Estimates of the In-situ Price of Non-Renewable Natural Resources

Transcription

1 Estimates of the In-situ Price of Non-Renewable Natural Resources Calvin Atewamba; University of Montreal, CIREQ March 16, 2010 Abstract The purpose of this paper is to estimate the in-situ price of nonrenewable natural resources using average extraction cost data as a proxy for marginal cost. Using the regime switching Generalized Method of Moment (GMM) estimation technique, the reduced form of the Hotelling rule is estimated with panel data of fourteen ores. I find five main results that strongly support the model. First, it appears that the Hotelling model has a good explanatory power of the observed market prices. Second, although the fitted prices seem to be time inconsistent, the time inconsistency does not have a significant impact on the explanatory power of the model. Third, marginal extraction cost can be approximated by the average extraction cost data. Fourth, the time consistent estimate of the in-situ price decreases or exhibits a U-shape form over time and appears to be positively correlated with the market price. Fifth, for 9 of the 14 minerals, the difference between the time consistent and the time inconsistent estimates of the in-situ price is a white noise process. Keywords: Non-renewable natural resource, Hotelling rule, in-situ and flow prices, panel data and Switching GMM analysis. 1 Introduction The purpose of this paper is to estimate the Hotelling rule, the fundamental principle of nonrenewable resource economics. The basic Hotelling rule states that the in-situ price, which is equal to the market price less the marginal extraction cost, should grow at the rate of interest, 1

2 (Hotelling, 1931). This rule has been unsuccessful tested (Farrow, 1985; Halvorsen and Smith, 1991; Young, 1992). Gaudet (2007) supported the idea that, in its complete form, the Hotelling rule is too complex to be tested, but this does not remove its value as an indicator to evaluate nonrenewable natural resources scarcity. The principal problem concerning the Hotelling rule, even it is too complex to be tested, is to determine whether or not it can be estimated or used to explain the evolution of non-renewable resource prices. Many authors have already addressed this issue with unsatisfactory results (Livernois, 2009). However, one point has to be underlined. The common approach used by those works is to make adequate assumptions on the behaviour of the marginal extraction cost and use the implication of the Hotelling rule to explain the evolution of the nonrenewable natural resource market price as time series (Slade, 1982; Berck and Roberts, 1995; Lee et al., 2006). Clearly, the implications of the Hotelling rule can be captured thought the time series s properties of the market price, but the history of the price cannot contain enough information to explain the evolution of the market price which is still consistent with the Hotelling rule. There is some information we lost by not including the in-situ price and the marginal extraction cost in our prediction of the market price. Sure, this information is not available because neither rent nor marginal extraction cost is observable. Therefore, to pretend to predict the evolution of the market price of a nonrenewable resource using Hotelling rule, one need first to obtain a suitable proxy of these two important components. In literature, Young (1992) had already tried to incorporate information of these components in the prediction of the market price. This author obtained an adequate proxy of the marginal extraction cost by specifying and testing a suitable cost function. However, the price estimated explains at most 1 percent of the observed market price. In fundamental microeconomics analysis, the marginal cost and the average extraction are related derivatives of the cost function. Even the marginal cost is not observable, the average cost is observable as a total cost divided by the production. The aim of this paper is to build a framework of resource extraction to estimate the Hotelling rule and predict the behaviour of nonrenewable resource prices using average extraction cost as a proxy for marginal cost. My methodology is straightforward: firstly, I assume a functional form for extraction cost in order to use average extraction as a proxy for marginal extraction cost; secondly, I use the theoretical behaviour of the in-situ price to estimate the market price which is consistent with the Hotelling rule, and finally, I derive the in-situ price ex-post. I find that the Hotelling model is explaining at least 88 percent of the observed market price. The re- 2

3 mainder of the paper is organized as follows. In section 2, I present the theoretical model of the resource extraction. In particular, I assume an increasing average extraction cost and focus on its impacts on the equilibrium price path. An empirical analysis is achieved in section 3, where I evaluate the theoretical equilibrium price path and propose a methodology to estimate the Hotelling rule. This empirical methodology is then used in section 4 to analyse fourteen natural resource market prices. In section 5, I conclude and make some remarks for future research. 2 Theoretical model of the resource extraction In this section, I present a theoretical model of the optimal nonrenewable resource extraction in a competitive market. 2.1 Basic Hotelling model of the resource extraction I consider a competitive firm extracting a known and a finite stock of a non-renewable resource. The manager chooses a time path of the resource extraction to maximize the present value of the stream of the net benefice subject to the constraint that the cumulative extraction is not greater than the initial resource endowment. The extraction cost, the principal characteristic of the firm, is determined by the technology progress, which causes the cost to decrease, and by the stock effect, which causes the cost to increase. My notation follows closely that used by Lin and Wagner (2007) and Krautkraemer (1998). At time t [0, + ], the supply of the mineral is given by q(t), the extraction flow in tons per unit at time t. The cost of extracting q(t) ton at time t is denoted by. C(z(t), q(t), S(t)), where S(t) is the remaining stock of the resource at time t and z(t) is the extraction technology. I assume that C z < 0, C s < 0, C ss > 0 C q > 0 and C qq > 0. Where C q denotes the partial derivative C/ q, etc. To be as simply as possible, I assume that the average extraction cost is an increasing function of the extraction rate, that is. C q (z, q, S) C(z, q, S), z, q, S (1) q 3

4 This assumption guarantees that each unit of the stock in ground is economically profitable for a firm. Therefore, it can be optimal to extract the entire resource stock. In making its extraction decision, the manager takes as given the market price p(t) of the minerals. The optimal control formulation of the problem is to choose a time path of the resource extraction q(t) to maximize: + 0 e δt [p(t)q(t) C(z(t), q(t), S(t))]dt (2) Subject to Ṡ(t) = q(t), t (3) q(t) 0, S(t) 0 t (4) S(0) = S 0 (5) where δ denotes the rate of discount. Let λ(t) denote the co-state variable for the resource stock, the current value Hamiltonian for this problem is: H(t, q(t), S(t), λ(t)) = p(t)q(t) C(z(t), q(t), S(t)) λ(t)q(t), (6) The Static efficiency condition is: H q = p(t) Cq(z, q, S) λ = 0 (7) where the time argument is implicit and where C q (z, q, S) denotes C/ q, etc. Let λ denote the time derivative of λ, the dynamic efficiency condition is given by: λ = δλ H S = δλ + Cs(z, q, S), (8) Eqs.(7) and (8) are known as the Hotelling rule. They state that, the in-situ price λ corrected by the stock effect C S should grow at a constant rate. With this form, the behaviour of the resource price cannot be estimated because the marginal cost C q and the stock effect C s are not 4

