increases in reported oil reserves among opec producers: a...
TRANSCRIPT
Increases in reported oil reserves among Opec producers: a
di�erential game approach
Mathias Berthod∗
Abstract
In the 1980's, the Opec's decision to set members' export quotas of oil in proportion of
individual reserves volumes led to a general rise of the proved reserves: between 1982 and
1988, they jumped from 467,39 to 760,48 billions of barrels. This gap highlights the fact
that nonrenewable resources reserves also depend on the market structure the producers
choose. The purpose of this article is twofold: �rst, we analyze the consequences of a change
of oligopoly on the level of reserves and second, we will explore what happen when countries
can mis-report some of their resources for strategical reasons.
In a di�erential game framework, oil producing countries can produce and explore as in
the seminal Pindyck model. But sharing the same global demand, they can choose between
two di�erent market structures. In the non cooperative oligopoly, they determine control
variables - extraction and exploration - in order to maximize their individual pro�ts. In the
"coopetitive" oligopoly - the term 'coopetition', �rst coined by Barry Nalebu� and Adam
Brandenburger, designates the phenomenon of �rms that cooperate and compete at the
same time -, they choose a global extraction that will maximize the global pro�t: individual
extractions are decided through a 'rule of quotas', that is each country will extract in
proportion of its individual reserves. Then, the choice of exploration is made in a competitive
way.
Through this framework we will show that, with homogeneous producers and under some
assumptions, a jump in the reserves still happen with a change of the oligopoly structure
from non cooperation to coopetition. Moreover, with heterogeneous producers, this change
is not exactly the same for every countries.
In a second framework, we will show that when producers are furthermore able to mis-
report some resources, the amount of "cheated reserves" will highly depend on the shape
of the marginal exploration cost. Then, by observing the new equilibrium situation we can
determine which countries over or under report some of its reserves.
Keywords: di�erential game, non-renewable resource, oligopoly, Opec cartel, stock mis-reporting.JEL classi�cation: C70; C71; Q35
∗Paris School of Economics, Université Paris 1 Panthéon-Sorbonne.
1
1 Introduction
There is a consensus on the fact that reserves estimates published by reporting and informationagencies are over-in�ated. Information agencies, such as the International Energy Agency, theU.S Energy Information Agency or the BP statistical review often reproduce the data fromreporting agencies such as Oil & Gas Journal, World Oil magazine and the Opec Secretariat. Asthe petroleum engineer Laherrère (2001) points out: "publishing is a political act and dependsupon the image the author wants to give". Reserves reported in o�cial publications oftenare politically or �nancially motivated so that there are too much optimistic and not reliable.Independent authors generally prefer using technical data: purchased from scouting companies,such as IHS energy or WoodMackenzie, it comes directly from individual �elds and is consideredas more objective.
Trying to explain the gap between public and technical data, Owen et al. (2010) proposedthat it could arises from: a lack of international standards about oil (grade, volume...), variousde�nitions of what is a 'proved reserve' (the point at which resources are commercially exploitablewith a certain probability), intentional mis-reporting due to �nancial or political reasons and lasttechnical assessment uncertainty. Often pointed as a perfect illustration of an intentional mis-
reporting, the Opec '�ght for quotas' occurred in the 1980s after OPEC countries agreement toset their export quotas in proportion to reserve volumes. It provided strategical behavior insidethe cartel: Opec members in�ated reported reserves in order to gain market shares. As a result,between 1982 and 1988, the Opec's proven reserves jumped from 467,39 Gb 1 to 760,48 Gb (see�gures 1 and 2). In 2008, the Agency (2008) acknowledged that: "proven reserves worldwidehave almost doubled since 1980. Most of the changes result from increases in o�cial �gures fromOpec countries [...]. They were driven by negotiations at that time over production quotas andhave little to do with the discovery of new reserves".
Laherrère Laherrère and Campbell (1998), insisting on the fact that these countries can reportthe reserves which correspond to their wish since it is never audited, estimated that, between 1986and 1990, more than 287 Gb were added without any signi�cant discoveries. The oil economistSalameh (2004) even suggested that Opec's proven reserves could have been overstated by 300Gb. The same result is obtained by De�eyes (2001) by subtracting out any abrupt jump during1982-1988 from the proven reserves. There is a wide literature on over-in�ated reserve estimates:a lot of independent authors thus try to revise downward global oil reserves in order to give anapproximation of the Ultimate Recoverable Reserves. But, most of the time, these authors studyOpec like a "black box" and work directly on statistics by correcting every suspicious jumps indatas. They can not explain for instance why some countries did not increase their reserves atall during this period (see for instance group 2 in �gure 2).
In this article, we wish to analyze the reserve gap, that occurred during the '�ght for quotas',as the result of inner non cooperative behaviors among members inside the cooperative structureof the Opec. We will thus consider oligopolistic producers who do not cheat, but have thepossibility to explore to add some new reserves, choosing thus the level of reserves they wantin order to increase their pro�t. We will study two types of competition between them andcompare what would be the impact on the level of reserves. Our model is typically an "oil-igopoly" model. Many authors built this kind of model to study exhaustible resource market:Loury (1986), Polasky (1996), Hartwick and Sadorsky (1990), Salant (1976). Our purpose isto study the particular structure of Opec cooperation where countries choose cooperatively aglobal production but then decide their own individual explorations. This mix of cooperation
1Billion of barrels.
2
and competition can be called "coopetition"2: this particular structure will have all our interest.We will show that the level of reserves highly depends on the competition structure that choose
the producers. We will also determine how this level evolves with the di�erent parameters ofcosts and the number of countries that compose the cartel.
Then in an other framework where countries can cheat on its reserves, we will show that wecan determine, by observing the extractions of the di�erent countries, which producers cheat onits reserves. We have also found that the real reserves of some countries might be even lowerthan before, which means that they cheated more than the jump of the reserves we can observe.This is an interesting result because it means that the studies we quoted before which tried tocorrect the level of reserves in order to know the real ones should be corrected by the fact thatsome of the countries may have not cheated and that some other may have cheated more thanwhat we thought.
