Luis H B Braido
email: luis.braido@fgv.br

V Filipe Martins-Da-Rocha

Piero Gottardi

Michael Magill

Her- Akles Polemarchakis

Martine Quinzii

Paolo Siconolfi We

Yves Balasko

Alessandro Citanna

Jan Eeckhout

John Geanakoplos

Jayant Ganguli

Christian Ghiglino

Zhiguo He

Felix Kubler

Karl Shell

Stephen Spear

Jan Werner

Getulio Fundação

Vargas

Output contingent securities and efficient investment by firms *

Keywords: CAPM, competitive equilibrium, production, investment, efficiency, endogenous risk, incomplete markets. JEL Classification: D51, D53, G11

We analyze competitive economies with risky investments. Unlike the classic Arrow-Debreu framing, firms and agents cannot contract upon the exogenous states underlying production risks. They can trade equities and any security written on the endogenous aggregate output. This financial structure is rich enough to promote efficient risk sharing among consumers. However, markets are incomplete from the production perspective, and the absence of prices for each primitive state of nature raises the question about the objective of firms. We show that output-contingent asset prices convey sufficient information to compute the competitive shareholder value that leads to efficient investment by firms.

Introduction

Following the work by Léon Walras in the 19th century, the general equilibrium literature focused on understanding how anonymous markets coordinate the about how alternative investment plans would affect the conditional expectation of each firm's output given the economy's aggregate production. 2 We start by stressing the main insights in a simple example with only two possible production levels. We describe the general model and formally define the competitive financial equilibrium in Section 3. Agents in this economy can only trade securities written on endogenous production levels. This financial structure is incomplete with respect to the underlying primitive states of nature but allows consumers to sell their endowment risks and make consumption plans that are contingent on the equilibrium aggregate output. In Section 4, we introduce our view on how competitive firms should compute the net present value of out-of-equilibrium investment plans. We argue that the financial equilibrium in which firms maximize our definition of market value is the counterpart of the Arrow-Debreu concept. In particular, we show that the associated consumption and investment allocation is Pareto optimal. In Section 5, we discuss some important topics related to our contribution. We stress how the results depend on our assumptions and then discuss existence of a competitive equilibrium for different specifications of the productive technology. We also use this section to compare our competitive notion of shareholder value to the alternative concepts in which firms exert market power by internalizing parts of the impact of their investment decisions.

Next, in Section 6, we describe our main contribution using the CAPM framework. Financial markets trade only stocks and a risk-free bond. Since agents hold mean-variance preferences, efficient risk sharing among consumers is achieved under this incomplete financial structure. Like before, this guarantees that firms can compute the competitive market value for different out-of-equilibrium investments and that profit-maximization generates Pareto efficiency. This section is self-contained and is particularly interesting for the reader who is familiar with the finance literature and is not concerned with general equilibrium details. Concluding remarks appear in Section 7. Appendix A is reserved for technical arguments.

An illustration

We borrow from Magill and Quinzii (2009) the following simple example with two periods t ∈ {0, 1}, one good, one firm, and one agent. At date 0, the firm 2 This competitiveness assumption on firms is consistent with the literature on the objective of corporations under incomplete markets-see for instance [START_REF] Drèze | Investment under private ownership: Optimality, equilibrium and stability[END_REF], [START_REF] Ekern | On the theory of the firm in an economy with incomplete markets[END_REF], [START_REF] Leland | Production theory and the stock market[END_REF], [START_REF] Ekern | On the theory of the firm in an economy with incomplete markets: an addendum[END_REF], [START_REF] Hart | On shareholder unanimity in large stock market economies[END_REF], [START_REF] Grossman | A theory of competitive equilibrium in stock market economies[END_REF], [START_REF] Makowski | Competitive stock markets[END_REF], and [START_REF] Bisin | Equilibrium corporate finance and intermediation[END_REF]. Our markets are also incomplete with respect to exogenous uncertainty, and the spanning condition of [START_REF] Ekern | On the theory of the firm in an economy with incomplete markets[END_REF] does not hold here (in the sense that investments do affect the individual ability to transfer wealth across primitive states of nature). However, differently from the aforementioned literature, agents share the same marginal rates of substitution for each given equilibrium aggregate output. Our contribution is to show that if managers internalize this equilibrium feature when forming their conjectures on firms' out-of-equilibrium market values, then efficiency is restored.

undertakes one of two possible investment levels a ∈ {0, 1}. This determines the probability over two possible date-1 output levels, namely, y l > 0 or y h > y l . The transition a → Q h (a) represents the probability of producing y h (success) under investment a. Investment is productive in the sense that Q h (1) > Q h (0). This technology has an alternative representation with three exogenous states of nature {ω h , ω m , ω l } and a random production production function f (a) := (f (ω h , a), f (ω m , a), f (ω l , a)), such that f (1) := (y h , y h , y l ) and f (0) := (y h , y l , y l ). The output is always high when the exogenous state is ω h , regardless the investment level. Analogously, it is always low in state ω l . When the state is ω m , the firm's production is high if, and only if, it has invested a = 1. State probabilities P (ω) are exogenous and satisfy

P (ω h ) = Q h (0), P (ω h ) + P (ω m ) = Q h (1) and, hence, P (ω l ) = 1 -Q h (1). 3
At date 0, the agent owns the firm and receives an initial endowment e 0 > 1. Preferences are represented by the expected utility

u 0 (x 0 ) + E (u 1 (x 1 )) ,
where the Bernoulli utility functions u 0 and u 1 are strictly increasing, continuously differentiable, strictly concave and satisfy the Inada condition at zero. 4 We also assume that

(2.1) u 0 (e 0 -1) < u 1 (y h )(y h -y l ) [Q h (1) -Q h (0)] .
Since Bernoulli functions are strictly concave, this condition ensures that investing is Pareto optimal.5 Two assets are traded in the financial markets: the firm's equity and a riskless bond. This market structure is incomplete with respect to exogenous uncertainty, as we have three states of nature but only two assets. Moreover, investment plans affect the individual ability to transfer wealth across primitive states, i.e., the spanning property of [START_REF] Ekern | On the theory of the firm in an economy with incomplete markets[END_REF] is not satisfied.6 Therefore, we cannot follow the standard approach of general equilibrium theory to define unambiguously shareholder value for out-of-equilibrium investment decisions by means of a non-arbitrage argument.

Markets are not complete with respect to exogenous uncertainty, but they are complete with respect to endogenous uncertainty. Formally, in the pure exchange economy defined by some arbitrarily given investment level a, trade on the equity and bond markets allow the agent to perfectly insure against production risks since we only have two output levels. In particular, the consumption allocation associated to any competitive equilibrium is efficient (given the arbitrary investment decision a). In this simple example, the entire production is consumed by the single agent at equilibrium. Since there is only one firm, the aggregate output is either z h := y h or z l := y l .

The limits of valuation by non-arbitrage Consider the pure exchange economy where the firm's investment is ā = 1 and denote by Ē its equilibrium equity price and by r its equilibrium risk-free interest rate. Since markets are complete with respect to production risks, the contract paying one unit of the good contingent to aggregate output being z can be implemented by trading the two available assets. Denoting by ρ(z) the non-arbitrage price of this contract, we have

ρ(z h ) = 1 z h -z l Ē - z l 1 + r and ρ(z l ) = 1 z h -z l z h 1 + r -Ē .
The standard definition of the competitive market value of the firm for the out-ofequilibrium investment a = 0 is

Π(0) := ω∈Ω p(ω)f (ω, 0) = p(ω h )y h + [p(ω m ) + p(ω l )]y l ,
where p(ω) is the non-arbitrage price of the contract paying one unit of the good contingent to the realization of the primitive state ω. However, since markets are incomplete with respect to primitive states of nature, the valuation of these contracts is ambiguous. Formally, we cannot recover all the prices p(ω) from the market prices Ē and r or, equivalently, from ρ(z h ) and ρ(z l ). Indeed, non-arbitrage only tells us that

(2.2) p(ω l ) = ρ(z l ) and p(ω m ) + p(ω h ) = ρ(z h ).
The value of a large corporation Building on the market completeness with respect to endogenous uncertainty, Magill and Quinzii (2009) suggested a definition for the out-of-equilibrium market value of the firm. They started from the following description of the equilibrium market value V := Ē -ā of the firm, that is

V = y h ρ(z h ) + y l ρ(z l ) -ā.
In order to make explicit the firm's investment decision in the above formula, we denote by χ(z) the equilibrium stochastic discount factor defined by the equation

ρ(z s ) = χ(z s )Q s (ā), for all s ∈ {l, h}. 7
Therefore, the equilibrium value of the firm takes the following form

V = y h χ(z h )Q h (ā) + y l χ(z l )(1 -Q h (ā)) -ā.
When analyzing the decision of a non-marginal firm, [START_REF] Magill | The Theory of Incomplete Markets[END_REF]Quinzii (2009, 2010) suggest to replace ā = 1 in the above formula by a = 0. This leads to the following conjecture

(2.3) M (a) := y h χ(z h )Q h (a) + y l χ(z l )(1 -Q h (a)) -a
of the firm's market value for the out-of-equilibrium investment a = ā. They show that the Pareto efficient level ā = 1 does not necessarily maximize M . In our example, this result follows from the fact that the condition (2.1) (which implies Pareto efficiency of investment) may hold simultaneously with M (0) > M (1) or, equivalently,

(2.4)

u 0 (e 0 -1) > y h u 1 (y h ) -y l u 1 (y l ) [Q h (1) -Q h (0)] . 8
This inefficiency is due to a particular type of market power exerted by the firm.

When choosing M (0) as its conjecture for the out-of-equilibrium value, the firm takes into consideration that its investment decision affects the distribution of aggregated variables.

Remark 2.1. Magill and Quinzii (2009) analyze the decision of a non-negligible firm which is aware of the impact of its own production on the level of aggregate output. They assume that the firm's manager takes the stochastic discount factor function z → χ(z) as given. They motivate this choice by arguing that "in the setting of capital markets, taking security prices as given is widely regarded as a good approximation, even for large corporations". We take the view that when corporations make their investment decisions, they take as given not only security prices but also the aggregate output.

A competitive conjecture for market value Defining a competitive conjecture for the out-of-equilibrium market value of the firm is not straightforward for an economy with output-contingent financial markets. Securities written on the aggregate output do not necessarily distinguish across primitive states of nature. Agents are aware that the equilibrium prices ρ(z) and the theoretical state prices p(ω) are related to each other as stated in condition (2.2). However, market prices alone do not convey sufficient information for splitting ρ(z h ) and ρ(z l ) into p(ω h ), p(ω m ),

8 Since u1 is strictly concave, we have u 1 (yh) < u 1 (yl). This implies that

yhu 1 (yh) -ylu 1 (yl) [Qh(1) -Qh(0)] < u 1 (yh)(yh -yl) [Qh(1) -Qh(0)] .
Therefore, we can always choose u0 and e0 such that u 0 (e0 -1) lies between the right-hand side and the left-hand side of the above inequality.

and p(ω l ) in an unambiguous way. To overcome this issue, we ask firms and agents to anticipate the following important equilibrium feature when analyzing investment plans out of the equilibrium. Markets are complete with respect to endogenous uncertainty. For any given investment level, the equilibrium ratio of marginal utilities are constant across primitive states of nature associated with the same aggregate output. This is to say that the agent's valuation of income streams contingent to a specific aggregate output is risk neutral. We propose to use this equilibrium feature to construct the shadow prices p(ω) from the market prices ρ(z). When the investment is ā = 1, we have p(ω l ) = ρ(z l ) and p(ω h ) + p(ω m ) = ρ(z h ).

