Three essays in macroeconomics and natural resources

Three Essays in Macroeconomics and Natural

Resources

Thèse

Daouda Belem

Doctorat en économique

Philosophiæ doctor (Ph. D.)

Québec, Canada

Three Essays in Macroeconomics and Natural

Resources

Thèse

Daouda Belem

Sous la direction de:

Stephen Gordon, directeur de recherche Kevin Moran, codirecteur de recherche

Résumé

Les ressources naturelles sont d’une grande importance pour l’activité économique. Malgré le lien étroit qui existe entre les flux de resources naturelles et le niveau de l’activité économique, l’attention accordée aux resources naturelles dans la modelisation et l’analyse macroecono-mique reste minime. Dans la majorité des cas, les resources sont ignorées, ou lorsqu’elles sont prises en compte, les modèles aboutissent à des conclusions dissociant l’activité économique et les flux de resource.

Cette thèse propose un cadre permettant d’incorporer les resources nonrenouvelables dans des modèles macroéconomiques. En particulier, une fonction de production est spécifiée pour l’extraction de la ressource. Aussi, nous intégrons la dynamique du stock de la resource qui incorpore le progrès technique, et la découverte et le développement de nouvelles reserves. Nous dérivons des conditions pour caractériser des chemins d’exploitation optimale, et d’extraction soutenable.

Ensuite, la thèse présente un modelè d’équilibre général stochastique (DSGE) dans lequel la production et l’utilisation du pétrole (ressource) sont analysées de manière exhaustive. Le modèle spécifie que le pétrole est extrait d’un stock dont le renouvellement est soit exogène, soit endogène et réagissant aux incitations économiques. Les simulations montrent que le fait de tenir compte des incitations économiques pour modéliser la production de la ressource affecte les implications à court et long termes du model.

Enfin, la thèse se termine par une analyse de la rareté du pétrole. Elle examine aussi l’influence de certains facteurs économiques sur les augmentations des réserves dans un panel de 37 pays de 1980 à 2017. Nous trouvons que le prix du pétrole et le cumul des découvertes de pétrole sont positivement corrélés à l’évolution des ajouts aux réserves mondiales de pétrole. Deux nouveaux facteurs ont aussi été intégrés dans l’analyse, à savoir la rente pétrolière en pourcentage du PIB et l’ouverture commerciale qui sont significativement corrélées avec les découvertes de pétrole.

Abstract

Natural resources, especially oil, are of tremendous economic importance. Despite the tight link between the flow of natural resources and measures of overall economic activity such as GDP, natural resources have not been given a commensurate attention in macroeconomic modelling and analysis through time. They are simply ignored in the vast majority of macroe-conomic models, and those models that do integrate natural resources end up with conclusions decoupling the evolution of economic activities and the flow of natural resources.

In this thesis, a framework for incorporating non-renewable resources into macroeco-nomic models is proposed. In particular, a simple production function is specified for the extraction process. In addition, we integrate a law of motion for the stock of nonrenewable resources that incorporates technical progress and the discovery and development of new re-serves. We derive conditions for optimal paths, and conditions for sustainable extraction of the resource.

Next, we develop a closed-economy stochastic general equilibrium (DSGE) model in which oil (resource) production and usage are analyzed extensively. We suppose that oil is extracted out of a stock whose replacement can either be exogenous, or endogenous and reacting to underlying economic incentives. The simulations indicate that modeling resource production as responding to economic incentives has important effects on both the business-cycle and long-term implications of the model.

Finally, we analyse oil scarcity and examine the influence of competing economic factors in driving oil reserve additions using a panel data model for 37 countries from 1980 to 2016. The results show a significant positive correlation of oil price and cumulative reserve additions with global oil reserve additions. Two new plausible economic factors have been considered, namely oil rent as per cent of GDP and trade openness which both present a significant correlation with the evolution of oil reserve additions.

Contents

Résumé iii

Abstract iv

Contents v

List of Tables vii

List of Figures viii

Remerciements ix

Avant-propos x

Introduction 1

Bibliography . . . 4

1 Depletion, Discovery and the Macroeconomics of Nonrenewable Re-sources 5 Résumé . . . 5

Abstract . . . 6

1.1 Introduction. . . 7

1.2 A model of resource extraction with discoveries . . . 10

1.3 Solving the resource-extracting firm’s problem . . . 18

1.4 Some results on convergence and stable paths . . . 23

1.5 Conclusion . . . 31

Bibliography . . . 33

Appendix . . . 35

2 Endogenous Oil Production and Fluctuations in a Closed Economy Framework 38 Résumé . . . 38 Abstract . . . 39 2.1 Introduction. . . 40 2.2 Model . . . 43 2.3 Calibration . . . 51 2.4 Impulse responses. . . 54

2.5 Alternative specification of the endogeneity of the resource stock replacement. 61 2.6 Conclusion . . . 65

Bibliography . . . 67

3 On the Determinants of Oil Reserves Additions: A Panel Data Analysis of Oil-Rich Countries 69 Résumé . . . 69 Abstract . . . 70 3.1 Introduction. . . 71 3.2 Reserve additions . . . 74 3.3 Previous research . . . 79

3.4 Exploration behaviour at the firm level . . . 80

3.5 The econometric specification . . . 83

3.6 Data . . . 86

3.7 Estimation and results . . . 87

3.8 Conclusion . . . 92

Bibliography . . . 93

Appendix . . . 95

List of Tables

2.1 Calibration of parameters and steady states . . . 52

3.1 1985-2015 period: Fixed Effects and Random Effects estimates. . . . 88

3.2 Random Effects: 5-year average and yearly data from 1990 to 2001. 89

List of Figures

1.1 Dynamics of st when ν = 1 + r . . . 25

1.2 Dynamics of s_t when ν > 1 + r and high profits . . . 26

1.3 Dynamics of st when ν > 1 + r and low profits . . . 27

1.4 Dynamics of st when 16 ν < 1 + r and high profits . . . 29

1.5 Dynamics of s_t when 1_{6 ν < 1 + r and low profits} . . . 30

2.1 Oil prices and world oil reserves . . . 41

2.2 World oil production and world oil reserves . . . 41

2.3 IRFs for 3 degrees of dependence of stock replacement to oil prices (1). . . 56

2.4 IRFs for 3 degrees of dependence of stock replacement to oil prices (2). . . 57

2.5 IRFs when an exogenous resource boom occurs . . . 60

2.6 IRFs for 3 degrees of dependence of replacement to 4-year price change (1) 63 2.7 IRFs for 3 degrees of dependence of replacement to 4-year price change (2) 64 3.1 Oil Proven reserves-production ratio . . . 72

3.2 Cumulative oil production versus net additions, 2008-2017. . . 75

3.3 Proven oil reserves in OPEC members at the beginning of 2018 . . . 76

3.4 Net additions to proven oil reserves . . . 77

3.5 Total discoveries or gross additions to proven oil reserves . . . 78

3.6 World, OPEC and Non-OPEC Proven oil reserves . . . 95

Remerciements

Tout d’abord, je rends grâce à Dieu pour m’avoir permis d’atteindre l’aboutissement de cette thèse. Je tiens tout particulièrement à remercier le Professeur Stephen Gordon pour avoir accepté de me diriger durant toutes ces années. Mes remerciements s’adressent aussi au Professeur Kevin Moran pour avoir accepté de codiriger cette thèse. Je le remercie de m’avoir éclairci sur la voie de recherche en modèle d’équilbre général. Les nombreuses discussions et collaborations qu’ils ont apporté ont été très enrichissantes dans mon cheminement.

