zero-destruction equilibrium.
In the economy with no
degradation, improvements in the destruction technology unambiguously
increase the capital stock.
Proposition 2:
More wasteful economies accumulate less capital and more
garbage.
Proof:
A more wasteful economy has a higher B.
characterizing the steady state,
.&
2r2<().
d/3
bA
% -+>o.
dS
‘
0
21
By the system of equations
In wasteful economies each unit of consumption generates a lot of garbage.
Thus, to equate the marginal utility of consumption with the marginal
disutility of garbage accumulation, individuals must dedicate more
resources to destruction, leaving less to save and consume.
Figure 1, the SSG curve lies further to the left.
In terms of
In a zero-destruction
equilibrium, increases in /3have no effect on the capital stock but still
result in increased garbage in steady state.
Whenb
- 0, the comparative
static results are unchanged.
Economies with high biodegradation rates accumulate more
Proposition 3:
capital and less garbage.
Proof:
The parameter b is higher in an economy with a higher natural rate
of biodegradation.
dk
db--
fi
bA
Differentiating, we obtain
> 0.
'
35
+<o.
db - - bh
0
The finding that the capital stock is increasing in the rate of degradation
is intuitive:
the higher is the natural rate of degradation, the smaller is'
the amount of destruction that need be undertaken by the agents in the
economy.
In Figure 1, the SSG line lies further to the right.
Once again,
changes in b have no effect on the capital stock in a zero-destruction
equilibrium.
As noted earlier, the biodegradation rate may be very small in reality.
Such slow rates may lead economies to find alternative forms of waste
destruction, such as incineration; to reduce waste generation, perhaps
22
through recycling; or to increase the rate of biodegradation, perhaps by
These decisions may in turn have
switching from foam to paper products.
implications for the rate of capital accumulation.
Proposition 4:
13
More productive economies have higher or lower equilibrium
stocks of capital and garbage than less productive economies.
Proof:
A more productive economy has a higher a.
dk
da-
74"P
- 0) - b
I
< 0
>
where (1 - u) is of either sign, and p $ 0.
&ii
da
Differentiating,
Further,
I
[jl(l-6)+ 7 + p')U'(l-a)r
2 0
since both terms are either positive or negative. o
Increases in productivity will increase the interest rate, increasing the
incentive to accumulate capital if (1 - o) > 0; thus the first term in the
first equation will be positive.
Increases in productivity also increase
total output, implying an increase in garbage if p > 0.
This gives agents an
incentive to destroy more and save less, implying that the second term is
then negative.
The response of garbage to changes in productivity is
ambiguous for similar reasons.
In a no-destruction equilibrium, improvements in productivity
increase the capital stock if w(k) is concave, but still have an ambiguous
13
We plan to examine recycling and landfill space in more detail in future
work.
23
effect on steady-state garbage.
14
Similarly, when b - 0, improvements in the
technology unambiguously increase the steady-state capital stock.
(In this
case, the first term in the expression for dk/da equals zero, and the second
term can be signed since no degradation implies that p() < 0.)
The effect
on garbage remains ambiguous.
Proposition 5:
Economies with higher rates of capital depreciation have
higher or lower equilibrium stocks of garbage and capital than economies
with lower rates of capital depreciation.
Proof:
The higher the rate of capital depreciation the higher is 6.
dk
1
d6-s
rb"Bk
- a) - b
I
> o
<'
since the first bracketed term is of either sign and [- Iqq
!!k5 l
d6 -hb
Then,
- /3kc+ U‘(l - a)tB(1-6)
< 0. Further,
I
+ -Y+ ~‘1 5 0,
since (1 - 0) 2 0. 0
The ambiguity of these comparative static results arises from the response
of saving to the interest rate.
If (1 - a) 5 0 (i.e., if the interest
elasticity of saving is non-positive), then dk/d6 is unambiguously positive.
We obtain the unusual result that increases in the depreciation rate will
increase the steady-state capital stock; further, this result can arise even
without a backward-bending savings function.
The intuition is that an
increase in the depreciation rate reduces the consumption externality, thus
14
If c'(k) > 0, however, the garbage stock increases when a increases. In a
zero-destruction equilibrium, c(k) - y(k) - 6k, so that c'(k) > 0 if the
economy is dynamically efficient.
24
-
discouraging destruction and encouraging saving.
