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DEPARTMENT OF ECONOMICS

ISSN 1441-5429

DISCUSSION PAPER 48/13

Growth with endogenous resource use and population growth

Ratbek Dzhumashev1 and Gennadi Kazakevitch

Abstract In growth models, natural resources are frequently treated as an exogenous factor. Moreover, the

relationship between non-renewable resources and population growth has not been examined

thoroughly. This paper develops a growth model where resources and the fertility rate are

determined endogenously. The availability of resources is treated as a function of labour employed

to explore new reserves and develop substitutes for depleted resources. The agents optimise their

fertility rate by trading-off the quantity of children for their quality. In the long run, it is found that

the inflow of newly available resources is an essential factor that determines per capita

consumption. We show that the long-run sustainability of growth does not rely on the Inada

condition with respect to resource inputs. It is also demonstrated that a quasi-Malthusian

population constraint is feasible, even with potentially unlimited resources.

1 Monash University Department of Economics E-mail: ratbek.dzhumashev@monash.edu

© 2013 Ratbek Dzhumashev and Gennadi Kazakevitch

All rights reserved. No part of this paper may be reproduced in any form, or stored in a retrieval system, without the prior written

permission of the author.

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1. Introduction

Hotelling (1931) raised concerns about exhaustibility of resources and their importance for

production and growth. Nevertheless, neoclassical growth models assume that resources are

not essential for production in the long run, and therefore, the long run growth outcomes are

not driven by resources (Ayres and Warr, 2009). The issues, raised by Hotelling (1931) and

later by Meadows et al. (1972), are addressed by the framework known as the DHSS (Das-

gupta and Heal, 1974; Solow, 1974a, 1974b; Stiglitz, 1974a, 1974b). The studies within the

DHSS framework explicitly assume that non-renewable resources are an essential input in

production. That is, the resources input should be positive, but can be negligibly small. The

main conclusion reached by these studies is that, for the long-run sustainability of growth,

there should be a substitution of natural capital (resources) with produced capital. Hartwick

(1977) generalizes the notion of the substitution by stating that a long-run consumption level

can be sustained by investing the resource rent in physical capital. This means that to achieve

sustainable growth resources should be entirely transformed into physical capital.

Daly (1997) based on Georgescu-Roegen (1979), argues that natural capital and manufac-

tured capital are complements rather than substitutes; hence, the sustainability of growth en-

visioned in the DHSS model is not feasible. Other researchers find that even with the re-

source and capital substitutability, long run growth may not be sustainable. For example,

Groth (2007) shows that theoretically the DHSS-type models allow growth to collapse if the

substitution is not sufficient to sustain growth. Arguments that there is a limit to the level of

substitution of physical capital by a non-renewable resource have been raised by Krautkra-

emer (1998) and Anderson (1987). Supporting this proposition, Baumgärtner (2004) has

shown that the Inada conditions on material inputs assumed in DHSS models are inconsistent

with the materials-balance-principle. That is, the marginal resource product is bounded from

above; hence, the resource input cannot be reduced below some minimum fraction of the out-

put, as it is proposed by the DHSS framework.

To overcome this problem, the DHSS framework proposes an existence of a backstop tech-

nology that allows switching to an alternative resource that is too costly to use while the tra-

ditional non-renewable resources is not close to exhaustion yet. However, with depletion of

the resources their price increases and at some point, due to the discovery of the backstop

technology, the use of alternative resources becomes viable (Endress et al., 2005; Krautkra-

emer, 1998; Dasgupta and Heal, 1974; Dasgupta and Stiglitz, 1981; Kamien and Schwartz,

3

1978). It is believed that resource scarcity drives innovations that lead to the backstop tech-

nology (Ayres and Warr, 2009). The modifications of the DHSS frameworks with explicit

formulation of innovations demonstrate that the results obtained in the original DHSS models

hold. For example, Robson (1980) considers an environment where innovation is determinis-

tic and continuous while Lafforgue (2008) considers a stochastic version of the model. Both

models give rise to similar results to that of a DHSS model.

The main advantage of the backstop technology assumption is that it eliminates the exhausti-

bility of non-renewable resources as a problem. This is because the backstop technology ap-

proach assumes the existence of inexhaustible resources that become available with the arri-

val of the backstop technology. This assumption finds some support by results of Goller and

Weinberg (1978), who based on their technical calculations argue that our planet holds enor-

mous (near inexhaustible), although diffused, amounts of most of the essential resources.

