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Young 1993 Sustainability Growth Kalecki UEA CSERGE Gec 1993 08
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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)
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
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.
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.
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
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
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.
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
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
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).
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.
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.