kirk hamilton (sustainable development the hartwick rule and optimal growth)

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SUSTAINABLE DEVELOPMENT,THE HARTWICK RULE

AND OPTIMAL GROWTH

BY

Kirk Hamilton

CSERGE Working Paper GEC 93-23



SUSTAINABLE DEVELOPMENT,THE HARTWICK RULE

AND OPTIMAL GROWTH

by

Kirk Hamilton

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)

The advice and comments of David Ulph, Malcolm Pemberton and David Pearce aregratefully acknowledged. The usual caveats apply.

ISSN 0967-8875



1

1. INTRODUCTION

Concern about damage to the environment and depletion of resources has made sustainable

economic development a concept with both wide currency and wide interpretation, as Pezzey's

(1989) exposition demonstrates. Although it is relatively easy to criticize various notions of

sustainable development (see, for instance, Nordhaus (1992)), it is the goal of this note to

explore a particularly simple definition, that per capita utility be non-declining, owing to Pezzey.

The key question to be answered is whether, or under what conditions, sustainable

development so defined is consistent with optimal growth and finite resources.

Minimal sustainability is defined to be constant utility over time. If the utility function is a

continuous and non-decreasing function of consumption, then minimal sustainability is

equivalent to constant consumption over time.

The initial problem to be examined is that of finding a development path with maximal

consumption that is minimally sustainable. Stated this way, it is clear that this is equivalent to a

maximin programme, which has been studied by Solow (1974), Hartwick (1977,1978),

Dasgupta and Heal (1979), Dixit, Hammond and Hoel (hereafter DHH) (1980), and Dasgupta

and Mitra (1983). The starting ethical position in this work is different, essentially a Rawlsian

framework in which welfare across time is equal to that of the least well-off generation, but the

end goal is the same: maximal constant consumption. A key result in this literature is Hartwick

(1977), which showed that the "Hartwick rule", to invest resource rents, is a sufficient condition

for a maximin programme.

The first part of this paper fills in several of the gaps in the literature on the Hartwick rule: (a) a

simple proof is given, for general production functions, that the generalized Hartwick rule

combined with the Hotelling rule is necessary and sufficient for maximal constant consumption;

(b) the behaviour of a Hartwick-Hotelling programme for CES production functions is thoroughly

explored, including a correction to one of the results in Hartwick (1978); and (c) a weaker

condition for maximal consumption than that in DHH (1980) is derived for CES production

functions. Some conclusions are drawn concerning the debate on "strong" versus "weak"

sustainability (Pearce, Markandya and Barbier, 1989).

The second part of the paper examines the conditions under which utility can be increasing in a

programme of optimal growth with finite resources.



2

2. THE GENERALIZED HARTWICK RULE

We assume that there is constant population (so that labour can be treated implicitly in the

production function), and no disembodied technological growth. The initial endowment is a

stock S 01 of resources and K 02 of capital. Output F 3 is produced from capital K 4 and

resources R 5 according to the production function,

),( RK F F = such that ,0, > RK F F .0, < RRKK F F

Consumption C 6 is defined by the following set of differential equations:

. RS

K F C

−=

−=

&

&

(1)

Any efficient programme of production, investment and consumption must satisfy two criteria:

,K

R

R F F

F =

&

(2)

and

.0lim =∞→ t

t S (3)

The first of these is the familiar Hotelling rule; in the economy postulated holders of naturalresource stocks must be indifferent between holding resources or the alternative asset, capital,

which yields F K 7. Any programme that left unexploited natural resources would clearly be

inefficient, hence expression (3) which says that the programme must exhaust the initial

resource stock.

