Time dependent spatial price equilibrium
problems
M. Milasi & C. Vitanza
Dipartimento di Matematica, Universitá di Messina, Italia
Abstract
We consider a time dependent economic market in order to show the existence of
the dynamical market equilibrium. Moreover, the Generalized Lagrange Multipliers and the Lagrangian theories are studied and, as an interesting consequence, we
obtain the Lagrangian variables. This theory plays an extraordinary role in the estimation of the dynamical market equilibrium. In our opinion this study makes these
problems more general and realistic than classical static spatial price equilibrium
problems.
Keywords: supply excess, demand excess, Lagrangian multipliers, subgradient
method.
1 Introduction
The aim of this paper is to consider a dynamic economic model. We see that this
problem can be incorporated directly in to a Variational Inequality models and
moreover we characterize the equilibrium solution by means of Lagrangian multipliers applying the duality theory in the case of infinite dimensional space. In
the static case the spatial price equilibrium problem has been formulated in terms
of a Variational Inequality by Nagurney and Zhao in [8] and A. Nagurney in [7].
Subsequently P. Daniele in [1] has been concerned with the spatial price equilibrium problem in the case of the quantity formulation under the assumption that the
data evolve in the time. In [6] the authors extended the result of [1] considering a
model with supply and demand excesses and with capacity constraints on prices
and on transportation costs. In the last years some papers have been devoted to the
study of the influence of the time on the equilibrium problems (see [2], [3] and
[4]). In fact, we cannot avoid to consider that each phenomenon of our economic
and physical world is not stable with respect to the time and that our static models
of equilibria are a first useful abstract approach.
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2 Evolutionary market model
Let us consider n supply markets Pi , i = 1, 2, .., n, and m demand markets
Qj , j = 1, 2, .., m involved in the production and in the consumption respectively
of a commodity during a period of time [0, T ], T > 0. Let gi (t), t ∈ [0, T ], i =
1, 2, .., n denote the supply of the commodity associated with supply market i at
the time t ∈ [0, T ] and let pi (t), t ∈ [0, T ], i = 1, 2, .., n denote the supply
price of the commodity associated with supply market i at the time t ∈ [0, T ]. A
fixed minimal and maximum supply price pi (t), pi (t) ≥ 0, respectively, for each
supply market, are given. Let fj (t), t ∈ [0, T ], j = 1, 2, .., m denote the demand
associated with the demand market j at the time t ∈ [0, T ] and let qj (t), t ∈
[0, T ], j = 1, 2, .., m, denote the demand price associated with the demand market
j at the time t ∈ [0, T ]. Let q j (t), q j (t) ≥ 0, for each demand market, be the
fixed minimal and maximum demand price respectively. Since the markets are
spatially separated, let xij (t), t ∈ [0, T ], i = 1, 2, .., n, j = 1, 2, ..., m denote the
nonnegative commodity shipment transported from supply market Pi to demand
market Qj at the same time t ∈ [0, T ]. Let cij (t), t ∈ [0, T ], i = 1, 2, .., n, j =
1, 2, .., m denote the nonnegative unit transportation cost associated with trading
the commodity between (Pi , Qj ) at the same time t ∈ [0, T ]. Let we suppose that
we are in presence of excesses on the supply and on the demand. Let si (t), t ∈
[0, T ], i = 1, 2, ..., n denote the supply excess for the supply market Pi at the time
t ∈ [0, T ]. Let τj (t), t ∈ [0, T ], j = 1, 2, .., m denote the demand excess for the
demand market Qj at the time t ∈ [0, T ]. We assume that the following feasibility
conditions must hold for every i = 1, 2, .., n and j = 1, 2, .., m a. e. in [0, T ]:
gi (t) =
m
xij (t) + si (t),
(1)
xij (t) + τj (t).
(2)
j=1
fj (t) =
n
i=1
Grouping the introduced quantities in vectors, we have the total supply vector
g(t) ∈ L2 ([0, T ], Rn ) and the total demand vector f (t) ∈ L2 ([0, T ], Rm ). Furthermore in order to precise the quantity formulation, we assume that two mappings
p(g(t)) and q(f (t)) are given:
p : L2 ([0, T ], Rn ) → L2 ([0, T ], Rn),
q : L2 ([0, T ], Rm ) → L2 ([0, T ], Rm ).
The mapping p assigns for each supply g(t) the supply price p(g(t)) and the
mapping q assigns for each demand f (t) the demand price q(f (t)). Analogously
x(t) ∈ L2 ([0, T ], Rnm ) is the vector of commodity shipment and the mapping
c : L2 ([0, T ], Rnm ) → L2 ([0, T ], Rnm )
assigns for each commodity shipment x(t) the transportation cost c(x(t)). Moreover let s(t) ∈ L2 ([0, T ], Rn ), τ (t) ∈ L2 ([0, T ], Rm ) be the vectors of supply
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and demand excess. Denoting by w(t) = (g(t), f (t), x(t), s(t), τ (t)), we set
= {w(t) = (g(t), f (t), x(t), s(t), τ (t)) : w(t) ∈ L2 ([0, T ], Rn )
L
×L2 ([0, T ], Rm) × L2 ([0, T ], Rnm ) × L2 ([0, T ], Rn ) × L2 ([0, T ], Rm)},
and
w(t)L = g(t)2L2 ([0,T ],Rn ) + f (t)2L2 ([0,T ],Rm ) + x(t)2L2 ([0,T ],Rnm )
+ s(t)2L2 ([0,T ],Rn ) + τ (t)2L2 ([0,T ],Rm )
12
.
Furthermore we assume that the feasible vector
w(t) = (g(t), f (t), x(t), s(t), τ (t)) satisfies the condition
w(t) ≥ 0
a. e. in [0, T ]
(3)
the set of feasible
Taking into account conditions (1), (2) and (3), we denote with K
vectors. K is a convex, closed, not bounded subset of the Hilbert space L.
Finally the presence of the capacity constraints on p, q, c can be expressed in the
following way, for each w(t) = (g(t), f (t), x(t), s(t), τ (t)) ∈ K:
p(t) ≤ p(g(t)) ≤ p(t),
q(t) ≤ q(f (t)) ≤ q(t),
c(t) ≤ c(x(t)) ≤ c(t).
Then the time-dependent market equilibrium condition in the case of the quantity
formulation takes the following form:
w∗ (t) is a
Definition 2.1. Let w∗ (t) = (g ∗ (t), f ∗ (t), x∗ (t), s∗ (t), τ ∗ (t)) in K.
market equilibrium if and only if for each i = 1, 2, .., n and j = 1, 2, .., m the
following conditions hold a. e. in [0, T ]:

∗
∗

 if si (t) > 0 ⇒ pi (g (t)) = pi (t)
(4)


∗
∗
if pi (t) < pi (g (t)) ⇒ si (t) = 0

∗
∗

 if τj (t) > 0 ⇒ qj (f (t)) = q j (t)


(5)
∗
if qj (f (t)) < q j (t) ⇒
τj∗ (t)
=0

∗
∗
∗
∗

 if xij (t) > 0 ⇒ pi (g (t)) + cij (x (t)) = qj (f (t))


(6)
∗
∗
∗
if pi (g (t)) + cij (x (t)) > qj (f (t)) ⇒
x∗ij (t)
=0
Moreover we are able to characterize time dependent market equilibrium as a solution to a Variational Inequality; in fact the following result holds:
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is a time dependent market equilibTheorem 2.1. u∗ = (g ∗ , f ∗ x∗ , s∗ , τ ∗ ) ∈ K
∗
rium if and only if u is a solution to Variational Inequality:
0
T
p(g ∗ (t))(g(t) − g ∗ (t)) − q(f ∗ (t))(f (t) − f ∗ (t)) + c(x∗ (t))(x(t) − x∗ (t))+
−p(t)(s(t) − s∗ (t)) + q(t)(τ (t) − τ ∗ (t)) dt ≥ 0
∀ w(t) = (g(t), f (t), x(t), s(t), τ (t)) ∈ K;
or, equivalently, the Variational Inequality:
0
T
p(x∗ (t), s∗ (t)) − q(x∗ (t), τ ∗ (t)) + c(x∗ (t)) x(t) − x∗ (t)
+(p(x∗ (t), s∗ (t)) − p(t))(s(t) − s∗ (t)) − (q(x∗ (t), τ ∗ (t))
−q(t))(τ (t) − τ ∗ (t)) dt ≥ 0
∀ u(t) = (x(t), s(t), τ (t)) ∈ K."
(7)
where: L = L2 ([0, T ], Rnm ) × L2 ([0, T ], Rn) × L2 ([0, T ], Rm ) and
K = {u(t) = (x(t), s(t), τ (t)) ∈ L : u(t) ≥ 0}.
Proof. See e. g. [6].
If we consider the function v : L → L defined setting for each u ∈ L:
v(u) = p(x, s) − q(x, τ ) + c(x), p(x, s) − p, q − q(x, τ ) ,
Variational Inequality (7) can be rewrite as:
"Find u∗ ∈ K such that:
< v(u∗ ), u − u∗ >≥ 0, ∀u ∈ K."
(8)
Now we can observe that it is possible to obtain the following existence result
of equilibria adapting a classical existence theorem for solution of a variational
inequality to our problem (for more details see e. g. [6]).
Theorem 2.2. Each of the following conditions is sufficient to ensure the existence
of the solution of (8):
1. v(u) = v(x(t), s(t), τ (t)) is hemicontinuous with respect to the strong topology and there exist A ⊆ K compact and there exist B ⊆ K compact, convex
with respect to the strong topology such that
∀u1 ∈ K \ A ∃u2 ∈ B : v(u1 ), u2 − u1 < 0;
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2. v is pseudomonotone, v is hemicontinuous along line segments and there exist
A ⊆ K compact and B ⊆ K compact, convex with respect to the weak topology
such that
∀p ∈ K \ A ∃p̃ ∈ B : v(p), p̃ − p < 0;
3. v is hemicontinuous on K with respect to the weak topology, ∃A ⊆ K compact,
∃B ⊆ K compact, convex with respect to the weak topology such that
∀p ∈ K \ A
∃p̃ ∈ B : v(p), p̃ − p < 0.
3 Example
