Tilburg University
Solving the nonlinear complementarity problem with lower and upper bounds
Kremers, J.A.W.M.; Talman, A.J.J.
Publication date:
1988
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Citation for published version (APA):
Kremers, J. A. W. M., & Talman, A. J. J. (1988). Solving the nonlinear complementarity problem with lower and
upper bounds. (Research memorandum / Tilburg University, Department of Economics; ???volumeLabel???
FEW 330). Unknown Publisher.
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SOLVING THE NONLINEAR COMPLEMENTARITY
PROBLEM WITH LOWER AND UPPER BOUNDS
Hans Kremers
Dolf Talman
~ 330
Department of Econometrics
Tilburg University
P.O. Box 90153
5000 LE Tilburg
The Netherlands
.Tuly 1988
This
research is part of the VF-program "Equilibrium and
Disequilibrium in Demand and Supply",
which has
been
approved by
the Netherlands Ministry of Education and
Sciences.
1
SOLVING THE NONLINEAR COMPLEMENTARITY PROBLEM
WITH LOWER AND UPPER BOUNDS
Hans KREMERS and Dolf TALMAN,
Abstract:
lower
In order to solve the nonlinear complementarity problem with
and
upper
bounds,
algorithm is introduced.
problem
is
chosen
Tilburg
defined
a
simplicial
variable
The algorithm subdivides
dimension restart
the set on which the
into simplices and generates from an arbitrarily
starting point a piecewise linear path of points
approximate
algorithm
solution.
can
simplicial
When
the
be restarted
subdivision.
at
The
accuracy
is
not
leading to an
sufficient
the
the approximate solution with a finer
piecewise
algorithm
is
dimension.
The path can be interpreted as
linear path generated by the
followed by a sequence of adjacent simplices of varying
the path of solutions of the
nonlinear complementarity problem with parametrized bounds.
1.
Introduction.
This
paper
algorithm
for
is
concerned
findíng
complementarity
problem
an
with
the
approximate
with
development of a simplicial
solution
for
the nonlinear
lower and upper bounds. The problem is
defined as follows.
Given two vectors a and b in Rn with ai C bi for all i E{1,...,n}
and a continuous function f:Cn ~ Rn, with Cn defined as
Cn -{x E Rn~ a C x C b}, find an xM E Cn such that for all
i E {1,
.
fi(x`)
,n}
C 0 if ai - xi
fi(x~) - 0 if ai C xi C bi
fi(x")
~ 0 if
xi - bi.
2
This
problem
is
complementarity
also
problem
known
(GNLCP)
as
the
and it
is
generalized
frequently met
nonlinear
in economic
problems.
The
GNLCP
encloses
mathematical
nonlinear
programming.
complementarity
complementarity
of
(1.1)
by
algorithm
problem
to
well-known
Among
problem
(GLCP).
taking ai
problems
in
the field of
these
problems
we
mention
(NLCP)
and the generalized linear
The NLCP can be seen
- 0 and bi -;m f'or all
solving the NLCP we refer to
special case of (1.1)
the
many
GLCP
can be found in
the
simplicial
The
paper
(2).
by assuming f to be
(3).
the
as a limit case
i E{1,...,n}.
For an
The GLCP can be seen as
linear.
Our algorithm
a
An algorithm solving
is a natural alternative
algorithm developed by van der Laen and Talman in
(4)
of
points
is organized as follows.
the
algorithm
are
described
in
subdivision
algorithm
follows
described
section
of
Cn.
in
2
the
Section 2
introduces the path
approximately.
section
3.
algorithm
To
makes
The
steps of the
approximate
use
the
path
of a simplicial
In section 4 we present an appropriate
simplicial
subdivision of Cn.
2. The path to be approximated by the algorithm.
Starting
in
an
arbitrarily
chosen
point
v E Cn the algorithm
follows approximately a path of points x in Cn such that for some p, 0
~ p C 1, x solves the GNLCP on CP :- (1-p){v} t pCn with respect to f,
i.e., for all i E{1,...,n}
fi(x) ( 0 if (1-p)vi t pai - xi
fi(x) - 0 if (1-p)vi t pai C xi ~(1-p)vi t pbi
fi(x) ~ 0 if
(2.1)
xi -(1-p)vi t pbi.
