Further Results for General Financial Equilibrium Problems via Variational Inequalities ()
1. Introduction
In the previous papers [1] and [2], a general equilibrium model of financial flows and prices is considered. The model is assumed evolving in time. The equilibrium conditions are considered in dynamic sense and the governing variational inequality formulation is presented. Precisely, the variational inequality we are working with is the following one:
where 
is the set of feasible assets and liabilities for each sector 
given by
is the total financial volume held by sector 
as assets, 
is the total financial volume held by sector 
as liabilities, 
is a measure of the risk of the financial agent, 
is the tax rate levied on sector
’s net yield on financial instrument 
is a nonnegative function, 
is the portion of financial transaction per unit employed to cover the expenses of the financial institutions, 
is the set of feasible instrument prices given by
where 
and 
are assumed to be in
.
Setting, for the sake of simplicity,
defined as:
(1)
(2)
then variational inequality (1) becomes the problem:
Variational inequality (1) proved to be a very useful tool which enables us to study the financial equilibrium of an economy evolving in time and in the previous papers [1] and [2] the authors provided an interesting and useful output as the deficit formula, a general balance law and the liability formula which could be of great importance for the theory of equilibrium problems evolving in time. For the reader’s convenience we recall such important formulas:
1) Deficit formula
where 
and 
are the Lagrange variables associated to the price bounds:
The meaning of 
is that it represents the deficit per unit whereas 
is the positive surplus per unit;
2) Balance law
3) Liability formula
where 
and 
are the averages of 
and
, namely 
and
, respectively.
These suggested formulas could be of topical utility for the management of the world economy and to this aim in [2] the authors give some suggestions for the achievement of the world financial equilibrium and for finding the necessary way to follow in order to reach an improvement of the economy. It is worth reminding such suggestions:
Proposition 1.1 When the prices are minimal, namely they coincide with the floor prices, the economy collapses.
Proposition 1.2 Minimal prices imply the increase in the public debt.
Proposition 1.3 Minimal prices produce an economic recession. There is no incentive to the economic efforts.
Proposition 1.4 Even if it could be shocking, the development of the economy and of the employment results from an increase in the prices.
The value of these propositions can be realized by taking into account that the increase in prices indicated in Proposition 1.4 has been forecasted many months before (June 2011) it happened in our days.
Moreover, in [2] an “Evaluation Index”, that we denoted by
, has been introduced as a useful and simple tool for the evaluation of an economy, given by
where we set
We remark that if 
is greater or equal to 1 the evaluation of the financial equilibrium is positive (better if 
is proximal to 1) whereas if 
is less than 1 the evaluation of the financial equilibrium is negative.
The aim of this paper is to provide new theoretical and numerical results about solutions of financial equilibrium problems. In particular, we will prove a continuity result with respect to time of the solution, namely:
Theorem 1.1 Let
, let
, let 
and let
be a strongly monotone mapnamely there exists 
such that, for
,
Then variational inequality (1) admits a unique continuous solution.
Furthermore, we prove the following Lipschitz continuity result:
Theorem 1.2 Let 
be strongly monotone, namely there exists 
such that, for 
Lipschitz continuous with respect to
, namely there exists 
such that, for 
and Lipschitz continuous with respect to
, namely there exists 
such that, for 
Let 
be two Lipschitz continuous functions and let 
be two Lipschitz continuous functions. Then, the unique financial equilibrium solution 
is Lipschitz continuous in
. Moreover, let
, the following estimate holds:
(3)
where 
and
The continuity and Lipschitz continuity of solutions to financial equilibrium problems is a very important property for applications. Indeed, it is fundamental to establish numerical approximate solutions. Such numerical solutions can be obtained making use of a modified version of the algorithms (see for instances [3-7]). It is worth remarking that the Lipschitz continuity allows us to calculate the error in approximating the solution.
In order to clearly illustrate theoretical results, some significant examples are provided and, in such a way, the impact that the components of the model have on the equilibrium are highlighted.