Under ā = 1, the conditional state probabilities given the aggregate output z h are

P (ω h |{ω h , ω m }) = P (ω h ) P (ω h ) + P (ω m ) and P (ω m |{ω h , ω m }) = P (ω m ) P (ω h ) + P (ω m )
.

Therefore, if we use a risk-neutral pricing conditional on z h , we obtain:

p(ω h ) = ρ(z h )P (ω h |{ω h , ω m }) and p(ω m ) = ρ(z h )P (ω m |{ω h , ω m }) .
This lead us to consider the conjecture Π(a) for the firm's market value under investment a ∈ {0, 1} defined by:

Π(0) := y h p(ω h ) + y l p(ω m ) + y l p(ω l ) -0 = ρ(z h ) [y h P (ω h |{ω h , ω m }) + y l P (ω m |{ω h , ω m })] + ρ(z l )y l and Π(1) := y h p(ω h ) + y h p(ω m ) + y l p(ω l ) -1 = y h ρ(z h ) + y l ρ(z l ) -1 = V .
It is then simple to verify that the Pareto optimal investment level ā = 1 does in fact maximize the function a → Π(a). We just need to notice that

Π(1) -Π(0) = u 1 (y h ) u 0 (e 0 -1) (y h -y l )P (ω m ) -1 > 0,
where the last inequality follows from assumption (2.1) and the fact that

P (ω m ) = Q h (1) -Q h (0).
There is an equivalent way to define price conjectures without relying explicitly on shadow (or conjectured) prices p(ω) for the exogenous states of nature. Let y(a|z) := E[f (a)|f (ā) = z] be the firm's average production, under investment a, across states of nature for which the equilibrium aggregate output is z. We define the competitive conjecture for firm's value as

V (a) := y(a|z h )ρ(z h ) + y(a|z l )ρ(z l ) -a.
In this illustration, we have that y(0|z h ) = y h P (ω h |{ω h , ω m }) + y l P (ω m |{ω h , ω m }), y(0|z l ) = y l , and then

V (0) = ρ(z h ) [y h P (ω h |{ω h , ω m }) + y l P (ω m |{ω h , ω m })] + ρ(z l )y l = Π(0).
Moreover, we also have y(1|z h ) = y h , y(1|z l ) = y l , and then V (1) = V = Π(1).

General model

Consider an economy with two periods t ∈ {0, 1}, a single good, a finite set K of firms, and a finite set I of consumers. At the initial date (t = 0), each firm k selects an investment level a k from a set A k ⊆ R + . Making no investment is always a possibility-i.e., 0 ∈ A k , for every k. At date 1, they are exposed to exogenous shocks ω drawn from a probability space (Ω, F, P ). Events B ∈ F represent primitive causes which odds are represented by the exogenous probability P (B). This probability is independent of consumers' and firms' actions.

Technology

The initial investment a k and the exogenous shock ω determine firm k's production

y k = f k (ω, a k ) at date 1 from a set Y k ⊆ R + .
The production possibilities of the economy are represented by the family f := (f k ) k∈K of non-decreasing random production functions

f k (ω, •) : A k → Y k .
We assume that, for each investment a k , the function ω → f k (ω, a k ) is measurable and essentially bounded on (Ω, F, P ).9

From this standard production-function framework, we derive the following alternative representation of the productive sector. Define the sets A := k∈K A k and Y := k∈K Y k , their respective elements a := (a k ) k∈K and y := (y k ) k∈K , and the transition probability a → Q(a) given by

Q(B, a) := P ({f (a) ∈ B}),
for every Borel set B ⊆ Y . 10 The investment profile a ∈ A undertaken at date 0 determines the joint probability Q(a) of firms' random outcomes at date 1.

To represent aggregate production, we define the σ-operator to be

σy := k∈K y k , for all y ∈ Y .
The random aggregate production is then represented by the function ω → σf (ω, a). We let Z := k∈K Y k denote the set of all possible aggregate outputs and derive the transition probability a → µ(a) by posing µ(B, a) := P ({σf (a) ∈ B}), for every Borel set B ⊆ Z. We assume that date-1 output is bounded away from zero, in the sense that there exists ε > 0 such that µ ([ε, ∞), a) = 1, for every a ∈ A.

Agents

Each agent i has initial resources consisting of an endowment e i 0 > 0 at date 0 and the ownership shares δ i k ∈ [0, 1] of each firm k, where i∈I δ i k = 1. Agents have no initial endowment at date 1, so that all consumption in that period comes from the firms' output. Preferences are represented by a utility function that is separable across time and has the expected utility form for future risky consumption. Let x i 0 0 denote agent i's consumption at date 0 and γ i be a probability measure on R + that represents random consumption at date 1. Agent i's expected utility function is given by

u 0 (x i 0 ) + R + u 1 (x i 1 )γ i (dx i 1 ),
where u 0 and u 1 are strictly increasing, continuously differentiable, and strictly concave functions which map R + into [-∞, ∞) and satisfy the Inada condition at zero.

Remark 3.1. The time separability of the expected utility and the cross-agent homogeneity of the Bernoulli utility functions are only assumed to simplify the notation.

It is straightforward to extend all results in this paper to the case with heterogeneous Bernoulli utility functions (x i 0 , x i 1 ) → ν i (x i 0 , x i 1 ).

Competitive equilibrium for a given investment

For sake of expositional clarity, we first define a competitive equilibrium for our output-contingent environment by taking the investment level of each firm as given. After understanding how agents insure each other, we analyze the problem of how firms choose their investments.

Different from the traditional Arrow-Debreu model, we do not consider contracts contingent on the realization of the primitive states of nature ω. We assume that the probabilities and the economic consequences of the events in F are well-understood by firms and agents, but the costs of describing ex-ante each primitive state of nature and enforcing ex-post state-contingent contracts are too large. The only traded contracts are those based on firms' output. We consider two types of assets: the equity of each firm k ∈ K traded in positive net supply; and securities in zero net supply representing bonds and all possible output-contingent derivatives.

Uncertainty only derives from production risks. Hence, an efficient allocation of risks among consumers (for a given vector of firms' investments) only requires that agents trade firms' equities and claims contingent on date-1 aggregate output. At date 0, for a given investment profile a, each agent i chooses current consumption

x i 0 ∈ R + , new equity holdings η i ∈ R K and a (Borel-measurable) contract θ i : Z → R contingent on aggregate output such that (3.1) x i 0 + Z θ i (z)ρ(dz) + E • η i e i 0 + (E -a) • δ i ,
where E stands for the vector of equity prices, and ρ is a positive measure on the Borel sets of Z such that ρ([0, z]) represents the date-0 price of the contract delivering one unit of consumption good contingent on the aggregate output being lower than or equal to z. At date 1, contingent on output profile y, agent i consumes

(3.2) x i 1 (y) := θ i (σy) + y • η i 0.
Each agent i maximizes the expected utility

u 0 (x i 0 ) + Y u 1 (x i 1 (y))Q(dy, a)
among all individual plans (x i 0 , x i 1 , η i , θ i ) satisfying the budget constraints (3.1) and (3.2), for Q(a)-almost every y.

A financial equilibrium associated with a given investment profile ā is a list ( Ē, ρ, (x 0 , x1 , η, θ)), where (x 0 , x1 , η, θ) := (x i 0 , xi 1 , ηi , θi ) i∈I is a consumptionportfolio allocation such that: (i) for every i ∈ I, the plan (x i 0 , xi 1 , ηi , θi ) solves agent i's optimization problem given ( Ē, ρ, ā); An allocation ((x 0 , x 1 ), a) is a pair composed of a consumption allocation (x 0 , x 1 ) = (x i 0 , x i 1 ) i∈I and an investment profile a = (a k ) k∈K . An allocation is said to be feasible if the consumption markets clear. An allocation ((x 0 , x 1 ), a) is said to Pareto dominate the allocation ((x 0 , x1 ), ā) whenever

u 0 (x i 0 ) + Y u 1 (x i 1 (y))Q(dy, a) u 0 (x i 0 ) + Y u 1 (x i 1 (y))Q(dy, ā),
for every agent i, with strict inequality for at least one agent i. An allocation ((x 0 , x1 ), ā) is Pareto optimal if ((x 0 , x1 ), ā) is feasible and there is no other feasible allocation ((x 0 , x 1 ), a) that Pareto dominates ((x 0 , x1 ), ā). A consumption allocation (x 0 , x1 ) is Pareto optimal for a given investment ā if ((x 0 , x1 ), ā) is feasible and there is no other feasible allocation ((x 0 , x 1 ), ā) with the same investment profile that Pareto dominates ((x 0 , x1 ), ā). The next result shows that our market structure implements a Pareto optimal distribution of resources among consumers for any given investment profile.12 Proposition 3.1. Fix a financial equilibrium ( Ē, ρ, (x 0 , x1 , η, θ)) associated with an arbitrary investment vector ā. The corresponding consumption allocation (x 0 , x1 ) is Pareto optimal given ā.

The equilibrium measure ρ(dz) is absolutely continuous with respect to µ(dz, ā). This is to say that there is a Borel-measurable function χ : Z → R + (called the stochastic discount factor) such that:

(3.6) ρ(dz) = χ(z)µ(dz, ā).
Pareto optimality of the consumption allocation (x 0 , x1 ) given ā implies the individual consumption xi 1 (y) to be constant across output vectors y generating the same aggregate output σy. Therefore, there exist Borel-measurable functions ci 1 : Z → R + such that xi 1 (y) = ci 1 (σy) for all y and i. Since date-1 aggregate production is bounded away from zero, our assumptions on the Bernoulli utilities imply that the ratio of marginal utilities equals the stochastic discount factor

(3.7) u 1 (x i 1 (y)) u 0 (x i 0 ) = u 1 (c i 1 (σy)) u 0 (x i 0 ) = χ(σy),
for every i and y. As a consequence, the market is risk neutral conditional on the aggregate output, and the equilibrium equity prices can be written as

(3.8) Ē = Y χ(σy)yQ(dy, ā).

