Thomas Fagart
email: thomas.fagart@psemail.eu.

Lack of Preemption Under Irreversible Investment

Keywords: Capacity investment, Cournot competition, Markov-perfect equilibrium, Real option games, Di¤erential games, JEL classi…cation: D43 L13 L25 Investissement en capacité, Concurrence à la Cournot, Equilibre markovien, Option réelle, Jeux di¤érentielle

This article considers the classic model of irreversible investment under imperfect competition and stochastic demand, and characterizes the markov perfect equilibrium.

To do so, I introduce a new way to de…ne strategies, permitting the players to create endogenous jumps in the state variable. The markov equilibrium is then similar to the open-loop equilibrium, meaning that the irreversibility of investment does not create a preemption e¤ect in this model. This is due to the form of investment's cost, which creates an incentive to invest as soon as possible, reducing the strategic interaction to the one of a static problem.

Introduction

Investment in capacity of production has usually three main features. It is irreversible, as a …rm cannot recover its installation fees if it decides to dismantle some unit of capacity.

It is done in a long run perspective, with an uncertain evolution of the environment. It is also often done in an imperfect competition framework. In that case, the commitment due to the irreversibility of investment has an impact not only on the …rm which invests, but also on its opponents. Indeed, the competitors of the investing …rm will have to deal with its new capacity and should reduce or delay their own capacity adoption. This usually creates a preemption e¤ect, as the investing …rm may invest not just for the project value, but also for its impact on opponents'future decisions.

This article considers the classic real option model of irreversible investment under uncertainty and imperfect competition. It shows that, in this model, there is no preemption e¤ect at the equilibrium. This is due to the form of investment's cost, which creates an incentive to invest as soon as possible and thus reduces the strategic interaction between …rms to the one of a static problem. This result shows that the irreversibility of investment alone is not su¢ cient to create preemption, and provides a better understanding of the reason driving preemption. This paper also introduces a re…nement of the theory of di¤erential games, allowing strategic jumps in the state variable.

More precisely, I consider a duopoly producing an homogenous good, and competing à la Cournot in a continuous time setting. The quantity produced by a …rm depends of its capacity. At each time, …rms can decide to increase their capacity at a linear cost. Demand evolves according to a continuous stochastic process, and capacities are smooth.

When there is a single …rm in the market, the optimal strategy is well known. (See for example [START_REF] Abel | Options, the Value of Capital, and Investment[END_REF] or [START_REF] Dixit | Markov interactions in a class of dynamic games[END_REF]). There exists a trigger, a level of demand depending of the capacity of the …rm, such that the monopoly invest only when the demand exceeds the trigger, and then install the smaller level of capacity which releases the trigger. As long as the demand is inferior to the trigger, the monopoly does nothing.

Under imperfect competition, this model has already been studied by several authors. [START_REF] Baldursson | Irreversible investment under uncertainty in oligopoly[END_REF] and [START_REF] Grenadier | Option exercise games: An application to the equilibrium investment strategies of …rms[END_REF] assume that …rms have the same behavior as the monopoly, i.e. invest when demand reaches a trigger. They characterize the Nash equilibrium in triggers'strategy and show that competition makes the …rms more reactive to demand. [START_REF] Back | Open-loop equilibria and perfect competition in option exercise games[END_REF] and [START_REF] Steg | Irreversible investment in oligopoly[END_REF] show that the behavior assumed by [START_REF] Baldursson | Irreversible investment under uncertainty in oligopoly[END_REF] and [START_REF] Grenadier | Option exercise games: An application to the equilibrium investment strategies of …rms[END_REF] is consistent with the concept of open-loop equilibrium. One of the disadvantages of open loop equilibria is that they usually fail sub-game perfection.

Indeed, in open loop equilibrium, …rms commit to their path of investment along time, and the open loop equilibrium corresponds to the Nash equilibrium in these paths.

In markov perfect equilibrium1 , …rms decide of their investment in function of the capacity repartition at the investment time. This equilibrium concept is a re…nement of subgame perfect equilibrium, and its comparison with open loop equilibria permits to identify the strategic e¤ects in a game (see for example Figuières (2009) or [START_REF] Genc | Capacity Investments in a Stochastic Dynamic Game: Good News Principle[END_REF]). However, in our case of irreversible investment, there exists a di¢ culty to de…ne a markovian strategy. Indeed, the linearity of the investment's cost creates a incentives to invest as soon as possible and the optimal path of capacity can presents some jumps2 . Such jumps are not allow in the usual theory of di¤erential games. [START_REF] Back | Open-loop equilibria and perfect competition in option exercise games[END_REF] and [START_REF] Chevalier-Roignant | Preemptive Capacity Investment under Uncertainty[END_REF] focus on markov perfect equilibrium. [START_REF] Back | Open-loop equilibria and perfect competition in option exercise games[END_REF] de…ne the strategy of a …rm in function of its opponent decisions 3 . Under that condition, they show that …rms can play strategies which ensure them the same pro…t as the open-loop equilibrium. In that case, a …rm can increase its pro…t by committing to an open loop strategy (according to the fact that its opponent will reacts to its new strategy). This has often be interpreted as the fact that the open loop equilibrium di¤ers from markov perfect equilibrium, as a …rm can preempt its opponent. This is misleading, as the possibility of preemption highly depends on the assumption that the opponent will change its path of capacity. If the opponent does not change its path of capacity, the preemption is not necessary pro…table. In fact, preemption is not pro…table, as open-loop and markov perfect equilibrium coincide. The right interpretation of this result of [START_REF] Back | Open-loop equilibria and perfect competition in option exercise games[END_REF] should be that Stackelberg equilibrium di¤ers from open-loop equilibrium, and that a …rm which has the possibility to commit to its open-loop strategy will have a higher pro…t than its markovian follower4 . [START_REF] Chevalier-Roignant | Preemptive Capacity Investment under Uncertainty[END_REF] use the usual theory of di¤erential games to show that …rms invest when the marginal revenue of capacity is higher than the price of investment, where the competitive externality and the e¤ect of uncertainty are taken into account in the marginal revenue of capacity. In that way, Tobin's q theory is still valid.

