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Economics of (oil) price politics: Penalizing price changes
Economics of (Oil) Price Pt;litics: Penalizing Price Changes Franz Wirl, Technical University of Vienna
This article studies intertemporal monopolistic pricing strategies when demand is dynamic and when price volatility may harm other producer objectives, e.g. the political good will for an international cartel. The major implications from this framework are that the introduction of a penalty for price changes may actually change an otherwise (at least locally) monotonic policy into a (transiently) oscillatory strategy; on the other hand, penalizing price changes smooths the price policy when demand is convex and the optimal strategy would otherwise be extremely volatile. But, then, all various types of solutions---cyclical and even unstable solutions--may occur.
1. INTRODUCTION
The traditional model of monopolistic pricing considers profits as the central criterion. This view has been criticized and extended by allowing for managerial discretion (see, e.g., Baumol, 1959). Leland (1980) tries to save the pure profit-maximizing hypothesis. Additionally, except for Leland's argument, most investigations are static. Therefore, the conventional model of a monopolist (or generally of a noncompetitive supplier) may provide a very inaccurate picture of many real-world commodity markets. The following analysis, although primarily theoretical, is inspired by the recent events of the world oil market. This market shows two distinct characteristics that contradict the textbook model of noncompetitive supply. OPEC, as the residual supplier of the world oil market and in particular Saudi Arabia, as the key member of OPEC, has definitely other objectives beyond pure profit maximization. Additionally, the nil market is very sluggish, that is, adjustment takes several years. Hence, dynamics and politics have to be included in an analysis of the oil market. The models of Pindyck (1978) and Marshalla and Nesbitt (1986) account for the dynamics of the oil market. However, the investigation of political aspects of the oil market have been left to date to the political scientists. Economists Address correspondence to rranz Wirl, lnstitut fiir Energiewirtschaft, Technische Universitiit Wien, Gusshausstr. 27-29, A-I040 Vienna, Austria Journal of Policy Modeling 13(4):515-527 ( 1991)
© Society for Policy Modeling, 1991
515 0161-8938/91/$3.50
516
F. Wirl
concentrate traditionally on narrow economic objectives, and see the defense in Marshalla and Nesbitt (1986). One approach to include political objectives is pursued in Wirl (1984), who computes "minimal" prices in order to maximize political goodwill subject to a profit constraint. He finds a considerable flexibility in (intertcmporal) OPEC crude oil pricing if a 10% reduction in revenues is acceptable. This approach is in line with a model proposed in Baumol (1959), who supposes that managers maximize revenues subject to a profit constraint. However, both approaches assume implicitly a lexicographical preference ordering that is not very likely; see Rosenberg (1971). The impact of dynamic demand relations on intertemporal monopolistic pricing has been studied in Wiri (1985). He finds that the dynamic solution may not exist, although the static problem using the steady state or equilibrium demand relation has a well-defined solution. For example, the optimal OPEC policy, subject to the world oil market representation according to the works of Marshalla and Nesbitt (! 986) and Amano (1987), would require a (discontinuously) chattering price strategy. Hence, the introduction of additional objectives could broaden the domain where intertemporal monopolistic pricing policies exist. This is the second motivation for the following investigation. The following section introduces the model and states the central proposition of this paper; the corresponding proof is relegated to an Appendix 1. Section 3 applies this proposition to OPEC pricing decisions by using an empirically estimated relation for the demand for OPEC oil. These examples will demonstrate the empirical relevance of the different price strategies that may result from the theoretical framework. A summary completes this investigation. 2. THE MODEL
Suppose that a monopolist attempts to maximize profits subject to the constraint that the price policy should be smooth. In particular assume that price changes rather than the price level create political turmoil, economic friction, and other distortions. Indeed, Saudi Arabia's prime goal was to smooth out, oil price volatility, as far as possible. During the periods of (perceived) shortages, 1979-1981, Saudi Arabia went to maximal capacity and broke sometimes away from the OPEC official selling pricer; but even towards 1986 Saudi 1In fact, Saudi Arabia, partly supported by other Gulf countries, charged deliberately lower crude oil prices during the first half of 1978, and at the beginning of the eighties, from 1981 until 1983.
PENALIZING OIL PRICE CHANGES
517
Arabia accepted cutbacks in production for a long time in order to support the prevailing oil price ($28/b at that time) and at least to avoid a dramatic price collapse. And recently the crisis due to Iraq's seizure of Kuwait in August 1990 confirmed this hypothesis. Saudi Arabia in particular (but also other Gulf countries) wanted to avoid another price shock and promised to increase output. This demonstrates that price changes create "real" costs for the Saudi government, largely for political reasons. Considering this political objective, suppose that the monopolist will maximize the present value of profits minus the loss that results from price variations. In order to simplify the exposition, suppose that a quadratic relation in price changes measures the associated political loss. This leads to the following optimal control problem: z
e'-"l(p(t) - c)x(t) - -~aq'(t)] dt;
max {q(t)}
(1)
0
Mt) = O[D(p(t)) - x(t)], x(O) = xo;
(2)
p ( t ) = q(t), p(O) = Po.
(3)
The monopolist applies the discount rate r to calculate the present value of benefits; c denotes constant marginal production costs; a is a positive constant that determines the penalty of a price change, q = j; D(p) is a conventional downward-sloping equilibrium demand function, D' < 0; 0 > 0 is the adjustment parameter: More precisely, (1/0) is the time constant of the differential equation (2). Dynamic demand relations of type (2) are frequently applied in empirical research. However, empirical economic applications draw on the discrete time pendent of (2) (see Section 3). Moreover, the relation (2) can be obtained from intertemporal consumer behavior if the consumers use myopic expectations to determine current adjustments (see Wirl, 1985). Assume that the steady-state revenue function (and hence, also the profit function, because of the assumption of linear production costs) is concave in both conceivable arguments: price (p) and quantity (x). This ensures that marginal revenues are downward-sloping vis h vis quantities, but upward-sloping vis ~ vi:x prices. Therefore, the static solution will yield a unique optimum from equating marginal revenues with marginal costs. However, Wirl (1985) shows that no intertemporal solution exists for the pure dynamic profit m~imization problem (maximize (1), setting a = 0 subject to (2), using price as control) if D is convex. More precisely, the optimal policy should chatter between two boundaries of admissible price policies once the "equilibrium"
518
F. Wirl
demand is reached. This points at another motivation: The inclusion of the loss resulting from a volatile price strategy could help to convert the otherwis~ volatile policy into a smoother one. Feichtinger and Sorger (1986) apply the Hopf bifurcation theorem to generate (stable) limit cycles for an originally chattering production-inventory problem. However, it will turn out that the introduction of a penalty in order to reduce volatility could ultimately create, oscillatory motions that would otherwise be absent. The following equations summarize the necessary optimality conditions of the control problem (1)-(3); Proposition 1 summarizes the major implications and results. H denotes the (current value) Hamiltonian; ~, and Ix denote the adjoint variables (also in current values) associated to the states x and p. The time subscript t and the other arguments are suppressed in order to simplify the notation. Subscript letters denote partial derivatives (for example, Hq = OH/Hq). !