The generalized Hartwick rule (to use the terminology of DHH) is given by:

),( v RF K R +=& v constant, (4)

where the return to resources, F R 8, is the resource rental rate. What follows is a proof, for

general production functions, that the combination of the Hotelling rule and the generalized

Hartwick rule is necessary and sufficient for constant consumption. Applying expressions (2)

and (4) we have: )( K F dt

d C && −=

))(( v RF F dt

d

R

+−=



3

RF v RF F R R&&& −+−= )(

RF v RF F F R RK

&& −+−= )(

RF K F F RK

&&& −−=

.0=

Alternatively, assuming C = 09 we have,

C F K &&&& −=

K F RF K R

&& +=

K F

F RF

R

R

R&

&& +=

).( RF K F

F RF RF

R

R

R

R R −++= &&&&

Now define , RF K Z R−= &

so that , RF K Z R−= &&&

and therefore the preceding expression for K 10 can be written as

. Z F

F Z

R

R&

& =

This equation has solution Z = vF R 11 for constant v 12, and therefore,

).( v RF K R +=&

It is clear from this derivation that the parameter v 13 is simply a constant of integration - there is

no obvious economic interpretation. Having established that the Hotelling rule and the

generalized Hartwick rule are together necessary and sufficient for constant consumption, the

next questions to be examined are the behaviour of the system under different assumptions

about v 14, the attainment of maximal consumption and the existence of solutions under this

programme.



4

The efficiency condition that 0→S 15 implies that 0→ R 16. If v <017, eventually K <018, and

therefore, assuming capital can be consumed and given the fixed initial endowment K 019, both

R 20 and K 21 will tend to 0; assuming that no output is produced purely by labour, constant

consumption is impossible.

A result that will be useful in what follows is that,

RF v RF C K F R R

&&&&&& ++=+= )(

)(v R

RF K K

++=

&&

).(v R

R

F

F K

R

R

++=

&&& (5)

To take the argument further requires more structure for the production function. Since it is

clearly the degree of substitutability between capital and resources that is of key importance in

models with exhaustible resources, a convenient functional form is the class of constant

elasticity of substitution (CES) production functions. Hartwick (1978) poses several questions

about the behaviour of the system under the Hartwick rule for different values of the elasticity of

susbstitution, but does not derive the answers.

Defining σ22 to be the elasticity of substitution, we have, assuming constant labour force and

normalizing per unit of labour,

.1,0,1,0,,)1( 1

11

≠><+>−−++= −

−−

σ σ β α β α β α β α σ

σ

σ

σ

σ

σ

RK F

For the case σ=123 this reduces to the familiar Cobb-Douglas form.

It will be convenient in what follows to define the following expressions:

),1(

11

β α β α σ

σ

σ

σ

−−++=

−−

RK X (6)

and

.1

1

<=

−

X

R σ

σ

β γ (7)

With these definitions we can derive,



5

.)(

11

1

1

σ σ σ β β β R

F R

X

F X F

R ===−

−

The examination of the behaviour of the system for v >024 will be divided into three parts,

according to the assumptions about the elasticity of substitution. For the Cobb-Douglas function

we have,

vF RF F C R R −−=

).1( R

vF

β β −−=

Therefore C 25 becomes negative as 0→ R 26, contradicting C =027.

If σ<128 then, vF RF F C R R −−=

σ β γ 1

−

−−= R X

F vF F

).

)1(

1(111

σ σ σ

σ

β α β α

β γ

R R RK

vF

−−+

−−=−

Again, C 29 becomes negative as R ®030 (note that −→1γ 31), contradicting C =032.

If σ>133 then resources are not essential for production. For large σ34, i.e. as

β σ →∞→ RF , 35, 36 and α →K F 37, so that the Hotelling rule is violated. Efficient

production is impossible when capital and resources are perfect substitutes in a CES production

function. As derived in the Appendix, the growth rate of output for general σ38 can be shown to

be given by:

).)(

)1((

v R

v R

v R

RK F

γ γ σ

σ

−−

−−

+= &&& (8a)

It will be convenient in what follows to use this expression for the case v =039:

.)(

)1(

γ σ

σ

−

−=

R

RK F

&&& (8b)

As β γ σ →→ +

,1 40, and, from expression (8a),



6

).)1(

(v R

v

v R

RK F

β β −−

−

+→

&&&

If in the initial period R 0(1-β)-βv >041, then for some time beyond this period,

,1

v R β

β

−=

at which point output is infinite (since the growth rate is positive and infinite - recall that R <042

because of efficiency condition (3)).