This paper is organized as follows: in the �rst section, we expose the general framework of themodel. The second section presents results for the case of a linear market demand, hyperbolicextraction costs and quadratic exploration costs. In the �nal section, we expose a di�erentframework in order to study a case of cheating producers.
2The term 'coopetition', coined by Nalebu� and Brandenburger (1996), designates the phenomenon of �rmsthat cooperate and compete at the same time.
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1:Opec
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ontheperiod1980-1990(1/2).
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2:Opec
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ontheperiod1980-1990(2/2).
5
2 General framework: a di�erential game
In our model, Opec members are in an oligopolistic competition. Each one is represented by anoil-extracting �rm and begins with a known initial stock of oil. Knowing the market demand andfacing average production costs increasing as the proved reserve is depleted, they make pro�t byextracting oil. As in the seminal paper of Pindyck Pindyck (1978), producers can also increasetheir proved reserves with exploration: the addition of new reserves x implies exploration costscβ(x). The exploration does not depend on cumulative discoveries - that is producers act as ifthe geological resources were in�nite -: this leads to a long-run steady state level of reserve whenextraction is exactly equal to exploration 3. This model is then not a "depletion" model4. Itis highly disputable to make such an assumption in the speci�c case of oil, but our concern israther to compare equilibrium situations of di�erent structures of oligopolies: these equilibriacan be seen as "golden rules" that the producers would tend to reach in a short term period. Thelevel of reserve becomes a variable that producers will indirectly control in order to maximizetheir pro�t. When the structure of the oligopoly changes or when the parameters are modi�ed(with new technologies of extraction or new techniques of exploration for instance), the level ofreserves will tend to an other equilibrium.
We will study two types of competitive structures5:
• The non cooperative oligopoly where each country chooses its extraction and exploratorye�ort to maximize its own pro�t function.
• The coopetitive oligopoly where extractions are chosen to maximize the global pro�t but,on the other hand, each country chooses its own exploration to maximize its individualpro�t. This is a partially cartelized organization.
This last structure, very particular, is similar to the current structure of the Opec. Members�rst decided production allocations at the 63rd meeting of the Opec Conference in 1982. Sincethen, they have regularly updated those allocations with the change in reserves of each countries.In this paper, we will consider that the gap occurred from a non cooperation situation to acoopetitive situation6.
Producer i's pro�t can be written as:
πi =
∫ ∞0
e−δt [p(q(t))qi(t)− qi(t)cα(Ri(t))− cβ(xi(t))] dt
(The "(t)" will be then omitted in the rest of the article). On the other hand, i's reserveswill follow the following dynamic equation:
3This case is actually studied by Pindyck for nonrenewable such as Bauxite.4Such issues as Peak Oil for instance cannot be analyzed with this approach.5In a two-period, two-player exploration and extraction game, Hartwick (1991) made a similar comparison
between three cases: price-taking producers, competition and the exact opposite of the case of 'coopetition' wediscuss below.
6We consider that before the set of quotas, Opec was not really a cooperative oligopoly even it is true thatbetween 1970 and 1980, it acted on price to in�uence the market.
6
Ri = −qi + xi
Where qi : country i's output.xi : country i's new additions to reserves.Ri : country i's reserves.cα(Ri) : country i's marginal extraction costs.cβ(xi) : country i's exploration costs.p(q) : inverse demand function with q the total output.δ : the discount rate.
The extraction cost is a decreasing and convex function of the level of reserves, that is c′α < 0and c′′α ≥ 0: a producer extracts at �rst the cheapest oil thus the cost of extraction will increaseas the proved reserve is depleted. When new wells are opened, it is equivalent to add new "cheapoil", then the extraction cost decreases. We also assume that cα(0) = +∞: it means that aproducer will never extract all the reserves. There is a choke price p(0) 6= +∞ after which theproducer will switch to a backstop technology. The price p is a decreasing and concave functionof q, that is p′(q) < 0 and p′′(q) ≤ 0. The exploration cost is an increasing function of x, that isc′β > 0.
In the next sections, we will present the two types of competition between producers and �ndthe equilibrium situations.
2.1 Non cooperative oligopoly
In this competitive structure, the N producers do not cooperate at all: they choose their indi-vidual extractions and exploration in order to maximize their own individual pro�ts:
{maxqi,xi
∫∞0e−δt [p(q)qi − qicα(Ri)− cβ(xi)] dt
s.t. Ri = −qi + xi ∀i ∈ [1, N ]
As in the Hotelling model, we get this static e�ciency condition:
p′(q)qi + p(q) = cα(Ri) + λi
It means that the marginal revenue should be equal to the addition of the marginal cost ofextraction and the shadow price of the resource extracted. We also get the following secondstatic condition:
c′β(xi) = λi
It adds the fact that the marginal discovery cost c′β should be also equal to the shadow price.
Then we get two dynamic e�ciency conditions. The �rst one is:
7
λi = δλi + qic′α(Ri)
It means that the shadow price evolves with the discount rate as in the Hotelling model. Butnow the term qic
′α(Ri) takes into account a stock e�ect on extraction costs: the fact that, because
of today's extractions, the producer will face higher extraction costs in future time period. Thesecond dynamic condition is:
Ri = −qi + xi
It is simply the fact that when extraction is greater than addition to the reserves, the reservesdecreases.
2.2 Coopetitive oligopoly
In this last oligopoly, the N producers only cooperate to decide the global production of thecartel. Then they chooses individually their own exploration decisions.
maxq
∫ ∞0
e−δt
[p(q)q − qCα((Rk)k∈[1,N ])−
N∑k=1
cβ(xk)
]dt
s.t. Ri = −qRiR + xi ∀i ∈ [1, N ]
Then, taking q as given from the Opec decision, each producer will chose his level of explo-ration in order to maximize his pro�t under his own resource constraint:
maxxi
∫ ∞0
e−δt[p(q)q
RiR− qRi
Rcα(Ri)− cβ(xi)
]dt
s.t. Ri = −qRiR + xi
The main di�erence here with the non cooperation case is the existence of di�erent shadowprices. It comes with the fact that producers defend at the same time the cartel's interest andtheir individual interests. Thus, the same stock of resource has two shadow price: from theviewpoint of the cartel and of the producer.
