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Technology adoption in nonrenewable resource management
Energy Economics 31 (2009) 235–239
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Energy Economics j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / e n e c o
Technology adoption in nonrenewable resource management Maria A. Cunha-e-Sá a,⁎, Ana Balcão Reis a,1, Catarina Roseta-Palma b,2 a b
Universidade Nova de Lisboa, Faculdade de Economia, Campus de Campolide, PT-1099-032, Lisboa, Portugal Departamento de Economia, ISCTE, Avenida das Forças Armadas, PT-1649-026, Lisboa, Portugal
a r t i c l e
i n f o
Article history: Received 8 June 2007 Received in revised form 18 September 2008 Accepted 23 September 2008 Available online 30 September 2008 JEL classification: O33 Q65
a b s t r a c t Technological change has played an important role in models of nonrenewable resource management, since its presence mitigates the depletion effect on extraction costs over time. We formalize the problem of a competitive nonrenewable resource extracting firm faced with the possibility of technology adoption. Based on a quadratic extraction cost function, our results show that the expected net benefits from adoption increase both with the size of the resource stock and with prices. A boundary that separates the region where expected net benefits are positive from the one where they are negative is derived. © 2008 Elsevier B.V. All rights reserved.
Keywords: Nonrenewable resource Technology adoption Quadratic cost Size of the stock
1. Introduction Simple models of nonrenewable resource extraction consider the case of a firm that has a fixed production process, implying that the firm's cost function does not change throughout the entire period of extraction activity. However, the assumption of no technical improvements in production is empirically inappropriate for most resources. The presence of an underlying process of technological development has an important role to play in those models, since its presence mitigates the depletion effect on extraction costs over time. Empirical research shows that the role played by technology in the natural resource industry has been crucial. Recently, in the case of oil and natural gas, Managi, Opaluch, Jin and Grigalunas (2004) have measured depletion effects and technological change for offshore oil production in the Gulf of Mexico based on a field-level data set from 1947–1998. This study supports the hypothesis that technological progress has mitigated depletion effects over the study period, and shows that diffusion had a significantly larger impact on total factor productivity than technological innovation.3 On the other hand, in Simpson (1999), where the impact of technological change for several natural resource industries in the US is examined, there is consider⁎ Corresponding author. Tel.: +351 213801660; fax: +351 213870933. E-mail addresses: [email protected] (M.A. Cunha-e-Sá), [email protected] (A. Balcão Reis), [email protected] (C. Roseta-Palma). 1 Tel.: +351 213801600; fax: +351 213870933. 2 Tel.: +351 217935000; fax: +351 217903933. 3 In a more general context, Hall and Khan (2003) argue that it is diffusion rather than invention or innovation that ultimately determines the pace of economic growth and the rate of change of productivity. 0140-9883/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.eneco.2008.09.008
able evidence that technological progress is intimately related to depletion of easily accessible reserves. The author concludes that “… costs of production have not increased because the inevitable effects of depletion have, to date, been more than offset by improvements in technology.”(Simpson, 1999)4 In particular, it is argued that depletion induces innovation. If this is the case, we should expect that those firms that benefit the most from adopting new technologies are those with more depleted stocks at the time the new technology becomes available. However, as Bohi notes in Simpson (1999), in the case of oil and natural gas, five (Amoco, Exxon, Arco, British Petroleum, and Shell) out of the six world's largest firms which have been pioneers in the deployment of new technologies are multinational organizations with global activities. Therefore, “…expensive innovations are conducted by firms large enough to use their results broadly”, in the author's own words. That indicates that the benefits from adoption increase with the size of the firm, or the amount of the resource stock. In contrast to many studies in the literature in which the potential of technology improvements to mitigate resource scarcity is examined as an empirical issue, we model the firm's optimal decisions both on resource extraction and on adoption of an exogenous incremental innovation. 5 Adoption is characterized by potential adopters 4
See Tilton and Landsberg (1997), as well as Krautkraemer (1998) and Dasgupta (1993). In the context of nonrenewable resources, as suggested by Lundstrom (2002), two types of technological innovation can be distinguished: incremental and drastic. While incremental innovations increase the efficiency of extraction and discovery of already familiar resource stocks, increasing the rate of exhaustion, drastic innovations are revolutionary, in the sense that they increase the quantity of familiar resource stocks, either by introducing an unexpected technology or by adding to the number of familiar resources. 5
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contemplating the use of a technology that reduces the marginal cost of production but has a known adoption cost.6 Adoption will only occur if its net benefits are positive. For a general quadratic extraction cost function, our main result is that expected benefits from adoption increase both with the level of the stock and with prices, whether those prices are considered deterministic or stochastic. Intuitively, for firms with more depleted stock levels at the time the new technology becomes available, the technological opportunities will be applied to a smaller stock, thus reducing benefits from adoption. Therefore, our findings are in contrast to the view according to which the firm will only incur in adoption costs when the stock is depleted enough. As for higher prices, they naturally increase expected adoption benefits. We derive a boundary that separates the region where expected net benefits are positive from the one where they are negative. In the case of deterministic prices and constant fixed costs in present value terms, the boundary embodies a decision rule for adoption. The results obtained clarify the importance of modeling the firm's decision problem, contributing to a more thorough understanding of the role of technology improvements on mitigating depletion in nonrenewable resource management. The remainder of the paper is organized as follows. In Section 2, the general problem of the firm with volatile prices is described. In Section 3, additional structure is imposed on the problem by specifying a quadratic cost function and the expected benefit from adoption is obtained. Section 4 looks at the behavior of expected net adoption benefits. Finally, the main conclusions of the paper are summarized in Section 5. Technical details are presented in the Appendices. 2. The firm's problem In this section, we set up a general model of a competitive nonrenewable resource extracting firm to analyze optimal extraction behavior. We begin by considering the problem for a given technology, and then determine the impact of technological improvements when they occur. The firm's problem consists of choosing the extraction path to maximize the expected present value of profits over time, given the evolution for market prices, the stock, and the technology, as follows: V ðS0 ; p0 ; a0 Þ = max E0
R∞
fet g
0
½pt et −C ðet ; St ; a0 Þe−rt dt
ð1Þ
1. twice continuously differentiable; C(e, S; a) b ∞, for all e, S, given a. 2. strongly jointly convex in (e, S), implying that the principal minors 2 2 of order one and two are strictly positive: AAeC2 N0, AASC2 N0 and A2 C A2 C A2 C − N0; Ae2 AS2 AeAS 3. it is expected that A2 C A2 C ASAa b0, AeAa b0.