H = tp - c)x - ~aq: + X0(D - x) + p#. H~ =
-aq
+ Ix = O==~ q = p.la.
= (r + O) k i~ = r p , -
x -
(p ~.0D'
c).
(4) (5) (6) (7)
PROPOSITION 1. The monopolist charges a higher price, at least in equilibrium, compared with the conventional static analysis (where marginal revenues must equal marginal costs). If the demand is convex, the equilibrium may be, in sequence of ascending penalties, an unstable focus, a limit cycle, a saddlepoint focus, or a saddlepoint. But surprisingly, the penalty for price variations may actually induce complex eigenvalues and hence (at least transient) oscillatory behavior, if D is concave. Two results of this proposition are of particular interest. First, and particularly striking, the introduction of a penalty for price changes may convert an otherwise monotonic policy into an oscillatory one. Second, the penalty does convert the chattering policy into a "smooth" policy, provided that the penalty is sufficiently high. Moreover, a very large penalty (for price changes) will result in a monotonic saddlepoint strategy even if demand is convex; without a penalty, the optimal strategy would be to apply a chattering price policy. This generalizes the proposition proven in Wirl (1985) that the equilibrium will be a saddlepoint if demand D(p) is concave and price changes are not penalized. On the other hand, lower penalties will result in a saddle-
Penalizing Oil Price C h a n g e s
519
point focus (i.e., transient oscillations but traditional stability) or even in an unstable focus (i.e., the price variations should grow over time, at a rate r/2). This transition from two roots with negative real parts to all roots having positive real parts may generate a Hopf bifurcation, that is, cyclical strategies, if the imaginary axis is crossed at a nonzero slope (see Guckenheimer and Holmes, 1983) for the Hopf bifurcation theorem. 2 Figure 1 shows the different equilibria within the stable manifold 3 (if existing; otherwise an arbitrary two-dimensional manifold from the four-dimensional state space) and for particular (nonlinear, local) coordinates. This discussion is restricted to the case that the elasticity of marginal price sensitivity, or: -- f"p/f", is not too negative. Otherwise, an unstable equilibrium of the following type is possible: One eigenvalue is negative and three eigenvalues are positive or have positive real parts. The evolution shown in Figure 1 proceeds vis ~tvis increases of the penalty parameter a. The following representations of the equilibria within the stable manifold are possible (again the above proviso applies): An unstable focus corresponds to complex eigenvalues where two have negative real parts; a stable node represents the conventional saddlepoint stability. The left-hand solution (i), an unstable focus, can occur only for convex demands. The solution (iii), stable focus (within the stable "saddlepoint manifold"), is possible under both a convex and a concave equilibrium demand relation. The transition from (i) to (iii) may include stable limit cycles, that is, a purely cyclical policy, if the equilibrium demand is convex, because of the Hopf bifurcation theorem. Feichtinger and Sorger (1986) consider a nonconcave production inventory problem and prove with the Hopfbifurcation theorem that adjustment costs may explain the optimality of cyclical production strategies. A necessary condition for limit cycles is that equation (A6) from Appendix 1 yield two pure imaginary roots, which requires that K > 0 and simultaneously II J II > K'/4; this follows immediately ~Fhis theorem states in lay terms that limit cycles exist in a one-sided surrounding of the bifurcation, if ~A6) yields two pure imaginary roots, if the derivative of the real part with respect to the b;.furc~.tion parameter (e.g., a) does not vanish at the bifurcation (i.e., where A6 yields two purely imaginary roots). Moreover, this cycle is stable if the coefficient of a quadratic Taylor approximation is negative. This last condition on stability is typically very difficult to check analytically. 3In case of saddlepoint stability two eigenvalues will have negative and two eigenvalues will have positive real parts. Hence, the general solution (of the linearized system) consists of linear combinations of the four exponentials (for properly chosen local coordinates). Now, a twodimensional manifold describes the condition that the coefficief,ts of the exponentially growing terms vanish.
520
F. Wirl
/\ %._
/
ii
I '
INCREASIN6
PE~IALTY
V
PARNe'.ETER
a
,
Eigenvalues (i)
(it)
(lii}
(iv)
Complex w i t h p o s i t i v e
COmplexw i t h p o s i t i v e r e e l
Couples and two have
Real and two
r e a l parts~
p a r t s : • paLr of lmeginat~/
negative r e a l p a r t s ,
negattve and two
eigenvaluee e x i s t s ,
are ere.
poelttve.
F i g u r e 1. Possible equilibria within a stable manifold, w h e n the elasticity o f marginal price sensitivity, o': = f"p/f' is such that o" > - 2 - (r/O).
from (A6) (see also Dockner, 1988)• For example, a small parameter a will help to establish that K > 0 and II J II > K214. A strongly convex demand, f " large, will make K large, but it will simultaneously increase K 2 - 4 II J II. Hence, small penalty a and a "medium" level off" could satisfy this (necessary) condition. Finally, a large parameter a will establish the conventional saddlepoint property (iv), even if demand is convex. 3. EXAMPLES (OPEC PRICING POLICIES) Discrete time versions of the dynamic consumer relation (2) are frequently applied, in particular to the energy market, (e.g., Pindyck 1978; Houthakker and Kennedy 1979; Amano 1987; and recently Hogan, 19894). In the following we will fit (ordinary least squares; OLS) a dynamic specification of (2) to the global demand for OPEC oil, supposing a quadratic approximation of the equ;.librium demand relationS:
4Hogan (1989) provides a thorough discussion and an extensive literature survey of the dynamics in energy marke;s. ~ h e main purpose of this reduced-form description of global demand for OPEC oil is to provide an empirical example complementary to the purely theoretical reasoning in Section 2. However, it is not claimed that (8) gives an accurate picture of all the forces influencing the pattern of global dependence on OPEC oil.
