Productive Efficiency and Contestable Markets

Jean-Pierre PonssardCNRS and Ecole Polytechnique,

Paris, Francemail address: jean-pierre.ponssard@polytechnique.edu

Revised December 2005

Abstract

This paper provides a new game theoretic model consistent with thepremises of contestable markets. Two firms repeatedly compete for anatural monopoly position. The limit price of the incumbent is disciplinedby a hit and run strategy of the entrant. In this model, contrarily to thewell known Maskin and Tirole model (1988): i) productive efficiency isencouraged, the more efficient firm gets a higher rent as an incumbentthan the one the less efficient firm would, ii) rent dissipation does notnecessarily prevails, even in the case of equally efficient firms. This opensthe way to a reassessment of the merits of contestable markets.

JEL classification: C73, D43, L10.Key words: contestable markets, productive efficiency, limit pricing,

Markov strategies.

1 Introduction1

This paper investigates if and how competition for a natural monopoly positionleads to productive efficiency and price discipline. This subject first appeared inthe limit pricing literature (Gaskins, 1971). Renewed attention occurred withcontestable markets (Baumol, Panzar and Willig, 1982). It is an importantantitrust issue in the cases of concentrated market structures.Such situations may be modelled as repeated entry games. A proper selection

process has to be introduced to eliminate collusive equilibria and to focus theattention on the more competitive ones. The main contribution in this areahas been made by Maskin and Tirole (1988), the selection process is based ona Markov hypothesis.

1 Special thanks are due to Rida Laraki, Robert Wilson, an associate editor and the refereesfor significant help in improving the content and presentation of this paper. I am also gratefulto Yves Balasko, Claude Henry, Laurent Linnemer, Michael Riordan, John Sutton, NicolasVieille and audiences of seminars at Core (Louvain), Columbia (New York), Cirano (Montreal),CMM (Santiago de Chile), Etape (Paris), Roy (Paris), Stanford, Hec Paris, Idei (Toulouse),IHP (Paris), LSE (London) and SSE (Stockholm) for valuable comments.

1

However, Maskin and Tirole did not explicitly address the productive effi-ciency issue, their paper only concerns equally efficient firms. When appliedto firms with different productive efficiencies, the Markov approach behavesstrangely (Lahmandi, Ponssard and Sevy, 1996). To see this, observe that todeter entry a firm has to limit its instantaneous profit and, the less efficient itis, the more severe the limitation, since this limitation is based on the profitfunction of the other firm and not on its own. In a finite horizon game, if thegame is long enough, entry deterrence induces the less efficient firm to be un-profitable, thus to give up its incumbency position (Gromb, Ponssard and Sevy,1997). The Markov approach, because of its circularity in an infinite horizon, issuch that an inefficient firm may remain a permanent incumbent. Furthermore,an increase in productive efficiency in favor of the incumbent leads to a lowerrent.This paper proposes a new approach directly based on the premises of con-

testable markets: firms differ in productive efficiencies, there is an incumbentand the entrant uses a hit and run strategy. It is proved that productive effi-ciency is encouraged. It is also proved that, contrarily to the celebrated rentdissipation property, this may not necessarily prevail in this model. It dependson the economic characteristics of the natural monopoly situation. This is con-sistent with a number of comments on the limits of potential competition. Ataxonomy is proposed to sum up the results and to revisit the merits of con-testable markets in the light of this model.The paper is organized as follows. Section 2 reviews the Markov approach,

the associated literature and provides the motivation for the new model. Theinfinitely repeated entry game under analysis is introduced in section 3. Thesubset of perfect Nash equilibria to be selected as its solution is precisely de-fined in section 4. The main properties of the selected equilibria are studied insection 5. Section 6 discusses their economic properties. The concluding sectiondiscusses limitations and further research.

2 Review of the literature and motivationPotential competition is ordinarily analyzed under the following framework:- the firms maximize their discounted profit over a long period of time,

possibly infinite- time is discrete and divided into stages- the economic context is a natural monopoly; within a stage, if more than

one firm are active, i.e. produce, the stage profits of at most one active firmmay be strictly positive due to high fixed costs; in this context, it is enough toconsider competition among two firms- competition takes the form of short run commitments; this embeds two

ideas: first, at any point of time, a firm can only commit to a short period and,second, during that short period, the other firm can react; this also means thatthe discount rate between stages is close to zero- the two firms may have different characteristics, one being strong and the

2

other weak, for instance the fixed costs may differ; when firms have identicalcharacteristics the are said to be symmetric- the proposed solution should be a subset of perfect Nash equilibria (which

are expected to be many)- the solution should be tested with regards to economic properties such as

rent dissipation and selection (productive efficiency).This general framework involves designing a game form, which depends on

the underlying situation featuring capacity, quantity, price, renewal policy com-petition ..., and a solution concept that should be general enough to apply toall relevant game forms.This section reviews the previous works on this subject, points out the diffi-

culties encountered so far and provides a preview of the results that will followin the paper.The discussion is carried out using an illustrative example. Price is the

decision variable and noted p ∈ [0, 1]. The pure monopoly profit is v(p) =p(1 − p) − f. The fixed cost f is incurred at each stage but only in case thefirm is active on the market. It is not a set up cost incurred once for all at thebeginning of the game. Firms differ only in their fixed cost with f1 ≤ f2, sothat firm 1 is more efficient than firm 2.Prices are set simultaneously. For a given price pj of firm j the demand

function of firm i is kinked using a positive constant switching cost denoted s.It is the pure monopoly demand 1 − pi on the range [0, pj − s], it is zero onthe range [pj + s, 1] and it decreases linearly on the range [pj − s, pj + s]. Itis easily checked that, as long as 3s ≤ pj , the best response of firm i is eitherpi(pj) = pj − s, in which case it is active, or pi(pj) ≥ pj + s, in which case itis inactive. It is assumed all along that the condition 3s ≤ pj holds for bothfirms on the relevant range of analysis: the fixed costs are relatively high andthe price competition is tough.Average cost pricing is characterized as the lower respective prices such that

vi(pi) = 0. Theses prices are denoted pac1 and pac2 . Observe that pac1 ≤ pac2 .

2.1 Static competition and selection

In a one stage game, any price p1 ∈ [pac1 , pac2 + s] is such that the best responseof firm 2 is to be inactive so that firm 1 static limit price is pac2 +s. Denote p

max1

this limit price. Similarly denote pmax2 the static limit price of firm 2. As longas pac2 ≤ pmax2 both (pmax1 , pmax1 + s) and (pmax2 , pmax2 + s) are the pure strategyNash equilibria associated with the usual notion of static limit pricing.But, if the difference in efficiency between the two firms is high enough,

pac2 > pac1 + s. Entry of the less efficient firm is blockaded. Static competitioninduces a selection process. Suppose it does not. The key question addressedin this paper concerns the existence and nature of a dynamic selection processin an infinitely repeated game.Note that this perspective is different from the dilemma between strategic

entry barrier versus accommodation ordinarily discussed in one shot Stackelberggames (Dixit, 1980).

