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THREE ESSAYS ONSUSTAINABLE DEVELOPMENT

by

ANDRES GOMEZ-LOBOKIRK HAMILTON

andCARLOS EDUARDO YOUNG

CSERGE WORKING PAPER GEC 93-08



Three Essays onSustainable Development

by

Andres Gomez-LoboKirk Hamilton

andCarlos Eduardo Young

The Centre for Social and Economic Researchon the Global EnvironmentUniversity College London

andUniversity of East Anglia

Acknowledgements:

The Centre for Social and Economic Research on the Global Environment (CSERGE) is adesignated research centre of the U.K. Economic and Social Research Council (ESRC)

ISSN 0967-8875



Abstract

This is a collection of three Papers on the economic issues encapsulated within the debate

surrounding the definition of sustainable development.

The first essay examines one issue in the debate concerned with the correct formulation of

sustainable income when resources are being depleted: in particular the treatment of outflows

due to debt service. It concludes that such outflows should be deducted in order to arrive at a

NNP measure that is more intuitively sustainable.

The second essay examines the difference between the concepts of optimal growth and

sustainable development. The concepts diverge when the discounting of future utility and otherissues are considered.

The third essay examines the implications of incorporating a sustainability constraint into a

macro-economic model of an economy, where optimal growth is defined in terms of full

employment of labour. In combination, the essays take further the debate over whether, and in

what form, trade-offs exist, between economic growth and sustainable development.



ESSAY 1

SUSTAINABLE DEVELOPMENT,

OPTIMAL GROWTH AND NATURAL RESOURCE ACCOUNTING

IN A SMALL OPEN ECONOMY

Andres Gomez-Lobo E.1

1. INTRODUCTION

Contrary to common belief, optimal economic growth and sustainable develop-ment are not

different nor antagonistic concepts. Sustainable development is just one type of "optimal"

growth, namely one where the rule by which to judge an "optimal" from a "non-optimal" path is

based on a particular, and strong, view of intergenerational equity.

It is important to establish what is meant by sustainable development and optimal economic

growth. Development is not synonymous with growth but is a much wider and encompassing

concept. To avoid this issue, in this paper sustainable development is understood to mean

sustainable growth, and sustainable growth is a growth path that allows future generations to

enjoy at least as much welfare as the current generation. In other words, a growth path is

sustainable if it does not entail future decreases in welfare.

To define optimal growth an objective function that ranks different growth paths is required. In

neoclassical growth theory this function is usually a present value utility function that weighs

utility levels at different moments in time by a discount factor. Since the discount factor becomes

smaller for periods further into the future it is possible to have large welfare costs in the distant

future that, once discounted, are not very significant. This possibility violates the ethical rule of

intergenerational equity and is the reason why sustainable growth has gained so much attention

lately.

However, the present value utility function is not the only possible objective function to use in

1 I would like to thank David Pearce, M. Pemberton, Kirk Hamilton, Carlos Young and Edward Barbier for their

excellent comments. I would also like to thank the Ford Foundation and the British Council for their financial

assistance.



calculating an optimal growth path. If instead, the objective function is of a Max-Min type where

the objective is to maximize the utility of the least well off generation then the "optimal" growth

path will be sustainable. In that case all generations should have the same consumption level

per capita. Solow (1974) shows that such a path exists even without technological change in a

simple model with an exhaustible resource and sufficient substitution possibilities between

reproducible capital and natural resources. The need to worry about intergenerational equity

arises when there is an exhaustible resource that the current generation may be using too fast.

Hartwick (1977) showed that, in a model with similar assumptions to Solow (1974), investing the

rents accruing from resource extraction in other types of capital is sufficient to maintain a

constant consumption path.

A key concept for understanding sustainable development is "economic income". The Hicksiannotion of income is the maximum amount an agent (or country in this case) can consume in the

present period and expect to consume the same amount in the future. Income is then

sustainable by definition. Therefore the relevant question when assessing the sustainability of a

growth path is whether economic growth is measuring true income growth.

The answer is no. The way the current system of national accounts works in practice does not

guarantee that aggregate economic indicators are true Hicksian income figures. One important

bias arises by the fact that variations in natural capital stocks are usually accounted for as

income instead of asset changes. Therefore a high growth figure may be hiding an underlying

process of asset reduction and may not be measuring true income. It is this fact that causes the

apparent paradox that national economic figures may show positive growth but which may not

be sustainable in the middle to long run. The important thing to notice is that the real problem is

the erroneous way in which economic growth is being measured and not a difference in

paradigm between optimal growth and sustainable development.

2. THE MEASUREMENT OF SUSTAINABLE GROWTH

The effort to correct the national accounts, known as natural resources accounting, has been a

lively research area in the last few years. There have been two basic methodologies to go about

this, the depreciation method (Repetto, et.al, 1989), and the user cost approach (El Serafy,

1989).

In this paper I will try to show the reasons why both methods differ and also extend the



depreciation method to a small open economy.

The depreciation method basically says that one should deduct from GDP an allowance for

natural capital depreciation equal to the rents obtained from the exploitation of natural

resources2. The rationale for this approach is Hartwick's rule, which states that to maintain a

constant consumption path one should reinvest in reproducible capital all rents accruing from

the extraction of natural resources. The reason for this is that rents reflect the asset value of the

resource and therefore is a measure of the asset loss when one unit of a natural resource is

drawn down3.

Hartwick's rule was developed in a closed economy context. This is reflected by the exclusion of

foreign trade and foreign assets. In a comment to Solow (1986), Svensson notes that

"For an open economy, intertemporal trade, i.e., borrowing and lending, implies thatconsumption need not equal output of the consumption good at each point in time.In particular, for a small open economy, the interest rate is given by the world capitalmarket, and consumption and investment decisions become separable. A smallopen economy should simply choose investment so as to maximize wealth. If thereis a social preference for a constant consumption path, it can simply be chosensubject to an intertemporal budget constraint, and is otherwise completelyindependent of the specific investment and production path. These circumstancescombined make me believe that Hartwick's rule, although a very neat theoretical

result, is of limited interest for discussing intergenerational equity in small openeconomies ."

I will try to show that in a small open economy Hartwick's rule of reinvesting all resource rents is

still valid. However, an allowance has to be made for the change in foreign assets.

