The problem of maximizing a function has been studied for several centuries

by means of differential calculus. Morse’s theory can be seen as a globalization of this problem. Comparatively recently, economists have considered in the special case the problem of “optimization” of several functions simultaneously, obtaining the so-called Pareto optimum.

Our aim is to indicate the place of this problem within the framework of global analysis,

or the theory of several real differentiable functions on a manifold. We will generalize the notion of a Pareto optimum and define a broader

broad set, which we will call the critical set Pareto Θ.

This set Θ is analogous and is a generalization of the set of critical points of a single differentiable function, while the Pareto optimum proper is an analog and generalization of the maximum of a single function. This generalization of the problem posed by economists makes it possible to systematically apply methods of global analysis to the procedure of optimization of several functions on a manifold.

For example, with this approach, we obtain a natural notion of dynamics generalizing the gradient flow. Our approach contrasts with the more conventional equilibrium, or static, approach that mathematical

economists use when studying the net exchange model.

For example, with this approach, we get a natural notion of dynamics that generalizes gradient flow. Our approach contrasts with the more conventional equilibrium, or static, approach that mathematical economists use when studying the net exchange model.

More precisely, the problem we are considering is as follows: let real differentiable functions ui: W→R, defined on a variety of W, where i = 1, …, m. What is the nature of the curves ϕ: R→W with positive derivatives

d/dt (uv ◦ ϕ)(t) for all i and t? For what x € W there are such ϕ, ϕ(0) = x?

Critical set Pareto Θ is defined as the set of points x € W or which there no such ϕ. The main problem is the study of sets of Θ

In other words, the way is to learn how and when to gradually increase the value of several functions at the same time. This subject can be seen as part of game theory.

These questions lead to attractive mathematical problems.

In particular, we get a new way of studying the Pareto set in economics by abandoning the traditional assumptions of convexity and monotonicity.

I believe that the question of optimizing multiple functions simultaneously goes beyond economics; in other social problems, optimizing multiple functions (instead of just one) allows us to go beyond the one-dimensional viewpoint. It is a matter of considering many values in private conflicts instead of maximizing a single value. We propose in the following to consider from a close perspective the price system in the economy.

I want to caution the reader about the somewhat presumptive nature of some of the statements in this paper.

1) Here we give an overview of the concepts of the pure exchange model in economics and the classical Pareto optimum.

The space of goods is an open set in the Cartesian space R^l.

The number of different goods in this model is l. The quantity of each good is measured by a real number (the unit of measure is fixed), which can be viewed as a coordinate in R^l.

We will be dealing with only a positive quantity of each good, and therefore define the space of goods as a positive ortant P in R^l. So, P = { x € R^l | all coordinates x is positive}. The point from P depicts a set of goods, lets say, that a consumer can possess.

Suppose there is a finite number of consumers, say, m. We denote the set of goods of the i-consumer by xi € P. The free state of the model in question will be the point x = (x1, …, xm) in the Cartesian product. Pm (in the variety of dimensionality ml). We will assume that the total resources in this model are fixed, say, given by point w € P. Thus, the space of admissible states forms subset W in P^m, defined as:

W is an open subset of an affine space with compact closure in (R^l)^m. W is the basic state space that we consider in this paper. It is assumed that each consumer has a preference represented by the function ui : P → R his utility function, which we will assume to be differentiable as many times as necessary. So the i-th consumer prefers a set of x`I set if xi if and only if

This i-th consumer is indifferent to the goods lying on the same level surface of the function ui. Therefore, the surfaces ui^(-1)(c), are called indifference surfaces. In fact, in economics is primarily set by these surfaces, and our analysis is ultimately is based on a consideration of these surfaces. But functions ui suitable like another means of communication.

Let

Projection пi(x) = xi. Define on W functions, like later call them ui, like mapping compositions:

onsider exchanges in W that increase each consumer’s utility function of each consumer, i.e., they increase each function ui on W. Condition

called optimum Pareto, if it has the following property: it does not exist

for all I and

for particular j. The idea is that if

is not a Pareto optimum. Pareto, there will be no economic equilibrium; there will be trade leading x to the Pareto optimum.

2) The purpose of this paragraph is to define the “hessian” in our situation.

Using it, we will obtain a stability criterion for the points of the critical of the Pareto set. In the case of a single function u: W→R it boils down to stating, that

is a local maximum and is stable, if the first differential turns to 0 and the second differential is negatively defined.