The spatial price equilibrium model before presented is now illustrated with a simple example consisting of 2 supply markets, P1 , P2 and 1 demand market, Q1 .
It follows from theorem 2.2 that we have existence of the time depending market
equilibrium. The supply price, transportation cost and demand price functions are,
respectively:
p1 (g1 (t), g2 (t)) = 2g1 (t) + g2 (t) + 5,
c11 (x(t)) = 5x11 (t) + x21 (t) + 9,
p2 (g1 (t), g2 (t)) = g2 (t) + 10.
c21 (x(t)) = 3x21 (t) + 2x11 (t) + 19.
q1 (f1 (t)) = −f1 (t) + 80.
The capacity constraints on price are:
p1 (t) ≥ 21,
p2 (t) ≥ 16,
q1 (t) ≤ 60.
From the equilibrium conditions we have the following:

p1 (g1 (t), g2 (t)) + c11 (x(t)) = q1 (f1 (t))










 p2 (g1 (t), g2 (t)) + c21 (x(t)) = q1 (f1 (t))






p1 (g1 (t), g2 (t)) = p1 (t)









p2 (g1 (t), g2 (t)) = p2 (t)








q1 (f1 (t)) = q 1 (t)
The time depending equilibrium is then given by:
g1∗ (t) = 5,
g2∗ (t) = 6,
s∗1 (t) = 0,
f1∗ (t) = 20,
x∗11 (t) = 5,
s∗2 (t) = 1,
τ1∗ (t) = 10,
x∗21 (t) = 5,
with equilibrium supply prices, costs and demand prices:
p1 (t) = 21,
p2 (t) = 16,
c11 (t) = 39,
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c21 (t) = 44,
q1 (t) = 60.
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4 Lagrangian theory
It is possible to apply the Lagrangian and the duality theory which plays an extraordinary role in economic theory in order to characterize the evolutionary market equilibrium conditions in terms of the Lagrangian multipliers.
We observe that if u∗ ∈ K is a solution to problem (8) then
min < v(u∗ ), u − u∗ >= 0.
u∈K
Let us introduce the functional
ψu∗ (u) =< v(u∗ ), u − u∗ >, ∀u ∈ K,
and let us observe that
ψu∗ (u) ≥ 0,
min ψu∗ (u) = 0.
u∈K
We associate to Variational Inequality (8) the following Lagrangian function:
∀u ∈ L, l = (α, β, γ) ∈ C ∗
L(u, l) = ψu∗ (u) −
0
T
α(t)x(t)dt +
0
T
β(t)s(t)dt +
0
T
γ(t)τ (t)dt
where
C ∗ = {l(t) = (α(t), β(t), γ(t)) ∈ L : α(t), β(t), γ(t) ≥ 0 a. e. in [0, T ]}
is the dual cone of L. We observe that the dual cone of L, from Riesz theorem, is
equal to the ordering cone of L : C ∗ = C. Then, using the infinite dimensional
Lagrangian and duality theory (see chapter 5 and 6 in [5]), we can obtain the following:
Theorem 4.1. Let u∗ ∈ K be a solution to problem (8). Then there exist three
functions α∗ ∈ L2 ([0, T ], Rnm), β ∗ ∈ L2 ([0, T ], Rn ), γ ∗ ∈ L2 ([0, T ], Rm)
such that:
i) α∗ (t), β ∗ (t), γ ∗ (t) ≥ 0 a.e. in [0, T ];
ii) α∗ · x∗ = 0, β ∗ · s∗ = 0, γ ∗ · τ ∗ = 0;
iii)

