limits and cycles of environmental policy

Environmental and Resource Economics 11: 155175, 1998. 155c

1998 Kluwer Academic Publishers. Printed in the Netherlands.

Limits and Cycles of Environmental Policy

THOMAS WAGNERGeorg-Simon-Ohm Fachhochschule Nurnberg, Hastverstrasse 31, D-90408 Nurnberg, Germany

Accepted 10 April 1997

Abstract. The OLG-model analyzes emissions of an accumulative pollutant in a laissez-faire economyand an economy regulated through a government controlled license market. The government eithertakes the price on the license market as given or sells the licenses demanded at the Cournot price. Thefirst type of regulation is called a liberal environmental policy, and the second type a monopolisticenvironmental policy. The forward looking temporary and the stationary equilibria as well as thepollution boundaries of the mechanisms are studied. If people can choose between laissez-faireand regulation (or between the liberal and the monopolistic environmental policy regime), then ingeneral no steady state exists. Instead endogenous policy cycles can alternate between laissez-faireand regulation or between liberal and monopolistic regulation.

Key words: overlapping generations, emission permits, pollution boundaries, environmental policycycles

1. Introduction

Global climatic changes, the extinction of species, and the fading capacity of theenvironment to assimilate pollutants are processes of the ecological system causedby anthropogenous and natural factors alike. The time dimension of these complexchains of causes and effects exceeds the planning horizons of social systems.Climatic changes are durable public bads. At the same time, they are internationalpublic bads, since they damage ecological systems around the globe. The regionaleffects of climatic changes, however, depend on the geography of a given country,its resources, its institutions and the level of its economic development (UNEP1997).

To model the intergenerational character of these processes, it seems rea-sonable to use the theory of overlapping generations (OLG) first developed bySamuelson (1958) and Diamond (1965). Models of the OLG type have recentlybecome popular, not only in the fields of growth theory and macroeconomic theory(Blanchard and Fischer 1989; Azariades 1993), but also within environmental andresource economics (Kemp and Long 1982; Hultkrantz 1991; Lofgren 1991; Burton1993). But within the theory of externalities as well as the neoclassical theory ofenvironmental policy they are still exceptions. Optimal growth models whichhave been developed in the context of anthropogenous climatic effects (Nordhaus

A version of his paper was delivered at the Seventh Annual Conference of the EARE, June2729, 1996, Lisbon.

156 THOMAS WAGNER

1991a, b; Uzawa 1991) are usually control theoretical approaches with continuoustime and perfectly informed individuals whose life expectancy is as infinite as thetime horizon of the allocative mechanism (Lines 1995). In such a context, there isno intergenerational moral hazard and no conflict between the generations whichcould not be solved, for example, by the specification of exclusive property rightsor by the assumption of bequest motives. Recently, however, some authors havechosen an overlapping generations approach. Howarth and Norgaard (1990, 1992,1995) utilize the time dimension of anthropogenous climatic effects to show thatPareto efficiency which within a static context is no guarantee for subsistence is not sufficient for the sustainability of a development path chosen by a socialplanner. John and Pecchenino (1994) and John et al. (1995) introduce a descriptivemodel with a long-lived pollutant which results from the old generations consump-tion and has an adverse effect on the welfare of present and future generations alike.The representative actor lives through two periods. During the first he works forhis income, during the second he consumes and suffers from the environmentalpollution. Each generation imposes a tax on the pollutant in such a way as tomaximize its own welfare. Wagner (1997) develops the Samuelson condition forintertemporal public bads with a constant rate of decay as well as the conditionfor steady state growth in a laissez-fair economy and in a myopically regulatedeconomy. An example compares the welfare levels of the representative individ-ual in the steady states of both allocative mechanisms. The Pareto superiority ofmyopic regulation depends especially on the residence time of the pollutant. Firmscan either buy emission licenses or avoid pollution through reduced production orabatement technologies. In this paper, we follow John and Pecchenino (1994) andJohn et al. (1995) in assuming that the pollution sink is not provided by the firms,but by government, which uses the license revenue to invest into the quality of theenvironment. The paper is structured as follows.

Part 2 describes the behavior of households, firms and government as well asthe market for emission permits. Part 3, which presents new results, shows theendogenous pollution boundaries of the different allocative mechanisms, namelythe laissez-faire economy, the liberal, and the monopolistic environmental policyregimes. Government offers emission licenses either taking the price on the licensemarket as given or using its market position to sell the amoung of licenses demandedat the Cournot price. The first type of regulation is called liberal environmentalpolicy, the second is called monopolistic environmental policy. The goods spaceof the model consists of two dimensions, one for the produced good and the otherfor the environmental quality. A survival limit in the goods space is characterizedby a critical amount of the produced good or of environmental quality below whichlife becomes impossible. This quasi-physiological minimum is independent of theactual allocative mechanism and is, for eonvenience, defined as zero consumptionor zero environmental quality. The three allocative mechanisms can be interpretedas decision structures transforming a given survival limit from goods space intothe state space of the model. The state space consists of two variables, namely the

LIMITS AND CYCLES OF ENVIRONMENTAL POLICY 157

per capita capital stock and environmental quality. In contrast to goods space, thepollution boundaries or survival limits in state space are not independent from theinstitutional structure of the prevailing allocative mechanism.

In Part 4, we construct the indifference curves (ID) containing all initial condi-tions where the generation in power (that is, the young generation) is indifferentbetween regulation and laissez-faire or between a liberal and a monopolistic inten-sity of regulation. Points on either side of the ID-curves represent states whereactors prefer either regulation or laissez-faire and, in the case of regulation, wherethey prefer either the monopolistic or the liberal policy regime. There already existanalytical concepts in John and Pecchenino (1994) and Wagner (1997) which aresimilar to the notion of the ID-curve.

In Part 5, we present some new results. We analyze the steady states of theallocative mechanisms and show that environmental policy can generate cyclesjust as fiscal and monetary policy can. Generally in democratic societies, whereeach generation can choose between laissez-faire and emission regulation and,in the case of regulation, between a liberal and a monopolistic environmentalpolicy, a stationary equilibrium does not exist. Instead, freedom of choice generatesendogenous policy cycles. Part 6 summarizes the results. Propositions are provenin the appendix.