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2.2 The Maskin and Tirole approach

In their 1988 paper, Maskin and Tirole analyze dynamic competition with largefixed costs in an infinite horizon. The proposed solution is based on a Markovapproach in which the state variable is the move to which the other playeris currently committed. Though the example discussed in that paper refersto quantity competition, as mentioned by the authors, the approach appliesas well to the case of price competition. Note also that an alternating movesituation taken from Cyert and DeGroot (1970) is used. The proposed Markovconstruction can be extended to other game forms. More importantly, Maskinand Tirole only consider the case f1 = f2.Let the discount factor between stages be denoted δ. Let p∗ be the unique

solution (with p∗ < 1/2, 1/2 is the unrestricted monopoly price) to the equationin p

v(p− s) + δv(p)/(1− δ) = 0

The Markov equilibrium strategies when δ is close to 1 are as follows:- along the equilibrium path only one firm, say firm i, is active and it selects

pi = p∗, it remains active all the time

- the reaction function of the outsider, say firm j 6= i, is such that if pi > p∗then pj =Min(pi − s, p∗) but if pi ≤ p∗ then pj = pi + s.This solution has many good economic properties. The exercise of monopoly

power by the incumbent is disciplined by potential competition. The equationthat defines the equilibrium path is easy to interpret: the first term may be seenas an entry cost that has to be recovered by a discounted future stream of profitsas an active firm, while preventing further re-entry. From an economic stand-point it is reminiscent of the limit pricing literature (Gaskins, 1971, Kamienand Schwartz, 1971, and Pyatt, 1971). This equation is known as the key recur-sive equation of dynamic entry games (Wilson, 1992). Eaton and Lipsey (1980)and Farrell (1986) had already derived similar equations through intuitive ar-guments based on rational expectations. The Markov assumption provides arigorous game analysis.As can be seen from this equation, it must be that v(p∗) → 0 as δ → 1,

which means that, as the length of the short term commitment decreases, theinstantaneous profit goes to zero. This is the celebrated rent dissipation property(Fudenberg and Tirole, 1987) which confirms the heuristic stories of Grossman(1981) and Baumol, Panzar and Willig (1982) on limit properties of contestablemarkets.However, the theory of contestable markets is mostly concerned about pro-

ductive efficiency, i.e. the selection of the most efficient firm through the com-petitive process. The Markov approach is totally inadequate to address to issue.Lahmandi et al. (1996) applied this approach to the case of asymmetric firms.The equilibrium now depends on the two prices p∗1 and p

∗2 (with p

∗i ≤ 1/2) which

solve a system of two equations:

v1(p2 − s) + δv1(p1)/(1− δ) = 0

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v2(p1 − s) + δv2(p2)/(1− δ) = 0

The Markov equilibrium strategies are as follows:- along the equilibrium path only one firm, say firm i, is active and it selects

pi = p∗i , it remains active all the time

- the reaction function of the outsider, say firm j 6= i, is such that if pi > p∗ithen pj =Min(pi − s, p∗j ) but if pi ≤ p∗i then pj = pi + s.In other words, the solution is almost not affected by the introduction of

efficiency differences among the two firms. Surprisingly, rent dissipation stillprevails as δ → 1. But the most striking feature of this approach is that the lessefficient firm may indefinitely remain as a permanent incumbent and that, for δclose to 1, its incumbency rent is greater than the one of the more efficient one.This is easy to prove analytically. Using the system of equations that defines p∗1and p∗2 one gets:

δ(v2(p∗2)− v1(p∗1)) = (1− δ)(v1(p

∗2 − s)− v2(p∗1 − s))

= (1− δ)((p∗2 − p∗1)(1 + 2s− (p∗2 + p∗1)) + f2 − f1)

which is strictly positive since p∗2 > p∗1 (for δ close to 1, p∗i is close to paci and

pac2 > pac1 ) and p∗i < 1/2.

This result can be illustrated numerically (for reference, if f1 = f2 =.15, p∗i = .186, vi(p

∗i ) = .0013):

s = .02 and δ = .9 efficient firm i = 1 inefficient firm i = 2fi .15 .152paci .184 .187pmaxi .207 .204p∗i .185 .189

vi(p∗i ) .0010 .0016

This is in contradiction with economic intuition: rent should increase withrelative efficiency and, if the more efficient firm cannot obtain the incumbencyposition from the competitive process, it should at least be able to profitablybuy it from the less efficient firm.It is also in contradiction with a finite horizon approach. In Gromb et al.

(1997), it is proved that, however small the difference in efficiency, as long as thediscount factor is close to 1, selection always prevails in a long enough game.Indeed, the entry preventing strategy of the less efficient firm is recursivelydetermined by the cumulative profit the more efficient firm would make as anincumbent. The cumulative profit of the less efficient firm eventually decreasesbelow zero. At that stage, the less efficient firm is blockaded from entry as insection 2.1. The Markov assumption creates a circularity in the infinite horizongame which is not consistent with a finite horizon approach.

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2.3 The approach followed in this paper

The key idea is to assume that the less efficient firm may not be able to deterentry for ever. It plays an explicit hit and run strategy. This can be illustratedusing the same numerical example, except that δ = 1 to make things morereadable.The proposed equilibrium is detailed in the following table

s = .02 and δ = 1 i = 1 i = 2time dependance stationary t = 4 t = 3 t = 2 t = 1 t = 0

pti .193 .204 .195 .186 .178 .170vi(p

ti) .0057 .010 .005 .000 -.006 -.011

vj(pti − s) -.009 .000 -.0057 -.0114 -.0171

Σt=4t=1(v2(pt2)) .010 .015 .015 .009 -.002

The more efficient firm (firm 1) uses the stationary strategy p1 = .193. Ifthat firm were to set a slightly higher price, say p01, the less efficient firm (firm2) would ”hit”, i.e. move in setting its price at p01 − s and then ”run” for fourstages setting its price respectively at .178, .186, .195 and finally at .204 (itsstatic limit price). Since p01 > .193 it must be that v2(p01 − s) > −.009, butΣt=4t=1(v2(p

t2)) = .009, this hit and run strategy is profitable.

Now, once the less efficient firm starts running, the best response of the moreefficient firm is to wait until it goes away. Indeed, along the path, for all t, onegets v1(pt2−s) = −(4− t)v1(.193) = −.0057(4− t). Moreover, firm 1 knows thatfirm 2 is running away along a four stage path since firm 2 cannot profitablydeter entry along a 5 stage path (Σt=4t=0(v2(p

t2)) = −.002).