5 observable. However, the implications for the resource price path can be derived under specific assumptions about the cost function. When marginal extraction cost, C q, is a constant or a decreasing function of time due to technology progress, an increasing function of extraction rate and a decreasing function of resource stock, one can find a U-shape price path,(slade, 1982). This behaviour of the market price has been testing unsuccessful, Berck and Roberts (1995). In the next section, I propose an approach to obtain a useful reduced form of the Hotelling rule which accounts for the average extraction cost as a proxy for marginal extraction cost. 2.2 Cost specification and the reduced form of the Hotelling rule To make Hotelling rule empirically tractable, I need to assume a functional form for the extraction cost. Let assume the following extraction cost function, which is closed to the one used in Lin and Wagner (2007). C(z, q, S) = z 1 q α S b ; ż = γz (9) with a nonnegative technology growth rate γ 0 and a nonnegative stock elasticity b 0. Technology progress causes cost to decrease and others thing been equal a one percent increase in technology causes costs to decrease by γ percent. Costs rise as more of the resource has been extracted and others thing been equal a one percent decrease in resource stock causes costs to increase by b percent. With this functional form, marginal cost can be expressed as a function of the average extraction cost. C(z, q, S) C q (z, q, S) = α q (10) The parameter α appears as an adjusted coefficient between the average extraction cost and the marginal extraction. To guarantee that the supply curve will be non decreasing, the marginal extraction cost elasticity given by. ε(z, q, S) = qc qq(z, q, S) C q (z, q, S) = α 1 (11) 5

6 must be nonnegative, that is α 1. If α = 1, the average extraction is equal to the marginal extraction cost. If α > 1, I obtain an increasing average extraction cost. The stock effect C s is not observable, but, with the cost function (9), it can be expressed as a function of observable quantities. Over time, resource stock available decreases respect to the extraction process. This exerts upward pressure on costs to increase. Denoting by X the cumulative resource extracted, the stock effect can be written on the form. q C s (z, q, S) = b S 0 X C(z, q, S) q (12) The function f(q, X) = b q S 0 X is an adjusted factor between stock effect and the average extraction cost. Note that, without the stock effect the parameter b is zero. Thus, the adjusted factor f(q, X) is zero too. Substituting eqs.(10) and (12) into static and dynamic efficiency conditions, I obtain the following reduced form of the Hotelling rule. λ(t) = p(t) αac(t) (13) λ(t) = q(t) δλ(t) b AC(t) S 0 X(t) (14) As the observed data is in discontinuous time, it will be suitable to discretize the above equations. The discrete form associated to the equation (13) is given by q t+1 λ t+1 λ t = δλ t b AC t+1 (15) S 0 X t+1 Substituting eq.(13) into eq.(15) and rearrange, I obtain the following expression of the market price. p t = (1 + δ)p t 1 + αac t α(1 + δ)ac t 1 b AC t (16) S 0 X t With this form, the Hotelling rule can be estimated to predict the behaviour of the market price. Note that, an estimation of the above equation provides the parameter α which can be used, ex post, to estimate the in-situ price given by the static efficiency condition (13). q t 6

7 3 Empirical analysis of the Hotelling rule In this section, I develop an empirical model to estimate and evaluate the reduced form of the Hotelling rule obtained in the previous section. 3.1 Data and the empirical model I use the same data as in Lin and Wagner (2007). This database is a compilation of the data on several countries producing a nonrenewable natural resource. It contains average annual world price, country s average cost and country s current stock data for 14 ores from previously unpublished World Bank data. The commodities are bauxite, copper, gold, hard coal, iron, lead, natural gas as well as nickel, oil, phosphate, brown coal, silver, tin. and zinc. The data cover 35 years from 1970 to The table 8, in the appendix, gives summary statistics on the ores which are analysed in this empirical section. The figure 3 and 5 present pattern of the market prices of some ores. These figures show that, the evolution of the market prices of oil and natural gas are close. For these ores, market price increases during the period and reaches its pick in From 1981 to 1999, it decreases. Since 2000, the market price exhibits a slight growth both for oil and natural gas. The similar behaviour is observed for hard coal and brown coal except that the market price is continuing to decrease since The similarity between the market price of oil and natural gas as well as the similarity of the market price of brown coal and hard coal can be explained by the fact that these ores are substitutes. Even the market price of gold and others ores cannot be compared, it remains that they also exhibit a sharp decrease during the period of study Even, the market price is common to all countries, the extraction cost depends on some country features. To account for these unspecified characteristics the better ways may be to introduce country effects, which are captured by a constant term. Let assume that countries have the same evaluation of future cash flows and also the same extraction technology. In other words, parameters δ, α, b and γ are constants which do not depend on country characteristics. Assuming that future market prices are well anticipated by each country, It follows that countries chose different extraction path only because they have different resource stocks. Therefore, the initial stock S 0 is an appropriate constant term which 7

8 should depend on country characteristics. I will use this initial stock to capture the country effects. So, let assume that the initial stock depends on each country characteristic. An estimate of S i0 should satisfy the following condition. T i S i0 q it (17) where T i is the sample size. To guarantee that this condition is satisfied, I can decompose the initial stock in two components. t=0 T i S i0 = β i + q it (18) where β i 0 is a constant which captures the last period stock. Let define the following residual t=0 T i R it = q it X it = t=0 T i τ=t+1 q iτ (19) Substituting the above expression in (16), the empirical model to be estimated is given by p t = (1 + δ)p t 1 + αac it α(1 + δ)ac it 1 b AC it + ɛ it (20) β i + R it subject to the constraints q it b 0; α 1; β i 0; δ 0 (21) where ɛ it is an error term which can be correlated. As the discount rate δ is positive then, the empirical model exhibits a nonstionnary trend for the market price. This trend can lead to spurious regression. However, looking closely to the empirical model to be estimated, one can note that the market price s trend is offsetted in the right hand side of the model. To see this clearly, I can rewrite the empirical model as follows. p t p t 1 = δ(p t 1 αac it 1 ) + α(ac it AC it 1 ) b AC it + ɛ it (22) β i + R it Under this form, the theoretical trend of the market price disappears and the error term seems to be the sum of stationary components. This other empirical representation show clearly that q it 8

9 the Hotelling model aims to explain the variation of the market price via the current in-situ price and the variation of the average extraction cost. It can have a mathematical interpretation. Note that, the deviation of the market price from the average extraction cost will be fed into its short dynamic. So the average extraction seems to be the long run relation of the market price. As δ is positive then, the Hotelling model appears to be a special case of an error correction model where the deviation of the market price from its long run relation increases its volatility. Interpreting the deviation of the market price from its long relation as the in-situ price, one can conclude that, the in-situ price increases the market price volatility. In other words, if the in-situ price is increasing then the deviation of the future market price from the current market price will also increase. This implies that the current in-situ price is positively correlated to the future market price. In this case, δ captures the speed of the adjustment between the current in-situ price and the future market price. Remember that, my mean target is to estimate the in-situ price which is not observable. According to the previous analysis, one can expect that the in-situ price will be positively correlated to the market price. Estimating equation (20) or (22) one obtains parameters α and β i which are used ex post to estimate the in-situ price λ it. To estimate the in-situ price λ it, one should care about the transversality condition. In the present work, the transversality condition requires that λ it 0 as sure. As the in-situ price λ it (α) = p t αac it is a decreasing function of α then it exists α it < such that for all α α it, λ it (α) < 0. Denote by ᾱ it = Inf(α it ), for given (p t, AC it ) and denote by ᾱ = Inf(ᾱ it ). It follows that, for all α > ᾱ, it exists (p t, AC it ) such that λ it (α) < 0. Therefore, the transversality condition may be satisfied only for a parameter α, less than ᾱ. That is, prob(λ it (α) < 0/α ᾱ) > 0. Now, the question is how to determine the appropriate value of ᾱ. To characterize the upper bound of α, I assume that. 0 < prob(λ it (α) < 0/α = ᾱ) prob(λ it (α) 0/α = ᾱ) (23) This condition states that when α is equal to its upper bound, the probability to obtain a positive in-situ price is different to 1 and is greater than the probability to obtain a negative in-situ price. Interpreting the above condition in terms of the stochastic variable p t /AC it, I derive the following 9

