Green accounting and the welfare gap

Ecological Economics 30 (1999) 161–175

ANALYSIS

Green accounting and the welfare gap

Paul Turner a, John Tschirhart b,*a CornerStone Energy Ad6isors, Illinois Institute of Technology, 29 S. LaSalle, c930, Chicago, IL 60603, USA

b Department of Economics and Finance, Uni6ersity of Wyoming, P.O. Box 3985, Laramie, WY 82071-3985, USA

Received 20 February 1998; received in revised form 21 December 1998; accepted 28 December 1998

Abstract

Although gross domestic product (GDP) is not intended to be a measure of societal welfare, it is often used as such.One shortcoming as a welfare measure is that it fails to account for the non-marketed value of natural resource flows.The difference between societal welfare and GDP is labelled the ‘welfare gap’. A model that accounts for both marketand non-market income flows from natural capital is used to examine this gap. Societal welfare depends on privategoods and the stock of natural capital. The latter is subject to a logistic growth relationship common to manynon-human species. Private goods are produced using human capital and flows of natural capital. An exogenouslygrowing human population either harvests the natural resource, produces human capital or produces the privategood. Optimal control theory and dynamic simulations provide steady-state harvest and human capital growth rateswhich determine the steady-state natural resource stock, GDP and societal welfare growth rates. The model illustratesthe feasibility of explicitly accounting for ecological relationships in economic growth models and shows that,depending on one’s preferences and the growth rate of human population and the intrinsic growth rate of naturalresources, GDP may diverge substantially from the growth rate of societal welfare, leaving a large welfare gap.© 1999 Elsevier Science B.V. All rights reserved.

Keywords: Gross domestic product; Green accounting; Welfare gap

1. Introduction

Gross domestic product (GDP) measures themoney value of goods and services produced in aneconomy during a specified time period; neither

GDP nor per-capita GDP is intended to be ameasure of societal welfare. One reason why GDPis not a measure of welfare is that it does notaccount for the value of the natural resourcesupon which all production ultimately depends.The natural resource stock, often referred to asnatural capital, includes soil and its vegetativecover, freshwater, breathable air, fisheries, forests,

* Corresponding author. Tel.: +1-307-766-2178; fax: +1-307-766-5090.

0921-8009/99/$ - see front matter © 1999 Elsevier Science B.V. All rights reserved.

PII: S 0921 -8009 (99 )00003 -8

P. Turner, J. Tschirhart / Ecological Economics 30 (1999) 161–175162

waste assimilation, erosion and flood control, pro-tection from ultra-violet radiation, and so on.GDP fails to account for the non-marketed valueof flows from these natural resources, includingmany cultural and recreational activities. Never-theless, GDP mistakenly is used often as an all-encompassing measure of societal welfare bygovernment policy makers and private enterprise.1

Because there is a difference between societal wel-fare and GDP, which we label the ‘welfare gap’,the gap is examined by introducing a model thatoffers a more inclusive welfare measure, one thataccounts for both market and non-market incomeflows from natural capital.

In the model societal welfare depends on pri-vate goods and the stock of natural capital. Pri-vate goods are produced using human capital (i.e.educated workers) and flows of natural capital.The annual growth rates of per-capita income (asmeasured by GDP) and societal welfare are com-pared along a balanced-growth path by employ-ing simulation modeling software (STELLA II).2

Simulations help identify the steady-state equi-librium of the system. The objectives are to: (1)depict the feasibility of explicitly accounting forecological relationships in economic growth mod-els; (2) illustrate the value of computer simulationexercises in extending the insights of theoreticaland empirical growth models; (3) formally showthat, depending on one’s preferences and thegrowth rate of human population and the intrin-sic growth rate of renewable resources, per-capitaincome growth may diverge substantially from thegrowth rate in per-capita welfare; and (4) usecomparative statistics to examine system responseto parametric changes.

This work is motivated by the debate betweenmainstream economists and natural scientists3

over the importance of natural capital to humanwelfare. Each group contends that the other doesnot understand its pedagogical perspective. Hop-ing to address the most ardent concerns of eachdiscipline, we construct a model based on theeconomists’ endogenous technological growth the-ory and constrained by the biologists’ logisticgrowth function for a renewable resource. It isassumed, as reflected in the theoretical model,that natural capital is essential to human welfare:‘‘zero natural capital implies zero human welfarebecause it is not feasible to substitute, in total,purely ‘non-natural’ capital for natural capital’’(Costanza et al. 1997).

Section 2 provides motivation for the problem,and Section 3 is a discussion of endogeneousgrowth theory. The difference between economicgrowth and economic development in the model iscovered in Section 4, and the model is introducedin Section 5. Section 6 presents the model resultsand a conclusion follows in Section 7. AppendixA presents the model in detail.

