Existence of Financial Equilibria in a General Equilibrium Model with Piece-Wise Smooth Production Manifolds

This paper considers a general equilibrium model with production and uncertainty. It is postulated that the firm finances its production capacity through the stock market and that its operational costs are covered through revenues. It is assumed that firms have linear technologies exhibiting constant returns to scale. Their production sets are piece-wise smooth convex manifolds. By method of regularization, it is shown using transversally theory that financial equilibria exists.


Introduction
General equilibrium theory is the study of simultaneous interaction between market demand and market supply.It aims at explaing the functioning of markets as a whole.The basic model, known as the Arrow-Debreu model is represented in Debreu [1].Its properties, such as the existence and efficiency of equilibria are well understood.However, these results are derived under a set of strong assumptions.For example, the model assumes a complete real asset structure and disregards any financial structure.This paper aims at moving towards a more realistic model that incorporates financial assets.Hence, a generalization of the private ownership Arrow-Debreu model to a model with financial assets is considered in this paper.Modelling the finance of production will lead to a complete contingent production model, where all production risks are covered.This, again, is not representing the real world satisfactorily.Hence, we assume that financial markets are incomplete.This implies that not all risk in the economy can be fully hedged.The aim of this paper is to develop such a model and to show that it is well defined.
The literature on general equilibrium modles with production mainly rests on the concepts introduced by Drèze [2] and Grossmann & Hart [3] which require to assign some sort of utility function to the firm.The two concepts as applied to many models slightly differ in the choice of average utility utilized (average utility of initial/final share holders).For a sample of the huge literature applying these concepts see [1] [2] [4] [5] [6] and the references therein.
At variance with the above literature, this paper introduces a model of the firm, where its activity is independent of any average utility of the stock holders [7].It postulates that firms maximize long run profits and make financial and real decisions sequentially over two periods.The assumption of long run profit maximization is justified by the sequential optimization structure of the firm introduced.Firms issue stocks in period one in order to acquire the cash needed to install production capacity.The optimal quantity of stocks issued by each firm is endogenously determined by the model.Once capacity is installed, after uncertain state of nature has occurred at the beginning of period two, firms produce real goods subject to capacity and technological constraints.The ownership structure introduced in this model eliminates the strategic choice problem of the firm present in the literature.Here, stock holders do not decide about the optimal input vector of the firm in period one.They invest in firms by purchasing stocks in order to transfer wealth across time and between uncertain states of nature.The total quantity of stocks demanded is equal to total quantity of stocks supplied by firms in the same period.The value of total stocks issued by a firm bounds the value of inputs a firm can purchase in period two.Real activities of the firm take place after uncertainty in period two has resolved.
These production activities correspond to finding the optimal net activity vector at given prices and revealed state of the world such that profits are maximized at given production capacity.
Stiefenhofer [7] shows that for convex smooth production manifolds with an endogenized price and technology dependent real asset structure, which is transvers to the reduced rank Grassmanian manifolds, equilibrium exists generically in the endowments by the application of Thom's parametric transversality theorem.The paper generalizes this result to the non-smooth convex production set case, where the piecewise linear production manifolds are regularized by convolution.Existence then follows from the smooth case.
The model of the firm is introduced in Section 2. Section 3 shows generic existence for linear production technologies with convex piecewise smooth production manifolds.Section 4 is a conclusion.

The Model of the Firm
We consider a two period { } . The consumption space for each i is ( ) , the strictly positive orthant.The associated price system is a collection of vectors represented by  , and denote , with associated spot price system  .We assume ( ) where by convention an element 0 k j y < denotes a factor of production and 0 k j y ≥ a good produced.Let Assumptions (F): (i) For each j, ( )  non-linear for all s S ∈ 1 .We replace the non-linearity assumption (1) in F with F(2).Imposing linearity on the technology of the firm enables the modelling of firms which exhibit constant return to scales.

Assumption F(2):
: ∈ piecewise linear j ∀ .Many economic applications deal with linear activity models.We therefore, consider the case of linear technologies.In order to apply previous existence proof to these models, we need to regularize the convex, piece-wise linear production manifolds nS j z Y =  by convolution and show that these con- volutes, denoted j Φ , are compact and smooth manifolds approximating the piecewise linear production manifolds.For that, we define the state dependent convolute for firm j ( where y U σ ∈ , and ∀ , is C ∞ and compact.Proof.For each j, denote the state dependent convolute Can restrict domain of integration to Int supp(λ).See (Dieudonnè [8]).Let Proof.Define for every s S ∈ ( ) by Cauchy inequality and Fubini's theorem, and since mass of λ is equal to one, and ζ ranges over its support, we obtain Thus it follows that , j y diam s φ λ . It converges to zero when ( ) where k is a constant of differentiation, and σ a distance.

