Thomas Fagart
email: fagart1@hotmail.com.

Markovian Equilibrium in a Model of Investment Under Imperfect Competition

Keywords: Capacity investment and disinvestment, Dynamic stochastic games, Markov perfect equilibrium, Real option games, JEL classi…cation: D43 L13 L25 Investissement en capacité, Jeux di¤érentiel, Equilibre markovien, Option réelle

This paper develops and analyzes a dynamic model of partially irreversible investment under cournot competition and stochastic evolution of demand. In this framework, I characterize the markov perfect equilibrium in which player's strategies are continuous in the state variable. There exists a zone in the space of capacities, named the no-move zone, such that if …rms capacity belongs to this area, no …rm invest nor disinvest at the equilibrium. Thereby, initial asymmetry between …rms capacity can be preserved. If …rms are outside this area, they invest in order to reached the no-move zone. The equilibrium as an e¢ciency property: the point of this area which is reached by the …rms minimizes the investment cost of the all industry.

Introduction

Capacity expansion or reduction under uncertainty is one of the most important decisions that …rms can make. It impacts their immediate pro…t and creates long-run commitment. In a dynamic setting, the investment pattern by a monopoly is well known. Because of uncertainty, the …rm has incentives to delay a pro…table project (in expectation) in order to wait for more information about the demand. This is the theory of real options. What happens in imperfect competition becomes the theory of real option games. (For recent surveys, see for instance [START_REF] Boyer | Real options and strategic competition: a survey[END_REF], [START_REF] Azevedo | Developing real option game models[END_REF], or Chevalier-Roignant et al (2011)).

In this literature several authors focused on capacity decision under uncertainty. In these models, at each time the pro…t made by a …rm depends on its size (i.e. its capacity), the size of its opponents, and a parameter which evolves randomly over time (which can be a parameter of demand or cost, the important point being that future evolutions are unknown). As this parameter evolves, …rms wish to adapt their sizes. Firms can either invest to increase their capacity, disinvest to reduce it, or let their capacity depreciate at its natural rate. Investment is said perfectly reversible when the cost of investing is equal to the cost of disinvesting, in which case …rms can perfectly adapt to the stochastic evolution of the parameter. Such repeated game framework is classic in industrial organization. However, in reality, increasing the size implies hiring new employees, building new factories or o¢ce, buying new equipment, and so on... These investments are usually at least partially sunk, so the …rm's size decisions are not perfectly reversible. Investment is said totally irreversible when …rms cannot reduce their sizes, and investment is partially irreversible when …rms can decrease their size by disinvesting (but with a scrap value inferior to the cost of investing) or by depreciation. In these cases, the theory of real option game links the hysteresis due to the irreversibility of investment and the imperfect competition. This paper studies capacity accumulation when competition is imperfect and investment partially irreversible. It shows that the irreversibility of investment implies a preservation of the asymmetry in …ms capacity: if a …rm owns more capacity than its opponent, the bigger …rm still have a capacity more important than its opponent at the equilibrium. The equilibrium has also an e¢cient capacity: the state of the industry at the equilibrium min-imizes the cost of investment of the all industry. This features are consistent with dynamics and uncertainty.

Severals authors study capacity's choices under imperfect competition and uncertain demand. [START_REF] Abel | Options, the Value of Capital, and Investment[END_REF] and Merhi and Zervos (2007) focus on the monopoly behavior under partially irreversible investment. They exhibit two boundaries, such that the …rm keep its capacity constant when the demand is between the two boudaries, invest when the demand is higher than the left-hand boundary, and disinvest otherwise. This creates an hysteresis on the investment decision. When competition is imperfect, Boyer et al (2004) and [START_REF] Boyer | A dynamic duopoly investment game without commitment under uncertain market expansion[END_REF] assume irreversible investment and lumpy investment. There exists some levels of demand where a …rm has no interest too buy a new unit of capacity if its competitors also install a new unit, but has an interest to do so if its competitor does not add a capacity. This creates di¤erent kinds of equilibria, some in which …rms want to preempt their opponents, and some where they behaves less agressively. When capacity is smooth and investment irreversible, [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] and Chevalier-Roignant, Huchzermeier and Trigeorgis (2011) focus on the choice of the duopoly. [START_REF] Grenadier | Option exercise games: An application to the equilibrium investment strategies of …rms[END_REF] assume that the strategy of a …rm is to invest until a boundary, which depends on the level of demand, and describes the Nash equilibrium with such strategies. [START_REF] Back | Open-loop equilibria and perfect competition in option exercise games[END_REF] show that this equilibrium is open-loop, i.e. it is the Nash equilibrium when the strategy of a …rm is its path of capacity and …rms can commit to their strategy. Usually, Open-loop equilibria are not sub-game perfect, and Chevalier-Roignant, Huchzermeier and Trigeorgis (2011) focuse on markovian equilibrium. The authors describe the optimal markovian best response of the …rms. However, the linearity of the investment cost implies in…nite value for the amount of investment (the capacity jumps, as there is no interest to delay purchases or sells of capital), which prevents the characterization of the markovian equilibrium. In order to …ll this gap, I introduce a possibility of jumps in the formulation of the strategies and characterizes the markovian equilibrium, in case partially irreversible investment.

To do so, I …rst study the simplest investment game. In a one shot model, …rms have initial capacities and can invest or disinvest in a partially irreversible way (…rms can buy or sell capacities at linear but di¤erent prices). I exhibit an area in the space of capacities, named the no-move zone. If the capacities of the …rms are inside this no-move zone, no …rm will neither invest nor disinvest. So any point of this no-move zone is a possible equilibrium, given some initial capacities. For some given initial capacities, the equilibrium is the point of the no-move zone which minimizes the costs of investment and disinvestment for the all industry. This e¢ciency result holds even though …rms have a priori no interest in coordinating their decisions. Furthermore, as long as the prices of investment and disinvestment are not equal, the no-move zone is not reduced to one point, and an initial asymmetry in capacities can be preserved.

In continuous time, the linearity of investment cost gives an incentive to invest as soon as possible. This creates a di¢culty of modeling the strategies of the …rms, as the desired ‡ow of investment is in…nite. In that context, I introduce a possibility of jump for the …rm and solve the markovian equilibrium of this game. There exists a no-move zone.

At each time …rms reach the point of the no-move zone which minimizes the industry costs of investment and disinvestment. In this sense, they behave as in the one shot game. This result is valid when demand evolves in a deterministic or a stochastic way.

The outline of this paper is straightforward. Section 2 studies the one-shot model.

Section 3 presents the dynamic model and characterizes the markovian equilibria. Section 4 concludes. All omitted proofs are reported in appendix.

2 Investment in the one-period game

The one-shot game model

The aim of this subsection is to abstract from dynamics and uncertainty issues, in order to focus on the e¤ect of partial irreversibility of investment. To do so, I present a simple static model of competition in capacity.

More precisely, consider a market with n …rms competing à la Cournot in capacities.

Each …rm i starts with some amount of capital k i , which can be extended or reduced through buying or selling some assets. Purchases are made at a linear price p + , and sales at a (also linear) price p (with p p + ). I call K i the capacity …nally installed by …rm i. Let k be the vector of industry's initial capacities and K the vector of installed capacities. For …rm i, the cost of installing a new capacity is:

C(K i ; k i ) = 8 < : p + (K i k i ) if K i k i p (K i k i ) if K i < k i 9 = ; : (1)
Firms produce and sell an homogenous good, at a price depending of the total quantity q = P n i=1 q i . Each …rm's production depends of its capacity, according to the technology q i = K i , 1 and has a cost, c i (q 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], Merhi and Zervos (2007) among others 2 . So, by selling the quantity K i , …rm i obtains a payo¤ of:

i (K) = P K K i c i (K i );
(2)

where K = P n j=1 K j . The pro…t function of …rm i is thus:

i (K; k) = i (K) C(K i ; k i ): (3)
Note that if p = p + , the investment decision is totally reversible, and the initial capacity has no impact. The smaller is p , the more irreversible are the capacities of the …rm, and, at the limit, when p = 1, investment is totally irreversible, as in [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], [START_REF] Boyer | A dynamic duopoly investment game without commitment under uncertain market expansion[END_REF], and others.

