The mass and energy-capital conservation equations are employed to study the time evolution of mass and price of nonrenewable energy resources, extracted and sold to the market, in case of no-accumulation and no-depletion, that is, when the resources are extracted and sold to the market at the same mass flow rate. The Hotelling rule for nonrenewable resources, that is, an exponential increase of the price at the rate of the current interest multiplied the time, is shown to be a special case of the general energy-capital conservation equation when the mass flow rate of extracted resources is unity. The mass and energy-capital conservation equations are solved jointly to investigate the time evolution of the extracted resources.
1. Introduction
The price evolution of nonrenewable energy resources is a very important problem in an economy based on nonrenewable energy resources. The economics of exhaustible resources was investigated by Hotelling [1], and reviewed in [2], with the conclusion that the price of a resource increases exponentially with the product of time and interest rate of capital.
After the oil crisis of 1973 the problem of the energy resources became a very important issue and several energy models and forecasts have been developed and employed. During the decade of 1970s, among the others, McCall [3] presented a linear program for the world oil industry which was employed by an oil company. Tewksbury [4] described a practical general approach to forecast foreign oil prices to the year 2000. Pindyck [5] described a new version of an econometric policy model of the natural gas industry. Adelman [6] forecasted a decade of rising real oil prices for the US. DuMoulin and Newland [7] came to the conclusion that a rapid take-off in demand is likely to be followed by a rapid increase in the price of oil.
During the 1980s, among the others, Pearce [8] presented an overall look at the world energy demand forecasts with the conclusion that the judgment, rather than explicit modelling, is used to suggest a world crude oil scenario in which oil prices rise at 10% per year. French [9] discussed the markets of world oil, natural gas, coal, electricity, and the energy demand, with the conclusion that from mid-decade onward energy prices will most likely increase more rapidly than the overall price level. Roberts [10] concluded his analysis that through the mid-1980s, real crude prices should be flat to down and thereafter increase at a rate equal to or less than inflation. Campbell and Hubbard [11] showed that forecasts of prices—including oil, natural gas, equipment and drilling costs, and money—affect the evaluation of all projects. Clark [12] focused on generalized price formation in the U.S. gas industry. David and Carson [13] focused on the banks which use the long term oil supply and demand situation as the underlying support for making oil and gas price projections. Lorentsen and Roland [14] outlined a very open model, which can easily impose different assumptions about oil supply and demand in order to indicate a range of feasible projections for oil prices. Curlee [15] concluded that overall assessment of forecasts and recent oil market trends suggests that prices will remain constant in real terms for the remainder of the 1980s, to increase by 2-3% during the 1990s and beyond. Anon [16] studied the U.S. demand for petroleum products in 1987, based on the assumption that world oil prices will average 13 US $/bbl through the third quarter of 1986 and rise gradually to 18 US $/bbl by the end of 1987. Dielwart and Coles [17] discussed the current market uncertainties and their impact on Canadian oil and natural gas prices. Yohe [18] presented an analytical technique based on an adaptive expectations model of incorporating current information into long-term forecast. Sekera [19] presented three price forecasting analyses with forecasts under each theory and the recommendation that all long term forecasts be based on fundamental/economic analysis, while short term forecasting needs to utilize other methods. Gately [20] reviewed recent and current opinions about prospects for prices in the world market for crude oil with two alternative views. Holden et al. [21] examined the accuracy and properties of forecasts by the OECD for 24 countries and 8 variables. Dougherty [22] presented some rules of thumb that may help to understand the evolution of oil prices, to identify factors to consider when deciding on the credibility of an oil price forecast, rather than to prophesize future oil price. Amano [23] developed an annual, small-scale econometric model of the world oil market to analyze oil market conditions and oil prices for the period 1986–1991.
During the 1990s, among the others, Garces [24] explored the effects of natural gas storage on short-term gas price forecasts, with preliminary results which indicate that by using natural gas storage as a causal variable, the final gas price forecasts can decrease by as much as 5% over the forecast period. Angelier [25] concluded his analytical framework with the forecast that, in the year 2000, oil prices will not be significantly different from those of 1990s. Bopp and Lady [26] concluded their analysis that when actual prices are employed, future prices did correctly anticipate the observed seasonal pattern. Abramson and Finizza [27] developed a system that forecasts crude oil prices via Monte Carlo analyses of the network. Charleson and Weber [28] forecasted energy consumption in Western Australia with Bayesian vector autoregressions to 2010. Fesharaki [29] believed that for the next three to five years oil prices will remain at lower levels than generally predicted, because price increases, even when they occur, will not be sustainable for very long. Huntington [30] reviewed forecasts of oil prices over the 1980s that were made in 1980, identifying the sources of errors due to such factors as exogenous GNP assumptions, resource supply conditions outside the cartel, and demand adjustments to price changes. Moosa and Al-Loughani [31], on the basis of monthly observations on spot and future prices of the West Texas Intermediate (WTI) crude oil, suggested that future prices are neither unbiased nor efficient forecasters of spot prices. Abramson [32] discussed a knowledge-based system that models the crude oil market as a belief network and uses scenario generation and Monte-Carlo analysis to forecast oil prices. Skov [33] reviewed some of the more significant forecasts of energy supply, demand, and oil prices made in the last 25 years and identifies the basic premises used and why they led to incorrect forecasts. Abramson and Finizza [34] described a case study in the use of inherently probabilistic belief network models to produce probabilistic forecasts of average annual oil prices. Santini [35] investigated the statistical model of oil prices examining issues and trends related to both the U.S. and world oil supply. Williams [36], in order to forecast the future, examined historical trends and presents the new idea that there is an inverse relationship between crude oil prices volatility and the strength of the relationship between the owners of crude oil (host governments) and the oil companies that develop, refine, market, and sell the products. Chaudhuri and Daniel [37] demonstrated that the nonstationary behavior of US dollar real exchange rates, over the post-Bretton Woods era, is due to the nonstationary behavior of real oil prices. Pindyck [38] discussed how price forecasts also influence investment decisions and the choice of products to produce. Silvapulle and Moosa [39] examined the relationship between the spot and futures prices of WTI crude oil using a sample of daily data with the result that both spot and futures markets react simultaneously to new information.
During the 2000s, among the others, Alba and Bourdaire [40] consider the actual situation as a “cohabitation” between oil and the other energies with the oil price, extremely volatile, reflecting the trial and error adjustment of the market share left to the other energies. The Centre for Global Energy Studies [41] covers several topics including oilfields set to begin production in 2000, non-OPEC production, demand in western OECD countries and the Asia Pacific region, and oil price forecasts. According to the US energy information [42] global crude oil prices in 2000 will rise to 24 US $/bbl, up by 2 US $/bbl from its earlier estimate, while spot prices for West Texas Intermediate crude will average more than 22 US $/bbl through 2001. Hare [43] reported that according to forecasts by the Energy Information Administration, increases in natural gas prices will continue all the way to the summer and winter of 2000, and that high oil prices will still be seen in 2001. According to the managing director of Shell’s exploration and production [44] crude oil prices will average in the mid-teens of US $/bbl over the next 10 years. The Energy Department of Energy Information Administration [45] forecasts the monthly average price of crude oil imported to the USA to stay above 28 US $/bbl for the rest of 2000. Walde and Dos Santos [46] identified factors that have to be considered in speculating about the future evolution of oil prices. Birky et al. [47] examined the potentiality of unconventional natural gas resources with all other fossil fuels combined with the conclusion of a widespread production and use of these unconventional reserves. The trend of world crude oil price in 1999 has been reviewed and analyzed by Yang [48] and the prospects on world crude oil price in 2000 were predicted. MacAvoy and Moshkin [49] used a model framework consisting of a simultaneous equations system for production from reserves and for demands for production in the residential, commercial, and industrial sectors, with a significant negative trend in the long-term price of natural gas. Morana [50] showed how the GARCH properties of oil price changes can be employed to forecast the oil price distribution over short-term horizons, with a semiparametric methodology based on the bootstrap approach. Tang and Hammoudeh [51] investigated the behavior of the nonlinear model, based on the target zone theory, which has very good forecasting ability when the oil price approaches the upper or the lower limit of the band. Alvarez-Ramirez et al. [52] studied daily records of international crude oil prices using multifractal analysis methods, where rescaled range Hurst analysis provides evidence that the crude oil market is a persistent process with long-run memory effects, demonstrating that the crude oil market is consistent with the random-walk assumption only at time scales on the order of days to weeks. Ye et al. [53] presented a short-term monthly forecasting model of West Texas Intermediate crude oil spot price using OECD petroleum inventory levels, which are a measure of the balance, or imbalance, between petroleum production and demand, and thus provide a good market barometer of crude oil price change. Sadorsky [54] used an ARMAX-ARCH model to estimate the conditional expected returns of petroleum futures prices under time-varying risk with results from a small forecasting experiment which indicates that the out-of-sample forecasts from an ARMAX-ARCH model generally outperform a random walk for all forecast horizons. Fong and See [55] examined the temporal behaviour of volatility of daily returns on crude oil futures using a generalised regime switching model that allows for abrupt changes in mean and variance, GARCH dynamics, basis-driven time-varying transition probabilities, and conditional leptokurtosis. Taal et al. [56] gave a summary of the most common methods used for cost estimation of heat exchange equipment in the process industry and the sources of energy price projections, showing the relevance of the choice of the right method and the most reliable source of energy price forecast used when choosing between alternative retrofit projects or when trying to determine the viability of a retrofit project. Cortazar and Schwartz [57] developed a parsimonious three-factor model of the term structure of oil future prices that can be easily estimated from available future price data. Cabedo and Moya [58] proposed to use value at risk (VaR) for oil price risk quantification, providing an estimation for the maximum oil price change associated with a likelihood level, which fits the continuous oil price movements well and provides an efficient risk quantification. Burg et al. [59] developed an econometric modeling of the world oil market suggesting that world oil prices are likely to fall in the latter part of 2004 and in 2005 as global demand pressures ease. Burg et al. [60] made a market forecast for various energy sources, including oil and gas and coal, with the conclusion that prices are forecasted to decline in 2005 to average 38 US $/bbl WTI. Mirmirani and Li [61] applied VAR and ANN techniques to make ex-post forecast of U.S. oil price movements. Abosedra and Bagesthani [62] evaluated the predictive accuracy of 1-, 3-, 6-, 9-, and 12-month-ahead crude oil future prices for the period from January 1991 until December 2001. Gori and Takanen [63] forecasted the energy demand in a specific country, where the analysis of the electricity demand was focused, for the first time, on the energy consumption and the possible substitution among the different energy resources by using a modified form of the econometric model EDM (Energy Demand Model). Feng et al. [64] predicted the oil price fluctuation by ARFIMA model, which takes the long memory feature into consideration, showing that ARFIMA model is better than ARMA model. Movassagh and Modjtahedi [65] tested the fair-game efficient-markets hypothesis for the natural gas future prices over the period 1990 through 2003. Ye et al. [66] presented a short-term forecasting model of monthly West Texas Intermediate crude oil spot prices using readily available OECD industrial petroleum inventory levels. Yousefi et al. [67] illustrated an application of wavelets as a possible vehicle for investigating the issue of market efficiency in future markets for oil. Sun and Lai [68] presented a forecast model of time series oil price about NYMEX with BP neural networks, analyzing the oil price fluctuation trend. Sadorsky [69] used several different univariate and multivariate statistical models to estimate forecasts of daily volatility in petroleum future price returns. Ye et al. [70] showed the effect that surplus crude oil production capacity has on short-term crude oil prices. Moshiri and Foroutan [71] modelled and forecasted daily crude oil future prices from 1983 to 2003, listed in NYMEX, applying ARIMA and GARCH models, tested for chaos using embedding dimension, BDS(L), Lyapunov exponent, and neural networks tests, and set up a nonlinear and flexible ANN model to forecast the series. The United States Energy Information Administration (EIA) [72] expects