The point of the present work will be to determine the marginal value of capacity and therefore to fully characterizes the dynamic equilibrium. 5When there is no demand uncertainty, this model is similar to some classic papers on dynamic oligopoly, as [START_REF] Fudenberg | Capital as a Commitment: Strategic Investment to Deter Mobility[END_REF], [START_REF] Reynolds | Dynamic oligopoly with capacity adjustment costs[END_REF]), or Figuières (2009), except in the form of investment's cost. [START_REF] Fudenberg | Capital as a Commitment: Strategic Investment to Deter Mobility[END_REF] assume an exogenous boundary on …rms'investments, and show the existence of strategic e¤ects. [START_REF] Reynolds | Dynamic oligopoly with capacity adjustment costs[END_REF] and Figuières (2009) assume quadratic cost of investment and characterize the impact of strategic e¤ects in function of the form of the competition. When there is complementarity (i.e. in price competition), commitment to open-loop strategies decreases the pro…t of the …rms, whereas when there is substitutability (i.e. in quantity competition), commitment to open-loop strategies decreases …rms'pro…t.

In both papers, the strategic e¤ect comes from the fact that each …rm has an unilateral incentive to delay investment, due to an exogenous bound in the case of [START_REF] Fudenberg | Capital as a Commitment: Strategic Investment to Deter Mobility[END_REF], and to an investment's cost increasing in the velocity of the investment in Figuières (2009) or [START_REF] Reynolds | Dynamic oligopoly with capacity adjustment costs[END_REF]. In order to get the intuition when …rms face a linear cost I start focusing on a two period model without uncertainty. In that framework, each …rm knows that its opponent's incentive to invest during the second period decreases with the amount of capacity invested at the …rst period. The strategic e¤ect should thus increases the incentive to invest in the …rst period. However, as …rms can make money on both periods, there is no incentive to delay investment. The game is thus reduced to a one shot game, and the strategic e¤ect of investment disappears. This highlights the fact that both commitment and an incentive to delay investment are important to the creation of strategic e¤ect in a simultaneous game.

When time is continuous, the optimal capacity path of the …rms is not continuous in time but presents some jumps. This is due to the linearity of investment cost, which creates an incentive to invest as soon as possible. In order to introduce the possibility for the …rm to jump, I introduce a new control variable, the desired capacity. When the capacity of the …rm (which is the state variable) di¤ers from the desired capacity, the …rm jumps to the desired level of capacity. With this de…nition of …rms'strategy, I characterize the markov perfect equilibrium. Both …rms invest to a Cournot level of capacity if both have capacity smaller than the Cournot level. When this is not the case, only the smallest of both …rms invests, except when it already has a capacity too large. Such investments take place at the …rst time of the game, and the markov perfect equilibrium coincides with the open-loop equilibrium.

When demand evolves stochastically, the equilibrium strategies are the same. The optimal levels of capacity depend on the solution of a di¤erential equation with smooth pasting and value matching conditions, due the real option e¤ect. However, the equivalence between open-loop and markov perfect equilibrium remains valid.

Section 2 describes the two period model. Section 3 presents the di¤erential game without uncertainty, and section 4 the di¤erential game with uncertainty. Section 5 concludes. All non-included proofs are given in appendix.

Discrete game 2.1 The model

In order to understand why there is no strategic interaction under irreversible investment, let's focus on the simplest game of irreversible investment, with two periods 6 and without uncertainty. Consider a duopoly competing à la Cournot in capacities. Each …rm i, for i 2 fA; Bg, starts with some amount of capacity k i 0 (k i 0 0), which can be extended through buying some assets. Purchases are made at a linear price p + (p + > 0). Let k i 1 be the capacity installed by …rm i in the …rst period, and k i 2 the capacity installed in period 2. Each …rm's production depends of its capacity, according to the technology q i = k i . Such technology is classic in dynamic investment models, and has been used by [START_REF] Fudenberg | Capital as a Commitment: Strategic Investment to Deter Mobility[END_REF], [START_REF] Grenadier | Option exercise games: An application to the equilibrium investment strategies of …rms[END_REF], [START_REF] Merhi | A model for reversible investment capacity expansion[END_REF] among others 7 .