The market value of each firm

We now analyze how investment levels are chosen by firms in equilibrium. Each firm is assumed to be small relative to the aggregate economy and does not seek to manipulate prices. Under these competitive conditions, a natural objective function for a firm to maximize is its market value. When markets are complete with respect to primitive states (like in the standard Arrow-Debreu framework), the Arrow prices p(B) associated with any (F-measurable) primitive event B ⊆ Ω are quoted in the market. Firm k's manager can take these prices as given and use them to compute the following conjectured equity value

E k (a k ) := Ω f k (ω, a k )p(dω)
associated with any out-of-equilibrium investment a k . In that case, maximizing the standard competitive market value

(4.1) Π k (a k ) := E k (a k ) -a k = Ω f k (ω, a k )p(dω) -a k
leads to Pareto optimality.

In the absence of securities whose payoffs are contingent on exogenous events, a deep issue arises: how firms should assess their equity value for production plans different from the equilibrium ones? Or equivalently, how can we define the conjecture E k (a k ) when the prices p(dω) are not available?

Competitive conjectures conditional on aggregate production

The equilibrium prices ρ(dz) allow us to price by non-arbitrage any (bounded) contingent claim written on the aggregate output. The non-arbitrage price of a claim represented by a bounded function h :

Z → R is Z h(z)ρ(dz).
However, we are also interested in pricing random variables that are not measurable with respect to the equilibrium aggregate output. 13 The question at issue is then how to extend this pricing formula to the space of bounded random variables.

We recall that markets are incomplete and, then, there are infinitely many stochastic discount factors on Ω that are consistent with the equilibrium prices ρ. These discount factors, however, do not generate the same value for random variables that are not σf (ā)-measurable.

We propose to assume that firms and agents rationally anticipate the equilibrium property that the market is risk neutral conditional on aggregate output-as follows from Proposition 3.1 and equation (3.7). Combining this with standard non-arbitrage valuation leads to the following definition of conditional risk-neutral valuation S. For any bounded random variable g : Ω → R on (Ω, F, P ), 14 In other words, among the many possible stochastic discount factors, we take the one that is constant across states ω that are associated (at equilibrium) with the same aggregate output z.

S(g) := Z h(z)ρ(dz), where h(z) := E[g|σf (ā) = z].
Under this conditional risk-neutral valuation, firms and agents hold the following conjecture for the value of the out-of-equilibrium claim f k (a k ):

(4.2) E k (a k ) := S(f k (a k )) = Z y k (a k |z)ρ(dz),
where

(4.3) y k (a k |z) := E[f k (a k )|σf (ā) = z]
is the conditional expected output associated with the out-of-equilibrium investment a k given the equilibrium aggregate output z. The value of the firm consistent with this pricing rule is given by

(4.4) V k (a k ) := E k (a k ) -a k = Z y k (a k |z)ρ(dz) -a k .
The key behavioral assumption behind Eqs. (4.2), (4.3), and (4.4) is that firms and agents take prices as given and form competitive beliefs about the conditional expected production under different out-of-equilibrium investment levels. They understand that firm k's output becomes the random variable ω → f k (ω, a k ) whenever it invests a k . However, they also believe that firm k's decisions do not affect the likelihood of aggregate production and, therefore, compute expected production y k (a k |z) conditional on the event {σf (ā) = z}. The conditioning event is evaluated at the equilibrium investment vector, which includes the investment choice of firm k. One could metaphorically think about this as if there was a continuum of firms so that the term y k (a k |z) represented the conditional expected output when a firm invested a k while all other infinite firms invested the equilibrium level.

Remark 4.1 (Correctness at equilibrium). Our competitive price conjectures coincide with the equilibrium prices. Formally, we have

E k (ā k ) = Z y k (ā k |z)ρ(dz) = Z y k (ā k |z) χ(z)µ(dz, ā) = Y χ(σy) y k (ā k |σy)Q(dy, ā) = Y χ(σy)y k Q(dy, ā) = Ēk ,
where these equalities follow from equations (3.8), (4.2), and (4.3).

Efficiency

We turn now to show the Pareto optimality of any allocation ((x 0 , x1 ), ā) derived from a financial equilibrium in which each investment āk maximizes the competitive market value V k . We have argued before that the impossibility to trade assets contingent on primitive states is an essential incompleteness of markets from the perspective of firms. Although consumers do not need to trade securities contingent on primitive states to perfectly share idiosyncratic risks, firms need the information embedded in the state-contingent prices p in order to compute the competitive present value Π k (a k ) associated with out-of-equilibrium investment plans a k = āk , as defined in equation (4.1). We show that this incompleteness can be overcome if firms believe the market is risk neutral conditional on aggregate output, or equivalently, if they hold the competitive price conjectures E k (a k ). To prove this, we first show that our definition of competitive conjectures provides a connection between the standard Arrow-Debreu concept of competitive equilibrium and our definition of financial equilibrium.

We recall that an Arrow-Debreu equilibrium is a list (p, ( ξ0 , ξ1 ), ā) composed of: (i) a positive measure p on (Ω, F) representing state-contingent prices; (ii) an allocation ( ξ0 , ξ1 ) := ξi 0 , ξi 1 i∈I of consumption plans, where ξi 0 0 and ξi 1 : Ω → R + is a random variable; and (iii) an investment vector ā := (ā k ) k∈K such that:

(a) the allocation (( ξ0 , ξ1 ), ā) is feasible, in the sense that (c) for each agent i, the consumption plan ( ξi 0 , ξi 1 ) maximizes the expected utility

u 0 (ξ i 0 ) + Ω u 1 (ξ i 1 (ω))P (dω)
subject to the present-value budget constraint (4.7)

ξ i 0 + Ω ξ i 1 (ω)p(dω) e i 0 + Π(ā) • δ i ,
where Π(ā) := (Π(ā k )) k∈K .

Proposition 4.1. There exists an Arrow-Debreu equilibrium (p, ( ξ0 , ξ1 ), ā) if, and only if, there exists a financial equilibrium ( Ē, ρ, (x 0 , x1 , η, θ)) associated with the investment vector ā such that, for each k, āk maximizes the competitive conjecture V k .

The proof of this proposition is given in Appendix A.2. We sketch the arguments here. It consists in constructing the elements of a given equilibrium concept from the elements describing the alternative equilibrium. If (p, ( ξ0 , ξ1 ), ā) is an Arrow-Debreu equilibrium, it follows from standard arguments that the consumption allocation at t = 1 only depends on aggregate resources, i.e., there are Borel-measurable functions ci 1 : Z → R + such that ξi 1 (ω) = ci 1 (σf (ω, ā)), for all ω and i. We define (x 0 , x1 ) by posing xi 0 := ξi 0 and xi 1 (y) := ci 1 (σy) for each y and i. Since ξi 1 is σf (ā)-measurable, the contingent consumption xi 1 can be implemented by some portfolio ( θi , ηi ). Asset prices are defined using the standard present value pricing rule: ρ(dz) := χ(z)µ(dz, ā) and Ē := Ω f (ω, ā)p(dω), where χ(z) = u 1 (c i 1 (z))/u 0 ( ξi 0 ) is the equilibrium stochastic discount factor. Finally, by the law of iterated expectations, we get that the Arrow-Debreu present value Π k (a k ) of the out-of-equilibrium equity coincides with our definition V k (a k ) of competitive conjecture:

Π k (a k ) := Ω f k (a k )p(dω) -a k = Ω E[f k (a k )|σf (ā)]p(dω) -a k = Z y k (a k |z)ρ(dz) -a k =: V k (a k ). (4.8)
5 Some important discussions

We use this section to discuss the limits of our results, comment on the existence of a competitive equilibrium, and compare our definition of competitive valuation with an alternative concept used in the literature. We start by stressing the importance of our key assumptions.

About our assumptions

Since the focus of this paper is the efficiency of firms' investment decisions, we restrict attention to security markets that implement an efficient consumption allocation in a competitive exchange equilibrium for fixed and known investment decisions by firms (efficiency of exchange). We have assumed that: (a) agents' preferences are represented by discounted expected utility with state-independent Bernoulli functions; (b) their endowments at the second date deterministic; (c) agents agree on the probability distribution over the exogenous shocks ω. Under these assumptions, efficiency of exchange follows through if financial markets allow consumers to trade firms' equities and make consumption plans contingent on aggregate output. Consumption risks become only associated with the variability of the aggregate output.

Efficiency of exchange is maintained if we relax these three assumptions and assume that financial markets are sufficiently rich to span the uncertainty in the outcomes of the firms, for each possible investment profile.15 However, our definition of the competitive conjecture for the value of the firm depends on assumption (c) and cannot be easily extended to environments in which agents' have heterogeneous expectations over the exogenous shocks. This is because using the conditional riskneutral valuation to infer shadow prices for exogenous states may lead to heterogeneous out-of-equilibrium conjectures among agents with heterogeneous expectations.

We have also focused on a model with a single good and two dates. These modeling choices are only for the sake of simplicity. Our results can be extended to environments with finitely many goods and periods as far as the financial structure ensures efficiency of exchange.

About existence

The objective of this paper is to investigate whether the maximization of a suitably defined conjecture of competitive market value leads to efficient investment decisions by firms, even if investors can only write contracts on observable output (which distribution of risk is endogenous). An important related issue is existence of a competitive equilibrium where firms maximize the corresponding competitive market value.

Given an arbitrary investment profile a, existence of a competitive financial equilibrium is assured by our assumptions on preferences and positiveness of date-0 endowments and of date-1 production outcomes. In addition to that, if the sets A k were convex and the production functions a k → f k (ω, a k ) were continuous and concave on A k (for every ω), then we could assure existence of a competitive financial equilibrium associated with an investment profile ā that maximizes each firm's competitive market value V k . This follows from Proposition 4.1 coupled with classic theorems on existence of Arrow-Debreu equilibrium-see [START_REF] Bewley | Existence of equilibria in economies with infinitely many commodities[END_REF].

Remark 5.1 (Finitely many states). When the support of the probability P is finite, the investment sets A k are convex, and the production functions f k are concave, then our assumption that securities' markets are complete with respect to aggregate output may (generically) yield complete markets with respect to the primitive states. 16 This feature is not generic when A k is finite and also does not appear when we have a continuum of primitive states of nature. We refer the reader to Appendix A.3 for a simple technology for which, given any output profile y and investment vector a, there is a continuum of states ω satisfying f (ω, a) = y.

Continuum of firms Among the many technologies that do not satisfy the general assumptions for existence of an Arrow-Debreu equilibrium, there is one that deserves particular attention. The benchmark production model in contract theory is such that each firm either succeeds or fails to produce a certain amount of output, and the probability of success increases with the firm's investment. This success-or-failure technology is represented by a non-convex production function and non-existence problems may arise.

A traditional approach to overcome this issue considers a continuum of ex-ante identical firms with i.i.d. production draws. In this case, all variation in output that underlies this particular technology is eliminated, and the objective of the firm ceases to be an issue. In a supplementary material, we present a model with a continuum of identical firms and perfectly correlated success-or-failure shocks. 17We explicitly compute an equilibrium under specific assumptions and also derive a general existence result. This illustrates an interesting way to keep variability in the (average) aggregate output while smoothing non-convexities through a continuum of firms. Notice also that modelling the productive sector with many small firms is consistent with the behavioral assumptions used along the paper.