2. Motivation

Disciplinary investigation of environmentalproblems by both economists and ecologists is atleast two-centuries old.4 There is ideologic frictionbetween the two groups as witnessed by the well-publicized dispute between Paul Ehrlich and thelate Julian Simon.5 This feud notwithstanding,

4 The writings of classical economist Thomas Malthus fo-cused on the effect of population growth on the availability ofresources necessary to sustain human existence (Malthus, 1960[1798]). Charles Darwin developed his theory of evolutionaround the same basic idea: species have the tendency tooverproduce, with the result that only those most ‘fit’ to theirenvironmental circumstances survive and reproduce (Darwin,1958 [1859]).

5 The well-publicized wagers between Ehrlich (1981, 1982)and Simon (1981b) on the direction of future welfare measureshighlight the philosophical rift between ecologists and main-stream economists (Shen, 1996). Simon’s exuberance to bet on‘‘...anything pertaining to material human welfare...’’ refersalmost exclusively to income-based welfare indices neglectingthe value of the natural capital on which all economic outputis ultimately based (Shen, 1996).

1 For example: ‘‘Today the two political parties differ some-what in regard to means, but neither disputes that the ultimategoal of national policy is to make the big gauge—the grossdomestic product—climb steadily upward.’’ (Cobb et al.,1995)

2 STELLA II is a simulation modeling software packageused to model complex systems including ecosystems andeconomic systems. See Hannon and Ruth (1994).

3 We refer to those natural scientists, including biologists,ecologists, and environmental scientists, who have tradition-ally criticized economists for their failure to account for thevalue of ecosystem functions, as ‘ecologists’.

P. Turner, J. Tschirhart / Ecological Economics 30 (1999) 161–175 163

interdisciplinary research between ecologists andeconomists will play an important role in address-ing social problems concurrent with an increasinghuman population whose consumption hastensthe rate of ecosystem degradation.

Mainstream neoclassical economists, slow toappreciate the importance of maintaining ahealthy and productive environmental resourcebase, are beginning to accept ecologists’ asser-tions. Nobel laureate economists Robert Solowand Kenneth Arrow recognize the inextricablelink between biological and economic systems,and urge a more careful consideration of account-ing for biological degradation in economic growthstatistics (Solow, 1994; Arrow et al., 1995). Whiledifficult, fraught with uncertainties, and opposedby many as immoral, valuing ecosystems is neces-sary when making choices. Ecologists, economistsand geographers have estimated the value of na-ture’s long list of ecosystem goods and services6

and argue that such an exercise is important inorder to ‘‘make the range of potential values ofthe services of ecosystems more apparent’’(Costanza et al., 1997).

While some economists are heeding the biologi-cal wake-up call, others, such as Simon (1981a,1995a), continue to downplay the economic andsociological importance of maintaining a healthyand viable natural resource base. According toSimon, society’s ability to advance technology asnecessary should obviate concern for the genius toconcoct, devise, and invent ways around barriersthat threaten production (and consumption). Be-cause humans have acquired the technology fornuclear fission and space travel, they now have allthe tools necessary to

‘‘…go on increasing our population forever,while improving our standard of living and

control over our environment’’. (Simon, 1995b)

Simon’s forecasts are optimistic: the environ-mental degradation associated with continuedpopulation growth and increasing per-capita con-sumption will test society’s collective intellect.7

Presuming that past trends of technological pro-gress can be relied on to overcome the environ-mental stresses concurrent with growingpopulations is a risky strategy. Humankind cur-rently appropriates approximately 40% of totalnet terrestrial primary production of the bio-sphere (Vitousek et al., 1986), 25–35% of coastalshelf primary production (Pauly and Christensen,1995), and half of the available global supply offresh water (Postel et al., 1996). There is ampleevidence that the carrying capacity8 of manyecosystems is being severely tested (Kirchner etal., 1985; Rees, 1990; Hardin, 1991; Brown, 1994,1995). Mapping a course through this unchartedecological territory should not be left solely tounfettered markets.9

Because more people means more land devotedto human habitat and less land devoted to habitatfor the millions of other species, Simon’s forecastsalso downplay the amenity value of nature. Peo-ple place positive value on many environmentalamenities (and negative value on disamenities)which are not easily traded in markets. The trickypart is placing an actual dollar value on theseamenities. Economists interested in quantifyinguse and non-use values for nature’s goods employa variety of measurement techniques such astravel-cost method, hedonics and contingent valu-ation method. The validity and applicability of

7 The precipitation of violent civil or international conflictcaused by renewable resource scarcity is another concern forresearchers of environmental issues (Homer-Dixon, 1994).

8 Carrying capacity is often defined as ‘‘the maximum popu-lation of a given species that can be supported indefinitely in adefined habitat without permanently impairing the productiv-ity of that habitat’’ (Rees, 1994).

9 The history of Easter Island provides a good example ofthe catastrophic social effects of exhausting a renewable re-source when the availability of adequate substitutes is severelyrestricted (Ponting, 1991).

6 Recognizing this pedagogical difference betweeneconomists and natural scientists, Costanza et al. (1997) quan-tify the monetary value of the services of ecological systemsand the natural capital stocks that produce them. The currenteconomic value of 17 ecosystem services for 16 biomes isestimated to be in the range of $16–54 trillion, with anaverage of $33 trillion/year. This compares with a world grossnational product of $18 trillion per year (1997 $).


the economic values obtained using these valua-tion techniques are sometimes suspect but thepremise holds: people value environmental re-sources, especially when few satisfactory substi-tutes exist.

