Working paper on the mineral prospecting, stackelberg with N players and Ricardo with growing costs

ECOLE NATIONALE SUPERIEURE DES MINES DE PARIS

Optimal strategies ofexploration and production foroil/mining companies under

different competitivesituations,

Price formation of mineralcommodities

Samy-Adrien AKOUM & Alexandre BENCHAOUINE,under the guidance of Pierre-Noël Giraud

2014/2015

1

Economics, Finance and Management ofCommodities (EFGC)

CERNA, Mines ParisTech

Acknowledgement & Thanks

Foremost, we would like to express our deepest thanks to our supervisor and tutor, Pierre-Noël Giraud, Professor and Researcher atMines ParisTech and Université Paris Dauphine, for his guidance and support as our tutor for this work, and for the fact that he never makes us feel as basic interns. Nothing would have been possible without him and his patience, encouragement, pedagogy and immense knowledge were key motivations throughout this internship. We are truly thankful for his great modesty, and selfless dedication to bothour personal and academic development. We cannot think of a better tutor to have, we owe him a great debt of gratitude.

We would like to thank Jean-Michel Lasry, Ph.D. in Mathematics,Professor and Researcher at Université Paris Dauphine, co-founder of MFG labs, and Antonin Pottier, Postdoctoral Researcher at CERNA, for offering thorough and excellent visions, extensions, and feedback about the models. They enabled us to work along side them without anysense of inferiority.

Special mention for Btissam El-Bahraoui and Jamal Hazizi, both research interns within the chair EFGC/OCP, for providing us with great data and helping us in its use.

We thank Mohamed Soual and Samia Charadi, respectively Senior Economist and chair's director in OCP.

2

Contextual SettingThis part-time research internship, under the guidance of

Pierre-Noël Giraud, aims to determine several theoretical results andtheir application to the raw materials market. This project has beenrealized within the chair: Economics, Finance and Management ofCommodities (EFGC), part-financed by Office Chérifien des Phosphates(OCP). We have been given the following questions and guidelines thathave been formulated by our adviser:

Price formation of commodities under both the visions ofRicardo and Hotelling, quantification of Hotelling's rent overRicardo's principle.

Extension of the Stackelberg duopoly to N actors, competitionaspects within this new framework with a descending leadershipand its prolongation over several periods

Exploration modelling and deposits portfolio optimization Legitimacy of price fluctuations by OPEC within Stackelberg

oligopoly with a competitive fringe under demand uncertainty

Those issues have either been raised with us in explicitmathematical forms - in particular for the first two of these topics- or in more qualitative formulations. In any case, the answers wehave provided to these problems have either been mathematical oralgorithmic.

The reporting of our work has been made every week with Pierre-Noël Giraud, in which we exposed our progress on the differenttopics. Through these daily meetings, we used to define short-termobjectives to get things going more efficiently. They were sometimesthe seat of deep discussions about the assumptions and methods ofwork we should provide. Some topics such as the modelling of optimaldeposit portfolios required more than one month of reflexion aboutthe hypothesis and the framework of the model. We then had theoccasion to be confronted with the difficulty of certain subjects at

3

the level of both the assumptions of the model and the informationavailable to us.

Furthermore, we had the great fortune during this internship tomeet highly skilled researchers who helped us to define the frameworkof the models, provided us with valuable data, or simply expanded ourunderstanding of the raw-material economy. It was also given to usthe opportunity to present a draft or our work to the sponsors of the chair in which we took partto take a closer look at the research community.

In conjunction with Pierre-Noël Giraud, we decided to carry onthis project until the end of the year so that our work culminate inthe preparation of working papers, and perhaps a publication in aneconomic journal, which justifies the shape of this research paper.

Samy-Adrien AKOUM & Alexandre BENCHAOUINE

Table of contents

Section A:On the theory of exhaustible resources, revising

Ricardo and Hotelling with increasing costs . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . p.6

A.1: Price formation within the framework of Ricardo in competitivemarkets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p.7

A.2: Price formation within the framework of Ricardo associated withthe Hotelling's rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p.9

A.3. Numerical results and comparison between Ricardo andHotelling's rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p.11

Section B:

4

Exploration-production modelling, deposits portfoliooptimization. . p.14

Section C: Competition between N actors within the framework of

Stackelberg with a descendingleadership . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p.17

C.1: Single-period extended Stackelberg competition with N economicagents. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p.21

C.2: Extended Stackelberg competition with N economic agents overtwo periods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p.27

C.3 Applications to the oil market . . . . . . . . . . . . . . . . . . . . . . p.34

Section D: Price fluctuation policy from OPEC, Stackelberg

oligopoly with competitive fringe under demanduncertainty . . . . . . . . . . . . . . . . . . . . .p.41

Appendix: Literature - State of the Art. . . . . . . . . . . . . . . . . . . . . . . . . .

p.48

A. On the theory of exhaustible resourcesRevising Ricardo an Hotelling with

increasing costs

A producer of exhaustible resources is placed in a competitivemarket. We can make the general hypothesis that the function cost ofthe marginal provider - producer with higher extraction costs -depends exponentially on the aggregate output Sc (Figure 1.1), and wetake a general inverse shape for the demand (Figure 1.2):

c=c0.eβSc

5

p=(D+D0 )−α

We obtain consecutively:

ps=D0−αetD0=ps

−1α

D=p−1α −ps

−1α

D(t,p(t))=e¿.(p−1α −ps

−1α )whileaddingatrendg

The specification c=c0.eβSc gives us:

Sc=1βlog( c

c0)etRu=

1βlog(

ps

c0)

It goes without saying that if the company wants to beprofitable, it is necessary that the difference between the price andthe marginal cost (extraction cost of its most expensive deposit)must be positive. In addition, given the intertemporal arbitrage,this quantity might increase with the interest rate i. Once updated,this amount is called scarcity rent. The maximum and logically acceptablerent would then be the difference between the marginal extractioncost and the present price of the substitute. Indeed, the hypothesisof Harold Hotelling states that the date T of the complete depletion- the time when the price of the market will reach the price of the

Sc: Aggregate productionRu: Ultimate recoverable reservesc: Production costs, with c<psps : Price of the substitute

Figure 1.1.Figure 1.2.

6

substitute - is known. Consequently, it is comparable to receive ascarcity rent now or producing a little before the time T. This lastreasoning gives us a way to describe the scarcity rent, orHotelling's rent, which will be precisely analysed later:

a (t )=(ps−c (t) )e−i(T−t)

It is to be noted that the functions price and cost do not haveany sense for periods exceeding the depletion time of the resourcesT.

In the way of thinking of Ricardo, i.e. in some kind ofcompetitive market: the price is equal to the extraction cost. Thecontribution of this paper is to study the differences and the limitsbetween those two perspectives of the rent, and the consequences ofcombining them together, considering then that:

p(t)=c(t)+a (t )=c(t)+(ps−c (t))e−i(T−t)

As we look over qualitatively all those equations, some issuesare to be raised: should discount rates α and price-elasticity α ofthe demand be quite easy to find, the value of β is more debatable.Regarding measurement units, the price dependence demand suggestsfrom the outset that the ratio volume unit/price unit is constant. Thisis really important because it may limit our possibility of action toset the parameters arbitrarily without altering the generality of themodel.

In addition, we defined the time T as the exhaustion time ofthe resources, which corresponds - with respect to the model - to thedate when the price of the market reaches the price of thesubstitute. Yet, in the vision of Ricardo, this price is neverreached because the constant function ps is solution to the firstorder differential equation associated to the model. It is then apriori difficult to set a value for T, except T=∞ (it can be shownactually that lim

t→∞p(t)=ps). If we could set T, the association of

both the models of Ricardo and Hotelling would change into a singleRicardo vision.

The argument that it can be found a guestimate value of T whichwould be refined with the Hotelling rent is also not the right way togo given that T would exactly be the moment when the price reachesthe price of the substitute.

Finally, this paper aims to define an exogenous criteria thatwould help us determine an exhaustion date T, and to find values of βrelated with the past of some kind of resources.

7

A.1. Price formation within the framework of Ricardo incompetitive markets:

____________

1Ricardo's theory is rather simple: lands with differentnatural conditions (fertility, soil type, micro climate...) yielddifferent amounts of product. The owner of the land with the bestnatural conditions can produce a unit of product by the lowestproduction price. Next owner, having some less favourable land, willhave higher production costs, and so on towards the further less andless favourable lands. Certainly, we have to suppose that otherfactors of production - same product, same method of production - areunified for all farmers.