Strategic conjectures

When analyzing the decision of non-marginal firms, [START_REF] Magill | The Theory of Incomplete Markets[END_REF]Quinzii (2009, 2010) define the following "competitive" conjecture for the value of a firm:

(5.1) M k (a k ) := Y χ(σy)y k Q(dy, (a k , ā-k )) -a k ,
where (a k , ā-k ) represents a vector in which the k-th entry of ā is replaced by a k . Each firm k takes as given the equilibrium stochastic discount factor χ(z) conditional on each aggregate output z and the investment vector ā-k of all other firms. We have that V k (ā k ) = M k (ā k ) at any given financial equilibrium ( Ē, ρ, (x 0 , x1 , η, θ)) associated with ā. However, M k (a k ) typically differs from our definition of competitive market value V k (a k ) for out-of-equilibrium investments a k = āk . The equilibrium in which each firm k sets an investment a k to maximize M k (a k ) is not necessarily Pareto optimal. This may be surprising since agents display some competitive behavior by taking as given stochastic discount factors. However, by analyzing equation (5.1), one realizes that firm k anticipates the impact of the investment a k over the probability distribution Q, which turns out to affect the value of all other firms through the probability Q(dy, (a k , ā-k )). To assess this issue from a different perspective, we use the exogenous shocks to write

(5.2) M k (a k ) = Ω f k (ω, a k ) p k (dω, a k ) -a k ,
where p k (dω, a k ) := χ(σf (ω, (a k , ā-k )))P (dω).

Recall that if state-contingent claims were introduced in the market, all consumers would just be indifferent to buying or selling them if prices satisfied p(dω) = χ(σf (ω, ā))P (dω). Therefore, we can interpret p(dω) as the competitive market shadow price for event dω ∈ F. By replacing p(dω) with p k (dω, a k ), firm k's manager implicitly anticipates the impact of her investment decisions over the aggregate output distribution and manipulates the underlying state prices that affect the market values of all firms. The inefficiencies associated to the conjecture M k are therefore due to the strategic behavior of the firm and are not related to market incompleteness as in the models analyzed by [START_REF] Geanakoplos | Existence, regularity and constrained suboptimality of competitive allocations when the asset market is incomplete[END_REF]. Actually, even if all state-contingent claims were available for trade, inefficiency would still arise if firms exerted the same market power through its investment decisions-using the price measure p k (dω, a k ) instead of the equilibrium price p(dω) to compute price conjectures.

6 Presenting the main point in the CAPM setup

Let us now illustrate our main contribution in an environment that is widely used in financial economics. For this, we adapt the CAPM environment described in [START_REF] Magill | The Theory of Incomplete Markets[END_REF], Chapter 3. There are two periods t ∈ {0, 1}, one good, and finitely many firms and investors. Uncertainty is represented by an objective probability P on a finite set Ω of primitive states of nature. We consider the simple case where only firm 1 has an exposure-to-risk decision. In the initial period, this firm chooses an investment level a ∈ [0, 1] that determines its output next period, according to a bounded random function f 1 (a). The production of all other firms is random and not related with the investment a. We use σf (a) to represent the sum of all productions. 18 The variance var(σf (a)) of the aggregate output is strictly positive, for every a.

Investors hold positive initial endowments and own shares of each firm. The current earning of investor i is given by a real number ξ i 0 and the future random earnings are represented by a random variable ξ i 1 . These earnings are ranked by

υ 0 ξ i 0 + E υ 1 ξ i 1 ,
where the functions υ 0 and υ 1 are strictly increasing (on the relevant part of the domain), continuously differentiable, strictly concave, and such that E υ 1 ξ i 1 has a mean-variance representation. 19The only traded assets are a riskless bond in zero net supply and firms' equities in positive net supply. Markets are incomplete as there are only three assets and possibly more than three states in Ω. Nonetheless, thanks to the mean-variance preference, for any arbitrary investment ā, the competitive equilibrium in the bondequity markets ensures a Pareto optimal distribution ( ξi 0 , ξi 1 ) i∈I of the available resources among investors. Moreover, there is also a discount factor χ that is an affine function of aggregate output. Formally, there exist two strictly positive constants γ and λ such that, for every investor i,

υ 1 ξi 1 υ 0 ξi 0 = χ(σf (ā)),
where χ(z) = γ -λz, for any possible equilibrium aggregate output z. The market value Ē1 of firm 1 is then given by the CAPM asset pricing equation

(6.1) Ē1 = E f 1 (ā) 1 + r -λ cov f 1 (ā), σf (ā) ,
where r is the equilibrium risk-free rate and λ is interpreted as the market price per unit of risk.20

In this context, we ask how firm 1 should make its investment decision. Will the objective of this firm lead to Pareto efficiency in the production side? If markets were complete, then firms and investors could use the information contained in asset prices to infer the equilibrium Arrow price p(ω), for each exogenous state ω. In this case, any investment that maximizes the Arrow-Debreu present value of firm 1, i.e., a → Π 1 (a) := ω∈Ω p(ω)f 1 (ω, a) -a, is Pareto efficient and unanimously supported by the stockholders. But how to compute a → Π 1 (a) when Arrow prices are not available? Since the bond-equity markets are incomplete, there are infinitely many stochastic discount factors leading to the same equilibrium prices but to different evaluations about out-of-equilibrium investment plans a = ā.

We propose to build on the CAPM asset pricing equation (6.1) to define conjectures about how the market would value the firm's equity under an alternative investment level a = ā. When making out-of-equilibrium price conjectures, we assume that the manager acts competitively in the following sense: he takes the risk-free interest rate r and the premium for aggregate risk λ as given, and he does not internalize the impact of his investment decision on the equilibrium aggregate production σf (ā). These behavioral assumptions lead to the following definition of "competitive" price conjecture: for each a ∈ [0, 1], (6.2)

E 1 (a) := E[f 1 (a)] 1 + r -λ cov f 1 (a), σf (ā) .
If the firm 1 maximizes the associated "competitive" market value function a → V 1 (a) := E 1 (a) -a, then the corresponding solution ā is Pareto optimal. To see this, recall from the Second Welfare Theorem that, after appropriate transfers, the efficient allocation of consumption ( ξi 0 , ξi 1 ) i∈I can be implemented as the equilibrium of a pure exchange economy with a complete set of Arrow securities contingent on primitive states ω. If we let m : Ω → R + be the Arrow-Debreu stochastic discount factor (i.e., m(ω)P (ω) = p(ω), for all ω), then we have

υ 1 ξi 1 υ 0 ξi 0 = m,
for every investor i. This implies the following relation between the stochastic discount factor χ from the bond-equity economy and the Arrow-Debreu factor m:

χ(σf (ā)) = m.
As a consequence, we obtain

E 1 (a) := E[f 1 (a)] 1 + r -λ cov(f 1 (a), σf (ā)) = E[f 1 (a)]E[ χ(σf (ā))] -cov(f 1 (a), χ(σf (ā))) = E[f 1 (a)]E[ m] -cov(f 1 (a), m) = E[ mf 1 (a)] = Π 1 (a) + a.
Therefore, an investment ā maximizes V 1 (a) = E 1 (a) -a if, and only if, it maximizes Π 1 (a). From the First Welfare Theorem for Arrow-Debreu economies, such an investment level is a Pareto efficient productive decision.

Relating to our main framing The CAPM price conjecture defined in equation (6.2) could be obtained from our main framing by developing equation (4.2) as follows:

E 1 (a) = Z y 1 (a|z)ρ(dz) = Y χ(σy) y 1 (a|σy)Q(dy, ā) = E χ(σf (ā))f 1 (a) = E f 1 (a) 1 + r -λ cov f 1 (a), σf (ā) ,
where [1 + r] -1 := E [ χ(σf (ā))] is the equilibrium risk-free discount factor. This is the CAPM version of our concept of competitive price conjectures. When making out-of-equilibrium price conjectures, firms and agents take the risk-free discount factor r and the premium λ for aggregate risks as given and do not internalize the impact of the investment decision on aggregate output. A similar decomposition for the market-value concept proposed by [START_REF] Magill | The Theory of Incomplete Markets[END_REF]Quinzii (2009, 2010) leads to:

M (a k ) -a k := Y χ(σy)y k Q(dy, (a k , ā-k )) = E χ σf (a k , a -k ) f k (a k ) = E f k (a k ) 1 + r(a k , ā-k ) -λ cov   f k (a k ), f k (a k ) + k =k f k (ā k )   where [1 + r(a k , ā-k )] -1 := E χ(σf (a k , ā-k ) is a modified risk-free discount factor, and f k (a k ) + k =k f k (ā k )
is the out-of-equilibrium aggregate output. In this approach, firm k is aware of its impact over the aggregate output and makes a conjecture about how its investment decision a k would affect the risk-free discount factor and the market risk premium.

Alternative strategic concepts for market value have been analyzed in the corporate finance literature. Important references includes [START_REF] Stiglitz | On the optimality of the stock market allocation of investment[END_REF], [START_REF] Jensen | Corporate investment under uncertainty and pareto optimality in the capital markets[END_REF], [START_REF] Fama | Perfect competition and optimal production decisions under uncertainty[END_REF], [START_REF] Leland | Production theory and the stock market[END_REF], [START_REF] Merton | The optimality of a competitive stock market[END_REF], [START_REF] Greenberg | The technology of risk and return[END_REF], [START_REF] Baron | Investment policy, optimality, and the mean-variance model[END_REF], [START_REF] James | The technology of risk and return: Comment[END_REF], Greenberg, Marshall and Yawitz (1981), among others. These papers use the CAPM framing. They assume that firms take as given the equilibrium risk-free discount but incorporate (in different ways) the impact of alternative investments over the aggregate output.

Conclusion

There are two alternative traditions in economics to represent the outcome of risky enterprises. On one hand, the reference model in macroeconomics, finance, and general equilibrium uses the state-of-nature approach, which relies on random production functions that map investments and random primitive states of nature (with fixed objective probabilities) into realized outputs. On the other hand, the literature on contract theory relies on the probability approach in which production is modeled through transition functions mapping investments into probability measures over the set of possible outcomes.

As far as the description of production possibilities is concerned, the two approaches are equivalent. However, the two approaches differ on the financial contracts that are used to share risks and direct investments. By keeping states of nature hidden, the probability approach reminds us that writing contracts on the primitive states is not realistic. It is sometimes difficult to describe these states in a contract or to verify them ex-post for execution. This is why most financial contracts available in practice are usually written on observed production outcomes or profits (e.g., stocks and options). It is then natural to ask whether this market incompleteness-generated by the lack of state-contingent claims-matters for efficiency.