worldwide oil demand to increase from 80 million bbl/day in 2003 to 98 million bbl/day in 2015 and to 118 million bbl/day in 2030. The latest forecast reference case calls for crude oil prices to climb from 31 US $/bbl in 2003 to 57 US $/bbl in 2030. Ghouri [73] analyzed qualitatively and quantitatively the relationship between US monthly ending oil stocks position with that of West Texas Intermediate (WTI) oil prices from February 1995 to July 2004, concluding that, if other things are held constant, WTI is inversely related to the petroleum products (PPP), combined petroleum products and crude oil (CPPP), crude oil alone (crude), total oil stocks including petroleum products, crude oil and strategic petroleum reserves SPR (total), total gasoline (TGO), and total distillate (TDO). Ye et al. [74] incorporated low- and high-inventory variables in a single equation model to forecast short-run WTI crude oil prices enhancing the model fit and forecast ability. The approach of using mass and energy-capital conservation equations to investigate the price evolution with time throughout the use of economic parameters was proposed in [75, 76] by the present author. The Hotelling rule was generalized in [75] with the conclusion that the price of the extracted resources increases exponentially with the product of the time and the difference between the inflation rate and the extraction rate of the resources, PIFE, “price increase factor of extracted resources.” A further generalization was done in [76] with the introduction of the price of the selling resources, which depends on PIFE and the new parameter PIFS, “price increase factor of selling resources,” that is, the difference between the prime rate of interest and the extraction rate of the resources. The political events and many complicated factors which happened in the last decades have made oil prices highly nonlinear and even chaotic, with irregularities and random oscillations, linked to the day-to-day events, giving reliability to forecast oil price and consumption in really short terms only, by using adaptive neural fuzzy inference systems (ANFIS), [77]. P. K. Narayan and S. Narayan [78] examined the volatility of the crude oil price using daily data for the period from 1991 to 2006, in various subsamples in order to judge the robustness of the results, implying that the behaviour of the oil prices tend to change over short periods of time. Knetsch [79] developed an oil price forecasting technique, based on the present value model of rational commodity pricing, suggesting the shifting of the forecasting problem to the marginal convenience yield, which can be derived from the cost-of-carry relationship. Deutsche Bank [80] reported that light sweet crude oil prices will average 80 US $/bbl on the New York Mercantile Exchange in 2008 while natural gas prices will average 7.75 US $/MMbtu and perhaps stay put there. It also reported that crude oil in this decade is likely to average nearly 55 US $/bbl. It seems that crude oil markets are following the reciprocal of the declining value of the US dollar and ignoring deteriorating oil fundamentals. MacAskie and Jablonowski [81] developed a decision-analytic model to value commodity price forecasts in the presence of futures markets and applied the method to a data set on crude oil prices. Askari and Krichene [82] observed markets expecting oil prices to remain volatile and jumpy and, with higher probabilities, to rise, rather than fall, above the expected mean. Xu et al. [83] investigated with ILS approach the forecast of the relationship between commodity inventory levels and crude oil spot prices effectively, with the conclusion that their empirical study suggests that both the ILS method and the confidence interval method can produce comparable quality forecasts. Yu et al. [84] proposed empirical mode decomposition (EMD) based on neural network ensemble learning paradigm for world crude oil spot price forecasting, decomposing the original crude oil spot price series into a finite, and often small, number of intrinsic mode functions (IMFs). Fan et al. [85] applied pattern matching technique to multistep prediction of crude oil prices and proposed a new approach: generalized pattern matching based on genetic algorithm (GPMGA), which can be used to forecast future crude oil price based on historical observations. This approach can detect the most similar pattern in contemporary crude oil prices from the historical data. Kang et al. [86] investigated the efficacy of a volatility model for three crude oil markets—Brent, Dubai, and West Texas Intermediate (WTI)—with regard to its ability to forecast and identify volatility stylized facts, in particular volatility persistence or long memory with the conclusion that CGARCH and FIGARCH models are useful for modeling and forecasting persistence in the volatility of crude oil prices. Hamilton [87] examined the factors responsible for changes in crude oil prices, reviewing the statistical behavior of oil prices, in relation to the predictions of theory, and looking in detail at key features of petroleum demand and supply, discussing the role of commodity speculation, OPEC, and resource depletion. Cuaresma et al. [88], using a simple unobserved components model, showed that explicitly modelling asymmetric cycles on crude oil prices improves the forecast ability of univariate time series models of the oil price. Ye et al. [89] predicted crude oil prices from 1992 through early 2004 by using OECD’s relative inventories and OPEC’s excess production capacity improving forecasts for the post-Gulf War I time period over models without the ratchet mechanism. Cheong [90] investigated the time-varying volatility of two major crude oil markets, the West Texas Intermediate (WTI) and the Europe Brent, with a flexible autoregressive conditional heteroskedasticity (ARCH) model to take into account the stylized volatility facts such as clustering volatility, asymmetric news impact, and long memory volatility. Ghaffari and Zare [91] developed a method based on soft computing approaches to predict the daily variation of the crude oil price of the West Texas Intermediate (WTI), with comparison to the actual daily variation of the oil price, and the difference is implemented to activate the learning algorithms. The energy supply curve, that is, price versus consumption, of nonrenewable energy resources was constructed in [92] where new parameters were introduced to forecast the price evolution of nonrenewable resources. The present theory has been reviewed and applied to the period from 1966 until 2006 in [93].
During the 2010s, among the others, de Souza e Silva et al. [94] investigated the usefulness of a nonlinear time series model, known as hidden Markov model (HMM), to predict future crude oil price movements, developing a forecasting methodology that consists of employing wavelet analysis to remove high frequency price movements, assumed as noise, and using the probability distribution of the price return accumulated over the next days to infer future price trends. He et al. [95] presented two forecasting models, one based on a vector error correction mechanism and the other based on a transfer function framework with the range taken as a driver variable, for forecasting the daily highs and lows, showing that both of these models offer significant advantages over the naïve random walk and univariate ARIMA models in terms of out-of-sample forecast accuracy. Miller and Ni [96] examined how future real GDP growth relates to changes in the forecasted long-term average of discounted real oil prices and to changes in unanticipated fluctuations of real oil prices around the forecasts. Forecasts were conducted using a state-space oil market model, in which global real economic activity and real oil prices share a common stochastic trend. Yaziz et al. [97] obtained the daily West Texas Intermediate (WTI) crude oil prices data from Energy Information Administration (EIA) from the 2nd of January, 1986, to the 30th of September, 2009, by using the Box-Jenkins methodology and generalized autoregressive conditional heteroscedasticity (GARCH) approach in analyzing the crude oil prices, which is able to capture the volatility by the nonconstant of conditional variance. Prat and Uctum [98] suggested a mixed expectation model, defined as a linear combination of these traditional processes, interpreted as the aggregation of individual mixing behavior and of heterogeneous groups of agents using these simple processes, consistent with the economically rational expectations theory. It was shown that the target oil price included in the regressive component of this model depends on the long-run marginal cost of crude oil production and on the short term macroeconomic fundamentals whose effects are subject to structural changes. Jammazi and Aloui [99] combined the dynamic properties of the multilayer back propagation neural network and the recent Harr A trous wavelet decomposition, implementing a hybrid model HTW-MPNN to achieve prominent prediction of crude oil price, by using three variants of activation function, namely, sigmoid, bipolar sigmoid, and hyperbolic tangent in order to test the model’s flexibility. Mingming and Jinliang [100] constructed a multiple wavelet recurrent neural network (MWRNN) simulation model, in which trend and random components of crude oil and gold prices were considered to capture multiscale data characteristics, while a real neural network (RNN) was utilized to forecast crude oil prices at different scales, showing that the model has high prediction accuracy. Baumeister and Kilian [101] constructed a monthly real-time dataset consisting of vintages for the period from January, 1991, to December, 2010, that is suitable for generating forecasts of the real price of oil from a variety of models and documenting that revisions of the data typically represent news, and introducing backcasting and nowcasting techniques to fill gaps in the real-time data. It was shown that the real-time forecasts of the real oil price can be more accurate than the nochange forecast at horizons up to 1 year. Azadeh [102] presented a flexible algorithm based on artificial neural network (ANN) and fuzzy regression (FR) to cope with optimum long-term oil price forecasting in noisy, uncertain, and complex environments, incorporating the oil supply, crude oil distillation capacity, oil consumption of non-OECD, U.S. refinery capacity, and surplus capacity as economic indicators. Wang et al. [103] investigated the interacting impact between the crude oil prices and the stock market indices in China and analyzed the corresponding statistical behaviors introducing a jump stochastic time effective neural network model and applying it to forecast the fluctuations of the time series for the crude oil prices and the stock indices, and studying the corresponding statistical properties by comparison. Hu et al. [104] attempted to accurately forecast prices of the crude oil futures by adopting three popular neural networks methods including the multilayer perceptron, the Elman recurrent neural network (ERNN), and the recurrent fuzzy neural network (RFNN), with the conclusion that learning performance can be improved by increasing the training time and that the RFNN has the best predictive power and the MLP has the worst one among the three underlying neural networks. Ruelke et al. [105] derived internal consistency restrictions on short, medium, and long-term oil price forecasts, by analysing whether oil price forecasts extracted from the Survey of Professional Forecasters (SPF), conducted by the European Central Bank (ECB), satisfy these internal consistency restrictions, finding that neither short-term forecast is consistent with medium-term forecasts nor that medium-term forecasts are consistent with long-term forecasts. A new category of cases, that is, the negative inflation rate, has been introduced, within the present theory, in [106], where some preliminary results have been presented. Pierdzioch et al. [107] found that the loss function of a sample of oil price forecasters is asymmetric in the forecast error, indicating that the loss oil price forecasters incurred when their forecasts exceeded the price of oil tended to be larger than the loss they incurred when their forecast fell short of the price of oil. Accounting for the asymmetry of the loss function does not necessarily make forecasts look rational. Shin et al. [108] proposed a study to exploit the method of representing the network between the time-series entities and to then employ SSL to forecast the upward and downward movement of oil prices by using one-month-ahead monthly crude oil price predictions between January, 1992, and June, 2008. Alquist [109] addressed some of the key questions that arise in forecasting the price of crude oil. Xiong et al. [110] proposed a revised hybrid model built upon empirical mode decomposition (EMD) based on the feed-forward neural network (FNN) modeling framework incorporating the slope-based method (SBM), which is capable of capturing the complex dynamic of crude oil prices. The results obtained in this study indicate that the proposed EMD-SBM-FNN model, using the MIMO strategy, is the best in terms of prediction accuracy with accredited computational load. Chang and Lai [111] attempted to develop an integrated model to forecast the cycle of energy prices with respect to the level of economic activity, showing that the oil price cycle and economic activities have bidirectional causality in the short run, and that the upwards (downwards) cycle of oil prices is accompanied by expansion (contraction) of economic activities, and vice versa, with a comovement trend in the long run. Conceptually, the model developed in this work is useful with regard to forecasting the level of economic activities using the oil price cycle, as most economic activities depend on energy. Salehnia et al. [112] used the gamma test for the first time as a mathematically nonparametric nonlinear smooth modeling tool to choose the best input combination before calibrating and testing models and developing several nonlinear models with the aid of the gamma test, including regression models, local linear regression (LLR), dynamic local linear regression (DLLR) and artificial neural networks (ANN) models. Shin et al. [113] exploited the method of representing the network between the time-series entities to employ semisupervised learning (SSL) to forecast the upward and downward movement of oil prices, by using one-month-ahead monthly crude oil price predictions between January, 1992, and June, 2008. Yang et al. [114] forecasted the international crude oil price by using the grey system theory and creating a MATLAB program to achieve it. The present theory has been confirmed in case of negative inflation rate [115], projecting the forecast to the following five months of 2013 [116], with a very recent verification [117].
