Growth with endogenous resource use and population growth

Transcription

1 DEPARTMENT OF ECONOMICS ISSN DISCUSSION PAPER 48/13 Growt wit endogenous resource use and population growt Ratbek Dzumasev 1 and Gennadi Kazakevitc Abstract In growt models, natural resources are frequently treated as an exogenous factor. Moreover, te relationsip between non-renewable resources and population growt as not been examined torougly. Tis paper develops a growt model were resources and te fertility rate are determined endogenously. Te availability of resources is treated as a function of labour employed to explore new reserves and develop substitutes for depleted resources. Te agents optimise teir fertility rate by trading-off te quantity of cildren for teir quality. In te long run, it is found tat te inflow of newly available resources is an essential factor tat determines per capita consumption. We sow tat te long-run sustainability of growt does not rely on te Inada condition wit respect to resource inputs. It is also demonstrated tat a quasi-maltusian population constraint is feasible, even wit potentially unlimited resources. 1 Monas University Department of Economics ratbek.dzumasev@monas.edu 2013 Ratbek Dzumasev and Gennadi Kazakevitc All rigts reserved. No part of tis paper may be reproduced in any form, or stored in a retrieval system, witout te prior written permission of te autor. 1

2 1. Introduction Hotelling (1931 raised concerns about exaustibility of resources and teir importance for production and growt. Neverteless, neoclassical growt models assume tat resources are not essential for production in te long run, and terefore, te long run growt outcomes are not driven by resources (Ayres and Warr, Te issues, raised by Hotelling (1931 and later by Meadows et al. (1972, are addressed by te framework known as te DHSS (Dasgupta and Heal, 1974; Solow, 1974a, 1974b; Stiglitz, 1974a, 1974b. Te studies witin te DHSS framework explicitly assume tat non-renewable resources are an essential input in production. Tat is, te resources input sould be positive, but can be negligibly small. Te main conclusion reaced by tese studies is tat, for te long-run sustainability of growt, tere sould be a substitution of natural capital (resources wit produced capital. Hartwick (1977 generalizes te notion of te substitution by stating tat a long-run consumption level can be sustained by investing te resource rent in pysical capital. Tis means tat to acieve sustainable growt resources sould be entirely transformed into pysical capital. Daly (1997 based on Georgescu-Roegen (1979, argues tat natural capital and manufactured capital are complements rater tan substitutes; ence, te sustainability of growt envisioned in te DHSS model is not feasible. Oter researcers find tat even wit te resource and capital substitutability, long run growt may not be sustainable. For example, Grot (2007 sows tat teoretically te DHSS-type models allow growt to collapse if te substitution is not sufficient to sustain growt. Arguments tat tere is a limit to te level of substitution of pysical capital by a non-renewable resource ave been raised by Krautkraemer (1998 and Anderson (1987. Supporting tis proposition, Baumgärtner (2004 as sown tat te Inada conditions on material inputs assumed in DHSS models are inconsistent wit te materials-balance-principle. Tat is, te marginal resource product is bounded from above; ence, te resource input cannot be reduced below some minimum fraction of te output, as it is proposed by te DHSS framework. To overcome tis problem, te DHSS framework proposes an existence of a backstop tecnology tat allows switcing to an alternative resource tat is too costly to use wile te traditional non-renewable resources is not close to exaustion yet. However, wit depletion of te resources teir price increases and at some point, due to te discovery of te backstop tecnology, te use of alternative resources becomes viable (Endress et al., 2005; Krautkraemer, 1998; Dasgupta and Heal, 1974; Dasgupta and Stiglitz, 1981; Kamien and Scwartz, 2