If R 0(1-β)-βv <043 then F <044. Note that

11

11

)1( −−

−−

−−++= R RK R

F σ

σ

σ

σ

σ

σ

β α β α

1

1

1

)1

)(( −

−

−−−

++= σ

σ

σ

σ σ

σ β α

β α

R R

K

is unbounded as 0→ R 45. Consumption is given by,

)( v RF F C R +−=

.)()1(

1

γ β γ R

F vF −−=

Therefore if R 0(1-β)-βv <046, consumption becomes negative as 0→ R 47.

To summarize the results concerning the value of v 48: (i) if it is negative then output and

consumption decline to 0; if it is positive then (ii) if the elasticity of substitution is less than or

equal to 1, consumption eventually is negative; or (iii) for elasticities greater than 1, the Hotelling

rule is violated for very large values of σ49, or, in the case of small σ50, either output is

eventually infinite or consumption becomes negative.



7

3. THE STANDARD HARTWICK RULE

In what follows we employ the more widely known form of the Hartwick rule,

RF K R=& (9)

i.e. that resource rents be re-invested.

An important question to be explored is the behaviour of output, consumption and investment

under the standard Hartwick-Hotelling programme for varying values of the elasticity of

substitution. We first consider the case σ<151.

In this instance the marginal product of resources is bounded since,

σ σ σ

σ

σ

σ

β α β α β 1

1

111

)1(−

−

−−

−−++= R RK F R

.)1

)(( 1

1

1

1

−

−

−−−

++= σ

σ

σ σ

σ β α

β α β

R R

K (10)

Thus, since K 52 is non-decreasing under the Hartwick rule,

1−→ σ

σ

β RF as .0→ R

This in turn implies that K 53 is bounded because

∫+=

T

RT Rdt F K K 0

0 and ∫∞

=

0

0S Rdt .

Consumption is given by,

RF F C R−=

R R X X σ σ σ

σ

β 1

1

1

1−

−− −=

).1()1(

)(

1

1

111

1

1

1

β α α β α β α

β

σ

σ

σ σ

σ

σ

σ

σ

σ

σ

−−+−−++=

−=

−

−

−−

−

−

K RK

R X X (11)

Therefore, since K 54 is bounded, consumption tends to 0 as R 55 tends to 0 when σ<156.



8

This derivation can be compared with Dasgupta and Heal (1979, ch. 7), who show that if the

elasticity of substitution is less than 1, then F / R 57 is bounded, implying total output is bounded,

and therefore that constant consumption is impossible.

The preceding derivation and the result from Dasgupta and Heal (1979) contradict Theorem 3

from Hartwick (1978). This is stated as,

0>F & if 1<σ

0=F & if 1=σ

0<F & if 1<σ

Recalling expression (8b), note that γ ®1-58 as R ®059, so that, while F 60 may initially be

increasing (i.e. for γ<σ<161), eventually it must decrease. In fact, since K 62 is bounded,

eventually K ®063, so that F ®0-64, contradicting the first part of the above theorem. Because

total output is bounded, we know that F ®065 in the long run.

Next we consider the case σ>166. Since resources are not essential in this instance, both

Solow (1974) and Dasgupta and Heal dismiss this case. It is still worth exploring, however, the

operation of the Hartwick-Hotelling programme under these conditions.

Recalling expression (8b), because σ>167 and γ<168, this expression implies that F <069.

Because consumption is constant, this in turn implies that 0→ RF R 70 as 0→ R 71, since

1

1

)1( −

−

−−+→ σ

σ

σ

σ

β α α K F as 0→ R .

A further conclusion from the preceding expression is that K 72 must be bounded since F <073.

This is in spite of RF 74 being unbounded as 0→ R 75, as is obvious from expression (10).

The behaviour of the system for differing values of σ>176 is also worth exploring. As was

alluded to earlier, β α β α −−++→ 1 RK F as .∞→σ

This makes the large σ77 case uninteresting because the Hotelling rule is violated.