2. Households, Firms and the Market for Pollution Licenses

In each period a new generation is born. The typical member of a generation has alifespan of two periods. During the first, she offers inelastically one unit of labor,spends an amount c1t of her incomewt on first period consumption, saves an amounts

t

such thatwt

= c1t+st, and invests her savings in the capital market at an interestrate z

t+1. The actor uses her rent income (1 + zt+1)st to pay for second periodconsumption c2t+1 = (1 + zt+1)st. The households lifecycle utility, Ut, is mea-sured by an intemporal CobbDouglas utility function U

t

= c

11t c

2t+1A1t

A

2t+1:

denotes the saving rate, 0 6 6 1. Both public goods (At

; A

t+1) have nonnegativeelasticities

i

> 0; i = 1; 2. The household maximizes its utility at the given factorprices and with respect to given amounts of the public goods. In the utility function,the income and the substitution effect of the interest rate compensate each other.Therefore, the households capital supply and consumption demand can be writtenas s

t

= w

t

; c1t = (1 )wt and c2t+1 = (1 + zt+1)wt respectively.The public goods are interpreted as natural capital or environmental quality

at time t and t + 1 respectively. Since there is neither exclusion nor rivalry inconsumption, all households enjoy them freely and equally. For example, A

t

withA

t

= AQ

t

can be interpreted as a measure of the remaining stock of unpollutedatmosphere. A stands for the natural amount of atmosphere which is not subjectto any autonomous growth processes. Q

t

is the stock of an anthropogenous pollu-tant which damages the protective and selfregulative capacity of the atmosphere

158 THOMAS WAGNER

(anthropogenous greenhouse gases). The generation born at time t inherits thestock Q

t

from its ancestors. The pollution follows the law of motion

Q

t+1 = X +Net(1 t) + Qt: (1)X denotes an exogenous flow of emissions (emissions of foreign countries). Nstands for the constant population, which is normalized at unity, N = 1: 2 [0; 1)is the parameter for the pollutions residence time, and (1 ) is its rate of decay.An assimilative substance has a residence time of = 0. For an accumulativesubstance, residence time is > 0 (Tietenberg, 1985). e

t

denotes the per capitaemission in period t which is added to the stock after a time-lag of one period.Under the laissez-faire regime (L), each firm has free access to the global commons,therefore,

t

= 0, where t

is the unit price at time t for pollution permits. Inthe regulated economy, government uses the license revenue e

t

t

to invest in thepreservation of the environmental quality (reforestation). denotes the productivityof the preservation activity. For simplicity, is assumed to be constant (John andPecchenino, 1994; John et al., 1995). If

t

> 1, the pollution sink has the qualitiesof a backstop technology: Although governments preservation activities do notsubstitute the endogenous source of pollution, the investment of the license revenueneutralizes their adverse effect as shown in Equation (1). If

t

> 1, the economyeven accumulates natural capital with a net investment of e

t

(

t

1) > 0.Emissions are treated as a third factor of production in addition to labor and

capital (Brock, 1973); as such they enter into the per capita production func-tion f(k

t

; e

t

) of the representative firm, where k stands for capital intensity.The state of technology is described by > 0. The production function isstrictly concave with f

i

> 0; fii

< 0; i = 1; 2; f(0; e) = f(k; 0) = 1, andf1(1; e) = f2(k;1) = 0. Firms acting as price takers maximize their per-capitaprofit

t

;

t

= f(k

t

; e

t

) z

t

k

t

(q+

t

)e

t

w

t

r. The emission generated inthe production process produces not only social costs but also private factor costsq. In addition, trading on the license market causes fixed per capita cost r > 0. Ata license price

t

> 0, firms invariably exhibit a positive demand for emissionsand emission permits, given the neoclassical characteristics of the technology. Aprofit maximizing firm employs emissions until their marginal product equals the(gross) unit price q

t

= q +

t

of the pollution, so that f2(kt; et) = qt holds in thefirms equilibrium. From that follows the firms direct demand for emissions andemission permits e

t

= e(k

t

; q

t

). The demand for licenses diminishes, if the pricefor emissions q

t

rises, and it increases if the capital stock grows, at least if capitaland emissions are cooperant factors of production, so that f12 > 0, an assumptionwhich is held throughout the rest of the paper. Moreover, for a profit-maximizingproduction program, the following conditions must also hold:

z

t

= f1(kt; e(kt)) z(kt) (2)w

t

= f(k

t

; e(k

t

)) k

t

z(k

t

) e(k

t

)q

t

r w(k

t

): (3)


With a term of office of one period, each government chooses a supplyof emission licenses e

t

, which maximizes the indirect utility function Vt

=

#w

t

(1 zt+1)

A

1t

A

2t+1; # = (1 )1 , subject to constraint Equation (1).

The regulating agent adjusts to the given factor prices, to the exogenous stockof the public good A

t

, and to the (indirect) demand function on the license mar-ket

t

(e

t

). If et

maximizes the constrained utility function of the government, weget the first order condition 1 (

t

+ e

t

0

t

) = 0. In the monopolistic envi-ronmental policy regime government regulates emissions with the Cournot price

= 1=(1 ), where = 1="e

is the degree of monopoly and "e

theprice elasticity of the demand for pollution licenses. For simplicity, we assumethat the degree of monopoly on the license market is constant, = const., so thatq

= q + 1=(1 ) is the unit price of pollution. In the liberal environmentalpolicy regime the regulating agent acts as a price taker ( = 0), and the licensemarket will be in equilibrium at the marginal cost price 0 = 1=, where 1/ is themarginal preservation cost.

Given the constant population, the firms capital demand kt+1, and the capital

supply of the households wt

, the capital market and, by Walras law, also thegoods market are in equilibrium if

k

t+1 = w(kt) S(kt): (4)From Equations (2)(4) and from the budget constraint of the household we can

deduce the demand functions for both private consumption goods:c1t = (1 )w(kt) c1(kt) (5)c2t+1 = (1 + z(S(kt)))S(kt) c2(kt): (6)

In the unregulated economy L, the factor price of emissions is qt

= q, since

t

= 0, and the cost of regulation are r = 0; apart from that, the equilibrium of Lcan be characterized by the same conditions of Equations (1)(4).

