Groot, F.

Withagen, C.A.A.M.

Zeeuw, A.J. de

English

Research Papers in Economics

Note on the Open-Loop von Stackelberg Equilibrium in the Cartel Versus Fringe
Model
Fons Groot; Cees Withagen; Aart de Zeeuw
The Economic Journal, Vol. 102, No. 415. (Nov., 1992), pp. 1478-1484.
Stable URL:
http://links.jstor.org/sici?sici=0013-0133%28199211%29102%3A415%3C1478%3ANOTOVS%3E2.0.CO%3B2-F
The Economic Journal is currently published by Royal Economic Society.
Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at
http://www.jstor.org/about/terms.html. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained
prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in
the JSTOR archive only for your personal, non-commercial use.
Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at
http://www.jstor.org/journals/res.html.
Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed
page of such transmission.
The JSTOR Archive is a trusted digital repository providing for long-term preservation and access to leading academic
journals and scholarly literature from around the world. The Archive is supported by libraries, scholarly societies, publishers,
and foundations. It is an initiative of JSTOR, a not-for-profit organization with a mission to help the scholarly community take
advantage of advances in technology. For more information regarding JSTOR, please contact support@jstor.org.
http://www.jstor.org
Fri Mar 7 03:39:03 2008The Economic  Journal,  102(.4Toz;ernber1992j,  1478-1 484 
Printed  in Great Britain 
NOTE ON THE OPEN-LOOP 

VON STACKELBERG EQUILIBRIUM IN THE 

CARTEL VERSUS FRINGE MODEL 

Fons Groot*, Cees  Withagen  and Aart de  Zeeuw 
In his  remarkable  article  'Oil prices,  cartels  and  the problem  of  dynamic 
inconsistency' Newbery  (1981)  shows that if the supply side of the oil market 
can be characterised as one with a dominant cartel and a large number of small 
producers  (called  the  fringe  as  a  group)  the  open-loop  von  Stackelberg 
equilibrium may give rise to dynamic inconsistency, and should therefore be 
rejected as an appropriate equilibrium concept. Dynamic inconsistency is also 
found to occur by Ulph (1982) in a similar model. 
The purpose of this note is not to dispute the validity of this result nor its 
relevance. What we wish to show is that the derivation of the equilibrium is not 
correct. Apart from the fact that the correct derivation yields qualitatively as 
well  as  quantitatively  different  equilibrium trajectories  (although the main 
conclusion  still stands), there is  some additional value in the present  paper 
because  one of  the objectives  in  Newbery's  article  was  'to demonstrate  a 
method of  analysis which makes the solution  of quite complex problems  . . . 
accessible  to  mathematically  unsophisticated  economists'  (p.  6I 9). It  is 
furthermore argued by lyewbery that 'This approach also provides scope for 
intuition so that the mathematically sophisticated can check the plausibility of 
their solutions, or, given various possible solutions, can choose the correct one' 
(p. 6I 9). The conclusion of the present note is, however, that in the case at 
hand intuition is not performing as an entirely reliable guide. 
I.  THE PROBLEM 
Let the world demand schedule of oil be given by 
x =p-p, 
where x denotes demand, p is the market price and pis a choke price. Demand 
is met by a coherent cartel with constant per unit extraction costs kc,  having an 
initial endowment of oil amounting to S;,  and a large number of identical small 
suppliers, each with constant per unit extraction costs kf and aggregate initial 
resources Sf,. The common discount rate is denoted by r, a positive constant. In 
order to have an interesting problem it will be assumed that p > max (6, kf). 
Each individual fringe member takes the market price as given1 and maximises 
*  Financial support from  the Cooperation Center.  Eindhoven University  and Tilburg University.  is 
gratefully acknowledged. 
It would be more elegant to start with a finite number of fringe members and then to see what happens 
to the equilibrium if that number goes to infinity. It  can be shown that indeed piice taking behaviour results. 
We follow Newbery and Ulph who assume price taking behaviour from the outset. 
[  '478  1 NOV.  19921  THE CARTEL VERSUS FRINGE MODEL  '479 
its total discounted profits. The cartel has the same objective and takes the 
reaction of the fringe into account in determining the optimal extraction path. 
Let Ef be the equilibrium aggregate extraction trajectory of the fringe and Ec 
be the extraction trajectory of the cartel. 
Then Ef maximises 
J:  eCt(!p-kf) Efdt, 
subject to  sf= -Ef,  Sf(0) = Si, 
The time  argument is  omitted  wherever  there  can  be  no confusion.  It is 
straightforward to see that we have as necessary conditions 
Efh, = 0, 
where hf(> o) is  the constant shadow price of the aggregate in situ oil stocks 
of the fringe. Since the objective functionals of the fringe members are concave, 
a trajectory satisfying the necessary conditions is  optimal, if hfSf approaches 
zero as t goes to infinity. 
The way Newbery and Ulph proceed is then to analyse the optimal strategy 
of  the cartel by  ruling out possible sequences of regimes, not using optimal 
control theory. In contrast, we argue that control theory is  a valuable tool in 
analysing the cartel's problem, be it that no standard theorems can be invoked. 
We  will  show  that  the  proper  analysis  with  optimal  control  theory  gives 
somewhat different results. 
The problem of the cartel can be formulated as follows. 
man 1;  e-lt [p- Ec-Ef-kc]Ecdt, 
E~, ~f 