3 1978. It is believed tat resource scarcity drives innovations tat lead to te backstop tecnology (Ayres and Warr, Te modifications of te DHSS frameworks wit explicit formulation of innovations demonstrate tat te results obtained in te original DHSS models old. For example, Robson (1980 considers an environment were innovation is deterministic and continuous wile Lafforgue (2008 considers a stocastic version of te model. Bot models give rise to similar results to tat of a DHSS model. Te main advantage of te backstop tecnology assumption is tat it eliminates te exaustibility of non-renewable resources as a problem. Tis is because te backstop tecnology approac assumes te existence of inexaustible resources tat become available wit te arrival of te backstop tecnology. Tis assumption finds some support by results of Goller and Weinberg (1978, wo based on teir tecnical calculations argue tat our planet olds enormous (near inexaustible, altoug diffused, amounts of most of te essential resources. Tey demonstrate tat substituting te depleted resources for ones tat are more diffused would be less costly tan re-cycling te resource near depletion. Along tese lines, Pindyck (1978 also igligts te existence of non-renewable resources wit virtually no depletion. Logically, te underlying assumption of inexaustibility of resources wit te backstop tecnology validates te results of neoclassical growt models. Te weakness of te backstop tecnology approac is tat, by disregarding resource exploration, it implicitly assumes tat tecnological advancements in te final output production lead to a resource substitution by making te backstop tecnology available. Tis assumption seems a bit ad oc. It would be more realistic to assume tat resource substitution is a result of a deliberate effort to explore new ways of extraction and use of non-traditional resource substitutes, along wit te final goods production tecnology improvements. Tis reasoning is exploited in tis paper. Te main innovation in our analysis is tat we explicitly model continuous resource creation instead of an assumed jump to a backstop resource use suggested by te afore-mentioned literature. In tis regard, it is wortwile to note tat te empirical fact tat te resource prices are not following te Hotelling s rule (Hotelling, 1931 as been related to te uncertainty about te available stock of resources (Gaudet, 2007 and te exploration activities tat is motivated by te cost of resources (Livernois and Uler, In oter words, te empirics indicate tat te available stock of resources are continuously evolving in line wit te proposition stated by Goller and Weinberg (

4 Based on a similar idea, Pindyck (1978 considers a model were e assumes tat te stock of non-renewable resources is not fixed but can be increased by means of exploration and production efforts. However, Pindyck (1978 focuses on te optimal production and exploration of resources depending on te possibility of decline in discoveries of resources. Our approac in modelling te resource stock evolution is similar to tat of Pindyck (1978, albeit we assume tat te new discoveries are independent of te cumulative stock of resource discoveries, and instead assume tat te marginal returns to te effort in resources discoveries are diminising. In ligt of tis, te model developed in tis paper accounts for te possibility of substitution for te backstop resource by means of producing knowledge of ow to extract and use it, but not troug advances in te final output production tecnology. Tat is, we syntesise te implicit substitution for backstop resources assumed in te DHSS models and exploration of resources envisioned by Pindyck (1978. In particular, we extend te approac of te traditional growt models by assuming tat potential resource reserves are unlimited, owever, as in Pindyck (1978, te available resources need to be discovered troug exploratory activities. Tis way we overcome te Georgescu-Reogen-Daly argument, as we consider te long run growt in a framework were sustainability does not depend on resource-capital substitution. Instead, te long-run growt (production is sustained due to resource substitution, tus, our model does not require binding Inada conditions on material inputs assumed in te DHSS framework. A main sortcoming of te DHSS and Pindyck (1978 is tat tey do not consider economic growt togeter wit population growt. Te endogeneity of population appears crucial for te DHSS framework from te endogenous growt perspective. In fact, building upon DHSS framework, Grot (2007 considers endogenous forms of te growt model wit resources, and finds tat fully endogenous models wit resources require unrealistic assumptions and appear unfeasible. He ascertains tat te only attractive form of endogenous growt model is te semi-endogenous one, were growt is driven by population growt. In our analysis, we extend te growt model wit resources along tis line and consider an interaction of population growt wit economic growt tat depends on resources. We find tat wen population growt is endogenous, a growt model wit resources leads to a static population, wic, given Grot (2007 results, implies tat endogenous growt may not be feasible wen production depends on non-renewable resources. 4