10 relation. ( ) ( ) pt pt 0 < prob < ᾱ prob ᾱ AC it AC it (24) Therefore, the stochastic variable p t /AC it captures the boundary constraint needed to obtain a positive in-situ price. Any important knowledge about the distribution of p t /AC it will be helpful to fix a proper upper bound for α. The boundary condition (24) suggests that Inf ( pt AC it ) ( ) pt < ᾱ Median, a.s. (25) AC it Table 1 summarizes some characteristics of p t /AC it. It appears that the mean is greater than the median for all ores. Therefore, the stochastic variable p t /AC it is asymmetric. As the median and the mean are closer to the minimum than the maximum then the distribution of p t /AC it has a long right tail. It follows from the empirical transversality condition (24) that the lower Table 1: Summary characteristics of the stochastic variable p t/ac it OIL NG GOL HC SC PHO BAU COP IRO LEA NIC SIL ZIN TIN Min Max Mean Median Std OIL, oil; NG, natural gas; GOL, gold; HC, hard coal; SC, brown coal; PHO, phosphate; COP copper; BAU, bauxite; IRO, iron; LEA, lead; NIC, nickel; SIL, silver; TIN, tin; ZIN, zinc; Source: Unpublished World Bank data bound 1 of α should have the high probability to explain the positive in-situ price. This is an important property because the value α = 1 has a suitable interpretation in terms of the extraction process. When α is equal to 1, the extraction cost process is proportional to the extraction rate. In other words, the average extraction cost may be a good approximation of the marginal extraction cost. 3.2 Market value of a unit of the resource in ground Let denote by λ t the market value of a unit of the resource in ground at time t. Let S t be, the total stock of resource in ground at time t. Given the total resource stock S t, the in-situ price λ t measures the scarcity of the resource when future markets prices are well anticipated. The above empirical model allows us to estimate each country evaluation of a unit of resource 10

11 in ground λ it given its own resource stock S it and when the country optimally anticipates future market prices (p t, p t+1,...). This implies that λ it follows the stochastic equation λ it = (1 + δ)λ it 1 b q itac it S it + ɛ it, with E(ɛ it /S it, p t, p t+1,...) = 0 (26) Where ɛ it is an error term. This equation is a compact form of the Hotelling rule (22). Assuming that there is only one market, so one price, the value of a unit of resource in ground obtained by country i, can be captured by the following stochastic equation. λ t = λ it + u it, with E(u it /S it, p t, p t+1,...) = 0 i, t (27) Where u it is an error term. This condition states that, the difference between λ it and λ t is due to the fact that country i have partial information S it of the stock of resource in ground. From stochastic equations (26) and (27), I derive the expression of the real market value of a unit of resource in ground and its dynamic as stated in the following proposition Proposition 1 Under the previous assumptions, the in-situ price λ t is characterized by the static and dynamic equations λ t = p t α lim n 1 n n i=1 S itac it 1 lim n n n i=1 S it λ t = (1 + δ)λ t 1 b lim n 1 n n i=1 q itac it 1 lim n n n i=1 S it (28) (29) The proof of this proposition is given in the appendix. An interpretation of this proposition is that, when there are many countries in a mineral market, the value of unit of the resource in ground is an aggregation of all countries information about this value. It follows that, in a resource market with m competitive countries (firms), the estimate of the real in-situ price is given by λ t = p t α m i=1 S it m k=1 S kt AC it (30) 11

12 3.3 Regime Switching Nonlinear GMM estimation The empirical model developed in this paper is nonlinear in b, α, δ and β i. The method of estimation depends on the nature of the explanatory variables and the further assumptions I would make for the behaviour of the error terms. In the simple case, if explanatory variables are exogenous and the error term follows a normal distribution then an appropriate estimation method is the Maximum Likelihood. The Maximum Likelihood Estimator has important properties of efficiency, but its well known limitation is the normality assumption of the error terms. The general and more used approach is to assume that the behaviour of the error terms is unknown. In that situation, if explanatory variables are exogenous then a convenient estimation method is the Nonlinear Least Squares (NLS). Looking closely to explanatory variables of the Hotelling model, it follows that the average extraction cost is a function of the extraction rate which is an endogenous variable. Furthermore, the residuals stock R it is a function of the upcoming resource extraction rates. Therefore, it will be interesting to treat the average extraction cost and the residuals stock as endogenous variables. This implies that the NLS method may not longer be an appropriate estimation method. It is imperative to incorporate this property to obtain a consistent estimator of the parameters. The estimation method called the Generalized Method of Moments (GMM), which includes the NLS as a special case, provides a solution ( see Matyas (1999)). Note that, the GMM estimator is a M.estimator, which is asymptotically normal under any distribution of the error terms. In this paper, I will discuss the way to use this estimation method to fit the Hotelling rule. GMM estimator Denote by θ = (b, α, δ, β 1,..., β n ) Θ the parameter to be estimated and by y it = (P it, AC it, AC it 1, q it, R it ) the observable variable. Let θ 0 Θ be the true parameter value. I assume f(y it ; θ 0 ) = ɛ it ; E(ɛ it ) = 0, θ 0 Θ (31) where f(y it ; θ) = p t (1 + δ)p t 1 αac it + α(1 + δ)ac it 1 + b β i +R it AC it is the elementary function or the residual. Grouping all these residuals in a T 1 vector f(y; θ), I assume. q it E [ f(y; θ 0 ) f(y; θ 0 ) ] = Ω, θ 0 Θ (32) 12

13 Where Ω is a positive definite matrix which is assumed to be unknown. Now, denote by w it = (P(AC it 1 ), P(q it 1 ), P(X it 1 )) the instrumental variable, where P(v it ) denotes de permutation of the data of the variable v it, among countries with the same sample size. This permutation of data among countries allows me to obtain at least, as many instruments as the parameters. Let W be a T k matrix of instruments, which are assumed to be predetermined, where k is the number of instruments. The theoretical moment condition is given by. E(W itf(y it ; θ 0 )) = 0 θ 0 Θ (33) Where W it is the it th row of W. Denote by g T (y; θ) = 1 T T t=0 W it f(y it; θ) and G(y; θ) = g(y; θ)/ θ, the nonlinear GMM estimator ˆθ is the value of θ, which minimizes the following criterion function. Q T (θ) = g T (y, θ) Σ 1 g T (y, θ) (34) Where Σ is the optimal covariance matrix of the moment variable W it f(y it, θ), Σ = W ΩW. It is shown that the estimate of the optimal covariance matrix of ˆθ is given by. V ar(ˆθ) = T 1 [ G(y; ˆθ) ˆΣ 1 G(y; ˆθ)] 1 (35) Where ˆΣ is the HAC estimator of the covariance matrix Σ = W ΩW. As the Hotelling model has many parameters ( 3 + number of countries producing an ores) then the number of instruments used to compute the GMM estimator is very large. With many instruments, the estimate of the covariance matrix with the usual procedure, the Newey-West HAC estimator of the covariance matrix ˆΣ, is generally not well conditioned. To obtain a well conditioned HAC estimator, I regularize the Newey-West HAC estimator with the regularization procedure of Ledoit and Wolf (2003) ( see appendix for more details). Computation of the GMM estimator ˆθ To compute the GMM estimator ˆθ described above, I use Newton s method for constrained nonlinear minimization ( see the appendix for a description of Newton s method). Because of the empirical transversality condition (24), a minimization of the objective function (34) can lead to a 13