2. Mass Conservation Equation of Extracted Resources
Assume the mass M of nonrenewable energy resources is in the reservoir, C, reported in Figure 1. The mass flow rate of extraction from the reservoir is, G, with dimensions (mass/annum).
Nonrenewable energy resources in the reservoir C.
The mass conservation equation of the extracted resources is
(1)dMdt=-G,
which, after integration, becomes
(2)M=M0-∫0tGdt,
where M0 is the mass of resources in the reservoir at the initial time t=0.
Assuming the extraction rate, α, at the time t, given by
(3)1GdGdt=α,
the mass flow rate of extraction with constant α in the time interval 0-t is
(4)G=G0exp(αt).
The evolution of the mass flow rate of extraction, given by (4) and presented in Figure 2, depends on the value of α. G is constant with time if α=0, G increases if α>0, and G decreases if α<0.
Time evolution of the mass flow rate of extraction, G.
The mass of nonrenewable resources in the reservoir C becomes, dividing by G0,
(5)MG0=M0G0+[1-exp(αt)]α,
and its time evolution is presented in Figure 3.
Time evolution of M/G0 (years) at the extraction rate α.
In case of a constant extraction rate, G=G0, that is, α=0, (1) gives
(6)M=M0-G0·t;
that is, the mass of resources in the reservoir C decreases linearly with time. Equation (6) gives the time tf=M0/G0 when M=0, that is, the time of exhaustion of the nonrenewable resources.
If, for example, the ratio M0/G0 is equal to 50 years, the time to exhaust the resources is tf=50 years. If α>0, the mass of resources in the reservoir C is exhausted faster than for α=0, that is, before 50 years. The time tf, which gives M=0, is given by (5) as
(7)tf=1αln[1+αM0G0].
Assuming M0/G0=50 years, the number of years for the exhaustion of the resources depends on the value of α. From (7) it can be found that
tf=40.55 years for α=0.01 (1/annum);
tf=17.92 years for α=0.1 (1/annum).
If α<0, (5) can be written as
(8)MG0=M0G0-[1-exp(|α|t)]|α|,
which, for an infinite time t→∞, gives
(9)M(∞)G0=M0G0-1|α|.
The extraction policy to exhaust the nonrenewable resources in an infinite time t→∞ is to have
(10)M(∞)G0=M0G0-1|α|=0,
which gives the critical extraction rate, αc,
(11)|αc|=G0M0.
For M0/G0=50 years the critical extraction rate is αc=-0.02 (1/annum).
If αc<α<0, the time to exhaust the resources is given by
(12)tf=-1|α|ln[1-|α|M0G0],
which depends on α. For M0/G0=50 years the time tf is
tf=69.31 years for α=-0.01 (1/annum);
tf=51.29 years for α=-0.001 (1/annum).
The owner of the nonrenewable resources does not find convenient to have
(13)M(∞)G0=M0G0-1|α|>0,
at an infinite time, t→∞, because the resources could remain nonextracted.
If α<αc<0, the nonrenewable resources will never be extracted and a certain amount of resources will remain in the reservoir as nonextracted. For example, if α=-0.04 (1/annum) the amount of resources which will remain nonextracted is M(∞)/G0=25 years.
3. Hotelling Rule
Assume the nonrenewable resource can be extracted without cost. Denote po the initial price per unit of resource at the initial time t=0 and suppose the interest rate, r, is constant in the time interval from t=0 to t. The owner of the resources unit has two options: to keep the unit of resources in the reservoir for the time interval t with the real expectation that the price at t will be higher, saying p(t), or to sell the unit of resources and put the money, obtained from the sale, in a bank for the same time interval t to get an interest return equal to p(t)tr. Under competitive conditions the owner is indifferent between these two options; then, the price of the unit at t, p(t), must be equal to the price p0 at the time t0, increased of the interest return p(t)tr. In other words
(14)p(t)=p0+p(t)tr,
or, taking the limit as t→0, it can be obtained
(15)dpdt=p·r.
After integration, (15) becomes
(16)p(t)=p0exp(rt),
also called Hotelling rule, [2], in honour of Hotelling [1], where, r, the interest rate of capital, is assumed constant in the time period 0-t.
Equation (16) states that the price of nonrenewable resources can only increase exponentially with time if the interest rate r is positive while only decreasing in case of negative inflation rate. The price evolution is independent of any other variable present in the energy game of nonrenewable resources.
4. Energy-Capital Conservation Equation of Extracted Resources
The energy-capital conservation equation can be written in unsteady state, similarly to the energy conservation equation of thermal engineering, for the nonextracted resources of the reservoir C, Figure 1, with interest rate rN (1/annum), as
(17)dKdt=Gk.
The left term expresses the time variation of the capital K, due to the resources of the reservoir C, while Gk is the capital flow rate entering into the reservoir C, which derives from the mass flow rate of extraction, G.
The capital K is written as
(18)K=G·p,
where p is the current price of a unity of extracted resources and G is the mass flow rate of extraction.
The right term of (17), that is, the capital flow rate entering into the reservoir, is written as
(19)Gk=K·rN=G·p·rN,
where rN is the interest rate of the nonextracted resources.
Equation (17) then becomes
(20)d(G·p)dt=G·p·rN,
which, for a mass flow rate of extraction being equal to 1, that is, G=1, it gives (15), that is, the Hotelling rule before integration. In conclusion, the Hotelling rule can be obtained from the energy-capital conservation equation with a mass flow rate of extraction being equal to G=1.
Differentiation of the capital conservation equation, (20), and division by the capital K gives
(21)1pdpdt+1GdGdt=rN.
Equation (21) can be used, with (3), to study the price evolution of the extracted resources with interest rate rN. Equation (21) becomes, indicating p as pE for the extracted resources and using (3),
(22)1pEdpEdt=rN-α,
which, after integration, gives
(23)pE=pE0·exp(β·t),
where,
(24)β=rN-α,that is, the difference between the interest rate of nonextracted resources, rN, and the extraction rate, α, is a new parameter called “price increase factor of extracted resources,” PIFE. The price evolution with time of the extracted resources, given by (23), is shown in Figure 4.
Time evolution of the price of extracted resources.
The price of extracted resources evolves with time according to the value of PIFE, β. If the extraction rate α is equal to the interest rate of nonextracted resources, rN, the price increase factor, PIFE is equal to zero, β=0, and the price pE remains constant with time. If the extraction rate α is higher than rN, the price increase factor, PIFE is negative, β<0, and the price pE decreases with time. If the extraction rate α is lower than rN, the price increase factor, PIFE is positive, β>0, and the price pE increases with time. In case of a conservative policy of extraction; that is, α<0, the price increase factor, is greater than zero, β>0, and the price pE increases with time.
5. Mass and Energy-Capital Conservation Equations of Selling Resources
Assume that nonextracted and extracted energy resources are in the two reservoirs, CN and CE, of Figure 5.
Nonextracted and extracted energy resources.
The nonextracted resources are in the reservoir CN with mass MN and interest rate rN. The mass flow rate of extracted resources from the reservoir CN is GE while the capital flow rate, relative to the mass flow rate GE, is GKE. The resources of the reservoir CN are extracted at the price of the extracted resources, pE, and allocated in the reservoir CE, where ME is the mass of extracted resources. The resources of the reservoir CE are sold at the price pS. The mass flow rate of the selling resources from the reservoir CE is GS while the capital flow rate, relative to the mass flow rate GS, is GKS.
The mass conservation equation of the extracted resources in the reservoir CE is
(25)dMEdt=GE-GS,
while the relative energy-capital conservation equation is
(26)ddt(KE)=GKS-GKE,
which becomes
(27)ddt(GS·pS)=GS·pS·rE-GE·pE·rN,
and finally
(28)1pS·dpSdt+1GS·dGSdt=rE-GE·pEGS·pS·rN.
In case of no-accumulation and no-depletion of the extracted resources, GE=GS, the right term of (25) is zero; that is, the mass ME of extracted resources in the reservoir CE is constant with time. The mass flow rates of extracted and selling resources have the same extraction rate α from the reservoirs CN and CE; that is,
(29)1GEdGEdt=1GSdGSdt=α,
which gives, after integration,
(30)GE=GS=GE0·exp·(α·t)=GS0·exp·(α·t),
and the time evolution is similar to that of Figure 2.
Equation (28) becomes
(31)1pSdpSdt+α=rE-pEpSrN,
or
(32)dpSdt=pS(rE-α)-pE·rN.
The substitution of (23) into (32) and the solution of the relative differential equation give the time evolution of the price of sold resources
(33)pS=pS0*exp(β·t)+[pS0-pS0*]exp(β′·t),
where
(34)β′=rE-α,
is called the “Price Increase Factor of Selling resources,” PIFS, and
(35)pS0*=rN·pE0rE-rN=rN·pE0β′-β,
the “critical initial price of selling resources,” CIPS.
The price of selling resources, (33), has an extreme (maximum or minimum) for the time tm given by
(36)tm=1(β-β′)ln[β′·(pS0*-pS0)β·pS0*].
The extreme is a maximum if
(37)β′(β′-β)(pS0-pS0*)<0,
otherwise it is a minimum.
The time tm of the extreme is zero if pS0 is equal to
(38)pS0**=pS0*(β′-β)β′=rNpE0β′,
which is called “critical initial price extreme of selling resources,” CIPES.
The time tm of the maximum is greater than zero if
(39)pS0>pS0**forβ′>β(i.e.,rE>rN),
or
(40)pS0<pS0**forβ′<β(i.e.,rE<rN).
For pS0=pS0* the time of the maximum is
(41)tm=+∞,
for β′>β, or
(42)tm=-∞,
for β′<β.
For rN=rE, that is, β′=β, pS is given by
(43)pS=(pS0-pS0**βt)exp(βt),
where CIPS, that is, pS0*, is not defined.
The price of selling resources has an extreme for the time tm given by
(44)tm=(pS0-pS0**)pS0**β′.
The extreme is a maximum if pS0**>0 and a minimum if pS0**<0. The time tm of the maximum is zero if the initial price pS0 is
(45)pS0=pS0**=rN·pE0β′.
The possible cases can be classified into six categories:
Category 0—rN=0;
Category 1: rN>α, that is, β=PIFE>0;
Category 2: α>rN, that is, β=PIFE<0;
Category 3: α=rN, that is, β=PIFE=0;
Category 4: rN=rE, that is, β′=β or PIFE=PIFS;
Category 5: rN<0.
On their side, each category presents the following possible cases.
Category 0 (rN=0). If the interest rate of nonextracted resources is rN=0, (i.e., pS0*=pS0**=0), the price evolution of selling resources is
(46)pS=pS0exp(β′t),
where β′ is the PIF of sold resources.
Figure 6 presents the price evolution of selling resources according to the relation between extraction rate, α, and interest rate of extracted resources rE.
Price evolution of selling resources for Category 0.
Case 0-A (α<rE, (i.e., β′>0)). The price pS increases with time at different rates, according to the values of α. The rate of increase is higher for α<0, β>0, intermediate for α=0, β=0, and lower for α<rE, β<0.
Case 0-B (α=rE, (i.e.,β′=0, β<0)). The price pS is constant with time at pS0.
Case 0-C (α>rE, (i.e.,β′<0, β<0)). The price pS is decreasing with time towards +0.
In conclusion, for rN=0 the criterium to evaluate the price evolution of selling resources, pS, is β′. If β′>0 the price pS increases, if β′=0 the price pS remains constant, and if β′<0 the price pS decreases towards +0.
Category 1 (rN>α, that is, β=PIFE>0). The price evolutions of pS are reported in Figure 7 according to the relation between rN and rE.
Price evolution of selling resources for Category 1.
Case 1-A (rE>rN>α, that is, β′>β>0). The price of selling resource increases with time if the initial price pS0 is pS0≥pS0*>pS0**>0, that is, equal to or greater than CIPS>CIPES>0.
