Communications in Applied and Industrial Mathematics,
ISSN 2038-0909, e-415
DOI: 10.1685/journal.caim.415
On solving dynamic oligopolistic market
equilibrium problems in presence of excesses
Annamaria Barbagallo1 and Paolo Mauro2
1
Department of Mathematics and Applications “R. Caccioppoli”,
University of Naples “Federico II”,
via Cintia - 80126 Naples, Italy
[email protected]
2
Department of Mathematics and Computer Science
University of Catania
viale Andrea Doria n. 6 - 95125 Catania, Italy
[email protected]
Communicated by Nicola Bellomo
Abstract
This paper deals with the dynamic oligopolistic market equilibrium problem in presence of excesses. In particular, besides presenting the problem in which only the production excess is considered, the demand excess is also discussed. In both cases, the
variational formulation of the equilibrium conditions is given. Taking into account the
continuity property of equilibrium solutions, it is possible to introduce a discretization
procedure in order to present a computational method for the dynamic oligopolistic market equilibrium problems in presence of excesses. Moreover, a convergence result is proved.
Finally, the performance of our algorithm is shown on some examples.
Keywords:
Dynamic oligopolistic market equilibrium problem, production
excess, demand excess, evolutionary variational inequality, discretization
procedure.
AMS Subject Classification: 49J40, 58E35, 65K10, 91A10
1. Introduction.
The main purpose of this paper is to provide a computational method
for solving oligopolistic market equilibrium problems in presence of excesses.
In the literature, these problems are modelled as evolutionary variational
inequalities as well as it happens for the most important time-dependent
equilibrium problems [1–8]. Moreover, many problems arising from the
theory of population dynamics and other physics phenomena are studied
not only by means of partial differential equations (for further details see
Received 2012 09 19. Accepted 2012 10 30. Published 2012 11 05.
Licensed under the Creative Commons Attribution Noncommercial No Derivatives
A. Barbagallo et al
e.g. [9–16]), but also by means of evolutionary variational inequalities (see
for instance [17,18]).
It is worth to perform a little history of the problem in question. The
oligopoly problem, namely the problem regarding the equilibrium between
a finite number of firms producing homogenous commodities in a noncooperative manner, has its origin in Cournot who first studied this problem in [19]. Cournot investigated competition between only two producers
while Nash, one century later, generalized the problem to n agents, each
acting in his own self-interest, which has come to be called a noncooperative game (see [20,21]). The time-dependent formulation of the oligopolistic
market equilibrium problem has been presented, for the first time, in [22].
Since “the time-dependent formulation of equilibrium problems allows one
to explore the dynamics of adjustment processes in which a delay on time
response is operating”, as M.J. Beckmann and J.P. Wallace pointed out
in [23], it is fundamental to study of the dynamic behaviour of the model.
Moreover, an other important aspect is that the delay on time response
always happens because the processes have not an infinite speed. Usually,
such adjustment processes can be represented by means of a memory term
which depends on previous equilibrium solutions according to the Volterra
operator (see, for instance [24]). In the study of evolutionary variational inequalities the search of the Lagrange variables is of topical importance and,
to this regard, we mention the paper [25] where the authors applied the
infinite-dimensional duality results developed in [26] to overcome the difficulty of the voidness of the interior of the ordering cone which defines the
constraints and so proved the existence of Lagrange variables which permit
to describe the behaviour of the market. Moreover, in [25] some sensitivity
results have been obtained each of them showing that small changes of the
solution happen in correspondence with small changes of the profit function.
In [27], an oligopolistic market equilibrium model with an explicit long-term
memory has been considered. In [1] and in [2] the variational formulation