L’expression de ma profonde gratitude s’adresse également aux Professeurs Lilia Karni-zova, Benoit Carmichael, et Markus Herrmann pour avoir généreusement accepté d’être mem-bres de ce Jury de thèse, témoignant ainsi leur intérêt pour mes travaux de recherche.

À tous mes amis où qu’ils se trouvent qui ont toujours cru en moi et m’ont soutenu et encouragé tout au long de ce long parcours, j’adresse mes sincères remerciements. Je pense également aux doctorants avec qui j’ai chaleureusement eu des échanges et collaborations enrichissants. Pour finir, j’aimerais rendre un hommage particulier à mes parents, mes frères et mes soeurs qui ont toujours fait preuve de patience durant ce long parcours et qui malgré la distance m’ont toujours soutenu et encouragé.

Avant-propos

Cette thèse s’articule autour de trois chapitres plus ou moins indépendants qui s’inscrivent dans les champs de la macroéconomie, l’économie des ressources naturelles et de l’énergie (pétrole). Je suis le principal auteur de chacun de ces trois articles. Le premier chapitre est un article réalisé avec mon directeur de thèse, Stephen Gordon. Le deuxième chapitre est un article réalisé avec mon co-directeur de thèse, Kevin Moran, et mon directeur de thèse, Stephen Gordon. Le troisième chapitre est un article que j’ai réalisé de manière plus ou moins indépendante avec la collaboration de mon directeur de thèse, Stephen Gordon. Ces trois articles font l’objet de quelques révisions pour être soumis à des revues scientifiques avec comité de lecture.

Introduction

Modern thought on economic growth stems from the basic model of Ramsey (1928), which endogenously determines a saving rate through optimal consumption behavior. This model is viewed as a generalization of the standard Solow (1956) model which considers the saving rate to be exogenously constant. Ramsey’s pioneering ideas were later adopted byCass(1965) and Koopmans (1965), and now form the basis of the workhorse model underlying much of modern macroeconomics. The vast majority of these models assumes production functions that feature decreasing marginal productivity of capital in order to have a steady state in which consumption per capita increases at the rate of the exogenously-unbounded technical progress. The AK model, which is one class of endogenous growth models, offers a framework that features a continuingly-increasing per capita consumption by assuming a world without technical progress, but a production function without diminishing returns to capital (Frankel,

1962;Romer,1986).

One major criticism about these models is that natural resources are left out. These models ignore the concern associated with the apparent incompatibility of the strong desire to sustain high economic growth in our societies and the fact that such growth was relying on a finite, nonrenewable stock of resources (Cook,1976). According toDaly(1977), there were no concerns to report about aggregate growth in the past when the world was relatively "empty". However, in the present full-world era, economic growth is likely to be uneconomical because it likely costs more than it is worth due to the accelerated natural capital depletion which turns out to be a limit in sustaining economic growth. In the empty world of the past, the limiting factor was capital; in today’s full world, the remaining natural resources are becoming scarcer and scarcer.

The choice to often leave aside natural resources when analysing growth is likely not due to a lack of interest in integrating natural resources in the macroeconomic analysis; but rather, taking natural capital into account is subject to technical difficulties. Due to the specificity of

natural resources which are most of the time finite or of low renewal rate, it is quite difficult to integrate them into a growth model in which everything is increasing.

Further growth-related models investigate whether an economy with constant popula-tion and without technical progress can have a sustained level of consumppopula-tion when faced with a finite stock of resources (Solow,1974;Hartwick,1977). In fact, Solow showed that a constant consumption policy will require society to unboundedly accumulate capital in order to counter the depletion of resources if the capital-elasticity of output is greater than that of natural resources. However, this mathematical solution to this scarcity problem seems rather counterintuitive due to the model’s own hypothesis stipulating that natural resources are essential to production.

With a Cobb-Douglas production function for the consumption good, the substitutability of the productive capital for natural resources is the main feature which allows for a sizeable production without natural resources in the long run. In addition, this strong substitutability hypothesis has given birth to the weak sustainability concept whose proponents argue that resource depletion can turn out not to be catastrophic if reproductive capital is accumulated in an appropriate way. On the other hand, there are the strong sustainability advocates who consider the weak sustainability concept to be unrealistic. They argue that the weak sustainability proponents’ statements regarding natural resources may be incorrect due to the fact that they rely on standard production functions that violate the laws of physics by treating resources no differently than other factors of production (Georgescu-Roegen, 1979;

Ayres and Nair, 1984; Cleveland et al., 1984) . They rather believe that natural resources and man-made capital are complements, and such complementarity is reflected through the ’material balance’ approach to the production process. The quantity of goods produced must reflects the quantity of raw materials that enter in the process.

In view of this debate, the problem of how to suitably incorporate natural resources in the macroeconomic models is still ongoing. In this thesis, we provide in Chapter 1 an analytical framework for integrating nonrenewable resources into macroeconomic models. In this analysis, a suitable resource extraction function is used and we allow for new reserve discoveries in order to depart from the traditional resource depletion models. Consequently, the law of motion for the stock of the resource depends on the quantity extracted and on the discovery of new reserves. This first chapter also provides some preliminary results about the

conditions under which extraction is sustainable over the long term.

A usual practice in macroeconomic analysis is to investigate the fluctuations of the macroeconomy around a steady state or around a balanced growth path. Economic fluctu-ations, which are then caused by changes in the growth rate in an economy, have also been given attention in macroeconomic theory and empirical studies. The second chapter provides a two-sector closed-economy dynamic stochastic general equilibrium model in which one of the sectors is the oil sector and the other is the consumption good sector. The main feature of this model is that it takes into account resource production and the evolution of the underlying resource stock, as modelled in Chapter 1. The consumption goods are produced by capital, labor and resource inputs which are determined by an endogenous extraction rate. Two key sources of macroeconomic fluctuations are considered in this model, namely shocks to the productivity in consumption goods’ production and to the stock of the resource (oil).

According to the estimates provided by the US Energy Information Administration (EIA), world oil production increased by 50 per cent between 1980 and 2018, while proved oil reserves more than doubled. In other words, the world oil industry has managed to decrease oil scarcity from 28 years of remaining proven reserves in 1981 to 47 years in 2018. The third chapter examines the influence of competing economic factors in driving reserve additions in oil-producing countries. It specifies and estimates a panel data model for 37 countries in the 1980–2016 period. The specificity in this study is that it accounts for new economic factors such as oil rent as per cent of GDP and trade openness.

This thesis contributes to the literature pertaining to the analysis of natural resources and the macroeconomy. Mainly, it provides an extensive modeling of the resource sector in Chapters1and2. The core of this resource sector modelling is the resource production function that we specified. This production function, admitting the available resource stock as an upper bound, is well-suited to describe resource production in both resource-rich and resource-scarce economies. In addition, the two-sector model developed in Chapter 2 contributes to the understanding of macroeconomic fluctuations, notably the behaviour of oil price after the occurrence of a shock to manufacturing productivity. Finally, this thesis contributes to the literature related to factors determining oil reserve additions. Specifically, the Chapter 3 has introduced into this literature two new economic factors susceptible to influence oil reserve additions; namely oil rent as percent of GDP and the economic openness.