If, by contrast, the
interest elasticity of saving is non-negative ((1 - a) 2 0), then increases
in the depreciation rate unambiguously decrease the stock of garbage.
In the no-destruction economy, increases in the depreciation rate have
no effect on the equilibrium capital stock and cause a decrease in the
steady-state level of garbage.
In the no-degradation economy, increases in
the depreciation rate unambiguously increase the capital stock, while the
effect on garbage remains ambiguous.
25
y
MULTIPLE EOUILIBRIA
It
is well-known that multiple equilibria can be obtained in
overlapping-generations models by assuming strong income effects (see, for
example, Azariadis and Guesnerie (1986)).
It has also been shown (Weil
(1989)) that external increasing returns can generate,multiple equilibria
(see also Chatterjee, Cooper and Ravikumar (1990)).
These results carry
over to this model, as illustrated in Figure 5.
I
k
Figure 5
In contrast to Figure 1, the first-order condition does not have
negative slope everywhere.
Assuming that steady-state garbage is still
monotonically increasing in k, it is evident from this figure that a
necessary condition for multiplicity is that the first-order condition
somewhere have positive slope.
g as a function of k.
15
The first-order condition implicitly defines
Differentiation of the first-order condition yields
15
In terms of the economics, it is natural to think of the first-order
condition's determining the level of saving, given the stock of garbage. As
Figure 5 shows, however, this relationship may be a correspondence, rather
than a function. By the assumptions on 40, we can write g as an implicit
function of k.
26
I
Recalling that,r - U‘() (1 - a)r,(k) - ac/k2 , a necessary condition for
multiplicity is (1-o)r'(k) > 0.
16
This in turn requires either strong income
effects ((1 - 0) > 0) or r,(> > 0.
The latter is possible only in the
presence of external increasing returns, since
r' (k) - $P(k)f"W
+ $'(k)f'(k),
where the first term is negative by the concavity of f().
In Weil's model, equilibria with higher k are Pareto-preferred to low-k
equilibria.
Such a result need not hold in this model.
Increases in the
capital stock are generally associated with greater utility,
17
but hLgh-k
equilibria also exhibit a larger steady-state garbage stock, which reduces
utility.
The disutility from higher garbage may more than offset the
increased utility from the higher capital stock.
Recalling the quote at the
start of this paper, this might be termed a Galbraithian view.
16
In Weil's model, multiplicity ds not possible when the interest elasticity
of savings is negative (see his Proposition 1). This is consistent with the
result here, since Weil's model imposes increasing returns.
17
The condition for utility to be increasing in k, given g, is
(l+r6)a$'()f() - (1:- L)r'(k)k > 0.
(In the absence of external increasing returns, $'() - 0, r'() < 0, and the
condition simply reduces to r > 6 (dynamic efficiency). In the presence of
external increasing returns the first term is positive but r'(k) may be
positive.) To see this, note that we can write the indirect utility of a
young agent as
V(k, g) - mzx U((w(k) - d)(l + r(k) - 6)) - d(g - yd)
*
av - U'()[r'(k)k + (1 + r - 6)w'(k)],
z
by the envelope theorem. The result follows from the fact that
w' (k) - y’ (k) - r'(k)k - r - a$'(k)f(k) - r'(k)k.
27
Multiplicity can arise in this model in the absence of both strong
income effects and external increasing returns if, at high values of the
capital stock, economies engage in sufficient destruction to reduce the
steady-state stock of garbage.
This
is illustrated in Figure 6.
SSG
L
k
Figure 6
At
the low-k, high-g equilibrium, agents have relatively more of an
incentive to destroy.
Since agents are putting resources into destruction,
they do not save much and the capital stock is low; this implies that agents
have low income and so cannot engage in much destruction, validating the
high garbage stock.
By contrast, at the high-k, low-g equilibrium, agents
have less of an incentive to destroy, leading to a higher capital stock and.
a greater ability to engage in destruction.
These complementarities lie
behind the multiple equilibria.
A necessary condition for multiplicity when the first-order condition
is downward sloping is that steady-state garbage is decreasing in k over
some range.
*
g(k) -
Recall that
;
01
M(l-6)
+ rlk + p(k)
1
fl(l-6)+ 7 + p'(k)
28
A necessary condition for multiplicity is thus that p'(k,) < 0.