of the oligopolistic market equilibrium problems in presence of production
excesses and both production and demand excesses are introduced. The
equilibrium conditions in terms of the well known dynamic Cournot-Nash
principle are analyzed, and then the equilibrium conditions will be expressed
also by using the Lagrange multipliers and the infinite-dimensional duality
theory. The equivalence between the two conditions is proved because they
are both expressed by an appropriate evolutionary variational inequality.
Moreover, thanks to the variational formulation, some existence and regularity results for the equilibrium solutions are shown.
In this paper we propose an algorithm for the computation of equilibria, by means of discretization methods. The continuity of the solution to
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dynamic oligopolistic market equilibrium problems in presence of excesses,
proved in [1] and in [2], allows us to consider a partition of the time interval and hence to reduce the infinite-dimensional problem to some finitedimensional problems that can be solved by means of a known method.
Since in literature some numerical schemes for the calculation of solutions
to dynamic equilibrium problems are available (see for instance [22,28–32]),
our results seem to have a particular relevance. More specifically, we propose
a discretization procedure which reduces the infinite-dimensional problem
to the calculus of solutions to finite-dimensional variational inequality. In
particular, taking into account of the continuity, we discretize the time interval [0, T ] and then we can compute, by means of the extragradient method
in Marcotte’s version, the solution of the finite-dimensional oligopolistic
market equilibrium problems obtained using the discretization. At last, we
construct an approximation solution with linear interpolation. In order to
obtain numerical results, we make use of a MatLab computation. Moreover,
we present some numerical examples which show the efficiency of the algorithm. In fact, we compute the equilibrium solutions by using also the direct
method (see [33–35]) and we obtain the same numerical results. At last, we
compare the numerical equilibrium solutions for the problem in presence
only of production excesses and the one in presence of both production and
demand excesses.
The paper is organized as follows: in Section 2 we recall the dynamic
oligopolistic market equilibrium problem in presence of production excesses
and then both production and demand excesses, then in Section 3 we describe an algorithm and we prove a convergence result, and finally in Section 4 we report the computational results.
2. Dynamic oligopolistic market equilibrium problem.
Let us consider, in a period time [0, T ], T > 0, m firms Pi , i = 1, . . . , m,
that produce one only commodity and n demand markets Qj , j = 1, . . . , n,
that are generally spatially separated.
Let pi (t), i = 1, . . . , m, be the nonnegative commodity output produced
by firm Pi at the time t ∈ [0, T ]. Let qj (t), j = 1, . . . , n, be the nonnegative
demand for the commodity at demand market Qj at the time t ∈ [0, T ]. Let
xij (t), i = 1, . . . , m, j = 1, . . . , n, be the nonnegative commodity shipment
between the supply market Pi and the demand market Qj at the time
t ∈ [0, T ]. In the following, we set xi (t) = (xi1 (t), . . . , xin (t)), i = 1, . . . , m,
t ∈ [0, T ] .
Let us group the production output into a vector-function p : [0, T ] −→
n
Rm
,
+ the demand output into a vector-function q : [0, T ] −→ R+ , the com-
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A. Barbagallo et al
modity shipments into a matrix-function x : [0, T ] −→ Rmn
+ .
We assume that the nonnegative commodity shipment belongs to
L2 ([0, T ] , Rmn
+ ). Furthermore, let us associate with each firm Pi a production cost fi , i = 1, . . . , m, and assume that the production cost of a firm Pi
may depend upon the entire production pattern, namely, fi = fi (t, x(t)).
Similarly, let us associate with each demand market Qj , a demand price
for unity of the commodity dj , j = 1, . . . , n, and assume that the demand
price of a demand market Qj may depend, in general, upon the entire