In order to ensure the existence of the equilibrium I make the following hypothesis:

H1: For each i = 1; ::; n, c i (:) is a twice-di¤erentiable positive function such that c 0 i 0, c 00 i 0. P (:) is also a twice-di¤erentiable positive function, with P 0 < 0, P 00 < 0 when P is 1 In the one-shot model, this technology can be seen as the result of an endogenized game, in which …rms buy capacities and then play a Cournot competition limited by the capacity previously bought. Indeed, …rms have no interest to invest in capacity which will not be used to produce, as its opponents only react to the …nal quantity. (Except if the disinvestment price is negative. In this case a …rm could whish to keep its unused capacity in order to avoid a disinvestment case. For example, this is sometimes the case for polluted production sites, for which the cost of decontamination is more important than the cost of conservation of this asset.) 2 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. However, when there is uncertainty, …rms have an incentive to keep their unused capacity for a possible further use, when 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.

strictly positive. 3 Furthermore, for all i = 1; ::; n and q 2 R n + , P 00 ( q)q i < c 00 i (q i ).

Best responses

This part describes the best response of …rm i. Assume that for all j 6 = i, …rm j installs a capacity K j . Obviously, the marginal revenue of capacity for …rm i depends of the choices of its opponents, and I note @ i @K i 1 (x) the inverse function of the marginal revenue of capacity, so

@ i @K i 1 (x) = K , @ i @K i K 1 ; ::; K i = K; ::; K n = x.
By concavity of in K i , such inverse function is well de…ned and increasing. Then, …rm i has three possible choices:

to invest (K i > k i ), so making a pro…t:

i = i (K) p + K i k i ;
leading to an optimal choice of capacity

@ i @K i 1 (p + k ). Hence, if k i > @ i @K i 1 (p + k )
, increasing its capacity decreases …rm's pro…t, and the …rm has no interest to invest.

to disinvest (K i < k i ), so making a pro…t:

i = i (K) + p k i K i ;
leading to an optimal capacity @ i @K i 1 (p ). Thus, the …rm has no interest in disinvesting if

k i < @ i @K i 1 (p ) (which is higher than @ i @K i 1 (p + ) ).
the last possibility is to do nothing (the …rm neither invest nor disinvest). Indeed, as @ i @K i 1 (:) is increasing, the initial capacity of the …rm can be greater than @ i @K i 1 (p + k ), so the …rm has no interest to invest more, but also smaller than @ i @K i 1 (p ), so the …rm has also no interest to disinvest.

Therefore there exists two thresholds, @ i @K i 1 (p ) and @ i @K i 1 (p + ) such that the …rm does not wish to invest nor disinvest if its capital is between this thresholds, invest until @ i @K i 1 (p + ) if its initial capacity is small, and disinvest until @ i @K i 1 (p ) if its initial capacity is large. This can be summarized in the following proposition:

Proposition 1: The best response of …rm i is:

K i BR = 8 > > > < > > > : @ i @K i 1 (p + ) if k i < @ i @K i 1 (p + ) k i if k i 2 h @ i @K i 1 (p + ); @ i @K i 1 (p ) i @ i @K i 1 (p ) if k i > @ i @K i 1 (p ) 9 > > > = > > > ; : (4)
Of course, this best response depends of the capacities of other …rms, as @ ~ i @K i depends of the capacity of all …rms. Graphic 1 represents the best response of …rm 2 in the space of capacity, for a duopoly with linear demand and no production costs.

[Insert G1]

In this graphic, we can see the existence of an area in the space of capacities, 2 , bounded by @ 2 @K 2 1 (p ) and @ 2 @K 2 1 (p + ), such that, it is never optimal for …rm 2 to be outside this area. Thus, if there is an equilibrium, it belongs to i for all …rm i, so it belongs to the intersection of these areas. Let H be this intersection. We know that all equilibria belong to H. Furthermore, assume that the initial distribution of capacity belongs to H. Then, if all players except i keep their capacity constant, then the best response of i is to keep its initial capacity. In the case of duopoly with linear demand and no production cost, this can be seen in graphic 2. The area H is called the no-move zone.

[Insert G2]

Characterization of the equilibrium

In this subsection I show the existence of an area in the set of space capacities, the no-move zone H, such that all equilibria belong to H. This no-move zone is de…ned by,

H = K 2 R n + : 8i 2 f1; ::; ng ; @ i @K i (K) 2 [p ; p + ] , (5)
which can rewritten as

H = K 2 R n + : 8i 2 f1; ::; ng ; P K + P 0 K K i c i (K i ) 2 [p ; p + ] . ( 6
)
If the initial capacities of the …rms are in H, the equilibrium is to keep the same capacities.

If the initial capacities do not belong to H, the equilibrium is described by theorem 1.

Theorem 1: Assume H1. Then, there exists only one Nash equilibrium K , which statis…es:

K = arg min K2H n X i=1 C(K i ; k i ): (7)
This condition is equivalent to the distance condition,

K = arg min K2H n X i=1 K i k i . ( 8
)
Theorem 1 provides existence and uniqueness for the equilibrium, and its characterization. To understand this characterization, let focus on the no-move zone. When investment is totally reversible (when p + = p ) the no-move zone is reduced to a unique point, and no industry e¢ciency appears. This is the usual Cournot competition 4 . When investment is not totally reversible, the no-move zone is a set, not reduced to a singleton. In this area the marginal revenue of each …rm is inferior to the price of adding a new capacity, but superior to the price of selling some capacity, so no …rms wish to invest nor disinvest. The size of the no-move zone is thus an increasing function of the irreversibility of investment. 5 By theorem 1, H is also the set of all possible equilibria, in the sense that all equilibrium belongs to H and all point of H can be an equilibrium for some initial value.

Inside the no-move zone, there is a property of no-reaction to competition. Indeed, assume that the …rms decide to install a vector of capacity which belongs to H. Then, if a …rm changes its level of capacity (and if the vector of capacity still belongs to H), its opponents will not invest nor disinvest in reaction to this change of capacity. This is due to the fact that if the evolution of the …rm capacity has an impact on its opponent's revenue, this impact is not su¢cient to generate a change in capacity due to the irreversibility of investment.

Inside the no-move zone, each …rm is thus myopic, and maximizes its pro…t assuming that the other …rms have constant capacities. Theorem 1 shows that this myopicity leads to a market e¢ciency. Indeed, the equilibrium is the point of the no-move zone which minimizes the industry cost of investment.

Graphic 3 presents which points of the no-move zone will be an equilibrium, in function of the initial capacities, for a duopoly with linear demand and no production costs. As it can be seen on the graphic, …rms with di¤erent initial capacities can still be asymmetric at the equilibrium, and there are di¤erent possible symmetric equilibria, even when …rms have the same pro…t function.

[Insert G3]
3 Investment in a dynamic game

The dynamic model

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. There exists a no-move zone and the …rms join it as soon as possible minimizing the investment cost of the all industry. This permits to characterize the equilibria.

Let k i t be the capital of …rm i at time t, time is continuous and capacity partially reversible. Let i (A t ; k t ) be the instantaneous payo¤ of …rm i, k t being the vector of …rms capacities and A t the parameter of uncertainty, following a di¤usion process:

dA t = (A t )dt + (A t )dW t ; (9)
where W t is a standard Wiener process. I assume Cournot competition. Let P At (:) be the inverse demand function, depending of the level of demand A t , and let c i (:) be the production cost of …rm i, then the instantaneous revenue of …rm i is

i (A t ; k t ) = P At k t k i t c i (k i t ), (10)
where k t = P n i=1 k i t , as previously. The interest rate is r and the purchase price of capacity p + , whereas the selling price is p . For each stochastic process k i t , let k i+ t and k i t be the respectively increasing and decreasing processes such that6 :

k i t = k i+ t + k i t . (11)
The total expected pro…t of …rm i at time 0 is thus:

i = E Z +1 0 e rt i (A t ; k t )dt p + Z +1 0 e rt dk i+ t + p Z +1 0 e rt dk i t j A 0 . ( 12
)
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 and the equilibrium concept used by the players. I focus on markov perfect equilibria. The strategy of a player is a function of a state variable, representing the level of the demand and …rms capacity at time t, which determines the futur evolution of the player capacities. Markov perfect equilibria have the advantage to be sub-game perfect, and to avoid time inconsistency.