subject to 
h,  = kf+hfert-(p-Ec-Ef)  2 o, 
Efh, = 0, 
So the cartel chooses not only its own extraction trajectory but the fringe's as 
well, subject to the condition that each fringe member is  a price taking profit 
maximiser. The cartel also takes care that the market is always in equilibrium 
(x = Ef+Ecis therefore inserted into the demand function). 
This optimal control problem does not allow for a standard application of 
the  Pontryagin  maximum  principle  because  in  general  the  constraint 
qualification  does  not  hold.  However,  Neustadt  (1976) gives  necessary I 480  THE ECONOMIC JOURNAL  [NOVEMBER 
conditions for  optimality in  this  case.  We follow  Seierstadt  and  Sydsaeter 
(1987),  who provide a useful formulation. It  would go too far to outline all the 
differences with  the standard  necessary  conditions. In general,  the co-state 
variables need no longer be continuous and the multiplier functions associated 
with the constraints are not necessarily piece-wise continuous. What remains 
however, and this is important in the present context, is that the Hamiltonian 
is maximised with respect to the control variables. The Hamiltonian reads 
Note that, since if= o, the corresponding co-state variable does not appear in 
the Hamiltonian. Furthermore, it is evident that the co-state variables A;  and 
A",  denoting the value the cartel attaches to a marginal increase in its own 
initial stock and the fringe's initial stock respectively, will be constants. Hence, 
although the co-states A>nd  A",eed  not be continuous in general, they are in 
the present problem. Along the lines set out by Ulph and Folie (1980)  we define 
the auxiliary prices 
p1=  kf +Af ert 
For reasons that will be clear P1is called the competitive price and P3is called 
the monopoly price.  P2can be interpreted  as  the marginal costs the cartel 
incurs when it supplies at the competitive price. P2consists of the marginal 
extraction costs kc, the opportunity costs  of extracting its own resource 
now rather than in the future and the costs -A",rt  the cartel incurs by the fact 
that producing an  additional amount now will cause a higher stock of the fringe 
(A:  is of course negative). Suppose that it is optimal to have an interval of time 
with hl = o. Then, along this interval of time, 
is maximised subject to 
o < E-p-P1, 
as can be seen from simple substitution. 
If it is optimal to have an interval of time with hl > o then, along this interval 
of time, the cartel maximises 
subject to  E$  0. 
This is so because if h,  > o we have Ef = o and P1>p. The following is now 
obvious. 
THEOREM 
(a) Suppose it is  optimal for  the  cartel to  have h,  = o.  Then 
zfP1>PzthenEc=max(p-P1,o),  Ef=o 
ifP2 > P1 then E"  0,  Ef = max (p- P1,0). 19921  THE CARTEL VERSUS FRINGE MODEL  I 48 I 
(b) Suppose  it  is  optimal  for  the  cartel  to  have  hl > o.  Then  Ef= o, 
EC= max ($-P3,0). 
From here the analysis can proceed in much the same way as in Newbery 
and  Ulph.  Let  us  concentrate  on  the  continuity  of  the  equilibrium  price 
trajectory. One of the results the authors mentioned above obtain is that in the 
rather plausible case where the cartel has a cost advantage over the fringe, the 
advantage however  not being  extreme  (kc <  kf <  +($+kc)),  and where  the 
cartel has an initial oil stock which is  sufficiently large relative to the initial 
stock of the fringe, there will at some instant of time occur a switch from the 
fringe supplying at the competitive price P1 to the cartel producing at the 
monopoly price P3. It  can be shown along the lines set out by these authors but 
using our formal control theoretic setting, that this result is correct. It has also 
been shown by the previous authors, and can be proved to be correct, that in 
the case at hand the cartel will start supplying at the monopoly price P3 only 
after it has supplied at the competitive price P1. 
Since the optimal value of the Hamiltonian should evidently be continuous 
