Tilburg University
The Generalized Extreme Value Random Utility Model for Continuous Choice
Dagsvik, J.
Publication date:
1989
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Citation for published version (APA):
Dagsvik, J. (1989). The Generalized Extreme Value Random Utility Model for Continuous Choice. (CentER
Discussion Paper; Vol. 1989-41). CentER.
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C,~c~ Discussion
`~
paper
~~o~~~~
:~ - iHiui uNinim~~ugiihnnuiui i igi
3414
1989
41
No. 8941
TI~ GENERALIZED EXTREME VALUE
RANDOM UTILITY MODEL FOR
CONTINUOUS CHOICE
~.'
by John K.
Dagsvik
September, ~989
~
330 ~? S ll
JDa~REF
December 1988
THE GENERALIZED EXTREME VALUE RANDOM UTILITY MODEL FOR CONTINUOUS CHOICE
by
John K.
DdgSvik
Central Bureau of Statistics, Oslo
Abstract
The
Generalized. Extreme Value Model was developed by McFadden for
the case with discrete choice sets.
Tlie present paper extends this model to
the case with continuous choice sets.
tation result for stochastic processe5
tions
of
the
multivariate
emerges as a special
The development relies on a represenwith
finite-dimensional
extreme value type.
distríbu-
The continuous Luce model
case.
The paper also discusses a testable property of the continuous
model which may be considered an extension of Luce Axiom.
Finally, an attempt
GEV
is given to justify the stochastic structure of
the utility function frOm theoretical
drgument5.
1
1.
Introduction
The
Extreme Value random utility model
Generalized
crete choice was developed by McFadden
(see
McFadden,
(GEV)
for di5-
The
motiva-
1981).
tion
was
extend Luce choice model
so as
to relax the IIA property and
stíll
retain this property as a special
case.
So far there
to
attempt5
nuum.
postiilating
a
(1985)
discrete
introduce a continuous
utility
random
model
version is established by a li~~iiting argument.
strate
tlie
existence
of
(1988)
logit model obtained by
from whiCh the Continuous
however, not demon-
They do,
a random utility structure consistent with this
li~niting case nor do they consider more general
Cosslett
considers a class
continuous GEV.
(although quite general) of GEV
models where he uses a particular approach to discuss the
continuous
ties.
GEV
and the implied functional
He also considers statistical
existence
Luce model
Tlie
structural
to link this
of random
type
this represen-
utility
motivation
for
studying
random
larly appealing property
of
of the revealed preference type.
to
preferences
be
Specifically, this model
While a similar type of characterization of
under quite general
assumptions,
a relaxed
version
IIA.
The
organisation
of
the
paper
is
as follows.
In Section 2 the
random utility choice setting is discussed and in Section 3 tlie
probabilities
are
derived.
cl,oice
model
is discussed and Section 5 is devoted to a theoretical
the
GEV
model.
That
correspon-
In Section 4 the continuous Luce
ding
ot
can
A particu-
I,as not been obtained we demonstrate here that the
GEV model
continuous GEV implies.
of
to
the Luce model is tliat it admits a behavioral
is equivalent to the IIA axiom.
discrete
models
utility models is to obtain
separated from parameters characterizing the choice environment.
the
con-
textbook models of consumer demand.
choice models where parameters related
characterization
multi-
as of the
can be obtained quite readily. Moreover,
tation result enable5 us
the more traditional
By apply-
from
value finite-dimensional distributions we demonstrate
that the choice probabilities of tlie continuous GEV as well
t~nuous
a
inference in such models.
ing a representation result of stocliastic processes generated
extreme
of
forms for the choice probabili-
The present approach take a different point of departure.
dimensional
few
been
to the case where the choice set is a conti-
to extend this model
Ben-Akiva et al
have
is, we provide theoretical
justification
as5umptions that
tl~e type of utility function5 postulated in Section 2.
The
final
imply
Section
2
d,sCUSSes briefly the application of the GEV settiny to a two-good consumer
demand exa~nple.
2.