In our framework, there is a di¢culty to de…ne the strategic variable. Indeed, the linearity of investment cost might create an incentive to install the desired capacity as soon as possible. This creates a jump in the capacity process of the …rms. However, the usual theory of di¤erential games does not allow the capacity processes to jump (see also [START_REF] Back | Open-loop equilibria and perfect competition in option exercise games[END_REF] for the same observation on the de…nition of strategies). Indeed, in di¤erential games, the evolution of the state variable is usually determined by I i t , the investment done by …rm i at date t, so that the capital of each …rm is determined by the following di¤erential equation:

@k i t @t = I i t . ( 13
)
This de…nition implicitly assumes that the capacity is a continuous function of the time.

A markovian strategy for …rm i being a function of the state variable (demand level and capacities),

I i t = Ĩi (A t ; k t ).
As it can be seen in Chevalier-Roignant, Huchzermeier and Trigeorgis (2011), the Bellman formula gives in that case:

r i (A; k) = sup I i 8 < : P A k k i p + (I i ) + p (I i ) + I i @ i @K i + (A) @ i @A + 2 (A) 2 @ 2 i (@A) 2 + P n j=1 j6 =i Ĩj (A; k) @ i @K j + P n j=1 j6 =i P n h=1 j6 =i Ĩh (A; k) Ĩj (A; k) @ 2 i @K j @K h 9 = ; . ( 14
)
The optimal investment policy maximizes @ i @K i I i p + (I i ) + p (I i ) . So, if @ i @K i 2 [p ; p + ], the …rm i has no interest to invest nor to disinvest. Otherwise, the optimal ‡ow of investment I i t is in…nite: the …rm installs its optimal capital (capital in the region

@ i @K i 1 ([p ; p + ])) instantly.
The optimal capital policy of the …rm is thus to jump in the area @ i @K i 1 ([p ; p + ]), and to do nothing as long as the capital stays in this area. So the optimal strategy cannot be de…ned by the investment variable, as the linearity of the cost of investment implies non-continuous capital strategies.

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 markovian state-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 n t ), is de…ned by the two equations:

@ + k i t @t = I i t , (15)
and

lim s!t s>t k i s = K i t , ( 16
)
where k 0 is the given initial level of capital, and @ + @t denotes the right-hand derivatives. 7 (iii) 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 (A t ; k t ) and I i t = Ĩi (A t ; k t ). (17)
Equation ( 16) states than the installed capacity k i t is continuous if and only if

K i t = k i t .
When the strategies are markovian, K i t is a function of the state variable, and …rm i jumps as long as k i t 6 = Ki (A t ; :). Graphic 4 presents the evolution of the capacity of …rm i when player's i strategy is

Ĩi (A t ; k t ) = 1, Ki (A t ; k t ) = 1 fk i t <2g k i t + 1 fk i t 2g ,
and A t and k j t j6 =i are constant. As long as the installed capacity, k i t , is inferior to two, the …rm invests continuously, and when installed capacity reaches two, the desired capacity, K i t di¤ers from the installed capacity, and the …rm disinvests instantly a unit of capacity.

[Insert G4]

The de…nition of the markovian state-control form extends the possibility of the stochastic processes of capacity from the class of continuous processes implicitly assumed by (13) to the class of the left-hand continuous stochastic processes with right-hand derivative.

However, this de…nition allow multiple jumps: a jump of …rm i can imply a jump of its 7 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 -2014.39

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.

C1: The strategies of the …rms i veri…es assumption C1 if, for all

(A; k) 2 R R n + , K A; K(A; k) = K(A; k). (18)
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 (:; :) ;

8(A; k) 2 R R n + ; i (A; k t ) i (A; k 0 t ), (19)
where k t is the process de…ned by ( 15) and ( 16) and the strategies K (:; :) ; Ĩ (:; :) , and k 0 t is the processes 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 markovian strategy. In this context, it permits to properly de…ne the best responses of the …rms. In the next section, proposition 2 veri…es that the best responses when …rms can jump are the same than when …rms can not jump. In addition, this de…nition permits to characterize the markov perfect equilibrium when I assume that the strategies are continuous functions of the state variable. Theorem 2 shows that the result of the one-shot game is preserved in this dynamic framework.

Characterization of the continuous markov equilibrium

In this subsection, I characterize the continuous markovian equilibria. I start by introducing technical assumptions. H2 is needed to prove proposition 2 (in order to use Ito's Lemma, to inverse the Ito's Lemma results and to apply theorem 1). H3 is classic to ensure the existence of a strong solution to (9). H4 ensures the existence of the stochastic integral determining the pro…t of the …rms.

H2:

For each i = 1; ::; n, c i (:) is a four times di¤erentiable positive function such that c 0 i 0, c 00 i 0, and for all A 2 R, P (:; :) is also four times di¤erentiable positive and strictly concave function in each variable. Furthermore, for all i = 1; ::; n, P 00 (A; q)q i < c 00 i (q i ). H3: (A) and (A) are continuous functions, and verify the Lipschitz conditions.

H4: 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.
These assumptions allow to state proposition 2, which gives the form of the best response in the markovian state-control form of the game.

Proposition 2: Assume H2, H3, H4 and C1. Let i 2 f1; ::; ng. In the markovian state-control form of the game, assume that for all j 6 = i, the strategy of …rm j Kj (:; :) is a continuous function of the state variable. Then there exists a continuous decreasing function : R + ! R + such that the best response for …rm i, Ki BR (:; :); Ĩi BR (:; :) veri…es:

Ki BR (A; k) = 8 > > < > > : (p ) if k i > (p ) k i if k i 2 [ (p + ); (p )] (p + ) if k i < (p + ) 9 > > = > > ; ; (20)
and Ĩi BR (A; k) = 0 if k i 2] (p + ); (p )[. The best response of …rm i is continuous in the state variable.

This proposition shows that the optimal capacity of …rm i can jump: if at some time t, k i t is strictly smaller than (p + ), then the …rm has interest in investing instantly up to (p + ). In this case the investment in period t is in…nite, so the markovian statecontrol form gives the same result as the regular form. However it also allows to go a step further and to characterize the equilibria, as presented in theorem 2. In fact, at each time, everything happens as in the one-shot game presented in the last section (with of course some modi…cation of the no-move zone H in order to take into account the future pro…t).

Firms always want to invest or disinvest forthwith in order to reach the no-move zone. As long as they are in the no-move zone, no …rms change its capacity.

The value of the zone H depends of the expected pro…t of the …rms. When …rms are inside the no-move zone, the evolution of …rms pro…ts only depends on the evolution of prices. The expected price, v, should be 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): (21)
The no-move zone is then de…ned by:

H v (A) = k 2 R n + j 8i 2 f1; ::; ng : v A; k + @v @k A; k k i 1 r c 0 i (k i ) 2 [p ; p + ] . ( 22
)
When a …rm reach the boundaries of the no-move zone, the marginal pro…t equals the price of investment (or disinvestment). This equation is the result of the optimization of the …rm, and I can derive this equation with respect to the level of demand. This gives a set of condition which should be veri…ed by the pro…t functions.

Assume that k 2 H v (A), and that inf i2f1;::;ng

v A; k + @v @k A; k k i 1 r c 0 i (k i ) = p , ( 23
) or sup i2f1;::;ng v A; k + @v @k A; k k i 1 r c 0 i (k i ) = p + . ( 24
)
Then, for each i which minimizes (23) or maximizes (24), the equation

@v @A A; k + k i @ 2 v @A@k A; k = 0 (25)
should be valid. In the theory of real option, these equations are known as smooth-pasting and value-matching condition. ( 23) is associated to the option to invest and (24) to the option to disinvest. The conditions are not necessarily active for all the …rms. Indeed, the value of waiting (i.e. the pro…t maid by a …rm if no one invest nor disinvest) depends on the capacity holds by the …rm and on its production cost.

Observe that equations ( 21), ( 23), ( 24) and ( 25) do not always de…ne an existing or a unique solution. However, when the solution is unique, theorem 2 implies that the continuous markovian equilibrium is unique. Furthermore, each continuous markov perfect equilibrium is characterized by theorem 2.

Theorem 2: Assume H2, H3, H4 and C1. Then, for all continuous markov perfect equilibrium K (:; :) ; Ĩ (:; :) , there exists a solution v to the equations ( 21), ( 23), ( 24) and ( 25), such that the equilibrium veri…es that, for all (A; k) 2 R R n + ; i 2 f1; ::; ng:

K (A; k) = arg min K2Hv(A) n X i=1 C(K i ; k i ), (26)
and

Ĩ (A; k) = 0 for all k 2 H v (A).
Condition ( 26) is equivalent to the distance condition,

K (A; k) = arg min K2Hv(A) n X i=1 K i k i . (27)
Furthermore, for each solution v, the strategies de…ned by ( 26) form a continuous markovian equilibrium.