3. Endogenous growth theory and population

Starting with the seminal work of Solow (1957),substantial economics literature, catalogued underthe title of ‘‘endogenous technological-growth the-ory’’, has sought to explain economic growth.10 Inthese models, physical (or human-made) capital,human capital (education) and the latest produc-tion technology are combined to generate outputwhich can be re-invested in human-capital (saved)or consumed. While conventional endogenousgrowth models track investment in physical andhuman capital stock, few models explicitly incor-porate the biological growth/stock relationship ofa renewable resource:11 the supply of natural re-sources is considered limitless. This assumption ischallenged and the role of natural capital in theproduction process elevated while discounting theexplicit importance of human-made capital.12 Re-quiring the production process to adhere to thebehavior of a natural system such as the fishery orforest subjects the economy to a biologicalconstraint.

Essentially all endogenous growth models ofknowledge predict that technological progress isan increasing function of population size. (Romer,1996). The larger the population, the more peoplethere are to make discoveries and, thus, the more

rapidly knowledge accumulates. Alternative inter-pretations hold that over almost all of humanhistory, technological progress has led mainly toincreases in population rather than increases inoutput per person (Kremer, 1993).13

Neglected in these stories is the impact of grow-ing populations on natural resource stocks. Moreefficient harvesting of natural capital along with agrowing human population has had a substantialeffect on reducing the resource stock of theworld’s major fisheries and rainforests (Dietz andRosa, 1994; Brown, 1995; Resources, 1996).Economists who believe population growth isconducive to economic growth cite data showingthat in most parts of the world, food productionand per-capita gross income have generally grownsince the end of World War II (Dasgupta, 1995a).These figures ignore the depletion of the naturalresource base.

4. Economic growth versus economic development

‘‘The general proposition that economicgrowth is good for the environment has beenjustified by the claim that there exists an empir-ical relationship between per-capita income andsome measures of environmental quality. …asincome goes up there is increasing environmen-tal degradation up to a point, after which envi-ronmental quality improves (The relation hasan ‘inverted-U’ shape)… in the earlier stages ofeconomic development, increased pollution isregarded as an acceptable side effect of eco-nomic growth. However, when a country hasattained a sufficiently high standard of living,people give greater attention to environmentalamenities.’’ (Arrow et al., 1995)

The inverted U-shaped curve has been shownto apply to a select set of pollutants (as quoted inArrow et al. (1995)). Some economists have sug-

10 Other than Arrow’s ‘learning-by-doing’ paper from 1962(Arrow, 1962), economic growth theory garnered little atten-tion until rejuvenated by work by Lucas (1988), Stokey (1988),Barro (1991) and Romer (1986, 1987). Sparked by the seminalwork by Barro (1991), there has been an explosion in empiricalwork on economic growth by authors such as Jones (1995),Sala-i-Martin (1996, 1997) and Sachs and Warner (1997).

11 This quadratic relationship between a biological stockand its instantaneous growth rate is often represented with theSchaefer curve (Gordon, 1954; Schaefer, 1957; Clark, 1990).

12 Daly (1991), Daly and Cobb (1994), Arrow et al. (1995),and Dasgupta (1995b).

13 Kremer (1993) concludes that endogenous technologicalmodels predict that over most of human history, the rate ofpopulation growth should have been rising.


gested that the notion of people spending propor-tionately more of rising incomes on environmen-tal quality also applies to the natural resourcebase. While economic growth is associated withimprovements in some environmental indicators,it is not a sufficient condition for general environ-mental improvements nor is it a sufficient condi-tion for the earth supporting indefinite economicgrowth. In fact, if this base were severely de-graded, economic growth itself could be jeopar-dized (Arrow et al., 1995).

Although GDP is commonly used as a measureof economic health and even personal well being,it is far from adequate as a measure of trueeconomic performance.14 Numerous authors pointout the need for a more comprehensive index thatincludes the flow of environmental services as wellas the value of net changes in the stocks ofnatural capital so that the true social costs ofgrowth are internalized. A better indicator wouldameliorate the conflict between growth and theenvironment by subtracting environmental deple-tion from the index. The United Nations hasrecommended accounting for depletion in ‘satel-lite accounts’ as opposed to including depletion inthe main tables. Repetto et al. (1989) have arguedforcefully against this approach as it continues theuse of misleading economic indicators, and thesesame authors have offered methodologies to im-prove national accounting.

Net domestic product (NDP), calculated bydeducting environmental degradation of the re-source base from GDP, is a more inclusive mea-sure of personal well-being today and tomorrow(as quoted by Dasgupta (1995b)).15 That NDP is

a better measure of the improvement in societalwell-being is similar to the claim of Daly andCobb (1994) that economic development is a bet-ter measure of well being than economic growth.(These authors also highlight the importance ofsustainability, but sustainable issues are not di-rectly addressed in this paper.16 Repetto et al.(1989) indicate that only by including environ-mental depletion into the national accounts will‘‘…economic policy be influenced toward sustain-ability’’. (p.12)) The analysis is concerned with anoptimal growth path that includes the unhar-vested natural resource stocks in the welfare func-tion, and it is, therefore, consistent with usingsome form of NDP.