We get the following static e�ciency conditions:
p′(q)q + p(q)− Cα((Rk)k∈[1,N ]
)=
N∑k=1
λopec,kRkR
And:
c′β(xi) = λi
Then, we get three dynamic condition due to the double optimization the producers make(with the cartel and for themselves). The �rst one is the following:
8
λopec,i = δλopec,i + q∂Cα
((Rk)k∈[1,N ]
)∂Ri
+q
R2
∑k 6=i
Rkλopec,i −q
R2
∑k 6=i
Rkλopec,k
It describes the evolution of the shadow price from the cartel viewpoint:
• q ∂Cα((Ri)i∈[1,N])∂Ri
: the stock e�ect on extraction costs is now triple: because of today'sextractions, the producer will face higher extraction costs but on the other hand withlower future reserves his quotas of extraction will become also lower and then it will tendto decrease his extraction costs. And third e�ect: the other producers' quotas of extractionwill all increase and then it will tend to increase their extraction costs. Through the cartelviewpoint, each producer has to take into account what would be the consequences onother producers.
• qR2
∑k 6=i
λopec,iRk: it takes into account that when the reserves of the country i decreases,
this tend also - by the rule of quotas - to decrease its production qi = qRiR . Then thephysical scarcity of the resource will be lower in future periods.
• − qR2
∑k 6=i
λopec,kRk: on the other hand, extractions of the other countries will tend - still
by the rule of quotas - to increase. This increases the physical scarcity of their resources.Through the cartel viewpoint, it has to be taken into account by country i.
The second dynamic e�ciency we get is the following:
λi = δλi + qRiRc′α(Ri) +
q
R2
∑k 6=i
Rkcα(Ri)− p(q)q
R2
∑k 6=i
Rk +q
R2
∑k 6=i
Rkλi
It describes the evolution of the shadow prices of the resources but view from each producer:
• qRiR c′α(Ri)+
qR2
∑k 6=i
Rkcα(Ri): the stock e�ect is di�erent because, individually, the producer
does not take into account the consequences on the other producers.
• −p(q) qR2
∑k 6=i
Rk: there is a new stock e�ect on the revenue because of the rule of quotas.
Indeed, from the cartel viewpoint, this e�ect did not exist because a change of individualreserves had no impact on the global production but rather on the repartition throughcountries.
• qR2
∑k 6=i
Rkλi: as we mentioned previously, when the country i's reserves are depleted, this
tend - by the rule of quotas - to decrease its extraction.
The last dynamic condition is always the same:
Ri = −qRiR
+ xi
9
2.3 Steady state
We can prove that (see A.1 and A.2) at the equilibrium situation:
∀i p′(q)qi + p(q)− cα(Ri) = − qiδ c
′α(Ri)
c′β(xi)
xi= − 1
δ c′α(Ri)
xi = qi
(In the non cooperative oligopoly).
∀i p′(q)q + p(q)− Cα((Ri)i∈[1,N ]) = − qδ
∑k
RkR
∂Cα((Ri)i∈[1,N])
∂Rk
c′β(xi)
xi+
qR
(1− Ri
R
)δ+ q
R
(1− Ri
R
) (p′(q) RRi
+ φi((Ri)i∈[1,N ]))
= − 1δ c′α(Ri)
xi = qi
With:
φi((Ri)i∈[1,N ]) =1
qi
(−Cα
((Ri)i∈[1,N ]
)+ cα(Ri)− q
RiR
1
δc′α(Ri) +
q
δ
N∑k=1
RkR
∂Cα((Ri)i∈[1,N ]
)∂Rk
)
(In the coopetitive oligopoly).
3 Linear demand, hyperbolic extraction costs and quadratic
exploration costs
In this section, we propose an analytical resolution of the basic model for N oil producing countrieswe exposed previously. We will study the case of a linear demand, hyperbolic extraction costsand quadratic exploration costs, that is :
p(q) = A−Bqcα(R) = α
Rcβ(x) = βx2
10
3.1 Homogeneous producers
Assuming that producers are homogenous, that is they share exactly the same parameters, we cancalculate equilibria in each cases we saw before in the general framework. In the non cooperativeoligopoly:
q = N A−√
2αδβ2β+(N+1)B
x = N A−√
2αδβ2β+(N+1)B
R = N√
α2δβ
And, in the coopetitive oligopoly:q = δR AR−αN
αN2+2NδBR2
x = δR AR−αNαN2+2NδBR2
2β − qN−1NR
δ+qN−1NR
NB = N2αδR2
The reserve gap In this speci�c case, we immediately verify that there is a gap in the reservesbetween the non cooperation situation to the coopetition situation. Indeed, as q ≥ 0, we get inthe coopetition case that:
N2α
δR2≤ 2β
And then:
R ≥ N√
α
2δβ
Which is the value of the reserves in the non cooperation case. Then we can explain thereserve gap with a change of the competition structure between producers.
We will see in the next section how each parameters in�uence this reserve gap.
In�uence of the parameters on the reserves and the reserve gap The in�uence ofthe parameters on the reserves in the non cooperation case is obvious and can be directly seenthrough the formula we have exposed before.
Yet, for the coopetitive case, it is more di�cult. We can calculate that the third equation inthe coopetitive oligopoly with homogeneous producers can be written as the following polynomial:
ψ(R) = R4+δA(N − 1)(2β −NB)
4βδ2BNR3+
αδN(2β −N(N + 1)B)
4βδ2BNR2−A(N − 1)αN2
4βδ2BNR− N3α2
4βδ2BN
The polynomial ψ has a unique real solution on R+. Moreover ψ is negative on [0, R] andpositive on [R,+∞[. as we proved in B.1.