consumption pattern, namely, dj = dj (t, x(t)).
Moreover, let gi , i = 1, . . . , m, denote the storage cost of the commodity
produced by the firm Pi and assume that this cost may depend upon the
entire production pattern, namely, gi = gi (t, x(t)).
Finally, let cij , i = 1, . . . , m, j = 1, . . . , n, denote the transaction
cost, which includes the transportation cost associated with trading the
commodity between firm Pi and demand market Qj . Here we permit the
transaction cost to depend upon the entire shipment pattern, namely,
cij (t) = cij (t, x(t)).
Hence, we have the following mappings,
2
m
f, g : [0, T ] × L2 ([0, T ] , Rmn
+ ) −→ L ([0, T ] , R+ ),
2
n
d : [0, T ] × L2 ([0, T ] , Rmn
+ ) −→ L ([0, T ] , R+ ),
2
mn
c : [0, T ] × L2 ([0, T ] , Rmn
+ ) −→ L ([0, T ] , R+ ).
The profit of the firm Pi , i = 1, . . . , m, at the time t ∈ [0, T ] is,
(1)
vi (t, x(t)) =
n
X
dj (t, x(t))xij (t) − fi (t, x(t))
j=1
−gi (t, x(t)) −
n
X
cij (t, x(t))xij (t),
j=1
namely, it is equal to the price that the demand markets are disposed to
pay minus the production costs, the storage costs and the transportation
costs.
The aim of the model for this problem in which the m firms supply
the commodity in a noncooperative fashion, is to maximize its own profit
function considered the optimal distribution pattern for the other firms.
We want to determine a commodity distribution matrix-function for which
the m firms and the n demand markets will be in a state of equilibrium as
defined later.
Suppose also that the following assumptions on the profit function v
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(given by (1)) and on ∇D v =
∂vi
∂xij
i = 1, . . . , m
j = 1, . . . , n
hold:
i. v(t, x(t)) is continuously differentiable a.e. in [0, T ],
ii. ∇D v is a Carathèodory function such that
(2) ∃h ∈ L2 ([0, T ]) : k∇D v(t, x(t))kmn ≤ h(t) kx(t)kmn a.e. in [0, T ] ,
iii. vi (t, x(t)) is pseudoconcave with respect to the variables xi , i = 1, . . . , m,
namely the following holds a.e. in [0, T ] :
∂v
(3)
(t, x1 , . . . , xi , . . . , xm ), xi − yi ≥ 0
∂xi
⇒ vi (t, x1 , . . . , xi , . . . , xm ) ≥ vi (t, x1 , . . . , yi , . . . , xm ).
In oligopolistic markets the pseudoconcavity assumption (3) is satisfied
mainly because explicit conditions on the data (i.e. production cost, demand
price, storage cost, transaction cost) are not given.
2.1. Dynamic oligopolistic market equilibrium problem in presence of production excesses.
At first, we want to consider the case in which some of the amounts of
the commodity available are sold out whereas for a part of the producers
can occur an excess of production (see [1]).
Let i (t), i = 1, . . . , m, be the nonnegative production excess for the
commodity of the firm Pi at the time t ∈ [0, T ]. Let us group the production
excess into a vector-function : [0, T ] −→ Rm
+ . The following feasibility
condition holds, a.e. in [0, T ] :
(4)
pi (t) =
n
X
xij (t) + i (t),
i = 1, . . . , m,
j=1
namely the quantity produced by each firm Pi at the time t ∈ [0, T ]
must be equal to the sum of the commodity shipments from that firm
to each demand markets plus the production excess at the same time
t ∈ [0, T ]. We suppose that ∈ L2 ([0, T ] , Rm
+ ). As a consequence, we have
2
m
p ∈ L ([0, T ] , R+ ).
In virtue of (4) since the production excess i (t) at the time t ∈ [0, T ]
is nonnegative, we have
(5)
n
X
xij (t) ≤ pi (t), ∀i = 1, . . . , m, a.e. in [0, T ] .
j=1
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A. Barbagallo et al
Taking into account (5), the feasible set for the distribution pattern is:
(
(6)
K=
x ∈ L2 ([0, T ] , Rmn ) :
xij (t) ≥ 0, ∀i = 1, . . . , m, ∀j = 1, . . . , n, a.e. in [0, T ] ,
)
n
X
xij (t) ≤ pi (t), ∀i = 1, . . . , m, a.e. in [0, T ] .
j=1
Now, we give the dynamic oligopolistic market equilibrium conditions
in presence of production excesses by using Lagrange variables associated
to the constraints.
Definition 2.1. x∗ ∈ K is a dynamic oligopolistic market problem equilibrium in presence of excesses if and only if for each i = 1, . . . , m, j = 1, . . . , n
and a.e. in [0, T ] there exists λ∗ij ∈ L2 (0, T ), µ∗i ∈ L2 (0, T ), such that
−
(7)
(8)
∂vi (t, x∗ (t))
+ µ∗i (t) = λ∗ij (t),
∂xij
∂vi (t, x∗ (t))
∗
−
+ µi (t) x∗ij (t) = 0,
∂xij
(9)
λ∗ij (t)x∗ij (t) = 0,
λ∗ij (t) ≥ 0,
(10)