Under some regularity and nondegeneracy conditions the set of points x
being a solution of (2.1)
curves.
Each
points.
One
for some p, 0( p( 1,
form piecewise smooth
of these curves is either a loop or a path with two end
of
these paths,
say P, has v as an end point for p- 0.
3
All
other
algorithm
end
points
follows
of
paths
approximately
in
Cn are solutions
the
to
(1.1).
The
path P from v to its other end
point.
By
increasing
pointing
and
p
towards
zi
from
0
the
path
the corner point
z
P
leaves v in the direction
of Cn where zi
-
ai
if
generality
we
assume that no component of f(v)
the
fi(v)
~
path P at a point x-(1-p)v t pz,
point in
while
the boundary of Cn,
z~
-
continues
p)v~
pb~).
(1-p)v~
b~),
decreasing
on
P,
p
pb~)
conditions
in
in
equal
(2.1)
(1.1)
reduce
equals
zero.
with p between 0 and
If along
1 and z a
becomes zero for some j
E{1,...,n}
either
or the path
(1-p)v~
f~(x)
to
to
1,
x
solves
. pa~
-
then,
Finally,
(1-
because Ci
if at a point x
- Cn and hence the
the point x is a solution to the
and thereby an end point of the path P in Cn.
way the path P leads
from
to (1-p)v, t
J
paJ
then the path P continues
0},
from zero.
(1.1),
(1.1)
(decreasing x~
x~ becomes equal
E{i~fi(x)
(increasing)
becomes
O
then
from
for a j
~
Without loss of
f~(x)
If at a point x on P,
t
by
GNLCP
(or
by increasing x~
t
(or
a~
- bi if fi(v)
0 for sll i E{1,...,n}.
In this
from v to a solution of (1.1).
3. The algorithm.
The
section
with
p.l.
algorithm
follows
2 by generating a piecewise linear
an
approximate
solution
x of
path we approximate the function
To define a p.l.
simplices.
Cn.
approximately
For
(1.1).
the
path
(p.l.)
P
described in
path P connecting v
For a description of this
f by a p.l.
approximation F.
approximation F of f we need to subdivide Cn into
So,
an
let Gn be a triangulation or simplicial subdivision of
appropriate simplicial subdivision of Cn we refer the
interested reader to section 4.
Definition
3.1:
The p.l.
approximation F of f with respect to the
simplicial subdivision Gn of Cn at a point x E Cn is given by
F(x)
- ~i}1 ~if(yi)
(3,1)
4
where
the
convex
hull
.yntl) of
6(yl
1
y.
..Y
nfl
in C
n
is an nor n-simplex in Gn containing x and where
~1,...,~ntl -~ 0
ntl
i
n41
are such that x - ~i-1
aiy and ~i-1 ~i - 1.
dimensional
The
applied
a
p.l.
with
results
obtained
in section 2 with respect to f can also be
to the p.l. approximation F of f. In particular, there exists
n
path
P
of points in C
connecting v and a solution to (1.1)
respect
to
between 0 and 1
F.
such
For each point x on the path P there exists a p
that
for all i E{1,...,n}
Fi(x)
~ 0 if (1-p)vi t pai - xi
Fi(x)
- 0 if (1-p)vi t pai t xi ((1-p)vi t pbi
Fi(x)
~ 0 if
(3.2)
xi -(1-p)~i , pbi,
Notice
that
in
DePinition
3.2:
A vector s E Rn i s a sign vector if,
(3.2)
the
sign
pattern of F(x) plays a very
important role. Therefore we introduce the notion of a sign vector.
for all i, si E
{-1,O,t1}.
