- Email: [email protected]
The transition to an oil contraction economy
EC O LO GIC A L E CO N O M ICS 6 4 ( 2 00 7 ) 1 8 6 –1 93
a v a i l a b l e a t w w w. s c i e n c e d i r e c t . c o m
w w w. e l s e v i e r. c o m / l o c a t e / e c o l e c o n
ANALYSIS
The transition to an oil contraction economy Andrei V. Bazhanov ⁎,1,2 Institute of Mathematics and Computer Science, Far Eastern National University, Vladivostok, Russia Department of Economics, Queen's University, Kingston, ON, Canada K7L 3N6
AR TIC LE I N FO
ABS TR ACT
Article history:
A normative analysis of the problem of optimal extraction of a nonrenewable resource is
Received 26 October 2006
considered. The economy depends on the essential nonrenewable resource and the rate of the
Received in revised form
resource extraction is increasing over time. At some point the government gradually switches
31 January 2007
to a sustainable (in sense of non-decreasing consumption over time) pattern of the resource
Accepted 11 February 2007
extraction. Different approaches are offered for the construction of the paths of switching to
Available online 29 March 2007
decreasing resource use. Some seemingly attractive short-run policies of switching to decreasing extraction can run counter to long-run criteria. If we consider the maximin
Keywords:
principle, applied to the negative shock on the output percent change, as the short-run
Nonrenewable resource
criterion, then the optimal transition path can be consistent with the long-run government
Intergenerational justice
goals. It is shown analytically and numerically that there are values of parameters for the
Hartwick rule
transition paths of extraction that consumption along these paths is asymptotically constant
Optimal path of extraction
or infinitely growing. A new approach to the Rawlsian maximin criterion which allows for
Generalized Rawlsian criterion
growth of consumption is offered. © 2007 Elsevier B.V. All rights reserved.
JEL classification: Q32; Q38
1.
Introduction
One of the important concerns of the United Nations' Agenda 21 is that “…the major cause of the continued deterioration of the global environment is the unsustainable pattern of consumption and production…” and that “Special attention should be paid to the demand for natural resources generated by unsustainable consumption…” (United Nations, 1993, Chapter 4). Observed world oil extraction has been increasing rapidly in recent years and some sustainable (in the sense of weak sustainability) programs proposed by economists require a contraction in quantities extracted. Here we consider the possible transition to a hypothetical sustainable path, a
path requiring non-decreasing consumption. We consider then the weak form of sustainable development. We note that a rapid re-direction can be extremely costly in terms of consumption foregone and this leads us to consider various more gradual transitions. This in turn leads us to reflect on welfare criteria for the case of significant changes in consumption levels across generations. We turn to an overview of the Solow–Hartwick approach to sustainable oil use. Dasgupta and Heal (1979, pp. 199–205 and pp. 288–308) remains a good exposition of the central Cobb– Douglas case. John Hartwick (1977) showed the saving for the Solow (1974) model must involve investing current exhaustible resource returns in reproducible capital in order to
⁎ Department of Economics, Queen's University, Kingston, ON, Canada K7L 3N6. Tel.: +1 613 533 2270; fax: +1 613 533 6668. E-mail addresses: [email protected], [email protected]. 1 Thanks to the Good Family Visiting Faculty Research Fellowship for financial support. 2 The author is grateful to John M. Hartwick and anonymous referee for very useful comments and advice. 0921-8009/$ - see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.ecolecon.2007.02.020
EC O L O G IC A L E C O N O M IC S 6 4 ( 2 0 07 ) 18 6 –1 93
maintain constant per capita consumption over time. We review this for the case of a Cobb–Douglas technology. For the case with zero population growth, no capital depreciation, no technological progress, and zero extraction cost, we have output q = f (k, r) = kα rβ where k is produced capital, r — current resource use, r = − S˙ , S — resource stock, S˙ = dS / dt, α, β ∈ (0, 1) are constants. Prices of capital and the resource are fk = αq / k, fr = βq / r where fx = ∂f / ∂x. Per capita consumption is c = q − ˙k . The Hartwick saving rule implies c = q − rfr or, substituting for fr, c = q(1 − β ), which means that instead of ċ = 0 we can check q˙ = 0. From Hotelling rule f˙r /fr =fk we have αβq /k +r˙ ( β − 1) /r =fk =αq / k which yields
¼ −aq=k: r=r
ð1Þ
Then
¼ a k=k þ b r=r ¼ bðaq=k þ r=rÞ ¼ 0; q=q
ð2Þ
which means that we really have q˙ = ċ = 0 or q = const. Then rfr = ˙ = βq = const for deriving k(t) and (1) for βq = const and we have k deriving r(t). We can find two constants of integration k0 for k (t) = k0 + βqt and the constant of equation r˙ / r = −1 / (k0 / αq + βt / α) using initial conditions r(0) = r0 and s(0) = s0, where s0 is the given resource stock Z lwhich must be used for production over infinite time: s0 ¼ rðtÞdt. Then we have 0
rðtÞ ¼ r0 ½1 þ r0 bt=s0 ða−bÞ−a=b ;
ð3Þ
where α N β (Solow condition) and
¼ −sðtÞ rðtÞ ̈ ¼ −ar20 =s0 ða−bÞ½1 þ r0 bt=s0 ða−bÞ−ðaþbÞ=b :
ð4Þ
Since we assume that our economy depends on the resource essentially, we obtain path r(t), asymptotically approaching zero and the path of extraction changes r˙ (t) (or negative acceleration of stock s(t) diminishing) also approaching zero, but starting from the negative value r˙0 = −αr20 / [s0(α − β)]. Note, that path (3), asymptotically approaching zero, is necessary, but not sufficient condition of following Hartwick rule for Cobb–Douglas economy under the Hotelling rule assumption. By definition of f(k, r) it can be seen, that if economy is extracting resource in accord with (3) and resource rent is being consumed (total investments are less than resource rent), then q(t) and c(t) are asymptotically approaching zero, but from a greater starting value c(0). Assuming that our economy has some “additional” saving, besides resource rent, it is possible to relax the assumption of zero population growth (as in Stiglitz, 1974; Asheim et al., 2005), or zero capital depreciation. But in any case, if we assume, that
resources and capital exceeds unity. This implies that the resource can become inessential. Other investigations (Fuss, 1977; Magnus, 1979, and partly in Halvorsen and Ford, 1979) show that energy and capital are strong complements rather than substitutes (elasticity is less than unity). Some researches find that the elasticity is close to unity (Griffin and Gregory, 1976; Pindyck, 1979). In any case empirical evidence is not a proof and as Dasgupta and Heal (1979, p. 207) noted “Past evidence may not be a good guide for judging substitution possibilities for large values of k/r”. We will assume that for the world economy oil is essential, especially taking into account that no adequate immediate substitutes are available for transportation fuels, a main area of oil use (Heinberg, 2003; Nemoto, 2005). However, as we can see, e.g., from oil extraction data in December issues of Oil and Gas Journal, rates of extraction are in fact both growing on the world level (see Fig. 1 before the year 2005) and for the leading oil producers, not declining. Consider now possible regime switches. Assume that the government after a period of oil rent consumption and increasing extraction decided to conform to the principle of intergenerational justice and to switch at t0 to some sustainable path of saving, e.g., to the Hartwick rule. An example with α = 0.3 and β = 0.05 gives us r(t) and r˙ (t) for world oil extraction in Figs. 1 and 2 after the year 2005 (dotted lines). An abrupt switch to the Hartwick rule means that people in oil-producing countries must instantly forget about this principal source of income and in a moment substantially re-structure the consumption and saving of their citizens. Moreover, countries must instantly reorganize their economies, because of the sharp decrease of consumption, which in turn leads to a decrease in production, a possible increase in unemployment, a further decrease of demand and so on. Thus, for an economy not following the Hartwick rule, the sudden invocation of intergenerational justice creates the dilemma of choosing between two awkward futures: diminishing consumption to zero in the future because of the inevitable shortage of essential exhaustible resources or diminishing consumption to a sustainable level right from the moment of switching to the Hartwick rule. Solow's model implies that oil rent is being invested from the very beginning and that there is no time gap between the moment of oil extraction and correspondent increase of reproducible capital according to the Hartwick rule. In reality,
1) economy at every instant of time depends on resource (even if we gradually introduce substitute technologies and this dependence asymptotically approaches zero), and 2) we really want to maintain non-decreasing per capita consumption, then rate of extraction r(t) must tend to zero. Capital–resource substitution is a fundamental topic in energy economics and there is empirical evidence (Nordhaus and Tobin 1972; Pindyck, 1979) which can support the assumption that the elasticity of substitution between natural
187
Fig. 1 – World oil extraction (mln t/year): historical data (before 2005); Hartwick curve (dotted); LA curve (solid).