n
X
µ∗i (t) 
x∗ij (t) − pi (t) = 0,
µ∗i (t) ≥ 0.
j=1
The terms λ∗ij (t) and µ∗i (t) are the Lagrange multipliers associated to the
n
X
constraints x∗ij (t) ≥ 0 and to
x∗ij (t) ≤ pi (t), respectively.
j=1
Making use of the infinite-dimensional duality theory (see [26,36–38]),
it is possible to prove the following equivalence (see Theorem 2.2 in [1]).
Theorem 2.1. x∗ ∈ K is a dynamic oligopolistic market equilibrium in
presence of production excesses if and only if it satisfies the evolutionary
variational inequality
Z TX
m X
n
∂vi (t, x∗ (t))
(11)
−
(xij (t) − x∗ij (t))dt ≥ 0
∀x ∈ K,
∂x
ij
0
i=1 j=1
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where K is given by (6).
Moreover, the equivalence between the evolutionary variational inequality (11) and the dynamic Cournot-Nash principle, which we recall in the
following, has been proved (see Theorem 3.1 in [22]).
Definition 2.2. x∗ ∈ K is a dynamic oligopolistic market equilibrium in
presence of production excesses if and only if for each i = 1, . . . , m and a.e.
in [0, T ] we have
(12)
vi (t, x∗ (t)) ≥ vi (t, xi (t), x̂∗i (t)), a.e. in [0, T ] , where
xi (t) = (xi1 (t), . . . , xin (t)), a.e. in [0, T ] and x̂∗i (t) = (x∗1 (t), . . . , x∗i−1 (t),
x∗i+1 (t), . . . , x∗m (t)).
Hence, we seek to determine a nonnegative commodity distribution matrix function x for which the m firms and the n demand markets will be
in a state of equilibrium, such that the m firms try to maximize their own
profit at the time t ∈ [0, T ].
2.2. Dynamic oligopolistic market equilibrium problem in presence of production and demand excesses.
Now we recall the dynamic oligopolistic market equilibrium problem in
presence of both production and demand excesses introduced in [2], namely
we allow that for a part of the producers can occur an excess of production
whereas for a part of the demand markets the supply can not satisfy the
demand. In order to clarify the presence of both excesses we consider some
concrete economic situations. During an economic crisis period the presence of production excesses can be due to a decrease in market demands
and, on the other hand, the presence of demand excesses may occur when
the supply can not satisfy the demand especially for fundamental goods.
Moreover, since the market model presented in this paper evolves in time,
the presence of both excesses can be consequence of the fact that the physical transportation of commodity between a firm and a demand market is
evidently limited, therefore there exist some time instants in which, though
some firms produce more commodity than they can send to all the demand
markets, some of the demand markets require more commodity.
In addition to what we have considered previously, let us denote by
δj (t), j = 1, . . . , n, the nonnegative demand excess for the commodity of
the demand market Qj at the time t ∈ [0, T ]. Let us group the demand
excess into a vector-function δ : [0, T ] −→ Rn+ and let us suppose that the
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A. Barbagallo et al
following feasibility conditions hold, a.e. in [0, T ] :
(13)
pi (t) =
n
X
xij (t) + i (t),
i = 1, . . . , m,
xij (t) + δj (t),
j = 1, . . . , n.
j=1
(14)
qj (t) =
m
X
i=1
Hence, the quantity produced by each firm Pi , at the time t ∈ [0, T ], must
be equal to the commodity shipments from that firm to all the demand
markets plus the production excess at the same time t ∈ [0, T ] and the
quantity demanded by each demand market Qj at the time t ∈ [0, T ] must
be equal to the commodity shipments from that firm to all the demand
markets plus the demand excess at the same time t ∈ [0, T ] .
Taking into account (13) and (14), since the production excess i (t) at
the time t ∈ [0, T ] and the demand excess δj (t) at the same time t ∈ [0, T ]
are nonnegative, we have
(15)
n
X
xij (t) ≤ pi (t), ∀i = 1, . . . , m, a.e. in [0, T ] ,
j=1
(16)
m
X
xij (t) ≤ qj (t), ∀j = 1, . . . , n, a.e. in [0, T ] .
i=1
Moreover, we assume that the nonnegative commodity shipment between the producer firm Pi and the demand market Qj has to satisfy
time-dependent constraints, namely, there exist two nonnegative functions
x, x ∈ L2 ([0, T ] , Rmn
+ ) such that
(17) xij (t) ≤ xij (t) ≤ xij (t), ∀i = 1, . . . , m, ∀j = 1, . . . , n, a.e. in [0, T ]
that define the limits in transportation.
2
n
Let us assume that ∈ L2 ([0, T ] , Rm
+ ), and δ ∈ L ([0, T ] , R+ ). As a
2
m
2
n
consequence, we have p ∈ L ([0, T ] , R+ ), and q ∈ L ([0, T ] , R+ ). According to (15), (16) and (17), the feasible set for the distribution pattern is:
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(
(18)
x ∈ L2 ([0, T ] , Rmn ) :
K=
xij (t) ≤ xij (t) ≤ xij (t),
∀i = 1, . . . , m, ∀j = 1, . . . , n, a.e. in [0, T ] ,
n
X
xij (t) ≤ pi (t), ∀i = 1, . . . , m, a.e. in [0, T ] ,
j=1
m
X
)
xij (t) ≤ qj (t), ∀j = 1, . . . , n, a.e. in [0, T ] .
i=1
Let us present the dynamic oligopolistic market equilibrium condition
in presence of production and demand excesses.
Definition 2.3. x∗ ∈ K is a dynamic oligopolistic market problem equilibrium in presence of excesses if and only if for each i = 1, . . . , m, j =
1, . . . , n and a.e. in [0, T ] there exists λ∗ij ∈ L2 (0, T ), ρ∗ij ∈ L2 (0, T ), µ∗i ∈
L2 (0, T ), νj∗ (t) ∈ L2 (0, T ) such that
(19)
−
∂vi (t, x∗ (t))
+ ρ∗ij (t) + µ∗i (t) + νj∗ (t) = λ∗ij (t),
∂xij
(20)
λ∗ij (t)(xij (t) − x∗ij (t)) = 0,
λ∗ij (t) ≥ 0,
(21)
ρ∗ij (t)(x∗ij (t) − xij (t)) = 0,
ρ∗ij (t) ≥ 0,