This paper establishes the basic properties of this equilibrium and exploresits most direct economic properties. It will be proved that productive efficiencyis encouraged. An interesting taxonomy of natural monopoly situations willemerge.

3 The repeated entry game Γδ∞

The game Γδ∞ involves two players, player 1 and player 2. Γδ∞ is the infiniterepetition of a stage game G. They maximize the sum of their discounted payoffsusing δ as the discount factor (0 < δ < 1). The attention is focused on δ closeto one.

3.1 The stage game G

The stage game G is a generalization of the illustration used in section 2. Recallthe notations. One player is denoted i ∈ {1, 2} and the other one j ∈ {1, 2}, j 6=i. A move of player i in G is a real number pi ∈ Ii where Ii =] −∞, ai] withai to be defined later on. This move may be interpreted as a price. The profitfunctions are denoted πi(pi, pj).The players move simultaneously and are restricted to pure strategies.

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Assumption 1 (natural monopoly): It is assumed that πi be increasing inpj and that if πi > 0 then πj ≤ 0.If πj(pi, pj) = 0 and if for all p0j ≥ pj, πj(pi, p0j) = 0, player j is said not to

be active. Otherwise player j is said to be active.For any pi define p

+j (pi) as the minimal pj for which player j is not active

that is:

p+i (pi) =Min{pj |for all p0j ≥ pj ,πj(pi, p0j) = 0}

Denote vi(pi) the pure monopoly profit function for player i. It writes:

vi(pi) = πi(pi, p+i (pi))

The move associated to average cost pricing is denoted paci . It is the smallestsolution to vi(pi) = 0.For any pi define p

−j (pi) as the best entry move of player j that is:

p−j (pi) = Arg{Maxpj<p+i (pi)πj(pi, pj)}

Denote Cj(pi) the entry cost function for player j. It writes:

Cj(pi) = −πj(pi, p−j (pi))

The one stage limit strategy is denoted pmaxi . It is such that Cj(pmaxi ) = 0.To make the analysis interesting, it is assumed that, for i ∈ {1, 2}, paci ≤

pmaxi .Assumption 2 (monotonicity): The functions p+i (.) and p

−j (.) are assumed

to be well defined and strictly increasing. The functions vi(.) are strictly increas-ing and the functions Cj(.) are strictly decreasing.This assumption simplifies the analysis in the sense that border conditions

need not be considered, otherwise the analysis may be more technical.Assumption 3 (Nash equilibria of G): Any pair of strategies (pi, p+j (pi))

such that paci ≤ pi ≤ pmaxi is a Nash equilibria of the game G.Take any p+j (pi) in which p

aci ≤ pi ≤ pmaxi , pi is a best response to p

+j (pi),

hence p−i (p+j (pi)) = pi, and so Ci(p

+j (pi)) = −vi(pi).

In the example used in section 2, as long as 3s ≤ pi, pi is the best entrymove to p+j (pi) and Ci(p

+j (pi)) = −vi(pi) whatever pi.

Here, contrarily to that example, it is not assumed that this relation remainsnecessarily true for pi < paci . The best entry move may not consist in excludingthe other player from the market. Yet, the monotonicity of Ci(.) implies that, ifpj < p

maxj , Ci(pj) > Ci(p

maxj ) = 0. Then, if player i selects pi < paci and player

j selects p+j (pi) < pmaxj , the best response of player i to p+j (pi) is certainlyunprofitable for player i. This assumption implies that competition remainsrelatively tough.In some applications it may be possible to relax assumption 3, see the con-

cluding comments.

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Assumption 4 (technical assumption): For mathematical convenience it isassumed that the functions vi and Ci have derivatives and that these derivativesare uniformly bounded away from zero and from infinity.In fact it may be that the functions Ci only have left and right hand size

derivatives when the respective best entry move switches from excluding theother player to allowing a duopoly. This will not change the result.The intervals Ii may now be precisely defined as Ii =]−∞, p+i (pmaxj )]. Note

that assumption 2 implies that p+i (pmaxj ) is lower than the unrestricted monopoly

price of player i.These assumptions allow the analysis to be carried on only with the functions

vi and Ci, except for theorem 3 in which assumption 3 is explicitly used.

4 The selected set of perfect equilibriaThe game Γδ∞ is a standard repeated game, the folk theorem applies and its setof perfect Nash equilibria is extremely large. The selection process introducedin section 2 is now formalized to focus on competitive equilibria consistent withthe premises of contestable markets.The two players are arbitrarily distinguished as a long term player, denoted

as player L, and a short term player, denoted as player S. At this point noassumption is made regarding the relative efficiency of the two players. PlayerL may be seen as the incumbent and player S as an entrant playing a hit andrun strategy.Given an integer n, when they exist, define the real number qL ∈ IL and the

sequence (qtS)t=nt=0 in IS that solve the following system to be denoted Φnδ :

for t ∈ {0, 1, 2...n} CL(qtS) = δ(1− δn−t)vL(qL)/(1− δ) (1)

CS(qL) = Σt=nt=1 δ

tvS(qtS) (2)

vL(qL) ≥ 0 (3)

for t ∈ {1, 2...n} Σt0=nt0=t δ

t0−tvS(qt0S ) ≥ 0 (4)

Σt0=nt0=0 δ

t0vS(qt0S ) < 0 (5)

The fact that the functions Ci(.) be strictly decreasing ensures the followinglemma.

Lemma 1 If Φnδ has a solution, the corresponding sequence (qtS)t=nt=0 is strictly

increasing in t.

Let H be the set of strategy pairs in G defined as

H = {(qL, p+S (qL)), (qtS , p+L(qtS))t=nt=1}.

To simplify notations, a pair h of H is identified by its first element that is,the active player and his move. Then

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H = {qL, (qtS)t=nt=1}.