They demonstrate that substituting the depleted resources for ones that are more diffused

would be less costly than re-cycling the resource near depletion. Along these lines, Pindyck

(1978) also highlights the existence of non-renewable resources with virtually no depletion.

Logically, the underlying assumption of inexhaustibility of resources with the backstop tech-

nology validates the results of neoclassical growth models.

The weakness of the backstop technology approach is that, by disregarding resource explora-

tion, it implicitly assumes that technological advancements in the final output production lead

to a resource substitution by making the backstop technology available. This assumption

seems a bit ad hoc. It would be more realistic to assume that resource substitution is a result

of a deliberate effort to explore new ways of extraction and use of non-traditional resource

substitutes, along with the final goods production technology improvements. This reasoning

is exploited in this paper. The main innovation in our analysis is that we explicitly model

continuous resource creation instead of an assumed jump to a backstop resource use suggest-

ed by the afore-mentioned literature.

In this regard, it is worthwhile to note that the empirical fact that the resource prices are not

following the Hotelling’s rule (Hotelling, 1931) has been related to the uncertainty about the

available stock of resources (Gaudet, 2007) and the exploration activities that is motivated

by the cost of resources (Livernois and Uhler, 1987). In other words, the empirics indicate

that the available stock of resources are continuously evolving in line with the proposition

stated by Goller and Weinberg (1978).

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Based on a similar idea, Pindyck (1978) considers a model where he assumes that the stock

of non-renewable resources is not fixed but can be increased by means of exploration and

production efforts. However, Pindyck (1978) focuses on the optimal production and explora-

tion of resources depending on the possibility of decline in discoveries of resources. Our ap-

proach in modelling the resource stock evolution is similar to that of Pindyck (1978), albeit

we assume that the new discoveries are independent of the cumulative stock of resource dis-

coveries, and instead assume that the marginal returns to the effort in resources discoveries

are diminishing. In light of this, the model developed in this paper accounts for the possibil-

ity of substitution for the backstop resource by means of producing knowledge of how to ex-

tract and use it, but not through advances in the final output production technology. That is,

we synthesise the implicit substitution for backstop resources assumed in the DHSS models

and exploration of resources envisioned by Pindyck (1978). In particular, we extend the ap-

proach of the traditional growth models by assuming that potential resource reserves are un-

limited, however, as in Pindyck (1978), the available resources need to be discovered through

exploratory activities. This way we overcome the Georgescu-Reogen-Daly argument, as we

consider the long run growth in a framework where sustainability does not depend on re-

source-capital substitution. Instead, the long-run growth (production) is sustained due to re-

source substitution, thus, our model does not require binding Inada conditions on material

inputs assumed in the DHSS framework.

A main shortcoming of the DHSS and Pindyck (1978) is that they do not consider economic

growth together with population growth. The endogeneity of population appears crucial for

the DHSS framework from the endogenous growth perspective. In fact, building upon DHSS

framework, Groth (2007) considers endogenous forms of the growth model with resources,

and finds that fully endogenous models with resources require unrealistic assumptions and

appear unfeasible. He ascertains that the only attractive form of endogenous growth model is

the semi-endogenous one, where growth is driven by population growth. In our analysis, we

extend the growth model with resources along this line and consider an interaction of popula-

tion growth with economic growth that depends on resources. We find that when population

growth is endogenous, a growth model with resources leads to a static population, which,

given Groth (2007) results, implies that endogenous growth may not be feasible when pro-

duction depends on non-renewable resources.

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By considering endogenous population growth within economic growth model with resources

we also link the DHSS framework to other strand of the literature. In particular, the unified

growth theory (UGT), developed in Galor and Weil (2000), Galor (2005, 2011), recognises

that demographic and economic dynamics are related and; thus, they need to be analysed

jointly.2 This literature explains how an economic transition from agrarian production to in-

dustrial production leads to a demographic transition that the world has been experiencing in

the last two centuries. In the UGT framework, resources are modelled as a fixed land en-

dowment; hence, assumed to be inexhaustible. The UGT shows that, at the Malthusian stage,

the benefits of technological progress are offset by population growth, which, in turn due to

the limits imposed by the fixed resource (land), result in a stagnation. However, this interac-

tion between the rate of technological progress and the size of the population boosts techno-

logical progress, and eventually gives rise to transition to the modern economy where human

capital becomes the driving force of economic growth. Assuming, a similar to the above au-

thors, inexhaustible but limited resource (land), Peretto and Valente (2011) consider a

Schumpeterian growth model and show that when labour and resources are substitutes, the

stable equilibrium will be one with constant population. However, they also find that if labour

and resources are complements then population dynamics may become unstable.

