changes in social welfare and sustainability: theoretical issues and empirical evidence
TRANSCRIPT
E C O L O G I C A L E C O N O M I C S 6 7 ( 2 0 0 8 ) 4 7 3 – 4 8 4
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ANALYSIS
Changes in social welfare and sustainability: Theoreticalissues and empirical evidence
Dimitra Vouvakia, Anastasios Xepapadeasb,⁎aUniversity of Crete, Department of Economics, University Campus, 74 100 Rethymno, GreecebAthens University of Economics and Business, Department of International and European Economic Studies, Greece
A R T I C L E I N F O
⁎ Corresponding author.E-mail addresses: [email protected]
0921-8009/$ – see front matter © 2008 Elsevidoi:10.1016/j.ecolecon.2007.12.029
A B S T R A C T
Article history:Received 18 October 2006Received in revised form30 December 2007Accepted 30 December 2007Available online 31 January 2008
We analyze the time derivative of a Ramsey–Koopmans social welfare function (R–K SWF),as an indicator of genuine investment and current change in social welfare (CSW)conditions, when feedback or arbitrary rules are used for selecting policy variables in non-optimizing economies. When policy variables are selected arbitrarily, their accountingprices should determine current CSW in addition to the accounting prices of the economy'sassets and genuine investment should be adjusted accordingly. We use our theoreticalframework to characterize CSW conditions for non-optimizing economies, based on directestimation of accounting prices.We use our theoretical model to provide empirical evidenceregarding the CSW conditions for the Greek economy.
© 2008 Elsevier B.V. All rights reserved.
Keywords:Current change in social welfareGenuine investmentSustainabilityAccounting pricesNon-optimizing economyFeedback ruleArbitrary rule
JEL classification:Q01
1. Introduction
Concerns about environmental deterioration and naturalresource depletion have advanced sustainable developmentas a key concept in policy design both at the national and theinternational levels. Sustainable development has been thecentral concept in the United Nations World ConservationStrategy and in the report of the World Commission onEnvironment and Development (United Nations, 1987), alsoknown as the Brundtland Report. Sustainability has alsobecome a central concept in the policy of the European Union.
(D. Vouvaki), xepapad@au
er B.V. All rights reserved
The most commonly used definition of sustainable devel-opment is that of the Brundtland Report which definessustainable development as “development which meets theneeds of the present without compromising the ability offuture generations to meet their own needs”. This definitionstresses the aspects of intertemporal distribution and inter-generational equity, but since it embeds many complexeconomic ideas it suffers from a lack of tractability, especiallywhen it comes to providing answers to applied questionsregarding the sustainability of economies, or the design andevaluation of sustainable development policies.
eb.gr (A. Xepapadeas).
.
2 Evidence provided by the World Bank 2006 suggest thatinvestments in produced capital, human capital, and governance,combined with saving efforts aimed at offsetting the depletion ofnatural resources, can lead to future welfare increases indeveloping countries.3 Asheim (1994), Hamilton and Clemens (1999), Pezzey (2004b),
show that negative genuine savings at t, that is declining socialwelfare, implies unsustainability in individual utility terms inoptimizing economy. This result however has not been shown to
474 E C O L O G I C A L E C O N O M I C S 6 7 ( 2 0 0 8 ) 4 7 3 – 4 8 4
In an attempt to make the definition of sustainabledevelopment operational and useful for the development ofsustainability criteria or sustainability indicators and for thedesign of sustainable policies, some auxiliary definitions havebeen developed. These definitions identify conditions underwhich an economy can be regarded as following a sustainabledevelopment path. Themost prevailing of these definitions, asreported by Pezzey (2004a), associate sustainability with:
1. Achieving constant utility (Solow, 1974; Hartwick, 1977).2. Having the representative agent's utility (well being) U(t),
not exceeding themaximum level of utility Um(t) which canbe sustained forever from t onwards given the capitalstocks existing at t (Pezzey, 2004b). This definition isimplied by, but does not imply, the well known conditionthat the agent's utility is forever non-declining from tonwards (Pearce et al., 1990; Pezzey, 1992, 1997).
3. Non-declining Social Welfare (NDSW), that is, avoiding anydecline in intergenerational social welfare defined in termsof a Ramsey–Koopmans social welfare functional (R–KSWF), either from time t forever onwards, or much lessdemandingly, just at time t (Riley, 1980; Dasgupta andMäler, 2001; Pemberton and Ulph, 2001; Arrow et al., 2003).
Arrow et al. (2003, p. 653) define a sustainable developmentpath at t as one with NDSW at t. Their Theorem 2 then showsthat in an autonomous economy, this criterion implies themaintenance of the economy's “productive base” at t, in thesense that NDSW at t is equivalent to non-negative genuineinvestment at t, defined as the sum of investments, valued ataccounting prices, in all productive assets such as manufac-tured capital, humancapital, natural capital andknowledge.1 Soin this paperwe callNDSWornon-negativegenuine investmentat time t “productive-base sustainability”. This can also beregarded as corresponding to the weak sustainability concept(Pearce and Atkinson, 1993; Hediger, 1999, 2000).
However, NDSW or positive genuine investment at time tdoes not imply that the utility of the representative agent attime t is at a level that can be sustained forever after t, whichhasbeen considered as a natural meaning of the economy beensustainable at time t. As shown by Asheim (1994) and Pezzey(2004b), there are optimizing economies where NDSW, and yetutility being unsustainable, could occur together during a finitetimeperiod. ThusNDSWor positive genuine investment at timet, does not imply ‘utility–sustainability’ that is, utility at a levelthat can be sustained forever after t.
In the present paper we choose to analyze both theoreticallyand empirically the concept of the time derivative of a R–K SWFata given time t, whichprovides, according toArrowet al. (2003),
1 The concept of genuine saving (also known as adjusted netsaving) was introduced by Pearce and Atkinson (1993) andHamilton (1994). Genuine saving provides a much broaderindicator of sustainability by valuing changes in natural re-sources, environmental quality, and human capital, in addition tothe traditional measure of changes in produced assets providedby net saving (World Bank 2006, Chapter 3). We use the termgenuine investment interchangeably with genuine saving in thispaper, since the former seems to be closer to the approach ofvaluing CSW in terms of changes in the values of the assetscomprising the productive base of the economy.
ameasure of the rate of change of the economy's current socialwelfare, or a measure of genuine investment at this time. Ourinterest in this time derivative stems from two facts: (i) If thetime derivative is positive, this implies that currently CSW ispositive and that genuine investment is positive,2 withoutimplying, as stated above, sustainability in individual utilityterms, (ii) If the time derivative is negative, then genuineinvestment is negative. As suggested by the World Bank (2006,Ch. 3), negative genuine saving rates imply that totalwealth is indecline and policies leading to persistently negative genuinesavings are unsustainable.3 Thus, using the World Bank'sterminology, we may interpret a negative current CSW, or anegative time derivative of the R–K SWF at a given time t, as anindication of currently unsustainable policies.