Definition 2 (The selection process) The selected strategies consist of sequences(hk)k=∞k=1 in H such that:Suppose hk = qL and suppose that (pL, pS) is observedif pL = qL for all pS ≥ p+S (qL) hk+1 = qL

for all pS < p+S (qL) hk+1 = q1S

if pS = p+S (qL) for all pL > qL hk+1 = q1S

for all pL ≤ qL hk+1 = qLif pL 6= qL and pS 6= p+S (qL) hk+1 = qLSuppose hk = qtS and suppose that (pL, pS) is observedif pS = qtS for all pL ≥ p+L(qtS) for all t < n hk+1 = qt+1S

if t = n hk+1 = qLfor all pL < p

+L(q

tS) for all t hk+1 = qL

if pL = p+L(q

tS) for all pS > qtS for all t hk+1 = qL

for all pS ≤ qtS for all t < n hk+1 = qt+1S

if t = n hk+1 = qLif pS 6= qtS and pL 6= p+L(qtS) for all t < n hk+1 = qt+1S

if t = n hk+1 = qL

Suppose the players never deviate, whatever h1, hk = qL for all k > n, playerL is active indefinitely.For t ∈ {1, ..n} denote Ht

S the sequence which starts with h1 = qtS and HL

the sequence in which h1 = qL.Consider a deviation in HL at some stage k. If player L is too ”greedy

”(pL > qL) or if player S is too ”tough” (pS < p+S (qL)) player S becomes the

active player at the next stage that is, hk+1 = q1S and the game is expected togo on as in H1

S .This is also true on the initial part of any path Ht

S , for k < n − t + 1,interchanging the roles of the players, with hk+1 = qL and the game beingexpected to go on as in HL.Denote by VL(.) and VS(.) the respective discounted payoffs of the two play-

ers as a function of H. It is easily seen that:

VL(HL) = vL(qL)/(1− δ)

for t ∈ {1, 2...n} VL(HtS) = δn−t+1VL(HL)

VS(HL) = 0

for t ∈ {1, 2...n} VS(HtS) = Σ

t0=nt0=t δ

t0−tvS(qt0S )

Conditions (3) and (4) of Φnδ ensures that these payoffs are non negative.

Theorem 3 For any h1 in H the strategy pair generated by the selection processis a perfect Nash equilibrium of Γδ∞.

Proof. Take h1 = qtS . The selected strategies generate the path HtS .

Consider a deviation from player L.

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If pL > p+L(q

tS) his payoff is not affected.

If pL < p+L(q

tS) it should be that πL(q

tS , pL) + δVL(HL) ≤ VL(Ht

S).Start with t < n, one may write:

δVL(HL) = δvL(qL)/(1− δ)

= δ(1 + ..+ δn−t−1 + δn−t...)vL(qL)= δ(1− δn−t)vL(qL)/(1− δ) + δn−t+1vL(qL)/(1− δ)

Condition (1) of Φnδ gives δVL(HL) = CL(qtS) + VL(H

tS). Since pL < p

+L(q

tS)

player L is active and his payoff is less or equal to the one associated to his bestentry move so that πL(qtS , pL) ≤ πL(q

tS , p−L (q

tS)) = −CL(qtS). Hence πL(qtS , pL)+

δVL(HL) ≤ VL(HtS).

If t = n, on the one hand, δVL(HL) = VL(HnS ) and on the other hand,

πL(qnS , pL) ≤ πL(q

nS , p−L (q

nS)) = −CL(qnS) = −CL(pmaxS ) = 0.

Consider a deviation from player S.If pS > qtS , it should be that πS(pS , p

+L(q

tS)) ≤ VS(Ht

S). If πS(pS , p+L(q

tS)) is

zero, the inequality holds since 0 ≤ VS(HtS). If it is not, it is certainly lower than

what he would get with his best entry move: πS(p−S (p

+L(q

tS)), p

+L(q

tS)). If p

acS ≤

qtS , by assumption 3 qtS is the best response to p

+L(q

tS), player S instantaneous

profit decreases. Since the sequence (qt0S ) is increasing, p

acS ≤ qtS implies vS(qtS) ≤

VS(HtS). The inequality holds. If q

tS < p

acS , assumptions 2 and 3 imply that the

entry cost associated to p+L(qtS) be positive: πS(p

−S (p

+L(q

tS)), p

+L(q

tS)) < 0. The

inequality holds.If pS < qtS , assumption 2 implies that his instantaneous payoff decreases,

while his continuation payoff is not affected.Similar arguments hold whatever h1 in H.An equilibrium obtained through the selected process is denoted a SPE(n)

which stands for a ”selected perfect equilibrium for a given n”.

5 The main properties of the selected set of per-fect equilibria and a further refinement

Some preliminary comments are in order.Our attention is focused on the case δ close to 1. Let Φn1 be the limit of Φ

nδ

as δ goes to 1. It is easily seen that the system Φn1 is

for t ∈ {0, 1, 2...n} −CL(qtS) + (n− t)vL(qL) = 0 (10)−CS(qL) +Σt=nt=1vS(q

tS) = 0 (20)

vL(qL) ≥ 0 (30)for t ∈ {1, 2...n} Σt

0=nt0=t vS(q

t0S ) ≥ 0 (40)

Σt0=nt0=0vS(q

t0S ) < 0 (50)

It may be useful to develop some intuition about the solution of system Φn1 tofollow the mathematical construction. Start with some pL so that vL(pL) > 0.Condition (1’) generates a strictly increasing sequence (ptS)

t=nt=t0 backwards from

an initial pnS = pmaxS . The expression Σt=nt=t0vS(ptS) is bell shaped and there is

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a t0 = t∗ such that condition (2’) almost holds (the adopted notation impliesthat the initial value of n has to be scaled down to n− t∗). A small change inpL results in a big change in Σt=nt=1 vS(p

tS) so that condition (2’) may be fixed.

Condition (4’) will hold by construction but condition (5’) may not. If it does,theorem 4 below proves that the solution is unique. If it does not, Φn1 hasno solution. In this construction, the smaller vL(qL), the closer qL to pacL , thelarger the value of n and the greater CS(qL). This gives theorem 5. If Φn1 hasa solution whatever the value of n, as n goes to infinity, the limit of vL(qL)goes to zero. Suppose this is the case and suppose that CL(p) = 1 − p, thenqt+1S − qtS = vL(qL) so that conditions (4’) and (5’) may be seen as

R qnSq0SvS(p)dp

andR qnSq1SvS(p)dp and so q0S and q

1S converges to p

∗ solution ofR 1p∗ vS(p)dp = 0.

This condition is restated in general terms in theorem 6.A technical difficulty arises regarding the fact that the limit solution of Φnδ

may not be the solution of Φn1 . Theorem 4 says that if δ is close enough to 1, ifΦnδ has a solution it is necessarily unique. This is also true for Φ

n1 . Because Φ

nδ

is continuous in δ, its unique solution when δ goes to 1 converges to the uniquesolution of Φn1 . The properties of the solution of Φ

n1 may then be used to infer

the limit properties of the solution of Φnδ .To avoid ambiguity, the respective solutions of Φnδ and Φ

n1 , when they exist,

are denoted as qL(n, δ) and (qtS(n, δ))t=nt=0 , on the one hand, and qL(n) and

(qtS(n))t=nt=0 , on the other hand.

Theorem 4 Φnδ admits at most one solution for large enough n and δ closeenough to 1. This is also true for Φn1 .

Proof of this theorem is in the appendix.