Penalizing Oil Price C h a n g e s 2.86 x, = (3.8) -
C.08~, (I.7} -
521 .00008~f 0,06t 0,92r~ ~0 I) + ~2.6) + (24.3)
'R'~ = 0.98.
(8}
The sample period is 1961-1986; "'end-of-year'" demand figures have been constructed, by interpolation of the annual figures, to comply with the underlying continuous time model." Similarly if, = (p, + p,_~)12 is introduced to account for delay (as opposed to dynamics from adjustment); otherwise an even larger time constant results. The figures in parentheses denote the absolute t statistics; R" gives the R-" corrected for degrees of freedom. OPEC production (according to OPEC Secretariat, 1988) minus stocks changes (according to OECD, 1988) proxy the demand for OPEC oil. The price series is obtained from the International Energy Agency (lEA, 1988) and OPEC Secretariat (1986) and proxies the real costs (in 1985 prices and 1985 exchange $US) for crude oil imports by lEA countries. The estimation of (8) reveals a large time constant (almost 12 years) for the demand for OPEC oil. Whether demand is concave or convex is indecisive because the point estimate (negative) of the quadratic term is highly insignificant.7 The following examples drop the time trend to study an autonomous demand relation: k = 0 . 0 8 3 4 (35.75 -
1.00p - 0 . 0 0 2 p " - x).
(9)
Figure 2 shows the (four) eigenvalues for the demand relation (9) and for variations of the penalty parameter, ae[ 100, 1,000]. This confirms the theoretical property that complex eigenvalues may result for a low penalty (for price adjustments) despite a concave equilibrium demand relation. However, a large penalty establishes the conventional characteristic of saddlepoint stability. Figure 3 shows the impact from variations of the second derivative of the equilibrium demand, ,nore precisely, variations in the degree of coavexity; the penalty parameter a is fixed in this exercise. The stability property (i.e., two eigenvalues with negative real parts) erodes as the convexity increases. But as expected from the theoretical in~:estigation in Section 2, the parameter a is more powerful, such that it generates all the different qualitative characteristics of optimal policies that may arise from the framework presented in this article; in 6Box and Jenkins (1976) show the correspo~ndence between continuous and discrete time models if the continuous model is generated by a pulsed input. Moreover. the parameter estimates that are obtained from the discrete data s t' -~- ' .: transformed into the parameters of the corresponding differt'ntial equation. 7Compare also Wirl ( 1990), which confirms this ambiguity about this crucial second der:"a~Ave of the equilibrium for OPEC oil.
522
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I ?
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0.05 0.04
P
0.03
e,
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b
0
i: o
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,
I"
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i
0.02 0.04 real ports
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I
0.08
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0. I
012
Fi~ure 2. Eigenvalues versus increases of a, a ¢ [ !00, 190001, concave demand.
12
1 08 0.6
0.4
/
L0 0
02
h c &
0 -0.2 -0.4 -0.6
L -
i
-0.8 --I
J
"
--
-I"2-
i -05
|
-05
II
II
-01 real
I
0.1
I
I
03
parts
Figure 3. Eigenvalues vis/t vis convexity, f ' ~ [0.002, 0.02].
I
I
0.5
Penalizing Oil Price Changes
523
0.,5 0.4 0.3 0.2 D to 0 O.
c 13n o
E
0.10 -01
~
/
-0.2 -0.3 -0.4 -O.5 ~ -008
-0.06
-0.04
-
.02
0.02
0 real
004
0.06
0.08
O. 1
O. 12
parts
Figure 4. Eigenvalues versus increases of a, a ¢ (0, ! ,000l, convex demand.
particular if demand is convex, the coefficient of the quadratic term is increased Io 0.0125. Figure 4 now shows, as claimed in Proposition 1, all differe t types of equilibria. Observe that the imaginary axis is crossed at a r ,nzero angle (at a = 2.38) so that (not necessarily stable) limit cycles ~^ist for a point that is sufficiently small, but close to the bifurcation point.
4. SUMMARY This article added political goals to the conventional profit objective of a monopolist. Additionally, a dynamic, rather than a static, demand relation was assumed. More precisely, the producer attempts to maximize profits but tries to avoid price changes and volatility in particular. These objectives are motivated by actual examples, in particular, Saudi Arabia's well-known and just recently proven intention to avoid dramatic price changes even if that means sacrificing revenues. Another motivation for this study is a theoretical puzzle: Dynamic demand implies a chattering policy if the demand relation is convex, even if the associated profit function is concave and a well-defined solution for the static framework exists. Now, a major surprise is the result that penalizing price changes may actually create (transient) oscillatory behavior when a (locally)
524
F. Wirl
monotonic price policy is optimal in the absence of such a penalty (i.e., for a concave equilibrium demand). A possible political explanation is the following: Saudi Arabia's attempt to "muddle through" between the conflicting objectives of maximizing revenues but smoothing price changes dampens current price shocks but may ultimately create a cyclical, albeit damped cyclical, policy. On the other hand, a very high penalty can convert a highly volatile price policy (chattering control resulting from a convex demand relation) into a (locally) monotonic saddlepoint strategy. However, low penalties and a convex demand could lead to price oscillations that are either persistent or increasing over time. The application to decisions of OPEC pricing proves that an oscillatory price behavior (possibly dampened) can be easily generated for plausible sets of parameters. Hence, the recent variations of oil prices, sometimes attributed to external political factors, could in fact comply with broader OPEC objectives. APPENDIX: PROOF OF PROPOSITION 1 q
Substitution of the maximum principle from equation (5) follows: = Ix~a, leads to the following canonical equations: = 0(D - x); =
ix~a;
= (r + 0)~. lfi. = rix - x -
(p -
c);
hOD'.
(AI)
The equilibrium (the index oo describes the steady-state solution) can be explicitly calculated as follows: x~ = D(p~); q~ = 0 ; h , = (p~ -
c)/(r + 0);
p,~ = 0.
(A2)
The optimal steady-state price policy follows from i.i, = 0, which implies the following relation: -D
-
(p -
c)OD'/(r + O) = O.
