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This paper shows existence and efficiency of equilibria of a production model with uncertainty, where production is modeled in the demand function of the consumer. Existence and efficiency of equilibria are a direct consequence of the catastrophe map being smooth and proper. Topological properties of the equilibrium set are studied. It is shown that the equilibrium set has the structure of a smooth submanifold of the Euclidean space which is diffeomorphic to the sphere implying connectedness, simple connectedness, and contractibility. The set of economies with discontinuous price systems is shown to be of Lebesgue measure zero.

This paper considers the Arrow-Debreu model with a complete set of contingent claims [

The aim of this paper is to consider a reformulation of the Arrow-Debreu model in terms of an exchange model with production in the utility function. Preliminary results are found in [

The paper is organized in three sections. Section 2 introduces the model. Section 3 establishes the results, and Section 4 is a conclusion.

We describe the two period private ownership production model

tion in a particular state

commodities is a set of normalized prices, denoted

are further endowed with a fraction

Consumers are endowed with a collection of vectors of initial resources

where initial endowments in a particular state

where

Producers are characterized by production sets and their smooth supply functions. The main property of the long run production model is that all activities of the firm are variable. An activity

state

Each consumer

be the market excess demand function in state

An equilibrium is a price vector

where

The model of the producer is to maximize profits. Each producer solves a constraint optimization profit maximization problem. Hence, each

where the state dependent production set

Definition 1. An equilibrium of the two period private ownership production model with uncertainty

An equilibrium allocation is a pair

A study of the qualitative equilibrium structure of the two period private ownership production model with uncertainty amounts to a study of the structure of the solution set of the equilibrium equation

Let

_{i}(s) is given by

denote the individual demand function of the two period “production adjusted” exchange model

in every

where

denotes the individual demand function. This follows immediately from rewriting the excess demand equation in terms of demand equal to supply. Rewriting

Hence, the equilibrium equation of the production model

since

This is the equilibrium equation of the exchange model with production adjusted demand functions

since

Theorem 1. For fixed

We have established a relationship between the production model with a long term time structure and uncertainty

The result suggests that the decentralized production model can be reformulated as a centralized model. It is efficiently applied in establishing many properties about production economies in the next section.

Let

and in the case of the production adjusted exchange model

Theorem 2. The set

Proof.

Theorem 3. The set

Proof. Consider the mapping

By theorem the regular value theorem

By simple algebraic manipulations we obtain the new matrices

Finally, we obtain

from which we extract the information required. Rank

The following theorem illustrates other economically interesting global properties of the equilibrium manifold. It says that by construction of a diffeomorphism

Theorem 4. The smooth equilibrium manifold

Proof. The aim of the proof is to define two smooth mappings between smooth manifolds such that we can apply the theorem (Hirsch [

be smooth mappings defined by

Then, let

denote smooth mappings defined by

Observe that the coordinates for the

Also observe that the coordinates for the

The application of theorem ([

and compute the inner product of (7) with

From that a reformulation of (7) readily follows in terms of the equilibrium equation

This is the equilibrium Equation (5), hence

from which it readily follows that

where

It remains to be shown that equilibria in the long run production model with uncertainty always exist. The strategy of the proof is to show that the natural projection mapping

Theorem 5. Equilibria of the two period production model with uncertainty

Lemma 1 (Smoothness)

Proof. Recall that

The next lemma makes use of theorem (see [

Lemma 2 (Properness)

Proof. Pick an arbitrary

and by non-satiation have also

which by monotonicity of

Clearly, there exists some

by boundedness of indifference mappings from below for every

where

Clearly,

for every

The number of equilibria of the long run production model with uncertainty is odd for any regular economy

I now define a subset of points on

Definition 2.

Proposition 1.

Proof. A necessary and sufficient condition for an equilibrium pair

Definition 3.

A singular value

Proposition 2. The set of singular economies

Proof. The proof follows from the application of Sards’s theorem which describes the set of singular values of a smooth mapping having the property of Lebesgue measure zero. Hence know that

This paper discusses local and global equilibrium properties of a production economy with a long-term time structure. Production is modeled in the demand functions of the consumers. The advantage of this way of modeling production is that it enables us to establish a relationship between production and pure exchange economies. Adding uncertainty to the production model is a further step towards realism. It is shown that the equilibrium set of all production economies with uncertainty has the structure of a smooth submanifold of the Euclidean space which is diffeomorphic to a sphere. These topological properties are of significant economic importance in terms of economic policy design since they imply connectedness and contractability of the set of solutions. It is also shown that the set of singularities of the catastrophe map is closed, and of Lebesuge measure zero. The practical implication of this result is that the probability of observing an economy with a discontinuous price system is close to zero.