Existence of Equilibria
In this section we show existence of equilibria.The strategy of the proof is to show that a pseudo equilibrium exists and that every pseudo equilibrium is also a financial markets equilibrium with production.It is known that pseudo equilibria exists for exchange economies.See Duffie, Shafer, Geanokopolos, Hirsh, Husseini, and others [9]- [14].Genakopolos et al. [4] showed that pseuedo equilibria exist for an economy with production for the case of exogenous financial markets.At variance with their model, where the firm's problem is to solve a Nash equilibrium, we show that a pseudo equilibrium for a more general price and technology dependent asset structure, permitting the modeling of production and its finance, exists Stiefenhofer [7].
We prove the follwing result.
Theorem 1 (Existence Equilibria).Let the assumptions T, P, F, F(2), and C hold.Then for any j z Y ∂  , there exists a pseudo FE with production , , , l S m l S n n S x y P L G for generic endowments.Moreover, by the relational propositions a FE with production , , , , l S m l S n nm n x y z p q Denote a long run equilibrium output vector associated with the production set boundary , . Each firm j is characterized by set of assumptions F (Debreu [1]).We modify Debreu's assumptions on production sets in order to allow the modeling of endogenous production capacity via financial assets.The

s y s p s y s p p S y S p S y S
where Z φ denotes the technology and capacity dependency of the payoff structure.We next introduce the consumer side of the economy.
The consumer: Each consumer is characterized by set of assumptions C of smooth economies.

Assumptions (C): (i)
( ) For each ( ) , s ∀ .For each ( ) Each i is endowed with ( ) Consumers want to transfer wealth between future spot markets.For that, they invest in firms in period 0 t = , receiving a share of total dividend payoffs which are determined in the next period in return.Denote the sequence of ( ) where 2 ownership structure is a ( ) vector defined by the mappings : , , where is the proportion of total payoff of financial asset j hold by consumer i I ∈ .In compressed notation, we write represents the full payoff matrix of order ( ) ( ) We introduce following prize normalization that the Euclidean norm vector of the spot price system p is a strictly po- sitive real number A financial markets equilibrium with production , , , , l S m l S n nm n x y z p q ; arg max ; : , , ; , arg max : ( ) (i) and (ii) are the optimization problems of the consumers and producers.(iii) and (iv) represent physical goods and financial markets clearance conditions.
( ) ˆ, , 0 For all consumers 2 i ≥ , the no arbitrage budget set consisting of a sequence of ( ) where ( ) .
Let p ∈S , such that : where T denotes the transpose, ( ) where  is a set of ( )

S n ×
matrices A of order ( ) We can now define the pseudo financial markets equilibrium with production.
We then state the relational propositions between a full rank FE with production and a pseudo FE with production.Definition 4. A pseudo financial markets equilibrium with production , , , l S m l S n n S x y P L G arg max : Stiefenhofer [7] shows that for every full rank FE with production ( ) ( ) , , , , x y z p q , there exists x y z p q is a ( ) , x y allocational equivalent FE with production.
Long run financial payoffs depend on the technology of the firm, its production capacity installed via financial markets, and on a set of regular prices.
Equilibrium does not exist for critical prices.The next step is therefore to introduce rank dependant payoff maps, and to exhibit a class of transverse price, technology, and capacity dependent maps.We will show that equilibria exists for this smooth rank dependent real asset structure, denoted ρ


. There are n financial assets traded in period 0. Denote the quantity vector of stocks purchased by consumer i, complete commodity markets and model producers' sequential optimization behavior in an incomplete financial markets environment.Incomplete markets is shown to be a consequence of the technological uncertainty hypothesis.Denote producer j's long run net activity vector associated feasible output vector.A state s net activity of the firm j is denoted the producers: Consider the sequential structure of the optimization problem of the firm.Firms build up long run production capacity in the first period, for that, they issue stocks.The value of total stocks issued in period one, denoted j j qz m = , where j m ∈  is a real number, bounds the quantity of goods a producer j can buy in state s S ∈ at input prices ( ) INP p s in period two.Once money is received through financial markets, firms install production capacity, and production activities take place subject to constraint long run production sets in the second period.Uncertainty in production is introduced by a random variable ): Firms maximize long run profits.
Denote the compact subset associated with any z, Φ .Proposition 2. For any j and C ∞ kernel λ , λ * is bounded and converges to identity φ , it satisfies ( ) ( ) ( ) ( ) C ∞ implies differentiability at any order required.The order depends on all transversality arguments employed.m denotes the inputs and n the output elements of the production set, and l = m + n.
exists for generic endowments.Proof.Production takes place in the second period, once capacity is installed and state s S ∈ occurred.At 0 t = , firms choose j z at price q such that long run profits are maximized in every state s S ∈ subject to long run technological feasibility j φ and capacity constraints j m .Denote the long run production set j z

=
 .The next step is to derive a normalized no arbitrage equilibrium definition[15].Let of agent 1, called the Arrow-Debreu agent.The Walrasian budget set for the Arrow-Debreu agent is a sequence of constraints denoted ( ) of normalized prices, and let ++ ∆ ∈  be a fixed strictly positive real number.This convenient normalization singles out the first good at the spot 0 s = as the numeraire.We introduce following definitions for the long run payoff maps associated with sets S and ′ S : Definition 3. (i) For any 1 x y P L is a pseudo FE with production.Moreover, if ( ) ( ) , , , x y P L is a pseudo FE with production then for every

π 5 .
Definition Define the rank dependent long run payoff maps : The set of reduced rank matrices A ρ of order ( ) following properties are well known.(i) For1 n ρ ≤ < , A ρ is a submanifold of A of codimension ( ) S n ρ ρ − + .(ii) for n ρ = the set { } ρ = ∅ is empty, and (iii) for 0 ρ = , ρ =   the set of reduced rank ma- trices is equivalent to the set of full rank matrices.These properties states that, for 1 n ρ ≤ < , the incomplete income transfer space is rank reduced.The rank dependent endogenized long run asset structure has following properties.is generic, since it is dense and open.Proof.(i) The linear map y D ρ π is surjective everywhere in Y. (ii) This