dSt = −et dt S0 given et z 0
dpt = μpt dt+σ p pt dwp
ð2Þ ð3Þ ð4Þ ð5Þ
The optimal value function at time t, V(St, pt;a0), represents the expected present value of the profits obtained from the extraction program operating with an (unchanged) technology level a0. Also, St is the existing stock of resource, pt is the market price of the resource and et is extraction, all evaluated at time t. S0 is the known endowment
or
Thus, marginal extraction cost is positive and increasing (reflecting diminishing returns to extraction); there are stock effects in both total and marginal cost; as for technology a, it is assumed to lower total and marginal extraction cost, and to decrease (or increase) the impact of stock effects on total cost. In what concerns the impact of the technological improvement on the marginal depletion cost, that is, A2 C ASAa, consider the case of a new energy-saving drilling method for oil that reduces the marginal extraction of going deeper into the well. In this case, the available amount of the stock becomes less important, A2 C implying that ASAa N 0. In alternative, we could consider the opposite case in which the technological improvement was more energy-saving in the surface area. In that case, as we go deeper the stock size becomes A2 C more important. so that ASAa b 0. Therefore, both cases are possible.8 Finally, conditions 2. above guarantee the existence of a unique solution to the firm's maximization problem. It should be stressed that the stated result explicitly assumes that the firm is a competitive agent in the market and thus takes prices as given. At the market level, price (and/or expected price) will be endogenous and, possibly, related to the level of resource stock. Moreover, technological change in this paper does not directly affect stock size. Looking at such effects requires a model where firms have a choice of developing additional discoveries into proven reserves, such as in Farzin (2001), since in that case the benefits from technological improvement may in themselves affect stock size.9
In this subsection, we describe the solution of problem (1), that is, the solution to the firm's problem for a given technology. This problem satisfies the associated Bellman equation, as follows: 1 rV= max pe−C ðe; S; a0 Þ−VS e+μpVp + Vpp σ 2p p2 ð6Þ e 2 If the right-hand side has an interior maximum, then the choice of e that satisfies the above partial differential equation must satisfy the Maximum Principle and p−
7
6
A2 C AC A2 C A2 C A2 C A2 C N0, AC AS b0, Aa b0, Aa2 N0, AS2 N0, AeAS b0, ASAa N0 Ae2
2.1. Solution to the firm's problem
s:t:
St z0;
of resource stock available to the firm, and a0 represents the quality of technology at t = 0. From the point of view of the firm, prices are exogenous. However, we take into account the existence of uncertainty surrounding the evolution of market resource prices over time. This uncertainty is driven by a geometric Brownian motion with drift, as described in Eq. (5), where dwp is the increment of the Wiener process, and σp is the volatility of market prices. Moreover, the extraction cost function is assumed to have the following properties:7
The problem of choosing the timing of adoption has been examined for a competitive firm in the context of the investment literature, as in Balcer and Lippman (1984), Doraszelski (2004), and Farzin, Huisman and Kort (1998). However, the presence of a nonrenewable resource stock changes the dynamics of the problem, as the firm has to decide simultaneously at each time period how much to extract and whether to adopt or not. This is also different from Dixit and Pindyck (1994). See Cunha-e-Sá, Balcão Reis and Roseta-Palma (2004) for a numeric example.
AC =VS Ae
ð7Þ
Subscripts are dropped whenever possible to simplify notation. These assumptions are similar to those found in the literature. See, for example, Krautkraemer (1998), and Farzin (1992, 1995). In Farzin's (1992), the properties of the cost function were first explicitly assumed and analyzed. We also assume that if nonextractive net benefits exist they are not reduced by an increase in the level of the stock, and that there are strictly positive net benefits from extracting the first unit of the resource. 9 We thank an anonymous referee for pointing this out. 8
M.A. Cunha-e-Sá et al. / Energy Economics 31 (2009) 235–239
Eq. (7) yields the optimal policy function for extraction, e⁎(S, p, Vs;a0) so that, evaluated at the optimum, Eq. (6)) can be written as: AC eT ð:Þ; S; a0 T 1 e ð:Þ+μpVp + Vpp σ 2p p2 rV=−C eT ð:Þ; S; a0 + 2 Ae
ð8Þ
Thus, the solution V(.) depends upon the stock of the resource as well as upon prices, for a given quality level a0. Using Itô's Lemma and Eq. (8), the expected rate of change in the opportunity cost of the resource can be stated as: 1 AC EdVS =rVS + dt AS
tY∞
eT ðp; S; VS Þ=
so that the following partial differential equation is satisfied: rV=
In this section, we characterize the benefit from adoption, namely, how it changes with the stock level and prices. We analyze the effect of an adoption of a new technology which becomes available at time t = τ. When the firm adopts a more advanced technology at t = τ, the quality increases according to
where υ represents the upgrade in technology. Let V(Sτ, pτ;a1) represent the maximum expected present value of profits at t = τ obtained from the extraction program with an upgraded technology of quality a1 for τ ≤ t ≤ ∞. In other words, V(Sτ, pτ;a1) is the solution of the following problem: R V ðSτ ; pτ ; a1 Þ= max Eτ τ∞ ½pt et −C ðet ; St ; a1 Þe−rðt−τÞ dt fet g
subject to the same constraints as in Eq. (1). Thus, the expected benefit from adopting at t = τ is defined as follows ð11Þ
To study the properties of the expected adoption benefits we need to impose additional structure. Thus, hereafter we consider a specific functional form for extraction costs. 3.1. Quadratic costs The extraction cost function is assumed to be quadratic in extraction and stock, as follows: ð12Þ
To ensure that Eq. (12) has the desirable properties, it is required that α1 N0, α2 N 0, α3 b 0, and 0 b 4α2α1 − α23 b ρα3.10
ðe4Þ 1 10 See Section 2. In fact, at the optimum, ACAe = a 2α 1 eT +α 3 S is always positive. Also, the positiveness of the second-order principal minor of the Hessian matrix of C(.), 2 2 A2 C ðe4Þ A2 C ðe4Þ C ðe4Þ − A AeAS = a12 4α 2 α 1 −α 23 , requires that 0 b 4α2α1 - α23. Moreover, a suffiAe2 AS2 ðe4Þ 1 cient condition for ACAS = a α 3 eT +2α 2 S b0 , is that 4α2α1 - α23 b 2ρα3, where α1 N 0, α2 N 0, α3 b 0, such that
2 A2 C ðe4Þ C ðe4Þ N0, A AS N0 2 Ae2
and
A2 C ðe4Þ AeAS b0.
ð14Þ
The solution is given by: α 1 ðr−μ Þ2 2 ρ 2 2ρ+α 3 p + S + Sp a0 C 2ρ+α 3 −2α 1 ðr−μ Þ
ð15Þ
In Eq. (15), ρ depends upon the parameters of the problem and it is negative.11 The sign of Γ is determined by the sign of (r − 2µ − σ2), which relates the discount rate, r, to the growth rate of the expected value of p2, (2µ + σ2), for the case of the geometric Brownian motion with drift.12 As it must be positive in order for lifetime expected profit in Eq. (1) to be finite, it is necessary that (r − 2μ − σ2) N 0, also implying that r N μ. Thus, Γ N 0 so that V(.) is convex with respect to p. From Eqs. (13) and (15), the optimal extraction decision is given by eT =−
ð10Þ
1 α 1 e2t +α 2 S2t +α 3 et St a
½a0 ðp−VS Þ−Sα 3 2 α 2 2 1 − S +μpVp + Vpp σ 2p p2 2 4α 1 a0 a0
C= r−2μ−σ 2 ð2ρ+α 3 −2α 1 ðr−μ ÞÞ2
3. The benefit from adoption
BðSτ ; pτ ; a0 Þ=V ðSτ ; pτ ; a0 ð1+υÞÞ−V ðSτ ; pτ ; a0 Þ
ð13Þ
where
tY∞
a1 =ð1+υÞa0
a0 ð p−VS Þ−Sα 3 2α 1
V ðp; S; a0 Þ=a0
where VSt represents the marginal user cost of the resource stock at t.