At each periods, …rms produce and sell an homogenous good, at a price P k i t + k j t depending of the total quantity sold on the market. The pro…t of …rm i is thus;

i = P k i t + k j t k i 1 p + k i 1 k i 0 + P k i t + k j t K i 2 p + k i 2 k i 1 , (1)
where 2 [0; 1] is the discount rate. As investments are irreversible, …rm's i capacities should verify:

k i 2 > k i 1 > k i 0 . (2)
In this game, I compare two kinds of equilibria concepts, open-loop equilibria and sub-game perfect equilibria. In open-loop equilibria, at the beginning of the game, …rms commit on their action for the two periods, and then played a classic Nash equilibrium where the strategy space is R 2 + . Formally, this gives:

6 Two periods are needed in order to have some possibility of dynamic interaction (the action of a player impacts the action of the other on the next period). 7 As it was shown in [START_REF] Reynolds | Capacity Investment, Preemption and Commitment in an In…nite Horizon Model[END_REF], this technology assumption is also the result of a dynamic games with limited Cournot competition, without uncertainty. Therefore we can relay this hypothesis in section 2 and 3 (for q i k i ). However when there is uncertainty, …rms have an incentive to keep their unused capacity for a possible further use in case of demand increases. Assuming that quantities are equal to capacities permits to avoid such adaptability e¤ects and focus on the direct e¤ect of uncertainty on capacity choice.

Open Loop Equilibrium: The group of strategies

(k i t ) i2fA;Bg t2f1;2g is an open loop equilib- rium if, for all i 2 fA; Bg, i k 1 ; k 2 ; k j 1 ; k j 2 = max K i 1 ;K i 2 i k i 1 ; k i 2 ; k j 1 ; k j 2 . ( 3
)
In sub-game perfect equilibria, at each time …rms choose their action knowing the history of all the previous actions. Here, the choice of capacity in the second period depends of the capacities installed in the …rst period. The formal de…nition follows.

Sub-game Perfect Equilibrium: The group of strategies

(k i 1 ; k i 2 ) i2fA;Bg , where k i 2 is a function from R 2 + to R + and k i 1 a real
, is a sub-game perfect equilibrium if, for all i 2 fA; Bg, the two following conditions are veri…ed: for all vectors of capacity k 1 2 R 2 + installed in the …rst period, k i 2 veri…es:

i h k i 1 ; k i 2 (k 1 ) ; k j 1 ; k j 2 (k 1 ) i = max ki 2 (:) i h k i 1 ; ki 2 (K 1 ) ; K j 1 K j 2 (K 1 ) i (4)
the vector of …rst period capacity, k i 1 , veri…es:

i h k i 1 ; k i 2 (k 1 ) ; k j 1 ; k j 2 (k 1 ) i = max k i 1 i h k i 1 ; ki 2 (k 1 ); k j 1 ; k j 2 (k 1 ) i . ( 5
)
Equation ( 4) represents the fact that the strategies should be optimal in all sub-game, i.e. whatever happens at the …rst period, and not just on the equilibrium path.

Open-loop equilibria usually fails sub-game perfection, but are used as a benchmark for discussing the e¤ects of dynamic strategic incentives, i.e. the incentives to change current play so as to in ‡uence the future play of opponents8 . In the game considered here, the openloop equilibrium coincides with the sub-game perfect equilibrium, and there is no strategic interaction.

An example

This sub-section focus on what happens with a linear price function

P k A t + k B t = 1 k A t k B t .
It solves the model by backward induction, in order to give intuition on the lack of strategic e¤ect under irreversible investment, and on the proof of proposition 1.

In the last period, given the capacity of period one, the best response of …rm A is given by

k A 2 (k B 2 ) = max 1 p + k B 2 2 ; k A 1 . ( 6
)
Therefore, there exist three possibilities, in function of the capacities installed at the …rst period. Both …rms are constrained (In the meaning that they wish to have a capacity in period 2 inferior to their capacity in period 1). One …rm is constrained and the other is not.

No …rms are constrained.

When no …rms are constrained, the equilibrium is straightforward, both …rms investing until the usual Cournot level. This happens when k A 1 and k B 1 are smaller than the Cournot equilibrium,

1 p + 3 . When …rm A is constrained and …rm B is not, …rm B installed a capacity k B 2 = 1 p + K A 1 2 . If k B 1 > 1 p + K A 1 2
, then …rm B has no interest to invest, and both …rms are constrained. Graphic 1 presents the area where …rms are constraint, in function of the capacities chosen in the …rst period, and Graphic 2 the equilibrium of second period.

