E C O L O G I C A L E C O N O M I C S 6 4 ( 2 0 0 7 ) 1 0 3 – 1 0 8

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METHODS

A step beside the maximin path: Can we sustain the economyby following Hartwick's investment rule?

Vincent MartinetINRA–UMR Economie Publique, 78850 Thiverval Grignon, FranceEconomiX — Université de Paris X-Nanterre, France

A R T I C L E I N F O

E-mail address: vincent.martinet@u-paris11 Hartwick (1977, 1978) argues that the rule

constant efficient consumption path. It corrmaintains both the total stock of capital andthat a constant net investment (positive or nnet investment is not equal to zero, the equi

0921-8009/$ - see front matter © 2007 Elsevidoi:10.1016/j.ecolecon.2007.07.019

A B S T R A C T

Article history:Received 12 March 2007Received in revised form 18 July 2007Accepted 20 July 2007Available online 7 September 2007

If sustainability is interpreted as the requirement to sustain consumption or utility at anoptimal level, a maximin objective appears to be relevant. The sustained economy ischaracterized by an optimal investment following Hartwick's investment rule. This paperexamines how the sustainability of a production–consumption economy with a non-renewable resource is modified in the neighborhood of the maximin path, i.e. when theconsumption and the resource price are not optimal. A Sustainable Consumption Indicatoris introduced in order to characterize the sustainability of constant consumption paths,defined as deviations from the maximin path. We describe how an over-consumptionjeopardizes future sustainability.

© 2007 Elsevier B.V. All rights reserved.

Keywords:Non-renewable resourcesSustainable consumptionHartwick's rule

JEL classification:Q01;Q30;D99

1. Introduction

If sustainability requires consumption or utility level to bepreserved (Solow, 1993) and if one requires the sustainabilityobjective to be included in the objective function (Krautkrae-mer, 1998; Pezzey and Toman, 2002), the natural framework toanalyze the sustainability of the economy is the maximinapproach (Solow, 1974a; Cairns and Long, 2006). This frame-work focuses on the definition of themaximal consumption orutility level that can be sustained by the economy given initialendowments, and defines the optimal consumption andinvestment decisions for the economy to be sustained.

0.fr.which consists in investinesponds to a zero net inthe consumption constanegative) is a sufficient contable path is not efficient

er B.V. All rights reserved

One of the main result of the literature on maximin andsustainable consumption level is the Hartwick investment rule(Hartwick, 1977, 1978; Dixit et al., 1980) which stipulates that theutility level is sustained only if the net investment of theeconomy is nil. Thenet investment includes positive investmentin manufactured capitals and negative investment linked to thedepletion of non-renewable natural resources.1 According toRobert Solow, “a society that invests aggregate resource rents inreproducible capital is preserving its capacity to sustain aconstant level of consumption” (Solow, 1993, p.170). However, ithas been shown that the Hartwick rule should be consideredmore like a descriptive property of themaximin path than like a

g the rents from the resource use in reproducible capital leads to avestment. This rule is often argued to lead to sustainability as itt along time. Dixit et al. (1980) generalize the Hartwick rule arguingdition to have an equitable path (constant consumption). But if the, as proven by Mitra (2002).

.

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prescriptive policy ensuring the sustainability of the economy(Asheim et al., 2003; Martinet, 2005). The key point of the issue isthat the Hartwick rule is associated with themaximin path onlyif the prices are optimal. In particular, the natural resource pricemust increase at a rate that equals the marginal productivity ofcapital, i.e. at the interest rate of the economy. This rule, knownas the Hotelling rule (Hotelling, 1931), leads to an optimal inter-temporal use of the natural resource. Moreover, a transversalitycondition must hold, ensuring that the rule will be satisfiedforever, including at an infinite time (Mitra, 2002;Martinet, 2005).

In this framework, an important concern is raised by Cairns(2007) in a recent contribution. If we consider neo-classicaleconomics, the discounted-utilitarian objective defines theNet National Income, which can be approximated by theincome from markets in a perfectly competitive economy.Observations of non-optimal economies can thus lead to ameasure of practical national income statistics, which aremeasures of the economy efficiency.2 It is not the same with amaximin objective. Amaximin problem gives rise to two typesof statistics: (a) the level of utility (which remains constant)and (b) the value of investment in stocks (which equals zero).Nevertheless, there is no theoretical relationship between theshadow values of the maximin problem (which are the forcesleading to equity and sustainability) and the prices observed inthe economy.3 It is thus impossible to define an observablestatistical indicator of whether a real economy is beingsustained, as the maximin criterion only characterizes theoptimal path. An important issue is then to examine whathappens in the neighborhood of a maximin path.

