Sustainable Economic Growth and Exhaustible Resources:

DiscreteDynamicsinNatureand Society, 2002Void.7(2), pp.79-92

i

Taylor &FrancisGroup

Sustainable Economic Growth and Exhaustible Resources:

A Model and Estimation the US

ALMUTH SCHOLL^a’*andWILLI SEMMLER^b’c

aDepartment of^Economics,Humboldt University, Berlin,Germany;bCenter_ofEmpiricalMacroeconomics,Bielefeld, Germany CNewSchool University,NY, USA

(Revised14 April2001)

This paper studies current models on sustainable economic growth withresource constraints and explores^towhatextent resourceconstraints canbeovercomebysubstitutionandtechnologicalchange.

Wealsostudytheproblemof intergenerational equityand the different criteria that have beensuggested in the literature. The central partof this paperis the presentation ofstylized facts on exhaustible resourcesand an estimation of a basicmodelwith resource constraintsforUStime seriesdata.The estimated years left until depletion and the empirical trends of the ratios of capital stock and consumption^to resources seem to indicatethat theremightbe a threat to sustainablegrowthinthe future.Inourestimation, we obtainparameter values,whichhelptointerpretthe extenttowhichgrowth with exhaustible resources is sustainable.

Keywords: Resource; Consumption; Intergenerational;Economicgrowth

INTRODUCTION

Since the early 1970s, there is a growing concern that human activity depletes resources and pollutes the environment.

As

economic decisions are restricted by the finiteness of natural resources, and by the limited capacity of the nature to absorb pollution, attention is devotedtothequestionwhether it ispossibleand desirable tocontinuepresentpatternsof economicgrowth.

In

1972,economistslikeMeadowsetal.

(1972)

^orDaly (1987, 1989) formulated pessimistic predictions about a

"sudden and uncontrollable decline in bothpopulationand industrial capacity"ifno "conditions forecological ^and economic stabilitythat is sustainable far into the future"

are

established.’

Other economists like Beckerman

(1974)

have theoptimistic viewthattechnological progress and the discovery of new substitutes make continued economic growth possible.

A

general consensus ofthe economicgrowthdebate is that there aretrade-offsamong environmental and economicgoals. Theagreementisthat economicactivity, whichignores^thebiological ^orsocial system, is not sustainable. There exist many different definitions of sustainability, ^{but all} of them have two pointsin common. First,they recognizethat resourceand

*Corresponding author.

tMeadowsetal., 1972, p. 23.

*seealso Nordhaus 1992.

*seealso Solow(1991,1992).

environmental constraints affect the patternsofdevelop- ment andconsumption inthelong-run. Second, they are concernedaboutequity^betweengenerations (intergenera- tionalequity). Oneof the most famous definitions is stated by the Brundtland Commission in 1987: "sustainable developmentisdevelopmentthatmeetsthe needs ofthe present without compromising the ability of future generations to meet their own needs." Similarly Solow

(1974)*

^definessustainablity^as

"an

obligationtoconduct ourselves so that we leave to future the option ^or the capacitytobeaswell off asweare." Pearceetal.

(1990)

point outthat "natural capital stock shouldnot decrease over time" whereas

Pezzey

(1989) defines sustainable economic growth as "non-declining output ^or consump- tion over time"and sustainable economicdevelopment^as

"non-declining utilityover

time."*

As to the sustainability of natural resources, one can distinguish between renewable and non-renewable resources. Dynamic models ^on renewable resources can befound in Clark

(1990),

Semmler andSieveking

(1994)

and Sieveking and Semmler

(1997).

In those papers, theorems on the sustainability of resource economics, whichhave beendeveloped by studyingoneresource,only are evaluated for the case when resources interact as an

ISSN 1026-0226(C)2002Taylor &Francis Ltd

ecological system. For the latter case, resource manage- ment problems and policies aiming at conserving resources are studied. Furthermore, in Semmler and Sieveking (1994),credit financed extractions of resources areconsideredand the fate of resourcesexplored.

The current paper deals with dynamic models with exhaustible resources.

We

discuss prototype growth models that incorporate and study the consequences of finitely available exhaustible resources. Some of the problems studied here will also arise in the case of renewable resources, so for example, the problem of intergenerational justice. The remainder ofthe paper is organized as follows.

In

the second section, we survey growthmodels with natural^resourceconstraints.The third section discusses theproblemofintergenerational justice.

The fourth section presents stylizedfacts on exhaustible resources pertainingtothe US.The fifth sectionpresents the estimation of ourgrowthmodel and the sixth section concludes thepaper.

MODERN GROWTH MODELS AND RESOURCE CONSTRAINTS

We

herereview growth models, whichtake into account the presence of exhaustible natural resources. The main interest ofthe analysisis thequestion towhat extent the processof economicgrowthis restrictedbythe finiteness ofresourcestocks and whether sustainedconsumptionand utilitylevels are feasible.

We

first considerabasic model wherethe exhaustible natural resourceisusedasan input for the production of^a good, which is then either consumed or added to the capital stock to enhance future production. Then, the consequences of different extensions concerning techno- logical progressare analyzed, and, finally, it is assumed that natural resourcesmay provide services inpreserved states.