(A3)
This relation can be rearranged in the following way: [ P + ~--77"7, uD~(Pp ')l,] ,
= c
OD'(p)" rD(p)
(A4)
Penalizing Oil Price Changes
525
The left hand side describes the (long-run) marginal revenues and the right-hand side exceeds the marginal costs. Hence, the monopolist will ultimately charge a higher price compared with the static framework. Computation of the Jacobi matrix (J) of the canonical equations (A 1) gives -0 0D' 0 0 ] 0 0 0 l/a 0 -1 r + 0 0 " -! -h0D' -0D" r
J=
(A5)
Dockner (1985) derives an explicit formula for the eigenvalues (e~, i = 1 , . . . ,4) of a linearized (around an equilibrium) dynamic system that is obtained from the necessary optimality conditions of an optimal control problem with two state variables. This formula: I
3 4
le2 =
(r12) + / ( r 1 2 ) "
-
V
(K/2)
I
_ - x / K -~ -
2
4 tt J I t ,
~..6)
where K can be exaressed as the sum of the following determinants: II
g =
Ox Oh
+
Op OIL
+ 2
ap 0~11
a~, 0£
af~ aft
oh a~,
ax Ok
ap al~
ap al.t
(A7) "
will be employed to analyze the stability of the optimal policy of the two-state variable control problem (1)-(3). The determinant of J, using the equilibrium values (A2), is given by the following expression: II J II = ( - O / a ) {O' (r + 20) + (p - c)0D"}.
(AS)
Now assume that demand is concave, D" < 0. Therefore, from (A8), the determinant of the Jacobi matrix must be positive; that is II J II > 0. However, the discriminant in the second root of Equation (A6) can easily become negative even if demand is concave, because K 2 - 4 II J II =
O(r + O) +
0 r+0
(p - c)D"
+
40D' (r + 0). a
(AI0)
For example, choose an only weakly concave relation (D" -~ 0 but is still negative) and a large time constant of adjustment (i.e., 0 is relatively small). This implies that the first term in (AI0), which is positive, can be made arbitrarily small such that the second and negative term dominates. Hence, complex eigenvalues are possible even if demand is concave. However, K must be always negative. Therefore
526
F. Wiri
(see Dockner, 1985) two eigenvalues will have positive and the other two eigenvalues will have negative real partsmthat is, the traditional stability of optimal control problems applies. Now suppose that the equilibrium demand is convex. Under these circumstances, basically anything can happen. First of all, the determinant will still be positive, unless the elasticity of marginal price sensitivity (o': = f"p/f") is much smaller than -2, or more precisely tr < -2 - (r/0); in this case one eigenvalue will be negative and the remainder will be positive. Hence, the following: Assume that II J II > 0. Now, the coefficient K can be made negative, and simultaneously the discriminant expressed in (A 10) can become positive such that only real eigenvalues (two positive and two negative) result. Hence, a sufficiently high penalty parameter a converts the originally chattering policy into a (locally) monotonic saddlepoint property but complex eigenvalues may result. A further reduction of the penalty parameter a will make K positive and only positive real parts remain: global instability. REFERENCES Amano, A. (1987) A Small Forecasting Model of the World Oil Market. Journal of Policy Modeling 9: 615-635. Baumol, W.J. (1959) Business Behavior, Value and Growth. New York: Macmillan. Benhabib, J., and Nishimura, K. (1979) The Hopf Bifurcation and the Existence and Stability of Closed Orbits in Multi-sector Models of Economic Growth. Journal of Economic Theory 21:421 ~A.. Box, G.E.P., and Jenkins, G.M. (1976) Time Series Analysis. San Francisco: Holden Day. Dockner, E. (1985) Local Stability Analysis, in Optimal Control Problems with Two State Variables. in Optimal Control Theory and Economic Analysis, Vol. 2 (G. Feichtinger, Ed.). New York: North-Holland. Dockner, E. (1988) On the Optimality of Cyclical Policies in Dynamic Economic Systems. Paper presented at the Optimization Days, Montreal, Canada, May 2-4. Feichtinger, G., and Sorger, G. (1986) Optimal Oscillations in Control Models: How can Constant Demand Lead to Cyclical Production? Operations Research Letters 5:277-281. Guckenheimer, J., and Holmes, P. (1983)Nonlinear Oscillations, Dynamical Systems, and Bifurcation of Vector Fields. New York: Springer. Hogan, W,W. (1989) A Dynamic Putty-Semi-putty Model of Aggregate Energy Demand. Energy Economics pp. 53-69. Houthakker, H.S., Kennedy, M. (1979) A Long-run Model of World Energy Demands, Supplies, and Prices. in Directions in Energy Policy: A Comprehensive Approach to Energy Resource Decision Making. (B. Kursunoglu and A. Peflmutter, Eds.). Cambridge, MA: Ballinger. International Energy Agency (lEA) (1988) Energy Prices and Ttttes, Third Quarter 1987. Paris: OECD/IEA. Leland, H.E. (1980) Alternative long-Run Goals and the Theory of the Firm: Why Profit Maximization May Be a Better Assumption Than You Think. in Dynamic Optimization and Mathematical Models (Pan-Tai Liu, Ed.). New York: Plenum Press.
Penalizing Oil Price Changes
527
Marshalla, R.A., and Nesbitt, D.M. (1986) Future World Oil Prices and Production Levels: An Economic Analysis. Energy Journal 7( i ): i-22. OECD (1988) Energy Statistics. Paris: Organization for Economic Cooperation and Development. OPEC Secretariat (1986) Petroleum Product Prices and their Components in Selected Countries. Vienna: OPEC Secretariat. OPEC Secretariat (1988) Annual Statistical Bul;etin 1987. Vienna: OPEC Secrelariat. Pindyck, R.S. (1978) Gains to Producers from the Cartelization of Exhaustible Resources. Review of Economi~'s and Statistics. pp. 238-25 l. Rosenberg, R. (1971) Profit Constraint Revenue Maximization; A Note. American Economic Review 61: 208-209. Wirl, F. (1984) Sensitivity Analysis of Opec Pricing Policies. Opec Review 8: 321-371. Wirl, F. (1985) Stable and Volatile Prices: An Explanation by Dynamic Demand. In Optimal Control Theory. and Economic Analysis, Vol. 2 (G. Feichtinger, Ed.). New York: NorthHolland. Wid, F. (1990) Dynamic Demand and Optimal Opec Pricing: An Empirical Investigation. Energy Economics ! 2:174-- 176.
