Annals of Operations Research 68(1996)409-422

409

The time path of the marginal cost of oil: The turning point and the subsequent upward drift S. T h o r e a, S.L. M c D o n a l d b, R. G o n z a l e z a, N.P. Ritchey c, T.W. Ruefli ~ and K.K. Sinha e

alC2 Institute, University of Texas at Austin, Austin, TX 78705-3596, USA bDepartment of Economics, University of Texas at Austin, Austin, TX 78705-3596, USA CDepartment of Mathematics, Youngstown State University, Youngstown, OH 44503, USA dDepartment of Management, IC2 Institute, University of Texas at Austin, Austin, TX 78705-3596, USA eDepartment of Operations and Management Science, University of Minnesota, Minneapolis, MN 55455, USA

The gradual exhaustion of existing deposits of a depletable non-renewable resource such as oil tends to shift the supply price curve of the resource upwards, increasing its marginal cost. Advances in technologies for exploration and production act as a brake on such upward shifts. Thus, there is a tug-of-war between the gradual exhaustion of existing deposits and technological progress. Using a recently developed constrained least-squares regression technique, we demonstrate that technological progress was the dominant force of the two during the first part of this century, causing a secular drop in marginal costs, but that this situation eventually was reversed, and that the gradual exhaustion of deposits gained the upper hand, causing marginal costs to increase. The turning point occurred around 1971-72. We also discuss the forecasting of the possible current upward drift of marginal costs. Keywords: Depletable non-renewable resources, marginal cost, technological progress, depletion of reserves, constrained least squares regression.

1.

Introduction

A standard proposition in the e c o n o m i c s o f exhaustible and non-replenishable resources states that the marginal cost o f a resource will gradually increase o v e r time, as the existing supplies are being depleted. Depletion o f the easily accessible deposits, and h i g h e r costs incurred due to the need to exploit deposits f r o m greater depths, remote locations, and regions with hostile environment, tend to cause upward shifts o f the marginal cost curve o f the resource.

9 J.C. Baltzer AG, Science Publishers

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S. Thore et al. / Time path of the marginal cost of oil

It is well appreciated, however, that technological progress in the means of locating and extracting fossil fuels such as oil can greatly extend the resource life of the substance, by making more and more of the resource base economically feasible to use. In the end, of course, depletion of the base must win out. But the end of the life of a resource is not a matter of abruptly exhausting its base. It is, rather, a matter of rising marginal costs until the latter equal those of one or more substitutes. For example, as marginal costs and the real price of oil rise, more and more substitution of other energy sources occurs; and as the marginal costs and the real prices of energy sources generally rise, more and more conservation occurs in the uses of energy. The long-run process of adjustment (generally, toward ultimate reliance on renewable energy resources) can be a smooth one if the appropriate trends in relative prices are permitted to reveal themselves to both producers and consumers of energy. A disaster can only occur if the underlying increase in true marginal costs is somehow masked for a time and then abruptly revealed, requiring a major adjustment in energy use, the lead-times for providing substitutes or conserving techniques being so long as to leave the economy severely handicapped for a significant, perhaps critical, period. It is, of course, possible that for extended periods technological progress of the sort indicated above can dominate depletion of the resource base, so that marginal costs of finding and extracting a material like oil remain constant over time or even decline. Adelman [1] studied finding-development costs in the U.S. over the time span 1918-1988 and provided evidence that the rate of technological progress was particularly rapid during the early part of the century. In a more recent publication [2], Adelman has argued that the supply curve of oil (plotting the supply price against reserves added) has recently shifted unfavorably. The purpose of the present study is to disentangle these two different mechanisms, using a recently developed constrained least-squares regression method for studying the systematic shifts over time of a statistical regression like that of a marginal cost curve, see Thore [9]. To the econometrician, the gradual depletion of the resource, or technological progress, constitutes a statistical "intervening factor" that may distort the estimation of the static curve. Assuming that depletion outweighs technological progress, the intervening factor moves the regression hyperplane monotonically upwards, parallel to itself. The new technique employs a least-squares format to determine the magnitude of these parallel shifts. Mathematically, one arrives at a quadratic programming problem with linear constraints. The objective function, to be minimized, equals the sum of all squared regression residuals. For further applications of the same technique, see McDonald and Thore [4, 5] and Thore et al. [12]. Our present work extends this econometric procedure by allowing for two consecutive regimes: first a regime of monotonic downward shifts (technological progress dominating exhaustion of reserves) and, later, a second regime of monotonic upward shifts (technological progress being swamped by the exhaustion of reserves). Varying the turning point between the two consecutive regimes parametrically, we determine the point in time that yields the best goodness of fit (the lowest overall sum