[Insert Graphic 1] [Insert Graphic 2]
In the …rst period, a …rm has two possibilities. Choosing a capacity which will constrained him in the second period, or choosing a capacity which will not. If …rm A chooses a capacity which does not constraint her, its best response is:

k A 1 (k B 1 ) = max 1 p + k B 1 2 ; k A 0 , (7)
so …rm A wishes to be constrained in the second period. This is due to the fact that a unit of capacity brings the same level of pro…t in the …rst and in the second period, and there is no interest to delay investment. Therefore, at the closed loop equilibrium, …rms have the same level of capacity in both periods. Such level maximizes For i 2 fA;

Bg, i = (1 + ) 1 k i 1 k j 1 k i 1 p + k i 1 k i 0 . (8)
Therefore, the equilibrium strategy of …rm A is to install the Cournot level if …rm B starts with a capacity inferior to the Cournot:

k A 2 = k A 1 = max ( 1 p + (1+ ) 3 ; k A 0 ) if k B 0 1 p + (1+ ) 3 , (9)
and to install the best response to …rm B's initial level when …rm B's capacity exceed the Cournot level:

k A 2 = k A 1 = max ( 1 p + (1+ ) k B 0 2 ; k A 0 ) if k B 0 > 1 p + (1+ ) 3 . ( 10
)
Firm B has the same equilibrium strategy. In some way, this game is equivalent to a one-shot game, where …rms invest for both periods. This is due to the linearity of investment cost, which overthrow the incentives to delay investment. The next section will show that the same e¤ect happens in a di¤erential game. Before that, the following sub-section presents this result for general demand and cost function.

The proposition

In order to present proposition 1, let k i Irr (K j ) be the best response of …rm i for i 2 fA; Bg, when …rm i has no initial capacity and …rm j has the same capacity in both periods, i.e. k j . k i Irr is de…ned as the implicit solution to the following equation:

P k i Irr + k j + P 0 k i Irr + k j k i Irr = p + 1 + : (11)
Let also k C be the Cournot level of capacity when both …rms has no initial value, i.e. the solution to

k A Irr (k B C ) = k A C and k B Irr (k A C ) = k B C . ( 12
)
In order to ensure the existence of k i Irr and k C , I make the following assumption: H1: P is a concave, twice di¤erentiable positive function.

This assumption permits to state proposition 1.

Proposition 1: Assume H1. Let (k i t )

i2fA;Bg t2f1;2g be the strategies de…ned by:

For i 2 fA; Bg ; k i 2 = k i 1 = 8 < : max k i Irr (k j 0 ); k i 0 if k j 0 > k j C max k j C ; k i 0 if k j 0 k j C . ( 13
)
Then, (k i t )

i2fA;Bg t2f1;2g is the unique open loop equilibrium of the game. It is also the only sub-game perfect equilibrium. As seen in the example, the linearity of the investment price overthrows the incentives to delay investment. This reduces the strategic interaction to the …rst period game, and explains the equivalence of the notion of open-loop and equilibrium in that game.

3 Dynamic game without uncertainty

Model and de…nition

This section presents a dynamic model of investment in capacity under imperfect competition. It shows that, under some smoothness condition, the markovian equilibria are similar to the one shot game previously considered, and there is no preemption e¤ect.

Let k i t be the capital of …rm i at time t, time is continuous and capacity partially reversible. Let i (k t ) be the instantaneous payo¤ of …rm i, k t being the vector of …rms capacities. I assume Cournot competition. Let P (:) be the inverse demand function. The instantaneous revenue of …rm i is

i (k t ) = P k t k i t , (14)
where k t = P n i=1 k i t , as previously. The interest rate is r and the purchase price of capacity is p + . As investment is irreversible, the path of investment, k i t , is assumed to be an increasing function of the time. The total pro…t of …rm i at time 0 is thus:

i = Z +1 0 e rt i (k t )dt p + Z +1 0 e rt dk i t . ( 15
)
The objective of …rm i is to choose the process k i t which maximizes its own expected pro…t, given the initial levels of capital and demand. Obviously, the optimal process will depend of the processes chosen by the other players. To properly de…ne the game, I must de…ne the strategic variable used by the players.

In this framework, the usual theory of di¤erential games assume that the path of investment of the …rm is described by the ‡ow of investment I i t :

@k i t @t = I i t , (16)
and that the strategic variable of the players is the ‡ow of investment. Coming from an initial value, the ‡ow of investment describe a unique capacity path. Equation ( 16) implicitly assume that this capacity path is a continuous function of the time (as it is di¤erentiable).

In most of the application of di¤erential game, such assumption is natural. However, in case of irreversible investment, the linearity of investment cost might create an incentive to install the desired capacity as soon as possible. Indeed, the Bellman formula gives:

r i = sup I i P k k i p + I i + I i @ i @K i + I i @ i @K i . ( 17
)
The optimal investment policy maximizes9 I i @ i @K i p + I i , and when @ i @K i < p + the …rm has no incentives to invest. When @ i @K i > p + , the optimal ‡ow of investment I i t is in…nite: the …rm wishes to install its new capacity as soon as possible. This creates a jump in the optimal capacity process of the …rms, which is not allowed by (16).

To address this di¢ culty, I introduce a new control variable, K i t , which is the capacity desired by …rm i at time t. If this desired capacity is equal to the installed capacity, k i t , the …rm continue to invest in a continuous way. If the desired capacity is di¤erent from the installed capacity, the …rm installs the desired capacity. Formally, this is expressed in the following de…nition.