Bibliography

Ayres, R. U. and I. Nair (1984): “Thermodynamics and economics,” Physics Today, 37, 62–71. Cass, D. (1965): “Optimum growth in an aggregative model of capital accumulation,” The Review of

economic studies, 32, 233–240.

Cleveland, C. J., R. Costanza, C. A. Hall, and R. Kaufmann (1984): “Energy and the US economy: a biophysical perspective,” Science, 225, 890–897.

Cook, E. F. (1976): “Man, energy, society,” Tech. rep., Texas A and M University, College of Geo-sciences, College Station, TX.

Daly, H. E. (1977): “Steady State Economics: The Economics of Biophysical and Moral Growth,” . Frankel, M. (1962): “The production function in allocation and growth: a synthesis,” The American

Economic Review, 52, 996–1022.

Georgescu-Roegen, N. (1979): “Comments on the papers by Daly and Stiglitz,” Scarcity and growth reconsidered, 95–105.

Hartwick, J. M. (1977): “Intergenerational equity and the investing of rents from exhaustible re-sources,” The American Economic Review, 67, 972–974.

Koopmans, T. C. (1965): “On the concept of optimal economic growth,” .

Ramsey, F. P. (1928): “A mathematical theory of saving,” The economic journal, 38, 543–559. Romer, P. M. (1986): “Increasing returns and long-run growth,” Journal of political economy, 94,

1002–1037.

Solow, R. M. (1956): “A contribution to the theory of economic growth,” The Quarterly Journal of Economics, 70, 65–94.

——— (1974): “Intergenerational equity and exhaustible resources,” The Review of Economic Studies, 41, 29–45.

Chapter 1

Depletion, Discovery and the

Macroeconomics of Nonrenewable

Resources

Résumé

Nous présentons un cadre théorique permettant d’incorporer les ressources non renou-velables dans des models macroéconomiques. Plus précisement, nous proposons une fonction de production simple pour l’extraction de la ressource, et également une loi de mouvement pour le stock des resources non renouvelables, laquelle loi incorpore le progrès technologique, ainsi que la découverte et développement de nouvelles réserves. Nous fournissons ainsi des conditions de chemins d’exploitation optimale, et identifions les conditions dans lesquelles l’extraction se fera de manière soutenable.

Classification JEL: L72, Q30, Q33.

Abstract

We present a framework for incorporating non-renewable resources into macroeconomic models. In particular, we propose a simple production function for the extraction process as well as a law of motion for the stock of nonrenewable resources that incorporates technical progress and the discovery and development of new reserves. We provide conditions for optimal paths, and identify conditions in which extraction is sustainable and when it is not.

JEL classification: L72, Q30, Q33.

1.1

Introduction

The prices of non-renewable resources - especially oil - play a crucial role in macroeconomic analysis: sharp movements in commodity prices are often associated with business cycle turn-ing points. But even if the non-renewables sector is seen as important, it generally plays a minor role in formal models of the macroeconomy. Commodities are typically modeled as a source of exogenous shocks, if they are included at all.

This under-representation is not due to a lack of interest in the interactions between oil and the rest of the macroeconomy. Rather, the problem is technical: conventional models of non-renewable resource extraction predict declining levels of reserves and production, and these predictions are difficult to reconcile with the usual practice of analyzing the fluctuations of the macroeconomy around a steady state or around a balanced growth path. On the other hand, more recent literature examines the role of oil production both in propagating and originating shocks (Andrés et al., 2006; Murchison et al., 2006; Dib, 2008; Lees et al.,

2009; Jääskelä and Nimark, 2011; Bodenstein et al., 2011; Natal, 2012; Lama and Medina,

2012). This strand of research generally treats the flow of oil into production as a stationary exogenous process, and is silent on the question of whether or not the underlying stock of reserves is replaceable or replaced. The predictions of these conventional models are difficult to reconcile with the stylized facts of increasing reserves and production in non-renewables sector.

Reconciling economic activity based on non-renewable resource with sustained economic growth is a challenge that goes back at least toMalthus(1798). A fixed supply of land, Malthus argued, could not sustain continuing exponential growth in human population. If people were unable or unwilling to restrain reproduction, then the population would be controlled by other means: war, diseases, and starvation. Interest in the topic was revived in the 1970’s, with the publication of the Club of Rome‘s warnings of the depletion of fossil and mineral resources.

Meadows et al. (1972), in their report on “Limits to Growth” to the Club of Rome, predicted that future world population levels, food production and industrialization would at first grow exponentially but then would collapse during the next century. The collapse would occur, so the argument went, because the world economy would attain its physical limits in terms of non-renewable resources, agricultural production, and excessive pollution. One relevant prediction of the study was that 11 vital minerals could be depleted by 2000. Leontief(1977),

on the behalf of the United Nations, adopted the same pessimistic assumptions as the “Limits to Growth” report, except that they incorporated the possibility that demand would respond to higher prices. This study projected that only two minerals - zinc and lead - were in danger of being exhausted.

Barnett and Morse (1963) offer a more optimistic view about resource scarcity and Malthusian predictions. They found that the prices and production costs of mineral, agricul-ture and renewable resources had fallen or remained unchanged over the 1870-1957 period, except in forestry where prices displayed an increasing trend. They attribute these findings to technological progress that made it possible to develop substitutes for scarce resources, to reduce extraction costs and to discover new economic reserves.

This study expands on this last point. Even though the nonrenewable resources - espe-cially oil - play an important role in the macroeconomy, incorporating them into macroeco-nomic models is problematic. As has already been noted, conventional models of the resource sector typically predict paths for output and reserves that decline over time, while much of macroeconomic analysis makes use of models that predict balanced growth paths. But perhaps more importantly, the basic predictions of declining production and reserves are difficult to reconcile with the data. According to the estimates provided by the US Energy Information Administration (EIA), world oil production increased by 50 per cent between 1980 and 2015, while proved oil reserves more than doubled. Calibrating a conventional resource-extraction model to these stylized facts is problematic.

While there are only so many minerals and fossil fuels on earth, the physical stock of nonrenewable resources is not the binding constraint over the length of a business cycle, or even over the span of several business cycles. The more relevant stock is the proved reserves - a measure of the resource that can be feasibly extracted using existing technology and under current prices. The combination of technical innovation and/or higher prices can increase estimates for proved reserves. When the EIA determined that exploitation of the Athabaska oil sands had become profitable under prevailing prices and technology, its estimate for Canada’s proved reserves increased from 4.8 billion bbls in 2002 to 180 billion bbls in 2003. A less dramatic increase has also occurred in the United States with the adoption of fracking techniques: US proved reserves in 2016 were fifty per cent higher than they were ten years earlier.

Although economists have studied the role of natural resources for centuries, modern thought on exhaustible resources is greatly influenced by the seminal work ofHotelling(1931). In a famous result, the optimal - and sustainable - strategy for a monopolistic owner of a resource will be to set extraction rates so that the net price of the exhaustible resource increases at the rate of interest. However, if there is no monopolistic owner that can control access, then the stock of resources will be depleted in finite time.