Before turning to the description ^{of the} models, it is important to have a clear distinction between renewable and exhaustible resources. The main feature of an exhaustible resource is that its growth rate is nil, and that itis notrecyclable. Furthermore, itis usedup when used as an input in production.

To

study a meaningful economicproblem,the natural resourcemustbe essential, i.e. productionisimpossiblewithout

it.

The BasicModel

Dasgupta

and Heal (1974), Heal

(1985),

Stiglitz

(1974;

1993)

and Solow

(1973)

analyzethe optimaldepletionof exhaustible natural resources in the contextof a growth model where the resource is used as an input for the production of^a composite commodity.* ^The production function

F

dependsontheflow of the exhaustibleresource

*Dasguptaand Heal, 1979.

*seealso Davison(1978).

atdate andonthestock ofareproducible good^atdatet.

In

order to obtain greatest possible social welfare, the presentvalueofutility Uderivedfrom consumption

Ct

^of

theproduced goodis maximized subjecttothe evolution of the reproducible capital stock

Kt

^and the constraints imposedbythe finiteness ofthe resource stock

St:

Max

^o

U(Ct)e -

^dt

⁽¹⁾

Subjectto

It ^F(Kt,

^Rt)

Ct

St=So- Rt

^dt

t -Rt

o

6 denotes the discount rate, and

Rt

is the flow of the exhaustible resource. The initialcapital stocks

Ko

^and

So

are strictly positive and given. The production function

F(Kt,Rt)

is assumed to be increasing, strictly concave, twice continuously differentiable and homogenous of degree unity.Theutilityfunction

U(Ct)

issupposedtobe strictly concave, and for

Ct-- ^O,

^its first derivative is infinity.

Here,

the extraction of the resource is assumedto be costless.

Solving the maximizationproblemandcombiningthe optimality conditions yields the following results (for details see "The BasicModel" section in theAppendix A):

first, along ^an optimal path, the rate of consumption depends ^on the discount rate 6, on the elasticity of marginal utility ^of consumption and on the marginal productivity^ofreproducible capital

FK:

?t FK

⁶

(2) with

rl(Ct) -CtU"(Ct)/U’(Ct)

^and

FK F(Kt,Rt)/i3Kt.

Equation

(2)

states that the higher the discount rate, themoretherateofconsumptionfalls over time along an optimal path. Second, along ^an optimal path, the rates ofreturn of exhaustible and reproducible capitalareequal:

OFR

¹

FK

_Ot ⁽³⁾

FR

with

FR )F(Kt,Rt)/Rt.

Ifthe production function is homogenousofdegreeone, it ispossibleto setxt

Kt/Rt

with

f(xt)=F(Kt/Rt,

1). Substituting

FR=f(xt) xtf(xt)

and

FK ^=f(xt)

ⁱⁿ

^{Eq. (3)}

^{yields the following}

capital-resourceratioalong^anoptimal path:

Yet f ^(xt)

tr (4)

xt xt

with

-f’(xt)ff

(x,)

xy’(x,))

as the elasticity of substitution between reproducible capital and the exhaustible resource. Equation (4) represents the rate at which reproducible capital is substitutedfor the exhaustible resource.

It

dependsonthe elasticity of substitution and on the average product per unitof fixedcapital.

In

^order to conclude whether a positive ^{level of} consumptionis sustainable over time,

Dasgupta

andHeal (1974) analyzeaneconomywhereoutputisproducedbya CES-production ^function, i.e. the case of a constant elasticityof substitution. There are three casestomention:

1. o"-- 1 (i.e. the Cobb-Douglas-production function).

The exhaustible resource is essential and infinitely valuable atthemargin, whereas theasymptoticvalue of marginal productivity ^of capital is zero. Solow (1973) concludes that sustainedper capitaconsump- tionis feasible ifthe share ofcapital ^exceedsthat of natural resources.

2. 0

-<

o"

<

1. The exhaustible resource is essential, but finitely valuable at the margin. Thus, a positive and non-decreasinglevel ofconsumption over an infinite time horizon isnotsustainable.

3. ^oo

>

o’> 1. Sustained consumption is feasible because in this case, the exhaustible resource is inessential.

Technology

The basic model can be augmented by introducing technical change, which makes it easier to find new substitutes in ordertorenderanessentialnatural resource inessential.

Dasgupta

and Heal

(1974)

^{assume that} technical progress is uncertain: the exact date of discovering a substitute and its detailed characteristics and usefulness are unknown. The new technique is supposedtooccuratanunknowndateTwhich isarandom number with an exogenously given probability density function

Probability(T t)= mr, _{ot dt=}

1,

@

>

0

0

In

order to express the situation of uncertainty, ^the objective is to maximize the expected present value of utility.After somemanipulation,one obtainsthefollowing maximization

problem:

Max [U(Ct)t +

^{ootw(gt,St)]e-tdt}

⁽⁵⁾

0

subjectto

[2t F(Kt,Rt)

o

Dasgupta,

1994pp.20.

See

alsoSievekingand Semmler, 1997.

withg2t

t

^{o)tdt and}

W(Kt,St) Max U(Ct)e^-t-rldr.

T

Kt, Ct, Rt

^and

St

^are all non-negative and the initial values

Ko

^and

So

^aregiven.