(22)
µ∗i (t) 
n
X

x∗ij (t) − pi (t) = 0,
µ∗i (t) ≥ 0,
j=1
(23)
νj∗ (t)
m
X
!
x∗ij (t) − qj (t)
= 0,
νj∗ (t) ≥ 0.
i=1
The terms λ∗ij (t), ρ∗ij (t), µ∗i (t), νj∗ (t) are the Lagrange multipliers associated
n
X
∗
∗
to the constraints xij (t) ≥ xij (t), xij (t) ≤ xij (t),
x∗ij (t) ≤ pi (t) and to
j=1
m
X
x∗ij (t) ≤ qj (t), respectively.
i=1
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A. Barbagallo et al
Also in this case, the previous equilibrium conditions is equivalent to
an evolutionary variational inequality (see Theorem 2.2 in [2]).
Theorem 2.2. x∗ ∈ K is a dynamic oligopolistic market equilibrium in
presence of both production and demand excesses if and only if it satisfies
the evolutionary variational inequality
Z TX
m X
n
∂vi (t, x∗ (t))
(xij (t) − x∗ij (t))dt ≥ 0
∀x ∈ K,
(24)
−
∂x
ij
0
i=1 j=1
where K is given by (18).
Finally, we have that the state of equilibrium of the dynamic oligopolistic market in presence of both production and demand excesses is the same
Cournot-Nash principle as the one considered in the previous model.
For existence and regularity results, consult [1,2,39,40].
3. Solution method.
Let us introduce a method to solve dynamic oligopolistic market equilibrium problems in presence of excesses expressed by evolutionary variational
inequalities.
We suppose that the assumptions which ensure the continuity of dynamic oligopolistic market equilibrium solution are satisfy. As a consequence, (11) and (24) hold for each t ∈ [0, T ], namely
h−∇D v(t, x∗ (t)), x(t) − x∗ (t)i ≥ 0,
(25)
∀x(t) ∈ K(t), ∀t ∈ [0, T ]
where
(
K(t) =
x(t) ∈ Rmn : xij (t) ≥ 0, ∀i = 1, . . . , m, ∀j = 1, . . . , n,
n
X
)
xij (t) ≤ pi (t), ∀i = 1, . . . , m
j=1
for the problem with only production excesses and
(
K(t) =
x(t) ∈ Rmn : xij (t) ≤ xij (t) ≤ xij (t),
∀i = 1, . . . , m, ∀j = 1, . . . , n,
n
X
xij (t) ≤ pi (t), ∀i = 1, . . . , m
j=1
m
X
)
xij (t) ≤ qj (t), ∀j = 1, . . . , n ,
i=1
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for the problem with both production and demand excesses.
In the following, applying a discretization procedure and making use
of the extragradient method in Marcotte’s version, we compute the solutions of the evolutionary variational inequality which expresses the dynamic
oligopolistic market equilibrium conditions in presence of excesses.
Let us consider a partition of [0, T ], such that:
0 = t0 < t1 < . . . < tr < . . . < tN = T. For each point tr , r = 0, 1, . . . , N ,
of the partition, we consider the finite-dimensional variational inequality
(26)
h−∇D v(tr , x∗ (tr )), x(tr ) − x∗ (tr )i ≥ 0,
∀x(tr ) ∈ K(tr ).
We compute now the solution to the finite-dimensional variational inequality (26) making use of a modified version of the extragradient method
introduced by Marcotte in [41].
The algorithm starting from any x∗0 (tr ) ∈ K(tr ) and a fixed number
α0 > 0, iteratively updates x∗k+1 (tr ) from x∗k (tr ) according to the following
projection formulas
(27)
x∗k+1 (tr ) = PK(tr ) (x∗k (tr ) − αk (−∇D (x̃∗k (tr )))),
(28)
x̃∗k (tr ) = PK(tr ) (x∗k (tr ) − αk (−∇D (x∗k (tr ))))
for k ∈ N, where αk is chosen as following
(
)
αk−1
kx∗k (tr ) − x̃∗k (tr )kmn
(29)
αk = min
,√
.
2
2k(−∇D (x∗k (tr ))) − (−∇D (x̃∗k (tr )))kmn
If −∇D v is a monotone and Lipschitz continuous mapping, then, the convergence of the scheme is proved. This method was improved by Tinti in [42].
After the iterative procedure, we get the dynamic equilibrium solution
through a linear interpolation of the computed static equilibrium solutions.
3.1. Convergence analysis.
In this section, we investigate on the convergence of algorithms presented in the previous section. We prove that under suitable assumptions,
