Production in General Equilibrium with Incomplete Financial Markets

This paper considers a general equilibrium model with incomplete financial markets where production sets depend on the financial decisions of the firms. In the short run, firms make financial choices in order to build up production capacity. Given production capacity firms make profit maximizing production decisions in period two. We provide the conditions of existence of equilibria.


Introduction
Classical general equilibrium literature on production with incomplete markets has focused on variations of the Arrow's seminal two-period model with exogenous financial assets [1] [2].In this framework, the firm's real sequential optimization structure is independent of its financial activities.Firms choose quantities of inputs of production in period one such that associated output choices in period two are optimal.This concept of the firm corresponds to the private ownership model of the firm introduced by Debreu [3], where the single argument of the firm's two period sequential optimization function is the real activity vector.In these recent models, influenced by Drèze [4] and Grossmann & Hart [5], optimality of the choice of a net real activity vector over two periods refers to the average utility of the group of owners of the firm, the stock holders.It is in that sense that the literature has assigned utilities to firms and that the firms' objective is to maximize some average utility of the share holders.The two concepts applied in most 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 [4]- [8].
This paper introduces a model of the firm, where its financial and real activities are independent of any average utility of the stock holders.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.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.
The sine qua non of the model is then to show that equilibrium exists.It is shown that, for an endogenized price and technology dependent real asset structure, which is transverse to the reduced rank manifolds, equilibrium exists generically in the endowments by the application of Thom's parametric transversality theorem.Finally, the non-smooth convex production set case is considered, where the piecewise linear production manifolds are regularized by convolution.Existence then follows from the smooth case.Bottazzi [9] demonstrated generic existence of equilibrium for an exchange economy for price dependent smooth assets.Equilibria exist for more general asset structures.
The model is introduced in Section 2. Section 3 shows generic existence for convex smooth production manifolds.

The Model
We consider a two period { } 0,1 t ∈ model with uncertainty in period 1 represented as states of nature.An element in the set of mutually exclusive and exhaustive uncertain events is denoted , where by convention 0 s = represents the certain event in period 0, and S denotes the set of all mutually exclusive uncertain events.This set denotes the overall description of uncertainty in the model, which is characterized by idiosyncratic and aggregate risk.The general uncertainty space is described by the Cartesian product S = ×    .For every production set j Y , there exists a set of states of nature { } The consumption space for each i is the strictly positive orthant.The associated price system is a collection of vectors represented by There are n financial assets traded in period 0. Denote the quantity vector of stocks purchased by consumer i, with associated spot price system We assume ( ) 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 Sequential behavior of 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 , 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 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 j s S ∈ for every j.We assume that there are less uncertain states of the world S than financial assets n available for wealth transfer.Hence n S < is out default asumption.

Assumption (T):
For every production set ( )

Assumption (P):
Firms maximize long run profits.Assumptions (F): and ( ) 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 Y This set is not independent of the firm's technology nor on its financial activities, denoted Z.More formally, the firm's sequential optimization problem is Denote a long run equilibrium output vector associated with the production set boundary , .
Each firm j is characterized by set of assumptions F (Debreu [3]).We modify Debreu's assumptions on production sets in order to allow the modeling of endogenous production capacity via financial assets.The 1 t = maps implied by equation ( 1), : , ∈ and all producers j define the ( )

S n ×
total long run payoff matrix, a collection of n vectors denoted ( ) 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 (Debreu [10]).
Assumptions (C): a) : , For each . 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 where represents the full payoff matrix of order ( ) ( ) We introduce following prize normalization such that the Euclidean norm vector of the spot price system p is a strictly positive real number ++ ∆ ∈  .

Definition 1.
A financial markets equilibrium with production , , , , l S m l S n nm n x y z p q

Generic Existence for Convex Smooth Production Manifolds
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] [11]- [16].Genakopolos et al. [8] 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.
is a no-arbitrage asset price relative to 1 p .
( ) Proof.Immediate consequence of the separation theorem for ( ) ( ) We can now rescale equilibrium prices without affecting equilibrium allocations, let 1 1 .

P p β = 
The next step is to derive a normalized no arbitrage equilibrium definition [17].Let S β ++ ∈  be ( ) the gradient vector from the optimization problem of agent 1, called the Arrow-Debreu agent.The Walrasian budget set for the Arrow-Debreu agent is a sequence of constraints denoted For all consumers 2, i ≥ the no arbitrage budget set consisting of a sequence of ( ) where ( ) is the span of the income transfer space of period one.Replace ( ) is the Grassmann manifold3 with its known smooth ( )   , ; , be the set 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 : 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 , , , x y P L G .
, f P w and ( ) Under assumptions F, supply mappings ( ) Proof.The details of this known result are omitted [11].However, note that smoothness of demand and supply functions follows from the setup of the model for smooth economies.□ Proposition 2. For every full rank FE with production ( ) ( ) , , , , ,  x y  such that (FE) spot prices at p can be rescaled such that , u x P ∇ = and ( ) 0 On the contrary, if have a ( ) ( ) , , , x y P L equilibrium, and 2 , , m z z  such that a) ( ) x y z solves 2 i ≥ maximization problem for constraints .
Under these conditions, a consumption bundle i x ( ) . Similarly, in the neighborhood of l , the vector space l in the orthogonal basis of S  is spanned by the columns of a ( ) . We also rewrite the financial return matrix ( ) .,. V in this basis, such that it becomes ( ) ( ) ( ) Need to find a matrix Q such that 0.
ρ × − matrix.Condition (2) can then be written in terms of Q and E: The final step is then to show that the pseudo equilibrium manifold , ρ Ψ parameterized by P and Q is locally 1 , The equivalence of a pseudo equilibrium with production and a financial markets with production then follows from similar arguments as in [16].□ Long 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 .
We thus have defined a fiber bundle ρ Ψ of codimension ( )


We can now state the main result.

∈
 the associated feasible output vector.A state s net activity of the firm j is denoted run net activity vectors.
and b) are the optimization problems of the consumers and producers.c) and d) represent physical goods and financial markets clearance conditions.
that each firm j is owned by the consumers.We now show that incomplete markets is a consequence of technological uncertainty and then move to the main section of the paper.and the issue of new securities such that in the limit as ( ) 0 p π → the number of firms j n S → < by assumption (T).Similar for negative long run profit prospects ( ) 0, n n − dimensional structure, and L an n-dimensional affine subspace of ( ) . i x y P L is a ( ) ( ) , , , , x y z p q equilibrium.Remark: Since agent 1 faces only the Arrow-Debreu constraints, his behavior is identical in both models.


i ≥ is feasible under the constraints b) in the ψ model if and only if i x ( ) i ∀ is feasible under the constraints holding with equality in a) in the (FE) model.The next step is then to show that such that in the neighborhood of ê , the vector space e is spanned by the columns of a

Definition 5 .
Define the rank dependent long run payoff maps : The set of reduced rank matrices A ρ of order ( ) of reduced rank matrices is equivalent to the set of full rank matrices.Proof.Consider the open set U of ( ) incomplete income transfer space is rank reduced.The rank dependent endogenized long run asset structure has following properties.Proposition 4. a) is generic, since it is dense and open.Proof.a) The linear map y D ρ π is surjective everywhere in Y. b) This property does not change for any .consequence of the transversality theorem for maps.Since each set containing the spot price system and income transfer space consisting of a base vector P ρ and fiber() .