Now, let for each sign vector s the set Cn(s)
n
n
C(s) -{x E C ~
for all i, xi - ai if si --1
and
If
n
v E C(s)
v and Cn(s),
A(s)
be defined by
we define A(s)
xi - bi if si -;1}.
- m,
otherwise A(s)
is
(3.3)
the convex hull
of
i.e.,
-{x E Cn~
for some p, 0( p~ 1,
and for all i
if si --1
(1-P)~i t Pai - xi
(1-p)vi t pai C xi ((1-p)vi t pbi if si - 0
(3.4)
x. -(1-p)v. . pb. if s. - tl}.
i
i
i
i
Clearly,
x E P satisfies x E A(s) with s the sign vector such that ssgn(F(x)).
5
The
simplicial
triangulates
dimension
subdivision
each
of
-
subdivision).
1
ttl
a(y ,..,y
)
of
Cn
nonempty subset A(s)
A(s),
{1,...,n}~si
Gn
is
0}
equal
(see
So,
if
in A(s)
to
section
E
x
has
~I~(s)~tl
4
for
A(s),
with
an
be such that
then
I~(s)
.-
it
the
{i
E
appropriate simplicial
there
and numbers ~1,...,~ttl
)
-
~iyl and ~i}i ~i - 1.
On
to
into t-simplices where t,
are
a
t-simplex
- ~t,l
0 such that x
i-1
the other hand, if sgn(F(x)) - s, then there exist y.h ) 0, h~
IG(s),
such
that
dimensional
unit
F(x)
-~hal0(s)
~she(h),
where e(h) is the n-
vector with ei(h) - 1 if h- i. Hence,
if x lies on
the
path P,
then
for some sign vector s there is a t-simplex
1
a(y ,..,y ttl )
in A(s) such that the system of linear equations given
by
~i}1 Ailf ( 11))
has
a
-
(3.5)
lOJ
nonnegative solution ai ) 0, i- 1,...,ttl, ~) 0, halG(s),
~ttl ~,~yi
~e vector 0 in (3.5) denotes the n-vector of
i-1
i
-
x-
with
zeros.
System
leaving
line
(3.5)
us
a
In
0
system
to
solutions
to
programming
the
of
ntl
freedom.
(3.5)
equations with nt2 unknowns
So,
assuming nondegeneracy,
lies
either
ap
a
exists which can be followed by
pivot step in
(3.5).
This line segment
linear piece of P in cs defined by the points x-
either ~p
some p E{1,...,ttl} or y,j - 0 for some j f~ IG(s).
point,
~iyl
a
an end point of a line segment of solutions to (3.5)
for
end
of
linear
corresponds
t}1
i
~i-1 ~iy '
is
with one degree of
segment
making
-
- ~h~IG(s) uhshle0h)J
- 0 for some p E{1,...,t41},
If at an
then the point x-~i~
in the facet T of a opposite the vertex yp. The facet T isp
also a facet of exactly one other t-simplex,
say 6,
in A(s) or
z lies in the boundary of A(s).
Suppose 6 exists. Then,
pivot
to
the
step
in order to continue the path P in A(s), a
is made in ( 3.5) with the column [f(y)T,1]T corresponding
unique
vertex
y
of
ct not contained in T.
continued by repeating the procedure described.
The
algorithm is
6
Suppose 6 does not exist and hence T lies in the boundary of A(s).
If t].ies in Cn(s), then the algorithm has found a point x E Cn(s)
with sign vector s equal
to sgn(F(x)) so that x is an approximate
solution for (1.1). Otherwise, T is a(t-1)-simplex in A(s) where s is
a sign vector such that s~ ~ 0 for some .~ E ID(s) while si - si for
all i~ i.
Then the algorithm continues in A(s) by pivoting the column
[s~e(,~)T,o)T into (3.5).
If
IC(s),
at an end point of solutions to
then at x--~i}1
vector
such
that s~
-
whereas sgn(F(x))
solution
Otherwise,
to
(1.1).
cs
in
A(s)
if A(s)
having
o
- s.
j.
- 0.