The approach ofmeasuring the current CSW could be usefulin providing an indication of sustainability in terms of genuineinvestmentandmaintenanceof theeconomy'sproductivebase,and not in utility terms, in a more general non-optimizingcontext. There isa cleardistinctionbetweenanoptimizingandanon-optimizing economy, as illustrated by Arrow et al. (2003).4
In a non-optimizing economy, the paths of the state variablescan also be used to define a value function for the economythrough theR–KSWFat agivenpoint in time.Thevalue functionat any given point t in time is taken to represent social welfarefrom time t onwards, and it is a function of the values of theeconomy's assets at time t.5 This makes it possible tocharacterize current CSW conditions in a general context andto provide a basis for empirical estimation of these changes.
We also believe that since, especially for developingcountries, there is no reason to assume that observed dataare generated by optimizing processes, the non-optimizingframework, properly defined, will be useful both for purposesof theoretical foundations of the current CSW conditionsunder alternative hypotheses about the structure and theobjectives of the economy, and for empirical estimations.
By using the non-optimizing theoretical framework, wecharacterize CSW conditions at time t — when controls arechosen according to some feedback rule.6 We also show thatwhen controls (or policy instruments) are chosen in an arbitrary
hold in a more general non-optimizing context.4 A non-optimizing economy is an economy in which the
government, whether by design or incompetence, does notchoose policies that maximize intergenerational welfare. Theterm sustainability acquires particular significance when it is putto work in imperfect economies, that is economies suffering fromweak or even bad governance.5 This is an interpretation of the value function similar to the
one given in Dynamic Programming. The main difference is thatno optimization needs to be assumed here.6 A feedback rule in this context is, for example, a behavioral
rule according to which instruments are determined in somerelation to the values of the state variables.
475E C O L O G I C A L E C O N O M I C S 6 7 ( 2 0 0 8 ) 4 7 3 – 4 8 4
way, which is independent of the stock of assets,7 the currentCSW depends not only on the growth of the assets and theircorresponding accounting prices, but also on the arbitrary pathsof the controls. In this case the value function for the economydepends bothon current stocks and current flows. These resultssuggest that the rate of change of the R–K SWF is not just thesum of genuine investments in specific assets, but that it mayalso include the rate of change of policy instruments valued attheir accountingprices. Thus, incertain casesofnon-optimizingeconomies with arbitrarily choices of controls, genuine invest-ment in assets might not be entirely appropriate for character-izing CSW conditions. In these cases genuine investmentshould be adjusted for the growth of the arbitrary chosen policyvariables, such as for example emission limits. This theoreticalframework is applied to data from an actual economy with thepurpose of providing estimates of the current CSW conditions.Our application refers to the Greek economy.
9 In this case the feedback function is chosen so that the steadystate is stable in the Lyapunov sense.10 In this case the feedback function is chosen so that the systemstarting from the initial point x0, reaches the terminal state xT at
2. Changes in current social welfare conditionsand sustainability in non-optimizing economies
Following Arrow et al. (2003) we assume that social welfare atany given time t is defined by the felicity functional:
Vt ¼Z l
te�q s�tð ÞU x sð Þ;u sð Þð Þds; s z t ð1Þ
where x=(x1, …, xn) denotes a vector of state variables whichcan be interpreted as stocks of assets and u=(u1, …, um)denotes a vector of control variables which can be interpretedas policy instruments. The function U(x(τ),u(τ)) can be inter-preted as the welfare of the generation living at time τ, underappropriate assumptions about the growth of the population,as will become clear in the following sections.
The evolution of the economy is described by a system oftransitionequations linking the state and the control variables,
:xs ¼ f u sð Þ;x sð Þð Þ;x tð Þ ¼ xt; sz t ð2Þ
In an optimizing economy the control paths u(τ) are chosento maximize (1) subject to the constraints imposed by thetransition equations (2). In a non-optimizing economy thechoice of the controls could be determined by a feedback ruleu(τ)=g(x(τ)) which might reflect behavioral characteristics ofthe economy, such as learning rules or imitation rules, orsome other intentional but not optimal feedback policy rule.8
For example, in the Solow model of economic growth (Solow1956), consumption, which can be interpreted as a controlvariable, is a constant fraction of output. Output is determined,through theaggregate production function, by the capital stockwhich is the state variable. This constant fraction is a be-havioral parameter. Thus in Solow's model, consumptionis determinedby a feedback rule. In addition, feedback controlscan be chosen to stabilize the economic system around some
7 This implies a non-feedback (arbitrary) way of choosing thecontrols.8 These types of controls could also be called closed loop controls.
desirable steady state,9 or can be chosen to steer the system toa certain state vector in finite time.10
Alternatively the choice of controls can be determined in acompletelyarbitraryway,byexogenous factors, suchasdomesticpolitical conditions,historic trendsor international conditions. Inthis case the control paths will be u(τ)=uP(τ). An arbitrary controlpath could be, for example, a path for which consumptionincreases x% per year, or CO2 emissions are reduced z% per year,without any relation to the evolution of a state variable.11 To putit in anotherway that is closer to the recent discussion about theKyoto protocol: choosing CO2 emissions as a proportion of globalCO2 stock is a feedback (closed loop) rule, while keepingemissions at the 1990 levels is an arbitrary (open loop) rule. Itseems that in actual economies most policy rules are arbitraryrather than feedback or optimal rules.
Consider the system of transition equations (2) under thefeedback rule or the arbitrary rule, respectively::xs ¼ f g x sð Þð Þ;x sð Þ;bð Þ;x tð Þ ¼ xt ð3Þ:xs ¼ f Pu sð Þ;x sð Þ;bð Þ;x tð Þ ¼ xt ð4Þ
where b is a vector of exogenous parameters. Solutions to thesesystems, provided they exist, will determine the paths of thestate variables as functions of their initial values, the exogenousparametersandpossibly thepathsof thearbitrary (or open loop)controls. In general these solutions will be of the form:
xs ¼ / s� t;xt;bð Þ; ð5Þ
xs ¼ w s� t;xt;Pu sð Þ;bð Þ ð6Þ
Substituting the solutions (5) or (6) into (1),weobtain thevaluefunction of the system as a function of the initial state vector xt,and possibly the vector of arbitrary controls uP(τ) . If the arbitrary(or open loop) control path can be written as: uP 0(τ)=u
P(τ−t, uPt),12
then the value function for the economy can be written for thefeedback and the open loop rules respectively, as:
VFt xt;bð Þ ¼
Z l
te�q s�tð ÞU g / s� t;xt;bð Þð Þ;/ s� t;xt;bð Þð Þds ð7Þ
VAt xt;
Put;bð Þ ¼Z l
te�q s�tð ÞU w s� t;xt;
Pu0 sð Þ;bð Þ;Pu0 sð Þð Þds ð8Þ
Accounting prices for asset xi or control (instrument) uPj attime t, are defined respectively as:
ptxi ¼AVl
tAxit
; l ¼ F;A;ptuj ¼AVA
tAPujt
; ð9Þ
Let:V
ltu
dVlt
dt , l=F, A denote the rate of change of the R–K SWFor the CSW at time t. Then,
:V
lt z 0; l ¼ F;A ð10Þ
implies non-negative genuine investment and can be re-garded, in the context of Arrow et al. (2003) arguments as an
finite time T. It is assumed in this case that the rank conditionsfor controllability are satisfied.11 Such types of controls could also be called open loop controls.12 This implies that the control is chosen according to somearbitrary time dependent rule, for example z% change relative tothe previous year.