C ðet ; St ; aÞ=
We solve the firm's problem, given by Eq. (1), with this specification. From the first order condition (7) and assuming an interior solution, optimal extraction is given by
ð9Þ
which is Hotelling's rule with stock effects. The transversality conditions for the stock are given by lim Et VSt expð−rt Þz0; lim Et VSt expð−rt ÞSt =0:
237
2ρ+α 3 a0 ðr−μ Þ p S− 2ρ+α 3 −2α 1 ðr−μ Þ 2α 1
for Vs N 0 and e⁎ b S.13 Thus, we observe that the optimal extraction decision is exactly the same as the one that solves the deterministic problem (σ2 = 0), as it does not depend on price volatility.14 Besides, it Ae4 is linear in the stock as well as in prices, and Ae4 Ap N0 and AS N0. From Eqs. (11) and (15), the expected benefits from adoption are given by BðSτ ; pτ ; a0 Þ=
a0 υðr−μ Þ2 α 1 2 ρυ pτ − S2 a0 ð1+υÞ τ C
ð16Þ
The behavior of Eq. (16) is examined in Proposition 1. Proposition 1. The expected benefits from adoption increase both with the stock and with prices. Proof. By differentiating Eq. (16) with respect to the stock we obtain AB −2υρSτ N0 = ASτ a0 ð1+υÞ
ð17Þ
Therefore, the expected benefits from adoption increase with the available stock and this is not influenced by uncertainty. The impact of prices on the expected benefits of adoption is given by AB 2a0 υðr−μ Þ2 α 1 pτ = Apτ C
ð18Þ
As Γ N 0, expected benefits also increase with prices. With deterministic prices, that is, σ2 = 0, the first term in Γ simplifies to (r − 2μ),
11
See Appendix 1 for the characterization of the optimal paths. The derivation is shown in Appendix 1. Moreover, from Appendix 3, the transversality condition for the stock also holds. 13 The specific ranges for pS for which these conditions are satisfied depend upon the A2 C sign of ASAa : This is shown in Appendix 2. 14 See Bertsekas (1987), page 106. 12
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which has to be positive, as in the stochastic case. Therefore, the results also apply to deterministic prices. □ Intuitively, the larger the stock the higher are the gains to be obtained from adopting a technological improvement that reduces costs, as the lower extraction costs are applied to more units of the resource. Thus, these results do not support the view according to which firms with more depleted stocks are more willing to adopt than firms with large stock levels at the time the improvement becomes available. Moreover, higher prices at t =τ increase expected benefits from adoption, whether prices are increasing (μ N 0) or decreasing (μ b 0). 4. Net benefits from adoption In this section we look at expected benefits from adoption net of adoption costs, at the time the new technology becomes available, t =τ. To this end, a specific functional form is considered for adoption costs. The general cost incurred by the firm when it decides to adopt, c(υ), depends upon the upgrading rate, υ, either directly or indirectly through the quality level, where c'(υ) N 0 and cq(υ) ≥ 0. It is possible to derive a boundary that separates the region of positive expected net benefits from the one where expected net benefits are negative. Expected net benefits from adoption at t = τ are non-negative as long as NBðSτ ; pτ ; a0 ; υÞ=V ðSτ ; pτ ; a0 ð1+υÞÞ−V ðSτ ; pτ ; a0 Þ−cða0 ð1+υÞ; υÞz0 ð19Þ This last condition establishes a boundary that separates the two above mentioned regions. From Eq. (19), the boundary is given by NBðSτ ; pτ ; a0 ; υÞ=0
ð20Þ
For a given a0 and υ, the location of those regions with respect to the boundary depends upon the behavior of the expected net benefit of adoption with respect to stock and prices. Expected benefits B(.) are given by (16). We assume that there is a fixed current cost of adoption at t = τ, k N 0. As for the firm υ is exogenously determined by the available technology, we look at the simple case in which cða0 ð1+υÞ; υÞ=k
ð21Þ
Using the above specifications, the boundary is characterized in Proposition 2. Proposition 2. The boundary that separates the region of positive expected net benefits from that of negative expected net benefits is given by a0 υðr−μ Þ2 α 1 2 ρυ pτ − S2 =k a0 ð1+υÞ τ C −2ρ 2ρ+α 3 −2α 1 ðr−μ Þ pτ b Sτ 2ρ+α 1 a0
for
ð22Þ
1 2ρ+α 3 −2α 1 ðr−μ Þ b 2ρ+α2α3 +2α . In the feasible region, −a0 ðr−μ Þ 1
defined by a cone, to the right of the boundary net benefits are positive, while to the left they are negative. Proof. From Eq. (22), we first obtain the intercepts both in the horizontal and in the vertical axis, in order to help when representing the boundary graphically. The intercept in the horizontal axis is given by S+τ =
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 −ka0 ð1+υÞ ρυ
ð23Þ
Fig. 1. Boundary for σ2 N 0 and σ2 = 0.
Eqs. (17) and (18). As Γ N 0, the boundary is quadratic and concave. Since expected benefits increase both with the stock level and prices, to the right of the boundary net benefits are positive, while to the left they are negative. □ In Fig. 1, a possible boundary is represented inside the feasible region defined by a cone, for a given σ2.15 Ap+ Given σ2 and r, the higher μ, the lower p+τ in Eq. (24), as Aμτ b0.16 Also, as S+τ in Eq. (23) does not change, the region of positive expected net benefits is enlarged, while the other region shrinks. Besides, a higher μ also enlarges the cone by increasing the upper bound and reducing the lower bound.17 Moreover, the boundary can be different depending upon the quality level operated by the firm at t = τ. In particular, p+τ and S+τ are lower for larger υ. When volatility changes, we observe that S+τ does not change. In Ap+ 2 + contrast, as Aσ 2 b0, pτ decreases with σ . Fig. 1 illustrates these results. Thus, volatility reduces the region of negative expected net benefits. In particular, when σ2 = 0, that is, with deterministic prices, p+τ is larger than in the stochastic case, ceteris paribus. As mentioned before, in the case of this paper, the optimal extraction decision is not affected by the presence of future uncertainty on prices.18 Given the convexity of V in p, implying that the firm is risk taker, the more uncertain is the price, that is, the larger is σ2, the larger is V. At t=τ, the firm may either keep its present technology, or adopt a new one at a cost. Unless the firm chooses to adopt a new technology, a does not change. As shown in Appendix 5, with deterministic prices and constant fixed costs in present value terms, the question of optimal timing of adoption does not arise. Thus, in this case, the region of positive net benefits is also the adoption region, while the region of negative net benefits is the non-adoption one, as in Fig. 1. In contrast, with uncertainty or decreasing adoption costs in present value terms, no implicit adoption decision rule is embodied in the stochastic boundary, and the firm could gain by waiting to obtain better information, as shown for a generic problem of investment under uncertainty by Dixit and Pindyck (1994). By inspection of Fig. 1, we observe that the adoption decision depends upon the location of prices and stocks at t = τ. Therefore, inside the feasible region, we conclude that: (a) when Sτ N S+τ or pτ N p+τ, the firm will decide to adopt; (b) otherwise, the decision to adopt depends on both the stock and on prices: when Sτ b S+τ, if prices are high enough at t = τ,
while the intercept in the vertical axis is p+τ =
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi kC 2 a0 υðr−μ Þ2 α 1
ð24Þ 15 16
To identify the two regions inside the cone, we make use of the two derivative conditions with respect to the stock and prices, given by
17 18
See Appendix 2 for the derivations of the cone zone in the Fig. 1. This is shown in Appendix 4. We thank an anonymous referee for calling our attention to this fact. See Abel (1983).
