Hotelling rule, economic responses and oil prices

Hotelling rule, economic responses and oil prices Gideon Fishelson

The traditional Hotelling model is applied to explain the stability of oil prices in the 1960s and in the second half of the I9 70s. Following this analysis a cycling of oil prices is predicted with fluctuations of O-15% per year. The effect of a continuous increase of oil prices on an economy is traced out. The only cure found to be feasible is that of technological progress. Keywords:

Oil; Prices; Modelling

In a recent article, Fishelson examined the effects of controlling energy supply to a small country importing all its energy.’ Quotas and prices were the controls examined. They were studied for different production structures ranging from fixed proportions to variable elasticity of substitution (translog). The main disadvantage of the study was its static nature. All events occurred once and stayed forever. Furthermore, no changes took place in the production function. In the present study, some dynamics are introduced. However, we limit the examination to reactions to price increases. In the first section we modify the Hotelling model to explain the behaviour of crude oil prices for the 1960-70 period in which oil prices did not change. In the second section, we hypothesize a Cobb-Douglas economy and allow for technological progress to offset the increase in oil prices. In the third section, we generalize the Cobb-Douglas function to a CES function and introduqe explicitly the notion of energy saving technological progress.

cent rule of Hotelling applies.3*4 This rule applies whether the industry that extracts crude oil is a competitive one or a monopoly. The only difference between the two is that for a monopoly the initial price would be higher and thus the first part of the path of prices would also be higher than that of a competitive industry.t The % rule did not materialize in the 1960s although OPEC already existed. On the contrary, the real price of crude declined towards 1970 and increased from then on. The issue is how can we reconcile this price behaviour with Hotelling’s rule, while still assuming rational behaviour of the producers of crude?$ The explanation we offer is that of simultaneous increases in reserves of oil, increases in the demand for oil, and changes in the costs of oil production. Each argument is introduced independently and then the effects are summed. We assume that the world market demand for crude oil is of constant price elasticity,

The determination

Let the known reserves at c = 0 be Ro. The cost of oil extraction is assumed at first to be zero. Thus, the solution to the problem OD Max PtQtdt set* OD Q,
of energy prices*

If crude oil is regarded as an exhaustible resource with finite known reserves and zero extraction costs, the r per l A recent study which is more detailed than the following section and which also discusses the role of uncertainty is Pindyck, Ref 2.

Q = BP-‘y

(1)

and ri, = -Q, The author is with the Foerder Institute for Economic Research, Faculty of Social Sciences, Tel-Aviv University, Ramat-Aviv, 69978 Tel-Aviv, Israel. Financial Research

assistance from the Foerdar is gratefully acknowledged.

Final manuscript

received 10 February

0140-9883/83/03015364

803.00

Institute

for Economic $ For purposes of simplification we will not consider here the distribution of the responsibility of price (quantity) determination among the oil producing countries, OPEC and the oil companies.

1963.

0

(2)

tRecall that MRt= MRo-8, while M&-P (1 + l/E) where E equals the own price elasticity of demand an d MRt = Pr( 1 + 1 /El. See Herfindahl, Ref 5; and Herfindahl and Kneese, Ref 6 for a graphical exposition of this behaviour.

1983

Butterworth

81 Co (Publishers)

Ltd

153

Hoteliing rule, economic responses and oil prices: G. Fishelson leads to an initial price PO=(a.r.l?,,/B)-“”

(3)

Hence, for any t in the future Pt = ((II. r - R,,/B)-“”

* eti

(4)

If reserves increase (ex post) at a rate of r% per year, ie RI = Roe7

(5)

(6)

The ratio PA/P, is e-/O’+’ I 1, ie the price, instead of increasing, might decline. If demand is increasing over time such that Q, = B’P*

(7)

where B’ = BOesr, then, (8) The ratio, pr /POis equal to k*‘la, ie the price increases by more than r per cent. If both reserves and demand increase, the price at year 1 would be (9)

In year t the price would thus be PI’ =poe~~++Y~~P

(10)

Between 1960 and 1971 reserves doubled and demand tripled while the price did not change. Hence, on average for each year in that period we expect r+6/(u-ylcu=O