The natural conditions of mineral resources - similar to arablelands - differ practically for each deposit (geographical settlement,geological conditions, technological, environmental conditions...).Mineral deposits for a certain commodity can then be ranked in orderof their production costs.

As a consequence, in a market where demand of the productincreases - increase of population, increase of customer's behaviour- product can be sold on the market at a higher price, therefore someof the less favourable lands, i.e. with higher extraction costs, willalso be included in the production. The farmer or producer with theless favourable deposit is called the marginal producer and heproduces at marginal cost c(t), that is the last unit of production thatcan be sold on the market. Ricardo's theory states that the marginalproducer does not earn economic rent, so that:

p(t)=c¿)

The supply-production equilibrium gives us:

Sc(p(t))=∫0

t

D(x,p(x))dxAnd:

1βlog(

p(t)c0

)=∫0

tegx(p(x)

−1α −ps

−1α )dx

We derive this last expression: p'(t)p(t)

=βe¿(p(t)−1α −ps

−1α )

1 Mineral Resources Management, Madai Ferenc, Foldessy Janos Credit: Financial Sense

8

dp

p(t)(p(t)−1α −ps

−1α )

=βe¿dt

∫pi

p dp

p(t)(p(t)−1α −ps

−1α )

=βg (e¿¿¿−1)¿

With: u=ln (p )etps=eus

∫ln¿ ¿¿

¿

Introducing power series, we obtain:

∫ln¿ ¿¿

¿

∫

ln¿ ¿¿

¿

We can switch the sums and integrals:

∑k=0

∞

∫ln¿ ¿¿

¿¿

∑k=0

∞ [ αe−1α

kuse1α(k+1)u

(k+1) ]ln¿ ¿¿

¿

So:

αps

−1α ∑

k=0

∞ (pk+1α −pi

k+1α )

(k+1)=βg

(e¿¿¿−1)¿

We note:

F(p)=α∑k=0

∞ ps

−1α p

k+1α

(k+1)

So that:

βg

(e¿¿¿−1)=F(p)−F(pi)¿

9

e¿=gβ (F(p)−F(pi))+1

p(t)=F−1¿

Or the time as a function of price:

t=1gln [gβ (F(p(t))−F(pi))+1]

This last expression is very useful to understand the globalbehaviour of the price function as a function of time. On the onehand, as exposed above, F(ps)=∞ which means that lim

t→∞p(t)=ps. On

the other hand, we can see the roles played by the differentparameters of the problem, even if only the time function as afunction of price is qualitatively exploitable.

A.2. Price formation within the framework of Ricardoassociated with the Hotelling's rule

____________

2By comparison with Ricardo's model, Harold Hotelling has shownthat since the quantity of the resource is limited (exhaustible), weshould consider that the resource extracted (and consumed) today willbe not available for future generations. Therefore during theevaluation of the currently extracted resource, it should also beconsidered the value that would have been reached if the resourcewould be extracted in the future. This future value, which is lostdue to extracting the resource today, is called the scarcity rent.According to Hotelling, the scarcity rent is the discounted presentvalue of the future profit that will be lost due to extracting theresource in the present. The owner of the marginal deposit with theextraction cost c (t ) requests then an extra scarcity rent defined as:

a (t )=(ps−c (t) )e−i(T−t)

The market's price becomes:

p(t)=c(t)+(ps−c(t))e−i (T−t)=(1−e−i(T−t ))c(t)+pse−i(T−t)

We introduce the parameter T, the depletion time of the reserve Ru:

2 Mineral Resources Management, Madai Ferenc, Foldessy Janos

10

Sc(c(T))=Ru=1βlog(ps

c0),depleted∈Twhenthemarket'spricereachesps

We then investigate on c (t ) such as:

Sc(c(t))=1β log(

cc0

)=∫0

t

D(t,p(t))dt

S.t.

Sc(c(T))=Ru=1βlog(

ps

c0)=∫

0

TD(t,p(t))dt

We have D(t,p(t))=e¿ (p−1α −ps

−1α ) and :

1βlog[ p(t)−pse−i(T−t )

c0 (1−e−i (T−t)) ]=∫0t egx(p(x)−1α −ps

−1α )dx

or:

1βlog[ c(t)

c0 ]=∫0te¿ [[ (1−e−i (T−t ) )c (t )+pse−i (T−t) ]

−1α −ps

−1α ]dt

Should a solution exist, we study its convergence and we define

(cn (t ))n∈N as:

1βlog[ cn+1 (t )

c0 ]=∫0te¿[ (cn (t )+pse−i(T−t))

−1α −ps

−1α ]dt

1βlog[ cn+1 (t )

cn (t ) ]=¿

∫0

te¿[ ((1−e−i (T−t ))cn (t )+pse

−i(T−t))−1α −((1−e−i (T−t) )cn−1 (t )+pse

−i(T−t))−1α ]dt

With the Taylor inequality, we get:

|x1

−1α −x2

−1α |≤max

x∈I |−1α x−1α −1|.|x1−x2|

11

|ln (cn+1 (t ))−ln (cn (t ))|≤β∫0

t e¿

αc0

−1α

−1|eln(cn(t))−eln(cn−1(t))|e−i(T−t)dt

≤βc0

−1α

−1

α ∫0

te¿|ln(cn(t))−ln(cn−1(t))|dt

≤βc0

−1α

−1

α ‖ln(cn (t ))−ln (cn−1 (t) )‖∞∫0

t

e¿dt

As a result:

‖ln(cn+1(t))−ln(cn(t))‖∞≤K(t).‖ln (cn(t))−ln(cn−1(t))‖∞With a recursive demonstration, we have:

‖ln(cn+1(t))−ln(cn(t))‖∞≤(K(t))n.‖ln (c1(t))−ln(c0(t))‖∞Let I be the following interval of time so that K(t)<1:

I=[0; 1g ln( αg

βc0

−1α

−1+1)]

On an interval closed to 0, in this case t∈I,the normalconvergencegivesthatln(cn)n∈N converges uniformly and then (cn)n∈Nconverges (pointwisely).

We managed to demonstrate a local convergence of a sequence offunctions converging towards a solution. Main results will be givenby numerical results.

A.3. Numerical results and comparison of Ricardo andHotelling's rules

____________

The world to which applies the Hotelling Rule in its modern interpretation and the world to which applies the classical,

12

especially Ricardian analysis are rather different. While the Hotelling Rule presupposes that a scarce natural resource is available in a known quantity and its extraction is not subject to any capacity constraints, Ricardo’s treatment of exhaustible depositsdoes not contemplate the case of the exhaustion of the resource as a whole and allows for capacity constraints that limit extraction per unit of time with respect to each deposit actually known at a given moment of time. Both types of analyses are valuable and improve our understanding of the properties of economic systems3.

In the Ricardo's price evolution model (Figure 1.3), the pricerises as the demand grows and due to capacity constraints, newdeposits are put into production, so as they become the new marginaldeposits. The price then reaches the new marginal extraction cost inorder to meet growing demand. Even though Ricardo does not take anytime of depletion in his analysis, one could ask the question of thepossible correlation of this model with the assumptions ofHotelling's rule, which takes into account a date of termination ofproduction, a time when a possible substitute would replace actualresources.

The scarcity rent defined by Hotelling is the discountedpresent value of the future profit that will be lost due toextracting the resource in the present, instead of waiting for theprice to reach the price of the substitute. Given the large timebetween the start of production and the depletion of the resources (inFigure 1.3. we could consider T=140 of the depletion date of theresources) the discounting operation atrophies the future profit in alimited period of time. Consequently, considering time scales ofseveral decades, scarcity rent quickly evaporates, meaning thatRicardo and Hotelling's rules reconcile (Figure 1.4, Figure 1.5).

3 Taylor inequality applied to the exponential function

Price

Time

Figure 1.3: Price evolution within the framework of Ricardo

,

13

4

It can be seen on Figure 1.6 that the reconciliation betweenRicardo and Hotelling's rule strongly depends on the time scale. Wecan see that it's deeply related with the time lag until the completedepletion of resources, and then the dependence with T of the weakrelative influence of Hotelling's rent over the Ricardian rent.

4 Mineral Resources Management, Madai Ferenc, Foldessy Janos

Figure 1.4.: Price evolution within the framework of Ricardo with Hotelling

,

Ricardo

Ricardo & Hotelling

T=40

T=60 T=100 T=120

Time

Price

Cost

Ricardo & Hotelling

Ricardo

Time

T=60

T=80T=100 T=110

Figure 1.5.: Cost evolution within the framework of Ricardo with Hotelling

,

14

It is interesting to see that the price evolution withinHotelling's framework rapidly fits with Ricardo's principle for highvalues of T, logically related with the fact that p(T)=¿ ps

5.However, the cost evolution does not follow that trend, as a

consequence of the model equations, that keeps unspecified the valueof cost at time T6.