M.A. Cunha-e-Sá et al. / Energy Economics 31 (2009) 235–239
(pτ N p+τ), the firm will decide to adopt, while when prices are low enough, (pτ b p+τ), the firm decides not to adopt, and viceversa for prices. In this problem, as adoption reduces marginal extraction costs, and the eventual adoption is never delayed, the major force influencing adoption behavior is the possibility of taking advantage of lower costs. Therefore, the incentive to adopt increases with the resource stock remaining at t = τ, as the benefits are larger. In fact, for firms with more depleted stock levels, the technological opportunities can only be applied to a smaller amount of the stock, reducing net benefits from adoption. Furthermore, higher prices at t = τ also increase the benefits from adoption. Hence, for a firm with a low stock level to optimally choose adoption, prices have to be sufficiently high. Finally, all previous results with respect to the impact on the boundary caused by changes in the price drift (μ), or in the quality level (υ) still apply to the deterministic case. 5. Conclusions Empirical research in the literature shows that the role played by technology in the natural resource industry is the determinant factor in extraction costs decrease, responding to the continuing search for lower costs in a competitive market. In particular, it is often stated that depletion has induced innovation. In contrast to this view, and based on a stochastic optimal control model of the firm's adoption decision with a nonrenewable resource, we claim that the larger the stock for which the new technology decreases extraction costs, the larger the benefit from adoption. In this sense, the incentive to adopt is larger the larger the stock. In the presence of a resource stock, with a quadratic cost function, our main result is that expected benefits from adoption increase with the stock level. In fact, for firms with more depleted stock levels, the technological opportunities can only be applied to a smaller amount of the stock, reducing expected benefits from adoption. Therefore, the firms with a large stock at the time the technological innovation is available will benefit more from adopting than those with more depleted stocks, as we should expect. Also, higher prices increase the expected benefit from adoption. Thus, it may also be the case that for “high enough” resource prices, even firms that are poorly endowed at the time the new technology becomes available decide to adopt. Recent developments in the North Sea oil fields are a good example: there smaller oil companies are taking over, benefiting from high prices and investments in fast-improving technology.19 While an improved technology reduces the marginal cost of production, it also has a known adoption cost. By introducing a fixed cost of adoption and based on previous results, we derive a boundary that separates the region where expected net benefits are positive from that where they are negative. In the deterministic case, the boundary embodies the decision rule for adoption, defining two regions: the adoption region and the non-adoption one. The results are illustrated graphically. These findings contribute to a better understanding of the interaction between depletion and the benefits of cost saving technologies in nonrenewable resource firms. In the future, it would
19
See The Economist (2004).
239
be interesting to consider more general adoption costs. As the adopted quadratic functional form determines a “certainty-equivalent” type of result, in a more general context, uncertainty will surely play a more important role on the optimal timing of adoption. Another interesting extension could consider the stock as endogenous. This is the case when the firm decides to develop discovered reserves into proven reserves, as studied by Farzin (2001). In this context, the benefit from adopting a technological improvement should also take into account its impact on the addition to proven reserves. This may qualify the result obtained in our paper. Depending on how the stock size is influenced by the expected price of the resource, different outcomes may result. Finally, it could be interesting to study other formulations for technological improvement, as, for instance, specific technologies for specific types of stock. Appendix A. Supplementary data Supplementary data associated with this article can be found, in the online version, at doi:10.1016/j.eneco.2008.09.008. References Abel, A., 1983. Optimal investment under uncertainty. American Economic Review 73, 228–233. Balcer, Y., Lippman, S.A., 1984. Technological expectations and adoption of improved technology. Journal of Economic Theory 34, 292–318. Bertsekas, D., 1987. Dynamic programming. Deterministic and Stochastic Models. Prentice-Hall. Cunha-e-Sá, Maria, Reis, Ana Balcão, Roseta-Palma, Catarina, 2004. Technology adoption in nonrenewable resource management. CentrA. Fundación Centros de Estudios Andaluces, Documento de trabajo, E2004/16. Dasgupta, P., 1993. In: Kneese, A.V., Sweeney, J.L. (Eds.), Natural resources in an age of substitutability. . Handbook of Natural Resource and Energy Economics, vol. III. Elsevier Science Publishers B.V. Dixit, A., Pindyck, R., 1994. Investment under Uncertainty. Princeton University Press, Princeton, NJ. Doraszelski, U., 2004. Innovations, improvements, and the optimal adoption of new technologies. Journal of Economic Dynamics and Control 28, 1461–1480. Farzin, Y.H., 1992. The time path of scarcity rent in the theory of exhaustible resources. The Economic Journal 102 (412). Farzin, Y.H., 1995. Technological change and the dynamics of resource scarcity measures. Journal of Environmental Economics and Management 29, 105–120. Farzin, Y.H., 2001. The impact of oil price on additions to proven reserves. Resource and Energy Economics 23, 271–291. Farzin, Y., Huisman, K., Kort, P., 1998. Optimal timing of technology adoption. Journal of Economic Dynamics and Control 22, 779–799. Hall, B., Khan, B., 2003. Adoption of new technology. Working-Paper #9730. NBER. May. Krautkraemer, J., 1998. Nonrenewable resource scarcity. Journal of Economic Literature XXXVI, 2065–2107. Lundstrom, S., Cycles of Technology, Natural Resources and Economic Growth, mimeo, Department of Economics, Goteborg University, June 2002. Managi, S., Opaluch, James J., Jin, Di, Grigalunas, T.A., 2004. Technological change and depletion in the offshore oil and gas industry. Journal of Environmental Economics and Management 47 (2) (March). Simpson, R.D., 1999. Technological innovation in natural resource industries. In: Simpson, R.D. (Ed.), Productivity in Natural Resource Industries. Resources for the Future. Tilton, J., Landsberg, H., 1997. Innovation, Productivity Growth, and the Survival of the U.S. Copper Industry. Discussion Paper 97-41. RFF. September. The Economist, 2004. Britain: small is better than nothing; North Sea oil. The Economist 373 (8400), 38 Nov. 6.