It is worth mensioning that even in this case variational inequalities are able to express the time-dependent equilibrium conditions. Then, applying delicate tools of nonlinear analysis (see [8-11]), it is possible to prove existence results and qualitative analysis. For other economic problems where the time plays an important role we refer to the papers devoted to the Walrasian equilibrium problem [12-15], to the oligopolistic market equilibrium problem [16,17], to the weighted traffic equilibrium problem [18,19], and to [20].
The paper is organized as follows. In Section 2 we present the general financial model. In Section 3 we study the continuity results of the solution to the variational inequality which characterizes the financial model. In Section 4 we provide a Lipschitz continuity result for the solution. Finally, in Section 5 we propose a numerical examples from which we deduce that the solution, computed by means of the direct method (see [21]), is Lipschitz continuous.
2. The Model
We consider a financial economy consisting of 
sectors, with a typical sector denoted by
, and of 
instruments, with a typical financial instrument denoted by
, in the time interval
. Let 
denote the total financial volume held by sector 
at time 
as assets, and let 
be the total financial volume held by sector 
at time 
as liabilities. Then, unlike previous papers (see [22-27]), we allow markets of assets and liabilities to have different investments 
and 
respectively. Since we are working in the presence of uncertainty and of risk perspectives, the volumes 
and 
held by each sector cannot be considered stable with respect to time and may decrease or increase depending on unfavorable or favorable economic conditions. At time
, we denote the amount of instrument 
held as an asset in sector
’s portfolio by 
and the amount of instrument 
held as a liability in sector
’s portfolio by
. The assets and liabilities in all the sectors are grouped into the matrices
and
We denote the price of instrument j held as an asset at time t by 
and the price of instrument j held as a liability at time 
by
, where h is a nonnegative function defined into 
and belonging to
. We introduce the term 
because the prices of liabilities are generally greater than or equal to the prices of assets so that we can describe, in a more realistic way, the behaviour of the markets for which the liabilities are more expensive than the assets. In such a way, this paper appears as an improvement in various directions of the previous ones [22-27]. We group the instrument prices held as assets into the vector
and the instrument prices held as liabilities into the vector
In our problem the prices of each instrument appear as unknown variables. Under the assumption of perfect competition, each sector will behave as if it has no influence on the instrument prices or on the behaviour of the other sectors.
In order to express the time-dependent equilibrium conditions by means of an evolutionary variational inequality, we choose as a functional setting the very general Lebesgue space
. Then, the set of feasible assets and liabilities for each sector
, is 
Now, in order to improve the model of competitive financial equilibrium described in [1], we consider the possibility of policy interventions in the financial equilibrium and incorporate them in form of taxes and price controls.
To this aim, denote the ceiling price associated with instrument 
by 
and the nonnegative floor price associated with instrument 
by
, with
, a.e. in
. The meaning of the constraint 
a.e. in 
is that to each investor a minimal price 
for the assets held in the instrument 
is guaranteed, whereas each investor is requested to pay for the liabilities not less than the minimal price
. Analogously each investor cannot obtain for an asset a price greater than 
and as a liability the price cannot exceed the maximum price
.
Denote the given tax rate levied on sector
’s net yield on financial instrument
, as
. Assume that the tax rates lie in the interval 
and belong to 
. Therefore, the government in this model has the flexibility of levying a distinct tax rate across both sectors and instruments.
Let us group the instrument ceiling prices 
into the column vector
, the instrument floor prices 
into the column vector
, and the tax rates 
into the matrix
The set of feasible instrument prices is 
In order to determine for each sector 
the optimal composition of instruments held as assets and as liabilities, we consider, as usual, the influence due to riskaversion and the process of optimization of each sector in the financial economy, namely the desire to maximize the value of the asset holdings and to minimize the value of liabilities. Then, we introduce the utility function
, for each sector
, in this way
where the term 
represents a measure of the risk of the financial agent and
represents the value of the difference between the asset holdings and the value of liabilities. We suppose that the sector’s utility function 
is defined on
, is measurable in 
and is continuous with respect to 
and
. Moreover we assume that 
and 
exist and that they are measurable in 
and continuous with respect to 
and
. Further, we require that 
and a.e. in 
the following growth conditions hold true:
(4)
and
(5)
where
, 
, 
are non-negative functions of
. Finally, we suppose that the function
is concave.