over time, a discontinuity occurs in the price trajectory at the points in time 
where a switch takes place from a phase with the fringe supplying to a phase 
with the cartel supplying at the monopoly price and vice versa. This can be 
seen as follows. 
Along intervals of time where the cartel is the sole supplier at the monopoly 
price P the value of the Hamiltonian is 
Along intervals of time where the fringe is the sole supplier at the competitive 
price P1 the value of the Hamiltonian is 
At the switch point, say s, these values should be equal implying that 
But, if the price trajectory were continuous in s we would have P1(s) = P(s) 
and hence P1(s) = P2(s)= P(s).  Now assume that at  s a switch occurs from the 
fringe supplying to the cartel supplying at the monopoly price. As  mentioned 
above, there should be an interval of time before s where the cartel is supplying 
at the competitive price. According to the theorem, P1> P2along that interval, 
whereas just before s we have P2>P1.There must therefore be an instant of 
time before s where P1= P2. But since kf $ kc the curves can intersect only 
once. So we obtain a contradiction. 
For the cost constellation at hand the equilibrium can be depicted as in Figs 
I, 2 and 3. Here the symbols C and C" mean that the cartel is the sole supplier 
at the competitive price and the monopoly price respectively. Fmeans that the 
fringe is the sole supplier. The  location of the curves P1, P2  and P3is determined 
by two equations for the exhaustion of the resources and a third equation for 
the optimal choice of the co-state variable hf.For the technical details of the 
derivation the reader is  referred  to Groot et  al. (1990). THE ECONOMIC JOURNAL  [NOVEMBER 
Fig. I. Equilibrium for SE,  'small' relative  to Sfo 
Fig. 2. Equilibrium for 'intermediate' case. 
Fig. 3.  Equilibrium for Si 'large' relative to S{ 
From these figures it can be seen that for more or less the same parameter 
values  as in Newbery and Ulph the conclusion  that dynamic inconsistency 
arises  remains  valid: with  the parameter values  we  departed from there is 
always an initial interval of time where the cartel supplies at the competitive 
price. Note also that, contrary to earlier findings, there are parameter values 
for which there is a transition from the cartel supplying at the monopoly price 
to the fringe supplying, accompanied by an  upward discontinuity. A numerical 
example can show the differences in the equilibrium trajectories obtained by us 
and Ulph and Newbery. '9g21  THE CARTEL VERSUS FRINGE MODEL  I 483 
Suppose fi=  40,  kf = 20,  kc = 15, r = 0.1 and Sf,= 15. For  Si = 50  the 
equilibrium will be as in Fig. 2, with t, = 0.47, tz = 8.1 and t3 = 11.8. Cartel's 
profits  are 596.8,  whereas  Newbery  would  obtain  594.2.  If,  ceteris paribus, 
S; = 150, then the equilibrium is as in Fig. 3, with t, = 6.2, t, = 7.2, t3 = 9.3 
and  t, = 21.4. In this  case  the cartel's  profits  are  1,096.4 where  Newbery 
would obtain only  I ,089.4. 
An interesting question is why the intuition of the previous authors has failed 
to reveal the correct solution. The main reason is  that it should be taken into 
account that the stock of the fringe has a negative shadow value to the cartel, 
i.e, Asis strictly negative. For the situation where the cartel is bound to supply 
at the competitive price the Hamiltonian reads 
ePtP1  (t)  EC  (t)  -ePt  (kc  +f h",  ert-A: ert)  Ec(t)  -A",fi- P1  (t)]  . 
The first term just gives the revenues. The first part of the second term are the 
direct production costs, the second part constitutes the opportunity costs of 
supplying now instead of in the future and the third part gives the price the 
cartel is willing to pay for a marginal increase of the stock of the fringe. If the 
cartel would produce one unit more, the fringe would produce one unit less and 
therefore the remaining stock of the fringe increases by one unit, which is to be 
considered  as a cost for the cartel. The final term is  a fixed cost factor: if the 