Case 1-B (rE>rN>α and pS0*>pS0>pS0**>0 or rN>rE>α and pS0>pS0**>0>pS0*). The price of selling resource increases temporarily with time up to tm, given by (19), and then decreases for rE>rN>α if the initial price pS0 has a value comprised between CIPS and CIPES, that is, pS0*>pS0>pS0**>0, or, for rN>rE>α, if the initial price pS0 is greater than CIPES, that is, pS0>pS0**>0>pS0*.
Case 1-C (rE>rN>α and pS0*>pS0**>pS0, or rN>rE>α and pS0**>pS0>0>pS0*, or rN>α>rE). The price of selling resource decreases with time for rE>rN>α if the initial price pS0 is smaller than CIPS and CIPES, that is, pS0*>pS0**>pS0, or, for rN>rE>α, if the initial price pS0 is smaller than CIPES, that is, pS0**>pS0>0>pS0*, or, for rN>α>rE for every initial price pS0 because pS0>0>pS0*>pS0**.
Category 2 (α>rN, that is, β=PIFE<0). The time evolution is presented in Figure 8 according to β and β′ or the relation between rN and rE.
Price evolution of selling resources for Category 2.
Case 2-A (rE>α>rN, that is, β′>β). The price of selling resource increases with time if the initial price pS0 is pS0≥pS0**>pS0*, that is, equal to or greater than CIPES>CIPS, pS0**>pS0*.
Case 2-B (rE>α>rN, that is, β′>β). The price of selling resource decreases temporarily with time up to tm, given by (19), and then increases if the initial price, pS0, has a value comprised between CIPS and CIPES, that is, pS0*>pS0>pS0**>0.
Case 2-C (α>rN, that is, β<0). In all other cases the price of selling resource decreases with time including rE=α>rN, or α>rN>rE, or α>rN>rE.
Category 3 (α=rN, that is, β=PIFE=0). The price evolution of selling resources, pS, is dependent on the relation between rN and rE, as reported in Figure 9.
Price evolution of selling resources for Category 3.
Case 3-A (rE>rN=α, that is, β′>β=0). The price of selling resource increases with time if the initial price pS0 is pS0>pS0**=pS0*, that is, greater than CIPS=CIPES, pS0**=pS0*.
Case 3-B (rE>rN=α, that is, β′>β=0). The price of selling resource remains constant with time if the initial price pS0 is pS0=pS0**=pS0*, that is, equal to CIPS=CIPES, pS0**=pS0*.
Case 3-C (rE>rN=α, that is, β′>β=0, or rN=α>rE, that is, β′<β=0). The price of selling resource, for rE>rN=α, decreases with time if the initial price pS0 is pS0<pS0**=pS0*, that is, smaller than CIPS and CIPES, pS0**=pS0*. In the other cases the price of the selling resource decreases with time for every value of pS0.
Category 4 (rN=rE, that is, β′=β). The time evolutions are presented in Figure 10.
Price evolution of selling resources for Category 4.
Case 4-A (rN=rE>α, that is, β′=β>0). The price cannot increase with time.
Case 4-B (rN=rE>α, that is, β′=β>0). The price of selling resource increases temporarily with time up to tm, given by (2), and then decreases if the initial price pS0 is pS0>pS0**>0, that is, greater than CIPES>0.
Case 4-C (rN=rE>α, that is, β′=β>0, or rN=rE<α, that is, β′=β<0, or rN=rE=α, that is, β′=β=0). The price of selling resource decreases with time if the initial price pS0 is pS0**≥pS0>0, that is, equal to or smaller than CIPES>0. In all the other cases the price of selling resource decreases with time for every value of pS0.
Category 5 (rN<0). A new category of cases has been introduced for negative inflation rate, rN<0, which was registered from March to October, 2009, after the economic crisis of 2008. This new category is interesting because the Hotelling rule forecasts a decrease of the oil price if the interest rate is assumed to be equal to the inflation rate, that is, negative, while the conclusions of the present approach are different.
The time evolutions are presented in Figure 11.
Price evolution of selling resources for Category 5.
Case 5-A (rE>α>0>rN, that is, β′>0>β). The price of selling resource increases with time for every value of the initial price pS0 because pS0>0>pS0*>pS0**, since CIPS and CIPES are negative, or 0>pS0*>pS0**.
Case 5-B (α>rE>0>rN, that is, 0>β′>β). The price of selling resource increases temporarily with time up to tm and then decreases if the initial price, pS0, is pS0**>pS0>0>pS0*, that is, smaller than CIPES>0>CIPS.
Case 5-C (α>rE>0>rN, that is, 0>β′>β). The price of selling resource decreases with time if the initial price, pS0, is pS0>pS0**>0>pS0*, that is, greater than CIPES>0>CIPS.
6. Price Difference between Selling and Extracted Resources
The difference between selling and extracted resources Δp=(pS-pE) is defined as(47)Δp=(pS-pE)=(pS0-pS0*)exp(β′t)-(pE0-pS0*)exp(βt)=(Δp0-Δ*p0)exp(β′t)+Δ*p0exp(βt),
where
(48)Δ*p0=pS0*-pE0=pE0(2rN-rE)(rE-rN),
is the critical initial price difference, CIPD, Δ*p0.
The price difference Δp has an extreme (maximum or minimum) for
(49)tm=1(β′-β)ln[β·(pS0*-pE0)β′·(pS0*-pS0)]=1(β′-β)ln[β·Δ*p0β′·(Δ*p0-Δp0)]=1(β-β′)ln[β′·(Δ*p0-Δp0)β′·Δ*p0].
The extreme is a maximum if
(50)β′(β′-β)(Δp0-Δ*p0)<0,
otherwise it is a minimum.
The time tm of the extreme is zero if Δp0 is equal to
(51)Δ**p0=Δ*p0[(β′-β)β′]=pE0(2rN-rE)β′,
where Δ**p0 is the critical initial extreme of the initial price difference, CIEIPD.
The time tm of the extreme is greater than zero if
(52)Δp0>Δ**p0forβ′>β,Δp0<Δ**p0forβ′<β.
For ΔpS0=ΔpS0* the time of the extreme is
(53)tm=+∞,
for β′>β, and
(54)tm=-∞,
for β′<β.
The time t0 is
(55)t0=1(β′-β)ln[Δ*p0(Δ*p0-Δp0)],
when ΔP=0.
The possible cases can be classified in five categories:
Category 0D: rN=0;
Category 1D: rN>α, that is, β=PIFE>0;
Category 2D: α>rN, that is, β=PIFE<0;
Category 3D: α=rN, that is, β=PIFE=0;
Category 4D: rN=rE, that is, β′=β or PIFE=PIFS.
On their side, each category presents the following possible cases.
Category 0D (rN=0). The difference between selling and extracted resources, Δp, is function of the initial difference Δp0=(pS0-pE0) and the extraction rate, α, as compared to rE. It can be remarked that for rN=0 the critical initial price and differences of sold resources are pS0*=pS0**=0, Δ*p0=-pE0, and Δ**p0=-rEpE0/β′.
The price difference is then given by
(56)Δp=(pS-pE)=pS0exp(β′t)-pE0exp(βt)=(Δp0-Δ*p0)exp(β′t)+Δ*p0exp(βt).
The time tm of the extreme is given by
(57)tm=1(β′-β)ln[β·pE0β′·pS0],
which is obtained for pS0*=0 and the extreme is a maximum for
(58)β′(β′-β)pS0<0.
The price difference is Δp=0 at the time t0(59)t0=1(β′-β)ln[pE0pS0].
Four initial conditions are investigated:
if the initial price difference Δp0 is Δp0≥0 or (pS0≥pE0);
if the initial price difference Δp0 is 0>Δp0≥Δ**p0;
if the initial price difference Δp0 is Δ**p0>Δp0>Δ*p0;
if the initial price difference Δp0 is Δ*p0=Δp0 or pS0=pS0*.
The time evolutions are presented in Figure 12.
Price difference evolution for Category 0D.
Case 0D-A (α≤0<rE, (i.e., β′>0, β′>β, (Δ**p0)a<0))
Case 0D-A1. If the initial Δp0≥0, Δp increases exponentially towards +∞ with a higher rate for α<0, (β>0), an intermediate one for α=0, (β=0), and a smaller one for rE>α>0, (β<0).
Case 0D-A2. If the initial Δp0 is 0>Δp0>Δ**p0 the evolution is similar to the previous case but now Δp increases from a negative value Δp0 and all the curves pass through the same point Δp=0 at t0. If the initial Δp0=Δ**p0, Δp increases exponentially with the minimum at tmin=0.
Case 0D-A3. If the initial Δp0 is Δ**p0>Δp0>Δ*p0 and β>0, Δp decreases initially to a minimum and then increases exponentially as in the previous cases.
Case 0D-B (α=rE, (i.e., β′=0>β, pS0** is not defined and Δ**p0=-∞<Δ*p0<0))
Case 0D-B1. If the initial Δp0≥Δ*p0, Δp tends towards pS0=Δp0-Δ*p0 as asymptote.
Case 0D-C (α>rE, (i.e., β<β′<0, (Δ**p0)c>0>(Δ*p0)c))
Case 0D-C1. If the initial value Δp0≥Δ**p0, the difference Δp decreases towards +0, with the maximum at t=0 for Δp0=Δ**p0.
Case 0D-C2. If the initial value Δ*p0<Δp0<Δ**p0, the difference Δp increases up to the time tmax when Δp reaches a maximum value, Δpmax, and then decreases towards +0.
Case 0D-C3. If the initial Δp0=Δ*p0, Δp increases towards 0.
Case 0D-C4. If the initial value Δp0=Δ*p0<Δ**p0, Δp decreases towards -∞ for β>0, tends to 0 for β<0, and remains constant for β=0.
In conclusion, for rN=0, the criteria to evaluate the evolution of Δp are the extraction rate α and the initial value Δp0. If α<rE, Δp increases towards +∞ if the initial Δp0 is Δp0>Δ*p0. If α=rE, Δp tends towards the asymptotic value pS0=Δp0-Δ*p0, for Δp0≥Δ*p0. If α>rE, Δp tends towards +0 if Δp0≥Δ*p0.
Category 1D (rN>α, that is, β=PIFE>0). The price evolution of the difference between selling and extracted resources, Δp, is investigated according to the initial value Δp0=(pS0-pE0) and the extraction rate α, as compared to rE.
Case 1D-A (α≤0<rN<rE, (i.e., β′>β>0, pS0*≥0)). Three relations between rE and rN can be investigated:
rE>2rN, that is, (Δ*p0)a<(Δ**p0)a<0;
rE=2rN, that is, (Δ*p0)a=(Δ**p0)a=0;
rE<2rN, that is, (Δ*p0)a>(Δ**p0)a>0.
If rE>2rN, that is, Δ*p0<Δ**p0<0, the following four initial conditions are investigated:
the initial price difference Δp0 is Δp0≥0 (or pS0≥pE0);
the initial price difference Δp0 is Δ**p0≤Δp0<0;
the initial price difference Δp0 is Δ**p0>Δp0>Δ*p0;
the initial price difference Δp0 is Δp0≤Δ*p0, or pS0≤pS0*;
and the price evolutions are presented in Figure 13.
Price difference evolution for Category 1D.
Case 1D-A1. If Δp0≥0 the difference Δp increases exponentially towards +∞ with a higher rate for α<0, an intermediate one for α=0, and a smaller one for 0<α.
Case 1D-A2. If 0>Δp0>Δ**p0 the difference Δp increases exponentially towards +∞ as in the previous case and the curves pass through the same point Δp=0 at t0. If Δp0=Δ**p0 the difference Δp increases exponentially with the minimum at t=0.
Case 1D-A3. If Δ**p0>Δp0>Δ*p0, Δp decreases initially down to a minimum and then increases exponentially as in the previous cases.
If rE=2rN, that is, Δ*p0=Δ**p0=0, the following cases are discussed.