The factor price functions in an (un-)regulated economyR

(L) can be describedby the following Lemma:1

LEMMA 1. If the per capita production function has neoclassical properties, then(i) in L the factor price functions are characterized by z(0) = 1; z(1) =

0; z(kt

)

0

< 0 and w(0) = 0; w(kt

)

0

> 0;(ii) in R

they are characterized by z

(0) =1; z

(1) = 0; z

(k

t

)

0

> 0 andw

(0) = r; w

(k

t

)

0

> 0.

3. Pollution Boundaries

3.1. LAWS OF MOTION OF THE POLLUTION STOCK

The accumulation of pollution in the environment is influenced by the laws ofnature as well as by the institutions of the allocative mechanism within which the

160 THOMAS WAGNER

sources of pollution operate. For instance, if we take into account Equations (1)and

t

= 0, the growth path of the pollution stock in an unregulated economy Lcan be expressed as:

Q

t+1 = X + e(kt) + Qt Q(kt; Qt): (7)The law of motion of the pollution stock in a regulated economy R

dependson the behavior of government on the license market. If the regulating agentacts as a price taker, and if, therefore, firms in equilibrium pay the marginal costprice 0 = 1= for the right to pollute, government investment in environmentalpreservation will be just enough to absorb any new emissions e0(kt), as Equation(1) points out: e0(kt)(1 0) = 0. Thus, for a liberal environmental policy R0,we can write:

Q

t+1 = X + Qt Q0(Qt): (8)If the regulating agent according to its market position acts as monopolist and

offers the emission permits at the Cournot price, then the installed capacity ofthe pollution sink absorbs not only all new emissions but beyond that also partof the accumulated stock of pollution. Net investments in the pollution sink willbe E

(k

t

) e

(k

t

)(

1) = e

(k

t

)=(1 ). Thus, under a monopolisticenvironmental policy R

, the law of motion of the pollution stock is

Q

t+1 = X E(kt) + Qt Q(kt; Qt): (9)Of course Equation (8) is a special case of Equation (9) and follows from

Equation (9) with = 0. With a high degree of monopoly or a large capital stock,net investmentE

(k

t

) inR

can cause pollution absorption to become so intensive,that Q

(k

t

; Q

t

) < 0. The pollutant becomes a utility augmenting good whichadds to the natural stock of A, so that A

t+1 = AQ(kt; Qt) > A.

3.2. COMPARATIVE ANALYSIS OF THE ALLOCATIVE MECHANISMS

If we compare the allocative mechanismsL andR

with respect to emissions, factorinput, factor prices, consumption, and pollution stock, we come to the followingprognosis.

LEMMA 2. If the initial condition (Qt

; k

t

) at time t with kt

> 0 is identical forboth allocative mechanisms L and R

and r > 0, then the following relationshipsbetween the endogenous variables in L and R

are to be expected:

(i) e(kt

) > e

(k

t

) and z(kt

) > z

(k

t

).

(ii) w(kt

) > w

(k

t

); c1(kt) > c1(kt) and S(kt) > S(kt), if "ek

(@e

t

=@k

t

)(k

t

=e

t

) < 1.(iii) c2(kt) > c2(kt), if "

ek

6 1 and "zk

(@z

t

=@k

t

)(k

t

=z

t

) > 1.(iv) Q(k

t

; Q

t

) > Q

(k

t

; Q

t

).


The comparison between a liberal (R0) and a monopolistic environmental policy(R

) leads to:

LEMMA 3. If the initial condition (Qt

; k

t

) with kt

> 0 is identical for both environ-mental policy regimes, then the following (in-)equalities hold:

(i) e0(kt) > e(kt) and z0(kt) > z(kt).(ii) w0(kt) > w(kt); c01(kt) > c1(kt) and S0(kt) > S(kt), if "

ek

< 1.(iii) c02(kt) > c2(kt), if "

ek

6 1 and "zk

> 1.(iv) Q0(Qt) = Q(0; Qt) and Q0(Qt) > Q(kt; Qt), if kt > 0.If we substitute Equations (2)(4) and Equations (7)(9) into the indirect utility

function, we obtain the lifecycle utility of a typical member of the generation bornin period t at the initial condition (Q

t

; k

t

) as the product of the utility drawn fromconsumption and the utility drawn from environmental quality:

V (k

t

; Q

t

) = (k

t

)a(k

t

; Q

t

)A

1t

: (10)The (indirect) utility from consumption is (k

t

) = #(1 + z(S(kt

)))

w(k

t

),

and the endogenous part of the utility from environmental quality is a(kt

; Q

t

) =

(A Q(k

t

; Q

t

))

2. In the same way, we decompose the actors indirect utility in

R

into their utility for consumption

(k

t

) = #(1+ z

(S

(k

t

)))

w

(k

t

) and theendogenous part of the utility from environment a

(k

t

; Q

t

) = (AQ

(k

t

; Q

t

))

2.

Lemma 2 points out that identical initial conditions lead to (kt

) >

(k

t

) anda(k

t

; Q

t

) < a

(k

t

; Q

t

), so that as a rule there is no Pareto dominance betweenthe allocative mechanisms L and R

. Comparing the liberal and the monopolisticenvironmental policy at identical initial conditions, we find a similar result, since aliberal regulation leads to a higher utility from consumption and to a lower utilityfrom environmental quality than a monopolistic regulation.

3.3. POLLUTION BOUNDARIES

We use the term subsistence level to denote conditions (Qt

; k

t

) where the utilityof the typical actor becomes zero. As we look only at those initial states (Q

t

; k

t

) atwhich the exogenous utility from environment is positive,A

t

> 0, at the subsistencelevel either the utility from consumption or the utility from environment is zero.As Lemma 1 shows, in a laissez-faire economy even a very small capital stockcreates a positive wage and rent income. In contrast, a regulated economyR

withregulation cost r and a capital stock smaller than the threshold value k

, wherew

(k

) = 0, is not viable. If the productivity of an economy is so low that for all kt

w

(k

t

) 6 0, or if the development level of the economy is such that kt

6 k

, actorshave no choice but to accept the laissez-faire regime. However, the laissez-faireregimes expansion is limited, the limiting factor being the environmental qualityrather than consumption. Taking into account the growth path of the pollutionstock, Equation (7), an unregulated economy with a capital stock k

t

reaches an

162 THOMAS WAGNER

ecological subsistence level as soon as the pollution stock at time t approaches theboundary Q for which A

t+1 A (X + e(kt) + Q) = 0. Thus, we can writefor the pollution boundary of the laissez-faire regime:

Q(k

t

) = fA (X + e(k

t

))g= (11)A generation which at the beginning of the first half of its life is faced with the

initial condition (Qt

; k

t

), for which Qt

>

Q(k

t

), cannot survive with a laissez-faire policy; the high pollution stock forces the actors to regulate emissions. ForQ

t

>

Q(k

t

) and kt

> k

, growth can only continue within the limits of a regulatedquantity of emission rights. Similar boundaries are relevant if we compare the tworegulated economies. Under a liberal environmental policy regime, the pollutionboundary Q

R0 , which does not depend on the capital stock at all, follows fromEquation (8) and A

t+1 A (X + QR0) = 0, so that

Q

R

o

= fAXg=: (12)A generation born at time t and faced with the initial condition (Q

t

; k

t

), forwhich Q

t

>

Q

R0 and kt > k, can only survive within the guidelines of amonopolistic environmental policy R

. Only if the government uses its monopo-listic power to ration the emission licenses at the Cournot price

, can the preser-vation investments financed by the license revenue absorb the new emissions andbeyond that a part of the inherited pollution stock. However, the monopolisticregulation of emission rights also meets an ecological boundary Q

R

, for whichA

t+1 A (X E(kt) + QR

= 0, so that

Q

R

(k

t

) = fA (X E

(k

t

))g=: (13)For ecological reasons, an economy can only grow beyond the pollution bound-

ary, Equation (13), if emissions as well as preservation investments are managed bya central planer. If we compare the three boundaries, Equations (11), (12) and (13),we find Q

R

(0) = Q(0) = QR0 and QR(kt) > QR0 > Q(kt) for all kt > 0. The

pollution boundary of the laissez-faire regime is strictly monotonically decreasingas k

t

increases, whereas the boundary of the monopolistic environmental policyregime increases strictly monotonically with k

t

(see Figure 1).

LEMMA 4. The boundaries(i) ( Q

R

;

Q; k

) divide the (Q; k)-plane in such a way that for initial condi-tions (Q

t

; k

t

) with Qt

0 there is a capital stock k

> 0 for whichw

(k

) = 0, then at initial condition (Qt

; k

t

) for which kt

> k

andQ

t

0 and At+1 = AQ(kt; Qt) > 0.

4. Indifference Curve

An actor facing an inherited situation (Qt

; k

t

) prefers pollution regulation if the gainin environmental quality more than compensates him for the loss in consumption.He prefers a laissez-faire regime if the loss in consumption is not compensated for.Or else, he is indifferent between regulation and laissez-faire if he expects the samelifecycle utility under both regimes. The set of initial conditions such that a typicalmember of the generation born at time t is indifferent between the allocations inR

and in L is called ID. (Qt

; k

t

) 2 ID, if and only if the utility difference is zero:V (k

t

; Q

t

)

(k

t

)a

(k

t

; Q

t

) (k

t

)a(k

t

; Q

t

) = 0:( Q(k

); k

) 2 ID, sinceat ( Q(k

); k

) the typical actor reaches the subsistence level in both R

and L in R

, because regulation costs exhaust the factor income so that consumptionis zero; in L, because the environmental quality becomes the limiting factor (seeFigure 1).

PROPOSITION 2. Consider a regulated economy R

with regulation costs r >0, a capital stock k

> 0 with w

(k

) = 0, and define the utility differenceV (k

t

; Q

t

) =

(k

t

)a

(k

t

; Q

t

) (k

t

)a(k

t

; Q

t

). Then

(i) @V (k;Q)=@Q > 0, if 2 6 1.(ii) For each Q0 with Q0 6 Q(k

) there exists a capital stock k0 > k

suchthat (Q0; k0) 2 ID, where

Q

0

= fA [X + e(k

0

)m(k

0

) + (m(k

0

) 1)E

(k

0

)]g=

Q

ID

(k

0

) (ID condition), (14)

where m(k) is given by m(k) = (1 h(k)1=2)1 and h(k) is the ratioof the utility from consumption in R

and L; h(k) =

(k)=(k), so thatm(k

) = 1 and m(k) > 1 if k > k

.

(iii) QID

(k

) =

Q(k

) and QID

(k) k

.


In view of the regulation cost, an actor born at time t with the inherited con-dition (Q

t

; k

t

) 2 ID is indifferent between a laissez-faire policy and emissionregulation. According to Proposition 2 (i), a marginal increase of the inheritedpollution stock makes him prefer R

whereas a marginal decrease makes himopt for L. As Proposition 2 (iii) shows, the ID-curve lies completely below thestrictly monotonically decreasing pollution boundary, Equation (11), of the laissez-faire regime. From the assumption that the ID function is injective it followsthat Q

ID

(k) is strictly monotonically decreasing and has one and only one zerok

ID

; Q

ID

(k

ID

) = 0 (Figure 1). If the economy on its growth path reaches acapital stock k

t

> k

ID

, generation t establishes a market for pollution licenses,irrespective of the accumulated pollution stock, in order to neutralize at least theirown emissions through preservation investments. Table I shows the comparativestatics for the threshold value k

ID

under the following assumptions, which arerelevant throughout the rest of this paper: d

(k

ID

)=dr < 0; d

(k

ID

)=d > 0and d

(k

ID

)=d > h(kID

)d(kID

)=d.

5. Steady States and Policy Cycles

While the allocative mechanisms L and R

are irreversible under regime W eachgeneration can choose between regulation and laissez-faire. In this section, wecharacterize the steady states of the allocative mechanisms L;R

, and W .

5.1. STEADY STATE OF THE LAISSEZ-FAIRE ECONOMY

Difference Equation (4) characterizes the equilibrium dynamics of the capital stockin the unregulated economyL. If the capital supply has a fixed point k = S(k) > 0,then (Q(k); k) is a (nontrivial) steady state of L with the stationary pollution stockQ(k). From Equation (7) we can infer thatQ(k) = (X+e(k))=(1). Existence,stability and uniqueness of the stationary equilibrium in L are the subject of thefollowing proposition (Azariadis 1993: 198).

