1 Introduction
This study investigates the problem of exhaustible resources using a dynamic input–output model with classical features. Although discussions on whether exhaustible resources set a limit to economic growth continue in debate to this day, one must acknowledge that exhaustible resources are and will remain of great importance to most economies. Solow’s opinion that the topic of exhaustible (or non-renewable) resources is “important, contemporary, and perennial” (Solow
1974, p. 1) in economics is indeed still very relevant.
The modern theory of exhaustible resources generally stems from the well-known Hotelling rule (Hotelling
1931). According to the Hotelling rule, the prices of exhaustible resources in situ must increase at a rate equal to that of the profit, provided that free competition prevails, due to the fact that the storage of exhaustible resources requires a normal rate of profit as do other production processes. Since resources are generally used in the production of many, if not all commodities, it is appropriate to conduct research on exhaustible resources within a multi-sectoral framework. In addition, since the Hotelling rule seems to imply that the prices of many, if not all, commodities are subject to change, a dynamic analysis is appropriate. Based on these two considerations, the dynamic input–output model is the best option for carrying out this study’s research.
The existing academic literature on exhaustible resources that utilises input–output methods focuses primarily on the interdependence of exhaustible resources or energy, economic growth, and the environment. Therein, the evolution of the price of exhaustible resources is usually not discussed or simply assumed to be exogenous. For example, much research has been devoted analysing the factors that affect energy consumption using structural decomposition analysis (SDA) via hybrid energy input–output models (e.g. Lin and Polenske
1995; Mukhopadhyaya and Chakraborty
1999; Kagawa and Inamura
2001,
2004; Dietzenbacher and Stage
2006), on the impact of non-renewable resources on economic growth (e.g. Dobos and Floriska
2005), on tracking energy paths (e.g. Treloar
1997), on reducing energy requirements (e.g. Wilting et al.
1999), and on the relationship between energy and the environment (e.g. Weber and Schnabl
1998; Karten and Schleicher
1999). In these models, the prices of exhaustible resources specifically are not typically discussed. Dejuan et al. (
2013) consider the impact of the prices of energy on technological coefficients, but in their analyses price evolvements are exogenous. In this article, the economic dynamics of both quantity and price will be put under comprehensive investigation in a dynamic input–output model built with classical features.
It should be noted that the dynamic input–output model used in this study differs slightly from the dynamic Leontief model. One of the most common ways to make a static input–output model dynamic is by introducing “capital” stock and the “stock-flow” matrix, as proposed by Leontief (
1953,
1970). However, this dynamic Leontief model has several theoretical problems due to the rigorous assumptions needed for empirical research. Specifically, these theoretical problems include the irreversibility of capital accumulation (Leontief
1953; McManus
1957), the singularity problem, causal indeterminacy (Dorfman et al.
1958; Takayama
1974), as well as problems of stability and dual stability/instability (Sargan
1958; Morishima
1959; Solow
1959; Jorgenson
1960,
1961; Steenge
1990).
Different remedies have been put forward for addressing these problems; for instance, Leontief himself (
1953) proposed a “multi-phase” process to deal with the problem of irreversibility. The models’ singularity problem can also be solved through many different means (e.g. Kendrik
1972; Livesey
1973; Luenberger and Arbel
1977; Jodar and Merello
2010). Causal indeterminacy can be avoided by allowing for extra production capacity, rather than resorting to full capacity utilisation (Duchin and Szyld
1985), or by changing the Leontief model into a planning model and introducing the non-negativity of output (Solow
1959). For the dual stability/instability problem, one can replace the assumption of perfect foresight with an assumption that firms make investments based on past experience (Aoki
1977). However, some notable imperfections in the dynamic Leontief model remain that must be acknowledged. First of all, the model cannot deal with the problem of joint production. Secondly, even though the static input–output model has a classical tradition (Kurz and Salvadori
2000a,
2006) regardless of illegitimate assumptions (e.g. value-added coefficients in price theory), its treatment of capital in the dynamic model is typically neo-classical which cannot survive the critiques raised by Sraffa’s work (Sraffa
1960) and the following capital controversies.
A second way to make the static input–output model dynamic is by introducing expenditure lags, as was done by Solow (
1952). However, in this kind of dynamic model, the input–output matrix represents the expenditure relations rather than production technology.
Given these considerations, for this study a dynamic input–output model with production lags and classical features is used. This specific model construction has the following three advantages. First, its input–output matrix still represents production technology. Second, it is possible to address joint production using this methodological framework. Third and more importantly, this model, which is Sraffa–von Neumann orientated and which has a classical tradition, does not suffer from the problem of “capital” as does the neo-classical theory. Hence, this study’s model is more logically consistent.