5 By considering endogenous population growt witin economic growt model wit resources we also link te DHSS framework to oter strand of te literature. In particular, te unified growt teory (UGT, developed in Galor and Weil (2000, Galor (2005, 2011, recognises tat demograpic and economic dynamics are related and; tus, tey need to be analysed jointly. 2 Tis literature explains ow an economic transition from agrarian production to industrial production leads to a demograpic transition tat te world as been experiencing in te last two centuries. In te UGT framework, resources are modelled as a fixed land endowment; ence, assumed to be inexaustible. Te UGT sows tat, at te Maltusian stage, te benefits of tecnological progress are offset by population growt, wic, in turn due to te limits imposed by te fixed resource (land, result in a stagnation. However, tis interaction between te rate of tecnological progress and te size of te population boosts tecnological progress, and eventually gives rise to transition to te modern economy were uman capital becomes te driving force of economic growt. Assuming, a similar to te above autors, inexaustible but limited resource (land, Peretto and Valente (2011 consider a Scumpeterian growt model and sow tat wen labour and resources are substitutes, te stable equilibrium will be one wit constant population. However, tey also find tat if labour and resources are complements ten population dynamics may become unstable. Tis paper differs from te existing literature by adopting a different perspective on te nexus between population, resources, and growt. Unlike te UGT and Peretto and Valente (2011, in our case, economic growt is affected by non-renewable resources tat potentially unlimited, but costly to explore and produce. Te main difference from bio-economic models is tat in our model population growt is not Maltusian, and our focus is on te non-renewable resources rater tan te renewable ones. We explicitly model fertility coice in a similar fasion to Peretto and Valente (2011, but instead of a dynastic family, we adopt a paternalistic family model as in Doepke (2004, Jones et al. (2010, and Mookerjee et al. (2012. Tat is, in our model fertility is endogenous and evolves as a trade-off between quantity and quality of cildren as well as parental consumption. 2 Anoter strand of literature (bio-economic models, devoted to analysis of te resource-growt nexus, considers exaustible and renewable resources and its interaction wit economic activity and population dynamics (Brander and Taylor, 1998; Pezzey and Anderis, 2003; Dalton et al., 2005; Good and Reuveny, Tese models explain ow a Maltusian-type growt regime can develop and in case of overarvesting can lead to a collapse of te eco-system and te economy wit it. We abstract from te environmental issues in our analysis. 5

6 Te main findings of tis study are as follows. Availability of resources is an crucial factor tat determines te level of per capita consumption in te long run. Improvements in resource expansion troug new reserves and substitutes drive te long run productivity of te final goods sector. Sustainability of long-run growt requires tat, in steady state, all used up resources sould be replenised by finding eiter new reserves or substitutes. Terefore, te long-run sustainability is driven not by resource-capital substitution envisioned by te DHSS framework, but by resource exploration and substitution. Te fertility rate increases wit te level of consumption, but te cost of education reduces it. Tis implies tat te rate of fertility depends on te level of income generating capacity of te parents and te cost of acquiring tis capacity. In tis sense, tis result is in line wit te literature. For example, Sculer (1979 used a similar approac to model fertility by assuming tat te cost of raising a cild is a function of per capita disposable income. Similarly, Fioroni (2010 based on empirical evidence also suggest a way to model endogenous cild survival as a concave function of uman capital. Yet, differing from tese autors, in our model, tere is a negative feedback from te size of te population to te stock of uman capital per worker. In tis sense, tis is akin to te interaction between te size of population and te tecnological cange modelled in te UGT. Consequently, increasing uman capital raises per capita consumption wic increases fertility. However, increasing population will create a drag on uman capital accumulation, wic ultimately limits bot economic and demograpic growt as soon as te return to uman capital in resource production is diminising. Given tis interplay between uman capital, fertility, and resource production, te economy reaces its steady state only wen te population becomes static. If it is te case, ten we observe a quasi-maltusian constraint on population growt. Hence, even wit potentially unlimited resources, a quasi-maltusian population growt limits are possible. It is sown tat only if te return to uman capital in te resource sector is non-diminising ten te population growt is not restricted. We also consider implications of te assumption tat te uman capital elasticity of resource production is equal or greater tan unity ( 1. We ave sown tat tis assumption does not seem realistic as results in explosive growt of bot te population and output. Overall, our paper contributes to te literature by i introducing a growt model wit endogenous fertility and an explicit mecanism of exploration and substitution of non-renewable resources tat are used as an input to final goods production; ii demonstrating tat te capacity, to dis- 6