The model herein allows the social planner tomake perfectly timed, precise adjustments to theharvest and human capital investment rates thatkeep the bionomic system at, or returns it to, asteady-state equilibrium along the balancedgrowth path. Assuming the social planner hasperfect information in such decisions is unrealis-tic. As pointed out by Stiglitz (1996), informa-tional uncertainty pervades economies makingidentification of and adherence to a balancedgrowth path and overly optimistic outcome. In-cluding a time lag to represent the reaction andadjustment process associated with data accumu-lation and analysis, or an uncertainty parameterto represent the inherent lack of analytical preci-sion would add realism to the model. Either mod-ification would almost certainly provide resultsdifferent from those presented here. Time lags anduncertainty may prevent adherence to the bal-anced growth path and possibly even precludemaintenance of a steady-state equilibrium. If suffi-ciently severe, the bionomic system could tendtoward a socially undesirably corner solution withcomplete exhaustion of the natural resource. Ex-ploration of these issues is left for furtherresearch.

14 The Bush Administration acknowledged this in the Coun-cil on Environmental Quality Annual Report (1992). ‘‘Ac-counting systems used to estimate GDP do not reflectdepletion or degradation of the natural resources used toproduce goods and services’’. Consequently, the more thenation depletes its natural resources, the more the GDP goesup.

15 El Serafy (1997) also recognizes the importance of green-ing the national accounts (particularly of developing countries)to include the flows from natural capital: ‘‘…it seems ratherelementary that economists should pay attention in their anal-ysis to the sustainability of natural resources’’. (1997 p. 219)

16 The model captures some notions of ‘weak sustainability’in that the welfare growth rate depends on the natural capitalstock as well as private good production.


5. Model

The social planner is responsible for maximiz-ing welfare of a representative agent from thepopulation of identical individuals. Welfare is afunction of the representative agent’s shares of therenewable natural resource stock and of the pri-vate good. The planner solves an optimal controlproblem with two state variables and two controlvariables. She chooses (i) a harvest effort rate thatdetermines the size of the renewable natural re-source stock; (ii) the number of people devoted tohuman capital production (education) that deter-mines the human capital stock, and the number ofpeople employed in private good production.

The private good, produced by combining thecurrent production technology with human capi-tal augmented labor and natural capital, is sharedequally among the population. Preferences fornatural resources and private good consumptionare captured by the relative size of the coefficientsin a Cobb–Douglas welfare function. Naturalcapital, therefore, enters the welfare function di-rectly as a stock and indirectly as an input used inthe production of the private good.

The model contains both rival and non-rivalknowledge. The choice of how many people todevote to human capital production determinesthe growth rate of non-rival technology. That is,devoting a greater proportion of labor to educa-tion can increase the human capital stock. Placinglabor in human capital production and applyingits skills to the existing technology creates thishuman capital. Patenting activity is used to repre-sent the growth rate in rival knowledge. Both thehuman capital and private good production func-tion are generalized Cobb–Douglas functions.The renewable natural resource is constrained bya biologic logistic growth function. The resourcemight be a forest that has value left standingincluding both use and non-use values. The natu-ral resource is characterized as a contestable pub-lic good in that the allocation to any individual isthe total stock divided by the population.

Finally, the welfare gap is used to determinehow well GDP tracks societal welfare. Defining ry

as economic growth measured by GDP growthand ru as economic development measured bywelfare growth, the welfare gap is DW=ru−ry..

The optimality conditions in Appendix Aprovide the basis for the STELLA simulations. Acomparative static analysis using STELLA is car-ried out, and the simulations show how per-capitaincome and welfare growth rates, the welfare gapand resource stock levels vary over time as thesocial planner adjusts the harvest rate and humancapital investment rate in order to keep the bio-nomic system on a balanced growth path follow-ing a transient, parametric perturbation. The fullmodel is presented in the Appendix A.

6. Results

The simulations require parameter values forthe social discount rate (r), the populationgrowth rate (m), and the growth rate of propri-etary technology (g). Selection of a social discountrate is contentious (Lind, 1982): here r is thelong-run real rate of growth in GDP in the USwhich is about 3% (see Kahn, 1998 p. 113). Ini-tially, it is assumed that the population grows atthe current average growth rate of world popula-tion (m) estimated at 1.5% (Reddy, 1996). The 3%growth rate of Fagerberg (1987) in patenting ac-tivity among OECD countries between 1974 and1983 is used as a proxy for technological growth(g).

