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The effect of uncertainty and aggregate investments on crude oil price dynamics
Energy Economics 24 Ž2002. 615᎐628
The effect of uncertainty and aggregate investments on crude oil price dynamics Jostein TvedtU Den norske Bank, P.O. Box 1171 Sentrum, 0107 Oslo, Norway The Norwegian School of Economics and Business Administration, Bergen, Norway
Abstract This paper is a study of the dynamics of the oil industry and we derive a mean reverting process for the crude oil price. Oil is supplied by a market leader, OPEC, and by an aggregate that represents non-OPEC producers. The non-OPEC producers take the oil price as given. The cost of non-OPEC producers depends on past investments. Shifts in these investments are influenced by costs of structural change in the construction industry. A drop in the oil price to below a given level triggers lower investments, but if the oil price reverts back to a high level investments may not immediately expand. In an uncertain oil demand environment cost of structural change creates a value of waiting to invest. This investment behaviour influences the oil price process. 䊚 2002 Elsevier Science B.V. All rights reserved. JEL classifications: D4; D81; Q49 Keywords: Oil price dynamics; Stochastic optimal control
1. Introduction In the long-run the oil price seems to follow a mean reverting pattern. That is, in the case of high oil prices the underlying price trend is downward and in the case of very low oil prices the trend is upward. In periods of a close to perfectly U
Corresponding author. DnB Markets, P.O. Box 1171 Sentrum, 0107 Oslo, Norway. Tel.: q47-22017664; fax: q47-2248-2983. E-mail address: [email protected] ŽJ. Tvedt.. 0140-9883r02r$ - see front matter 䊚 2002 Elsevier Science B.V. Al rights reserved. PII: S 0 1 4 0 - 9 8 8 3 Ž 0 2 . 0 0 0 1 6 - 6
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J. T¨ edt r Energy Economics 24 (2002) 615᎐628
competitive oil market the mean reversion in prices can be explained almost solely by restricted supply flexibility in an environment of stochastic demand. In the case of an unexpected jump in oil demand it takes time to increase production. Hence, the price of oil increases until new oil fields are developed and transport capacity is provided to a sufficient degree in order to revert the oil price towards a long-run average price level. In a perfectly competitive market the long-run average price level is determined by long-run marginal costs, which in the oil market is mainly determined by available oil reserves and technology. Some of the reasons for restricted supply flexibility are irreversibility of investment, time-to-build and restricted construction capacity. The cost of reducing these restrictions may in many cases be substantial. In the oil industry this is especially the case for large-scale offshore oil field developments. High oil prices may trigger a decision to develop new oil fields, but restricted construction capacity implies that it will take time from when a project is initiated to first oil. To adjust the capacity of the construction industry is costly. To increase the capacity entails substantial investments and labour force training. To decrease capacity may cause significant regional unemployment and loss of know-how. Since the early 1970s the oil price has been heavily influenced by the market power of the OPEC cartel. Besides internal rivalry, OPEC’s main challenge has been the trade-off between price and market share. High prices have triggered development of non-OPEC oil fields that by no means would have been profitable in a perfectly competitive market. During the high oil price regime from 1979 to 1985, OPEC production was reduced by approximately 45% due to a loss of market shares and reduced demand. Because of lower oil prices after 1986, OPEC production during the late 1990s is back at the same level as before the two oil price shocks in 1973 and 1979. A reduction in the marginal cost for the non-OPEC producers due to technological progress has challenged OPEC’s market power during the 1990s. Hence, despite lower oil prices than in the early 1980s there have been substantial oil field developments outside OPEC during the 1990s. The drop in oil prices in 1998 to pre-OPEC levels, despite improved oil prices during 1999, curbed investments for 2000 and 2001. At present, OPEC produces slightly above 40% of the total world consumption of crude oil. Back in the mid-1980s the OPEC market share was only marginally above 30%, which was down from above 50% in the first part of the 1970s. The market share of OPEC is in strong contrast to OPEC’s share of known oil reserves. OPEC’s share of known oil reserves is approximately 77% and the Middle East OPEC countries have alone 65% of known reserves. The production cost of OPEC and in particular, that of the Middle East countries is significantly below that of the rest of the world. According to rough estimates ŽThe Economist, 1999. production cost of oil, including costs related field development, is approximately US$2 per barrel in the Middle-East in contrast to US$10 in the US-Gulf and US$11 in the North Sea. Hence, OPEC operates as a leader in the crude oil market and restricts production to a significant degree in order to keep prices at a preferred level from an OPEC perspective.
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2. Overview of the paper At any point of time the equilibrium in the oil market is modelled as a leaderrfollower model. The leader, OPEC, has the lowest marginal costs whereas the marginal costs of the non-OPEC producers, the follower, depend on past and present investments. Present investments are restricted by the size of the construction industry. To increase or reduce the capacity of this industry entail costs. The dynamic of the market is therefore, modelled as a stochastic optimal control problem. Hence, uncertainty in the demand for oil, via the demand for oil industry investments, creates hysterisis effects in the adjustment of the capacity of the construction industry. In a perfectly competitive market, changes in investments are solely triggered by the current oil price, and the oil price is determined by the current demand and production capacity. However, the cartel members know that they influence the oil price by their production strategy. Hence, when choosing an oil production path the cartel may take into account the effect this path may have on the investment decision of the non-cartel producers. Hence, the cartel production strategy is a trade-off between high oil prices and the risk of triggering non-cartel oil industry investments.
3. A model of the crude oil market The static equilibrium of the oil market is modelled fairly straightforward in order to reduce the complexity of the dynamic model. It is assumed that OPEC operates as a cartel without internal rivalry, whereas the rest of the world’s oil producers take market prices as given, that is, they operate as if the market was perfectly competitive. The non-OPEC producers, or fringe producers, have an aggregated total variable production cost function given by C s ky␣ Q 2f
Ž1.
where k is the total capital employed by the fringe producers in oil production and Q f is the total oil production of the fringe producers, at a given point of time.  ) 0, ␣ ) 0 are constants. Given the cost function wEq. Ž1.x short-term marginal cost equal the oil price gives the linear inverse supply function of the fringe producers cQ f s X
Ž2.
where c s 2ky␣ and X is the price of oil. Total oil demand, Y, is assumed totally inelastic to the oil price. This assumption is fairly unrealistic in the long run, but simplifies the model to a significant degree and does not influence the main conclusions. A demand function like the one used by Dixit Ž1991. may be a possible
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extension to this model. Demand is assumed to be stochastic with incremental change given by a geometric Brownian motion dYt s Y ˆ t dt q Y ˆ t d Bt
Ž3.
where ˆ and ˆ are constants and have the standard interpretations, and d Bt is the increment of a standard Brownian motion. In the model there is no oil stocks and total supply of oil, Q, must at any time be equal demand. Therefore, the cartel will at any time produce the residual demand, and the cartel is assumed to have unlimited production capacity. The cartel’s oil production, Q l , will then be Ql s Y y
X c
Ž4.