7 cover new reserves and substitutes drives te long run consumption per capita; iii sowing te possibility of a quasi-maltusian trap even wit potentially unlimited resources; iv establising tat wit resource substitution, te long-run sustainability does not require tat te Inada conditions are binding as assumed in te DHSS framework; v sowing tat long-run endogenous growt is not sustainable. Te rest of te paper is structured as follows. Section 2 presents te set-up of te model. Section 3 lays out te optimisation problem and presents its solution. In Section 4, evolution of resources and ow it affects te steady state capital stock is discussed. In Section 5, we discuss te implications of increasing returns to uman capital input in te resource sector. Section 6 concludes te paper. 2. Te model 2.1. Basic setup Te basic setup used is similar to Endress et al. (2005, and Scou (2000 tat extends te Uzawa-Lucas-type model were uman capital accumulation serves as te growt engine of te economy (Uzawa, 1965; Lucas, Tat is, we assume tat te production tecnology employed to produce a single omogenous good tat combines capital and labour, as well as a flow input of natural resources, wic is extracted from te currently available aggregate resource stock. Tus, te production function is in te form of F( K, H, R, were K, H, and R are pysical and uman capital and resources correspondingly. However, different from Endress et al. (2005 and Scou (2000, we are not assuming eiter population or endowment wit resources to be exogenous or fixed. Instead, we make a rater more empirically-based assumption tat te population increases in time, and we treat bot aggregate resource stock and flows are endogenous. We also ignore any uncertainties in te economic activities and abstract from te use of exaustible and renewable resources. Following, Galor and Weil (2000, de la Croix and Doepke (2003, and Kolmann (1997, a paternalistic utility function is assumed. Tat is, te parents derive utility from te quantity and quality of cildren, but not from teir future welfare. Terefore, te utility function of te representative agent is given by u( c, n, were c is consumption, n is te number of surviving cildren, is teir uman capital per capita. Following te above-mentioned autors, we adopt an utility function of te following specification: 7

8 u exp( t log( c log( qb. (1 It is assumed tat te number of surviving cildren is found as n bq, 0 q 1, (2 were b is te number of birts, and q is te probability of survival. Te agent incurs costs wile aving a cild and ten as to bear te cost of education of te surviving cildren. Tat is, te cost related to cild rearing is given by [ qe( t ] b( t, (3 were cost of aving a cild given in terms of consumption goods, et ( is expenditure on education. Te law of motion of te adult population is were d is te deat rate given exogenously. L( t L( t( qb( t d, ( Production and explorations sectors Te ouseolds own te firms, and tey supply uman and pysical capital to firms. Since two sectors of production are considered, we need to specify te tecnology employed by eac of tem. We assume tat te production function for te consumption good is given as: 1 ( 1 Y AR K vh (5 were A stands for te tecnology coefficient and v is te fraction of labour engaged in te final consumption goods sector, 0 1, and 0 1. Accounting for H L, were is uman capital per capita, te production function is written in te intensive form as: y Ar ( k ( v 1 1. (6 Since we assume tat te resource stock is expandable, tis also requires some costly activity. Te expansion depends on te exploration, researc and development effort tat makes new resources available. We denote te incoming flow of resources by Z, and assume tat resource-creating activity is captured by a production function: Z E(1 v H, (7 8

9 were E is a tecnological coefficient, 0 1, and (1 v is te fraction of labour engaged in tis industry. In section 6, we will consider a case wit 1. In per worker terms, we write as: E(1 v z. (8 1 L Expression (7 involves te assumption tat te formation of incoming resource flow requires uman capital only. Terefore, te dynamics of te aggregate resource stock is defined as te net inflow of te aggregate resource and evolves according to te following differential equation: x r z (9 We follow Hartwick (1977, 1978 in modelling te resource use. Tat is, te resources are not owned privately, so cost of extraction is a pure cost to te economy. Given tis environment, te capital accumulation process in per worker terms is governed by: k y k ( X r c ( qe b, (10 were is te unit cost of extracting te resource. Following Endress et al. (2005, we model resource extraction as cost in terms of final goods. Suc a simplification is especially suitable for our model as it focuses on resource expansion rater tan extraction. Tis cost is a decreasing function of te resource stock, X. 3 d Tat is, 0. For furter simplicity, we as- dx sume tat tis function is given as: were is a cost parameter. ( X, (11 X Bot sectors consist of te firms maximising teir profits. We assume tat bot types of firms take te interest rate, te wage rate, and te cost of resource extraction as given. Tis optimisation ten yields te rate of return to pysical capital, i, te rate of return to uman capital in te production sector, w Y, and in resources sector, inputs (te outgoing resource flow, r, as follows: w Z, and te optimal amount of resource 3 Tis assumption is basically te Hotelling s rule. We reconcile te fact tat tis rule may not be being supported by te empirical observations (e.g. Gaudet, 2007 by te possibility tat te known stock of resources being altered continuously troug exploration and substitution. 9