p(x∗ , s∗ ) − q(x∗ , τ ∗ ) + c(x∗ ) = α∗ ,







p(x∗ , s∗ ) − p = β ∗ ,







q − q(x∗ , τ ∗ ) = γ ∗ .
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Proof. See e. g. [6].
Remark 4.1. Moreover it is easy to show that if u ∈ K and there exist α∗ , β ∗ , γ ∗
as in theorem (4.1) such that conditions i), ii) and iii) are fulfilled, then u verifies
the Variational Inequality (8).
5 Calculus of the solution
We start to observe that in our assumptions on the capacity constraint we can assume, in natural way, that u is upper limited by a function u ∈ L u(t) ≥ 0 a. e. in
[0, T ]; then we get:
K = {u(t) ∈ L : 0 ≤ u(t) ≤ u (t)}.
Now, we suppose that the pseudomonotonicity and hemicontinuity conditions hold
in order to ensure the existence of the solution to our variational inequality. By
pseudo-monotonicity hypothesi on v,
< v(u) − v(w), u − w >≥ 0
∀u, w ∈ K :
Variational Inequality (8) is equivalent to the following Minty Variational Inequality:
"Find u∗ (t) ∈ K such that:
< v(u), u − u∗ ≥ 0 ∀u ∈ K.
(9)
Let us set, for all u ∈ K:
ψ(u) = max < v(w), u − w > .
w∈K
ψ is well defined, because the operator
w →< v(w), u − w >
is weakly upper semicontinuous, K is a bounded convex subset of L and hence
weakly compact set. Moreover ψ(u), being the maximum of a family of continuous
affine functions, is convex and weakly lower semicontinuous. ψ(u) is also:
ψ(u) ≥ 0, ψ(u) = 0 ⇔ u is a solution to VI (8).
We can consider the subdifferential of ψ(u):
∂ψ(u) = {τ ∈ L : ψ(w) − ψ(u) ≥< τ, w − u >, ∀w ∈ K}.
We will show that ψ(u) has a nonempty subdifferential for all u ∈ K. The subgradient method for finding a solution of Variational Inequality (8) runs as follows:
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choose u0 ∈ K arbitrary. Given un ∈ K, ψ(un ) > 0, let
un+1 = ProjK (un − ρn τn ),
where
τn ∈ ∂ψ(un ) τn = 0 because ψ(un ) > 0 ,
ρn =
ψ(un )
.
||τn ||2
We shall see below that τn can be chosen in such a way that ||τn || remains bounded.
If τn remains bounded and ψ(un ) > 0 for all n, then we have the following result:
Theorem 5.1. There holds that lim ψ(un ) = 0. The sequence {un }n∈N has weak
n→∞
cluster points and in every weak cluster point ψ is equal to zero.
Proof. Let us w∗ a solution to Variational Inequality (8); by inequality:
< τn , w∗ − un >≤ ψ(w∗ ) − ψ(un ) = −ψ(un );
and by nonexpansivity of the projection mapping, we have:
||un+1 − w∗ ||2 = ||P roj(un − ρn τn ) − P roj w∗ ||2 ≤ ||un − ρn τn − w∗ ||2
= ||un − w∗ ||2 + ||ρn τn ||2 + 2ρn < τn , w∗ − un >
≤ ||un − w∗ ||2 + ρ2n ||τn ||2 − 2ρn ψ(un )
= ||un − w∗ ||2 + ρ2n ||τn ||2 − 2ρ2n ||τn ||2 = ||un − w∗ ||2 − ρ2n ||τn ||2 .
Since the sequence {||un − w∗ ||} is decreasing and bounded, and since ||τn || are
bounded from above, it follows that:
lim ψ(un ) = 0,
n→∞
and in particular
||un − w∗ || ≤ ||u0 − w∗ ||,
∀n ∈ N.
This show that {un } is bounded and therefore has a weak cluster point. Since
ψ(un ) → 0 and ψ is weakly lower semicontinuous, it follows that, if u is a weak
cluster point:
0 = lim inf ψ(unk ) ≥ ψ(u) ≥ 0,
k→∞
namely ψ(u) = 0.
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We show that ψ(u) has a nonempty subdifferential for all u ∈ K:
we call u∗ the element of K such that:
max < v(w), u − w >=< v(u∗ ), u − u∗ >= ψ(u),