It is well-known that the ability to contract upon primitive states of nature is not essential for an efficient allocation of resources in exchange economies with endowment risks only. If agents can sell their endowed stocks and trade claims written on the aggregate output, then the equilibrium consumption is efficient and only varies with aggregate risks. The interesting question concerns the ability of financial markets to efficiently direct firms' investments. We show that the difficulty raised by the lack of state prices can be overcome if firms and agents competitively anticipate out-of-equilibrium scenarios. The corresponding market value conjecture is then consistent with an appropriate notion of competitive beliefs out of the equilibrium. It describes the way each firm assesses the impact of alternative out-of-equilibrium investments without anticipating the effect of this decision over the distribution of aggregate variables. As a result, we obtain the classic recipe that shareholder value maximization leads to Pareto efficiency even for the more realistic financial structure in which all contracts are written on firms' outcomes.

Since the portfolio allocation (η, θ) satisfies the market clearing condition (3.5), we can then deduce that ( Ē, ρ, (x 0 , x1 , η, θ)) is a financial equilibrium associated with ā.

To conclude, we recall that āk maximizes Π k and show that the objective function V k is identical to Π k given the equilibrium prices derived before. Indeed, standard first-order conditions applied to each agent's problem in the Arrow-Debreu equilibrium imply that p(dω) = χ(σf (ω, ā))P (dω), where χ(z) := u 1 (c i 1 (z))/u 0 ( ξi 0 ) for every z. We have ρ(dz) = χ(z)µ(dz, ā), by construction, and then

V k (a k ) = Z y k (a k |z)ρ(dz) -a k = Z χ(z) y k (a k |z)µ(dz, ā) -a k = Z χ(z) Y k y k Q(dy k , a k |z) µ(dz, ā) -a k = Z χ(z) Ω f k (ω, a k )P (dω|σf (ā) = z) µ(dz, ā) -a k = Ω χ(σf (ω, ā))f k (ω, a k )P (dω) -a k = Π k (a k ).
From FE to AD We now take a financial equilibrium ( Ē, ρ, (x 0 , x1 , η, θ)) associated with investments ā such that āk that maximizes V k , for every k, and construct an Arrow-Debreu equilibrium (p, ( ξ0 , ξ1 ), ā). Recall from the proof of Proposition 3.1 that there is a Borel-measurable function χ : Z → R + representing the stochastic discount factor in the sense that χ(σy) = u 1 (x i 1 (y))/u 0 (x i 0 ) for every y and i. We then pose p(dω) := χ(σf (ω, ā))P (dω), ξi 0 := xi 0 , and ξi 1 (ω) := xi 1 (f (ω, ā)).

The allocation (( ξ0 , ξ1 ), ā) satisfies the market clearing conditions (4.5) and (4.6).

As for budget feasibility, we notice that, for each i, there is a Borel-measurable function ci 1 : Z → R + satisfying ci 1 (z) = xi 1 (σy), for all y. and, then, the plan ( ξi 0 , ξi 1 ) satisfies the present-value budget constraint (4.7).

21 See the proof of Proposition 3.1.

For each given equilibrium investment ā ∈ [0, 1] and production z > ā, there are infinitely many possible shocks ω ∈ Ω that generate f (ω, ā) = z. To see this, take z > ā, ω1 := z/ā, and ω2 := (1 -z)/(1 -ā). We obtain f (ω, ā) = z for any ω such that: (a) ω 1 = ω1 and ω 2 ω2 , or (b) ω 1 ω1 and ω 2 = ω2 . Figure 2 illustrates these cases. We conclude that observing the output z is not sufficient to pin down exactly the underlying productive shock.

[PLACE FIGURE 2

  (ii) the consumption markets clear, i.e., ) = σy, for Q(ā)-almost every y; 10 (iii) the financial markets clear, i.e., (3.5) i∈I ηi = 1 and i∈I θi (z) = 0, for µ(ā)-almost every z. 11

  k∈K f k (ω, āk ), for P -almost every ω ∈ Ω; (b) for each firm k, the investment āk maximizes the present-value function Π k (a k ) := Ω f k (ω, a k )p(dω) -a k ;

Figure 1 :

 1 Figure 1: The production function f ((ω 1 , ω 2 ), •)

The probability space (Ω, P ) with Ω := {ωh, ωm, ωl} represents the exogenous uncertainty since the probability measure P does not depend on endogenous actions of the firm or the agent. The probability space (Y, Q(a)) with Y := {yh, yl} represents the endogenous uncertainty since the distribution Q(a) depends on the investment level a.

By that we mean limx t →0 u t (xt) = ∞, for t = 0, 1.

Strict concavity implies u0(e0) -u0(e0 -1) < u 0 (e0 -1) and u 1 (yh)(yh -yl) < u1(yh) -u1(yl). By combining theses inequalities with Equation (2.1), we obtain that u0(e0) -u0(e0 -1) < (u1(yh) -u1(yl)) [Qh(1) -Qh(0)].

 6 When the equilibrium investment is ā = 1, the equity and bond payoff vectors are respectively (yh, yh, yl) and (1, 1, 1). The production vector (yh, yl, yl) associated with the out-ofequilibrium investment a = 0 does not belong to the marketed space, in the sense that it cannot be generated by the assets payoff vectors (yh, yh, yl) and (1, 1, 1) respectively.

Consumption optimality implies χ(zs) = u 1 (ys)/u 0 (e0 -ā), for s ∈ {l, h}.

To fix ideas, we can take Ω to be the product space k∈K (Y k ) A k and F to be the product ⊗ k∈K B k of each Borelian σ-algebra B k defined by the product topology of the space (Y k ) A k . The support of the probability P is then assumed to be a subset of k∈K N (A k , Y k ), where N (A k , Y k ) is the set of non-decreasing functions from A k to Y k .

The set {f (a) ∈ B} stands for {ω ∈ Ω : f (ω, a) ∈ B}. Similar notation omitting ω is used throughout the paper.

The term 1 represents the K-dimensional vector of ones.

The proof follows from standard arguments and the details are postponed to Appendix A.1.

For instance, since the spanning property of[START_REF] Ekern | On the theory of the firm in an economy with incomplete markets[END_REF] is not necessarily satisfied, the out-of-equilibrium production claim f k (a k ), may not be σf (ā)-measurable, when a k = āk .

By this we mean that h : Z → R is a Borel function such that E[g|σf (ā)] = h(σf (ā)), almost everywhere.

This condition is called "complete spanning" inMagill and Quinzii (2010). It means that it is possible (at a cost) to find a portfolio of bonds, equity contracts and derivatives whose payoff is one unit if a given profile of outcomes for the firms is realized, and nothing otherwise. As[START_REF] Ross | Options and efficiency[END_REF] showed, in a two-period model this is always possible if a sufficient number of options are introduced.

Indeed, concavity of the production function leads to a continuum of possible outcomes. Completeness with respect to aggregate output then requires (infinitely) many more securities than primitive states. We thank Martine Quinzii and Michael Magill for pointing this out.

 17 For reference, see[START_REF] Braido | Supplement to 'Output contingent securities and efficient investment by firms[END_REF].

Formally, if we have K firms, then σf (a) := f 1 (a) + f 2 + . . . + f K , where f k is a random variable, for any k = 1.

That is, there exists a function φ : R ×R+ → R such that E υ1 ξ i 1 = φ E ξ i 1 , var ξ i 1, where φ is increasing in the first variable and decreasing in the second.

These results follow fromMagill and Quinzii (1996), Theorem 17.3.

Conversely, let ( Ē, ρ, (x 0 , x1 , η, θ)) be a financial equilibrium associated with the investment vector ā such that, for each k, āk maximizes the competitive conjecture V k . Recall from Proposition 3.1 that there are Borel-measurable functions ci 1 : Z → R + such that xi 1 (y) = ci 1 (σy), for all y and i. We then define the state-contingent consumption allocation ( ξ0 , ξ1 ) by posing ξi 0 := xi 0 and ξi 1 := ci 1 (σf (ā)). The novelty of our approach is to use conditional risk-neutral valuation in order to define the state prices p by posing dp dP (ω) := dρ dµ(ā)

(σf (ω, ā)), or, equivalently, p(dω) = χ(σf (ω, ā))P (dω), where χ(•) is the equilibrium stochastic discount factor defined by the equation ρ(dz) = χ(z)µ(dz, ā). This definition of p allows us to go from the bottom to the top in equation (4.8).

The First Welfare Theorem holds true in the Arrow-Debreu framework, given the baseline assumptions of our model. It then follows from Proposition 4.1 that the allocation composed of an investment vector that maximizes each firm's competitive conjecture for out-of-equilibrium market value and the consumption profile from the corresponding financial equilibrium is Pareto optimal. This is proven in our next result.

Theorem 4.1. Let ( Ē, ρ, (x 0 , x1 , η, θ)) be a financial equilibrium associated with an investment vector ā. If āk maximizes V k , for all k, then the allocation ((x 0 , x1 ), ā) is Pareto optimal.

Proof. Take the financial equilibrium ( Ē, ρ, (x 0 , x1 , η, θ)) associated with ā. Consider the corresponding Arrow-Debreu equilibrium (p, ( ξ0 , ξ1 ), ā) defined by Proposition 4.1. Assume then, by way of contradiction, that there is an alternative outputcontingent feasible allocation ((x 0 , x 1 ), a) which Pareto dominates ((x 0 , x1 ), ā). For each i, define the state-contingent consumption plan (ξ i 0 , ξ i 1 ) by posing ξ i 0 := x i 0 and ξ i 1 (ω) := x i 1 (f (ω, a)), for every ω. The state-contingent allocation ((ξ 0 , ξ 1 ), a) satisfies the feasibility constraints (4.5) and (4.6). Moreover, we have

where the inequality is strict for at least one agent i. This means that the allocation ((ξ 0 , ξ 1 ), a) Pareto dominates (( ξ0 , ξ1 ), ā), which contradicts the First Welfare Theorem applied to the Arrow-Debreu equilibrium (p, ( ξ0 , ξ1 ), ā).

A Appendix

A.1 Proof of Proposition 3.1

Assume by way of contradiction that there is a feasible allocation ((x 0 , x 1 ), ā) that Pareto dominates ((x 0 , x1 ), ā). Since the consumption market clearing condition (3.4) only involves aggregate output and agents are risk-averse, the allocation ((x 0 , x1 ), also Pareto dominates ((x 0 , x1 ), ā), where

and Q(B, ā|z) := P ({f (ā) ∈ B}|σf (ā) = z) is the probability that firms' production profile belongs to a Borel set B ⊆ Y , under the investment vector ā and conditional on the aggregate production being z. Since the function xi 1 only varies with the aggregate output σy, this consumption level is implemented by the portfolio (θ i , η i ) := (c i 1 , 0), in the sense that (x i 1 , θ i , η i ) satisfies equation (3.2). Since the plan (x i 0 , xi 1 , ηi , θi ) solves agent i's optimization problem given ( Ē, ρ, ā), we must have (A.1)

for every agent i, with a strict inequality for at least one i. By non-arbitrage, we know that

Therefore, by adding inequality (A.1) over i, we find i∈I

which contradicts the date-0 consumption market clearing condition (3.3).