De…nition: The investment game previously considered is in its jump-control form if:

(i) a strategy of player i is a pair (K i t ; I i t ) t2R + , where for each t, (K i t ;

I i t ) 2 R 2 + . (ii) the state variable at time t, k t = (k 1 t ; k 2 t )
, is de…ned by the two equations:

@ + k i t @t = I i t , (18)
and

lim s!t s>t k i s = K i t , ( 19
)
where k 0 is the given initial level of capital, and @ + @t denotes the right-hand derivatives. 10 (iii) the strategic variables of player i is its level of investment, I i t , and the desired capacity

K i t .
Equation ( 19) states than the installed capacity k i t is continuous if and only if

K i t = k i t .
When the strategies are functions of the state variable, …rm i jumps as long as k i t 6 = Ki (k t ). Graphic 3 introduces two examples of strategies than can be follow, and their implication in term of capacity installation.

[Insert Graphic 3]

In such framework, the strategy of player i is markovian if its strategy is only function of the state variable (…rms capacity and level of demand), i.e. if there exists Ki (:) and Ĩi (:) such that

K i t = Ki (k t ) and I i t = Ĩi (k t ). ( 20
)
A strategy is an open-loop strategy if K t and I t only depends on the time, and not on the installed capacity k t .

The de…nition of the jump-control form permits to extend the class of capacity path from the class of continuous functions implicitly assumed by ( 16) to the class of the lefthand continuous function with right-hand derivative. However, this de…nition allows multiple jumps: a jump of …rm i can imply a jump of its opponents in reaction, which can bring a new jump of …rm i, and so on... In order to prevent such multiple jumps, I introduce the following assumption.

H2: The strategies of the …rms i veri…es assumption H2 if, for all

k 2 R n + , K K(k) = K(k). ( 21
)
10 As process k t can be discontinuous in t, the right-hand derivative is de…ned by:

@ + k i t @t = lim n!+1 l k i t+h h , where l = lims>t s!t k i s .
13 Documents de travail du Centre d'Economie de la Sorbonne -2015.79

A markov perfect equilibrium in the markovian state-control form is de…ned as a vector of functions K (:) ; Ĩ (:) = K 1 (:) ; Ĩ 1 (:) ; ::; K n (:) ; Ĩ n (:) such that, for all …rm i 2 f1; ::; ng and for all other markovian strategy of …rm i, K0i (:) ; Ĩ0i (:) ;

8k 2 R n + ; i (k t ) i (k 0 t ), ( 22
)
where k t is the path de…ned by ( 18) and ( 19) and the strategies K (:) ; Ĩ (:) , and k 0 t is the path created when …rm i uses the strategy K0i (:) ; Ĩ0i (:) instead of K i (:) ; Ĩ i (:) , and the other …rm does not change their strategies. Furthermore, a markovian strategy for …rm i is said continuous if the function Ki is continuous, and a continuous markov perfect equilibrium is a markov perfect equilibrium with continuous strategy.

To my knowledge, this is a new way of modeling di¤erential games. In this context, it permits to properly de…ne the possibility to jump.

Characterization of the continuous markov equilibrium

As in the case of the discrete game, let k i Irr (k j ) be the best response of …rm i when …rm i has no initial capacity and its opponent install the level k j at the beginning of the game and maintain the same level afterwards. k i Irr is de…ned by:

P (k i Irr + k j ) + P 0 (k i Irr + k j )k i Irr = rp + , (23)
and the Cournot level, k C , is given by:

k A Irr (k B C ) = k A C and k B Irr (k A C ) = k B C . ( 24
)
Assumption H1 is required in order to be sure that equations ( 23) and ( 24) are well de…ned.

Proposition 2 shows that …rms have interest to invest at the beginning of the game, and to keep their capacity forever after that. This is the situation in both the continuous markov perfect equilibrium and the open-loop equilibrium. Indeed, as the linearity of investment cost overthrows the incentives to delay investment, everything happens like in a one-shot game, and there is no preemption e¤ect.

Proposition 2: Assume H1 and H2. Let ( K i ; Ĩ ) i2fA;Bg be the markovian strategies de…ne by:

For i 2 fA; Bg ; K i (k t ) = 8 < : max k i Irr (k j 0 ); k i 0 if k j 0 > k j C max fk i C ; k i 0 g if k j 0 k j C and Ĩ i (A t ; k t ) = 0. (25)
Then, ( K i ; Ĩ ) i2fA;Bg is the unique continuous markov perfect equilibrium of the game. It is also the unique open loop equilibrium.