the sequence, generated by the algorithm, converges in L1 -sense to the dynamic oligopolistic market equilibrium solution in presence of excesses.
Let us assume that all hypotheses which ensure the continuity of solution
to (25) and the convergence of the Marcotte’s extragradient method to
compute solutions to finite-dimensional variational inequalities hold. Let
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A. Barbagallo et al
us introduce a sequence {πs }s∈N of (not necessarily equidivided) partitions
0
s
of the time interval [0, T ] such that πs = (t0s , t1s , . . . , tN
s ), where 0 = ts <
1
N
ts < . . . < ts s = T . We consider a sequence of equidivided partitions, in
the sense that ks := max{|trs − tr−1
s | : r = 1, 2, . . . , Ns }, approaches zero
for s → +∞.
By the interpolation theory, we know that if we construct the approximate solution to (25) by means of Hermite’s polynomial, using known
values of the solution x∗ (t), the sequence converges uniformly to the exact
solution. We do not use Hermite’s polynomial, but we consider an approximation by means of piecewise constant functions and we can prove that
the convergence is in L1 -sense.
Let us consider the approximate solutions to (25) given by the following
formula:
x∗s (t) =
(30)
Ns
X
x∗ (trs )χ[tr−1
,tr [ (t),
s
s
r=1
where x∗ (trn ) is the solution to the finite-dimensional variational inequality
which is obtained to (25) for t = trs , which can be compute by means of the
Marcotte’s extragradient method.
Let us estimate the following integral
Z T
Ns
X
∗
∗ r
x (t) −
x
(t
)χ
r−1 r (t)
s
[ts ,t [
dt
0
s
r=1
Z
T
=
0
≤
r=1
Ns Z
X
r=1
mn
Ns
Ns
X
X ∗
∗ r
x (ts )χ[tr−1
x (t)χ[tr−1
,trs [ (t)
,trs [ (t) −
s
s
tsr
tsr−1
dt
mn
r=1
kx∗ (t) − x∗ (trs )kmn dt
.
Being x∗ uniformly continuous, it results that for every ε > 0 there exists
r
r
δ > 0 such that if t ∈ [tr−1
s , ts ] satisfies the condition |t − ts | < δ it results
ε
kx∗ (t) − x∗ (trs )kmn < , for r = 1, 2, . . . , Ns , ∀s ∈ N.
T
Choosing s large enough in such way that ks < δ, we reach
Z T
Ns
Ns
X
X
∗
ε s
∗ r
(31)
x (ts )χ[tr−1
dt <
(tr − tsr−1 ) = ε.
,trs [ (t)
x (t) −
s
T
0
mn
r=1
r=1
Taking into account (31), we conclude that the sequence (30) converges
in L1 -sense to the dynamic oligopolistic market equilibrium solution in presence of excesses.
12
DOI: 10.1685/journal.caim.415
4. Numerical results.
Let us consider some numerical examples of the dynamical oligopolistic
market equilibrium problem in presence of excesses consisting of three firms
and four demand markets, as in Figure 1. At first, we suppose that there
are only production excesses.
P1
•PP
P2
P3
•
•P
@PP
@
@
@ P
P
P
PP
PP @
PP@
PP
@@
P
@
P@
R
R
R
)
)
q
P
q
P
P
@
@•
P•
@•
•
Q1
Figure 1.
Q2
Q3
Q4
Network structure of the numerical dynamic spatial oligopoly problem.
Let p ∈ C([0, 1], R3 ) be the production function such that, in [0, 1],
T
p(t) = 4t 3t 4t ,
As a consequence, the feasible set is
n
K = x ∈ L2 ([0, 1], R3×4 ) : xij (t) ≥ 0, i = 1, 2, 3, j = 1, 2, 3, 4, a.e. in [0, 1],
4
X
o
xij (t) ≤ pi (t), i = 1, 2, 3, a.e. in [0, 1] .
j=1
Let v ∈ C 1 (L2 ([0, 1], R3×4 ), R3 ) be the profit function defined by
v1 (t, x(t)) = −4x211 (t) − 4x212 (t) − 6x213 (t) − 6x214 (t) − 4x11 (t)x12 (t) − 6x13 (t)x14 (t)
+3tx11 (t) + 4tx12 (t) + tx13 (t) + tx14 (t),
v2 (t, x(t)) = −5x221 (t) − 2x222 (t) − 2x223 (t) − 2x224 (t) − 2x21 (t)x22 (t) − 2x23 (t)x24 (t)
+2tx21 (t) + 2tx22 (t) + 3tx23 (t) + 2tx24 (t),
v3 (t, x(t)) = −10x231 (t) − 4x232 (t) − 4x233 (t) − 5x234 (t) − 2x11 (t)x32 (t) − 2x33 (t)x34 (t)
−2x12 (t)x31 (t) + tx31 (t) + 2tx32 (t) + 10tx33 (t) + 3tx34 (t).
Then, the operator ∇D v ∈ C(L2 ([0, 1], R3×4 ), R3×4 ) is given by