13 In our stylized economy we do not consider natural resourcesand their contribution to production. This is because we want tokeep the model relatively simple in order to obtain the repre-sentations of value functions and accounting prices which willhelp to provide some insights into the structure and thedeterminants of these concepts. The introduction of naturalresources in this context is undoubtedly an area for furtherresearch.
476 E C O L O G I C A L E C O N O M I C S 6 7 ( 2 0 0 8 ) 4 7 3 – 4 8 4
indicator of current productive-base sustainability.:V
ltb0; l ¼
F;A or a negative CSW implies negative genuine investment.
Proposition 1. Consider a non-optimizing economy with xi, i=1, …n assets and uj, j=1, …, m policy instruments. (i) If policyinstruments are chosen following feedback rules associated withthe assets of the economy, then
:V
Ft depends on the assets' growth
and their corresponding accounting prices. (ii) If policy instrumentsare chosen arbitrarily then,
:V
At depends both on the assets' and
the policy instruments' growth and their corresponding accountingprices.
Proof. (i) Differentiating (7) totally with respect to time weobtain:
SFtu:V
Ft ¼
Xni¼1
AVFt
Axitdxitdt
þ AVFt
Atð11Þ
(ii) Differentiating (8) totally with respect to time we obtain:
SAt u:V
At ¼
Xni¼1
AVAt
Axitdxitdt
þXmj¼1
AVAt
APujt
dPujt
dtþ AVA
tAt
ð12Þ
It should be noticed that part (ii) of the above proposi-tion shows that in arbitrary (open loop) non-optimizing econ-omies — that is, economies where instruments are chosenwithout any relationship to assets � :
VAt depends on the
growth of these instruments too. Thus the growth of theinstruments affects the change in social welfare in addition tothe growth of the assets. Since the term
Pni¼1
AVAt
Axitdxitdt represents
genuine investment, our result implies that in time autono-mous economies, where AVA
tAt ¼ 0, positive genuine investment
does not provide indications about the change in socialwelfare at time t. To fully assess this change, the impacts ofinstruments should also be taken into account. In this sense,part (ii) of Proposition 1 extends previous results about non-optimizing economies where the change in current socialwelfare depended on genuine investment alone. This resultcan be associated, for example, with the introduction of en-vironmental policy, which in actual economies can be re-garded most of the times as arbitrary. For example, let uPj
denote an arbitrary upper limit on emissions, then AVAt
APujt
can beinterpreted as the accounting price for this limit and the termAVA
tAPujt
dPujtdt can be interpreted as the impact of a changing emission
limit on the current social welfare conditions.If, in the arbitrary instrument choice case, instruments are
constant so that dPujtdt ¼ 0, the value function (8) depends on the
vector of parameters uP and is written as VAt xt;
Puð Þ. In this casewe can still define the accounting price for the instrument,although the current CSW does not depend directly on uP butindirectly, through the accounting prices for the assets, whichcan be written as: ptxi
Puð Þ ¼ AVAt xt;
Puð Þ=Axit.Criteria (11) and (12) along with the definitions of account-
ing prices (9) can be used to suggest a broad rule for evaluatingcurrent policies according to their impact on changes incurrent social welfare.
Consider two alternative feedback policies (rules) (g1(x(τ)),g2(x(τ))), or two arbitrary policies (uP1(τ), u
P2(τ)). The time de-
rivative of the SWF at time t can be defined through (11) underthe feedback rule as (SF1t, SF2t), or through (12) under the
arbitrary rule as (SA1t, SA2t) Then the following comparisonscould be useful for policy evaluation:
1. If S1tNS2tN0, policy 1 leads to higher change in social wel-fare at time t relative to policy 2.
2. If S1tN0, S2tb0, policy 1 can be regarded as consistent with‘current productive-base sustainability’ at time t, whilepolicy 2 implies negative genuine saving rates. S2t-typepolicies leading to persistently negative genuine saving areunsustainable in the World Bank's (2006) context.
3. If policies 1 and 2 result in welfare paths V1(t)≶V2(t), but:V1(t)≷
:V2(t) then there is a potential conflict in ranking two
policies, when one leads to a higher welfare path relative tothe other, but is inferior, relative to the other, in terms ofchanges in current social welfare. This is part of a moregeneral open issue in the evaluation of development cri-teria, which reflects a potential conflict between optimalityand sustainability as criteria for development. The resolu-tion of this conflict is beyond the purpose of the presentpaper.
3. Defining value functions and accountingprices in non-optimizing economies
Having defined the value function and the associated ac-counting prices at time t, we proceed to consider a structuredmodel of an economywith the purpose of providing exact, andwhenever possible, closed form representations of theseconcepts. These representations will provide more insightsinto our approach, as well as a basis for empirical estimations.
We consider therefore an aggregate model of a growingeconomy where output is produced by capital and labor. Theproduction process is affected by exogenous labor augmentingtechnical change, while the total labor force is determined bydomestic population growth and migration inflows (or out-flows). Output is divided among consumption and investmentand consumption generates utility. On the other hand outputproduction generates emissions which affect utility nega-tively. Thus, although we are dealing with a stylized model,important characteristics of modern economies such astechnical change, environmental pollution and migration aretaken into account in exploring the CSW conditions at time t.13
Capital accumulation in our stylized economy is describedby using the standard Solow model (Solow 1956). Assumingexogenous labor augmenting technical change, the aggregateproduction function can be written as Y=F(K, AN), where asusual Y is aggregate output, K is capital stock, N is labor input,:AA ¼ g is the rate of exogenous technical change so that L=ANis effective labor. The standard Cobb–Douglas production
477E C O L O G I C A L E C O N O M I C S 6 7 ( 2 0 0 8 ) 4 7 3 – 4 8 4
function Y=Ka(AN)1− a, 0bab1, can then be expressed in pereffective worker terms as ŷ= ka, where y ¼ Y
AN ; k ¼ KAN.