Theorem 5 If there exists a solution respectively in Φmδ and in Φnδ with m > nthen qL(m, δ) ≤ qL(n, δ).Proof. By contradiction. Suppose qL(m, δ) > qL(n, δ) then vL(qL(m, δ)) >

vL(qL(n, δ)). Since CL is strictly decreasing this implies for all t ∈ {0, 1, 2...n}:qm−tS (m, δ) < qn−tS (n, δ)

so that

Σt=0t=nδn−tvS(qm−tS (m, δ)) < Σt=0t=nδ

n−tvS(qn−tS (n, δ))

For t ∈ {n+ 1, ..m} we still have qm−tS (m, δ) < q0S(n, δ) and, because of (5)we also certainly have vS(q0S(n, δ)) < 0 then

Σt=0t=m−1δm−1−tvS(qm−tS (m, δ)) ≤ Σt=0t=nδ

n−tvS(qm−tS (m, δ))

Then

Σt=0t=m−1δm−1−tvS(qm−tS (m, δ)) < Σt=0t=nδ

n−tvS(qn−tS (n, δ))

By construction the left hand side should be greater or equal to zero whilethe right hand side should be strictly negative thus a contradiction.

11

Theorem 6 If δ < 1,∃nδ such that ∀n > nδ the system Φnδ has no solution. Ifδ = 1 and if the system Φn1 has a solution for all values of n then:

limn→∞ vL(qL(n)) = 0

limn→∞Σ

n1vS(q

tS(n)) = CS(p

acL )

limn→∞ q

1S(n) = p

∗

in which p∗ is uniquely defined as:Z pmaxs

p∗vS(p)

dCLdp

(p)dp = 0

Proof of this theorem is in the appendix.

Theorem 7 If there is a solution in Φn1 for arbitrarily large values of n it isnecessary that:

vS(p∗) + CS(pacL ) ≤ 0

IfvS(p

∗) + CS(pacL ) < 0

there is a solution in Φn1 for arbitrarily large values of n.

Proof. Consider the first part. Using (5’), for all n we have vS(q0S(n)) +CS(qL(n)) < 0 so that at the limit we certainly have vS(p∗) + CS(pacL ) ≤ 0.As for the second part, theorem 5 proves in fact that in the construction

of theorem 3 for n large enough q̂1S converges to p∗ as q̂L goes to pacL . Since

vS(p∗) + CS(pacL ) < 0 it must be that (5’) will be satisfied and a solution is

obtained.

Theorem 8 Suppose vS(p∗) + CS(pacL ) < 0, then limδ→1 vL(qL(nδ, δ)) = 0.

Proof. Since vS(p∗)+CS(pacL ) < 0, Φn1 has a solution for arbitrarily large n,

this proves that limδ→1 nδ =∞. But limn→∞ vL(qL(n)) = 0. Since limδ→1 nδ =∞, whatever n the solutions of Φn1 and of Φnδ exist and can be made arbitrarilyclose so that limδ→1 vL(qL(nδ, δ) = 0.

Definition 9 (The solution) The selection process is further refined so that thesolution of Γδ∞ is taken as the SPE(nδ) in which nδ in the maximal n for whichΦnδ has a solution.

The motivation to select this SPE(nδ) among the other SPE(n)’s as thesolution comes from theorem 5, which may be interpreted in two ways. Comparethe strategies in SPE(nδ) and in any SPE(n). Firstly, player S maximizes histotal rent over his incumbency time (Σt

0=nt0=1 vS(q

t0S (n))). Secondly, the long term

incumbent L deters the most aggressive entry from player S. In the concludingsection this particular selection is further discussed.

12

6 The merits of contestable markets revisitedThe economic properties of SPE(nδ) when δ goes to 1 are now explored usingthe results obtained in the preceding section. Attention is on the case when thetwo firms have different efficiencies. Note that the solution is defined whetherplayer L is the more efficient or the less efficient firm. Observe that the higherthe instantaneous incumbency rent of player L, the lower the cumulative rentof player S and vice versa. Productive efficiency is intuitively associated to theidea that it pays to be more efficient. This can be investigated in two ways:firstly it should be that the incumbency rent of player L is increasing relativeto the difference in efficiency, secondly, contrarily to the MT model, that theincumbency rent of player L as the efficient firm is greater than the one he wouldget as the less inefficient firm. Rent dissipation is associated to the idea thatthe incumbency rent of player L is zero, i.e. nδ →∞ when δ goes to 1. It mayor it may not prevail in the case the two firms are symmetric. If it does not,does this depend on the degree of toughness of competition, as some economistsclaimed it should?The exploration of these issues may be more or less complicated depending

on the payoff functions πi(pi, pj). A simple benchmark is proposed. Supposethe situation can be approximated by a case in which the vi and Ci functionsare linear: in that case the solution of Φn1 can be derived analytically, whetheror not nδ →∞ when δ goes to 1 can be characterized, this leads to a taxonomyof situations. This simple benchmark is used to explore more meaningful modelsof competition such as the one introduced in section 2.

6.1 A benchmark: the linear case

Take a simple linear two parameter (∆f and λ) model defined as follows:

v1(p1) = λp1 C1(p2) = 1− p2v2(p2) = λp2 −∆f C2(p1) = 1 +∆f − p1

Firm 1 is more efficient than firm 2 if and only if ∆f ≥ 0. The lower λ,the lower the incumbency profit for a given entry cost, i.e. the tougher thecompetition.This game may be played with player L either as firm 1 or as firm 2.Average cost pricing and static limit moves are respectively:

pac1 = 0pmax1 = 1 +∆fpac2 = ∆f/λpmax2 = 1

Assumption 3 implies that:

−1 ≤ ∆f ≤ λ

13

Whoever is player L, the system Φn1 may be solved analytically. Index by 1or 2 its solution.

Proposition 10 The selected equilibrium is such that:i) with firm 1 as player L and firm 2 as player S

if ∆f < −1 entry of firm 1 is blockadedif −1 < ∆f ≤ (λ− 1)/2 qL1 = p

ac1 = 0

if (λ− 1)/2 < ∆f < λ qL1 = 1− λ+ 2∆fif λ < ∆f qL1 = p

max1 = 1 +∆f

ii) with firm 2 as player L and firm 1 as player Sif ∆f < −1 qL2 = p

max2 = 1

if −1 < ∆f ≤ λ(1− λ)/(1 + λ2) qL1 = 1− λ(1 +∆f)

if λ(1− λ)/(1 + λ2) < ∆f ≤ λ qL2 = pac2 = ∆f/λ

if λ < ∆f entry of firm 2 is blockaded

Proof. Consider the first part.By construction

R pmax2

p∗ vS(p)dCLdp (p)dp =

R pmax2

p∗ v2(p)dC1dp (p)dp =

R 1p∗(λp2 −

∆f)dp. It follows that p∗ = 2∆f/λ− 1.To obtain rent dissipation, it must be that vS(p∗) + CS(pacL ) = v2(p