PROPOSITION 3. If S(0) = 0; limk!1

S(k)=k = 0; S0(k) > 0, and S0(0) > 1,then the capital supply in L has at least one fixed point k

L

= S(k

L

) > 0, and(Q(k

L

); k

L

) is a stable steady state of L, provided that the stationary pollutionstock does not violate the pollution boundary of the laissez-faire regime:Q(k

L

) 0 the inequality S0(k) < 1 holds,then the steady state (Q(k

L

); k

L

) is unique.

5.2. STEADY STATE OF THE REGULATED ECONOMY

Let S

(k; r) be the capital supply of a regulated economy R

at regulation costr > 0. First we consider the case where r = 0. If the capital supply S(k) in L

166 THOMAS WAGNER

has the features mentioned in Proposition 3, then S

(k; 0), the capital supply inthe regulated economy with zero regulation cost, has the same properties as S(k),and the growth path of R

has one and only one steady state (RR0; k

R0). For thissteady state, the potentially negative stationary pollution stock from Equation (9)is Q

R0 = [X E(k

R0)]=(1 ) Q(kR0). A comparison of the steady statesin L andR

yields (QR0; k

R0) < (Q

L

; k

L

). In spite of the zero regulation cost, thehigher factor cost in R

causes not only a lower pollution stock but also a lowerstationary capital stock.

Given that S(k) has the properties mentioned in Proposition 3 for r 2 (0; r),the capital supply S

(k; r) has exactly two fixed points, the larger of which equalsthe capital stock k

Rr

of the stable steady state in the economyR

with nonnegativeregulation cost. The higher is the regulation cost, the lower is the stationary valueof the capital stock, so that k

Rr

6 k

R0. Boundary r, where r = maxk>0 '(k) (S

(k; 0)k)=, is the upper limit of the regulation cost interval for which steadystates exist. From '(0) = 0 it follows that r > 0. Furthermore, for all r 6 r thereexists a capital stock k

such that w

(k

) = 0.

5.3. STEADY STATE OF W

Under regime W each generation has the option to issue emission licenses and toinvest the revenue into environmental preservation. Since the allocative mechanismW differs from L and R

only with respect to the actors freedom of choice, thesteady state of W , if it exists at all, is either identical with the steady state of Lor with the steady state of R

. To find out whether W has a stable steady state,we take another look at the phase lines of the pollution stock. From Equation (7)it follows that in LQ

t

Q

t+1 Qt = 0 holds, if Q = (X + e(k))=(1 ) Q(k); for the regulated economy we can infer Q

t

= 0 from Equation (9), ifQ = [X E

(k)]=(1 ) Q

(k). A comparison of the phase lines in R

andL shows: Q(0) = Q

(0) = X=(1 ), and Q(k) > Q

(k), for k > 0. While thephase line in L increases strictly monotonically with the capital stock, Q

(k) iseither constant ( = 0) or decreases strictly monotonically as k increases ( > 0).Figure 2 shows the phase lines and their intersections (k

L

; k

R

) with the ID-curve,Equation (14). Whether or not regimeW has a steady state depends on the positionof the threshold values (k

L

; k

R

).

The existence of the capital stock kL

is guaranteed if 6 1 (X + e(k

))=A

holds for the residence time of the pollutant. With a capital stock kL

; Q(k

L

) =

(X + e(k

L

))=(1 ) is the point in L at which the pollution stock becomesstationary. For k

L

, the pollution stock at which the actors are indifferent betweenR

andL isQID

(k

L

). SinceQ(kL

) = Q

ID

(k

L

), the steady state ofL; (Q(kL

); k

L

),

is a stationary equilibrium of W , if and only if kL

6 k

L

. Otherwise (Q(kL

); k

L

) isa stationary state in W . However, it is not an equilibrium, because with Q(k

L

) >

Q

ID

(k

L

) the individuals prefer emission regulation.


Figure 2. Threshold values of the steady states in W .

If an intersection exists between the phase line in R

and the ID-curve, itsposition is either in the first quadrant of the (Q; k)-plane at a position pollutionstock as in Figure 2 or in the forth quadrant at a negative pollution stock. Thesituation in Figure 2 is relevant if the residence time of the substance fulfills thecondition mentioned above, and if the degree of monopoly on the license marketis small. At k

R

the regulated pollution stock is stationary at the value Q

(k

R

).

For kR

; Q

ID

(k

R

) is the pollution stock at which the actors are indifferent betweenR

and L. Since Q

(k

R

) = Q

ID

(k

R

), the steady state of R

at given regulationcost r is a stationary equilibrium of W , if and only if k

Rr

> k

R

. Otherwise,(Q

(k

Rr

); k

Rr

) is a stationary state in W at which the actors prefer a laissez-faireregime and therefore not an equilibrium. Considering k

L

< k

R

and kL

> k

Rr

, we

arrive at the following proposition:

PROPOSITION 4.(i) If the residence time of the pollution stock fulfills the condition mentioned

above, and if the degree of monopoly on the license market is smallenough, then there exists a stable steady state in W at regulation cost rif and only if k

L

2= (k

L

; k

R

+ k

L

k

Rr

). For kL

6 k

L

the steady stateof W is (Q(k

L

); k

L

). If kL

> k

R

+ k

L

k

Rr

, then (Q

(k

Rr

); k

Rr

) is thesteady state of regimeW . The allocative mechanismW has no stationaryequilibrium if k

L

k

L

< 0 < kR

k

Rr

.

(ii) Table II shows the results of the comparative-static analysis of the thresh-old values (k

L

; k

R

) and of the stationary capital stocks (kL

; k

Rr

).

168 THOMAS WAGNER

Table II. Effects of regulation costs, exogenous emissions, residence time, productivityof the pollution sink, and level of development

r X

k

L

+

k

R

+= =+ =+ (=+) (=+)

k

L

0 0 0 0 +

k

Rr

0 0 + +

The effects of the parameters on the threshold value kR

are generally ambiguous(see Table II), because an increase in k

R

decreases not only QID

but also thestationary pollution stock Q

( > 0). Under the assumption that the first effect is(absolutely) dominating, the comparative-static effects of the parameters (r;X; )are unambiguous and equal the first sign in Table II. However, changes in theproductivity of the pollution sink and changes in the level of development stillhave ambiguous signs, since such changes give rise to parallel adjustments of Q

ID

andQ

. If the reaction ofQID

is stronger, we get the first sign in Table II. As TableII shows, a decrease of the regulation cost lowers the probability that (Q(k

L

); k

L

)

or (Q

(k

Rr

); k

Rr

) are equilibria of W . In contrast, an increase of the exogenousemissions, of the residence time, the productivity of the pollution sink, or thedevelopment level reduces the probability that (Q(k

L

); k

L

) is an equilibrium ofW , while it increases the probability of the regulation equilibrium (Q

(k

Rr

); k

Rr

).