Theorem 2 is the analog of theorem 1, but in a continuous time setting. It characterizes the continuous markovian equilibria. At each time, …rms invest (or disinvest) in order to join the no-move zone at the smallest possible cost for the industry. The e¢ciency of investment observed in the one-shot model is thus still valid in a dynamics and uncertain framework.

This theorem focuses on the continuous markovian equilibrium. Nevertheless, one can ask about other markovian equilibria. Indeed, there is a priori no reason that collusion can not be sustained by markovian strategies in a di¤erential game. In a companion paper on the same investment game, [START_REF] Fagart | Evolution of Firms Size Under Imperfect Competition and Collusion[END_REF], I exhibit a markovian equilibrium with tit-for-tat strategy implementing the monopoly pro…t.8

Conclusion

In this work, I characterize the continuous markovian equilibria of a model of partially irreversible investment in capacity under uncertainty and Cournot competition. There exists an area in the space of …rms capacities, the no-move zone, such that …rms invest or disinvest in order to join this area as soon as possible, and keep their capacity constant when they are inside this area. The existence of this area is due to the irreversibility of investment, and when investment is perfectly reversible, the no-move zone is reduced to a single point, as in usual Cournot competition. When …rms have a similar pro…t function, but di¤erent initial capacities, their asymmetry can be preserved, due to the irreversibility of investment.

At the equilibrium, …rms install the capacities vector of the no-move zone which minimizes the cost of investment of the all industry. The intuition on this result is that the no-move is the area where other …rms' actions do not impact the optimal action of a …rm, so the e¢ciency of the equilibrium comes from the absence of reaction to competitor action inside the no-move zone.

Appendix

Proof of theorem 1:

First, note that the pro…t function (3) veri…es the hypothesis of theorem 3 of [START_REF] Novshek | On the Existence of Cournot Equilibrium[END_REF], so there exists an unique Nash equilibrium. Let

K = arg min K2H n X i=1 K i k i : (28)
Then, if I show that K is an equilibrium, it is unique. As (28) has always a solution, this will prove theorem 1.

The …rst step of this proof introduces some notation. For any i 2 f1; ::; ng, I will assume that Ki is not the best response to Kj j6 =i

. The second step assume that Ki > k i and introduce a vector of capacity K, which belongs to H and is closer to k than K.

This contradicts (28) and implies that Ki is a best response when Ki > k i . The third step presents the same reasoning when Ki < k i and concludes. The last step shows the equivalence between the two conditions ( 7) and (8).

Step 1: Some de…nition.

Let x i = G i (x); G + i (x) be the implicit functions (these functions are well de…ned and C 1 by assumption H1) de…ned by:

P (x) + P 0 (x)x i c 0 i (x i ) = p , P (x) + P 0 (x)x i c 0 i (x i ) = p + . ( 29
)
Observe that the best response of …rm i veri…es:

K i = 8 > > < > > : G + i ( K) if k i < G + i ( K) k i if G + i ( K) k i G i ( K) G i ( K) if k i > G i ( K) 9 > > = > > ; . ( 30
)
Thus, K is an equilibrium if and only if (30) holds for all i. The no-move zone can be rewritten as

H = K 2 R n + j 8i 2 f1; ::; ng ; K i 2 G + i ( K); G i ( K) . ( 31
)
Step 2: Ki is …rm's i best response when Ki > k i .

This step is subdivided in three intermediate stages.

K j = 8 > > < > > : Ki " if j = i Kj + " z+1 if j 2 Kj elsewhere 9 > > = > > ; : (32)
For " small enough to have K i > k i , observe that:

n X h=1 K n k n = n X h=1 Kn k n " z + 1 . ( 33
)
To contradict the fact that Ki is not …rm's i best response, I just need to show that Ki > G + i (

P n h=1 K h ) implies that K 2 H. Observe that n X h=1 K n = n X h=1 Kn " z + 1 . ( 34
)
By hypothesis H1, G u0 j (x) = P 0 (x)+P 00 (x)x j P 0 (x) c 00 i (x j ) < 0 and, as P 00 (x)x j c 00 j (x j ), G u0 j > 1, u 2 f+; g. By Taylor, there exists u j 2]0; 1[ such that

G u j n X h=1 K h ! = G u j n X h=1 Kh ! + u j " z + 1 . ( 35
)
Furthermore, when " is small enough,

G j n X h=1 K h ! > G + j n X h=1 Kh ! + " z + 1 . ( 36
)
Thus, K belongs to H if and only if

G + j n X h=1 K h ! K j G j n X h=1 K h ! , (37)
for all j 2 f1; ::; ng.

When j = 2 , I have K j Kj G j P n h=1
Kh < G j P n h=1 K h . If j 2 ; from (36) and ( 35),

K j = Kj + " z+1 = G + j P n h=1 Kh + " z+1 G j P n h=1 Kh G j P n h=1 K h . Thus K j G j P n h=1 K h for all j. When j 2 , K j = Kj + " z + 1 = G + j n X h=1 Kh ! + " z + 1 = G + j n X h=1 K h ! + " z + 1 u j " z + 1 G + j n X h=1 K h ! . ( 38
)
The …rst equality comes from (32), the second one from the de…nition of , and the third one from (35). The inequality holds as u j 2]0; 1[. Finally, when j = 2 , by construction of

, Kj > G + j P n h=1
Kh , so, for " small enough, I have:

Kj G + j n X h=1 Kh ! + + j " z + 1 + ".
Thus, by ( 35), K j G + j P n h=1 K h , and K 2 H.

Step 3: Ki is …rm's i best response when Ki < k i .

As step 2, this step is subdivide in three intermediate stage.

When Ki < k i , Ki is …rm's i best response if and only if Ki = G i ( P n h=1 Kh ).

Assume that Ki is not …rm's i best response. Then, as K 2 H, Ki < G i ( P n h=1 Kh ).

Let = n j : Kj = G j P n h=1
Kh o , z = card( ) and K de…ned by:

K j = 8 > > < > > : Ki + " if j = i Kj " z+1 if j 2 Kj elsewhere 9 > > = > > ; : (39)
Observe that, for small ":

n X h=1 K n k n = n X h=1 Kn k n " z + 1 ; (40) and n X h=1 K n = n X h=1 Kn + " z + 1 : (41)
Now, ( 35) and ( 36) became, for u j 2]0; 1[:

G u j n X h=1 K h ! = G u j n X h=1 Kh ! u j " z + 1 . ( 42
)
Furthermore, when " is small enough, 39) and (42) gives:

G j n X h=1 K h ! > G + j n X h=1 Kh ! + " z + 1 . ( 43
) When j = 2 , we have K j Kj G j P n h=1 Kh > G j P n h=1 K h . If j 2 ; from (43) and (42), K j = Kj " z+1 = G j P n h=1 Kh " z+1 G + j P n h=1 Kh G j P n h=1 K h . Thus K j G j P n h=1 K h . When j 2 , (
K j = Kj " z + 1 G j n X h=1 Kh ! " z + 1 = G j n X h=1 K h ! " z + 1 + u j " z + 1 G j n X h=1 K h ! ,
whereas, when j = 2 , for " small enough, we have:

Kj G j n X h=1 Kh ! + j " z + 1 ",
and thus, K j G j P n h=1 K h . Therefore K 2 H. Observe that if Ki = k i , k i 2 H as Ki 2 H, and ( 30) and (31) implies that Ki is …rm i best response. This prove that the K is an equilibrium.

Step 3: Equivalence between the two equilibrium characterization ( 7) and (8).

To show the equivalence between ( 7) and ( 8), it su¢ces to remark that equations ( 33) and ( 40) become respectively

C( K; k) = C( K; k) p + 1 z + 1 " (44)
and

C( K; k) = C( K; k) p 1 z + 1 " (45)
with the investment cost function. Thus, the same reasoning applies.

In the following proofs, 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, K i = Ki (A; k) 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.

Proof of proposition 2:

This proof is made in three step. The properties of the pro…t function exhibited in step 1 are used in step 2 to …nd the optimal control when …rm i does not jump. Step 3 describes the optimal jump and concludes.

Step 1: Properties on i .