Dynamic game with uncertainty

In this subsection, I characterize the continuous markovian equilibrium when demand evolves randomly. A t is the parameter of demand, following a di¤usion process:

dA t = (A t )dt + (A t )dW t ; (26)
where W t is a standard Wiener process. The price at time t is now a function of the parameter of demand at time t and of the capacities of the …rm at time t. The revenue of …rm i at time t is thus:

i (k t ) = P A t ; k 1 t + k 2 t k i t . (27)
In this real option game model, similar to the one of [START_REF] Grenadier | Option exercise games: An application to the equilibrium investment strategies of …rms[END_REF], [START_REF] Back | Open-loop equilibria and perfect competition in option exercise games[END_REF] or Chevalier-Roignant, Huchzermeier and Trigeorgis (2011) there is no strategic e¤ects. As previously, this is due to the linearity of investment's cost, which overthrows the incentives to delay investment. The introduction of uncertainty does not change the strategic interaction. In order to prove this result, I make the following assumptions:

H3: P (:) is a positive function, four times di¤erentiable, such that P 0 0, P 00 0.

H4: (A) and (A) are continuous functions, and verify the Lipschitz conditions.

H5: There exists a function G : R

+ ! R + , such that, 8(A; x) 2 R R + , xP (A; x) < G(A), and R +1 0 e rt G(A t )dt < +1.
The high order of di¤erentiability in assumption H3 is needed in order to use Ito's Lemma and to inverse Ito's Lemma results. H4 is classic to ensure the existence of a strong solution to (26). H5 ensures the existence of the stochastic integral determining the pro…t of the …rms.

In order to states proposition 2, let v be the expected price when no …rms change their capacity. v is a solution to the di¤erential equation:

8x > 0; rv(A; x) = P (A; x) + (A) @v @A (A; x) + 2 (A) 2 @ 2 v (@A) 2 (A; x), (28)
and should verify that, for all (A; k) such that

sup i2fA;Bg v A; k + @v @k A; k k i = p + , ( 29
) then @v @A A; k + k i @ 2 v @A@k A; k = 0. ( 30
)
The conditions ( 29) and ( 30) are the value matching and smooth pasting conditions. As previously, I de…ne k i Irr (A; k j ), the optimal level of investment of …rm i for a level of demand A, by

v A; k i Irr + k j + @v @k A; k i Irr + k j k i Irr = p + (31)
and k C , the Cournot equilibrium, by

k A Irr (k B C ) = k A C and k B Irr (k A C ) = k B C . ( 32
)
Remark that the Cournot level, k C , depends on the level of demand, A. With these de…nition, …rms jump at the beginning of the game and make their capacity evolves continuously with the continuous evolution of demand. This continuity of …rms'decision overthrow the possibility of strategic e¤ect, as it is seen in theorem 2.

Proposition 3: Assume H2, H3, H4 and H5. Let ( K i ; Ĩ ) i2fA;Bg be the markovian strategies de…ne by: For i 2 fA; Bg ; K i (A t ; k t ) =

<

:

max k i Irr (A t ; k j t ); k i t if k j t > k j C max fk i C (A t ); k i t g if k j t k j C and Ĩ i (A t ; k t ) = 0. (33)
Then, ( K i ; Ĩ ) i2fA;Bg is the unique continuous markov perfect equilibrium of the game. It is also the unique open loop equilibrium.

An important point concerning proposition 2 and 3 is that it focus only on continuous markov perfect equilibrium. However there exists a range of other markov perfect equilibrium, as the state variable contains enough information on the past of the game to implement punishment strategies. The continuous markov perfect equilibrium has some interesting properties. It assumes some regularity on the decisions of the …rms: a small change in the state variable should not imply an important change in …rms'actions. It is unique.

When the …rms are initially symmetric, it is equivalent to the Cournot equilibrium. For these reasons, I …ll con…dent enough to claim that the continuous markov perfect equilibrium describes the regular equilibrium of the market. It plays the same role as the stage game in a repeated game framework.

Conclusion

This work shows that, in the classic model of irreversible investment in capacity, there is no preemption e¤ect. This is due to the linearity of investment price which destroys any incentive to delay investment, and thus the possibility to in ‡uence future decisions. The introduction of uncertainty does not change the nature of strategic interactions between players (at least as long as the uncertainty is a generalized Brownian motion and evolves continuously). These results contradict the intuition of the real option game literature. The possibility of preemption, found in models of entry (models of investment in a single project), may not be valid for investment in capacity, as long as the capacity is su¢ ciently smooth.

These results are highly dependent on the nature of the investment's cost. In the industrial organization literature, there exists two other assumption which permits to recover the strategic e¤ects. The …rst one is to assume quadratic cost of investment, as in Figuières (2009). Quadratic investment cost creates incentives to always postpone a part of the remaining investment. A monopoly with quadratic cost will never achieve its stationary capacity, even in a static environment. The second possibility is to assume linear investment's cost but also an exogenous boundary on the ‡ow of investment of the …rms, as in [START_REF] Fudenberg | Capital as a Commitment: Strategic Investment to Deter Mobility[END_REF]. In that case both …rms invest as much as they can, until the marginal value of their investment is inferior to the investment cost. The path of capacities is thus highly dependent on the exogenous boundary, representing the …nancial constraint of the …rm. It is this …nancial constraint which creates the strategic e¤ects. Recent empirical papers, as [START_REF] Angelini | On the Evolution of Firm Size Distribution[END_REF] and [START_REF] Bottazzi | Financial constraints and …rm Dynamics[END_REF] shows that small …rms are more restricted than large …rms on the capital market. It suggests that strategic e¤ects should be more accurate one market composed of small …rms than on market composed of large competitors.