Let s be a sign
Suppose
Hence,
~ ó,
as
is zero for some j(t
u~
- s~u~
~iyl
0 and sh - sh for h~
Then x lies in Cn(s)
(ttl)-simplex
(3.5),
we have F~(x)
that A(s)
then there is
a
facet.
continues by pivoting the column [f(y)T,1]T into
-~.
x is an approximate
exactly one
Now the algorithm
(3.5),
where y is the
vertex of o not contained in o.
in
Now
we have described how the algorithm proceeds along the path P
the
different
subsets
A(s) of Cn, we still have to describe the
initialization of the algorithm at v. At v the system (3.5) becomes
allf(i)
having
a
J
- ~h-1 s0~re(hC)1
unique
solution
- r01
al
(3.6)
- 1, uh - s~fh(v)
~ 0, h E{1,...,n},
where
s0 - sgn(f(v)). If A(s~) - rá, then v E Cn(s~) and the algorithm
stops
with an exact solution at v. Otherwise,
a
facet
of
a
unique
1-simplex
the starting point v is
o(yl,y2) in A(s~) with yl - v. The
algorithm then pivots the column [f(y2)T,1]T into (3.6).
Since
all steps are unique,
number
of
finite
number
simplices
of
is
returning to v is
finite,
the
impossible,
algorithm terminates within a
steps with an approximate solution x of
accuracy of the approximation f(x)
and the
(1.1). The
can be measured by the smallest E~
0 for which for all i E{1,...,n}
Fi(X)
-6 ~ fi(x)
:
E
( s
if al - xi
if ai ~ xi ~ bi
(3.7)
-e ( fi(x)
If
f(x)
if
xi - bi.
is not accurate enough,
i.e.
if e
is
too large,
the algorithm
is repeated being started at v- x with a finer simplicial
of Cn.
This
in the hope
to
subdivision
find a more accurate approximation within a
relative small number of steps.
In this way,
within a finite number of
steps an approximate solution with any accuracy can be found.
4. A simplicial subdivision of Cn.
In order
The
only
underly
of
into
unit
triangulate Cn one can use any simplicisl subdivision,
restriction
the
triangulate
fits
to
A(s)
described
this framework is
simplices
in
section
3
is
describe
j or -j
to
that perfectly
in
(1).
In this
section we adapt
the
triangulation of Cn.
the
a permutation of the
that it has
the V-triangulation of the product space
developed
triangulation
into subsets A(s,y(T))
either
n
has to pose on the triangulation of Cn to
all nonempty subsets A(s). A triangulation
V-triangulation to a
To
one
algorithm
with y(T)
we first subdivide each nonempty
-(~ll,,,,,~t-1),
t-
~ID(s)~tl,
t-1 elements of a set T such that for all jEI~(s)
belongs
to T.
Zf we define the projection p(s)
of v on
C(s) as the vector with elements
ah if sh - -1
ph(s)
-
bh if sh -;1
h E{1,...,n},
(4.1)
vh if sh - 0
then
h
p(s
A(s,y(T)) is defined as the convex hull of v and the projections
),
h E {1,...,t},
where
sh - s t ~~.h e(Yh),
with
ei(,!~)
Cn.
for
all
i,
h E {1,...,t},
(4.2)
ei(~j) - fl if ,yj - i, ei(yj) --1 if ~j --i, and
- 0 otherwise. Notice that st - s and that p(sl) is a vertex of
a
For
some
triangulated
positive
integer
m,
into t-simplices 6(yl,rt)
each
nonempty
A(s,~r(T))
with vertices yl,
..,yttl
is now
in Cn
such that
i)
1
t
y- v. ~k-I a(k)m
0 C
ii)
a(t)
C...
rt-(nl,...,rtt)
C
-1
a(1)
q(k)
with integers a(k)
satisfying
C m-1;
is a permutation of the elements of
{1,...,t} such that for all i E{1,...,t-1} holds
p) p' if n,- i, rt- itl, and a(rt ,)
iii)
Y
itl
where q(1)
we
i
- p(sl)
denote
is
p
1,...,t,
p
- v and
- p(sk-1).
k- 2.....t.
this triangulation by Gm(s,y(T)),
triangulated by
Cn
p-1
p
t m 9(rti). i-
- p(sk)
9(k)
If
- Y
:
- a(n );
the union Gm(s)
triangulated by
of Gm(s,~(T))
(4.3)
then the set A(s)
over all 7~(T).
the union Gm of Gm(s) over all s,
m-1
is
Moreover,
being the
grid size.
