Each consumer chooses a utility maximizing consumption bundle at fixed and satisfying his budget constraints. Each producer chooses profit maximizing net activities at competitive prices. Let

be the market excess demand function in state. Then, market clearance requires demand to equal supply in each market and uncertain state of the world. Hence

An equilibrium is a price vector which satisfies this equation for a fixed distribution of initial resources and exogenously given ownership structure. An equilibrium pair is an equilibrium price vector with associated. An equilibrium allocation is an allocation associated with an equilibrium price. The model of the consumer is to solve a constraint optimization problem. This requires a consumer to maximize utility subject to a sequence of budget constraints. Hence, each consumer

where is the consumer’s smooth3 utility function. The new production adjusted budget set is now defined by

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 for all satisfies the assumptions of Debreu [1] also stated in the pervious section for the deterministic case.

Definition 1. An equilibrium of the two period private ownership production model with uncertainty is a price vector at fixed pair if for utility maximizing consumers and profit maximizing producers

An equilibrium allocation is a pair associated with an equilibrium price vector for fixed parameters. Let , denote the mathematical operation defined by a state by state inner product. There are equilibrium equations less equations satisfying Walras’ law, hence we have a system of linearly independent equations. This amounts to the number of unknowns, given the number of normalized prices of.

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. The first result is an equivalence relation between the two period exchange model with uncertainty and the two period production model with uncertainty. The relation between these models follows from the definition of a two period exchange model with production adjusted Marshallian demand functions.

Let for any price system and uncertain state of the world

. Let defined by, where h_i(s) is given by

denote the individual demand function of the two period “production adjusted” exchange model, where for every ownership structure is fixed, and total wealth defined by

in every. Now, let the equilibrium equation of the production model be given by

where

denotes the individual demand function. This follows immediately from rewriting the excess demand equation in terms of demand equal to supply. Rewriting in terms of ownership, summing over, and using yields

Hence, the equilibrium equation of the production model writes

since. This can be rewritten as

This is the equilibrium equation of the exchange model with production adjusted demand functions. Hence, by definition of the production adjusted demand function obtain

since for all. This is the equilibrium equation of the production adjusted exchange model in terms of. This concludes the proof of theorem (1).

Theorem 1. For fixed, is an equilibrium of the long run production model with uncertainty if and only if is an equilibrium of the two period exchange model with uncertainty and production adjusted demand functions.

We have established a relationship between the production model with a long term time structure and uncertainty, and a pure exchange model with a long term time structure and uncertainty with production adjusted demand functions.

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 denote the set of equilibrium solutions of the production adjusted exchange model or the set of solutions of the long run production model.4 This set consists of pairs satisfying the equilibrium equations. Formally, for the case of the production model we have

and in the case of the production adjusted exchange model, we have

Theorem 2. The set of model is a closed subset of the Euclidean space defined by.

Proof. is defined by pairs satisfying the equilibrium Equation (5). is the preimage of the vector by the smooth mapping for all, and closed by the closed map lemma closed ([9] , p. 553). The closed map lemma requires the excess demand mapping to be continuous and the domain to be a compact set and the range a Hausdorff space. Recall that the excess demand mapping is differentiable at any order required. This is a consequence of subtracting differentiable aggregate supply mappings from differentiable aggregate demand mappings. Differentiability of demand and supply mappings is in turn a consequence of the assumptions of the model discussed earlier ([1] [8] ).

Theorem 3. The set of model is a smooth manifold of dimension.

Proof. Consider the mapping into defined by the smooth mapping

.

By theorem the regular value theorem is the preimage of. We need to prove that this mapping does not contain critical points. This follows by showing that the linear tangent map is onto. The onto property follows directly from the rank property of the Jacobian matrix chosen for any arbitrary individual and state of nature. By the chain rule, we obtain

By simple algebraic manipulations we obtain the new matrices

Finally, we obtain

from which we extract the information required. Rank is equal to in every state. By the regular value theorem is a smooth manifold. This manifold is parameterized by smooth coordinate functions. From the regular value theorem it also follows that its dimension is equal to the dimension of minus, hence

.

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

that is diffeomorphic to the sphere in implying that the equilibrium manifold is arc-connected, simply connected, and contractible. These properties are particularly useful in applied work such as economic policy equilibrium analysis.5 For example, economic policy is often concerned with finding a path between a current point on and a desired point on.

Theorem 4. The smooth equilibrium manifold of model is diffeomorphic to the sphere of dimension.

Proof. The aim of the proof is to define two smooth mappings between smooth manifolds such that we can apply the theorem (Hirsch [10] , pp. 15-16). Hence, let

be smooth mappings defined by

Then, let

denote smooth mappings defined by

Observe that the coordinates for the good of the m − 1 consumers in, are defined

Also observe that the coordinates for the consumer of the goods in, are defined by

The application of theorem ([10] ) requires to show that and that. The first part of the proof requires to calculate two inclusions, (i) and (ii). We start by showing the se- cond part first. Now, to show that (i), take any consumption bundle

,

and compute the inner product of (7) with , and apply Walras’ law to obtain

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

This is the equilibrium Equation (5), hence. Next, need to show that (ii). Take. It is then trivial to do the computations proving following equality

from which it readily follows that. Clearly we have constructed the two smooth relations such that

where is the identity map defined on. We have shown that the smooth mapping restricted to the equilibrium manifold defines a diffeomorphism between and the sphere of dimension.

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 is smooth and proper. Existence of long run equilibria of the production model with uncertainty follows immediately from the smoothness lemma (1) and the properness lemma (2) below.

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

Lemma 1 (Smoothness) is smooth.

Proof. Recall that is a smooth submanifold of. It follows from the definition of a smooth manifold that its natural embedding is itself smooth. The projection mapping being itself smooth, it follows that the restriction of the natural projection to as the composition of two smooth mappings is therefore smooth.

The next lemma makes use of theorem (see [11] , p. 174).

Lemma 2 (Properness) is proper.

Proof. Pick an arbitrary for. Let be an element in a compact set. Now, for every and need to show (i) that is bounded from below. It follows that

and by non-satiation have also

which by monotonicity of implies that

Clearly, there exists some for every and satisfying

by boundedness of indifference mappings from below for every. (ii) We now show that for every and, is also bounded from above. For have

where

Clearly, , is bounded above by some, since for, is bounded from above for every. Hence have established upper and lower bounds defining a compact set

for every. Let G be a compact set defined by the preimage of the diffeomorphism projected onto. Now, by continuity of, is closed in, which by theorem (2) is a closed subset of. Closedness of follows from closedness of.

The number of equilibria of the long run production model with uncertainty is odd for any regular economy. The modulo 2 degree of is. See Guillemin and Pollack for example [12] .

I now define a subset of points on at which pairs are not regular.

Definition 2. is the set of singular equilibria given by the singular points of.

Proposition 1. is closed.

Proof. A necessary and sufficient condition for an equilibrium pair to be singular is that the determinant of the Jacobian matrix of, denoted by is equal to zero. Now, the set of critical points defined by the preimage of is closed by the closed mapping lemma ([9] ), since, , and the coefficients of, are all continuous, from which the result follows.

Definition 3..

A singular value is the image of of a singular point into. The set of regular values is defined by. It follows that represents the sets of regular economies. The next proposition states the is closed and of measure zero. This means that the probability of observing an economy with this property is “close” to zero. Hence, its complement is an open dense set.

Proposition 2. The set of singular economies is closed and of Lebesgue measure zero in.

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 is a set of Lebesgue measure zero. Closedness of follows from the properness of.