11
In the coopetitive case, we can show that:
Proposition 1. Assuming a linear demand, hyperbolic extraction costs, linear exploratory e�ort
costs and linear additions to reserves, the reserve in the coopetitive oligopoly:
• Increases with α.
• Decreases with β.
• Increases with A.
• Increases with B.
• Increases with N .
Proof. See B.3.
With the help of the following lemma:
Lemma 1. When 2β > NB we get that :
R2 <N2α
δ(2β −NB)
Proof. See B.2.
Now we want to study how the reserve gap between these oligopolies evolves with the param-
eters. We will then write the polynomial ψ making appear R0 = N√
α2δβ into it. We call this
new polynomial χ whose the reserve gap is a solution:
χ(R
R0) =
(R
R0
)4
+A(N − 1)(2β −NB)
2BN2√
2αδβ
(R
R0
)3
+2β −N(N + 1)B
2BN2
(R
R0
)2
− A(N − 1)β
BN2√
2αδβ
(R
R0
)− β
BN2
As with the polynomial ψ, we get that the polynomial χ, whose the equilibrium reserve gapRR0
is a root, has a unique real solution on R+. Moreover χ is negative on [0, RR0
] and positive on
[ RR0,+∞[.As for ψ, we can show that:
Proposition 2. Assuming a linear demand, hyperbolic extraction costs, linear exploratory e�ort
costs and linear additions to reserves, the reserve gap :
12
• Decreases with α.
• Decreases with β.
• Increases with A.
• Increases with B.
• Increases with N .
Proof. See B.4.
With the help of the following lemma:
Lemma 2. When 2β > NB we get that :
R
R0<
√2β
2β −NB
Proof. Directly from the lemma 1
The very last property of the in�uence of N is interesting. We saw that in the non cooperative
case, the individual equilibrium reserves, R0
N =√
α2δβ , do not depend on N thus, when a member
leaves the cartel, there is no change for the other members. In the coopetitive oligopoly, thingsare di�erent: as we have just seen that N in�uences positively the reserve gap, it means thatindividual equilibrium reserves are modi�ed when the number of members is changed. The levelsof individual reserves re�ect then the strategical interactions that occur into the coopetitiveoligopoly.
3.2 Heterogeneous case
Now, let's assume that each producer has his own parameters for every cost functions: extractionand exploration (coe�cients α and β will be indexed by i). In the non cooperative oligopoly, weget that:
qi = p(q)−√
2αiβiδ2βi+B
xi = p(q)−√
2αiβiδ2βi+B
Ri =√
αi2δβi
With p(q) =A+B
N∑k=1
√2αkβkδ
2βk+B
1+N∑k=1
B2βk+B
And in the coopetitive oligopoly:
13
q = δR AR−Nα
Nα+2BδR2
x = δR AR−NαNα+2BδR2
2βi +qR
(1− Ri
R
)δ+ q
R
(1− Ri
R
) [−B RRi
+ φ((Ri)i∈[1,N ])]
= αiδR2
i
Denoting α = 1N
N∑k=1
αk and 1/β = 1N
N∑k=1
1/βk.
Where the function φ here is equal to:
φi((Ri)i∈[1,N ]) =1
δqi
(δ +
q
R
)(αiRi− Nα
R
)
In the coopetitive oligopoly, we can notice that:
N∑i=1
(RiR
)2
φi((Ri)i∈[1,N ]) = 0
Thus this implies that φi will be positive for some producers and negative for others. Theconsequence is that when it is positive (it means when Ri
R< αi
N∑k=1
αk
), the reserves can then be
sometimes very close to the level of the non cooperative oligopoly as we can see on the �gures3 and 4. These �gures represent the reserves of four countries with di�erent value of α or β.We can see that some of them have almost the same level of reserves than before (β = 0.3 forinstance). This could explain why some of the countries did not choose to increase their reservesduring the �ght for quotas period.
14
Figure 3: In�uence of α on the reserve gap.
Figure 4: In�uence of β on the reserve gap.
15
4 The case of cheating
We will now study a case where countries can mis-report their reserves. Indeed they will beable now to choose between two ways of changing their levels of reserves: exploration, as before,and an alternative way which is adding some new oil, but without any e�ect on the extractioncost. Then, there are �real reserves� which have an impact on extraction cost and �false reserves�which have not but are used only for strategical reasons.
4.1 About reserves declaration and misreporting
Having a closer look to data published about oil reserves, we can often notice great di�erencesbetween sources. According to Owen et al. (2010), there is three types of sources. First, thereare the reporting agencies (Oil & Gas Journal, World Oil magazine and Opec Secretariat):most of the available public data comes from the surveys they conduct. These sources arepolitically sensitive and often give optimistic estimates since, as Simmons (2007) pointed out,if a few countries have accepted a third party audit, none of the Opec producers did. Thesecond type of sources are the information agencies such as International Energy Agency, theUS Energy Information Agency and BP Statistical Review: they often take their data fromreporting agencies with small corrections. The last source are technical data directly taken fromindividual �elds by oil expert companies such as IHS or Wood Mackenzie.
At the end, most of the public information is coming from reporting agencies and there is aconsensus that this kind of sources publish over-in�ated reserve estimates. Owen et al. (2010)highlights four types of ambiguities that can explain di�erences in data through these sources:
1. A problem of de�nition of reserves as commercially extractable resource.
2. A lack of binding international standards to report oil reserve volume and grade.
3. Technical uncertainty.
4. Intentional mis-reporting for political reasons.
(1) The notion of reserves is indeed based on a probability system which splits resources intowhat is undiscovered, what is discovered but not economically extractable and reserves. OpecSecretariat does not include for instance unconventional oil such as tar sands in Canada.
(2) Reserves themselves are split into three categories (1P, 2P and 3P). Each of these cate-gories correspond to a probability of successfully producing oil, 1P-reserves means for instancethat you will be able to produce successfully 90 % of the amount considered. Laherrére (2009)points out that in Russia, companies declare 3P-reserves when US companies had to declare1P-reserves until 2010. There is also a problem on measure unities: depending on the density wetake, one barrel can represent many possible volumes.