Assume that the optimal control, K i (:) is a continuous function. Then, as Kj (:) is also continuous, the path k t is continuous (exept in 0). Indeed, when h ! 0,

k t+h = K(k t+h ) ! K(k t ) = k t . (46)
Thus, if there is a jump, it is done only at time 0, and after that the …rm uses it continuous control, I i , to adapt continuously the state variable k i t . Let J be the set of intial conditions such that …rm i does not jump,

J = n (A; k) j K i (A; k) = k i o . ( 47
)
By assumption C1, this space is non empty. If (A; k) belongs to J, Bellmans veri…cation theorem applies as the state space variable evolves continuously, and the objectif function, i , is C 2 . Furthermore, i is an increasing function of k i . Indeed, assume that k and k 0 are two initial conditions such that k i > k 0i and k j = k 0j if j 6 = i. Then, let k 0 i t be the optimal path induced by k 0 . If the initial condition are (A; k), …rm i can always imitate the optimal strategy coming from (A; k 0 ), investing according to k 0i instantly and following k 0 i t after. In this case,

i (A; k) i (A; k 0i ) + p k i k 0 i . ( 48
)
Thus, i is an increasing function.

Step 2: Optimal strategy when the …rm does not jump.

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

r i (A; k) = max I i i (A; k) p + I i + p I i + I i @ i @K i + D i , (49)
where D i does not depend of the control I i . If @ i @K i 2 [p ; p + ], then the optimal control is I i = 0. If @ i @K i = 2 [p ; p + ], then there is no optimal control available in R, and in that case, (A; k) = 2 J, i.e. …rm i wishes to jump.

Step 3: Optimal strategy when the …rm does jump.

Assume now that the optimal strategy of …rm i is to jump at the initial point (A; k).

Let K i be the level of the jump and ki be the vector of capacity at the instant of the jump of …rm i:

k i = K i , k j = Kj (k) for j 6 = i. (50)
Then, the pro…t of …rm i is equal to:

i (A; k) = i A; k C ki ; k i 0 . ( 51
)
After the jump, A; k belongs to J and thus i is derivable in A; k . As i is increasing, the solution to (51) is unique and given by:

K i = 8 > > < > > : (p ) if k i > (p ) k i if k i 2 [ (p + ); (p )] (p + ) if k i < (p + ) 9 > > = > > ; , (52)
As g + w veri…es assumption H1, I can apply theorem 1. This gives the characterization of the equilibrium (26), and the form of zone H: H(A) = k 2 R n + j 8i 2 f1; ::; ng : g(A) + w(A; k) + w 0 (A; k)

k i 1 r c 0 i (k i ) 2 [p ; p + ] . ( 77
)
Of course, by ( 74), this can be rewritten as:

H(A) = k 2 R n + j 8i 2 f1; ::; ng : v(A; k) + v 0 (A; k)k i 1 r c 0 i (k i ) 2 [p ; p + ] . ( 78
)
Step 5: Equation ( 21), ( 23), ( 24), ( 25) and ( 26) de…ne an equilibrium.

Assume that there exists a solution v to the equations ( 21), ( 23), ( 24), and (25), and let K(:; :) be the equilibrium de…ned by (26) (with Ĩ = 0). For all (A; k) 2 R R n + , let K = K(A; k) be the vector of desired capacities. Then, for all i 2 f1; ::; ng, I show that K i is the best response of …rm i to (K j ) j6 =i . Indeed, assume that (K j ) j6 =i is …xed, then, by ( 26), the strategy of …rm i veri…es, Ki (A; k) = arg max

K i 2H i v (A) C(K i ; k i ), (79)
where

H i v (A) = K i 2 R + j v(A; K) + v 0 (A; K)K i 1 r c 0 j (K i ) 2 [p ; p + ] : (80)
As v is a solution to (21), v veri…es H1 by step 3. Thus, for all x 2 [p ; p + ], there exists a unique solution to the implicit equation,

v(A; K) + v 0 (A; K)K i 1 r c 0 j (K i ) = x, (81)
as a unique solution K i = (x). In that case, the strategy of …rm i is de…ned by

K i = 8 > > < > > : (p ) if k i > (p ) k i if k i 2 [ (p + ); (p )] (p + ) if k i < (p + ) 9 > > = > > ; : (82)
Furthermore, even if K(:; :) is not an equilibrium, …rms does not invest nor disinvest inside the no-move zone, and the reasoning of step 1 and step 2 applies. Thus, for k 2 H(A), @ i @k i = v(A; k) + v 0 (A; k)k i 1 r c 0 j (k i ).

(83)

Graphic 1: best response of firm 2

0, 0 K 2 K 1 K BR 2 K 1   2   2 K 2 1 p k     2 K 2 1 p k  
Documents de Travail du Centre d'Economie de la Sorbonne -2014.39

Graphic 2: No-move zone and equilibrium

0, 0 k 1 k 2 H  2 K 2 K 1 K BR 2 K 1  K BR 1 K 2   1 K  

  The …rst stage constructs the candidate point for the contradiction, K. The second stage shows that K 2 H if and only if (37) is valid. The third stage veri…es that inequalities of (37) are valid.First, remark that, as Ki > k i , Ki is …rm's i best response if and only if Ki =

	G + i ( G + i ( Let = P n Kh ). Assume that Ki is not …rm's i best response. Then, as K 2 H, Ki > h=1 P n h=1 Kh ). n j : Kj = G + j P n h=1 o Kh , z = card( ) and K de…ned by:

Documents de Travail du Centre d'Economie de la Sorbonne -2014.39

Documents de Travail du Centre d'Economie de la Sorbonne -2014.39

Theorem 1 (presented page 9) holds under less restrictive conditions. Indeed, our proof rests on the third theorem of[START_REF] Novshek | On the Existence of Cournot Equilibrium[END_REF], and the linearity of cost of investment and disinvestment. However, as the point of interest is the dynamic game, and as some regularity is needed for the existence of the dynamic di¤erentiable equations, I place ourselves in the assumption made by[START_REF] Szidarovszky | A new proof of the existence and uniqueness of the cournot equilibrium[END_REF].

With a cost c i (K i ) + p + K i .

More precisely, when p + increases and p decreases, the size of no-move zone increases according to (5).When both p + and p increases (or decreases), the evolution of the size of the no-move zone is unknown even if the di¤erence p + p increases, as it depends of the marginal revenue of each …rm.

This de…nition is valid for all left-continuous A t -adapted stochastic process with …nite variation. The assumption of …nite variation is natural with the investment cost considered. Indeed, a …rm with an in…nite variation of its capacity will pay an in…nite cost of investment and disinvestment (as p + > p ). However, its future revenue is …nite (due to hypothesis H4), and thus a strategy with in…nite variation would lead to a negative and in…nite pro…t, which is obviously not optimal.

Such collusive equilibria arise because we assume that the strategy of the players rests on the state variable. In the classic modeling of di¤erential games, the strategies of the players leads on the derivative of the state variable, which imposes that the state variable will evolve continuously with time. As shown by theorem 2 in our case, this continuity assumption provides the …nding of collusive equilibria when the strategy of the player rests on the derivative of the state variable. If the …nding of collusive equilibria is an argument in favor of our modeling, it also asks why the monopoly pro…t can not be implemented by continuous strategies.

where is the implicit function de…ned by

for k i = (x) and k j = k j . In that case, J = [ (p + ); (p )], and the strategy is

Remark that if i is C 2 for all its variables, D i is given by

Proof of Theorem 2:

The idea of proof is to characterizes the pro…t function inside the no-move zone (where the optimal …rms strategies are to keep their capacities constant), and to apply theorem 1 in order to characterize the equilibria.

Let K (:; :) ; Ĩ (:; :) be a continuous markov perfect equilibrium. For all A 2 R, let H(A) be the no-move zone de…ned by,

By assumption C1, this no-move zone is not empty. Proposition 2 states that when the installed capacity k does not belong to H, …rms wish to invest or disinvest in order to install a vector of capacity which belongs to H. The …rst two steps of the proof characterize the pro…t of the …rm inside the no-move zone. The …rst step gives the dynamic equation veri…ed by the pro…t functions. The second step presents the smooth-pasting and value matching conditions. The third step establishes that the application requirements for theorem 1 are valid.

Step 4 applies theorem 4, gives the form of the no-move zone and characterizes the equilibrium. Finally, step 5 shows that every strategies de…ned by equations ( 21), ( 23), ( 24), ( 25) and ( 26) is an equilibrium.