∗) +C2(p

ac1 ) < 0. That is, ∆f < (λ− 1)/2.If there is no rent dissipation, using (1’) one gets qtS = 1−(n− t)λqL, so that

Σt=nt=1 qtS = n− λqLn(n− 1)/2. Condition (2’) writes −CS(qL) +Σt=nt=1vS(q

tS) = 0

which gives −(1− qL)−∆f − n∆f + λn− λ2qLn(n− 1)/2 = 0.The maximal duration nδ when δ → 1 is obtained using (5’). Denote n1 =

limδ→1 nδ. Assuming away integer problems, n1 solves Σt=n1t=0 vS(qtS) = 0. But

Σt=nt=0 vS(qtS) = −(n + 1)∆f + λ(n + 1) − λ2qLn(n + 1)/2, so that n1 = 2(λ −

∆f)/λ2qL.Substituting n by 2(λ−∆f)/λ2qL in the expression of qL gets qL = 1−λ+

2∆f.Hint : first use the fact that Σt=nt=0vS(q

tS) = 0 to eliminate ∆f in

−(1− qL)−∆f − n∆f + λn− λ2qLn(n− 1)/2 = 0to get 1 + λ = qL(1 + λ2n) and then substitute n by 2(λ−∆f)/λ2qL.The proof of the second part is obtained through similar calculations.

Corollary 11 Productive efficiency prevails.

Proof. This is easily done by inspection. Figure 1 visualizes the case λ =1/2. It depicts qL1 and qL2 as functions of ∆f . The respective average costs arealso drawn so that the incumbency rents may be read graphically as λ(qLi−paci ).They are of course identical for ∆f = 0. Then λ(qL1 − pac1 ) is seen to increasewith ∆f while λ(qL2 − pac2 ) is seen to decrease. This proves that it pays tobe more efficient either as player L or as player S, and that a more efficientfirm gets a higher incumbency rent as player L than the one a less efficient firmwould. Note a nice feature of the model: the continuity of qL1 and qL2 as entrybecomes blockaded. It is easily seen that this remains true for all values of λ.

14

- 1 -1/4 0 1/5 1/2 ∆f

The entry of firm 1 is blockaded

The entry offirm 2is blockaded

qL1qL2

1

pac1

pac2

0

Figure 1: The incumbency rents in the linear case

15

The impact of productive efficiency on the incumbency rent can be quanti-fied. Say λ = 1, then qL2(−∆f) = qL1(∆f) = 2∆f. An innovation that gives acompetitive advantage in average cost of ∆f generates an increase in instanta-neous rent not of ∆f but of 2∆f ! The factor 2 may be seen as a ”Shumpeterianmultiplier”. This multiplier varies with λ. This is to be contrasted with the MTapproach in which there is no rent whatever the competitive advantage.A taxonomy of natural monopoly situations is introduced to discuss further

how productive efficiency works in this model.

Definition 12 A natural monopoly is said to be a situation of:

• under-competition if a less efficient incumbent can deter entry forever andmake stationary positive profits;

• selection if a less efficient incumbent is not able to deter entry forever andmake stationary positive profits, but a more efficient incumbent may;

• excess-competition if a more efficient incumbent were to choose to deterentry forever, it would have to dissipate all of its profits.

The results of proposition 10 may be combined to depict the taxonomy in asingle diagram (figure 2).

6.2 Discussion

As will be shown shortly, examples such as the one introduced in section 2 aresuggestive of λ ≤ 1. It corresponds to tough price competition.To be a little bit more general than in section 2, let R(p) be the revenue

function with R0 > 0 (at least in the relevant range for p) and R00 < 0. Thefixed cost is denoted f. The symmetric entry game is thus defined with:

v(p) = R(p)− fC(p) = −(R(p− s)− f )

Proposition 13 In this simple price competition game: i) there is no rentdissipation if the switching cost s is small enough ii) there is rent dissipation asthe switching cost s goes to zero.

Proof. For small s, we certainly have pmax close to pac. Linear approx-imations of the v and C functions may be used. According to proposition10 the ratio v(pmax)/C(pac) relative to 1 characterizes the situation. Writev(pmax)/C(pac) = −(v(pmax)− v(pac)/(pmax− pac))/(C(pac)−C(pmax))/(pac−pmax)) so that v(pmax)/C(pac) is close to −v0(pac)/C 0(pac) = R0(pac)/R0(pac −ε) < 1 since R00 < 0.To obtain illustrations suggestive of λ ≥ 1, the entry cost function C(.)

should be lowered relative to the monopoly profit function v(.). Entry shouldnot be so tough. Consider the price model with a switching cost s such thatpmax > 3s > pac, the best entry move p−(p) is to exclude the incumbent for

16

- 1 - 1/2 0 ∆f

λ

∆f = λ

∆f = (λ - 1)/2 ∆f = λ(1 - λ)/(1 + λ2)

Under Competition

Selection for 2

Selection for 1

Excess Competition

0

1

The entry offirm 2 is blockaded

The entry offirm 1 is blockaded

Figure 2: A taxonomy of natural monopoly situations

17

p > pmax but to obtain an (unprofitable) duopoly for p < pac (assumption 3holds). At some point p between pac and pmax, the entry cost C(p) would belower than −v(p− s), this would make entry less tough. The Eaton and Lipseymodel of durable capital as an entry barrier (1980) provides another illustrationof such cases. There, entry leads to a drop of the instantaneous revenue until theduration of the capital of the incumbent is exhausted, the higher the duopolyrevenue, the softer the competition (see Gromb et al., 1997, section 4.3 for areinterpretation of the Eaton and Lipsey model as an entry game with v andC functions). The linear model simply means that as the entry cost function islowered, entry is facilitated and, at equilibrium, the incumbency rent decreaseseventually to zero after some threshold.With this in mind, come back to the literature on contestable markets. The

standard game theoretic idea models contestable markets through short termcommitments. Rent dissipation prevails as the discount factor goes to 1 (section2.2). With this new approach proposition 13 says that it is not enough that thediscount factor goes to 1, it must be that the switching cost s, which alsoprovides an incumbency advantage, goes to zero as well. This view is in linewith Weizman (1983): if contestable markets simply mean average cost pricingand if this average cost is independent of the underlying time structure as it isin this model, they correspond to a frictionless (and meaningless) situation.But this model allows a meaningful discussion in a world of small frictions