This article studies intertemporal monopolistic pricing strategies when demand is dynamic and when price volatility may harm other producer objectives, e.g. the political good will for an international cartel. The major implications from this framework are that the introduction of a penalty for price changes may actually change an otherwise (at least locally) monotonic policy into a (transiently) oscillatory strategy; on the other hand, penalizing price changes smooths the price policy when demand is convex and the optimal strategy would otherwise be extremely volatile. But, then, all various types of solutions---cyclical and even unstable solutions--may occur.
1. INTRODUCTION
The traditional model of monopolistic pricing considers profits as the central criterion. This view has been criticized and extended by allowing for managerial discretion (see, e.g., Baumol, 1959). Leland (1980) tries to save the pure profit-maximizing hypothesis. Additionally, except for Leland's argument, most investigations are static. Therefore, the conventional model of a monopolist (or generally of a noncompetitive supplier) may provide a very inaccurate picture of many real-world commodity markets. The following analysis, although primarily theoretical, is inspired by the recent events of the world oil market. This market shows two distinct characteristics that contradict the textbook model of noncompetitive supply. OPEC, as the residual supplier of the world oil market and in particular Saudi Arabia, as the key member of OPEC, has definitely other objectives beyond pure profit maximization. Additionally, the nil market is very sluggish, that is, adjustment takes several years. Hence, dynamics and politics have to be included in an analysis of the oil market. The models of Pindyck (1978) and Marshalla and Nesbitt (1986) account for the dynamics of the oil market. However, the investigation of political aspects of the oil market have been left to date to the political scientists. Economists Address correspondence to rranz Wirl, lnstitut fiir Energiewirtschaft, Technische Universitiit Wien, Gusshausstr. 27-29, A-I040 Vienna, Austria Journal of Policy Modeling 13(4):515-527 ( 1991)
© Society for Policy Modeling, 1991
515 0161-8938/91/$3.50
516
F. Wirl
concentrate traditionally on narrow economic objectives, and see the defense in Marshalla and Nesbitt (1986). One approach to include political objectives is pursued in Wirl (1984), who computes "minimal" prices in order to maximize political goodwill subject to a profit constraint. He finds a considerable flexibility in (intertcmporal) OPEC crude oil pricing if a 10% reduction in revenues is acceptable. This approach is in line with a model proposed in Baumol (1959), who supposes that managers maximize revenues subject to a profit constraint. However, both approaches assume implicitly a lexicographical preference ordering that is not very likely; see Rosenberg (1971). The impact of dynamic demand relations on intertemporal monopolistic pricing has been studied in Wiri (1985). He finds that the dynamic solution may not exist, although the static problem using the steady state or equilibrium demand relation has a well-defined solution. For example, the optimal OPEC policy, subject to the world oil market representation according to the works of Marshalla and Nesbitt (! 986) and Amano (1987), would require a (discontinuously) chattering price strategy. Hence, the introduction of additional objectives could broaden the domain where intertemporal monopolistic pricing policies exist. This is the second motivation for the following investigation. The following section introduces the model and states the central proposition of this paper; the corresponding proof is relegated to an Appendix 1. Section 3 applies this proposition to OPEC pricing decisions by using an empirically estimated relation for the demand for OPEC oil. These examples will demonstrate the empirical relevance of the different price strategies that may result from the theoretical framework. A summary completes this investigation. 2. THE MODEL
Suppose that a monopolist attempts to maximize profits subject to the constraint that the price policy should be smooth. In particular assume that price changes rather than the price level create political turmoil, economic friction, and other distortions. Indeed, Saudi Arabia's prime goal was to smooth out, oil price volatility, as far as possible. During the periods of (perceived) shortages, 1979-1981, Saudi Arabia went to maximal capacity and broke sometimes away from the OPEC official selling pricer; but even towards 1986 Saudi 1In fact, Saudi Arabia, partly supported by other Gulf countries, charged deliberately lower crude oil prices during the first half of 1978, and at the beginning of the eighties, from 1981 until 1983.
PENALIZING OIL PRICE CHANGES
517
Arabia accepted cutbacks in production for a long time in order to support the prevailing oil price ($28/b at that time) and at least to avoid a dramatic price collapse. And recently the crisis due to Iraq's seizure of Kuwait in August 1990 confirmed this hypothesis. Saudi Arabia in particular (but also other Gulf countries) wanted to avoid another price shock and promised to increase output. This demonstrates that price changes create "real" costs for the Saudi government, largely for political reasons. Considering this political objective, suppose that the monopolist will maximize the present value of profits minus the loss that results from price variations. In order to simplify the exposition, suppose that a quadratic relation in price changes measures the associated political loss. This leads to the following optimal control problem: z
e'-"l(p(t) - c)x(t) - -~aq'(t)] dt;
max {q(t)}
(1)
0
Mt) = O[D(p(t)) - x(t)], x(O) = xo;
(2)
p ( t ) = q(t), p(O) = Po.
(3)
The monopolist applies the discount rate r to calculate the present value of benefits; c denotes constant marginal production costs; a is a positive constant that determines the penalty of a price change, q = j; D(p) is a conventional downward-sloping equilibrium demand function, D' < 0; 0 > 0 is the adjustment parameter: More precisely, (1/0) is the time constant of the differential equation (2). Dynamic demand relations of type (2) are frequently applied in empirical research. However, empirical economic applications draw on the discrete time pendent of (2) (see Section 3). Moreover, the relation (2) can be obtained from intertemporal consumer behavior if the consumers use myopic expectations to determine current adjustments (see Wirl, 1985). Assume that the steady-state revenue function (and hence, also the profit function, because of the assumption of linear production costs) is concave in both conceivable arguments: price (p) and quantity (x). This ensures that marginal revenues are downward-sloping vis h vis quantities, but upward-sloping vis ~ vi:x prices. Therefore, the static solution will yield a unique optimum from equating marginal revenues with marginal costs. However, Wirl (1985) shows that no intertemporal solution exists for the pure dynamic profit m~imization problem (maximize (1), setting a = 0 subject to (2), using price as control) if D is convex. More precisely, the optimal policy should chatter between two boundaries of admissible price policies once the "equilibrium"
518
F. Wirl
demand is reached. This points at another motivation: The inclusion of the loss resulting from a volatile price strategy could help to convert the otherwis~ volatile policy into a smoother one. Feichtinger and Sorger (1986) apply the Hopf bifurcation theorem to generate (stable) limit cycles for an originally chattering production-inventory problem. However, it will turn out that the introduction of a penalty in order to reduce volatility could ultimately create, oscillatory motions that would otherwise be absent. The following equations summarize the necessary optimality conditions of the control problem (1)-(3); Proposition 1 summarizes the major implications and results. H denotes the (current value) Hamiltonian; ~, and Ix denote the adjoint variables (also in current values) associated to the states x and p. The time subscript t and the other arguments are suppressed in order to simplify the notation. Subscript letters denote partial derivatives (for example, Hq = OH/Hq). !