Since then, the literature on natural resource economics can be classified into two main groups. One branch investigates the feasibility of sustained consumption under limited natural resources by addressing the question of whether or not an economy could maintain a constant level of consumption if there is no technical progress and if the production of consumption goods is possible only by using limited nonrenewable resources, such as oil (Solow, 1974;

Dasgupta and Heal,1974;Solow and Wan,1976;Hartwick,1977;Neumayer,2013). Another strand of the literature addresses the question of optimal resource extraction and includes analyses pertaining to the effect of profitability on resource depletion (Puu,1977), the effect of capital intensity on optimal extraction rate (Campbell,1980), the effect of taxation, percentage depletion allowance and non-internalized externalities on depletable resource use (Sweeney,

1977; Lasserre and Daubanes, 2011), the existence of a backstop technology in a model of exhaustible resource consumption (Oren and Powell,1985) and the order of resource extraction (Solow and Wan,1976;Chakravorty and Krulce,1994;Holland,2003).

This present study rather provides an analytical framework for incorporating nonre-newable resources into models of the macroeconomy, in which a suitable resource extraction function is used and the law of motion for the stock of the resource depends on the quantity extracted and on the discovery of new reserves. It also provides some preliminary results about the conditions under which extraction is sustainable over the long term, and offers some suggestions for extensions in future work.

The remainder of this paper is organised as follows: Section 1.2 develops the model of resource extraction; Section 1.3 presents the procedure followed to solve the model; Section

1.2

A model of resource extraction with discoveries

The key to reconciling the observed increases in productions and proven reserves with the predictions of the standard models is to model the discovery of new reserves. The framework developed here introduces new discoveries into the model with a single parameter. It also introduces a functional form for extraction technology that is both linearly homogeneous in capital, labour and in the resource stock, and which the available stock is an upper bound for current extraction.

1.2.1 A law of motion for the resource stock with discoveries

The relevant measure for the resource stock is not the total amount that exists on Earth -a qu-antity th-at is obviously non-incre-asing - but proven reserves. Denote by St the stock of proved reserves at the beginning of period t, and let Yt represent the resource extracted from proved reserves during period t. In the absence of the development or discovery of new reserves, the law of motion for the stock is simply

St+1= St− Yt (1.1)

As noted earlier, proven reserves are a measure of the resource stock that can potentially be profitably exploited, and are not necessarily always in decline: developments that make it easier or more profitable to access previously-inaccessible resources can increase the stock of proven reserves, even if extraction continued unabated.

We generalize (1.1) as in d’Albis and Ambec (2010) and incorporate the discovery of new reserves by introducing the parameter νt≥ 1, the gross discovery rate of new reserves:

St+1= νt(St− Yt) (1.2)

If νt > 1, then there will be new proved reserves at the beginning of period t + 1, so that St+1 > St− Yt. If there are no new reserves made available in period t, then νt = 1 and simplifies to (1.1). Note also that in order for extraction Yt to be strictly positive, (1.2) also implies that νt(St/St+1) > 1; this condition will play an important role in the discussion below.

The gross discovery rate ν_t can be interpreted as a reduced-form solution to a deeper structural model of firms’ exploration and development, varying with changes in technology and in the price of the resource:

νt= ν _A t+1 At , Pt+1 Pt  (1.3) where A_t represents the level of technology - either extraction or exploration - available at t, and Pt is the price. Everything else being equal, we expect ν(·, ·) to be increasing in its inputs, and that exploration and development activity will cease if prices and technology remain constant. If this is the case, then ν(1, 1) = 1 and (1.2) again reduces to (1.1).

1.2.2 A resource extraction production function

Extraction output Y_t depends on the level of capital (K_t) and labour (L_t) inputs, as well as the stock of proved reserves (St); denote this relationship by

Yt= G(Kt, Lt, St) (1.4)

For tractability and internal coherence, the function G(·, ·, ·) should satisfy the following conditions

1. Bounded production: 0 < G(K, L, S) < S

A distinguishing feature of the resource sector is that production at any period is bounded above by the available stock of proved reserves.

2. Linear homogeneity: G(·, ·, ·) is homogeneous of degree 1

We adopt the standard replicability argument here: if the same amounts of capital and labour are put at work on identical deposits, then both projects will extract the same amount. In other words, doubling all of the inputs of G(K, L, S) will also double production Y .

3. Inada conditions

A candidate production function that satisfies all of the above features can be specified by the following adaptation of the logistic function:

G(Kt, Lt, St) = St 

1 − e−F (Kt,Lt)St



(1.5)

In this specification, the function F has the usual features of a neoclassical produc-tion funcproduc-tion, i.e homogeneous of degree one in K and L, positive and decreasing marginal productivity and respecting the Inada conditions:

∂F ∂K ≥ 0 ∂F ∂L ≥ 0 ∂2_F ∂K2 ≤ 0 ∂ 2_F ∂L2 ≤ 0 lim K→0 ∂F ∂K = +∞ _K→+∞lim ∂F ∂K = 0 lim L→0 ∂F ∂L = +∞ _L→+∞lim ∂F ∂L = 0 (1.6)

Subsequently, G is homogeneous of degree one in (K, L, S) and we have: ∂G ∂K = ∂F ∂K. e −F (Kt,Lt) St _{≥ 0} ∂G ∂L = ∂F ∂L. e −F (Kt,Lt) St _{≥ 0} ∂2G ∂K2 = h ∂2F ∂K2 −_S1_{t ∂K}∂F
2i e−F (Kt,Lt)St _{≤ 0} ∂2G ∂L2 = h ∂2F ∂L2 −_S1_t ∂F_∂L
2i e−F (Kt,Lt)St _{≤ 0} lim K→0 ∂G ∂K = +∞ _K→+∞lim ∂G ∂K = 0 lim L→0 ∂G ∂L = +∞ _L→+∞lim ∂G ∂L = 0 (1.7) The function G has the same features as F , except that G is not homogeneous of degree one in K and L 1_{. Note also that when expressed as a share of the available reserves, the}

extraction technology is homogeneous of degree zero. Doubling extractive inputs and the stock of reserves does not affect the share of the resource that will be extracted.

Another attractive feature of this production function is that given the levels of capital and labor employed in resource extraction, larger reserves are associated with higher produc-tion: ∂G ∂S = 1 − h 1 +F (Kt,Lt) St i e−F (Kt,Lt)St _{> 0 (1.8)} 1

The verification of the Inada conditions at infinity is not necessary because the specification of the function G takes them into account in that when inputs are increasing, the production will be bounded above by the available stock.

Hence, given the quantity of capital and labor, we have2_: lim S→+∞ ∂G ∂S = 0 _S→0lim ∂G ∂S = 1 (1.9)

When the resource endowment is large enough relative to inputs, production is effectively unconstrained, and a further increase of proved reserves will not result in an significant increase of production. In contrast, when the resource endowment is very scarce, the upper bound on resource extraction becomes more binding, and a marginal increase in the stock will translate into increased extraction.