Max-staóle random ufilitv oro
~~ c
Let
us
start by recalling
some fundamental
properties of the
multivariate extreme value distributionl which is the basis
for GEV.
Let
-(U~. Uz. ..... Um) be a randoin vector with probability distributi0n function F(ui,up,
(2.1)
for
...,
u~,).
loyF(u~. uZ...., u~,) - e-eY
some
constant
distribution (MEV).
the
If for all y ER
univariate
a ~ 0.
logF(ui
- y, uz
- y.
...,ux, - Y)
then F is a type III multivariate extreme value
From this characterization it fOllows immediately
mdrginals
have distribution exp (-be-"~),
that
b ~ 0, whiCh
is
the type lil extreme value distribution (see Galambos, 1978).
Now consider stochastic processes
{U(x),
x
2 0}
for wl~ich the
fin,te-dimensional
distributions are
processes
called
processes
m x-stable
since it follows
from (2-1)
max-stable processes
(cf.
MEV.
de Haan,
that the
is max-stable.
This
1984).
maximum
of
class
of
is
It is denoted max-stable
independent
copies
We shall now state a very useful
of
repre-
sentation result for max-stable processes given by de Haan (1984).
Theorem
and continuous
function v(.)
1
Ide
Haan): Assume that U-{U(x),
in probability.
x 2 0}
is max-stable
Then there exists a(measurable and
finite)
and a finite measure 1` such that U has the same finite-dimen-
sional distributions as
max{v(x,T(z))
z
where
{T(z).c(z))
is
4 e(z)}
an enumeration of a Po,sson process on [0.1]xR with
;ntensity measure
a(dt)
.
i)
This
class
diStributi0ns.
e Ede.
~s
sometimes
called
the
class
of
generalized
extreme
value
3
Recall that a Po~550n process on [0.1]XR is completely analogous to
a Po~sson process on R.
coord~nates
Here the realizations occur independently and
respectively.
(T(z).E(z)),
The
probability
have
that there is a
point within
(t,tfdt)
is
x
(approximately)
( E,EtdE)
equal
to
1`(dt)e EdE
and the expected number of points within an area A c[0,1]XR is given by
A(A)
- J1`(dt)e EdE.
A
Let
us
universe is R,
individual s
now
and
the choice context.
introduce
Assume that the choice
{U(x),x20}
let K c R, denote a choice set. Let
be
the
util,ty assigned to x. U is perceived as a stochastic process
by the observer due to
opportunities
across
unobserved
heterogeneity
consumers.
We
probability (see appendix, Lemma A2).
in
preferences
shall assume that U
and
in
is continuous in
In that case it follows that
we
rnay
without loss of generality define the utility process by
U(x) - max U(x,z)
z
where
(2.2)
and
U(x.z) - v(x,T(z)) } E(z)~a
(T(z).E(z))
represents
Furthermore
we
The representation (2.2)
may think of
the points
the Poisson
is
process
Continuou5
t~iat
has an
interesting behavioral
z as an indexation
natives where T(z)
of
v(x.t)
a55ume
in
defined above.
x
for
because consumer5 differ
t.
We
of a set of latent countable choice alter-
is the attribute value assigned to z and E(z)
unobservables that affect tastes.
given
interpretation.
Specifically,
for given z, E(z)
in their taste for alternative z.
The
represent
is random
Collection
4
of
T-values is perceived as randorn because conswners face diffe-
feas~ble
rent
(latent) opportunity sets.
There are many examples
vablP chorce
i;
uften
affected
set5.
releted
by
the
From
to
that
consumer demand
the choice
ama~y
choice of job-type as
(2.2)
of the utility
Proof:
this
for
example,
d~fferent
well
as
of
interpretation
the Choice
van ants
of
non-market
of
of
unobserquantity
a product
or
distributions
U.
1
- exp{- J exp[max(av(x~,t)
~
~Sm
shall
We
case
present
the
-
aut)]1`(dt)}.
proof for the bivariate case,
is completely analogous.