10 1 1 1 XA r k ( v, (12 i A r k v (1 1 (1 (1 (1 (, (13 w Y (1 (1 (1 A(1 (1 r k ( v, (14 w Z E(1 v. (15 1 ( L Observing equations (13 and (14, we conclude tat te rate of return to capital and te wage rate depend on te optimal amount of resources used in production. Tat is, in an environment wit iger optimal levels of resource input, te rental prices of factors of production are iger. Terefore, it is crucial to ascertain wat drives te optimal levels of resource input. By observing (12, we establis tat te optimal level of te resource use, not surprisingly, increases wit te larger aggregate resource stock, X, tecnology coefficient (productivity, A, and wit a decrease in te exogenous cost of extraction,. One can also see tat te optimal amount of te aggregate resource used in te production of te aggregate final consumption good increases not only wit te level of per worker capital stock, but also wit te sare of labour engaged in te production of final consumption good. Intuitively, an increase in te sare of labour in te final consumption good production is possible only if te sare of labour engaged in resource expansion sector is srinking. Optimality requires tat any decrease in te labour input is offset by an increase in labour productivity. We will analyse tis link after determining te equilibrium values for te proportions of labour used in bot sectors. 3. Te representative agent s problem Te representative agent maximizes inter-temporal utility given by: c, v, b, e 0 maxu exp( t log( c log( qb dt (16 10

11 were exp( t is te coefficient discounting te utility of consumption over time, and subject to te following constraints: k y k ( X r c ( qe b, k(0 k, (17 0 x r z, (18 x( t 0, k( t 0, ( t 0, (19 e. (20 We write te Hamiltonian of te problem as follows: J u( c, qbexp( t y k ( X r c ( qe b E(1 v r ( e. Te first-order conditions for tis optimal control problem are given as: J u c exp( t 0, c (21 (22 (1 (1 0, v v (23 (1 (1 J (1 ( v Ar k E (1 1 J (1 Ar [( v k ], k k (24 J X X 2, (25 1 k v (1 J ( E(1 v u exp( t (1 (1 Ar, 1 J u b exp( t ( qe( t 0, b J qb 0. e (26 (27 (28 Accounting for tat u exp( t log( c log( qb, and combining (22 and (24 we obtain tat te growt rate of consumption: 11

12 (1 (1 (1 A (1 v c ( t r k. (29 c( t From (27 and (22 we obtain te birt rate function: ct ( bt (. (30 qe( t Anoter equilibrium condition is tat te marginal product of pysical capital sould equal te marginal product of uman capital. Tat is, (1 (1 (1 (1 Ar k (1 (1 (1 ( v ( v (1 (1 Ar k, k Solving wic we obtain te pysical-to-uman capital ratio in equilibrium: k 1. (31 Tat is, te stocks of pysical and uman capital sould grow at te same rate in equilibrium. Tese findings immediately give rise to te following lemma. Lemma 1. Te per capita consumption growt rate in te transition to te steady state positively depends on te evolution of te aggregate resource stock, te sare of workers engaged in final good production, te uman to- pysical capital ratio, and negatively on te cost of resource extraction. Proof. Te growt rate given by (29 can be re-written as: (1 (1 c XA 1 (1 (1. g A v c k g It is straigtforward to verify tat 0 X, g v 0, g k 0 g and 0. Tis result implies tat in an environment were initial stock of resources ig, te growt rates also sould be relatively iger. However, te steady state growt rate depends also on ow effective te country is in terms of maintaining te stock of resources. Tis implies tat if te economy is more effective in extending and extracting resources, tat is, a smaller sare of te labour force is employed in te resource sector, and a greater sare of te labour force is engaged in te production of te final goods, ence, suc an economy as a faster transition to steady state. 4. Te steady state analysis 12