S. Thore et al. / Time path of the marginal cost of oil

411

of all squared deviations). See also Thore and Gonzalez [10]. That is, the turning point is determined endogeneously. Furthermore, once this base point has been established, we also gain access to a mechanism for forecasting marginal costs in the future. Current research into the mechanisms of the supply of oil and other exhaustible resources tends to be directed toward spelling out in some detail the multiple levels that are involved (exploration, development, production). These efforts at disaggregation have been pursued both in the construction of optimization models and in econometric estimation, see the literature surveys in Bohi and Toman [3] and in Epple and Londregan [6]. The present investigation takes a step in the opposite direction, returning to the simplistic aggregate marginal cost schedule and its shifts. In doing so, we deliberately renounce the opportunity to portray the various engineering characteristics of technical progress, and the ongoing exhaustion of reserves in various locations. Instead, the advantage of our approach is that the shifts of the marginal cost schedule reflect the cumulated effects of all technical progress and all exhaustion of reserves, whether identified by origin or not. The question posed here is a simple one: when did the gradual exhaustion of reserves eventually overtake the rapid technical progress that occurred during the first part of the century, leading to the present regime of a gradual upward drift of the marginal cost curve for oil? This is a topic that is important and enduring and that has far-reaching implications for energy policy. Section 2 explains the statistical techniques. Section 3 investigates the possible drift over time of the supply curve of oil, measuring supply as reserve additions. Section 4 examines the episodes of the two oil shocks (in 1973 and 1978) and discusses the problem of forecasting marginal costs. Section 5 sums up the results. There is an appendix, describing a simple procedure for reducing the particular quadratic program with linear constraints encountered in the econometric work into one whose only constraints are nonnegativity constraints. .

A least-squares technique for determining the turning point of a shift variable

In the estimation of a single-equation linear regression from time series data, one often encounters situations where one may suspect the presence of a systematic intervention of some hidden causal factor and yet is unable to observe this factor directly. We now discuss a least-squares technique for estimating the effects of such an unknown intervening factor, assuming only that it intervenes monotonically over time (i.e., it shifts the intercept of the regression hyperplane monotonically over time). Looking first at the case of a regression shifting monotonically upwards over time, and limiting the notation to the case of one single independent variable, the task is to estimate the regression model

yt=at+~Xt+Et,

t = l ..... T.

(1)

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S. Thore et al. / T i m e path of the marginal cost of oil

subject to the constraints O~1 __~ a 2 - ~ a 3 ~

... < a T.

(2a)

The estimation takes the form of solving the constrained quadratic programming problem m i n ~.~ (Yt - OCt -- ~ X t ) 2 ,

(3)

t

subject to the constraints (2a). The unknowns in this minimization problem are the parameters at, t = 1..... T and ft. For numerical solution, any standard software package for nonlinear programming may be employed. Note that no hypothesis is required regarding the mathematical form of the intervention of the unknown factor over time. The data themselves dictate the estimated effects of the factor. These effects may occur at a few discrete points of time, with intermittent time periods of no change. Relaxing the assumptions even further, one may permit the presence of monotonic shifts, but leave it to the data to determine whether these shifts are upwards or downwards. That is, one may introduce the alternative assumption of a monotonic downward shift a 1 >--O: 2 -> a 3 >_ ... _> a T (2b) and leave it to the statistical estimation procedure to determine which of the two competing approaches (2a) and (2b) fits the data best (i.e., delivers the lowest sum of squares (3)). Next, weakening the assumptions step by step, assume now that there are two phases: first a monotonic lowering of the regression hyperplane, eventually replaced by a monotonic increase. The turning point is denoted ~. The regression model (1)-(2) then reads Yt = O~t + flXt + e t , t = 1. . . . , T, (4) subject to the constraints a 1 > - - a 2 --> " " > - - 0 : , _ 1 --> a ~ : ,

(Sa)

O~v _< a ~ + 1 --< a ~ + 2 --< ... _< a T .