Let
M - {(t,E)~E
and
~ max(av(xt,t)
- aut,av(xz,t)
- auz)}
let M denote the complement of M. Evidently
P4u(xr) s ur. u(xZ) s uZ}
- P{max(av(xt,T(z)) t e(z)) s aut, max(av(xz.T(z)) i c(z)) s auz}
z
z
- P((T(z),
e(z)) e M. vz}
- P{7here are ne points of the process
in M}
The expected number of points in M is given by
~(M)
is
activities.
The finite-dimensional distributions of U have the form
m
P{ n(U(xt)su,)}
t-1
s~nce the 9eneral
allow
we can now derive the finite-dimensional
functian,
Tlieorem 2:
(2.3)
In
- fMl`(dt)e ede
1
- J exp[a max(v(xl,t)
u
- ul.v(xz.t)
-
uz))]1,(dt).
n~-2,
5
By
the
PoiSSOn
law the probability of no point5 within M i5 given
by exp(-l1(M)) which yields the above theorem.
Q. E. D.
3. Choice nrobabilitiPc
Let
x E K,
i.e..
(3.1)
x'(K) be the randoin variable that maximizes utility subject to
x'(K) is determined by
U(x'(K))
-
where K is a Borel
m(A~K)
(3.2)
where
A
is
-
a
sup U(x)
xEK
set
P{sup
xEA
Borel
in R,.
Let ~(A~K) be the probability measure
U(x)
Sup
xEK
-
set, AcK.
U(x)}
Our first concern
is whether m is a well
def,ned choice probability.
Theorem
and
the
choice
3:
set
If
K
is
the
family
compact
{v(x,t)}
then
(indexed by
x'(K)EK
with
t)
is equicontinuous
probability one.
Moreover tlie choice probability measure is given by
P{x'(K)EA}
- m(A~K)
if A and K are compact, AEK.
sample
it
Proof:
By Lemma A2 in the Append~x equicontinuity
paths
of U are continuous with probability one.
implies that
the
Since K is compact
fellows that maximum is attained within K with probability one.
Q.E.D.
Mote that equicontinuíty follows
and
if v(x,t)
i5 dífferentidble
in
x
6
Sup
t
ThiS follOws
dv
x
dx
t
I
immedidtely 5ince
-
(v(Y.t)
v(x.t)~
S
{y-x{supldvdX
t
t
Now let
vlt.K)
-
sup
xEK
When K is compact,
v(x.t).
then SinCe v
is continuou5 there exi5ts d point within K
at which the supremum is attdined,
sdy x(t.Y,).
Thus
v(x(t.K),t) - v(t.K).
We are now ready to state the following theorem.
Theorem 4:
m(A~K)
-
The choice probability measure is given by
K)exp(av(t,K))a(dt)
fó exp(av(t,K))~(dt)
JQ(A
where A and K are Borel
Q(A,K)
-
{t
~
sets,
v(t,A)
AcK,
and
- v(t.K)}.
The proof of this theorem is given in the appendix.
Corollarv 1: A55ume that x(t.K)
one-to-one
is Continuously differentidble
as a function of t for t in the interior of S2(K,K).
let 1` be absolutely continuous with respect to the Lebesgue
the density of in exists and is given by
~p(x~K)
-
exp(av(x,t(x.K))1`'(t(x,K))at(x,K)lax
j~ exp(av(t,K))a'(t)dt
and
Furthermore
measure.
Then
7
where t(x.K)
is
Proof:
the
inverse
The
conditions
ously differentiable.
(3.3)
m(B~K)
function
x(t,K)
of
and
x
w.
istif.he
interior
imply tliat t is welldefined and
Put 8-[xt~x,x].
of
K.
is continu-
Then by Theorem 4
- ~x exp(av(x,t(x,K)))a"(t(x,K))at(x,K)~ax
fó exp(av(t,K))1`~(t)dt
. o(px)
since
v(t(x,K),K)
Hence
dividing
- v(x,t(x,K)).
both
sides
of
(3.3)
by ~x
and
letting px - 0
the corollary
follows.
Q.E.D.
Since
the
opportunity
the measure a is not
identified.
lity by letting 1,(t)
- t.