The importance people place on consumptionrelative to natural amenities and the importanceof human capital relative to natural capital in theproduction function are captured by the coeffi-cients f and a, respectively. Initially, the relativeimportance of the two goods are weighted equallyin the welfare function (f=0.5), as are the rela-tive importance of two inputs in the productionfunction (a=0.5). The natural resource carryingcapacity (S), is unitized which allows all steady-state harvest and resource stock values to beexpressed as a percentage of S. The intrinsicbiological growth rate was calculated using thetime–growth–volume relationship for a stand ofDouglas Fir (Hartwick and Olewiler, 1986, 1986,p. 353). For a forest that is less than 70 years ofage, the average biological growth rate (g) is 25%.

Steady-state equilibrium values are determinedusing STELLA. These steady-state values include


Fig. 1. Base case steady-state equilibrium.

per-capita harvest effort (6*), total harvest effort(V*), total harvest (h*), resource stock (X*) andhuman capital growth rate (rA*). The asteriskdenotes steady-state equilibrium values. The vari-ables 6, V, h, and X are measured in terms of ‘S ’and described by the ‘catch-per-unit effort hy-pothesis’ (Hartwick and Olewiler, 1986).

Modeling Appendix A Eqs. (A.11), (A.13),(A.14), (A.16), (A.17), (A.18) and (A.19) inSTELLA allows investigation of the behavior ofthe rates of human capital growth, economicgrowth, economic development and the welfaregap along the balanced growth path in bothsteady-state and transient periods. Fig. 1, which isgenerated by STELLA, establishes the base casescenario where the system is initially in a steady-state equilibrium. Curves 1, 2, 3, 4 and 5 in Fig. 1show that if the initial resource stock equals thesteady-state stock, economic growth or GDP (ry),economic development (ru), the welfare gap (DW),the natural resource stock (X) and the humancapital growth rate (rA) start and remain at theirsteady-state levels. The resource stock initiallystarts at its steady-state level, X* and the socialplanner makes the requisite adjustments to keep

the bionomic system in a steady-state equilibriumalong the balanced growth path. To obtain Fig. 1,the social planner maximizes intertemporal wel-fare by choosing optimal steady-state harvestrates and human capital accumulation rates thatresult in an optimal steady-state stock of80.875%17 and a human capital accumulation rateof 4.50%. Consequently, per-capita GDP in-creases at an annual rate of 3.75% with per-capitawelfare growing at 0.5625% annually. A negativewelfare gap (DW *= −3.1875%) implies thatgiven the model’s parameters, per-capita GDPoverstates the rate at which people’s lot in life isimproving by 3.1875%.

Simulations identify the steady-state equi-librium, and they are also helpful in examininghow parametric changes from the status quo af-fect the transient and steady-state behavior of thesystem. Consider two cases in which: (1) prefer-ences for preserving the natural resource stock

17 This refers to the percentage of the carrying capacity, S,of the biological resource. This steady-state stock is right ofmaximum sustained yield, which is the ‘stable’ half of thebiological growth function.


Fig. 2. Effect of step-change in preference for natural resource (f) at t=10.

and (2) population growth change over a discretetime period. In the first case, the relative weightplaced on the private good in the welfare function(1−f) is decreased while the weight placed onthe natural resource stock (f) is increased. Spe-cifically, f is increased at an annual rate of 1%over the time period t=10–40. The increasingweight on the natural resource stock reflects the‘inverted-U’ hypothesis discussed above whichmaintains that as a society enjoys economic pros-perity, it will place more emphasis on the environ-ment. In the second case, the current populationgrowth rate (m) is increased at an annual rate of1% over the time period t=10–40, after which itremains at its new level for the remaining analysisperiod.

Fig. 2 pertains to case (1). The system is ini-tially at the steady-state equilibrium shown in Fig.1. At t=10, the preference for natural resourcesbegins to increase at an annual rate of 1%. Asexpected, the natural resource stock begins toclimb owing to lower harvests. To compensate forthe lower harvest rate, the social planner increasesthe rate of human capital accumulation (rA rises).This trend continues until the point where, given

the decreased flow from the natural capital stock,the marginal returns from additional human capi-tal investment begin to decline at approximatelyt=25 (ry falls). At this point, lower harvest levelsencourage a decrease in the human capital invest-ment rate. At t=40, the preference for naturalresources stops increasing and holds steady atapproximately f=0.67. The resource stock in-creases to a new steady-state level of 90.73% withthe human capital growth rate eventually return-ing to its initial steady-state level of 4.5%.

During this time when the preference for natu-ral resources is increasing, the social planner ad-justs the harvest rate and human capitalaccumulation rate to remain on the balanced-growth path. In turn, this adjustment processaffects GDP and welfare growth rates (ry and ru).Initially in a steady-state equilibrium, GDPgrowth rate begins to climb in conjunction withthe rise in the human capital growth rate.18 How-ever, as the growth rate in human capital falls, theGDP growth rate eventually returns to its pre-per-turbation steady-state level.19 The transient and

18 See Eq. (A.13).19 See Eq. (A.14).


Fig. 3. Effect of step-change increase in population growth rate (m) at t=10.