Solvingthe maximizationproblem, combiningthe first order conditions, andarguing^thatatthediscoverydate of the substitute, the then existing stocks ofreproducibleand naturalcapitalhave no economic valueanymorebecause the newtechnologyismoreefficient(i.e. W/

Ws ^0),

allows for the following conclusions (for details, see

"Technology" section in Appendix

A):

first, along an optimal path, the rate of consumption depends ^{on a} modified discountrate:

Ct (6)

with

fit o)t/12t

^as the conditional probability of the technologicalbreakthroughatdatet, given the substitute has not been discovered earlier. The discount rate is modified by the addition of the factor

Pt

^{showing the}

probabilityof the essential resourcebecominginessential as a result of technical progress. Thus, in a situation of uncertainty, the discountrateis higher than in a situation ofcertainty. Obviously,^the equation describing^{the ratio} of capital-resource input is thesame asbefore:

.ict f

^(xt

tr

(7)

Xt Xt

Stiglitz (1974,

1993)

examines an economy where output is produced by a Cobb-Douglas-production function.

He

concludes that sustained per capita consumptionis feasible, if there isaresourceaugmenting technicalchange atany positiverate(for o-

>-

1).

Toman

etal.

(1993)

point outthat for the case o-

< 1,

sustained per capita consumption is possible if technological progressishighenough.

BackstopTechnology

So far, it is assumed that the natural resource is exhaustible, i.e. once it is used up, it is impossible to find more, and that the extraction is costless. As an extension, it is nowsupposedthatthe resource is available in unlimitedquantities, ^butatvariousgradesand various costs.

For

example,theoresof anumber of metals can be extracted from the deposits currently used, which are exhaustible.Iftheyareusedup,the metals themselves can be extracted from the seaorfrom rock formations, which ismuch moreexpensive. Thus,athigher prices,the natural resource may be of unlimited availability. Heal (1993)

calls this a backstop technology.*

We

can incorporateit into the basic modeldescribedin"The BasicModel." The totalamountof the resource used atdate isdenoted as follows:

z(t)

Rt

^dt

o

It

is assumed thatat date T, the conventional deposits areexhausted andabackstop technologytakesover.

Up

^to the level zr, the extraction costs rise with cumulative extraction, then thebackstop technologyisavailable at a constantcostperunit b.g(zt)denotes theextraction costs perunit atdate with

Og/Ozt g(zt) >

0 for0--< _zt <--

z

andg(z) b

>

^0forzt

>-- z.

The maximizationproblemissolved intwo steps (see

"Backstop Technology" section inAppendixA): first, the situation is examined before current deposits ^are exhausted (maximization problem (8)), second, the situation is examined after the backstop technology has takenover(maximization problem

(9)):

Max U(Ct)e

^-6tdt

(8)

0

subjectto

Condition (12) is a generalization of condition (4).

Here,

the equation is augmented by the term

(f(xt)/xtf"(xt))(g(zt)/xt)

which reflects the cumulative costsof extraction.

Following ^Heal (1993), we can draw the following conclusions. During the initial period, the lower-cost stocksof the natural resourceareexhausted,and thepath oftheeconomyis describedby problem (8)and conditions (10) and (11). The difference between prices and extraction costs, i.e. the user costs, decline until they reach zeroatdate

T,

when thebackstop technologytakes over because the lower-cost stocks are totally used up.

From

then on, theeconomy^behavesaccording^toproblem (9) thus,the extractioncostsof the natural resourcealways equalits price.

INTERGENERATIONAL EQUITY

As

apparent from the above models, the depletion of resources generates externalities for future generations.

Therefore, the problem of intergenerational equity arises. In order to study this problem, we will first show how natural resources may affect the welfare of the society.

[(t F(Kt,

Rt)

Ct

^g(zt)Rt

St So Rt

^dt,

t

o

whereg(zt)Rt representsthetotalextraction costs.

Max U(Ct)e^-6tdt (9)

0

subjectto

t ^F(Kt,Rt) Ct bRt

The initialcapital stocks

K0

^and

So

^arestrictly positive and given. Computing the conditions along an optimal pathofproblem

(8)

yields"

?t FK

⁽³

(0)

The Amenity Value of a natural

Resource

Therearetwoways of howanaturalresourcecontributes tosociety’s welfare. The models describedsofar referto the firstway:the resourceisutilizedas aninput factor for the production of a composite commodity. The second way of a natural^resourceserving for thewell-being^{of the} society is that it may provide valuable services in preserved states, that is scientific, recreational, and aesthetic values. To take into account these so-called amenity values

of

natural resources, the resource stock

St

is included in the utility function

(Krautkraemer,

1985).