2 Background
The issue of exhaustible resources in modern classical theory is still in a state of intense debate. Sraffa only mentioned natural resources in passing: ‘Natural resources which are used in production, such as land and mineral deposits...’ (Sraffa
1960 p. 74). Several scholars have subsequently investigated the theory of exhaustible resources using Sraffian or classical frameworks (e.g. Parrinello
1983,
2001,
2004; Bidard and Erreygers
2001a,
b; Schefold
1989,
2001; Lager
2001; Ravagnani
2008; as well as Kurz and Salvadori
1995,
1997,
2000b,
2001,
2009,
2011). A survey on the research contributions made by many researchers on the topic was compiled by Kurz and Salvadori (
2015). However, there has been no consensus among these models, as exemplified at the proceedings of a symposium presented by Metroeconomica in 2001. To this day, the issue seems to be far from settled.
The research conducted in this study is built on and inspired primarily by the work of Kurz and Salvadori. The preliminary results from initial their studies on exhaustible resources (Kurz and Salvadori
1995,
1997) were later revised and improved using a dynamic input–output model (Kurz and Salvadori
2000b,
2001), which enables the tracking of the time paths of prices, royalties, and other endogenous variables. In their subsequent work, Kurz and Salvadori (
2009) offer new interpretations to the treatment of exhaustible resources by Ricardo. They clarify that Ricardo’s analysis of exhaustible resources starts from his discussion on the differences between rent and profit in Chapter II of his
Principles (Ricardo
1817), and that what we call “royalties” are in actuality comprised in profits in Ricardo’s analysis.
Based on their systematic investigation of Ricardo’s work, Kurz and Salvadori (
2009) point out that although the Hotelling rule is not elaborated upon by Ricardo, it does not necessarily mean that Ricardo’s analyses are wrong, incomplete, or insufficient. The fundamental differences between the analyses of Ricardo and those of Hotelling are found in their varied assumptions. From Ricardo’s point of view, there are searching activities such that each exhausted mine is replaced by a newly discovered mine with the same quality and quantity, and the searching costs in terms of labour and commodities are constant. In addition, there is capacity constraint on extraction in each mine. Under this theory, royalties are treated as a sub-category of profits, and are not introduced explicitly. Rather, the rents of mines caused by different fertilities are a considered wholly different concept. On the other hand, from Hotelling’s point of view, the amount of one homogeneous and exhaustible resource is known and given at the outset. The extraction of resource at each time is constrained only by the remaining pool of the resource left over from the preceding point in time.
Both of the arguments developed by Ricardo and Hotelling on exhaustible resources are helpful for improving our understanding on this issue. The ideas put forward by Ricardo and Hotelling can be incorporated into a single framework. This was done by Kurz and Salvadori (
2009), who provide a formalisation of exhaustible resources with explicit capacity constraints on extraction, clearly distinguishing three types of property incomes: profits, royalties, and rents. A numerical example was subsequently provided by them (Kurz and Salvadori
2011) to shed more light on this issue.
This study, following the work of Kurz and Salvadori (
2009,
2011), seeks to contribute to the reconciliation of Ricardo’s and Hotelling’s analyses on exhaustible resources by introducing resource-searching activities. More specifically, this study is aims to expand on the following aspects. First, it seeks to introduce searching activities and to provide a sufficient and necessary condition for the existence of solutions to the model. Given a real wage rate and a consumption vector, the model indicates that the paths of prices, royalties, rents, intensities of commodity production, and resource-searching processes can be determined once a sequence of profit rates as well as the initial amounts of commodities and resources are given. Second, it is found that some well-defined circumstances under which commodity prices are constant, as initially discussed by Kurz and Salvadori (
2009,
2011), can be represented by the study’s model as well.
Two points need to be stressed further. First, the method used in this study is similar to that of Solow in his work on the generalisation of the dynamic input–output system (Solow
1959). However, the theory diverges in some notable respects. Due to the difficulties faced by neo-classical capital theory, the model in this study does not assume that distributive variables are determined by the demand for, or the supply of, factors. For this study, one distributive variable (the real wage rate) is assumed to be given exogenously (i.e. its determination lies outside the system), while the remaining distributive variables are determined endogenously. It is because of this that the model used in this study can be said to preserve some classical features.
Second, in order to make the model more manageable, there is an assumption of perfect foresight. That is to say, when prices are bound to change over time, agents are aware of it, and are accurately aware of how prices change. This assumption is very strong. Since the decisions made by firms and individuals are usually based on their expectations for the future, and because all future states cannot be known with certainty, a discussion on the expectation cannot be avoided in principle. However, introducing expectations not only complicates the issues to be investigated in this study, but also makes it difficult, if not impossible, to achieve confident results. This is because any results will depend on the assumptions made on the formation of expectations. A detailed discussion on expectations lies beyond the scope of this study. To resolve this impasse, this study adopts one of the simplest expectations: perfect foresight. As such, this paper’s analysis can be considered the preliminary result of a more satisfactory investigation on the subject.
