Oligopoly on the market for a nonrenewable

resource: open-loop and feedback Nash equilibrium�

Cees Withagena, Alex Halsemab and Hassan Benchekrounc

March 18, 2007

Abstract

We give a full characterization of the open-loop Nash equilibrium of a non-

renewable resource game. We provide an analysis of the cartel-versus-fringe

model when agents play simultaneously. We show that contrary to the case

where the cartel is a leader and the fringe a follower, there exists a phase where

both supply simultaneously positive quantities. When the number of fringe

�rms is �nite we show that there is no closed-loop equilibrium that can result

in the extraction path prescribed by an open-loop Nash equilibrium. However,

in the limit case of an in�nite number of purely price taking fringe �rms we

claim that the open-loop Nash and the feedback Nash equilibrium coincide.

Key words: nonrenewable resources, cartel-versus-fringe, Nash equilib-

rium, open-loop, feedback.

1 Introduction

Although there exists an established and very large literature addressing the problem

of exploitation of a nonrenewable resource such as oil, there is still no complete

treatment of the case where several agents interact at the market level, each owning

�Acknowledgements: The �rst author is indebted to NWO for providing funding for this project.

The third author is a visiting scholar at Tilburg University and received a grant from NWO. Cor-

responding author: Cees Withagen, Department of Economics, Tilburg University, P.O. Box 90153,

5000 LE Tilburg, The Netherlands. Email: c.withagen@uvt.nl

1

a stock of the resource. Probably the reason is the technical complexity of treating

such a problem in a general formulation. To make progress, this question has been

addressed under di¤erent assumptions. The assumptions typically specify the type

of competition prevailing, the type of information and commitment possibilities, the

market conditions and �rms�characteristics. The aim of he present paper is to make

a further conribution to this literature

Interest in nonrenewable resource has been at least partly driven by the importance

of the oil market worldwide. Salant (1976) and later Ulph and Folie (1980) model the

oil market as a market with one coherent cartel and a fringe, consisting of many small

oil suppliers acting as price takers. Lewis and Schmalensee (1980) and Loury (1986)

justify the assumption of price taking behavior of the fringe and show that when the

number of fringe members becomes arbitrarily large, the equilibrium converges to the

equilibrium obtained under price taking behavior assumption. In these papers the

cartel and the fringe act simultaneously. Gilbert (1978), Groot, Withagen and de

Zeeuw (1992, 2003) consider the case where the cartel has a �rst mover advantage

and the fringe members are followers.

In this literature �rms are assumed to choose a production path and therefore

are able to commit to a production path at the initial time, they choose open-loop

strategies. The alternative modelling choice is to allow the �rms to adjust their

extraction rates at each moment to the level of the stock at that moment and to

time, �rms choose closed-loop strategies. Commitment can be for example achievable

in the presence of a perfect futures market for the resource. In other circumstances

information about the resource stocks of competitors may not be available. In that

case modelling as a choice by each �rm of an open-loop extraction strategy that

maximizes this �rms discounted pro�ts is a reasonable assumption. The drawback is

that if �rms have information about all stocks at future dates and have the �exibility

to adjust their production, the equilibrium obtained with open-loop strategies may

not be subgame perfect.

Under simultaneous play of the cartel and the fringe, Eswaran and Lewis (1985)

show, in discrete time, that with a �nite number of oligopolists, each extracting at

zero cost and facing isoelastic demand, the two equilibria coincide. More generally, if

extraction costs are independent of the existing resource stock, and if the decision rule

applied by each �rm in the feedback equilibrium does not depend on the stocks owned

2

by other �rms, the feedback equilibrium is also an open-loop equilibrium. Moreover,

if the initial stocks of the �rms are equal, if the �rms�cost functions are identical and

quadratic in the extraction rate, and if demand is linear, then also the feedback and

the open-loop equilibrium coincide. However, if the initial stocks do not coincide, the

two equilibria di¤er, as is shown by means of a numerical example. Attention is also

paid to the case of the cartel-versus-fringe model, where the fringe is a price taker. It is

shown that in the case of zero extraction costs the two equilibria coincide, at least for

a speci�c set of parameter values. Hartwick (2006) studies the feedback equilibrium

for the case of a �nite number of players also with extraction costs depending on the

extraction rate in a quadratic way and shows that the open-loop Nash equilibrium is

subgame perfect. Under hierarchical play where the cartel is a leader and the fringe

follower, Groot et al. (1992) characterize a time consistent open-loop Stackelberg

equilibrium. Groot et al. (2003) show that the open-loop Stackelberg equilibrium

can qualify as a strongly time-consistent or subgame perfect equilibrium.

The objective of this paper is threefold. First fully characterize the open-loop Nash

equilibrium in the case where there is an aribitrary number of �rms in two groups

where �rms can di¤er in both their resource stocks and their constant marginal costs.

We thereby extend the analysis by Loury (1986), where the players are identical

with regard to extraction costs. We also extend the work by Lewis and Schmalensee

(1980) by including more than one supplier in each cost category. Secondly, we derive

the open-loop Nash equilibrium for the cartel-versus-fringe model from the oligopoly

model by considering the case of one coherent cartel and the number of fringe members

going to in�nity. Thereby we provide a sound basis for price-taking behaviour by

the fringe in the cartel-versus-fringe model. Thirdly we investigate whether such an

equilibrium is subgame perfect.

Before describing the results we �rst clarify what we mean by subgame perfection

of an open-loop equilibrium in the context of a nonrenewable resource di¤erential

game. A �rst approach is to demand that the extraction paths determined in the

open-loop Nash equilibrium remain equilibrium extraction path for any set of stock

levels and that at a given moment, the truncation of the open-loop equilibrium paths

constitutes an equilibrium of the game that starts at that moment for all possible

stock levels. This criterion is clearly too demanding in the context of a natural

resource game and an open-loop Nash equilibrium will not be subgame perfect under

3

standard assumptions. A second and less demanding criterion is to determine if, given

initial stocks and the resulting open-loop extraction path, there exists a closed-loop

Nash equilibrium that generates the same extraction path as the open-loop Nash

equilibrium. This is the criterion we use.

We show that for a �nite number of players there exists no closed-loop Nash

equilibrium that replicates the extraction path of the open-loop equilibrium. Thus

for the result of Hartwick (2006) to hold, the assumption of quadratic extraction costs

is essential, and in Groot et al. (2003), it is the hierarchical play assumed that is

essential for the open-loop Stackelberg equilibrium to be strongly time consistent.