This paper differs from the existing literature by adopting a different perspective on the nexus

between population, resources, and growth. Unlike the UGT and Peretto and Valente (2011),

in our case, economic growth is affected by non-renewable resources that potentially unlim-

ited, but costly to explore and produce. The main difference from bio-economic models is

that in our model population growth is not Malthusian, and our focus is on the non-renewable

resources rather than the renewable ones. We explicitly model fertility choice in a similar

fashion to Peretto and Valente (2011), but instead of a dynastic family, we adopt a paternal-

istic family model as in Doepke (2004), Jones et al. (2010), and Mookherjee et al. (2012).

That is, in our model fertility is endogenous and evolves as a trade-off between quantity and

quality of children as well as parental consumption.

2 Another strand of literature (bio-economic models), devoted to analysis of the resource-growth nexus, consid-

ers exhaustible and renewable resources and its interaction with economic activity and population dynamics

(Brander and Taylor, 1998; Pezzey and Anderis, 2003; Dalton et al., 2005; Good and Reuveny, 2006). These

models explain how a Malthusian-type growth regime can develop and in case of overharvesting can lead to a

collapse of the eco-system and the economy with it. We abstract from the environmental issues in our analysis.

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The main findings of this study are as follows. Availability of resources is an crucial factor

that determines the level of per capita consumption in the long run. Improvements in resource

expansion through new reserves and substitutes drive the long run productivity of the final

goods sector. Sustainability of long-run growth requires that, in steady state, all used up re-

sources should be replenished by finding either new reserves or substitutes. Therefore, the

long-run sustainability is driven not by resource-capital substitution envisioned by the DHSS

framework, but by resource exploration and substitution.

The fertility rate increases with the level of consumption, but the cost of education reduces it.

This implies that the rate of fertility depends on the level of income generating capacity of the

parents and the cost of acquiring this capacity. In this sense, this result is in line with the lit-

erature. For example, Schuler (1979) used a similar approach to model fertility by assuming

that the cost of raising a child is a function of per capita disposable income. Similarly, Fioroni

(2010) based on empirical evidence also suggest a way to model endogenous child survival as

a concave function of human capital. Yet, differing from these authors, in our model, there is

a negative feedback from the size of the population to the stock of human capital per worker.

In this sense, this is akin to the interaction between the size of population and the technologi-

cal change modelled in the UGT. Consequently, increasing human capital raises per capita

consumption which increases fertility. However, increasing population will create a drag on

human capital accumulation, which ultimately limits both economic and demographic growth

as soon as the return to human capital in resource production is diminishing. Given this inter-

play between human capital, fertility, and resource production, the economy reaches its

steady state only when the population becomes static. If it is the case, then we observe a qua-

si-Malthusian constraint on population growth. Hence, even with potentially unlimited re-

sources, a quasi-Malthusian population growth limits are possible. It is shown that only if the

return to human capital in the resource sector is non-diminishing then the population growth

is not restricted.

We also consider implications of the assumption that the human capital elasticity of resource

production is equal or greater than unity ( 1 ). We have shown that this assumption does

not seem realistic as results in explosive growth of both the population and output. Overall,

our paper contributes to the literature by i) introducing a growth model with endogenous fer-

tility and an explicit mechanism of exploration and substitution of non-renewable resources

that are used as an input to final goods production; ii) demonstrating that the capacity, to dis-

7

cover new reserves and substitutes drives the long run consumption per capita; iii) showing

the possibility of a quasi-Malthusian trap even with potentially unlimited resources; iv) estab-

lishing that with resource substitution, the long-run sustainability does not require that the

Inada conditions are binding as assumed in the DHSS framework; v) showing that long-run

endogenous growth is not sustainable.

The rest of the paper is structured as follows. Section 2 presents the set-up of the model. Sec-

tion 3 lays out the optimisation problem and presents its solution. In Section 4, evolution of

resources and how it affects the steady state capital stock is discussed. In Section 5, we dis-

cuss the implications of increasing returns to human capital input in the resource sector. Sec-

tion 6 concludes the paper.