9

What value of σ78 maximizes consumption? This can be examined as follows, following from

expression (11):

2

11

2

1

11

)ln(()1(

)1()ln(

σ α α

σ σ σ

σ

σ

σ

σ K K K X X C

−−

−

−

−=

∂

∂

).)1(

)ln()1()

)1(

)ln()ln(((

222

1

1

1

−−−−

−−=

−

−

σ β α

σ σ α σ

σ

σ X X K

K X

For small values of σ>179 we can approximate ln(X )³ln(σ)80. Using l'Hôpital's rules we can

see that,

∞→−

2)1(

)ln(

σ

x as .1+

→σ

Consumption is therefore a declining function of σ>181. Constant consumption under the

Hartwick rule is consequently maximized as σ ®1+82, i.e. the Cobb-Douglas production function

yields maximal consumption.

Because resources are not essential for elasticities of substitution greater than 1, one strategy

for achieving maximal consumption might be to consume all of the resource in the initial period.

The derivation in the Appendix shows, however, that such a strategy will not yield constantconsumption under the Hartwick rule.

The solution of the system for a Hartwick-Hotelling programme is remarkably simple in the

Cobb-Douglas case:

, β α RK F = ,0, > β α .1<+ β α

F RF K R β ==

implying,

F C )1(0 β −= constant F ⇒ constant K F

R

R−=⇒

&

from (5)

so that

,1

0

0 t C

K K β

β

−+=

and therefore,

.)1()1(0

0

1

0 β

α

β

β

β

β

−

−+−= t

C

K

C

R



10

The condition for the existence of a solution to the system is therefore α>β83, i.e. the elasticity

of output with respect to capital must be greater than that with respect to resources, since R 84

must have a finite integral equalling S 085. Performing the integration yields the value for

maximal consumption,

.))(1( 10

10

1

0 β

β α

β

β β

β

β α β −

−

−−

−−= K S C (12)

To tie some of the literature together, it is worth describing the solutions of Solow (1974) and

Dasgupta and Heal (1979) to the maximin problem, which did not use the Hartwick rule

explicitly. Both choose the Cobb-Douglas production function after rejecting CES functions

where total output is bounded (σ<186) and where resources are not essential (σ>187). For thisproduction function maintaining constant consumption C 088 implies that,

β

α

β −

++=

+=

)()(

1

0

0

mt K mC R

mt K K

o

(13)

where m =K 89 is a constant. Both point out that efficiency requires that the integral of the above

expression for R 90 exist and be equal to S 091, so that the condition α>β92 is required.

Performing this integration yields:

.)( 000 mK S mC −−

= − β α β β

β

β α

Dasgupta and Heal then maximize this expression with respect to m 93 to yield the optimal

constant consumption,

,))(( 10

10

11

1

1

0 β

β α

β

β β

β

β β

β

β

β α β β −

−

−−−− −

−= K S C

which simplifies to expression (12).

Solow (1974) takes a different approach to deriving the optimum. By constructing phase

diagrams for the problem he arrives directly at the following expression for the parameter m 94 in

the system of equations (13),

β

β

−==

1

0C K m & (14)



11

for some fixed C 095. He then integrates the equation for R 96 in expression (13), and sets this

equal to S 097 to arrive at the maximal level of consumption, as given by expression (12).

Hartwick's key insight, that expression (14) embodies the rule "invest resource rents", is not

derived explicitly in his 1977 paper; instead he proves the sufficiency of this rule for a

programme of constant consumption.

It is also worth linking what has been presented so far to the much more general framework

employed by Dixit, Hammond and Hoel (1980)1. If we assume the existence of a competitive

output price p t 98 and a positive constant exhaustion rent µ99 (i.e. assuming efficient resource

extraction), then expression (4) may be re-written as,

)( v RF pK p Rt t +=&

)( v R += µ

,vS µ µ +−= &

where µv 100 is constant. Rearranging terms then gives the analog of the "generalized Hartwick

rule" of DHH:

vS K pt µ µ =+ &&

The authors prove that this rule is necessary and sufficient for constant utility in their general

framework. They go on to show that any path with v <0101 is infeasible.