(3) Technical problem can be a source of ambiguity in reserves estimates.(4) This last point is the type of cheating on data we will focus on. Opec countries are indeed
highly suspected to have increased their reserves during the period we focus on. As we say inintroduction, a lot of studies estimate that 300 Gb could be withdrawn from the current reserves.Since there is no audit, countries can add the amount of reserves they want. Why won't they
16
add in�nite reserves then ? Increasing global reserves has an impact on many variables as wesaw in the case with no cheating: it implies a new distribution of quotas. A country has then notinterest to increase inde�nitely its reserves with cheated ones: there will be a level after whichit will not be interesting anymore for its own pro�t.
4.2 Framework
The new pro�t function for a producer i will be:
πi =
∫ ∞0
e−δt [p(q(t))qi(t)− qi(t)cα(Ri(t)− Zi(t))− cβ(xi(t))] dt
(The "(t)" will be then omitted in the rest of the article). The reserves will follow the followingdynamic equation:
Ri = −qi + xi + ziZi = zi
zi(t) is the amount a producer chooses to add or remove from his reserves (this variable canbe negative). The idea then is that through this variable, a producer can change his reservesuncostly but also with no impact on the extraction costs (because those reserves are not realones). He will have thus to arbitrate between increasing his reserves by exploration - in orderto decrease his extraction cost - or by cheating - in order to change his position in the cartelthrough the rule of quotas.
The incentive to cheat comes when reserves becomes a strategical variable. Then, we willonly apply this framework to the coopetitive case. In the non cooperation case, producers haveno incentive to cheat because their reserves are not linked two any control variables.
4.2.1 Dynamic equations
Now in the coopetitive oligopoly, we will see that producers have interest to cheat in order togain market shares. The maximization program will be the following for the Opec:
maxq
∫ ∞0
e−δt
[p(q)q − qCα((Rk)k∈[1,N ])−
N∑k=1
cβ(xk)
]dt
s.t. Ri = −qRiR + xi + zi ∀i ∈ [1, N ]
In this case, the Opec takes the information which is given by producers. Then they calculateglobal production with the reserves the producers give, taking it as real reserves. The quotas ofproduction are also calculated from the false reserves qi = qRiR .
In the same time, taking q as given from the Opec decision, each producer will chose hisexploratory e�ort and cheating in order to maximize his pro�t under his own resource constraint:
17
maxxi,zi
∫ ∞0
e−δt[p(q)q
RiR− qRi
Rcα(Ri − Zi)− cβ(xi)
]dt
s.t. Ri = −qRiR + xi + ziZi = zi
The producer makes his calculations with the real reserves he has in the ground Ri−Zi whenhe declared Ri to the Opec.
The static and dynamic e�ciency conditions for the Opec will be the same than before:
p′(q)q + p(q)− Cα
((Rk)k∈[1,N ]
)=
N∑k=1
λopec,kRkR
λopec,i = δλopec,i + q∂Cα((Rk)k∈[1,N])
∂Ri+ q
R2
∑k 6=i
Rkλopec,i − qR2
∑k 6=i
Rkλopec,k
And almost as before, we get for an individual producer:
{λi = δλi + qRiR c
′α(Ri − Zi) + q
R
(1− Ri
R
)cα(Ri − Zi)− p(q) qR
(1− Ri
R
)+ q
R
(1− Ri
R
)λi
c′β(xi) = λi
This is actually the same condition than in the no cheating model except that the extractioncost and marginal extraction cost take as argument the �real reserves� (Ri − Zi).
But now, there is also a new shadow price µi due to the new dynamic constraint Zi = zi. Weget then this new dynamic condition:
µi = δµi − qRiRc′α(Ri − Zi)
And also the following static condition:
0 = µi + λi
18
4.2.2 Steady state
In the non cooperative case, the steady states will be exactly the same because countries will notcheat as we have seen before.
In coopetition, at the equilibrium, we can prove that (see C.1):
∀i p′(q)q + p(q)− Cα((Ri)i∈[1,N ]) = − qδ
∑k
RkR
∂Cα((Ri)i∈[1,N])
∂Rk
c′β(xi)
xi= − 1
δ c′α(Ri − Zi)
c′β(xi) = −cα(Ri − Zi) + p(q)
xi = qi
4.3 Linear demand, hyperbolic extraction cost and linear marginal ex-
ploration cost
With the same speci�c functions we have taken in the analytical case as before, we can get that:
q = δR AR−NαNα+2BδR2
x = δR AR−NαNα+2BδR2
RiR
= 1q
[A−Bq
2βi− δ√
αi2δβi
]R2
[δA(2β −NB) + 2δ2Bβ
N∑k=1
√αi
2δβi
]− R [Nα(2β +NB)]−Nα
[NA− 2δβ
N∑k=1
√αi
2δβi
]= 0
Zi = Ri −√
αi2δβi
With α = 1N
N∑k=1
αk and 1/β = 1N
N∑k=1
1/βk.
The last equation Zi = Ri−√
αi2δβi
is interesting because it means that, in this particular case
of a linear marginal exploration cost, the increase of reserves from no cooperation to coopetitionis only cheated. We can indeed recognize the equilibrium reserves of the non cooperation case√
αi2δβi
then it means that the whole gap in reserves between non cooperation to coopetition is
cheated. This would give support to De�eyes (2001), Salameh (2004) and others authors whothink that the so-called �ght for quotas is a purely political period.
Then to correct the reserves and know the real ones, we would just have to remove everyjumps that occurred during the �ght for quotas. This would give us the real reserves of oil.
19
4.4 The case of non linear marginal exploration cost
In every analytical cases, we have assumed a linear marginal exploration cost (cβ(x) = βx2).What happen if this cost is not linear, that is, if:
c′β(x) = βxθ
We will assume that θ ≥ 27. We can show that:
Proposition 3. if and only if:
• If qi|coopetition ≤ qi|no cooperation, the real reserves of a cheating producer will be greater in
coopetition than in no cooperation.