In
section
making
sequence
After
of
a
follow the path P through Cn by
adjacent simplices 6 in A(s)
having
describe
of
3 we described how to
pivot steps in the system of equations
introduced
how,
a
specific
6,
the
with respect to a
triangulation of Cn we will now
g iven the parameters y,
1
t-simplex
(3.5)
for varying sign vectors s.
rr,
and a(h),
for h-
1,...,t,
parameters of a simplex á adjacent to 6 are
obtained.
The
movement from a t-simplex ~(yl,n)
sim p lex
6 (y-1 ,n)
-
t-simplex
vertex
cs
by
listed
ttl,
.
in
is
in A(s,~r(T))
to an adjacent
called a replacement step when a(yl,rr)
is also a
A(s,y(T)).
Making a replacement step we replace the
yp, p E{1,...,ttl}, of 6 opposite the common facet T of 6 and
the vertex y of á not belonging to i. The possibilities are
in
,n.
Table
1,
where
ah
- a(h),
h- 1,...,t, and ah - 0, h-
9
Table
l:
Replncement, step.
p - 1
-1
Y
rt
a
yl}m-lq(R1)
( rt2,....nt,n1)
ate(nl)
(n1,...,Rp-2'Rp'rtp-1'Rptl~...,nt)
a
( nt,rt1,...,Rt-1)
a-e(Rt)
1
1 ~ p ~ ttl
Y
P - ttl
yl-m-lq(rtt)
In
case
the
performed,
the
A(s,y(T)).
Lemma
replacement
step
facet T of ~(yl,n)
4.1
describes
with
respect
to
yp
cannot be
opposite yp lies in the boundary of
when
T
lies
in
the
boundary of
and T
the facet of
A(s,~r(T) ) .
Lemma 4.1:
6
opposite
A(s,~(T))
Let 6(yl,n)
be a t-simplex
vertex
1~ p C
yp,
P- 1, nl - 1,
2)
1~
and
In
Then T lies in the boundary of
and a(rtl) - m-1;
p( ttl, np-1 - i and np - itl for some i E{1,...,t-1},
a(rtp-1) - aÍRp);
nt - t, and a(nt)
P- ttl,
case
ttl.
if and only if one of the following cases holds:
1)
3)
in Gn(s,y(T))
1 of Lemma 4.1,
T
- 0.
lies in Cn(s).
shares i with an adjacent t-simplex 6(yl,rr)
In case 2 and when iin A(s,y(T))
1,
6
where T- T`
{yl}
u{-ól}.
ë(T)
-(-ál,à~2....,ët-1)' and R-(A1,... n 2,rr
rr },.. n
Otherwise
in case 2,
6 (Yl,rt)
shares T, with
pp'
an
p-1' p 1'
' t)'
adjacent t-simplex 6(yl,n) in A(s,y(T)) where ~r(T) -(~r ,.. ~r
~r
rr
1
~i-1,?fitl..O..ót-1)
and
when I(s)
and n -
~ ó represents
vertex yt}1 of 6 is
the
'
'
(nl,....rtp-2'Rp'rtp-1'np}1,...,Rt).
the case where
(t-1)-simplex ~(yl,n)
i-2' i'
Case 3
the facet T opposite
in A(s,~r(T))
the
where s-
s
t e(ót-1). T- T`{yt-1}. à'(T) -(ól.-...Yt-2) and n -( nl,....Rt-1).
Otherwise in case 3, we have that t- 1 and e(1) - 0 which means that
T - {v}.