2. The model

2.1. Basic setup

The basic setup used is similar to Endress et al. (2005), and Schou (2000) that extends the

Uzawa-Lucas-type model where human capital accumulation serves as the growth engine of

the economy (Uzawa, 1965; Lucas, 1988). That is, we assume that the production technology

employed to produce a single homogenous good that combines capital and labour, as well as

a flow input of natural resources, which is extracted from the currently available aggregate

resource stock. Thus, the production function is in the form of ( , , ),F K H R where ,K ,H and

R are physical and human capital and resources correspondingly. However, different from

Endress et al. (2005) and Schou (2000), we are not assuming either population or endowment

with resources to be exogenous or fixed. Instead, we make a rather more empirically-based

assumption that the population increases in time, and we treat both aggregate resource stock

and flows are endogenous. We also ignore any uncertainties in the economic activities and

abstract from the use of exhaustible and renewable resources.

Following, Galor and Weil (2000), de la Croix and Doepke (2003), and Kolmann (1997), a

paternalistic utility function is assumed. That is, the parents derive utility from the quantity

and quality of children, but not from their future welfare. Therefore, the utility function of the

representative agent is given by ( , )u c nh , where c is consumption, n is the number of surviv-

ing children, h is their human capital per capita. Following the above-mentioned authors, we

adopt an utility function of the following specification:

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exp( ) log( ) log( )u t c qbh . (1)

It is assumed that the number of surviving children is found as

, 0 1n bq q , (2)

where b is the number of births, and q is the probability of survival. The agent incurs costs

while having a child and then has to bear the cost of education of the surviving children. That

is, the cost related to child rearing is given by

[ ( ) ] ( ),qe t b t (3)

where cost of having a child given in terms of consumption goods, ( )e t is expenditure on

education.

The law of motion of the adult population is

( ) ( )( ( ) ),L t L t qb t d (4)

where d is the death rate given exogenously.

2.2. Production and explorations sectors

The households own the firms, and they supply human and physical capital to firms. Since

two sectors of production are considered, we need to specify the technology employed by

each of them. We assume that the production function for the consumption good is given as:

1

1( )Y AR K vH

(5)

where stands for the technology coefficient and is the fraction of labour engaged in the

final consumption goods sector, 0 1 , and 0 1 . Accounting for H hL , where h is

human capital per capita, the production function is written in the intensive form as:

1 1( ( ) )y Ar k vh . (6)

Since we assume that the resource stock is expandable, this also requires some costly activity.

The expansion depends on the exploration, research and development effort that makes new

resources available. We denote the incoming flow of resources by ,Z and assume that re-

source-creating activity is captured by a production function:

(1 )Z E v H , (7)

A v

9

where is a technological coefficient, 0 1 , and (1 )v is the fraction of labour engaged

in this industry. In section 6, we will consider a case with 1 . In per worker terms, we

write as:

1

(1 )E v hz

L

. (8)

Expression (7) involves the assumption that the formation of incoming resource flow requires

human capital only. Therefore, the dynamics of the aggregate resource stock is defined as the

net inflow of the aggregate resource and evolves according to the following differential equa-

tion:

x r z (9)

We follow Hartwick (1977, 1978) in modelling the resource use. That is, the resources are

not owned privately, so cost of extraction is a pure cost to the economy. Given this environ-

ment, the capital accumulation process in per worker terms is governed by:

( ) ( ) ,k y k X r c qe b (10)

where is the unit cost of extracting the resource. Following Endress et al. (2005), we model

resource extraction as cost in terms of final goods. Such a simplification is especially suitable

for our model as it focuses on resource expansion rather than extraction. This cost is a de-

creasing function of the resource stock, X .3 That is, 0

d

dX

. For further simplicity, we as-

sume that this function is given as:

( )XX

, (11)

where is a cost parameter.

Both sectors consist of the firms maximising their profits. We assume that both types of firms

take the interest rate, the wage rate, and the cost of resource extraction as given. This optimi-

sation then yields the rate of return to physical capital, i , the rate of return to human capital

in the production sector, Yw , and in resources sector, Zw , and the optimal amount of resource

inputs (the outgoing resource flow), ,r as follows:

3 This assumption is basically the Hotelling’s rule. We reconcile the fact that this rule may not be being sup-

ported by the empirical observations (e.g. Gaudet, 2007) by the possibility that the known stock of resources being altered continuously through exploration and substitution.