H = tp - c)x - ~aq: + X0(D - x) + p#. H~ =
-aq
+ Ix = O==~ q = p.la.
= (r + O) k i~ = r p , -
x -
(p ~.0D'
c).
(4) (5) (6) (7)
PROPOSITION 1. The monopolist charges a higher price, at least in equilibrium, compared with the conventional static analysis (where marginal revenues must equal marginal costs). If the demand is convex, the equilibrium may be, in sequence of ascending penalties, an unstable focus, a limit cycle, a saddlepoint focus, or a saddlepoint. But surprisingly, the penalty for price variations may actually induce complex eigenvalues and hence (at least transient) oscillatory behavior, if D is concave. Two results of this proposition are of particular interest. First, and particularly striking, the introduction of a penalty for price changes may convert an otherwise monotonic policy into an oscillatory one. Second, the penalty does convert the chattering policy into a "smooth" policy, provided that the penalty is sufficiently high. Moreover, a very large penalty (for price changes) will result in a monotonic saddlepoint strategy even if demand is convex; without a penalty, the optimal strategy would be to apply a chattering price policy. This generalizes the proposition proven in Wirl (1985) that the equilibrium will be a saddlepoint if demand D(p) is concave and price changes are not penalized. On the other hand, lower penalties will result in a saddle-
Penalizing Oil Price C h a n g e s
519
point focus (i.e., transient oscillations but traditional stability) or even in an unstable focus (i.e., the price variations should grow over time, at a rate r/2). This transition from two roots with negative real parts to all roots having positive real parts may generate a Hopf bifurcation, that is, cyclical strategies, if the imaginary axis is crossed at a nonzero slope (see Guckenheimer and Holmes, 1983) for the Hopf bifurcation theorem. 2 Figure 1 shows the different equilibria within the stable manifold 3 (if existing; otherwise an arbitrary two-dimensional manifold from the four-dimensional state space) and for particular (nonlinear, local) coordinates. This discussion is restricted to the case that the elasticity of marginal price sensitivity, or: -- f"p/f", is not too negative. Otherwise, an unstable equilibrium of the following type is possible: One eigenvalue is negative and three eigenvalues are positive or have positive real parts. The evolution shown in Figure 1 proceeds vis ~tvis increases of the penalty parameter a. The following representations of the equilibria within the stable manifold are possible (again the above proviso applies): An unstable focus corresponds to complex eigenvalues where two have negative real parts; a stable node represents the conventional saddlepoint stability. The left-hand solution (i), an unstable focus, can occur only for convex demands. The solution (iii), stable focus (within the stable "saddlepoint manifold"), is possible under both a convex and a concave equilibrium demand relation. The transition from (i) to (iii) may include stable limit cycles, that is, a purely cyclical policy, if the equilibrium demand is convex, because of the Hopf bifurcation theorem. Feichtinger and Sorger (1986) consider a nonconcave production inventory problem and prove with the Hopfbifurcation theorem that adjustment costs may explain the optimality of cyclical production strategies. A necessary condition for limit cycles is that equation (A6) from Appendix 1 yield two pure imaginary roots, which requires that K > 0 and simultaneously II J II > K'/4; this follows immediately ~Fhis theorem states in lay terms that limit cycles exist in a one-sided surrounding of the bifurcation, if ~A6) yields two pure imaginary roots, if the derivative of the real part with respect to the b;.furc~.tion parameter (e.g., a) does not vanish at the bifurcation (i.e., where A6 yields two purely imaginary roots). Moreover, this cycle is stable if the coefficient of a quadratic Taylor approximation is negative. This last condition on stability is typically very difficult to check analytically. 3In case of saddlepoint stability two eigenvalues will have negative and two eigenvalues will have positive real parts. Hence, the general solution (of the linearized system) consists of linear combinations of the four exponentials (for properly chosen local coordinates). Now, a twodimensional manifold describes the condition that the coefficief,ts of the exponentially growing terms vanish.
520
F. Wirl
/\ %._
/
ii
I '
INCREASIN6
PE~IALTY
V
PARNe'.ETER
a
,
Eigenvalues (i)
(it)
(lii}
(iv)
Complex w i t h p o s i t i v e
COmplexw i t h p o s i t i v e r e e l
Couples and two have
Real and two
r e a l parts~
p a r t s : • paLr of lmeginat~/
negative r e a l p a r t s ,
negattve and two
eigenvaluee e x i s t s ,
are ere.
poelttve.
F i g u r e 1. Possible equilibria within a stable manifold, w h e n the elasticity o f marginal price sensitivity, o': = f"p/f' is such that o" > - 2 - (r/O).
from (A6) (see also Dockner, 1988)• For example, a small parameter a will help to establish that K > 0 and II J II > K214. A strongly convex demand, f " large, will make K large, but it will simultaneously increase K 2 - 4 II J II. Hence, small penalty a and a "medium" level off" could satisfy this (necessary) condition. Finally, a large parameter a will establish the conventional saddlepoint property (iv), even if demand is convex. 3. EXAMPLES (OPEC PRICING POLICIES) Discrete time versions of the dynamic consumer relation (2) are frequently applied, in particular to the energy market, (e.g., Pindyck 1978; Houthakker and Kennedy 1979; Amano 1987; and recently Hogan, 19894). In the following we will fit (ordinary least squares; OLS) a dynamic specification of (2) to the global demand for OPEC oil, supposing a quadratic approximation of the equ;.librium demand relationS:
4Hogan (1989) provides a thorough discussion and an extensive literature survey of the dynamics in energy marke;s. ~ h e main purpose of this reduced-form description of global demand for OPEC oil is to provide an empirical example complementary to the purely theoretical reasoning in Section 2. However, it is not claimed that (8) gives an accurate picture of all the forces influencing the pattern of global dependence on OPEC oil.