A.2 Proof of Proposition 4.1

From AD to FE Let us take an Arrow-Debreu equilibrium (p, ( ξ0 , ξ1 ), ā) and construct a financial equilibrium ( Ē, ρ, (x 0 , x1 , η, θ)) associated with the investment vector ā. We define the equity and output-contingent prices as

for any Borel set B ⊆ Z. It follows from strict concavity of u 1 that ξ1 (ω) is constant across states ω associated with the same equilibrium aggregate output σf (ω, ā). This is to say that, for each agent i, there exists a Borel-measurable function ci 1 : Z → R + satisfying ci 1 (z) = ξi 1 (ω), for every ω such that σf (ω, ā) = z. We pose (x 0 , x1 (y)) := ( ξ0 , c1 (σy)), for every y.

The individual consumption plan ( ξi 0 , ξi 1 ) must satisfy the Arrow-Debreu budget constraint ξi

Given the equity prices defined before, we have

This allows us to rewrite the previous inequality as

Therefore, by taking ηi := (1/I)1 and θi (z) := ci 1 (z)-z/I, we have that (x i 0 , xi 1 , ηi , θi ) satisfies the budget constraints (3.1) and (3.2) at the equilibrium financial prices ( Ē, ρ).

The consumption plans (x i 0 , xi 1 ) are optimal among all plans that can be financed by equity holdings and output-contingent contracts. To see this, let (x i 0 , x i 1 , η i , θ i ) be an alternative plan satisfying the budget constraints (3.1) and (3.2) at the equilibrium prices ( Ē, ρ). We construct the state-contingent plan ξ i 0 := x i 0 and ξ i 1 (ω) := x i 1 (f (ω, ā)) and notice from equation (3.2) that

By replacing this relation into equation (3.1) and using equation (A.2), we deduce that the consumption plan (ξ i 0 , ξ i 1 ) satisfies the present-value budget constraint (4.7). Therefore, by optimality of ( ξi 0 , ξi 1 ), we have

Feasibility of the allocation (( ξ0 , ξ1 ), ā) as defined by equations (4.5) and (4.6) implies feasibility of the allocation ((x 0 , x1 ), ā) as defined by equations (3.3) and (3.4).

To prove that ( ξi 0 , ξi 1 ) is individually optimal, we fix an arbitrary consumption plan (ξ i 0 , ξ i 1 ) satisfying the present-value budget constraint

By posing

we get from the definitions of Ē and ρ that (A.3) writes as follows

By fixing x i 0 := ξ i 0 , x i 1 (y 1 ) := c i 1 (σy 1 ), θ i (z) := c i (z) and η i := 0, we have that the plan (x i 0 , x i 1 , θ i , η i ) must satisfy the budget restrictions (3.1) and (3.2). The optimality of (x i 0 , xi 1 , θi , ηi ) then implies that

Finally, since āk maximizes V k and V k = Π k , the Arrow-Debreu profit maximization condition is also satisfied.

A.3 Technology example

We consider the following technology to illustrate that completeness with respect to aggregate output does not yield (even generically) complete markets with respect to the states of nature. There is a single firm, and the primitive states ω = (ω 1 , ω 2 ) lie in Ω := [1, ∞) × [0, 1). The production function (see figure 1) maps investment a ∈ [0, 1] to output f (ω, a) ∈ [0, 1] and displays piece-wise constant returns to scale as follows:

where the threshold is defined by a ω := (1 -ω 2 )/(ω 1 -ω 2 ). In words, our firm earns a high marginal productivity ω 1 ∈ [1, ∞) for low investment levels and a low marginal return ω 2 ∈ [0, 1] for high investment levels.

[ PLACE FIGURE 1 HERE]


Bernd Friesenbichler

Julia F Item-Glatthorn

Fabian Neunstöcklin

Nicola C Casartelli

Gaël Guilhem

Nicola A Maffiuletti

Differences in trunk and thigh muscle strength, endurance and thickness between elite sailors and non-sailors

Keywords: sailing, strength, hiking, endurance, muscle thickness

Dinghy sailors lean their upper body over the windward side of the boat ('hiking') to keep the boat's balance and maximise its speed. Sustaining the hiking position is essential for competitive performance and this study examined sport-specific differences of muscles relevant for hiking in elite sailors. Knee extensor muscle strength as well as trunk muscle strength, muscle endurance and muscle thickness were assessed in elite dinghy sailors (n = 15) and compared to matched, nonsailing controls (n = 15). Isometric extensor strength was significantly higher in sailors at 60° (+14%) but not at 20° knee flexion. Sailors showed significantly higher trunk flexor (but not extensor) strength under isometric (+18%) and eccentric (+11%) conditions, which was associated to greater muscle thickness (rectus abdominis +40%; external oblique +26%) and higher endurance for ventral (+66%) and lateral (+61%) muscle chains compared to non-sailors. Greater muscles thickness and the particular biomechanical requirements to maintain the hiking position may drive the increases in isometric and eccentric muscle strength as well as ventral and lateral trunk endurance. The current findings identified sport-specific muscle function differences and provide performance benchmarks for muscle strength and endurance in elite sailors.

Introduction

Dinghy sailing can be hard physical work, depending on wind conditions, crew position and competitive level (Bojsen-Moller, [START_REF] Bojsen-Moller | Yacht type and crewspecific differences in anthropometric, aerobic capacity, and muscle strength parameters among international Olympic class sailors[END_REF][START_REF] Larsson | Exercise performance in elite male and female sailors[END_REF][START_REF] Niinimaa | Characteristics of the successful dinghy sailor[END_REF]. In moderate and high wind conditions, dinghy sailors adapt the 'hiking' position by leaning the upper body over the windward side of the boat (Figure 1). This position prevents the boat from tipping over and concomitantly maximises lift force and thus boat speed [START_REF] Castagna | Assessment of energy demand in Laser sailing: Influences of exercise duration and performance level[END_REF]. Hiking requires prolonged isometric thigh and trunk muscle contractions, including small amplitude flexion and extension movements. These movements allow to adjust the hiking position to time-varying environmental influences, such as wind gusts and waves [START_REF] Mackie | The physical demands of Olympic yacht racing[END_REF]. Besides strategic components, the duration a sailor can sustain the hiking position is a crucial factor for performance, as the boat speed can be maintained at high levels for extended periods of time [START_REF] Legg | Temporal patterns of physical activity in Olympic dinghy racing[END_REF][START_REF] Maïsetti | Specific neuromuscular responses of high skilled laser sailors during a multi-joint posture sustained until exhaustion[END_REF][START_REF] Tan | Indicators of maximal hiking performance in Laser sailors[END_REF]. The anterior side of the body and in particular the knee extensor and trunk flexor muscles were reported to be important for this position (Chicoy-García & Encarnación-Martínez, 2015; [START_REF] Shephard | The biology and medicine of sailing[END_REF]. Previous research found that strength is crucial especially in up-wind sailing, which required about 73-87% of predicted maximal knee extensor strength in international-level sailors [START_REF] Mackie | The physical demands of Olympic yacht racing[END_REF]. Correspondingly, top-level sailors showed higher knee and trunk extensor strength in addition to higher knee extensor and trunk flexor endurance compared to club level sailors or control participants [START_REF] Aagaard | Isokinetic muscle strength and hiking performance in elite sailors[END_REF][START_REF] Larsson | Exercise performance in elite male and female sailors[END_REF][START_REF] Vangelakoudi | Anaerobic capacity, isometric endurance, and Laser sailing performance[END_REF].

Considering the importance of trunk and thigh muscles in elite sailing, the available information about some elementary muscle properties of elite dinghy sailors is limited. For example, eccentric trunk flexor and extensor strength or trunk muscle thickness have never been evaluated in elite sailors. The assessment of those properties is important to gain a deeper understanding of the sport-specific muscular differences of elite dinghy sailors. It provides muscle performance benchmarks for aspiring sailors and may aid coaches in identifying strength deficits in their athletes. In order to have an understanding of what can be considered average (baseline) strength in those muscle groups, non-sailors were compared to elite sailors. The primary aim of this study was to investigate if there are specific differences in concentric, isometric and eccentric trunk extensor, trunk flexor and knee extensor muscle strength in elite sailors compared to non-sailors. Secondary aims were to compare estimates of ventral, lateral and dorsal trunk muscle endurance as well as of trunk and knee muscle thickness between the two groups. It was hypothesised that elite dinghy sailors will demonstrate higher muscle strength as well as ventral muscle endurance and thickness when compared to non-sailors.

Methods

Participants

A total of 15 professional, Olympic-level dinghy sailors (470er and Laser class; 8 female) and 15 controls matched for sex, age, height and body mass were included in this study. Mean (±SD) age was 26 ± 6 and 28 ± 6, height was 178 ± 10 cm and 175 ± 9 cm and body mass was 72 ± 15 kg and 70 ± 14 kg for sailors and non-sailors, respectively. Controls were participating in recreational sport including both aerobic exercise and resistance training on approximately three occasions per week but did not have any previous experience in operating a sailing boat. Both sailors and controls did not present with overt cardiorespiratory, orthopaedic, neurological or general diseases. The study protocol was approved by the Cantonal Ethics Committee Zurich, participants were informed of the benefits and risks of the investigation, and written informed consent was given prior to participation.

Procedures

In this cross-sectional study, elite sailors and non-sailors were evaluated in a single session. Concentric, isometric and eccentric muscle strength, muscle thickness and muscle endurance of trunk and knee muscles were investigated. The same general testing sequence was used for all participants: muscle thickness (20 min)-warm up on a cycle ergometer (10 min)-muscle strength (80 min)-break (30 min)-muscle endurance (45 min). Within each sequence, the testing conditions were presented in randomised order.

Trunk and thigh muscle strength

All strength measurements were conducted using an isokinetic dynamometer (Con-Trex MJ, CMV AG, Dübendorf, Switzerland). The dynamometer was coupled with a specific trunk module moving in the sagittal plane (Con-Trex TP-1,000, CMV AG, Dübendorf, Switzerland) to assess trunk muscle strength. Reliability, validity and visual representations of this device have been shown elsewhere [START_REF] Guilhem | Validity of trunk extensor and flexor torque measurements using isokinetic dynamometry[END_REF][START_REF] Maffiuletti | Reliability of knee extension and flexion measurements using the Con-Trex isokinetic dynamometer[END_REF]. Participants were familiarised using a standardised warm-up procedure including submaximal concentric and eccentric contractions of trunk and thigh at different angular velocities for 5 min prior testing. Participants were given standardised verbal encouragement during contractions.