(5b)

Assuming for a moment that the turning point 7: were known, the estimation problem would reduce to minimizing (3) subject to the constraints (5a-b). To determine "rendogeneously, we let the computer solve the same programming problem (3) subject to (5a-b) for a number of alternative turning points "r = 1..... T, choosing the one that delivers the lowest sum of squared deviations (3). That is, the computer will search through all possible partitionings of the total time span t = 1..... T into two successive regimes, the first characterized by the regression drifting monotonically downward, and the other the regression drifting monotonically upward, choosing the turning point that yields the best goodness of fit.

S. Thore et al. /Time path of the marginal cost of oil

413

Note that "r= 1 produces a version of model (4) and (5a-b) that in effect coincides with model (1) with (2a). Similarly, ~= T produces a version of the same model that coincides with model (1) with (2b). That is, the one-directional monotonic model is actually a special case of the turning point model with the turning point being the very first time period, or the very last time period, respectively. Finally, there is not even any need to assume in advance that the regime of downward drift precedes the regime of upward drift. It could also be the other way around: Of1 ~ Of2 s "'" s a T _ 1 ~ a T, (6a) tZ~ >__ 0 ~ + 1 _> a r

(6b)

2 _> ... _> O~T,

where, again, one would determine by estimation which of the two alternatives (5) and (6) produces the best fit. The only assumption, then, is that there may be present a monotonic shift (in either direction) that either remains sustained during the entire time span or may possibly reverse itself at one point or other (the turning point). Everything else is left to the data and to the estimation procedure. 3.

The marginal cost of reserve additions: The turning point

The concept of the supply price of oil is arrived at through a thought experiment along the lines of conventional textbook microeconomics. The supply price is the lowest price that a producer needs to make him willing to supply that quantity. Furthermore, supply is related to costs. At neoclassical equilibrium, no profits are permitted beyond the payments to all productive factors and the coverage of all risk incurred. The supply price equals marginal cost. The supply price schedule (plotting the supply price against the supplied quantity) has a positive inclination, see the schedule A - A in figure 1. price, marginal cost C

C

B

A B quantity Figure 1. The supply price schedule: original position ( A - A ) and possible shifts.

414

s. Thore et al. / ~ m e path of the marginal cost of oil

Exhaustion of existing resources will ceteris paribus shift the supply price schedule (and the marginal cost schedule) upwards, to a position like C - C in figure 1. The reason is that existing and known deposits will always be tapped in a certain order, motivated by economic reasoning - easier accessible deposits with lower exploration and drilling costs will be tapped earlier, and more remote locations and regions with hostile environment will be explored and drilled later. Therefore, exhaustion means moving from lower-cost operations to higher-cost operations. Technological advances in the oil and gas industry occur in many forms. They are related to the exploration and drilling, making it possible to drill to greater depths in environmentally hostile and remote areas, such as deep waters or Arctic regions, and to tap new reserves of oil and gas. They are also related to recovering additional oil from the known reservoirs via improved geological reservoir characterization followed by geologically targeted infill drilling, or enhanced oil recovery. In all these cases, technological advances will ceteris paribus shift the supply curve downwards over ranges of output which have not been attained. For instance, in figure 1, technological advance would cause the initial supply price schedule A - A to shift to position B-B.

The concept of a turning point can be illustrated in figure 1 in the following fashion. There is a tug-of-war between the gradual exhaustion of existing deposits and technological progress. Initially, technological progress is assumed to have the upper hand, leading to a net downward shift of the supply price schedule (such as from A - A to B-B). Eventually, however, the effects of technological progress are swamped by those of exhaustion of reserves and the supply price schedule starts shifting upward again, back to A - A , and then shifting even further upward, to position C - C . We now discuss the concept of "reserve additions", and the use of reserve additions to measure supply and to explain marginal costs. One usually distinguishes between two categories of oil reserves: proved reserves and the resource base. The base is made up of probable reserves andstatic reserves, see e.g. Lamberton [7]. The upgrading of the resource base into proved reserves depends upon many factors, including current price, price expectations, and investment in infrastructure. The upgrading of earlier established reserves into proved reserves is thus a dynamic process. Sometimes the process decelerates, with resulting low statistical figures for "reserve additions"; sometimes it accelerates, generating higher figures. In any case, ultimately recoverable reserves in an oil reservoir, while always less than oil in place, are a function of the price of oil and the costs of various means of enhancing recovery beyond the stage of flowing production, as well as the rate of interest. Recovery stops when the marginal cost of extraction is equal to price, but the former can often be lowered repeatedly by adopting progressively more intensive methods of extraction. It is always a question of whether the expected enhancement of net cash flow from extraction has a present value greater than that of the necessary capital outlays, given the current rate of discount.