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Energy Economics j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / e n e c o
Technology adoption in nonrenewable resource management Maria A. Cunha-e-Sá a,⁎, Ana Balcão Reis a,1, Catarina Roseta-Palma b,2 a b
Universidade Nova de Lisboa, Faculdade de Economia, Campus de Campolide, PT-1099-032, Lisboa, Portugal Departamento de Economia, ISCTE, Avenida das Forças Armadas, PT-1649-026, Lisboa, Portugal
a r t i c l e
i n f o
Article history: Received 8 June 2007 Received in revised form 18 September 2008 Accepted 23 September 2008 Available online 30 September 2008 JEL classification: O33 Q65
a b s t r a c t Technological change has played an important role in models of nonrenewable resource management, since its presence mitigates the depletion effect on extraction costs over time. We formalize the problem of a competitive nonrenewable resource extracting firm faced with the possibility of technology adoption. Based on a quadratic extraction cost function, our results show that the expected net benefits from adoption increase both with the size of the resource stock and with prices. A boundary that separates the region where expected net benefits are positive from the one where they are negative is derived. © 2008 Elsevier B.V. All rights reserved.
Keywords: Nonrenewable resource Technology adoption Quadratic cost Size of the stock
1. Introduction Simple models of nonrenewable resource extraction consider the case of a firm that has a fixed production process, implying that the firm's cost function does not change throughout the entire period of extraction activity. However, the assumption of no technical improvements in production is empirically inappropriate for most resources. The presence of an underlying process of technological development has an important role to play in those models, since its presence mitigates the depletion effect on extraction costs over time. Empirical research shows that the role played by technology in the natural resource industry has been crucial. Recently, in the case of oil and natural gas, Managi, Opaluch, Jin and Grigalunas (2004) have measured depletion effects and technological change for offshore oil production in the Gulf of Mexico based on a field-level data set from 1947–1998. This study supports the hypothesis that technological progress has mitigated depletion effects over the study period, and shows that diffusion had a significantly larger impact on total factor productivity than technological innovation.3 On the other hand, in Simpson (1999), where the impact of technological change for several natural resource industries in the US is examined, there is consider⁎ Corresponding author. Tel.: +351 213801660; fax: +351 213870933. E-mail addresses: [email protected] (M.A. Cunha-e-Sá), [email protected] (A. Balcão Reis), [email protected] (C. Roseta-Palma). 1 Tel.: +351 213801600; fax: +351 213870933. 2 Tel.: +351 217935000; fax: +351 217903933. 3 In a more general context, Hall and Khan (2003) argue that it is diffusion rather than invention or innovation that ultimately determines the pace of economic growth and the rate of change of productivity. 0140-9883/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.eneco.2008.09.008
able evidence that technological progress is intimately related to depletion of easily accessible reserves. The author concludes that “… costs of production have not increased because the inevitable effects of depletion have, to date, been more than offset by improvements in technology.”(Simpson, 1999)4 In particular, it is argued that depletion induces innovation. If this is the case, we should expect that those firms that benefit the most from adopting new technologies are those with more depleted stocks at the time the new technology becomes available. However, as Bohi notes in Simpson (1999), in the case of oil and natural gas, five (Amoco, Exxon, Arco, British Petroleum, and Shell) out of the six world's largest firms which have been pioneers in the deployment of new technologies are multinational organizations with global activities. Therefore, “…expensive innovations are conducted by firms large enough to use their results broadly”, in the author's own words. That indicates that the benefits from adoption increase with the size of the firm, or the amount of the resource stock. In contrast to many studies in the literature in which the potential of technology improvements to mitigate resource scarcity is examined as an empirical issue, we model the firm's optimal decisions both on resource extraction and on adoption of an exogenous incremental innovation. 5 Adoption is characterized by potential adopters 4
See Tilton and Landsberg (1997), as well as Krautkraemer (1998) and Dasgupta (1993). In the context of nonrenewable resources, as suggested by Lundstrom (2002), two types of technological innovation can be distinguished: incremental and drastic. While incremental innovations increase the efficiency of extraction and discovery of already familiar resource stocks, increasing the rate of exhaustion, drastic innovations are revolutionary, in the sense that they increase the quantity of familiar resource stocks, either by introducing an unexpected technology or by adding to the number of familiar resources. 5
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contemplating the use of a technology that reduces the marginal cost of production but has a known adoption cost.6 Adoption will only occur if its net benefits are positive. For a general quadratic extraction cost function, our main result is that expected benefits from adoption increase both with the level of the stock and with prices, whether those prices are considered deterministic or stochastic. Intuitively, for firms with more depleted stock levels at the time the new technology becomes available, the technological opportunities will be applied to a smaller stock, thus reducing benefits from adoption. Therefore, our findings are in contrast to the view according to which the firm will only incur in adoption costs when the stock is depleted enough. As for higher prices, they naturally increase expected adoption benefits. We derive a boundary that separates the region where expected net benefits are positive from the one where they are negative. In the case of deterministic prices and constant fixed costs in present value terms, the boundary embodies a decision rule for adoption. The results obtained clarify the importance of modeling the firm's decision problem, contributing to a more thorough understanding of the role of technology improvements on mitigating depletion in nonrenewable resource management. The remainder of the paper is organized as follows. In Section 2, the general problem of the firm with volatile prices is described. In Section 3, additional structure is imposed on the problem by specifying a quadratic cost function and the expected benefit from adoption is obtained. Section 4 looks at the behavior of expected net adoption benefits. Finally, the main conclusions of the paper are summarized in Section 5. Technical details are presented in the Appendices. 2. The firm's problem In this section, we set up a general model of a competitive nonrenewable resource extracting firm to analyze optimal extraction behavior. We begin by considering the problem for a given technology, and then determine the impact of technological improvements when they occur. The firm's problem consists of choosing the extraction path to maximize the expected present value of profits over time, given the evolution for market prices, the stock, and the technology, as follows: V ðS0 ; p0 ; a0 Þ = max E0
R∞
fet g
0
½pt et −C ðet ; St ; a0 Þe−rt dt
ð1Þ
1. twice continuously differentiable; C(e, S; a) b ∞, for all e, S, given a. 2. strongly jointly convex in (e, S), implying that the principal minors 2 2 of order one and two are strictly positive: AAeC2 N0, AASC2 N0 and A2 C A2 C A2 C − N0; Ae2 AS2 AeAS 3. it is expected that A2 C A2 C ASAa b0, AeAa b0.