Step 1: The pro…t functions as a solution to a unique di¤erential equation, (58).

Assume that k 2 H. In this case, 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 f1; ::; ng,

Furthermore, when k belongs to H, the equilibrium strategies are K (A; k) = k, and thus, by ( 49):

Cauchy theorem shows that the space of solutions to a second degree di¤erential equation is a two-dimensional vector space. The set of solution to ( 57) is thus a priori two-n-dimensional.

However, (57) can be reduced to a unique equation, and the set of solutions to a twodimensional space. Indeed, if v(:; :) is a solution to

the pro…ts de…ned by 8i 2 f1; ::

is solution to the equations (57). Reciprocally, a solution to (57) de…nes a solution to (58).

Step 2: 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 sup i2f1;::;ng

which can be written as:

The di¤erentiation of (60) gives the optimal matching condition:

for all i which maximizes (62), which can be written

The same reasoning applies for disinvesting when the level of demand decreases, and in that case the smooth-pasting and value matching conditions are:

Equations ( 63), ( 65), ( 66) and ( 67) characterizes the solution of the di¤erential equation ( 58).

At this point of the proof, we have seen that the pro…t function of …rm i, i 2 f1; ::; ng, can be rewritten:

where v is a solution to the di¤erential equation ( 58), which veri…es ( 63), ( 65), ( 66) and (67).

Step 3: Veri…cation of assumption H1.

The principal di¢culty of that step is that (58) de…nes v only on H. So I will use a general solution to (58), w, and show that assumption H1 is valid for this function, before to bring me to v.

Let w(A; x) be a solution to (58) for all (A; x) 2 R 2 + . Using Itô Lemma on the function e rt w(A t ; k) gives:

By (58), this equation becomes d e rt w(A t ; k) = P A t ; k e rt dt + (A t )e rt @w @A (A t ; k)dW t , (69) which can be rewritten, for h > 0,

and thus

When h ! +1, (71) becomes,

Assumption H4 implies that this integral is well de…ned. Assumption H2 implies that v 0 < 0, v 00 < 0 and:

So w veri…es assumption H1. Furthermore, on H, w(A; k) is a particular solution to the di¤erential equation ( 58), and, as v is also a solution to (58), there exists a function g(:)

such that, on H:

where g is a solution to the di¤erential equation:

Outside the zone H, v is not necessarily equal to g + w. However, the pro…t functions do not depend on v outside H, and equation ( 68) is equivalent to:

Step 4: Applying theorem 1 and showing the unicity of the equilibrium.

Equations ( 82) and ( 83) are equivalent to ( 52) and ( 53), so by proposition 2 the strategy of …rm i is its best response to the opponents strategy, and K(:; :) is a markov perfect equilibrium.

29

Documents de Travail du Centre d'Economie de la Sorbonne -2014.39


Dr Baptiste Sandoz
email: baptiste.sandoz@ensam.eu

Alina Badina

Sébastien Laporte

Karene Lambot

David Mitton

Wafa Skalli

Quantitative geometric analysis of rib, costal cartilage and sternum from childhood to teenagehood

Keywords: child, rib, cartilage, thorax, sternum

Better understanding of the effects of growth on children's bones and cartilage is necessary for clinical and biomechanical purposes. The aim of this study is to define the 3D geometry of children's rib cages: including sternum, ribs and costal cartilage. Three-dimensional reconstructions of 960 ribs, 518 costal cartilages and 113 sternebrae were performed on thoracic CT-scans of 48 children, aged four months to 15 years. The geometry of the sternum was detailed and nine parameters were used to describe the ribs and rib cages. A "costal index" was defined as the ratio between cartilage length and whole rib length to evaluate the cartilage ratio for each rib level. For all children, the costal index decreased from rib level one to three and increased from level three to seven. For all levels, the cartilage accounted for 45 to 60% of the rib length, and was longer for the first years of life. The mean costal index decreased by 21% for subjects over three years old compared to those under three (p<10 -4 ). The volume of the sternebrae was found to be highly age dependent. Such data could be useful to define the standard geometry of the paediatric thorax and help to detect clinical abnormalities.

Introduction

The thoracic anatomy of children is clinically important in spinal deformities such as scoliosis, and anatomical measurements can be used to identify normal geometry, quantify the severity of deformity, evaluate pulmonary capacity or build models for orthopedic or surgical treatment. Moreover, the thorax contains and protects vital organs and can be injured when subjected to impact, as in motor vehicle accidents. For children, the thorax is the second most often injured segment in crash events [START_REF] Brown | Patterns of severe injury in pediatric car crash victims: Crash Injury Research Engineering Network database[END_REF].

Child external morphology is well known, and some specific databases have been created to design child dummies [START_REF] De Jager | Assessing new child dummies and criteria for child occupant protection in frontal impact[END_REF][START_REF] Roger | Description and performance of the Hybrid III three-year-old, six-year-old and small female test dummies in restraint system and out-of-position air bag environments[END_REF][START_REF] Van Ratingen Michiel | Biomechanically based design and performance targets for a 3-year-oldchild crash dummy for front and side impact[END_REF]. However costal cartilage and sternal anatomy are generally not assessed, even though they can be of primary importance. The rib ossification process progressively increases the stiffness of the thorax and sets final thoracic geometry. During ribcage ossification, the large difference in material properties between cartilage and bone affects the stiffness of the rib cage, which is important when considering the orthotic brace effect or response to an impact in children.

Child thorax geometry is often described to highlight specific abnormalities (scoliosis, pectus carinatum and pectus excavatum), but quantitative descriptions of child ribcages are rare, while they are essential to build numerical models or to identify normal patterns for different age groups. Only a few existing studies provide descriptive parameters of the child rib cage. Derveaux et al. used 2D measurements on lateral X-rays to evaluate the anteroposterior width of the thorax [START_REF] Derveaux | Preoperative and postoperative abnormalities in chest x-ray indices and in lung function in pectus deformities[END_REF]. On CT-Scan slices, Haller et al. defined an index to describe the ratio between the transversal and anteroposterior diameters [START_REF] Haller | Use of CT scans in selection of patients for pectus excavatum surgery: a preliminary report[END_REF], but did not find any correlation with age in a group of 19 patients. Using the same method on 574 child CT-scans, Daunt et al. found a smaller Haller index for children under two, but a higher index for girls of 0-6 and 12-18 years compared to boys of the same ages [START_REF] Daunt | Age-related normal ranges for the Haller index in children[END_REF]. In 1989, Stokes et al. studied the 3D geometry of 71 scoliotic rib cages, compared to 10 controls composed of six cadavers and four volunteers (aged 26 to 54), using stereoradiography reconstruction modeling with 0-20° incidences [START_REF] Stokes | Rib cage asymmetry in idiopathic scoliosis[END_REF].

Costal cartilage dimensions were estimated from direct measurements on four adult cadavers. Using a similar technique, Delorme et al. studied the effect of surgical correction on the shape of the ribcages of 29 adolescents (mean age 15  1.5 years) by calculating the 3D rotations of the ribs [START_REF] Delorme | Preoperative and early postoperative three-dimensional changes of the rib cage after posterior instrumentation in adolescent idiopathic scoliosis[END_REF]. Costal cartilages were not included in the reconstructions. Bertrand et al. performed more precise 3D rib cage reconstructions on 15 asymptomatic adults (mean age 27 ±8 years), using two simultaneous perpendicular planar X-rays from the EOS system (Biospace Instruments, Paris, France) [START_REF] Bertrand | Three-dimensional reconstruction of the rib cage from biplanar radiography[END_REF]. While various geometric parameters described the ribs, the costal cartilage and sternum were not investigated, due to the lack of visibility. Due to the superimposition of bone structures and the high quantity of radiotransparent cartilage, it is difficult to have a precise quantitative description of child cartilage using standard X-rays.

Another anatomic part of the thorax is the sternum, which influences the global stiffness of the rib cage, and can be used as an osseous age estimator. In 1967, Riach found a high age correlation with the surface of the sternebrae in23 specimens aged between 26 weeks of pregnancy and six-years-old [START_REF] Riach | Ossification in the sternum as a means of assessing skeletal age[END_REF].