3. HARTWICK’S RULE IN A SMALL OPEN ECONOMY? 4

The approach of this section is based on Solow (1986) and Hartwick (1990). They use a result

due to Weitzman (1976) which states that the Net National Product is equal to the current value

Hamiltonian of the corresponding optimal growth problem. In an economy with natural resources

2 In theory, even the strong sustainability school of thought may be included in this argument since nonsubstitutable

natural capital will have a marginal productivity tending towards infinity. Therefore any growth path that draws down

these resources will be unsustainable because the depreciation allowance will be infinite.

3 As Hartwick has correctly pointed out, rent should be defined as the price of the resource minus the extraction costs of

the marginal unit. The use of an average rent (price minus average extraction costs) in most empirical work probably

causes an overestimation of the depreciation if there are increasing marginal costs of extraction.

4

Asheim (1986) shows that Hartwick's rule does not hold in an open economy because, due to resource depletion, theterms of trade will improve for future generations. In contrast to Asheim (1986), in the present model the country is

assumed to be small and thus is unable to affect its terms of trade.



this Net National Product will be equal to GDP minus the rents from natural resource extraction5.

Solow (1986) shows that if Hartwick's rule is imposed on such an economy then consumption

will be constant. Therefore, NNP is the income measure we are seeking.

The extension to an open economy is straight forward. The maximization problem is,

(1) dt t C U e Max t )((

0

δ −∞

∫

St: (2) rAS R f I C RK Y rA M X A +−−−=+−= ),(),(&

(3) I K =&

(4) RS −=&

where A is the stock of foreign assets, X is exports, M is imports, r is the international interest

rate (exogenous for this small economy), Y(K,R) is the production function of the composite

good with the capital stock, K, and resource extraction, R, as arguments (labour is assumed

fixed), C is the aggregate consumption, I is investment, f(R,S) is the cost of extraction

(measured in terms of the composite good), S is the stock of the natural resource, and δ is the

discount rate6.

The current value Hamiltonian for this problem is,

and the corresponding first order conditions are,

5 For simplicity we assume that the man-made capital stock does not depreciate. If this was not so the depreciation of

this stock would also have to be deducted from GDP to arrive at a NNP figure.

6 Time subscripts have been omitted to simplify notation. However, except for the discount rate and the international

interest rate, each variable above is a function of time.

(5) Η = U (C )+λ1[I ]+λ2[-R ]+λ3[Y (K ,R )-C -I -f (R ,S )+rA]



Substituting (6), (7) and (8) into (5) yields the value of the Hamiltonian in an optimal path

expressed in utility units,

Using a linear approximation to utility, U(C)=UcC, and dividing (12) by Uc we get a monetary

value of the Hamiltonian in an optimal path, or the Net National Product,

This result is similar to the closed economy case. To derive NNP, resource rents have to be

deducted from GDP, but, in addition, the interest earned on the foreign asset stock has to be

added7. In most developing countries, where the small open economy hypothesis is relevant, A

will be negative due to net foreign indebtedness. Thus, to arrive at NNP foreign debt services

have to be deducted from GDP.

7 The adjustment to GDP for the change in foreign assets is limited to the interest earned on these assets only. The

other components of the change in foreign assets, exports minus imports, are already accounted for in the traditional

measure of GDP.

(6)¶Η¶I

= λ1-λ3 = 0

(7)¶Η¶R

= -λ2+λ3(Y R -f R ) = 0

(8)¶Η¶C

= U c -λ3 = 0

(9) λ1 = δλ1-λ3Y K

(10) λ2 = δλ2+λ3f S

(11) λ3 = δλ3-λ3r

(12) Η = U (C )+U c I -U c (Y R -f R )R +U c (Y (K ,R )-C -I -f (R ,S )+rA)

(13)Η

U c = NNP = C +I -(Y R -f R )R +A

(14) NNP = C +I -(Y R -f R )R +X -M +rA

(15) NNP = GDP -(Y R -f R )R +rA



This result is intuitively simple and is not new. Solow (1986) expresses Hartwick's rule in a

setting with multiple types of capital as p(t)(δK/ δt)=0, where p(t) is a vector of shadow prices for

investment goods and (δK/ δt) is a vector in which each element is the change in one type of

capital. Since foreign assets are just another type of capital, the change in this asset has to be

accounted for. Since the traditional measure of GDP already includes foreign trade, only a

correction for interest earnings or payments has to done.

To anyone familiar with national accounting concepts it will be immediately clear that the above

procedure is equivalent to deducting resource rents from GNP, since this last figure includes

foreign assets interest payments. In fact Hartwick (1990) correctly considers GNP as the

measure that has to be modified to arrive at NNP. However, in this paper the role of foreignassets is stressed for several reasons.

In the first place, empirical studies such as Repetto, et.al (1989) use GDP instead of GNP,

possibly because GNP figures are not always calculated as often as GDP by the statistical

offices of developing countries. In many developing countries severely affected by the foreign

debt crisis during the eighties, the change in foreign assets is probably a more important

determinant of sustaina-bility than the loss of natural resource assets. Therefore, it is important

to stress that the change in foreign assets must be explicitly accounted for when deriving

sustainability indicators for a developing economy.

More importantly, by formulating the role of foreign assets explicitly in the above optimization

problem, some insight is gained as to the difference between the depreciation method and the

user cost approach. This will be made clear in the next section.

Following Solow (1986) it can be shown that an extended Hartwick rule (invest in reproducible

capital an amount equal to the rents from resource extraction minus the change in foreign

assets) will ensure a constant consumption path for the small open economy case. This is

shown in the appendix of this paper.

The aim of natural resource accounting is to have a better indicator of National Income, in a

Hicksian sense, and therefore evaluate the sustainability of an economic growth process. As far

as I know, all empirical studies on natural resource accounting have ignored the change in

foreign assets. Therefore, an important element in determining whether an economy is on a



sustainable path has been omitted8.

3. THE DIFFERENT APPROACHES TO NATURAL RESOURCE ACCOUNTING

The result in the previous section shows that to arrive at the Net National Product all rents from

resource extraction have to be deducted from GDP (as well as interest earnings or payments of

foreign assets). This is consistent with the depreciation approach to natural resource accounting

as in Repetto, et al. (1989). However, El Serafy (1989) criticizes the depreciation approach and

instead proposes an alternative method whereby only a fraction of current rents are deducted

from GDP to arrive at a sustainable growth indicator.