10
Finally,
one
-
-
c(yl,n)
in A(s,~r(T))
.
where sk
si for all other i E{1,...,n}.
tl
and h--k if sk --1,
A(s,y(T))
Ínl,
t-simplex
(ttl)-simplex 6 in a nonempty A(s)
and si
sk
a
where
T-
T
More precisely,
then ~ is
u{h},
is a facet of exactly
- 0 for some k~ IO(s)
~(T)
the
let h-
tk
if
(ttl)-simplex 6(yl,n)
in
~d n-
-(JJl,,,,~~t-l~h)~
.nt,ttl).
References.
(1) T.M
DOUP,
Dimension
A.J.J.
Unit Simplices.
(2)
M.
KOJIMA
(1974),
University,
G.
to
(1987),
Find
A
New
Computational Methods
Problem.
Yokohama,
VAN DER LAAN
Variable
Keio
319-355.
for Solving the Nonlinear
Engineering
Reports
27,
Keio
Japan.
,A.J.J TALMAN
Complementarity
Problem
Memorandum
200,
FEW
Simplicial
Equilibria on the Product Space of
Mathematical Programming 37,
Complementarity
(3)
TALMAN
Algorithm
(1985),
with
Upper
An Algorithm
and
Tilburg University,
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1
IN 1987 REEDS VERSCHENEN
242
Gerard van den Berg
Nonstationarity in job search theory
243
Annie Cuyt,
Brigitte Verdonk
Block-tridiagonal linear systems and branched continued fractions
244
J.C. de Vos, W. Vervaat
Local Times of Bernoulli Walk
245
Arie Kapteyn, Peter Kooreman, Rob Willemse
Some methodological issues in the
of subjective poverty definitions
implementation
246
J.P.C. Kleijnen, J. Kriens, M.C.H.M. Lafleur, J.H.F.
Sampling
for Quality
Inspection
and Correction:
Criteria
247
D.B.J. Schouten
Pardoel
AOQL Performance
Algemene theorie van de internationale conjuncturele
afhankelijkheden
en
strukturele
248
F.C. Bussemaker, W.H. Haemers, J.J. Seidel, E. Spence
On (v,k,~.) graphs and designs with trivial automorphism group
2e}9
peter M. Kort
The Influence of a Stochastic Environment on the Firm's Optimal Dynamic Investment Policy
250
R.H.J.M. Gradus
Preliminary version
The
reaction
of the
approach
251
firm on governmental policy:
a game-theoretical
J.G. de Gooijer, R.M.J. Heuts
Higher order moments of bilinear
cally distributed errors
time series
processes with symmetri-
252
P.H. Stevers, P.A.M. Versteijne
Evaluatie van marketing-activiteiten
253
H.P.A. Mulders, A.J. van Reeken
DATAAL - een hulpmiddel voor onderhoud van gegevensverzamelingen
254
P. Kooreman, A. Kapteyn
On the identifiability of household production functions with joint
products: A comment
255
B.
van Riel
Was er een profit-squeeze in de Nederlandse industrie?
256
R.P. Gilles
Economies with coalitional
cepts
structures and core-like equilibrium
con-
11
25~
P.H.M.
258
W.H. Haemers, A.E. Brouwer
Association schemes
259
G.J.M.