In order to determine the equilibrium prices, we establish the equilibrium condition which expresses the equilibration of the total assets, the total liabilities and the portion of financial transactions per unit 
employed to cover the expenses of the financial institutions including possible dividends, as in [1]. Hence, the equilibrium condition for the price 
of instrument 
is the following:
(6)
In other words, the prices are determined taking into account the amount of the supply, the demand of an instrument and the charges
, namely if there is an actual supply excess of an instrument as assets and of the charges 
in the economy, then its price must be the floor price. If the price of an instrument is greater than
, but not at the ceiling, then the market of that instrument must clear. Finally, if there is an actual demand excess of an instrument as liabilities and of the charges 
in the economy, then the price must be at the ceiling.
Now, we can give different but equivalent equilibrium conditions, each of which is useful to illustrate particular features of the equilibrium.
Definition 2.1 A vector of sector assets, liabilities and instrument prices 
is an equilibrium of the dynamic financial model if and only if 
and a.e. in 
it satisfies the system of inequalities
(7)
(8)
and equalities
(9)
(10)
where 
are Lagrange functions, and verify conditions (6) a.e. in
.
Let us explain the meaning of the above conditions. To each financial volumes 
and 
held by sector
, we associate the functions
, related, respectively, to the assets and to the liabilities and which represent the “equilibrium disutilities” per unit of the sector
. Then, (7) and (9) mean that the financial volume invested in instrument 
as assets 
is greater than or equal to zero if the 
-th component
of the disutility is equal to
, whereas if
, then
. The same occurs for the liabilities and the meaning of (6) is already illustrated.
The functions 
and 
are Lagrange functions associated a.e. in 
with the constraints
and
, respectively. They are unknown a priori, but this fact has no influence because we will prove in the following theorem that Definition is equivalent to a variational inequality in which 
and 
do not appear.
Theorem 2.1 A vector 
is a dynamic financial equilibrium if and only if it satisfies variational inequality (1).
Moreover, we recall the result about Lagrange multipliers (see [2]):
Theorem 2.2 Let 
be a solution to variational inequality (1). Then there exist
such that a.e. in
,
(11)
(12)
(13)
(14)
3. Continuity Results for Financial Equilibrium Solutions
In order to show the continuity result for the financial equilibrium problem, first of all, let us recall the wellknown property of set convergence due to K. Kuratowski (see [28]), that is a generalization of the classical Hausdorff definition of a metric for the space of closed subsets of a (compact) metric space.
Let 
be a metric space and let 
be a sequence of subsets of
. Recall that
where eventually means that there exists 
such that 
for any
, and frequently means that there exists an infinite subset 
such that 
for any
. Finally we can remind the set convergence in Kuratowski’s sense.
Definition 3.1 We say that 
converges to some subset 
in Kuratwoski’s sense, and we briefly write
, if
. Thus, in order to verify that
, it suffices to check that
(K1)
, i.e. for any
, there exists a sequence 
converging to x in X such that 
lies in 
for all
;
(K2)
, i.e. for any subsequence
converging to x in X, such that 
lies in 
for all
, then the limit 
belongs to
.
Now, let us prove that the set of feasible vectors satisfies the property of the set convergence in Kuratowski’s sense.
Proposition 3.1 Let
, let
, let 
and let
be a sequence such that
, as
. Then, the sequence of sets
, converges to
as
, in Kuratowski’s sense.