B. Exploration-production modellingDeposits portfolio optimization

5 The different numerical resolutions have been made with WindPlot®6 Hotelling's hypothesis

15

When a portfolio of escalating costs deposits is available toan explorer-producer agent, it first puts into production the depositwith lower costs of extraction. Indeed, the aim is to delay the startof production on the most expensive deposits available. In thisregard, when the agent eventually empties its cheapest deposit, theremay be particularly strong interest in undertaking an explorationprogram in order to find a cheaper deposit than the one he would haveput into production without any exploration, so that he could run itin a subsequent period. Nevertheless, these exploration programs havenon-negligible costs, and consecutive losses must be anticipated. Thequestion then arises of an optimal portfolio of deposits, where"portfolio" refers to the quantity of deposits and their costs ofextraction.

In this model, which has been given to us by Jean-Michel Lasry,we assume that there are either "good" deposits (i.e. with relativelycheap extraction costs) or "bad" deposits (i.e. with relatively highextraction costs). We also assume existing reserves Rtotal includenumber of "good" and "bad" deposits.

In his portfolio, the explorer-producer agent can increase itsx(t) good deposits which cost c1 and its y(t) bad deposits which costc2 by making exploratory effort α which costs c0. During theexploration, finding good or bad deposits is equiprobable. The agentproduces β from the good deposits and γ on bad ones. Mathematically,this can be written as:

R=−αdt

x=12α−β

y=12α−γ

The expenditure function verifies:

D=c0α+c1β+c2γ+rD

The problem becomes:

minα(t)

D(T)∨minα(t)

D(T)e−rT

s.t.:

16

y(0)=y(T)=R(T)=0,∀t, x(t)=0,β+γ=1

x≥0,y≥0,α≥0,β≥0,γ≥0,R≥0

Underlying assumptions reflect the fact that the portfolio isempty at the beginning and an explorer-producer agent first puts gooddeposits into production without incurring any.

x=12α−β=0⇒1

2α=β

β+γ=1⇒y=12α−γ=α−1

But:(De−rT)'=(c0α(t)+c1β(t)+c2γ(t))e−rT

D (T)e−rT=∫0

T

(c0α(t)+c1β(t)+c2γ(t))e−rtdt

¿∫0

T

α(t)(c0+c1−c22 )e−rtdt+∫

0

T

c2e−rtdt

Therefore, the problem becomes:

minα(t) [(c0+

c1−c22 )∫

0

Tα(t)e−rtdt ]

Whether c0−c2−c1

2≥0 (resp. c0−

c2−c1

2≤0), we shall minimize

(resp. maximize) the quantity:

∫0

T

α(t)e−rtdt

In the first case, let's consider α0(t)=1, α0 meet all the criteriaand:

∫0

T

¿¿

17

¿∫0

T

y(t)e−rtdt=[y(t)e−rt]0T+r∫

0

T

y(t)e−rtdt

¿y(T)e−rT+r∫0

T

y(t)e−rtdtThus:

∫0

T

¿¿

∀α(t),∫0

T

α(t)e−rtdt≥∫0

T

α0(t)e−rtdt

And α0(t) is then the minimum.

In the second case, let's consider α1(t)=2×l[0, T

2], α1(t) meets all

the criteria and:

∫0

T

¿¿

∫0

T

¿¿

≥(∫0T2

(2−α(t))dt−∫T2

Tα(t)dt)e−r T

2=(∫0

T2

2dt−∫0

Tα(t)dt)e−r T

2=0

Indeed:

∫0

T

α(t)dt=∫0

T

(x+y)(t)+(β+γ)dt=y(T)−y(0)+∫0

T

1dt=T

∫0

T

α1(t)e−rtdt≥∫0

T

α(t)e−rtdt

And α0(t) is then the minimum.

In conclusion:

18

If c0−c2−c1

2≥0, then α0(t)=1 is the optimal extraction effort

If c0−c2−c1

2≤0, then α1(t)=2×l

[0, T2

] is the optimal extraction effort

C. Competition between N actors within theframework of Stackelberg with a descending

leadership

Stackelberg duopoly, also called Stackelberg competition is amodel of imperfect competition based on a non-cooperative game. Itwas developed in 1934 by Heinrich Stackelberg in his "Market Structureand Equilibrium" and represented a breaking point in the study of marketstructure, particularly the analysis of duopolies, since it was amodel based on different starting assumptions and gave differentconclusions to those of the Cournot's and Bertrand's models.

In game theory, a Stackelberg duopoly is a sequential game (notsimultaneous as in Cournot's model). There are two firms, which sellhomogeneous producers, and are subject to the same demand and cost

19

functions. One firm, the leader, is perhaps better known or hasgreater brand equity, and is therefore better placed to decide firstwhich quantity q1

¿ to sell, and the other firm, the follower, observesthis and decides on its production quantity q2

¿.

We take a demand function such as D=α−βp with β=1, theoptimization of the profit of 2 with the extraction cost c2 becomes:

Π2=(α−c2−q1−q2)q2

∂Π2∂q2

=−2q2¿+(α−c1−q1)=0

q2¿=1

2(α−c2−q1)

We then calculate the optimal quantity q1¿:

Π1=(α−c1−q1−q2¿ )q1

¿(α2

−c1+c22

−q12

)q1

∂Π1∂q1

=−q1¿+(α2−c1+

c2

2 )=0q1

¿=α2

−c1+c2

2

In order to extend this theory with N actors, we first tried tomonitor these productions with recursive algorithms. Here are theresults for N= 5, 8, 13, 18, 24 actors:

N = 5

20

N = 13

N = 24

21

___________________________________

Thus, the symmetry of those results led us to think that ananalytical solution should exist. We tried to observe globalsequences of solutions from those calculus, in order to get anoverall idea over the conclusions our demonstration would led to:

qN¿=

12N D−

2N−12N

c1+12N−1 c2+

12N−2 c3+....................+

122 cN−1+

12 cN

qN−1¿ =

12N−1 D+

12N−1 c1−

2N−122N

4

c2+1

2N−3 c3+..................+12cN−1+cN

22

qN−2¿ =

12N−2 D+

12N−2 c1+

12N−3 c2−

2N−122

2N

42

c3+.................+cN−1+2cN

...

...

qN−k¿ =

12N−k D+

12N−k c1+..............−

2N−12k

2N

4k

ck+1+...+2k−2cN−1+2k−1cN

...

q1¿=1

2D+1

2c1+c3+2c4+..................+2N−4cN−2+2

N−3cN−1+2N−2cN

When 2k=N−1sok0=12

(N−1), the bold term becomes 2N−1−12k0

ck0 and then evolve as:

2N−1−12k0

ck0

(2N−1−12k0−1

−2)ck0+1

(2(2N−1−12k0−1 −2)−23)ck0+2

(2(2(2N−1−12k0−1

−2)−23)−25)ck0+3

23

C.1. Single-period extended Stackelberg with Nactors

_____________________

We will first think about the mechanism of the extension of theStackelberg duopoly with three competitors, with 3 which exercisesleadership on 2, itself exercising leadership on 1, so that 1 > 2 > 3.

We take a demand function such as D=α−βp with β=1, theoptimization of the profit of 3 with the extraction cost c3 becomes:

Π3=(α−c3−q1−q2−q3 )q3

∂Π3∂q3

=−2q3¿+(α−c3−q1−q2 )=0

q3¿=1

2(α−c3−q1−q1)

What you have to understand is that we obtain q3¿ (q1,q2). We

then calculate the optimal quantity:

Π2=(α−c2−q1−q2−q3¿ (q1,q2))q2

¿(α2

−c2+c32

−q1+q2

2)q2

∂Π2∂q2

=−q2¿+(α2−c2+

c3

2−q1

2 )=0q2

¿=(α2−c2+c32

−q1

2 )=q2¿ (q1)

Thus, after this new calculus, q3¿ is not in the same form

anymore: after the second computation step, it is now:

24

q3¿=1

2(α−c3−q2

¿ (q1)−q1)=α4

−3c34

+c2

2−q1

4=q3

¿ (q1)

It must be noted that though q3¿ and q2

¿ retain the same value,their form is going to be modified overtime. Eventually, we get:

q3¿=

18α−

7c3

8+c2

4+c12

q2¿=

14α+

14c3−

3c22

+c1

q1¿=

12α+

12c3+c2−2c1

We then think about the extension of the Stackelberg duopolywith N competitors which 1 exercises leadership on 2, itselfexercising leadership on 3, itself exercising leadership on 4 etc.,so that 1 > 2 > 3 > ... > N.