14 boundary solution. To avoid this situation one can use the Newton s method with a trust region algorithm which must force an interior solution. Note that, before running Newton s method to minimize the objective function Q T (θ), I should replace the covariance matrix Σ by Ledoit and Wolf (2003) HAC estimator obtained from an initial estimation of the Hotelling model with a NLS method. Like other procedures that start from preliminary estimate, this one is iterated. Indeed, the GMM estimator residuals is used to calculate a new estimate of the covariance matrix Σ, which is then used to obtain a second GMM estimator, which is then used to obtain another GMM estimator until the procedure converges relative to given criteria. This iterative procedure is called the continuously updated GMM and was investigated by (Hansen et al., 1996). Analysis of the effect of jumps in prices As it will be discussed in the next section, the estimate of the in-situ price present many conjunctural breaks or Jumps. Note that, one of the mean theoretical assumptions of the resource price behaviour is the continuity over time. Therefore, the present of jumps can be explained by the existence of some exogenous shocks, which are not captured in the basic model estimated. To take these conjunctural breaks into account, I create a dummy variable which takes a value 1 at a jumping date. Let denote by J it this dummy variable. Now, replacing the elementary function f(y it ; θ) of the theoretical moment condition (33) with the new elementary function f(y it ; θ) ηj it, one obtains a moment condition with jump effects. In the next section, a model with jump effects will also be estimated and the impact of these jumps will be checked out with the null assumption H 0 : η = 0. If this null assumption is rejected, then the appropriate specification of the Hotelling which must be used to estimate the in-situ price will be the model with jump effects. Goodness of fit and analysis of the time dimension in parameters One important point of this paper, is to determine the consistency of the Hotelling model with the observed data. For this purpose, I use two mean approaches. First, I check the predictive power of the model by plotting the observed market prices versus the fitted market prices. If the model performs well, the curve obtained will be close to the 45 degree line. Second, I check the structural stability or the time consistency of the Hotelling model. As it will be discussed in the next section, the estimate of the in-situ prices shows two regimes. In the first regime, the in-situ price increases and in the second regime, it decreases. My purpose will be to verify whether or not the two regimes are consistent with the Hotelling model. To test the structural stability 14

15 with the GMM estimation method, one should make a distinction between the identifying and the over identifying stability restrictions. While the first restriction concerns the variation of the parameters between the two regimes, the second restriction focuses on the predictive stability. In this paper, I will deals particularly on the parameter stability. The idea is to compare two estimators, which are assumed to be consistent under different regimes. For Hotelling model, the stability investigation needs a special treatment because countries enter and quit the market at the different dates. Indeed, it is not easy to control the parameter β capturing country fix effects. Note that, for each country i, it corresponds a particular parameter β i. As the dates at which countries enter or quit the market are independent between them, it may exist a sub-sample where some countries will not be represented. This situation implies that the vector of parameters β may be different from one sub-sample to another. Therefore, to test the parameter stability of the Hotelling model, one should restrict the analysis to countries acting in the two regimes. To overcome this situation, one can drop from sample, countries which are not represented in two sub-sample. However, I prefer to solve this problem differently. I restrict the analysis of the stability to the fixe part (b, α, δ) of the vector of parameters θ. Let A be a k k matrix of rank 3, which satisfies (b, α, δ) = Aθ, with k = dim(θ). Denote by L i, i {1, 2}, the sub-samples designed to test the stability of the Hotelling model. L i is an interval of N and L 1 L 2 = O. Let φ 0 = (θ 1, θ 2 ) Φ the true parameter value. The theoretical moment condition (33) becomes. E(W itf(y it ; φ 0 )) = E [ d t (L 1 )W itf(y it ; θ 1 ) + (1 d t (L 1 ))W itf(y it ; θ2) ] = 0 φ 0 Φ(36) Where W it is the it th row of W and d t (L 1 ) is a dummy variable which equals one when t L 1. I call the above moment condition, the " Switching Hotelling model" to make a distinction with the "Basic Hotelling model" characterized by the moment condition (33). Note that the moment condition ( 36) generalizes the moment condition ( 33) by incorporating the time dimension in the Hotelling model parameters. This new specification will be helpful to analyse the effect of time on the behaviour of in-situ price. From Andrews and Fair (1988), I derive the Wald statistic. [ 1 W stat = T (A 1 ˆθ1 A 2 ˆθ2 ) π A 1 V ar(ˆθ 1 )A ] 1 1 π A 2 V ar(ˆθ 2 )A 2 (A 1 ˆθ1 A 2 ˆθ2 ) (37) 15

16 Where πt = #L 1 and ˆθ i is the GMM estimator based on the sub-sample L i and V ar(ˆθ i ) is a Ledoit-Wolf HAC estimator of Σ based on the L i. This statistic has a limiting χ 2 (3) distribution under the identiflying restrictions of the two sub-samples and will be used to test the time consistency of the Hotelling model. The structural stability of the Hotelling model developed above must be interpreted with caution. Indeed, if there is no evidence to reject the null assumption of the structural stability, this does not mean that the Hotelling model is time consistent. It simply means that the vector of parameters (b, α, δ) = Aθ is time consistent and not the vector θ. The second bloc β of θ can change over the two sub-periods. It is useful to denote that, even one country can extract a resource in the two sub-periods, the parameters β 1i and β 2i capturing the last period resource stock are conceptually different. Indeed, by definition, β 1i = S i[πti ], β 1i = S iti and in the case of the time consistency respect to the last period resource stock, the following relation must be satisfied. β 1i = T i s=[πt i ] q is + β 2i (38) Therefore, to test the time consistency of Hotelling model respect to the last period resource stock β i, one need an additional test. However, if the null assumption of the structural stability is rejected then the Hotelling model is really time inconsistent. 4 Application and results This section presents results obtained by implementing the above methodology to analyse data of fourteen nonrenewable natural resources: bauxite, copper, gold, hard coal, iron, lead, natural gas as well as nickel, oil, phosphate, brown coal, silver, tin. and zinc. In the first subsection, I present the results of the Basic Hotelling model where the parameter is assumed to be constant over time. These first results are used as a benchmark to study the structural stability of the Hotelling model. The following results are based on the continuously update GMM with the convergence criteria fixed on the objective function. 16

17 4.1 Basic Hotelling model To obtain the benchmark model, I assume that parameters are constants over time and the theoretical moment condition is given by eq. (33). Two models are estimated. The second model is an extension of the first model in which I take into account jump s effects in the behaviours of the in-situ price. As I have already mentioned in the previous section, I create a dummy variable to capture the jump s effects. This dummy variable is associated to an additional parameter η introduced in the Basic Hotelling model and used to check the conjunctural stability. First, let present the results of the Hotelling model yielding a data set. I focus particularly on Figure 1: Measure of the goodness of fit the explicative power of the model. As the Hotelling model is estimated with a technique different to the Nonlinear Least Square (NLS) then, the usual R 2 which measures the goodness of 17