Proof In order to prove that the sequence 
converges to 
in Kuratowski’s sense, for any sequence 
such that
, as
, it is enough to show that conditions (K1) and (K2) hold.
Let 
be fixed and let us consider the sequence
, such that
, 
, 
,
and
, 
,
Let us verify that
,
. Taking into account that
there exist two index 
and 
such that for 
we get
and for 
we have
Since
,
, it results
,
. Moreoverbeing
,
, it follows
,
. Finally, it results
Then we can consider a sequence 
such that for
, 
, 
,
and for
, 
, 
,
where 
denotes the Hilbertian projection on
.
We have 
and for 
Then the first condition has been shown.
For the second one, let 
be a fixed sequence, with
,
, such that 
in
,
in
, 
in
. We want to prove that
. Since
, 
, it results
(15)
(16)
(17)
Passing to the limit as 
in (15), (16) and (17), we obtain
Then
.
The claim is, now, achieved.□
For what follows, it is convenient to recall that variational inequality (1) can be rewritten in the equivalent parameterized form:
(18)
where the constraint set
, 
, is a closed convex and nonempty subset of
,
is a mapping and 
denotes the scalar product in
. Moreover, we recall that under general assumptions existence theorems have been proved in [2] (see Section 6).
Taking into account the general continuity result for solutions to parameter variational inequalities in reflexive Banach spaces (see [29], Theorem 4.1) and Proposition, we obtain Theorem 1.1 of Section 1.
Theorem 1.1 Let
, let
, let 
and let
be a strongly monotone map, namely there exists 
such that, for
,
Then variational inequality (1) admits a unique continuous solution.
4. Lipschitz Continuity Result
The aim of this section is to provide a Lipschitz continuity result for the financial equilibrium solution. For this reason, we recall a general result proved in [30] for the solutions to the parameterized variational inequality (18). More precisely, the following result holds (see [30], Theorem 1):
Theorem 4.1 Let 
be strongly monotone, Lipschitz continuous with respect to
, Lipschitz continuous with respect to
, and there exists 
such that, for
,
where
, 
, denotes the projection onto the set
. Then, the unique solution
, 
, to (18) is Lipschitz continuous in
, 
, the following estimate holds:
where
.
For the sake of simplicity, we set
.
Before applying the previous result to our dynamic financial equilibrium problem, it is necessary to estimate the variation rate of projections onto time-dependent constraint set 
describing the problem. It is useful to note that 
can be rewritten as the Cartesian product of the following set:
namely
Making use of Proposition 1 in [30], we can show that assuming
, 
, is Lipschitz continuous with Lipschitz constant
, for
, it results
(19)
Moreover, under the assumption that
, 
, is Lipschitz continuous with Lipschitz constant
, for
, we have
(20)
Now, taking into account Proposition 4.1 in [31] and assuming that 
and
, 
, are Lipschitz continuous with Lipschitz constants 
and
, respectively, for
, we can prove
(21)
where
.
We can conclude that Proposition 4.1 Let
, let 
be two Lipschitz continuous functions and let 
be two Lipschitz continuous functions. Let 
be an arbitrary point in
. Then it results to be
where 
is the positive constant as in (3).
As a consequence, it results
Hence, applying Theorem 4.1, we get the following result.
Theorem 1.2 Let 
be strongly monotone (with constant
), Lipschitz continuous with respect to 
(with constant
), Lipschitz continuous with respect to 
(with constant
), and let 
be two Lipschitz continuous functions and let 
be two Lipschitz continuous functions. Then, the unique financial equilibrium solution 
is Lipschitz continuous in
. Moreover, let
, the following estimate holds:
where
.
5. Some Numerical Examples
Let us study some numerical financial examples in which we consider as the risk aversion function the one suggested by H.M. Markowitz in [32] and [33], which expresses at each instant 
the risk aversion by means of variance-covariance matrices denoting the sector’s assessment of the standard deviation of prices for each instrument. We will see that the solutions of the examples are regular and we illustrate the impact of the components of the model on the financial equilibrium, especially for what regards 
and 
and the deficit and the surplus.