188
EC O LO GIC A L E CO N O M ICS 6 4 ( 2 00 7 ) 1 8 6 –1 93
do not react on “natural experiments” such as the Reagan cuts in taxes (Poterba, 1988). Hence, the problem of switching realistically to a sustainable path of essential resource extraction must take into account that:
Fig. 2 – Extraction accelerations: historical data (before 2005); Hartwick curve (dotted); LA curve (solid).
in order to sustain non-decreasing output with the same structure, we must invest at least part of oil profit into development of oil-substituting technologies. In other words, we must create an “anti-oil market” with the oil rent. And under this assumption the model of instant investment cannot be really adequate because of the difficulties of a rapid re-structuring. Historical examples show that the development and the introduction of coal-based technologies took decades despite the obvious benefit of the new technologies for economy. The same can be said about the switch from a coal to an oil economy. Now we must consider the problem of switching to technologies, based on renewable resources not because they are economically more preferable but just because of anticipated shortage of profitable but exhaustible raw materials. And this process will occur over decades, not months. The second dimension of the difficulty of an instant switch to the Hartwick rule is the requirement of an abrupt and very substantial change of saving patterns for oil producing countries. For background, we can compare nonrenewable resource profit only from oil with the total amount of investments for a selection of countries. For example, oil gives Kuwait about 50% of GDP but gross fixed investments are only 6.6% of GDP. For Saudi Arabia these numbers are 45% and 16.3%, United Arab Emirates — 30% and 20.7%, Venezuela — 33% and 23.8%.3 Of leading oil producers, only Norway can boast numbers (about 18.6%4) almost coincident, because of investing oil rent to Petroleum Fund. There is however no direct connection between this Fund and investment in the development of oil-substituting technologies. There is evidence that at least in the short run saving rate is very stable, and it is much more difficult to change it instantly, than to change a government policy. The well-known empirical research of Kuznets (1946) tells us that consumer behavior is very persistent over time despite changes of governments and government policy. Subsequent analyses, for example, the work of Duesenberry (1949), tried to explain this phenomenon, and later papers examined why consumers
1) the path must have a period of a gradual slow-down in the rate of extraction; 2) there is a time lag between the moment of resource rent investment and correspondent increase in capital; 3) there is a non-zero period length for changing saving patterns from resource rent consumption to resource rent investment. In this paper we suppose, for simplicity, that the third problem is already solved (as in Norway), and also we will temporarily neglect the influence of the second factor. So, we will concentrate on the question of the construction the trajectories for the transition period using various optimality criteria and examine consumption behavior along the paths.
2.
Formulations of the problem
It is natural for the government, switching to a decreasing path of the resource extraction, to try to construct this path in such a way that the short-run negative social impacts along the path are minimal. According to (2) GDP percent change for our economy is
¼ a k=k þ b r=r: q=q
ð5Þ
Since the economy is going to enter the period with r˙ b 0, the government can try to minimize the negative influence of the decreasing extraction on output q. We can apply a mechanical analogy, comparing the rate of extraction with the speed of some vehicle. The most unpleasant feelings in a vehicle trip occur when the vehicle has to reduce the speed very quickly. Then, using this analogy, we can consider 1) the criterion of “smooth breaking”, which gives the optimal path with the minimal in absolute value peak of negative acceleration A(t) = r˙ (t). In other words we have a problem Z l rðtÞdt ¼ s0 ; ð6Þ F1 ðrðtÞÞ ¼ mint AðtÞY max; s:t: rð0Þ ¼ r0 ; rðtÞ
where the last condition means that the resource is essential. However, the negative acceleration r˙ (t) influences the GDP percent change not directly but as a numerator of the fraction r˙ (t) / r(t). Then it looks more relevant for economics consider 2) the criterion of “minimum shock on GDP”, namely
max F2 ðrðtÞÞ ¼ min½ rðtÞ=rðtÞY t
3
Source of information: http://www.cia.gov/cia/publications/ factbook/docs/profileguide.html (March 2006). 4 Source of information: http://www.ssb.no/en/indicators/ (March 2006).
0
rðtÞ
ð7Þ
subject to the same initial conditions. Finally, we can take into account the first term in Eq. (5) which can shift the point of maximum influence of negative acceleration, or, in other words, we can apply the maximin
EC O L O G IC A L E C O N O M IC S 6 4 ( 2 0 07 ) 18 6 –1 93
criterion to the whole expression q˙ /q. Hence, we will consider as the third criterion 3) the criterion of the “maximin GDP percent change”:
max : F3 ðrðtÞÞ ¼ min½a kðtÞ=kðtÞ þ b rðtÞ=rðtÞY t
ð8Þ
rðtÞ
with the same constraints.
3.
Solving the problem
The transition path can be constructed in the same class of rational functions as the Hartwick curve (3). The difference is in the numerator, which in the expression for acceleration A = r˙ must depend on t with a negative coefficient to control “smooth breaking” in the neighborhood of t = 0. Namely, A(t) must be in the form of
¼ ðA0 þ btÞ=ð1 þ ctÞd ; Aðt; b; c; dÞ ¼ rðtÞ
ð9Þ
where b b 0, c N 0, d N 1 (for convergence A(t) → − 0 with t → ∞). Corresponding to Eq. (9) r(t) has a dependence on b, c, and d in rðtÞ ¼ f−½A0 þ b=½cðd−2Þ=½cðd−1Þ þ bt=½cð2−dÞg=ð1 þ ctÞd−1 : Note, that a constant of integration for r˙ (t) = A(t) must be Rl zero for the convergence of 0 rðtÞdt and also for the convergence, d actually must be greater than 3. Then we have r0 = − [A0 + b / [c(d − 2)]] / [c(d − 1)], which can be used to express b: bðc; dÞ ¼ −cðd−2Þ½r0 cðd−1Þ þ A0 ;
ð10Þ
and then the transition curve has a dependence on c and d in rðtÞ ¼ r0 f1 þ ½cðd−1Þ þ A0 =r0 tg=ð1 þ ctÞd−1 :
ð11Þ
Coefficient c can be Z lexpressed from the condition that rðtÞdt: resource is finite s0 ¼ 0
Z
l
s0 =r0 ¼
1−d
ð1 þ ctÞ
Z
dt þ ½cðd−1Þ þ A0 =r0
0
l
t=ð1 þ ctÞd−1 dt
0
¼ ½1 þ fr0 cðd−1Þ þ A0 g=fr0 cðd−3Þg=½cðd−2Þ; which means that c is a solution of quadratic equation c2 s0 =r0 −2c=ðd−3Þ−A0 =½r0 ðd−3Þðd−2Þ ¼ 0: The only relevant root (because we are looking for c N 0) is 0:5
cðdÞ ¼ ½r0 =ðd−3Þ þ fr20 =ðd−3Þ2 þ s0 A0 =½ðd−3Þðd−2Þg
=s0 :
ð12Þ
Hence, we have a single independent parameter d which defines the shape of the curve (including its peak) and we can use this parameter as a control variable in some selected optimization problem F½rðt; dÞY max
189
1 ton of crude oil = 7.3 barrel. For our short-run criteria F1 [r(t, d)], F2 [r(t, d)] and F3 [r(t, d)] we constructed the following “optimal” paths of extraction: 1) the “smooth breaking” or the Least Acceleration (LA) curve (the solution of problem (6), see Bazhanov, 2006) has d = 36.8837 which implies c = 0.001459 and b = − 0.0136. Plots of r(t) and A(t) are on the Figs. 1 and 2 after the year 2005 (solid lines). This path has a peak of extraction at tmax = 5.87 with the rmax = 3.8016 bln t /year. Numerical estimation of the qualitative behavior of GDP percent change (Eq. (5)) along this path yields results similar to those for in r˙(t) on Fig. 2 after the year 2005 (solid line). It has a minimum [q˙ / q]min = − 0.122% at tmin = 271, then it is negative in the long run, approaches zero, but never exceeds it. Note, that for the LA curve d → ∞ with A0 → + 0 and we failed to construct this curve for A0 ≤ 0.06. 2) The “minimum shock on GDP” or the MS curve (the solution of problem (7)) can be obtained by solving the first order condition d(r˙(t) / r(t)) / dt = 0 which gives us the only relevant root (since we are looking for t⁎ N 0) corresponding to a minimum in t: t⁎ ðdÞ¼−ffA0 þ½A0 −bðcðdÞ; dÞ=cðdÞgfA0 þr0 cðdÞðd−2Þg0:5 g=bðcðdÞ;dÞ: Then we substitute for t⁎(d) and solve the first order condition d [r˙(t⁎(d)) / r(t⁎(d))] / d [d] = 0. Numerical investigation gives us the single root which corresponds to the maximum in d: d = 6.1178 with c and b defined by Eqs. (10) and (12). This curve qualitatively resembles the LA curve, but it has a peak of extraction earlier, at tmax = 3.52 with the rmax =3.7162 bln t /year. The MS curve, unlike the LA curve, can be constructed for A0 b 0. Another important difference is that our numerical estimations yield q˙ /q asymptotically approaching zero, always remaining positive. 3) The “maximin GDP percent change” or the MM curve can be constructed as a numerical solution of problem (8) only for the cases when we have an inevitable short-run minimum in GDP percent change (period of crisis). For our numerical examples we had 2 different cases: a) GDP percent change is always positive, monotonically decreasing, and asymptotically approaching zero; b) GDP percent change decreases to some negative value (minimum) and then asymptotically approaches zero always remaining negative. In the first case there is no solution for problem (8) because the minimum does not exist. In the second case, if we have the constraints (technical restrictions), which do not allow us to realize the sustainable path, then problem (8) has a corner solution along which the consumption decreases to zero.