E

10

1

11( )

XAr k vh

, (12)

(1 ) 1 (1 )(1 )(1 ) ( ) ,i A r k vh (13)

(1 ) (1 )(1 )(1 )(1 ) ( )

Y

A r k vhw

h

, (14)

1

(1 )

( )Z

E vw

Lh

. (15)

Observing equations (13) and (14), we conclude that the rate of return to capital and the wage

rate depend on the optimal amount of resources used in production. That is, in an environ-

ment with higher optimal levels of resource input, the rental prices of factors of production

are higher. Therefore, it is crucial to ascertain what drives the optimal levels of resource input.

By observing (12), we establish that the optimal level of the resource use, not surprisingly,

increases with the larger aggregate resource stock, X , technology coefficient (productivity),

,A and with a decrease in the exogenous cost of extraction, .

One can also see that the optimal amount of the aggregate resource used in the production of

the aggregate final consumption good increases not only with the level of per worker capital

stock, but also with the share of labour engaged in the production of final consumption good.

Intuitively, an increase in the share of labour in the final consumption good production is

possible only if the share of labour engaged in resource expansion sector is shrinking. Opti-

mality requires that any decrease in the labour input is offset by an increase in labour produc-

tivity. We will analyse this link after determining the equilibrium values for the proportions

of labour used in both sectors.

3. The representative agent’s problem

The representative agent maximizes inter-temporal utility given by:

, , ,

0

max exp( ) log( ) log( )c v b e

U t c qbh dt

(16)

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where exp( )t is the coefficient discounting the utility of consumption over time, and sub-

ject to the following constraints:

( ) ( ) ,k y k X r c qe b

0(0) ,k k (17)

x r z , (18)

( ) 0, ( ) 0, ( ) 0,x t k t h t (19)

h e h . (20)

We write the Hamiltonian of the problem as follows:

( , )exp( ) ( ) ( )

(1 ) ( ).

J u c qbh t y k X r c qe b

E v h r e h

(21)

The first-order conditions for this optimal control problem are given as:

exp( ) 0,c

Ju t

c

(22)

(1 )(1 )(1 ) ( )

(1 )(1 ) 0,J vh

Ar k Ehv v

(23)

(1 ) 1(1 ) [( ) ],

J Ar vh k

k k

(24)

2

,J

X X

(25)

(1 )1

1

( ) (1 )exp( ) (1 )(1 ) ,h

k vhJ E vu t Ar

h h h

(26)

exp( ) ( ( )) 0,b

Ju t qe t

b

(27)

0.J

qbe

(28)

Accounting for that exp( ) log( ) log( )u t c qbh , and combining (22) and (24) we ob-

tain that the growth rate of consumption:

12

(1 )

(1 )(1 )( )(1 )

( )

c t r kA v h

c t h h

. (29)

From (27) and (22) we obtain the birth rate function:

( )

( )( )

c tb t

qe t

. (30)

Another equilibrium condition is that the marginal product of physical capital should equal

the marginal product of human capital. That is,

(1 ) (1 )(1 )

(1 )(1 ) (1 )(1 ) ( )( ) (1 )(1 )

Ar k vhvh Ar k

k h

,

Solving which we obtain the physical-to-human capital ratio in equilibrium:

1

k

h

. (31)

That is, the stocks of physical and human capital should grow at the same rate in equilibrium.

These findings immediately give rise to the following lemma.

Lemma 1. The per capita consumption growth rate in the transition to the steady state posi-

tively depends on the evolution of the aggregate resource stock, the share of workers engaged

in final good production, the human –to- physical capital ratio, and negatively on the cost of

resource extraction.

Proof. The growth rate given by (29) can be re-written as:

(1 ) (1 )1

(1 )(1 ) .c XA h

g A vc k

It is straightforward to verify that 0g

X

, 0

g

v

,

0

hk

g

and 0

g

. ∎

This result implies that in an environment where initial stock of resources high, the growth

rates also should be relatively higher. However, the steady state growth rate depends also on

how effective the country is in terms of maintaining the stock of resources. This implies that

if the economy is more effective in extending and extracting resources, that is, a smaller

share of the labour force is employed in the resource sector, and a greater share of the labour

force is engaged in the production of the final goods, hence, such an economy has a faster

transition to steady state.

