1
PARETO OPTIMUM IN THE MATHEMATICAL ECONOMY
Zdravko D. Slavov
Varna Free University, Department of Mathematics, 9000 Varna, Bulgaria
e-mail: [email protected]
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:
(i) Equalizing the demand and offer of goods between the agents;
(ii) Obtaining the maximum utility of the agents within the budgetary limitations
determined by the prices on the goods.
Under certain conditions it is proved that the equilibrium distributions of goods are
Pareto optimum.
2
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,..,vji,..,vmi)  m 0 , where the number vji0 shows the quantity of gjG property of aiA.
n
The vector =  v i we will name the vector of common goods, where (1,2,...,m)  m .
i 1
n
The set D={X(x1,x2,...,xn)  L 0 :  x i   } we will name the set of distributions,
i 1
where aiA is owner of xi  m 0 , let’s mark Pi(X)= x i ( x 1i , x i2 ,..., x im ) and Pi,j(X)= x ij . 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  L 0 .
Let each agent aiA has an utility function ui:D  0 with the following
characteristics:
(i) The function ui is continuous in D;
(ii) If X,YD and Pi(X)=Pi(Y), then ui(X)=ui(Y);
(iii) If X,YD, Pi(X)Pi(Y) and Pi(X)Pi(Y), then ui(X)>ui(Y);
(iv) If X,YD, Pi(X)Pi(Y) and (0;1), then ui(X+(1-)Y)>min(ui(X),ui(Y)).
The function U:D  n 0 we will call collective utility function then and then only,
when  XD U(X)=(u1(X),u2(X),...,un(X)), where the functions ui for i=1..n are the utility
functions of the agents. From the continuity of the functions ui 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  aiA ui(Y)ui(X) and  akA uk(Y)>uk(X). The set of
distributions of D, which are Pareto optimum, we will mark with P.
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Therefore XP then and then only, when
{YD :  aiA ui(Y)ui(X) and  akA uk(Y)>uk(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
n
n
n
i 1
i 1
i 1
D*={XD : {ci0 : i=1..n}  ,  c i  1 and  c i u i (X)   c i u i (Y)  YD}
we will call set of maximum distributions of D.
Theorem 1. The set D* is a compact sub-set of D.
n
Proof: Let’s examine the function f:D  0 where  DW f(X)=  c i u i (X) , for ci0,
i 1
n
 c i  1 and ui are the utility functions of the agents for i=1..n.
i 1
From the continuity of the functions ui 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  aiA
ui(Y)ui(X) and  akA uk(Y)>uk(X).
n
Let f(X)=  c i u i (X) , therefore f(Y)f(X).
i 1
If ck>0, then f(Y)>f(X), which is in contradiction with XD*.
n
n
i 1
i 1,i  k
If ck=0, then f(Y)=  c i u i (Y) =  c i u i (Y) . From uk(Y)>uk(X)0 it follows that
n
Pk(Y)0. From  c i  1 it follows that  ajA such that cj>0. From Pk(Y)0 it follows that
i 1
Pj(Y). Therefore  ZD such that Pj(Z)>Pj(Y), Pk(Z)<Pk(Y) and Pi(Z)=Pi(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.
4
Characteristics of Pareto optimum distribution
Definition 4. If XD, then the set Ri(X))={YD : ui(Y)ui(X)} we will call set of
preferences of aiA.
n
It is clear that Ri(X) is compact  aiA, therefore  R i (X) is compact and
i 1
n
X  R i (X) .
i 1
n
Theorem 3. Let XD. There is XP then and then only, when {X}=  R i (X) .
i 1
n
Proof: Let XP. There is XRi(X) aiA, therefore {X}  R i (X) .
i 1
n
Let Y  R i (X) , (0;1) and Z=X+(1-)Y.
i 1
Let’s assume that XY.
If aiA and Pi(X)=Pi(Y), then Pi(Z)=Pi(X). Therefore ui(Z)=ui(X).
If aiA and Pi(X)Pi(Y), then ui(Z)>min(ui(X),ui(Y))=ui(X).
From XY it follows that  akA such that Pk(X)Pk(Y). In result, there is  aiA
ui(Z)ui(X) and uk(Z)>uk(X), which contradicts the condition XP. Therefore X=Y, i.e.
n
n
i 1
i 1
 R i (X ) {X}. Finally, there is {X}=  R i (X ) .
n
Let {X}=  R i (X ) . Let’s assume that XP, therefore  YD such that  aiA
i 1
n
ui(Y)ui(X) and  akA uk(Y)>uk(X). In result there is Y  R i (X ) ={X}, therefore X=Y.
i 1
This contradicts uk(Y)>uk(X), therefore XP. 