4.
Consumption along the transition curves
d
which can be connected with the short- or long-run policy in output or in consumption. In our numerical examples we used A0 = 0.08 and as world oil reserves and extraction on January 1, 2006 (Oil & Gas J., 2005, 103, 47: p. 25.): r0 = 71, 793.8 [1000 bbl/ day] × 365 = 26,204,737 [1000 bbl/year] (or 3.58969 bln t/year); s0 = 1,292,549,534 [1000 bbl] (or 177.06 bln t). We use coefficient
We are going to check if our short-run criteria are consistent with our long-term goal, namely, non-decreasing consumption. We will examine, for simplicity, the case when all resource rent is always invested in capital and there are no time lags between the moments of investment and the corresponding capital increase. The only reason for the change
190
EC O LO GIC A L E CO N O M ICS 6 4 ( 2 00 7 ) 1 8 6 –1 93
in the pattern of extraction is that sustainable path of the essential resource extraction must be decreasing and asymptotically approaching zero. Note, that the constant per capita consumption over time for our formulation is the result of 1) total investment of oil rent in capital (with no time lag) and 2) fulfillment of the Hotelling rule. The rational transition curves (11) satisfy only the first condition unlike the Hartwick curve (3) which is derived from the Hotelling rule and so satisfies it identically. Hence, to examine consumption behavior along some path we should check the fulfillment of the Hotelling rule along this curve. For ˙ +frr˙. Then f˙r =βd (q /r) / dt =β [ fkk ˙ /r +frr˙ /r] − the standard case q˙ =fk k ˙ /kr+βqr˙ /r 2] / (βq )− βr˙q /r2. Dividing on fr =βq /r we have f˙r/fr =rβ [αk ˙ /k − (1−β )r˙/r or r˙/r =αk
f r =fr ¼ fk ½ k=q−ð1−bÞk r=ðaqrÞ:
ð13Þ
Just to check, we can see, that for the Hartwick curve [·] ≡ 1, because r˙ / r = − αq / k and ˙k = βq. Hence, if [·] b 1, then q˙ N 0, because f˙r / fr b fk, which follows −r˙ / r b αq / k or αq / k + r˙ / r N 0. And the latter, using expression in the left hand side of Eq. (2), means q˙ N 0. In the same way, [·] N 1 follows q˙ b 0 and, in general, sgn q˙ = sgn{1 − [·]}. So, to examine the long-run consumption c = (1 − β)q along the transition curve, we can check asymptotic behavior of [·] in Eq. (13).
Proposition 1. If an economy with technology q =kαrβ is such that 1) resource rent is completely invested in capital; 2) there is no time lag between the moment of investment and the corresponding increase in capital; 3) rate of extraction r(t) is such that
¼ ðA0 þ btÞ=ð1 þ ctÞd ; b b 0; c N 0; d N 3; rðtÞ then the asymptotic behavior in output q for different β is: bðd−2Þz1; ¼ −1; lim sgn qðtÞ sgnLðd; a; bÞ; bðd−2Þ b 1; tYl
ð14Þ
where L(d, α, β) = [α − β(d − 2)] / [α − αβ(d − 2)]. Proof of the Proposition is in Bazhanov (2006, Appendix). The simplified expression for L(d, α, β ) was obtained by direct substitution of the expressions (10) and (12) for b(c(d), d) and c(d). Corollary 1. Under the assumption of the Proposition 1 the consumption c(t) is 1) asymptotically decreasing if d N α / β + 2; 2) asymptotically constant if d = α / β + 2; 3) asymptotically growing if 3 b d b α / β + 2 where d is the control variable for the transition curve (Eq. (11)). Proof. Note that for β (d − 2) b 1 or d b 1 / β + 2 denominator of L(d, α, β ) is positive. Then the sign of L(d, α, β ) is defined by nominator. Since c(t) = (1 − β )q(t) and sgn ċ = sgn q˙ then substituting the expressions for d into L(d, α, β ) in Eq. (14) we obtain the assertion of the Corollary. □ In the case when d ≥ 1 /β + 2 or β(d − 2) ≥ 1 we define the sign of dc/dt by the first line in Eq. (14) which is included in the first case of the Corollary.
5.
Numerical examples
In order to construct numerical estimations of the consumption paths in different cases we used the procedure described in Bazhanov (2007). For the example with given r0, A0 for the world oil extraction, α =0.3, β =0.05, and optimal (in sense of criterion F1 ) d⁎, b(d⁎), and c(d⁎) we have β (d −2) = 1.74 N 1 which means, that consumption and output decrease in the long run along the LA curve (Fig. 3). For α =0.2, β =0.05 (estimations from Nordhaus and Tobin, 1972) we also have consumption decreasing to zero in finite time. We can see from the Corollary 1 that there are sets of α and β for which, given d⁎, we have asymptotically constant consumption along the LA curve. For example, L(d⁎, 0.697, 0.02) =L (d⁎, 0.872, 0.025) =0. Selection of different values for (α, β) for which limt → ∞ ċ = 0 makes some sense if we wish to get a feeling for how far some real extraction path can be from the stable one, given that we do not know the true values of α and β. In fact it is unrealistic to speak about the short-term regulation of these magnitudes by the government's decisions. Our examples make the path of resource extraction look plausibly controllable. Corollary 1 shows that we can fit our transition curve (the single free parameter d) using some long-run welfare criterion, e.g., constant consumption over time (asymptotically constant consumption) instead of the criterion F1. An example with α = 0.3 and β = 0.05 gives us d = 8.0, c = 0.00967, b = −0.01873. In this case the oil peak must be closer, namely, at t = 4.27 with rmax = 3.7433. The behavior of GDP percent change along this path is qualitatively similar to the one along the LA curve. It has the minimum [q˙ / q]min = −0.046% at tmin = 63.15 after which it increases and asymptotically approaches zero, always remaining negative. To check that the level of consumption along this curve, which we will call the “Transition Constant Consumption” (TCC) curve, is far enough from zero, we can solve the differential equation for k(t) numerically (the numerical solution was obtained in Maple by the procedure rkf 45) and then plot c(t) (Fig. 4). The constant consumption for the t, big enough, is cconst = 2.42801. GDP percent change along this curve is always positive and asymptotically approaches zero. Consumption along the MS curve, according to the Corollary, must be asymptotically increasing since for these curves we have d b α / β + 2 = 8. Numerical solution gives us the unlimited monotonically growing consumption (Fig. 5) with the same c0, but with cmax less than for TCC curve: cmax = 3.088 at t = 14.55. After cmin = 2.956 at tmin = 306 consumption grows and exceeds c0 after t = 1037. The behavior of consumption along the MM curve is similar. It has cmax = 3.084 at t = 12.45 and cmin = 2.9986 at tmin = 176.5. GDP percent change decreasing to zero but with slower rate than for the TCC curve. The only “cost" of this growth along these curves, compared with the TCC curve are: 1) cmax is a little bit less, than for the TCC curve (3.088 and 3.084 vs. 3.0955); 2) is that the peak of oil extraction must be earlier (t = 3.52 vs. t = 4.27) and the maximum extraction must be less than for TCC curve (3.7162 vs. 3.7433).
EC O L O G IC A L E C O N O M IC S 6 4 ( 2 0 07 ) 18 6 –1 93
Fig. 3 – Consumption decrease along the LA curve.