s. Thore et al. / Time path of the marginal cost of oil

415

It follows that proved reserves are not indicators of scarcity or abundance, but rather reflect optimizing choices of operators. Similarly, changes in proved reserves may tell us nothing useful about sustainability of production, but rather reflect changes in relative prices and the continued tug-of-war between depletion of oil in place and technological progress. Long-term sustainability depends on the size of the resource base (proved reserves plus estimated oil or gas remaining to be discovered, given a specified state of technology and level of price) which has always been a large multiple of proved reserves and can grow further absolutely for some time, despite continuous extraction and consumption, with sufficient technological progress. To reflect what has now been said, we report in this section on the results of two separate regression runs, in both cases employing the monotonic regression technique with turning points: (1) regressing the real price of oil on annual reserve additions; (2) regressing the real price of oil on "average" annual reserve additions, the average being formed through statistical smoothing over time. We use the following notation: Yt = the natural logarithm of price of crude oil in 1983 dollars, t = 1918 ..... 1988;

xt = the natural logarithm of crude oil reserve additions (excluding associated gas and natural gas liquids), in millions of barrels, t = 1918 ..... 1988. For data sources, we are indebted to M.A. Adelman. (Data on current prices were obtained from Twentieth Century Petroleum Statistics, and from the EIA Annual Energy Review. The price deflator is from the U.S. Department of Commerce; data on reserve additions from the U.S. Department of Energy, excluding associated gas and natural gas liquids.) See also Adelman [2]. Calculating first the conventional least-squares regression of Yt on x t, things look fairly hopeless: a loose scatter of points yielding the regression Yt = 2.560 + 1.247 • 1 0 - 3 x t ,

(7)

with R 2 = 8.978 • 10 -4. Price and quantity seem to be entirely unrelated. But when a monotonic shift of the regression is permitted, order is introduced into this apparent jumble. Permitting an upward monotonic shift over the entire 70-year period, one finds Yt = O~t - 39.193 • 1 0 - 3 x t , (8) where a t = 2.805 for t = 1918-78, at = 2.872

for t = 1979,

a t = 3.265

for t = 1980-88.

R 2 has now been raised to 0.293. But things still do not look good. The supply curve has negative inclination, which makes little sense. And the curve remains static until 1979.

416

S. Thore et al. // Time path of the marginal cost of oil

T h e next step is to try a m o n o t o n i c d o w n w a r d drift instead. N o w the result is Yt = ~ t + 138.99 x l O - 3 x t ,

(9)

where a t = 2.160

for t = 1 9 1 8 - 2 0 ,

a t = 1.995

for t = 1 9 2 1 - 2 2 ,

a t = 1.862

for t = 1 9 2 3 - 2 6 ,

a t = 1.550

for t = 1927,

at

for t = 1928,

=

1.528

a t = 1.525

for t = 1 9 2 9 - 3 2 ,

at

1.457

for t = 1 9 3 3 - 3 6 ,

a t = 1.452

for t = 1 9 3 7 - 8 7 ,

a t = 1.297

for t = 1988.

=

This time, R 2 = 0.337. T h e d o w n w a r d drift p r o c e e d s fairly e v e n l y during the 1920s and early 1930s, but stops in 1937. C o m p a r i n g (8) and (9), the obvious conclusion is that there was a net d o w n w a r d drift during the first t w o decades, but that this situation eventually r e v e r s e d itself and that a net u p w a r d drift resulted toward the end o f the t i m e span. T h u s prepared, the stage is set for fitting the least-squares m o d e l (4) with the t w o constraint sets ( 5 a - b ) . T h a t is, the entire time span t = 1918 ..... 1988 is n o w split into t w o regimes, one earlier r e g i m e t = 1918 .... , "r during which a net d o w n w a r d drift o f the m a r g i n a l cost c u r v e is permitted, and a second r e g i m e t = "r, ~'+ 1 ..... 1988 during which a net u p w a r d drift is permitted. F u r t h e r m o r e , as explained, w e let the turning p o i n t 7: v a r y p a r a m e t r i c a l l y "r= 1918,...,1988 to d e t e r m i n e the p a r t i c u l a r partitioning o f the entire t i m e span that yields the best g o o d n e s s o f fit. T h e calculations show that the turning point occurred at some time in 1 9 6 9 - 1 9 7 3 . E a c h one o f these four years p r o d u c e d the best goodness o f fit, R z = 0.749. R e p o r t i n g here on the results for 1971, we found Yt = O~t "-I-146.82 • 1 0 - 3 x t ,