14
In our stylized economy we seek to incorporate the impactof migration into the change in the total labor force. Given theimportance that migration flows have played in the history ofeconomic development, it is interesting to determine thecontribution of migration to the current CSW conditions of aneconomy, along with technical change and environmentalpollution. Migration is a phenomenon that affects an econ-omy's population and labor supply. It represents gains inpopulation for the destination economy and at the same timelosses for the source economy. The movement of a personcould also entail the movement of human capital and that isthe reason why migration also implies some degree of capitalmobility.15
Following Barro and Sala-i-Martin (2004, pp. 384–5), letM(t)⪌0 be the flow of migrants into the domestic economy.Let the domestic population and labor force, N(t), grow at theconstant rate n, then the overall growth rate of the domesticlabor is::NN
¼ nþMN
¼ nþm ¼ n
where m=M /N is the net migration rate. Assuming that eachmigrant does not bring along any capital into the domesticeconomy 16 the accumulation of capital in the domesticeconomy, measured in per effective worker terms, is given by:
kt ¼ skat � nþ dþ gð Þkt �mkt ð13Þ
where s is the constant saving rate and δ is thedepreciation rate.This is a Bernoulli differential equation which can be solved toobtain:17
ks k1�at � s
x
� �e� 1�að Þx s�tð Þ þ s
x
� � 11�a
; szt;x ¼ nþ dþmþ gð Þ ð14Þ
Let C denote aggregate consumption and c ¼ CAN denote
consumption per effective worker. Since in the Solow modelconsumption is a fixed proportion of output,18 we have, in pereffective worker terms:
cs ¼ 1� sð Þkas ð15ÞUsing (14) we obtain:
cs ¼ 1� sð Þ k1�at � s
x
� �e� 1�að Þx s�tð Þ þ s
x
� � a1�a
ð16Þ
Environment is introduced into the model by assumingthat pollution, denoted by P, which is a by-product of
14 An alternative approach would be to specify the productionfunction in the context of an endogenous growth model, by usingan AK function or more generally a production function withknowledge externalities or human capital. This approach isanother area of further research, once the structure of the valuefunction and accounting prices is understood in the context of thetraditional Solow model.15 See, for example, Barro and Sala-i-Martin (2004), pp. 384–5.16 This can be interpreted as migration which does not supportany capital movement.17 For the solution, see Appendix A.18 In the terminology of the previous section, consumption is afeedback control.
production, affects utility in a negative way. Then the utilityfunction becomes a function of per capita consumption cτ andtotal pollution Pτ and is assumed, as is common in this type ofanalysis, to have the following separable specification:
U cs;Psð Þ ¼ �c� r�1ð Þs � D Psð Þ ð17Þ
In (17) −σ is the elasticity of marginal utility, with σN1, andD(Pτ) can be interpreted as a damage function assumed strict-ly increasing and convex. We specify the damage function asD(Pτ)=θPτγ with θN0 and γ≥1.19
Since the production structure is determined in pereffective worker terms, we need to specify the utility function(17) in per effective worker terms. Since ĉ=C /AN, from thedefinition of per capita consumption we obtain:
Cs
Nsucs ¼ csAteg s�tð Þ; u csð Þ ¼ �c� r�1ð Þ
s ¼ � csAteg s�tð Þ� �� r�1ð Þ
where At is some initial technology level, and the utility func-tion (17) becomes:
U cs;Psð Þ ¼ � csAteg s�tð Þ� �� r�1ð Þ
�hPgs ð18Þ
We assume that pollution is of the flow type and that theflow of emissions, since it is a by-product of production, isrelated to output by a strictly increasing function Pτ=µ(Yτ). Interms of the discussion in Section 2, pollution can be regardedas a form of a feedback control, since by using the productionfunction to substitute for output, emissions can bewritten as afunction of the capital stock. This feedback rule can beassociated with technical conditions and production practiceswhich determine completely, in the absence of environmentalpolicy, the evolution of emissions.20 The µ(·) function can befurther specified as:
Ps ¼ AYbs e
xt; AN0; bN0 ð19Þwhere x reflects technical change in pollution generation.21
A negative x reflects pollution reducing technical change.In per effective worker terms, Ys ¼ ysAsNs ¼ k
asAtNte gþnð Þ s�tð Þ;
n ¼ nþm, with Nt the initial value for the domestic labourforce. By substituting Yτ in (19) and using (14), (16), and (18), theutility flow in per effective worker terms is specified as:
U kt;Nt;At
� �¼ � csAteg s�tð Þ
� �� r�1ð Þ�h A k
asAtNte gþ nð Þ s�tð Þ
� �bex s�tð Þ
� �gð20Þ
The flow of total utility in the economy is NτU(Cτ,Pτ) and thevalue function for the economy can be defined, using (20), as:
Vt kt;Nt;At
� �¼Z l
te�q s�tð ÞNsU ks;Ns;As
� �dt; Ns ¼ Nte
n s�tð Þ ð21Þ
19 This specification is consistent with empirical work related tothe formulation of damages from pollutants such SO2, NOx andparticulates. (See for example Barker and Rosendahl, 2000;Eyckmans and Bertrand, 2000).20 For example in the absence of environmental policy or anyother environmental constraint, a firm will emit as much aspossible for a given level of output and technical conditions, sinceemissions can be regarded as an unpaid factor.21 This type of technical change can be induced by environ-mental policy. We do not model this process here, but in theempirical application we try to make inferences about theexistence of this type of technical change from data.
478 E C O L O G I C A L E C O N O M I C S 6 7 ( 2 0 0 8 ) 4 7 3 – 4 8 4
where the value function depends only on the current valuesof state variables of the problem kt, Nt, At and the parametersdescribing the structure of the economy.
This formulation of the social welfare, with utility of percapita consumption multiplied by the total number ofindividuals, which is the criterion function proposed byArrow and Kurz (1970), should be contrasted to anotherpossible formulation where per capita utilities are summed.There is an old controversy about the choice of the criterion(Koopmans, 1977). In our formulation, the growth of popula-tion helps welfare and this reflects the idea that “if morepeople benefit, so much the better”. In this context migrationhelps welfare. On the other hand, if population size is adecision variable, and in our case this could have been thecase if the migration rate was not exogenous but a controlvariable, then the choice of the criterion function wouldacquire more importance.
Using (21) the current accounting prices are defined as:
ptkt ¼AVt
Akt;ptNt ¼
AVt
ANt; ptAt ¼
AVt
AAtð22Þ
Since k ¼ kA ¼ K
AN ; k ¼ KN, the accounting price of capital in
physical units and per capita units is defined respectively as:
ptKt ¼AVt
Akt
AktAKt
¼ 1AtNt
ptkt ð23Þ
22 Zero profits for any given wage w require that f ks� �
�h
ksf ′ ks� �
þ k/′f ′ ks� �
ex s�tð Þ�e�g s�tð Þ ¼ w.23 Inada conditions state that limkY0 f ′ kð Þ ¼ þl and limkYlf ′ kð Þ.When they are combined with a concave production functionthen f ′(k) is monotonically declining and intersects r+δ onlyonce, providing a unique solution.
ptkt ¼AVt
Akt
AktAkt
¼ 1At
ptkt ð24Þ
It should be noted that in this case there is no specificaccounting price for pollution since pollution is not a stock,but the impact of pollution is realized through the accountingprice of capital pt kt
¼ AVt=Akt which depends on the para-meters of the damage function.
3.1. Changes in current social welfare in the presence ofenvironmental policy
In the previous section emissions were considered as a by-product of output production, determined by technical condi-tions alone. In this section we explicitly introduce environ-mental policy which is expressed through a performancestandard that determines an upper limit for the emissions ofthe firms. The emission function of the representative firmcan be written as:
Ps ¼ AYbs e
x s�tð Þ ¼ A ysAN� bex s�tð Þ ¼ / f kt
� �AN
� �ex s�tð Þ ð25Þ
where f ks� �
¼ ys is the production function in per effectiveworker terms.