A point of particular interest in the Dixit, Hammond and Hoel paper is their proof that, given any

efficient path such that v =0102, any other path with v >0103 and a larger capital stock for all t >0

104 will yield a lower level of utility under the generalized Hartwick rule. Expressing their

proposition in terms of the CES production function analysis presented so far, we can say that

00' 00)'( RF v RF R R >+ for ,0>v

1

I am grateful to David Ulph for pointing out this connection.



12

is a necessary condition for C ¢

0<C 0105, where the "unprimed" variables represent their values

for v =0106 (this is a necessary condition because the capital stock can only be greater than its

previous value for all t >0107 if its rate of change in the initial period is greater).

Changes in v 108 will clearly affect R 0109 since the capital stock level will be altered under the

generalized Hartwick rule and the efficient path must both exhaust the resource and satisfy the

Hotelling rule. For infinitesimal changes in v 110 it is shown in the Appendix that the DHH

condition for maximal utility is equivalent to

,1

0

σ γ

σ

−−<

∂

∂

v

R (15)

while a more direct derivation of this condition, using the machinery of CES production functions

presented so far, yields

.1

0

γ

σ

−<

∂

∂

v

R (16)

Recalling that the case σ>1111 was the most problematic in terms of analyzing its behaviour

with respect to changes in v 112, and recalling that γ<1113, we can conclude that the DHH

condition is stronger than necessary for this case, since expression (15) is negative and

expression (16) is positive. At least for infinitesimal changes in v 114 the weaker condition (16)

yields maximal consumption under the Hartwick-Hotelling programme when the elasticity of

substitution is greater than 1.

If development is to be minimally sustainable, the Hartwick rule is the key.



13

4. RISING UTILITY

Are there optimal programmes under which utility is increasing? This could be defined to be

strict sustainability . To answer this question requires a different criterion of optimality than that

employed to this point.

Given two programmes for which U >0115, it would be rational to prefer the programme for

which the sum (or, in continuous time, the integral) of utility over time is the greater. Thus, at

least for the purposes of exploring some simple models, we will assume that sustainability is

consistent with a Utilitarian ethic as applied to current and future generations. The interesting

question, it turns out, is whether there is a pure rate of time preference in the models. If there is,

the Utilitarian ethic would suggest that we wish to maximize the present value of utility, wherefuture utility is discounted at some rate r 116.

In what follows we will assume an explicit form of the utility function,

.1,)()1(

>−= −− η η C C U (17)

This is one of the class of functions for which the elasticity of marginal utility, η117, is constant.

In the analysis that follows the Cobb-Douglas production function will be used and the efficiency

of any programme will again require that the resource be exhausted, as given in expression (3).

Finding the optimal programme can now be expressed as an optimal control problem:

∫∞

−

0,

)(max dt eC U rt

RC

Subject to: C F K −=&

. RS −=&

The Hamiltonian function for this problem is,

H S K Ue rt &&21 γ γ ++=

−

,)( 21 RC F Ue rt γ γ −−+= −

where γ1118 and γ2119 are the co-state variables. The optimum programme must satisfy the

following first order conditions:



14

0' 2 =−=∂

Η ∂ − γ rt eU

C (18)

021 =−=

∂

Η ∂

γ γ RF R (19)

11 γ γ &−==∂

Η ∂K F

K (20)

20 γ &−==∂

Η ∂

S (21)

From expression (19) we see that

.2

1

RF

γ γ =

By substituting this into expression (20) we obtain

).( 22

R R

K

F dt

d

F

F γ γ −=

Since expression (21) implies that γ2120 is constant, this simplifies to

).( 22

R R

K

F dt

d

F

F γ γ −=

or,

. R

R

K F

F F

&

=

This is just the Hotelling rule. It is derived directly as an efficiency condition for the optimal

control problem.

It is easily verified that, from expression (17),

.'