post-perturbation behavior of the growth rate inwelfare differs significantly from the growth ratein GDP. Initially in steady-state at approximately0.56%, the welfare growth rate first climbs inresponse to the increase in the resource stock andGDP growth rate20 reaching a maximum of about0.63%. As the rate of increase in GDP growth ratebegins to diminish, the welfare growth rate beginsto fall in conjunction with the increase in prefer-ences for the natural resource. This is attributableto natural resource preferences increasing fasterthan the stock of natural resources, so that eventhough the resource stock and its relative weightin the welfare function is increasing, the net effectcannot overcome an increasing GDP growth ratewhose relative weight is declining. As seen in Eq.(A.14), because the steady-state GDP growth ratereturns to its pre-perturbation level but is nowweighted less in the welfare function, the newsteady-state value for the welfare growth rate isless than its pre-perturbation level. Thus, eventhough the natural resource stock and its relativeweight in the welfare function are larger, the net

effect is a lower steady-state welfare growth rate.The welfare gap, as measured by DW, initially at−3.1875%, continues to widen as the preferencefor natural resources increases. When the welfareweight placed on natural resources reaches a newsteady-state at 0.675 at t=40, the widening trendceases and the steady-state welfare gap increasesslightly to −3.65%. The new steady-state welfaregap is wider than its pre-perturbation level. Thesechanges capture the notion that as preference forthe natural resource increases relative to GDP,the welfare gap widens.

In Fig. 3, system sensitivity is examined to anannual 1% increase in the population growth ratefrom its original steady-state value of 1.5% att=10 to a new-steady-state value of 2.02% att=40. When the population growth rate begins toincrease, the added population results in an in-crease in the number of people involved in humancapital production, private good production andresource harvesting. The increase in the steady-state human capital growth rate from 4.50 to5.02% is wholly a result of the increased popula-tion growth rate with none of the increase due toa change in the percentage of labor working in20 See Eq. (A.16).


private good production and that devoted to hu-man capital production.21 Even after the popula-tion growth rate levels off at 2.02% at t=40, thehuman capital growth rate does not reach its newsteady-state equilibrium until t=70. It takes thesystem time to accommodate the changes insti-gated by the parameter change.

The natural resource stock sees its steady-statestock decline from 80.875 to 77.857%. Reflectiveof the increased demand for private good concur-rent with a larger population, total output in-creases as the increased human capital stockrequires additional natural resources. The addedpopulation each period results in an increase intotal consumption, which requires an increase inthe steady-state harvest rate and a decline in thenatural resource stock to a lower steady-stateequilibrium.22

The decline in resource stock that begins att=10 results in the income growth rate initiallydeclining until the human capital growth rateincreases to compensate. The rapid accumulationof human capital stock leads to an increase in theincome growth rate until t=40. As the popula-tion growth rate stabilizes at 2.02%, the incomegrowth rate continues to increase but at a decreas-ing rate. In a sense, the social planner, recognizingthat the perturbation has ended, is no longerrequired to take the drastic steps necessary tokeep the system on the balanced growth path. Thenew steady-state per-capita income growth rate isnow at a higher level as a result of the highersteady-state population growth rate.23

As a result of the initial decline in both theresource stock and income growth rate, and theneed to now share the stock and income withmore people, the welfare growth rate witnesses aprecipitous decline at t=10. Even after the in-come growth rate recovers and begins to increase,

the protracted decline in the per-capita share ofresource stock results in a continued decline inwelfare growth reaching a low point of 0.428% att=40. Heretofore, the stock, declining at an in-creasing rate, continues to decline after t=40 butat a decreasing rate.24 The net result of thischange is a partial recovery for the welfare growthrate from its minimum of 0.428% to a new steady-state equilibrium of 0.497%. The new steady-statewelfare growth rate of 0.497% is lower than itspre-perturbation level of 0.5625%.

The gap between the growth rates in GDP andwelfare widens as the growth rate in populationincreases. That is, the faster population multiplies,the greater the divergence between per-capitaGDP and welfare growth rates. The reason forthis divergence is primarily the reduced naturalresource stock. As mentioned before, the stockdeclines from 80.875 to 77.857% while concur-rently the number of people who share it in-creases. The net effect is less public goodper-capita, which leads to a decline in the welfaregrowth rate and an increase in the welfare gap.

7. Conclusion

Standard methods employed to solve dynamicproblems such as those seen in the endogenousgrowth literature are often unable to accommo-date modifications such as the parabolic behaviorof renewable resources. The clever method ofRomer (1990) of identifying the balanced growthpath was not, by itself, sufficient to solve ourproblem of accounting for the behavior of a non-linear, biological growth function. The simulationsoftware allowed the retention of the most inter-esting characteristics of the problem while alsoproviding a tractable solution. It is felt that simu-lations can also be employed to provide insightsand possibly help solve problems that heretoforehave been beyond the scope of traditional solu-tion methods.

Using simulations, the growth rate expressionsfor GDP and welfare are used to examine why a

21 This can be seen by comparing the percentage changesbetween m and rA.

22 This is different from the situation depicted in Fig. 2where the increase in f eventually led to diminishing returnsto additional human capital investment which resulted in ryreturning to its original steady-state level.