The main feature of this function is that it generalizes the production of the resource by pooling the scarce resource stock and the abundant resource stock cases together. In a resource-scarce economy, resource extraction will not show constant return to scale, when resource stock is abundant the extraction increases proportionally with an increase of labor and capital inputs. At the limit, we have:

lim

S→+∞G(K, L, S) = F (K, L) (1.10)

That is to say, when the resource stock is infinitely abundant so that its nonrenewable nature is no longer economically relevant, the constrained production function G converges to the unconstrained function F .

In order to obtain a well-defined and realistic investment function, it will sometimes be useful to incorporate adjustment costs. Everything else held constant, investment in new capital goods can be expected to disrupt production. In this case, the production in the resources sector can be represented by :

Yt= FR[Kt, Lt, St, It] (1.11) with I_tthe investment in the resource producing firm and where FR is decreasing in I_t 3.

2

It can be shown that 0 < ∂G_∂S < 1. 3

1.2.3 Behavior of a myopic resource-producing firm

In order to develop intuition, it is worthwhile to consider the case where the firm is only concerned with events in the current period. In a given period, and knowing the available stock of the resource, the resource sector firm will choose levels of inputs that minimize the costs of production of any given quantity of the resource. The instantaneous problem of the resource producer is :

min

{L,K} wL + rkK subject to G(K, L, S) ≥ Y

Where w and rk are respectively the unit cost of labor and capital; and Y is the quantity of

the resource to be produced.

The Lagrangian of this problem is:

L = wL + rkK − λ [G(K, L, S) − Y ] , (1.12)

where λ is the shadow price of the resource. The first order conditions are:

λGL= w λGK = rk G(K, L, S) = Y (1.13) This implies: GL GK= ∂F ∂L

/

In the special case in which the unconstrained production function F has the Cobb-Douglas form F (K, L) = KαL1−α, we obtain:

(1−α)Kα_L−α α.Kα−1_L1−α= _rw k (1.15) S  1 − e−KαL1−αS  = Y (1.16)

At time t, given the ratio of wage to capital cost (_rw

k), the quantity of resource to be

be used in resource sector are: L(w, r_k, S, Y )= − 1−α_α
α_w rk −α S log 1 −Y_S
(1.17) K(w, rk, S, Y )= − 1−α_α
α−1_w rk 1−α S log 1 −Y_S
(1.18)

These two relations are the instantaneous labor and capital demand functions of the resource firm. The labour demand function is decreasing in the wage w and increasing in capital cost r_k, whereas the capital demand function is decreasing in r_k and increasing in w. These relationships simply reflect the substitution possibilities between labor inputs and capital in the Cobb-Douglas technology. When the price of any input increases, the firm simply reduces its demand in favor of the other input.

Note also that both functions are homogeneous of degree one in S and Y . That reflects the fact that, given the same input prices, if a country is two times richer in terms of the stock of resources than another, this country must be using two times more labor and capital in order to have the same resource production-resource stock ratio. In other words, the resource-rich country must have an resource industry two times larger than that of the resource-poor country. Moreover, the labor and capital demands are both increasing with the quantity of resources to be produced Y , with 0 ≤ Y ≤ S.

With no resource production, the resource industry is of course nonexistent as labor and capital demands are both zero. But if at any period, a country aims to extract all of its resource stock, it must have an infinite-sized resource industry as it takes an infinite quantity of labor and capital to extract the entire stock. Because it is so costly to extract all of the resource stock, compete depletion will never occur in an economy where there are limited inputs.

It can be shown that labor and capital demands are both decreasing in S. This feature is consistent with the standard theory of resource production in that it is easier to extract when the resource stock is large, and extraction becomes more difficult as the resource stock becomes scarcer. Therefore, given the quantity of resource to be extracted, the closer the resources stock is to planned production, the more inputs must be used. That is to say, that when the resource stock is approximately infinite relative to extraction, the economy uses the

least capital and labor inputs possible. These quantities are 4_: L(w, rk, S, Y )= 1−α_α
α_w rk −α Y (1.19) K(w, rk, S, Y )= 1−α_α
α−1_w rk 1−α Y (1.20)

These relations are independent of S and are also what would be obtained by simply using the unconstrained Cobb-Douglas resource production function F .

1.2.4 Intertemporal behavior of a resource-producing firm

Let us now consider the problem as one of maximizing the present value of the stream of profits generated by the extraction of a nonrenewable resource. As noted earlier, resource production may be subject to adjustment costs when it comes to bringing capital to its desired level. The firm decides what level of capital to own in the next period by investing during the current period: current investment becomes productive in the next period. These costs of adjusting the capital stock may be measured as current forgone output due to current capacity building for the next period. We specify a production function net of adjustment costs as:

Yt= G[Kt, Lt, St].e−c(log[Kt+1]−log[(1−δ)Kt])

2

(1.21) ≡ ˜F (St, Kt, Kt+1, Lt)

where δ is the rate of depreciation of capital. Note also that (1.21) incorporates the law of motion for the capital stock:

Kt+1 = (1 − δ)Kt+ It (1.22)

A firm in the resource sector chooses paths of capital and labor inputs to maximize its total discounted profits during the life of the extraction project. Their problem is:

max {Yt,Lt,It} −I₀+ ∞ X t=1 Rt[PtYt− wtLt− It] (P.2) subject to (1.2), (1.22) and (1.21)

Where R_t represents the discount factor, P_t the relative price of the resource (price in terms of the capital good). Equation (1.2) implies that:

Yt= St−St+1_νt (1.23) (1.21) and (1.23) imply: G(St, Kt, Lt)e−c(log[Kt+1]−log[(1−δ)Kt]) 2 = St−St+1_νt (1.24) ⇒ St 1 − e −Kαt L1−αt ) St ! e−c(log[Kt+1]−log[(1−δ)Kt])2_{= S} t− St+1_νt (1.25) ⇒ e− Kα_{t L}1−α_t ) St −c(log[Kt+1]−log[(1−δ)Kt]) 2 = St+1 νt.St + e −c(log[Kt+1]−log[(1−δ)Kt])2_{− 1} (1.26) Equation (1.26) requires that:

0 < St+1

νt.St + e

−c(log[Kt+1]−log[(1−δ)Kt])2 _{− 1 ≤ 1 (1.27)}

In the case where (1.27) holds, (1.26) implies that:

Kα tL1−αt St = − log _S t+1 νt.St + e −c₍log[Kt+1]−log[(1−δ)Kt])2 − 1− c (log[Kt+1] − log[(1 − δ)Kt])2 (1.28)

Another constraint stemming from equation (1.28) is: logSt+1

νt.St + e

−c(log[Kt+1]−log[(1−δ)Kt])2 _{− 1}



+ c (log[Kt+1] − log[(1 − δ)Kt])2≤ 0 (1.29) And then, (1.28) finally leads to:

˜ Lt = S 1 1−α t K − α 1−α t h − logSt+1 νt.St+ e −c(log[Kt+1]−log[(1−δ)Kt])2 − 1− c (log[Kt+1] − log[(1 − δ)Kt])2 i1−α1 (1.30) ≡ L(St, St+1, Kt, Kt+1) ˜

Lt is the minimal labor inputs necessary for production in period t when the capital input and the resource stocks are given. Production in period t is determined by the choice of St+1.