4. The steady state analysis

13

Intuitively, in the long run the extraction and creation of resources in per capita terms should

be equal. Otherwise, either the economy would run out of resources or it would accumulate

excessive resources. Clearly, both strategies are not optimal. This discussion results in the

following lemma.

Lemma 2. In steady state, r z holds.

Proof. Let us recall the equation describing the evolution of the aggregate resource stock giv-

en by x z r . In general, we cannot assume that . If is true, then it will lead to

an increase in the stock of resources, X . Then given that

1

11( )

XAr k vh

, the opti-

mal level of resource inputs also increases at a growing rate with the stock of resources, the

system will move towards . If holds then the opposite would happen, given that the

stock of resource would be diminishing. This consideration implies that, in the long run,

steady state is achieved only if 0x holds.■

This result implies that, in the long run, resource use should not depend on the substitution

between capital and natural resources; it rather depends on the productivity of the exploration

sector and the share of the labour force engaged there. The long run resource extraction cost

is constant because the stock of resources becomes constant, unlike exogenously given back-

stop technology driven cost of assumed in Endress et al. (2005). Unlike in Hartwick (1977),

this equilibrium condition does not imply that resource rents should be entirely invested into

physical capital, hence, the sustainable growth does not depend on the Inada condition with

respect to resource inputs. This result allows to overcome the Georgescu-Reogen-Daly criti-

cism and addressing the shortcoming of the DHSS framework pointed out by Anderson (1987)

and Baumgärtner (2004). Moreover, for a fixed amount of resource extraction, equation (8)

implies that a larger population size results in lower resources use in production. This condi-

tion deals with the “sustainability bias” argued by Groth (2007), as the resource creation in

this model (and extraction in steady state) exhibits diminishing returns to human capital, and

has a negative scale effect on the size of population.

We can find the optimal share of labour engaged in exploration from the optimality condi-

tions in the labour market and the inputs market. First, the optimality in the labour market

leads to the equality of the returns to human capital in both sectors. That is,

Z Yw w . (32)

z r z r

z r z r

14

The wage rate, Yw , is given by (14),(1 ) (1 )(1 )(1 )(1 ) ( )A r k vh

wh

, while the wage

rate in the resource sector is equal to the marginal product of labour, hence, Z

zw

h

. In

steady state, z r , therefore, Y

rw

h

. Taking these ideas into account, we write

1 1(1 )(1 ) ( ( ) )r A r k vh

h h

. (33)

By solving (33) for v we obtain:

1

(1 )(1 )

(1 )

(1 )(1 )

r

hv

kA

h

. (34)

Here, the bar denotes the steady state value. We notice that due to r

wh

the following

holds,

1

(1 ).

E v hr

L

(35)

That is, with h , r

h will be marginally decreasing ( i.e.

2

20, 0

r r

h h

). We can find the

steady state value of human capital from the condition that both (35) and (12) equations

should be satisfied simultaneously. That is,

1

11

1

(1 )( )

E v h XAk vh

L

. (36)

We recall that 1

k h

. Then taking this into account, from (36) we find:

1 1

1 (1 )(1 )

1

1 (1 ) (1 )E vh

L v XA

. (37)

Proposition 1. In steady state, the stock of human capital is static only if the population is

static.

Proof. It is straightforward from(37). ∎

One can also analyse comparative statics for h and draw the following conclusion.

15

Corollary 1. The steady state human capita per worker is decreased in the size of population

and the steady state stock of resources, and increased in the productivity of the resource sec-

tor and cost of resource extraction.

Proof: Taking respective derivatives of expression(37) yields the following: 0h

L

,

0h

E

, 0

h

X

, 0

h

.∎

4.1. Steady-state consumption and long run population growth

Using the population growth equation, ( )

( )( )

L tb t q d

L t ,

and the birth rate equation in equi-

librium, ( )

( )( )

c tb t

qe t

, one can write the steady-state consumption function:

( )d qe

cq

. (38)

The findings to this end allow us to determine the long run population growth. This result is

formulated as the following proposition.

Proposition 2. In steady state, the population is static.

Proof. Recall that ( )

( )( )

L tb t q d

L t and

( )( )

( )

c t db t

qe t q

* ( )d qe

cq

. Hence,

when ( ) ,d

b tq

and *c c ( )

0( )

L t

L t . If per capita consumption is *c c then 0

L

L .