It is easily
confirmed that
p'(k) - (B + -r)r(k)Ll - rlrkl- ry'W
where t) - - r,(k)k/r(k).
rk
f
In the absence of external increasing returns,
y'(k) - r(k), and the necessary condition for multiplicity becomes
r(W[B(l
- s,> - rtlr,l< 0.
External increasing returns are thus not required for this type of
multiplicity.
ia
External increasing returns imply increases in y'() and
decreases in Q=~, and so have,an ambiguous effect on p'().
This model is consistent with observations of relatively poor economies
with serious pollution problems; many countries in Eastern Europe currently
fit this description.
Contrary to the popular perception that pollution is
associated with high GNP, it may rather be the case that it is only rich
countries that are able to spare the resources to combat environmental
problems -- a conclusion more Solovian than Galbraithian.
Multiplicity:
Examples
In order to demonstrate the possibility of multiplicity more formally, we now
consider two simple examples.
Proposition 6:
$(k) - k'.
a - 3.
Suppose that U(c) - h(c),
4(g) - (6/2)g2, f(k) - k" and
Let t - 3/2, Y - l/2, p - 5, y - 7, 6 - 1, B - l/56, b - l/2, and
Then there are two equilibria.
18
This is explicitly proved (by example) in Proposition 7 below.
29
Proof:
The steady-state first-order condition from this model is g - l/r0k.
Further, it is easily shown that p(k) - [BY - r(l-v>]ak'+".
The steady-state
solution for k is defined by
Z(k) - (flr/b>l(P(l-J>+ r)k + p(k)1 - l/k - 0,
which, for the parameter values specified, implies
Z(k) - (k/4)(7 - 3k) - l/k - 0.
This equation has two positive roots:
k - (1, 2).
It is easily confirmed
that destruction is positive at these values of k. o
Proposition 7:
$(k) - 1 V k.
Suppose that U(c) - h(c),
d(g) = (O/2>g2, f(k) - k/(1 + k),
Let 7 - 6 - 1, b/B - 18/2S, @ - 257/313, and a - 313/SO.
Then
there are three equilibria.
Proof:
As in Proposition 6, the first-order condition is g - l/Byk.
For
the CES production function specified, it can be shown that
g(k)- ; [8(1-6)
+ rlk+ (BGiven the parameter values specified, the equilibrium values of k are
defined by
Z(k)-k+[&k][$#-&]-g-0.
This equation has three positive roots:
k - (1, 3/2, 2).
Destruction is
positive at these values of k. o
Note that the functional forms assumed for these examples are
completely standard (log utility, quadratic costs, CES production
functions).
Note also that; since the first-order condition slopes downward
in k-g space, multiplicity must arise through the garbage effect just
discussed.
In the first example, multiplicity arises because of external
30
increasing returns combined with a negative value of p(k).
In the second
example, multiplicity arises because labor's share of output increases as k
increases; this effect is sufficiently strong to cause garbage to be a
decreasing function of k over some range.
Multiple equilibria can arise under the special cases discussed
earlier.
Considering first zero degradation, there will be multiple
equilibria if equation (26) (reproduced here) has multiple roots:
p(k) - -
[8(1-6)
+ rlk
(26)
It is easy to construct examples where this is true.
Multiple equilibria
with zero destruction will arise if equation (23) (w(k) - k) has multiple
roots,
and if the steady-state garbage line lies below the zero-destruction
locus at these values of k.
Again, examples are easily constructed.
31
E
WELFARE ANALYSIS
Since environmental damage may outlive its perpetrators, overlappinggenerations models provide the most appropriate demographic structure for
analysis of environmental externalities.
direct implications for welfare analysis.
Adoption of this framework has
First, intergenerational
externalities are intrinsically hard to internalize:
those imposing the
externalities are not alive at the same time as those who enjoy or suffer the
consequences.
Second, an overlapping-generations structure complicates the
analysis of Pareto-improving policies, since the welfare of multiple
generations must be considered.
We first consider the golden-rule allocation, solving the problem
of a social planner who treats all generations symmetrically.
The social
planner solves
Maximize U(c) - b(g)
(28)
subject to
y+(l-6)k-c+d+k
(29)
bg - PC - yd
(30)
Y - a$WfW,
(31)
where (29) and (31) combined represent economic feasibility and (30) is the
steady-state stock of garbage.