If we replace ˜Lt, Yt and It, the maximization problem becomes :

max {Kt+1,St+1} −I0+ ∞ X t=1 Rt  Pt  St− St+1 νt  − wt. ˜Lt−

(

)

The last three inequalities are, respectively, the non-negativity constraints for resource production, capital and the stock of the resource.

The constraint (1.29) implies that : log St+1 St.νt+1 + e −c(log[Kt+1]−log[(1−δ)Kt])2 _{− 1}  ≤ −c (log[K_t+1] − log[(1 − δ)Kt])2 (1.31) ⇒ log St+1 St.νt+1 + e −c(log[Kt+1]−log[(1−δ)Kt])2_{− 1}  ≤ 0 (1.32) ⇒ St+1 St.νt+1 + e −c(log[Kt+1]−log[(1−δ)Kt])2_{− 1 ≤ 1 (1.33)}

This means that when (1.29) holds, the second inequality in (1.27) must hold as well. The problem of the extractive firm then becomes:

max {Kt+1,St+1} −I0+ ∞ X t=1 Rt  Pt  St− St+1 νt  − wt. ˜Lt−

(

)

This problem (P.4) is a fairly general specification, taking into account both investment adjustment costs and a freely-varying path for the gross discovery rate νt. It will often be the case - as is done below - to simplify (P.4) in some dimensions in order to learn more about certain particular features of interest.

1.3

Solving the resource-extracting firm’s problem

In what follows, we make the following simplifying assumptions:

1. c = 0

The key role of the adjustment cost parameter is to capture the dynamics of how the firm responds to shocks: higher adjustment costs imply slower rates of adjustment to the new long-term path. In the analysis below, we are principally concerned with

the properties of the long-term paths, so there is little loss in setting the adjustment cost parameter to zero.

2. ν_t= ν > 1

This is perhaps less restrictive than it seems. From (1.3), we might expect that discovery rates vary with changes in technology and prices. But if technology and prices evolve according to fixed rates of growth, then the arguments of ν(·, ·) in (1.3) would be constant, which in turn implies that the gross discovery rate would also be constant.

With these simplifying assumptions, the problem becomes:

max {Kt,St} −I0+P ∞ t=1Rt  Pt  St− St+1 ν  − wtS 1 1−α t K − α 1−α t h − logSt+1 St.ν i_1−α1 − (Kt+1− (1 − δ)Kt)  subjet to : St+1 St.ν> 0 (1.34) − logSt+1 St.ν  ≥ 0 (1.35) St−St+1_ν ≥ 0 ; Kt≥ 0 ; St≥ 0

(1.35) is equivalent to the non-negativity of resource output constraint, and (1.34) pre-vents resources from being completely exhausted in a given period. This non depletion con-straint is a reflection of the resource production technology, which makes it infinitely costly for the firm to exhaust the available resource stock at any date.

Then, taking into account these constraints and supposing that the firm will always have a strictly positive output once it starts producing (that is, inputs will remain strictly positive), the maximization problem finally becomes:

max {K_t+1,St+1} −I0+ ∞ X t=1 Rt " Pt  St− St+1 ν  − wtK − α 1−α t  −St. log  St+1 St.ν _1−α1 − (Kt+1− (1 − δ)Kt) # subjet to : St− St+1_ν > 0 ; Kt> 0 ; St> 0

The problem consists of choosing the paths of the stocks of reserves and capital that maximize total discounted profits. The path of resource stock will determine all production

quantities along the period of extraction, and, together with the path of the capital, will give the quantity of labor to be employed in extraction activities. The path of capital will determine the investments to be made over the period of extraction.

1.3.1 Optimality conditions

The first order conditions for St+1 and Kt+1 are respectively :

Rt  −Pt ν + 1 1−αwt.K −_1−αα t  St St+1  h −St. log  St+1 St.ν i α 1−α  + Rt+1  Pt+1− _1−α1 wt+1.K − α 1−α t+1 . h − log St+2 St+1.ν  + 1i h−St+1. log  _S t+2 St+1.ν i_1−αα  = 0 (1.36) Rt[−1] + Rt+1  α 1−α.wt+1.K − 1 1−α t+1 h −S_t+1. log St+2 St+1.ν i_1−α1 + (1 − δ)  = 0 (1.37) Given that Rt= t Y s=1 1 1 + rs

, we can write these conditions as

1 1+rt+1  Pt+1−_1−α1 wt+1.K − α 1−α t+1 . h − log St+2 St+1.ν  + 1i h−St+1. log  _S t+2 St+1.ν i_1−αα  = Pt ν − 1 1−αwtK − α 1−α t  St St+1  h −Stlog  St+1 Stν i_1−αα (1.38) 1 1+rt+1  α 1−αwt+1K − 1 1−α t+1 h −St+1. log _S t+2 St+1ν i_1−α1 + (1 − δ)  = 1 (1.39)

The condition (1.38) says that the net marginal benefits from not extracting an extra unit of the resource which is the same as leaving an extra unit of the resource for the future -are equal to its net marginal costs. On the left-hand side, the benefits include the (discounted) value of having an extra unit of the resource in the future, net of extraction costs. On the right-hand side, the costs are the sacrificed current price, net of the cost of current extraction. Equation (1.39) has a similar interpretation. When the firm decides to invest in one more unit of capital for the next period, it has to incur a current investment cost, represented by the right hand term of (1.39). But in the next period, this extra unit of capital in the next period makes it possible to substitute capital for labor, thus reducing future wage costs. In addition, the extra capital in place reduces the need for future investment spending.

1.3.2 The constant-price case

Again, in order to develop understanding of the properties of this model, we now make the simplifying assumption that all prices are constant: rt= r, Pt= P and wt= w, ∀t.

In this case, the optimality conditions become:

1 1+r  P −_1−α1 w K− α 1−α t+1 h − logSt+2 ν St+1  + 1i h−St+1log  St+2 ν St+1 i_1−αα  = P_ν − 1 1−αw K − α 1−α t  St St+1  h −Stlog  St+1 ν St i_1−αα (1.40) 1 1+r  α 1−αw K − 1 1−α t+1 h −St+1log  St+2 ν St+1 i_1−α1 + (1 − δ)  = 1 (1.41) ⇒ 1 1−α w 1+r _K t+1 St+1 −_1−αα h logSt+1 St+2  + 1 + log(ν)i hlogνSt+1 St+2 i_1−αα − 1 1−αw  Kt St −_1−αα  St St+1  h logν St St+1 i_1−αα =−P ν+ P 1+r (1.42) _K t+1 St+1 −_1−α1 h logνSt+1 St+2 i_1−α1 = 1−α α
r+δ w
(1.43)

In order to further simplify these conditions, define k_t= Kt/Stand st= St/St+1. This latter expression can be interpreted as a gross depletion rate: if the stock of reserves declines, then S_t+1< St and st> 1. Substituting kt and st into equations (1.42) and (1.43) leads to:

1 1−α w 1+rk − α 1−α t+1

[

][

]

[

]

[

]

[

]

Finally, optimal extraction is given by these two equations: log(st+1)= −1 + (1 + r)st− log(ν) +  w 1−α α−1 r+δ α
−α ν−1−r ν P (1.47) kt=  α 1−α 1−α w r+δ 1−α [log(ν st)] (1.48)

Note that the dynamics of the optimal extraction program are completely captured in (1.47), which provides a law of motion for st, while (1.48) is a static condition that provides the cost-minimizing levels of capital (and, implicitly, labour) in each period.