Theobjectiveis tomaximizepresentvalue ofutility (see

"The Amenity Value ofa Natural

Resource"

section in Appendix A):

Max U(Ct,

St)e^-gtdt

(13)

o and

FR

^{1 FKg(Zt)}

FK

^-]- (11)

Ot

FR FR

Substituting

FK f(xt)

^and

FR f(xt) xd(xt)

with

f(xt)--F(Kt/Rt, 1)results

in the following capital-resource ratioalong ^anoptimalpath:

yc_, ^of(x,) ₊ ^f’(x,)

^g(z,)

₍₁₂₎

Xt Xt XtjCll(xt) X

subject ^to

t

^{F(K,, Rt)}

Ct St So Rt

^dt

t -Rt

0

where

Ct, Kt, Rt, St

^are all non-negative. U(Ct, St) is assumed to be twice continuously differentiable with UC

OU(Ct,St)/OCt >

0 and

Us

^OU(Ct,

St)/OSt) >

O, Ucc (02U(Ct, St)/OC2t) <

0 and

Uss (O2U(Ct,St)/

*seealsoOrenetal.(1993).

and

limUc

Ct.--O

The production function

F(Kt,Rt)

^{has the} same properties ^as before. Solving the maximization problem and combining the first order conditions yield ^the following conditions for the rate of consumption and for thecapital-resourceratioalonganoptimalpath:

t FK - ^{(Ucs/Uc)et (14)}

Ct

^q

2,

f(x,) Us/Uc

o-+ (5)

xt xt

xtf"(xt)

The amenity ^services, of a natural resource affect the extraction and the consumption mainly through their contribution to the rate of return to the resource stock.

Resource

amenities enhance the value of the resource stock. Therefore,the initialpriceof the resourcemustbe higher than in the previous models. Furthermore, since

ftt(xt) < O,

the amenity services lower the^rateofchangein thecapital-resourceratiobytheterm

((Us/Uc)/x2tf"(xt)).

The higher ^the marginal rate of substitution between amenities andconsumption,the higher the reduction of the rateofchangeof the inputratio.

In this framework where natural environments are valued in their own rights, the so-called Green Golden Rule can be introducedwhichismotivatedbythe Golden Rule

of

^{EconomicGrowth.#}The Golden Rule of Economic Growthgivesthegrowth pathwiththehighest indefinitely maintainable level of consumption, whereas the Green Golden Rule focusesonthehighestindefinitely maintain- able level of instantaneousutility. Thus,^theGreenGolden Ruleincorporates^{the aim of}sustainability. Formally, the rulecanbe writtenas

maXfeasiblepathslimU(Ct St)

If the resource is used as an input factor for the productionofacompositecommodity,in thelong-run,the only constant level of resource input is zero. Since theresource isessential, no output^canbe produced and consumption is zero.

Hence,

the highest indefinitely maintainable utility level is feasible, if the total initial stock is conserved.

Intergenerational Equity

Standardgrowth models,whichincorporate^theconcept^of sustainability, ^{focus on the} consequences ^{of natural} resource constraints onthelong-run pattern ofeconomic

IIChichilnisky,1996; BeltrattietaL 1994, 1993, 1995, 1996.

#Phelps, 1961.

**Heal,1998, p. 43.

ttRawls,1972.

*SeealsoDixit etal.(1980),^Hartwick(1977).

development and consumption.

In

order to determine inter-temporal welfare,recentgrowth theoryhas used the concept of discounted utilitarism that is the future is discounted in comparison with the present.

Ramsey (1928)

^states that discounting is "ethically indefensible and arises merelyfrom the weakness of theimagination"

becausea positivediscountrateresults in anasymmetric treatment of present and future generations. Thus, discounted future utility neglects intergenerational equity as the second important point of the concept of sustainability. In the following, we want to give a brief review ofsome alternative concepts, which try ^to meet the objective of a fair treatment of different generations.

A simple way ^{to account for} intergenerational equity istoassumethecaseofa zeroutilitydiscount rate, that is present and future generations ^are given ^{the same} weight. Another alternative is to apply the "overtaking criterion" as proposed by Weizicker

(1967),

which states that one consumption path is better than another, if from some date ontotal utility of thatpath isgreater.

Formally,if

TU(C)dt

^>_

^U(C2t

^)dt

o o

But applying ^these approaches gives rise to technical problems. For a zero discount rate, the set of attainable values of theintegral may^beopen,and thewayofranking consumptionpaths accordingtotheovertakingcriterionis incomplete.

Accordingtothe Rawlsian Criterion

**

intergenerational equity ^means: maximize the welfare of the less advantagedgeneration.Formally,

maXfeasiblepaths

mingenerations

^(Welfaret)

Theconsequenceof this decision rule is that the welfare level should be the same for all generations. If a later generation enjoys higher welfare, an earlier generation should increase its own welfare attheexpenseof the later generationand vice versa. Solow(1973) pointsoutthat in comparison with the discounted solution based on utilitarism, the Rawlsian Criterionwill useupthe natural resource stock faster. Since the utilitarian rule demands higher savings, earlier generations will have a lower standard oflivingthan theconstantmax-min rulewould

generate.*

The Rawlsian Criterion has two main difficulties: first, a society needs an initial capital stock high enoughtomakeadecent standard ofliving possible, but theexplanationof its existence ismissing. Second,^the ruledoesnotyield^areasonableresult,ifongoingtechnical progressisassumed.^##

##Amoreelaborateversionof the Rawlscriterionisproposedin SemmlerandSieveking,2000.

More

recently, Chichilnisky

(1996)

defines two axioms for sustainability, which deal with the problem of intergenerational equity. The first axiom states that the present generation should not dictate the outcome in disregardfor the future. Thesecondaxiom statesthat the long-run future should not dictate the present. Welfare criteria, which do satisfy ^the two axioms, are called sustainablepreferences.