The cartel is aware of how their oil production influences the oil price and that the fringe producers take the oil price as given. For simplicity assume that the cartel has zero production costs. The profit of the cartel in then given by l s XQ l s c Ž YQl y Q l2 .
Ž5.
Maximising instantaneous profit gives the optimal oil production of the cartel Ql s
Y 2
Ž6.
That is, given the zero cost assumption for the cartel, the totally inelastic demand and the linear supply function of the fringe producers, the cartel simply supply half the demand for oil. Historically, OPEC has produced approximately 30᎐50% of the total world consumption of oil. Given the 50r50 split of the oil market between the fringe producers and OPEC the equilibrium oil price is given by X s 12 cY
Ž7.
A competitive market assumption requires that the present value of the future instantaneous total market surpluses less the investment costs is maximised. See Lucas and Prescott Ž1971.. This model describes an imperfectly competitive market. However, the fringe producers and the consumers are assumed to behave as if they were operating in a perfectly competitive market. Hence, the competitive nature of the non-OPEC part of the market implies that investments should be modelled in order to maximise total consumer surplus and the profit of the fringe producers. This follows since the cartel is assumed to have an infinite production capacity and therefore, has no incentives to make additional investments. Given the oil price wEq. Ž7.x, the production split wEq. Ž6.x and the production cost of the fringe wEq. Ž1.x, the instantaneous profit of the fringe producers and the cartel is f s 18 cY 2 and l s 14 cY 2 , respectively. Given totally inelastic demand the consumer surplus is infinite. To make consumer surplus finite, assume an
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upper price level, P, at which other types of energy sources are substituted for oil. Then the consumer surplus plus the profit of the fringe producers are given by S s PY y 38 cY 2
Ž8.
Observe that the negative part of this relation is simply given by the instantaneous profit of the cartel plus the variable production costs of the fringe producers g s 38 cY 2
Ž9.
By changing the capital stock, k, of the fringe producers the future consumer and fringe producer surpluses are influenced via changes in c, i.e. via changes in the slope of the supply function of the fringe producers. Hence, to maximise the surplus, S, in Eq. Ž8. is equivalent to minimising the level of the cartel’s surplus and the production cost of the fringe producers via an optimal path of the production capital. Capital, k, is produced by the construction industry. Let the incremental change in the capital stock of the oil industry be linear in the total level of the existing capital stock of the fringe producers, i.e. d k t s a t k t dt
Ž 10 .
where a t is the marginal relative change of the capital stock at time t. Let the production of new capital be given by the marginal relative net addition of capital, t . Hence, t represents the activity level of the construction industry. Let the cost of producing new capacity at time t, be given by pt where p is a constant. The activity level of the construction industry can be changed, but at a cost. Let there be two alternative levels of capital production, 1 and 2 . Let the cost of moving from the low activity level, 1 , to the high activity level, 2 , be given by q2 and let the cost of moving from the high to the low activity level given by q1. Existing capital deteriorates at a fixed rate ␦. This may either be due to general depreciation of capital or it may be interpreted as a depletion rate of oil wells. It follows that a t s t y ␦. Given the dynamics of the capital stock, the dynamics of the slope of the supply curve, c s 2 ky␣ , will be given by dc t s ya t ␣ c t dt
Ž 11 .
That is, an increase in the capital stock will make the supply function more price elastic, i.e. c is reduced, and the equilibrium price drops. Consequently, the profit of the cartel is reduced as k increases. However, the 50r50 split of production will remain unchanged given the specification of the model. From Eqs. Ž3., Ž9. and Ž11. it follows by Ito’s lemma that the dynamics of the production cost of the fringe and the profit of the cartel, g, is given by a geometric Brownian motion d g t s g t dt q g t d Bt where s 38 ŽyŽ y ␦ . ␣ q 2 ˆ q ˆ 2 . and s 34 . ˆ
Ž 12.
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4. Dynamic equilibrium of the crude oil market To derive the dynamic equilibrium is restricted to regulate the capital stock in such a manner that the present value of the cartel’s profit and the production cost, plus the cost of producing capital and regulating the construction level, is minimised. Let ey t be a discount factor where is constant, and let X t be the state of the economic system, X t s w s q t,k t ,Yt ,a t xy1 . Then, it follows from the discussion above that the present value, at time zero, of the cartels profit, the production cost and the cost of new capital, at time t, is given by F Ž X . s ey t 38 cY 2 s ey t Ž g q p.
Ž 13 .
and that the present value, at time zero, of the cost of adjustment, at time t, is
¡q ; ~q ; ¢0;
K X j , j s ey t
ž
/
1
j - 0
2
j ) 0
Ž 14 .
j s 0
where j is the jump in at time j . The minimum of the present value of the cartel’s profit, the production cost and the costs of keeping an optimal investment path is given by the criteria function ⌰Ž x .. ⌰ Ž x . s inf E x
N
⬁
H0 F Ž X . dt q Ý K ž X t
js1
j , j
/
Ž 15 .
where the controls are given by , where s Ž 1 , 2 , . . . , N ; 1 , 1 , . . . , N ., N - ⬁, and 1 is the time of the first control and 1 is the size of the first jump in , etc. The incremental change in the state of the system, X t , between each change in , is given by
d Xs s
1 Ž s y ␦ . Ys 0
0 0 dt q d Bs Ys 0
Ž 16.
It follows from the assumptions above that the process X t ; t G 04 is a Markov process and hence, has an infinitesimal generator A, As
⭸ ⭸t
q gt
⭸ ⭸g
q 12 2 g t2
⭸2 ⭸g2
Ž 17 .
The minimising problem of Eq. Ž15. can be transformed into a maximising problem by the fact that infŽ a. s ysupŽya.. To simplify the discussion let ⌰Ž x . s
J. T¨ edt r Energy Economics 24 (2002) 615᎐628
⬁
⌽ Ž x . s sup E x
H y F Ž X . dt y Ý K ž X t
621
N j , j
js1
0
/
. The impulse control problem
may be handled by applying the approach of formulating the quasi-variational inequalities A⌽ q F F 0
Ž 18 .
⌽ Ž x . G M⌽ Ž x .
Ž 19 .
A⌽ q F Ž x .4 ⌽ Ž x . y M⌽ Ž x .4 s 0
Ž 20 .
where M is the shift operator, defined by M⌽ Ž x . s inf ⌽ Ž s,k , y,a q . y K Ž x, .4
Ž 21 .