Case 1D-A1. If Δp0>Δ*p0=0 the difference Δp increases exponentially towards +∞ with a higher rate for α<0, an intermediate one for α=0, and a smaller one for 0<α.
If rE<2rN, that is, Δ*p0>Δ**p0>0, the following cases are discussed.
Case 1D-A1. If Δp0≥Δ*p0>0 the difference Δp increases exponentially towards +∞ with a higher rate for α<0, an intermediate one for α=0, and a smaller one for 0<α.
Case 1D-B (Δp0=Δ*p0=0). Δp remains constant at Δ*p0=0 with time.
Case 1D-C1 (α=rE<rN, (i.e., β′=0, β>0, (pS0*)b<0<(pS0**)b=∞, and (Δ*p0)b<0<(Δ**p0)b=∞)). If Δp0 is -pE0<Δp0<Δ*p0, Δp decreases towards -∞. If Δp0 is Δ*p0>Δp0>Δ**p0, Δp increases up to a maximum and then decreases towards -∞. If Δp0≤Δ**p0, Δp decreases towards -∞.
Case 1D-C2 (rE<α<rN, (i.e., β′<0<β, (pS0*)c<0, (Δ**p0)c<(Δ*p0)c<0)). The difference Δp decreases towards -∞ for every initial Δp0.
In conclusion, for α<rN, (i.e., β>0) the criteria to evaluate the price difference evolution are the extraction rate α, the initial condition, and the relation between rE and rN. For β′>0 and rE>2rN, Δp increases towards +∞ if Δp0>Δ*p0. For β′>0 and rE=2rN, Δp increases towards +∞ if Δp0>Δ*p0. For β′>0 and rE<2rN, Δp increases towards +∞ if Δp0≥Δ*p0, while Δp increases up to a maximum and then decreases if Δ*p0>Δp0>Δ**p0. For β′≤0, Δp decreases towards -∞.
Category 2D (α>rN, that is, β=PIFE<0). The price evolution of the difference Δp is investigated for β<0 according to the initial value Δp0 and the relation between rN and rE.
The price evolutions are presented in Figure 14.
Price difference evolution for Category 2D.
Case 2D-A (rE>α>rN, (i.e., β′>0>β)). If 2rN<rE, that is, Δ**p0<Δ*p0<0, the following situations are discussed.
Case 2D-A1. If Δp0>Δ*p0 the difference Δp increases towards +∞.
If 2rN>rE, that is, Δ**p0>Δ*p0>0, the following situations are discussed.
Case 2D-A1. If Δp0≥Δ**p0 the difference Δp increases towards +∞. If Δp0 is Δ**p0>Δp0>Δ*p0>0 the difference Δp increases towards +∞ after a minimum at tmin.
If 2rN=rE, that is, Δ**p0=Δ*p0=0, the following situations are discussed.
Case 2D-A1. If Δp0>Δ*p0=0 the difference Δp increases towards +∞.
Case 2D-B (α=rE>rN, (i.e., β′=0>β, Δ**p0=∞, Δ*p0>0 for 2rN>rE, Δ*p0=0 for 2rN=rE, and Δ*p0<0 for 2rN<rE))
Case 2D-B1. If Δp0=Δ*p0=0, the difference Δp remains constant at Δp0=Δ*p0=0. If Δp0=Δ*p0>0 the difference Δp tends to +0.
Case 2D-B2. If Δp0>Δ*p0, the difference Δp decreases towards (Δp0-Δ*p0). For rE=2rN, Δp remains constant at (Δp0-Δ*p0).
Case 2D-C (α>rE>rN, (i.e., 0>β′>β, Δ*p0>0>Δ**p0 for 2rN>rE, Δ*p0<0<Δ**p0 for 2rN<rE, and Δ**p0=Δ*p0=0 for 2rN=rE))
Case 2D-C1. If Δp0 is Δ**p0<Δp0<Δ*p0<0 the difference Δp decreases towards -∞ after a maximum at tmax.
Case 2D-C2. If Δp0<Δ*p0=0, the difference Δp decreases towards -∞. If Δp0≤Δ**p0 the difference Δp decreases towards -∞. If Δp0<Δ*p0 the difference Δp decreases towards -∞.
For rE>2rN, Δp tends towards 0, if Δp0≥Δ**p0>0, Δp increases up to a maximum and then tends towards +0, if Δ**p0>Δp0>Δ*p0>0, and Δp tends towards +0, if Δp0≤Δ*p0<0. For rE=2rN, Δp decreases to +0 if Δp0>Δ**p0, remains constant at Δp0 if Δp0=Δ**p0, and increases towards −0 if Δp0<Δ**p0. For rE<2rN, Δp decreases towards 0 for every Δp0.
Case 2D-D (α>rN>rE, (i.e., β′<β, Δ*p0<Δp0**<0)). For every Δp0 the difference Δp decreases down to a minimum and then increases towards −0.
In conclusion, for α>rN, (i.e., β<0) the criteria to evaluate the evolution of Δp are the extraction rate α and the initial condition. For α<rE, Δp increases towards +∞ if Δp0>Δ*p0≤0, or Δp0>Δ**p0>Δ*p0>0 or Δ**p0>Δp0>Δ*p0, Δp decreases towards +0 if Δp0=Δ*p0> or <0. For α=rE, Δp tends towards (Δp0-Δ*p0). If α>rE, Δp tends towards +0 for every initial Δp0 except for rE>2rN and Δ**p0>Δp0>Δ*p0 when Δp tends towards 0 after a maximum.
Category 3D (α=rN, that is, β=PIFE=0). The price evolution of the difference Δp is investigated for β=0, (i.e., Δ*p0=Δ**p0) according to the initial value Δp0 and the relation between α=rN and rE.
The price evolutions are reported in Figure 15.
Price difference evolution for Category 3D.
Case 3D-1 (rE>rN=α, that is, β′>β=0)
Case 3D-1A. If Δp0>Δ*p0 the difference Δp increases exponentially with time towards +∞.
Case 3D-1B. If Δp0=Δ*p0 the difference Δp remains constant at the initial value Δp0.
Case 3D-1C. If Δp0<Δ*p0 the difference Δp decreases towards -∞.
Case 3D-2 (α=rN>rE, (i.e., β′<β=0)). The difference Δp decreases towards Δ*p0 for every initial Δp0.
In conclusion, for β=0, the criteria to evaluate the evolution of Δp are the relation between rN and rE and the initial condition Δp0. For rN<rE, Δp increases towards +∞ if Δp0>Δ*p0, Δp remains constant if Δp0=Δ*p0, and Δp decreases towards -∞ if Δp0<Δ*p0. For rN>rE, Δp tends towards the asymptotic value Δ*p0.
Category 4D (rN=rE, that is, β′=β). The price difference Δp is given by
(60)Δp=(pS-pE)=[(pS0-rEpE0t)-pE0]exp(β′t)=(Δp0-rEpE0t)exp(β′t)=(Δp0-pS0**β′t)exp(β′t).
The critical initial price difference Δ*p0 and the critical initial extreme price difference Δ**p0 are not defined and the only critical price defined is the critical initial price extreme of the sold resources pS0**, already defined in the second part of (38) or (45).
The price difference Δp has an extreme for the time tm given by
(61)tm=(Δp0-pS0**)β′·pS0**.
The extreme is a maximum if β′>0 and a minimum if β′<0.
The time of maximum tmax is zero if
(62)Δp0=pS0**=rEpE0β′,
and tmax is greater than zero if
(63)Δp0>pS0**,pS0**β′>0,Δp0<pS0**,pS0**β′<0.
The difference Δp is zero for the time t0 equal to
(64)t0=(pS0-pE0)(rEpE0)=Δp0(rEpE0)=Δp0(pS0**β′).
The price evolution of the difference Δp is investigated according to the initial value Δp0=(pS0-pE0) and the extraction rate α.
The price evolutions are reported in Figure 16.
Price difference evolution for Category 4D.
Case 4D-1 (rN=rE>α, (i.e., β′=β>0))
Case 4D-1A. The price cannot increase with time.
Case 4D-1B. If Δp0>pS0** the difference Δp initially increases up to the maximum time tmax and then decreases towards -∞.
Case 4D-1C. If Δp0=pS0** the difference Δp decreases towards -∞ with the maximum at t=0. If Δp0<pS0**, the difference Δp decreases towards -∞.
Case 4D-2 (α>rN, (i.e., β′=β<0, pS0**<0))
Case 4D-2A. The price cannot increase with time.
Case 4D-2B. If Δp0>pS0**, the difference Δp decreases with time, reaches a minimum at tmin, and then increases towards −0. If Δp0≤pS0** the difference Δp increases towards 0.
Case 4D-3 (α=rE=rN, (i.e., β′=β=0, pS0**=∞)). The difference Δp decreases linearly towards -∞ for every initial Δp0.
Discussion. The discussion on the price evolutions of sold resources, pS, and price difference, Δp=(pS-pE), in case of no-accumulation and no-depletion of the extracted resources, is carried out summarizing the results in the following Tables 1–8.
Price trends to t→∞ of selling resources, pS, extracted resources, pE, and difference Δp=(pS-pE) for rN=0, (i.e., pS0* = pS0** = 0, β′>β, Δ*p0 = -pE0, and Δ**p0=-rEpE0/β′).
Price trends to t→∞ of selling resources, pS, extracted resources, pE, and difference Δp=(pS-pE) for a rate of extraction 0<α=rN, (β=0, pS0*=pS0**, and Δ*p0=Δ**p0).
Extraction rate:α=rN,β=0β′=rE-α,β=rN-α
Initial condition
pS0 or Δp0
Selling resources,
pS (∞)
Extracted resources,
pE (∞)
Price difference
Δp (∞)
rN<rE, β′>β, (pSmin*=pS0*=pSmax*=pS0**>0) (Δ*pmin=Δ*p0=Δ**p0=Δ*pmax> or = or < 0)
Final conclusions of the price trends at t→∞ of selling resources, pS, for rN=0, pS0*=pS0**=0.
Relation between
β′ and β
β
pS0>pS0*=pS0**=0
A
β′>β
A.1
β′>β>0
+∞
A.2
β′>β=0
+∞
A.3
β′>0>β
+∞
A.4
β′=0>β
pS0
A.5
0>β′>β
+0
Final conclusions of the trends to t→∞ of the price difference Δp for rN=0, Δ*p0=-pE0<0, Δ**p0=-pE0(β′-β)/β′.
Relation between
β′ and β
β
Δp0>Δ*pmax
Δp0=Δ*pmax
Δ*pmin<Δp0<Δ*pmax
Δp0=Δ*pmin
A
β′>β
A.1
β′>β>0, (0>Δ*pmax=Δ**p0>Δ*p0=Δ*pmin)
+∞
+∞
+∞ (Min)
−∞
A.2
β′>β=0, (0>Δ*pmax=Δ**p0=Δ*p0=Δ*pmin)
+∞
Δp0
Δp0
A.3
β′>0>β, (0>Δ*pmax=Δ*p0>Δ**p0=Δ*pmin)
+∞
0
A.4
β′=0>β, (0>Δ*p0=Δ*pmax>Δ**p0=Δ*pmin=-∞)
pS0=Δp0-Δ*p0
pS0=Δp0-Δ*p0
A.5
0>β′>β, (Δ*pmax=Δ**p0>0>Δ*p0=Δ*pmin)
0
0
0 (Max)
0
Final conclusions of the trends to t→∞ of selling resources, pS, for rN≠0.