(11)

If we let r = 0.04, cu= 0.5, y = 0.07 and S = 0.10, the sum is 0.055, ie we would expect the price to rise by 5.5% per year. The fact that it did not might be due to the declining (and expected to decline) cost of extraction, C. Note that if C > 0, then the r% rule implies that (Pt - C,) = er(Pt+ - C,_,)

or,

Pr = erP,_l + (C, - e’&)

(12)

Hence, PJP,_, Be’

as

C&C,_, ZZe’

(13)

Over the 1960-71 period C,/C,_, of drilling was about e”*02 which is less than e’, thus pushing P19,, towards PIgeO(for additional analysis, see Pindyck, Ref 2).

The Cobb-Douglas

economy

Let the economy in which we are interested have at first a constant oil price. The firms there (or the central planner) assumed (ex post wrongly) that this stability would continue forever. The production function of this economy is of the Cobb-Douglas type, where the only two inputs are labour and energy. Output is

154

E, =A”ae~“aLo(l

Q,

Ph = ((Y - r - ROey/B)-“” *d

_& = PO. p8lff-w

Let the price of Q be 1. Then the quantity demanded at year t is

of energy

_(y)r~~P;r~~

(15)

where P denotes the price of energy. Output (the indirect production function) is

then (view year 1 as the base year for the future path) the price for that year 1 (reevaluated, given the actual reserve level at year 1) is

p;’ = (a . r . RoeY/&6)-lf~

denoted by Q. For simplicity, we assume a fixed labour force, L = Lo. Energy is denoted as E. Technological progress is neutral (by definition) at a rate A. (Y l-o! Q, =Ae hfLoE, (14)

=Allaehf/aLo(l

_OL)l/Qp;(l-+

(16)

As long as Pt was stable the economy grew at a rate of X/or%per year (A/& > h since OL< 1). Consider the case where this small economy was suddenly faced with continuously increasing prices at a rate of J/% per year. 8 The elasticity of output with respect to oil prices is -( 1 - cr)/cr. Thus, the growth rate of the economy declined to A/& - JI - (1 - &Y). The only way the economy could return to its previous path of growth (offset the increasing price effect) is to increase its technological progress X, (a: is assumed to be unchanged) such that Ah = \L( 1 - a). In other words, the neutral technological change should increase but by less than the increase in energy prices.+ For example, if A = 0.03, $ = 0.05 and QL= 0.90 then Axh = 0.20 (A.h = 0.005) which is not unrealistic. In order to get AX > 0, resources need to be diverted from production to the research and development of new technology for production. The explanation for this change of resource allocation is that research which is an indirect production process was less efficient than direct production before the increase in energy prices, but became more efficient due to the increase of energy prices. In a Cobb-Douglas production function framework the above statement leads to an internal inconsistency when related to the analysis. It can be shown that for a Cobb-Douglas economy the profitability of investment in research and development has nothing to do with P nor with J, > 0. Thus, if it is profitable to allocate resources to research and development when J/ > 0, it should also have been profitable when J/ = 0.l Thus, if research and development is observed when J, > 0 while it was not observed before, and the production function did not change over time, then either the economy is not Cobb-Douglas, or it was not an optimizing economy.** QExcept for the 1973-74 price jumps, the r&s/ price of oil has been relatively stable. The reasons for this are analogous to the ones listed in the first section although the direction of change is different. *We talk about neutral technological change given the specific production function for which a biased change and a neutral change are identical. ( In the simple economy that \NBS postulated labour saving technological progress was needed since the fixity of labour implies that its price (equating demand and supply) was continuously increasing at a rate of h/o. The policymakers ware not concerned, however, probably because this increase constituted returns to a domestic factor of production (did not leek out). l * For empirical studies this implies identification difficulties not just for parameters but for function forms as wall, eg is the originating production function C-D or CES?