The illustration above, though it implies only three actors,emphasizes the interest of the computation step, within which thecoefficients' values under each ci reside. They will never be thesame over the calculus. At the step k, qi,i≥k have been calculated andqi is a function of (qi+1

¿ ,qi+2¿ ,qi+3

¿ ,...qn¿). The aim of this work

is to determine the coefficients under each ci when all the optimalproductions are calculated. Thus, the calculus of the coefficients ismade overtime, at every single step of the process.

Each coefficient is specific to: the production qi, the step k,

and other actors' production qj

Let ai,k,j be these coefficients_____________________________________________________

The production qi,k then accounts for the production qiexpressed through the step k, and consequentlyi≥k :

qi,k=ai,k,0−∑j=1

k−1ai,k,jqj

Πk=(α−ck−∑j=1

k−1qj−qk− ∑

i=k+1

nqi)qk

25

Πk=(α−ck−∑j=1

k−1qj−qk− ∑

i=k+1

n [ai,k+1,0−∑j=1

k−1ai,k+1,jqj])qk

Πk=(α−ck−∑j=1

kqj− ∑

i=k+1

nai,k+1,0+∑

j=1

k

∑i=k+1

nai,k+1,jqj)qk

¿(α−ck− ∑i=k+1

nai,k+1,0+∑

j=1

k [( ∑i=k+1

nai,k+1,j)−1]qj)qk

∂Πk∂qk

=0=(α−ck− ∑i=k+1

nai,k+1,0+∑

j=1

k [( ∑i=k+1

nai,k+1,j)−1 ]qj)+[( ∑

i=k+1

nai,k+1,k)−1 ]qk

2qk[ ∑i=k+1

nai,k+1,k−1]= ∑

i=k+1

nai,k+1,0−α+ck+∑

j=1

k−1 [1−( ∑i=k+1

nai,k+1,j)]qj

Thus,qk,k=∑

i=k+1

nai,k+1,0−α+ck

2[ ∑i=k+1

nai,k+1,k−1 ]

−

∑j=1

k−1 [ ∑i=k+1

nai,k+1,j−1]

2[ ∑i=k+1

nai,k+1,k−1]

qj

Identifying with (1), we get the following recurrence relation:

ak,k,0=∑

i=k+1

nai,k+1,0−α+ck

2 [ ∑i=k+1

nai,k+1,k−1]

ak,k,j=∑

i=k+1

nai,k+1,j−1

2 [ ∑i=k+1

nai,k+1,k−1 ]

________________________________________________________

26

It can be written:

i>k⟹i≥k+1,qi,k=qi,k+1=ai,k+1,0−∑j=1

k−1ai,k+1,jqj−ai,k+1,kqk,k

¿ai,k+1,0−∑j=1

k−1ai,k+1,jqj−ai,k+1,k(ak,k,0−∑

j=1

nak,k,jqj)

We get a second property:

ai,k,0=ai,k+1,0−ai,k+1,kak,k,0

ai,k,j=ai,k+1,j+ai,k+1,kak,k,j

That brings us to make the assumption: ai,k,j=ai,k, so that thecoefficients do not depend on j. Let's verify it with induction on k :

Fork=n,ai,n,j=12

(seetheexamplewiththreeplayersbelow )

Weassumethatthepropertyistruefork∈N,∧wedemonstratefork−1

ai,k−1,j=ai,k,j−ai,k,k−1ak−1,k−1,j=ai,k−ai,kak−1,k−1,j

Butak−1,k−1,j=∑i=k

nai,k,j−1

2[∑i=k

nai,k,k−1−1 ]

≝∑i=k

nai,k−1

2 [∑i=k

nai,k−1]

=12parH.R.

We also get ai,k+1,j=12ai,k+1,j.

Consequently, the relation becomes:

qk,k=ak,k,0−∑j=1

k−1 qj2

Let's determine the coefficients as:

27

(an,1,0

.

.

.

.

.a1,1,0

)=(1 ⋯12

⋮ ⋱ ⋮0 ⋯ 1

)(qn.....q1

)¿(qn.....q1

)=(1 ⋯12

⋮ ⋱ ⋮0 ⋯ 1 )

−1(an,1,0

.

.

.

.

.a1,1,0

)At the step 0, we get:

ak,k,0=∑

i=k+1

nai,k+1,0−α+ck

2 [ ∑i=k+1

nai,k+1,k−1]

withai,k=12ai,k+1fori>kwithai,k,jindependentofj

¿ai,k=ai,k+1−12ai,k+1=

12ai,k+1

Asaresult,ai,k=(12 )i−k

ai,i=(12 )i+1−k

stilltruefork=i

ai,k+1,0 et ai,k+1,k are determined as follows:

But:ai,k,0=ai,k+1,0−ai,k+1,kak,k,0fori>k

¿ai,k+1,0−(12 )i−k

ak,k,0

28

soitai,k,0−ai,k+1,0=−(12 )i−k

ak,k,0

∑r=k

i−1ai,r,0−ai,r+1,0=−∑

r=k

i−1

(12 )i−r

ar,r,0

ai,k,0−ai,i,0=−∑r=k

i−1

(12)i−r

ar,r,0

And we get:

ai,k,0=−∑r=k

i−1

(12 )i−r

ar,r,0

+ai,i,0i>kstilltrueifi=k

We carry out the determination of ak,k,0 :

ak,k,0=∑

i=k+1

nai,k+1,0−α+ck

2 [ ∑i=k+1

nai,k+1,k−1]

2[ ∑i=k+1

nai,k+1,k−1]=2[ ∑i=k+1

n

(12)i+1−k−1

−1]¿2 [1−(12)

n−k−1 ]=−2(12 )

n−k

We note:

∀k∈‖1,n‖,γk=−2(12 )n−k

So that:

γkak,k,0= ∑i=k+1

nai,k+1,0−α+ck

¿ ∑i=k+1

n (− ∑r=k+1

i−1

(12 )i−r

ar,r,0

+ai,i,0)−α+ck

¿ ∑i=k+1

n (− ∑r=k+1

i

(12 )i−r

ar,r,0

+2ai,i,0)−α+ck

¿− ∑r=k+1

n

∑i=k

n

(12 )i−r

ar,r,0

+2 ∑i=k+1

nai,i,0−α+ck

29

γkak,k,0=− ∑r=k+1

nar,r,0(2−(12)

n−r

)+2 ∑i=k+1

nai,i,0−α+ck

¿− ∑r=k+1

nar,r,0

γk

2−2 ∑

r=k+1

nar,r,0+2 ∑

i=k+1

nai,i,0⏟

−α+ck

= 0

We eventually deduce:

ck−α=γkak,k,0+ ∑r=k+1

nar,r,0

γk

2Which can be written as:

(1 ⋯ 0⋮ ⋱ ⋮12

⋯ 1)(γnan,n,0

.

.

.

.

.γ1a1,1,0

)=(cn−α.....

c1−α)

30

With:

(γnan,n,0

.

.

.

.

.γ1a1,1,0

)=(γn ⋯ 0⋮ ⋱ ⋮0 ⋯ γ1

)(an,n,0.....

a1,1,0

)Then:

(1 ⋯ 0⋮ ⋱ ⋮12

⋯ 1)(γn ⋯ 0⋮ ⋱ ⋮0 ⋯ γ1

)(an,n,0

.

.

.

.

.a1,1,0

)=(cn−α.....

c1−α)

Eventually:

(qn

.

.

.

.

.q1

)=(1 ⋯12

⋮ ⋱ ⋮0 ⋯ 1

)−1

(−γn ⋯ 0⋮ ⋱ ⋮0 ⋯ −γ1

)−1(

1 ⋯ 0⋮ ⋱ ⋮12

⋯ 1)−1(

α−cn.....