dSt = −et dt S0 given et z 0
dpt = μpt dt+σ p pt dwp
ð2Þ ð3Þ ð4Þ ð5Þ
The optimal value function at time t, V(St, pt;a0), represents the expected present value of the profits obtained from the extraction program operating with an (unchanged) technology level a0. Also, St is the existing stock of resource, pt is the market price of the resource and et is extraction, all evaluated at time t. S0 is the known endowment
or
Thus, marginal extraction cost is positive and increasing (reflecting diminishing returns to extraction); there are stock effects in both total and marginal cost; as for technology a, it is assumed to lower total and marginal extraction cost, and to decrease (or increase) the impact of stock effects on total cost. In what concerns the impact of the technological improvement on the marginal depletion cost, that is, A2 C ASAa, consider the case of a new energy-saving drilling method for oil that reduces the marginal extraction of going deeper into the well. In this case, the available amount of the stock becomes less important, A2 C implying that ASAa N 0. In alternative, we could consider the opposite case in which the technological improvement was more energy-saving in the surface area. In that case, as we go deeper the stock size becomes A2 C more important. so that ASAa b 0. Therefore, both cases are possible.8 Finally, conditions 2. above guarantee the existence of a unique solution to the firm's maximization problem. It should be stressed that the stated result explicitly assumes that the firm is a competitive agent in the market and thus takes prices as given. At the market level, price (and/or expected price) will be endogenous and, possibly, related to the level of resource stock. Moreover, technological change in this paper does not directly affect stock size. Looking at such effects requires a model where firms have a choice of developing additional discoveries into proven reserves, such as in Farzin (2001), since in that case the benefits from technological improvement may in themselves affect stock size.9
In this subsection, we describe the solution of problem (1), that is, the solution to the firm's problem for a given technology. This problem satisfies the associated Bellman equation, as follows: 1 rV= max pe−C ðe; S; a0 Þ−VS e+μpVp + Vpp σ 2p p2 ð6Þ e 2 If the right-hand side has an interior maximum, then the choice of e that satisfies the above partial differential equation must satisfy the Maximum Principle and p−
7
6
A2 C AC A2 C A2 C A2 C A2 C N0, AC AS b0, Aa b0, Aa2 N0, AS2 N0, AeAS b0, ASAa N0 Ae2
2.1. Solution to the firm's problem
s:t:
St z0;
of resource stock available to the firm, and a0 represents the quality of technology at t = 0. From the point of view of the firm, prices are exogenous. However, we take into account the existence of uncertainty surrounding the evolution of market resource prices over time. This uncertainty is driven by a geometric Brownian motion with drift, as described in Eq. (5), where dwp is the increment of the Wiener process, and σp is the volatility of market prices. Moreover, the extraction cost function is assumed to have the following properties:7
The problem of choosing the timing of adoption has been examined for a competitive firm in the context of the investment literature, as in Balcer and Lippman (1984), Doraszelski (2004), and Farzin, Huisman and Kort (1998). However, the presence of a nonrenewable resource stock changes the dynamics of the problem, as the firm has to decide simultaneously at each time period how much to extract and whether to adopt or not. This is also different from Dixit and Pindyck (1994). See Cunha-e-Sá, Balcão Reis and Roseta-Palma (2004) for a numeric example.
AC =VS Ae
ð7Þ
Subscripts are dropped whenever possible to simplify notation. These assumptions are similar to those found in the literature. See, for example, Krautkraemer (1998), and Farzin (1992, 1995). In Farzin's (1992), the properties of the cost function were first explicitly assumed and analyzed. We also assume that if nonextractive net benefits exist they are not reduced by an increase in the level of the stock, and that there are strictly positive net benefits from extracting the first unit of the resource. 9 We thank an anonymous referee for pointing this out. 8
M.A. Cunha-e-Sá et al. / Energy Economics 31 (2009) 235–239
Eq. (7) yields the optimal policy function for extraction, e⁎(S, p, Vs;a0) so that, evaluated at the optimum, Eq. (6)) can be written as: AC eT ð:Þ; S; a0 T 1 e ð:Þ+μpVp + Vpp σ 2p p2 rV=−C eT ð:Þ; S; a0 + 2 Ae
ð8Þ
Thus, the solution V(.) depends upon the stock of the resource as well as upon prices, for a given quality level a0. Using Itô's Lemma and Eq. (8), the expected rate of change in the opportunity cost of the resource can be stated as: 1 AC EdVS =rVS + dt AS
tY∞
eT ðp; S; VS Þ=
so that the following partial differential equation is satisfied: rV=
In this section, we characterize the benefit from adoption, namely, how it changes with the stock level and prices. We analyze the effect of an adoption of a new technology which becomes available at time t = τ. When the firm adopts a more advanced technology at t = τ, the quality increases according to
where υ represents the upgrade in technology. Let V(Sτ, pτ;a1) represent the maximum expected present value of profits at t = τ obtained from the extraction program with an upgraded technology of quality a1 for τ ≤ t ≤ ∞. In other words, V(Sτ, pτ;a1) is the solution of the following problem: R V ðSτ ; pτ ; a1 Þ= max Eτ τ∞ ½pt et −C ðet ; St ; a1 Þe−rðt−τÞ dt fet g
subject to the same constraints as in Eq. (1). Thus, the expected benefit from adopting at t = τ is defined as follows ð11Þ
To study the properties of the expected adoption benefits we need to impose additional structure. Thus, hereafter we consider a specific functional form for extraction costs. 3.1. Quadratic costs The extraction cost function is assumed to be quadratic in extraction and stock, as follows: ð12Þ
To ensure that Eq. (12) has the desirable properties, it is required that α1 N0, α2 N 0, α3 b 0, and 0 b 4α2α1 − α23 b ρα3.10
ðe4Þ 1 10 See Section 2. In fact, at the optimum, ACAe = a 2α 1 eT +α 3 S is always positive. Also, the positiveness of the second-order principal minor of the Hessian matrix of C(.), 2 2 A2 C ðe4Þ A2 C ðe4Þ C ðe4Þ − A AeAS = a12 4α 2 α 1 −α 23 , requires that 0 b 4α2α1 - α23. Moreover, a suffiAe2 AS2 ðe4Þ 1 cient condition for ACAS = a α 3 eT +2α 2 S b0 , is that 4α2α1 - α23 b 2ρα3, where α1 N 0, α2 N 0, α3 b 0, such that
2 A2 C ðe4Þ C ðe4Þ N0, A AS N0 2 Ae2
and
A2 C ðe4Þ AeAS b0.
ð14Þ
The solution is given by: α 1 ðr−μ Þ2 2 ρ 2 2ρ+α 3 p + S + Sp a0 C 2ρ+α 3 −2α 1 ðr−μ Þ
ð15Þ
In Eq. (15), ρ depends upon the parameters of the problem and it is negative.11 The sign of Γ is determined by the sign of (r − 2µ − σ2), which relates the discount rate, r, to the growth rate of the expected value of p2, (2µ + σ2), for the case of the geometric Brownian motion with drift.12 As it must be positive in order for lifetime expected profit in Eq. (1) to be finite, it is necessary that (r − 2μ − σ2) N 0, also implying that r N μ. Thus, Γ N 0 so that V(.) is convex with respect to p. From Eqs. (13) and (15), the optimal extraction decision is given by eT =−
ð10Þ
1 α 1 e2t +α 2 S2t +α 3 et St a
½a0 ðp−VS Þ−Sα 3 2 α 2 2 1 − S +μpVp + Vpp σ 2p p2 2 4α 1 a0 a0
C= r−2μ−σ 2 ð2ρ+α 3 −2α 1 ðr−μ ÞÞ2
3. The benefit from adoption
BðSτ ; pτ ; a0 Þ=V ðSτ ; pτ ; a0 ð1+υÞÞ−V ðSτ ; pτ ; a0 Þ
ð13Þ
where
tY∞
a1 =ð1+υÞa0
a0 ð p−VS Þ−Sα 3 2α 1
V ðp; S; a0 Þ=a0
where VSt represents the marginal user cost of the resource stock at t.