In

orderto formulateacriterion, which doesbelongtothe class of sustainablepreferences, positive weightisplaced on thepresent and onthe very long-runproperties of^a growth path. Formally,

x

U(Ct, St)A(t)dt +

⁽¹

c0

_p--*OO^{limU(Ct St)}

0

wherec

(0,

1).

A(t)

isanymeasurewith

f0 ^A(t)dt

^1.

If

A(t)

^e

-6t,

^thefirst termisjustthe discountedintegrals of utilities. The secondtermreflects thelimiting properties of theutilitystream, and it hasalreadybeen mentionedas the Green Golden Rule solution. The Chichilnisky Criterion places more weight on the future than the standardapproachofdiscounting utility,but less than the Green Golden Rule. It is possible to apply the Chichilnisky Criterion to neoclassical growth models at the aforementionedtype ^butfindingthe solution isquite complicated. We therefore leave aside detailed discussions.

Thisveryshort review of differentwelfare criteria has shown that it isverydifficulttofindapproaches,whichdo meetthe objective ofpermitting intergenerational equity and whicharetechnicallyoperable atthe same time.For that reason, discountedutilityisstilldominant, asit isthe technicallymostconvincingapproach, thoughitfavors the present over the future.

STYLIZED FACTS

Economic theory states that substitution possibilities, technological progress and the value of the resource in preservedstatesmay preventthe totaldepletion^{of natural} capital.

In

this section, thepatterns^{of some} selectednon- renewable resources of the US economy are analyzed from1960to1995. This then servesasbackgroundforour estimations in the next section.

Here,

extraction rates, resource quantities available today and in the future are examined withrespect tothe question whether there ^are reasonstoarguethatthe resources will besoonexhausted or that improved technology and the development of reproducible substitutes make a sustainable economic development possible.

Two

different kindsofnatural resources fulfill themain characteristicsofdepletion: fuel minerals such asenergy resources, and non-fuel minerals such as metals and industrial minerals. When talking about the limited availability of natural resources, it is important to have

For

^adetailed analysisseeHeal, 1995.

For

definitionsseeEnergyInformationAdministration,(1996;1997).

RESOURCE

STOCK

nonrecoverable resource

recoverable resource

undiscovered resource

discovered resource

cumulative

production ^reserves

other reserves provedreserves

FIGURE The components of resource stock.

clear definitions of the differentcomponentsof which the total resource stock consists. Figure ¹ explains ^the differentcomponents.

The reserves of the discovered resources consist of proved reserves and other reserves such as inferred reserves (field growth), measured reserves and indicated reserves. Proved reserves are those amounts of the resourcethatgeologicalandengineeringdata demonstrate withreasonablecertainty ^to be recoverable in the future from known reservoirs under existing economic and technological conditions. The other reserves consist on one hand of that part of the identified economically recoverableresourcethat will be addedtoprovedreserves in the future through extensions, revisions and the discovery of new fields in already discovered regions, and on the other hand, on those quantities of the resource that may become economically recoverable in the future from existing production reservoirs through the application of currently available, but as-yet uninstalled recovery technology. For details on the data sources for the subsequent summaries, see Appendix B.

As

notedbefore,fuel mineralsareenergy sources such ascrudepetroleum,coal and naturalgas.Sinceduringthe last 30years,theUS economyhasexperiencedcontinued economicgrowth, that is arising level of realGDP,total energy consumption has increased by roughly 35%. To satisfy increasing energy demand, the production of

c5

yeors

FIGURE2 Fuel:mineralproduction per1990^$GDP (coal production measured short tons, crude petroleumproduction measured in barrels, natural gas production measured in millions of cubicfeet).

especially coal and naturalgashave risen.

In

ordertodraw conclusions whether the resources are used more efficiently over time, it is interesting to analyze the patterns ofproduction rates per dollar of real

GDR

^see

Fig. 2.Table

I

summarizestheresults.

For every energy resource, the production rates per GDP are falling, i.e. crude petroleum, coal and natural gas are used more efficiently today, which may be the result of improvedtechnology. Non-fuel minerals are on one-hand metals and on the other hand, industrial minerals.

Here,

only some selected metals such as copper, iron ore, lead and zinc are analyzed. The productionratesofcopper,iron ore, lead and zinc behave quite irregularlyfrom 1960to 1995 andnotrends can be determined.

Figure3 andTable

II

show that theproductionratesper dollar real GDP are decreasing. Exhaustible energy resources canbe substitutedbyrenewableenergysources likeforexample,wind andsunpower.

Figure 4 shows the shares of the different energy sources. Today, nuclear power and renewable energy togetherarejust 20% of totalenergyproduction.Butthe trend is that theproductionof crudepetroleum, coal and natural gas is decreasing, while nuclear and renewable energy productionisincreasing.

Eh

-

,-I- ’x

;,

o| ’-7.,."

_... ...

"" ... "."

’.

... _’

_",

^.,..,..

-’,:2:.,...,....:...".2.’...

yecrs

FIGURE3 Metal:production per GDP(metalproduction in shorttons).

Copper,

lead and zinc are metals, which can be substitutedbyaluminumand plastic.Forironore, there do notexistanysubstitutes, but as it will be shownlater,the reservesof ironore arevery highandwilllast for thenext centuries.

In

ordertodraw conclusions aboutfuture availability of exhaustible resources,itisnecessarytohaveacloserlook onthe amountofprovedand otherreservesofaresource incomparisonwith itsproductionrates.