Moreover, Groot et al. (1992) show that along an open-loop Stackelberg the cartel

and the fringe never extract simultaneously. We show that this is no longer true when

the cartel and the fringe play simultaneously: there always exist a phase where both

players act simultaneously. When the fringe �rms cost disadvantage is large enough

the cartel exhausts its stock before the fringe regardless of its initial stock. If its initial

stock is large enough the cartel is, for an initial period time, the sole supplier in the

market. When the cost di¤erence is not large, the equilibrium starts with both type

of players supplying simultaneously regardless of the level of the stocks. If the number

of fringe members is in�nite and if they are treated as pure price takers the situation

becomes di¤erent. Groot et al. (2000) consider the open-loop Stackelberg equilibrium

for this case, and show that for certain parameter values it is time inconsistent. Hence

open-loop and feedback do not coincide. We claim in the present paper that if the

fringe is modelled as a pure price taker, the open-loop Nash and the feedback Nash

equilibrium coincide.

We would like to point out that the predominance of the analysis of open-loop

strategies in the literature is not purely motivated by realism. One of the reasons

given by Lewis and Schmalensee who concentrate on the open loop Nash equilibrium,

and not on the feedback equilibrium, is that, "The latter concept seems intractable

in this model" as they note (p477). A few exceptions are Eswaran and Lewis (1985),

Salo and Tahvonen (2001) and Polasky (1990).

In section 2 we present the model and the equilibrium concepts. In section 2 we

fully characterize the open-loop Nash equilibrium. Section 4 treats the limit case

where the number of fringe �rms is in�nite thereby characterizing the equilibrium

when the fringe �rms are price takers. In section 5 we derive a necessary condition

4

that a closed-loop Nash equilibrium must satisfy in order to generate the extraction

path followed under the open-loop Nash equilibrium. We show that such a necessary

condition is never satis�ed. Section 6 treats the limit case where fringe �rms are price

takers.

2 The model and equilibrium concepts

There are n �rms and two type of mines h and l. Each �rm exploits one and only one

mine. There are nh �rms that exploit the mines of type h and nl �rms that exploit

the mines of type l with nh + nl = n: Firm i of type j (i = 1; 2; :::; nj; j = l; h) is

endowed with an initial stock Sj0i and has a constant marginal cost of extraction kj.

Demand for the resource is stationary and linear with a choke price �p : p(t) = �p�x(t);where p(t) is the price, x(t) is demand at time t and �p > kj for j = l; h. We work in

continuous time, which starts at time 0. Extraction rates at time t � 0 are denoted

by qli (t) and qhi (t) respectively and are non-negative: De�ne q

j (t) =njXi=1

qji (t) and

Sj0 =njXi=1

Sj0i for j = l; h: At each moment t � 0 the price of the resource is given by

p(t) = �p � ql(t) � qh(t): All agents within a group are identical with regard to their

stocks: Sj0i =Sj0njfor j = l; h. An extraction path qji (i = 1; 2; :::; n

j; j = l; h) is called

feasible if

1Z0

qji (s)ds � Sji0 for all i = 1; 2; :::; nj with j = l; h (1)

Firms are oligopolist in the resource market and the objective of each �rm i is to

maximize the discounted sum of its pro�ts

1Z0

e�rs[�p� ql(s)� qh(s)� kj]qji (s)ds (2)

subject to (1). We denote this game by ��~S0; 0

�where the second argument 0 refers

to the initial instant of time of the game and where ~S0 ��Sh01; :::; S

h0nh; Sl01; :::; S

l0nl

�.

The game ��~S (t) ; t

�is a game that starts at time t with initial stocks ~S (t) ��

Sh1 (t) ; :::; Shnh(t) ; Sl1 (t) ; :::; S

lnl(t)�.

5

We consider two classes of strategies: open-loop and closed-loop strategies. An

open-loop strategy ji for �rm i that owns a stock of type j prescribes an extraction

rate of �rm i as a function of the calendar only:

qji (t) = ji (t) for all t � 0 and for all i = 1; 2; :::; nj with j = l; h:

A closed-loop strategy for �rm i that owns a stock of type j is a decision rule �ji that

gives the extraction rate at t as a function of t and the vector of stocks at time t,~S (t) :

qji (t) = �ji

�t; ~S (t)

�:

De�nition: Open-loop Nash Cournot equilibrium (OLNE)

A vector of open-loop strategies ~ � ( h1 ; ::; hnh ; l1; ::; lnl) is an open-loop Nash-Cournot equilibrium if

i. (1) is satis�ed for all i = 1; 2; :::; nj; j = l; h where qji (t) = ji (t) for all t � 0ii. for all i = 1; 2; :::; nl

1Z0

e�rs[�p� ql(s)� qh(s)� kl]qli(s)ds �1Z0

e�rs[�p�Xj 6=i

qlj(s)� qli(s)� qh(s)� kl]qli(s)ds

for all feasible l

i (i.e. such that (1) is satis�ed)

iii. for all i = 1; 2; :::; nh

1Z0

e�rs[�p�qh(s)�ql(s)�kh]qhi (s)ds �1Z0

e�rs[�p�Xj 6=i

qhj (s)� qhi (s)�ql(s)�kh]qhi (s)ds

for all feasible h

i (i.e. such that (1) is satis�ed).

De�nition: Closed-loop Nash-Cournot equilibrium (CLNE)

A vector of closed-loop strategies ~� ���h1 ; ::; �

hnh ; �

l1; ::; �

lnl

�is a closed-loop Nash-

Cournot equilibrium if

i. (1) is satis�ed for all (i = 1; 2; :::; nj; j = l; h) where qji (t) = �ji

�t; ~S (t)

�i. for all i = 1; 2; :::; nl

6

1Z0

e�rs[�p� ql(s)� qh(s)� kl]qli(s)ds �1Z0

e�rs[�p�Xj 6=i

qlj(s)� qli(s)� qh(s)� kl]qli(s)ds

for all feasible �l

i (i.e. such that (1) is satis�ed).

iii. for all i = 1; 2; :::; nh

1Z0

e�rs[�p�qh(s)�ql(s)�kh]qhi (s)ds �1Z0

e�rs[�p�Xj 6=i

qhj (s)� qhi (s)�ql(s)�kh]qhi (s)ds

for all feasible �h

i (i.e. such that (1) is satis�ed).

3 Open-loop Nash equilibrium with a �nite num-

ber of players

Each �rm i takes the strategy pro�le of its n� 1 competitors as given and maximizes(2) subject to (1). The Hamiltonian associated with the problem of �rm i therefore

reads

Hji (q

ji ; �

j; t) = e�rt��p� ql � qh � kl

�qji + �ji

��qji

�(3)

Writing the necessary conditions for a solution to each of these n problems and using

symmetry among the players within a group, i.e. qji = qj=nj and �ji = �j for j = l; h

and all i = 1; ::; nj gives the following.

Along intervals of time [t1; t2] with t2 > t1 � 0 where, for all t 2 [t1; t2]; qli(t) > 0for all i = 1; ::; nl and qhi (t) = 0 for all i = 1; ::; n

h we have

e�rt��p� ql(t)� 1

nlql � kl

�= �l (4)

e�rt��p� ql(t)� kh

�� �h (5)

p(t) =1

nl + 1

��p+ nl

�kl + �lert

��(6)

7

The �rst condition follows from the maximization of the Hamiltonian of player

i from category l assuming an interior solution. The second condition is necessary

in order for players from category h not to supply. The third condition gives the

resulting market price.