• If qi|coopetition ≥ qi|no cooperation, the real reserves of a cheating producer will be lower in
coopetition than in no cooperation.
Proof. (See C.2.)
Thus, equilibrium productions together with the form of the marginal exploration cost revealinformation about cheated reserves. A contraction of the production of a country together withan elasticity of marginal exploration cost greater than one reveals for instance that the increaseof reserves is not totally cheated: a part of it really comes from exploration. At the opposite,if the production is, the real reserves of coopetition is lower than in non cooperation, it meansthat the increase of the reserves is totally cheated and, even worse, the producer has in factdecrease his real reserves. This can be seen on the �gure 5 that summarizes this result. Whenthe production of the coopetition case is greater than non cooperation production, the ratio ofreal reserves ∆R = R|coopetition − R|non cooperation is negative.
If the assumption we made on θ ≥ 2 is veri�ed, it means for instance that Saudi Arabiacheated on its reserves because its production increased (see �gure 6) between 1988 and 1990.At the opposite, Kuwait did not cheat on its reserves because its production remained nearly thesame.
7Without this assumption, the problem is much harder: we found multiple equilibria.
20
Figure
5:Variationofrealreserves
revealedbyequilibrium
production.
21
0
20
00
40
00
60
00
80
00
10
00
0
12
00
0
19
80
19
81
19
82
19
83
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84
19
85
19
86
19
87
19
88
19
89
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90
Production (Mb/day)
OP
EC
me
mb
ers
pro
du
ctio
ns
on
th
e p
eri
od
19
80
-19
90
in
th
ou
san
d b
arr
els
da
ily
-fi
rst
gro
up
Ira
n
Ira
q
Ku
wa
it
Sa
ud
i A
rab
ia
Un
ite
d A
rab
Em
ira
tes
Ve
ne
zue
la
Figure
6:Opec
mem
bersproductionontheperiod1980-1990(1/2).
22
0
500
1000
1500
2000
2500
1980
1981
1982
1983
1984
1985
1986
1987
1988
1989
1990
Production (Mb/day)
OP
EC
me
mb
ers
pro
du
ctio
ns
on
th
e p
eri
od
19
80
-19
90
in
th
ou
san
d b
arr
els
da
ily
-se
con
d g
rou
p
Ecuador
Qatar
Algeria
Angola
Libya
Nigeria
Figure
7:Opec
mem
bersproductionontheperiod1980-1990(2/2).
23
5 Conclusion
In this article, we showed �rst that the level of reserves of a non renewable resource highlydepends on the structure of the competition between producers. We found that in a coopetitivestructure, like the one that runs currently in the Opec cartel, the level of reserves will be higherthan in a non cooperative oligopoly for strategical reasons. We detailed how each parametersin�uence the level of reserves that will be optimum for producers and we saw that if some ofthem have interest to highly change their reserves with a switch of competition structure, otherwill keep the same level.
Then, in a second part of the paper, we found that the incentive to add some cheated reservesto the total amount depends on the shape of the marginal discovery cost function. Then, knowingthis shape for a country and observing his choice of production, we can know what will be theamount of cheated reserves. Then it means that in order to correct the reserves to know the realones, we should take into account that some of countries did not cheat and that at the oppositesome of them even decrease their real reserves.
24
A General framework: a di�erential game
A.1 Non cooperative oligopoly equilibrium situation
We saw that the di�erential game is described in this case by the following system:
p′(q)qi + p(q)− cα(Ri) = λi (1)λi = c′β(xi) (2)
λi = δλi + qic′α(Ri) (3)
Ri = −qi + xi (4)∀i ∈ [1, N ]
At the equilibrium, we get that:
λi = 0⇔ λi = − qiδ c′α(Ri)
Ri = 0⇔ qi = xi∀i ∈ [1, N ]
And then, directly, we �nally get that:
∀i p′(q)qi + p(q)− cα(Ri) = − qiδ c
′α(Ri)
c′β(xi)
xi= − 1
δ c′α(Ri)
xi = qi
A.2 Coopetitive oligopoly equilibrium situation
We saw that the di�erential game is described in this case by the following system:
p′(q)q + p(q)− Cα((Rk)k∈[1,N ]
)=
N∑k=1
λopec,kRkR
(1)
λi = c′β(xi) (2)
λopec,i = δλopec,i + q∂Cα((Rk)k∈[1,N])
∂Ri+ q
R2
∑k 6=i
Rkλopec,i − qR2
∑k 6=i
Rkλopec,k (3)
λi = δλi + qRiR c′α(Ri) + q
R
(1− Ri
R
)cα(Ri)− p(q) qR
(1− Ri
R
)+ q
R
(1− Ri
R
)λi (4)
Ri = −qRiR + xi (5)∀i ∈ [1, N ]
At the equilibrium, we get that:
λopec,i = 0⇔ λopec,i(δ + q
R) = −q ∂Cα((Ri)i∈[1,N])
∂Ri+ q
R
N∑k=1
λopec,kRkR
λi = 0⇔ λi
[δ + q
R
(1− Ri
R
)]= −q Ri
Rc′α(Ri)− q
R
(1− Ri
R
)cα(Ri) + p(q) q
R
(1− Ri
R
)Ri = 0⇔ q Ri
R= qi = xi
∀i ∈ [1, N ]
25
By multiplying by RiR and then summing the expression of λopec,i, we get that:
N∑i=1
RiRλopec,i(δ +
q
R) = −q
N∑i=1
RiR
∂Cα((Ri)i∈[1,N ]
)∂Ri
+q
R
N∑i=1
RiR
N∑k=1
λopec,kRkR
And then:
N∑k=1
λopec,kRkR