El Serafy's main criticism to the depreciation approach is that because all rents are deductedfrom GDP a country with a large endowment of natural resources would not seem to have an

income (ie. permanent consumption) edge over other countries. This result would obviously be

flawed.

As an alternative, El Serafy (1989) proposes the following9,

where I is income, R(0) is the rent generated in the current period, n is the number of years that

the resource will last given a constant extraction equal to the current extraction, and r is an

exogenously given interest rate.

Equation (16) transforms a finite cash flow (from resource extraction rents) to an infinite income

flow. Since I and R(0) are constant we can integrate both sides of (16) and arrive at,

Equation (17) gives the proportion of current rents (if maintained for n years) that can be

transformed into income and thus consumed. Therefore, not all resource rents are asset

8 It is interesting to note that the inclusion of foreign assets not only has implications for Hartwick's rule for reinvesting

resource rents, but also for the optimal rate of resource depletion, particularly if the stock also has value. On this see

Barbier and Rauscher (forthcoming).

9 El Serafy (1989) derives his formula in discrete time. However, in this work it is derived in continuous time.

(16) ò¥

0e -rt Idt = ò

n

0e -rt R (0)dt

(17)I

R (0) = (1-

1

e rn )



depreciation but only a fraction (1-I/R(0)).

Before discussing El Serafy's approach we will pursue a point that was not mentioned in the

optimal control problem of section (b). Namely, that there is an additional restriction to that

optimal control problem that might shed some light as to the reason why El Serafy's method

differs from the results of that section.

What is the maximum constant sustainable consumption that an economy characterized by

equations (1) to (4) may enjoy? As Svensson mentions:"...it can simply be chosen subject to an

intertemporal budget constraint, and is otherwise completely independent of the specific

investment and production path".

The intertemporal budget constraint is given by the following condition10,

which states that in the long run a country cannot be a net lender or borrower in present value

terms. This implies that a country's foreign debt should not grow faster than the interest rate ithas to pay.

The first restriction of the optimization problem (equation (2)) was,

Integrating this differential equation yields,

and using the limit condition (18) yields11,

10

The following derivations follow Sachs (1982) very closely.11

For simplicity it is assumed that A(0)=0.

(18)limn ®¥

e -rt A = 0

(2) A = Y (K ,R )-C -I -f (R ,S )+rA

(19) A(t ) = A(0)e rt +e rt òt

o e -r ξ(Y (K (ξ),R (ξ))-C (ξ)-I (ξ)-f (ξ))d ξ



This last equation is the intertemporal budget constraint. It states that the present value of trade

deficits has to equal the present value of trade surpluses.

We can transform equation (20) into an alternative form that will be more useful. Let us define

the "permanent" or "perpetual equivalence" of a variable as,

Xp is the constant value of the variable that over the horizon will give the same present value as

the changing variable path X(τ)12.

Defining Yp(0), Cp(0), Ip(0), fp(0) as the permanent values of Y(t), C(t), I(t) and f(t) respectively,

the budget constraint after integrating is,

Therefore the maximum consumption feasible in this economy is the interest from the wealth

that is possible to generate with the initial endowments.

Now we are ready to discuss El Serafy's formula. Implicitly, his method is only valid for a small

open economy, because the interest rate is exogenous. In addition, the flavour of his argument

implies a small open economy since the intuition behind his result is that part of the resource

rents that can be invested in the international market and the return on these assets will be used

to finance consumption after the resource is exhausted.

From the previous discussion it should be clear that El Serafy's calculation of "consumable"

12 As an example, Weitzman (1976) showed that NNP at time t is the perpetuity equivalent of the optimal and variable

(20) ò¥

0e -rt (Y (t )-C (t )-I (t )-f (t ))dt = 0

ò¥

0e -r (τ-t )X p (t )d τ = ò

¥

t e -r (τ-t )X (τ)d τ

(21)C p (0)

r =

Y p (0)-I p (o )-f p (0)

r = Wealth

(22) C p (0) = rW



income, I, is just the perpetuity equivalent of the wealth generated by the natural resource rents.

This is what equation (16) does. Therefore El Serafy is correct in the sense that consumption is

a fraction of total wealth. However, in the open economy case consumption and investment

decisions are independent. Therefore the level of consumption is independent of the amount of

resource rents generated in the current period. The extraction path of the resource should be

such that wealth is maximized.

We have seen that in an efficient path to construct an NNP indicator we have to deduct all of

resource rents from GDP as well as any foreign debt interest payments. In the case where the

country is exploiting its resource base to built up foreign assets, rA will be positive. Therefore the

amount of investment necessary to maintain a constant consumption path is less than the total

resource rents of the period, confirming El Serafy's intuition. However, the correct way toaccount for such an effect is to deduct all resource rents and then incorporate the change in

foreign assets.

Since El Serafy (1989) does not postulate a behavioral model of resource extraction it is not

clear why rents are generated in his formulation in the first place13. Therefore it is not possible to

see whether the correcting factor derived from our optimization model, ((YR-fR)R+rA), would be

equal to the depreciation allowance of El Serafy method, (R(0)/ern). We do know however that in

an optimal growth path, where resource extraction is efficient, NNP is equal to GDP minus

resource rents and net interest payments.

The main thrust of El Serafy's criticism of the depreciation approach is no longer valid.

Deducting rents from GDP will not reduce NNP to zero since the current account will adjust to

make NNP equal to permanent consumption (if Hartwick's extended rule is imposed).

4. CONCLUSIONS

The conflict between sustainable development and optimal economic growth is really a problem

of how to measure growth. In this paper an approach to measuring national income for a small

open economy was discussed. The result shows that the two principal methods for accounting

consumption path.

13 His assumption of a constant rent stream R(0) for n periods is a highly unlikely outcome of any model in which

resource owners are somehow optimizing intertemporally. If they are not optimizing intertemporally then resource rents

should not arise in the first place.



natural resource depreciation are not valid in a small open economy context. However, the

depreciation approach would be the correct method if it is extended to include interest earnings

or payments on foreign assets.

