ABSTRACT

This paper shows existence and efficiency of equilibria of a two period production model with uncertainty as a consequence of the catastrophe map being smooth and proper. Its inverse mapping defines a finite covering implying finiteness of equilibria. Beyond the extraction of local equilibrium information of the model, the catastrophe map renders itself well for a global study of the equilibrium set. 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.

1. Introduction

This paper considers a two period production model with uncertainty. The time structure and associated uncertainty is described by a finite number of uncertain states of the world. It is assumed that all firms are owned by the consumers according to an exogenously determined ownership structure. This economic scenario describes the private ownership model discussed in Debreu [1] where the objective of each firm is to maximize profits. The seminal paper of this model without uncertainty dates back to the path breaking paper by Arrow and Debreu [2].

In this paper, we show that many economically interesting equilibrium properties of the two period production model with uncertainty can be derived from the catastrophe map. For that purpose we follow the mathematical approach discussed in Balasko [3] and in Dierker [4].

More specifically, we describe the set of solutions of all two period production economies and explore its structure. It is shown that this set is a smooth submanifold of the Euclidean space which is diffeomorphic to the sphere. A study of some of the properties of the catastrophe map enables us to characterize the set of economies into sets with various properties, such as economies with singular equilibria, economies with multiple equilibria, and economies with catastrophes, where equilibrium behavior is more difficult to study. Most of these properties have been studied in the context of exchange economies [5] or simple production economies [6-10] or Balasko (Preprint 2011) for example¹. This paper generalizes the economic scenario by adding more structure to the model of the firm and thus moving towards a more realistic model where time and uncertainty is present.

The structure of the paper is as follows: Section 1 is an introduction. Section 2 introduces the economic scenario and states a definition of economic equilibrium. Section 3 explores the topological structure of the equilibrium set of all two period production economies with uncertainty. The next section states equilibrium properties of the model such as existence, efficiency and finiteness of equilibria. The final section is a conclusion.

2. The Long Run Private Ownership Production Model with Uncertainty

We describe the two period private ownership production model introduced in Debreu ([1], chapter 7). Uncertainty is defined by a finite set of mutually exclusive and exhaustive states of nature denoted by, where is the certain event in time period one and are the uncertain events in time period two. In total there are states of nature. There are consumers, producers, and physical goods. For all consumers

, a consumption bundle is a collection of vectorswhere consumption in a particular state is a vector. Associated with physical commodities is a set of normalized pricesdenoted.

Consumers are further endowed with a fraction of the profits of each firm. represents the exogenously determined ownership structure of the private ownership production economy. It satisfies for each and, 0 ≤ θ_ij ≤ 1, and. Denote the set of ownership structures

.

Consumers are endowed with a collection of vectors of initial resources denoted by

, where initial endowments in a particular state is a vector. Consumer

is further characterized by a smooth Marschallian demand function, where

is defined for price vector and wealth level, [11], where for all.

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 is a collection of vectors, where an activity in state is a vector of inputs

, and

is the associated vector of outputs in state. Let denote the smooth supply function of firm, where is defined on the set of normalized prices. Standard assumptions of smooth production economies introduced in [1] hold for each production set. In particular is convex, , and has a strictly positive Gaussian curvature for every. These assumptions imply that supply functions are smooth.

Equilibrium

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

(1)

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 smooth² utility function. The production adjusted consumer budget set is 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].

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

(2)

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 l(S + 1) − (S + 1) 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 (2).

3. Equilibrium Structure of the Model

Let denote the set of equilibrium solutions of the two period production model with uncertainty. This set consists of pairs satisfying the equilibrium equations for all. Formally, we have

For the proof of the next theorem we need the following result.

Lemma 1 (Properness of a mapping). Suppose M(s) is a compact space and is a Hausdorff space for every. Then every continuous map for all is proper.

Proof. We need to show that for every compact set the inverse image is compact for every.

1) Let us show that the direct image of any closed subset of is closed in for all. To show this let

, for all, where

belongs to the set. From the convergence property of the sequence we see that the set is compact. From that it follows that is compact for every.

2) Let us show that inverse image is compact. We take in such that

. Clearly, the sequence

belongs to the compact set defined by the inverse image

. Therefore, there exists a subsequence

for all such that

([12], p. 41), where

. Since is the limit of a subsequence of elements belonging to, we have. By continuity of the mapping we have

This proves that for every . ■

Theorem 1. The set of model is a closed subset of the Euclidean space defined by.