α−c1)

C.2. Stackelberg with N economic agents over twoperiods

_____________________

Now, we turn the model into two periods in which producersdeplete their reserve. Each actor must then maximize its presenttotal profit, subject to yi,1

¿+yi,2¿=Ri, where yi,1

¿ is the optimalproduction at time 1 and yi,2

¿:

∀i∈ {1,2 },Πi=Πi,1+Πi,2

∀i∈ {1,2 },Πi=(Ps−ci−q1,1−q2,1)qi,1+1

1+i. (Ps−ci−q1,2−q2,2)qi,2

32

s.t.∀i∈ {1,2 },qi,1+qi,2=Ri

∀i∈ {1,2 },Πi=(Ps−ci−q1,1−q2,1)qi,1+1

1+i.¿

Taking N actors, each present total profit becomes:

∀i∈ {1,..,N },Πi=(Ps−ci−∑j=1

Nqj,1)qi,1+

11+i

.¿

¿(Ps−ci−∑j=1

Nqj,1)qi,1+ 1

1+i.(Ps−ci−∑

j=1

NRj+∑

j=1

Nqj,1)(R¿¿i−qi,1)¿

∀i∈ {1,..,N },Πi=((1−1

1+i)Ps−(1−

11+i

)ci+1

1+i∑j=1

NRj−(1+

11+i

)∑j=1

Nqj,1)qi,1+ 1

1+i¿

Πi=((1−1

1+i )Ps+Ri

1+i−(1−1

1+i )ci+1

1+i∑j=1

NRj−(1+

11+i )∑

j=1

Nqj,1)qi,1+ 1

1+i (Ps−ci−Ri− ∑j=1,j≠i

Nqj)Ri

We will indifferently note qi,1∨qi in what follows. The production qi,k accounts for the production qi expressed at thestep k, and consequentlyi≥k:

qi,k=ai,k,0−∑j=1

k−1ai,k,jqj

Πk=(Akα+BkR+BkRk−Akck−Ck∑j=1

k−1qj−Ckqk−Ck ∑

i=k+1

nqi)qk+BkRk(∑j=1

k−1qj+ ∑

i=k+1

nqi)+cte

Πk=(Akα+BkR+BkRk−Akck−Ck∑j=1

kqj−Ck ∑

i=k+1

nai,k+1,0+Ck∑

j=1

k

∑i=k+1

nai,k+1,jqj)qk+BkRk(∑j=1

k−1

qj+ ∑i=k+1

n

(ai,k+1,0−∑j=1

kai,k+1,jqj))

Πk=(Akα+BkR+BkRk−Akck−Ck ∑i=k+1

nai,k+1,0+Ck∑

j=1

k [( ∑i=k+1

nai,k+1,j)−1 ]qj)qk+BkRk( ∑

i=k+1

nai,k+1,0−∑

j=1

k−1 [( ∑i=k+1

nai,k+1,j)−1 ]qj−qk ∑

i=k+1

nai,k+1,k)

∂Πk∂qk

=0=(Akα+BkR+BkRk−Akck−Ck ∑i=k+1

nai,k+1,0+Ck∑

j=1

k [( ∑i=k+1

nai,k+1,j)−1]qj)−BkRk ∑

i=k+1

nai,k+1,k+Ck[( ∑

i=k+1

nai,k+1,k)−1]qk

33

2Ckqk[ ∑i=k+1

nai,k+1,k−1]=Ck ∑

i=k+1

nai,k+1,0−Akα−BkR−BkRk+Akck+Ck∑

j=1

k−1 [1−( ∑i=k+1

nai,k+1,j)]qj+BkRk ∑

i=k+1

nai,k+1,k

soitqk,k=(Ck+BRk ) ∑

i=k+1

nai,k+1,0−Akα−BkR−BkRk+Akck

2Ck[ ∑i=k+1

nai,k+1,k−1]

−

Ck∑j=1

k−1 [ ∑i=k+1

nai,k+1,j−1]

2Ck[ ∑i=k+1

nai,k+1,k−1]

qj

Identifying with (1), we get the following recurrence relation:

ak,k,0=(Ck+BkRk) ∑

i=k+1

nai,k+1,0−Akα−BkR−BkRk+Akck

2Ck[ ∑i=k+1

nai,k+1,k−1]

ak,k,j=

Ck( ∑i=k+1

nai,k+1,j−1)

2Ck[ ∑i=k+1

nai,k+1,k−1]

________________________________________________________

We can write:

i>k⟹i≥k+1,qi,k=qi,k+1=ai,k+1,0−∑j=1

k−1ai,k+1,jqj−ai,k+1,kqk,k

¿ai,k+1,0−∑j=1

k−1ai,k+1,jqj−ai,k+1,k(ak,k,0−∑

j=1

k−1ak,k,jqj)

34

We get a second property:

ai,k,0=ai,k+1,0−ai,k+1,kak,k,0

ai,k,j=ai,k+1,j+ai,k+1,kak,k,j

That brings us to make the assumption that: ai,k,j=ai,k so thatthe coefficient do not depend on j. Let's verify it by induction on k :

Fork=n,ai,n,j=12

(vseetheexamplewiththreeplayersbelow )

Weassumethatthepropertyistruefork∈N,∧wedemonstratefork−1

ai,k−1,j=ai,k,j−ai,k,k−1ak−1,k−1,j=ai,k−ai,kak−1,k−1,j

ak−1,k−1,j=

Ck[ ∑i=k+1

nai,k,j−1]

2Ck[ ∑i=k+1

nai,k,k−1−1]

≝

Ck[ ∑i=k+1

nai,k−1 ]

2Ck [ ∑i=k+1

nai,k−1]

=12

We also get from that: ai,k,j=12ai,k+1,j.

Thus, we get the relation:

qk,k=ak,k,0−∑j=1

k−1 12qj

We then determine the coefficients as:

(an,1,0

.

.

.

.

.a1,1,0

)=(1 ⋯12

⋮ ⋱ ⋮0 ⋯ 1

)(qn.....q1

)35

soit(qn

.

.

.

.

.q1

)=(1 ⋯12

⋮ ⋱ ⋮0 ⋯ 1 )

−1(an,1,0

.

.

.

.

.a1,1,0

)At the step 0, we have:

ak,k,0=(Ck+BRk) ∑

i=k+1

nai,k+1,0−Akα−BkR−BkRk+Akck

2Ck[ ∑i=k+1

nai,k+1,k−1]

withai,k=12ai,k+1fori>kbecauseai,k,jareindependent¿j

¿ai,k=ai,k+1−12ai,k+1=

12ai,k+1

¿ai,k=(12 )i−k

ai,i=(12 )i+1−k

stilltruefork=i

We shall determineai,k+1,0 et ai,k+1,k:

ai,k,0=ai,k+1,0−ai,k+1,kak,k,0fori>k

¿ai,k+1,0−(12 )i−k

ak,k,0

soai,k,0−ai,k+1,0=−(12)i−k

ak,k,0

∑r=k

i−1ai,r,0−ai,r+1,0=−∑

r=k

i−1

(12 )i−r

ar,r,0

ai,k,0−ai,i,0=−∑r=k

i−1

(12)i−r

ar,r,0

Then we get:

36

ai,k,0=−∑r=k

i−1

(12 )i−r

ar,r,0

+ai,i,0i>kstilltrueifi=k

Let's determine ak,k,0 :

ak,k,0=(Ck+BRk) ∑

i=k+1

nai,k+1,0−Akα−BkR−BkRk+Akck

2Ck[ ∑i=k+1

nai,k+1,k−1]

2Ck[ ∑i=k+1

nai,k+1,k−1]=2Ck[ ∑i=k+1

n

(12 )i+1−k−1

−1]¿2Ck[1−(12)

n−k−1]=−2Ck(12 )

n−k

We note:

∀k∈‖1,n‖,γk=−2Ck(12 )n−k

So that:

γkak,k,0=(Ck+BkRk ) ∑i=k+1

nai,k+1,0−Akα−BkR+Akck

¿ (Ck+BkRk ) ∑i=k+1

n (− ∑r=k+1

i−1

(12 )i−r

ar,r,0

+ai,i,0)−Akα−BkR−BkRk+Akck

¿ (Ck+BkRk ) ∑i=k+1

n (− ∑r=k+1

i

(12 )i−r

ar,r,0

+2ai,i,0)−Akα−BkR−BkRk+Akck

¿−(Ck+BkRk ) ∑r=k+1

n

∑i=k

n

(12)i−r

ar,r,0

+2 (Ck+BkRk ) ∑i=k+1

nai,i,0−Akα−BkR−BkRk+Akck

γkak,k,0=−(Ck+BkRk ) ∑r=k+1

n (2−(12)n−r

)ar,r,0+2 (Ck+BkRk ) ∑

i=k+1

nai,i,0−Akα−BkR−BkRk+Akck

37

¿−(Ck+BkRk ) ∑r=k+1

nar,r,0

γr2Cr

−2 (Ck+BkRk ) ∑r=k+1

nar,r,0+2 (Ck+BkRk ) ∑

i=k+1

nai,i,0⏟

−Akα−BkR−BkRk+Akck=0

We eventually deduce that:

−Akα−BkR−BkRk+Akck=γkak,k,0+(Ck+BkRk ) ∑r=k+1

nar,r,0

γr

2Cr

It can be written as:

(1 ⋯ 0⋮ ⋱ ⋮

12Cr

(Ck+BkRk ) ⋯ 1)(γnan,n,0

.