(such as δ close to 1 and s close to 0). There the results are in line with theidea that the intensity of competition in case of entry should play a role in pricediscipline. This point is seen as crucial by Dasgupta and Stiglitz (1986), itsimpact has been formalized in a two stage game in Henry (1988), Sutton (p.35, 1991) provided ample empirical evidence that ”a very sharp fall in pricesuffices to deter entry and maintain a monopoly outcome”. This correspondsto the fact that it makes a clear difference whether λ < 1 or λ > 1. In the firstcase, competition is tough and there is no rent dissipation, in the second case,competition is soft and rent dissipation prevails. Interestingly, according to thisview, rent dissipation always prevails in the durable capital model of Eaton andLipsey.Consider now the important issue relative to productive efficiency. For large

differences in efficiency, the competitive process indeed selects the more efficientfirm. For low differences in efficiency, it may not. Two cases may occur. Inthe zone of under-competition, the more efficient firm cannot directly obtainthe incumbency position but it can profitably acquire it from the less efficientfirm. In the zone of excess-competition, an incremental competitive advantageis worth nothing because the competition for the incumbency position is sointense that it is worth zero anyway. These results are not only consistent witheconomic intuition, they also provide interesting avenues for future research.It seems worthwhile to carry on this discussion, not through the simple linear

model, but through a direct analysis of more substantial economic models. Thiswill be done in a companion paper.

18

7 Concluding commentsThe approach proposed in this paper requires some further theoretical work. Inthis last section some open questions are pointed out.Question 1: The results should be extended to other game forms. The

extension to the Stackelberg model with endogenous leadership introduced inGromb, Ponssard and Sevy (1997) is straightforward, assumption 3 may berelaxed to simply paci ≤ pmaxi . The interested reader will also observe that theassumption introduced in GPS, such that vi+Ci be a decreasing function, playsno role in this paper. The extension to the alternate move quantity model usedby Maskin and Tirole (1988) requires to relax assumption 3 and to deal withborder conditions.Question 2: It would certainly be helpful to clarify the rationality assump-

tions used in the proposed refinement of Nash equilibria. A starting point maybe in extending the forward induction approach introduced in Ponssard (1991)to the case of infinitely repeated games. This is certainly not a simple matter.Question 3: The fact that a player plays a stationary strategy when he

is active gives him a tremendous advantage. This explains why a weak playermay stay as a permanent incumbent with the MT approach. This advantageis not completely wiped out in our approach (player L enjoys it). It would beinteresting to define an approach in which it is, such as a game in which thereis a preliminary stage where each player decides how long he could stay, say n1for player 1 and n2 for player 2. The choices n1 and n2 would be assumed tobe revealed and a repeated game Γn1,n2∞ would be played in which each playercan only use SPE strategies according to the number of stages announced atthe preliminary stage. It is suspected that the equivalent of theorem 5 holds(i.e., given ni the best response nj(ni) is the highest nj for which the entrygame Γn1,n2∞ has an equilibrium). If this were indeed the case, the taxonomyof natural monopoly situations introduced in section 6 would lie on a strongerground namely:- selection: only the strong player would select to stay infinitely (the limit

equilibria in Γn1,n2∞ with respect to large values of (n1, n2) would have n∗1 =∞, n∗2 <∞, where player 1 is the strong player) ;- under-competition: either player could select to stay infinitely but if one

does, the other would not wish to, the preliminary game would be similar toa battle of the sexes game (there would be two limit equilibria Γn1,n2∞ withn∗1 =∞, n∗2 <∞ and n∗1 <∞, n∗2 =∞ ) ;- excess-competition: either player would select to stay infinitely whatever

the other one does, the preliminary game would be similar to a prisoner dilemmagame (formally Γn1,n2∞ would have no limit equilibrium with respect to largevalues of (n1, n2), the best response n1(n2) being ∞ and vice versa, while bothequilibrium payoffs in a game Γn1,n2∞ would decrease as (n1, n2) increases).

19

8 REFERENCESBaumol, W., Panzar J. C. and R. D. Willig (1982). Contestable Markets andTheory of Industry Structure, New York : Harcourt Brace, Jovanovich.Cyert, R. and M. DeGroot (1970). Multiperiod decision models with al-

ternating choice as the solution of the duopoly problem, Quarterly Journal ofEconomics, 84, 419-429.Dasgupta, P. and J. Stiglitz (1986). Welfare and Competition with Sunk

Costs, CEPR, London.Dixit, A. K. (1980). The role of investment in entry deterrence, Economic

Journal, 90:95-106.Eaton, B. and R. Lipsey (1980). Exit barriers are entry barriers : the

durability of capital as an entry barrier, Bell Journal of Economics, 10:20-32.Farrell, J. (1986). How effective is potential competition, Economic Letters,

20, 67-70.Fudenberg, D. and Tirole J. (1987). Understanding rent dissipation : on

the use of game theory in industrial organization, American Economic Review77:176-183.Gaskins, D. (1971). Dynamic limit pricing: optimal pricing under threat of

entry, Journal of Economic Theory, 2, 306-322.Gromb, D., Ponssard J-P. and D. Sevy (1997). Selection in dynamic entry

games, Games and Economic Behavior, 21:62-84.Grossman, S. (1981). Nash equilibrium and the industrial organization of

markets with large fixed costs, Econometrica, 49, 1149-1172.Henry, C. (1988). Concurrence potentielle et discrimination dans un mod-

èle de duopole avec différenciation verticale, Mélanges économiques, essais enl’honneur d’Edmond Malinvaud, 289:313, Economica-EHESS, Paris.Kamien, M. and N. Schwartz (1971). Limit pricing and uncertain entry,

Econometrica, 39, 441-454.Lahmandi, R.,Ponssard J-P. and D. Sevy (1996). Efficiency of dynamic

quantity competition : a remark on Markov equilibria, Economic Letters, 50:213-221.Louvert, E. (1998). Solving entry games through extended Markov equilib-

ria, in Modèles dynamiques de la concurrence imparfaite, doctoral dissertation,Ecole Polytechnique.Maskin, E. and J. Tirole (1988). A theory of dynamic oligopoly, Part I :

overview and quantity competition with large fixed costs, Econometrica, 56-3:549-569.Pyatt, G. (1971). Profit maximization and the threat of new entry, Economic

Journal, 81, 242-255.Ponssard, J-P. (1991). Forward Induction and Sunk Costs Give Average

Cost Pricing, Games Economic Behavior, 3:221-236.Sutton, J. (1991). Sunk Costs and Market Structure, Cambridge, Mass.:

MIT Press.Wilson, R. (1992). Strategic Models of Entry Deterrence, in Handbook of

Game Theory, vol. 1 (R.J. Aumann and S. Hart, eds.), Amsterdam, Elsevier

20

Science.Weizman, M. L. (1983). Contestable Markets: An Upraising in the Theory

of Industry Structure: Comment, American Economic Review 73, 3:486-487.