Proof. Note that continuity of the mapping

for all is sufficient to show closedness of the set of model. is the preimage of the vector by the smooth mapping

for all which is closed by Lemma 1. Continuity of the equilibrium equation is satisfied by the assumptions of differentiability of demand and supply mappings [1,11]. ■

Theorem 2. The set of model is a smooth manifold of dimension.

Proof. We consider the mapping defined by the smooth mapping

.

By the regular value theorem (Guillemin and Pollack [13], p. 21) 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 ([13], p. 21) 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 a further economically interesting global property of the equilibrium manifold. It says that by construction of a diffeomorphism restricted to the equilibrium manifold into

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. For example, economic policy is often concerned with finding a path between a current point on and a desired point on. The following theorem proves that such a path always exists. In order to prove this result, we use a theorem given in (Hirsch [14], pp. 15-16).

Theorem 3. The smooth equilibrium manifold of model is diffeomorphic to.

Proof. The aim of the proof is to define two smooth mappings between smooth manifolds such that we can apply the theorem given in (Hirsch [14], 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 consumers in, are defined

(3)

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

(4)

The application of the theorem in ([14]) requires to show that and that. The first part of the proof requires to calculate two inclusions, 1) and 2). We start by showing the second part first. Now, to show that 1), take any consumption bundle , and compute the inner product of (4) with , and apply Walras’ law to obtain

From that a reformulation of (4) readily follows in terms of the production equilibrium equation

hence. Next, we need to show that 2). Take any arbitrary. 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 f restricted to the equilibrium manifold defines a diffeomorphism between and the sphere of dimension. ■

4. Existence, Efficiency, and Finiteness of Equilibria

We now show that equilibria in the two period production model with uncertainty always exist. The strategy of the proof is to show that the catastrophe mapping is smooth and proper. Existence of equilibria of this production model with uncertainty follows immediately from the smoothness proposition (1) and the properness proposition (2) below. The result of properness of provides a deep insight into the definition of economics itself. It implies that economic resources are scarce. The diffeomorphism for all between the spaces and suggests that the vector tends to infinity in norm if prices tend to zero. It tends to zero if prices tend to infinity.

Axiom 4 (Bounded and strictly convex preferences). 1) The set of consumptions bundles indifferent or preferred to consumption bundle for all is bounded from below for every for all. The preordering is then said to be bounded from below; 2) The set of consumptions bundles indifferent or preferred to consumption bundle for all is strictly convex for every for all . The preordering is then said to be strictly convex.

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

Definition 2. The catastrophe map is defined by the. It is the restriction of the projection of the set of equilibria into the space of economies.

Proposition 1 (Smoothness). of model is smooth.

Proof. From Theorem (3) we know that of model is a smooth submanifold of which is diffeomorphic to the sphere of dimension. It follows from the definition of a smooth submanifold ([15], p. 174) that its natural embedding is smooth. It is clear that the projection mapping is itself smooth. It then follows that the restriction of the natural projection to as the composition of two smooth mappings is therefore smooth. ■

Proposition 2 (Properness). of model is proper.

Proof. The strategy of the proof is to define the economic scenario such that lemma (1) can be applied to the model. Hence, we need to show that for all the inverse image, where is a compact set in the space of initial resources, , is compact.

We show that individual consumer demand is bounded below in every uncertain state of the world. To show this, consider any and define the projection of initial resource into the coordinate and state, defined by

Pick an arbitrary for. Let be an element in a compact set. Note that is compact by the projection of a compact set on the coordinate space. Compactness of in implies for every that

1) Now, for every and and need to show that is bounded from below. It then follows from standard assumptions of consumer theory that for all

where

and

for all.

By non satiation we also have

which by monotonicity of implies that

Clearly, there exists some for every and for all satisfying

by boundedness (Axiom 4) of indifference mappings from below for every.

2) We now show that for every, and, is also bounded from above. Consider the equilibrium price vector for any. Then for all pairs we have

where³

Clearly, , is bounded above by some, since for is bounded from above for every. Hence, we have established the upper and lower bounds for every consumer given by

for every.