For the exhaustible energyresources crude petroleum and natural gas, it is possible to plot the reserve- productionratiofor the observed timeperiod, ^seeFig. 5.

The smaller the ratio the ^scarcer is the natural ^resource.

For petroleumand naturalgas,the trend isdeclining, thus, proved reserves will be depleted soon if for example,

extensions/discoveriesofnewfields inalreadydiscovered regionsornew discoveriesdonotmake reserve additions possible.

Every

year,the

Energy

Information Administration and the US Geological

Survey

estimate quantities of technically recoverable resource ^amounts that could be added to the already proved reserves of the US. It is interestingtoask about the number ofyearsit willtaketo exhaust the today estimated technical recoverable resources quantities.

As

the production rates of

the

energy resources, coal and natural gas are steadily TABLE Fuelmineralproduction per dollar realGDP

Crudepetroleumbarrels Coal in shorttons

Naturalgasin millions of cubic feet

1960 1.2680x10^-3 2.0526X10^-4 6.3155X10^-6

1995 3.8447 10^-4 1.6589 10^-4 3.1316 10^-6

Change (%) -70 19 -48

TABLE II Non-fuel mineralproduction perdollar realGDP

Copperin shorttons Ironorein shorttons Lead in shorttons Zincin shorttons 1960

1995 Change(%)

5.3287x10^-4 2.9700x10^-3

44

4.0952x10^-2 9.8124x10^-3

-76

1.2187x10^-4 6.1990X10^-5

--49

2.1463X10^-4 9.8605X10^-5

54

.t_:

o

years

rene;obte

FIGURE4 Energyproductionsourcesin percent of total.

o

\ ./

\.\./ . ^s

_..:’

\..,_, 1976

FIGURE5 Reserve-productionratio.

increasing duringthe observed time period,it isassumed thattheycontinue to increasewithanaverage production growth ^rate. The otherresources do not show any clear trend in their production rates and, therefore, it is supposedthat productionwillcontinuetofollowastable pattern during ^the next years. ^Tables

III-V

summarize the results.

Although the results of the empirical analysis are restrictedbythe lengthof the examined timeperiodand there is a clear decline and finiteness of resources, yet, theremaybe technologicalprogressthat makes afurther development of renewable substitutes for exhaustible resources possible.

As

the US economyhas experienced continued economic growth during the second half of this century, the production ^rates of exhaustible resources have risen and for all analyzed minerals, the production per dollar real GDP has declined. Increased efficiency, as the consequence of advanced technology and the use of close substitutesarelikely explanationsfor that fact.

Assuming no changes in production trends yields predictions of the number of years left until present estimated resource reserves are depleted. The estimated reserves of coal and naturalgaswill last for thenext two orthree centuries.

As

the present estimated reserves of crude petroleumwould be depleted in 40 years, the US economy isdependent ^on oilimports mainly from Arab countries. Substitutes of coal, natural gas and crude petroleum ^are nuclear and renewable power. Since the share ofthese alternative energy production sources has increased from 1960to 1995,it seems likely^thatfurther

TABLE III Estimatedreservesofpetroleumand natural gas 1995

Crudepetroleum^{in bill}bbl Natural gas intrill(cu. ft) Proved reserves

Fieldgrowth

Undiscovered resources Totalreserves

Averageannual production/average productiongrowthrate Yearsleft untilexhaustion

20.2 60.0 30.3 110.5

3.0076 40

135 322 259 716

0.012 300

TABLEIV Estimatedreservesof coal1996

Provedreserves Otherreserves Totalreserves Averageproductiongrowthrate Yearleft untilexhaustion

19,428 507,740 527,168 0.026 250

TABLE V Estimatedreserves ofsomeselected metals 1997

Copperin1000 Ironore in1000 Lead in 1000 Zinc in1000

metrictons metrictons metric tons metrictons

Proved reserves Otherreserves Total reserves

Averageannual production Yearsleft until exhaustion

45 000 10 000 000

90 000 23 000 000

135 000 33 000 000

1414.25 68844.44

95 480

6 500 20 000 26 500

439.72 60

25 000 80000 105 000

437.14 240

research and development will enhance this trend. The reserves of iron ore and zinc are very high while lead andcopper^{seem to}beratherscarce.

Yet,

anincreased useof particularly plasticsasarenewable substitutemayrelax the constraintsthatasustainabledevelopmentfaces.

These predictions ^are all made under the assumption that only currently available technology is applied. It seems to be likely ^that technological progress improves for example, mining and refining ^{methods or makes} discoveries of new resources fields possible, and, there- fore,enhancespresentestimationsof available reserves.

In

orderto ensuresustainablegrowthin the sense of minimal degradationof naturalcapitalstocks andintergenerational equity, it appears to be important to develop further renewable substitutes.

ESTIMATION OF THE MODEL

In

^order to estimate the growth model as described in

"Moderu Growth Models and Resource Constraints,"

first, consumer preferences and the technology of producinggoodshavetobe specified, and, second,adata sethastobe constructed.

In

^ourestimation,wefocusonthe standard modelaspresentedin "The Basic Model."