β
pS0>pSmax*
pS0=pSmax*
pS0**>pS0>pS0*
pS0*>pS0>pS0**
pS0=pSmin*
0<pS0<pSmin*
A
β′>β
A.1
β′>β>0,(pSmax*=pS0*>pS0**=pSmin*>0)
+∞
+∞
−∞ (Max)
−∞
−∞
A.2
β′>β=0,(pSmin*=pS0*=pS0**=pSmax*>0)
+∞
pS0
−∞
A.3
β′>0>β,(pSmax*=pS0**>pS0*=pSmin*>0)
+∞
+∞
+∞ (Min)
0
−∞
A.4
β′=0>β,(pS0**=pSmax*=∞>pSmin*=pS0*)
pS0-pS0*
pS0-pS0*=0
pS0-pS0*
A.5
0>β′>β,(pSmax*=pS0*>0>pS0**=pSmin*)
0 (Min)
B
β′=β
B.1
β>0,(pSmin*=pS0**>0)
−∞ (Max)
−∞ (Max)
−∞
B.2
β=0,(pSmin*=pS0**=∞)
−∞ (Lin)
B.3
β<0,(pSmin*=pS0**<0)
0 (Min)
C
β′<β
C.1
β>0,(pSmax*=pS0**>0>pS0*=pSmin*)
−∞ (Max)
−∞ (Max)
−∞
C.2
β=0,(pSmin*=pS0*=pS0**=pSmax*<0)
pS0*
C.3
β<0,(pSmin*=pS0**<0<pS0*=pSmax*)
0 (Min)
D
β′=0
D.1
β>0,(pSmin*=pS0*<0<pS0**=∞)
−∞
Final conclusions of the trends to t→∞ of price difference Δp for rN≠0.
β
Δp0>Δ*pmax
Δp0=Δ*pmax
Δ*pmin=Δ*p0<Δp0<Δ*pmax=Δ**p0
Δ*pmax=Δ*p0>Δp0>Δ*pmin=Δ**p0
Δp0=Δ*pmin
Δp0<Δ*pmin
A
β′>β
A.1
β′>β>0,
(0>Δ**p0=Δ*pmax>Δ*p0=Δ*pmin)
+∞
+∞
+∞ (Min)
−∞
−∞
β′>β>0,
(Δ*p0=Δ*pmax>Δ**p0=Δ*pmin>0)
+∞
+∞
−∞ (Max)
−∞
−∞
β′>β>0,
(Δ*pmax=Δ*pmin=Δ*p0=Δp0**=0)
+∞
Δ*p0
−∞
A.2
β′>β=0,
(Δ*pmax=Δ*pmin=Δ**p0=Δ*p0>or=or<0)
+∞
Δ*p0
−∞
A.3
β′>0>β,
(0<Δ*pmin=Δ*p0<Δp0**=Δ*pmax)
+∞
+∞
+∞ (Min)
0
−∞
β′>0>β,
(0>Δ*pmax=Δ*p0>Δ**p0=Δ*pmin)
+∞
0
−∞ (Max)
−∞
−∞
β′>0>β,
(Δ*pmax=Δ*p0=Δ**p0=Δ*pmin=0)
+∞
Δ*p0
−∞
A.4
0>β′>β,
(Δ*pmin=Δ*p0<0<Δ*pmax=Δ**p0)
0
0
0 (Max)
0
0
A.5
0>β′>β,
(Δ*pmin=Δ*p0>0>Δ*pmax=Δ**p0)
0
0
0
0
0
B
β′=β
B.1
β>0,
(Δpmin*=pS0**>0)
−∞ (Max)
−∞
−∞
B.2
β=0(Δ*pmin=pS0**=∞)
−∞ (Lin)
B.3
β<0(Δ*pmin=pS0**<0)
0
0
0
C
β′<β
C.1
β>0,
(Δ*p0=Δ*pmin<Δ**p0=Δ*pmax<0)
−∞
−∞
−∞
−∞
−∞
C.2
β=0,
(Δ*pmax=Δ*p0=Δ*pmin=Δ**p0<0)
Δ*p0
Δ*p0
Δ*p0
Δ*p0
C.3
β<0,
(Δ*p0=Δ*pmin<Δ**p0=Δ*pmax<0)
0 (Min)
0 (Min)
0 (Min)
0 (Min)
0 (Min)
D
β′=0
D.1
β>0,
(Δ*pmin=Δ*p0<0<Δ**p0=∞)
−∞
−∞
−∞
D.2
β=0(Δ*pmin=pS0**=∞)
−∞ (Lin)
D.3
β<0,
(Δ*p0=Δ*pmin>or=or<0,Δ**p0=∞)
Δp0-Δ*p0
Δp0-Δ*p0
Δp0-Δ*p0
Table 1 presents the price trends of pS, pE, and Δp, for nonrenewable resources without interest rate, rN = 0, that is, β′>β, pS0*=pS0**=0, Δ*p0=-pE0, and Δ**p0=-rEpE0/β′ with the following conclusions:
the price pE of extracted resources increases towards +∞ if α<0, (i.e., β>0), pE remains constant at pE0 if α=0, (i.e., β=0), and pE decreases towards +0 if β<0;
the price pS of sold resources increases towards +∞ if α≤0<rE, (i.e., β′>0), pS remains constant at pS0 if α=rE, (i.e., β′=0), and pS decreases towards +0 if α>rE, (i.e., β′<0);
if the extraction rate α<0, (i.e., β′>β>0), the trends of the price difference Δp depend on the initial value, Δp0. If Δp0≥Δ**p0=Δ*pmax, Δp increases towards +∞, if Δ*pmin=Δ*p0<Δp0<Δ**p0=Δ*pmax, Δp increases towards +∞ after a minimum, and if Δp0=Δ*p0=Δ*pmin, Δp decreases towards -∞;
if the extraction rate is α=0, (i.e., β′>β=0), the price difference Δp increases towards +∞ if Δp0>Δ**p0=Δ*p0=Δ*pmin=Δ*pmax=-pE0 while tending towards Δp0 if Δp0=Δ*pmin. It can be noticed that Δp0 cannot be smaller than Δ*p0 because pS0 cannot be smaller than 0;
if the extraction rate α<rE, (i.e., β′>0>β), the price difference Δp increases towards +∞ if Δp0>Δ*p0=Δ*pmax=-pE0 while tends towards 0 if Δ*p0=Δ*pmin;
if the extraction rate α=rE, (i.e., β′=0>β), the price difference Δp tends towards pS0=Δp0-Δ*p0.;
if the extraction rate α>rE, (i.e., 0>β′>β), the price difference Δp tends towards +0 for any Δp0 but with a maximum at tmax if Δ*pmax>Δp0>Δ*pmin.
Table 2 presents the price trends of pS, pE, and Δp, for α<rN, that is, β>0, with the following conclusions:
the price pE increases towards +∞, because β>0;
the price pS increases towards +∞ if rE>rN, (i.e., β′>β>0), and pS0≥pS0*=pSmax*;
the price pS initially increases, reaches a maximum at tmax, and then decreases towards -∞ in the following cases: rN<rE and pSmax*=pS0*>pS0>pS0**=pSmin*, rN=rE and pS0≥pS0**, and rN>rE and pS0≥pS0**=pSmax*;
the price, pS, decreases towards -∞, in all the other cases;
the price difference Δp has different trends for rE>rN according to the relation between 2rN and rE. If rE>2rN, the price difference Δp increases towards +∞ if Δp0≥Δ*pmax=Δ**p0, Δp increases towards +∞ after a minimum if Δ**p0>Δp0>Δ*p0, and Δp decreases towards -∞ if Δp0≤Δ*pmin. If rE=2rN, the price difference Δp increases towards +∞ if Δp0>Δ*pmax=Δ*p0=0, Δp remains constant at Δ*p0=0 if Δp0=Δ*p0=0, and Δp decreases to -∞ if Δp0<Δ*p0=0. If rE<2rN, the price difference Δp increases towards +∞ if Δp0≥Δ*pmax=Δ*p0, Δp decreases towards -∞, after a maximum, if Δ*p0>Δp0>Δ**p0, and Δp decreases to -∞ if Δp0≤Δ**p0=Δ*pmin;
the price difference Δp for rN=rE initially increases up to a maximum at tmax and then decreases towards -∞ if Δp0>pS0**;
the price difference Δp decreases towards -∞ in all other cases.
Table 3 presents the price trends of pS, pE, and Δp, for α=rN, that is, β=0, with the following conclusions:
the price pE is constant at pE0, because β=0;
the price pS increases towards +∞, for rN<rE, and pS0>pS0*=pSmax*=pS0**=pSmin*;
the price pS is constant at pS0* for rN<rE and pS0=pS0*=pSmin* or for rN>rE and every pS0>0;
the price pS decreases towards -∞ for rN<rE and 0<pS0<pS0*=pSmax* or decreases linearly towards -∞ for rN=rE and every pS0>0;
the price difference Δp increases towards +∞ for rN<rE, and Δp0>Δ*p0=Δ*pmax;
the price difference Δp is constant at Δ*p0, for rN<rE, and Δp0=Δ*p0 or for rN>rE and every Δp0;
the price difference Δp decreases towards -∞, for rN<rE, and Δp0<Δ*p0 or linearly to -∞ for α=rN=rE and every Δp0.
Table 4 presents the price trends of pS, pE, and Δp for α>rN>0, that is, β<0, with the following conclusions:
the price pE decreases towards +0, because β<0;
the price pS increases towards +∞ for rE>α>rN (i.e., β′>0>β) and pS0>pS0**=pSmax* or after a minimum for pS0**>pS0>pS0*;
the price pS tends to (pS0-pS0*) for rE=α>rN;
the price pS decreases towards -∞ for rE>α>rN and pS0<pS0*=pSmin*;
the price pS decreases towards 0 without or with a minimum, in the other two cases;
the price difference Δp increases towards +∞ for rE>α>rN and Δp0>Δ*pmax, if rE>2rN, if rE=2rN, if rE<2rN and Δp0=Δ*pmax. If rE<2rN a minimum is present if Δ**p0>Δp0>Δ*p0;
the price difference Δp tends to (Δp0-Δ*p0) if rE=α>rN for every Δp0;
the price difference Δp decreases towards -∞ for rE>α>rN and Δp0<Δ*pmin, or Δp0=Δ*pmin for rE>2rN while Δp passes through a maximum if Δ*p0>Δp0>Δ**p0 and for rE>2rN;
the price difference Δp remains constant at Δ*p0 for rE>α>rN, rE=2rN and Δp0=Δ*p0;
the price difference Δp tends to 0 in the other three cases.
Table 5 resumes the trends of pS for rN=0, that is, β′>β, pS0*=pS0*=0.
The price pS increases towards +∞ if β′>0, remains constant at pS0 if β′=0, and decreases to +0 if β′<0.
Table 6 resumes the trends of Δp for rN=0, that is, β′>β. The trends can be divided according to the initial conditions which are grouped into four categories.
The greater value between Δ*p0 and Δ**p0 is indicated as Δ*pmax while the smaller one as Δ*pmin. Consider the following:
for Δp0>Δ*pmax, Δp increases to +∞ if β′>0, Δp tends to pS0=Δp0-Δ*p0 if β′=0, and Δp tends to +0 if β′<0;
for Δp0=Δ*pmax, Δp increases to +∞ if β′>β>0, Δp remains constant at Δp0 if β′>β=0, Δp tends to 0 if β′>0>β, and β′<0, Δp tends to pS0=Δp0-Δ*p0 if β′=0;
for Δ*p0=Δ*pmin<Δp0<Δ*pmax=Δ**p0, Δp increases to +∞ after a minimum if β′>0, and Δp tends to +0 after a maximum if β′<0;
for Δ*p0=Δ*pmin>Δp0, Δp decreases to -∞ if β′>β>0, Δp remains constant at Δp0 if β′>β=0, and Δp tends to 0 if β′<0.
Table 7 resumes the trends of pS for rN different from 0. The trends can be divided into four classes, A-B-C-D, according to the relation between β′ and β, and β′=0. In each class the trends are depending on the value of β or β′, and the initial value pS0. The greater between pS0* and pS0** is indicated as pSmax* and the smaller one as pSmin*.