3) We now apply Lemma 1. For any arbitrary consumer, we have established the compact set. Let be a compact set defined by the preimage of the diffeomorphism ([11]) projected onto. Hence, we observe that is a subset of the compact set. Lemma (1) requires to show that is closed in.

Now, by continuity of, , it follows that is closed in, which by Theorem (1) is a closed subset of. Closedness of follows from closedness of. ■

Lemma 2 (Individual demand: Diffeomorphism of) For every the individual demand mapping is a diffeomorphism for all.

Proof. The strategy of the proof is to show that is smooth, bijective, and that is also smooth.

The problem of the consumer is to solve the constraint optimization given by

where

We can use the Lagrangean method to solve this problem. Hence the solution of this problem satisfies the first order conditions of the optmimzation problem and is given by for all. Hence the pair, where is the Lagrangian multiplier is a solution of the Lagrangian problem. Hence, to show smoothness of requires to show that is a smooth function of and. This is a consequence of the implicit function theorem applied to the solutions of the Lagrangian. Hence, we calculate the bordered Hessian matrix, for all. Thus,

and the inverse of at exists since

We now show that is also smooth. Let defined by

By assumptions of Debreu [11] all ingredients of this formula are smooth. Also the inner product of smooth functions is smooth. Hence we conclude that is also smooth.

We now show that and are inverse mappings for all. Hence 1) We calculate the individual composite mapping for all and show that. This condition is satisfied since

As required, we have established

.

2) We calculate the individual composite mapping for all and show that

. This condition is satisfied since by definition of we have

As required, we have established

. We have proved the bijection property of the individual demand function⁴. ■

This proves existence of equilibria.

Definition 3. A feasible allocation

associate with equilibrium price vector and economy

is Pareto efficient for all

if there is no other feasible allocation

such that for all

and

with at least one strict inequality.

Theorem 6 (Pareto efficiency of model). Every economy of the model is Pareto efficient for all.

Proof. We proceed by contradiction. We show that if at equilibrium price the economy, where is an allocation of consumption and production which is not efficient, then it must be that firms do not maximize profits. This contradicts the assumption that all firms maximize profits (Debreu, [1] Chapter 5) and implies that not all economies are Pareto efficient.

We have for all

Hence,

Hence we obtain the equilibrium equation given by

We can now establish a contradiction.

Now, let be an equilibrium price vector for any arbitrary and an associated feasible equilibrium allocation which is not Pareto efficient. Since is feasible we have

hence

(5)

Since by assumption is not Pareto efficient, there exists a feasible allocation associated with with and such that

with at least one strict inequality. This implies that

with at least one strict inequality. Aggregating consumption bundles we obtain together with the inner product the strict inequality

(6)

Substituting Equation (5) into strict inequality (6) and using the feasible allocation we obtain

But this strict inequality says that for some that for feasible

. Hence a violation that firms maximize profits. Clearly, since for at least one, is a Pareto inefficient economy. ■

Theorem 7. of model is a finite covering for every, for all.

Proof. Let consist of a single element of for all. Consider the tangent map of elements of not contained in the set of singular points,. Then as a non singular point in there exists a bijective map which by the inverse function theorem implies that is locally a diffeomorphism. By the inverse function theorem there exists an open set of and an open set of such that the restriction of the natural projection to, is a diffeomorphism for all. It follows from the one-to-one property of this map that . Since is open in it follows from the definition of open sets of as intersections with of open sets of that the subset is open in. The union of all open subsets define an open covering of. Compactness of the set follows from compactness of the preimage of a compact set by the proper mapping. It follows from compactness of that the open covering has a finite subcovering defined by the unique element of. The union of a finite number of elements defines the set which is therefore a finite set. This proves finiteness of the number of equilibria. ■

5. Conclusion

This paper discusses local and global equilibrium properties of a production economy with a two period time structure and uncertainty. 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. Beyond that, the paper shows that equilibria always exist, and that they are efficient and finite.

REFERENCES

NOTES

¹Discussion paper: The natural projection approach to smooth production economies, 2011.

²“Smoothness” follows from the assumptions stated in [11]. It essentially means that all functions are differentiable at any order required.

³−i is standard notation used in economic theory. It is equivalent to saying such that, hence

⁴We have assumed that supply functions are smooth. Hence