13 Intuitively, in te long run te extraction and creation of resources in per capita terms sould be equal. Oterwise, eiter te economy would run out of resources or it would accumulate excessive resources. Clearly, bot strategies are not optimal. Tis discussion results in te following lemma. Lemma 2. In steady state, r zolds. Proof. Let us recall te equation describing te evolution of te aggregate resource stock given by x z r. In general, we cannot assume tat z r. If z r is true, ten it will lead to 1 XA an increase in te stock of resources, X. Ten given tat r k ( v 1 1, te optimal level of resource inputs also increases at a growing rate wit te stock of resources, te system will move towards z r. If z rolds ten te opposite would appen, given tat te stock of resource would be diminising. Tis consideration implies tat, in te long run, steady state is acieved only if x 0 olds. Tis result implies tat, in te long run, resource use sould not depend on te substitution between capital and natural resources; it rater depends on te productivity of te exploration sector and te sare of te labour force engaged tere. Te long run resource extraction cost is constant because te stock of resources becomes constant, unlike exogenously given backstop tecnology driven cost of assumed in Endress et al. (2005. Unlike in Hartwick (1977, tis equilibrium condition does not imply tat resource rents sould be entirely invested into pysical capital, ence, te sustainable growt does not depend on te Inada condition wit respect to resource inputs. Tis result allows to overcome te Georgescu-Reogen-Daly criticism and addressing te sortcoming of te DHSS framework pointed out by Anderson (1987 and Baumgärtner (2004. Moreover, for a fixed amount of resource extraction, equation (8 implies tat a larger population size results in lower resources use in production. Tis condition deals wit te sustainability bias argued by Grot (2007, as te resource creation in tis model (and extraction in steady state exibits diminising returns to uman capital, and as a negative scale effect on te size of population. We can find te optimal sare of labour engaged in exploration from te optimality conditions in te labour market and te inputs market. First, te optimality in te labour market leads to te equality of te returns to uman capital in bot sectors. Tat is, w Z w. (32 Y 13

14 Te wage rate, w, is given by (14, w Y A(1 (1 r k ( v (1 (1 (1 rate in te resource sector is equal to te marginal product of labour, ence,, wile te wage w Z z. In steady state, z r, terefore, r w. Taking tese ideas into account, we write Y 1 1 r A(1 (1 r ( k ( v. (33 By solving (33 for v we obtain: r v k A(1 (1 (1 1 (1 (1. (34 r Here, te bar denotes te steady state value. We notice tat due to w te following olds, r E(1 v. (35 1 L Tat is, wit, r 2 will be marginally decreasing ( i.e. r r 0, 0. We can find te 2 steady state value of uman capital from te condition tat bot (35 and (12 equations sould be satisfied simultaneously. Tat is, k ( v 1 E (1 v XA L. (36 We recall tat k. Ten taking tis into account, from (36 we find: (1 (1 1 E(1 v (1 1 L v XA. (37 Proposition 1. In steady state, te stock of uman capital is static only if te population is static. Proof. It is straigtforward from(37. One can also analyse comparative statics for and draw te following conclusion. 14

15 Corollary 1. Te steady state uman capita per worker is decreased in te size of population and te steady state stock of resources, and increased in te productivity of te resource sector and cost of resource extraction. Proof: Taking respective derivatives of expression(37 yields te following: 0 L 0, 0, 0. E X, 4.1. Steady-state consumption and long run population growt Lt ( Using te population growt equation, b( t q d, and te birt rate equation in equilibrium, bt (, one can write te steady-state consumption function: Lt ( ct ( qe( t d( qe c. (38 q Te findings to tis end allow us to determine te long run population growt. Tis result is formulated as te following proposition. Proposition 2. In steady state, te population is static. Proof. Recall tat Lt ( b( t q d and Lt ( c( t d bt ( qe( t q * d( qe c. Hence, q d wen bt (, and q c c * Lt ( 0. If per capita consumption is c Lt ( L L. * c ten 0 Falling population leads to increasing per capita uman capital in equilibrium, due to (37. Tis, in turn, leads to more resources per capita being used (see (35. Given tese increases, te production tecnology implies tat income per capita y rises. Tis will lead to rise in te level of consumption. Tus, consumption will rise, wic lifts te birt rate and te economy will be adjusting till it reaces te point wen c c *. In case, wen c c *, te adjustment will occur in te opposite direction. Terefore, te equilibrium wen population is static is stable equilibrium. Tis finding immediately leads to te following conclusion. 15

16 k c Corollary 2. In steady state 0 old. k c Proof. Wen c * c, and ence 0 c L, (37 implies tat in steady state 0. Since, c k, a fixed uman capital implies tat te steady state per capita pysical capital is 1 k also constant, or 0 k. Furtermore, te fixed uman capital in steady state implies tat education spending per cild is given as follows: e. (39 Overall, tese results indicate tat even wit potentially unlimited resources, an economy will find itself in a quasi-maltusian state, were bot te population and consumption per capita turn static Transition Dynamics We can consider te following transformed variables to analyse te transition dynamics: k, c, k After taking into account tat r. Ten te following equations k k y k ( X r c ( qe b, 1 c XA 1 k (1 A (1 v, c 1 r v XA k 1 1, (40 te growt of pysical capital can be re-written as follows: (1 (1 1 1 k v XA v XA A (1, (41 k (1 1 c v XA A (1. (42 c 16