A rise of or speeds up the adjustment process that makes the stationary state(Q

(k

Rr

); k

Rr

) an equilibrium of W , and it does so in two ways. Both thresholdvalues (k

L

; k

R

) decrease simultaneously, while at the same time the stationarycapital stock of R

grows.

5.4. POLICY CYCLES

1. If W has no stationary equilibrium, then, after a certain period of time, thegrowth path inW reaches the interval (k

Rr

; k

L

) bounded by the steady state capitalstocks of R

and L and is possibly subject to endogenous cycles. In democracies,endogenous policy cycles are a symptom for changing majorities. Voters react toconditions which are the consequences of their own or their forerunners behavior,thus generating a cyclical pattern of political decisions. The model shows that suchcyclical behavior is to be expected within the context of environmental policy. Forexample, in a two-period cycle L 7! R

7! L with the fixed point (kLR

; Q

LR

)

shown in Figure 1, every other period preferences alternate between laissez-faireand regulation. If the first generation prefers consumption, the next generation iswilling to sacrifice consumption in order to invest into the natural capital with


the result that their heirs meet with a relatively clean environment and opt forlaissez-faire.

PROPOSITION 5. For the transformation L 7! R

7! L a stable fixed point(k

LR

; Q

LR

) of period two exists if the natural quantity of the public good Avaries within certain limits (as defined in the proof, see appendix).

2. Similar cycles can occur within regulated economies. Environmental policygenerates trajectories with the liberal policy R0 and the monopolistic policy Ras periodically alternating regimes. In analogy to Proposition 2, we can showthat the state plane of an economy contains regions in which the actors prefereither the liberal or the monopolistic environmental policy; the correspondingID-curve, which is calculated just like Equation (14), contains all those initialstates, at which the actors are indifferent between the liberal intensity of regulationR0 and the monopolistic intensity of regulation R. Now suppose that at theinitial state (k

RR

; Q

RR

), the actors, for example, prefer the liberal governmentintervention where emission rights are sold at marginal preservation cost. Moreover,let k

RR

2 (k

R

; k

R0), where k

R

and kR0

are the stationary capital stocks undera monopolistic and a liberal environmental policy regime respectively, for whichk

R

< k

R0. A liberal regulation policy implies that the pollution sink will absorb

exactly the emissions of the young generation in period t, so that by using Equation(8) we have Q

t+1 = X+ Q

RR

Q0(Q

RR

). If this policy, which is rational fromthe perspective of generation t, generates a pollution stock leading the growthpath into the region where monopolistic regulation is preferred, then the regulatingagent of the next generation will supply emission rights at the Cournot price.Production in t + 1 with the capital stock S0(k

RR

) generates an income, whichleads to a capital supply S

(S0(k

RR

)) of generation t+ 1. If kRR

= S

(S0(k

RR

))

and if the net preservation investmentE

(S0(k

RR

)) of generation t+1 reduces thepollution stock to Q

t+2 = X E(S0(k

RR

)) + Q0(Q

RR

), so that QRR

= Q

t+2,then (k

RR

; Q

RR

) is a periodic point. Generation t+ 2 will again prefer the liberalregulation intensity and will supply emission rights at the marginal cost price. Thusthe policy cycle R0 7! R 7! R0 starts again.

PROPOSITION 6. There are stable periodic points (kRR

; Q

RR

) of two-period cyclesR0 7! R 7! R0, the trajectories of which alternate between the liberal and themonopolistic government intervention.

6. Summary and Conclusions

The OLG model describes the dynamics of an accumulative pollutant with aconstant rate of decay. The growth patterns of the substance and the pollutionboundaries limiting the sustainable growth paths depend on the residence time,the reaction lag of the emissions, and endogenous variables with regime-specific

170 THOMAS WAGNER

values. In a certain region of the state plane because of the regulation cost only thelaissez-faire regime (L) can survive; in another region for ecological reasons onlya regulated economy is viable. Within the emission boundaries of the regulatedeconomies there are states where growth can only continue with the instruments ofa monopolistic environmental policy (R

). Only in R

, where emission rights aresupplied at the Cournot price, are there positive net investments into the naturalcapital. In contrast, under liberal regulation (R0), government acts as a price takenand adapts its supply of emission licenses to the market price, which reaches itsequilibrium at the marginal preservation cost, where the license revenue is justsufficient to absorb the emissions of the generation in power.

The ID-curve containing all states where actors are indifferent between L andR

divides the state plane in regions where the actors either prefer regulation orlaissez-faire. If the steady state of L lies within the regulation region, or if thesteady state of R

lies within the laissez-faire region, then, ceteris paribus, aneconomy where each generation can choose between laissez-faire and regulationhas no stationary equilibrium. In such a system, endogenous policy cycles canoccur alternating between L and R

with a frequency determined by the two-period lifespan of the individuals. Moreover, under the conditions of regulation,oscillations between the liberal and the monopolistic environmental policy arepossible. In the first period, for example, the actors may prefer the instruments ofR0, since they impose a smaller sacrifice in terms of foregone consumption. Butwith the right to discharge valued at marginal preservation cost, emissions fromthe exogenous source are not completely controlled and generate an increase of thepollution stock which causes the next generation to ration emissions imposing theCournot price. As a result, the initial state of the policy cycle is reestablished, andthe third generation will again opt for a liberal environmental policy to keep thepollution stock in check.

According to the above OLG-model, every economy goes through an era ofseveral generations where the actors live on the natural capital stock using naturalcapital for consumption and production and accumulating only produced capital.Only when wealth and pollution have reached certain threshold values, will a gen-eration find it worthwhile to regulate its emissions and to restrict consumption inorder to invest into the stock of natural capital. Those critical values depend onthe state of technology, on preferences, policy options, and particularly on the costof regulation institutions. Myopic policy makers always chose the instrument ofregulation that maximizes the welfare of the current generation without consideringthe consequences for generations to come. The proposed model can be extendedto include different options of the myopic policy approach. In addition to marketoriented and indirect methods of pollution regulation, command and control mea-sures as well as liability rules can be integrated into the model in order to analyzetheir regulation efficiency and their temporary and stationary equilibria. In such


a model environmental policy and planning are described as two-stage decisionswhere the preferred type of regulation instrument is chosen first and the intensityof its application afterwards.