Definition 5. If XD, then the set Mi(X)={YD : ui(X)=ui(Y)} we will call set of
indifference of aiA.
n
Theorem 4. If XP, then {X}=  M i (X) .
i 1
n
Proof: There is XMi(X)  aiA, therefore {X}  M i (X) .
i 1
n
n
i 1
i 1
Let Y  M i (X) , therefore Y  R i (X ) ={X}. In result there is X=Y. 
5
Definition 6. Let aiA. The utility function ui of aiA is convex then and then only,
when if  X,YD, Pi(X)Pi(Y) and [0;1], then ui(X+(1-)Y)ui(X)+(1-)ui(Y).
Theorem 5. If the utility functions of the agents ui for i=1..n are convex, then D*=P.
Proof: Let XP and S={U(X)+s : s  n 0 \{0}}. It is clear that S.
We will prove that the set S is convex. Let S1,S2S и  [0;1], there are 
s1,s2  n 0 \{0} such that S1=U(X)+s1 and S2=U(X)+s2. After due multiplication and addition
we have S1+(1-)S2=U(X)+(s1+(1-)s2), therefore S1+(1-)S2S, i.e. the set S is convex.
Let B={U(Y)-b : YD and b  n 0 \{0}. It is clear that B.
We will prove that the set B is convex. Let B1,B2B and  [0;1]. There are 
Y1,Y2D and  b1,b2  n 0 \{0} such that B1=U(Y1)-b1 and B2=U(Y2)-b2. After due
multiplication and addition we have B1+(1-)B2 =(U(Y1)+(1-)U(Y2))-(b1+(1-)b2).
Let Y=Y1+(1-)Y2. From the convexity of the set D it follows that YD.
If aiA and Pi(Y1)=Pi(Y2) , then ui(Y)=ui(Y1)=ui(Y2).
If aiA and Pi(Y1)Pi(Y2), then from the convexity of the utility function ui it follows
that ui(Y)ui(Y1)+(1-)ui(Y2).
Finally, there is U(Y)U(Y1)+(1-)U(Y2).
Let b=U(Y)-(U(Y1)+(1-)U(Y2)), therefore b  n 0 \{0}. In result there is B1+(1)B2=U(Y)-(b1+(1-)b2), therefore B1+(1-)B2B, i.e. the set B is convex.
We will prove that S  B=.
Let’s assume that S  B, therefore  YD and  s,b  n 0 \{0} such that
U(X)+s=U(Y)-b. From s,b0 it follows that  aiA ui(Y)ui(X) and  akA uk(Y)>uk(X),
which contradicts the condition XP, therefore S  B.
From the theorem for detachability of sets it follows that  {qi : i=1..n}  such that
n
n
2
 q i  0 and  c  such that  s(s1,s2,..,sn),b(b1,b2,…,bn)   0 \{0} there is
i 1
n
n
i 1
i 1
 q i (u i (X)  s i )  c   q i (u i (Y)  b i )  YD.
n
n
i 1
i 1
Therefore  q i u i (X)  c   q i u i (Y)  YD and we have in result an equality at
X=Y.
n
We will prove that qi0  i[1;n]. Let’s examine the inequality  q i (u i (X)  s i )  c ,
i 1
which is true  s(s1,s2,..,sn)  n 0 \{0}.
6
n
Let’s assume that qk<0, therefore  q i (u i (X)  s i )  c is true at
i 1
n
sk 
c   q i ( u i ( X)  s i )
i 1,i  k
qk
 u k ( X) .
In result we have
n
u k (X)  s k 
c   q i ( u i ( X)  s i )
i 1,i  k
qk
n
q k (u k (X)  s k )  c   q i (u i (X)  s i )
i 1,i  k
n
 q i (u i (X)  s i )  c .
i 1
This leads to a contradiction, therefore qi0  i[1;n].
n
n
qi
i 1
i 1
 qi
From  q i2  0 it follows that  q i  0 . Let c i 
n
n
0 for i=1..n, therefore  c i  1
i 1
i 1
n
n
i 1
i 1
and  c i u i (X)   c i u i (Y)  YD.
Finally, there is XD*, i.e PD*. From theorem 2 it follows that D*=P. 
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