Ruys, G. van der Laan
Computation of an industrial equilibrium
van den Boom
Some modifications and applications of Rubinstein's perfect equilibrium model of bargaining
260
A.W.A. Boot,
Competition,
A.V. Thakor, G.F. Udell
Risk Neutrality and Loan Commitments
261
A.W.A. Boot, A.V. Thakor, G.F. Udell
Collateral and Borrower Risk
262
A. Kapteyn, I. Woittiez
Preference Interdependence and Habit Formation in Family Labor Supply
263
B. Bettonvil
A formal description of
pcrturbation ana]ysis
discrete
event
dynamic
systems
including
264
Sylvester C.W. Eijffinger
A monthly model for the monetary policy in the Netherlands
265
F. van der Ploeg, A.J. de Zeeuw
Conflict over arms accumulation in market and command economies
266
F. van der Ploeg, A.J. de Zeeuw
Perfect equilibrium in a model of competitive arms accumulation
26~
Aart de Zeeuw
Inflation and reputation: comment
268
A.J. de Zeeuw, F. van der Ploeg
Difference games and policy evaluation:
269
a conceptual framework
Frederick van der Ploeg
Rationing in open economy and dynamic macroeconomics:
a survey
2~0
G. van der Laan and A.J.J. Talman
Computing economic equilibria by variable dimension algorithms: state
of the art
2~1
C.A.J.M. Dirven and A.J.J. Talman
A simplicial algorithm for finding
linear production technologies
272
Th.E. Nijman and F.C. Palm
Consistent estimation of regression models with incompletely observed
exogenous variables
2~3
Th.E. Nijman and F.C. Palm
Predictive accuracy gain from disaggregate sampling in arima - models
equilibria
in
economies
with
111
2~4
Raymond H.J.M.
Gradus
The net present value of governmental policy:
the Stackelberg solutions
275
277
find
Jack P.C. Kleijnen
A
DSS for production planning:
optimization
2~6
a possible way to
A.M.H. Gerards
A short proof of
matrices
Tutte's
a case study including simulation and
characterization
of
totally
unimodular
Th. van de Klundert and F. van der Ploeg
Wage
rigidity and capítal mobility in an optimizing model of a small
open economy
2~8
Peter M. Kort
The net present value in dynamic models of the firm
2~9
280
281
282
Th. van de Klundert
A Macroeconomic Two-Country Model with
lists
Price-Discriminating
Arnoud Boot and Anjan V. Thakor
Dynamic equilibrium in a competitive credit market:
contracting as insurance against rationing
Monopo-
intertemporal
Arnoud Boot and Anjan V. Thakor
Appendix:
"Dynamic equilibrium
in
a competitive
credit
intertemporal contracting as insurance against rationing
Arnoud Boot, Anjan V. Thakor and Gregory F. Udell
Credible
commitments,
contract
enforcement problems
intermediation as credibility assurance
market:
and
banks-
283
Eduard Ponds
Wage bargaining and business cycles a Goodwin-Nash model
284
Prof.Dr. hab. Stefan Mynarski
The mechanism of restoring equilibrium and stability in polish market
285
P.
Meulendijks
An exercise in welfare economics (II)
286
S.
Jfórgensen,
P.M.
Optimal investment,
tial game
28~
288
Kort,
G.J.C.Th.
van Schijndel
financing and dividends:
E. Nijssen, W. Reijnders
Privatisering en commercialisering;
verzelfstandiging
a Stackelberg
differen-
een oriëntatie ten aanzien van
C.B. Mulder
Inefficiency of automatically lialcing unemployment benefits to private sector wage rates
1V
2B9
290
M.II.C. Paardekooper
A
Quadratically convergent parallel Jacobi
nal matrices with distinct eigenvalues
process for almost diago-
Pieter H.M. Ruys
Industries with private and public enterprises
291
J.J.A. Moors 8~ J.C. van Houwelingen
Estimation of linear models with inequality restrictions
292
Arthur van Soest, Peter Kooreman
Vakantiebestemming en -bestedingen
293
Rob Alessie, Raymond Gradus, Bertrand Melenberg
294
The problem of not
expenditure survey
observing
small
expenditures
in
a
consumer
F. Boekema, L. Oerlemans, A.J. Hendriks
Kansrijkheid en economische potentie:
Top-down en bottom-up analyses
295
Rob Alessie, Bertrand Melenberg, Guglielmo Weber
Consumption,
Leisure and Earnings-Related Liquidity Constraints: A
Note
296
Arthur van Soest, Peter Kooreman
Estimation of the indirect translog demand system with
negativity constraints
binding
non-
V
IN 1988 REEDS VERSCHENEN
297
Bert Bettonvil
Factor screeninq by sequential bifurcation
298
Robert P. Gilles
On perfect competition in an economy with a coalitional structure
299
Willem Selen,
Ruud M.