Note that
al~nost
everywhere
correspond5
if
here is typiCally unObserved
Consequently there is no loss
there exists a Borel
( 1`) then 4(A,K)
to "shadowing"
(almost certainly)
vdridble T(z)
set
A
of
genera-
for which v(t,A) ~ v(t,K)
becomes empty and thus m(A~K)
in the terminology of Cosslett:
- 0.
A p0int
This
in A is
never chosen because A is in the " shadow" of some nearby
peak of the function v.
An
interestíng
question
"revealed preference type"
next
result
specifies
Theorem 5:
of
is
whether
a nonparametrically
the
sirnilarly
testable
Let K~ and Kz be co~~~pact Borel
dis~oint compact Borel sets that
v(x.t)
is
properties
lie in the
continuous GEV
to
satisfies
model
The
property.
sets and let A~
interior
strictly quasiconcave in x for fixed t
the Luce
of
K1nKz.
and Az be
Then
if
8
N(A~~K~)
m(ql(Kz)
m(Az~K~)
- ~(Az~Kz)
Since
v(x,t)
the interior of K it
is strictly quasiconcave in x then if x(t,K)
is determined by local
criteria.
Therefore
lies in
x(t,K)
as
as v(t.K) are independent of K, and it follows that also 4(A,K)
inis
dependent of K if A lies in the interior of K.
Consequently, by Theorem
4 the ratio
well
m(A~~Kt)
m(AzIKt)
is
independent
of
K~
Q.E.D.
This
cally
,t
tions".
Theorem represents a relaxation of the
states tliat IIA holds for all
that
IIA property.
choices that are not
Specifi-
solufor alternatives that are contained in the boundary of the
is.
"corner
choice set.
4.
The continuout
rrcP mod 1
In
this
section we demonstrate that tl,e continuous Luce ~nodel is
consistent with the maximization of a random utility function without
applying a finite choice set type of approximation.
Let now T(z) be a(continuous) observable attribute assigned to tl~e
Tliis universe is assumed to consist
alternative z in the choice univer5e.
of
a countable collection of alternat~ves.
The utility function is defined
by
(4.1)
where
~(Z)
v(.)
-
v(T(z))
is
t E(Z)~d
continuous and
{T(z),E(z))
process on [0.1]XR with inten5ity medsure
1`(dt)
.
e EdE
are the points in the Poisson
9
Tl,e corresponding choice probability is
P{T'(K)EA}
where
T'(K)
is
tl,e
optimal
choíce
of
T from K and A,K are Borel
sets,
AcKc[0.1]. Analogously to the preceeding section define
m(A~K)
- P{sup
(av(T(z))
T(z)EA
Theorem
~:
P(T`(K)EA)
t e(z))
- sup
(av(T(z))
T(z)EK
t e(z))}.
We have
- m(AlK)
and
JAexp(av(t))a(dt)
m(A~K)
-
Kexp(av(t))7,(dt)
where A and K are Borel
Proof:
sets, AcKc[U.1].
By substituting the domain [U,1] of a by a( Borel) set B
we
get by Theorem 2 with m-1 that
(4.2)
P{max
U(z)
T(Z)EB
Since
~
and
S u} - exp(-e-a~ jBea~it)l,(dt)),
K-A
are
disjoint
sets and
the realizations of the
Poisson process are stochastically independent we have that
sup
U(z)
T(z)EA
are indeoendent.
and
sup
U(z)
T(z)EK-A
Since they by
(4.2) also are extreme value distributed
it
10
follows readily (see for instance Maddala.
1983)
that
jA exp(av(t))~(dt)
P{
1(z)EAD(~) ~ T(z)eK-AD(z)) -
exp(av(t))a(dt)f
A
K-A
exp(av(t))1~(dt)
Q.E.D.
From Theorem 5 it follow5 immedidtely
Corollarv 2: Assume that ~
the Lebesgue measure.
is absolutely continuous with respect to
Then the probability density of in exists and is given
by
,-0(t~K)
oa~(t)a"
~av(t)
(t)
-
fKe
Corollarv 3:
perty
tEK.