.

.

.

.γ1a1,1,0

)=(−Anα−BnR−BnRn+Ancn

.

.

.

.

.−A1α−B1R−B1R1+A1c1

)With:

(γnan,n,0

.

.

.

.

.γ1a1,1,0

)=(γn ⋯ 0⋮ ⋱ ⋮0 ⋯ γ1

)(an,n,0.....

a1,1,0

)Then:

(1 ⋯ 0⋮ ⋱ ⋮

12Cr

(Ck+BkRk ) ⋯ 1)(γn ⋯ 0⋮ ⋱ ⋮0 ⋯ γ1)(

an,n,0.....

a1,1,0)=(

−Anα−BnR−BnRn+Ancn.....

−A1α−B1R−B1R1+A1c1)

Eventually:

38

(qn

.

.

.

.

.q1

)=(1 ⋯12

⋮ ⋱ ⋮0 ⋯ 1

)−1(

an,n,0.....

a1,1,0)

(qn

.

.

.

.

.q1

)=(1 ⋯12

⋮ ⋱ ⋮0 ⋯ 1 )

−1

(−γn ⋯ 0⋮ ⋱ ⋮0 ⋯ −γ1

)−1(

1 ⋯ 0⋮ ⋱ ⋮

12Cr

(Ck+BkRk ) ⋯ 1)−1(

−Anα−BnR−BnRn+Ancn.....

−A1α−B1R−B1R1+A1c1

)_____________________________

39

C.3. Applications to the oil market_____________________

Price evolution:

We can deduce from this modelling of concurrence with anextended Stackelberg oligopoly with N actors a price-formation model.Indeed, at any moment, the price will be equal to p=α−∑qi

¿ wherethe qi

¿ are the optimal productions determined previously.To calculate this progression, we assumed that, for the sake of

simplicity, every producer have the same initial reserves, and wechose α big enough to obtain significant results. Thus, after acertain amount of time, the producer with the highest optimalextraction stream would deplete his reserves first. We then move fromN actors to N-1 actors, optimal extraction streams are essentiallychanged, another producer will deplete his remaining resources and soon.

Here is the price evolution curve Figure C.3.1 (we just focused onthe trend), working with 10 economic agents, with the same initialreserves: 500.000, whose production costs regularly range from 20$ to110$.

40

Positioning of the actors on the marketplace:

Single period model:

We apply the previous model (Figures C.3.2 and C.3.3), buildingrepresentative costcurves of the market. Production volumes areexpressed in millions of barrels / day:

Figure C.3.2: Costcurve in the reality 7

Figure C.3.3: Costcurve in the extended single-period Stackelberg model

7 p(T)=c(T)+(ps−c (T ))e−i(T−T)=ps, c(T) cannot be determined

Saud

iArab

ia

Othe

rOPEC

Russ

ia

U.S wt.

Shal

eKa

zakhst

anChin

a

Figure C.3.1: Price evolution curve

41

Two periods model:

We apply the two periods model, and we consider for the sake ofsimplification that every actor has the same initial reserves and thesame interest rate:

Figures C.3.4: Costcurve in the reality 8

Figure C.3.5: Costcurve in the two-periods Stackelberg model

The comparison between these two costcurves show that theintroduction of two periods, and so turning the static model into thebeginning of something dynamic, enable other oil producers toparticipate equally in the global stage production. Saudi Arabia andgenerally OPEC, in the static situation, should increase itsextraction steam so that to maximize its present profit. However, in

8 Credit: Financial Sense

Saudi

Arabia

Othe

rOP

EC

Russ

ia

U.S

wt.

Shal

eKa

zakhst

an Chi

naNorw

ayOthe

r No

n-OPEC Braz

ilU.

S Sh

ale

Mexi

coCa

nada

Oil

Sands

42

the dynamic model, they should still increase their production butless than the static situation. That could be an explanation whySaudi Arabia does not produce today as much as they can (Figure C.3.6).

It is to be noted that we considered the market as being onlyshared by the economic agents in the tables below, which gives themnon-real market shares.

Middle-East25$

Conventional70$

Non-Conventional

90$

Substitutes 100$

Marketshares model

52,7% 26,8% 13,4% 7,1%

Real marketshares with

40% 30% 20% 10%

Figure C.3.6

Here is a more precise analysis of the influence of thedifferent parameters at stake: for instance, we take into accountthree producing which costs go from 25$/b to 90$/b, which couldcorrespond to the 25$/b conventional oil for Middle-East, 50$/bconventional oil in the rest of the world and eventually 90$/bAmerican shale oil.

Firstly, we take different interest rates, ten years betweenthe two periods, and that Middle-East has twice more initial reservesthan the other actors, and we get the following results for the firstperiod production (Figure C.3.7):

Middle-East 25$ Conventional 50$ Shale U.S 90$Real market

shares57,9% 28,4% 13,7%

Market shareswith r=3%

50,1% 25,0% 24,8%

Market shareswith r=6%

50,5% 25,2% 24,3%

Market shareswith r=13%

51,4% 25,6% 23,0%

Market shareswith r=20%

52,5% 26,0% 21,5%

Figure C.3.7

We can see with the two periods model that oil producers withlower extraction costs have to produce less in the first period andthen have less market shares. As might be expected, we also see thatinterest rates have a crucial role in this game. With large discount

43

rates, the companies have a big interest in producing a lot todaywithout waiting. Nevertheless, if we take real interest rates, wenote it is valuable for lower costs companies to produce less inorder to maximize their inter-temporal profits in the dynamic model.The single-period extended Stackelberg model tends to show thatMiddle East does not produce enough to increase its present profit,but this interpretation is tempered when we consider the issue in adynamic situation.

Let us consider other parameters' influence: relative initialreserves of the different economic agents are varied as follows: weconsider that each actor has the same initial reserves except for theMiddle-East, with reserves evolving from the same as others to fourtimes more (Figure C.3.8):

Middle-East 25$ Conventional 50$ Shale U.S 90$Market shares

reality57,9% 28,4% 13,7%

Market sharesx1

38,7% 32,3% 29,0%

Market sharesx2

52,5% 26,0% 21,5%

Market sharesx3

59,7% 21,2% 19,1%

Market sharesx4

65,6% 18,1% 16,3%

Figure C.3.7

__________________We also had the opportunity to think about more evolutions of

the extended Stackelberg duopoly model. Still in the inter-temporalside of the model, instead of setting initial reserves as parameters,we decide de take into account the different production capacitiesduring each period of time.

As a result, if the single-period extended Stackelberg duopolyimposes extraction streams larger than its production capacity, heshould strive to produce at full capacity in order to maximize itsprofit. During this period - because of a certain degree ofinformation asymmetry between due to the cost ranking of the economicagents related to the constitution of the extended Stackelberg model-, lower rank competitors will have adapted their production as ifupstream actors have been behaving in accordance with the Stackelberg

44

without capacity constraints model. However during the next period,these same agents will have taken note of each producer's restrictionand will adapt their extraction stream, which upstream players willhave taken into account in accordance to the Stackelberg ranking.

Those economic behaviours can be shown as follows:

Time t=0:

c1

c2

c3

c4

cn-2 cn-1 cn

q1

q2

q3 > C3

q4

qn-2 qn

qn-1 > Cn-1

Agents

Expression of the profit

1Π1=(α−c1−q1

¿−q2¿−C3−q4

¿...−qn−2¿−Cn−1−qn

¿)q1¿

2Π2=(α−c2−q1

¿−q2¿−C3−q4

¿...−qn−2¿−Cn−1−qn

¿)q2¿

3Π3=(α−c3−q1

¿−q2¿−C3−q4

¿...−qn−2¿−Cn−1−qn

¿)C3

4Π4=(α−c4−q1

¿−q2¿−q3

¿−q4¿...−qn−2

¿−Cn−1−qn¿)q4

¿

i Πi=(α−ci−q1¿−q2

¿−q3¿−q4

¿...−qn−2¿−Cn−1−qn

¿)qi¿

n-2 Πn−2=(α−cn−2−q1¿−q2

¿−q3¿−q4

¿...−qn−2¿−Cn−1−qn

¿)qn−2¿

n-1 Πn−1=(α−cn−1−q1¿−q2

¿−q3¿−q4

¿...−qn−2¿−Cn−1−qn

¿)Cn−1¿

n Πn=(α−cn−q1¿−q2

¿−q3¿−q4

¿...−qn−2¿−qn−1

¿−qn¿)qn

¿

45

We could propose a recursive resolution algorithm to solve thisproblem:

#Initialization the profit functions

Πn=(α−cn−∑i=1

nqi)qn

Πn−1=(α−cn−1−∑i=1

nqi)qn−1

.