Substitute (30) into (28) and form the
Lagrangian
maximize I - U(c) c, d, k
+ X(a$(k)f(k) - 6k - c - d).
(32)
The-first-order conditions are
are
B - x - 0
ac - U' - 4'6
(33)
a!e
~-t$'i;r-x-o
(34)
32
-
ait?
- x(a[$f' + Ip'f] - 6) - 0
z
(35)
aif2
- a$(k)f(k) - 6k - c - d - 0.
ax
(36)
-
From (3S), the planner sets capital at the level at which net output is
maximized:l'
a[$f' + $'f] - r + a$'f - 6.
(37)
Equation (37) is essentially the familiar condition for the golden-rule
level of capital.
Note that the planner's choice internalizes the
externality from the external increasing returns.
From (33) and (34), the
planner sets the marginal utility from consumption equal to the marginal
utility from destruction:
U' - pgw
p;.
(38)
Comparing (38) with (5) establishes that the planner internalizes the
consumption externality (the +'B/b term) and the destruction externality
(the l/b in the 4’7/b
term).
Intuitively, the planner's problem decomposes in this way because, for'
any given value of the capital stock, the planner can divide output between
consumption and destruction to achieve any constant garbage stock.
Thus the
planner can translate an increase in output into increased consumption while
keeping garbage constant, unambiguously increasing utility.
The planner
therefore chooses k to maximize output, and then divides it optimally.
As
Proposition 8 shows, the social planner can achieve this social optimum by
means of taxes and/or subsidies on wage and rental income.
19
Note that if there are sufficiently strong external increasing returns net
output might be unbounded.
33
The decentralized economy with taxes on wages and capital
Proposition 8:
income can achieve the first-best (golden-rule) steady-state allocation.
Suppose the planner taxes the net return on capital and wages at the
Proof:
rates r
k
respectively.
and r
u'
In the presence of taxes, a representative
agent's choice problem is to choose c
t+1'
dtp
and st to solve
u(ct+l) - Hg,,,)
subject to
~$1
- rJ
- st + dt
- (1 +
C
g t+1 -
(r
t+1
t+1
(1
- 6)(1 - rk))st + T
- b)gt + Bc
t
t+1
- 7dt
dt, st > 0
C
t+1'
where Tt+l is a transfer (lump-sum tax).
A steady-state equilibrium for the economy with taxes is now a vector
(c, d, g, w, f, s, k, T) such that agents optimize, factor and goods
-*
markets clear, the stock of garbage is constant, and the government budget
constraint is satisfied: r,w + rk(r - 6)k - T.
At a steady state the
representative agent's indirect utility can be defined as a function of the
tax parameters:
Wk, r”) - U([l + (r(rk,rw) - 6)Ik(rk,rw) + ~~w(r~;,r~)) 4 t([l+(r(r
t
k,rw)-6)lk(rk,rw)
+ rww(rk,rr)) - $w(rk.rw)(l
- r-1 - k(rk,rw))
where r(rk,rw), k(rk,rw), and w(rk,rW) are the implicit functions defining
the interest rate, capital, and the wage rate, respectively, in steady-state
equilibrium.
34
The planner chooses r and rk to maximize the representative agent's
"
indirect utility function.
The first-order conditions of the planner's
constrained problem are
avo
-777
1
ak
- u'()w + u'() (1 + r - 6) + kr'+ r [y' - r - kr'] r
I
de()r+]w
-
- dt()[g((l + 1: - 6)-+ kr' + rW[y' - 1*- kr']] -
11*
[y' - r - r'k](l - rW) - 1
1
ak
av() = IJ'() (1 + '1:- 6) + kr'+ rW[y' - r - kr'] ar
ark
4’ 0
(1 + r - 6) + kr' + r,[y' L r - kr']
g
- 0
k
1
-
11
[y' - r - r'k](l - rW) - 1
$$- - 0,
k
which simplify to
U’O -
ay(rks
rW)/ak - 6.
These equations define rk and r .
*
Note that ay/ak - a[$f' + $'f].
Thus,
the first-order conditions for the constrained and unconstrained planner's
problems are identical:
the planner can choose rk and r to achieve the
w
social optimum.0
The planner requires two tax parameters to direct the economy to the
social optimum because there are two market distortions:
one in the goods
market, the other in the capital market.