Penalizing Oil Price C h a n g e s 2.86 x, = (3.8) -
C.08~, (I.7} -
521 .00008~f 0,06t 0,92r~ ~0 I) + ~2.6) + (24.3)
'R'~ = 0.98.
(8}
The sample period is 1961-1986; "'end-of-year'" demand figures have been constructed, by interpolation of the annual figures, to comply with the underlying continuous time model." Similarly if, = (p, + p,_~)12 is introduced to account for delay (as opposed to dynamics from adjustment); otherwise an even larger time constant results. The figures in parentheses denote the absolute t statistics; R" gives the R-" corrected for degrees of freedom. OPEC production (according to OPEC Secretariat, 1988) minus stocks changes (according to OECD, 1988) proxy the demand for OPEC oil. The price series is obtained from the International Energy Agency (lEA, 1988) and OPEC Secretariat (1986) and proxies the real costs (in 1985 prices and 1985 exchange $US) for crude oil imports by lEA countries. The estimation of (8) reveals a large time constant (almost 12 years) for the demand for OPEC oil. Whether demand is concave or convex is indecisive because the point estimate (negative) of the quadratic term is highly insignificant.7 The following examples drop the time trend to study an autonomous demand relation: k = 0 . 0 8 3 4 (35.75 -
1.00p - 0 . 0 0 2 p " - x).
(9)
Figure 2 shows the (four) eigenvalues for the demand relation (9) and for variations of the penalty parameter, ae[ 100, 1,000]. This confirms the theoretical property that complex eigenvalues may result for a low penalty (for price adjustments) despite a concave equilibrium demand relation. However, a large penalty establishes the conventional characteristic of saddlepoint stability. Figure 3 shows the impact from variations of the second derivative of the equilibrium demand, ,nore precisely, variations in the degree of coavexity; the penalty parameter a is fixed in this exercise. The stability property (i.e., two eigenvalues with negative real parts) erodes as the convexity increases. But as expected from the theoretical in~:estigation in Section 2, the parameter a is more powerful, such that it generates all the different qualitative characteristics of optimal policies that may arise from the framework presented in this article; in 6Box and Jenkins (1976) show the correspo~ndence between continuous and discrete time models if the continuous model is generated by a pulsed input. Moreover. the parameter estimates that are obtained from the discrete data s t' -~- ' .: transformed into the parameters of the corresponding differt'ntial equation. 7Compare also Wirl ( 1990), which confirms this ambiguity about this crucial second der:"a~Ave of the equilibrium for OPEC oil.
522
F. Wirl 0.07 0.06
I ?
y~,
0.05 0.04
P
0.03
e,
0.02 0,01 -
b
0
i: o
E
L_
L ....
AAA
-" -"".~."
A
-001
-0.02 -0.03 -0.04 - 0 05 -0.06 -0.07
,
i
-0.06
-0,08
,
,
l
,
,
I"
-0.04 -0.02
0
I
I
i
0.02 0.04 real ports
I
I
I
I
0.06
I
I
0.08
I
I
0. I
012
Fi~ure 2. Eigenvalues versus increases of a, a ¢ [ !00, 190001, concave demand.
12
1 08 0.6
0.4
/
L0 0
02
h c &
0 -0.2 -0.4 -0.6
L -
i
-0.8 --I
J
"
--
-I"2-
i -05
|
-05
II
II
-01 real
I
0.1
I
I
03
parts
Figure 3. Eigenvalues vis/t vis convexity, f ' ~ [0.002, 0.02].
I
I
0.5
Penalizing Oil Price Changes
523
0.,5 0.4 0.3 0.2 D to 0 O.
c 13n o
E
0.10 -01
~
/
-0.2 -0.3 -0.4 -O.5 ~ -008
-0.06
-0.04
-
.02
0.02
0 real
004
0.06
0.08
O. 1
O. 12
parts
Figure 4. Eigenvalues versus increases of a, a ¢ (0, ! ,000l, convex demand.
particular if demand is convex, the coefficient of the quadratic term is increased Io 0.0125. Figure 4 now shows, as claimed in Proposition 1, all differe t types of equilibria. Observe that the imaginary axis is crossed at a r ,nzero angle (at a = 2.38) so that (not necessarily stable) limit cycles ~^ist for a point that is sufficiently small, but close to the bifurcation point.
4. SUMMARY This article added political goals to the conventional profit objective of a monopolist. Additionally, a dynamic, rather than a static, demand relation was assumed. More precisely, the producer attempts to maximize profits but tries to avoid price changes and volatility in particular. These objectives are motivated by actual examples, in particular, Saudi Arabia's well-known and just recently proven intention to avoid dramatic price changes even if that means sacrificing revenues. Another motivation for this study is a theoretical puzzle: Dynamic demand implies a chattering policy if the demand relation is convex, even if the associated profit function is concave and a well-defined solution for the static framework exists. Now, a major surprise is the result that penalizing price changes may actually create (transient) oscillatory behavior when a (locally)
524
F. Wirl
monotonic price policy is optimal in the absence of such a penalty (i.e., for a concave equilibrium demand). A possible political explanation is the following: Saudi Arabia's attempt to "muddle through" between the conflicting objectives of maximizing revenues but smoothing price changes dampens current price shocks but may ultimately create a cyclical, albeit damped cyclical, policy. On the other hand, a very high penalty can convert a highly volatile price policy (chattering control resulting from a convex demand relation) into a (locally) monotonic saddlepoint strategy. However, low penalties and a convex demand could lead to price oscillations that are either persistent or increasing over time. The application to decisions of OPEC pricing proves that an oscillatory price behavior (possibly dampened) can be easily generated for plausible sets of parameters. Hence, the recent variations of oil prices, sometimes attributed to external political factors, could in fact comply with broader OPEC objectives. APPENDIX: PROOF OF PROPOSITION 1 q
Substitution of the maximum principle from equation (5) follows: = Ix~a, leads to the following canonical equations: = 0(D - x); =
ix~a;
= (r + 0)~. lfi. = rix - x -
(p -
c);
hOD'.
(AI)
The equilibrium (the index oo describes the steady-state solution) can be explicitly calculated as follows: x~ = D(p~); q~ = 0 ; h , = (p~ -
c)/(r + 0);
p,~ = 0.
(A2)
The optimal steady-state price policy follows from i.i, = 0, which implies the following relation: -D
-
(p -
c)OD'/(r + O) = O.