1`"(t)dt~
The choice probabilities of Theorem 6 satisfy the pro-
independence frnm irrelevant alternatives.
Let
9(t)
- 1,'(t)Il~(1)
The function g represents
(randon~ly selected)
.
the
density
decision-maker faces.
of the choice opportunities that a
The probability density g can
be
expressed as
(4.3)
~~(tIK) -
fKe
eav(t)9(t)
av(t) g(t)dt
With separate information about the distribution of the opportunity variables T(z)
we realize that it is possible to identify g(nonparametrically).
11
5
Justification
of
max-stable
the
utilitv
function
from theoretical
assumotions
Here
present
space
latent
ef
mapping T:S -[0,1],
that
that
imply
the max-stable utility
alternatives.
z- T(z).
assume
a
coun-
S. Also suppose that there exists a
The interpretation is that T(z)
is an
index
(unobservable) qualitative characteristic5 of alternative
summarizes
z. Let
assumptions
In addition to the continuous choice variable,
structure.
table
we
PB(~~x)) be a probability space
([U.1].~,
where ~
is
the
Borel
field and
PB(A~x)
for
A,BE~.
- P{sup
U(x,z)
T(Z)EA
Ac8
and
where
interpretation of PB(A~x)
U(x
.z")
and U(x,z)
Assumption
1:
is the utility function of (x,z). The
U(x,z)
í5 as the
alternative z witl, T(z)EA when x
9ccumotion
- sup
U(x,z)}
T(Z)EB
probability
of
choosing
the
latent
is given and the choice set for T is B.
For each x. U(x,z), z- 1,2,
are i.i.d.
Furthermore
are independent when z~ z".
1
states
that to the observer all
latent alternatives
apart from purely random disturare "orthonor,nal" and "look the sa~ne"
is,
there are no hierarchical difference between the latent
bances.
That
alternatives.
A~~~mo ion 2:
PB(A,Ix)
A1cAzc8 then
- PB(Az~x)PAZ(Ailx)
P[U 1](A~x)
Moreover
If A1, AZ, BE~,
is
absolutely
continuous
with respect to a finite
measure ~(say).
We
PB(A~x)
recognize
this
assumption as a version of IIA.
~s assumed to be a conditional probability measure.
Let us now consider the implications of Assumptions
structure of the utility function.
Let {Al} be a finite partition of [0,1], AIE~.
In other words
1 and 2 for the
From ASSUmption 2
it
12
follows that there ex~sts
a
measure.
p(~,x),
that
is
proportional
to
PB(A~~x) such that
p(A~.x)
PB(AIIx)
-
F V(A „ x)
A~EB
By A55uinption 1 and from Yellott
(5.1)
max
U(x,z) Q log p(Aj,x)
T(z)EA~
where Q means equality
drawS
fr0in
(1977)
t n~(x)
in distribution and n~,
ex p(-e n ).
it follows that
Since p(..x)
j-1,2,...,
are
independent
is absolutely continuous with respect
to 1` we nave
p(Al,x)
- f exp(av(x,t))~(dt)
A~
for some function v(x,t).
ThuS (5.1)
implies tliat
{max
U(x,z) S u} - exp(-e-au f exp(av(x,t))h(dt))
r(z)Ea
A
for A ~.
(5.2)
where
But this means that we have
max
U(x.z) Q max
{v(x.T(z))
T(z)EA
T(z)EA
{T(z),c(z)},
z-1,2,..,
is
i Eaz
}
an enumeration of the points in a Poisson
process on [0,1]XR with intensity measure a(dt)
holds let us
m(t)
for notational
-
~~
-m
-
v(x.t)
Using th~s notation we get
convenience define
for
tEA
otherwise.
. e Ede.
To see that
(5.2)
13
P{max
(v(x,T(z))
T(z)EA
t Eaz
)
~ u}
- P{E(z) s au - av(x,T(z)), vz. T(z)eA}
-
P{
There are
m(.) }
no
points
- exp(- f
ae-dede
m(t)~e
which proves (5.2).
Eq.