According to the model, wealth can reach a threshold value beyond whicha myopic policy will regulate the firms emission activities independently of theaccumulated stock of natural capital and of environmental pollution. Policy makerswill initiate pollution-reducing measures such as the purification of emissions,reforestation, declaration of nature reserves, renaturalization, etc., which are justsufficient to absorb the new emissions of the generation in power. Because of theexogenous ability of the environment to regenerate itself, this reinvestment policyresuscitates and increases the stock of natural capital, even though in each phaseenvironmental policy is exclusively oriented towards the egotistical interests ofthe current generation. In many countries, measures and successes in the areasof reafforestation, prevention of air pollution, and improvement of water qualityfollow the pattern of reinvestment policy predicted by the OLG-model.

Even when the possibilities of Pareto-improving interventions are exhaustedfrom the perspective of the myopic policy approach, a policy maker with a planninghorizon of several generations still has the possibility to increase welfare throughintergenerational reallocation. A long-run environmental policy approach takinginto account the interests of a sequence of future generations is faced with twomain problems, which are closely interconnected. The first problem is the designof a planning and decision mechanism that is time-consistent and protected by theconstitution and that ensures the necessary incentives in order to free environmentalpolicy from the narrow time horizon of the current generation and to conceptualizeit in such a way that the interests of the different generations are balanced out. Thesecond question is how to compensate the current generation for the consumptionwhich it must sacrifice without any prior effort of the generation before inorder to initiate the balancing out of interests within the long chain of overlappinggenerations. It is very likely that a certain minimum level of intergenerationalaltruism is necessary in order to solve those problems.

Note1. In the following, we mostly use the model of a monopolistic environmental policy regime to

represent the regulated economy. In the context of this model, a liberal environmental policy canbe seen as the special case with = 0. Differences between the types of regulation are mentionedwherever necessary. Endogenous variables are marked with for either the regulated economyor the monopolistic policy regime and with 0 for the liberal environmental policy.

172 THOMAS WAGNER

Appendix

Lemma 1.(i) z(0) = f1(0; e) =1; z(1) = f1(1; e) = 0, and z0

t

= (f11f22f212)=f22 0, if kt

> 0.(ii) The proof is analogous to (i).

Lemma 2. From t

= 0 follows e(kt

) = e

(k

t

); z(k

t

) = z

(k

t

), and w(kt

) = w

(k

t

) + r.

Thus one obtains:

(i) Implicit differentiation of the necessary condition f2(kt; et) q = 0 givesde

t

=dq

< 0. Together with Equation (2) we can infer that dzt

=dq

=

det

=dkt

< 0, so that the statement with q

> q follows in both cases.(ii) Differentiation of Equation (3) yields dw

t

=dq

= e

t

(1 "ek

) < 0, so thatthe statement follows from q

> q or from r > 0.(iii) (a) First we show that c

2(kt) 6 c2(kt), if t = 0. The assumption means thatk

t+1 = w(kt) = [w(kt) r] = kt+1 r, so that dkt+1=dr = anddw

t

=dr = 1. Thus we obtain dc2=dr = [1 + zt+1(1 + "zk)] < 0 and

therefore the statement. (b) From (a) and from dc2=dq = kt+1@zt+1=@q +

[1 + zt+1(1 + "zk)]dwt=dq < 0 we can infer the statement.

(iv) Q(kt

; Q

t

) X + e(k

t

) + Q

t

> X E

(k

t

) + Q

t

Q

(k

t

; Q

t

).

Lemma 3. The proofs of the statements (i), (ii) and (iii) are analogous to Lemma 2.(iv) Q0(Qt) X + Qt > X E(kt) + Qt Q(kt; Qt).

Lemma 4.

(ii) (a) k

: Implicit differentiation of w

(k

) = 0 yields

dk

dr = 1

k

@z

=@k

> 0; dkd = e

(k

)(1 "ek

)dq

=dk

@z

=@k

< 0;

dk

d =r + q

e

(k

)(1 "ek

)

k

@z

=@k

< 0:

Further we can say that dk

=dX = 0 and dk

=d = 0.(b) Q: The statements can be obtained through differentiation of Equation (11).(c) Q

R

: The statements can be obtained through differentiation of Equation (13). For theproductivity of the pollution sink we can say that

dE

(k)

d =

1 @e

@q

dq

d > 0;dE

(k)

d =

1 @e

@

> 0:


(d) kID

: Taking into account QID

(k

ID

) = 0, the statements can be obtained throughdifferentiation of Equation (14), where

dm(kID

)

dr =m

2h

1=2

2

d

dr < 0;dm(k

ID

)

d =m

2h

1=2

2

d

d > 0;

dm(kID

)

d =m

2h

1=2

2

d

d hdd

> 0:

Proposition 2.

(i) Using

@V=@Q =

@a

=@Q @a=@Q

= 2

[AQ(k;Q)]

21

[AQ

(k;Q)]

21

> 0;

the statement can be inferred from 2 6 1; (k) > (k) and Q(k;Q) > Q(k;Q).

(ii) Using

(k) = #[1 + z

(S

(k))]

w

(k)

(k) = #[1 + z(S(k))]w(k);

we can derive fromw

(k

) = 0 that

(k

) = 0 and therefore h(k

) = 0 andm(k

) = 1.If k > k

; (k) >

(k) > 0 follows from Lemma 1 and 2, so that 0 < h(k) < 1 andm(k) > 1.

For Q0 < QL

(k

) there exists a capital stock k00 > k

for which Lemma 2 leads to

a(k

00

; Q

0

) = [AQ(k

00

; Q

0

)]

2= 0

a

(k

00

; Q

0

) = [AQ

(k

00

; Q

0

)]

2> 0;

so that V (k00; Q0) > 0. In addition, since V (k

; Q

0

) = (k

)a(k

; Q

0

) < 0, wecan infer from the intermediate value theorem the existence of a k0 2 (k

; k

00

), for whichV (k

0

; Q

0

) = 0. After several rearrangements, we obtain from V (k0; Q0) = 0 the IDcondition, Equation (14).Proposition 4.