Given this structure, we can also derive 5 the firm’s total discounted profit: Π∗ =  P −  w 1−α 1−α r+δ α
α 1 + log(νs1) _S 1 1+r (1.49)

The total discounted profit depends on the initial depletion rate s₁ and the initial total stock S1.

We now consider the conditions in which Π∗ is positive. If Π∗ > 0, then P >  w 1−α 1−α r+δ α
α +  w 1−α 1−α r+δ α
α log(ν s1) (1.50)

Recall that from (1.2), a positive level of extraction requires

ν s1> 1 (1.51)

Therefore, the second term on the right-hand side of (1.50) is strictly positive.

A necessary condition - but not sufficient - for extraction to be profitable is that the resource price must be greater than a threshold price P defined by

P > P ≡_1−αw  1−α r+δ α
α (1.52)

The right-hand side of the inequality (1.52) can be interpreted as an index of the cost of inputs for extraction: this index is increasing in both w and r. When the wage rate or the interest rate is high relative to the resource price, condition (1.52) may not hold and there will be no profitable extraction path. However, when condition (1.52) holds, there will always be an initial extraction rate that can allow for profitable extraction of the resource. The closer s1 is to _ν1, the higher is the maximized total discounted profit. It can also be shown that there will be a critical value (smax₁ ) that leaves the firm with zero profit. If initial production

5

is large enough so that s₁ > smax

1 , the total discounted profit becomes negative. This is due to the fact that the firm has to use a large amount of inputs in order to extract a large share of the stock of the resource at any period. In other words, the firm earns more profits if the initial production is small, so that extraction will be spread out over a longer horizon.

1.4

Some results on convergence and stable paths

In order to develop some insight into the role of the gross discovery rate in the dynamic properties of the optimal paths described by (1.47) and (1.48), this section presents some graphical representations for various values of ν and P . The analysis assumes that the neces-sary condition for profitability (1.52) holds.

Substituting (1.52) into (1.47) provides a simplified version of the law of motion for s_t: log(st+1) = −1 + (1 + r)st− log(ν) +ν−(1+r)_ν P_P (1.53) where P is the threshold price defined in (1.52). The dynamics of s_t will depend on the positions of the curves representing the logarithmic function on the left-hand side of (1.53) and the linear relation in the right-hand side.

There is no general closed-form solution for the stationary state in (1.53), but with a graphical analysis, it is possible to show whether a steady state exists or not and to describe the dynamics away from the steady state. Let g(s_t) represent the (linear) function on the right-hand side of (1.53), so that

log(st+1) = g(st)

When graphed as a function of s, a steady state - if any exist - will be characterized by the intersection of the curves log(s) and g(s): if log(¯s) = g(¯s) then s1 = s2 = s3 = · · · = ¯s. But if log(s) and g(s) do not intersect at s₁, then s₂ will not equal s₁. For a given s₁, the value of s2 is obtained by drawing a horizontal line through g(s1) and seeing where this line intersects log(s). The value of s₂ is simply the value of s associated with the intersection of log(s) and the horizontal line through g(s1). As can be seen in the figures below, there are two cases to consider:

- The intersection of log(s) with the horizontal line through g(s₁) lies to the left of g(s). The value of s2 is therefore less than s1. In the region where the curve log(s) lies to the left (or above) the line g(s), s_t+1< stand s falls over time.

2. The curve log(s) is to the right of (or below) the point where g is evaluated at s₁ - The intersection of log(s) with the horizontal line through g(s1) lies to the right of

g(s). The value of s2 is therefore greater than s1. In the region where the curve log(s) lies to the right (or below) the line g(s), st+1> st and s increases over time.

1.4.1 Gross discovery rate equal to the gross interest rate

Suppose that discoveries are occurring at the rate of interest, so that ν = 1 + r and the resource is replenishing at the interest rate. If this is the case, the (1.47) simplifies to:

log(st+1) = −1 + (1 + r)st− log(1 + r) (1.54)

Note that the price terms in (1.53) cancel out along with the gross discovery rate ν. In this special case, an analytic solution for the steady state can be obtained: ¯s = 1/(1 + r). As graphed in Figure1.1, the two curves are in fact tangent at this point. Recall that st≡ St/St+1 is the (gross) depletion rate; a value of s_t < 1 implies St+1 > St and an increasing stock of proved reserves: newly-discovered reserves exceed current extraction. While the tangent between these curves suggests a possible steady state in which reserves continually increase, this is not a feasible outcome.

In order for production to take place, the condition (1.51) - which requires that ν s₁ > 1 - must be satisfied. If ν = 1 + r, then (1.51) implies s1 > 1/(1 + r), so the steady state in which s₁ = 1/(1 + r) is infeasible. Feasible values for s1 must lie to the right of the point of tangency, in the region where the curve log(s) is always to the right of g(s). The gross rate of depletion s_twill continually increase, and reserves will be exhausted at an accelerating rate.

1.4.2 Discovery rate higher than the interest rate (ν > 1 + r)

The gross discovery rate only affects the intercept of the line g(st) on the right-hand side of (1.53); the slope of g is (1 + r) and does not depend on the value of ν. Since the slopes of neither log (st+1) nor g(st) depend on ν, the point of tangency still occurs at s = 1/(1 + r).

g ; log s g log s1 s2 s3 • 1 1+r

Figure 1.1: Dynamics of st when ν = 1 + r

Let D denote the vertical distance between g(st) and the tangent of log(st+1) at st+1= 1/(1 + r); if D > 0 then g(st) lies above the tangent and if D < 0, the line lies below the tangent. From (1.53) and (1.54), we have

D = − log(ν) +ν−(1+r)_ν P P + log(1 + r) = log 1+r_ν
− (1+r_ν − 1) P_P = 1+r_ν − 1
 log(1+r ν ) 1+r ν −1 − P_P  (1.55)

As we have already seen, it is possible to obtain D = 0 for a value of ν = 1 + r and gross depletion rates will increase without bound. (This is largely a restatement of the discussion in Section1.4.1 above.)

When ν > 1 + r, there is a unique P such that D equates 0. Let us denote this value by ¯ P1 as defined by : ¯ P1= log 1+r_ν
1+r ν − 1 P (1.56)

At price ¯P1, g(s) and log(s) are once again tangent at the point 1/(1+r). ¯P1 is a critical price

resource stock. The ‘high profitability’ scenario refers to situation in which the price of the resource is higher than this critical price (P > ¯P1). The case where the price of the resource is lower or equal to the critical price (P ≤ ¯P1) is referred to as the ‘low profit’ scenario.

(a) High profits (P > ¯P1)

Higher resource prices shift the line g(s) to the left (or above). When the price is high enough (P > ¯P1) so that the high profits case prevails, D > 0, and then the

line g(s) will lie above the point of tangency.