. As is well-known, the competitive equilibrium of a Diamond (1965) type,
overlapping-generations model can be dynamically inefficient (see, for
example, Blanchard and Fischer (1989, p. 103)).
35
If so, all generations would
be better off if they saved less (accumulated less capital) and consumed
more.
Not surprisingly, such a result carries over to this model.
There is
an analogous possibility of inefficiency in terms of the garbage stock.
That
is, agents may underaccumulate garbage, implying that all generations could
be made better off by destroying less and consuming more.
To see this,
consider Figure 7.
k
sr
Figure 7,
k
Equation (37) defines the golden-rule level of the capital stock (k,=).
Equation (38) can be interpreted as defining the optimal level of garbage
for a given value of the capital stock (g*(k)), in steady state.
these divide k-g space into four regions.
20
Together,
In regions II and IV, the capital
stock exceeds its golden-rule level, the economy is dynamically inefficient,
and it is possible to increase all generations' utility by increasing
consumption and decreasing saving.
In regions III and IV, below g*(k),
agents are underaccumulating garbage, and it is possible to increase all
20 *
Using the facts that, in steady state, bg - @c - yd and y(k) - 6k - c + d,
we can show that c - [r(y(k) - 6k) + bg]/(@ + 7). Substituting this into
equation (38), it is easily shown that g*(k) attains a minimum at the
golden-rule capital stock.
36
21
generations' utility by increasing consumption and'decreasing destruction.
In the absence of external increasing returns, the economy cannot be in
equilibrium in region III; that is, there are no dynamically efficient
steady-state equilibria where agents underaccumulate garbage.
In the
presence of external increasing returns, however, equilibria in region III
are possible, as established by the following proposition.
Underaccumulation of garbage in a dynamically efficient
Proposition 9:
equilibrium is possible only in the presence of external increasing returns.
Proof:
The first-order condition from the agent's maximization problem is
given by equation (22):
U'(c)(l + r - 6) - 74'(g) - 0.
(22)
In any steady state,
c , r(y(k) - 6k) + bg
B+r
'
so equation (22) can be interpreted as defining the steady-state value of g
for a given k.
Now g*(k) is implicitly defined by (38):
U’ (c> - 4’ (g’)[ - d’q
*
U’(c) -
(38)
(q$ 14’d>,
with consumption given by the same expression as before.
Since U() is
concave and I$() is convex, comparison of (22) and (38) reveals that g < g* if
and only if
b
L3+7
>l+r-6
7
21
If the economy is in region II, III or IV, Pareto-improving policies thus
entail transfers from the young to the old. Such Pareto improvements could
be supported by social contracts of the type discussed by Kotlikoff, Persson
and Svensson (1988).
37
*
l+r-6<%
/3+7-
since b s 1 and ,9> 0.
r < 6.
It follows that a necessary condition for g < g* is
In the absence of external increasing returns, this condition
implies dynamic inefficieicy.
But in the presence of external increasing
returns, the economy is dynamically inefficient iff r + a$'f < 6, so this
condition does not contradict dynamic efficiency. o
In region I, simple Pareto-improving policies are not so easily found.
Pareto improvements are possible in general if agents are at an equilibrium
with positive destruction or if there are no external increasing returns.
This is proved in Proposition 10.
Proposition 10:
Pareto-improving policies are generically possible in any
equilibrium with positive destruction or with no external increasing returns.
Proof:
For the sake of economy of notation, we suppose that the economy is
initially in steady-state equilibrium (k, g), although the method of proof
is applicable to any equilibrium.
this steady state.
Consider the following perturbation to
At time r, increase saving by one unit and decrease
destruction by one unit.
At time r+l, set s~+~ - k and let dr+i - d + a.
time r+2, set s~+~ - k and let dr+z - d + f.
At
Choose 6 such that gr+3 - g and
choose a such that the utility of generation r+l is 3 (steady-state utility).
Since kr+r - kr+j - k, and gr+a - g, the utility of all generations born
at r-+2and after can be maintained at c.
The utility of all generations up
to and including r-l is unchanged, and the utility of generation r+l is
unchanged by construction.
It thus remains to consider the effect of this
38
Note first that gr+l - g + 7 and Y,+~ - y + r.
perturbation on generation r.
a,
Since dr+l - d +
C
r+l
feasibility at r+l implies
-c+(l+r-
6)-a.