That is, Eq. Ž19. holds with equality at the optimal time of control, and Eq. Ž18. holds with equality between the controls. Informally in words, if the value function in the present state is larger then what it would have been after a control, i.e. Eq. Ž19. does not hold with equality, then a control cannot be optimal. The value function is equal to the negative of the discounted value of cartel’s future profits and the production costs of the fringe plus the net value of the controls, given an optimal control strategy. If there is no control the change in the value should be equal to the cartel’s profits and the production costs. Hence, if Eq. Ž18. does not hold with equality then a control is optimal. But in that case, the value function after the control must be at least as large as before the control. Try a solution of the form ⌽ Ž x . s ey t ⌿ Ž x . for the value function, where ⌿ Ž x . is a time homogeneous function. Consequently, we may write Eqs. Ž18. and Ž19. as y⌿ q g t
⭸⌿ ⭸g
q
1 2
g t2
⭸2 ⌿ ⭸g2
y g t y p F 0
Ž 22 .
⌿ Ž x . G sup ⌿ Ž k, y,a q . y Ž q2 ) 0 q q1 - 0 . 4
Ž 23 .
where A is the indicator function of the event A. Between each point of time of adjustment, Eq. Ž22. must hold with equality. Trying the functional form ⌿ s g ␥ for the homogenous part of the relation gives the following solution
¡¨
¨i s
~
¢
s A1 g ␥ 1 q B1 g ␥ 2 y 1
1
1
¨ 2 s A 2 g ␥ 1 q B2 g ␥ 2 y 2
2
g y Ž a1 . g y Ž a2 .
y y
p1 p2
;
a s a1
;
a s a2
Ž 24.
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where
␥ij
s
1 2
2 y Ž aj . "
'Ž y Ž a . . q 2 1 2
2
2
2
j
2
;
j s 1,2
Ž 25.
Let ␥1j ) 0 and ␥ 2j - 0. This assumption makes it easier to derive the optimal controls. Observe that ␥ 1 and ␥ 2 depend on a. The optimal level of capital increase, aU , will then be a function, though in most cases not continuous, of g. For very low values of g, i.e. low demand or a very large capital stock, it is optimal to keep production of new capital at a low level. The value function is then given by ¨ 1. In the case that g approaches zero then the value of ¨ 1 should be finite. Hence, it follows that B1 s 0. For very high demand or low levels of existing capital, production of new capital is kept at a high level and the value function is given by ¨ 2 . If g becomes very high the option to reduce the production of new capacity becomes small. From this it follows that A 2 s 0. Consequently, the value function is reduced to
¡¨
¨i s
~
¢¨
s A1 g ␥ 1 y 1
1
2
s B2 g
␥ 22
y
g y Ž a1 . g y Ž a2 .
y y
p1 p2
;
a s a1 Ž 26.
;
a s a2
In order to satisfy Eq. Ž21. of the optimisation problem, the value matching relations must hold, that is ¨ 1 Ž g i . s ¨ 2 Ž g i . y qi
Ž 27 .
and ¨ 1 Ž g r . s ¨ 2 Ž g r . y qr
Ž 28 .
Further, for the optimal controls to be optimal the high contact principle must be satisfied. That is, d¨ 1Ž g i . dg d¨ 1Ž g r . dg
s
s
d¨ 2 Ž g i . dg d¨ 2 Ž g r . dg
Ž 29.
Ž 30.
Hence, Eqs. Ž27. ᎐ Ž30. gives four equations to determine the four unknowns, g i , g r , A and B.
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5. Oil price dynamics Empirically, there seems to be a long run mean revering pattern in the oil price when observations of the crude oil price for the last 100 years are studied. See, e.g. Pindyck and Rubinfeld Ž1991., Dixit and Pindyck Ž1994.. From Eqs. Ž3., Ž7. and Ž11. it follows from Ito’s lemma that the dynamics of the price of oil, X, is given by a stochastic differential equation d Xt s Ž ˆ y Ut q ␦ . X t dt q ˆ X t d Bt
Ž 31.
where U is the optimal percentage increase of new capital in the oil industry. Between any change in U wEq. Ž31.x is a geometrical Brownian motion. If the production of new capacity is low, i.e. ˆ ) U y ␦, then the trend of the process is positive. If the production of new capital is high, i.e. ˆ - U y ␦, then the trend of the process is negative. High demand and high oil prices trigger high oil investments and vice versa. Consequently, the overall dynamics of the oil price, given by Eq. Ž31. and the optimality conditions for U is a mean reverting pattern. The standard deviation of the relative change in the oil price is identical to the standard deviation of the relative change in demand, . ˆ This property is related to the chosen shape of the supply and demand functions and is not a general result. Generally, the volatility in prices is related to the volatility of demand via the price elasticity of supply and demand. The oil price process wEq. Ž31.x is closely related to other partial differential equations that have been used for valuation of financial oil derivatives as well as for real option evaluation in the oil industry. The percentage increase of new capital U is a function of the oil price, X, though not continuous. For asset valuation purposes it is more convenient to assume that U is a continuous function of X. A popular version in this respect is to let U be linear in X. Let X ˆ y␦ U s where is a constant, and let ˆ xs . Then it follows that the oil price process wEq. Ž31.x is reduced to a geometrical mean reversion process of the form d Xt s Ž ˆ x y X t . X t dt q ˆ X t d Bt
Ž 32 .
For this process the long run oil price convergence level is given by ˆ x and the degree of convergence is given by . For applications of the process wEq. Ž32.x to the valuation of oil industry assets see, e.g. Paddock et al. Ž1988., Dixit and Pindyck Ž1994. who also present estimates for the parameter values. An alternative mean reverting process applied to commodity prices is d Xt s Ž ˆ x y ln X t . X t dt q ˆ X t d Bt
Ž 33.
In this case, the percentage increase of new capital is implicitly assumed to be ln X U s . The log of Eq. Ž33. is an Ornstein᎐Uhlenbeck process. For stochastic
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properties and applications to commodity markets see Schwartz Ž1997., Tvedt Ž1997, 1999.. For arguments why a geometric Brownian motion cannot be an equilibrium price process for exhaustible resources see Lund Ž1993.. For discussions of the relevance for valuation of assuming either random walk or mean reversion see, e.g. Metcalf and Hasset Ž1995., Schwartz Ž1997..