(A3)
This relation can be rearranged in the following way: [ P + ~--77"7, uD~(Pp ')l,] ,
= c
OD'(p)" rD(p)
(A4)
Penalizing Oil Price Changes
525
The left hand side describes the (long-run) marginal revenues and the right-hand side exceeds the marginal costs. Hence, the monopolist will ultimately charge a higher price compared with the static framework. Computation of the Jacobi matrix (J) of the canonical equations (A 1) gives -0 0D' 0 0 ] 0 0 0 l/a 0 -1 r + 0 0 " -! -h0D' -0D" r
J=
(A5)
Dockner (1985) derives an explicit formula for the eigenvalues (e~, i = 1 , . . . ,4) of a linearized (around an equilibrium) dynamic system that is obtained from the necessary optimality conditions of an optimal control problem with two state variables. This formula: I
3 4
le2 =
(r12) + / ( r 1 2 ) "
-
V
(K/2)
I
_ - x / K -~ -
2
4 tt J I t ,
~..6)
where K can be exaressed as the sum of the following determinants: II
g =
Ox Oh
+
Op OIL
+ 2
ap 0~11
a~, 0£
af~ aft
oh a~,
ax Ok
ap al~
ap al.t
(A7) "
will be employed to analyze the stability of the optimal policy of the two-state variable control problem (1)-(3). The determinant of J, using the equilibrium values (A2), is given by the following expression: II J II = ( - O / a ) {O' (r + 20) + (p - c)0D"}.
(AS)
Now assume that demand is concave, D" < 0. Therefore, from (A8), the determinant of the Jacobi matrix must be positive; that is II J II > 0. However, the discriminant in the second root of Equation (A6) can easily become negative even if demand is concave, because K 2 - 4 II J II =
O(r + O) +
0 r+0
(p - c)D"
+
40D' (r + 0). a
(AI0)
For example, choose an only weakly concave relation (D" -~ 0 but is still negative) and a large time constant of adjustment (i.e., 0 is relatively small). This implies that the first term in (AI0), which is positive, can be made arbitrarily small such that the second and negative term dominates. Hence, complex eigenvalues are possible even if demand is concave. However, K must be always negative. Therefore
526
F. Wiri
(see Dockner, 1985) two eigenvalues will have positive and the other two eigenvalues will have negative real partsmthat is, the traditional stability of optimal control problems applies. Now suppose that the equilibrium demand is convex. Under these circumstances, basically anything can happen. First of all, the determinant will still be positive, unless the elasticity of marginal price sensitivity (o': = f"p/f") is much smaller than -2, or more precisely tr < -2 - (r/0); in this case one eigenvalue will be negative and the remainder will be positive. Hence, the following: Assume that II J II > 0. Now, the coefficient K can be made negative, and simultaneously the discriminant expressed in (A 10) can become positive such that only real eigenvalues (two positive and two negative) result. Hence, a sufficiently high penalty parameter a converts the originally chattering policy into a (locally) monotonic saddlepoint property but complex eigenvalues may result. A further reduction of the penalty parameter a will make K positive and only positive real parts remain: global instability. REFERENCES Amano, A. (1987) A Small Forecasting Model of the World Oil Market. Journal of Policy Modeling 9: 615-635. Baumol, W.J. (1959) Business Behavior, Value and Growth. New York: Macmillan. Benhabib, J., and Nishimura, K. (1979) The Hopf Bifurcation and the Existence and Stability of Closed Orbits in Multi-sector Models of Economic Growth. Journal of Economic Theory 21:421 ~A.. Box, G.E.P., and Jenkins, G.M. (1976) Time Series Analysis. San Francisco: Holden Day. Dockner, E. (1985) Local Stability Analysis, in Optimal Control Problems with Two State Variables. in Optimal Control Theory and Economic Analysis, Vol. 2 (G. Feichtinger, Ed.). New York: North-Holland. Dockner, E. (1988) On the Optimality of Cyclical Policies in Dynamic Economic Systems. Paper presented at the Optimization Days, Montreal, Canada, May 2-4. Feichtinger, G., and Sorger, G. (1986) Optimal Oscillations in Control Models: How can Constant Demand Lead to Cyclical Production? Operations Research Letters 5:277-281. Guckenheimer, J., and Holmes, P. (1983)Nonlinear Oscillations, Dynamical Systems, and Bifurcation of Vector Fields. New York: Springer. Hogan, W,W. (1989) A Dynamic Putty-Semi-putty Model of Aggregate Energy Demand. Energy Economics pp. 53-69. Houthakker, H.S., Kennedy, M. (1979) A Long-run Model of World Energy Demands, Supplies, and Prices. in Directions in Energy Policy: A Comprehensive Approach to Energy Resource Decision Making. (B. Kursunoglu and A. Peflmutter, Eds.). Cambridge, MA: Ballinger. International Energy Agency (lEA) (1988) Energy Prices and Ttttes, Third Quarter 1987. Paris: OECD/IEA. Leland, H.E. (1980) Alternative long-Run Goals and the Theory of the Firm: Why Profit Maximization May Be a Better Assumption Than You Think. in Dynamic Optimization and Mathematical Models (Pan-Tai Liu, Ed.). New York: Plenum Press.
Penalizing Oil Price Changes
527
Marshalla, R.A., and Nesbitt, D.M. (1986) Future World Oil Prices and Production Levels: An Economic Analysis. Energy Journal 7( i ): i-22. OECD (1988) Energy Statistics. Paris: Organization for Economic Cooperation and Development. OPEC Secretariat (1986) Petroleum Product Prices and their Components in Selected Countries. Vienna: OPEC Secretariat. OPEC Secretariat (1988) Annual Statistical Bul;etin 1987. Vienna: OPEC Secrelariat. Pindyck, R.S. (1978) Gains to Producers from the Cartelization of Exhaustible Resources. Review of Economi~'s and Statistics. pp. 238-25 l. Rosenberg, R. (1971) Profit Constraint Revenue Maximization; A Note. American Economic Review 61: 208-209. Wirl, F. (1984) Sensitivity Analysis of Opec Pricing Policies. Opec Review 8: 321-371. Wirl, F. (1985) Stable and Volatile Prices: An Explanation by Dynamic Demand. In Optimal Control Theory. and Economic Analysis, Vol. 2 (G. Feichtinger, Ed.). New York: NorthHolland. Wid, F. (1990) Dynamic Demand and Optimal Opec Pricing: An Empirical Investigation. Energy Economics ! 2:174-- 176.