In class A, β′>β, the trends of pS are to increase towards +∞ for β′>β>0 if pS0≥pSmax*=pS0*, for β′>β=0 if pS0>pS0*=pSmax*, and for β′>0>β if pS0≥pSmax* and if pS0*>pS0>pS0** after a minimum. A temporary increase, followed by a decrease towards -∞, is present if pS0*=pSmax*>pS0>pSmin*=pS0** and β>0. In all the other cases pS tends towards asymptotes equal to pS0, 0, pS0-pS0*, or decreases towards -∞.
In class B, (β′=β), for β>0 a temporary increase, followed by a decrease towards -∞, is present if pS0≥pS0**. In all the other cases pS tends towards an asymptote, equal to +0, or decreases towards -∞.
In class C, (β′<β), for β>0 a temporary increase, followed by a decrease towards -∞, is present if pS0≥pSmax*=pS0**. In all the other cases pS tends towards asymptotes equal to pS0*, 0, or decreases towards -∞.
In class D, (β′=0), pS decreases towards -∞.
Table 8 resumes the trends of Δp for rN different from 0. The trends can be divided into four classes, A-B-C-D, according to the relation between β′ and β, and β′=0. In each class the trends depend on the value of β or β′, and the initial value Δp0. The greater between Δ*p0 and Δ**p0 is indicated as Δ*pmax and the smaller one as Δ*pmin.
In class A, (β′>β), the trends of Δp are to increase towards +∞ in the following cases:
for β′>0 and Δp0>Δ*pmax;
for β′>β>0 and Δp0=Δ*pmax≠0;
for β′>0>β and Δp0=Δ*pmax>0;
for β′>β>0, β′>0>β and Δ**p0>Δp0>Δ*p0 after a minimum.
A temporary increase, followed by a decrease towards -∞, is present for β′>β>0 and β′>0>β if Δ*p0=Δ*pmax>Δp0>Δ*pmin=Δ**p0. In all the other cases Δp decreases towards asymptotes equal to Δ*p0, 0, or decreases towards -∞.
In class B, (β′=β), for β>0 a temporary increase, followed by a decrease towards -∞, is present if Δp0>pS0**. In all the other cases Δp decreases towards the asymptote, equal to 0, or decreases towards -∞.
In class C, (β′<β), Δp decreases towards asymptotes, equal to Δ*p0, +0, or decreases towards -∞.
In class D, (β′=0), Δp decreases towards the asymptotes, equal to (Δp0-Δ*p0), or decreases towards -∞.
7. Energy Supply Curve7.1. Supply Curve of Extracted Resources
The mass conservation equations of extracted and sold resources, under the hypothesis of no accumulation nor depletion of the resources, can be written in dimensionless form as
(65)GE′=GEGE0=GS′=GSGS0=exp(αt).
The price evolution with time of extracted resources is, in dimensionless form,
(66)pE′=pEpE0=exp(βt).
The extraction rate, α, can be obtained from (65) on the base of the mass flow rates of extraction in two successive years,
(67)α=lnGS′t.
The elimination of the time variable, t, between (65) and (66), allows obtaining the relation, called supply curve of extracted resources, between the dimensionless price, pE′, and the dimensionless mass flow rate, GE′, of extracted resources,
(68)pE′=GE′(β/α)=GE′(y-1),
where the new variable, y,
(69)y=rNα,
is called rate of interest of nonextracted resources on the extraction rate, RINE.
The variation of pE′ with GE′ is presented in Figure 17 where the only variable affecting the evolution is y. Consider the following:
for y=0 (i.e., rN=0 or α=±∞), (68) is an equilateral hyperbole: pE′ decreases with the increase of GE′ or increases with the decrease of GE′;
for 0<y<1, pE′ decreases with the increase of GE′;
for y=1 (i.e., α=rN), pE′ is constant with the increase of GE′;
for y>1, pE′ increases with the increase of GE′;
for y=±∞ (or α=0), (68) is a vertical line: pE′ increases at constant GE′=1;
for y<0, pE′ increases with the decrease of GE′.
Dimensionless price of extracted resources, pE′, versus GE′.
7.2. Supply Curve of Sold Resources
The price evolution with time of sold resources becomes, in dimensionless form,
(70)pS′=pSpS0=exp(β′t)-pS0′*[exp(β′t)-exp(βt)],
where
(71)pS0′*=pS0*pS0=rNpE0pS0·(β′-β)=pE0·ypS0(x-y)
is the dimensionless critical initial price of sold resources, DCIPS, and the new variable, x,
(72)x=rEα,
is the rate of interest of sold resources on the extraction rate, RISE.
For x≠y (i.e., rN≠rE), (70), combined with (65), becomes
(73)pS′=pSpS0=GS′(β′/α)-pS0′*(GS′(β′/α)-GS′(β/α))=GS′(x-1)-pS0′*(GS′(x-1)-GS′(y-1))=GS′(x-1)(1-pS0′*)+pS0′*GS′(y-1).
An extreme value of pS′ is present for
(74)GS′={[y(1-y)][(x-1)(x-2y)]}1/(x-y),
when pE0=pS0, and
(75)GS′={[pS0′*(1-y)][(x-1)(1-pS0′*)]}1/(x-y),
when pE0≠pS0.
The maximum of pS′ is present at GS′=1 for pS0′(76)pS0′=pS0′**=pS0**pS0=pS0′*(β′-β)β′=rNpE0pS0·β′=pE0·ypS0(x-1),
which is the dimensionless critical initial price extreme of sold resources, DCIPES.
7.2.1. Supply Curve of Sold Resources for rN=0, That Is, pS0′*=0
For rN=0, that is, pS0′*=0, (73) becomes
(77)pS′=pSpS0=exp(β′t),
which, combined with (65), gives
(78)pS′=GS′(β′/α)=GS′(x-1).
The variation of pS′ with GS′ is presented in Figure 18 for y=0 (i.e., rN=0).
Dimensionless price of selling resources, pS′, versus GS′, for rN=0.
The only variable affecting the evolution is x. Consider the following:
for x=0 (i.e., rE=0 or α=±∞), (78) is an equilateral hyperbole: pS′ decreases with the increase of GS′ or increases with the decrease of GS′;
for 0<x<1, pS′ decreases with the increase of GS′;
for x=1 (i.e., α=rE), pS′ is constant with the increase of GS′;
for x>1, pS′ increases with the increase of GS′;
for x=±∞ (or α=0), (78) is a vertical line: pS′ increases at constant GS′=1;
for x<0, pS′ increases with the decrease of GS′.
7.2.2. Supply Curve of Selling Resources for pS0′*=1
For pS0′*=1, (73) reduces to
(79)pS′=pSpS0=GS′(y-1),
and the variation of pS′ is only dependent on y, RINE.
The variation of pS′ with GS′ for pS0′*=1 (i.e., x=2y if pE0=pS0=1) is presented in Figure 19. The only variable involved in the evolution is y. Consider the following:
for y=0 (i.e., rE=0 or α=±∞), (79) is an equilateral hyperbole: pS′ decreases with the increase of GS′ or increases with the decrease of GS′;
for 0<y<1, pS′ decreases with the increase of GS′;
for y=1, pS′ is constant with the increase of GS′;
for y>1, pS′ increases with the increase of GS′. A special case is y=2 when the variation of pS′ is a linear increase with the increase of GS′;
for y=±∞ (or α=0), (79) is a vertical line: pS′ increases at constant GS′=1;
for y<0, pS′ increases with the decrease of GS′.
Dimensionless price of selling resources, pS′ versus GS′, for pS0′*=1, (i.e., x=2y if pS0=pE0=1).
7.2.3. Supply Curve of Sold Resources for |x|>|y|>1
The variation of pS′ with GS′ is presented in Figure 20 for |x|>|y|>1 (i.e., rE>rN>α).
Dimensionless price of selling resources, pS′, versus GS′, for |x|>|y|>1.
The variables involved in the evolution are here two, that is, pS0′* and pS0′**. The main role is played by pS0′* when pS0′*≤1, but, when pS0′*>1, it is determinant the second variable pS0′**. Consider the following:
for pS0′*≤1, pS′ increases with the increase or the decrease of GS′;
for pS0′*>1>pS0′**, pS′ increases up to a maximum and then decreases with the increase or the decrease of GS′;
for pS0′**=1, pS′ decreases with the increase or the decrease of GS′ and the maximum is at GS′=1;
for pS0′*>pS0′**>1, pS′ decreases with the increase or the decrease of GS′.
7.2.4. Supply Curve of Sold Resources for |y|>|x|>1 or y>1>x
The variation of pS′ with GS′ is presented in Figure 21 for |y|>|x|>1 (i.e., rN>rE>α and pS0′*<0<pS0′**).
Dimensionless price of sold resources, pS′, GS′, for |y|>|x|>1 or y>1>x.
The only variable influencing the evolution is pS0′** because pS0′*<0. Consider the following:
for pS0′**<1, pS′ increases up to a maximum and then decreases with the decrease or the increase of GS′;
for pS0′**=1, pS′ decreases with the decrease or the increase of GS′ and the maximum is at GS′=1;
for pS0′**>1, pS′ decreases with the increase or the decrease of GS′.
The variation of pS′ with GS′ is presented in Figure 21 also for y>1>x (i.e., rN>α>rE and 0>pS0′*>pS0′**). The price pS′ decreases with the increase of GS′.
The conclusion of Figure 21 is that pS′ cannot increase indefinitely for |y|>|x|>1 or y>1>x.
7.2.5. Supply Curve of Sold Resources for y=1 or x=1
The variation of pS′ with GS′ is presented in Figure 22 for y=1 (i.e., α=rN>0 and pS0′*=pS0′**).
Dimensionless price of sold resources, pS′, versus GS′, for y=1 or x=1.
The main variable influencing the evolution is pS0′**=pS0′*. Consider the following:
for x>y=1 (i.e., α=rN<rE and pS0′**=pS0′*>0), pS′ increases with the increase of GS′ if pS0′*<1; pS′ remains constant with GS′ if pS0′*=1; pS′ decreases with GS′ if pS0′*>1;
for x<y=1 (i.e., α=rN>rE and pS0′**=pS0′*<0), pS′ decreases asymptotically to pS0′* with the increase of GS′.
The evolution of pS′ with GS′ is presented in Figure 22 also for x=1<y (i.e., rN>rE=α and pS0′*<0, pS0′**=+∞). The price pS′ decreases with the increase of GS′.
7.2.6. Supply Curve of Sold Resources for y<1
The variation of pS′ with GS′ is presented in Figure 23 for y<1 (i.e., α>rN).
Dimensionless price of sold resources, pS′, versus GS′, for y<1.
The main variables influencing the evolution are pS0′* and pS0′**. Consider the following:
for x>1>y (i.e., rE>α>rN and pS0′**>pS0′*>0) five cases are possible: pS′ increases with GS′ if pS0′**≤1; pS′ increases with GS′, after a minimum, if pS0′**>1>pS0′*; pS′ decreases asymptotically to +0 with GS′ if pS0′**>pS0′*=1; pS′ decreases with GS′ if pS0′**>pS0′*>1;
for x=1>y (i.e., rE=α>rN and pS0′**=+∞, pS0′*>0) pS′ decreases asymptotically to (1-pS0′*) with the increase of GS′;
for 1>x>y (i.e., α>rE>rN and pS0′*>0>pS0′**) two cases are possible: pS′ decreases asymptotically to +0 with GS′ if pS0′*≤1; pS′ decreases asymptotically to −0 with GS′, after a minimum, if pS0′*>1;
for 1>y>x (i.e., α>rN>rE and 0>pS0′**>pS0′*) pS′ decreases asymptotically to −0 with GS′ after a minimum.
7.2.7. Supply Curve of Sold Resources for x=y (i.e., rN=rE)
The price evolution of sold resources, for x=y (i.e., rN=rE), can be written in dimensionless form as
(80)pS′=pSpS0=(1-pS0′**βt)exp(βt).