On a game theory perspective, this work contributes to extend the theory of di¤erential games, introducing a way to de…ne strategies which enables players to create endogenous jumps in the state variable. In this framework, there exists a multiplicity of markovian equilibrium, as in discrete dynamic games with in…nite time-horizon, but only one of them is describe by a strategy continuous in the state variable.

Appendix

Proof of proposition 1:

In step 1 and 2, I show that in any sub-game perfect equilibrium (k i t )

i2fA;Bg t2f1;2g , …rms have the same capacity in period 1 and 2. Existence, unicity and characterization of this equilibrium follow. Step 3 focuses on open-loop equilibrium.

Step 1: In this step, I show that for all choices of capacity made in the …rst period, the equilibrium is:

For i 2 fA; Bg ; k i 2 = 8 < : max n ki Irr (k j 1 ); k i 1 o if k j 1 > kj C max n kj C ; k i 1 o if k j 1 kj C , (34)
where ki Irr (k j 1 ) is implicitly de…ned by

P ki Irr + k j + P 0 ki Irr + k j k i Irr = p + , (35)
and kC by kA Irr ( kB C ) = kA C and kB Irr ( kA

C ) = kB C . ( 36
)
The existence of kIrr is due to assumption 2. The existence and unicity of kC comes from [START_REF] Novshek | On the Existence of Cournot Equilibrium[END_REF]. [START_REF] Novshek | On the Existence of Cournot Equilibrium[END_REF] also implies that the equilibrium of the last stage game exists and is unique (the pro…t function of the last stage veri…es the assumption of theorem 3).

The best response of …rm i is

k i 2 = max n ki Irr (k j 1 ); k i 1 o . ( 37
)
Therefore, if opponent of …rm i have a capacity inferior to the Cournot level, the equilibrium strategy of …rm i will be to install the Cournot level, except if its capacity already exceed this level. If the capacity of its opponents exceed the Cournot level, …rm i will invest until its best response level, knowing that the other …rm will not invest (if …rm j invest until its best response level, both …rms install the Cournot level).

Step 2: This step shows that, at the equilibrium, …rms are constraint in the second period. Assume that k i 2 > k i 1 , for one i. Then, there is two possibility: …rst k j 1 > kj C and k i 2 = ki Irr (k j 1 ) > k i 1 . In that case, the pro…t of …rm i in the …rst period is

i j …rst period = P k i 1 + k j 1 k i 1 p + k i 1 k i 0 (38)
which is a concave function of k i 1 , with a maximum in ki Irr (k j 1 ). However, k i 1 < ki Irr (k j 1 ), and …rm 1 does not invest until its best response level, which contradict the fact that (k i t )

i2fA;Bg t2f1;2g is an equilibrium.

The second possibility is that k j 1 < kj C and k i 2 = kj C > k i 1 . In that case, the pro…t of …rm i is the same as (38), and, as (k i t )

i2fA;Bg t2f1;2g is an equilibrium, k i 1 = ki Irr (k j 1 ). Using the theorem of implicit function to (35) implies that ki Irr (:) is a decreasing function of the capacity of …rm i's opponent. Therefore, as k j 1 < kj C , k i 1 > ki Irr ( kj C ) = kj C , which contradict the fact that …rm i is not constrained.

At equilibrium, …rms do not invest in the second period. The sub-game perfect equilibrium can thus be seen as a one shot game, and the characterization of the equilibrium, as presented in proposition 1, comes from the same reasoning as step 1.

Step 3: Assume that (k

  ĨA t , k t   1, andK  A t , k t   k t i  1 k t i  .Documents de travail du Centre d'Economie de la Sorbonne -2015.79

					i
							i
				K
					t		k
							t
								
				0, 0		0, 0	
	i
	K
	t
				i
				k
				t
	3
				3
	4
				4
	2 3 2 1			2 3 2			ĨA K  A t , k t , k t   1, and t   k t i  1 2 1	k	t i k	t i .
				1
	0, 0	1	3 2	0, 0	1	3
						2
				Graphic 3: Examples
				i t ) i2fA;Bg t2f1;2g is an open loop equilibrium, and that …rm A is not
		constrained in the second period (k A 2 > k A 1 ). Then, by de…nition,
				k A 2 = arg max P k i 2 + k j 2	k i 2	p + k i 2	k i 0	(39)

In this case, the notions of markov perfect equilibrium and of closed loop equilibrium coincide, as the closed loop information structure is the state variable of markov perfect strategies.

At the equilibrium, these jumps are made at the very …rst moment of the game. However there exist subgames, which are not on the equilibrium path, in which …rms make several jumps. It is therefore important, for the determination of the markov perfect equilibrium, to permits the …rm to jump at any time, and not just on the …rst moment of the game

.3 In[START_REF] Back | Open-loop equilibria and perfect competition in option exercise games[END_REF],

page 4538, the capacity choosed by …rm i at time t depends of the capacity of its opponents at the same time t.

