Laan, G. van der

Talman, A.J.J.

English

Research Papers in Economics

CBM
R
7626
1981
106
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subfaculteit der econometrie
RESEARCH MEMORANDUM
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TOLBURGNOTE OF THE PATH FOLLOWING APPROACH OF
EQUILIBRIUM PROGRAMMING
by
G. van der Laanz
A.J.J. Talman~
~ Department of Actuarial Sciences and Econometrics
Free University, Amsterdam, The Netherlands
x~ Department of Econometrics
Tilburg University, Tilburg, The Netherlands.
December 1981.NOTE ON THE PATH FOLLOWING APPROACH OF EQUILIBRIUM PROGRAMMING
~~
A.J.J. Talman
ABSTRACT.
Recently Zangwill and Garcia introduced a general formulation of equilibrium
problems. To prove the existence of an equilibrium they discussed a path
following procedure. In this note we consider the application to the exchange
economy problem. An economic equilibrium may be found by applying a simplicial
variable dimension algorithm developed by Van der Laan and Talman.
We will show that when an appropriate triangulation and labelling rule is taken
the limiting path of this algorithm coincides with the adjustment process
induced by the procedure of Zangwill and Garcia.
KEY WORDS: Equili.brium programming, er.change c~conomy, simplicial algorithms,
limiting path.
December 1981.
~ Department of Actuarial Sciences and Econometrics
Free University, Amsterdam, The Netherlands
~~ Department of Econometrics
Tilburq University, Tilburq, The Netherlands.NOTE ON THE PATH FOLLOWING APPROACH OF EQUILIBRIUM PROGRAMMING
by
G. van der Laan and A.J.J. Talman
1. The economic equilibrium problem.
In this section we consider a path following procedure to obtain an
economic equilibrium. This procedure has been given by Zangwill and Garcia
[5] as an application to a general approach of equilibrium programming. In
the next section we show that the procedure is very similar to the variable
dimension approach of Van der Laan and Talman (see [1]-[9]).
Consider an exchange economy of m agents with n commodities. Let
w1-(wi,...,wn)~0 be the endowment of agent i, and let the utility function
of agent i be given by fi: Rn-~R. Let w-E~-lwl be the total endowments.
Definition 1.1.
A competitive equilibrium is a pair of vectors (x,p) where
x-(xl,...,xm)eR~ and peRn such that
(a) fi(xl) - max fl(xl), i-1,...,m,
for pxi~pwl and Osxl~w,
(b) Lm-lxi~w,
(c) p?0 and En p.-1.
~-1 ~
To show the existence of a competitive equilibrium, Zangwill and Garcia [5]
introduced the following equilibrium program.
(a) For i-1,...,m, given p,
max fl(xl)
for pxlspwl and O~xisw, (1.2)-2-
(b) Given x,
max p(Em-1xl-w)
for
peSn-1(t) - {pESn-llp~P-(tte)e},
n-1 n n
where S -{peR}~E, 1 P~-1}, c~0 is very c,mall, e is a vector with all components 1,
J-
Ostsl, and ~S is some arbitrarily chosen initial price vector. For given p,
let z(p) be the excess demand, i.e. z(p)-Em-ixl(p)-w, where xl(p) solves (1.2a).
n-1 n It is assumed that z:S -~R is a continuous function.
Without loss of generality we can assume that the excess demand in p has a
maximum at a unique index, say k. Clearly, for t--e, (1.2) has a unique solution
(x(1S),~), where xi(~) solves (1.2a) given p-p.
For E~0 small enough we have that zk(p) is still the unique maximum excess
demand for t-0 and p satisfying the conditions of (1.2b). Therefore, given x(p)
with peSn-1(t), the solution of (1.2b) is
0
pk - pkt(n-1)e
0
pj - Pj-e j~k
(1.3)
0 0
when t-0. Hence ( x(p),p) is the unique solution of (1.2) at t-0. In their paper
Zangwill and Garcia prove the following theorem.
Theorem 1.2.
Suppose for all i, fi is 3-differentiable and strictly concave. If the
equilibrium program (1.2) is regular, then starting from (x,p,t)-(x(p),p,0)
there is a path of solutions to (1.2) that reach a solution to (1.1) at t-1.
As discussed by Zangwill and Garcia, the economic interpretation of this
path is as follows. For small e~0, when t is increased from zero, the price
of qood k having the largest excess demand is increased, whereas the other
prices are decreased, each with the same amount. In general the price vector
p and the variable t are adapted in such a way that the usage of all goods
with highest demand is tried to decrease. So, at any p, the process (1.2) works
on the worst cases, i.e. on the markets with highest excess demand. As soon as
all goods have the same excess demand, by Walras' law we must have z(p)50 and
an equilibrium is reached.-3-
Lookinq at the pzocess in more detail, we first note that obviously t is
not necessarily increasing monotonically during the process. The possibility
of decreasing t has not made clear by Zangwill and Garcia, who state "the
specialist slowly adjusts thu 1~rices to biqger and biyger r~rice .,ets...".
In fact, considering the projection of the path on the set of prices
Sn-1 the adjustment proces behaves as follows. Define, for TcIn{1,...,n},
C(T) - {peSn-llzk(p) - max zj(p), keT}
j
and,for O~tsl,
P(T,t) -{pEgn-1(t)~P- E a.p~(t), a.?0, E a.-1}.
jeT ~ ~ jeT ~
where p~(t) is the vertex of Sn-1(t) such that the j-th component is maximal.
Then, a point (x,p,t) on the path of solutions to (1.2) has the property that