18 fit is not longer valid. Generally, the R 2 is less than 0 or greater than 1 depending on the use of the residuals or the fitted data for calculation. However, in the present work, even with the GMM method used, I obtain the most value between 0.88 and Probably, one can conclude that the Hotelling model is doing a good job in approximating the observed data. To see this clearly, Iet plot the observed data versus the fitted data. The results are presented in figures 1 and 5. When the dots are around the 45 degrees line, this suggests a good prediction of the model. From these figures, one can conclude that the Hotelling model has a good explicative power regarding natural resource market prices. This is an important result in the sense that it allows the use of the Hotelling model to focus on natural resource commodities. As the purpose of this paper is to estimate the in-situ price which is not observable, then the result that the fitted prices are close to the observed prices suggests that the outcome will be close to the reality. Comparing this result with the one obtained in Young (1992), one can deduce that the average extraction cost approximates the marginal extraction cost better than an econometric estimation of the marginal extraction cost. Table 2 presents results based on an implementation of the GMM to estimate the Hotelling model with and without jump s effects. The empirical model has many parameters. For the brevity, I present results only for the key ones. However, the parameter β which is not reported in the table is summarized with a box plot. Note that, the results of the Hotelling model with jumps and without jumps are slightly different. This suggests that the exogenous shocks which affect the natural resource market do not have a significant impact on the behaviour of the resource prices. To formally justify this point I compute the test of the null assumption that the parameter η associated to the jumps is equal to 0. The results of this test are reported in the table 2. Except for gold, hard coal and iron, I find that there is no evidence against η = 0. Therefore, the Basic Hotelling model captures resource price fluctuations. This can be explained by the fact that the Basic Hotelling model is a particular form of an error correction model ( see eq.(22)). That is, the empirical model is designed to explain the fluctuation of the market price instead off the market price itself. With this result, one can use the Basic Hotelling model to estimate the in-situ price without care about the conjunctural breaks or discontinuities. However, it will not matter if the model with jump effects is used as well. To be as general as possible, my preference goes to the model with jump effect because it captures the conjunctural breaks found in the case of the gold and the hard coal 18

19 Table 2: Estimated results OIL NG GOL HC SC PHO BAU COP IRO LEA NIC SIL ZIN TIN Model without jump effects b (0.20) (0.17) (15.6) (0.00) (0.50) (0.65) (0.69) (0.67) (0.13) (0.54) (0.11) (245.18) (0.25) (0.00) α (1.33) (1.45) (3.69) (0.47) (6.98) (1.73) (1.58) (0.54) (0.71) (1.09) (2.03) (2.53) (2.93) (0.00) δ (0.01) (0.01) (0.21) (0.04) (0.08) (0.07) (0.05) (0.03) (0.01) (0.30) (0.09) (0.57) (0.14) (0.08) Model with jump effects b (0.39) (0.13) (5.64) (0.01) (0.54) (0.53) (0.99) (0.13) (0.09) (0.77) (0.58) (14.16) (0.10) (0.00) α (1.90) (1.08) (4.50) (0.10) (8.32) (1.49) (8.00) (2.32) (1.02) (2.62) (2.36) (7.06) (3.02) (2.04) δ (0.02) (0.02) (1.40) (0.00) (0.15) (0.33) (0.08) (0.11) (0.01) (0.42) (0.18) (4.12) (0.21) (0.08) Test of the conjunctural breaks (H 0 : η = 0) η χ χ 2 p.v The value between ( ) is the standard deviation. There is no evidence against H 0 at level 1% ( resp. 5%) when the p.value (p.v)is greater than 0.01 (resp. 0.05). OIL, oil; NG, natural gas; GOL, gold; HC, hard coal; SC, brown coal; PHO, phosphate; COP copper; BAU, bauxite; IRO, iron; LEA, lead; NIC, nickel; SIL, silver; TIN, tin; ZIN, zinc; Source: Unpublished World Bank data. The following analysis is then based on the model with jump s effects. It appears that the stock elasticity b is different to zero excepted for hard coal and nickel. Thus, the market prices of the hard coal and nickel are not significantly affected by the variation of the resource stock. The marginal extraction cost elasticity α is close to one for all ores. Parameter α gives the magnitude at which the marginal extraction cost deviates from the average extraction cost. With a value close to one, the parameter α indicates that the average extraction cost and the marginal extraction cost are equal or close to each other. Therefore, the use of the average extraction cost data as a proxy for marginal extraction cost is not mattered. To convince the reader that this statement is statistically satisfactory, I perform a test to check whether or not average extraction cost is equal to marginal cost. The results are reported in table 3. It appears that the evidence against a constant marginal extraction cost is weak for all minerals. Regarding the condition of transversality, this result suggests that, the Hotelling model gives a high probability to a positive in-situ price than the negative in-situ price. Therefore, setting the upper bound of the parameter α closed to one will not matter. Even, there is a strong support to the value 1 of the parameter α, it is important to compute its exact value in order to obtain an adequate estimate of the in-situ price. 19

20 The Hotelling model is an optimal intertemporal allocation of a known stock of a nonrenewable natural resource. Thus, the discount rate δ is an essential element of the analysis. This paper Table 3: Test of a constant marginal extraction cost α = 1 OIL NG GOL HC SC PHO BAU COP IRO LEA NIC SIL ZIN TIN χ χ 2 p.v There is no evidence against H 0 at level 1% ( resp. 5%) when the p.value (p.v) is greater than 0.01 (resp. 0.05) OIL, oil; NG, natural gas; GOL, gold; HC, hard coal; SC, brown coal; PHO, phosphate; COP copper; BAU, bauxite; IRO, iron; LEA, lead; NIC, nickel; SIL, silver; TIN, tin; ZIN, zinc; The value between ( ) is the standard deviation; Source: Unpublished World Bank data indicates that discount rate is between 0 and 0.10 with most values close to zero. This suggests that, in natural resource economics, one can give the same valuation to the present and the future cash flow from the extraction of the stock of a nonrenewable natural resource. As it has been noted earlier, the parameter δ also captures the speed at which the market price adjusts from the in-situ price. With δ close to 0, one can imagine that the transmission of the information from the in-situ price to the market price is slow. That is, when the resource becomes scare, the market price takes time to incorporate this information of the scarcity. In table 2, I have not reported the last period stock β i. The distribution of β i is very important because it gives an idea about the depletion of the resource and the share of the remaining stock among countries. Figure 2 presents, the box plot and the histogram of the resource stock of the end of the period, β i of oil. The long right tail and plus signs show the lack of symmetry in the sample values. This indicates that the large resource stock belongs to few countries. The same patterns are observed for other ores as shows by the figures 5, 6 and 7 in the appendix. A good estimate of the last period resource stock β i is useful to obtain an estimate of the current resource stock, S it given by. S it = β i + R it (39) Recall that, the aim of this paper is to estimate the in-situ price of a nonrenewable natural resource, which governs the evolution of the observed market price. As the Hoteling model estimated in this paper has a high explanatory power, I hope that this estimate will be close to the reality. The in-situ price λ t is the price at which a unit of the current resource stock S it i can be sold. With an estimate of the parameter α and the current resource stock S it, the value 20

