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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.

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 [

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, equili- brium 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 [

The model is introduced in Section 2. Section 3 shows generic existence for convex smooth production manifolds.

We consider a two period

convention

idiosyncratic and aggregate risk. The general uncertainty space is described by the Cartesian product

The economic agents are the

issue stocks which are traded at

There are n financial assets traded in period 0. Denote the quantity vector of stocks purchased by consumer i,

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

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

Assumption (T):

For every production set

Assumption (P):

Firms maximize long run profits.

Assumptions (F):

(i) For each j, ^{1}.

Production takes place in the second period, once capacity is installed and state

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

firm j is characterized by set of assumptions F (Debreu [

where

The consumer: Each consumer

Assumptions (C): a)

Consumers want to transfer wealth between future spot markets. For that, they invest in firms in period

where^{2} ownership structure is a

where

where

We introduce following prize normalization

Definition 1. A financial markets equilibrium with production

a)

b)

c)

d)

a) and b) are the optimization problems of the consumers and producers. c) and d) represent physical goods and financial markets clearance conditions.

consumers. We now show that incomplete markets is a consequence of technological uncertainty and then move to the main section of the paper.

Proposition 1

Proof. Let

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 [

Definition 2. if

Lemma 1.

Proof. Immediate consequence of the separation theorem for

We can now rescale equilibrium prices without affecting equilibrium allocations, let

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

where ^{3} with its known smooth

Denote the pseudo opportunity set

Let

Definition 3. For any

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

a)

b)

c)

e)

e)

Lemma 2. Under assumptions C, demand mappings

Proof. The details of this known result are omitted [

Proposition 2. For every full rank FE with production

Proof. By lemma 1, there exists

On the contrary, if have a

Remark: Since agent 1 faces only the Arrow-Debreu constraints, his behavior is identical in both models.

Observation (2): Suppose

Under these conditions, a consumption bundle

The next step is then to show that

Let

Condition (1):

Translate

Condition (2):

Need to find a matrix Q such that

Q is a

The final step is then to show that the pseudo equilibrium manifold

Proposition 3. If

Proof. Using (Definition 3), let

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

Definition 5. Define the rank dependent long run payoff maps

Lemma 3. a) For

Proof. Consider the open set U of ^{4}. □

The lemma states that, for

Proposition 4. a)

Proof. a) The linear map

Definition 6. Denote

We thus have defined a fiber bundle

Theorem 5. There exists a pseudo FE with production

Proof. By (Proposition 4) and using (Definition 6) define an evaluation map

For the Arrow-Debreu agent have

The evaluation map is a submersion, since

where ^{5}, it follows that the subset

For all

This paper links the real and the financial sector in a general equilibrium model with incomplete financial markets. Production capacity available to a firm is endogenized and depends on the financial decisions of the firm in period one. At varianve to utility maximizing objective functions of firms, the model developed here considers a long run profit maximization objective function. This rehabilitates the decentralization property of the standard Arrow-Debreu model. It is shown by a parametric transversality theorem that equilibria exists.

Pascal Stiefenhofer, (2016) Production in General Equilibrium with Incomplete Financial Markets. Journal of Mathematical Finance,06,293-302. doi: 10.4236/jmf.2016.62025