17 c k Ten from 0 one obtains te condition 0 c k. 1 (1 1 v v XA 1 A(1 (1. (43 1 Te steady state for will be determined wen, wic will determine. Ten 1 te dynamics of te system can be sown as in Figure 1. Te figure sows tat wen consumption and productive capital are in steady state, te population turns static. 5. Te case wit 1 Since, in te above analysis, we assumed tat 0 1, wic obviously drives te steadystate results obtained. To see wat will be different if we alter tis assumption, let us consider te case wen te return to uman capital in resource production sector is constant. Tat is, 1. In tis case, from(35, we can find tat r E(1 v. (44 Figure 1. Pase diagram 17

18 Tis implies tat in steady state, resource use in production, r, will increase wit te stock of per capita uman capital. Moreover, in tis case, we will not get a negative relationsip between uman capital and te size of te population suc as(37. Instead, per capita capital will be independent of te size of population. Let us take a closer look at te expression for growt rate in tis case: (1 r k (1 (1 g A (1 v. (45 Te main difference from 0 1 case is tat te stock of uman capital on te balanced growt pat can grow unboundedly, wen 1. Terefore, te growt rate given by (45 can also grow witout a limit as soon as keeps growing. Tis result also implies tat growt of consumption will lead to iger population growt rates. Since, tere is no feedback from te size of te population to uman capital accumulation; te size of te population will not be limited as soon as te level of per capita consumption keeps growing. All in all, in te environment wit 1, we will see explosive growt of bot te economy and population. Obviously, it does not sound realistic. 6. Conclusions By assuming te endogeneity of bot te resource stock wic is dependent over time on te efforts of exploration, researc and development, and of te population wic ultimately depends on te dynamics of income stream, we ave made a few conclusions. Availability of resources made troug exploration and substitution is an important factor tat drives te long-run level of per capita consumption. Tat is because improvements in resource expansion troug new discoveries of reserves and substitutes determine te productivity of final good sector in te long run. Differing from te extant literature, we address te sustainability bias in te resource sector by assuming diminising returns to scale in te resource sector. We sow tat as soon as te stock of resources is expandable, te long-run sustainability does not depend on te binding Inada condition wit respect to resource inputs as in te DHSS framework. We find tat sustainability of long-run growt requires tat in steady state all used up resources sould be replenised by eiter finding new reserves or new substitutes. Given tat te productivity of uman capital in te resource sector is diminising, te economy reaces its steady state only wen te population reaces its steady state. If it is te case, ten we observe a quasi-maltusian constraint on population growt. 18

19 Environment and renewable resources are not considered ere. Accounting for tese factors may impose binding constraints tat will make sustained economic growt callenging. We also abstract from te uncertainty of resource discoveries. Tese questions are left for future researc. References Anderson, C. L. (1987, Te Production Process: Inputs and Wastes, Journal of Environmental Economics and Management 14, Ayres, R. U. B. Warr (2009. Te Economic Growt Engine: How Energy and Work Drive Material Prosperity, Edward Elgar Publising, Celtenam, UK. Baumgärtner, S. (2004. Te Inada Conditions for Material Resource Inputs Reconsidered, Environmental and Resource Economics 29, Brander, J.A. and. Taylor, M. S (1998. Te Simple Economics of Easter Island: A Ricardo- Maltus Model of Renewable Resource Use, te American Economic Review 88(1, Dalton, T. R., R. M. Coats, and B.R. Astrabadi (2005. Renewable resources, property- rigts regimes and endogenous growt, Ecological Economics 52, Daly, H. E., (1997. Georgescu-Roegen versus Solow/Stiglitz, Ecological Economics 22, Dasgupta, P. and G. M. Heal (1974. Te Optimal Depletion of Exaustible Resources, Review of Economic Studies 41, Doepke, M. (2004, Accounting for Fertility Decline During te Transition to Growt, Journal of Economic Growt 9, De La Croix, D. and M. Doepke (2003, Inequality and Growt: Wy Differential Fertility Matters," American Economic Review 93, Jones, L. and A. Scoonbroodt (2010, Complements versus Substitutes and Trends in Fertility Coice in Dynamic Models, International Economic Review 51(3, Endress, LH., Roumasset, JA & Ting Z (2005. Sustainable growt wit environmental spillovers, Journal of Economic Beavior & Organization, 58, Fioroni, T. (2010. Cild mortality and fertility: public versus private education, Journal of Population Economics 23, Galor, O. (2005. From stagnation to growt: unified growt teory. In P. Agion and S.N. Durlauf (eds Handbook of Economic Growt, Vol. 1A, Elsevier: Amsterdam. Galor, O. (2011. Unified Growt Teory. Princeton NJ: Princeton University Press. Galor, O., Weil, D.N. (2000. Population, tecnology and growt: From te Maltusian regime to te demograpic transition. American Economic Review 110,