(10)

where a t = 2.107

for t = 1918,1919,

a t = 1.108

for t = 1 9 7 2 - 7 3 ,

a t = 1.944

for t = 1920,1921,

a t = 1.430

for t = 1 9 7 4 - 7 8 ,

a t = 1.808

for t = 1 9 2 2 - 2 6 ,

a t = 1.477

for t = 1979,

a t = 1.488

for t = 1927,

a t = 1.795

for t = 1980,

a t

1.475

for t = 1 9 2 8 - 3 2 ,

a t = 1.811

for t = 1 9 8 1 - 8 8 .

a t = 1.402

for t = 1 9 3 3 - 3 6 ,

at=1.347

for t = 1 9 3 7 ,

=

S. Thore et al. / Time path of the marginal cost of oil

a t = 1.323

for t = 1938-58,

a t = 1.309

for t = 1959-63,

at=1.276

for t = 1 9 6 4 ,

a t=1.208

for t = 1 9 6 5 ,

o~t = 1.195

for t = 1966,

s t = 1.170

for t = 1967,

a t = 1.166

for t = 1968-69,

a t = 1.106

for t = 1970-71,

417

To simplify the reading of the table of a-values, the regime of net downward drift has been entered in the left hand colunm and the regime of net upward drift in the right hand column. Please also refer to the top curve in figure 2.

2.5 . ,. o r i g i n a l d a t a ........... r e s e r v e

additions smoothed

/

.= 1.5

.% .-\

W e-.

8

e. o

L-..~

f ............... ~

"x

0.5

,

, , , , i t

I

I

I

I

I

, , l i t , I

I

r

i

i

i

/

, , t t , t i

i

i

i

i

i

l l l l t l i

i

i

l

I

~

, , 1 , , , I

I

i

i

9

q

| , ~ , 1 i

i

i

4

i

J u l a a * i

i

i

I

I

1

I I i = l = l a , , t a I

I

i

R i

i

i

i

i

i

t

t

:.

,/

i

* l l t l * I

I

I

i

1

i

i

' ' a l l l e

B

I

U

U

I

a l l *

I

[

I

U

Figure 2. M o n o t o n i c regressions. T u r n i n g point: 1971.

Notice that, since all variables are measured as logarithms, the decrease from one year to the next of the at intercept (or increase, respectively) directly measures the percentage shift of the marginal cost of oil. To illustrate, looking at the estimates (10), between 1919 and 1920 marginal cost fell by 2.107 - 1.944 = 0.163, or 16.3%. Between 1980 and 1981, marginal cost increased by 1.811 - 1.795 = 0.016, or 1.6%.

418

S. Thore et al. / T i m e path of the marginal cost of oil

Returning to the misgivings about the reserve additions variable that we have already aired, we next run an alternative regression, smoothing the reserve additions over time. Hopefully, we should then be able to eliminate some of the volatility in reserve additions distorting the long-run marginal cost curve. To illustrate the kind of corrections that one may undertake for this purpose, we simply form a 3-year moving average of the reserve additions. In other words, an average of past, present, and expected reserve additions is formed. (It is assumed that expectations are rational, so that the expected reserve additions can be taken as an unbiased estimate of actual future reserve additions.) The turning point this time comes out as occurring some time in 1971-72. R 2 is raised to 0.834. Again reporting on the results for 1971, we obtained Yt = a t + 262.07 x 10-3Xt .