This paper is organised as follows. Basic definitions and the study’s model are provided in Sect.
3. The dynamics of quantity and price are explored in Sect.
4, which gives the conditions for the solutions of the study’s model. Section
5 discusses the circumstances under which prices are found to be constant based on the model presented. Notably, the circumstances seem contradictory to the Hotelling rule. Section
6 concludes.
3 Basic definitions and the study’s model
The formalisation of this study’s model is based on the following assumptions. It is assumed that there are n perfectly divisible commodities in the economy, which are produced by \(m_{1}\,(m_{1}>n)\) constant-returns-to-scale processes. There are s kinds of resources provided by nature, but only part of the total amount is known at the outset. The remaining resource pool is progressively discovered with \(m_{2}\,(m_{2}>s)\) processes using commodities and labour.
Instead of extracting the resources, the owners can choose to store them. Hence, there are
s processes for storage. Each process
i producing commodities is represented by a quintuplet
\((\varvec{a}_{i}, \varvec{c}_{i}, l_{1i}, \varvec{b}_{i}, \varvec{0})\), where
\(\varvec{a}_{i}^{\mathrm{T}}=(a_{i1}, \ldots , a_{in})\) is the commodity inputs vector,
1 \(\varvec{c}_{i}^{\mathrm{T}}=(c_{i1}, \ldots , c_{is})\) is the resource inputs vector,
\(l_{1i}\) is the labour input scalar, and
\(\varvec{b}_{i}^{\mathrm{T}}=(b_{i1}, \ldots , b_{in})\) is the commodity outputs vector. Each process
j searching resources is represented by
\((\varvec{f}_{j}, \varvec{0}, l_{2j}, \varvec{0}, \varvec{d}_{j})\), where
\(\varvec{f}_{j}^{\mathrm{T}}=(f_{j1}, \ldots , f_{jn})\) is the commodity inputs vector,
\(l_{2j}\) is the labour input scalar, and
\(\varvec{d}_{j}^{\mathrm{T}}=(d_{j1}, \ldots , d_{js})\) is the resource outputs vector. The whole technology at time
t is represented by the following matrices:
$$\begin{aligned} \mathbf{A}&= {} [\varvec{a}_{1}, \varvec{a}_{2}, \ldots , \varvec{a}_{m_{1}}]^{\mathrm{T}}\\ \mathbf{C}&= {} [\varvec{c}_{1}, \varvec{c}_{2}, \ldots , \varvec{c}_{m_{1}}]^{\mathrm{T}}\\ \varvec{l}_{1}&= {} [l_{11}, l_{12}, \ldots , l_{1m_{1}}]^{\mathrm{T}}\\ \mathbf{B}&= {} [\varvec{b}_{1}, \varvec{b}_{2}, \ldots , \varvec{b}_{m_{1}}]^{\mathrm{T}}\\ \mathbf{F}&= {} [\varvec{f}_{1}, \varvec{f}_{2}, \ldots , \varvec{f}_{m_{2}}]^{\mathrm{T}}\\ \varvec{l}_{2}&= {} [l_{21}, l_{22}, \ldots , l_{2m_{2}}]^{\mathrm{T}}\\ \mathbf{D}&= {} [\varvec{d}_{1}, \varvec{d}_{2}, \ldots , \varvec{d}_{m_{2}}]^{\mathrm{T}} \end{aligned}$$
The processes are ordered so that the first
\(m_{1}\) processes produce commodities, the
\(m_{2}\) processes thereafter search for resources, and the remaining
s processes store resources. All processes are listed in Table
1.
Table 1
Input–output patterns
Commodities | A | C |
\(l_{1}\)
|
\(\rightarrow\)
| B | 0 |
Seaching | F | 0 |
\(l_{2}\)
|
\(\rightarrow\)
| 0 | D |
Storage | 0 | I | 0 |
\(\rightarrow\)
| 0 | I |
Technology producing commodities is assumed to be time invariant, meaning \(\mathbf{A}\), \(\mathbf{C}\), \(\varvec{l}_{1}\) and \(\mathbf{B}\) are constant. However, technology of searching resources is not necessarily time invariant. The costs for searching for one unit of a certain resource may increase as a result of the increased difficulties in locating any remaining resources. The costs for searching for one unit of a certain resource may also decrease due to increased experience or technological progress as relevant to searching for resources.
For this study, and in order to simplify analysis, it is assumed that searching costs are non-decreasing. In order to compare the searching costs among different periods, \((\mathbf{F}, \varvec{l}_{2})\) are assumed to be time invariant, while non-decreasing searching costs are manifest in the changes in \(\mathbf{D}\) over time. In addition, note that the joint searching of resources is not assumed (i.e. there is only one positive component in \(\varvec{d}_{j}\), and the others are zero). Formally, this can be stated as follows.