23 See Eq. (A.14). 24 See curve 4 in Fig. 3.


commonly used indicator such as per-capita GDPis a poor proxy for per-capita welfare owing tothe welfare gap. In steady-state, the per-capitaincome growth rate exceeds the per-capita welfaregrowth rate.

Simulations generated with STELLA were em-ployed to examine how the variables in the modelare affected by a slight but persistent parametricperturbation. This exercise helps to illustrate howthe social planner adjusts the harvest rate andhuman capital accumulation rate in response tothe parametric perturbation. The parameter thatchanges influences the size of DW *. However, asshown above, the effect on ry and ru is not alwayspermanent, illustrating that some exogenousparameter shifts cause only transient changes.Once the perturbation ceases, the income or wel-fare growth rate may return to its original steady-state levels.

These results have important policy implica-tions. For example, a policy that encourageshigher population growth does increase per-capitaincome; however, the policy also diminishes wel-fare growth resulting in an increase in the welfaregap. As expected, a policy designed to improvetechnology will have the intended effect of nar-rowing the welfare gap. And policymakers indeveloping countries may question the ‘inverted-U’ relationship between income growth and envi-ronmental quality; while an increase in preferencefor the natural resource does increase the naturalresource stock, it leads to a decline in the steady-state welfare rate.

Growth theoretic models are highly abstract,extremely aggregated representations of the realworld. Nevertheless, empirical applications ofgrowth models are used by economists in attemptsto explain the wide divergences in living standardsacross nations, and why some countries seem tobe making the transition from less to more devel-oped, while other countries remain in poverty.Absent from these models is any recognition thateconomies function in and depend on a biologicalworld that places limits on physical growth. WhenSolow first introduced his growth model in the1950s, these biological constraints may haveseemed more remote because the world’s popula-tion was half of what it is today. The results

suggest that models which continue to ignore thebiological constraints are desperately incomplete;they will yield misleading guidance owing to theirfocus on income growth instead of a more com-plete welfare measure.

Appendix A

The per-capita growth expressions for incomeand welfare are derived from a one-region en-dogenous technological growth model that is con-strained by a natural capital production function.Welfare for a representative individual is based onher consumption of a public and private good. Allpeople in the region (m+n) share a private good,Y, produced using the harvested renewable re-source. The individual also derives welfare fromthat portion of renewable resource that remainsunharvested, X. The renewable resource repre-sents a public good such as a forest or fishery andis shared among everyone in the region (m+n).

In the instantaneous welfare function, (A.1), h

determines the consumer’s attitude toward in-tertemporal consumption. The smaller is h, themore slowly marginal welfare falls with a rise inconsumption. It is assumed that h=0.5 in themodel. The relative weight that one places on theunharvested resource stock (private good) is rep-resented by f (1−f).

U� X

m+n,

Ym+n

�=

11−h

� � Xm+n

�f� Ym+n

�1−fn1−h

(A.1)

The goal is to maximize intertemporal welfareby identifying the optimal per-capita consumptionpath for the resource and private good. Welfare isconstrained by a biological growth function andhuman capital stock accumulation relationship.

Maximize

6.HA

&�0

11−h

�� Xm+n

�f� Ym+n

�1−fn1−h

e−rt dt

subject to X: =gX(1−X/S)−6mX

A: =sAHA


Y= (A(n−HA))a(6mX)1−a(G)a−1

and A(0)=A0 6mX(0)= (6mX)0

The relative importance of the productive in-puts is captured in the Cobb–Douglas productioncoefficient (a). The larger is a, the more importantis labor in the production process and the lessimportant is natural capital. The subjective dis-count weight on consumption is r. The constrainton the renewable resource stock is represented bythe biological growth relationship F(X)=gX(1−X/S) where g depicts the intrinsic biologicalgrowth rate and S the carrying capacity of theresource (Hartwick and Olewiler, 1986). The hu-man capital constraint, A: , is a function of educa-tional success, s, the current stock of humancapital, A, and the number of people involved inhuman capital production, HA.

A frequently used assumption in endogenousgrowth models is to have population growing at aconstant rate (Romer, 1996). All population termsare assumed to grow at the constant rate m (m=m emt, n=n emt, m+n= (m+n) emt, and HA=HA emt). Also, proprietary production technologythat converts natural capital to a durable good ofany design grows at a constant rate, g (G=G egt),where G represents the current state of technologyby denoting the amount of natural capital neces-sary to produce one unit of private good. Becausethe exponent on G in the production function forY is negative, g must be negative in order torepresent improving production technology withlarge negative values representing relatively bettertechnology. An increase in g represents a reduc-tion in (or worsening of) the growth rate ofproduction technology; it takes more natural cap-ital to produce one unit of private good. Makingthese substitutions, m, n, HA, and G representinitial condition levels and the problem becomes

Maximize

6.HA

&�0

11−h

�� Xm+n

�f � Ym+n

�1−fn1−h

e−Ct dt

subject to X: =F(X)−6m emtX

A: =sAHA emt

and A(0)=A0 6mX(0)= (6mX)0

where C=r− (1−h)(1−a)(1−f)g

+ (1−h)fm.