Along intervals of time [t1; t2] with t2 > t1 � 0 where, for all t 2 [t1; t2]; qli(t) = 0for all i and qhi (t) > 0 for all i we have

e�rt��p� qh(t)� 1

nhqh(t)� kh

�= �h (7)

e�rt��p� qh(t)� kl

�� �l (8)

p(t) =1

nh + 1

��p+ nh

�kh + �hert

��(9)

Along intervals of time [t1; t2] with t2 > t1 � 0 where, for all t 2 [t1; t2]; qli(t) > 0for all i and qhi (t) > 0 for all i we have

e�rt��p� ql(t)� qh(t)� 1

nlql(t)� kl

�= �l (10)

e�rt��p� qh(t)� ql(t)� 1

nhqh(t)� kh

�= �h (11)

For ql (t) and qh (t) we then obtain

nl + nh + 1

nlql(t) = �p+ nh

�kh + �hert

���nh + 1

� �kl + �lert

�(12)

nl + nh + 1

nhqh(t) = �p+ nl

�kl + �lert

���nl + 1

� �kh + �hert

�(13)

The price path is then given by

p(t) =1

nl + nh + 1

��p+ nl

�kl + �lert

�+ nh

�kh + �hert

��(14)

By S; C l; Ch we denote intervals of time with simultaneous supply, sole supply

by the l�type mines and sole supply by the h�type mines, respectively. At points oftransition from one regime to another the price trajectory is continuous, a property

that will be exploited below. We denote by T the date of exhaustion of all stocks;

8

when there is more than one phase we denote by t1 the transition date to the last

phase and when there are more than two phases we denote by t0 the transition date

from the �rst phase to the second phase. We show below that an OLNE has at most

three phases.

The direct procedure to solve for an OLNE is to solve the above system of neces-

sary conditions. Given all the possible alternatives such an approach is very tedious.

Instead we use an approach where we examine di¤erent sequences of phases of ex-

ploitation and rule out some that would violate the above set of conditions along

with the continuity of the price path. This turns out to be a powerful instrument to

characterize the sequence of exploitation along the equilibrium path and it greatly

facilitates the characterization of the exploitation path.

Continuity of the price path at the di¤erent possible transitions gives:

- a transition at t from S to C l or vice versa requires

�p+ nl�kl + �lert

�=�nl + 1

� �kh + �hert

�(15)

- a transition at t from S to Ch or vice versa requires

�p+ nh�kh + �hert

�=�nh + 1

� �kl + �lert

�(16)

- a transition at t from C l to Ch or vice versa requires

�nh + 1

��p+ nl

�nh + 1

� �kl + �lert

�=�nl + 1

��p+ nh

�nl + 1

� �kh + �hert

�(17)

We establish several lemmata, that are helpful in the derivation of a complete

characterization of the equilibrium.

Lemma 1:

i. For nl < 1 or nh < 1 there cannot be a transition from C l to Ch or vice

versa.

ii. For nl <1 or nh <1 there exists a phase S:

Proof:

i. Suppose a transition from C l to Ch or vice versa takes place at time t: Then

equation (17) holds. It follows from (5) and (6) that

9

1

nl + 1

��p+ nl

�kl + �lert

��� kh + �hert (18)

Moreover, it follows from (8) and (9) that

1

nh + 1

��p+ nh

�kh + �hert

��� kl + �lert (19)

Hence, rewriting (18) to obtain

nl�kl + �lert

���nl + 1

� �kh + �hert

�� �p

and substituting into (19) gives �p � kh + �hert. Similarly we have �p � kl + �lert:

Therefore

p(t) =1

nl + 1

��p+ nl

�kl + �lert

��� 1

nl + 1

��p+ nl�p

�= �p

which implies a too high price to have a positive quantity demanded after the tran-

sition.

ii. This follows immediately from statement (i) of the lemma.

Lemma 2:

i. Suppose �p+ nhkh ��nh + 1

�kl > 0: Then Ch cannot precede S:

ii. Suppose �p+ nhkh ��nh + 1

�kl < 0: Then S cannot precede Ch:

iii. Suppose �p+ nlkl ��nl + 1

�kh > 0: Then C l cannot precede S:

iv. Suppose �p+ nlkl ��nl + 1

�kh < 0: Then S cannot precede C l:

Proof:

i. and ii. Along Ch equations (8) and (9) hold. Hence, after substution of qh from

(8) into (9) we have

�p+ nhkh ��nh + 1

�kl �

��nh + 1

��l � nh�h

�ert (20)

A transition at t1 from Ch to Sor vice versa requires (16) to hold, stated otherwise

�p+ nhkh ��nh + 1

�kl =

��nh + 1

��l � nh�h

�ert1

Now suppose that �p+nhkh��nh + 1

�kl > 0 and that at time t1 a transition occurs

from Ch to S: We note that��nh + 1

��l � nh�h

�ert is decreasing over time since it

is a monotonic function of time and it is larger than �p+ nhkh ��nh + 1

�kl before t1

10

and equal to �p + nhkh ��nh + 1

�kl at t1. This implies that

�nh + 1

��l � nh�h < 0;

which along with �p+ nhkh ��nh + 1

�kl > 0 contradicts (20).

If �p+ nhkh ��nh + 1

�kl < 0 and if a transition occurs from S to Ch; say at time

t1; then similarly��nh + 1

��l � nh�h

�ert must increase after t1; but this is ruled out

since�nh + 1

��l � nh�h < 0:

iii) and iv). Due to symmetry the proofs are similar to the proofs of i) and ii).

Lemma 3:

i. There is simultaneous supply throughout only if

Sl0=nl

Sh0 =nh=�p+ nhkh �

�nh + 1

�kl

�p+ nlkl � (nl + 1) kh (21)

ii. If �p + nlkl ��nl + 1

�kh < 0 or if �p + nhkh �

�nh + 1

�kl < 0 there is no

simultaneous supply just before total exhaustion.

Proof :

i. If there is simultaneous exploitation just before T , then ql(T ) = qh(T ) = 0,

and, from (12), (13) and (14), �p = kh + �herT = kl + �lerT or �p � kh = �herT and

�p�kl = �lerT : Let the interval with simultaneous exploitation start at t1: Integration

of (12) and (13) gives

nl + nh + 1

nl

Z T

t1

ql(t)dt =

Z T

t1

��p+ nh

�kh + �hert

���nh + 1

� �kl + �lert

��dt

That is

nl + nh + 1

nlSl(t1) =

��p+ nhkh �

�nh + 1

�kl�(T � t1) +

�nh�h �

�nh + 1

��l�erT�1� ert1�rT

�r

Substituting �p� kh = �herT and �p� kl = �lerT gives after algebraic manipulation

nl + nh + 1

nlrSl(t1) =

��p+ nhkh �

�nh + 1

�kl� �rT � rt1 � 1 + ert1�rT

�(22)

Similarly we have

11

nl + nh + 1

nhrSh(t1) =

��p+ nlkl �

�nl + 1

�kh� �rT � rt1 � 1 + e�rT+rt1 ]

�(23)

Setting t1 = 0 and dividing (22) by (23) gives (21).

ii. This is immediately clear from (22) and (23).