−8x11 − 4x12 + 3t −8x12 − 4x11 + 4t −12x13 − 6x14 + t −12x14 − 6x13 + t
∇D v =  −10x21 − 2x22 + 2t −4x22 − 2x21 + 2t −4x23 − 2x24 + 3t −4x24 − 2x23 + 2t  .
−20x31 − 2x12 + t −8x32 − 2x11 + 2t −8x33 − 2x34 + 10t −10x34 − 2x33 + 3t
Moreover, it is possible to verify that −∇D v is a strongly monotone operator
(for the proof, you can see similar calculations in [1] and [2]).
The dynamic oligopolistic market equilibrium distribution in presence
of excesses is the solution to the evolutionary variational inequality:
Z 1X
3 X
4
∂vi (t, x∗ (t))
(32)
−
(xij (t) − x∗ij (t))dt ≥ 0, ∀x ∈ K.
∂x
ij
0
i=1 j=1
13
A. Barbagallo et al
Taking into account the direct method
lowing system
 ∗
8x11 (t) + 4x∗12 (t) − 3t = 0,




12x∗13 (t) + 6x∗14 (t) − t = 0,



10x∗21 (t) + 2x∗22 (t) − 2t = 0,
(33)
4x∗23 (t) + 2x∗24 (t) − 3t = 0,





20x∗ (t) + 2x∗12 (t) − t = 0,

 ∗ 31
8x33 (t) + 2x∗34 (t) − 10t = 0,
(see [33–35]), we consider the fol4x∗11 (t) + 8x∗12 (t) − 4t = 0,
6x∗13 (t) + 12x∗14 (t) − t = 0,
2x∗21 (t) + 4x∗22 (t) − 2t = 0,
2x∗23 (t) + 4x∗24 (t) − 2t = 0,
8x∗32 (t) + 2x∗11 (t) − 2t = 0,
2x∗33 (t) + 10x∗34 (t) − 3t = 0
We get the following solution, in [0, 1],
 1
t
 6

 1
∗
t
x (t) = 
 9

 1
t
120
5
t
12
4
t
9
5
t
24
1
t
18
2
t
3
47
t
38
1 
t
18 

1 
t ,
6 

1 
t
19
that belongs to the constraint set K, then it is the equilibrium solution.
Moreover, the production excesses, in [0, 1], are given by
T
119 29 2843
.
(t) =
t t
t
36 18 1140
Now, we solve the numerical problem using the generalized Marcotte’s
version of the extragradient method. This method is convergent for the
properties of −∇D v. Then,
k we can compute anapproximate curve of equilibria, by selecting tr ∈ 20 : k ∈ {0, 1, . . . , 20} . With the help of a MatLab computation and choosing the initial point
2 1 1

3 tr 3 tr 2 tr tr
1 1 1 1 


x∗0 (tr ) =  2 tr 10 tr 2 tr 8 tr 


tr 13 tr 25 tr 81 tr
in order to start the iterative method, we obtain the equilibrium solutions
for every time instant. Setting R(x∗k (tr )) = x∗k (tr ) − x∗k−1 (tr ) the difference between two approximations of the equilibrium
solution
in the time
∗k
instant tr and making use of the stopping criterion R(x (tr ))3×4 ≤ 10−6 ,
for r = 0, 1, . . . , 20, we obtain the approximate curve of equilibria shown in
Figure 2.
Now, we analyze the dynamic oligopolistic market equilibrium problem
in presence of both production and demand excesses.
14
DOI: 10.1685/journal.caim.415
1.4
x*11(t)
x*12(t)
x*13(t)
x* (t)
14
x*21(t)
x* (t)
22
x*23(t)
x*24(t)
x* (t)
31
x*32(t)
x*33(t)
x*34(t)
1.2
Numerical solution
1
0.8
0.6
0.4
0.2
0
Figure 2.
0
0.1
0.2
0.3
0.4
0.5
Time
0.6
0.7
0.8
0.9
1
Curves of equilibria of problem with only production excesses.
Let x, x ∈ C([0, 1], R3×4 ) be the capacity constraints such that, in [0, 1],