To simplify subsequent notation we set / fð Þ ¼ Afb; f ¼f ks� �
AN.The emission limit will take the form:
Ps VPPs ð26Þ
The profit function of the representative firm can bewritten in per effective worker terms as:
AN f ks� �
� rþ dð Þks �we�g s�tð Þh i
ð27Þ
The firm considers the interest rate r and the wage rate was fixed and chooses capital, for any fixed level of effective
labor AN, to maximize (27) subject to (26). The Lagrangian forthe problem is:
L ¼ AN f ks� �
� rþ dð Þks �we�g s�tð Þh i
þ kPPs � / fð Þex s�tð Þh i
ð28Þ
The Kuhn–Tucker conditions for an interior solution to theproblem imply:
f ′ k4s� �
1� k/′ex s�tð Þh i
¼ rþ d; k4s N0 ð29Þ
kPPs � / f k4s
� �AN
� �ex s�tð Þ
h i¼ 0; kz0 ð30Þ
If the emission constraint is not binding then λ=0 and thesolution k4s is obtained by the usual condition f ′ ks
� �¼ rþ d.22
Under concavity of the production function and Inadaconditions, a unique solution always exists.23
If λN0 then the constraint is binding and the capital stock isdetermined as a function of the emission limit by the solutionof:
PPs ¼ / f ks
� �AN
� �ex s�tð Þ; as ð31Þ
k4s ¼ wPPs;AN; ex s�tð Þ� �
¼ k4sPPs� �
;withdksdPPs
N 0 ð32Þ
Thus a more stringent emission limit will reduce theequilibrium stock of capital. This can also be seen from (29). Apositive λ shifts themarginal product curve f ′ ks
� �to the left. As
a result, k4
sb ks and the binding performance standard reducesthe equilibrium stock of capital. It can also be noticed that if xb0so that we have emission saving technical change then thereduction of the equilibrium stock of capital under the per-formance standard will be smaller, the larger this type oftechnical change is. Sincecapital stock is reduced fromabindingperformance standard or equivalently from a more stringentperformance standard, output is also reduced ceteris paribus.
This reduction is determined as f ks� �
� f k4sPPs� �� �
, wheref ks� �
is the is the output of the economy without theperformance standard, and f k4s
PPs� �� �
is the output of theeconomy under the binding performance standard,
PPs. Con-
sumption in per effective worker terms is defined ascs ¼ 1� sð Þyt, and since y ¼ f k4s
PPs� �� �
,we have:
cs ¼ 1� sð Þf ks4PPs� �� �
¼ csPPs� �
ð33Þ
Thus the per capita utility flow in the economy under theperformance standard will be:
U cs;PPs
� �¼ � csAteg s�tð Þ
� �� r�1ð Þ�h
PPg
s
� �ð34Þ
with cs determined by (33). In empirical applications, wherethe main purpose is to examine the impact of a performancestandard on the current CSW conditions of the economy,a reliable estimate of f k4s
PPs� �� �
is unlikely due to data
479E C O L O G I C A L E C O N O M I C S 6 7 ( 2 0 0 8 ) 4 7 3 – 4 8 4
limitations. In this case one approach could be to assume thatthe reduced output under the binding standard is approxi-mately proportional to the output obtained without a limit onemissions. This means that we set:
f k4sPPs� �� �
c 1� zPP
� f ks� �
ð35Þ
which implies that f ks� �
can be interpreted as full capacityoutput without environmental constraints and zP
P is a new pa-rameter introduced here which reflects the reduction in outputdue to the upper emission limit
PPs.24 Under the Cobb–Douglas
assumption, we have, y ¼ 1� zPP
� ka. In this case the accumula-
tion of capital equation in per effective worker terms is:
:
kt þ nþ sþmþ gð Þkt ¼ s 1� zPP
� kat ð36Þ
The solution of this Bernoulli equation is:25
ks ¼ k1�at � s 1� zP
P
� x
� �e� 1�að Þx s�tð Þ þ s 1� zP
P
� x
� � 11�a
ð37Þ
x ¼ nþ dþmþ g ð38Þ
Therefore cs ¼ 1� sð Þ 1� zPP
� kas ¼ cs ks; zP
P
� �, and the value
function for the economy becomes:
Vt ¼R l
te�q s�tð ÞNtU ks;Ns;As;
PPs; zP
P
� �ds; or
Vt ¼ �R l
te�q s�tð ÞNt cs ks; zP
P
� �Ateg s�tð Þ
� �� r�1ð Þ�h
PPg
s
� �ds
ð39Þ
The current accounting price for the performance standardPPt can be calculated as:
ptPP ¼ AVt
APP t
¼Z l
te�q s�tð Þ A
APPt
NtU ks;Ns;As;PPs; zP
P
� �h ids ð40Þ
Thus there is a specific accounting price for the arbitrarycontrol
PPt, as was anticipated by Proposition 1. There is also
an accounting price associatedwith the parameter zPP, which is
defined as:
ptzPP¼ AVt
AzPP
ð41Þ
4. Current changes in social welfare in anon-optimizing economy
The previous section obtained representations of valuefunctions and accounting prices. Combining these represen-tationswith Proposition 1, it follows that our stylized economyis characterized by non-declining current CSWwhen feedbackrules are followed, if:
:V
Ft ¼ pKt
:Kþ pNt
:Nþ pAt
:Az0
Dividing by Nk where k ¼ KN , using the fact that
:k ¼ d K=Nð Þ
dt ¼K:
N � K:
N k and that the accounting price for capital in physical
24 For an estimate of the proportion of output loss due toenvironmental regulation in the US economy see Jorgenson andWilcoxen (1998).25 See Appendix A for details.
terns is related to the accounting price of capital in pereffective worker terms, by (23) we obtain:
SFt ¼
:V
Ft
Ntkt¼
ptktAtNt
k:
kþ N
:
N
!þ ptNt
N:
N1kt
þ ptAt
A:
AAt
Ntkt
where SFt measures the change in the value of the economy per
unit of produced capital stock at time t. Thus SFt could be
interpreted as the rate of return on produced capitalmeasuredin terms of social welfare. It is clear that by multiplying S
Ft by
the current stock of capital we obtain a measure of currentgenuine investment. Using as before
:A=A ¼ g; n ¼ nþm, with
m⪌0, and denoting the rate of growth of capital per workerby
:k=k ¼ v, we have that social welfare is currently non-
declining if:
SFt ¼
pt ktAtNt
vþ n� þ ptNt n
1kt
þ ptAt g1kt
At
Ntz0 ð42Þ
When an arbitrary environmental policy in the form of theemission limit
PPs is present, the criteria become:
:V
At ¼ pKt
:Kþ pNt
:Nþ pAt
:Aþ pP
PtdPPsdt
or ð43Þ
SAt ¼
pktAtNt
vþ n� þ pNt n
1kt
þ pAtg1kt
At
Ntþ pP
Ptp1kt
PPtNt
ð44Þ
where π is the rate of growth of the emission limit, with πb0indicating that environmental policy becomes gradually morestringent and πN0 indicating that environmental policy isgradually becoming laxer. As before, by multiplying S
At by the
current stock of capital we obtain a measure of currentgenuine investment. In this case genuine investment isadjusted for the changes in environmental policy, a requiredadjustment that has not been noticed in earlier literature.