There are many other topics in natural resource accounting that merit discussion. One

empirically relevant point is the correct way to account for discoveries. Hartwick (1990) includes

a discovery function in the optimal growth problem and concludes that the correct depreciation

measure is the value of the net change in assets (extraction minus discoveries).

It is unlikely that real world discoveries follow the deterministic smooth function that Hartwick

uses. In particular, some natural resource accounting exercises show that the variance of

proved or probable reserves is many times larger than the variance of GDP (see Young, 1992).Therefore the NNP figure will be very volatile and useless as an income or sustainability

indicator.

A different approach might be to assume discoveries to be unexpected. The optimal control

problem of equations (1) to (4) is solved subject to the initial stock of the resource. If discoveries

are unexpected (or in unexpectedly high discrete quantities) then the optimal control problem

would be solved again and a new optimal extraction path will arise.

The interesting aspect of this approach is that only resource extraction should be considered as

depreciation. No correction should be made for discoveries because the wealth enhancing

effect would be automatically captured through a rise in the conventional measure of GDP. In

other words, when there is an unexpected discovery wealth increases and this should induce an

increase either in consumption, investment or exports. For example, in the case where the

economy is maximizing a constant consumption path, then equation (19) states that

consumption should rise by a similar proportion as wealth. Then the only correction needed to

derive NNP is to subtract rents from the (new) extraction path.

One final point worth mentioning is that the models used to derive constant consumption paths

are extremely simplified versions of an economy and therefore care has to be taken when

extracting policy implications. In particular, if there is technological change then Hartwick's rule

is too conservative as a way of guaranteeing intergenerational equity. Hartwick's rule, more than

a precise theoretical result, should be viewed as a rule of thumb to help us think about

sustainability. In Solow's words: "...I could see the rule as a rebuttable presumption, as a way ofconstantly reminding ourselves that there are considerations other than immediate utility to be



taken into account" (Solow, 1986).

References

Asheim, G., (1986) "Hartwick's rule in open economies", Canadian Journal of Economics XIX

No. 3, 395-402.

Barbier, E.B. and M. Rauscher, (forthcoming) "Trade, Tropical Deforestation and PolicyInterventions", Environmental and Resource Economics.

El Serafy, S., (1989) "The Proper Calculation of Income from Depletable Natural Resources", inAhmad, Y., S. El Serafy and E. Lutz, Environmental Accounting for SustainableDevelopment, The World Bank, Washington D.C.

Hartwick, J., (1977) "Intergenerational Equity and the Investing of Rents from ExhaustibleResources", American Economic Review 66, 972-4.

Hartwick, J., (1990) "Natural Resources, National Accounting and Economic Depreciation", Journal of Public Economics 43, 291-304.

Repetto, R., McGrath, W., Wells, M., Beer, C., and Rossini, F., (1989) Wasting Assets: NaturalResources in the National Income Accounts, World Resources Institute, Washington,D.C.

Sachs, J., (1982) "The Current Account in the Macroeconomic Adjustment Process", Scandinavian Journal of Economics 84 (2), 147-159.

Solow, R., (1974) "Intergenerational Equity and Exhaustible Resources", Review of EconomicStudies (Symposium), 29-45.

Solow, R., (1986) "On the Intergenerational Allocation of Natural Resources", ScandinavianJournal of Economics 88 (1), 141-149.

Weitzman, M., (1976) "On the Welfare Significance of National Product in a Dynamic Economy, Quarterly Journal of Economics 90, 156-62.

Young, C.E., (1992) Renda Sustentavel da Extracaon Mineral no Brazil, Tesis de Maestria,Instituto de Economia Industrial, USRJ, Rio de Janeiro.

Appendix

The following appendix shows that the extended Hartwick rule implies a constant consumption

path. To see this we use Weitzamn's result regarding the welfare meaning of NNP. Weitzman

(1976) shows that the net present value of the optimal consumption path is exactly equal to thenet present value of a constant consumption stream equal to the current NNP. In other words,

NNP in period t is the perpetual equivalence of the optimal consumption path from t onwards.

Mathematically,

where NNP(t) is the current period Net National Product and C*

(s) is the consumption level at

each moment of time given by the optimal growth path.

(A.1) ò¥

t e -δ(s -t )NNP (t )ds = ò

¥

t e -δ(s -t )C *(s )ds



The extended Hartwick rule would be,

By equation (14) we see that if the extended Hartwick rule is imposed on the economy in period

t then NNP(t) will just be the consumption level C*(t). Therefore we can write equation (A.1) as,

taking the derivative of equation (A.3) with respect to t we have,

Therefore, the extended Hartwick rule guarantees that consumption is constant over time.

(A.2) I = (Y R -f R ) -A = (Y R -f R ) -(X -M +rA)

(A.3) C *(t ) = δò¥

t e -δ(s -t )C *(s )ds

(A.4)¶C *(t )

¶t = δ[δe δt ò

¥

t e -δs C *(s )ds + e δt -e -δt C *(t )]

(A.5)¶C *(t )

¶t = δ[C *(t ) - C *(t )] = 0



ESSAY 2

OPTIMAL GROWTH AND SUSTAINABLE DEVELOPMENT

Kirk Hamilton

In what ways, if any, do the concepts of optimal growth and sustainable development differ? As

Pezzey (1989) remarked, the notion of optimality as maximizing the present discounted value of

utility is quite widely accepted in economics. Similarly, the idea of sustainability as the

preservation of capital has had wide currency at least since Hicks wrote "Value and Capital"(1939). What has given many economists the impetus to re-examine these concepts in recent

years is the growing concern about environmental quality and the role that this plays in

determining the welfare of both current and future generations. Issues of resource depletion,

the public good nature of environmental resources, and the public bad nature of pollution have

led to a re-appraisal of the meaning of sustainability.

This article will argue, along with Dasgupta and Heal (1979), that discounting future utility can

lead to a divergence between optimality and sustainability. The degree of substitutability of

human-made and natural resources is also a key consideration. Finally, a few thoughts on the

ethical basis of sustainable development and how this relates to the degree of binding of

sustainability constraints will be offered.