In the assumption presented above, if \(\alpha =1\), the searching costs (in terms of commodities and labour) are constant. Otherwise, the searching costs are increasing.
Three different types of property income are distinguished: royalties, rents, and profits. Specifically, royalties are the profits that the owners of exhaustible resources will earn in order to keep their capital ‘crystallised’ in their respective resource mines. Rents are the income earned by these owners generated by the differences in fertilities of resource mines. Profits are the income earned by capitalists for using their capital. These distinctions are important because the laws that regulate these types of incomes are different.
More formally, \(\varvec{p}_{t}\), \(\varvec{y}_{t}\), \(\varvec{q}_{t}\) (\(t \in N_{0}\), the set of all non-negative integers) denotes the prices of commodities, royalties and rents paid to the owners of the resources. Let \(r_{t}\) denote the nominal rate of profit at time t, and \(\varvec{w}_{t}\) a bundle of wage goods which is assumed to be exogenously given and constant.
Moving on to quantity side of the model,
\(\varvec{x}_{t}\) and
\(\varvec{s}_{t}\,(t \in \mathbb {N})\) represent the intensities of processes that produce commodities and the processes for searching for resources, respectively. Let
\(\varvec{z}_{t}\) (
\(t \in \mathbb {N}_{0}\)) serve as the amount of exhaustible resources known at time
t. It is assumed that workers consume all their incomes (a typical classical assumption) and that the annual consumption by non-workers is proportional to
\(\varvec{\delta }\), a bundle of commodities, which is given and constant. More specifically, the non-workers’ consumption is assumed to be
\(\gamma\) units of consumption vector
\(\varvec{\delta }\), where
\(\gamma\) is endogenously determined. Let a vector
\(\varvec{h}\) serve as the capacity constraints on extraction, whose elements represent the maximum amount of resources that can be extracted at any given time.
2 Finally, the initial amounts of resources and commodities are known and given as
\(\bar{\varvec{z}}\) and
\(\varvec{v}\), respectively.
3
Based on the assumptions noted above, and in the condition of free competition, the following inequalities and equations hold.
$$\begin{aligned}&\mathbf{B}\varvec{p}_{t+1} \leqq (\mathbf{A}\varvec{p}_{t} +\mathbf{C}\varvec{y}_{t}+\mathbf{C}\varvec{q}_{t})(1+r_{t}) +\varvec{l}_{1}\varvec{w}^{\mathrm{T}}\varvec{p}_{t+1} \end{aligned}$$
(1a)
$$\begin{aligned}&\varvec{x}_{t+1}^{\mathrm{T}}\mathbf{B}\varvec{p}_{t+1}=\varvec{x}_{t+1}^{\mathrm{T}}\left[ (\mathbf{A} \varvec{p}_{t}+\mathbf{C}\varvec{y}_{t}+\mathbf{C}\varvec{q}_{t})(1+r_{t}) +\varvec{l}_{1}\varvec{w}^{\mathrm{T}}\varvec{p}_{t+1}\right] \end{aligned}$$
(1b)
$$\begin{aligned}&\varvec{y}_{t+1} \leqq (1+r_{t})\varvec{y}_{t} \end{aligned}$$
(1c)
$$\begin{aligned}&\varvec{z}_{t+1}^{\mathrm{T}}\varvec{y}_{t+1}=(1+r_{t})\varvec{z}^{\mathrm{T}}_{t+1}\varvec{y}_{t} \end{aligned}$$
(1d)
$$\begin{aligned}&\mathbf{D}_{t+1}\varvec{y}_{t+1} \leqq \mathbf{F}\varvec{p}_{t}(1+r_{t})+\varvec{l}_{2}\varvec{w}^{\mathrm{T}}\varvec{p}_{t+1} \end{aligned}$$
(1e)
$$\begin{aligned}&\varvec{s}^{\mathrm{T}}_{t+1}\mathbf{D}_{t+1}\varvec{y}_{t+1}=\varvec{s}^{\mathrm{T}}_{t+1}\left[ \mathbf{F} \varvec{p}_{t}(1+r_{t})+\varvec{l}_{2}\varvec{w}^{\mathrm{T}}\varvec{p}_{t+1}\right] \end{aligned}$$
(1f)
$$\begin{aligned}&\varvec{v}^{\mathrm{T}} \geqq \varvec{x}^{\mathrm{T}}_{1}\mathbf{A}+ \gamma \varvec{\delta }^{\mathrm{T}}+\varvec{s}^{\mathrm{T}}_{1}\mathbf{F} \end{aligned}$$
(1g)
$$\begin{aligned}&\varvec{v}^{\mathrm{T}}\varvec{p}_{0}= (\varvec{x}^{\mathrm{T}}_{1}\mathbf{A}+ \gamma \varvec{\delta }^{\mathrm{T}}+\varvec{s}^{\mathrm{T}}_{1}\mathbf{F})\varvec{p}_{0} \end{aligned}$$