Documents de travail du Centre d'Economie de la Sorbonne -2015.79

Such interpretation should be taken carefully, as the de…nition of a Stackelberg equilibrium in such a continuous setting is not straightforward.

The results of[START_REF] Chevalier-Roignant | Preemptive Capacity Investment under Uncertainty[END_REF] is established in a more general setting than the model of this work, as they consider a general investment's cost function,with a disinvestment possibility. 5 Documents de travail du Centre d'Economie de la Sorbonne -2015.79

See for example Figuières (2009) or[START_REF] Genc | Capacity Investments in a Stochastic Dynamic Game: Good News Principle[END_REF].

This is the case under markovian or open-loop equilibrium. The choice of the equilibrium concept impacts only the di¤erential equation determining …rms'pro…t. 12 Documents de travail du Centre d'Economie de la Sorbonne -2015.79

and

Let X(k j ; y) be the solution to the implicit equation

By assumption H1, (41) is well de…ned, and, by application of the theorem of implicit function, decreasing in k j and y. Furthermore, k A 2 = X(k j 2 ; p + ) and k A 1 = X(k j 1 ;

(1 ) p + ).

As by assumption k j 2 k j 1 ,

In the following proof, I put the time in index when I consider a variable as a stochastic process and nothing in index when I consider the variable as a constant. Thereby,

is the capacity implemented by …rm i for a level of demand A and a vector of installed capacity k, whereas K i t = Ki (A t ; k t ) is the stochastic process implemented during a period of time.

Theorem 1 is a particular case of theorem 2. In order to prove theorem 2, I de…ne H, the set of state variables such that no …rms jump:

Proof of theorem 2:

The proof is devised in four steps. The …rst step characterizes the optimal strategy and the pro…t function when both …rms belong to H. The second step ensures that the implicit functions expressed in (31) are well de…ned. To do so, I show that the pro…t functions are concave (in H). Third step described the equilibrium, and the last step fully characterizes the pro…t functions using the smooth pasting and value-matching conditions.

Step 1: Characterization of the pro…t functions when (A; k) belongs to H i .

If the initial point (A; k) belongs to H, Bellman theorem applies and:

where D i does not depend of the control I i . If @ i @K i p + , then the optimal control is I i = 0.

, then there is no optimal control available in R, and …rm i wishes to jump. The same works for …rm j, and, H veri…es

Inside H, the best strategy of each …rm is to maintain its capacity constant and thus the …rms pro…t depends only on the evolution of the uncertainty. Therefore, for all i 2 fA; Bg,

The solutions to this di¤erential equation have the form

where v is a solution to:

This permits to pass from two di¤erential equations in ( 46) to a unique one in (48). However, this di¤erential equation can still have many solutions.

Step 2: Concavity of the pro…t functions

The principal di¢ culty of that step is that (48) de…nes v only on H. So I will use a general solution to (48), w, and show that assumption H3 implies the concavity of v.

Let w(A; x) be a solution to (48) for all (A; x) 2 R 2 + . Using Itô Lemma on the function e rt w(A t ; k) gives:

By (48), this equation becomes

which can be rewritten, for h > 0,

and thus

When h ! +1, (51) becomes,

Assumption H5 implies that this integral is well de…ned. Assumption H3 implies that v 0 < 0 and v 00 < 0, so w veri…es assumption H1. Furthermore, on H, w(A; k) is a particular solution to the di¤erential equation ( 48), and, as v is also a solution to (48), there exists a function g(:) such that, on H:

thus v also veri…es assumption H1.

Step 3: Equilibrium outside H.

Assume now that, for one …rm, the optimal strategy is to jump at the initial point (A; k). Let ki be the vector of capacity just after the jump:

If ki = k i , …rm i does not jump. By assumption H2, …rms do not jump just after a jump, and k 2 J. Then, the pro…t of …rm i is equal to:

As v veri…es assumption H1, the implicit theorem ensures the existence of (31). Then, there is two possibilities:

If both …rms have capacity inferior to the Cournot level, as de…ned by (32), …rms invest until the Cournot level.

If …rm i have a capacity higher than the Cournot level, k i > k i C , then its opponents will jump until kj = k j Irr (k i ) whereas …rm i keep its capacity constant. If …rm j has a capacity higher than its best response to …rm i's capacity, k j > k j Irr (k i ) then no …rms invest, and k 2 H. This gives the unicity and the form of equilibrium for any solution of (48). However, v should also verify the smooth-pasting and value matching conditions.

Step 4: Smooth pasting and value-matching condition.

Let k 2 H(A). Let A + be the level of demand such that one …rm invests if the demand increases over this level. Then, if …rm i is a …rm which invests just after A + , the pro…t function of …rm i should verify,

If …rm i does not invest just after A + ,

The smooth pasting condition is thus

which can be written as:

The di¤erentiation of (55) gives the optimal matching condition:

for all i which maximizes (57), which can be written

Equations ( 58), (60), characterizes the solution of the di¤erential equation (48).

At this point of the proof the pro…t function of …rm i can be rewritten:

where v is a solution to the di¤erential equation ( 48 Graphic 1: Second period equilibrium

H

Graphic 2: Second period equilibrium