Pezzey has collected some 51 different quotations from the literature, each offering a definition

or a shade of meaning in the definition of sustainable development. This abundance

notwithstanding, he offers a succinct and serviceable definition for the economist: that utility not

decline over time or, mathematically,

If the rate of change is zero, then utility is constant over all time - this will be referred to as

minimal sustainability in what follows.

It is straightforward to demonstrate how discounting can lead to a divergence between optimalgrowth and sustainability. In Figure 1 we see one path for utility that is constant, and therefore

u ³ 0.



minimally sustainable, and another that grows at some rate g then collapses to 0 at time T. The

discount rate is assumed to be r > 0.

SupposeT ò0

u 0e gt e -rt dt >¥ò0u 0e -rt dt , 1

Þu 0g -r

(e (g -r )T - 1) >u 0r

,2

Þ (g -r )T > lng - lnr .3

The mathematical derivation shown next to Figure 1 shows that for either an appropriate choice

of the growth rate g or for

T >(lng - lnr )

(g - r )



then the present value of utility for the "grow and crash" scenario will be greater than that for the

"minimal sustainability" scenario. This result is only possible if utility is being discounted.

As Pezzey points out, sustainable development as defined above is a constraint rather than an

optimizing criterion. This means that it is still possible in principle to define optimal sustainable

growth as the solution to:

Scenarios such as the "grow and crash" example above are optimal if the sustainability criterion

is ignored. Note that utility is still discounted in this formulation. As Dasgupta and Heal (1979)

argue, while early writers such as Ramsey (1928) felt that discounting future utility was "ethically

indefensible", there is at least one rational motivation for discounting: to represent uncertainty

with respect to the existence of future generations14. Norgaard (1991, p.30) argues that we

cannot ignore rates of return (and therefore discounting) in looking at sustainability from the

point of view of intergenerational equity, when he says that "if we are concerned with the

distribution of welfare across generations, then we should transfer wealth, not engage in

inefficient investments". Mäler (1989) arrives at similar conclusions by more formal means

when he argues that the relevant discount rate in judging the endowment of capital that one

generation leaves another is the return on capital in the future - this is important because it

makes the discount rate endogenous and, moreover, it is a rate that is not necessarily identical

to the pure rate of time preference of the current generation, which is one of the standard

criticisms of discounting in an intergenerational context.

This being said, the analysis of Dasgupta and Heal (1979, ch. 10) of optimal consumption in

economies with exhaustible resources puts severe limits on the relationship between

discounting, optimality and sustainability. Their model contains three critical assumptions:

i. production follows a Cobb-Douglas functional form, with constant elasticity of

substitution between resources and reproducible capital equal to 1.

14 As David Pearce pointed out in an earlier version of this paper, however, sufficiently high discount rates may be one

way of ensuring that future generations do not exist.

max¥ò0u t e -rt dt

subject u ³ 0.



ii. elasticity of output with respect to capital is greater than that with respect to resources.

iii. elasticity of marginal utility with respect to consumption is constant and greater than 1.

The basic model for utility discount rate r, utility u, stock of resources S, resource extraction R,

capital stock K, output F and consumption C, is therefore:

Under these conditions Dasgupta and Heal show that efficiency of an optimal consumption path

implies the following relationships:

(1)

F R

F R = F K , the Hotelling rule,

(2) r +ηC C

= F K , the Ramsey rule.

For the Cobb-Douglas production function, expression (2) reduces to

(3)C C

=

α(R K

)β-r

η.

Since efficiency requires that the resource be exhausted over infinite time, this implies that

R → 0 as t → ∞. For consumption (and therefore utility) to be non-declining, expression (3)

therefore requires that K → 0 as well, which implies that output F must also decline. This in turn

requires that consumption C decline in the long run, as must utility. A positive rate of discount

for utility implies that optimal growth is not sustainable.

If the rate of discounting utility is zero, Dasgupta and Heal show that for suitably large elasticities

of marginal utility of consumption then optimal paths are sustainable, with increasing utility over

max¥ò0ue -rt dt

given K = F -C , S = -R

where:

u(C ) = -C -(η-1), η > 1

F(K ,R ) = K αR β

α,β > 0, α+β = 1, α > β.



time. For this class of model, some of whose precepts will be discussed further, discounting of

utility and sustainability are apparently incompatible.

However, there is a nuance to this story that the literature does not bring out. Hartwick (1977) is

working with essentially the same model in examining consumption paths with exhaustible

resources. He shows that if the economy follows the rule that investment equal rents on

resource extraction, then a constant consumption level (and therefore level of utility for the utility

function employed by Dasgupta and Heal) is optimal. The Hartwick rule is simply:

(4) K = F R R .

The rate of change of consumption is derived as:

)( K F dt

d C &&

−=

)( RF F dt

d R−=

RF RF F R R

&&& −−=

RF RF F F R RK && −−= [from (1)]

.0=

We have therefore the result that the optimal growth model, when augmented with a dynamic

constraint in the form of the Hartwick rule, will lead to minimal sustainability as its optimal path.

The nuance is that Solow (1974) and Dasgupta and Heal start from a Rawlsian maxi-min

principle for intergenerational equity, show that this implies constant consumption over time,

then derive conditions for this to be an optimal growth path in the face of exhaustible resources.

The degree of substitutability of capital and resources is critical in this analysis of optimal growth

with exhaustible resources. Dasgupta and Heal demonstrate that if the elasticity of substitution

between capital and resources is greater than 1, then resources are not essential: positive

output is possible with zero resource input, so resources do not constrain consumption -

sustainability would be equivalent to maintaining non-negative net investment levels over time.

If this elasticity is less than 1, then output (and therefore consumption and utility) must be finite -

sustainability is impossible. Only for an elasticity of substitution exactly equal to 1 are resources

essential and output potentially unlimited.



If we assume this degree of substitutability, the 1986 paper by Solow places the Hartwick rule in

a particularly elegant light: if the rule "investment equals resource rents" is followed, this is

equivalent to living on the (constant) flow of interest from a fund of capital whose value is

maintained constant over time. The capital fund in this instance consists of the values of

reproducible capital and natural resources - as natural resources are used up, an identical value

of investment in reproducible capital must take place.