Lemma 4:

Consider the sequence S ! C l; with C l the �nal phase before exhaustion and

where the transition takes place at instant of time t1 and exhaustion at T: Then

nl + nh + 1

nhrSh0 =

��p+ nlkl �

�nl + 1

�kh� �rt1 � 1 + e�rt1

�(24)

nl + nh + 1

nlrSl0 = � nh

nl + 1

��p+ nlkl �

�nl + 1

�kh� �rt1 � 1 + e�rt1

�+ (25)

nl + nh + 1

nl + 1

��p� kl

� �rT � 1 + e�rT

�Proof: See Appendix A.

Lemma 5:

Consider the sequence S ! Ch; with Ch the �nal phase before exhaustion and

with the transition taking place at instant of time t1 and exhaustion at T: Then

nl + nh + 1

nlrSl0 =

��p+ nhkh �

�nh + 1

�kl� �rt1 � 1 + e�rt1

�(26)

nl + nh + 1

nhrSh0 = � nl

nh + 1

��p+ nhkh �

�nh + 1

�kl� �rt1 � 1 + e�rt1

�+ (27)

nl + nh + 1

nh + 1

��p� kh

� �rT � 1 + e�rT

�Proof: Due to symmetry the proof is exactly the same as the proof of the previous

lemma.

It will prove useful to rewrite (27) in the following form:

r

�Sh0 +

nh

nh + 1Sl0

�=

nh

nh + 1

��p� kh

� �rT � 1 + e�rT

�(28)

12

We can now exploit the lemmata above to characterize the sequence along an OLNE

depending on the parameter values.

Proposition 1

Suppose nl <1 or nh <1: Suppose �p+ nlkl ��nl + 1

�kh < 0: For a given Sh0 ,

there exists ~Sl0 > 0 such that the equilibrium sequence reads Cl ! S ! Ch if Sl0 > ~Sl0

and S ! Ch if Sl0 � ~Sl0.

Proof:

Given the assumption on the costs and the choke price, �p > max�kl; kh

, we

cannot have C l alone or Ch alone: both type of mines eventually exploit and exhaust

their stock. According to lemma 1 ii), there must be a phase S: According to lemma

3 ii), for �p+nlkl��nl + 1

�kh < 0 the �nal phase of the exploitation cannot be S. A

phase S cannot precede C l (lemma 2). Therefore, the last phase of the exploitation is

Ch: From Lemma 1 i) there cannot be a transition C l ! Ch. Therefore the last two

phases will be S ! Ch: It remains to show that the only phase that can precede S is

C l. This can easily be seen from (16), that implies that there is at most one transition

S ! Ch or vice versa. Since there must one transition S ! Ch, the equilibrium reads

C l ! S ! Ch or S ! Ch.

We show that when the initial stock Sl0 is large enough we must have a phase

C l that precedes S ! Ch. Suppose that the equilibrium reads S ! Ch with the

transition at t1 and �nal time T: It follows from (13) that

nl + nh + 1

nhqh(0) = �p+ nlkl �

�nl + 1

�kh + nl�l �

�nl + 1

��h (29)

When Sl0 becomes arbitrarily large it follows from (26) and (27) that t1 and T also

become arbitrarily large. Using �h = e�rT��p� kh

�and rewriting (16) to obtain

�l =nh

nh + 1�h +

e�rt1��p+ nhkh �

�nh + 1

�kl�

nh + 1(30)

we see that �l and �h tend to zero when Sl0 becomes arbitrarily large. Equation

(29) then implies a negative value of qh (0) which is not admissible. Therefore, when

Sl0 is large enough the equilibrium reads C l ! S ! Ch. Moreover, using (30) and

�h = e�rT��p� kh

�we can state that qh(0) is a continuous strictly decreasing function

of Sl0 with@qh(0)

@Sl0= r

nh

nh + 1

�1

erT � 1 �1

ert1 � 1

�< 0: (31)

13

Thus, there exists a threshold stock level for Sl0 that we denote Sl0; beyond which the

equilibrium reads C l ! S ! Ch.

We need to show that for Sl0 positive but small enough, the equilibrium reads

S ! Ch, hence the phase C l collapses. Given Sh0 , for any Sl0 > Sl0 the equilibrium

reads C l ! S ! Ch. Consider the stock Sl (t0) and then consider the game that starts

at time 0 with stocks�Sl (t0) ; S

h0

�. The equilibrium must read S ! Ch because the

open-loop Nash equilibrium is by construction time consistent. Thus given Sh0 , for

any Sl0 > Sl0 the transition date t0 is positive and depends on Sl0; the equilibrium must

be such that Sl�t0�Sl01��= Sl

�t0�Sl02��. The stock exploited by the �rm l during

the period from t0 to t0 + t1 is then independent of the initial stock Sl0 provided that

Sl0 > Sl0 (again from the time consistency of the open-loop Nash equilibrium). Denote

this quantity by ~Sl0. Given Sh0 , we then have, for any S

l0 > ~Sl0 the equilibrium reads

C l ! S ! Ch and Sl0 < ~Sl0 the equilibrium reads S ! Ch:

Remark 1: The manifold in the space�Sl0; S

h0

�along which qh(0) = 0 divides

the state space in two separate regions with one region where the equilibrium reads

C l ! S ! Ch and the other region where the equilibrium reads S ! Ch. This

manifold is upward sloping. Indeed we show in appendix B that along this manifold

dSh0dSl0

= �@qh(0)

@Sl0@qh(0)

@Sh0

= � nh

nh + 1erT

�1

erT � 1 �1

ert1 � 1

�> 0.