21 Figure 2: Distribution of the resource stock, β i, of oil The box plot contains 7 handles for the smallest observation, the lower quartile, median, upper quartile, the largest observation and the outliers of the in-situ price is obtained from the following expression. λ t = m i=1 S it m k=1 S kt (P t αac it ) (40) Figures 3, 4 and 6 plot the evolution of the in-situ price from the above functional form. These figures show that, the evolution of the in-situ prices of oil and natural gas are close. For these ores, in-situ price increases during the period and reaches its top in From 1981 to 1999, it decreases. Since 2000, in-situ price exhibits a slight growth both for oil and natural gas. The similar behaviour is observed for hard coal and brown coal except that the in-situ price is continuing to decrease since As in the case of the market prices, the similarity between the in-situ price of oil and natural gas as well as the similarity of the in-situ price of brown coal and hard coal can be explained by the fact that these ores are substitutes. Even the in-situ price of the gold and others ores cannot be compared, it remains that they also exhibit a sharp decrease during the period of study. In figure 3, I also represent the evolution of 21

22 the market price in order to compare with the evolution of the in-situ price. As expected, the two prices are highly correlated. This correlation suggests that, the market price is a projection of the in-situ price in the real world. Therefore, even the in-situ price is not observable, the market price can be used as an indicator of the scarcity of the non-renewable resource. However, as it has been noted in the previous analysis, the speed at which the information is transmitted from in-situ prices to market prices is weak. That is, when the market price indicates that the resource is becoming scare, this has happened one time before. Therefore, it seems imperative to estimate the real in-situ price to be sure that the scarcity of the resource will be observed instantaneously when it occurs. Figures 3, 4 and 6 show that, the in-situ is decreasing since 1980, and this for any ores. In theoretical works, the decrease of the in- situ price is explained by the technology progress, which exerts downward pressure on the extraction cost to decrease. But, it is unimaginable that this technology progress can continue to shift down the in-situ price in the long run. Perhaps, the depletion effect will occur in certain moment to offset the technology progress and turn up in-situ price. However, with the Basic Hotelling model estimated in this paper, I cannot make any projection to determine whether or not the in-situ price should increase in the long run. 4.2 Regime Switching Hotelling model The results obtained in the previous section are fundamental in the sense that, they justify the use of the Hotelling model to focus on the nonrenewable natural resource commodities. However, the evolution of the in-situ prices obtained, show the existence of two regimes (an increasing and a decreasing regime for in-situ price). Now, the problem is to determine whether or not the regression parameters are significantly affected by these structural breaks. In other words, I want to determine the time consistency of the Hotelling model. I take as a breakpoint, the date at which the in-situ price estimated from the basic Hotelling changes the trend or the regime. These dates are reported in table 4. As one can see, most of the structural breaks occur between 1975 and This period is characterized by the world recession and important adjustment policies, particularly in the energy importing countries 1. To capture these regimes switching, I build a special Hotelling model denoted a " Switching 1 sees World Economic Survey , Unites Nations, New York 1981, for more details 22

23 Figure 3: Evolution of the resource prices Hotelling model" to distinguish with the "Basic Hotelling model" discussed in the previous section. This Switching Hotelling model assumes that the moment condition given in eq.(36) hold. This moment condition allows the parameters to differ between the two regimes. It appears that, the Basic Hotelling model is a particular case where the parameters are the same (θ 1 = θ 2 = θ). With this new design, it becomes easy to verify the time consistency of the Hotelling model using the Wald statistic described in eq.(37). The results are reported in tables 4. It follows that the stability of the Hotelling is rejected for all ores. Therefore, the results obtained in the previous section for the Basic Hotelling model are not time consistent. An appropriate specification of the Hotelling model will be to allow the parameters to change between the two regimes. Tables 5 presents the results obtained from the Switching Hotelling model. As in the case of the Basic model, the Switching Hotelling model also performs well in approximating the observed market prices ( see figure 7 in the appendix). The same, a constant marginal extraction may explain 23

24 Table 4: Test of the structural stability OIL NG GOL HC SC PHO BAU COP IRO LEA NIC SIL ZIN Breaks W (10 3 ) W p.v There is no evidence against H 0 at level 1% ( resp. 5%) when the p.value (p.v) is greater than 0.01 (resp. 0.05) OIL, oil; NG, natural gas; GOL, gold; HC, hard coal; SC, brown coal; PHO, phosphate; COP copper; BAU, bauxite; IRO, iron; LEA, lead NIC, nickel; SIL, silver; TIN, tin; ZIN, zinc Source: Unpublished World Bank data the extraction process, regardless of the regime. This is tested and the results are reported in table 6. From this table, one can note that, the marginal extraction cost elasticity α may be time consistent. This important result suggests that, a manager of a stock of a resource does not need to reallocate his investment in other to shift down the marginal extraction cost when the expected marginal benefice from extraction is decreasing. While the parameter α is approximately constant over the two regime, the discount rate appears to vary significantly. Note that Table 5: Results of the switching model OIL NG GOL HC SC PHO BAU COP IRO LEA NIC SIL ZIN Sub-period where the in-situ price is increasing b (0.33) (0.16) (0.41) (0.16) (0.75) (0.78) (1.83) (0.21) (0.12) (0.38) (1.12) (0.32) (0.20) α (0.87) (0.42) (0.32) (0.73) (5.28) (1.01) (2.13) (0.70) (0.41) (0.11) (0.67) (0.48) (0.30) δ (0.03) (0.02) (0.21) (0.15) (0.62) (0.52) (0.98) (0.04) (0.02) (0.34) (0.50) (0.56) (0.43) Sub-period where the in-situ price is decreasing b (0.02) (0.008) (4.44) (0.008) (0.20) (0.07) (1.37) (0.45) (0.11) (0.10) (1.10) (7.13) (0.06) α (0.27) (0.03) (6.28) (0.10) (4.34) (0.73) (3.97) (2.13) (0.94) (2.64) (3.12) (10.4) (1.02) δ (0.00) (0.03) (5.28) (0.00) (0.11) (0.09) (0.55) (0.09) (0.01) (0.15) (0.38) (0.11) (0.09) The value between ( ) is the standard deviation. OIL, oil; NG, natural gas; GOL, gold; HC, hard coal; SC, brown coal; PHO, phosphate; COP copper; BAU, bauxite;iro, iron; LEA, lead; NIC, nickel; SIL, silver; TIN, tin; ZIN, zinc. Source: Unpublished World Bank data when the in-situ price is increasing the discount rate is between 0.10 and 0.50, and it shifts to 0 and 0.10, with most value around zero when the In-situ price is decreasing. The discount rate would be one explanation of the non-consistency of the Hotelling model. The fact that the discount rate is greater in the regime where in-situ price is increasing than in the regime where in-situ price is decreasing suggests that, agents give less value to the future cash flow when their marginal benefice is increasing and more value to the future cash flow when their marginal 24