-
of
the
Poisson
process
implies that
. E z
a
U(x,z) Q v(x,T(z))
(2.2).
We state this below.
We have thus provided a set of assumptions
Theorem 7-
5
of
1`(dt)) - exp(-e-au j exp(av(x,t))a(dt)).
A
(5.2)
(5.3)
distributed as
above the graph
If Assu~nptions
that are consistent with
1 and 2 hold then the utility function is
(5.3)
Aoolication
of
the
GEV framework to consumer-demand.
continuous
The
two-qQod case.
Let U'(x,y,z) be the utility function of the three
where
(x,y)ERt
variable z is
alternatives
are
interpreted
that
and
continuous
affect
as
an
goods
(x,y,z),
z is countable and unobservable.
indexation
of
all
unobserved
The
choice
the preferences for the observable goods (x,y).
Assu~ne that
U"(x.y.z)
where {T(z).e(z)}
defined above.
-
v"(x.y.T(z))
are the
points
t
E(z)la
of
the
bivariate
Let the budget constraint be given by
px 4 qy - 1
Poisson
process
as
19
where p and q are the respective normalized prices.
Define
U(x,z)
- U'(x,(1-px)Iq,z)
v(x,t)
- v'(x,(1-px)Iq,t).
and
Let
us
assume
that v'(x,y,t)
for fixed t is
quasiconcave and continuously differentiable.
x
conditional
on
~
is
well
defined
dnd it
unconStrained demand function Conditional on z,
the aPPlication of Roy~s
found
by
follows
the
application
that
the
of
4.
for
The
x(T(z),R4), can be found by
utility function
the unconditional
Theorem
strictly
is given by x(T(z),K).
identity to the indirect
v(T(z),R4) and the distribution of
increasing,
Then the demand function
demand,
x'(R{),
i5
Under the assumption above it
corresponding constrained demand distribution satisfies
Theorem 5.
From
Theorem
4
it
follOws
also
when
that
a
-
0 then the
distribution m reduce5 t0 the "conventional" demdnd distribution
which
i5
expressed as
lim m(A~K)
a - o
i.e.,
- f
Q(A,K)
~ ~ dt
( 0.1
}
the demand distribution equals the probability mass of all
{T(z)} for
which the Marshallian demand x(T(z),K)eA.
Tl,e
intuition
is as
follows.
From (2.2) we see
that when a
then the effect of the "systematic" part v(x.T(z)) will
a value of a near zero the optimal value of z.z.
the
sequence
{C(z)}
which
utility mdximization problem when T(z)
z
is independent of the rest
function
it
follows
that
of
the
{T(z)}
be
Given z,r.`
is purely random.
maximizing v(x,T(z)) with respect to x.
would
be small.
But this
is given.
is
a
is small
Tlius
determined
for
by
is determined by
standard
textbook
Since the determination of
variables
can be viewed
that
enter
the
utility
as a draw from the same
distribution as tl,e one that generated the sequence {T(z)}.
Thus the GEV framework
contains
the
standard
textbook
consumer
15
demand
model
econometric
int?rpreted
as
a
as
a
conventional
special
case
where
now
{T(z)}
may be
heterogeneity parameter with distribution
funct,on
a
a(
t
U.1
)
An examole
Let the conditional
v(t.R~)
(unconstrained)
' c(l~q)
- -tb(P~q)
where b'(x) ~ 0,
~ 0,
c'(x)
b'~(x)
By the application of Roy~s
x(t)
'
indirect utility have the form
~ 0.
identity we get
t
D(D.q)
where
C
D(p.q)
-
(q)
b~lp)
Hence
exp(v(t(x),R~))
- exp(-bxD(p.q) ; c(llq))
from Corollary 1 we get the density of the demand:
tD(x~R,) -
exp(-abxD(P,q))~~(xD(D.q))B(P,q)
fpexp(-abt)7`(dt)
16
Auoendix
Lemma
A1:
Assume that f and g are two bounded functions on [0,1].