6. OPEC production and the effect on investments As discussed in the first part of this paper, OPEC controls almost 80% of the world oil reserves, but the OPEC production varies from approximately 30% to slightly above 50% of the world oil production. For the basic model it is assumed a 50r50 sharing of production between OPEC and the other oil producers. However, it may be optimal for the cartel to deviate from the 50r50 sharing of the market. OPEC prefers low oil investments in non-OPEC countries. Hence, it may be optimal for the cartel to produce more than 50% of the world demand such that the trigger price x r Ž g r ., is reached earlier and the trigger price x i Ž g i . is reached later than what is the case in the basic model. If the fringe producers were totally myopic, that is, they mistake a fall in the oil price from an increase in the cartel’s production for a fall in demand, a short-term OPEC production increase will be sufficient to trigger a reduction in oil investments. However, since production statistics are easily available, such myopic behaviour is a fairly unrealistic assumption. Nevertheless, there will be a trade-off between the period of high OPEC production that is necessary to trigger lower development of new fields in nonOPEC areas, and the future higher profit that is gained from higher oil prices due to a different path of U . To illustrate the effect of OPEC’s production response to a change in the investment levels in the fringe producers the basic model is extended. Let the deviation from the 50% market share at time t be given by t . The production of Yt OPEC at time t is then given by Q l t s Ž1 q t . and the production of the fringe 2 Yt producers is Q f t s Ž1 y t .. Price equals marginal cost implies from Eq. Ž2. that 2 the oil price is given by X t s 12 cYt Ž 1 y t .
Ž 34 .
From this it follows that the profit of OPEC is given by l t s 41 cYt 2 Ž1 y t .Ž1 q t ., the production cost of the fringe producers is Ct s 18 cYt 2 Ž1 y t . 2 and the aggregated profit of the fringe producers is f t s 18 cYt 2 Ž1 y t . 2 . The sum of the profit of OPEC and the production cost of the fringe producers are then given by g t s 18 cYt 2 Ž 1 y t .Ž 3 q t .
Ž 35 .
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For increased production, i.e. 0 - F 1, it follows from Eqs. Ž9. and Ž35. that g t - g t . The lower price of oil implies that the OPEC profit decrease more than the reduction in the production cost of the fringe producers. However, total producer and consumer profits increase since OPEC is assumed to have lower production costs than the fringe producers. The dynamics of g t is, as for g t in Eq. Ž9., given by Eq. Ž12.. The trend is, nevertheless, changed since the dynamics of oil industry investments will be different. The capital stock will in this case, be regulated in order to minimise g t , i.e. the cartels profit and the production cost of the fringe producers in the case of a deviation from the 50r50 sharing of the market, plus the cost of producing capital and the costs of changing the level of investments. To introduce / 0 does not change the basic structure of the optimal control problem. The only change is that F Ž X t . s ey t 38 cY 2 Ž1 y t .Ž3 q t . s ey t Ž g t y pt . in Eq. Ž15.. To simplify further, assume that OPEC keeps their market share mark-up at a fixed s l as long as non-OPEC investments are low and at s h as long as investments are high. The leader changes market share at the same time as the fringe producers change investment levels. The change in the leader’s production gives a jump in the oil price at the same time as the investment strategy changes. The relation between the oil prices for different market shares of the leader and a given level of demand is x th s x tl
Ž1 y h . Ž1 y l .
s 12 cYt Ž 1 y h .
Ž 36.
The jump in the oil price at the thresholds implies that the options to change investment levels are influenced. The value matching and high contact conditions are given by ¨ 1Ž g il . s ¨ 2 Ž g ih . y qi
Ž 37.
¨ 1Ž g rl . s ¨ 2 Ž g rh . q qr
Ž 38.
d¨ 1Ž g il . dg d¨ 1Ž g rl . dg
s
s
d¨ 2 Ž g ih . dg d¨ 2 Ž g rh .
Since g h s g l
dg Ž 1 y h .Ž 3 q h .
Ž 39.
Ž 40.
, Eqs. Ž37. ᎐ Ž40. give four equations to solve for Ž 1 y l .Ž 3 q l . the four unknowns A, B, g li and g lr.
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Fig. 1. Simulated path of a price process given optimal levels of investment and without any dynamic production strategy on behalf of the leader.
7. Simulations Fig. 1 shows the sample path of the oil price together with the increase and decrease price levels, x i and x r , in the case of s 0, and production levels of capital at h s 0.4 and l s 0.1.1 Fig. 2 shows the sample path of the oil price for the identical demand path Yt but with l s 0 and h s 0.1. From Figs. 1 and 2 it is evident that the oil price is mean reverting. A high oil price triggers increased investments in the oil industry. Consequently, the oil price then follows a downward trend. However, the trigger prices are not reflecting barriers, only the trend is changed. Hence, the oil price may overshoot the trigger prices. In Fig. 2, the leader increases production as soon as the fringe producers increase the level of investments. Hence, the oil price falls immediately. This makes it less attractive for the fringe producers to increase the investment level. Consequently, the trigger levels increase. That is, the fringe producers wait longer before increasing the investment level but return to a low investment level earlier. Consequently, total aggregated investment is reduced and this increases the long run average oil price. Hence, higher leader production in the case of high investment levels is a trade-off between reduced profit from reduced oil prices
1
The other parameter values in the simulation are as follows: p s 4; ␦ s 0.05; s 0.2; s 0.2; qi s 1; qr s 1;  s 1; ␣ s 1; k 0 s 1; and Y0 s 1.
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Fig. 2. Simulated path of a price process given optimal levels of investment with increased leader production during periods of high investments.
during the period of high oil investments vs. a shorter period of high investments and higher long-term prices. The effect on total efficiency of higher leader production is positive. High leader production reduces the investment of the fringe producers. In the model this has no effect on total production. The effect is solely on the price of oil and therefore, the only effect is on the distribution of total surplus between the leader, the fringe producers and the consumers. In the model, fringe producer investments are inefficient from a global perspective since the leader is assumed to have infinite capacity and the lowest production costs, whereas demand is totally inelastic.
8. Conclusion and final comments This industry level model gives an oil price process that is mean reverting. This is in accordance with empirical findings. The model may be seen as an argument for using a single stochastic differential equation for the oil price process as a starting point for valuing oil-related financial assets as well as real options, despite the complexity of the oil market. However, uncertainty is not only due to shifts in demand and technological shocks, but also due to strategic changes in OPEC’s production policy. Hence, when valuing oil-related options the market power of OPEC should be taken into account.
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Acknowledgements The paper has benefited substantially from suggestions to the model from Bernt Øksendal and Knut Aase, and from comments to an earlier version of the paper from participants at the Third Annual International Conference on Real Options in WassenaarrLeiden, the Netherlands, June 1999.