Proposition 2

Suppose nl <1 or nh <1: Suppose �p+ nlkl ��nl + 1

�kh > 0 and �p+ nhkh ��

nh + 1�kl > 0

i. If

Sl0=nl

Sh0 =nh=�p+ nhkh �

�nh + 1

�kl

�p+ nlkl � (nl + 1) kh (32)

then the equilibrium reads S

ii. If

Sl0=nl

Sh0 =nh<�p+ nhkh �

�nh + 1

�kl

�p+ nlkl � (nl + 1) kh

then the equilibrium reads S ! Ch

14

iii. If

Sl0=nl

Sh0 =nh>�p+ nhkh �

�nh + 1

�kl

�p+ nlkl � (nl + 1) kh (33)

then the equilibrium reads S ! C l:

Proof:

From Lemma 1 there always exists an S phase. If �p + nlkl ��nl + 1

�kh > 0;

by Lemma 2 iii the S phase cannot be preceded by C l and from Lemma 2 i, when

�p + nhkh ��nh + 1

�kl > 0 the S phase cannot be preceded by Ch: Therefore the

equilibrium starts with S: Since a transition from C l to Ch or vice versa is ruled

out by Lemma 1, the equilibrium reads S ! Ch or S ! C l: If condition (32) is

satis�ed we have simultaneous supply throughout. If conditions (26) and (27) are

satis�ed with T > t1 > 0 then we must have S ! Ch: Otherwise the equilibrium

reads S ! C l:

Proposition 3

Suppose nl < 1 or nh < 1: Suppose �p + nhkh ��nh + 1

�kl < 0. For a given

Sl0, there exists ~Sh0 > 0 such that the equilibrium sequence reads Ch ! S ! C l if

Sh0 >~Sh0 and S ! C l if Sh0 � ~Sh0 .

Proof: This is similar to the proof of proposition 1 and is omitted.

These propositions fully characterize the open-loop Nash equilibrium for the case

of a �nite number of players. It can be checked that in each of the cases above we

have T > t1:

Remark 2: Consider the case where the OLNE consists of the sequence S ! C l:

Then t1 depends only the stock Sh0 ; it does not depend on Sl0

Remark 3: Consider the case where the OLNE consists of the sequence S ! C l:

Then using (24), (25) can be rewritten as

nl + nh + 1

nlrSl0 = � nh

nl + 1

nl + nh + 1

nhrSh0 +

nl + nh + 1

nl + 1

��p� kl

� �rT � 1 + e�rT

�or

r�Sl0 + Sh0

�=

nl

nl + 1

��p� kl

� �rT � 1 + e�rT

�15

The �nal exhaustion time depends on�Sl0 + Sh0

�and not on the distribution

�Sl0; S

h0

�.

It depends on kl and nl and not on kh and nh.

4 Open-loop Nash equilibrium in the cartel-versus-

fringe case

In this section we consider the case of an in�nite number of fringe members and

a single cartel. This what we call the cartel-versus-fringe model. We denote all

variables for the cartel by the superscript c and those for the fringe by superscript

f: Moreover, in the notation of the previous section kl = kc; nl = 1; kh = kf and

nh = 1: An equilibrium of the cartel-fringe model is given by a riplet of functions�p (:) ; qf (:) ; qc (:)

�such that:

(i) the fringe takes the price path (p (:)) as given and chooses an extraction path�qf (:)

�that maximizes its discounted sum of pro�ts subject to the resource constraint

(ii) the cartel takes the fringe extraction path and the demand function as given

and chooses an extraction strategy. In this section we treat the case where the cartel

chooses an open-loop extraction path qc (:) that maximizes its discounted sum of

pro�ts subject to its resource constraint.

(iii) the quantity of resource extracted qf (t)+qc (t) equals the quantity demanded

�p� p (t) at each moment.

It can be shown that if qf (:) is the optimal control of the fringe then along an

interval of time where the fringe produces a positive quantity there exists �f such

that

p (t) = kf + �fert (34)

during that interval of time.

Remark: a triplet�p (:) ; qf (:) ; qc (:)

�where the price path that never follows (34)

cannot qualify as an equilibrium since the fringe would the use a bang-bang impulse

control which would violate the market equilibrium condition (iii) at the moment the

impulse control is applied.

In order to derive the Nash equilibrium we build on the results of the previous

section. We �rst consider the case where the cartel has a cost advantage over the

fringe: kc < kf : By F we denote a phase with the fringe supplying alone. By Cm

16

we mean a phase with the cartel supplying alone. It follows from Lemma 1 that

a transition between F and Cm cannot occur. Moreover, there exists an S phase.

During an S phase, the price path for nh + nl < 1 is given by (14). In the case

where nl = 1 and nh = 1 the price path corresponds exactly to (34). The limit of

the OLNE when nh !1 is the equilibrium of the cartel-versus-fringe model.

Recall that for the equilibrium sequence S ! Cm necessary conditions are (24)

and (25). The couple (t1; T ) is implicitly determined by

rSf0 =��p+ kc � 2kf

� �rt1 � 1 + e�rt1

�and

2rSc0 + rSf0 = (�p� kc)�rT � 1 + e�rT

�:

For S ! F conditions (26) and (27) can be rewritten as

rSc0 =�kf � kc

� �rt1 � 1 + e�rt1

�(35)

rSf0 = ��kf � kc

� �rt1 � 1 + e�rt1

�+��p� kf

� �rT � 1 + e�rT

�(36)

Propositions 1 and 2 of the previous section suggest that a distinction should

be made between 12(�p+ kc) being larger than, equal to or smaller than kf : Let us

consider the di¤erent possibilities in turn.

If 12(�p+ kc) < kf ; then according to proposition 1 the Nash equilibrium is of the

type Cm ! S ! F; with the Cm phase collapsing if Sf0 is large.

If 12(�p+ kc) = kf ; then the solution of (35) and (36) is such that

qf (0) =1

2

��p� kf

� �e�rt1 � e�rT

�> 0

so that the equilibrium is S ! F:

If 12(�p+ kc) > kf > kc; then we can apply proposition 2. For nl = nc = 1 and

nh = nf =1 what matters is

Sc0

Sf0compared to

kf � kc

�p+ kc � 2kf

IfSc0

Sf0=

kf � kc

�p+ kc � 2kf

17

then the Nash equilibrium is S throughout.

IfSc0

Sf0>

kf � kc

�p+ kc � 2kf

then the equilibrium is S ! Cm:

IfSc0

Sf0<

kf � kc

�p+ kl � 2kh

then the equilibrium is S ! F:

The only case remaining is kf � kc: According to proposition 1 (with nl = 1)the equilibrium reads F ! S ! Cm; with the phase F possibly collapsing.

5 Open-loop versus feedback for a �nite number

of players

In the literature essentially two approaches to exploring the feedback equilibrium can

be found. The �rst is to postulate a functional relationship between extraction rates

and existing resource stocks, and then to see whether the Hamilton Jacobi Bellman

(HJB) conditions are satis�ed. This works in several cases (e.g., Salo and Tahvonen,

Eswaran and Lewis). However, no such simple relationship works in the particular

model at hand with linear demand and constant but di¤erent marginal extraction

rates. We willemploy an alternative approach, described below. In this section we

do not pursue a full treatment but as an example we consider the case where there

is one high cost �rm, with marginal extraction costs kf and one low cost �rm with

marginal extraction costs kc; such that kf > 12[�p+ kc]: As in the previous section we

employ the symbols f and c here for convenience and they will also refer to the fringe

and the cartel. We know from the section 3 that the equilibrium gives the following

sequence S ! F: if the initial resource stock of the cartel is not too large.

