Abstract

In the presented work analyzes the Pareto optimum distributions in exchange economy. We are examining a mathematical model of economy with n? 2 agents and m? 2 goods. The examining theorems are not using the prices of the goods and budgetary limitations of the agents. The theorems are based only on the utility of the goods and the status quo of agents. Measure for the status quo of the agent here will be his or her utility function.

Introduction

A basic characteristic of the Equilibrium State in the mathematical economy is given with the Pareto optimum concept. A distribution of goods is Pareto optimum then and then only, when in the process of transition into another distribution there is no deterioration of the status quo of agents. Measure for the status quo of the agent here will be his or her utility function. Searching for optimum Vilfredo Pareto lays out the criterion: if in the process of distribution of goods between the agents in an economic system the welfare of one single agent increases, without decreasing the welfare of all the other agents, then the welfare of the system as a whole increases. Therefore, we have in result the definition: the distribution of goods is Patero optimum then and then only, when it is not possible for the welfare of a certain agent to be improved without involving the worsening of the welfare of another agent.

The exchange of goods is realized in an equilibrium, which is characterized by:

Under certain conditions it is proved that the equilibrium distributions of goods are Pareto optimum.

We will examine a number of characteristics of the Pareto optimum distributions using neither the fact of equilibrium, nor the prices on the goods. This is an important issue because there are no prices or budgetary limitations used with the Pareto optimum. The Pareto optimum is based only on the utility of the goods.

Mathematical model

We are examining a mathematical model of economy with n? 2 agents. Let’s mark the set of agents with A and they exchange between each other m? 2 goods. Let G be the set of goods and let L=nm.

Let each agent own initial property demonstrated with the vector vⁱ(v₁ⁱ,..,v_jⁱ,..,v_mⁱ)?, where the number v_j^i?0 shows the quantity of g_j?G property of a_i?A. The vector W = we will name the vector of common goods, where W (W₁,W₂,...,W_m)?.

The set D={X(x¹,x²,...,xⁿ)? : } we will name the set of distributions, where a_i?A is owner of x^i?, let’s mark P_i(X)= and P_i,_j(X)=. The initial property of the agents is demonstrated with the vector V(v¹,v²,…,vⁿ)? D. It is clear that D?? is a convex and compact set in .

Let each agent a_i?A has an utility function u_i:D® with the following characteristics:

(i) The function u_i is continuous in D;

(ii) If X,Y? D and P_i(X)=P_i(Y), then u_i(X)=u_i(Y);

(iii) If X,Y? D, P_i(X)? P_i(Y) and P_i(X)? P_i(Y), then u_i(X)>u_i(Y);

(iv) If X,Y? D, P_i(X)? P_i(Y) and a? (0;1), then u_i(a X+(1-a )Y)>min(u_i(X),u_i(Y)).

The function U:D® we will call collective utility function then and then only, when " X? D U(X)=(u₁(X),u₂(X),...,u_n(X)), where the functions u_i for i=1..n are the utility functions of the agents. From the continuity of the functions u_i it follows that the function U is continuous in D.

Definition 1. The economy stands here for the set {A,G,V,D,U}.

Definition 2. We will say that the distribution X? D is Pareto optimum then and then only, when  Y? D such that " a_i?A u_i(Y)? u_i(X) and $ a_k?A u_k(Y)>u_k(X). The set of distributions of D, which are Pareto optimum, we will mark with P.

Therefore X? P then and then only, when

{Y? D : " a_i?A u_i(Y)? u_i(X) and $ a_k?A u_k(Y)>u_k(X)}=? .

It becomes clear from the definition that the Pareto optimum is not related to the prices of goods, but is defined only by the utility functions of the agents.

Existence of Pareto optimum distribution

Definition 3. The set

D*={X? D : $ {c_i?0 : i=1..n}?,  and ? " Y? D}

we will call set of maximum distributions of D.

Theorem 1. The set D*?? is a compact sub-set of D.

Proof: Let’s examine the function f:D® where " D? W f(X)=, for c_i?0,  and u_i are the utility functions of the agents for i=1..n.

From the continuity of the functions u_i it follows that the function f is continuous in D. The set D is compact, therefore $ X? D such that f(X)=sup{f(Y) : Y? D}. There is D*?? .

For the continuity of the function f it follows that the set D* is a closed sub-set of the compact set D. Therefore D* is a compact sub-set of D. ™

Theorem 2. D*? P.

Proof: Let X? D*. Let’s assume that X? P, therefore $ Y? D such that " a_i?A u_i(Y)? u_i(X) and $ a_k?A u_k(Y)>u_k(X).

Let f(X)=, therefore f(Y)? f(X).

If c_k>0, then f(Y)>f(X), which is in contradiction with X? D*.

If c_k=0, then f(Y)==. From u_k(Y)>u_k(X)? 0 it follows that P_k(Y)? 0. From  it follows that $ a_j?A such that c_j>0. From P_k(Y)? 0 it follows that P_j(Y)?W . Therefore $ Z? D such that P_j(Z)>P_j(Y), P_k(Z)<P_k(Y) and P_i(Z)=P_i(Y) for i? j and i? k. Finally there is f(Z)>f(Y)? f(X), which is in contradiction with X? D*.