References Dixit, A., Pindyck, R.S., 1994. Investment under Uncertainty. Princeton University Press. Dixit, A., 1991. Irreversible investment with price ceiling. J. Polit. Econ. 99 Ž3., 541᎐557. Lucas, R.E., Jr., Prescott, E.C., 1971. Investment under uncertainty. Econometrica 39 Ž5., 659᎐681. Lund, D., 1993. The log-normal diffusion is hardly an equilibrium price process for exhaustible resources. J. Environ. Econ. Manage. 25, 235᎐241. Metcalf, G.E., Hasset, K.A., 1995. Investment under alternative return assumptions. Comparing random walks and mean reversion. J. Econ. Dyn. Control 19, 1471᎐1488. Pindyck, R., Rubinfeld, D., 1991. Econometric Models and Economic Forecasts, 3rd ed. McGraw-Hill. Paddock, J., Siegel, D., Smith, J., 1988. Option valuation of claims on real assets: the case of offshore petroleum leases. Q. J. Econ. 103, 479᎐508. Schwartz, E., 1997. The stochastic behaviour of commodity prices: implications for valuation and hedging. J. Financ. LII Ž3., 923᎐973. Tvedt, J., 1997. Valuation of VLCCs under income uncertainty. Marit. Policy Manage. 24 Ž2., 159᎐174. Tvedt, J., 1999. Valuation of a European futures option in the BIFFEX market. J. Futures Mark. 18 Ž2., 167᎐175.
The effect of uncertainty and aggregate investments on crude oil price dynamics Jostein TvedtU Den norske Bank, P.O. Box 1171 Sentrum, 0107 Oslo, Norway The Norwegian School of Economics and Business Administration, Bergen, Norway
Abstract This paper is a study of the dynamics of the oil industry and we derive a mean reverting process for the crude oil price. Oil is supplied by a market leader, OPEC, and by an aggregate that represents non-OPEC producers. The non-OPEC producers take the oil price as given. The cost of non-OPEC producers depends on past investments. Shifts in these investments are influenced by costs of structural change in the construction industry. A drop in the oil price to below a given level triggers lower investments, but if the oil price reverts back to a high level investments may not immediately expand. In an uncertain oil demand environment cost of structural change creates a value of waiting to invest. This investment behaviour influences the oil price process. 䊚 2002 Elsevier Science B.V. All rights reserved. JEL classifications: D4; D81; Q49 Keywords: Oil price dynamics; Stochastic optimal control
1. Introduction In the long-run the oil price seems to follow a mean reverting pattern. That is, in the case of high oil prices the underlying price trend is downward and in the case of very low oil prices the trend is upward. In periods of a close to perfectly U
Corresponding author. DnB Markets, P.O. Box 1171 Sentrum, 0107 Oslo, Norway. Tel.: q47-22017664; fax: q47-2248-2983. E-mail address: [email protected] ŽJ. Tvedt.. 0140-9883r02r$ - see front matter 䊚 2002 Elsevier Science B.V. Al rights reserved. PII: S 0 1 4 0 - 9 8 8 3 Ž 0 2 . 0 0 0 1 6 - 6
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competitive oil market the mean reversion in prices can be explained almost solely by restricted supply flexibility in an environment of stochastic demand. In the case of an unexpected jump in oil demand it takes time to increase production. Hence, the price of oil increases until new oil fields are developed and transport capacity is provided to a sufficient degree in order to revert the oil price towards a long-run average price level. In a perfectly competitive market the long-run average price level is determined by long-run marginal costs, which in the oil market is mainly determined by available oil reserves and technology. Some of the reasons for restricted supply flexibility are irreversibility of investment, time-to-build and restricted construction capacity. The cost of reducing these restrictions may in many cases be substantial. In the oil industry this is especially the case for large-scale offshore oil field developments. High oil prices may trigger a decision to develop new oil fields, but restricted construction capacity implies that it will take time from when a project is initiated to first oil. To adjust the capacity of the construction industry is costly. To increase the capacity entails substantial investments and labour force training. To decrease capacity may cause significant regional unemployment and loss of know-how. Since the early 1970s the oil price has been heavily influenced by the market power of the OPEC cartel. Besides internal rivalry, OPEC’s main challenge has been the trade-off between price and market share. High prices have triggered development of non-OPEC oil fields that by no means would have been profitable in a perfectly competitive market. During the high oil price regime from 1979 to 1985, OPEC production was reduced by approximately 45% due to a loss of market shares and reduced demand. Because of lower oil prices after 1986, OPEC production during the late 1990s is back at the same level as before the two oil price shocks in 1973 and 1979. A reduction in the marginal cost for the non-OPEC producers due to technological progress has challenged OPEC’s market power during the 1990s. Hence, despite lower oil prices than in the early 1980s there have been substantial oil field developments outside OPEC during the 1990s. The drop in oil prices in 1998 to pre-OPEC levels, despite improved oil prices during 1999, curbed investments for 2000 and 2001. At present, OPEC produces slightly above 40% of the total world consumption of crude oil. Back in the mid-1980s the OPEC market share was only marginally above 30%, which was down from above 50% in the first part of the 1970s. The market share of OPEC is in strong contrast to OPEC’s share of known oil reserves. OPEC’s share of known oil reserves is approximately 77% and the Middle East OPEC countries have alone 65% of known reserves. The production cost of OPEC and in particular, that of the Middle East countries is significantly below that of the rest of the world. According to rough estimates ŽThe Economist, 1999. production cost of oil, including costs related field development, is approximately US$2 per barrel in the Middle-East in contrast to US$10 in the US-Gulf and US$11 in the North Sea. Hence, OPEC operates as a leader in the crude oil market and restricts production to a significant degree in order to keep prices at a preferred level from an OPEC perspective.
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2. Overview of the paper At any point of time the equilibrium in the oil market is modelled as a leaderrfollower model. The leader, OPEC, has the lowest marginal costs whereas the marginal costs of the non-OPEC producers, the follower, depend on past and present investments. Present investments are restricted by the size of the construction industry. To increase or reduce the capacity of this industry entail costs. The dynamic of the market is therefore, modelled as a stochastic optimal control problem. Hence, uncertainty in the demand for oil, via the demand for oil industry investments, creates hysterisis effects in the adjustment of the capacity of the construction industry. In a perfectly competitive market, changes in investments are solely triggered by the current oil price, and the oil price is determined by the current demand and production capacity. However, the cartel members know that they influence the oil price by their production strategy. Hence, when choosing an oil production path the cartel may take into account the effect this path may have on the investment decision of the non-cartel producers. Hence, the cartel production strategy is a trade-off between high oil prices and the risk of triggering non-cartel oil industry investments.