In this paper, we shed a first glimpse at the “sustainability”of an economy in the neighborhood of themaximin path. Suchan approach is a first step on the definition of a frameworkallowing the analysis of the sustainability of real ‘sub-optimal’economies. For this purpose, we will examine the sustainabil-ity of economic paths when both the statistics defined in amaximin path, i.e. (a) a constant level of consumption, and(b) a net nil investment (Hartwick's rule), are observed.

We examine the sustainability of intertemporal economicpaths by studying the evolution along time of a so-called Sus-tainable Consumption Indicator defined as the maximal sustain-able consumption reachable from a given economic state.

We apply this approach to the canonical model used byDasgupta and Heal (1974) and Solow (1974a) and examine howsustainability evolves when prices are sub-optimal and lead toan over-consumption. A numerical illustration is given to helpthe reader to quantify the qualitative results.

The paper is organized as follows. The Sustainable Con-sumption Indicator we use to analyze the evolution of thesustainability of the economy is described in Section 2. Section3 presents the standard consumption–production model wedevelop to apply the approach to an illustrative theoreticalcase. The analysis is developed in Section 4. Illustrations of theresults are provided in Section 5. Section 6 concludes.

2 Close to the optimal path, given enough continuity, prices canbe assumed to be close from the shadow values of the optimiza-tion problem.3 In a maximin problem, the shadow value of equity is nil when

the equity constraint is not bounded. Moreover, this shadowvalue cannot be observed in real economies.

2. The Sustainable Consumption Indicator

We define sustainability as follows. At a given time t, a level ofconsumption 4 is sustainable if it can be attained forever fromtheeconomic state at time t. Economicdecisions are sustainableif they do not reduce the set of sustainable consumption levels,i.e. the maximal sustainable consumption.5 This definition ofsustainability is not related to the actual consumption levelsalongapath. Inparticular,wedonot require the consumption tobe either equal to the maximal sustainable level, or non-decreasing.

Consider an economy with n capital goods, represented bya vector K∈ R <n of n capital stocks. The economy evolvesunder the dynamics

�K ¼ FðK;u; tÞ ð1Þwhere u is a decision vector (e.g. consumption, extractionrate...).

To describe the sustainability of the economy, we intro-duce the Sustainable Consumption Indicator which is defined asthe maximal consumption that could be sustained foreverfrom a given economic state K.

Definition 1. The Sustainable Consumption Indicator

SCI ¼ CðKÞ ð2Þrepresents the maximal level of consumption that could besustained from economic state K.

This indicator is equal to the constant consumption levelthat would result from a maximin program starting from theconsidered economic state. It is a potential sustainableconsumption level that can be computed for any economicstate, which means that C(K) is defined by a map Rn↦R.

We are interested in the evolution of the indicator along agiven trajectory K(.). At time t, that indicator is equal to CðKtÞ. Ifit decreases, the “time t” sustainable consumption level is nomore sustainable after time t. In that sense, decisions at time tare not considered sustainable as the set of future sustainableconsumptions is reduced. For this purpose, we introduce χ,the time derivative of the indicator:

v ¼ dCðKtÞdtjs:t: �K¼FðK;u;tÞ: ð3Þ

In our notation, if χb0 (resp. χN0), sustainability decreases(resp. increases) at time t, along the trajectory K(.).

Eq. (3) can be written

v ¼Xni¼1

�KiACðKÞAKi

jKt

: ð4Þ

As �Ki represents the investment in capital i, ACðKÞAKi

can beinterpreted as the shadow value of an i-capital unit withrespect to the sustainability indicator SCI , at the economic

4 The whole analysis can be developed using utility instead ofconsumption. We use consumption to match with the example innext section.5 It means that the decisions of a generation (including

consumption and investment) must not reduce the set ofsustainable consumptions of following generations.

Fig. 1 –Value of the Sustainable Consumption Indicator with respect to the capital stocks K and S.

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state Kt. These shadow-values can be represented by theslopes of the map CðKÞ at the economic state Kt.

More generally, in the maximin framework, the level ofmaximal sustainable consumption depends on the stocks ofcapital, including manufactured capital and natural resourcestocks. Maximin paths are trajectories following isovalues ofthe map CðKÞ. Actually, the SCI is constant along a maximinpath.

3. Maximin path in the Solow's model

For illustrative purpose, we develop the analysis in a standardtwo stock model with a reproducible manufactured capital Kt,and a non-renewable natural resource St.6 This simple illustra-tion case makes it possible to match the results with existingliterature, and to represent the results in a 3-D mapping.