The change in utility of generation r is thus given by
- a +
(55171
- - au'(c),
using the first-order condition from the agent's problem.
Using the
expressions for c~+~ and dr+l, we obtain
g r+2 - g + p(l + r - 6) + 7(1-b) -
a(/3
+ 7).
Since Y,+~ - y + atg'f and dr+2 - d + (F,feasibility at r+2 implies
C
r+2
- c + a$'f - c.
It then follows that
gr+3 - g + (1-b)[/3(1+ r - 6) + 7(1-b) -
+ 7)]
a@
+ fia$'f - (@ + 7)~.
Choosing c such that gr+3 - g implies
1
8(1 + r - 6) - 7(1-b)
- a(l-b).
The change in utility of generation r+l is given by
U’ (c)(c,+2
- cl
- 4’ (g> (gr+2 - g)
- U'(c) a$'f - c + [' + : - 6](~(1 + r - 6) + 7(1-b) - a(p + 7)]].
Choosing a to set this equal to zero and substituting in for c yields finally
a , B(1 + r - 6) + 7(1-b)
f3+7
a$'f7'
(B + r)[(B
+ r)(l
+ r
-
6)
- r(l-b)l’
If the external increasing returns are sufficiently strong, so that the
second term in this expression dominates and
will be Pareto-improving.
a
< 0, then this perturbation
Since the proposed perturbation entails decreasing
destruction, it is only feasible in an equilibrium with positive destruction.
In the absence of external increasing returns, the second term in this
expression equals zero and
a
is positive.
39
In this case the proposed
perturbation would unambiguously lower the utility of generation r;
increasing destruction by one unit and lowering saving by one unit would be
Pareto-improving. 0
Proposition 10 shows that, even when agents do not overaccumulate
capital or underaccumulate garbage, Pareto-improving policies may be possible
because of the externalities present in the model.
With sufficiently strong
external increasing returns, the reallocation suggested in the proposition
entails a reduction in destruction, which is not feasible in a
zero-destruction equilibrium.
Increasing saving in period r while leaving
destruction unchanged is not a Pareto-improving policy, since, by
feasibility, this reduces the consumption, and hence the utility, of the old
alive at time 7.
This possibility is a direct consequence of the
overlapping-generations structure.
Note, finally, that Proposition 10 does
not prove that no Pareto-improving reallocations are possible when the
suggested perturbation fails, although we conjecture this to be the case.
40
-
SECTION n
CONCLUSIONS
Actions that affect environmental quality both influence and respond to
macroeconomic variables, and many environmental and macroeconomic
consequences of agents' actions have uncompensated effects that outlive the
actors.
This paper has examined an overlapping-generations model of
environmental externalities and capital accumulation in which young agents
either invest in capital or in destruction of garbage, and where the
consumption of the old augments the stock of garbage.
The model in this paper demonstrates that it may be misleading to
address environmental and macroeconomic concerns in isolation.
Changes in
parameters describing the evolution of the stock of waste also have effects
on the accumulation of capital.
In particular, we find that increases in
the natural rate of degradation of waste encourage the accumulation of
capital, and that improvements in society's ability to dispose of waste may
reduce capital accumulation.
Also, a higher depreciation rate of capital may
be associated with a higher equilibrium capital stock.
Multiple Pareto-ranked equilibria -- coordination failures -- are
possible in our model.
Such multiplicities can arise not only from the
well-understood sources of strong income effects or external increasing
returns, but also from the interaction of garbage and capital accumulation.
An economy may get stuck in a low-output, high-garbage equilibrium, where
the.high level of garbage reduces agents' incentives to invest, and
the resulting low level of income prevents them from destroying the stock of
garbage, even though a better (low-garbage high-output) equilibrium exists.
41
We plan to extend the model of this paper in a number of directions.
First, we are investigating the dynamic behavior of the model in order to
understand the effects of the environmental externality on growth paths.
Second, we intend to generalize the model to allow for consumption in youth
and old age, so that there are distinct saving and destruction decisions.
Third, we wish to explore the implications of introducing interest-bearing
government debt, in order to gain insight into the effects of macroeconomic
policies on decisions that affect the environment.
Finally, we will apply
our model to specific environmental concerns, such as recycling and the
shortage of landfill space.
42
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