3. A model of the crude oil market The static equilibrium of the oil market is modelled fairly straightforward in order to reduce the complexity of the dynamic model. It is assumed that OPEC operates as a cartel without internal rivalry, whereas the rest of the world’s oil producers take market prices as given, that is, they operate as if the market was perfectly competitive. The non-OPEC producers, or fringe producers, have an aggregated total variable production cost function given by C s ky␣ Q 2f
Ž1.
where k is the total capital employed by the fringe producers in oil production and Q f is the total oil production of the fringe producers, at a given point of time.  ) 0, ␣ ) 0 are constants. Given the cost function wEq. Ž1.x short-term marginal cost equal the oil price gives the linear inverse supply function of the fringe producers cQ f s X
Ž2.
where c s 2ky␣ and X is the price of oil. Total oil demand, Y, is assumed totally inelastic to the oil price. This assumption is fairly unrealistic in the long run, but simplifies the model to a significant degree and does not influence the main conclusions. A demand function like the one used by Dixit Ž1991. may be a possible
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extension to this model. Demand is assumed to be stochastic with incremental change given by a geometric Brownian motion dYt s Y ˆ t dt q Y ˆ t d Bt
Ž3.
where ˆ and ˆ are constants and have the standard interpretations, and d Bt is the increment of a standard Brownian motion. In the model there is no oil stocks and total supply of oil, Q, must at any time be equal demand. Therefore, the cartel will at any time produce the residual demand, and the cartel is assumed to have unlimited production capacity. The cartel’s oil production, Q l , will then be Ql s Y y
X c
Ž4.
The cartel is aware of how their oil production influences the oil price and that the fringe producers take the oil price as given. For simplicity assume that the cartel has zero production costs. The profit of the cartel in then given by l s XQ l s c Ž YQl y Q l2 .
Ž5.
Maximising instantaneous profit gives the optimal oil production of the cartel Ql s
Y 2
Ž6.
That is, given the zero cost assumption for the cartel, the totally inelastic demand and the linear supply function of the fringe producers, the cartel simply supply half the demand for oil. Historically, OPEC has produced approximately 30᎐50% of the total world consumption of oil. Given the 50r50 split of the oil market between the fringe producers and OPEC the equilibrium oil price is given by X s 12 cY
Ž7.
A competitive market assumption requires that the present value of the future instantaneous total market surpluses less the investment costs is maximised. See Lucas and Prescott Ž1971.. This model describes an imperfectly competitive market. However, the fringe producers and the consumers are assumed to behave as if they were operating in a perfectly competitive market. Hence, the competitive nature of the non-OPEC part of the market implies that investments should be modelled in order to maximise total consumer surplus and the profit of the fringe producers. This follows since the cartel is assumed to have an infinite production capacity and therefore, has no incentives to make additional investments. Given the oil price wEq. Ž7.x, the production split wEq. Ž6.x and the production cost of the fringe wEq. Ž1.x, the instantaneous profit of the fringe producers and the cartel is f s 18 cY 2 and l s 14 cY 2 , respectively. Given totally inelastic demand the consumer surplus is infinite. To make consumer surplus finite, assume an
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upper price level, P, at which other types of energy sources are substituted for oil. Then the consumer surplus plus the profit of the fringe producers are given by S s PY y 38 cY 2
Ž8.
Observe that the negative part of this relation is simply given by the instantaneous profit of the cartel plus the variable production costs of the fringe producers g s 38 cY 2
Ž9.
By changing the capital stock, k, of the fringe producers the future consumer and fringe producer surpluses are influenced via changes in c, i.e. via changes in the slope of the supply function of the fringe producers. Hence, to maximise the surplus, S, in Eq. Ž8. is equivalent to minimising the level of the cartel’s surplus and the production cost of the fringe producers via an optimal path of the production capital. Capital, k, is produced by the construction industry. Let the incremental change in the capital stock of the oil industry be linear in the total level of the existing capital stock of the fringe producers, i.e. d k t s a t k t dt
Ž 10 .
where a t is the marginal relative change of the capital stock at time t. Let the production of new capital be given by the marginal relative net addition of capital, t . Hence, t represents the activity level of the construction industry. Let the cost of producing new capacity at time t, be given by pt where p is a constant. The activity level of the construction industry can be changed, but at a cost. Let there be two alternative levels of capital production, 1 and 2 . Let the cost of moving from the low activity level, 1 , to the high activity level, 2 , be given by q2 and let the cost of moving from the high to the low activity level given by q1. Existing capital deteriorates at a fixed rate ␦. This may either be due to general depreciation of capital or it may be interpreted as a depletion rate of oil wells. It follows that a t s t y ␦. Given the dynamics of the capital stock, the dynamics of the slope of the supply curve, c s 2 ky␣ , will be given by dc t s ya t ␣ c t dt
Ž 11 .
That is, an increase in the capital stock will make the supply function more price elastic, i.e. c is reduced, and the equilibrium price drops. Consequently, the profit of the cartel is reduced as k increases. However, the 50r50 split of production will remain unchanged given the specification of the model. From Eqs. Ž3., Ž9. and Ž11. it follows by Ito’s lemma that the dynamics of the production cost of the fringe and the profit of the cartel, g, is given by a geometric Brownian motion d g t s g t dt q g t d Bt where s 38 ŽyŽ y ␦ . ␣ q 2 ˆ q ˆ 2 . and s 34 . ˆ
Ž 12.
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4. Dynamic equilibrium of the crude oil market To derive the dynamic equilibrium is restricted to regulate the capital stock in such a manner that the present value of the cartel’s profit and the production cost, plus the cost of producing capital and regulating the construction level, is minimised. Let ey t be a discount factor where is constant, and let X t be the state of the economic system, X t s w s q t,k t ,Yt ,a t xy1 . Then, it follows from the discussion above that the present value, at time zero, of the cartels profit, the production cost and the cost of new capital, at time t, is given by F Ž X . s ey t 38 cY 2 s ey t Ž g q p.
Ž 13 .
and that the present value, at time zero, of the cost of adjustment, at time t, is
¡q ; ~q ; ¢0;
K X j , j s ey t
ž
/
1
j - 0
2
j ) 0
Ž 14 .
j s 0
where j is the jump in at time j . The minimum of the present value of the cartel’s profit, the production cost and the costs of keeping an optimal investment path is given by the criteria function ⌰Ž x .. ⌰ Ž x . s inf E x
N
⬁
H0 F Ž X . dt q Ý K ž X t
js1
j , j
/
Ž 15 .
where the controls are given by , where s Ž 1 , 2 , . . . , N ; 1 , 1 , . . . , N ., N - ⬁, and 1 is the time of the first control and 1 is the size of the first jump in , etc. The incremental change in the state of the system, X t , between each change in , is given by
d Xs s
1 Ž s y ␦ . Ys 0
0 0 dt q d Bs Ys 0
Ž 16.
It follows from the assumptions above that the process X t ; t G 04 is a Markov process and hence, has an infinitesimal generator A, As
⭸ ⭸t
q gt
⭸ ⭸g
q 12 2 g t2
⭸2 ⭸g2
Ž 17 .
The minimising problem of Eq. Ž15. can be transformed into a maximising problem by the fact that infŽ a. s ysupŽya.. To simplify the discussion let ⌰Ž x . s
J. T¨ edt r Energy Economics 24 (2002) 615᎐628
⬁
⌽ Ž x . s sup E x
H y F Ž X . dt y Ý K ž X t
621
N j , j
js1
0
/
. The impulse control problem
may be handled by applying the approach of formulating the quasi-variational inequalities A⌽ q F F 0
Ž 18 .