The stocks evolve under dynamics

�K ¼ FðKt; rtÞ � ct ð5Þ

�S ¼ �rt ð6Þ

where F is the production function, ct is the consumption attime t and rt is the extraction of natural resources. For the sakeof simplicity, and to be consistent with the Solow (1974a) and

6 This model is a benchmark in the literature on sustainable useof exhaustible resources since the 1974 symposium on theEconomics of exhaustible resources in the Review of EconomicStudies. In this framework, Dasgupta and Heal (1974) define theoptimal path of the economy with respect to the neoclassicaldiscounted criterion, and Solow (1974a) defines the optimal pathof the economy with respect to a maximin criterion.

Hartwick (1977) analysis, F is defined as a Cobb–Douglasproduction function, i.e. F ¼ Karb, with βb1 and αNβ. With sucha production function, marginal productivities of inputs aredecreasing. Initial stocks K0 and S0 are given.

In this model, the Sustainable Consumption Indicator is equalto the maximal sustainable consumption from state (K, S) andis defined 7 by

CðK; SÞ ¼ ð1� bÞðSða� bÞÞb

1�bKa�b1�b: ð7Þ

This Sustainable Consumption Indicator defines a shape in the(K, S) map, with isovalue curves as illustrated in Fig. 1.

If we consider themaximin path starting from state (K0, S0),it follows the isovalue CðK;SÞ ¼ CðK0; S0Þ of the map. Thismaximin path is characterized by a constant consumptionc⁎t ¼ CðK0; S0Þ.

Along a constant consumption path, Solow (1974a) showedthat the resourceextractionmust beoptimal and satisfy the rule

r⁎ðcÞ ¼ c1� b

� �1b K

�ab

t ð8Þ

in order to minimize the quantity of resource used along thepath, which is a necessary condition for a maximin path. Eq. (8)is the same as

c ¼ ð1� bÞr⁎bKa: ð9Þ

It means that, along a constant consumption path with anefficient resource use, a part (1−β) of the production is

7 See Solow (1974a) for the computation of the maximalsustainable consumption level in a maximin framework, andMartinet and Doyen (2007) for its computation in the viabilityframework.

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consumed and then that a part β of the production is invested,which reads

�K ¼ bKarb ⇔ �K ¼ rbrb�1Ka ⇔ �K ¼ rFVr: ð10Þ

As the marginal productivity of the resource FVr can be

interpreted as the resource price, Eq. (10)means that the resourcerent is invested inreproduciblecapital.This resultwasgeneralizedbyHartwick (1977) and is known as theHartwick investment rule.

Moreover, along an optimal path, the resource price mustsatisfy the Hotelling rule and increase at a rate that equals theinterest rate (Hotelling, 1931; Solow, 1974a). In our model, thisrule reads

dFVr

dtFVr¼ FV

K: ð11Þ

Wesee in Eqs. (10) and (11) that both the rules dependon theresource price, or equivalently on themarginal productivity ofthe resource, FV

r. Hotelling's rule gives information on theevolution of that price along time, but not on its optimal value.The optimal resource price is defined thanks to a terminal costcondition (Mitra, 2002) inspired by the Michel's optimalitycondition (Michel, 1982). In our case study, this conditionensures that the resource stock is exhausted at infinite time. Ifthe resource price follows the Hotelling rule from time to time,but without ensuring the terminal condition, the resourcestockwon't be exhausted in an infinite time.As an inputwill beused in production until its marginal productivity equals itsprice, and given the fact that the marginal productivity of thenatural resource is decreasingwith aCobb–Douglasproductionfunction, the lower the price, the more resource used toproduce. We can then distinguish two cases

• If the price is lower than the optimal one,more resourcewill beused inproduction.Theproduction ishigher than in theoptimalcase and the resource stock will be exhausted in a finite time.

• On the contrary, a high resource price will result in a loweruse of the resource in production. The production will belower than in the optimal case and the resource stock willnever be exhausted (a part of it will be preserved forever).

As the consumption is a part (1−β) of the production, bothcases will lead to a consumption that is different from theoptimal one: lower if the resource price is too high, and higherif this price is too low. In the sameway, if one wants to sustaina consumption level that is different from the maximalsustainable one, the extraction rule will be defined by Eq. (8),which will result in a greater or lower resource use inproduction leading to a sub-optimal resource price.

If the Sustainable Consumption Indicator is constant along amaximin path, we can wonder how it evolves when resourceprice and consumption level are not optimal.

8 One reason can be that the resource price is different from theoptimal one.9 Using the fact that Eq.(7) is also CðK; SÞ1�b

b ¼ ð1� bÞ1�bb Sða� bÞKa

b�1.