w∈K
and we pose τ = v(u∗ ). We prove that τ = v(u∗ ) ∈ ∂ψ(u), that is:
ψ(v) − ψ(u) ≥< v(u∗ ), v − u >,
∀v ∈ K.
< v(u∗ ), v − u >=< v(u∗ ), v − u∗ + u∗ − u >=< v(u∗ ), v − u∗ >
+ < v(u∗ ), u∗ − u >≤ max < v(w), v − w >
w∈K
∗
∗
− < v(u ), u − u >= ψ(v) − ψ(u).
Then:
τ = v(u∗ ) ∈ ∂ψ(u) ⇒
∂ψ(u) = ∅.
Let us u∗n ∈ K such that:
max < v(w), un − w >=< v(u∗n ), un − u∗ >= ψ(un ),
w∈K
and we choose τn = v(u∗n ) ∈ ∂ψ(un ) ∀n ∈ N. Because v is bounded, then
||τn || remains bounded. Finally the approximating sequence is so defined:
maxw∈K < v(w), un − w >
un+1 = P rojK un −
v(un )
∗
2
||v(un )||
∗
< vn , un − u∗n >
∗
v(u
)
= P rojK un −
n
||v(u∗n )||2
6 Conclusions
The model we are concerned with is the spatial price equilibrium model in the presence of excess of the supplies and of demands. We stress that it is surprising to see
that this kind of network problem is incorporated into Variational Inequality model
as in the static case. An existence theorem for this Variational Inequality is provided and the same Variational Inequality formulation is used in order to provide
a computation of the equilibrium patterns and to study the generalized Lagrange
multipliers. The existence of the equilibrium solution of the time depending model
seems to be important because it allows us to follow the evolution in time of prices
and commodities. Moreover the importance of the functions α∗ , β ∗ , γ ∗ , studied in
section [4], derives from the fact that they are able to describe the behavior of the
∗
evolutionary market. In fact the set Aij
+ = {t ∈ [0, T ] : αij (t) > 0} indicates the
time when there is not trade between the supply marker i and the demand market
j
= {t ∈ [0, T ] : βi∗ (t) > 0} indicates the time when there is
j. Analogously B+
a zero supply excess of the market i. The same holds for γj∗ which indicates when
the demand market j has not demand excess.
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References
[1] P. Daniele, Time-Dependent spatial Price Equilibrium Problem: Existence and
Stability results for the Quantity Formulation Model, Journal of Global Optimization, to Appear.
[2] P. Daniele and A. Maugeri, On Dynamical Equilibrium Problems and Variational Inequalities Equilibrium Problems: Nonsmooth Optimization and Variational Inequality Models, Kluwer Academic Publishers, F. Giannessi- A.
Maugeri- P. Pardos Eds.,59-69 (2001).
[3] P. Daniele, A. Maugeri and W. Oettli, Variational Inequalities and timedependent traffic equilibria, C. R. Acad. Sci. Paris t.326, serie I, 1059-1062
(1998).
[4] P. Daniele, A. Maugeri and W. Oettli, Time Dependent traffic Equilibria, Jou.
Optim. Th. Appl., Vol. 103, No 3, 543-555 (1999).
[5] J. Jahn, Introduction to the theory of nonlinear Optimization, Springer (1996).
[6] M. Milasi and C. Vitanza, Variational Inequality and evolutionary market disequilibria: the case of quantity formulation, in Variational Analysis and Applications, F. Giannessi and A. Maugeri Editors, Kluwer Academic Publishers.
[7] A. Nagurney, Network Economics - A Variational Inequality Approach,
Kluwer Academic Publishers, 1993.
[8] A. Nagurney and L. Zhao, Disequilibrium and Variational Inequalities, I Comput. Appl. Math. 33 (1990), 181-198.
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