The example is an illustration of the answer to the question, “what is worse?”: a small decrease of consumption in the present or the depriving of oneself and (or) one's descendants of any prospects for improving their lives in the future. According to Rawls's maximin principle, extraction along the MS curve in combination with the Hartwick saving rule is obviously a pattern of overinvestment. But actually Rawls (1971, p. 291) objected to applying his maximin principle to the questions of justice among generations because of unacceptable consequences. In Bazhanov (2006) we offer a generalized approach for the defining a “relevant position” in Rawls's theory which implies that we must take into account not only the values of some indicators of life quality in the present but also their time changes or differences in con-
Fig. 4 – Consumption along the TCC curve.
191
Fig. 5 – Unlimited consumption growth along the MS curve.
sumption from previous years. The review of literature on using the “comparison utility” can be found, e.g., in Carroll et al. (1997). If the recent past has a big weight in the consumption prehistory and we substitute the derivative ċ for the difference c(t) − c(t − 1) then the utility in its simplest form is u = u ( c, ċ ). Applying the maximin principle, e.g., for u in additive form we have u (c, ċ) = wc(t) + (1 − w)ċ(t) = γ = const for any t N 0, w ∈ [0, 1] which with c0 = c(0) follows cðtÞ ¼ ½g−expf−wt=ð1−wÞgðg−c0 wÞ=w
ð15Þ
or we have a case of limited growth (Fig. 6) for γ N c0w and Eq. (15) is desirable in the sense “…that an extra bit of consumption at t is more valuable than the same extra bit at t + 1, since individuals will, in any case, have more consumption at t + 1” (Dasgupta and Heal, 1979, p. 284). Observe also that Eq. (15) describes a limited decline for γ b c0w and a constant consumption (as in the Hartwick rule) for γ =c0w. We do not claim that everybody favors this type of just path, particularly when it is apparent that rather small sacrifices in present can bring slow but unlimited growth in the long run (Fig. 5). For those, who prefer this form of intertemporal distribution, the more appropriate consumption utility function would be the function with essential factors, e.g., the Cobb–Douglas case. Then the rule of intertemporal distribution is cwċ1−w = γ = const which gives us c(t) =c0 (1 + μt)φ where c0 = c(0), μ = (γ / c0)1/φ / φ, φ = 1 − w or a pattern of unlimited (quasiarithmetic, Asheim et al., 2005, p. 5) growth which (for w close to 1) looks like the curve on Fig. 5. If we consider u(c, ċ) = cw1ċw2 and assume that w1, w2 can be negative, e.g., w1 = − 1, w2 = 1 (then u is the consumption percent change ċ / c which in our case is equal to GDP percent change q˙ / q), then the maximin principle implies q˙ / q = const which gives us q = q0et as an optimal GDP path. Since exponential path for GDP is unrealistic with finite exhaustible essential resources (with no technological progress), it raises a question about using GDP percent change as an indicator for sustainable economic development.
192
EC O LO GIC A L E CO N O M ICS 6 4 ( 2 00 7 ) 1 8 6 –1 93
There are some questions raised by and limitations of the model we have presented.
Fig. 6 – Generalized “fair Rawlsian growth” (Eq. (15)), w = 0.5.
In general, utility can be written as a CES function, or as a function with a variable elasticity where the elasticity parameter and w are to be chosen by the government. Then the specific just saving principle can be deduced for the specific utility function and the transition path of extraction can be adjusted to approach as close as possible (depending on constraints) the asymptotically optimal (in the long run) pattern of the intertemporal distribution of consumption.
6.
Concluding remarks
Numerical analysis of long-run consumption along the rational transition paths supports the main assertions of our Proposition and Corollary. The Corollary claims that depending on the shape of the transition curve of the resource extraction, which in our formulation is defined by the single control variable d, the long-run consumption can exhibit various patterns of behavior. It can decrease to zero, asymptotically approach some positive constant, or grow infinitely. This means that the transition curve can be adjusted to satisfy any desirable qualitative behavior of consumption in accord with the various optimality criteria for the long run. However, the example for the LA curve shows that some, at first glance plausible, short-run criteria can run counter to the government's primary goal of pursuing intergenerational justice and sustainable development in the long run. It is interesting that another formulation of the short-run criterion, namely, the minimum short-run shock on GDP, leads to the long-run unlimited growth of consumption, which is consistent with the long-run government concerns. The possibility of unlimited consumption growth for the Cobb–Douglas economy was considered in Dasgupta and Heal (1979). We have shown that growing consumption and unlimited growth particularly are consistent with the Rawls's maximin principle if we take into account the consumption prehistory in the form of a multiplicative utility function, one defined on the “relative personal position”.
(1) We examined transition curves as interior solution neglecting restrictions imposed by technical possibilities and difficulties connected with the saving rate changing. Constraints on the speed of changing saving behavior can restrict us from implementing the paths of extraction with asymptotically constant or growing consumption in the long run. (2) There remains the question of the path stability when technology is changing. (3) Transition curves can be constructed for a different class of functions. We also assumed that: (4) Cost of extraction is zero and population is constant though it would be interesting to consider the problem of transition when these values are allowed to vary. (5) There is no time lag between the moment of oil extraction and corresponding addition to capital. A lag would be present if we had oil rent invested in the development of alternative technologies. (6) All oil rent is invested in reproducible capital. In the real world, this is not observed and we should consider some period of increasing investments along some smooth (maybe hysteresis-like) curves and examine the influence of this curve on long-run consumption behavior. Each of these questions merits careful consideration in separate papers.
REFERENCES Asheim, G.B., Buchholz, W., Hartwick, J.M., Mitra, T., Withagen, C., 2005. Constant savings rates and quasi-arithmetic population growth under exhaustible resource constraints. CESifo Working Paper No 1573. October 2005. Bazhanov, A.V., 2006. Decreasing of Oil Extraction: Consumption behavior along transition paths. MPRA Paper No. 469, posted October 16, 2006. Online at http://mpra.ub.unimuenchen.de/ 469/. Bazhanov, A.V., 2007. The peak of oil extraction and a modified maximin principle. MPRA Paper No. 2019, posted March 07, 2007. Online at http://mpra.ub.uni-muenchen.de/2019/01/ MPRA_paper_2019.pdf. Carroll, C.D., Overland, J., Weil, D.N., 1997. Comparison utility in a growth model. J. Econ. Growth 2, 339 – 367. Dasgupta, P., Heal, G., 1979. Economic Theory and Exhaustible Resources. Cambridge University Press, Cambridge, Eng. 501 pp. Duesenberry, J., 1949. Income, Saving and Theory of Consumer Behavior. Harvard University Press, Cambridge, MA. 128 pp. Fuss, M.A., 1977. The demand for energy in Canadian manufacturing. J. Econom. 5, 89 – 116. Griffin, J.M., Gregory, P.R., 1976. An intercountry translog model of energy substitution responses. Am. Econ. Rev. 66, 845 – 857. Halvorsen, R., Ford, J., 1979. Substitution among energy, capital and labor inputs in American manufacturing. In: Pindyck, R. (Ed.), Advances in the Economics of Energy and Resources, vol. 1, pp. 51 – 75. Hartwick, J.M., 1977. Intergenerational equity and the investing of rents from exhaustible resources. Am. Econ. Rev. 67, 972 – 974. Heinberg, R., 2003. The Party's Over. New Society Publishers, Gabriola Island, Canada. 274 pp.
EC O L O G IC A L E C O N O M IC S 6 4 ( 2 0 07 ) 18 6 –1 93
Kuznets, S., 1946. National Product Since 1869. National Bureau of Economic Research, NY. 239 pp. Magnus, J.R., 1979. Substitution between energy and non-energy inputs in the Netherlands 1950–1976. Int. Econ. Rev. 20, 465 – 484. Nemoto, K., 2005. High oil prices dampening Asia-Pacific product demand. Oil Gas J. 103, 58 – 62. Nordhaus, W.D., Tobin, J., 1972. Is economic growth obsolete? Economic Growth, 5th Anniversary Colloquium, vol. V. National Bureau of Economic Research, New York, pp. 1 – 80. Pindyck, R.S., 1979. Interfuel substitution and the demand for energy: an international comparison. Rev. Econ. Stat. 61, 169 – 179.