If at least some elements of natural resources do not have ready substitutes, or, more precisely,

if their elasticity of substitution is less than 1 with reproducible capital, then this sustainable

programme is in some difficulty. This is essentially the argument of Pearce, Barbier and

Markandya (1990), although it may appear in stronger forms, for instance that the total value of

stocks of natural resources must be maintained constant (as opposed to the sum ofreproducible and natural capital as appears in the Solow article). It is certainly arguable that

particular critical functions of the natural environment (for example the ozone layer or

biodiversity) have limited substitution possibilities, and that therefore sustainability is threatened

by their depletion.

Pearce et al.'s notion of maintaining capital intact is not necessarily inconsistent with optimal

growth, particularly as Solow (1986) presents it. Solow conceives the total capital stock of an

economy as being represented by a vector of individual items, beginning perhaps with

categories of reproducible capital, such as buildings, machines and infrastructure, followed by

categories of natural capital - preserving capital means preserving the total of this vector. While

the particular context in which Solow was writing would probably have limited natural capital to

forests, fish, energy and minerals, there is no reason not to extend the list to include air, water,

wildlife, etc., provided consistent valuations could be constructed. What is required to make this

interpretation of the capital stock consistent with sustainability (and optimal growth) is that the

uses made of the critical elements of the natural stock not be depleting; in general this can be

achieved for the exploitation of renewable resources or for the enjoyment of amenity values

(although even eco-tourism has its limits in this regard).

This casting of the optimal growth problem as one of maintaining capital suggests,

parenthetically, an approach to "green national accounting" that has not been fully exploited: the

expansion of wealth accounts to include environmental resources. Hamilton (1991) argues that

total wealth per capita, so defined, is a superior national accounts indicator of sustainability.

So far in this argument it has been assumed that there is no technical progress or population



growth. Solow (1986) describes how if the simple model of optimal growth is extended in the

following way,

F = e mt K αR β,

where F, K, and R are output, capital and resource extraction per unit of labour, labour grows at

rate n, the rate of technological change is m, and 1-α-β4 is the Cobb-Douglas elasticity for

labour, then investing rents according to the Hartwick rule will lead to optimal growth at the rate

(m -αn )(1-β)

.

In this instance, therefore, optimal growth will be sustainable at a positive rate, given the very

plausible restriction m >αn 5. It is clear that exponential growth in technological progress canoffset the limit to growth that is imposed if the elasticity of substitution of resources for capital is

less than 1. However, for any critical elements of natural capital which have no substitutes

technological progress cannot be the answer: how many widgets are required to substitute for

declines in the life-supporting functions of the biosphere?

Finally, it is worth reflecting on a more sophisticated example than the "grow and crash"

scenario of Figure 1, in order to examine more closely the ethical principles that underlie

sustainability. Figure 2 presents two alternative development paths.



Along path B the generation at time T reduces consumption and increases investment,

leading to greater growth for future generations. Along this path, u ³06 for t > T.

Along path A, u ³07 for all t.

The question we wish to pose is this: is path B consistent with our notions of sustainability?

Clearly the path is sustainable from time T onwards, and the level of consumption soon

overtakes and remains greater than on path A. The answer to the question would appear to be:

path B is sustainable if the generation at time T voluntarily agreed to reduce consumption and

increase investment in order to produce greater welfare in the future. If this choice wereimposed by a previous generation15 then we would be inclined to say that this is not a

sustainable path. If this argument is correct, then we cannot insist blindly that sustainability

means u ³08 for all time, but rather (or perhaps, in addition) that no generation can force

involuntary hardship on a future generation. This is clearly a long way from the classical

Utilitarianism that underlies the optimal growth models, since by this notion path B would be

unequivocally superior to path A.

Conclusions

By assuming a very simple definition of sustainability, that utility be non-declining, it is clear that

discounting of utility plays a critical role in determining whether optimal growth with exhaustible

resources is sustainable. It is minimally sustainable if the Hartwick rule, that resource rents be

invested, is followed - otherwise, only if the discount rate for utility is 0 or if there is sufficient

(and continual) technological progress can optimal growth be sustainable. Preserving wealth,

and critical elements of the natural environment in particular, is consistent with this notion of

sustainability.

The desirability of sustainable development is, of course, an ethical question rather than an

economic principle. It is nonetheless illuminating to explore its ramifications in traditional

economic models of optimal growth. Ramsey's view on the ethical defensibility of discounting

utility bears repetition in this context. The discussion of the strict adherence to the principle that

utility be non-declining suggests, however, that sustainability as an ethical position requires a

15 It is not hard to conceive of examples: suppose earlier generations profligately consumed conventional energy, so

that the generation at T had to go on a crash programme of development of alternative energy sources.



more sophisticated representation in our models than has been provided to date.

References

Dasgupta, P., and Heal, G., (1979) Economic Theory and Exhaustible Resources, Cambridge

University Press, Cambridge.

Hamilton, K.E., (1991) Proposed Treatments of the Environment and Natural Resources in theNational Accounts: A Critical Assessment, National Accounts and Environment Division,Discussion paper no. 7, Statistics Canada, Ottawa.

Hartwick, J.M, (1977) Intergenerational Equity and the Investing of Rents from ExhaustibleResources, American Economic Review, 67, No. 5, 972-4.

Hicks, J.R., (1939) Value and Capital, Oxford University Press, Oxford.

Mäler, K.-G., 1989) Sustainable Development, (mimeo), Stockholm.

Norgaard, R.B., (1991) Sustainability as Intergenerational Equity, Report IDP 97, Asia Regional

Series, The World Bank.

Pearce, D.W., Barbier, E., and Markandya, A., (1990) Sustainable Development, Earthscan,London.

Pezzey, J., (1989) Economic Analysis of Sustainable Growth and Sustainable Development, Environment Dept. Working Paper No. 15, The World Bank.

Ramsey, F., (1928) A Mathematical Survey of Saving, Economic Journal, 38, 543-59.

Solow, R.M., (1974) Intergenerational Equity and Exhaustible Resources, Review of EconomicStudies (Symposium) 29-46.

Solow, R.M., (1986) On the Intergenerational Allocation of Natural Resources, Scandinavian

Journal of Economics, 88(1), 141-49.



ESSAY 3

SUSTAINABILITY, ECONOMIC GROWTH AND EMPLOYMENT16

Carlos Eduardo Frickmann Young17

1. INTRODUCTION

The trade-off between sustainability and economic growth is still the subject of controversy. The

use of macroeconomic growth models subject to environmental constraints is an interesting

theoretical way to treat the problem.