25 benefice is decreasing. This seems to be a rational behaviour because one usually takes less care to the future revenue if one expects to earn more tomorrow. Regarding the discounting, this result suggests that, it is not consistent, in natural resource economics, to use a single discount rate for all future cash flows. It will be convenient to take into account the evolution of the in-situ price to determine the appropriate value of the future gain. Regarding the information flow in a natural resource market, this result tells us that, the information about the scarcity of a non-renewable resource is quickly transmitted to the market price when the in-situ price is increasing. This is an important result. Note that, one of the principal targets of non-renewable resource economies is to determine an appropriate indicator to measure the scarcity of a non-renewable resource. In-situ price, the formal indicator, is not observable. As the relation between the in-situ price and the market price is high when the resource is becoming scare, then using the market price in place of the in-situ price would not matter too much as a measure of the resource scarcity. As in the case of the Basic Hotelling model, I have not reported the result of the last period resource stock β. However, it is difficult to obtain a snapshot of its distribution because this Table 6: Test of a constant marginal extraction cost OIL NG GOL HC SC PHO BAU COP IRO LEA NIC SIL ZIN H 0 : α 1 = α 2 χ 2 (10 2 ) χ 2 p.v H 0 : α 1 = 1 χ χ 2 p.v H 0 : α 2 = 1 χ χ 2 p.v There is no evidence against H 0 at level 1% ( resp. 5%) when the p.value (p.v) is greater than 0.01 (resp. 0.05) OIL, oil; NG, natural gas; GOL, gold; HC, hard coal; SC, brown coal; PHO, phosphate; COP copper; BAU, bauxite; IRO, iron; LEA, lead; NIC, nickel; SIL, silver; TIN, tin; ZIN, zinc; The value between ( ) is the standard deviation; Source: Unpublished World Bank data distribution is not stationary. Denote by β it the last period resource stock. A good estimate of this last period resource stock is useful to obtain an estimate of the current resource stock, S it given by. S it = β it + R it ; β it = d t (L 1 ) β 1i T i s=[πt i ] q is + (1 d t (L 1 ))β 2i (41) 25

26 The above equation shows that the current resource stock S it is the sum of a continuous process R it and a jump process β it. Therefore, the regime Switching Hotelling model may also be explained by a jump of the state variable S it. In this paper, the Switching Hotelling model is designed to obtain a time consistent estimate of the in-situ price. Therefore, a similar formula to (40) obtained in the Basic Hotelling model is given. λ t = m i=1 S it m k=1 S kt (P t α t AC it ); α t = d t (L 1 )α 1 + (1 d t (L 1 ))α 2 (42) This formula allows different distribution for the in-situ price in each regime and consequently, provides a time consistent estimate of the in-situ price. general than the one obtained in the Basic Hotelling model. Theoretically, this formula is more However, comparing the two outcomes by plotting the evolution of the in-situ price ( see figures 4 and 6), one can note that, the behaviour of the in-situ price is slightly different for Switching and Basic Hotelling model. So Figure 4: Evolution of the in-situ prices 26

27 it is a natural question to ask whether this difference is statistically significant. For this purpose, denote by λ b (resp. by λ s ) the in-situ price obtained from the Basic Hotelling model ( resp. from the Switching Hotelling model). If the two distributions of the in-situ price are not statistically different then it must have the same mean. Therefore, one can compare the two outcomes by testing the null hypothesis. ( ) H 0 : E λ s t λ b t = 0, t (43) Note that, the above null hypothesis can be tested using the t student statistic. However, the requirement is that, the stochastic process λ s t λ b t should be normal distributed. The normal distribution seems to be a very strong assumption in the present framework due to the GMM method used. To be as general as possible, let assume that the stochastic variable λ s t λ b t is stationary and ergodic. Using Lyapunov condition of the central limit theorem, one can show that the Wald statistic. W stat = m[ 1 m m ( t=1 λ s t λ b ) ( t E λ s t λ b 2 t)] V ar ( λ s t ) (44) λb t follows the χ 2 (1) distribution. This statistic can be used to test if the differences between the insitu price of Basic Hotelling model and the Switching Hotelling model are statistically significant. Table 7 gives results of this test. It appears that for 8 of 13 minerals, there is no evidence to reject the null hypothesis of the equal mean at level one percent. Therefore, for most ores, the Table 7: Test of equal mean OIL NG GOL HC SC PHO BAU COP IRO LEA NIC SIL ZIN H 0 : E(λ s t ) = E(λb t ) χ χ2p.v H 0 : E(λ s it ) = E(λb it ) χ 2 ( 10 4 ) χ2p.v H 0 : E(λ s it ) = E(λ0 it ) χ 2 ( 10 3 ) χ2p.v There is no evidence against H 0 at level 1% ( resp. 5%) when the p.value (p.v) is greater than 0.01 (resp. 0.05) OIL, oil; NG, natural gas; GOL, gold; HC, hard coal; SC, brown coal; PHO, phosphate; COP copper; BAU, bauxite; IRO, iron; LEA, lead; NIC, nickel; SIL, silver; TIN, tin; ZIN, zinc Source: Unpublished World Bank data time consistent estimate of the in-situ price λ s t has the same mean with the time inconsistent estimate λ b t. However, this does not mean that the two in-situ prices have the same distribution. It simply implies that the stochastic relation between λ s t and λ s t can be described by the following 27

28 stochastic equation. λ s t = λ b t + u t ; E(u t ) = 0 (45) The above relation states the difference between the time consistent and the time inconsistent estimate of the in-situ price is a white noise process. For country i, the value of a unit of the resource in ground is given by the formula λit = pt αac it. Denote by λ 0 it the approximation of λ it for α = 1. It follows from table 7 that, the stochastic relation obtained above for aggregate in-situ price (45) is not satisfied by the country λ it even with α = 1. This suggests that, parameter α estimated in this paper captures primarily the industry marginal cost elasticity. It can be explained by the method of fixe effects panel data used in this paper. This can also be obtained by rewriting the aggregate formula (42) of in-situ price as follows. λ t = P t α t AC t (46) AC t = n S it m k=1 S AC it kt (47) i=1 α t = d t (L 1 )α 1 + (1 d t (L 1 ))α 2 (48) Where AC t is the industry marginal cost obtained as a weighting average of country s average extraction cost. 5 Conclusion Hotelling (1931) proposed a framework to explain theoretically the optimal extraction of nonrenewable natural resources. One fundamental results was that the in-situ price which is an economic measure of the resource scarcity should grow at the rate of interest. This principle, known as the basic Hotelling rule has been unsucceful tested since the in-situ price is not observable. Gaudet (2007) supported the idea that in is a complete form, the Hotelling rule is too complex to be tested, but this complexity does not remove its value as an indicator to measure the scarcity of nonrenewable natural resources. The fundamental question is to determine whether or not the Hotelling principle can be used to explain the observed behaviour of the market price and as possible to estimate the unobserved behaviour of the in-situ price 28