C ðet ; St ; aÞ=
We solve the firm's problem, given by Eq. (1), with this specification. From the first order condition (7) and assuming an interior solution, optimal extraction is given by
ð9Þ
which is Hotelling's rule with stock effects. The transversality conditions for the stock are given by lim Et VSt expð−rt Þz0; lim Et VSt expð−rt ÞSt =0:
237
2ρ+α 3 a0 ðr−μ Þ p S− 2ρ+α 3 −2α 1 ðr−μ Þ 2α 1
for Vs N 0 and e⁎ b S.13 Thus, we observe that the optimal extraction decision is exactly the same as the one that solves the deterministic problem (σ2 = 0), as it does not depend on price volatility.14 Besides, it Ae4 is linear in the stock as well as in prices, and Ae4 Ap N0 and AS N0. From Eqs. (11) and (15), the expected benefits from adoption are given by BðSτ ; pτ ; a0 Þ=
a0 υðr−μ Þ2 α 1 2 ρυ pτ − S2 a0 ð1+υÞ τ C
ð16Þ
The behavior of Eq. (16) is examined in Proposition 1. Proposition 1. The expected benefits from adoption increase both with the stock and with prices. Proof. By differentiating Eq. (16) with respect to the stock we obtain AB −2υρSτ N0 = ASτ a0 ð1+υÞ
ð17Þ
Therefore, the expected benefits from adoption increase with the available stock and this is not influenced by uncertainty. The impact of prices on the expected benefits of adoption is given by AB 2a0 υðr−μ Þ2 α 1 pτ = Apτ C
ð18Þ
As Γ N 0, expected benefits also increase with prices. With deterministic prices, that is, σ2 = 0, the first term in Γ simplifies to (r − 2μ),
11
See Appendix 1 for the characterization of the optimal paths. The derivation is shown in Appendix 1. Moreover, from Appendix 3, the transversality condition for the stock also holds. 13 The specific ranges for pS for which these conditions are satisfied depend upon the A2 C sign of ASAa : This is shown in Appendix 2. 14 See Bertsekas (1987), page 106. 12
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which has to be positive, as in the stochastic case. Therefore, the results also apply to deterministic prices. □ Intuitively, the larger the stock the higher are the gains to be obtained from adopting a technological improvement that reduces costs, as the lower extraction costs are applied to more units of the resource. Thus, these results do not support the view according to which firms with more depleted stocks are more willing to adopt than firms with large stock levels at the time the improvement becomes available. Moreover, higher prices at t =τ increase expected benefits from adoption, whether prices are increasing (μ N 0) or decreasing (μ b 0). 4. Net benefits from adoption In this section we look at expected benefits from adoption net of adoption costs, at the time the new technology becomes available, t =τ. To this end, a specific functional form is considered for adoption costs. The general cost incurred by the firm when it decides to adopt, c(υ), depends upon the upgrading rate, υ, either directly or indirectly through the quality level, where c'(υ) N 0 and cq(υ) ≥ 0. It is possible to derive a boundary that separates the region of positive expected net benefits from the one where expected net benefits are negative. Expected net benefits from adoption at t = τ are non-negative as long as NBðSτ ; pτ ; a0 ; υÞ=V ðSτ ; pτ ; a0 ð1+υÞÞ−V ðSτ ; pτ ; a0 Þ−cða0 ð1+υÞ; υÞz0 ð19Þ This last condition establishes a boundary that separates the two above mentioned regions. From Eq. (19), the boundary is given by NBðSτ ; pτ ; a0 ; υÞ=0
ð20Þ
For a given a0 and υ, the location of those regions with respect to the boundary depends upon the behavior of the expected net benefit of adoption with respect to stock and prices. Expected benefits B(.) are given by (16). We assume that there is a fixed current cost of adoption at t = τ, k N 0. As for the firm υ is exogenously determined by the available technology, we look at the simple case in which cða0 ð1+υÞ; υÞ=k
ð21Þ
Using the above specifications, the boundary is characterized in Proposition 2. Proposition 2. The boundary that separates the region of positive expected net benefits from that of negative expected net benefits is given by a0 υðr−μ Þ2 α 1 2 ρυ pτ − S2 =k a0 ð1+υÞ τ C −2ρ 2ρ+α 3 −2α 1 ðr−μ Þ pτ b Sτ 2ρ+α 1 a0
for
ð22Þ
1 2ρ+α 3 −2α 1 ðr−μ Þ b 2ρ+α2α3 +2α . In the feasible region, −a0 ðr−μ Þ 1
defined by a cone, to the right of the boundary net benefits are positive, while to the left they are negative. Proof. From Eq. (22), we first obtain the intercepts both in the horizontal and in the vertical axis, in order to help when representing the boundary graphically. The intercept in the horizontal axis is given by S+τ =
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 −ka0 ð1+υÞ ρυ
ð23Þ
Fig. 1. Boundary for σ2 N 0 and σ2 = 0.
Eqs. (17) and (18). As Γ N 0, the boundary is quadratic and concave. Since expected benefits increase both with the stock level and prices, to the right of the boundary net benefits are positive, while to the left they are negative. □ In Fig. 1, a possible boundary is represented inside the feasible region defined by a cone, for a given σ2.15 Ap+ Given σ2 and r, the higher μ, the lower p+τ in Eq. (24), as Aμτ b0.16 Also, as S+τ in Eq. (23) does not change, the region of positive expected net benefits is enlarged, while the other region shrinks. Besides, a higher μ also enlarges the cone by increasing the upper bound and reducing the lower bound.17 Moreover, the boundary can be different depending upon the quality level operated by the firm at t = τ. In particular, p+τ and S+τ are lower for larger υ. When volatility changes, we observe that S+τ does not change. In Ap+ 2 + contrast, as Aσ 2 b0, pτ decreases with σ . Fig. 1 illustrates these results. Thus, volatility reduces the region of negative expected net benefits. In particular, when σ2 = 0, that is, with deterministic prices, p+τ is larger than in the stochastic case, ceteris paribus. As mentioned before, in the case of this paper, the optimal extraction decision is not affected by the presence of future uncertainty on prices.18 Given the convexity of V in p, implying that the firm is risk taker, the more uncertain is the price, that is, the larger is σ2, the larger is V. At t=τ, the firm may either keep its present technology, or adopt a new one at a cost. Unless the firm chooses to adopt a new technology, a does not change. As shown in Appendix 5, with deterministic prices and constant fixed costs in present value terms, the question of optimal timing of adoption does not arise. Thus, in this case, the region of positive net benefits is also the adoption region, while the region of negative net benefits is the non-adoption one, as in Fig. 1. In contrast, with uncertainty or decreasing adoption costs in present value terms, no implicit adoption decision rule is embodied in the stochastic boundary, and the firm could gain by waiting to obtain better information, as shown for a generic problem of investment under uncertainty by Dixit and Pindyck (1994). By inspection of Fig. 1, we observe that the adoption decision depends upon the location of prices and stocks at t = τ. Therefore, inside the feasible region, we conclude that: (a) when Sτ N S+τ or pτ N p+τ, the firm will decide to adopt; (b) otherwise, the decision to adopt depends on both the stock and on prices: when Sτ b S+τ, if prices are high enough at t = τ,
while the intercept in the vertical axis is p+τ =
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi kC 2 a0 υðr−μ Þ2 α 1
ð24Þ 15 16
To identify the two regions inside the cone, we make use of the two derivative conditions with respect to the stock and prices, given by
17 18
See Appendix 2 for the derivations of the cone zone in the Fig. 1. This is shown in Appendix 4. We thank an anonymous referee for calling our attention to this fact. See Abel (1983).