In general, studies on this theme use neoclassical macroeconomic models, where the objective

is to find the optimal path of growth. The optimal growth path means that path which maximises

social welfare in a certain period of time. Sustainability appears as a constraint, imposing an

additional intertemporal goal. For example, utility per capita (or consumption per capita, or

capital stock level) cannot be decreasing (for a review in this issue see Pezzey, 1989).

However, a very important assumption is usually made: 'conventional' factors of production -

man-made capital and labour - are fully employed. The justification for this procedure is derived

from the belief that there is no involuntary unemployment in the long term. But there is no

definitive evidence (empirical or theoretical) supporting this belief. On the contrary, long term

unemployment characterizes the majority of economies, especially in developing countries.

Hence, the determination of effective demand, and the functioning of an economy out of the full-

employment equilibrium, is the main issue for several approaches not considered mainstream

(Keynes, 1973; Kalecki, 1991a). Economic growth is not thought of as oscillating around an

assured long run equilibrium path since full-employment is not an assumption but a desired

objective.

The aim of this paper is to present a simplified non-conventional macroeconomic model which

relates effective demand and the sustainable use of natural resources. It supposes a small non-

developed economy based on the exports of a natural resource and with a heavy income

16

I am grateful for the comments of Prof. David W. Pearce.17

Universidade Federal do Rio de Janeiro and visiting fellow at CSERGE-UCL.



concentration. 'Optimal growth' is understood as maintaining labour full-employed18 and

sustainability is defined as maintaining a non-decreasing capital stock level. The model is based

in a modified Kaleckian equation of income determination (Kalecki, 1991b) where exports of the

natural resource substitute for the role of investment as autonomous expenditure. Therefore,

employment is a direct function of resource depletion and is negatively related to sustainability.

The paper shows the effects of changes in some macroeconomic variables (income distribution,

terms of trade, external debt, exchange rate) on sustainability. The paper concludes that the

management of macroeconomic variables influences the sustainability of natural resource

based economies.

2. THE MACROECONOMIC MODEL

We suppose an economy is characterized by a huge disproportion in the income distribution. As

a simplification, there are two basic classes, 'poor' and 'rich'19, each one with very distinct

patterns of income and consumption. Four distinct uses for final products are possible:

investment, consumer goods for the 'rich', consumer goods for the 'poor', and exports. Thus, this

economy can be divided into four departments, each one representing the integrated production

(including all intermediate consumption) of final demand categories (see Table 1).

Table 1: Production and Income Generation

P(I) + P(Cr) + P(Cp) + P(X) = P

W(I) + W(Cr) + W(Cp) + W(X) = W

M(I) + M(Cr) + M(Cp) + M(X) = M

--------------------------------------------------------

I + Cr + Cp + X = Yg+M

where: P: gross profits (including rents)

W: wages

M: total imports (including imported inputs)

I: gross investment

Cr: consumer goods for the 'rich'

Cp: consumer goods for the 'poor'

18

This condition itself may not warrant optimality in neoclassical terms, but it is certainly a necessary condition.19

In the original model 'workers' and 'capitalists', respectively (Kalecki 1991b).



X: exports

Yg: gross income

The 'poor' receive wages and spend all their income. Hence, the production of consumer goods

to the poor is equal to the total amount of wages.

Cp = W (1)

The 'rich' receive the profits obtained from domestic production and exports. Since the economic

importance of the domestic market is small, the consumption level of the rich is a function of

exports.

Cr = r.X (0<r<1) (2)

The investment in man-made capital is assumed to be equal to its depreciation and is done with

imported equipment. Since there is no domestic production of capital goods, P(I) and W(I) are

equal to zero, and there are no multiplier effects from it.

Exports are totally based on the natural resource. Their value in domestic prices is a function of

the international price of the resource (pi), the exchange rate (e) and the quantity depleted (Q).

X = pi.e.Q (3)

Kn(t) is the economic value of the stocks of the natural resource in the period t. The rate of

growth of Kn if the resource is not depleted is g, and expresses changes both in economic and

physical terms. The rate of actual depletion of the resource is d. The variation of Kn in time can

be expressed as:

Q = d.Kn (4)

S(t) = dKn(t) = (g-d).Kn(t) (5)

dt

By assumption, there is no change in man-made capital stock. Hence, S(t) is the 'sustainability

indicator'20.

20 Following the definitions presented in Pearce and Atkinson (1992), it is a 'weak-sustainability indicator'.



The imports are proportional to the net product level (Y).

M = m.Y (0<m<1) (6)

Analogously to Kalecki (1991b), the net product can be determined as follows (the complete

development of the model is shown in the Appendix):

Y = 1 .(1+r+ r.wr+ wx ).X = α.X (7)

1+m 1-wp

where: wr: the share of wages in the total income of consumer goods to the 'rich'wx: the share of wages in the total income of exports department

wp: the share of wages in the total income of consumer goods to the 'poor'

α: income-multiplier of autonomous expenditure (X)

The product level is a direct function of the exports, i.e., the depletion of the natural resource.

Assuming m, r, wr, wx, wp and the labour-productivity as constants, there is one level of natural

resource exports (X*) which assures full-employment.

Y* = α.X* (7a)

The full-employment extraction level and correspondent rate of depletion are:

Q* = X* (8)

pi.e

d* = Q* (9)

Kn

Consequently, there are three possibilities for the sustainability of this economy:

(i) d* < g Full-employment is compatible with an increasing level of sustainability (S>0).

(ii) d* = g Full-employment implies a constant level of sustainability (S=0).

(iii) d* < g Full-employment implies a decreasing level of sustainability (S<0).

Figure 1 presents these results:



WY

S

t

Kn i

ii

iii

S

i

ii

iii

0

Figure 1

In situations (i) and (ii), the ambition of full-employment in the present does not compromise thesame objective in the future. However, in situation (iii), short-run full-employment means

unemployment in the long run. In this case, changes in the macroeconomic conditions are

strictly necessary to reconcile sustainability and full-employment.

It becomes clear that the macroeconomic environment has important impacts on sustainability.