cartel does not produce at all, the fringe supplies p- P1 and hence the stock of 
the fringe decreases by that amount. 
One might not like outcomes with discontinuous price trajectories, because 
they can be deemed to open arbitrage opportunities in reality. However, if one 
wants  to avoid this phenomenon  the forces behind  the absence of arbitrage 
should be modelled explicitly. For the present model price discontinuities are 
not excluded and can be beneficial for the cartel. 
11.  CONCLUSION 
It has  been  shown  that  the  derivation  of  the  open-loop  van  Stackelberg 
equilibrium in the cartel versus fringe model of the oil market by Newbery and 
Ulph has not been entirely correct. In several cases the true price trajectories 
will display one or two discontinuities. The main conclusion that the open-loop 
von  Stackelberg equilibrium can be dynamically inconsistent  is  not affected 
and its consequences regarding the appropriateness of this equilibrium concept 
are not under dispute. However, our claim is that in deriving the equilibrium 
one should not only rely on intuition and let formal mathematical analysis do 
its work in checking for the merits of intuition. 
Free  University Amsterdam 
Eindhoven  University of  Technology 
Tilburg University 
Date of  receipt ofjinal typescript:  April 1992 Groot, F., Withagen, C. and de Zeeuw, A.  I 1990). 'The binding-contracts Stackelberg equilibrium in the 
cartel-versus-fringe model reconsidered.' Center Discussion Paper 8924, Tilburg University. 
Neustadt, L. W. (1976).Optinzizatzon. -4  Theory of 'Vecejjary Conditionj. New Jersey: Princeton University Press. 
Newbery. D. (I  98I ). 'Oil prices, cartels, and the problem of dynamic inconsistency. ' Ecorvonnc JOURNAL, 
vol. 91, pp  617-46. 
Seierstadt, A. and Sydsaeter, K. (1987).Optzmal Control  Theory u~itlz  Economic .-Ipplication~.  Amsterdam: North- 
Holland. 
Ulph, A.  (I  982). 'Modeling partially  cartelized markets for exhaustible resources. ' In Economzc  Theory oj 
.4Tatural Re~ourcei  (ed. W. Eichhorn et a1.j. Wurzburg: Physica  Verlag, pp. 269-91. 
Ulph, A. and Folie, G. (1980!. 'Exhaustible resources and cartels: an intertemporal Nash-Cournot  model.' 
Canadian Journal of  economic^, vol.  13, pp. 645.. 58. You have printed the following article:
Note on the Open-Loop von Stackelberg Equilibrium in the Cartel Versus Fringe Model
Fons Groot; Cees Withagen; Aart de Zeeuw
The Economic Journal, Vol. 102, No. 415. (Nov., 1992), pp. 1478-1484.
Stable URL:
http://links.jstor.org/sici?sici=0013-0133%28199211%29102%3A415%3C1478%3ANOTOVS%3E2.0.CO%3B2-F
This article references the following linked citations. If you are trying to access articles from an
off-campus location, you may be required to first logon via your library web site to access JSTOR. Please
visit your library's website or contact a librarian to learn about options for remote access to JSTOR.
References
Oil Prices, Cartels, and the Problem of Dynamic Inconsistency
D. M. G. Newbery
The Economic Journal, Vol. 91, No. 363. (Sep., 1981), pp. 617-646.
Stable URL:
http://links.jstor.org/sici?sici=0013-0133%28198109%2991%3A363%3C617%3AOPCATP%3E2.0.CO%3B2-Y
Exhaustible Resources and Cartels: An Intertemporal Nash-Cournot Model
A. M. Ulph; G. M. Folie
The Canadian Journal of Economics / Revue canadienne d'Economique, Vol. 13, No. 4. (Nov.,
1980), pp. 645-658.
Stable URL:
http://links.jstor.org/sici?sici=0008-4085%28198011%2913%3A4%3C645%3AERACAI%3E2.0.CO%3B2-J
http://www.jstor.org
LINKED CITATIONS
- Page 1 of 1 -

Cartels, and the Problem of Dynamic Inconsistency

http://links.jstor.org/sici?sici=0008-4085%28198011%2913%3A4%3C645%3AERACAI%3E2.0.CO%3B2-J http://www.jstor.org LINKED CITATIONS -

http://links.jstor.org/sici?sici=0013-0133%28198109%2991%3A363%3C617%3AOPCATP%3E2.0.CO%3B2-Y Exhaustible Resources and Cartels: An Intertemporal Nash-Cournot Model

Note on the open-loop von Stackelberg equilibrium in the cartel versus fringe model.

Abstract

Similar works

Full text

Available Versions

Research Papers in Economics