⌽ Ž x . G M⌽ Ž x .
Ž 19 .
A⌽ q F Ž x .4 ⌽ Ž x . y M⌽ Ž x .4 s 0
Ž 20 .
where M is the shift operator, defined by M⌽ Ž x . s inf ⌽ Ž s,k , y,a q . y K Ž x, .4
Ž 21 .
That is, Eq. Ž19. holds with equality at the optimal time of control, and Eq. Ž18. holds with equality between the controls. Informally in words, if the value function in the present state is larger then what it would have been after a control, i.e. Eq. Ž19. does not hold with equality, then a control cannot be optimal. The value function is equal to the negative of the discounted value of cartel’s future profits and the production costs of the fringe plus the net value of the controls, given an optimal control strategy. If there is no control the change in the value should be equal to the cartel’s profits and the production costs. Hence, if Eq. Ž18. does not hold with equality then a control is optimal. But in that case, the value function after the control must be at least as large as before the control. Try a solution of the form ⌽ Ž x . s ey t ⌿ Ž x . for the value function, where ⌿ Ž x . is a time homogeneous function. Consequently, we may write Eqs. Ž18. and Ž19. as y⌿ q g t
⭸⌿ ⭸g
q
1 2
g t2
⭸2 ⌿ ⭸g2
y g t y p F 0
Ž 22 .
⌿ Ž x . G sup ⌿ Ž k, y,a q . y Ž q2 ) 0 q q1 - 0 . 4
Ž 23 .
where A is the indicator function of the event A. Between each point of time of adjustment, Eq. Ž22. must hold with equality. Trying the functional form ⌿ s g ␥ for the homogenous part of the relation gives the following solution
¡¨
¨i s
~
¢
s A1 g ␥ 1 q B1 g ␥ 2 y 1
1
1
¨ 2 s A 2 g ␥ 1 q B2 g ␥ 2 y 2
2
g y Ž a1 . g y Ž a2 .
y y
p1 p2
;
a s a1
;
a s a2
Ž 24.
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where
␥ij
s
1 2
2 y Ž aj . "
'Ž y Ž a . . q 2 1 2
2
2
2
j
2
;
j s 1,2
Ž 25.
Let ␥1j ) 0 and ␥ 2j - 0. This assumption makes it easier to derive the optimal controls. Observe that ␥ 1 and ␥ 2 depend on a. The optimal level of capital increase, aU , will then be a function, though in most cases not continuous, of g. For very low values of g, i.e. low demand or a very large capital stock, it is optimal to keep production of new capital at a low level. The value function is then given by ¨ 1. In the case that g approaches zero then the value of ¨ 1 should be finite. Hence, it follows that B1 s 0. For very high demand or low levels of existing capital, production of new capital is kept at a high level and the value function is given by ¨ 2 . If g becomes very high the option to reduce the production of new capacity becomes small. From this it follows that A 2 s 0. Consequently, the value function is reduced to
¡¨
¨i s
~
¢¨
s A1 g ␥ 1 y 1
1
2
s B2 g
␥ 22
y
g y Ž a1 . g y Ž a2 .
y y
p1 p2
;
a s a1 Ž 26.
;
a s a2
In order to satisfy Eq. Ž21. of the optimisation problem, the value matching relations must hold, that is ¨ 1 Ž g i . s ¨ 2 Ž g i . y qi
Ž 27 .
and ¨ 1 Ž g r . s ¨ 2 Ž g r . y qr
Ž 28 .
Further, for the optimal controls to be optimal the high contact principle must be satisfied. That is, d¨ 1Ž g i . dg d¨ 1Ž g r . dg
s
s
d¨ 2 Ž g i . dg d¨ 2 Ž g r . dg
Ž 29.
Ž 30.
Hence, Eqs. Ž27. ᎐ Ž30. gives four equations to determine the four unknowns, g i , g r , A and B.
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5. Oil price dynamics Empirically, there seems to be a long run mean revering pattern in the oil price when observations of the crude oil price for the last 100 years are studied. See, e.g. Pindyck and Rubinfeld Ž1991., Dixit and Pindyck Ž1994.. From Eqs. Ž3., Ž7. and Ž11. it follows from Ito’s lemma that the dynamics of the price of oil, X, is given by a stochastic differential equation d Xt s Ž ˆ y Ut q ␦ . X t dt q ˆ X t d Bt
Ž 31.
where U is the optimal percentage increase of new capital in the oil industry. Between any change in U wEq. Ž31.x is a geometrical Brownian motion. If the production of new capacity is low, i.e. ˆ ) U y ␦, then the trend of the process is positive. If the production of new capital is high, i.e. ˆ - U y ␦, then the trend of the process is negative. High demand and high oil prices trigger high oil investments and vice versa. Consequently, the overall dynamics of the oil price, given by Eq. Ž31. and the optimality conditions for U is a mean reverting pattern. The standard deviation of the relative change in the oil price is identical to the standard deviation of the relative change in demand, . ˆ This property is related to the chosen shape of the supply and demand functions and is not a general result. Generally, the volatility in prices is related to the volatility of demand via the price elasticity of supply and demand. The oil price process wEq. Ž31.x is closely related to other partial differential equations that have been used for valuation of financial oil derivatives as well as for real option evaluation in the oil industry. The percentage increase of new capital U is a function of the oil price, X, though not continuous. For asset valuation purposes it is more convenient to assume that U is a continuous function of X. A popular version in this respect is to let U be linear in X. Let X ˆ y␦ U s where is a constant, and let ˆ xs . Then it follows that the oil price process wEq. Ž31.x is reduced to a geometrical mean reversion process of the form d Xt s Ž ˆ x y X t . X t dt q ˆ X t d Bt
Ž 32 .
For this process the long run oil price convergence level is given by ˆ x and the degree of convergence is given by . For applications of the process wEq. Ž32.x to the valuation of oil industry assets see, e.g. Paddock et al. Ž1988., Dixit and Pindyck Ž1994. who also present estimates for the parameter values. An alternative mean reverting process applied to commodity prices is d Xt s Ž ˆ x y ln X t . X t dt q ˆ X t d Bt
Ž 33.
In this case, the percentage increase of new capital is implicitly assumed to be ln X U s . The log of Eq. Ž33. is an Ornstein᎐Uhlenbeck process. For stochastic
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properties and applications to commodity markets see Schwartz Ž1997., Tvedt Ž1997, 1999.. For arguments why a geometric Brownian motion cannot be an equilibrium price process for exhaustible resources see Lund Ž1993.. For discussions of the relevance for valuation of assuming either random walk or mean reversion see, e.g. Metcalf and Hasset Ž1995., Schwartz Ž1997..