193
Poterba, J., 1988. Are consumers forward looking? Evidence from fiscal experiments. Am. Econ. Rev. 78, 413 – 418. Rawls, J., 1971. A Theory of Justice. Belknap Press of Harvard University Press, Cambridge, MA. 607 pp. Solow, R.M., 1974. Intergenerational equity and exhaustible resources. Rev. Econ. Stud. 41, 29 – 45. Stiglitz, J.E., 1974. Growth with exhaustible natural resources: efficient and optimal growth paths. Rev. Econ. Stud. 41, 123 – 137. United Nations (UN), 1993. Agenda 21. United Nations, New York.
a v a i l a b l e a t w w w. s c i e n c e d i r e c t . c o m
w w w. e l s e v i e r. c o m / l o c a t e / e c o l e c o n
ANALYSIS
The transition to an oil contraction economy Andrei V. Bazhanov ⁎,1,2 Institute of Mathematics and Computer Science, Far Eastern National University, Vladivostok, Russia Department of Economics, Queen's University, Kingston, ON, Canada K7L 3N6
AR TIC LE I N FO
ABS TR ACT
Article history:
A normative analysis of the problem of optimal extraction of a nonrenewable resource is
Received 26 October 2006
considered. The economy depends on the essential nonrenewable resource and the rate of the
Received in revised form
resource extraction is increasing over time. At some point the government gradually switches
31 January 2007
to a sustainable (in sense of non-decreasing consumption over time) pattern of the resource
Accepted 11 February 2007
extraction. Different approaches are offered for the construction of the paths of switching to
Available online 29 March 2007
decreasing resource use. Some seemingly attractive short-run policies of switching to decreasing extraction can run counter to long-run criteria. If we consider the maximin
Keywords:
principle, applied to the negative shock on the output percent change, as the short-run
Nonrenewable resource
criterion, then the optimal transition path can be consistent with the long-run government
Intergenerational justice
goals. It is shown analytically and numerically that there are values of parameters for the
Hartwick rule
transition paths of extraction that consumption along these paths is asymptotically constant
Optimal path of extraction
or infinitely growing. A new approach to the Rawlsian maximin criterion which allows for
Generalized Rawlsian criterion
growth of consumption is offered. © 2007 Elsevier B.V. All rights reserved.
JEL classification: Q32; Q38
1.
Introduction
One of the important concerns of the United Nations' Agenda 21 is that “…the major cause of the continued deterioration of the global environment is the unsustainable pattern of consumption and production…” and that “Special attention should be paid to the demand for natural resources generated by unsustainable consumption…” (United Nations, 1993, Chapter 4). Observed world oil extraction has been increasing rapidly in recent years and some sustainable (in the sense of weak sustainability) programs proposed by economists require a contraction in quantities extracted. Here we consider the possible transition to a hypothetical sustainable path, a
path requiring non-decreasing consumption. We consider then the weak form of sustainable development. We note that a rapid re-direction can be extremely costly in terms of consumption foregone and this leads us to consider various more gradual transitions. This in turn leads us to reflect on welfare criteria for the case of significant changes in consumption levels across generations. We turn to an overview of the Solow–Hartwick approach to sustainable oil use. Dasgupta and Heal (1979, pp. 199–205 and pp. 288–308) remains a good exposition of the central Cobb– Douglas case. John Hartwick (1977) showed the saving for the Solow (1974) model must involve investing current exhaustible resource returns in reproducible capital in order to
⁎ Department of Economics, Queen's University, Kingston, ON, Canada K7L 3N6. Tel.: +1 613 533 2270; fax: +1 613 533 6668. E-mail addresses: [email protected], [email protected]. 1 Thanks to the Good Family Visiting Faculty Research Fellowship for financial support. 2 The author is grateful to John M. Hartwick and anonymous referee for very useful comments and advice. 0921-8009/$ - see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.ecolecon.2007.02.020
EC O L O G IC A L E C O N O M IC S 6 4 ( 2 0 07 ) 18 6 –1 93
maintain constant per capita consumption over time. We review this for the case of a Cobb–Douglas technology. For the case with zero population growth, no capital depreciation, no technological progress, and zero extraction cost, we have output q = f (k, r) = kα rβ where k is produced capital, r — current resource use, r = − S˙ , S — resource stock, S˙ = dS / dt, α, β ∈ (0, 1) are constants. Prices of capital and the resource are fk = αq / k, fr = βq / r where fx = ∂f / ∂x. Per capita consumption is c = q − ˙k . The Hartwick saving rule implies c = q − rfr or, substituting for fr, c = q(1 − β ), which means that instead of ċ = 0 we can check q˙ = 0. From Hotelling rule f˙r /fr =fk we have αβq /k +r˙ ( β − 1) /r =fk =αq / k which yields
¼ −aq=k: r=r
ð1Þ
Then
¼ a k=k þ b r=r ¼ bðaq=k þ r=rÞ ¼ 0; q=q
ð2Þ
which means that we really have q˙ = ċ = 0 or q = const. Then rfr = ˙ = βq = const for deriving k(t) and (1) for βq = const and we have k deriving r(t). We can find two constants of integration k0 for k (t) = k0 + βqt and the constant of equation r˙ / r = −1 / (k0 / αq + βt / α) using initial conditions r(0) = r0 and s(0) = s0, where s0 is the given resource stock Z lwhich must be used for production over infinite time: s0 ¼ rðtÞdt. Then we have 0
rðtÞ ¼ r0 ½1 þ r0 bt=s0 ða−bÞ−a=b ;
ð3Þ
where α N β (Solow condition) and
¼ −sðtÞ rðtÞ ̈ ¼ −ar20 =s0 ða−bÞ½1 þ r0 bt=s0 ða−bÞ−ðaþbÞ=b :
ð4Þ
Since we assume that our economy depends on the resource essentially, we obtain path r(t), asymptotically approaching zero and the path of extraction changes r˙ (t) (or negative acceleration of stock s(t) diminishing) also approaching zero, but starting from the negative value r˙0 = −αr20 / [s0(α − β)]. Note, that path (3), asymptotically approaching zero, is necessary, but not sufficient condition of following Hartwick rule for Cobb–Douglas economy under the Hotelling rule assumption. By definition of f(k, r) it can be seen, that if economy is extracting resource in accord with (3) and resource rent is being consumed (total investments are less than resource rent), then q(t) and c(t) are asymptotically approaching zero, but from a greater starting value c(0). Assuming that our economy has some “additional” saving, besides resource rent, it is possible to relax the assumption of zero population growth (as in Stiglitz, 1974; Asheim et al., 2005), or zero capital depreciation. But in any case, if we assume, that
resources and capital exceeds unity. This implies that the resource can become inessential. Other investigations (Fuss, 1977; Magnus, 1979, and partly in Halvorsen and Ford, 1979) show that energy and capital are strong complements rather than substitutes (elasticity is less than unity). Some researches find that the elasticity is close to unity (Griffin and Gregory, 1976; Pindyck, 1979). In any case empirical evidence is not a proof and as Dasgupta and Heal (1979, p. 207) noted “Past evidence may not be a good guide for judging substitution possibilities for large values of k/r”. We will assume that for the world economy oil is essential, especially taking into account that no adequate immediate substitutes are available for transportation fuels, a main area of oil use (Heinberg, 2003; Nemoto, 2005). However, as we can see, e.g., from oil extraction data in December issues of Oil and Gas Journal, rates of extraction are in fact both growing on the world level (see Fig. 1 before the year 2005) and for the leading oil producers, not declining. Consider now possible regime switches. Assume that the government after a period of oil rent consumption and increasing extraction decided to conform to the principle of intergenerational justice and to switch at t0 to some sustainable path of saving, e.g., to the Hartwick rule. An example with α = 0.3 and β = 0.05 gives us r(t) and r˙ (t) for world oil extraction in Figs. 1 and 2 after the year 2005 (dotted lines). An abrupt switch to the Hartwick rule means that people in oil-producing countries must instantly forget about this principal source of income and in a moment substantially re-structure the consumption and saving of their citizens. Moreover, countries must instantly reorganize their economies, because of the sharp decrease of consumption, which in turn leads to a decrease in production, a possible increase in unemployment, a further decrease of demand and so on. Thus, for an economy not following the Hartwick rule, the sudden invocation of intergenerational justice creates the dilemma of choosing between two awkward futures: diminishing consumption to zero in the future because of the inevitable shortage of essential exhaustible resources or diminishing consumption to a sustainable level right from the moment of switching to the Hartwick rule. Solow's model implies that oil rent is being invested from the very beginning and that there is no time gap between the moment of oil extraction and correspondent increase of reproducible capital according to the Hartwick rule. In reality,
1) economy at every instant of time depends on resource (even if we gradually introduce substitute technologies and this dependence asymptotically approaches zero), and 2) we really want to maintain non-decreasing per capita consumption, then rate of extraction r(t) must tend to zero. Capital–resource substitution is a fundamental topic in energy economics and there is empirical evidence (Nordhaus and Tobin 1972; Pindyck, 1979) which can support the assumption that the elasticity of substitution between natural
187
Fig. 1 – World oil extraction (mln t/year): historical data (before 2005); Hartwick curve (dotted); LA curve (solid).