It is assumed that the natural resource contributes to economic activity only in one way: it is used as an input factor for theproductionofacommodity, which is either consumed or added to the capital stock. The present ^{value of} utility received from consumption is givenby

C]-

1

_,

o

]-rl

^{e dt}

where, is the elasticity of marginal utility. The technology of goods is described by ^a Cobb-Douglas- production function, which depends ^on reproducible capital

Kt

and the exhaustible resource flow

Rt.

^The

elasticityof substitution betweenreproduciblecapital and the exhaustible resource o-equals 1. The evolution of capitalisdeterminedby

K R

/3

denotes the share ofreproducible capitalinproduction.

Maximizing present^valueofutility^andsetting Yt

Ct/Rt

yields ^the following estimable system: ^{for a} detailed study of the solution, see "Estimation Section" in the Appendix A.

Yt 3Xt -1-

b ⁽¹⁶⁾

Yt rl

:it x_

⁽¹⁷⁾

xt

t --= ⁽¹⁸⁾

Rt

TABLE VI Estimationresults

Parameter Value Standarderror

0.32 3.4668

0.03 0.0369

0.5 0.2488

0.002 0.1856

with

b

^{as the growth rate} of the exhaustible resource flow.

As

time seriesdata,weneedconsumption, reproducible capital stockIIII and the exhaustible resource flow.^## The reproducible capital stock,

Kt,

is gross realprivate fixed capital stock and

Ct

^is private consumption. The time series for the resource,

R,

is based on our own computation. Since the total mineral productionvalue is the amount of extracted exhaustible resources times average prices, it is used to measure the exhaustible resource flow. All time series are deflated by the GDP- price^{index 1990 100.}

Equations

(16)-(18)

are estimated directly by using non-linear least squares techniques

(NLLS).*** In

the estimation, wehaveprefixedthediscountrate,

,

^{and the}

parameterof relative risk aversion, /. The reason for this procedure isthat the modelwe areconsidering--leaving aside substitution, technological change and the role of otherinputs--isin itscurrentform,ratherincompleteand reliable estimates for the discount rate as well as for relative riskaversion, cannotbe expected. Therefore, we prefix ^them at levels that have been obtained by other recent studies (see Semmler and

Gong,

1997). This procedure can be understood as a calibration exercise rather than estimation. The results ^are summarized in Table

VI.

The estimated capital share in income,

/3,

^{and the} estimated growth rate of the resource flow,

b,

^are reasonable.

Althoughthere is avery irregularbehavior of the total mineralproduction^{value over} the observed timeperiod, the data show an enormous increase of the value of resources in the years 1972-1981 caused by the price effects of theoilcrisis.

Figures 6 and7 showthe estimated (fitted) and actual time series for the ratio of capital stocktoresources and the ratio ofconsumptiontoresources. Bothfigures show that there is, in particular, since 1980, ^a strong trend toward thedepletionof resources relativetocapitalstock andconsumption.

As

the figures show, the model with our estimated parameters matches the data well.

As

alreadynoted, the very simple structure of the model may explain the observable slight correlation of the error terms.

We

also have supposed that population remains constant which implies that labor as an input factor in production is

IIIIData

onconsumption andcapitalstockareobtainedfrom Citibase(1998).

##Source:USDepartmentofCommerce,Economics and Statistics,Bureauof theCensus1965-1997.

***ComputedwithGAUSS--OPTMUMversion 3.00.

yeers

FIGURE6 Capital-resource ratio.

1996 yeors

FIGURE 7 Consumption-resourceratio.

constant, too. This assumption seems justified when analyzing the very short-run, but not appropriate when examiningatime period of 35 years. It^seemstobelikely that incorporatingthe factor labor into the model would improve the estimation results. Such animprovement is particularly critical for our estimation result

on/3,

^since

ourestimategives^avalue

of/3

0.32. This would mean, according to Solow (1973), that sustained per capita growth is not feasible (see "The Basic Model").

Improvement

of our estimations by including further factors would most likely give us the result that exhaustibleresource share in production, 1

-/3,

^islikely tobe smaller.Furthermore,^asthestylizedfactssupport,it would be reasonable^toallowtechnological progressand substitution.Finally,thequalityof non-linear estimations depends stronglyonthenumber of observations.Sincethe analyzedtime period consist ofonly 35 data points, it is difficulttoachieve sufficient robustness in the estimations.

Thisproblemwasclearly observable,whenweattempted to estimate the prefixed parameters and $. Their estimation in factturned outtobe non-robust with respect tothealgorithmused.Wethereforekeptthemprefixed.

In summary, although our preliminaryresults may be illuminating,future research should take into account the

factor labor and technological progress and substitution, andtoestimatesuchamodel overalongertime periodor withdata withahigherfrequency.

CONCLUSIONS

This paper attempts to study in a formal model, the growing concern that human activity and economic growth depletesnatural resources.Although both,renew- able as well as exhaustible resources are threatened by extinction in the process of economic

growth****

this paper in particular focuses on exhaustible economic resources.

We

pursuethequestionof whether thecurrent rateofextraction of exhaustibleresources is sustainable giventhe presentpatternof economicgrowth.

We

present time seriesdata andgive roughestimatesofdepletiontime for exhaustible resources. We also studythe problemof intergenerational equityand the different criteria thathave beensuggestedin the literaturetoconserveresourcesand dojustice to future generations. In a particular growth model,^westudytowhatextent resourceconstraintscanbe overcome by technological change and substitution.