Measures of current CSW (42) and (44) are basically short-term measures since they reflect measurements at time t.These conditions will change if basic parameters, such asgrowth rates of assets or choices of instruments, change. Sincethe economy is not on an optimal path these changesespecially in the case of arbitrary choice of controls mightactually take place. Therefore, if the basic parameters arelikely to change, then recalculations and updating of (42) or(44) are necessary. We believe that this observation isimportant, especially for applied work.
5. Exploring current changes in social welfareconditions for the Greek economy
The stylized model developed above is used to explore thecurrent social welfare conditions for the Greek economy. Toapply themodelwe need estimates of the parameters requiredto define value functions like those defined in (21) or (39).
Our approach was to estimate, using econometric estima-tions, the parameters that correspond to structural relationsand to assign plausible values to those parameters for whicheconometric estimation was not possible. For these para-meters we used sensitivity analysis to explore the robustnessof our results.
The parameters required in order to estimatemeasures (42)and (44) are: n the rate of growth of the domestic population
28 We did not include human capital in our production function.However, the value of estimated a can be regarded, under certainassumptions, as incorporating human capital effects (Barro andSala-i-Martin, 2004).29 This method of estimating the labour augmenting technicalchange from a Cobb–Douglas production function is similar towhat is proposed by Barro and Sala-i-Martin (2004).30 All estimations were performed using the software packageEViews 5.0.31 Lignite fired power plants in Greece produced 63% of totalelectricity in 2003, and are concentrated mainly in two locationsin the Northern and in the Southern part of the country.32 The source of the data was the European Environment Agency(http://www.eea.europa.eu/).33 Estimates were corrected for first order serial correlation,which turned out to be highly significant. Details are presented inAppendix B.34 Data were taken from “The Greek Economy in Figures,” (2002,page 105).35 The value of 3% has been used by a number of researchers for the
Table 1 – Average growth rates of capital, output and laborforce for the Greek economy, 1965–1990
% per year Per worker terms % per year
Capital:K=K�
5.55 Capital:k=k� �
4.95GDP
:Y=Y�
3.64 GDP :y=yð Þ 3.035Labor force (n) 0.60 – –
480 E C O L O G I C A L E C O N O M I C S 6 7 ( 2 0 0 8 ) 4 7 3 – 4 8 4
and labor force, m the net migration rate; v the rate of growthof capital per worker; g the rate of growth of labor augmentingtechnological change; s which expresses savings as a propor-tion of the Greek GDP in the period analyzed; a which is theparameter of the production function reflecting the elasticityof capital input; ρ which represents the discount rate; σ theelasticity of marginal utility the value of which reflectsintertemporal preferences towards equality in income dis-tribution; δ which is the depreciation rate; θ and γ which arethe parameters of the postulated damage function D(Pτ)=θPτγ;µ, β and x which are the parameters of the emission functionPτ=µYτ
βext; and finally, when we need to examine the impact ofan emission limit, the potential reduction in GDP due to thisemission limit is required, which is the parameter zP
P.The fundamental data for the Greek economy were GDP,
capital, and labor, measured in 1990 million $ and thousandsof workers respectively, taken from the Penn World Table(Mark 5.6) for the period 1965–1990 (Table 1). We obtain theaverage annual growth rates of these variables in physicalunits and in per capita terms during the sample period byestimating the relationship ln xt=ao+a1t, where xt is thevariable of interest and t takes values t=1, …, TS during thesample period, with TS=26.26
The estimates of the growth rates for the variables ofinterest in physical and in per worker terms are shown in thetable below.
The basic structural relationship for the Greek economy isthe aggregate production function (19), since estimates fromthe production function will be used to determine theelasticity of capital with respect to output, which is theparameter a, and the rate of labor augmenting technicalchange g. For this estimation we assume the existence of aconstant return to scale Cobb–Douglas long run aggregateproduction function for the Greek economy, defined over manmade capital and effective labor input, which takes the form:
Yt ¼ BKa Negt� 1�a
or in per worker terms:27
yt ¼ Bkat eqt; q ¼ g 1� að Þ
The statistical model can be written as:
lnyt ¼ lnBþ a lnkt þ qtþ et; t ¼ 1; N ;TS ð45Þwhere εt is the usual error term. The production function (45)can be interpreted as a long run equilibrium relationship thatshifts in time as it is affected by technical change. To test for
26 Relationship ln xt=ao+a1t corresponds to the standard ex-ponential growth model xt=Aoea1t.27 It is clear that in per worker terms this function becomesyt ¼ Bk
a
t , which is the function used in the previous sections withB≡1.
the existence of such an equilibrium relationship we test forthe existence of a cointegrating relationship. The Johansencointegration test suggests that both the trace and themaximum eigenvalue tests indicate one cointegrating rela-tionshipwith constant and deterministic trend at the 5% level.When a cointegrating relationship exists, ordinary leastsquare (OLS) estimation is superconsistent, that is theestimated coefficients are consistent and asymptoticallynormal (Stock, 1987). Using therefore OLS to estimate (45) weobtain that the elasticity of capital input is a=0.4025,28 whilethe rate of labor augmenting technical change is g ¼ q
1�a ¼0:009 or 0.9% annually.29 The details of the cointegration testand the OLS estimation results are presented in Appendix B.30
To model environmental pollution we consider sulfurdioxide emissions (SO2) as the main flow pollutant. Sulfurdioxide emissions in Greece are mainly localized because themajority of them are created in the processes of powergeneration.31 These emissions were related to output, assum-ing an emission function of the constant elasticity form (19),which was regarded as a technological relationship and wasestimated using data of annual emissions in kilotons coveringthe period 1980–1999.32 The estimated elasticity of SO2
emissions with respect to aggregate output was 0.225. Atrend term which could indicate technical change associatedwith SO2 emissions was highly insignificant.33
For themigration rate, a recent study (Lianos, 2003) indicatesthat between 1991 and 2001 the number of immigrants whoentered the Greek economy was around 630,000. Assuming anaverage annual flow of 60,000 immigrants, the average netmigration rate is approximately 1.5%, and n+m=0.021. For themarginal propensity to save, we use the average value for theperiod 1970–1990 of savings as a proportion of GDP, withs=0.21.34 The depreciation rate was set at δ=3% followingMankiw et al. (1992); the utility discount rate at ρ=3%;35 andthe elasticity of marginal utility at σ=3 which reflects relatively
estimationofmarginal social costs ofCO2 emissions (see, for example,surveys by Fankhauser and Tol, 1997; Tol, 2005). The values of 1% and2%, along with time declining rates, have also been used in thesestudies. We perform a sensitivity analysis of our results by using avalue of 1% for the utility discount rate. There is an increase in theabsolutevaluesof theaccountingprices reported inTables2and3, butthere is no change in the signs of the S
Ft and S
At .
Table 2 – Accounting prices and changes in social welfarefor Greece in 1990⁎
M θ pK pN pA SFt
0 0 0.0011216 −0.0464493 315.511 0.000078390.015 0 0.00238486 −0.125464 852.225 0.001429680.015 0.2856·10−6 0.00238306 −0.12792 850.693 0.00014060.015 10−5 0.00237046 −0.134064 846.860 0.000134610.015 10−4 0.00224252 −0.21147 798.574 0.000059500.015 10−3 0.00096316 0.985523 315.712 −0.0006916
⁎The accounting prices pK, pN, pA, are defined in (22), while SFt is
defined in (42).