188
EC O LO GIC A L E CO N O M ICS 6 4 ( 2 00 7 ) 1 8 6 –1 93
do not react on “natural experiments” such as the Reagan cuts in taxes (Poterba, 1988). Hence, the problem of switching realistically to a sustainable path of essential resource extraction must take into account that:
Fig. 2 – Extraction accelerations: historical data (before 2005); Hartwick curve (dotted); LA curve (solid).
in order to sustain non-decreasing output with the same structure, we must invest at least part of oil profit into development of oil-substituting technologies. In other words, we must create an “anti-oil market” with the oil rent. And under this assumption the model of instant investment cannot be really adequate because of the difficulties of a rapid re-structuring. Historical examples show that the development and the introduction of coal-based technologies took decades despite the obvious benefit of the new technologies for economy. The same can be said about the switch from a coal to an oil economy. Now we must consider the problem of switching to technologies, based on renewable resources not because they are economically more preferable but just because of anticipated shortage of profitable but exhaustible raw materials. And this process will occur over decades, not months. The second dimension of the difficulty of an instant switch to the Hartwick rule is the requirement of an abrupt and very substantial change of saving patterns for oil producing countries. For background, we can compare nonrenewable resource profit only from oil with the total amount of investments for a selection of countries. For example, oil gives Kuwait about 50% of GDP but gross fixed investments are only 6.6% of GDP. For Saudi Arabia these numbers are 45% and 16.3%, United Arab Emirates — 30% and 20.7%, Venezuela — 33% and 23.8%.3 Of leading oil producers, only Norway can boast numbers (about 18.6%4) almost coincident, because of investing oil rent to Petroleum Fund. There is however no direct connection between this Fund and investment in the development of oil-substituting technologies. There is evidence that at least in the short run saving rate is very stable, and it is much more difficult to change it instantly, than to change a government policy. The well-known empirical research of Kuznets (1946) tells us that consumer behavior is very persistent over time despite changes of governments and government policy. Subsequent analyses, for example, the work of Duesenberry (1949), tried to explain this phenomenon, and later papers examined why consumers
1) the path must have a period of a gradual slow-down in the rate of extraction; 2) there is a time lag between the moment of resource rent investment and correspondent increase in capital; 3) there is a non-zero period length for changing saving patterns from resource rent consumption to resource rent investment. In this paper we suppose, for simplicity, that the third problem is already solved (as in Norway), and also we will temporarily neglect the influence of the second factor. So, we will concentrate on the question of the construction the trajectories for the transition period using various optimality criteria and examine consumption behavior along the paths.
2.
Formulations of the problem
It is natural for the government, switching to a decreasing path of the resource extraction, to try to construct this path in such a way that the short-run negative social impacts along the path are minimal. According to (2) GDP percent change for our economy is
¼ a k=k þ b r=r: q=q
ð5Þ
Since the economy is going to enter the period with r˙ b 0, the government can try to minimize the negative influence of the decreasing extraction on output q. We can apply a mechanical analogy, comparing the rate of extraction with the speed of some vehicle. The most unpleasant feelings in a vehicle trip occur when the vehicle has to reduce the speed very quickly. Then, using this analogy, we can consider 1) the criterion of “smooth breaking”, which gives the optimal path with the minimal in absolute value peak of negative acceleration A(t) = r˙ (t). In other words we have a problem Z l rðtÞdt ¼ s0 ; ð6Þ F1 ðrðtÞÞ ¼ mint AðtÞY max; s:t: rð0Þ ¼ r0 ; rðtÞ
where the last condition means that the resource is essential. However, the negative acceleration r˙ (t) influences the GDP percent change not directly but as a numerator of the fraction r˙ (t) / r(t). Then it looks more relevant for economics consider 2) the criterion of “minimum shock on GDP”, namely
max F2 ðrðtÞÞ ¼ min½ rðtÞ=rðtÞY t
3
Source of information: http://www.cia.gov/cia/publications/ factbook/docs/profileguide.html (March 2006). 4 Source of information: http://www.ssb.no/en/indicators/ (March 2006).
0
rðtÞ
ð7Þ
subject to the same initial conditions. Finally, we can take into account the first term in Eq. (5) which can shift the point of maximum influence of negative acceleration, or, in other words, we can apply the maximin
EC O L O G IC A L E C O N O M IC S 6 4 ( 2 0 07 ) 18 6 –1 93
criterion to the whole expression q˙ /q. Hence, we will consider as the third criterion 3) the criterion of the “maximin GDP percent change”:
max : F3 ðrðtÞÞ ¼ min½a kðtÞ=kðtÞ þ b rðtÞ=rðtÞY t
ð8Þ
rðtÞ
with the same constraints.
3.
Solving the problem
The transition path can be constructed in the same class of rational functions as the Hartwick curve (3). The difference is in the numerator, which in the expression for acceleration A = r˙ must depend on t with a negative coefficient to control “smooth breaking” in the neighborhood of t = 0. Namely, A(t) must be in the form of
¼ ðA0 þ btÞ=ð1 þ ctÞd ; Aðt; b; c; dÞ ¼ rðtÞ
ð9Þ
where b b 0, c N 0, d N 1 (for convergence A(t) → − 0 with t → ∞). Corresponding to Eq. (9) r(t) has a dependence on b, c, and d in rðtÞ ¼ f−½A0 þ b=½cðd−2Þ=½cðd−1Þ þ bt=½cð2−dÞg=ð1 þ ctÞd−1 : Note, that a constant of integration for r˙ (t) = A(t) must be Rl zero for the convergence of 0 rðtÞdt and also for the convergence, d actually must be greater than 3. Then we have r0 = − [A0 + b / [c(d − 2)]] / [c(d − 1)], which can be used to express b: bðc; dÞ ¼ −cðd−2Þ½r0 cðd−1Þ þ A0 ;
ð10Þ
and then the transition curve has a dependence on c and d in rðtÞ ¼ r0 f1 þ ½cðd−1Þ þ A0 =r0 tg=ð1 þ ctÞd−1 :
ð11Þ
Coefficient c can be Z lexpressed from the condition that rðtÞdt: resource is finite s0 ¼ 0
Z
l
s0 =r0 ¼
1−d
ð1 þ ctÞ
Z
dt þ ½cðd−1Þ þ A0 =r0
0
l
t=ð1 þ ctÞd−1 dt
0
¼ ½1 þ fr0 cðd−1Þ þ A0 g=fr0 cðd−3Þg=½cðd−2Þ; which means that c is a solution of quadratic equation c2 s0 =r0 −2c=ðd−3Þ−A0 =½r0 ðd−3Þðd−2Þ ¼ 0: The only relevant root (because we are looking for c N 0) is 0:5
cðdÞ ¼ ½r0 =ðd−3Þ þ fr20 =ðd−3Þ2 þ s0 A0 =½ðd−3Þðd−2Þg
=s0 :
ð12Þ
Hence, we have a single independent parameter d which defines the shape of the curve (including its peak) and we can use this parameter as a control variable in some selected optimization problem F½rðt; dÞY max
189
1 ton of crude oil = 7.3 barrel. For our short-run criteria F1 [r(t, d)], F2 [r(t, d)] and F3 [r(t, d)] we constructed the following “optimal” paths of extraction: 1) the “smooth breaking” or the Least Acceleration (LA) curve (the solution of problem (6), see Bazhanov, 2006) has d = 36.8837 which implies c = 0.001459 and b = − 0.0136. Plots of r(t) and A(t) are on the Figs. 1 and 2 after the year 2005 (solid lines). This path has a peak of extraction at tmax = 5.87 with the rmax = 3.8016 bln t /year. Numerical estimation of the qualitative behavior of GDP percent change (Eq. (5)) along this path yields results similar to those for in r˙(t) on Fig. 2 after the year 2005 (solid line). It has a minimum [q˙ / q]min = − 0.122% at tmin = 271, then it is negative in the long run, approaches zero, but never exceeds it. Note, that for the LA curve d → ∞ with A0 → + 0 and we failed to construct this curve for A0 ≤ 0.06. 2) The “minimum shock on GDP” or the MS curve (the solution of problem (7)) can be obtained by solving the first order condition d(r˙(t) / r(t)) / dt = 0 which gives us the only relevant root (since we are looking for t⁎ N 0) corresponding to a minimum in t: t⁎ ðdÞ¼−ffA0 þ½A0 −bðcðdÞ; dÞ=cðdÞgfA0 þr0 cðdÞðd−2Þg0:5 g=bðcðdÞ;dÞ: Then we substitute for t⁎(d) and solve the first order condition d [r˙(t⁎(d)) / r(t⁎(d))] / d [d] = 0. Numerical investigation gives us the single root which corresponds to the maximum in d: d = 6.1178 with c and b defined by Eqs. (10) and (12). This curve qualitatively resembles the LA curve, but it has a peak of extraction earlier, at tmax = 3.52 with the rmax =3.7162 bln t /year. The MS curve, unlike the LA curve, can be constructed for A0 b 0. Another important difference is that our numerical estimations yield q˙ /q asymptotically approaching zero, always remaining positive. 3) The “maximin GDP percent change” or the MM curve can be constructed as a numerical solution of problem (8) only for the cases when we have an inevitable short-run minimum in GDP percent change (period of crisis). For our numerical examples we had 2 different cases: a) GDP percent change is always positive, monotonically decreasing, and asymptotically approaching zero; b) GDP percent change decreases to some negative value (minimum) and then asymptotically approaches zero always remaining negative. In the first case there is no solution for problem (8) because the minimum does not exist. In the second case, if we have the constraints (technical restrictions), which do not allow us to realize the sustainable path, then problem (8) has a corner solution along which the consumption decreases to zero.