In the high-profit case, the line g(s) does not intersect the curve log(s), and there is no steady state. This situation can be represented in Figure1.2in which depletion rates increase without bound for any initial feasible extraction rate.

g ; log s g log s1 s2 s3 s4 s5 s6 1 ν

Figure 1.2: Dynamics of s_t when ν > 1 + r and high profits

(b) Low profits (P ≤ ¯P1)

Lower resource prices shift the line g(s) to the right (or below). When the price is low enough (P ≤ ¯P1) so that the low profits case prevails, D < 0 and the line g(s)

intersects the curve log(s) at two points, denoted by ¯s_low and ¯s_high, as in Figure1.3, where ¯s_low ≤ ¯s_high. For the special case in which P = ¯P1, ¯s_low = ¯s_high = 1/(1 + r).

g ; log s g log s₁ s2 s 0 1 s0₂ ¯ s_low s00₁ s00₂ ¯ s_high 1 ν

Figure 1.3: Dynamics of st when ν > 1 + r and low profits

Under the low-profit scenario, there are two possible steady states: ¯s_low and ¯s_high. Neither steady state is globally stable. If s > ¯shigh, then depletion rates increase

without bound. In addition to not being globally stable, ¯s_high is also locally unsta-ble. For values of s less than but ’close to’ ¯shigh, g(s) is below log(s) and s decreases

towards ¯s_low. And if s greater than but ’close to’ ¯s_high, g(s) is above log(s) and s increases without bound.

However, the steady state ¯s_low is locally stable. For values of s less than but ’close to’ ¯s_low, g(s) is above log(s) and s increases towards ¯s_low. And if s is greater than but ’close to’ ¯s_low, g(s) is below log(s) and s decreases towards ¯s_low. Note also that ¯

s_low ≤ 1/(1 + r), so this locally-stable steady state is associated with paths in which the stock of proved reserves increases over time at a rate greater than or equal to the interest rate .

1.4.3 Discovery rate lower than the interest rate (ν < 1 + r)

When ν is less than (1 + r), D is negative for any profitability ratio P/P > 1, and the line g(s) again intersects the curve log(s) at two points, denoted by ¯s_low and ¯s_high, where ¯s_low < ¯s_high. In this case, the only feasible - albeit unstable - steady state is ¯s_high 6, which is the least feasible optimal depletion rate and either is less than 1, equal to 1, or greater than 1. For a given combination of the other parameters, let us denote by ¯P2the price for which ¯shigh equals

1. From equation (1.53), it can be shown that ¯P2 is defined by: ¯ P2= r − log(ν) 1+r ν − 1 P (1.57)

When the price of the resource is equal to the upper threshold ¯P2, ¯shigh = 1 and the

resource stock remains constant over the extraction period. Depending on whether the price of the resource is less than or greater than ¯P2, one of the two following scenarios will prevail.

(a) High profits (P > ¯P2)

Higher resource prices now shift the line g(s) to the right (or below). When the price is high enough (P > ¯P2), the high-profits scenario prevails. Subsequently, we

have ¯shigh> 1 as in Figure1.4and the resource stock will be depleted at a constant

rate.

For values of s₁ > ¯s_high, depletion rates increase without bound, and for values of s1 < ¯shigh, depletion rates fall to the lower bound 1/ν. Since this lower bound

characterizes the positivity constraint for extraction, this path is one in which extraction levels fall towards zero in a finite period of time.

6_{See in Proof2that steady state ¯s}

high is profitable and this ensures the existence of profitable extraction

g ; log s g log s1 s2 ¯ shigh ¯ s_low 1 1+r 1 ν

Figure 1.4: Dynamics of s_t when 1_{6 ν < 1 + r and high profits}

(b) Low profits (P ≤ ¯P2)

Lower resource prices now shift the line g(s) to the left (or above). When the price is low enough (P ≤ ¯P2), the low-profits scenario prevails. Subsequently, we have

¯

shigh ≤ 1 as in Figure1.5and the resource will be accumulated at a constant rate.

For values of s1 > ¯shigh, depletion rates increase without bound, and for values

of s1 < ¯shigh, depletion rates fall to the lower bound 1/ν. Since this lower bound

characterizes the positivity constraint for extraction, this path is one in which extraction levels fall towards zero in a finite period of time.

g ; log s g log 1 1+r 1 ν ¯ slow ¯shigh s₁ s₂

Figure 1.5: Dynamics of s_t when 1_{6 ν < 1 + r and low profits}

1.4.4 Effects of royalties on resource sustainability

As explained above, profitability levels are the key determinant of resource sustainability. For any discovery rate, if prices - and hence profitability - are sufficiently high, the resource-extracting firm has an incentive to rapidly deplete the resource stock. Given the discovery or replacement rate of a resource, there is a critical price level which defines the threshold between the high- and low-profitability scenarios. When the price of the resource is less than or equal to the threshold price, the resource stock will be increasing or constant. Conversely, whenever the price of the resource is greater than the threshold price, the resource stock will be declining.

Usually, natural resources are publicly-owned, and complete exhaustion is never optimal for society. Therefore, governments may wish to adopt a taxation system in order to rule out the depletion scenarios. It is suggested that the interplay between resource owners and extrac-tive companies should be deeply examined and royalties should be appropriately determined (Ravagnani, 2008). Suppose that a royalty rate (tax rate) t is imposed on each unit of the resource extracted or sold. What matters to the resource extracting firm now is the price after tax (1 − t)P . The first concern is that the royalty rate is not set so high that it is equivalent to a ban on extraction activities. In other words, the net price after royalties must still satisfy

the necessary profitability condition (1.52):

(1 − t)P > P (1.58)

In addition to being high enough to satisfy (1.58), the net of royalty price should be low enough to ensure that the optimal extraction path is sustainable, that is, less than or equal to the critical price ¯P 7_:

(1 − t)P ≤ ¯P (1.59)

Taken together, these two above conditions together imply that the royalty rate must satisfy the following condition:

1 − ¯ P P ≤ t < 1 − P P (1.60)

Imposing a royalty rate makes it possible for the government to transform an unsus-tainable high-profit path to a low-profit path in which the resource is not depleted over time. However, setting the royalty rate at too high a level will completely shut down extraction of the resource.

1.5

Conclusion

Even though the data show secular increases in both production and reserves of such crucial non-renewable resources such as oil, conventional models predict secular declines. This study offers a framework for the non-renewables sector - and oil in particular - that can generate predictions that fit the broad stylized facts, and which can also generate the sort of balanced paths that can be incorporated into a model of the macroeconomy.

The key to reconciling the model with the data is an explicit modeling of the discovery of new reserves. While the introduction of discoveries of new deposits can be seen as a necessary condition for paths in which output and proved reserves increase over time, it is not sufficient. Secular increases in output and reserves are not a foregone conclusion in this framework; stable paths are only obtained under certain conditions.

7

The discussion of the conditions for sustainable resource extraction - and the potential role for government action - has implications for the political economy of low-income resource-exporting countries. If resource extraction is highly profitable, an appropriate royalty rate is required to ensure sustainability. But if governance is weak - as is often the case - then it may find it difficult to enforce those royalties in the face of opposition from foreign-based resource extraction firms.

While this study limits attention to the case where the gross discovery rate of non-renewable resources is constant, it can readily be extended to consider cases where discoveries increase when prices are high or increasing. For example, integrating this model of the resource sector along with the manufacturing sector in a dynamic stochastic general equilibrium model may help explain the dynamics in the resource sector and its interactions with the rest of the economy. Other extensions and application will be the subject of future work.

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