We have in result D*? P. ?

From the above two theorems it follows that in the economy exists a Pareto optimum distribution.

Characteristics of Pareto optimum distribution

Definition 4. If X? D, then the set R_i(X))={Y? D : u_i(Y)? u_i(X)} we will call set of preferences of a_i?A.

It is clear that R_i(X) is compact " a_i?A, therefore  is compact and X?.

Theorem 3. Let X? D. There is X? P then and then only, when {X}=.

Proof: Let X? P. There is X? R_i(X) " a_i?A, therefore {X}?.

Let Y?, a? (0;1) and Z=a X+(1-a )Y.

Let’s assume that X? Y.

If a_i?A and P_i(X)=P_i(Y), then P_i(Z)=P_i(X). Therefore u_i(Z)=u_i(X).

If a_i?A and P_i(X)? P_i(Y), then u_i(Z)>min(u_i(X),u_i(Y))=u_i(X).

From X? Y it follows that $ a_k?A such that P_k(X)? P_k(Y). In result, there is " a_i?A u_i(Z)? u_i(X) and u_k(Z)>u_k(X), which contradicts the condition X? P. Therefore X=Y, i.e. ? {X}. Finally, there is {X}=.

Let {X}=. Let’s assume that X? P, therefore $ Y? D such that " a_i?A u_i(Y)? u_i(X) and $ a_k?A u_k(Y)>u_k(X). In result there is Y?={X}, therefore X=Y. This contradicts u_k(Y)>u_k(X), therefore X? P. ?

Definition 5. If X? D, then the set M_i(X)={Y? D : u_i(X)=u_i(Y)} we will call set of indifference of a_i?A.

Theorem 4. If X? P, then {X}=.

Proof: There is X? M_i(X) " a_i?A, therefore {X}?.

Let Y?, therefore Y?={X}. In result there is X=Y. ?

Definition 6. Let a_i?A. The utility function u_i of a_i?A is convex then and then only, when if " X,Y? D, P_i(X)? P_i(Y) and "a? [0;1], then u_i(a X+(1-a )Y)?a u_i(X)+(1-a )u_i(Y).

Theorem 5. If the utility functions of the agents u_i for i=1..n are convex, then D*=P.

Proof: Let X? P and S={U(X)+s : s?\{0}}. It is clear that S?? .

We will prove that the set S is convex. Let S₁,S_2?S и a? [0;1], there are $ s₁,s_2?\{0} such that S₁=U(X)+s₁ and S₂=U(X)+s₂. After due multiplication and addition we have a S₁+(1-a )S₂=U(X)+(a s₁+(1-a )s₂), therefore a S₁+(1-a )S_2?S, i.e. the set S is convex.

Let B={U(Y)-b : Y? D and b?\{0}. It is clear that B?? .

We will prove that the set B is convex. Let B₁,B_2?B and a ? [0;1]. There are $ Y₁,Y_2?D and $ b₁,b_2?\{0} such that B₁=U(Y₁)-b₁ and B₂=U(Y₂)-b₂. After due multiplication and addition we have a B₁+(1-a )B₂ =(a U(Y₁)+(1-a )U(Y₂))-(a b₁+(1-a )b₂).

Let Y=a Y₁+(1-a )Y₂. From the convexity of the set D it follows that Y? D.

If a_i?A and P_i(Y₁)=P_i(Y₂) , then u_i(Y)=u_i(Y₁)=u_i(Y₂).

If a_i?A and P_i(Y₁)? P_i(Y₂), then from the convexity of the utility function u_i it follows that u_i(Y)?a u_i(Y₁)+(1-a )u_i(Y₂).

Finally, there is U(Y)?a U(Y₁)+(1-a )U(Y₂).

Let b=U(Y)-(a U(Y₁)+(1-a )U(Y₂)), therefore b?\{0}. In result there is a B₁+(1-a )B₂=U(Y)-(a b₁+(1-a )b₂), therefore a B₁+(1-a )B_2?B, i.e. the set B is convex.

We will prove that SB=? .

Let’s assume that SB?? , therefore $ Y? D and $ s,b?\{0} such that U(X)+s=U(Y)-b. From s,b? 0 it follows that " a_i?A u_i(Y)? u_i(X) and $ a_k?A u_k(Y)>u_k(X), which contradicts the condition X? P, therefore SB=? .

From the theorem for detachability of sets it follows that $ {q_i : i=1..n}? such that  and $ c? such that " s(s₁,s₂,..,s_n),b(b₁,b₂,…,b_n)?\{0} there is

" Y? D.

Therefore  " Y? D and we have in result an equality at X=Y.

We will prove that q_i?0 " i? [1;n]. Let’s examine the inequality , which is true " s(s₁,s₂,..,s_n)?\{0}.

Let’s assume that q_k<0, therefore  is true at

.

In result we have

.

This leads to a contradiction, therefore q_i?0 " i? [1;n].

From  it follows that . Let ? 0 for i=1..n, therefore  and ? " Y? D.

Finally, there is X? D*, i.e P? D*. From theorem 2 it follows that D*=P. ?

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