4.
Consumption along the transition curves
d
which can be connected with the short- or long-run policy in output or in consumption. In our numerical examples we used A0 = 0.08 and as world oil reserves and extraction on January 1, 2006 (Oil & Gas J., 2005, 103, 47: p. 25.): r0 = 71, 793.8 [1000 bbl/ day] × 365 = 26,204,737 [1000 bbl/year] (or 3.58969 bln t/year); s0 = 1,292,549,534 [1000 bbl] (or 177.06 bln t). We use coefficient
We are going to check if our short-run criteria are consistent with our long-term goal, namely, non-decreasing consumption. We will examine, for simplicity, the case when all resource rent is always invested in capital and there are no time lags between the moments of investment and the corresponding capital increase. The only reason for the change
190
EC O LO GIC A L E CO N O M ICS 6 4 ( 2 00 7 ) 1 8 6 –1 93
in the pattern of extraction is that sustainable path of the essential resource extraction must be decreasing and asymptotically approaching zero. Note, that the constant per capita consumption over time for our formulation is the result of 1) total investment of oil rent in capital (with no time lag) and 2) fulfillment of the Hotelling rule. The rational transition curves (11) satisfy only the first condition unlike the Hartwick curve (3) which is derived from the Hotelling rule and so satisfies it identically. Hence, to examine consumption behavior along some path we should check the fulfillment of the Hotelling rule along this curve. For ˙ +frr˙. Then f˙r =βd (q /r) / dt =β [ fkk ˙ /r +frr˙ /r] − the standard case q˙ =fk k ˙ /kr+βqr˙ /r 2] / (βq )− βr˙q /r2. Dividing on fr =βq /r we have f˙r/fr =rβ [αk ˙ /k − (1−β )r˙/r or r˙/r =αk
f r =fr ¼ fk ½ k=q−ð1−bÞk r=ðaqrÞ:
ð13Þ
Just to check, we can see, that for the Hartwick curve [·] ≡ 1, because r˙ / r = − αq / k and ˙k = βq. Hence, if [·] b 1, then q˙ N 0, because f˙r / fr b fk, which follows −r˙ / r b αq / k or αq / k + r˙ / r N 0. And the latter, using expression in the left hand side of Eq. (2), means q˙ N 0. In the same way, [·] N 1 follows q˙ b 0 and, in general, sgn q˙ = sgn{1 − [·]}. So, to examine the long-run consumption c = (1 − β)q along the transition curve, we can check asymptotic behavior of [·] in Eq. (13).
Proposition 1. If an economy with technology q =kαrβ is such that 1) resource rent is completely invested in capital; 2) there is no time lag between the moment of investment and the corresponding increase in capital; 3) rate of extraction r(t) is such that
¼ ðA0 þ btÞ=ð1 þ ctÞd ; b b 0; c N 0; d N 3; rðtÞ then the asymptotic behavior in output q for different β is: bðd−2Þz1; ¼ −1; lim sgn qðtÞ sgnLðd; a; bÞ; bðd−2Þ b 1; tYl
ð14Þ
where L(d, α, β) = [α − β(d − 2)] / [α − αβ(d − 2)]. Proof of the Proposition is in Bazhanov (2006, Appendix). The simplified expression for L(d, α, β ) was obtained by direct substitution of the expressions (10) and (12) for b(c(d), d) and c(d). Corollary 1. Under the assumption of the Proposition 1 the consumption c(t) is 1) asymptotically decreasing if d N α / β + 2; 2) asymptotically constant if d = α / β + 2; 3) asymptotically growing if 3 b d b α / β + 2 where d is the control variable for the transition curve (Eq. (11)). Proof. Note that for β (d − 2) b 1 or d b 1 / β + 2 denominator of L(d, α, β ) is positive. Then the sign of L(d, α, β ) is defined by nominator. Since c(t) = (1 − β )q(t) and sgn ċ = sgn q˙ then substituting the expressions for d into L(d, α, β ) in Eq. (14) we obtain the assertion of the Corollary. □ In the case when d ≥ 1 /β + 2 or β(d − 2) ≥ 1 we define the sign of dc/dt by the first line in Eq. (14) which is included in the first case of the Corollary.
5.
Numerical examples
In order to construct numerical estimations of the consumption paths in different cases we used the procedure described in Bazhanov (2007). For the example with given r0, A0 for the world oil extraction, α =0.3, β =0.05, and optimal (in sense of criterion F1 ) d⁎, b(d⁎), and c(d⁎) we have β (d −2) = 1.74 N 1 which means, that consumption and output decrease in the long run along the LA curve (Fig. 3). For α =0.2, β =0.05 (estimations from Nordhaus and Tobin, 1972) we also have consumption decreasing to zero in finite time. We can see from the Corollary 1 that there are sets of α and β for which, given d⁎, we have asymptotically constant consumption along the LA curve. For example, L(d⁎, 0.697, 0.02) =L (d⁎, 0.872, 0.025) =0. Selection of different values for (α, β) for which limt → ∞ ċ = 0 makes some sense if we wish to get a feeling for how far some real extraction path can be from the stable one, given that we do not know the true values of α and β. In fact it is unrealistic to speak about the short-term regulation of these magnitudes by the government's decisions. Our examples make the path of resource extraction look plausibly controllable. Corollary 1 shows that we can fit our transition curve (the single free parameter d) using some long-run welfare criterion, e.g., constant consumption over time (asymptotically constant consumption) instead of the criterion F1. An example with α = 0.3 and β = 0.05 gives us d = 8.0, c = 0.00967, b = −0.01873. In this case the oil peak must be closer, namely, at t = 4.27 with rmax = 3.7433. The behavior of GDP percent change along this path is qualitatively similar to the one along the LA curve. It has the minimum [q˙ / q]min = −0.046% at tmin = 63.15 after which it increases and asymptotically approaches zero, always remaining negative. To check that the level of consumption along this curve, which we will call the “Transition Constant Consumption” (TCC) curve, is far enough from zero, we can solve the differential equation for k(t) numerically (the numerical solution was obtained in Maple by the procedure rkf 45) and then plot c(t) (Fig. 4). The constant consumption for the t, big enough, is cconst = 2.42801. GDP percent change along this curve is always positive and asymptotically approaches zero. Consumption along the MS curve, according to the Corollary, must be asymptotically increasing since for these curves we have d b α / β + 2 = 8. Numerical solution gives us the unlimited monotonically growing consumption (Fig. 5) with the same c0, but with cmax less than for TCC curve: cmax = 3.088 at t = 14.55. After cmin = 2.956 at tmin = 306 consumption grows and exceeds c0 after t = 1037. The behavior of consumption along the MM curve is similar. It has cmax = 3.084 at t = 12.45 and cmin = 2.9986 at tmin = 176.5. GDP percent change decreasing to zero but with slower rate than for the TCC curve. The only “cost" of this growth along these curves, compared with the TCC curve are: 1) cmax is a little bit less, than for the TCC curve (3.088 and 3.084 vs. 3.0955); 2) is that the peak of oil extraction must be earlier (t = 3.52 vs. t = 4.27) and the maximum extraction must be less than for TCC curve (3.7162 vs. 3.7433).
EC O L O G IC A L E C O N O M IC S 6 4 ( 2 0 07 ) 18 6 –1 93
Fig. 3 – Consumption decrease along the LA curve.