(1h)
$$\begin{aligned}&\varvec{x}^{\mathrm{T}}_{t+1}\left( \mathbf{B}-\varvec{l}_{1}\varvec{w}^{\mathrm{T}}\right) -\varvec{s}^{\mathrm{T}}_{t+1}\varvec{l}_{2}\varvec{w}^{\mathrm{T}} \geqq \varvec{x}^{\mathrm{T}}_{t+2}\mathbf{A}+ \gamma \varvec{\delta }^{\mathrm{T}}+\varvec{s}^{\mathrm{T}}_{t+2}\mathbf{F} \end{aligned}$$
(1i)
$$\begin{aligned}&\left[ \varvec{x}^{\mathrm{T}}_{t+1}\left( \mathbf{B}-\varvec{l}_{1}\varvec{w}^{\mathrm{T}}\right) -\varvec{s}^{\mathrm{T}}_{t+1}\varvec{l}_{2}\varvec{w}^{\mathrm{T}}\right] \varvec{p}_{t+1} = \left( \varvec{x}^{\mathrm{T}}_{t+2}\mathbf{A}+ \gamma \varvec{\delta }^{\mathrm{T}}+\varvec{s}^{\mathrm{T}}_{t+2}\mathbf{F}\right) \varvec{p}_{t+1} \end{aligned}$$
(1j)
$$\begin{aligned}&\varvec{z}_{0}^{\mathrm{T}} \geqq \varvec{x}_{1}^{\mathrm{T}}\mathbf{C}+\varvec{z}_{1}^{\mathrm{T}} \end{aligned}$$
(1k)
$$\begin{aligned}&\varvec{z}_{0}^{\mathrm{T}}\varvec{y}_{0} = \left( \varvec{x}_{1}^{\mathrm{T}}\mathbf{C}+\varvec{z}_{1}^{\mathrm{T}}\right) \varvec{y}_{0} \end{aligned}$$
(1l)
$$\begin{aligned}&\varvec{z}^{\mathrm{T}}_{t}+\varvec{s}^{\mathrm{T}}_{t}\mathbf{D}_{t} \geqq \varvec{x}^{\mathrm{T}}_{t+1}\mathbf{C}+\varvec{z}^{\mathrm{T}}_{t+1} \end{aligned}$$
(1m)
$$\begin{aligned}&\left( \varvec{z}^{\mathrm{T}}_{t}+\varvec{s}^{\mathrm{T}}_{t}\mathbf{D}_{t}\right) \varvec{y}_{t} = \left( \varvec{x}^{\mathrm{T}}_{t+1}\mathbf{C}+\varvec{z}^{\mathrm{T}}_{t+1}\right) \varvec{y}_{t} \end{aligned}$$
(1n)
$$\begin{aligned}&\varvec{x}_{t+1}^{\mathrm{T}}\mathbf{C} \leqq \varvec{h}^{\mathrm{T}} \end{aligned}$$
(1o)
$$\begin{aligned}&\varvec{x}_{t+1}^{\mathrm{T}}\mathbf{C}\varvec{q}_{t}= \varvec{h}^{\mathrm{T}}\varvec{q}_{t} \end{aligned}$$
(1p)
$$\begin{aligned}&\varvec{z}_{0}= \bar{\varvec{z}} \end{aligned}$$
(1q)
$$\begin{aligned}&\sum _{t=0}^{\infty } \frac{\varvec{\delta }^{\mathrm{T}}\varvec{p}_{t}}{\prod _{\tau =0}^{t-1}(1+r_{t})} = 1 \end{aligned}$$
(1r)
$$\begin{aligned}&\gamma > 0, \quad \varvec{p}_{t} \geqq \varvec{0}, \quad \varvec{y}_{t} \geqq \varvec{0}, \quad \varvec{q}_{t} \geqq \varvec{0},\quad \varvec{z}_{t} \geqq \varvec{0}, \quad \varvec{x}_{t} \geqq \varvec{0}, \quad \varvec{s}_{t} \geqq \varvec{0} \end{aligned}$$
(1s)
Inequality (
1a) means that no individual can obtain extra profits by producing commodities at time
t. Equation (
1b) means that if there is a process incurring extra costs, then the process is not operated at time
\(t+1\). Inequality (
1c) presents how no extra profits can be obtained from the storing of resources from
t to
\(t+1\). Equation (
1d) means that if the storing activity of one resource cannot obtain the nominal rate of profit at time
t, then the resource is no longer available at time
\(t+1\). Inequality (
1e) means that no extra profits can be obtained by discovering resources at time
t. Equation (
1f) shows that if one searching process incurs extra costs, then this process is not operated at time
\(t+1\). Inequalities (
1g) and (
1i) mean that the amount of commodities at time
t cannot be smaller than the amount of commodities required for production and consumption at time
\(t+1\). Equations (
1h) and (
1j) mean that if an amount of one commodity at time
t is larger than needed, then the corresponding price of this commodity is zero. Inequalities (
1k) and (
1m) show that the amount of resources known at time
t, plus the resources discovered at time
t, cannot be less than the amount of resources known and the resources utilised to produce commodities at time
\(t+1\). Equations (
1l) and (
1n) illustrate that if the amount of one kind of resource is strictly larger in the preceding period of time, the royalty of this resource is zero. Inequality (
1o) presents how the amount of extracted resources cannot be larger than
\(\varvec{h}\).