The current-value Hamiltonian can be ex-pressed as:

He=11

1−h

�� Xm+n

�f�(A(n−HA))a(6mX)1−aGa−1

m+n�1−fn1−h

+lX(gX(1−X/S)−6m emtX)

+lA(sAHA emt) (A.2)

There are four first-order conditions: one forthe control variable, 6, ((Hc/(6)=0, and its corre-sponding state variable, X, l: X= − ((Hc/(X)+ClX ; and one for the control variable, HA,((Hc/(HA)=0, and its state variable, A, l: A= −((Hc/(A)+ClA. Solving the first-order conditionfor the control variable 6 in terms of lX, theshadow price of the resource, gives

lX=(1−f)(1−a)6m emtX

�� Xm+n

�f� Ym+n

�1−fn1−h

(A.3)

The first-order condition for X is

l: X= −�� X

m+n�f� Y

m+n�1−fn

1−h�f+ (1−a)(1−f)X

�+lX(C−F %+6m emt) (A.4)

Dividing (A.4) by (A.3) provides

l: XlX

=C−F %−f

(1−a)(1−f)6m emt (A.5)

Solving the first-order condition for the controlvariable HA in terms of lA, the shadow price ofhuman capital investment, provides

lA=a(1−f)sAHY emt

�� Xm+n

�f� Ym+n

�1−fn1−h

(A.6)

Differentiating (A.6) with respect to time pro-vides the following expression.


l: A=a(1−f)sAHY emt

�� Xm+n

�f� Ym+n

�1−fn1−h

ÃÃ

Ã

Æ

È

(1−h)(f+ (1−a)(1−f))X:X

+ (a(1−f)(1−h)−1)A:A

−m

ÃÃ

Ã

Ç

É(A.7)

Dividing (A.7) by (A.6) gives

l: AlA

= (1−h)(f+ (1−a)(1−f))X:X

+ (a(1−f)(1−h)−1)A:A

−m (A.8)

Solving the system explicitly for its dynamics iscomplex. To simplify the analysis, the discussionis focused on the properties of the balancedgrowth equilibrium inherent in the model. Eq.(A.9) illustrates a basic feature of the steady-stateequilibrium: along the balanced growth path, thegrowth rate of the human capital stock (rA=A: /A) equals the sum of the growth rates of naturalcapital, population and proprietary productiontechnology.

rA=X:X

+m+g (A.9)

As Romer shows in his paper (1990), in steady-state, l: A/lA=l: X/lX. Equating (A.5) and (A.8)and using (A.9) provides the steady-state expres-sions for the optimal harvest effort level (V) andthe optimal human capital growth rate (rA) as afunction of the various system parameters, Cobb–Douglas production and welfare coefficients andresource stock level (see A.10 and A.12).

With the resource stock variable X still present,Eq. (A.10) and Eq. (A.11) do not accurately rep-resent the first-order conditions for V and HA.

The expression for the biological growth functioncould be used to eliminate X from (A.10) and(A.11). However, solving this quadratic equationfor X requires assumptions that detract from theissues of interest.25 For this reason, dynamic sim-ulations are used to determine the optimal steady-state equilibrium for the bionomic system.

The expressions for per-capita income and wel-fare growth rates are obtained from (A.1). In Eq.(A.12), y represents the equal income share to allagents in the region.

y=Y

m+n=

(A(n−HA)a(6mX)1−aGa−1)m+n

(A.12)

Totally differentiating (A.12) with respect totime, (A.13) represents the general expression forthe growth rate in per-capita income along thebalanced growth path

r y;y=arA+ (1−a)

�X:X

+gn

(A.13)

with the steady-state growth rate of per-capitaincome represented by (A.14)

ry*=am+g (A.14)

where the ‘*’ represents a steady-state equilibriumvalue.26

Knowing that per-capita welfare is representedby

V=6m emt=

�r+ (1− (1−h)(1−f))g+ (2+ (1−h)(f−a(1−f)))m−F %+h

FXn

h+f

(1−a)(1−f)

(A.10)

rA=FX

+g+m−

�r+ (1− (1−h)(1−f))g+ (2+ (1−h)(f−a(1−f)))m−F %+h

FXn

h+f

(1−a)(1−f)

(A.11)

25 For example, assuming that is valid for steady-state con-ditions but not accurate for transient periods when X ischanging.

26 At steady-state.


u=1

1−h

�� Xm+n

�f� Ym+n

�1−fn1−h

=1

1−h

�� Xm+n

�f

(y)1−fn1−h

(A.15)

the growth rate expression for per-capita welfareis

ru= (1−h)�

8�X:

X−m

�+ (1−f)ry

n(A.16)

In steady-state, (A.16) becomes

ru*= (1−h)[(1−f)ry*−8m ] (A.17)

The difference between ru and ry is the welfaregap, DW (DW=ru−ry). Using (A.13) and (A.16),the general welfare gap along a balanced growthpath can be written as

DW=f�X:

X−rA−am

n(A.18)

with (A.19) representing the welfare gap insteady-state.

DW*= −f [rA* +am ] (A.19)

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