The example is an illustration of the answer to the question, “what is worse?”: a small decrease of consumption in the present or the depriving of oneself and (or) one's descendants of any prospects for improving their lives in the future. According to Rawls's maximin principle, extraction along the MS curve in combination with the Hartwick saving rule is obviously a pattern of overinvestment. But actually Rawls (1971, p. 291) objected to applying his maximin principle to the questions of justice among generations because of unacceptable consequences. In Bazhanov (2006) we offer a generalized approach for the defining a “relevant position” in Rawls's theory which implies that we must take into account not only the values of some indicators of life quality in the present but also their time changes or differences in con-
Fig. 4 – Consumption along the TCC curve.
191
Fig. 5 – Unlimited consumption growth along the MS curve.
sumption from previous years. The review of literature on using the “comparison utility” can be found, e.g., in Carroll et al. (1997). If the recent past has a big weight in the consumption prehistory and we substitute the derivative ċ for the difference c(t) − c(t − 1) then the utility in its simplest form is u = u ( c, ċ ). Applying the maximin principle, e.g., for u in additive form we have u (c, ċ) = wc(t) + (1 − w)ċ(t) = γ = const for any t N 0, w ∈ [0, 1] which with c0 = c(0) follows cðtÞ ¼ ½g−expf−wt=ð1−wÞgðg−c0 wÞ=w
ð15Þ
or we have a case of limited growth (Fig. 6) for γ N c0w and Eq. (15) is desirable in the sense “…that an extra bit of consumption at t is more valuable than the same extra bit at t + 1, since individuals will, in any case, have more consumption at t + 1” (Dasgupta and Heal, 1979, p. 284). Observe also that Eq. (15) describes a limited decline for γ b c0w and a constant consumption (as in the Hartwick rule) for γ =c0w. We do not claim that everybody favors this type of just path, particularly when it is apparent that rather small sacrifices in present can bring slow but unlimited growth in the long run (Fig. 5). For those, who prefer this form of intertemporal distribution, the more appropriate consumption utility function would be the function with essential factors, e.g., the Cobb–Douglas case. Then the rule of intertemporal distribution is cwċ1−w = γ = const which gives us c(t) =c0 (1 + μt)φ where c0 = c(0), μ = (γ / c0)1/φ / φ, φ = 1 − w or a pattern of unlimited (quasiarithmetic, Asheim et al., 2005, p. 5) growth which (for w close to 1) looks like the curve on Fig. 5. If we consider u(c, ċ) = cw1ċw2 and assume that w1, w2 can be negative, e.g., w1 = − 1, w2 = 1 (then u is the consumption percent change ċ / c which in our case is equal to GDP percent change q˙ / q), then the maximin principle implies q˙ / q = const which gives us q = q0et as an optimal GDP path. Since exponential path for GDP is unrealistic with finite exhaustible essential resources (with no technological progress), it raises a question about using GDP percent change as an indicator for sustainable economic development.
192
EC O LO GIC A L E CO N O M ICS 6 4 ( 2 00 7 ) 1 8 6 –1 93
There are some questions raised by and limitations of the model we have presented.
Fig. 6 – Generalized “fair Rawlsian growth” (Eq. (15)), w = 0.5.
In general, utility can be written as a CES function, or as a function with a variable elasticity where the elasticity parameter and w are to be chosen by the government. Then the specific just saving principle can be deduced for the specific utility function and the transition path of extraction can be adjusted to approach as close as possible (depending on constraints) the asymptotically optimal (in the long run) pattern of the intertemporal distribution of consumption.
6.
Concluding remarks
Numerical analysis of long-run consumption along the rational transition paths supports the main assertions of our Proposition and Corollary. The Corollary claims that depending on the shape of the transition curve of the resource extraction, which in our formulation is defined by the single control variable d, the long-run consumption can exhibit various patterns of behavior. It can decrease to zero, asymptotically approach some positive constant, or grow infinitely. This means that the transition curve can be adjusted to satisfy any desirable qualitative behavior of consumption in accord with the various optimality criteria for the long run. However, the example for the LA curve shows that some, at first glance plausible, short-run criteria can run counter to the government's primary goal of pursuing intergenerational justice and sustainable development in the long run. It is interesting that another formulation of the short-run criterion, namely, the minimum short-run shock on GDP, leads to the long-run unlimited growth of consumption, which is consistent with the long-run government concerns. The possibility of unlimited consumption growth for the Cobb–Douglas economy was considered in Dasgupta and Heal (1979). We have shown that growing consumption and unlimited growth particularly are consistent with the Rawls's maximin principle if we take into account the consumption prehistory in the form of a multiplicative utility function, one defined on the “relative personal position”.
(1) We examined transition curves as interior solution neglecting restrictions imposed by technical possibilities and difficulties connected with the saving rate changing. Constraints on the speed of changing saving behavior can restrict us from implementing the paths of extraction with asymptotically constant or growing consumption in the long run. (2) There remains the question of the path stability when technology is changing. (3) Transition curves can be constructed for a different class of functions. We also assumed that: (4) Cost of extraction is zero and population is constant though it would be interesting to consider the problem of transition when these values are allowed to vary. (5) There is no time lag between the moment of oil extraction and corresponding addition to capital. A lag would be present if we had oil rent invested in the development of alternative technologies. (6) All oil rent is invested in reproducible capital. In the real world, this is not observed and we should consider some period of increasing investments along some smooth (maybe hysteresis-like) curves and examine the influence of this curve on long-run consumption behavior. Each of these questions merits careful consideration in separate papers.
REFERENCES Asheim, G.B., Buchholz, W., Hartwick, J.M., Mitra, T., Withagen, C., 2005. Constant savings rates and quasi-arithmetic population growth under exhaustible resource constraints. CESifo Working Paper No 1573. October 2005. Bazhanov, A.V., 2006. Decreasing of Oil Extraction: Consumption behavior along transition paths. MPRA Paper No. 469, posted October 16, 2006. Online at http://mpra.ub.unimuenchen.de/ 469/. Bazhanov, A.V., 2007. The peak of oil extraction and a modified maximin principle. MPRA Paper No. 2019, posted March 07, 2007. Online at http://mpra.ub.uni-muenchen.de/2019/01/ MPRA_paper_2019.pdf. Carroll, C.D., Overland, J., Weil, D.N., 1997. Comparison utility in a growth model. J. Econ. Growth 2, 339 – 367. Dasgupta, P., Heal, G., 1979. Economic Theory and Exhaustible Resources. Cambridge University Press, Cambridge, Eng. 501 pp. Duesenberry, J., 1949. Income, Saving and Theory of Consumer Behavior. Harvard University Press, Cambridge, MA. 128 pp. Fuss, M.A., 1977. The demand for energy in Canadian manufacturing. J. Econom. 5, 89 – 116. Griffin, J.M., Gregory, P.R., 1976. An intercountry translog model of energy substitution responses. Am. Econ. Rev. 66, 845 – 857. Halvorsen, R., Ford, J., 1979. Substitution among energy, capital and labor inputs in American manufacturing. In: Pindyck, R. (Ed.), Advances in the Economics of Energy and Resources, vol. 1, pp. 51 – 75. Hartwick, J.M., 1977. Intergenerational equity and the investing of rents from exhaustible resources. Am. Econ. Rev. 67, 972 – 974. Heinberg, R., 2003. The Party's Over. New Society Publishers, Gabriola Island, Canada. 274 pp.
EC O L O G IC A L E C O N O M IC S 6 4 ( 2 0 07 ) 18 6 –1 93
Kuznets, S., 1946. National Product Since 1869. National Bureau of Economic Research, NY. 239 pp. Magnus, J.R., 1979. Substitution between energy and non-energy inputs in the Netherlands 1950–1976. Int. Econ. Rev. 20, 465 – 484. Nemoto, K., 2005. High oil prices dampening Asia-Pacific product demand. Oil Gas J. 103, 58 – 62. Nordhaus, W.D., Tobin, J., 1972. Is economic growth obsolete? Economic Growth, 5th Anniversary Colloquium, vol. V. National Bureau of Economic Research, New York, pp. 1 – 80. Pindyck, R.S., 1979. Interfuel substitution and the demand for energy: an international comparison. Rev. Econ. Stat. 61, 169 – 179.
193
Poterba, J., 1988. Are consumers forward looking? Evidence from fiscal experiments. Am. Econ. Rev. 78, 413 – 418. Rawls, J., 1971. A Theory of Justice. Belknap Press of Harvard University Press, Cambridge, MA. 607 pp. Solow, R.M., 1974. Intergenerational equity and exhaustible resources. Rev. Econ. Stud. 41, 29 – 45. Stiglitz, J.E., 1974. Growth with exhaustible natural resources: efficient and optimal growth paths. Rev. Econ. Stud. 41, 123 – 137. United Nations (UN), 1993. Agenda 21. United Nations, New York.