9 Appendix : ProofsProof. 2 (Theorem 4) The proof is given for δ = 1. The argument can beextended to δ close to 1. It runs as follows. Firstly prove that conditions (1’-2’-3’-4’) of Φn1 have a unique solution. Secondly, check whether condition (5’)is satisfied: if it is, the unique solution of Φn1 is obtained; if it is not, Φ

n1 has no

solution.To prove the first part, for all pL ∈ [pacL , pmaxL ] , define the function W (pL) =

CS(pL)−Σn1vS(ptS) in which the sequence (ptS) is derived from pL through (1’)that is,

−CL(ptS) + (n− t)vL(pL) = 0 for t ∈ {0, 1, 2...n}

then, show that W (pL) is negative (step 1) then positive (step 2) and that itsderivative is strictly positive (step 3) so that there is a unique solution to theequation W (pL) = 0.Step 1: if pL = pacL then W (pL) < 0In that case ptS = C

−1L (0) for all t so that W (pacL ) = CS (p

acL )− nvS (pmaxS )

by assumption vS (pmaxS ) > 0 so that for n large enough W (pacL ) < 0.Step 2: if pL = pmaxL then W (pL) > 0Since CL is strictly decreasing, the sequence (ptS) is a strictly increasing

sequence bounded by pmaxS . Since vS is strictly increasing this implies thatΣn1 vS(p

tS) is certainly negative for n large enough so thatW (pmaxL ) = −Σn1vS(ptS)

is certainly positive.Step 3: dW

dpL> 0

We have

dW

dpL=dCSdpL

− Σt=nt=1 (dvSdptS

· dptS

dpL)

Using (1’) we get:dptSdpL

= (n− t)dvLdpL

/dCLdptS

By substitution it follows that:

dW

dpL=dCSdpL

− dvLdpLΣt=nt=1 ((n− t)

dvSdptS

/dCLdptS

)

2 I am indebted to Rida Laraki for providing the argument for this proof.

21

By assumption −dvSdptS

/dCLdptS

is uniformly bounded away from zero by ε so that

dW

dpL≥ dCSdpL

+dvLdpL

n(n− 1)2

ε

SincedvLdpL

is bounded away from zero and sincedCSdpL

is bounded away from

−∞ we certainly havedW

dpL> 0 for n large enough.

Hence for a given n large enough there is a unique solution to W (pL) = 0that is, to (2’). This solution is in ]pacL , p

maxL [ so that (3’) is also satisfied. Denote

q̂L this solution and (q̂tS) for t ∈ {0, 1, 2...n} the associated sequence obtainedthrough (1’). Observe that (4’) is satisfied as well: since vS is increasing thefunction Σt

0=nt0=t vS(q̂

t0S ) is bell shaped with respect to t so for all t we have:

Σt0=nt0=t vS(q̂

t0S ) ≥Min(Σn1 vS(q̂tS), vS(q̂nS)) =Min(CS(q̂L), vS(pmaxS ) > 0

because q̂L < pmaxL implies CS(q̂L) > 0 and v(pmaxL ) > 0 by construction.It is now a simple matter to check whether (5’) holds or not. If it does a

complete solution to Φn1 is obtained, if it does not there cannot be a solutionfor that value of n since conditions (1’) through (4’) have a unique solution.

Proof. (Theorem 6)Suppose vL(qL(n, δ)) ≥ ε > 0 for all n, then using (1) the sequence (qtS(n, δ))

t=nt=0

is a strictly increasing sequence defined backwards from qnS(n, δ) = pmaxS . Conse-

quently, for an arbitrarily large number of items in this sequence vS(qtS(n, δ)) < 0whereas for a finite number of them, vS(qtS(n, δ)) ≥ 0. It follows that (4) cannothold. Hence either there cannot be a solution for arbitrarily large n or, if thereis one, limn→∞ vL(qL(n, δ)) = 0.Suppose limn→∞ vL(qL(n, δ)) = 0, it is now proved that, if δ < 1, we have

a contradiction. Indeed, since vL(qL(n, δ)) is close to zero, using (1) it is seenthat the whole sequence (qtS(n, δ))

t=nt=0 can be made arbitrarily close to p

maxS . By

assumption pacS < pmaxS , so that (5) cannot hold. This completes the first partof the theorem. Furthermore we proved that if Φn1 has a solution for arbitrarilylarge n, limn→∞ vL(qL(n) = 0.Combining this result with (2’) we get limn→∞Σn1vS(qtS(n)) = CS(p

acL ).

Consider now the last point. First of all, given that dCLdp is bounded away

from infinity and from zero and that vS(p) is bounded away from zero, thereexists a unique p∗ < pacS such thatZ pacS

p∗vS(p)

dCLdp

(p)dp = 0

For all p ≤ pacS define F (p) =R pacSp

vS(u)dCLdu (u)du, the function F is such

that F (p) > 0 iff p < p∗.

22

We now show the convergence of q1S(n) to p∗.

Using (1’) and (2’) we get :

CS(qL(n))vL(qL(n)) = Σt=nt=1 vS(qtS(n))vL(qL(n))

= Σt=nt=1 vS(qtS(n))

£CL¡qt−1S (n)

¢− CL ¡ qtS(n)¢¤When vL(qL(n) is small this non negative expression is close to F (q1S(n)).To see this, make the change of variable from pS to u = CL (pS) . As t goes

from 1 to n, pS increases from q1S(n) to qnS(n) and u from u

1(n) = CL¡q1S(n)

¢to

un(n) = CL (qnS(n)) = 0 but u

t−1(n)− ut(n) remains t independent and equalsvL(qL(n)), let ∆u(n) = vL(qL(n)).We may then write

vL(qL(n))Σt=nt=1vS(q

tS(n)) = Σ

t=nt=1 vS(C

−1L (ut(n)))∆u(n)

For large values of n we have

Σt=nt=1vS(C−1L (ut(n)))∆u(n) ≈

Z 0

u1(n)

vS(C−1L (u))du =

Z pacS

q1S(n)

vS(p)dCLdp

(p)dp.

This proves that q1S(n) cannot be far below p∗. Using (2’) and (5’) for

the two sequences n and n + 1, it is clear that q1S(n + 1) and q1S(n) cannot be

far apart either. More precisely:

¯̄q1S(n+ 1)− q1S(n)

¯̄ ≤ −Min(dCLdp

(q1S(n)),dCLdp

(q0S(n)))vL(qL(n))

Since dCLdp is bounded away from infinity, limn→∞

¯̄q1S(n+ 1)− q1S(n)

¯̄= 0,

this is enough to prove that q1S(n) converges to some limit and this limit canonly be p∗ sincelimn→∞CS(qL(n))vL(qL(n)) = limn→∞CS(qL(n)) limn→∞ vL(qL(n)) = CS(0).0 =

0.

23