In the next session, the model is used to highlight these impacts in sustainability due to changes

in some macroeconomic variables.

3. CHANGE IN MACROECONOMIC VARIABLES

3.1 Improvement in income distribution

In that case, wages represent a bigger share of total income. As a consequence of the increase

in wr, wp and wx, the multiplier α becomes bigger. It means that a smaller depletion of the

natural resource is necessary to maintain full-employment. Due to the reduction of d*,

sustainability increases when the share of the 'poor' in total income increases (see Figure 2).

Figure 2



S

Y per capita

WY

Y per capita

If the income distribution along the time behaves as a Kuznets Curve, i.e., with a U-shape for

the ratio wages-total income, the sustainability function will also have a U-shape format (see

Figure 3).

Figure 3

Conventional Kuznets Curve ‘Environmental’ Kuznets Curve

There is an important assumption for these results. It is supposed that there is unemployed

capacity in the production of 'poor' consumer goods. If not, the excessive demand will start an

inflationary process and a fall of real wages, which would restore the income distribution to its

original situation.

3.2 Change in terms of trade (price of the resource in the international market)

There is a controversy about the price path of natural resources in international trade. There are

two opposing positions: the 'Hotelling Rule', which points to an increasing price due to growing

scarcity of the resource, and the 'Prebisch-Singer Rule', which points to a decrease of natural

resources prices in the long term. Both situations can be examined and their results are

symmetric.

In the first case, there is an increase in the value of the exports measured in domestic prices (d*'

< d*). Therefore, a smaller depletion can lead to full-employment. On the other hand, a loss in

terms of trade implies the necessity to expand depletion (d*' > d*). Figure 4 illustrates these

results.



S

e

S

pi

Figure 4

3.3 Structural adjustment and external indebtedness

In general, developing countries have relatively high levels of external indebtedness. Thepayment of the service of the debt implies the necessity to increase exports. Hence, adjustment

programmes to stimulate exports are very frequent. In our hypothetical case, two solutions are

possible.

The first one is a shortage of the supply, considering that the country has a monopolistic power

in the international market for the resource. The shortage produces a rise in the price of the

resource and a reduction in the depletion rate, as discussed above.

But the exporter country usually has a small share of the international trade of the resource. A

devaluation of the exchange rate (i.e. increasing e) is often recommended to stimulate exports.

The resource can be sold at a lower internat-ional price without damage to the exporter's profit

measured in domestic prices.

The devaluation also changes the income distribution against the 'poor', since real wages

decrease21. Therefore, the fall in sustainability should be bigger than in the case discussed in

section 3.2 (see Figure 5).

Figure 5

21

If wages in domestic prices are held constant, real wages fall through a rise in the prices of imported consumergoods.



e I : revaluation e I : devaluation

4. CONCLUSION

The purpose of this paper has been to present a dilemma: the pursuit of current full-employment

in economies based on natural resources depletion can bring future unemployment. In poorcountries, the degrees of freedom to 'postpone' welfare are strongly reduced since people live

very close to subsistence.

However, the management of macroeconomic variables influences decisions related to the

exhaustion of the resources. In some circumstances the results on sustainability are positive, in

others negative. Table 2 reviews the principal hypothetical cases discussed in this paper.

Table 2

Macroeconomic circumstance Effect on sustainability

Improvement in the income distribution Increase

Loss in terms of trade Decrease

Devaluation of exchange rate Decrease

In more mathematical terms:

+ - + + -S(t) = f{g(t), d(t), w(t), pi(t), e(t)}

Nevertheless, it is important to highlight two of the assumptions adopted. The most important is

the absence of domestic production of capital goods. Withdrawing this assumption can lead to

different results, since current depletion can be used to enlarge the stock of man-made capital,

with positive effects on long-run employment.

The second one is the belief about the future economic importance of the resource. Forecasts



are generally based on current trends but unexpected changes in technology and consumers'

preferences may make depletion economically not feasible. Past experiences (latex in Brazil,

saltpetre in Chile, etc.) are not sufficient to prevent new cases, since the course of technical

progress and human behaviour are uncertain.

The extreme simplicity of the model requires that results in this paper should not be viewed as

definitive statements. Empirical case study analysis is essential for a better comprehension22.

References

Jonish, J. (1992) Sustainable Development and Employment: Forestry in Malaysia. WorkingPaper No. 234. International Labour Office, Geneva

Kalecki, M. (1991a) 'Theory of Economic Dynamics: An Essay on Cyclical and Long-Run

Changes in Capitalist Economy', in Osiantynski, J. (ed.) Collected Works of MichalKalecki , v.2, Oxford University Press, Oxford. pp.205-348.

Kalecki, M. (1991b) 'The Marxian Equations of Reproduction and Modern Economics', inOsiantynski, J. (ed.) Collected Works of Michal Kalecki , v.2, Oxford University Press,Oxford. pp.259-466.

Keynes, J.M. (1973) 'The General Theory of Employment, Interest and Money'. CollectedWritings of John Maynard Keynes , v.7. Macmillan, London.

Pearce, D.W. and Atkinson, G. (1992) Are National Economies Sustainable? MeasuringSustainable Development . GEC Working Paper 92-11, Centre for Social and EconomicResearch in the Global Environment, University College London and University of East

Anglia.

Pezzey, J. (1989) Economic Analysis of Sustainable Growth and Sustainable Development .(Environment Department Working Paper No.15) World Bank, Washington.

22 A good example is Jonish, 1992.



Appendix

Equation (1) can be re-written as:

Cp = W(I) + W(Cr) + W(Cp) + W(X)

By assumption, W(I) is null. Assuming wr=W(Cr)/Cr; wp=W(Cp)/Cp and wx=W(X)/X:

Cp = wr.Cr + wp.Cp + wx.X

Cp = wr.Cr + wx.X (10)

1 - wp

The general macroeconomic equation is:

Yg + M = I + Cr + Cp + X

Investment is just equal to depreciation, so the net income is:

Y + M = Cr + Cp + X

Using equations (2), (6) and (10):

Y + m.Y = r.X + wr.(r.X) + wx.X + X

1 - wp

Y = 1 .(1 + r + r.wr + wx).X

1+m 1 - wp

Young 1993 Sustainability Growth Kalecki UEA CSERGE Gec 1993 08

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