Let us consider an economy with two sectors and two financial instruments, as shown in Figure 1, but this setting is not restrictive since we can consider a general economy by an iteration of this significant case, and assume that the variance-covariance matrices of the two sectors are the following:
Figure 1. Two sectors and two financial instruments network a.e. in [0,1].
Let us choose as the feasible set
The variational inequality
which expresses the financial equilibrium conditions, becomes
(22)
Following the direct method (see [21]) in order to find the solution, we can derive from the constraints of the convex set 
the values of some variables in terms of the others, namely we obtain, a.e. in
,
Then, setting
variational inequality (22) can be expressed in the equivalent form:
Let us set
As the first step of the direct method suggests, let us search solutions obtained from the system
We get the values of the variables 
and 
in terms of 
and
, namely
For the sake of simplicity, we assume
We get
In the following, we study various examples of financial equilibrium for which the deficit (namely
) and surplus (namely
) are different from zero in certain time intervals. Such examples depend on the choose of suitable values of the data. In the first example, we consider values of 
such that
as Figures 2 and 3 show.
In particular, we assume
There exist 
and
, such that the vector
is solution of the variational inequality in the interval
, because 
fulfil the constraints and
for 
small enough and for any 
non negative. In fact, if we consider
we obtain that they are negative and positive respectively, as it can be verified in Figure 4 where the graphs of the numerators are represented.
Let us remember that, by virtue of Theorem 2.2, 
, the following relationships are satisfied
(23)
(24)
Taking into account that, in our case,
from (23), we obtain
So, from (24), we get
provided that 
is small enough and for any 
non negative.
The importance of this example remains in the fact that, in the interval
, for 
small enough, price 
is maximum and the financial market has a surplus given by
Whereas, for the instrument
, we have minimal price and the deficit is given by
Now, we would like to calculate the Evaluation Index for this financial problem. More precisely, we have
As a consequence, the Evaluation Index is given by:
In the interval
, if
the economy has a positive average evaluation. The same situation happens if
in the interval 
(where
).
Now let us consider the case where the values of 
are such that
as Figures 5 and 6 show.
In particular, we assume
There exist 
and
, such that the vector
is solution of the variational inequality in the interval
, because 
fulfil the constraints and
for any 
non negative and for 
small enough. In fact, if we consider
we obtain that they are positive and negative respectively, as it can be verified in Figures 7 and 8 where the graphs of the numerators are represented in detail.
If we remember that, in this case,
from (23), we obtain
So, from (24), we get
provided that 
is small enough and for any 
non negative.
In this situation, we can assert that, in the interval
, for the instrument
, we have minimal price and the deficit is given by
Whereas, for 
small enough, price 
is maximum and the financial market has a surplus given by
Now let us proceed to the calculation of the Evaluation Index. We have
Then the Evaluation Index is
As a consequence, in the interval
, if
the economy has a positive average evaluation. The same situation happens if
in the interval 
(where
). From this example we see that we can get useful information from the
, which requires simple calculations.
Now, we would like to study another example in which we assume
Under these assumptions, there exist 
and
, such that the vector
is solution of the variational inequality in the interval
, because 
fulfil the constraints and
for 
and 
non negative. In fact, if we consider
we obtain that they are both positive, as it can be verified in Figure 9 where the graphs of the numerators are represented.
We observe that, in our case
so, from (23), it follows that
Then, from (24), we obtain
for any 
and 
non negative.
As consequence of the meaning of
, in the interval
, for both the instruments, we have minimal price and the deficit is given, respectively, by
As regards the calculation of the Evaluation Index, we have
As a consequence, the Evaluation Index is
Then, in the interval
, if
the economy has a positive average evaluation because
. From this example we see that it is easier to achieve financial balance if 
and 
are more or less equal.