(11)

The txt's are plotted as the lower of the two curves in figure 2. As the reader can see, the smoothing of the independent variable produces a steeper slope but almost perfectly synchronized swings of the constant term. Actually, it is easy to see that, given any scatter of points (xt, Yt), t = 1 ..... T, and any process of statistical smoothing of the xt variable, moving each observed xt closer to its average ( ~ , t x t ) / T , must result in a steeper (or possibly, in degenerate cases, an unchanged) estimated regression slope. The economics of the matter is worth noting: To the extent that annual statistics exaggerate the movements of the true long-run reserve additions variable, we are underestimating the slope of the true static marginal cost curve. But the upward or downward drift of that curve is not affected.

4.

The development since 1971: the oil shocks and forecasting future trends

Referring again to the estimated monotonic regression (10) and the top curve in figure 2, let us now take a more detailed look at the development of marginal costs since the turning point in 1971. There are two big jumps, a 32% jump from 1973 to 1974 and a prolonged jump between 1978 and 1981 amounting to no less than 1.811 - 1.430 = 38%. Each such jump was followed by a period of consolidation (a horizontal stretch of the curve in figure 2). Note that these upward jumps coincided in time with the oil price shocks, resulting from OPEC actions. At any given time, large oil companies (and the industry at large) have a certain inventory of prospects. These prospects, which range from pure exploration prospects to enhanced recovery prospects, vary as to expected profitability in a given state of technology and real price level. As long as the real price is modest and not expected to rise substantially in future, only the best of these prospects are exploited in a given year. At the same time, the geological and engineering staff are engaged in identifying new prospects to replace those exploited in the inventory. If, now, there is a large,

S. Thore et al. // Time path of the marginal cost of oil

419

unexpected rise in the real price, it suddenly becomes profitable to go "deeper" into the inventory of prospects and undertake higher cost, more risky prospects that "normally" would be rejected and simply held in reserve. This means that the shortrun marginal cost curve is steeper than the intermediate-run marginal cost curve. This allows us to interpret the stairstep behavior associated with the oil crises as first a movement up the steep curve, then a movement horizontally to the appropriate place on the less steep intermediate-run curve. In the period of the horizontal movement, the successful projects are in the development phase in which most reserves are proved and the technical staff has time to re-stock the inventory of prospects with some new low-cost ones. With the second price shock late in 1978, the process was repeated, but with a still "deeper" dip into the inventory of prospects. The movements of marginal costs are also tied to the perception of risk in the industry. Market risk arises out of price volatility. The oil price shocks that occurred in 1973 and in 1978 presumably increased market risk dramatically. The necessary risk premium that producers would feel that they need in order to protect themselves against market risk shot up in the wake of each oil shock. We emphasize market risk, because there is no evidence that other risk elements, such as drilling risk, increased. With the proposed concept of market risk, we can now discuss the time path of marginal costs. The marginal cost curve shifted upwards in the wake of each oil price shock. Our understanding is that the risk premium shot up each time the market price of oil entered a new regime of increased price volatility. Our calculations then provide an empirical measure of this increased premium. Mathematically, the argument can be presented as follows. Returning to the regression model (1), assume for simplicity that (varyt)lxt is constant. (In words: the variance of the dependent variable Yt, given any given value of the independent variable xt, is assumed to remain constant.) Write the risk premium r var Yt, where r is the rate of premium per unit of the variance and one gets E(y t ) = a t + r v a r y t + f l x t ,

t = l ..... T.

(12)

The risk premium is incorporated in the estimated constant term. If var Yt suddenly increases, the constant term increases with it, causing a monotonic shift upward of the regression. How did management respond to increased market risk and an increased risk premium? Each time the marginal cost curve was elevated to a higher plateau, it stayed unchanged until the next price shock was encountered. For one thing, management may have been able to contain increased risk through the exercise of prudent risk management, such as hedging operations in the currency markets and the diversification of their holdings of leases. Also, it seems reasonable to assume that the steep increases in costs prompted many corporations to accelerate their quest for technological advance. All in all, management apparently was able to hold the cost line until the next price shock came along.

S. Thore et al. / lime path of the marginal cost of oil

420

The mechanism for generating forecasts from monotonic regressions was discussed in Thore [9]. For the purpose of forecasting, the very last monotonic regression should be used, in this case

Yt

=

1.811 + 262.07

x 10-3xt

.