''

C U

U η −=

From expression (18) we see that



15

.' 1

rt eU γ =

Differentiating with respect to time gives

rt rt

reeC U 11'' γ γ += &&

rt rt

K reeF 11 γ γ +−= (from (20))

)(1 K

rt F r e −= γ

).('K

F r U −=

We therefore have the optimality condition

.K F C

C r =+

&

η (22)

This is the "Ramsey rule" (after Ramsey (1928)). Optimality requires that the social rate of

return on investment equal the marginal product of capital. For the Cobb-Douglas production

function the percentage rate of change in consumption is therefore given by,

)(1

1r

K

R

C

C −=

−

−

α

β

α η &

(23)

Because resources are essential for production and in finite supply, resource use R 121 will tend

asymptotically to 0 in an optimal programme. The only way that the percentage rate of change

in consumption, as given in expression (23), can be non-decreasing is if capital stock K 122

eventually tends to 0 as well - i.e. eventually capital must be consumed. But if both capital and

resources are declining, then so is output. Since the initial capital stock is fixed, consumption

must eventually fall. The optimal path is not sustainable.

By equating the expression for the Hotelling rule and expression (22) it is possible to solve thedifferential equation to give,

δ η =

−− C F e R

rt

for constant of integration δ123, and by re-arranging terms,

.)(

1

1

η

β

α

λ −

−=

R

K eC rt



16

If the discount rate for utility, r 124, is 0 then it is clear from this expression that consumption will

grow indefinitely as resource use R 125 declines. As Dasgupta and Heal (1979, ch. 10) point

out, however, the optimum path will only exist if the non-discounted integral of utility is finite,

which requires that,

.1

β α

β η

−

−≥

If the elasticity of marginal utility η126 approaches infinity, then consumption will approach the

constant λ127.

This latter point bears some elaboration. If any level of consumption yields the same level of

utility (i.e. η ®¥128), then the optimal programme will be to maintain consumption constant.From the characterization of this programme given earlier, we know that investment will just

equal resource rents and, moreover, that the constant λ129 will be given precisely by

expression (12), the maximum constant consumption level given the initial endowment of capital

and resources.



17

5. CONCLUSIONS

The definition of sustainability as non-declining utility entails strong inferences about optimal

growth with finite resources. A central issue concerns the discounting of utility: if the pure rate of

time preference is greater than 0, then the traditional Utilitarian maximand, the present value of

utility, leads to an optimal programme that is not sustainable. It is worth recalling, therefore,

Ramsey's view of discounting utility as "ethically indefensible". If, however, the discount rate for

utility is 0 then, under fairly weak restrictions, the Utilitarian maximum yields continually

increasing utility, and therefore strict sustainability.

If utility is constant for any level of consumption, or if the stated goal is minimal sustainability

(perhaps for reasons of intergenerational equity) then the Hartwick rule, to invest resource rents,is the keystone. Solow (1986) refers to this as a "rule of thumb" for growth policy.

This study has drawn together several strands from a diverse literature on the Hartwick rule.

The analysis has shown that, given virtually unrestricted production functions, the generalized

Hartwick rule in combination with the Hotelling rule is both necessary and sufficient for

consumption to be constant. The Cobb-Douglas production function (out of the class of CES

production functions), in which the elasticity of substitution between capital and resources is

exactly 1, yields consumption that is constant, positive and maximal when a standard Hartwick

programme is followed in combination with two efficiency conditions, the Hotelling rule and

complete resource exhaustion.

The magnitude of the elasticity of substitution for CES production functions has major

implications for the operation of a Hartwick-Hotelling programme. These implications are

summarized in Tables 1 and 2.

Table 1



18

Results for the Generalized Hartwick RuleK = F

R(R+v )130

σ<1131 σ=1132 σ>1133

σ≥1134

σ→∞135

v <0136 F →0137

C →0138

as t→∞

139

F →0140

C →0141

as t→∞

142

F →0143

C →0144

as t→∞145

Hotelling

Rule

violated

v >0146 C T <0147

for T <∞

148

C T <0149

for T <∞

150

Either

F (K T ,R

T )= ∞

151

for T <∞152

orC T <0153

for T <∞154

Hotelling

Rule

violated

Table 2

Results for the Standard Hartwick Rule

K = F RR155

σ<1156 σ=1157 σ>1158

σ≥1159

σ→∞160

F →0161

C →0162

as t→∞163

F =0164

C =0165

C is166

maximal167

F <0168

C =0169

C is 170

maximal171

Hotelling

Rule

violated

The results for σ<1172 in Table 2 contradict Theorem 3 in Hartwick (1978). Two results when

the elasticity of substitution is greater than 1 are of note. The programme that exhausts the

resource in the initial period (which is feasible since resources are not essential in this model)

yields non-constant consumption. And a weaker condition than that given by Dixit, Hammond

and Hoel exists for the standard Hartwick rule to yield greater (constant) consumption than the

generalized rule.