M.A. Cunha-e-Sá et al. / Energy Economics 31 (2009) 235–239
(pτ N p+τ), the firm will decide to adopt, while when prices are low enough, (pτ b p+τ), the firm decides not to adopt, and viceversa for prices. In this problem, as adoption reduces marginal extraction costs, and the eventual adoption is never delayed, the major force influencing adoption behavior is the possibility of taking advantage of lower costs. Therefore, the incentive to adopt increases with the resource stock remaining at t = τ, as the benefits are larger. In fact, for firms with more depleted stock levels, the technological opportunities can only be applied to a smaller amount of the stock, reducing net benefits from adoption. Furthermore, higher prices at t = τ also increase the benefits from adoption. Hence, for a firm with a low stock level to optimally choose adoption, prices have to be sufficiently high. Finally, all previous results with respect to the impact on the boundary caused by changes in the price drift (μ), or in the quality level (υ) still apply to the deterministic case. 5. Conclusions Empirical research in the literature shows that the role played by technology in the natural resource industry is the determinant factor in extraction costs decrease, responding to the continuing search for lower costs in a competitive market. In particular, it is often stated that depletion has induced innovation. In contrast to this view, and based on a stochastic optimal control model of the firm's adoption decision with a nonrenewable resource, we claim that the larger the stock for which the new technology decreases extraction costs, the larger the benefit from adoption. In this sense, the incentive to adopt is larger the larger the stock. In the presence of a resource stock, with a quadratic cost function, our main result is that expected benefits from adoption increase with the stock level. In fact, for firms with more depleted stock levels, the technological opportunities can only be applied to a smaller amount of the stock, reducing expected benefits from adoption. Therefore, the firms with a large stock at the time the technological innovation is available will benefit more from adopting than those with more depleted stocks, as we should expect. Also, higher prices increase the expected benefit from adoption. Thus, it may also be the case that for “high enough” resource prices, even firms that are poorly endowed at the time the new technology becomes available decide to adopt. Recent developments in the North Sea oil fields are a good example: there smaller oil companies are taking over, benefiting from high prices and investments in fast-improving technology.19 While an improved technology reduces the marginal cost of production, it also has a known adoption cost. By introducing a fixed cost of adoption and based on previous results, we derive a boundary that separates the region where expected net benefits are positive from that where they are negative. In the deterministic case, the boundary embodies the decision rule for adoption, defining two regions: the adoption region and the non-adoption one. The results are illustrated graphically. These findings contribute to a better understanding of the interaction between depletion and the benefits of cost saving technologies in nonrenewable resource firms. In the future, it would
19
See The Economist (2004).
239
be interesting to consider more general adoption costs. As the adopted quadratic functional form determines a “certainty-equivalent” type of result, in a more general context, uncertainty will surely play a more important role on the optimal timing of adoption. Another interesting extension could consider the stock as endogenous. This is the case when the firm decides to develop discovered reserves into proven reserves, as studied by Farzin (2001). In this context, the benefit from adopting a technological improvement should also take into account its impact on the addition to proven reserves. This may qualify the result obtained in our paper. Depending on how the stock size is influenced by the expected price of the resource, different outcomes may result. Finally, it could be interesting to study other formulations for technological improvement, as, for instance, specific technologies for specific types of stock. Appendix A. Supplementary data Supplementary data associated with this article can be found, in the online version, at doi:10.1016/j.eneco.2008.09.008. References Abel, A., 1983. Optimal investment under uncertainty. American Economic Review 73, 228–233. Balcer, Y., Lippman, S.A., 1984. Technological expectations and adoption of improved technology. Journal of Economic Theory 34, 292–318. Bertsekas, D., 1987. Dynamic programming. Deterministic and Stochastic Models. Prentice-Hall. Cunha-e-Sá, Maria, Reis, Ana Balcão, Roseta-Palma, Catarina, 2004. Technology adoption in nonrenewable resource management. CentrA. Fundación Centros de Estudios Andaluces, Documento de trabajo, E2004/16. Dasgupta, P., 1993. In: Kneese, A.V., Sweeney, J.L. (Eds.), Natural resources in an age of substitutability. . Handbook of Natural Resource and Energy Economics, vol. III. Elsevier Science Publishers B.V. Dixit, A., Pindyck, R., 1994. Investment under Uncertainty. Princeton University Press, Princeton, NJ. Doraszelski, U., 2004. Innovations, improvements, and the optimal adoption of new technologies. Journal of Economic Dynamics and Control 28, 1461–1480. Farzin, Y.H., 1992. The time path of scarcity rent in the theory of exhaustible resources. The Economic Journal 102 (412). Farzin, Y.H., 1995. Technological change and the dynamics of resource scarcity measures. Journal of Environmental Economics and Management 29, 105–120. Farzin, Y.H., 2001. The impact of oil price on additions to proven reserves. Resource and Energy Economics 23, 271–291. Farzin, Y., Huisman, K., Kort, P., 1998. Optimal timing of technology adoption. Journal of Economic Dynamics and Control 22, 779–799. Hall, B., Khan, B., 2003. Adoption of new technology. Working-Paper #9730. NBER. May. Krautkraemer, J., 1998. Nonrenewable resource scarcity. Journal of Economic Literature XXXVI, 2065–2107. Lundstrom, S., Cycles of Technology, Natural Resources and Economic Growth, mimeo, Department of Economics, Goteborg University, June 2002. Managi, S., Opaluch, James J., Jin, Di, Grigalunas, T.A., 2004. Technological change and depletion in the offshore oil and gas industry. Journal of Environmental Economics and Management 47 (2) (March). Simpson, R.D., 1999. Technological innovation in natural resource industries. In: Simpson, R.D. (Ed.), Productivity in Natural Resource Industries. Resources for the Future. Tilton, J., Landsberg, H., 1997. Innovation, Productivity Growth, and the Survival of the U.S. Copper Industry. Discussion Paper 97-41. RFF. September. The Economist, 2004. Britain: small is better than nothing; North Sea oil. The Economist 373 (8400), 38 Nov. 6.