for some TcIn, pEC(T)nP(T,t). The projection of this path on the price space
Sn-1 is illustrated in figure 1. In figure la the procedure starts in C({2})
and hence p2 is increased until C({3}) is reached.Then the 1-manifold C({2,3})-
C({2})nC({3}) is followed until the equilibrium price p is obtained. ObServe
that t increases monotonically. In figure Sb again we have that the procedure
starts in C({2}), however in a subset of C({2}) which is surrounded by C({3}).
Now t increases on the path from p(p) to pl, decreases from pl to p2 and increases
again on the path from p2 to the equilibrium price p.
e(3) e(3)
e(2)
a
Figure 1. The paths of solution points projected on Sn-1 have been drawn
heavily, e(i) is the i-th unit column.-9-
2. The variable dimension approach.
In this section we show that the projection on the price space Sn-1 of
the adjustment process of 7.angwill and Garcia coincides with the limiting path
of the simplicial variable dimension algorithm developed by Van der Laan and
Talman, provided an appropriate labelling rule and triangulation underly the
algorithm.
To do so, for TcIn, let the sets A(T) be defined by
A(T) - {pESn-1IP-Pt]ÉTaju(j), aj?0}
where u(j), j- 1,...,n, is the j-th column of the nxn matrix
(n-1) 1 . . . 1
1 (n-1) .
U -
1 . . . 1 (n-1)
(2.1)
(see Van der Laan and Talman C3, page 2827). Since, for T~O,n, C(T) is an(n-
- ITI}dimensional subset of Sn-1 and,for peint Sn-1, A(T) is a IT~-dimensional
subset of Sn-1, the set of points p with
p e u (A (T) nC (T) )
TcI
n
is a collection of paths and loops. The set of endpoints of the paths is the
set of points pe u (A(T)nC(T)). Clearly, the point p is the unique endpoint
IT~-O,n
for T-pJ. For all other endpoints p we have T-In and hence, , pEC(In),i.e..p is
an equilibrium price. So, under some regularity conditions, a path from ~ can be
followed, leading to an economic equilibrium.Clearly this path yields the
projection of the adjustment process described by Zangwill and Garcia on the set
Sn-1 when e~0 small.
On the other hand, the path in u(A(T)nC(T)) originated in p is the limiting
path of the simplicial variable dimension algorithm on Sn-1 developed by
Van der Laan and Talman f27, when the U triangulation of Sn-1 proposed by the same
authors in f37 underlies the algorithm and the followinq labelling rule is used.
In case of vector labelling each point peSn-1 is labelled according to z(p) and in
case of integer labelling p is labelled with the index k of the commodity with the-5-
highest demand. In both cases the same limiting path in u(A(T)nC(T)) is obtained~
as discussed in Van der 7.aan fl, pp. 72, 73 and 83, 841. Observe thit the
computational results in ~1} were obtained for these ].abelling rul~s and in
both fll and [4] also for the U triangulation.
Therefore the adjustment process of Zangwill and Garcia can be interpreted in
the following way. Starting in an arbitrarily chosen price vector pEP with T-(D
the process generates for varying T a path of prices p in A(T) such that all
goods j have highest excess demand zj(p),for jeT. As soon as good i, i[T, has
an excess demand zi(p) equal to the highest one, the process continues in
A(TU{i}) with prices such that zh(p)-maxj zj(p) for all hETU{i}. If, however,
the process generates a price p in A(T`{k}) for some keT, the process continues
in A(T`{k}) with prices p such that zh(p)-maxjzj(p) for all heT`{k}. The latter
step happens when ak in (2.1) becomes equal to zero. So,let (x,p,t) be a
solution generated by the adjustment process and let T. ~Tl~n, be the unique
index set such that
p- p t F, al u(j), a}~0.
~, ,~,
Then, peA(T)nC(T), i.e. we have the following complementarity
aj-0 and zj(p)~maxizi(p) j~ÉT
71j~0 and zj(p)-maxizi(p) jeT.
As a final remark we note that it can easily be shown that t is equal to
E x., i.e. roughly speaking t denotes how far the process is from the starting
jET ~
point p.
Concluding.the variable dimension alqorithm can be utilized as a simplicial
path following scheme for the projection on Sn-1 of the path of solutions to
(1.2). Moreover, the limiting path of the algorithm generates the path of points
of the adjustment process proposed by Zangwill and Garcia.References.
[1] G. van der Laan, "Simplicial fixed point algorithms", Mathematical Centre
Tract 129, Amsterdam, The Netherlands, ( 1980).
[2] G. van der Laan and A.J.J. Talman, A restart algorithm for computing fixed
pc~ints without an extra dimension, Mathematieal Proclramming 17, (1979),
79--84.
[.37 G. van der Laan and A.J.J. Talman, An improvement of fixed point algorithms
by using a good triangulation, Mathematical Programming 18, (1980), 274-285.
[4] A.J.J. Talman, "Variable dimension fixed point algorithms and triangulations",
Mathematical Centre Tract ]28, Amstcrdam, The Nctherlands,(1980).
[5] W.I. Zangwill and C.B. Garcia, Equilibrium programminq: The path following
approach and dynamics, Mathematical Programming 21,(1981), 262-289.uiuuo~iiiou~áiiiiuiiiiii~i~i~nuu~i~~ui

Note of the path following approach of equilibrium programming

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