= − qδ
N∑k=1
RkR
∂Cα((Ri)i∈[1,N ]
)∂Rk
We can the write that:
p′(q)q + p(q)− Cα((Ri)i∈[1,N ]) = − qδ
∑k
RkR
∂Cα((Ri)i∈[1,N ])
∂Rk
At the equilibrium, we also get that:
λi
[δ +
q
R
(1− Ri
R
)]= −q Ri
R
1
δc′α(Ri)
[δ +
q
R
(1− Ri
R
)]− qR
(1− Ri
R
)[cα(Ri)− p(q)− q
RiR
1
δc′α(Ri)
]
Thus:
λi = −q RiR
1
δc′α(Ri)−
qR
(1− Ri
R
)δ + q
R
(1− Ri
R
) [cα(Ri)− p(q)− qRiR
1
δc′α(Ri)
]
By substituting the expression of p(q) in this last expression, we can write that:
λi=−qRiR
1δ c′α(Ri)−
qR
(1− Ri
R
)δ+
qR
(1− Ri
R
)[p′(q)q+cα(Ri)−Cα((Ri)i∈[1,N])+qδ
∑k
RkR
∂Cα((Ri)i∈[1,N])
∂Rk−q Ri
R1δ c′α(Ri)
]
And then, remembering that λi = c′β(xi):
c′β(xi)
xi=− 1
δ c′α(Ri)−
qR
(1− Ri
R
)δ+
qR
(1− Ri
R
)[p′(q) RRi
+ 1qi
(cα(Ri)−Cα((Ri)i∈[1,N])+
qδ
∑k
RkR
∂Cα((Ri)i∈[1,N])
∂Rk−q Ri
R1δ c′α(Ri)
)]
26
B Analytical case: linear demand, hyperbolic extraction
cost and linear marginal exploration cost
B.1 ψ has a unique positive real solution
The polynomial that veri�es R can be written as:
R4+δA(N−1)(2β−NB)
4βδ2BNR3+
αδN(2β−N(N+1)B)
4βδ2BNR2−A(N−1)αN2
4βδ2BNR− N3α2
4βδ2BN
Thus the signs of the coe�cients give us informations on solutions. There are three types ofsolutions for this kind of polynomial:
• Four complex solutions. Here it is impossible because of the negative constant which meansthat the product of the solutions is negative.
• Two complex and two real solutions. This case is possible and therefore the negativeproduct of solutions implies that the real solutions have opposite signs (then there is aunique positive solution).
• Four real solutions. There are two possibilities: a unique positive solution and three nega-tive or three positive solutions and a unique negative. We will now show that this last caseis actually impossible.
We have to study the derivative of the polynomial:
4R3 +3δA(N − 1)(2β −NB)
4βδ2BNR2 +
2αδN(2β −N(N + 1)B)
4βδ2BNR− A(N − 1)αN2
4βδ2BN
The negative constant allows us to say that the product of solutions would be this timepositive. In the case of 'four real solutions with a unique negative' we focus on, there should bethen three positive real solutions to this derivative polynomial.
But if the coe�cient in R2 is positive (that is 2β −NB > 0), then the sum of the solutionsis negative and there is a contradiction with the fact that there all are positive.
If the coe�cient in R2 is negative (that is 2β − NB < 0), then the coe�cient in R is alsonegative (because 2β −N(N + 1)B < 2β −NB < 0) and the 2 by 2 solutions products shouldbe negative which is also in contradiction with the fact that there all are positive.
In conclusion, this case is impossible. The two cases that remain are:
• One positive real solution, one negative real solution and two complex solutions.
• One positive real solution and three negative real solutions.
In these two cases, there is a unique positive real solution.
27
B.2 Proof of lemma 1
We can calculate that :
ψ
(N
√N2α
δ(2β −NB)
)= N5α2B
2β +NB
(2β −NB)2> 0
Then with the property of ψ, we can conclude that R2 < N2αδ(2β−NB) .
B.3 Proof of proposition 1
In�uence of α We can use the implicit functions theorem to know how the extraction costin�uences the equilibrium level of reserves. We know that :
dR
dα= −
∂ψ∂α∂ψ∂R
We already know that ∂ψ∂R
> 0 because we previously showed that ψ is increasing in the
neighborhood of its unique positive real root. Thus the sign of dRdα is the opposite sign of :
∂ψ
∂α=[δN(2β −NB)R2 −N3α
]− δN3BR2 −A(N − 1)N2R−N3α
• When 2β < NB, we directly get that ∂ψ∂α < 0 and thus R
dα > 0.
• When 2β > NB, we know from lemma 1 that R2 < N2αδ(2β−NB) then dR
dα > 0.
Finally the parameter α in�uences positively the equilibrium reserves in the coopetitive case.
dR
dα> 0
28
In�uence of β As previously we know that the sign of dRdβ is the opposite sign of :
∂ψ
∂β= 4δ2BNR4 + 2δA(N − 1)R3 + 2αδNR2
This last expression is always positive. Then the coe�cient β in�uences negatively the equi-librium reserves in the coopetitive case.
dR
dβ< 0
In�uence of A As previously we know that the sign of dRdA is the opposite sign of :
∂ψ
∂A= R
[δ(N − 1)(2β −NB)R2 − (N − 1)αN2
]This is the same reasoning than for the in�uence of α, we can use the lemma 1 to show �nally
that A in�uences positively the reserve gap.
dR
dA> 0
In�uence of B As previously we know that the sign of dRdB is the opposite sign of :
∂ψ
∂B= 4βδ2NR4 − δANR3 − α‘δN2(N + 1)R2
Knowing that ψ(R) = 0, we can write that this is equal to :
∂ψ∂B = 1
B
[−δA(N − 1)2βR3 − αδN2βR2 +A(N − 1)αN2R+N3α2
]= RA(N−1)+Nα
B (αN2 − 2δβR2)
We know that R > R0, thus we have αN2 − 2δβR2 < 0. This last expression is then always
negative. The coe�cient B in�uences positively the equilibrium reserves in the coopetitive case.