In our case we have a smooth mapping u: W→Rm , W manifold in Euclidean space Rm. We are comfortable making the following assumption about rank.

Definition.

Dot

satisfies the assumption about the rank if the rank Du(x) >= m-1 and u is satisfies the assumption of rank if it is satisfied for all

Let us make a few remarks. It is clear that if

rank of Du(x)<= m-1. Consequently, you can write equality instead of inequality in the rank assumptions. We will make a small digression concerning the use of the expressions “almost all” and “the property of general position”. Let us introduce in the space of mappings u: W→Rm natural Cɣ topology which transforms this space into a Baer space. The Baer space has the following property: the countable intersection of dense sets open everywhere is a dense set and is called a Baer set. “Almost all” means all elements of some Baer set. By “the property of general position” for and we will mean the property fulfilled for all elements of some Baer set. For “almost all” and in this Baer sense almost all points Θ satisfy the rank assumption. However, it cannot be said that the property of satisfying the rank assumption is a “property of the general position” for u.

On another side if

almost all u satisfy the assumption of rank. In the case of the pure exchange model, this condition on dimensionality is always satisfied (as a simple calculation shows). Thus, the ranking assumption does not seem to be a serious limitation. We will now define the Hessian for u: W→Rm in dot

Imagine that

satisfies the assumption of rank. Then Du(x): Tx(W)→Rm have rank m-1. The second derivative defines a certain invariant (independent of the map) symmetric bilinear mapping Hx tangent space Tx (w) with values in one-dimensional vector space Rm/Im Du(x) has a canonically definite positive direction (or ortant, etc.) because the image Du(x) don’t cross

This last fact is basic to our whole theory and allows us to define a negative definiteness, an index, a zero subspace, etc. for x. This idea does not apply to the theory of singularities of mappings, because in the general case there is no natural definition of the positive part of Rm/Im Du(x).

Theorem

Let the mapping u: W→Rm – class Cr, where r is large enough and x belongs to the critical Pareto set. Let x also satisfy the rank assumption and the generalized Hessian Hx negatively define. Then x belongs to the stable set Pareto. The proof can be obtained, for example, from Levin’s normal forms. Note that if x satisfies the conditions of the theorem and the admissible curve ϕ: [0, 1]→ W have x in closing set, so clearly say ϕ(t)→x with t→1. Note, finally, that if

satisfies the rank condition, we define for x the index, zero subspace, nondegeneracy as index, zero subspace, nondegeneracy respectively for the bilinear form Hx.

3)The purpose of this paragraph is to try to obtain a global theory of the Pareto critical set and Pareto stable points.

Theorem 1. Suppose that u: W→Rm satisfies the rank assumption, the condition of transversality of jets and the condition of transversality of A and that the manifold W is compact. Then dots

For solving that you need to use Morse Theory.

Theorem 2.

Suppose that u: W→Rm have no cycles satisfies the assumption of rank, the condition of transversality of jets, condition of transversality of A and that W is compact. Let

with coefficients in an arbitrary field. Then Mi satisfies the Morse inequalities, i.e., if Bi means i-th number of the Batty manifold W, then

Note that the Morse function u: W→R, obviously satisfies the hypotheses of “theorem” 2. Thus, “theorem” 2 contains the usual theory of Morse. Let us now see what all this means at m = 2. Note that

This stems from the fact that Θ one-dimensional and the index is strictly less than n. Then we should look all different situations fo

Let’s outline an idea of how we could prove “theorem” 2. Let us choose a gradient system u on W. Then we define for each stratum

as the set of points tending to

in a gradient dynamical system.

Then

From this follows Theorem 1.

Perhaps a more sophisticated version of Morse’s theory can account for the existence of cycles. We end this paper with the following remarks. First, how far can one go without the ranking assumption? Clearly, in the general case, the complexities of the theory of singularities of mappings make it difficult to advance.

Finally, one might ask what is the relation of the concept of the kernel of theoretical economics to what I made here. I do not think that the concept of the kernel is well captured by this approach. The reason is that we see exchange, many exchanges, as a core element of our approach to Pareto theory. Essentially, we are looking at a dynamic approach. On the other hand, it seems to me that the concept of the kernel refers essentially to a static, or equilibrium approach. For example, in order to define the kernel, we need to know the initial resources of each consumer. But these resources change after the first transaction, and after a few transactions they can be forgotten. Thus, after a few initial, but not final, transactions, the idea of a coalition to describe the kernel loses force.

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