481E C O L O G I C A L E C O N O M I C S 6 7 ( 2 0 0 8 ) 4 7 3 – 4 8 4
strong preferences towards equal income distribution. Theparameterγ of the damage functionwas set atγ=1. This impliesa linear damage function in which θ, the damage costcoefficient, reflects constant marginal damages from SO2.Since the units of output and consumption were million US$, θreflects the environmental damages in Greece, in million US$,from the emissions of 1 kt of sulphur dioxide in a year. Because,as mentioned above, SO2 emissions are mainly localized, thevalue of θ in ourmodel can be interpreted as capturingmarginaldamages averaged over the whole population. As noted by Sáezand Linares (1999), damages from SO2 emissions are associatedwithhealth damages fromSO2and sulfates exposure alongwithdamages inflicted on buildings, crops and natural habitats.Estimates of SO2 damages and associated damage cost coeffi-cients, θ, obtained by Sáez and Linares (1999) and Barker andRosendahl (2000) for Greece, range from 0.12946 to 0.5128 US$per kiloton of SO2 a year.36 In our estimations we consideredvalues of θ in the interval [0.2856·10−6, 10−3] indicating damagesfrom 0.28256 US$ to 1000 US$ per kiloton of SO2 a year.37 For theparameter zP
P there is no information for the Greek economy.Jorgenson and Wilcoxen (1998), using a computable generalequilibrium approach, estimated the cost of all environmentalrestrictions for the US economy to be 2.592% of real GNP, so weset zP
P at a conservative value of 1%.The parameter values used are summarized in the follow-
ing table:
3
3
Rabo5eeth3
3
c7shre
Parameter
6 Estimates7 The valuosendahl (2lso be noticeen estimatf statistical0%. We usestimates, inxtreme valuat is the va
8 Numerica9 The fundaonvergence0 years. Thteady stateorizon. Of cecalculatedstimations c
ñ
in eure of 0000) ofed thated as tlife (VOhigheorder
e of 10lue ofl resultmentato a sus, theperiod.ourse, if thhange
v
os were.2856 isthe damin thes
he valueSL) cour valueto chec00 US$ fθ for whs were ol paramteady sttime hThe resas notede funda.
g
converthe page c
e estimof yeald haves of θk theor θ isich cribtaineeters ofate wiorizonults areabovementa
s
ted toointoefficiatesrs lostincre
, in asensitused tteria Sd by uthe G
ll takechoserobuaccol gro
a
1990 Uestimatent forpremat(VOYL)ased thdditionivity ofo identiFt and Ssing Mreek ecoplacen extenst to chuntingwth ra
ρ
S$.e ofGreecure m. Use oese eto thour rfy a swAt turnathemanomyin appds wangespricestes us
δ
Barkere. It sortalitf the
stimate avaesultsitchnegatica.implroximell intin theneeded in
σ
Value 0.021 0.0495 0.009 0.21 0.4025 0.03 0.03 3Parameter
β µ x γ θ zPPValue
0.225 4.146 0 1 [0.2856 ·10−6, 10−3] 0.01Using the above parameters, accounting prices werecalculated with numerical integration of the derivatives ofthe value function.38 We used a time horizon of 100 years as anecessary approximation for the numerical estimation.39 Twosets of results were obtained. The first set which correspondsto emissions determined by a feedback rule through theemission function is obtained using (42). The second set isobtained using the 1999 sulphur dioxide emissions as anupper emission limit and (44). Table 2 below shows accountingprices and the estimated changes in social welfare fordifferent marginal damages in Greece in 1990.
andhouldy hasvaluees byilable. Thepoint,tive.
y thatatelyo thetimeto bethe
40 It should be noticed that the positive effect of migration isassociated with the choice of criterion function (21). Thealternative formulation of using per capita utility in the criterionfunction and the impact on the estimated accounting priceschanges in social welfare could be an area for further research.
We can observe from Table 2 that for marginal environmentaldamages below 1000US$ per kiloton of SO2 the Greek economywas characterized by a positive change in social welfare in1990. Given the very high value of θ relative to the estimatesfor Greece, for which criterion S
Ft changes sign, it seems that in
the context of our analysis it might be claimed that theevolution, for the examined sample period, of the Greekeconomy is characterized by positive changes in socialwelfare, which can be regarded as an indication that it iscurrently productive-base sustainable. Furthermore, it is clearthat migration has played an important role in the currentconditions of the Greek economy, since the positive change insocial welfare is reduced substantially when we set m=0.40 Inaddition, the accounting prices have the expected signs andthe rate of change of social welfare is declining in environ-mental damages as expected.
In Table 3 we present accounting prices and estimatedchanges in social welfare under a binding environmentalpolicy. Thus Table 3 shows accounting prices and changes insocial welfare for different marginal damages in Greece in1990 as if the emission limit for sulphur dioxide had been setat the 1999 emission level, whichwas 541 kt. Values have beencalculated for m=0.015 and zP
P ¼ 0:01.In Table 3 the column pP
P refers toAVAPPs
which is the accountingprice for the emission standard. This price is negative asexpected, since an increase in
PPs, that is a laxer environmental
policy, is expected to reduce the economy's socialwelfare,whenzPP remains constant. The column pzP
Prefers to AV
AzP
Pwhich is
negative as expected. Thismeans that if the cost of the standardin termsofoutput foregone increases, then theeconomy'ssocialwelfare is reduced ceteris paribus. Since a lax standard isexpected to reduce zP
P, the final effect of a change in theperformance standard on social welfare depends on theexpression AV
APPsdPPs þ AV
AzPPdzP
P. Again, as expected, the criterion isdeclining in marginal environmental damages.
6. Concluding remarks
This paper analyzed current change in social welfare (CSW)conditions under a non-optimizing framework. The main
Table 3 – Accounting prices and changes in social welfarefor Greece in 1990 under an emission limit⁎
θ pK pN pA pPP pzP
PSAt
0.2856 ·10−6
0.002449 −0.144 873.9 −0.09694 −1107.6 0.00013
10−3 0.002449 −54.23 873.9 −339.6 −1107.6 −0.0409
⁎Accounting prices (pK, pN, pA) are defined in (22) pPP; pzP
P
� �are defined
in (40), (41) respectively, while SA
t is defined in (44).