29 (Livernois, 2009). This paper builds a framework to evaluate the Hotelling model and estimate the unobserved in-situ price. In contrast to previous works, this paper has three main contributions. Instead of using time series s property of the market price or an estimate of the in-situ price to evaluate the Hotelling model (Slade, 1982; Berck and Roberts, 1995; Lee et al., 2006), I combine the first order conditions of the optimal resource extraction to estimate a market price which is consistent with the Hotelling rule. Instead of an econometric approximation of the marginal extraction cost (Young, 1992), I use average extraction cost data as a proxy of the marginal cost to estimate the unobserved in-situ price. These two contributions can be summarized in three steps. First, I assume a functional form of the extraction cost in order to use average extraction cost as a proxy for marginal cost. Second, I combine optimal conditions to estimate a market price which is consistent with the Hotelling rule. Finally, I deduce the corresponded in-situ price. The last contribution of this paper is the empirical technique used to investigate the Hotelling model of a nonrenewable resource extraction. I use Regime Switching GMM estimation with panel data. This robust estimation technique seems to be very useful to handle the endogenous property of the average extraction cost and discuss the time consistency of the Hotelling model. Applying my methodology to analyse fourteen non-renewable natural resource prices, I find many results that strongly support the use of the Hotelling model to focus on the natural resource commodities. It appears that the Hotelling model explains about 88 to 99 per cent of the observed market prices. Note that, Young (1992) estimated a model close to the one developed in this paper using econometric investigation technique to obtain an adequate proxy of the marginal extraction cost and found that Hotelling model explained at most 1 percent of the observed market prices. Therefore, the approach used in the present work seems to be satisfactory. Using appropriate breakpoints to evaluate the stability of my estimations I find that, even conjunctural breaks do not have a significant impact on the fitted prices the Hotelling model is time inconsistent under two regimes. The first regime is characterized by the increase of the in-situ price and the second regime by the decrease of the in-situ price. Using a regime switching model to reconcile the Hotelling model with the change of regime, I find that the time inconsistency does not have a significant impact on explanatory power of the model. However, this new setting of the model seems to be very informative to understand the behaviour of a competitive 29

30 firm in a natural resource market. It appears that the rate at which agents discount the future cash flow is high in the regime where in-situ price is increasing and low with most value close to zero in the regime where in-situ price is decreasing. This suggests that agents given less valuations to the future cash flow when their marginal benefit is increasing and more when their marginal benefice is decreasing. Furthermore, I find that the average extraction cost data is a good proxy for the marginal extraction cost and a constant marginal extraction should explain the extraction process regardless the regime. I find that the stock of a resource in ground may be pareto distributed among countries. The stock of a non-renewable resource is unequally distributed and few countries would hold the greatest part of the resource stock in ground. The time consistent estimate of in-situ price decreases or exhibits a U-shape form over time and is highly correlated to the market price. Furthermore, the difference between a time consistent and a time inconsistent estimate of the in-situ price is a simply a white noise process. The results obtained in this paper are of the great importance in the sense that they agree the use of the Hotelling model to focus on natural resource commodities. However, the Hotelling rule estimated has some limitations. In particular, I cannot make any projection to see whether or not the in-situ price should increases in the long run as predict by Hotelling (1931). Even, this paper deal with time consistency of the Hotelling model, it remains that the time effect can have a multi forms. So, I have just introduced this feature of the Hotelling model. Furthermore, I have assumed the existence of the equilibrium of natural resource market and use the implication of a competitive firm behaviour to explain the market and the in-situ prices. A complete analysis should solve for the equilibrium to obtain the equilibrium prices. This will be done in the future works. 6 Appendix Proof of proposition1 From the static equation (27), one obtains E(S it u it ) = E (E(S it u it /S it, p t, p t+1,...)) = E (S it E(u it /S it, p t, p t+1,...)) = 0 (49) 30

31 Drawing the expression of the error term u it from (27) and substituting in (49), I obtain E(S it u it ) = E(S it (λ t λ it )) = 0 (50) Using the law of large number, It follows that 1 lim n n n S it (λ t λ it ) = E(S it (λ t λ it )) = 0 (51) i=1 Extracting the value of λ t from above equation, the expression of the market value of a unit of resource in ground is given by λ t = lim n 1 n n i=1 S itλ it 1 lim n n n i=1 S it = p t α lim n 1 n n i=1 S itac it 1 lim n n n i=1 S it (52) As desired. To obtain the dynamic of λ t, extract the expression of λ it from static equation (27) and substitute in dynamic equation (26). Using the similar approach as previous, I obtain the dynamic of λ t. Newey West HAC estimator of matrix Σ The asymptotic covariance matrix of the variable W it f(y it; θ 0 ) is given by. Σ = lim T T 1 ( (j)) Γ(0) + Γ(j) + Γ (53) The Newey-West estimator of this covariance matrix takes the form Σ NW = Γ(0) + Γ(j) = 1 T T t=j+1 p j=1 j=1 ( 1 j ) ) ( Γ(j) + Γ (j) p + 1 (54) ɛ t ɛ t j W t W t j (55) Note that the optimal cutoff p is given by T 1/4. Ledoit-Wolf well conditioned HAC estimator of Σ Let consider the Frobenius norm X = (tr (XX ) /k, X is a k T matrix, whose associated 31

32 inner product is: X 1 X 2 = tr (XX ) /k. The Ledoit-Wolf HAC estimator is given by Σ LW = ˆb 2 â2 ˆmI + ΣNW (56) ˆd 2 ˆd 2 where Σ NW is a Newey West HAC estimator of Σ, I is k k identity matrix and the coeffecients are given by: ˆm = Σ NW, I (57) ˆd 2 = Σ NW ˆmI (58) b2 = 1 p ( p 1 j ) ) ( Γ(j) + Γ (j) p + 1 Σ NW (59) ˆb2 j = min ( b2, ˆd 2) (60) â 2 = ˆd 2 ˆb 2 (61) Quasi-Gauss-Newton- Algorithm Denote by Q(θ) the objective function. Since Q(θ) is twice continuously differentiable for M- estimators, there is a second order Taylor expansion Q T (θ) Q T (ˆθ j ) + s T (ˆθ j ) (θ ˆθ j ) (θ ˆθ j ) H T (ˆθ j )(θ ˆθ j ) (62) where ˆθ j is the estimate in the j th round of the iterative procedure to be described in a moment, and s T and H T are the gradient and the Hessian of the objective function: s T (θ) = Q T (θ) θ ; H T (θ) = 2 Q T (θ) θ θ (63) The (j+1)-the round estimator ˆθ(j + 1) is the maximizer of the quadratic function on the righthand side of (63). It is given by ˆθ j+1 = ˆθ j [H T (ˆθ j ] 1 s T (ˆθ j ) (64) This iterative procedure is called the Newton-Raphson algorithm. If the objective function is 32

33 concave, the algorithm often converges to the global minimum. This algorithm works well if the matrix H T (ˆθ(j) is positive definite. ˆθ j+1 = ˆθ j + α j [D T (ˆθ j ] 1 s T (ˆθ j ), θ 0 is given (65) where α j is a scalar which is determined at each step to be minimize Q T (θ j +α j [D T (ˆθ j ] 1 s T (ˆθ j )), D T (ˆθ j is a matrix which approximates the Hessian(matrix of second derivatives of the objective function ) near the maximum but it is constructed so that it is always positive definite and s T (ˆθ(j)) is the gradient ( vector of first derivatives of the objective function). An approximation is given by D T (ˆθ j ) = 1 T G (ˆθ j )G(ˆθ j ) (66) and the second term of the quasi-newton algorithm becomes [D T (ˆθ j ] 1 s T (ˆθ j ) = [ ] 1 i G (ˆθ j )G(ˆθ j ) G (ˆθ j )g(ˆθ j ) (67) 33

34 Figure 5: Evolution of the resource prices References D.W.K. Andrews and R. Fair. Infrerence in econometric models with structural change. Review of Economic Studies, Peter Berck and Micheal Roberts. Natrural resource prices: Will they ever turn up? Journal of Environmental Economics and Management, Scott Farrow. Testing the efficiency of extraction from a stock resource. The Journal of Political Economy, Gerard Gaudet. Natural resource economics under the rule of hotelling. The Canadian Journal of Economics,