.

Π1=(α−c1−∑i=1

nqi)1

#Launch of the loop

For t from 1 to T do :

qn¿=solve(

∂Πn∂qn

,qn);qn−1¿=solve(

∂Πn−1

∂qn−1,qn−1);...;q1

¿=solve(∂Π1∂q1

,q1);

r:=1;

ifqn¿<Cn∧qn−1

¿<Cn−1∧...q1¿<C1

thenSingle-period Stackelberg until complete depletion

elseforj¿1¿ndo:

ifqj¿<Cjthenqj

¿=qj¿

elseqj¿=Cj;pr:=j;r:=r+1;fi;od;

For i from n to pr+1 do:

46

Πi=(α−ci− ∑k=1,k≠pi

nqk−∑

k=1

rCpk)qi;od;

For i from pr to pr-1+1 do:

Πi=(α−ci− ∑k=1,k≠pi,i≤r−1

nqk−∑

k=1

r−1Cpk)qi;od;

For i from pr-1 to pr-2+1 do:

Πi=(α−ci− ∑k=1,k≠pi,i≤r−2

nqk−∑

k=1

r−2Cpk)qi;od;

.

.

.For i from p1 to 1 do :

Πi=(α−ci−∑k=1

nqk)qi;od;

fi ; od ;

D. Price fluctuation policy from OPECStackelberg oligopoly with competitive

fringe under demand uncertainty__________________________________

Figure D.1

47

A producers' oligopoly that we will call the "core" has reallyrich deposits with lowest extraction costs on the market. These twocharacteristics can lead to a supply with the same and constantaverage and marginal costs. If they compete with each other, theproducers of this first group could satisfy the whole demand with thesame price as their average costs. The second group, which gathersall other companies of the industry, has less abundant deposits withhigher extraction costs: increasing average and marginal costs(price-takers) Figure D.1.

We consider a price-elastic market demand, and we assume thatdemand is driven by a known random law (Figure C.2):

Y(p)=ω−pwithω=knownlawof densityf

Figure C.2

Y(p)=ω−p Ccore(yc)=cycCfrin≥¿(yf)=ayf+byf

2¿

The competitive fringe behaves as a price-taker, it must equalits marginal cost to the market price:

a+2byf=psoyf=p2b

−a2b

Which is made possible only if p>a. If p<a, then yf=0 andthe core satisfies the whole demand.

48

Aggregate supply of the competitive fringe:

Yf(p)={ p2b− a2b

sip>a

0sip<a

If p<a,Yf(p)=0thenYc(p,ω)=¿ Y(p)=Ps+ω−p

If p<a,Yf (p)= p2b

−a2b

thenYc (p,ω)=¿ Y (p,ω )−Yf (p )=A(ω)−Bp

avecA(ω)=a2b

+ωetB=1+12b

Which is made possible only if p<A(ω)B . If p<

A(ω)B , alors yc=0

, et la frange prend tout le marché.

Aggregate supply of the core: Yc(p,ω)={A(ω)−Bpifp<A(ω)B

0if p>A(ω)B

The leader behaves as a monopoly, and we show that its profit

Πc,n is maximum when a<p<A(ω)B with:

p (ω )=A (ω)+cB2B ,q(ω)=

A (ω)−cB2

;

Πc(ω)=A(ω)+cB

2B (A(ω)−cB2 )−c( A(ω)−cB

2 )= 14B

(A (ω)−cB)2

________________________________

How does the core can cause prices to fluctuate so that to getmaximum benefit?

If this fluctuation policy follows the randomness of thedemand, we will obtain an expected profit :

49

E(Πc)=∫Πc(ω).f(ω)dω=∫ 14B (A(ω)−cB )2f(ω)dω

If the core decides not to cause the prices to fluctuate, itwill choose the price p0 which will maximize its expectedprofit :

E(Πc)=maxp0∈R

(∫Πc(p0,ω)f(ω)dω)avecΠc(p0,ω)=(p0−c)(A(ω)−Bp0)

If the fluctuation policy consists in adding a voluntaryfluctuation :

________________________________

Inter-temporal modelling:

The core focuses now on the maximization of its inter-temporalpresent profit over the different price fluctuation policies:

Jc=E (Πc1+αΠc2+α2Πc

3+...+αT−1ΠcT ):αisthediscountrate,0<α<1

If this fluctuation policy follows the randomness of thedemand, we will obtain an expected profit:

Jc(1)=∑

i=1

Tαi∫Πc

i(ω).f(ω)dωi

Ω follows a Bell distribution with mean m and variance σ, so that :

f(ω)=1

2π√σe

−(ω−m)2

4σ2

From the calculus we get:

Jc(1)=∑

i=1

Tαi 14B [K+m2+m(ab−2cB)]

Jc(1)=α 1−αT

1−α14B [K+m2+σ2+m(ab−2cB)]

oùK=a24b2+c

2B2−caBb

50

And the profit of the fringe will be:

Jf(1)=∑

i=1

Tαi (p−a−byf )yf=∑

i=1

Tαi[p−a−b( p2b−

a2b )]( p

2b−

a2b )

¿α 1−αT

1−α14b [K'+ 1

4B2 (m2+σ2+(ab+2cB−4aB)m)]Jf

(1)=α 1−αT

1−α14b [K'+

14B2 (m2+σ2+(ab+2cB−4aB)m)]

oùK'=a2+c2

4+

a216B2b2

+ac4Bb−

a22Bb−ac

If the core decides not to cause the prices to fluctuate, itwill choose the price p0 which will maximize its expectedprofit.

Jc=maxp0∈R

(∑i=1

Tαi∫Πci(ω,p0).f(ω)dω)

Then:

Jc(2)=max

p0∈R(∑i=1

Tαi∫ (p0−c ) (A (ω )−Bp0)f(ω)dω)

¿maxp0∈R

((p0−c )∑i=1

Tαi [∫A (ω )fi (ω )dω−Bp0 ])

dJc(2)

dp0=∑

i=1

Tαi [∫A (ω )f(ω)dω−Bp0 ]−B (p0−c)∑

i=1

Tαi=0

p0¿=

1

2B∑i=1

Tαi

(∑i=1

Tαi∫A (ω )f(ω)dω+Bc∑

i=1

Tαi)

51

¿12B ( a2b+m+Bc)

And:

Jc(2)=α 1−αT

1−α (p0¿−c)([ a2b+m ]−Bp0¿)Jf

(2)=∑i=1

Tαi (p0¿−a−byf )yf

Jc(2)=∑

i=1

Tαi[p0

¿−a−b(p0¿

2b−

a2b )](p0

¿

2b−

a2b )

Jf(2)=α 1−αT

1−α12b

(p0¿−a)2=α 1−αT

1−α12b [ 12B ( a2b+m+Bc)−a ]

2

If the fluctuation policy consists in adding a voluntaryfluctuation :

The core here adds a cumulative noise over the fluctuatingdemand, i.e. a voluntary random fluctuation ϵ with density g,meanM, varianceS of its production over the optimal productiondetermined above:

qc=(A (ω)−cB2 )+ϵ=(

~A(ω,ϵ)−cB2 )où~A(ω,ϵ)=A(ω)+2ϵ

And:

p(ω,ϵ)=A(ω)+cB

2B−ϵB

Πc1(ω,ϵ)=( A(ω)−cB

2+ϵ)( A(ω)

2B−c2−ϵB)

The present expected profit will be:

Jc(3)=∑

i=1

Tαi∫∫Πc

i (ω,ϵ)f (ω ).g (ϵ )dωdϵ

52

Jc(3)=¿α 1−αT

1−α × 14B [γc−4(M2+SZ) ]

γc=( a2b

−cB)2

+2m( a2b−cB)+m2+σZ

The competitive fringe acted as a price-taker, considering theprevious price p(ω,ϵ=0)without the unsaid fluctuation ϵ.

a+2byf=p(ω,ϵ=0)soityf=p(ω,ϵ=0)

2b−

a2b

The profit of the fringe becomes:

Πf (ω )=p(ω,ϵ)(p(ω,ϵ=0)2b

−a2b )−a(p(ω,ϵ=0)

2b−

a2b)−b(p(ω,ϵ=0)

2b−

a2b )

2

¿14b (p(ω,ϵ=0)−a )2−p(ω,ϵ=0)−a

2bBϵ

And an expected present profit:

Jf(3)=∑

i=1

Tαi∫∫Πf

i (ω,ϵ )f (ω ).g (ϵ )dωdϵ

Jf(3)=α 1−αT

1−α× 14b [γf− M

B2 ( a2B

+m−cB−aB2b )]

γf=14B2

(m2+σ2 )+( a4bB

+c2

−a)2

+1Bm( a4bB

+c2−a)

In the borderline cases where M=S=0∧ϵ=0, it can be notedthat we logically find Jc

(3)=Jc(1 )∧Jf

(3)=Jf(1).