1
1
1
t 50
t 100
t
0 12
2t 4t 6t t
1
1 
t 0 0 20
t , x(t) =  4t 3t 5t t  .
x(t) =  100
1
1
t t t 3t
0 24 t 2 t 0
Let p be the same production function considered in the previous example
and let q ∈ C([0, 1], R4 ) be the demand function such that, in [0, 1],
q(t) = 6t 4t 4t 8t
T
.
Hence, the feasible set is
(
K=
x ∈ L2 ([0, 1] , R3×4 ) :
xij (t) ≤ xij (t) ≤ xij (t),
4
X
i = 1, 2, 3, j = 1, 2, 3, 4, a.e. in [0, 1] ,
xij (t) ≤ pi (t), i = 1, 2, 3, a.e. in [0, 1] ,
j=1
3
X
)
xij (t) ≤ qj (t),
j = 1, 2, 3, 4, a.e. in [0, 1] .
i=1
Let us consider the same profit function v. In order to compute the
solution to (32) we make use of the direct method (see [33–35]). We consider
again the system (33) and we get the same solution. But we observe that x∗
15
A. Barbagallo et al
does not belong to the constraint set K because x∗33 (t) > t. Let us consider
now the set
(
e=
K
x ∈ L2 ([0, 1] , R3×4 ) : xij (t) ≤ xij (t) ≤ xij (t),
i = 1, 2, 3, j = 1, 2, 3, 4, (i, j) 6= (3, 3), a.e. in [0, 1] ,
x33 (t) = t, a.e. in [0, 1] ,
4
X
xij (t) ≤ pi (t), i = 1, . . . , 3, a.e. in [0, 1] ,
j=1
3
X
)
xij (t) ≤ qj (t),
j = 1, . . . , 4, a.e. in [0, 1]
i=1
and the system
 ∗
∗
∗ (t) + 8x∗ (t) − 4t = 0,

12
 8x11∗(t) + 4x12∗(t) − 3t = 0, 4x11

∗ (t) + 12x∗ (t) − t = 0,

12x
(t)
+
6x
(t)
−
t
=
0,
6x

13
14
13
14


10x∗21 (t) + 2x∗22 (t) − 2t = 0, 2x∗21 (t) + 4x∗22 (t) − 2t = 0,
(34)
4x∗23 (t) + 2x∗24 (t) − 3t = 0, 2x∗23 (t) + 4x∗24 (t) − 2t = 0,





20x∗ (t) + 2x∗12 (t) − t = 0, 8x∗32 (t) + 2x∗11 (t) − 2t = 0,

 ∗ 31
x33 (t) = t, 2x∗33 (t) + 10x∗34 (t) − 3t = 0
We can observe that
 1
t
 6

 1
∗
t
x (t) = 
 9

 1
t
120
5
t
12
4
t
9
5
t
24
1
t
18
2
t
3
1 
t
18 

1 
t ,
6 

1 
t
t
10
is a solution, in [0, 1], since 8x∗33 (t) + 2x∗34 (t) − 10t < 0.
Now, we are able to compute the production and the demand excesses
of each firm in [0, 1] :
T
T
119 29 161
2057 211 41 691
(t) =
, δ(t) =
,
t t
t
t
t t
t
36 18 60
360 72 18 90
With the same method and initial data of previous example it is possible
to compute the numerical solution, so we get the approximate curve of
equilibria shown in Figure 3.
Comparing the numerical results, we can remark that the equilibrium
solutions are different and dependent on the presence of the excesses. The
difference of the equilibrium solutions between the two examples is pointed
16
DOI: 10.1685/journal.caim.415
1
x*11(t)
x*12(t)
x*13(t)
x* (t)
14
x*21(t)
x* (t)
22
x*23(t)
x*24(t)
x* (t)
31
x*32(t)
x*33(t)
x*34(t)
0.9
0.8
Numerical solution
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
Figure 3.
0
0.1
0.2
0.3
0.4
0.5
Time
0.6
0.7
0.8
0.9
1
Curves of equilibria of problem with both production and demand excesses.
out by the presence of capacity constraints. If we did not consider the
capacity constraints, the equilibrium solutions computed through the direct
method, would be the same and belonging to the same set K. The presence
of the capacity constraints, like we see, forces the solution to belong to the
e In this examples we observe that the numerical method fits very
face K.
well the approximate solution with the exact solution computed through
the direct metod. This verifies even the good convergence of the numerical
method.
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