19

The derivationa in this paper emphasize the "knife edge" nature of the Cobb-Douglas production

function. Although the generalized Hartwick rule promises constant consumption for very

general production functions, the requirement that a maximal constant consumption path exist

places severe limits on the type of production function.

These results also shed light on the debate concerning "strong" versus "weak" sustainability.

The weak sustainability position, as typified by Solow, and Dasgupta and Heal, is essentially

that the elasticity of substitution between capital and resources is greater than or equal to 1.

The results in Table 2 show that for weak sustainability, therefore, the Hartwick-Hotelling

programme yields constant maximal consumption if the elasticity is exactly 1, constant sub-

maximal consumption when it is greater than 1, and, in the limit as capital and resources

become perfect substitutes, that efficient extraction is impossible (because the Hotelling rule isviolated). The strong sustainability position of Pearce, Markandya and Barbier is that the

elasticity of substitution between capital and resources (especially certain critical resource

stocks) is less than 1. The preceding results show that in this case the Hartwick-Hotelling

programme leads to declining output and consumption. For the proponents of strong

sustainability, therefore, the rule "invest resource rents" is not sufficient to ensure sustainable

development.



20

Appendix

Recalling the definitions of X 173 (expression (6)) and γ174 (expression (7)), a few basic results

follow directly from the definition of the CES production function:

σ σ σ σ β β β

111

1

1

===

−−−

R

F R

X

F R X F

R (A1)

K K RK F

RF

X

RK R X F γ

σ

β

σ σ

αβ σ σ σ σ 11

111

11

1

===

−

−−

−

− (A2)

.0)1(1

<−= γ σ R

F F R

RR (A3)

Note as well that γ<1175 and that β γ → 176 as 1→σ 177.

The first item to be derived is the expression for F & 178. We begin with,

.K F RF F RK RR R &&& +=

Therefore,

)()1(1

v RF R

R

F

F RK

R

R++−= γ

σ

&&

K F v R R R

R)(

1)1(

1++−=

γ

σ γ

σ

&

')(1

)1(1

R

R

F

F v R

R R

R &&

++−= γ

σ γ

σ

so that,

.)(

))(1()(

v R R

v R

v R

R

F

F

R

R

+−

+−

+=

γ σ

γ &&

From expression (5) we therefore derive,



21

−−

−−

+=

+

+−

+−

+=

v R

v R

v R

RK

v R R

v R

v R

RK F

γ γ σ

σ

γ σ

σ

)(

)1(

1)(

))(1(

&&

&&&

and thus,

−−

−

+→

v R

v

v R

RK F

β β )1(

&&& as .1+

→σ

The next issue to be considered is the behaviour of the Hartwick-Hotelling system when σ>1

179 and, since resources are not essential for this value of the elasticity of substitution, when all

of the resource is extracted in the base period. 0=C & 180 if and only if, given F =F (K ,R ) 181,

.),(),()0),,(( 00000000 S S K F S K F S K F K F Rr −=+ (A4)

It will simplify the algebra considerably, without unduly affecting the generality of the argument, if

we assume for this derivation that α+β=1182, so that

.

11

σ

σ

σ

σ

β α −−

+= RK X

Expression (A4) may now be re-written as,

),())((1

01

1111

01

1

0 σ

σ σ σ

σ

σ

σ

σ

σ σ β β α

−

−

−−−

− −=+ S X X S X K

which implies that,

),((1

01

1)

1

01

1

01

σ

σ σ σ

σ σ σ

σ

α β α −

−−

−− =+ K X S X K

and therefore that,

).(1

01

01

1

01

σ

σ σ

σ

σ σ

σ

β α α α −

−−− −= S K X K

For consumption to be constant this latter expression should be an identity. However, there are

clearly choices of K 0183 and S 0184 for which the right hand side is less than or equal to 0, so

the identity does not hold.