Falling population leads to increasing per capita human capital in equilibrium, due to (37).

This, in turn, leads to more resources per capita being used (see (35)). Given these increases,

the production technology implies that income per capita y rises. This will lead to rise in the

level of consumption. Thus, consumption will rise, which lifts the birth rate and the economy

will be adjusting till it reaches the point when *c c . In case, when *c c , the adjustment will

occur in the opposite direction. Therefore, the equilibrium when population is static is stable

equilibrium. ∎

This finding immediately leads to the following conclusion.

16

Corollary 2. In steady state 0h k c

h k c hold.

Proof. When *c c , and hence 0L , (37) implies that in steady state 0c h

c h . Since,

1

hk

, a fixed human capital implies that the steady state per capita physical capital is

also constant, or 0k

k .■

Furthermore, the fixed human capital in steady state implies that education spending per child

is given as follows:

e h . (39)

Overall, these results indicate that even with potentially unlimited resources, an economy will

find itself in a quasi-Malthusian state, where both the population and consumption per capita

turn static.

4.2. Transition Dynamics

We can consider the following transformed variables to analyse the transition dynamics:

k

h ,

c

k ,

r

k . Then the following equations

( ) ( ) ,k y k X r c qe b

11

(1 )(1 ) ,c XA k

A vc h

After taking into account that

11

1r v XA

k

, (40)

the growth of physical capital can be –re-written as follows:

(1 ) (1 )1 1

(1 )k v XA v XA

Ak

, (41)

(1 )1

(1 )c v XA

Ac

. (42)

17

Then from 0c k

c k

one obtains the condition 0 .

1 (1 )11

(1 (1 ))1

v v XAA

. (43)

The steady state for will be determined when 1

, which will determine . Then

the dynamics of the system can be shown as in Figure 1. The figure shows that when

consumption and productive capital are in steady state, the population turns static.

5. The case with 1

Since, in the above analysis, we assumed that 0 1 , which obviously drives the steady-

state results obtained. To see what will be different if we alter this assumption, let us consider

the case when the return to human capital in resource production sector is constant. That is,

1 . In this case, from(35), we can find that

(1 )r E v h . (44)

Figure 1. Phase diagram

18

This implies that in steady state, resource use in production, r , will increase with the stock

of per capita human capital. Moreover, in this case, we will not get a negative relationship

between human capital and the size of the population such as(37). Instead, per capita capital

will be independent of the size of population. Let us take a closer look at the expression for

growth rate in this case:

(1 )

(1 )(1 )(1 )r k

g A v hh h

. (45)

The main difference from 0 1 case is that the stock of human capital h on the balanced

growth path can grow unboundedly, when 1 . Therefore, the growth rate given by (45) can

also grow without a limit as soon as h keeps growing. This result also implies that growth of

consumption will lead to higher population growth rates. Since, there is no feedback from the

size of the population to human capital accumulation; the size of the population will not be

limited as soon as the level of per capita consumption keeps growing. All in all, in the envi-

ronment with 1 , we will see explosive growth of both the economy and population. Ob-

viously, it does not sound realistic.

6. Conclusions

By assuming the endogeneity of both the resource stock which is dependent over time on the

efforts of exploration, research and development, and of the population which ultimately de-

pends on the dynamics of income stream, we have made a few conclusions.

Availability of resources made through exploration and substitution is an important factor

that drives the long-run level of per capita consumption. That is because improvements in

resource expansion through new discoveries of reserves and substitutes determine the produc-

tivity of final good sector in the long run. Differing from the extant literature, we address the

“sustainability bias” in the resource sector by assuming diminishing returns to scale in the

resource sector. We show that as soon as the stock of resources is expandable, the long-run

sustainability does not depend on the binding Inada condition with respect to resource inputs

as in the DHSS framework. We find that sustainability of long-run growth requires that in

steady state all used up resources should be replenished by either finding new reserves or new

substitutes. Given that the productivity of human capital in the resource sector is diminishing,

the economy reaches its steady state only when the population reaches its steady state. If it is

the case, then we observe a quasi-Malthusian constraint on population growth.

19

Environment and renewable resources are not considered here. Accounting for these factors

may impose binding constraints that will make sustained economic growth challenging. We

also abstract from the uncertainty of resource discoveries. These questions are left for future

research.

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