Nevertheless, the number and time of appearance of the sternebrae show high variations from one child to another; so it seemed not to be a relevant bone-age indicator [START_REF] Ashley | The relationship between the pattern of ossification and the definitive shape of the mesosternum in man[END_REF][START_REF] O'neal | Postnatal development of the human sternum[END_REF][START_REF] Ogden | Radiology of postnatal skeletal development -II. The manubrium and sternum[END_REF][START_REF] Riach | Ossification in the sternum as a means of assessing skeletal age[END_REF][START_REF] Rush | Missing" sternal ossification center: Potential mimicker of disease in young children[END_REF].

To take into account growth of cartilage and bone, it is essential to have a better geometrical description of the child ribcage. Thus, the aim of this study is to quantify the 3D geometry and to study the age effect of the child rib cage: including sternum, ribs and costal cartilage, using reconstructions from CT-Scan data.

Methods

Population

Forty-eight thoracic CT-scans of children aged from four months to 15 years (22 girls, 26 boys) were collected and anonymized in the Necker Hospital (Paris, France). The CT-scans had previously been performed on medical prescriptions with consecutive slices of 4 or 5 mm thickness. The clinical prescriptions for CTscans of the thorax were: severe asthma, acute respiratory distress syndrome, investigation of intrathoracic lymph nodes, inhaled foreign body, trauma with no bone lesion, staging of primary extrathoracic malignancies. CT-scans in children with syndromes or heart congenital lesions were excluded. CT-scans showing thorax abnormalities or recent surgery were not included. Four groups of 12 children were defined according to age: four months to three years (A group), four to seven years (B group), eight to 11 years (C group) and then 12 to 15 years old (D group).

3D reconstruction method

An automatic segmentation and reconstruction of the ribs and ossified sternebrae was performed on each transversal plane using Avizo software (V5, VSG, USA), with further manual corrections at the boundaries of the sternebrae. A manual segmentation of the costal cartilages was accomplished to assess both their shape and the junction to the sternum. This thorough segmentation was performed by a pediatric orthopedic surgeon previously trained in radiological identifications and supervised by a pediatric radiologist. A total of 113 sternal sternebrae, 960 ribs and 518 costal cartilages were reconstructed. The sternal cartilage and some costal cartilage were not considered because incomplete or too difficult to discern on the CT-scan images (Table 1). In order to assess the reliability of the resulting sternebrae volume, 102 sternebrae were reconstructed twice, four weeks apart.

Data processing and calculated parameters

From the 3D reconstructions, each rib was modeled by its mid-line, according to the following steps (Figure 1):

1. The least square circle of the rib was calculated, defining center O and plane A.

2. From the anterior to the posterior extremities, fifty equidistant-angle planes P i rotating through O, orthogonal to plane A, were created.

3. Fifty corresponding cross sections S i were calculated as the intersection of planes P i and the external 3D surface of the rib.

4. The rib mid-line was constructed as the geometric centroid of all S i sections.

Applied to all left and right ribs, a wireframe of the thorax was then constructed (Figure 2). A similar method was applied to calculate the costal cartilage mid-line. Unlike ribs, costal cartilages are not curved in the transversal plane, so the intersection planes were then defined as parallel to the sagittal plane.

Rib cage morphometry was described by three parameters: maximum anteroposterior width, maximum lateral width, and thoracic index, their ratio. The local quantitative description of the ribs and the costal cartilage was calculated using rib mid-line length, chord length, enclosed area, maximum width, frontal and lateral orientations of the rib [START_REF] Bertrand | Three-dimensional reconstruction of the rib cage from biplanar radiography[END_REF][START_REF] Jean | Measurements of the threedimensional shape of the rib cage[END_REF][START_REF] Jolivet | Fast 3D reconstruction of the rib cage from biplanar radiographs[END_REF][START_REF] Mitton | 3D reconstruction of the ribs from lateral and frontal X-rays in comparison to 3D CT-scan reconstruction[END_REF]. In order to estimate the relative length of the cartilage, the costal index was defined: for each rib, it describes the ratio between the cartilage mid-line length and the whole costal segment, i.e. rib and cartilage mid-line lengths.

Statistical analysis

Because no assumption was made regarding the distribution of the data, the Kruskal-Wallis test was used to assess the statistical significance of differences in terms of gender, laterality, age group and rib level; with a threshold p-value below 0.05 being used to denote significance.

Results

Reconstruction assessment

The reproducibility study performed on 102 sternal elements showed a mean volume difference of 2.7 % (max 9.9 %, Standard Deviation 2.3 %), i.e. 0.2 cm 3 (max 2.0 cm 3 , SD 0.4 cm 3 ).

Sternebrae distribution

Most of the sternums presented a manubrium composed of one sternebra and a mesosternum composed of three sternebrae. A high variability of anatomical configurations was found. For example, Figure 3 shows one immature sternum (a. While the distribution of the sternebrae was found to be highly variable, Figure 4 shows the evolution of the sternebrae volumes of each sternum during growth. A global increase was observed and an exponential equation was fitted to describe sternum volume versus age relation of the studied population. The volumes were significantly different between all age groups (p < 10 -4 ) and the volume of the 15year-old sternum is about 10 times the volume at birth. Furthermore, dispersion is higher for the oldest patients. No significant difference was found between girls and boys with regard to the sternal volume distribution (p > 0.8).

Ribs and rib cage geometry

Statistical tests did not show significant differences for any parameter, either for gender (p > 0.15) or laterality (p > 0.95). Consequently, in the first approach, no distinction is made between girls and boys, left and right ribs or costal cartilages.

The global parameters of the rib cages are summed up in Table 1. The lateral (LAT) and anteroposterior (AP) widths increase with age, but the thoracic index shows a very small increase during growth.

In Figure 5, the mean and SD costal index is plotted for each rib level by age group. For all groups, the costal index decreases from rib level 1 to 3 and increases from level 3 to 7. Furthermore, the costal index is higher for the first years of life, with an almost equal length of cartilage and bone for levels 1 and 6 (ratio of 47 % and 45 % respectively) for group A. Level 7 has the longest cartilage region in the youngest group with a ratio of 60 %. For all levels, the mean costal index decreases significantly by 21 % between groups A (0 to 3 years old) and B (4 to 7 years old) (p < 10 -4 ).

In Figure 6, the progression of ribcage parameters is plotted by age group and rib level. Rib area, rib mid-length, maximum width and chord length increase with age. Rib angles show a small variation in the present population. All the parameters have a similar pattern of evolution regarding rib level. The statistical significance of differences between adjacent groups (A-B, B-C and C-D) have been calculated (Table 2): the geometric rib parameters are significantly different (p<10 -4 ) between two adjacent groups, except for the frontal and lateral angles.

With respect to costal and cartilage parameters, Table 3 (supplementary material) sums up the mean values and standard deviations of all calculated parameters. Except for the frontal and lateral orientations of the rib, the mean values of all costal parameters increase from level 1 to level 6, and then decrease from level 7 to level 10. Cartilage length increases with the rib level, from level 1 to 7.

Discussion

This study characterizes the bony and cartilaginous structures of the child thorax during its growth, from CT-scan data. Even if the subjects in this cohort were not fully healthy, the CT-scans were prescribed for a list of indications that do not affect the ribcage geometry. Performing CT-scans in healthy children without clinical indication is not possible due to ethical considerations. In this study, only CT-scans performed for limited pathologies were collected, while patients with chest malformations or chronic diseases with potential consequences for the child's growth were excluded. The dataset is therefore believed to be pertinent to represent non-pathological geometries of the child ribcage. The number of subjects is large (48) and covers a wide range of ages, from four months to 15 years, with three subjects per year. The gender distribution of the study population is well balanced. The reconstructions made using the Avizo software have been validated by an intra-observer reproducibility study on 102 sternebrae. Although the detection of bone pixels was automatic, an operator correction was required when two different bone structures were in contact. It was then necessary to distinguish boundaries manually.

As expected, the results show an increase of all parameters with growth.

The originality of this study lies in its quantitative approach. The thoracic index of the present study shows a slight evolution during growth, up to 11 years; it then stabilizes (Table 1). Furthermore, the thoracic index shows no significant evolution between children and adults: [START_REF] Bertrand | Three-dimensional reconstruction of the rib cage from biplanar radiography[END_REF] evaluated this parameter at a mean value of 0.63 (SD 0.07) on 15 adults (mean age: 27 years, SD 8 years), whereas the present child population shows a mean thoracic index of 0.65 (SD 0.05). The thicknesses of the ribs were not taken into account in the present study; therefore the maximum anteroposterior and lateral widths of the thorax morphometric parameters have been slightly underestimated.