4 Equation (
1p) shows that if the amount of extraction for one resource
i is less than
\(h_{i}\) (the
\(i\text{th}\) element of
\(\varvec{h}\)), then no rent is paid to the owner of the resource. Equation (
1q) states that the initial resources are given as
\(\bar{\varvec{z}}\). Equation (
1r) serves as the
\(\text{num}\acute{e}\text{raire}\) equation. The meaning of inequality (
1s) is obvious.
Since it is impossible to determine the dynamics of the nominal rates of profits in this model, the sequence of the nominal rates of profits
\(\{r_{t}\}\) is assumed to be given. This sequence being given means that the determination of dynamics in
\(r_{t}\) is externalised from the model. However, the given sequence
\(\{r_{t}\}\) does not influence relative actualised prices. This is because if the sequences
\(\{\varvec{p}_{t}\}\),
\(\{\varvec{y}_{t}\}\),
\(\{\varvec{q}_{t}\}\),
\(\{\varvec{z}_{t}\}\),
\(\{\varvec{x}_{t+1}\}\),
\(\{\varvec{s}_{t+1}\}\) serve as a solution to System (1) for the given sequence
\(\{r_{t}\}\), then the sequences
\(\{\varvec{p}_{t}^{\prime }\}\),
\(\{\varvec{y}_{t}^{\prime }\}\),
\(\{\varvec{q}_{t}^{\prime }\}\),
\(\{\varvec{z}_{t}\}\),
\(\{\varvec{x}_{t+1}\}\),
\(\{\varvec{s}_{t+1}\}\) serve as a solution to the same system for another given sequence
\(\{\rho _{t}\}\), provided the following.
$$\begin{aligned} \varvec{p}_{t}^{\prime }= & {} \prod _{\tau =0}^{t-1} \frac{1+\rho _{t}}{1+r_{t}} \varvec{p}_{t}\\ \varvec{y}_{t}^{\prime }= & {} \prod _{\tau =0}^{t-1} \frac{1+\rho _{t}}{1+r_{t}} \varvec{y}_{t}\\ \varvec{q}_{t}^{\prime }= & {} \prod _{\tau =0}^{t-1} \frac{1+\rho _{t}}{1+r_{t}} \varvec{q}_{t} \end{aligned}$$
Based on the reasoning presented above, the sequence
\(\{r_{t}\}\) is assumed to be constant and
\(r_{t}=0\) for the sake of simplifying the expression. Therefore, System (1) can be simplified as follows.
$$\begin{aligned}&\left( \mathbf{B}-\varvec{l}_{1}\varvec{w}^{\mathrm{T}}\right) \varvec{p}_{t+1} \leqq \mathbf{A}\varvec{p}_{t}+\mathbf{C}\varvec{y}_{t}+\mathbf{C}\varvec{q}_{t} \end{aligned}$$
(2a)
$$\begin{aligned}&\varvec{x}^{\mathrm{T}}_{t+1}\left( \mathbf{B}-\varvec{l}_{1}\varvec{w}^{\mathrm{T}}\right) \varvec{p}_{t+1}=\varvec{x}^{\mathrm{T}}_{t+1}(\mathbf{A}\varvec{p}_{t}+\mathbf{C}\varvec{y}_{t}+\mathbf{C}\varvec{q}_{t}) \end{aligned}$$
(2b)
$$\begin{aligned}&\varvec{y}_{t+1} \leqq \varvec{y}_{t} \end{aligned}$$
(2c)
$$\begin{aligned}&\varvec{z}_{t+1}^{\mathrm{T}}\varvec{y}_{t+1}=\varvec{z}^{\mathrm{T}}_{t+1}\varvec{y}_{t} \end{aligned}$$
(2d)
$$\begin{aligned}&\mathbf{D}_{t+1}\varvec{y}_{t+1} \leqq \mathbf{F} \varvec{p}_{t}+\varvec{l}_{2}\varvec{w}^{\mathrm{T}}\varvec{p}_{t+1} \end{aligned}$$
(2e)
$$\begin{aligned}&\varvec{s}^{\mathrm{T}}_{t+1}\mathbf{D}_{t+1}\varvec{y}_{t+1}=\varvec{s}^{\mathrm{T}}_{t+1}\left( \mathbf{F} \varvec{p}_{t}+\varvec{l}_{2}\varvec{w}^{\mathrm{T}}\varvec{p}_{t+1}\right) \end{aligned}$$
(2f)
$$\begin{aligned}&\varvec{v}^{\mathrm{T}} \geqq \varvec{x}^{\mathrm{T}}_{1}\mathbf{A}+ \gamma \varvec{\delta }^{\mathrm{T}}+\varvec{s}^{\mathrm{T}}_{1}\mathbf{F} \end{aligned}$$
(2g)
$$\begin{aligned}&\varvec{v}^{\mathrm{T}}\varvec{p}_{0}=\left( \varvec{x}^{\mathrm{T}}_{1}\mathbf{A}+ \gamma \varvec{\delta }^{\mathrm{T}}+ \varvec{s}^{\mathrm{T}}_{1}\mathbf{F}\right) \varvec{p}_{0} \end{aligned}$$