20 Gaudet, G. (2007. Natural resource economics under te rule of Hotelling, te CaNADIAN Journal of Economics 40(4, Georgescu-Reogen, N., (1979 Comments on te papers by Daly and Stiglitz. In: Smit, V. K. (Ed., Scarcity and Growt Reconsidered. RFF and Jon Hopkins Press, Baltimore, MD. Goeller, H. E. and Weinberg, A. M., (1978 Te Age of Substitutability. Te American Economic Review. 68(6, Good, D.H., and Reuveny, R. (2006. Te fate of Easter Island: Te limits of resource management institutions, Ecological Economics 58, Grot, C. (2007. A New-Growt Perspective on Non-Renewable Resources in Sustainable Resource Use and Economic Dynamics, eds. Bretscger, L. and Smulders, S., Springer Neterlands Hartwick J.M. (1977. Intergenerational Equity and te Investing of Rents from Exaustible Resources, American Economic Review 77( 5, Hotelling, H., (1931. Te Economics of Exaustible Economics, Journal of Political Economy. 39( 2, Kamien, M. I. and N. L. Scwartz (1978. Optimal Exaustible Resource Depletion wit Endogenous Tecnical Cange, Review of Economic Studies 45 (1, Krautkraemer, J. (1998. Nonrenewable Resource Scarcity, Journal of Economic Literature 36(4, Lafforgue, G. (2008. Stocastic tecnical cange, non-renewable resource and optimal sustainable growt, Resource and Energy Economics 30, Livernois, J. R. and R. S. Uler (1987. Extraction costs and te economics of nonrenewable resources, Journal of Political Economy 90, Lucas, R.E. (1988. On te Mecanics of Economic Development, Journal of Monetary Economics 22, Meadows, D.H. et al. (1972. Te Limits to Growt, New York: Universe Books. Mookerjee, D., S. Prina and D. Ray (2012, A Teory of Occupational Coice wit Endogenous Fertility", American Economic Journal,4(4, Peretto, P. and S. Valente (2011. Growt on a Finite Planet: Resources, Tecnology and Population in te Long Run, Center of Economic Researc (CER-ETH, ETH Zuric working paper series, 11/147, ttp:// Pezzey, J. C.V. and J. M. Anderies (2003. Te effect of subsistence on collapse and institutional adaptation in population-resource societies, Journal of Development Economics 72, Pindyck, R. (1978. Te Optimal Exploration and Production of Nonrenewable Resources, Journal of Political Economy 86(5,

21 Robson, A.(1980. Costly innovation and natural resources, International Economics Review 21(1, Scou, P. (2000. Polluting Non-renewable Resources and Growt, Environmental and Resource Economics 16, Sculer, R. (1979. Te Long Run Limits to Growt: Renewable Resources, Endogenous Population, and Tecnological Cange, Journal of Economic Teory 21, Solow, R. M. (1974a. Intergenerational Equity and Exaustible Resources, Review of Economic Studies 41, Solow, R. M. (1974b. Te Economics of Resources or te Resources of Economics, American Economic Review Papers and Proceedings 64, Stiglitz, J. (1974a. Growt wit Exaustible Natural Resources: Efficient and Optimal Growt pats, Review of Economic Studies Symposium Issue, 41, Stiglitz, J. (1974b. Growt wit Exaustible Natural Resources: Te Competitive Economy, Review of Economic Studies 41, Uzawa, H. (1965. Optimum Tecnical Cange in Aggregative Model of Economic Growt, International Economic Review 6,