6. OPEC production and the effect on investments As discussed in the first part of this paper, OPEC controls almost 80% of the world oil reserves, but the OPEC production varies from approximately 30% to slightly above 50% of the world oil production. For the basic model it is assumed a 50r50 sharing of production between OPEC and the other oil producers. However, it may be optimal for the cartel to deviate from the 50r50 sharing of the market. OPEC prefers low oil investments in non-OPEC countries. Hence, it may be optimal for the cartel to produce more than 50% of the world demand such that the trigger price x r Ž g r ., is reached earlier and the trigger price x i Ž g i . is reached later than what is the case in the basic model. If the fringe producers were totally myopic, that is, they mistake a fall in the oil price from an increase in the cartel’s production for a fall in demand, a short-term OPEC production increase will be sufficient to trigger a reduction in oil investments. However, since production statistics are easily available, such myopic behaviour is a fairly unrealistic assumption. Nevertheless, there will be a trade-off between the period of high OPEC production that is necessary to trigger lower development of new fields in nonOPEC areas, and the future higher profit that is gained from higher oil prices due to a different path of U . To illustrate the effect of OPEC’s production response to a change in the investment levels in the fringe producers the basic model is extended. Let the deviation from the 50% market share at time t be given by t . The production of Yt OPEC at time t is then given by Q l t s Ž1 q t . and the production of the fringe 2 Yt producers is Q f t s Ž1 y t .. Price equals marginal cost implies from Eq. Ž2. that 2 the oil price is given by X t s 12 cYt Ž 1 y t .
Ž 34 .
From this it follows that the profit of OPEC is given by l t s 41 cYt 2 Ž1 y t .Ž1 q t ., the production cost of the fringe producers is Ct s 18 cYt 2 Ž1 y t . 2 and the aggregated profit of the fringe producers is f t s 18 cYt 2 Ž1 y t . 2 . The sum of the profit of OPEC and the production cost of the fringe producers are then given by g t s 18 cYt 2 Ž 1 y t .Ž 3 q t .
Ž 35 .
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For increased production, i.e. 0 - F 1, it follows from Eqs. Ž9. and Ž35. that g t - g t . The lower price of oil implies that the OPEC profit decrease more than the reduction in the production cost of the fringe producers. However, total producer and consumer profits increase since OPEC is assumed to have lower production costs than the fringe producers. The dynamics of g t is, as for g t in Eq. Ž9., given by Eq. Ž12.. The trend is, nevertheless, changed since the dynamics of oil industry investments will be different. The capital stock will in this case, be regulated in order to minimise g t , i.e. the cartels profit and the production cost of the fringe producers in the case of a deviation from the 50r50 sharing of the market, plus the cost of producing capital and the costs of changing the level of investments. To introduce / 0 does not change the basic structure of the optimal control problem. The only change is that F Ž X t . s ey t 38 cY 2 Ž1 y t .Ž3 q t . s ey t Ž g t y pt . in Eq. Ž15.. To simplify further, assume that OPEC keeps their market share mark-up at a fixed s l as long as non-OPEC investments are low and at s h as long as investments are high. The leader changes market share at the same time as the fringe producers change investment levels. The change in the leader’s production gives a jump in the oil price at the same time as the investment strategy changes. The relation between the oil prices for different market shares of the leader and a given level of demand is x th s x tl
Ž1 y h . Ž1 y l .
s 12 cYt Ž 1 y h .
Ž 36.
The jump in the oil price at the thresholds implies that the options to change investment levels are influenced. The value matching and high contact conditions are given by ¨ 1Ž g il . s ¨ 2 Ž g ih . y qi
Ž 37.
¨ 1Ž g rl . s ¨ 2 Ž g rh . q qr
Ž 38.
d¨ 1Ž g il . dg d¨ 1Ž g rl . dg
s
s
d¨ 2 Ž g ih . dg d¨ 2 Ž g rh .
Since g h s g l
dg Ž 1 y h .Ž 3 q h .
Ž 39.
Ž 40.
, Eqs. Ž37. ᎐ Ž40. give four equations to solve for Ž 1 y l .Ž 3 q l . the four unknowns A, B, g li and g lr.
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Fig. 1. Simulated path of a price process given optimal levels of investment and without any dynamic production strategy on behalf of the leader.
7. Simulations Fig. 1 shows the sample path of the oil price together with the increase and decrease price levels, x i and x r , in the case of s 0, and production levels of capital at h s 0.4 and l s 0.1.1 Fig. 2 shows the sample path of the oil price for the identical demand path Yt but with l s 0 and h s 0.1. From Figs. 1 and 2 it is evident that the oil price is mean reverting. A high oil price triggers increased investments in the oil industry. Consequently, the oil price then follows a downward trend. However, the trigger prices are not reflecting barriers, only the trend is changed. Hence, the oil price may overshoot the trigger prices. In Fig. 2, the leader increases production as soon as the fringe producers increase the level of investments. Hence, the oil price falls immediately. This makes it less attractive for the fringe producers to increase the investment level. Consequently, the trigger levels increase. That is, the fringe producers wait longer before increasing the investment level but return to a low investment level earlier. Consequently, total aggregated investment is reduced and this increases the long run average oil price. Hence, higher leader production in the case of high investment levels is a trade-off between reduced profit from reduced oil prices
1
The other parameter values in the simulation are as follows: p s 4; ␦ s 0.05; s 0.2; s 0.2; qi s 1; qr s 1;  s 1; ␣ s 1; k 0 s 1; and Y0 s 1.
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Fig. 2. Simulated path of a price process given optimal levels of investment with increased leader production during periods of high investments.
during the period of high oil investments vs. a shorter period of high investments and higher long-term prices. The effect on total efficiency of higher leader production is positive. High leader production reduces the investment of the fringe producers. In the model this has no effect on total production. The effect is solely on the price of oil and therefore, the only effect is on the distribution of total surplus between the leader, the fringe producers and the consumers. In the model, fringe producer investments are inefficient from a global perspective since the leader is assumed to have infinite capacity and the lowest production costs, whereas demand is totally inelastic.
8. Conclusion and final comments This industry level model gives an oil price process that is mean reverting. This is in accordance with empirical findings. The model may be seen as an argument for using a single stochastic differential equation for the oil price process as a starting point for valuing oil-related financial assets as well as real options, despite the complexity of the oil market. However, uncertainty is not only due to shifts in demand and technological shocks, but also due to strategic changes in OPEC’s production policy. Hence, when valuing oil-related options the market power of OPEC should be taken into account.
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Acknowledgements The paper has benefited substantially from suggestions to the model from Bernt Øksendal and Knut Aase, and from comments to an earlier version of the paper from participants at the Third Annual International Conference on Real Options in WassenaarrLeiden, the Netherlands, June 1999.
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