(2h)
$$\begin{aligned}&\varvec{x}^{\mathrm{T}}_{t+1}\left( \mathbf{B}-\varvec{l}_{1}\varvec{w}^{\mathrm{T}}\right) -\varvec{s}^{\mathrm{T}}_{t+1} \varvec{l}_{2}\varvec{w}^{\mathrm{T}} \geqq \varvec{x}^{\mathrm{T}}_{t+2}\mathbf{A}+ \gamma \varvec{\delta }^{\mathrm{T}}+\varvec{s}^{\mathrm{T}}_{t+2}\mathbf{F} \end{aligned}$$
(2i)
$$\begin{aligned}&\left[ \varvec{x}^{\mathrm{T}}_{t+1}\left( \mathbf{B}-\varvec{l}_{1}\varvec{w}^{\mathrm{T}}\right) -\varvec{s}^{\mathrm{T}}_{t+1} \varvec{l}_{2}\varvec{w}^{\mathrm{T}}\right] \varvec{p}_{t+1}=\left( \varvec{x}^{\mathrm{T}}_{t+2}\mathbf{A}+ \gamma \varvec{\delta }^{\mathrm{T}}+\varvec{s}^{\mathrm{T}}_{t+2}\mathbf{F} \right) \varvec{p}_{t+1} \end{aligned}$$
(2j)
$$\begin{aligned}&\varvec{z}_{0}^{\mathrm{T}} \geqq \varvec{x}_{1}^{\mathrm{T}}\mathbf{C}+\varvec{z}_{1}^{\mathrm{T}} \end{aligned}$$
(2k)
$$\begin{aligned}&\varvec{z}_{0}^{\mathrm{T}}\varvec{y}_{0}=\left( \varvec{x}_{1}^{\mathrm{T}}\mathbf{C}+\varvec{z}_{1}^{\mathrm{T}}\right) \varvec{y}_{0} \end{aligned}$$
(2l)
$$\begin{aligned}&\varvec{z}^{\mathrm{T}}_{t}+\varvec{s}^{\mathrm{T}}_{t}\mathbf{D}_{t} \geqq \varvec{x}^{\mathrm{T}}_{t+1}\mathbf{C}+\varvec{z}^{\mathrm{T}}_{t+1} \end{aligned}$$
(2m)
$$\begin{aligned}&\left( \varvec{z}^{\mathrm{T}}_{t}+\varvec{s}^{\mathrm{T}}_{t}\mathbf{D}_{t}\right) \varvec{y}_{t}= \left( \varvec{x}^{\mathrm{T}}_{t+1}\mathbf{C}+\varvec{z}^{\mathrm{T}}_{t+1}\right) \varvec{y}_{t} \end{aligned}$$
(2n)
$$\begin{aligned}&\varvec{x}_{t+1}^{\mathrm{T}}\mathbf{C} \leqq \varvec{h}^{\mathrm{T}} \end{aligned}$$
(2o)
$$\begin{aligned}&\varvec{x}_{t+1}^{\mathrm{T}}\mathbf{C}\varvec{q}_{t}=\varvec{h}^{\mathrm{T}}\varvec{q}_{t} \end{aligned}$$
(2p)
$$\begin{aligned}&\varvec{z}_{0}=\bar{\varvec{z}} \end{aligned}$$
(2q)
$$\begin{aligned}&\sum _{t=0}^{\infty } \varvec{\delta }^{\mathrm{T}}\varvec{p}_{t}=1 \end{aligned}$$
(2r)
$$\begin{aligned}&\gamma >0, \quad \varvec{p}_{t} \geqq \varvec{0}, \quad \varvec{y}_{t} \geqq \varvec{0}, \quad \varvec{q}_{t} \geqq \varvec{0}, \quad \varvec{z}_{t} \geqq \varvec{0}, \quad \varvec{x}_{t} \geqq \varvec{0}, \quad \varvec{s}_{t} \geqq \varvec{0} \end{aligned}$$
(2s)
In order to avoid “the end of the world” scenario, it is assumed that the consumption \(\varvec{\delta }\) required annually can be produced using a backstop technology (which only uses non-exhaustible resources). Formally, a backstop technology, as denoted by \((\bar{\mathbf{A}}, \mathbf{0}, \bar{\varvec{l}_{1}}, \bar{\mathbf{B}})\), is defined by the processes which are obtained from \((\mathbf{A}, \mathbf{C}, \varvec{l}_{1}, \mathbf{B})\) by deleting all the processes using exhaustible resources directly, (i.e. a process \((\varvec{e}^{\mathrm{T}}_{i}\mathbf{A}, \varvec{e}^{\mathrm{T}}_{i}\mathbf{C}, \varvec{e}^{\mathrm{T}}_{i}\varvec{l}_{1}, \varvec{e}^{\mathrm{T}}_{i}\mathbf{B})\) is in \((\bar{\mathbf{A}}, \mathbf{0}, \bar{\varvec{l}_{1}}, \bar{\mathbf{B}})\) if and only if \(\varvec{e}^{\mathrm{T}}_{i}\mathbf{C}=\varvec{0}\), where \(\varvec{e}_{i}\) is the ith unite vector). The remaining processes are denoted by \((\tilde{\mathbf{A}}, \tilde{\mathbf{C}}, \tilde{\varvec{l}_{1}}, \tilde{\mathbf{B}})\). The existence of a backstop technology is summarised as follows.