53

As a result, it is necessary to compare whether the thirdfluctuation policy is better than the first one, comparing thedifferent present expected profit of the fringe and then build ϵ sothat:

Jf(3)−Jf

(1)≤0

Jf(3)−Jf

(1)=−α 1−αT1−α

× 14b

MB2 ( a2B+m−cB− aB

2b )M is then chosen depending on the sign of ( a2B+m−cB−

aB2b ).

Whatever the law of ϵ, present profit will not be as high as inthe first fluctuation policy for the core. Indeed:

Jc(3)−Jc

(1)=−α 1−αT

1−α × 1B (M2+S2)≤0

The core will have to choose between weakening the fringe anddecreasing its own profit. Of course, there is no doubt that thecore, very powerful over the fringe, has sufficient cash and cashequivalents to justify this kind of strategy.

We take the following numeric values for the parameters: the

interest rate observed by the core is α=3%, their marginal cost isabout 30$/b. Concerning the fringe, we take a=60, b=20. The randomlaw respects m=120with different variances. We worked within aperiod of ten years.

σ=5 Jc(1) Jc

(2) Jc(3)

61.83 61.65 61.83-0.03M2-0.03S2

Jf(1) Jf

(2) Jf(3)

0.14 0.28 0.14-0.08M

σ=15 Jc(1) Jc

(2) Jc(3)

63.31 61.65 63.31-0.03M2-0.03S2

Jf(1) Jf

(2) Jf(3)

0.18 0.28 0.18-0.08M

54

σ=25 Jc(1) Jc

(2) Jc(3)

66.25 61.65 66.25-0.03M2-0.03S2

Jf(1) Jf

(2) Jf(3)

0.25 0.28 0.25-0.08M

σ=35 Jc(1) Jc

(2) Jc(3)

70.67 61.65 70.67-0.03M2-0.03S2

Jf(1) Jf

(2) Jf(3)

0.35 0.28 0.38-0.08M

Appendix: Literature and State of the Art______________

Extended Stackelberg Duopoly:

Stackelberg-Nash-Cournot Equilibria: Characterizations and Computations Author(s):

55

Hanif D. Sherali, Allen L. Soyster, Frederic H. MurphySource: Operations Research, Vol.31, No. 2, (Mar. - Apr., 1983), pp. 253-276

A Stochastic Version ofa Stackelberg-Nash-Cournot Equilibrium Model∗. Daniel De Wolf, GREMARS, Université de Lille 3, France and Yves Smeers, CORE, Université Catholique de Louvain, Belgium

An Existence Result for Hierarchical Stackelberg v/s Stackelberg Games Ankur A. Kulkarni and Uday V. Shanbhag

Coalition Formation in n-Person Stackelberg Games∗

, Wayne F. Bialas, Department of Environmental Engineering Cornell UniversityIthaca, New York 14853, Mark N. Chew, Department of Industrial Engineering State University of New York at Buffalo, New York 14260

STACKELBERG COMPETITION WITH ENDOGENOUS ENTRY*, Federico Etro, The Economic Journal, 118 (October), 1670–1697.

IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 29, NO. 2, MARCH 2014, Equilibria in an Oligopolistic Market With Wind Power Production, S. Jalal Kazempour, Member, IEEE, and Hamidreza Zareipour, Senior Member, IEEE

IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 29, NO. 2, MARCH 2014, Equilibria in an Oligopolistic Market With Wind Power Production, S. Jalal Kazempour, Member, IEEE, and Hamidreza Zareipour, Senior Member, IEEE

Finding Optimal Strategies in Multi-period Stackelberg Games Using an Evolutionary Framework, Anton Frantsev, Ankur Sinha, Pekka Malo, Department of Information and Service Economy Aalto University School of Economics

A Stochastic Multiple-Leader Stackelberg Model: Analysis, Computation, and Application, Victor DeMiguel, Department of Management Science and Operations, London Business School, Huifu Xu, School of Mathematics, University of Southampton

Finding Optimal Strategies in a Multi-Period Multi-Leader-Follower Stackelberg Game

Using an Evolutionary Algorithm, Ankur Sinhaa,∗, Pekka Maloa, Anton Frantseva, Kalyanmoy Deb, Department of Information and Service Economy, Aalto University School of Business

Global Equilibria of Multi-leader Multi-follower Games with Shared Constraints,

Ankur, A. Kulkarni Uday V. Shanbhag∗ An Existence Result for Hierarchical Stackelberg v/s Stackelberg Games, Ankur A.

Kulkarni and Uday V. Shanbhag Existence and uniqueness of equilibrium points for concave N-person games, J. B.

Rosen, Econometrica, Vol. 33, No. 3 (July, 1965)

Multi-Firm Mergers with Leaders and Followers, Gamal Atallah1

University of Ottawa

Stackelberg Uncertain Demand:

Endogenous Shiftsin OPEC Market Power – A Stackelberg Oligopoly with Fringe, Daniel Huppmann, Deutsches Institut für Wirtschaftsforschung

Evidence of Market Power in the Atlantic Steam Coal Market Using Oligopoly Models with a Competitive Fringe, Clemens Haftendorn, Deutsches Institut für Wirtschaftsforschung

56

Global Oil Markets Revisited – Cartel or Stackelberg Market? Daniel Huppmann, Franziska Holz*, German Institute for Economic Research (DIW Berlin)

Nonrenewable resource oligopolies and the cartel-fringe game∗

, Hassan BenchekrounDepartment of Economics, CIREQ. McGill University, Cees WithagenDepartment of Spatial Economics, VU University Amsterdam Department of Economics, Tilburg University

Note on the open-loop von Stackelberg equilibrium in the cartel versus fringe model, Groot, F.; Withagen, C.A.A.M.; de Zeeuw, A.J.Published in: Economic Journal

Stackelberg equilibria in a multiperiod vertical contracting model with uncertain price-dependent demand, Leif K. Sandal AND Jan Ubøe

Ricardo & Hotelling:

A model of exhaustible resource exploitation with a ricardian rent. Robert D. Cairns, JEEM 13, 313-324 (1986)

On comparing monopoly and competition in exhaustible resource exploitation, Gérard Gaudet, JEEM 15, 412-418 (1988)

Hotelling H. (1931). The economics of exhaustible resources, Journal of Political Economy, 39, pp. 137-75.

Krautkraemer, J. A. (1998). Nonrenewable resource scarcity, Journal of Economic Literature, XXXVI, pp. 2065-2107.

Ricardo on Exhaustible resources, and the Hotelling Rule, Heinz D. Kurz and Neri Salvadori1

Exploration & Production portfolios:

Hubbert's oil peak revisited by a simulation model, Pierre-Noël Giraud, Aline Sutter,Timothée Denis, Cédric Léonard, Laure Bossy, Zouhair Fard, CERNA Mines ParisTechand EDF-R&D

Mean Field Games (MFG):

A mean field game approach to oil production, R. Carmona,F. Delarue, A. Lachapelle, Journal of Mathematical Economics, Volume 47, Issue 7, 2011

Robert Aumann. Markets with a continuum of traders. Econometrica, 32(1/2), 1964.

Olivier Guéant. Mean Field Games and applications to economics. PhD thesis, Université Paris-Dauphine, 2009.

57

Olivier Guéant, Jean-Michel Lasry, and Pierre-Louis Lions. Mean field games and applications. in Paris-Princeton Lectures in Quantitative Finance, 2009.

Jean-Michel Lasry and Pierre-Louis Lions. Mean field games. Japanese Journal of Mathematics, 2(1), Mar. 2007

Jean-Michel Lasry and Pierre-Louis Lions. Mean field games. Cahiers de la Chaire Finance et Développement Durable, (2), 2007.

Pierre-Louis Lions. Théorie des jeux `a champs moyen et applications. Cours au Collège de France, http://www.college-defrance.fr/default/EN/all/equ der/cours et seminaires.htm, 2007-2008.

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