ENERGY ECONOMICS July 1983

Hotelling rule, economic responses and oil prices: G. Fishekon

The CES economy The economists in the constant elasticity of substitution, CES, production function economy have a harder life than those in the Cobb-Douglas economy, since the former have to resort to linear approximations. Let output be a CES function of labour and energy. Q, =

e*‘(&jp + [l - &]~!?;~)-l’~

(17)

Also, let the demand for energy follow the optimality condition, that of value of marginal product equals price, aQ/aE = P. Hence, (Q/E)jIO( 1 - 6) . eWO-l)‘O = p,

(18)

and E,

= Q,( 1 -

~)OeA’(u-l)p~u

(19)

where u = l/( 1 + p) is the constant elasticity of substitution between energy and labour. Substitution of Equation (17) for Q in Equation (19) leads to the specification of the demand for energy with respect to the exogenous variables. E =8-1’p(l -S)“*eAM*LO _ t (1 _ e-hru [l _ g]u)-l/p pt u =AtPT

(20)

Note that for Pt = POthe rate of growth of energy demanded is not constant. However, with respect to its price the demand for energy is of a constant elasticity -u. Substituting Equation (20) into Equation (17) yields Q, = eh’(6L;p t (1 - G)(A&‘)-p)-l’p

(21)

From Equation (20) it is clear that A/A declines over time and approaches ha from above as t gets larger, assuming Pt = PO.Thus when energy prices are constant, output grows at a rate greater than lhl. It is at least X( 1 t u). However, when the price of energy increases, output growth rate declines by less than uJl where J/ is the rate of growth of energy prices since the first term on the right in Equation (21) is not affected by the energy price increase. Hence, the lowest growth rate is h(1 + u - JI). The price of oil has to more than double each year in order to cause zero growth. The interesting case for investigation is that for which p > 0 (u < 1). For p < 0 (u > 1) it might have paid to invest in R&D for increasing both the efficiency of labour and energy before the energy price increase. This would be less likely in the case for which p > 0. To show this we use the extreme example of p = 00 ((I = 0). The produc tion is of fixed proportion (the first case dealt with in Fishelson, Ref 1). Denote the rate of improved efficiency of labour by /I and of improved efficiency of energy by 7. Accordingly, the inputs measured in efficiency units are

Thus, unless (after reaching the proper ratio for full employment) the inputs measured in efficiency units grow at equal rates, units of one input will be wasted.?? Furthermore, for u = 0 and L, = Lo the quantity of energy that would be employed at year t is Et = L,-,(b/a)e@-‘)’ when Pr = P < ertlb

(25)

when Pr = p > eTtlb

0

Also for u = 0 regardless of whether /I 5 7, the economy converges to a neutral technological progress path at a growth rate of /I. Hence, even for u = 0 and certainly for u > 0, the economy should attempt to increase the efficiency of its constrained factor - labour. When the research production functions are p=hL(LL)

and

7=hE(L&

(26)

then technological progress is not a free good. Labour must be diverted from production to research if positive technological progress is desired. When P =P, the diversion of labour from L to LE becomes less profitable while the diversion of labour from L to LL becomes more profitable as u + 0 (with regard to the latter there is risk of running into ‘Perpetuum Mobile’ thus assuming internal equilibria constraints have to be imposed upon the properties of the production of improved technology, P and 7). For Pt = POey’ and u + 0, the diversion of labour from production to improving energy technology is more strongly justified. If initially 7 = 0, then the minimal /I needed to maintain the economy growth rate of 7 equals $ and its level increases as p diminishes (u increases). Using various approximations, the optimal investment in /I and 7 can be determined. The issues that should be considered are as follows: Since labour is fixed at Lo, the diversion of labour to R&D implies losses of Q that equal the marginal product of labour. If technological progress does not deteriorate, once positive fl or 7 are produced, they will last forever. The benefits from diverting labour to R&D is the present value of the flow of increases in Q due to the technological progress. The increase in energy prices could not continue forever. Eventually a back stop technology would emerge (exogenous to the small country) that would produce energy at some constant price. At that price the economy will converge to a new path of growth obviously lower than the initial one, but over which the rate of growth would be the same as that before the price increases (Figure l)?