4. Evolution of the Sustainable ConsumptionIndicator through time

Consider that the social planner wants to follow the maximinpath. The consumption is set at the constant level c0 and both

Hartwick's and Hotelling's rules are satisfied from instant toinstant in order to keep the consumption constant.

We examine what happens when c0pc⁎ (the consumption isdifferent from the optimal level, i.e. there is a mistake incomputing the optimal level of consumption, for any reason 8).We thus study the evolution of the SCI along a constantconsumptionpath that followsbothHotellingandHartwick rules.

To examine how sustainability evolves in the neighbor-hood of the maximin path, we consider the sign of χ, definedby Eq. (3).

Differentiating Eq. (7) with respect to time, we get

v ¼ CðKt;StÞ b1� b

�SS�1t þ a� b

1� b�KK�1

t

� �ð12Þ

¼ CðKt; StÞ b1� b

rtSt

ða� bÞb

�KK�1t

Strt

� 1� �

� ð13Þ

For constant consumption ct ¼ c0p0, using Hartwick invest-ment rule (Eq. (10)), the extraction rule Eq. (8) and thedynamics Eq. (5), we have

�K ¼ bc01� b

:

and

rt ¼ c01� b

� �1b

K�ab

t :

Eq. (13) then becomes

v ¼ CðKt;StÞ bð1� bÞ

rtSt

ða� bÞ c01� b

� �1�1b

Kab�1t St � 1

" #ð14Þ

which is equivalent to9

v ¼ CðKt;StÞ bð1� bÞ

rtSt

CðKt;StÞc0

� �1� bb � 1

� �: ð15Þ

We obviously deduce from this equation that the Sustain-able Consumption Indicator SCI increases when c0bCðKt; StÞ anddecreases otherwise. The quantity χ represents the rise/fall ofsustainable consumption level. It can be interpreted as thegains/losses on future sustainability, i.e. for all future genera-tions. The rate of change of the SCI depends on

• the resource extraction ratiorS: the greater this ratio is, the

higher the effect (positive or negative) on the SCI is.• the ratio between the maximal sustainable consumptionCðKt;StÞ and the actual consumption c0.

In order to determine the magnitude effect of a mistake inthe consumption computation on the sustainability of theeconomy, we study χwith respect to c0. Taking the expressionof rt defined by Eq. (8), Eq. (15) becomes

v ¼ CðKt;StÞ b

ð1� bÞ1þbb

K�ab

t S�1t c0CðKt; StÞ

1�bb � c

1b

0

� �: ð16Þ

Fig. 2 –Value of χ (the time derivative of CðKt ;StÞ with respectto the actual consumption c0.

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Taking the derivative of Eq. (16) w.r.t. c0, we get

dvdc0

¼ CðKt; StÞ b

ð1� bÞ1þbb

K�ab

t S�1t CðKt; StÞ

1�bb � 1

bc1�bb

0

� �: ð17Þ

As βb1, the sign of this derivative depends on that of theexpression into brackets. It is negative if c0NkCðKt; StÞ wherek ¼ b

b1�bb1 and positive otherwise. Fig. 2 represents the value of

χ with respect to the consumption level c0 and illustrates thisresult.

We can distinguish two cases.

• Over consumption case ðc0NCðKt;StÞÞ: If the consumption isgreater than the maximal sustainable one, the greater c0 is,the higher the losses χ for all future generations are.Moreover, as along the constant consumption path c0 thatwe consider the over-consumption10 reduces the level of theSustainable Consumption Indicator, the difference betweenactual consumption and sustainable one is increasing. The“loss-effect” will increase along time.11

• Sacrifice case ðc0bCðKt;StÞÞ: On the contrary, the gains from asmaller consumption than the maximal sustainable one arebounded. It means that a larger restriction of consumptioncan even be less efficient to increase the indicator level as χis not always increasing as c0 decreases.

An over-consumption is then more armfull than a restric-tion on consumption is beneficial for sustainability.12

5. Illustrations

For an illustrative purpose of previous results, we consider acontinuous time dynamic model defined by Eqs. (5) and (6),

10 Such an over-consumption period can represent the con-sumption choice of a generation, over a couple of decades forexample.11 This cumulative effect is illustrated in the numerical examplein next section (Fig. 3).12 Note that the Sustainable Consumption Indicator could alsodecrease when consumption is lower than the maximal sustain-able one if the extraction and investment decisions are notadequately determined.

with three time intervals. On each interval, consumption isconstant and extraction is (myopically) efficient (HartwickInvestment Rule and Hotelling efficiency Rule are satisfied ateach instant).