dR
dB> 0
29
In�uence of N As previously we know that the sign of dRdN is the opposite sign of :
∂ψ
∂N=
The condition of existence of a steady state is :
A >√
2αδβ
Moreover, we know that RR0
> 1, then we get that 2β(N − 2)− (2β(N − 2) +NB)(RR0
)2
< 0
because 2β(N−2)2β(N−2)+NB < 1. Finally we can write that :
∂ψ
∂N<
1
2BN3
[2β(N − 2)− (2β(N − 2) +NB)
(R
R0
)2]
+1
2BN3
[4β − (4β −NB)
(R
R0
)2]
Thus :
∂χ
∂N< 0
Then the number of members in�uences positively the reserve gap.
dR
dN> 0
B.4 Proof of proposition 2
In�uence of α We can use the implicit functions theorem to know how the extraction costin�uences the reserve gap. We know that :
d RR0
dα= −
∂χ∂α∂χ∂ RR0
We already know that ∂χ∂ RR0
> 0 because we previously showed that χ is increasing in the
neighborhood of its unique positive real root. Thus the sign of∂ RR0
∂α is the opposite sign of :
∂χ
∂α=
R
R0
A(N − 1)
4BN2√
2α3βδ
[2β − (2β −NB)
(R
R0
)2]
30
• When 2β < NB, we directly get that ∂χ∂α > 0 and thus
d RR0
dα < 0.
• When 2β > NB, we know from lemma 2 that(RR0
)2
< 2β2β−NB then
d RR0
dα < 0.
Finally the parameter α in�uences negatively the reserve gap.
d RR0
dα< 0
In�uence of β As previously we know that the sign ofd RR0
dβ is the opposite sign of :
∂χ
∂β=
A(N − 1)
2BN2√
2αβδ
R
R0
[2β +NB
2β
(R
R0
)2
− 1
]+
1
BN2
[(R
R0
)2
− 1
]
As we have shown that RR0
> 1, this last expression is then always positive. Then thecoe�cient β in�uences negatively the reserve gap.
d RR0
dβ< 0
In�uence of A and B We know that R0 does not depend on A and B. Then the variation ofthe reserve gap with respect to these variables will be the same as R :
d RR0
dA> 0
d RR0
dB> 0
31
In�uence of N As previously we know that the sign ofd RR0
dN is the opposite sign of :
∂χ
∂N=
A
2BN3√
2αβδ
R
R0
[2β(N − 2)− (2β(N − 2) +NB)
(R
R0
)2]
+1
2BN3
[4β − (4β −NB)
(R
R0
)2]
The condition of existence of a steady state is :
A >√
2αδβ
Moreover, we know that RR0
> 1, then we get that 2β(N − 2)− (2β(N − 2) +NB)(RR0
)2
< 0
because 2β(N−2)2β(N−2)+NB < 1. Finally we can write that :
∂χ
∂N<
1
2BN3
[2β(N − 2)− (2β(N − 2) +NB)
(R
R0
)2]
+1
2BN3
[4β − (4β −NB)
(R
R0
)2]
Thus :
∂χ
∂N< 0
Then the number of members in�uences positively the reserve gap.
d RR0
dN> 0
C About cheating
C.1 Coopetitive oligopoly with cheating
We saw that the di�erential game is described in this case by the following system:
32
p′(q)q + p(q)− Cα((Rk)k∈[1,N ]
)=
N∑k=1
λopec,kRkR
0 = µi + λic′β(xi) = λi
λopec,i = δλopec,i + q∂Cα((Rk)k∈[1,N])
∂Ri+ q
R2
∑k 6=i
Rkλopec,i − qR2
∑k 6=i
Rkλopec,k
λi = δλi + qRiR c′α(Ri − Zi) + q
R
(1− Ri
R
)cα(Ri − Zi)− p(q) qR
(1− Ri
R
)+ q
R
(1− Ri
R
)λi
µi = δµi − qRiR c′α(Ri − Zi)
Ri = −qRiR + xi + ziZi = zi
As in ?? and A.2, we get directly:
p′(q)q + p(q)− Cα((Ri)i∈[1,N ]) = − qδ
∑k
RkR
∂Cα((Ri)i∈[1,N ])
∂Rk
And at the equilibrium:
λi = 0⇔ λi
[δ + q
R
(1− Ri
R
)]= −q Ri
Rc′α(Ri − Zi)− q
R
(1− Ri
R
)cα(Ri − Zi) + p(q) q
R
(1− Ri
R
)µi = 0⇔ δµi = q Ri
Rc′α(Ri − Zi)
Ri = 0⇔ q RiR
= qi = xi + ziZi = 0⇔ zi = 0∀i ∈ [1, N ]
Then:
λi = −cα(Ri − Zi) + p(q)
Finally, we get that:
{c′β(xi)
xi= − 1
δ c′α(Ri − Zi)
c′β(xi) = −cα(Ri − Zi) + p(q)
33
C.2 Real reserves gap
Remembering that: {c′β(xi)
xi= − 1
δ c′α(Ri − Zi)|coopetition
c′β(xi)
xi= − 1
δ c′α(Ri)|no cooperation
Then, knowing that the marginal extraction cost is an increasing function, we get that:
(Ri − Zi
)|coopetition ≥ Ri|no cooperation ⇔ c′α(Ri − Zi)|coopetition ≥ c′α(Ri)|no cooperation
It means that �the real reserves� - reserves that really come from exploration - in the coopet-itive oligopoly are greater that in the no cooperative oligopoly if and only if marginal extractioncosts are ordered in the same way at the equilibrium, and then:
(Ri − Zi
)|coopetition ≥ Ri|no cooperation ⇔
c′β(xi)
xi|coopetition ≤
c′β(xi)
xi|no cooperation
Then if for instance the functionc′β(x)
x is an increasing function, we get that:
c′β(xi)
xi|coopetition ≤
c′β(xi)
xi|no cooperation ⇔ qi|coopetition ≤ qi|no cooperation
Last, we can notice that:
d
dx
(c′β(x)
x
)≥ 0⇔
c′′β(x)x− c′β(x)
x2≥ 0⇔
c′′β(x)x
c′β(x)≥ 1
Thenc′β(x)
x is increasing if and only if its elasticity is greater than one.
34
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