Trunk flexor and extensor strength were evaluated in the standing position with the lower limbs fixed below and above the patella using inflexible straps to stabilise the lower extremities. The upper body was fixed to the moving part of the dynamometer using inflexible straps over the chest and shoulders. In the sagittal plane, the dynamometer's axis of rotation for flexion and extension was aligned to 1/3 of the line from the left anterior superior iliac spine to the ipsilateral posterior superior iliac spine. In the frontal plane, the axis was aligned at the level of the fifth lumbar vertebra [START_REF] Guilhem | Validity of trunk extensor and flexor torque measurements using isokinetic dynamometry[END_REF]. Trunk flexor and extensor strength was measured under concentric and eccentric conditions. Three maximal, consecutive contractions at an angular velocity of ±60°/s and a range of motion (ROM) between -10° trunk extension to 50° trunk flexion (0° = vertical position) were conducted [START_REF] Guilhem | Validity of trunk extensor and flexor torque measurements using isokinetic dynamometry[END_REF]; Figure 2(A)). The isometric conditions were conducted at 25° trunk flexion and a total of 2 maximal contractions were performed. A rest period of 60 s was interspersed between the different trials.

Knee extensor strength was evaluated unilaterally for the dominant limb, which was fixed to the dynamometer's lever arm about 2 cm above medial malleolus using an inelastic strap. The dynamometer's axis of rotation was aligned to the lateral femoral epicondyle. Participants were in semi-supine position (135° trunk-thigh angle). Shoulders and hips were fixed with belts to minimise body movements. They were asked to complete three maximal, consecutive contractions under concentric conditions at 120°/s and under eccentric conditions at -30°/s [START_REF] Aagaard | Isokinetic muscle strength and hiking performance in elite sailors[END_REF] with a ROM between 105° and 5° of knee flexion (0° = full extension, Figure 2(B)). Isometric knee extensor strength was measured at two knee flexion angles of 60° and 20° using 3 maximal contractions per angle interspersed by 60 s of rest.

The torque signal was visualised in real-time and sampled at 100 Hz, corrected for gravity and normalised to the participants' body mass (Nm/kg). Peak torque from each trial was provided by the dynamometer's internal software, and the mean peak torque out of 3 attempts was calculated separately for each muscle group, contraction mode, and postural position.

Trunk muscle endurance

Trunk muscle endurance was tested by measuring the time until exhaustion while performing three standardised core exercises. The exercises were (a) pillar bridge front (ground contact with one foot, alternating every second), (b) pillar bridge side (one second to lower the hip until touching the ground, one second to move the hip up again) and (c) back extension in the prone position (one second to flex the spine 30° downwards and one second to move back upwards to horizontal) for the ventral, lateral and dorsal muscle chains (in that order), respectively [START_REF] Petersen | Reliability of a new test battery for fitness assessment of the European Astronaut corps[END_REF]. The test was terminated by the investigators if participants were unable to maintain the required position after a maximum of two warnings or after volitional fatigue. A stopwatch was used to record the time to exhaustion and a metronome was used to respect the exercise cadence. Each test was interspersed with 10 min of rest.

Trunk and thigh muscle thickness

Muscle thickness of four trunk flexors (rectus abdominis, internal/external oblique, transversus abdominis), one trunk extensor (erector spinae longissimus) and two knee extensors (vastus lateralis and vastus intermedius) were measured unilaterally on the dominant side using B-mode ultrasound (Mylab 25, Esaote, Florence, Italy) equipped with a linear array transducer (frequency band 7.5-12 MHz) [START_REF] Dupont | Real-time sonography to estimate muscle thickness: Comparison with MRI and CT[END_REF]. The measurements were conducted according to methodology described elsewhere [START_REF] Fukunaga | Determination of fascicle length and pennation in a contracting human muscle in vivo[END_REF][START_REF] Ikezoe | Effects of age and inactivity due to prolonged bed rest on atrophy of trunk muscles[END_REF]. In short, recordings of trunk flexor thickness were made in the supine position and at the end of a relaxed expiration and participants were in the prone position for trunk extensor measurements. For knee extensors, participants were asked to sit with the hip and knee joints flexed at 90° and to relax the quadriceps muscles. Muscle thickness was analysed with an imageediting program (ImageJ 1.36b, National Institutes of Health, Bethesda, USA) and defined as the largest Euclidean distance between two points that were placed orthogonally to the centreline between the superficial and the deep aponeurosis [START_REF] Minetto | Ultrasound-based detection of low muscle mass for diagnosis of sarcopenia in older adults[END_REF]. A total of three images were acquired for each muscle and the mean peak distance was retained for further analysis.

Statistical analysis

Means and standard deviations were calculated for all variables and normality was verified using Shapiro-Wilk tests. Percentage differences between the sailor and non-sailor group were calculated for each variable as (sailor/non-sailor) × 100-100 and the 95% confidence intervals (CI) were provided. Matched sailors and non-sailors were compared using dependent t-tests [START_REF] Bland | Matching[END_REF]. The level of significance was set to p < 0.05 and adjusted for multiple comparisons (Bonferroni). Effect sizes were calculated by Cohen's d and considered as trivial, small, moderate and large at the 0.25, 0.5, 1 and >1 level, respectively [START_REF] Rhea | Determining the magnitude of treatment effects in strength training research through the use of the effect size[END_REF]. All statistical analyses were conducted using Statistica software (Version 7, StatSoft., Tulsa, USA).

Results

Trunk flexor torque was significantly higher in sailors compared to non-sailors for isometric (+18% (+5 to +30%); p = 0.027; d = 0.6) and eccentric (+11% (+2 to +20%); p = 0.040; d = 0.4) contractions, but not for concentric contractions (+4% (-4 to +12%); p = 0.464; d = 0.2; Figure 3(A)). Trunk extensor torque did not differ between groups for concentric (+1% (-8 to +10%); p = 1.0; d = 0.0), isometric (+5% (-7 to +16%); p = 0.639; d = 0.2) or eccentric conditions (+5% (-9 to +18%); p = 0.740; d = 0.2; Figure 3(B)).

Knee extensor torque was higher in sailors compared to non-sailors under eccentric (+20% (+7 to +34%); p = 0.012; d = 0.8) but not concentric conditions, although the effect size was moderate (+8% (-1 to +16%); p = 0.087; d = 0.5; Figure 4(A)). Isometric knee extensor torque was significantly higher in sailors compared non-sailors at 60° knee flexion (+14% (+3 to 25%); p = 0.025, d = 0.7), but no significant difference and a small effect size was found at 20° (+10% (-7 to +2%); p = 0.266; d = 0.4; Figure 4(B)).

Trunk muscle endurance was significantly higher in sailors than in non-sailors for ventral (+66% (+36 to +92%); p < 0.001; d = 1.4) and lateral conditions (+61% (+40 to +94%); p < 0.001; d = 1.6), but not for the dorsal condition, yet the effect size was moderate (+28% (+2 to +55%); p = 0.082; d = 1.0; Figure 5).

No significant differences between sailors and non-sailors were observed for trunk extensor and knee extensor muscle thickness, while sailors showed significantly larger thickness than non-sailors for the rectus abdominis and external oblique muscles (Table 1).

Discussion and implications

The aim of this study was to quantify the differences between elite sailors and non-sailing active controls with respect to trunk and thigh muscle strength, muscle endurance and muscle thickness. The results of this study revealed (a) higher trunk flexor and knee extensor strength in isometric and eccentric conditions, (b) higher trunk muscle endurance of ventral and lateral muscle chains and (c) higher trunk flexor muscle thickness in sailors compared to non-sailors.

The finding of higher trunk flexor but similar trunk extensor strength for sailors compared to non-sailors in the current study diverges from previous work, which found higher trunk extensor strength for sailors but no difference in trunk flexors [START_REF] Aagaard | Isokinetic muscle strength and hiking performance in elite sailors[END_REF]. This discrepancy may be related to several differences in measurement techniques and protocol, such as the use of different dynamometers (Con-Trex vs. Kin-Com), measurement positions (standing vs. seated), trunk angle excursions (-10° to 50° vs. -15° to 30°) and strength data analysis (body weight-normalised vs. absolute values). To the best of our knowledge, this is the first study to investigate trunk flexor and extensor strength under eccentric conditions. Interestingly, we showed that trunk flexor torque was higher for sailors compared to non-sailors when performing eccentric (+11%) and isometric (+18%) contractions, while concentric flexor torque (+4%) did not differ between groups. When employing the hiking position at constant wind conditions and low speed, isometric contractions (for trunk flexors) may be the predominantly used contraction mode due to the quasi-constant behaviour of sail and boat. However, during gusty wind conditions and high boat speed, the boat may be exposed to large vertical displacements [START_REF] Slooff | Forces, moments and motions[END_REF], so that the trunk may be pushed waterwards by its own inertia in addition to gravity when the boat switches from moving down to moving upwards. Isometric and eccentric trunk flexor muscle actions could thus avoid trunk overextension and aid to sustain the proper hiking position. Concentric trunk flexion may be less important, as trunk inertia points against gravity when the boat switches from moving upwards to moving waterwards again, thus supporting the flexors. Although the described dynamics of boat and trunk during sailing in the hiking position are speculative and need verification, they may explain the observed specific differences of trunk flexors in sailors.

Eccentric knee extensor strength was 20% higher in sailors compared to non-sailors, which is in line with previous research [START_REF] Aagaard | Isokinetic muscle strength and hiking performance in elite sailors[END_REF]. Interestingly, the current study also showed higher isometric knee extension strength at 60° knee flexion in sailors, which is contrasting the findings of Aagaard and colleagues (1998). A noteworthy difference between the two studies is that isometric knee extension torque was measured in the semi-supine position (135° trunk-thigh angle, Figure 2(B)) in the current study, compared to the seated position (100° trunk-thigh angle) in the other study. The semi-supine position resembles the hiking position more closely and suggests a sport-specific adaptation at muscle lengths similar to those when in the hiking position. Such potential adaptations appear to occur gradually with skill level, since national level sailors showed higher isometric knee extension strength compared to club-level sailors [START_REF] Vangelakoudi | Anaerobic capacity, isometric endurance, and Laser sailing performance[END_REF].

The differences in maximal strength between sailors and non-sailors may be due, at least in part, to differences in muscle thickness. For example, the larger thickness of rectus abdominis and external oblique muscles for sailors was associated with greater trunk flexor strength in this group. Knee extensor muscle thickness was not different between groups, but a moderate effect on vastus lateralis thickness was observed, which may partly explain the higher knee extensor strength observed in sailors. The trunk extensor (erector spinae) thickness was comparable between the two participant groups and corresponded to the similar trunk extensor strength.

In addition to strength, sailors also showed greater trunk muscle endurance compared to non-sailors for ventral and lateral conditions. No differences in isometric endurance of abdominal and back muscles were reported in a previous study [START_REF] Larsson | Exercise performance in elite male and female sailors[END_REF], which compared sailors to non-sailors, however, the non-sailor group consisted of male physical education students, which may have stronger abdominal muscles than the average population. This could have underestimated the typical difference between the sailor and the non-sailor groups. Conversely, when the hiking position was simulated on a hiking bench in the same study [START_REF] Larsson | Exercise performance in elite male and female sailors[END_REF], endurance time was significantly longer in an isolated group of participants consisting of 'hikers' only, compared to the non-sailors. Similar results were observed when comparing national-level sailors to club-level sailors [START_REF] Vangelakoudi | Anaerobic capacity, isometric endurance, and Laser sailing performance[END_REF], or specialised hikers to non-hikers and controls [START_REF] Maïsetti | Specific neuromuscular responses of high skilled laser sailors during a multi-joint posture sustained until exhaustion[END_REF]. The current findings in muscle endurance need to be taken with caution, as the validity of the used test battery needs yet to be fully confirmed.