482 E C O L O G I C A L E C O N O M I C S 6 7 ( 2 0 0 8 ) 4 7 3 – 4 8 4
purpose was to develop an applicable and operationalapproach to measure current CSW, and to contribute to thedevelopment of a framework for the evaluation of policieswith respect to their impact on current social welfareconditions. For this purpose we tried to determine a measurefor CSW which would fit into a non-optimizing economicframework, sincewe consider such a framework to adequatelyrepresent current economic conditions, at least in developingcountries. By considering two different approaches for choos-ing policy instruments, a feedback rule and an arbitrary rule,we determined two corresponding criteria for measuringchanges in current social welfare conditions which canprovide numerical results for actual economies and could beapplied in empirical work. Since current changes in socialwelfare can be associated with current changes in theproductive base of the economy valued at accounting prices,positive changes imply positive genuine investment andcurrent productive-base sustainability. On the other handnegative changes in social welfare and therefore negativegenuine investment can be regarded as an indicator ofcurrently unsustainable policies. In doing this, we extendedcurrent results about genuine investment by showing thatwhen policy rules are chosen in an arbitrary way, thengenuine investment should be adjusted accordingly. Giventhe arbitrary nature of most government policies in practiceenvironmental policies included this observation might haveimportant implications for empirical applications. We provideexact representations and closed form solutions for valuefunctions and accounting prices, by considering a ”Solow”economy, where domestic population growth, migration,labor augmenting technical change, and environmentaldamages associated with pollutant flows generated by eco-nomic activities are taken into account in determining thecurrent social welfare conditions.
The criteria developed in this paper were applied to thecase of the Greek economy and empirical estimates wereobtained. Our findings seem to support the idea that ourtheoretical framework can be used for empirical purposes. Inparticular, our results show thatmigration inflows, exogenoustechnical change, growth of capital per worker, and SO2
emissions damages, are important factors characterizing thecurrent changes in social welfare conditions of the Greekeconomy. Our approach allows for the estimation of thecontribution of these factors which is undoubtedly usefulinformation for the design and evaluation of policies. Themain empirical finding is that the Greek economy seems to becurrently ‘productive-base sustainable’, given the currentestimates of SO2 emission damages, which was the onlypollutant considered in our analysis. Furthermore, and asexpected, taking into account environmental damages has a
negative effect on the current social welfare conditions. Thus,our empirical results for the case of Greece support theperception that pollution damages are a factor affectingconditions associated with changes in social welfare condi-tions. Our approach not only provides an empirical confirma-tion of this result, but can be used to quantify, at leastapproximately, environmental impacts on social welfare,another important piece of information for policy design. Amore precise quantification of these effects is an openresearch area.
Admittedly sustainable development as a general defini-tion does not provide a systematic framework for empiricalestimations and for policy design. The present paper is amodest attempt to make the definition operational andcapable of providing empirical estimates which are based onthe structure of the economy, and which can be associatedwith concepts of current changes in social welfare and currentproductive-base sustainability. Thus important fundamen-tals, such as the elasticity of the production function, the rateof technical change, migration, environmental damages, andassets' rates of growth, play a key role in the measurement ofcurrent CSW conditions. The model developed in this papercan be extended and made more realistic by includingtransition equations for stocks of pollutants, naturalresources (depletable or renewable), human capital, or byintroducing uncertainty in the evolution of the economy.These extensions will provide better insights regarding thechanges in social welfare conditions of economies and willenhance our ability to provide meaningful estimates of suchchanges.
Acknowledgements
We gratefully acknowledge valuable comments received by ananonymous referee on earlier drafts of this paper. D. Vouvakiacknowledges financial support under the 03ED375 researchproject, implemented within the framework of “Reinforce-ment Programme of Human Research Manpower” (PENED)and co-financed by National and Community Funds (25% fromthe Greek Ministry of Development — General Secretariat ofResearch and Technology and 75% from EU— European SocialFund). A. Xepapadeas acknowledges support fromMarie CurieDevelopment Host Fellowship of the European Community'sFifth Framework Programme (HPMD-CT-2000-00036) and the03ED375 PENED research project.
Appendix A
Solutions of the Bernoulli equations for the capital stock.
A.1. Solution of Eq. (13)
The Bernoulli equation is solved in the following way: Multi-plying by k
�at we have:
:
ktk�at þ nþ dþmþ gð Þktk
�at ¼ sk
�at k
at ð46Þ
:
ktk�at þ nþ dþmþ gð Þk1�a
t ¼ s ð47Þ
483E C O L O G I C A L E C O N O M I C S 6 7 ( 2 0 0 8 ) 4 7 3 – 4 8 4
If g ¼ k1�at and
:y ¼ 1� að Þ
:
ktk�at
� �, then we have:
:g þ nþ dþmþ gð Þg 1� að Þ ¼ 1� að Þs;which is linear in g ð48Þ
with solution:
gt ¼ g0 �s
nþ dþmþ g
� �e� 1�að Þ n�dþmþgð Þt þ s
nþ dþmþ gð49Þ
Setting g ¼ k1�at , we have:
kt ¼ k1�a0 � s
nþ dþmþ g
� �e� 1�að Þ nþdþmþgð Þt þ s
nþ dþmþ g
� � 11�a
ks ¼ k1�at � s
gþ dþmþ g
� �e� 1�að Þ nþdþmþgð Þ s�tð Þ þ s
nþ dþmþ g
� � 11�a
A.2. Solution of Eq. (36)
Using function y ¼ 1� zPP
� kainstead of y ¼ k
a, the accumula-
tion of capital in per effective worker terms becomes:
:
kt ¼ s 1� zPP
� kat � nþ dþ gð Þkt �mkt þ z
Working as before we obtain:
kt ¼ k1�a0 � s 1� zP
P
� nþ dþmþ g
� �e� 1�að Þ nþdþmþgð Þt þ s 1� zP
P
� nþ dþmþ g
� � 11�a
Appendix B
B.1. Johansen cointegration test
Trend assumption: Linear deterministic trend (restricted)Series: ln yt, ln ktLags interval (in first differences): 1:1
Unrestricted cointegration rank test (Trace)
Hypothesized
0.05 criticalNo of CE(s)
Eigenvalue Trace statistic Value Value prob ⁎⁎None ⁎
0.649227 30.97507 25.87211 0.0106 At most 1 0.347789 8.975136 12.51798 0.1819Trace test indicates 1 cointegrating equation(s) at the 0.05 level.⁎ Denotes rejection of the hypothesis at the 0.05 level.⁎⁎ MacKinnon–Haug–Michelis (1999) p-values.
Unrestricted cointegration rank test (maximum eigenvalue)
Hypothesized
0.05 criticalNo of CE(s)
Eigenvalue Trace statistic Value Prob ⁎⁎None ⁎
0.649227 21.99993 19.38704 0.0204 At most 1 0.347789 8.975136 12.51798 0.1819Max-eigenvalue test indicates 1 cointegrating equation(s) at the0.05 level.⁎ Denotes rejection of the hypothesis at the 0.05 level.⁎⁎ MacKinnon–Haug–Michelis (1999) p-values.
B.2. Econometric estimations
The production function
Variable
Coefficient Std. error t-statisticln B
1.438115 0.187444 7.672226 ln k 0.402501 0.080129 5.023150 t 0.005392 0.003080 1.750943 R-squared 0.957372 Adjusted R-squared 0.952636 Durbin–Watson stat 1.175040The emission function
Variable
Coefficient Std. error t-statisticConstant
4.156018 2.289511 1.815243 ln Y 0.225308 0.241803 1.931786 AR(1) 0.745520 0.084558 8.816671 R-squared 0.906529 Adjusted R-squared 0.894845 Durbin–Watson stat 2.176287R E F E R E N C E S
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