(ii) (a) kL

: Implicit differentiation of QID

(k

L

)Q(k

L

) = 0 yields the statements.(b) k

R

: The derivative of H(kR

; r;X; ; ; ) Q

ID

(k

R

)Q

(k

R

) = 0 with respect tothe threshold value is

@H

@k

R

=

@Q

ID

@k

R

+

1(1 )

@E

(k

R

)

@k

R

R 0:

Similarly we can say that

@H

@(@)

=

@Q

ID

@(@)

+

1(1 )

@E

(k

R

)

@(@)

R 0:

174 THOMAS WAGNER

(c) kL

: The statement follows from kL

w(k

L

) = 0 through implicit differentiation.(d) k

R

: The statement follows from kR

w

(k

R

) = 0 through implicit differentiation.

Proposition 5. (a) First we will show that S

(S()) has a fixed point kLR

with kR

S

(k), if k > 0, the statement follows from S

(S(k

L

)) = S

(k

L

) 0: kR

< k

L

leads under the assumptionthat L has one and only one (non-trivial) stable steady state to S(k

R

) > k

R

, so thatS

(S(k

R

)) > S

(k

R

) = k

R

follows with respect to dS

(k)=dk > 0. As H(kR

) > 0 andH(k

L

) < 0, from kR

< k

L

follows the existence of the fixed point kLR

(intermediatevalue theorem), for which H(k

LR

) = 0; second, it is obvious that the graph of S

(s()) inthe (k

t+1; kt)-plane cuts the 45-line from above, which establishes the stability.(b) Next we have to show the existence of a stock Q

LR

, for which QLR

=

Q

(S(k

LR

); Q(k

LR

; Q

LR

)). Substituting Equations (7) and (9) into this relation we obtain

Q

LR

= f(1 + )X + e(kLR

)E

(S(k

LR

))g =(1 2):

Finally we have to prove that QLR

is feasible. This is true, if QLR

< Q

ID

(k

LR

;A) andQ(k

LR

; Q

LR

) > Q

ID

(S(k

LR

);A). Since S(kR

) > k

R

, we can say that QID

(k

LR

;A) >Q

ID

(S(k

LR

);A); furthermore from the above definition ofQLR

and Equation (7) it is truethat Q

LR

< Q(k

LR

; Q

LR

). Since @QID

(k;A)=@A = 1= > 1, there exist an A0 and A00withA00 > A0, for whichQ

LR

= Q

ID

(k

LR

;A0) andQ(kLR

; Q

LR

) = Q

ID

(S(k

LR

);A00).Using the monotonicity of Equation (14) we can infer for all A 2 (A0; A00], that Q

LR

Q

ID

(S(k

LR

);A).

Proposition 6. The proof is analogous to Proposition 5.

ReferencesAzariadis, C. (1993), Intertemporal Macroeconomics, Oxford.Blanchard, O.J. and S. Fischer (1989), Lectures on Macroeconomics, Cambridge.Brock, W. A. (1973), A Polluted Golden Age, in V. L. Smith, ed., Economics of Natural and

Environmental Resources.Burton, P. S. (1993), Intertemporal Preferences and Intergenerational Equity Considerations in

Optimal Resources Harvesting, Journal of Environmental Economics Management 24, 119132.

Diamond, P. A. (1965), National Debt in a Neoclassical Growth Model, American Economic Review55, 11261150.

Howarth, R. B. and R. B. Norgaard (1990), Intergenerational Resource Rights, Efficiency, and SocialOptimality, Land Economics 66, 111.

Howarth, R. B. and R. B. Norgaard (1992), Environmental Valuation under Sustainable Develop-ment, American Economic Review 82, 473477.

Howarth, R. B. and R. B. Norgaard (1995), Intergenerational Choices under Global EnvironmentalChange, in D. W. Bromley, ed., The Handbook of Environmental Economics, Cambridge, MA.

Hultkrantz, L. (1991), Forestry and the Bequest Motive, Journal of Environmental EconomicsManagement 22, 164177.

John A. and R. Pecchenino (1994), An Overlapping Generations Model of Growth and the Environ-ment, Economic Journal 104, 13931410.

John A., R. Pecchenino, D. Schimmelpfennig and S. Schreft (1995), Short-lived Agents and theLong-lived Environment, Journal of Public Economics 58, 127141.


Kemp, M. C. and N. V. Long (1980), The Underexploitation of Natural Resources: A Model withOverlapping Generations, in M. C. Kemp and N. V. Long, eds., Exhaustible Resources, Optimalityand Trade, Amsterdam.

Lines, M. (1995), Dynamics and Uncertainty, in H. Folmer, H. L. Gabel and H. Opschoor, eds.,Principles of Environmental and Resource Economics, Aldershot.

Lofgren, K. G. (1991), Another Reconciliation Between Economists and Forestry Experts: OLG-arguments, Environmental and Resource Economics 77, 95111.

Nordhaus, W. D. (1991a), To Slow or Not to Slow: The Economics of the Greenhouse Effect,Economic Journal 101, 920937.

Nordhause, W. D. (1991b), Economic Approaches to Greenhouse Warming, in R. Dornbusch andJ. M. Poterba, eds., Global Warming, Economic Policy Responses, Cambridge.

OECD (1994), The Economics of Climate Change.Samuelson, P. A. (1958), An Exact Consumption-Loan Model of Interest with or without the Social

Contrivance of Money, Journal of Political Economy 66, 467482.Tietenberg, T. (1985), Emissions Trading: An Exercise in Reforming Pollution Policy, Washington.UNEP (1997), Global Environmental Outlook-1, Oxford.Uzawa, H. (1991), Global Warming Initiatives: The Pacific Rim, in R. Dornbusch and J. M. Poterba,

eds., Global Warming, Economic Policy Responses, Cambridge.Wagner, T. (1997), Dauerhafte offentliche (Un-)Guter: Ist eine myopische Regulierung von CO2-

Emissionen besser als keine?, Jahrbucher fur Nationalokonomie und Statistik 216/1, 92112.

limits and cycles of environmental policy

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