Such processes operated at intensity \(\bar{\varvec{x}}\), which is obtained from \(\varvec{x}^{*}\) by augmenting it with zeros, will be referred to as ‘cost-minimising backstop processes’ and are denoted by \((\hat{\mathbf{A}}, \mathbf{0}, \hat{\varvec{l}}_{1}, \hat{\mathbf{B}})\). The backstop technology and cost-minimising backstop processes are assumed to have the following characteristics.
Assumption 3 is presented in order to separate the problem of convergence (or gravity) from the study’s analysis. The theory that market prices continue to gravitate to the natural prices as determined by cost-minimising technique is well elaborated upon by classical economists, and it is also advocated by some early neo-classical economists, such as Marshall, Walras, and Wicksell. Even though there are debates on how to formally explain the convergence problem in modern classical theory (e.g. see a survey by Bellino
2011), it is still legitimate to use it as a reasonable assumption, given the fact that plenty of empirical observations support the idea of convergence (Petri
2011). In this study, Assumption 4 is made in order to simplify analysis from the complexities presented when considering pure joint production,
5 and it certainly holds if only single production prevails.
After establishing the model’s basic framework in the context of this study, an explanation is offered into how Ricardo and Hotelling’s ideas on exhaustible resources are represented in the model itself. As explained earlier, Ricardo and Hotelling’s analyses on exhaustible resources are based upon different assumptions.
To elaborate further, and in the context of this study’s model, we start with Ricardo’s perspective.
(R1)
After each mine is exhausted, another mine with the same characteristics can be found. In other words, there exist sequences
\(\{\varvec{x}_{t}\}\) and
\(\{\varvec{s}_{t}\}\), such that the following inequality holds.
$$\begin{aligned} \varvec{s}_{t}^{\mathrm{T}}\mathbf{D} \geqq \varvec{x}_{t+1}^{\mathrm{T}}\mathbf{C}. \end{aligned}$$
(5)
(R2)
Searching costs are constant. That is \(\alpha =1\), or \(D_{t}=D_{t+1}\) for \(t>0\).
(R3)
There exists a capacity constraint on extraction in each mine, which is represented here by the corresponding element of \(\varvec{h}\).
In Hotelling’s world:
(H1)
There exists only one kind of resource, with a quantity that is given and known. That is to say \(s=1\) and \(\bar{\varvec{z}}\) is given and known.
(H2)
There exist no capacity constraints on extraction: \(\varvec{h}\) is close to infinity.
(H3)
There are no searching activities. For instance, \(D_{t}=0\) for all \(t>0\).
Hence, the Hotelling rule does not hold in all circumstances. On the other hand, Ricardo’s analyses on exhaustible resources are found to be neither incomplete nor inferior. It can be seen that both Ricardo and Hotelling’s ideas on exhaustible resources are incorporated into this study’s model. The implications behind the Hotelling rule as well as Ricardo’s analyses will be analysed more comprehensively in Sect.
5. Nevertheless, we must first investigate the model’s dynamics of quantity and price.