Using (65), (80) becomes finally
(81)pS′=pSpS0=GS′(β/α)[1-pS0′**β(ln(GS′))α]=GS′(y-1)[1-pS0′**(y-1)(ln(GS′))].
The maximum value of pS′ corresponds to
(82)GS′=exp[(x-y-1)y·(y-1)],
for pE0=pS0, and to
(83)GS′=exp[(1/pS0′*-1)(y-1)],
for pE0≠pS0.
The maximum of pS′ is present at GS′=1 for
(84)pS0′=pS0′**=pS0**pS0=pS0′*(β′-β)β′=rNpE0pS0·β′=pE0·ypS0(x-1),
which is the dimensionless critical initial price extreme of sold resources, DCIPES.
The price evolution of sold resources for x=y=1 (i.e., rN=rE=α) becomes, in dimensionless form,
(85)pS′=pSpS0=ln(GS′)1-pE0pS0.
The evolution of pS′ with GS′ is presented in Figure 24 for x=y (i.e., rN=rE). Consider the following:
for |x|=|y|>1 (i.e., rE=rN>α and pS0′**>0, pS0′*=+∞) the only variable affecting the evolution is pS0′**>0. The price pS′ decreases with the increase or the decrease of GS′ for pS0′**≥1. The price pS′ increases up to a maximum and then decreases with the decrease or the decrease of GS′ for pS0′**<1;
for x=y<1 (i.e., rE=rN<α and pS0′**<0, pS0′*=+∞) the price pS′ decreases asymptotically to −0 with the increase of GS′ after a negative minimum;
for x=y=1 (i.e., rE=rN=α and pS0′*=pS0′**=+∞) the price pS′ decreases with the increase of GS′.
Dimensionless price of sold resources, pS′, versus GS′, for x=y.
8. Supply Curve of the Difference between Selling and Extracted Resources
The difference Δp between the price of selling and extracted resources is given by
(86)Δp=pS-pE=(Δp0-Δ*p0)GS′(x-1)+Δ*p0GS′(y-1),
where
(87)Δ*p0=pS0*-pE0=pE0(2rN-rE)(rE-rN),
is the critical initial price difference, CIPD. The price difference Δp has an extreme for
(88)GS′={[ΔpS0′*(1-y)][(x-1)(1-ΔpS0′*)]}1/(x-y),
which is equal to 1 for Δp0 equal to
(89)Δ**p0=Δ*p0(x-y)(x-1)=Δ*p0(β′-β)β′=pE0(2·rN-rE)β′,
which is the critical extreme of the initial price difference, CEIPD. The relative evolutions of Δp with GS′ are presented in Figures 25–28.
Price difference versus GS′, for y=0 and |x|>1.
Price difference versus GS′, for |x|>2|y|≥1, or y>x=1, or y>1>x.
Price difference versus GS′, for |y|>1 and |x|<2|y|.
Price difference versus GS′, for x=y.
8.1. Supply Curve of Δp for y=0 and |x|>1
Figure 25 presents the price difference Δp for y=0 (i.e., rN=0) and |x|>1 (i.e., α<rE, Δ*p0=-pE0<0 and Δ*p0<Δ**p0<0). Consider the following:
for Δp0>0 the price difference Δp increases with the increase or the decrease of GS′, that is, for GS′<1 or GS′>1. For GS′<1 the difference Δp increases exponentially while the difference Δp increases at a lower rate for GS′>1;
for Δp0 in the range from 0 to Δ**p0, that is, Δ**p0<Δp0<0, the price difference Δp increases with the increase or the decrease of GS′. The increase is exponential for GS′<1 while having a lower rate of increase for GS′>1;
for Δp0=Δ**p0, the price difference Δp increases with the increase or the decrease of GS′. For GS′<1 the difference Δp increases exponentially with the minimum at GS′=1 while the increase has a lower rate for GS′>1;
for Δp0 in the range from Δ*p0 to Δ**p0, that is, Δ*p0<Δp0<Δ**p0, the price difference Δp increases exponentially, after a minimum, with the decrease of GS′. For GS′>1 the difference Δp increases with a lower rate;
for Δp0=Δ*p0, the price difference Δp increases asymptotically towards −0 for GS′>1, while Δp decreases exponentially for GS′<1.
8.2. Supply Curve of Δp for y=0 and 0<x≤1
The price difference Δp for y=0 (i.e., rN=0, Δ*p0=-pE0<0) and 0<x≤1 (i.e., α≥rE>0) has the following trends:
for x=1 (i.e., α=rE and Δ**p0=-∞), the price difference Δp increases asymptotically towards pS0 for any value of Δp0 (i.e., Δp0>0, 0>Δp0>Δ*p0 or Δp0=Δ*p0);
for x<1 (i.e., α>rE>0 and Δ**p0>0) the price difference Δp tends to 0 for any value of Δp0.
8.3. Supply Curve of Δp for |x|>2|y|≥1, or y>x=1, or y>1>x
Figure 26 presents the price difference Δp for: |x|>2|y|≥1 (i.e., rE>2rN≥α and Δ*p0<Δ**p0<0); y>x=1 (i.e., rN>α=rE and Δ*p0<0); y>1>x (i.e., rN>α>rE and Δ**p0<Δ*p0<0. Consider the following:
for |x|>2|y|≥1 and Δp0>0 or Δp0>Δ**p0 the price difference Δp increases exponentially with the increase or the decrease of GS′. For Δp0=Δ**p0 the price differenceΔp increases exponentially with a minimum at GS′=1. For Δ**p0>Δp0>Δ*p0 the price difference Δp increases exponentially, after a minimum, with the decrease or the increase of GS′. For Δp0=Δ*p0 the price difference Δp decreases exponentially with the decrease or the increase of GS′.
Figure 26 presents also the price difference Δp for y>x=1 (i.e., rN>α=rE and Δ*p0<0) and for y>1>x (i.e., rN>α>rE and Δ**p0<Δ*p0<0). The price difference Δp decreases exponentially with GS′.
8.4. Supply Curve of Δp for |y|>1 and |x|=2y
The price difference Δp for |y|>1 and |x|=2|y| (i.e., α<rN, rE=2rN, and Δ*p0=Δ**p0=0) has the following trends. For Δp0>0 the price difference Δp increases exponentially with the increase or the decrease of GS′. For Δp0<Δ*p0 the price difference Δp decreases exponentially with the increase or the decrease of GS′.
8.5. Supply Curve of Δp for |y|>1 and |x|<2y
Figure 27 presents the price difference Δp for |y|>1 and |x|<2|y| (i.e., α<rN, rE<2rN and Δ*p0>Δ**p0>0).
For Δp0≥Δ*p0 the price difference Δp increases exponentially with the increase or the decrease of GS′. For Δ*p0>Δp0>Δ**p0 the price difference Δp increases up to a maximum and then decreases exponentially with the increase or the decrease of GS′. For Δp0=Δ**p0 the price difference Δp decreases exponentially with the increase or the decrease of GS′ and the maximum is at GS′=1. For Δp0<Δ**p0 the price difference Δp decreases exponentially with the increase or the decrease of GS′.
8.6. Supply Curve of Δp for y=1
The price difference Δp for y=1 (i.e., α=rN) has the following trends:
for x>y=1 (i.e., 0<α=rN<rE and Δ*p0=Δ**p0), three cases are possible. For Δp0>Δ*p0 the price difference Δp increases with the increase of GS′; for Δp0=Δ*p0 the price difference Δp remains constant with the increase of GS′; for Δp0<Δ*p0 the price difference Δp decreases with the increase of GS′.
for x<y=1 (i.e., α=rN>rE and Δ*p0=Δ**p0), the price difference Δp decreases with the increase of GS′.
8.7. Supply Curve of Δp for y<1
The price difference Δp for y<1 (i.e., α>rN) and Δ**p0>Δ*p0>0 has the following trends. For Δp0>Δ**p0 the price difference Δp increases with the increase of GS′. For Δp0=Δ**p0 the price difference Δp increases with the increase of GS′ and the minimum is at GS′=1. For Δ**p0>Δp0>Δ*p0 the price difference Δp increases with the increase of GS′ after a minimum. For Δp0=Δ*p0 the price difference Δp decreases asymptotically to +0 with the increase of GS′. For Δp0<Δ*p0 the price difference Δp decreases with the increase of GS′.
The price difference Δp for y<1 (i.e., α>rN>0) and Δ**p0<Δ*p0<0 has the following trends. For Δp0>0>Δ*p0 the price difference Δp increases with the increase of GS′. For Δp0=Δ*p0 the price difference Δp increases asymptotically to −0 with the increase of GS′. For Δ*p0>Δp0>Δ**p0 the price difference Δp increases up to a maximum and then decreases with the increase of GS′. For Δp0≤Δ**p0 the price difference Δp decreases with the increase of GS′.
The price difference Δp for the other three cases which are possible according to the relation between x and y (i.e., α and rE) has the following trends:
for 1≥x>y (i.e., α≥rE>rN) the price difference Δp decreases asymptotically to +0 with the increase of GS′;
for 1>y>x (i.e., α>rN>rE) the price difference Δp decreases asymptotically to −0 after a negative minimum with the increase of GS′.
8.8. Supply Curve of Δp for x=y
The price difference, Δp, is, in dimensionless form,
(90)Δp=pS-pE=(Δp0-pS0**β′t)exp(β′t).
For x=y the critical initial price difference Δ*p0 and the critical initial extreme price difference Δ**p0 are not defined and the only critical price defined is the critical initial price extreme of the sold resources pS0**.
The price difference Δp, as a function of GS′, is then
(91)Δp=[Δp0-pS0**(y-1)ln(GS′)]GS′(y-1).
The price difference Δp has an extreme for
(92)GS′=exp[(Δp0-pS0**)pS0**·(1-y)],
which is equal to 1 if Δp0 is equal to
(93)Δp0=p0**.
Figure 28 presents the price difference Δp for x=y (i.e., rN=rE). Consider the following:
for x<0 (i.e., α<0<rE and p0**>0), three cases are possible with the decrease of GS′. For Δp0>p0** the price difference Δp decreases after a maximum; for Δp0=p0** the price difference Δp decreases with the maximum at GS′=1; for Δp0<p0** the price difference Δp decreases exponentially;
for x<1 (i.e., α>rE and p0**<0), three cases are possible with the increase of GS′. For Δp0>0>p0** the price difference Δp decreases asymptotically to −0 after a negative minimum; for 0>Δp0> or = or <p0** the price difference Δp increases asymptotically to −0;
for x=y=1 (i.e., α=rE=rN) the price difference Δp, which has the following dependence on GS′,
(94)Δp=Δp0-ypE0ln(GS′),
decreases with the increase of GS′.
NomenclatureLatinGk=Gpr:
Capital flow rate
G:
Mass flow rate
K=Gp:
Capital flow
M:
Mass
p:
Price
pS0*:
Critical initial price of selling resources, CIPS
pS0**:
Critical initial extreme price of selling resources, CIPES
pS0′*:
Dimensionless critical initial price of sold resources, DCIPS
pS0′**:
Dimensionless critical initial price extreme of sold resources, DCIPES
r:
Interest rate (annum−1)
t:
Time
x=rE/α:
Rate of interest of sold resources on extracted rate, RISE
y=rN/α:
Rate of interest of nonextracted resources on extracted rate, RINE.
Greekα=(1/G)(dG/dt):
Extraction rate
β=rN-α:
Price-increase factor of extracted resources, PIFE
β′=rE-α:
Price-increase factor of selling resources, PIFS
Δp=pS-pE:
Difference between price of sold and extracted resources
Δ*p0:
Critical initial price difference, CIPD
Δ**p0:
Critical extreme initial price difference, CEIPD.
Subscript0:
Initial
c:
Critical
E:
Extracted resources
f:
Final
k:
Capital
m:
Maximum or minimum
max:
Maximum
min:
Minimum
N:
Nonextracted resources
S:
Selling resources.
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.
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