We seek to determine if there exists a feedback Nash equilibrium, that is therefore

subgame-perfect, that replicates the exploitation path of the OLNE, given a vector

of initial stocks.

The cartel takes the closed-loop strategy of the fringe as given �f�Sc; Sf ; t

�and

chooses a closed-loop strategy �c�Sc; Sf ; s

�that maximizes its discounted sum of

18

pro�ts1Zt

e�rs��p� qc(s)� �f

�Sc (s) ; Sf (s) ; s

�� kc

�qc(s)ds (37)

subject to1Zt

qc(s)ds � Sc (38)

and1Zt

�f�Sc (s) ; Sf (s) ; s

�ds � Sf (39)

for all non-negative couples�Sc; Sf ; t

�; with qc(s) = �c

�Sc (s) ; Sf (s) ; s

�:

The Hamiltonian associated with the cartel�s problem is given by

Hc = e�rs��p� qc(s)� �f

�Sc (s) ; Sf (s) ; s

�� kl

�qc(s)��ccqc (s)��cf�f

�Sc (s) ; Sf (s) ; s

�where �cc is the costate variable associated with S

c and �cf is the costate variable asso-

ciated with Sf . Applying the Maximum Principle gives the following set of necessary

conditions for an interior solution (i.e. qf > 0 and qc > 0):

e�rs��p� 2qc(s)� �f

�Sc (s) ; Sf (s) ; s

�� kl

�� �cc = 0

_�cc = �@Hc

@Sc=�e�rsqc + �cf

� @ ��f �Sc (s) ; Sf (s) ; s��@Sc

(40)

_�cf = �@Hc

@Sf=�e�rsqc + �cf

� @ ��f �Sc (s) ; Sf (s) ; s��@Sf

(41)

For a CLNE to result in the extraction path of the OLNE, we must have �cc (s) = �c

for all instants s � t for all t � 0: Therefore we must have �cc constant. From (40) it

follows that then

�e�rsqc + �cf

� @ ��f �Sc (s) ; Sf (s) ; s��@Sc

= 0

This implies that either (i) e�rsqc + �cf = 0 where qc is the OLNE equilibrium path

of the cartel and therefore �cf (s) = �e�rsqc (s) or (ii)

@��f�Sc (s) ; Sf (s) ; s

��@Sc

= 0:

19

We note that this latter condition does not require@(�f(Sc;Sf ;s))

@Sc= 0 for all

�Sc; Sf

�but only along the OLNE paths of the stock.

Condition (i) is in contradiction with (41) since it implies that _�cf = 0; but

e�rsqc (s) is not constant along the OLNE. Therefore, for a CLNE to result in the

extraction path of the OLNE, we must have@(�f(Sc(s);Sf (s);s))

@Sc= 0 along the OLNE

paths of the stock.

The production path of the fringe along the OLNE is given by

3qf (t) = �p+�kc + �cert

�� 2

�kf + �fert

�(42)

So

3@�f (t)

@Sc=@��c � 2�f

�ert

@Sc

where

�f = e�rT��p� kf

��c =

1

2�f +

e�rt1��p+ kf � 2kc

�2

and where

3rSc(t) =��p+ kf � 2kc

� �rt1 � rt� 1 + ert�rt1

�(43)

3rSf (t) = �12

��p+ kf � 2kc

� �rt1 � rt� 1 + ert�rt1

�(44)

+3

2

��p� kf

� �rT � rt� 1 + ert�rT

�We start by writing

�c � 2�f = �32e�rT

��p� kf

�+e�rt1

��p+ kf � 2kc

�2

so

@��c � 2�f

�@Sc

= r3e�rT

2

@T

@Sc��p� kf

�� r

@t1@Sc

e�rt1��p+ kf � 2kc

�2

We get @T@Sc

and @t1@Sc

from (43) and (44)

3r =��p+ kf � 2kc

� �rt1 � rert�rt1

� @t1@Sc

20

and

0 = �12

��p+ kf � 2kc

� �rt1 � rert�rt1

� @t1@Sc

+3

2

��p� kf

� �rT � rert�rT

� @T@Sc

Therefore

0 = �32r +

3

2

��p� kf

� �rT � rert�rT

� @T@Sc

Or1

(T � ert�rT )=��p� kf

� @T@Sc

and3

(t1 � ert�rt1)=��p+ kf � 2kc

� @t1@Sc

Substituting into@(�c�2�f)

@Scgives

@��c � 2�f

�@Sc

= r3e�rT

2

1

(T � ert�rT )� r

3

(t1 � ert�rt1)

e�rt1

2

@��c � 2�f

�@Sc

=3r

2

�e�rT

(T � ert�rT )� e�rt1

(t1 � ert�rt1)

�@��c � 2�f

�@Sc

=3r

2

�1

(TerT � ert)� 1

(t1ert1 � ert)

�For any t we have

f (X) =1

(XerX � ert)

strictly decreasing in X and therefore@(�c�2�f)

@Sc6= 0 since T > t1. This implies that

the necessary condition for the CLNE to yield the OLNE extraction path is not met.

We have thus shown that for any equilibrium that reads C ! S ! F or S ! F

the necessary condition for the CLNE to yield the OLNE extraction path is not

met. Using the same approach, this result can be extended to any possible OLNE

equilibrium characterized in the previous section.

21

6 CLNE in the cartel-versus-fringe case

In this section we consider the limit case with a cartel�nl = nc = 1

�and the number of

fringe �rms tends to in�nity�nf !1

�. We �rst clarify what is meant by the cartel-

versus-fringe when �rms are allowed to use closed-loop strategies. The cartel chooses

a closed-loop strategy that gives an extraction rate at each moment as a function of

the stocks and time. Fringe �rms are assumed to be very small and therefore price

takers. However, it is unclear whether each �rm should be allowed to condition its

supply on the vector of stocks. In the existing literature, Groot et al. (2000 and

2003) consider fringe �rms that are price takers and assume that fringe �rms do not

condition their supply on the stock of the cartel (nor on any other stock). In order

to investigate the potential equivalence between open-loop and feedback Stackelberg

equilibrium they proceed as follows. The open-loop equilibrium depends on the initial

stock of the fringe and the cartel. This allows them to derive the extraction rate of

the cartel as a function of the existing stocks. This representation then serves as a

candidate for the feedback equilibrium. Whether this indeed constitutes a feedback

equilibrium is checked by inspection of the Hamilton Bellman Jacoby equation for the

cartel. In some cases it is found that the representation does not constitute a feedback

equilibrium. We apply the same procedure for the open-loop Nash equilibrium. It

involves the calculation of the marginal value of the cartel�s stock as well as the

fringe�s stock. It also involves the maximization of

e�rt[�p� qf (t)� qc � kc � ert@V c

@Sc]qc

where @V c

@Scis the marginal value, along the path, of the cartel�s stock. Suppose we are

in an S phase, where the (realized) price is given by p(t) = kf + ert�f ; for some �f :

Typically, it is found that the maximization entails an extraction rate qc of the cartel,

such that the resulting price is higher than kf + ert�f . Moreover, the HJB equation

is not satis�ed. This would then lead to a rejection of the claim that open-loop and

feedback coincide. However, it is natural to assume that the cartel knows beforehand

that it is contrained in setting the price. If it sets the price too high, the fringe will

supply the entire market and cartel�s supply will be zero. Hence, the maximization

should be a constrained maximization, of which the cartel can be assumed to be

aware. If this is taken into account we claim that the open-loop and feedback Nash

22

equilibrium coincide for all cost constellations. It should be noted that this result

does not hold for the Stackelberg equilibrium.