The evolution of all the costal parameters for each rib level are compared to a young asymptomatic adult population [START_REF] Bertrand | Three-dimensional reconstruction of the rib cage from biplanar radiography[END_REF] (Figure 6). The mean growth speed, observed as the distance between the different curves of the same parameter, is not the same between groups, depending on the parameter considered. The differences between 2 adjacent age groups are significant except for frontal and lateral angles (Table 2). Because various body parts are known to have different growth timing [START_REF] Beusenberg | Status of injury biomechanics for the development of child dummies[END_REF][START_REF] Burdi | Infants and children in the adult world of automobile safety design: pediatric and anatomical considerations for design of child restraints[END_REF], no assumptions were made with respect to the growth shape or the variations between the various parameters, even if global growth was expected. Therefore due to the similar proportions in the ribcage dimensions between children and adults, child geometric data can be built from that of adults using an appropriate scaling factor.

Comparison of frontal and lateral angles with adults is difficult due to differences in methodology: while Bertrand et al. studied biplanar X-rays in standing position, the present study is based on CT-scan data, performed on children in a lying position [START_REF] Bertrand | Three-dimensional reconstruction of the rib cage from biplanar radiography[END_REF]. This is the main limitation of this study: CT-scans were performed in a lying position, together with the unknown respiratory phase during acquisition. These conditions require a careful interpretation of the angular parameters of ribs in frontal and sagittal planes, as the effect of the posture on the thoracic structures (supine versus upright versus seated) has already been underlined [START_REF] Beillas | The effects of posture and subject-to-subject variations on the position, shape and volume of abdominal and thoracic organs[END_REF][START_REF] Lafon | Combination of a modeldeformation method and a positional MRI to quantify the effects of posture on the anatomical structures of the trunk[END_REF]. Similarly, for the youngest children, the presence of clothing or diaper can change the natural angle of the ribs, as well as the anteroposterior diameter of the rib cage, as calculated. However, the other parameters are not affected by the lying position as they are calculated using the rib and cartilage mid-lines. The costal index presented in this study gives the cartilage length for the corresponding rib length, for a specific age and rib level. This result -rarely reported in the literature -is essential in a model design because cartilage is a chest component with different mechanical properties and behavior from ribs that are ossified [START_REF] Forman | Modeling costal cartilage using local material properties with consideration for gross heterogeneities[END_REF]. The current results allow estimates based on in vivo measurements.

The sternum is a structure that gradually ossifies. According to the literature, the number and distribution of sternebrae vary with growth. In 1967, by taking anteroposterior radiographs of specimens excised at necropsy, Riach found a correlation between the sum of the sternebrae surfaces and age, especially for the youngest children, less than six years old [START_REF] Riach | Ossification in the sternum as a means of assessing skeletal age[END_REF]. Considering the total volume of the sternebrae (Figure 4), the present study completes this trend for older children, up to age 15. Contrary to the number of sternebrae, the volume of the bony parts of the sternum is found to be a good age-predictive parameter, especially for the youngest. An exponential trend fits with age for the population studied. The consequences of growth will probably decline after puberty. Due to the high sensitivity of the results to the age, any conclusion on a gender effect would be uncertain: the age effect will overwhelm the gender effect.

Results from the present study could help to complete missing data or to validate numerical child models. Indeed, various models from the literature used X-rays to reconstruct the 3D geometry of the bony structures of the child rib cage [START_REF] Dworzak | 3D reconstruction of the human rib cage from 2D projection images using a statistical shape model[END_REF][START_REF] Jolivet | Fast 3D reconstruction of the rib cage from biplanar radiographs[END_REF]. The costal cartilage, not visible on standard X-rays, is then often approximated. When geometry is unknown in a child model, it is often scaled from the adult ones [START_REF] Annette | Biomechanical bases for the CRABI and Hybrid III child dummies[END_REF][START_REF] Schneider | Mass, center of mass, and moment of inertia estimates for infant limb segments[END_REF][START_REF] Wang | Design of a biofidelic instrumented 3.4 kg infant dummy[END_REF]. Besides the known ratio variation of the geometry and mass of the body segments with growth [START_REF] Burdi | Infants and children in the adult world of automobile safety design: pediatric and anatomical considerations for design of child restraints[END_REF][START_REF] Sandoz | Subject-specific body segment parameters' estimation using biplanar X-rays: a feasibility study[END_REF], children's ribcages contain a substantial quantity of growth cartilage, which has different material properties. The present results provide new quantitative data on child ribcage geometry which will assist in building more relevant child numerical models.

To refine the interpretation of the results, it would have been interesting to distinguish between girls and boys according to their respective puberty growth. However, it then would have been necessary to obtain data on a higher number of patients that had passed the peak of puberty, until 17 or 18 years old. The same protocol could be used in further studies for such assessment.

Finally, the present study highlighted the cartilaginous preponderance and the evolution of the young thorax geometry compared to that of adults. Thus such data could help to improve the biofidelity of child models for the thoracic segment. Future applications could be considered, like the improvement of Finite Element Models [START_REF] Mizuno | Development of a threeyear-old child FE model[END_REF], physical dummies for CPR training [START_REF] Akira | Effect of mattress deflection on CPR quality assessment for older children and adolescents[END_REF], forensic analysis, or to define the standard geometry of the paediatric thorax and help to detect clinical abnormalities.

Supplementary material -

  subject 10; 3-years-old), two sternums with merged sternebrae (b. subjects 33 and 18; 10 and 5-years-old respectively), and two early-adult sternums (c. subjects 26 and 28; 8 and 9-years-old respectively). In three cases, the manubrium had two ossification centers in a vertical disposition. The uppermost part of the mesosternum was always composed of a single ossification center. The lowest part of the mesosternum often exhibited lateral or/and longitudinal bifid ossification centers. The xiphoid process was already ossified in seven cases before six years old (out of 18 cases).

Figure 1 :Step 3 :

 13 Figure 1: Rib mid-line calculation steps.

Figure 2 :

 2 Figure 2: Calculated mid-lines rib cage (patient 10).

Figure 3 :

 3 Figure 3: Various sternum configurations, high variability not correlated to age. a. immature sternum (3 years old); b. merged sternebrae (10 and 5 yo); c. early-adult sternums (8 and 9 yo).

Figure 4 :Figure 5 :

 45 Figure 4: Measured sternebrae volume evolution with age, and boxplots by age groups. Group A: 0.3 to 3 years old; Group B: 4-7 yo; Group C: 8-11 yo; Group D: 12-15 yo. The volumes are significantly different between all age groups (p < 10 -4 ).

Figure 6 :

 6 Figure 6: Evolution of parameters and comparison with the literature, function of rib level and age group (mean and standard deviation). Group A: 0.3 to 3 years old; Group B: 4-7 yo; Group C: 8-11 yo; Group D: 12-15 yo; Bertrand et al. (2008): 27 yo (SD 8 yo).

Table 3 :

 3 Mean (SD) rib and cartilage calculated parameters, 3 children per age group.

	Data Rib level	< 1	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15
		45.0 (5.6)	57.5 (2.9)	58.1 (3.1)	62.9 (1.6)	79.4 (14.1)	76.1 (5.9)	76.1 (3.6)	81.7 (8.5)	86.3 (8.9)	85.5 (7.4)	85.6 (8.8)	94.9 (3.5)	89.6 (9.1)	103.0 (4.7)	100.8 (7.4)	100.8 (7.5)
		80.7 (9.4)	100.3 (9.7)	107.3 (5.4)	112.5 (5.3)	131.8 (13.3)	140.4 (8.9)	134.1 (5.1)	146.7 (11.6)	151.4 (12.2)	146.1 (8.9)	159.0 (6.1)	170.1 (3.7)	166.1 (15.0)	189.4 (10.4)	183.9 (9.0)	182.3 (21.1)
		99.9 (10.4)	124.9 (10.3)	135.4 (4.7)	140.4 (5.7)	161.6 (13.1)	170.8 (9.6)	166.4 (4.3)	181.6 (10.2)	186.8 (13.9)	184.0 (9.8)

Acknowledgements

The authors gratefully acknowledge the kind review of C. Adam. This research was partly funded by a grant from the ANR (SECUR_ENFANT 06_0385) and supported by the GDR 2610 "Biomécanique des chocs" (CNRS/INRETS/GIE PSA Renault).