Let
Ur ' max (f(T(t)) ' c(z)). UZ - max (9(T(z)) t c(t))
z
z
where
{T(z),c(z)}
is
an enumeration of a Poisson process on (U,1]XR with
intensity measure
a(dt)
. e ede.
If f~ g almost everywhere a then U~
a.e.,
then U1
) Uz
with probability
~ Uz with probability zer0.
Prpof:
From Theorem 2 it follow5
that when f~ g
P(UrSu1.U25uz} - P(U~Su~}
for urSuZ whiCh is eyuivalent t0
P{Ursu~,UZ~uz} - 0.
But then
P(Ur~Uz)
- 1.
Assume next that fsg a.e.a.
Let
A ` itif(t) - 9(t)}
and define
Urr
-
max(f(T(z))
F E(z)).
T(t)EA
Uo
-
max(f(T(z))
T(z)EA
UZr
-
inax(g(T(Z))
T(2)EA
}
e(z)).
t
t(t))
one.
If
f S g,
17
Where A ~s the complement of A.
We have
P(Ut~Uz) - P{(Utt ~ max (Uzt.Uo))U (Uo ~ max ( Uzt,Uu))}
- P(Utt~ max (Uzt.Uo)) s P(Utt ? max (Uz1,Ua))
`- P(Uit
On Á
1-P(Ult
' Uzt) -
f)g a.e.. which
'. U21).
implies that U11~Uzt with probability one.
Thus U1~Uz
witli probability zero.
p.E.D.
l-emina
t)
A2:
If tlie family of functions {v(x,t),tE[0,1~}
( indexed by
i5 equiCOntinuou5 almost everywhere 1, then
U(x)
- max(v(x,T(z))
z
' E
z
)
a
is cOntinuous with probability one.
Proef:
~U(x)
whiCh
hOldS
Censider the event
- U(y)~
~ Ó
if
max(v(x,T(z)) ' b' E z ) ~ max(v(y,T(z)) ' E z)
a
z
a
z
and
max(v(y,T(2))
Z
'
b'
Z)~
E
d
mdx(v(x.T(Z))
Z
'
E
Z).
d
By applying Lemma A1 we realize that for x and y sufficiently close the two
inequal;ties above hold with probability one since by equicontinuity there
exists a K~o such that
18
~v(x.t) - ~(y.t)I ~ b
w~~en Ix-ylt K
axrent on a set of 7~ measure zero.
O.E.D.
Proof of Theorem ~:
Let
(A.1.)
fi(t)
-
amax
xEA
v(x.t)
-
av(t.A),
(A.Z.)
fz(t)
-
amax
xEK-A
v(x.t)
-
av(t.K-A),
AcK,
and
E
-
{t~f~(t)
? fz(t)).
The event
{max(fl(T(z))
z
` eíz))
? max(fz(T(z))
z
t e(z))}
~ e(z))
? max(fz(T(z))
t e(z))}
is equivalent to
(A.3.)
(max(fl(T(z))
T(z)EE
U{mdxÍfl(T(z))
z
{
EÍz))
?
max(fz(T(Z))
T(z)EE
where E is the complement of E.
{max(f~(T(z))
T(z)EE
t
C(z))}
z
' E(z))
Moreover
) mdx(fz(T(z)) t C(z))}
z
19
c{max(fi(T(z)) ' E(z)) y max(f2(T(z)) ' E(z))}.
T(z)EÈ
T(z)EE
ay
when tEE.
(A.4.)
A1
Lemma
th? lact event has probability zero since f,(t)Sf,(t)
Similarly
{max(fi(T(z)) ' E(z)) ~ maxlfz(T(z)) ' E(z))}
T(z)EE
z
~-, {max(f~(T(z)) ; E(z)) y max(fz(T(z)) ' e(z))}
T(z)EE
T(z)EE
because
{max(fi(T(z)) 4 e(z)) ~ max(fz(T(z)) ~ E(z))}
T(z)EE
T(z)EE
has
probability one by Lenxna A1.
frpm theorem 2
Since EnE - 0 it follows directly
that
max (f~(T(z)) 4 E(z))
and max(fz(T(z))
T(z)EE
t e(z))
7(z)EE
are stochastically
independent with joint distribution function
exp{-e u~fexp(fi(t))~(dt)
- e uzfexp(fz(t))~(dt)).