One can argue that the assumption on pure price taking is a modelling choice.

In other circumstances it might be appropriate to consider that fringe �rms have

information about all stocks and can condition their supply on that information. We

therefore also treat the case where fringe �rms are assumed to be price takers and can

condition their supply on the stocks. We only develop the case where �p+kc�2kf < 0.However a similar treatment can be replicated in all the other cases. A necessary

condition for a closed-loop Nash equilibrium to induce the open-loop production path

is@(�f(Sc(s);Sf (s);s))

@Sc= 0 along the OLNE paths of the stock. We show below that this

condition is not satis�ed.

The production path of the fringe along the OLNE, under simultaneous extraction

is then given by

qf (t) = �p+�kc + �cert

�� 2

�kf + �fert

�(45)

So@�f (t)

@Sc=@��c � 2�f

�ert

@Sc

where

�f = e�rT��p� kf

��c = �f + e�rt1

�kf � kc

�and where

rSc (t) =�kf � kc

� �r (t1 � t)� 1 + e�r(t1�t)

�(46)

r�Sf (t) + Sc (t)

�=��p� kf

� �r (T � t)� 1 + e�r(T�t)

�(47)

We start by writing

�c � 2�f = �e�rT��p� kf

�+ e�rt1

�kf � kc

�so

@��c � 2�f

�@Sc

= re�rT@T

@Sc��p� kf

�� r

@t1@Sc

e�rt1�kf � kc

�

23

We get @T@Sc

and @t1@Sc

from (46) and (47)

1 =�kf � kc

� �t1 � ert�rt1

� @t1@Sc

and

1 =��p� kf

� �T � ert�rT

� @T@Sc

Substituting @T@Sc

and @t1@Sc

into@(�c�2�f)

@Scgives

@��c � 2�f

�@Sc

= r

�1

TerT � ert� 1

t1ert1 � ert

�For any t, we have

f (X) =1

(XerX � ert)

strictly decreasing in X and therefore@(�c�2�f)

@Sc6= 0 since T > t1. This implies that

the necessary condition for the CLNE to yield the OLNE extraction path is not met.

7 Conclusions

We have fully characterized the open-loop Nash equilibrium in the case of �oil�igopoly

in a general case where �rms may have di¤erent extraction costs and initial stocks

of the resource. We have then used this equilibrium to derive the open-loop Nash

equilibrium for the cartel-versus-fringe model by considering one coherent cartel and

the number of fringe members going to in�nity.

When the number of �rms is �nite there exists no closed-loop Nash equilibrium

that replicates the extraction path of the open-loop equilibrium. Thus for the result

of Hartwick (2006) to hold, the assumption of quadratic extraction costs is essential,

and in Groot et al. (2003), it is the hierarchical play assumed that is essential for

the open-loop stackelberg equilibrium to be strongly time consistent. Moreover, in

contrast with the case with hierarchical play we have shown that when the cartel and

the fringe play simultaneously, there always exists a phase where both players act

simultaneously. When the fringe �rms cost disadvantage is large enough the cartel

exhausts its stock before the fringe regarless of its initial stock. If its initial stock is

large enough the cartel is, for an initial period time, the sole supplier in the market.

24

When the cost di¤erence is not large enough, the equilibrium starts with both type

of players supplying simulatneously regardless of the level of the stocks. In the limit

case of an in�nite number of fringe �rms we claim that the open-loop Nash and the

feedback Nash equilibrium coincide.

The characterization of an equilibrium in this more general context brings us closer

to more realistic modelling of the complex reality of the oil market and nonrenewable

resource markets in general. This full characterization can be exploited in future

research to study implications of di¤erent policies such as competition policies or

trade policies.

References

[1] Eswaran, M. and Lewis, T. (1985), �Exhaustible resources and alternative equi-

librium concepts�, The Canadian Journal of Economics 18, pp. 459-473.

[2] Gilbert, R. (1978), �Dominant �rm pricing policy in a market for an exhaustible

resource�, Bell Journal of Economics 9, pp. 385-395.

[3] Groot, F., Withagen, C. and A. de Zeeuw (1990), �Theory of natural exhaustible

resources: the cartel-versus-fringe model reconsidered�, Center Discussion Paper

8924, Tilburg University.

[4] Groot, F., Withagen, C. and A. de Zeeuw (1992), �Note on the open-loop von

Stackelberg equilibrium in the cartel versus fringe model�, The Economic Journal

102, pp. 1478-1484.

[5] Groot, F., Withagen, C. and A. de Zeeuw (2000), "Open-loop von Stackelberg

equilibrium in the cartel-vs.-fringe model", Energy Economics 22, pp. 209-233.

[6] Groot, F., Withagen, C. and A. de Zeeuw (2003), "Strong time-consistency in

the cartel-versus-fringe model", Journal of Economic Dynamics and Control 28,

pp.287-306.

[7] Hartwick, J. (2006), "The quadratic oil extraction oligopoly", Working Paper,

Queen�s University, Kingston Canada.

25

[8] Lewis, T. and Schmalensee, R. (1980a), �On oligopolistic markets for non-

renewable natural resources�, Quarterly Journal of Economics 95, pp. 475-491.

[9] Lewis, T. and Schmalensee, R. (1980b), �Cartel and oligopoly pricing of non-

replenishable natural resources�, in P. Liu (ed.), Dynamic Optimization and Ap-

plication to Economics, Plenum Press, New York, pp. 133-156.

[10] Loury, G. (1986), "A theory of oiligopoly: Cournot equilibrium in exhaustible

resource markets with �xed supplies", International Economic Review 27, pp.

285-301.

[11] Polasky, S. (1990), "Exhaustible resource oligopoly: Open-loop and Markov per-

fect equilibria", Boston College Working Paper 199.

[12] Polasky, S. (1992), "Do oil producers act as �Oil�igopolists?", Journal of Envi-

ronmental Economics and Management 23, pp. 216-247.