Although it would be advisable to consider worldwide trends in the exhaustion of resources,webecause ofdata problems, restrain our study to empirical ^{trends and} stylized^factsonexhaustibleresources of theUS economy.

We

estimate a standard growth model with resource constraintsforUStime series data.Wecanobserve from ourtime seriesdata that there is,disruptionbythe strong increase ofthe value of resources duetothe oil crisis in the 1970s, astrongdecreaseof theuseof resources relativeto capital stock and consumption since 1980. Our econo- metric estimationsconcerning^thecapitalshare point into theoppositedirection.

Yet,

ourestimates, in particular our estimate of the capital share, which indicatesmif one follows Solow(1973)mnon-sustainable growth,^hastobe interpreted with some care since the estimated growth model lacks other variables, uses limitedtime seriesand low frequency data.

Moreover,

the lack of data on technological change and substitutes for exhaustible resources prevents^usfromdrawingthe strongconclusion that there is a threatto sustainable growth inthe future.

Yet,

^theratioofcapitalstockorconsumption^toresources has tripled, possibly indicating ^futureproblems concern- ing the availability of exhaustible resources orthis also may indicate that technical change and substitution has reduced theuseofresources.

Acknowledgements

We would like to thank Toichiro Asada for his helpful comments.

Forstudies ontheextinctionof renewable resources,seeClark, 1990; Semmler and Sieveking 1994, 1997, 2000.

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APPENDIX A

The Basic Model

The followingmaximizationproblemmustbe solved:

Max

U(Ct)e

^-rtdt

0

(A1)

subject to Computing thefirstorder conditionsyields

t

^F(Kt,Rt)

^Ct, St So Rt

^dt

o

The currentvalue Hamiltonian isdefined as

Hc ^U(Ct) +

^pt(-Rt)

+

qt(F(Kt,Rt) Ct)

(A2)

where Pt denotes the shadow price of the resource constraint and qt denotes the shadow price of capital accumulation.Computingthe first order conditionsyields thefollowing equations,whichmustbe fulfilledalong^an optimalpath:

Dt

^{6pt (A3)}

Ut(Ct)-

qt

(A4)

Pt

qtFR (A5)

(It- 6qt

--qtFK (A6)

with

FR OF(Kt,Rt)/aRt

^and

FK aF(Kt,Rt)/aKt.

Differentiating

Eq. (A4)

with respect to time and substituting

Eq. (A6)

yields the consumption rate along anoptimalpath:

t FK

⁶

Ct

rl

(A7)

Differentiating

Eq. (A5)

withrespecttotime and using

Eqs. (A3)

and(A6)results in

aFR

¹

FK ^(A8)

at

FR

Substituting

FR =f(xt) -Xtfl(Xt)

and

FK ^=f1(xt)

^with

f(xt) F((Kt/Rt),

1) _{and xt}

Kt/Rt

ⁱⁿ

Eq. (A8)

yields the capital-resourceratioalonganoptimalpath:

2t f(xt)

o-

(A9)

xt xt

1)t 6pt (A12)

OtU’(Ct)

qt

(A13)

Pt

ootWs

^nt-

qtFg (A14)

(It

6qt

o)tWK

qtFK (A15)

Differentiating

Eq. (A13)

with respect to time and substituting

Eq. (A15)

results in

d;t FK-

⁶

+ ^d/t(WK- UI(Ct))/U’(Ct)

(A16) Ct

with

t oot/l’t.

^If

W Ws ^0,

^{onegets the}following consumptionratealonganoptimalpath:

d;t FK (3 + ^d/t) _(A17)

Ct

rl

BackstopTechnology

The followingmaximizationproblem^mustbe solved

Max U(Ct)e^-atdt

(36)

0

subjectto

F(gt,Rt)

Ct

^g(zt)Rt,

St- So Rt ^dt,

o

-R

where g(zt)Rt represents the total extraction costs. The currentvalue Hamiltonian is

Hc ^U(Ct)

^{-k-pt(-Rt)-4c-}qt(F(Kt, Rt)

Ct

g(zt)Rt) (A19)

Computing the first order conditionsyields

6Ptqtg

(Zt)Zt

fit

^6pt

+

^qtg

^(zt)Rt

Technology

Thefollowing maximizationproblemmustbe solved:

Max

[U(Ct),lt

n^t-(.otW(Kt, St)]e^-6tdt o

(A10)

subjectto

t

^{F(Kt Rt)}

Ct St So Rt

^dt

t -Rt

0

The currentvalue Hamiltonianis

Hc U(Ct)t

"[-"ootW(Kt, St) -+-pt(-Rt)-t- qt(F(Kt, Rt) Ct)

(All)

ag(zt)

6ptqt

(A20)

Ut(Ct)

qt

(A21)

--Pt qt(FR g(zt)) (A22)

(It- 6qt

--qtFK (A23)

Differentiating

Eq.

(A22)withrespecttotimeandusing

Eqs.

(A20)and

(A23)

results in

aFR

^{1 FKg(Zt)}

FK

^{-!- (A24)}

at

FR FR

Substituting

FR =f(xt)- Xt(Xt)

and

FK--fl(xt)

in

Eq. (A24)

results in thefollowing capital-resourceratio