22

Finally, we consider how the level of consumption varies with v 185 in the generalized Hartwick

rule. The analysis will be based on infinitesimal positive changes dv 186, to examine the

transition from the standard to the generalized Hartwick rule - negative values have already

been ruled out because they lead to declining consumption. In order for the capital stock to be

greater for all time (after the initial period) in the transition to the generalized rule, as Dixit,

Hammond and Hoel hypothesize, a necessary condition is that K 187 be greater under the

generalized rule. This may be written as,

0

'

0 C C < if 0>dv and .)'('00 0 o R R RF dv RF >+

However, for infinitesimal dv 188 we may write,

,' dv

v

F F F o

oo

R

R R ∂

∂+=

and

.' 0

00 dvv

R R R

∂

∂+=

Therefore,

))1()(()'(' 0

000

00dv

v

R Rdv

v

F F dv RF

R

R R +∂

∂+

∂

∂+=+

,)1( 000 0

0

0 dvv

RF dv R

v

F RF R

R R +

∂

∂

+∂

∂

+=

where terms in dv 2189 have been dropped because they will go to zero in the limit. The

question is therefore reduced to whether the sum of the second two terms in the preceding

expression is greater than 0. Note that, since we are dealing with the initial period, K 0190 is

independent of v 191, and therefore the relationship we wish to test for these terms may be

written as,

.0)1( 0

0

0

00>+

∂

∂+

∂

∂dv

v

RF dv R

v

RF R RR

Recalling expression (A3), this may be written for CES production functions as,

,0)1(1 00

>+∂

∂+

∂

∂−dv

v

Rdv

v

R

σ

γ



23

so that, after dividing by dv 192, the Dixit, Hammond and Hoel condition reduces to

.1

0

σ γ

σ

−−<

∂

∂

v

R (A5)

Because K 0193 is given and independent of v 194, both C 0195 and R 0196 are functions of v 197

under the generalized Hartwick rule. A more direct attack on this problem is therefore to

evaluate

))(( 00

0

0dv RF F

vv

C R +−

∂

∂=

∂

∂

.)(00 0

0

R RR F dv Rv

RF −+

∂

∂−=

In order for C 0198 to vary negatively with v 199 we require the latter expression to be less than

0, or, again employing expression (A3) and dividing by F R 0200,

.1)(11

0

0

0

<+∂

∂−− dv R

v

R

Rσ

γ

Taking the limit as dv 201 tends to 0, this reduces to the condition

.1

0

γ

σ

−<

∂

∂

v

R (A6)

For the case σ>1202, expression (A6) is less restrictive than (A5) because its right-hand side is

positive, while that of (A5) is negative.



24

References

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

Dasgupta, S., and Mitra, T., 1983, Intergenerational Equity and Efficient Allocation ofExhaustible Resources, International Economic Review, Vol. 24, No. 1, 133-53.

Dixit, A., Hammond, P, and Hoel, M., 1980, On Hartwick's Rule for Regular Maximin Paths ofCapital Accumulation and Resource Depletion, Review of Economic Studies, Vol. XLVII,551-6.

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

Hartwick, J.M., 1978, Substitution Among Exhaustible Resources and Intergenerational Equity,Review of Economic Studies, Vol. XLV(2), No 140, 347-54.

Nordhaus, W.D., 1992, Is Growth Sustainable? Reflections of the Concept of SustainableEconomic Growth, International Economic Association, (mimeo).

Pearce, D.W., Markandya, A., and Barbier, E., 1989, Blueprint for a Green Economy, London:Earthscan.

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

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Solow, R.M., 1986, On the Intergenerational Allocation of Natural Resources, ScandinavianJournal of Economics, 88(1), 141-49.

kirk hamilton (sustainable development the hartwick rule and optimal growth)

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