[13] Salant, S. (1976), �Exhaustible resources and industrial structure: a Nash-

Cournot approach to the world oil market�, Journal of Political Economy 84,

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[14] Salant, S. (1982), �Imperfect competition in the international energy market: a

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Appendix A: Proof of Lemma 4

We have, from (8)

p(T ) =1

nl + 1

��p+ nl

�kl + �lerT

��= �p

so that

26

kl + �lerT = �p (48)

At the transition we have from (15)

�p+ nl�kl + �lert1

�=�nl + 1

� �kh + �hert1

�(49)

It follows from straightforward integration of (13) that

nl + nh + 1

nhrSh0 =

��p+ nlkl �

�nl + 1

�kh�rt1 +

�nl�l �

�nl + 1

��h� �ert1 � 1

�(50)

Equations (48) and (49) imply

�l = e�rT��p� kl

�nl�l �

�nl + 1

��h = �e�rt1

��p+ nlkl �

�nl + 1

�kh�

Hence, after substitution into (50) we obtain after simpli�cation (24).

We now derive (25). Integration of ql from (12) yields

t1Z0

rql(s)ds =nl

nl + nh + 1

���p+ nhkh �

�nh + 1

�kl�rt1 +

�nh�h �

�nh + 1

��l� �ert1 � 1

�and

TZt1

rql(s)ds =nl

nl + 1

���p� kl

�(rT � rt1)� �l

�erT � ert1

�Summing up the two cumulative extractions gives

t1Z0

rql(s)ds+

TZt1

rql(s)ds = rSl0 =nl

nl + nh + 1

��p+ nhkh �

�nh + 1

�kl�rt1

+nl

nl + nh + 1

�nh�h �

�nh + 1

��l� �ert1 � 1

�+

nl

nl + 1

���p� kl

�(rT � rt1)� �l

�erT � ert1

�27

Hence

rSl0 =nl

nl + nh + 1

��p+ nhkh �

�nh + 1

�kl�rt1

+nl

nl + nh + 1

�nh�h �

�nh + 1

��l� �ert1 � 1

�+

nl

nl + 1

��p� kl

�(rT � rt1)�

nl

nl + 1�l�erT � ert1

�Substituting in the last term �l from �p�

�kl + �lerT

�= 0 yields

rSl0 =nl

nl + nh + 1

��p+ nhkh �

�nh + 1

�kl�rt1 +

nl

nl + nh + 1

�nh�h �

�nh + 1

��l� �ert1 � 1

�+

nl

nl + 1

��p� kl

�(rT � rt1)�

nl

nl + 1

��p� kl

�e�rT

�erT � ert1

�Factorise nl

nl+1

��p� kl

�in the last two terms

rSl0 =nl

nl + nh + 1

��p+ nhkh �

�nh + 1

�kl�rt1 +

nl

nl + nh + 1

�nh�h �

�nh + 1

��l� �ert1 � 1

�+

nl

nl + 1

��p� kl

� �rT � 1 + ert1�rT � rt1

�Adding and substracting e�rT in the last term

rSl0 =nl

nl + nh + 1

��p+ nhkh �

�nh + 1

�kl�rt1 +

nl

nl + nh + 1

�nh�h �

�nh + 1

��l� �ert1 � 1

�+

nl

nl + 1

��p� kl

� �rT � 1 + e�rT +

��1 + ert1

�e�rT � rt1

�Multiplying both sides by nl+nh+1

nland splitting the last term

nl + nh + 1

nlrSl0 =

��p+ nhkh �

�nh + 1

�kl�rt1 +

�nh�h �

�nh + 1

��l� �ert1 � 1

�+nl + nh + 1

nl + 1

��p� kl

� ���1 + ert1

�e�rT � rt1

�+nl + nh + 1

nl + 1

��p� kl

� �rT � 1 + e�rT

�Note that the last term only includes T and not t1 and corresponds to the last term

in (25). Algebric manipulation of the �rst three terms yields (25).

Appendix B:

28

We show in this appendix that along the curve�Sl0; S

h0

�where qh(0) = 0 we have

dSh0dSl0

= �@qh(0)

@Sl0@qh(0)

@Sh0

> 0

We use the fact that

nl + nh + 1

nhqh(0) = �p+ nlkl �

�nl + 1

�kh + nl�l �

�nl + 1

��h (51)

where �h = e�rT��p� kh

�and

�l =nh

nh + 1�h +

e�rt1��p+ nhkh �

�nh + 1

�kl�

nh + 1(52)

We �rst determine @qh(0)

@Sh0: We have

nl + nh + 1

nh@qh(0)

@Sh0= nl

@�l

@Sh0��nl + 1

� @�h@Sh0

(53)

substituting �l gives

nl + nh + 1

nh@qh(0)

@Sh0= nl

@

�nh

nh+1�h +

e�rt1(�p+nhkh�(nh+1)kl)nh+1

�@Sh0

��nl + 1

� @�h@Sh0

(54)

using @t1@Sh0

= 0 gives

nl + nh + 1

nh@qh(0)

@Sh0= �n

l + nh + 1

nh + 1

@�h

@Sh0(55)

where �h = e�rT��p� kh

�that is @�h

@Sh0= �re�rT

��p� kh

�@T@Sh0

. We use (28) to deter-

mine @T@Sh0:

nh + 1

nhr =

�1� e�rT

�r��p� kh

� @T@Sh0

> 0 (56)

and subtitute into (55) to obtain

@qh(0)

@Sh0= re�rT > 0: (57)

We now determine @qh(0)

@Sl0. We have

nl + nh + 1

nh@qh(0)

@Sl0= nl

@�l

@Sl0��nl + 1

� @�h@Sl0

(58)

29

Subtituting �l gives

nl + nh + 1

nh@qh(0)

@Sl0= nl

@

�nh

nh+1�h +

e�rt1(�p+nhkh�(nh+1)kl)nh+1

�@Sl0

��nl + 1

� @�h@Sl0

(59)

or

nl + nh + 1

nh@qh(0)

@Sl0= �n

l + nh + 1

nh + 1

@�h

@Sl0� nlr

e�rt1��p+ nhkh �

�nh + 1

�kl�

nh + 1

@t1@Sl0(60)

with @�h

@Sl0= �re�rT

��p� kh

�@T@Sl0.

From (26) and (28) we obtain

nl + nh + 1

nl1

(1� e�rt1)r = r

��p+ nhkh �

�nh + 1

�kl� @t1@Sl0

(61)

and

1 =�1� e�rT

� ��p� kh

� @T@Sl0

(62)

Substituting both terms into (60) gives

nl + nh + 1

nh@qh(0)

@Sl0=nl + nh + 1

nh + 1re�rT

1

(1� e�rT )�nl e�rt1

(nh + 1)

nl + nh + 1

nl1

(1� e�rt1)r

(63)

or@qh(0)

@Sl0=

nh

nh + 1r

�e�rT

1� e�rT� e�rt1

1� e�rt1

�< 0 (64)

We have thus shown that

dSh0dSl0

= �@qh(0)

@Sl0@qh(0)

@Sh0

= � nh

nh + 1erT

�1

erT � 1 �1

ert1 � 1

�> 0:

30