Heuts
Capacitated Lot-Size Production Planning in Process Industry
300
J. Kriens, J.Th. van Lieshout
Notes on the Markowitz portfolio selection method
301
Bert Bettonvil, Jack P.C. Kleijnen
Measurement scales and resolution IV designs:
302
a note
Theo Nijman, Marno Verbeek
Estimation of time dependent parameters in lineair models
using cross sections, panels or both
303
Raymond H.J.M. Gradus
A differential
approach
30~1
Leo W.G.
game between government and firms:
Strijbosch,
Ronald J.M.M.
a non-cooperative
Does
Comparison of bias-reducing methods for estimating the
dilution series
305
Drs. W.J. Reijnders, Drs. W.F.
Strategische
concept
306
307
bespiegelingen
parameter
in
Verstappen
betreffende het Nederlandse kwaliteits-
J.P.C. Kleijnen, J. Kriens, H. Timmermans and H. Van den Wildenberg
Regression sampling in statistical auditing
Isolde Woittiez,
Arie Kapteyn
A Model of Job Choice,
Labour Supply and Wages
308
Jack P.C. Kleijnen
Simulation and optimization in production planning: A case study
309
Robert P. Gilles and Pieter H.M. Ruys
Relational constraints in coalition formation
310
Drs. H. Leo Theuns
Determinanten van de vraag naar vakantiereizen:
materiële en immateriële factoren
een
311
Peter M. Kort
Dynamic Firm Behaviour within an Uncertain Environment
312
J.P.C. Blanc
A numerical
approach
to cyclic-service queueing models
verkenning
van
vi
313
Drs.
Does
N.J. de Deer, Drs.
Morkmon Matter?
314
Th, van de Klundert
Wage differentials
labour market
A.M.
van Nunen,
Drs.
Aart de Zeeuw, Fons Groot, Cees Withagen
On Credible Optimal Tax Rate Policies
316
Christian B. Mulder
Wage moderating effects of corporatism
Decentralized versus centralized wage
government context
setting
31~
Jdrg Glombowski, Michael KrUger
A short-period Goodwin growth cycle
318
Theo Nijman, Marno Verbeek, Arthur van Soest
319
320
321
Nijkamp
and employment in a two-sector model with a dual
315
The optimal design
variance model
M.O.
of
rotating
panels
in
a
in
a union,
simple
analysis
Th. van de Klundert
Wage Rigidity, Capital Accumulation and Unemployment in a Small
Economy
M.H.C.
Th.
de
Open
Paardekooper
to a nearby
ten Raa, F. van der Ploeg
A statistical
analysis
323
of
Drs. S.V. Hannema, Drs. P.A.M
Versteijne
De
toepassing en
toekomst
~an public private partnership's bij
grote en middelgrote Nederlandse gemeenten
An upper and a lower bound for the distance of a manifold
point
322
firm,
approach to the problem of
negatives
in
input-output
P. Kooreman
Household Labor Force Participation as a Cooperative Game;
cal Model
an Empiri-
324
A.B.T.M. van Schaík
Persistent Unemployment and Long Run Growth
325
Dr. F.W.M. Boekema, Drs. L.A.G. Oerlemans
De lokale produktiestructuur doorgelicht.
Bedrijfstakverkenningen ten behoeve van regionaal-economisch onderzoek
326
J.P.C. Kleijnen, J. Kriens, M.C.H.M. Lafleur, J.H.F. Pardoel
Sampling for quality inspection and correction:
AOQL performance
criteria
V11
32~
Theo E. Nijman, Mark F.J.
Exclusion
328
Steel
restrictions in instrumental variables equations
B.B. van der Genugten
Estimation
in
linear regression under
ticity of a completely unknown form
329
Raymond H.J.M. Gradus
The employment policy of government:
create?
the presence of heteroskedas-
to create jobs or
to
let
them
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