The results from previous studies, in which top-level sailors were compared to lower level or non-sailors [START_REF] Aagaard | Isokinetic muscle strength and hiking performance in elite sailors[END_REF][START_REF] Maïsetti | Specific neuromuscular responses of high skilled laser sailors during a multi-joint posture sustained until exhaustion[END_REF][START_REF] Vangelakoudi | Anaerobic capacity, isometric endurance, and Laser sailing performance[END_REF], suggest gradual muscular adaptations to dinghy sailing. Future studies should consider a longitudinal design together with the addition of a third group, such as non-competitive or club level sailors, to provide a definite answer to this question.

In the context of sailing performance, the observed muscular differences appear relevant for sustaining the advantageous hiking position for extended periods of time. This relationship was shown by positive correlations between knee extensor strength and hiking performance [START_REF] Aagaard | Isokinetic muscle strength and hiking performance in elite sailors[END_REF], and negative correlations between knee extensor strength and neuromuscular fatigue [START_REF] Bourgois | Isometric quadriceps strength determines sailing performance and neuromuscular fatigue during an upwind sailing emulation[END_REF]. The same may hold true for trunk flexor strength when considering the current results, although Aagaard and colleagues (1998) found weak correlations between hiking performance and trunk flexor strength, with the exception of static hiking in females.

It was shown that sustaining the hiking position progressively reduced quadriceps oxygen availability due to restricted blood flow to the muscle tissue, leading to an earlier onset of fatigue [START_REF] Sjogaard | Muscle blood flow during isometric activity and its relation to muscle fatigue[END_REF][START_REF] Vogiatzis | Quadriceps oxygenation during isometric exercise in sailing[END_REF][START_REF] Vogiatzis | Quadriceps muscle blood flow and oxygen availability during repetitive bouts of isometric exercise in simulated sailing[END_REF]. The advantage of high maximal strength of knee extensors and trunk flexors is that a smaller percentage of maximal strength needed to produce enough force to sustain the hiking position. Reducing the percentage of maximal strength needed to fulfil a task leads to a reduction in intramuscular pressure [START_REF] Lind | Cardiovascular adjustments to isometric contractions: Static effort[END_REF] and would allow for more blood and concomitantly oxygen to be delivered to the active muscle. Therefore, it appears to be of great importance for dinghy sailors to have sufficient knee extensor and trunk flexor strength. Future studies need to evaluate which combination of strength and endurance training for trunk flexors and knee extensors has the potential to optimise hiking performance most effectively.

It can be concluded that elite dinghy sailors have higher and direction-dependent (a) trunk flexor muscle strength, (b) knee extensor muscle strength and (c) trunk muscle endurance compared to non-sailors. The current results may be used as muscle performance benchmarks specific to dinghy sailing and can help coaches to identify strength deficits of their athletes.

Figure 1 .

 1 Figure 1. Hiking position in a laser radial dinghy. Permission obtained from Maud Jayet and copyright granted by loris von siebenthal.

Figure 2 .

 2 Figure 2. lateral view of the con-trex tP-1000 module (a) for dynamic (trunk angle: -10° to 50°) and isometric (25°) trunk strength assessments. Knee extensor strength assessments (B) were conducted in a semi-supine position for dynamic (knee angle: 105° to 5°) and isometric (20° and 60°) conditions.

Figure 3 .

 3 Figure 3. trunk flexor (a) and extensor (B) torque by contraction modality in sailors and non-sailors (mean ± sD). asterisks indicate significant group differences at p < 0.05.

Figure 4 .

 4 Figure 4. Knee extensor torque by contraction modality (a) and knee flexion angle (B) in sailors and nonsailors (mean ± sD). asterisks indicate significant group differences at p < 0.05.

Figure 5 .

 5 Figure 5. trunk endurance by test conditions in sailors and non-sailors (mean ± sD). asterisks indicate significant group differences at p < 0.05.

Table 1 .

 1 trunk and knee extensor muscle thickness (cm) in sailors and non-sailors. Values are means ± sD. asterisks indicate significant group differences at p < 0.01 (**) and p < 0.05 (*).

			Sailors	Non-sailors	% Difference	p-value	Effect size
					mean
			(cm)	(cm)	(95% CI)		(d)
	trunk flexors	rectus abdominis 1.46 ± 0.30**	1.04 ± 0.19	40 (22 to 58)	0.002	1.7
		external oblique	0.77 ± 0.15*	0.62 ± 0.15	26 (8 to 43)	0.024	1.1
		internal oblique	1.01 ± 0.26	0.99 ± 0.19	2 (-13 to 17)	1.000	0.1
		transversus	0.37 ± 0.11	0.42 ± 0.17	-11 (-28 to 6)	0.432	-0.3
		abdominis
	trunk extensor	erector spinae	3.21 ± 0.62	3.17 ± 0.67	1 (-8 to 10)	1.000	0.1
	Knee extensors	Vastus lateralis	2.63 ± 0.43	2.42 ± 0.46	8 (0 to 16)	0.056	0.5
		Vastus interme-	1.95 ± 0.38	1.89 ± 0.42	3 (-8 to 14)	0.604	0.2
		dius
	note:

Acknowledgement

We want to thank the Swiss Sailing team and Loris von Siebenthal.

Disclosure statement

No potential conflict of interest was reported by the authors.


C Giroux

G Guilhem

D Chollet

G Rabita

Computer Methods in Biomechanics and Biomedical Engineering Muscle coordination in loaded squat jump

Keywords: squat jump, force platform, forcevelocity relationship, EMG

Introduction

Power-generating capacity is essential in explosiveoriented physical tasks [START_REF] Newton | Effects of ballistic training on preseason preparation of elite volleyball players[END_REF]. Its assessment could help us to understand the basic properties of the neuromuscular system [START_REF] Cronin | Challenges in understanding the influence of maximal power training on improving athletic performance[END_REF]. Vertical jump is one of the most common and simple means used to evaluate maximal power [START_REF] Vandewalle | Standard anaerobic exercise tests[END_REF]. However, the loading condition that maximises the resulting power output in squat jump exercise is still a matter of debate in the literature [START_REF] Cormie | Developing maximal neuromuscular power: part 2training considerations for improving maximal power production[END_REF]. Although previous investigations have reported that the mean electromyographical (EMG) activity remains relatively stable as load increases during the concentric phase (i.e. push-off) of a vertical jump [START_REF] Eloranta | Effect of postural and load variation on the coordination of the leg muscles in concentric jumping movement[END_REF][START_REF] Nuzzo | The effect of loading and unloading on muscle activity during the jump squat[END_REF], the activation timing could be affected differently by external load. Therefore, this study aimed to investigate the neuromuscular coordination of the lower limb muscles involved during squat jumps performed under different isoinertial loading conditions.

Methods

A total of 20 (7 females, 13 males) trained athletes (age: 25.1 ± 4.5 years; height: 177.8 ± 7.4 cm; body mass: 74.1 ± 11.9 kg) participated in this study. They performed squat jumps in seven loading conditions: 0%, 10%, 20%, 30%, 40%, 50% and 60% of the maximal additional load the participants were able to lift concentrically once (1RM). Mechanical parameters were recorded using a force platform in time with EMG activity, recorded wirelessly from seven muscles (soleus (SOL), gastrocnemius lateralis (GL), tibialis anterior (TA), vastus lateralis (VL), rectus femoris (RF), semitendinosus (ST) and gluteus maximus (GMax)) of both legs. From the vertical ground reaction force, instantaneous force, acceleration, velocity, position and power were determined over the entire push-off phase duration. All EMG data were analysed as the root mean square (RMS) and normalised to the RMS obtained during maximal voluntary isometric contractions performed for each joint on a Con-Trex MJ (CMV AG, Duebendorf, Switzerland) isokinetic dynamometer. Analyses of variance were performed to compare mechanical and EMG patterns obtained during the jump push-off phase between loading conditions.

Results

Mechanical patterns

While peak force signifi increased (from 1962 ± 397 to 2559 ± 525 N; p = 0.0001), peak velocity (from 2.5 ± 0.2 to 1.6 ± 0.1 m.s (1 ; p = 0.0001), peak acceleration (16.2 ± 2.0 to 6.6 ± 1.4 m.s (2 ; p = 0.0001) and peak power (3770 ± 899 to 3491 ± 935 W; p = 0.006) decreased as external load increased. Peak force and peak acceleration occurred significantly later as load increased (from 54% to 69% of the push-off phase duration; p = 0.0001), whereas peak velocity and peak power occurred significantly earlier for heavy loads compared to light loads ( p = 0.0001).

EMG patterns

Although a significant main effect of muscle (mean activity: p = 0.004; peak activity: p = 0.58) was observed on muscle timing and amplitude, these parameters were not affected by the load.

Conclusions

This study showed that in a ballistic movement carried out maximally, the changes in mechanical outputs with increasing load do not result from a reorganisation of the underlying neuromuscular activity. These results are in accordance with those reported in the previous studies [START_REF] Eloranta | Effect of postural and load variation on the coordination of the leg muscles in concentric jumping movement[END_REF][START_REF] Nuzzo | The effect of loading and unloading on muscle activity during the jump squat[END_REF] but provide *Corresponding author. Email: caroline.giroux@insep.fr Figure 1. Time-course of vertical acceleration, velocity, position, force and power of the centre of mass during the pushoff phase of squat jumps performed at 0% (red), 10% (orange), 20% (yellow), 30% (green), 40% (light blue), 50% (dark blue) and 60% (purple) of the 1RM. For the sake of clarity, SDs are presented for 0% and 60% of 1RM. †, significant main effect of load on peak value. *, significant main effect of load on peak time occurrence (Colour online). supplemental information on EMG patterns of more muscles. This lack of load effect on the EMG patterns could originate from the fact that within the central nervous system, template motor programmes exist for specific classes of movements whose output are determined by the setting of specific parameters (i.e. timing, amplitude). A specific neuromuscular command would then be dedicated to maximal explosive ballistic efforts [START_REF] Rodacki | Vertical jump coordination: fatigue effects[END_REF]Van Zandwijk et al. 2003). The present findings yield a new insight into the design of optimal isoinertial muscle training for power-oriented activities and in performance or rehabilitation settings.

Acknowledgements

C. Giroux was supported by a scholarship funded by the French Ministry of Research.

Figure 2. The EMG activity patterns during the push-off phase of squat jumps performed at 0% (red), 10% (orange), 20% (yellow), 30% (green), 40% (light blue), 50% (dark blue) and 60% (black) of 1RM. Data are pooled for both limbs and normalised to the maximal isometric root mean square value (RMSmax) for GMax, VL, RF, ST, GL, SOL and TA muscles and presented as mean þ SE (Colour online).