(13)

That is, the last period of unchanged constant terms (in this case encompassing the years 1981-1988) is projected into the future. Note in particular that it is not permitted to conclude that there is present some kind of long-term "trend", slowly pushing the constant term upward. There is no guarantee that future observations will leave the timing of the period 1981-1988 intact; if future observations happen to lie above the regression (12), they will form a separate regime of their own (a new and higher step of the step function at, t = 1..... T), but if those future observations fall below (12), then the earlier calculations of the monotonic regressions have to be revised, typically leading to a lower constant term in 1981-1988. In simple words, the last estimated step of the step function must be used for forecasting. True enough, the gradual exhaustion of reserves is now - after the turning point in 1971 - assumed to swamp the effect of technological progress. But we do not know when that exhaustion of reserves will again push the constant term upward. And in the absense of such information, the forecasting must be based on the assumption of no change. 5.

Conclusions

The behavior of the price of oil and gas has been difficult to predict. This paper analyzes the monotonic shifts of the marginal costs curve for oil during a 70-year period. As existing deposits are being depleted and the drilling efforts are extended to geologically inferior fields, the marginal cost of bringing up a barrel of oil is bound to increase. Historically, pessimists in the industry have always foreseen the imminent depletion of existing reserves. The history of the oil industry since its very first days is of a series of technological breakthroughs that have foiled the doomsayers. Dramatic advances in technology have occurred, including the development of seismic techniques, reservoir characterization, enhanced oil recovery methods, and improved pipeline transportation systems. Such advances open up new possibilities and taken by themselves, tend to lower marginal costs. For a while, then, technological progress has been able to offset the long-term depletion of reserves. But in a secular perspective, there is no way of avoiding the eventual course of events because there is only so much oil and gas to find. The calculations reported here are based on the use of a new constrained leastsquares regression technique. This technique makes it possible to estimate some static relationship such as a marginal cost curve that is subject to monotonic change over time. In the present instance, application is based on the assumption that the development of new technology to produce new oil may during some time period during the

S. Thore et al. /Time path of the marginal cost of oil

421

century partly or wholly have offset the gradual exhaustion of existing reserves. Note the weak formulation of this assumption. There is no assumption that marginal costs actually rose, or fell, during some particular time period. The data themselves will determine if any such changes occurred. The estimations show that the 80-year period can be divided into two regimes: an initial period lasting up to 1971 when technological advance outweighed the gradual exhaustion of reserves, and the period after 1971 when the depletion of reserves has been the dominant factor. The timing of the turning point is significant, of course, and it agrees with conventional wisdom. Many authors have pointed out that the formation of the OPEC in the early 1970s and the flexing of its political muscles were only the political expression of underlying deeper economic changes. The novelty of our own approach is the length of the time perspective. The year 1971 marked a turning point not just in the post-war period, but in the history of oil since the very first days of the industry.

Appendix: A procedure for estimating monotonic regressions This appendix describes a simple mathematical procedure for solving the constrained quadratic programming problem that occurs in Thore [9]. In the main text above, this problem is given by (3) subject to constraints (2a). Extensions necessary to handle (5a-b) (or 6 a - b ) are immediate and need not concern us here. The procedure reduces the given constrained quadratic programming problem into a quadratic programming problem whose only constraints are the simple nonnegativity constraints on T - 1 of the variables. This is accomplished as follows. Define new variables A1, A2. . . . . A T _ 1 by A 1 = 5 2 - 51 , A 2 = 53 - 52,

AT_ 1 = 5 T-

5T_l,

so that the constant terms are al,al+AI,tZI+AI+A2

..... a1+Al+A2+A3

+...+AT_

l,

respectively. The programming problem (3) subject to (2a) then simply reads min(yl

- 5 1 - fix1

)2 q_ (Y2 -- al -- AI -

+ "'" + ( Y T - a l -

Al - A2 . . . . .

f i x 2 )2 d-

AT-I -

(Y3 - al - AI

- A 2 - fix3 )2

flXT) 2,

with the unknowns al and fl being unrestricted in sign and At, t = 1..... T - 1 nonnegative. Any standard algorithm for solving this type of quadratic program may then be used. See also Ritchey [8, chap. 3].

S. Thore et al. / Time path of the marginal cost of oil

422

Acknowledgement The authors would like to thank M.A. Adelman, Massachusetts Institute of Technology, for providing both intellectual stimulus and the data base that was employed.

References [1] [2] [3] [4]

[5]

[6]

[7] [8] [9] [10]

[11]

[12]

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