Consider that consumption is constant according to

• c(t)=ca on interval Ia=[0; ta], with caNCðK0; S0Þ• c(t)=cb on interval Ib= [ta; tb], with cbbCðKta ; Sta Þ• c(t)=cc for all tN tb, with cc ¼ CðKtb ; Stb Þ:

We thus have an unsustainable initial consumption on Ia,and a restriction of the consumption (w.r.t. the maximalsustainable one) on Ib. After tb, consumption is the maximalsustainable one, as it would be defined by a maximin pathstarting from (Ktb, Stb).

During the first period Ia, the consumption is greater thanthemaximal sustainable one CðKt;StÞ. Such a situation reducesthe level of the indicator SCI . After ta, consumption is lowerthan C(Kt, St). The indicator SCI increases.

Fig. 3 illustrates this example. We set ta=20 and tb=40.Production function parameters are α=2/3 and β=1−α. Initialstocks are K0=2 and S0=10. At t=0, the maximal sustainableconsumption is then cþ0g1:72. Consumption is ca ¼ 1:8Ncþtduring the first period. At t=20, the sustainable consumptionis cþ20g0:77. Consumption on the second period is cb ¼ 0:5bcþt .

First, we get in Fig. 3 an illustration of the “cumulativeeffect” described in the previous section. In the first timeinterval, along the constant over-consumption ca, the differ-ence between actual consumption and the maximal sustain-able one increases. The decreasing effect on the indicator isgreater and greater along time and the looses on sustainabilityare cumulative. We can also see in Fig. 3 that, although timeperiods Ia and Ib have equal length and the restriction onconsumption during the second period is greater than theinitial over-consumption, the Sustainable Consumption Indi-cator does not reach its initial value. Other simulations with

Fig. 3 –Evolution of the Sustainable Consumption Indicatorwhen consumption is different from themaximal sustainableone.

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different parameters values (changing β, the length of Ia withrespect to Ib, and the values of ca and cb with respect to thesustainable levels for example) lead to similar qualitativeresults. It informs us that an over-consumption is veryharmful for future sustainability (and then for future genera-tions) as it would require a longer restriction period tocompensate for a “prodigal” one. Anyway, we don't focushere on the relationship between the sacrifice level and thetime it takes to make up for a deviation c0NCðKt; StÞ. Let usmention that this issue of recovering from an over-exploita-tion can be addressed in the viability framework using theconcept of time of crisis (Martinet et al., in press).

6. Discussions and conclusion

This paper examined how sustainability evolves in theneighborhood of the maximin path when all observablestatistics are close to optimal characteristics of the maximinpath: the consumption is constant, and the investment is nilwith respect to real prices. For this purpose, we have studiedthe evolution of the Sustainable Consumption Indicator whichis defined for any economic state as the maximal level ofconsumption that could be sustained forever from that state.

Along the maximin path, the maximal sustainable con-sumption is sustained, which requires both Hotelling andHartwick rules to be satisfied forever. But having both therules satisfied from instant to instant, along a constantconsumption path is not sufficient to ensure sustainability(Mitra, 2002). And the observation of the rules in the short-rundoes not ensure that they could be satisfied forever. Asmentioned by Solow (1974b, p.12):

“many patterns of exploitation of the exhaustible-resourcepool obey Hotelling's fundamental principle myopically,from moment to moment, but are wrong from a very longrun point of view. Suchmistaken paths may even stay verynear the right path for a long time, but eventually theyveer off and become bizarre in one way or another.”

We examined the long-run sustainability of a dynamiceconomic production–consumption system with non-renew-able resources in the presence of such a “mistake” by studyingthe evolution of the Sustainable Consumption Indicator whenconstant consumption deviates from the maximin level. Wethus consider sustainability beyond first best and describehow future sustainability is jeopardized by present over-consumption.

We have described how future sustainability is reducedwhen the consumption is too high. This study emphasizes thefact that consuming too much now would require a heavysacrifice for future generations to restore the initial level ofsustainable consumption (if one wants to restore it).

Using an illustrative example, we show that even if pricesare close to the shadow value of capital goods, the empirical

path quickly diverges from the sustainable path defined by amaximin program. Genuine saving (a non-negative netinvestment) is not sufficient for sustainability if real pricesare not exactly the optimal one.

As a concluding remark, we argue that the maximinapproach should be enlarged in order to define a frameworkto study sustainability of complex real economic systems. Thegeneralization of the present approach and the use ofSustainable Consumption Indicator can be a way to copewith the sustainability issue away from optimal trajectories.

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