Conclusions

EL, = e@@. Lo = e@L,

(& =&p = fi)

(22)

EE, = el^ltdt- E, = ey’E,

(?f = 7r’ = 7)

(23)

Technological progress is economically justified to be biased (input saving) for the input whose quantity is limited by ‘nature’ (zero population growth) or due to

(24)

tt If Lr = Lge”‘, biased technological change at a rate of n is already called-for energy when Et is constrained to equal ED

The fixed proportion

technology

implies

Q, = Min (ELJa, EE,/b)

ENERGY ECONOMICS July 1983

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.Hotelling rule, economic responses and oil prices: G. Fishelson

t’ t* Figure 1. The growth paths of the economy. t’: energyprice started to increase at a rate of r%. P: energy price reached the backstop

t

technology.

political actions (oil quotas). The need for technological progress becomes obvious when input prices increase (the historical labour-saving induced technological changes.)$$ The CES model, despite its analytical inconvenience, proves useful for illustrating this equivalence. The intensity of research for improved technology is frequently related positively to the marginal productivity of labour in research and negatively to the productivity of labour in production. Furthermore, it is positively related to the rate of change of energy prices and to the elasticity of substitution between energy and labour in production. The analysis and conclusions presented above are the first stage in the determination of the optimal allocation of resources between direct and indirect production activities where the objective function is the maximization of the social welfare function (eg 10mB(t)U(Q,)dt).$§ The specific element we introduce into this traditional maximization model is the continuously rising energy prices which accelerate the economic justification for investment in research for energy saving technologies. The model we have analysed does not optimize. It is rather a satisfying model, ie it leads back to the path of growth that was experienced with constant energy prices. It should also show that research for improving energy and labour utilization was justified before the increase in energy prices. If so, the discussion above is to be viewed in terms of intensifying it. $$ The fixed input case and the increasing input prices are actually the same. For the former, the relevant price is its shadow price. 5 0 The analogy for investment

156

is perfect.

The drive towards increasing energy efficiency, ie to get more units of service from the same quantity of energy, started to be observed after energy shortages and higher energy prices were realized (eg more efficient air conditioners, petrol efficient cars in terms of miles/ gallon, more efficient space heat and water heating furnaces, and better housing insulation). Furthermore the increase in energy prices led to an increase of exploration activities and of the known reserves. The Hotelling % rule is still effective. Disregarding the shocks introduced in 1979 and 1980 because of the Iranian revolution and the Iran-Iraq war respectively, between 1975 and 1982 the real price of energy did not change while production costs have increased (the Alaska pipeline and more offshore and deeper drilling). Equation (11) explains this result. The same interest rate (0.04) and demand elasticity (0.5) with an increase of reserves at a rate of 0.02 and a decline of demand at a rate of 0.01 (world recession is another factor that lowered demand) should lead to a decline of oil prices at a rate of 0.02. The increase of the costs of production neutralized it to no change in price. The interesting question is how long would the nochange-in-price hold? A slowdown of explorations and of energy saving technological progress, coupled with an upturn in the world economy, might lead to continuous increases in oil prices at rates between 0.10 and 0.15 per year. Whether such increases are sufficient to stimulate energy saving and explorations, or to slow economic growth, is questionable. At the margin, however, they would obviously occur. Hence, one can predict that in the future we are likely to observe cycling of oil prices at real rates between 0.05 and 0.15. References G. Fishelson, ‘Control on energy supplies, the damage to a small economy’, Energy Economics, Vol2, No 3, July 1980, pp 166-171. R. S. Pindyck, Models of resource markets and the explanation of resource price behavior, MIT-EL 7-9 062 WP, MIT Energy Laboratory, June 1979. H. Hotelling, ‘The economics of exhaustible resources’, Journal ofPoliticalEconomy, Vo139, 1931, pp 137-175. G. Fishelson, op tit, Ref 1. 0. C. Herfindahl, ‘Depletion and economic theory’, in M. Gaffney, ed, Extraction Resources and Taxation, University of Wisconsin Press, 1967, pp 63-90. 0. C. Herfindahl and A. V. Kneese, Economic Theory of Natural Resources, Merrill, Columbia, Ohio, 1974. G. Fishelson, Dynamics in the Hotelling model, a note, Working Paper 3582, Foerder Institute for Economic Research, Tel Aviv University, Israel.

ENERGY ECONOMICS July 1983