E
E
Hence by straight forward calculus and (A.1.)
P{max(f~(T(z))
~
E(Z))
~
~
max(fz(T(z))
to (A.4.)
t
JEexp(f~(t))a(dt)
Jexp(f~(t))a(dt) ~ fexp(fz(t))1`(dt) -
E
E(z))}
z
É
fQ(A K)exp(av(t,K))1`(dt)
J~exp(av(t,K))a(dt)
Q.E.D.
zo
Reference~
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Choice:
N.
Densities".
Cosslett,
S.R.
Utility
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Galambos,
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Judgment and the Double Exponential
15.
109-146.
Discussion Paper Series, CentIIt, Tilburg University, The Netherlands:
No.
Author(s)
Title
8801
Th. van de Klundert
and F. ven der Plceg
Fiscal Policy and Finite Lives in Interdependent Economies with Real and Nominal Wage
Rigidity
8802
J.R. Magnus and
B. Pesaran
The Bias of Forecasts from a First-order
Autoregression
8803
A.A. Weber
'1'he Credibliity cf Monetary Policies, Policymakers' Reputation and the EMS-Hypothesis:
Empirical Evidence from 13 Countries
8804
F. van der Plceg and
A.J. de Zeeuw
Perfect Equilibrium in a Model of Competitive
Arms Accumulation
8805
M.F.J. Steel
Seemingly Unrelated Regression Equation
Systems under Diffuse Stochastic Prior
Information: A Recursive Analytical Approach
8806
Th. Ten Raa and
E.N. Wolff
Secondary Products end the Measurement of
Productivity Growth
880~
F. van der Ploeg
Monetary and Fiscal Policy in Interdependent
Economies with Capital Accumulation, Death
and Population Growth
8901
Th. Ten Rsa and
P. Kop Jansen
The Choice of Model in the Construction of
Input-Output Ccefficients Matrices
8902
Th. Nijman and F. Palm
Generalized Least Squares Estimation of
Linear Models Containing Rational Future
Ezpectations
8903
A. van Soest,
I. Woittiez, A. Kapteyn
Labour Supply, Income Taxes and Hours
Restrictions in The Netherlands
8904
F. van der Plceg
Capital Accumulation, Inflation and LongRun Conflict in International Objectives
8905
Th. van de Klundert and
A. van Schaik
Unemployment Persistence and Loss of
Productive Capacity: A Keynesien Approach
8906
A.J. Markink and
F, van der Ploeg
Dynamic Policy Simulation of Linear Models
with Rational Expectations of Future Events:
A Computer Package
890~
J. Osiewalski
Posterior Densities for Nonlinear Regreasion
with Equicorrelated Errors
8908
M.F.J. Steel
A Bayesian Analysis of Simultaneous Equation
Models by Combining Recursive Analytical and
Numerical Approaches
No.
Author(s)
Title
8909
F. van der Ploeg
Two Essays on Political Economy
(i) The Political Economy of Overvaluation
(ii) Election Outcomes and the Stockmarket
8910
R. Gradus and
A. de Zeeuw
Corporate Tex Rate Policy and Public
and Private Employment
8911
A.P. Barten
Allais Characterisation vf Preferer.ce
Structures and the Structure of Demand
8912
K. Kamiye and
A.J.J. Talman
Simpliciel Algorithm to Fínd Zero Points
of a Function with Special Structure on a
Simplotope
8913
G. van der Laan and
A.J.J. Talman
Price Rigídities and Rationing
8914
J. Osiewalski and
M.F.J. Steel
A B~yesian Analysis of Exogeneity in Models
Pooling Time-Series and Cross-Section Data
8915
R.P. Gilles, P.H. Ruys
and J. Shou
On the Existence of Networks in Relational
Models
8916
A. Kapteyn, P. Kooreman
and A. van Soest
Quantity Rationing and Concavity in a
Flexible Household Labor Supply Model
8917
F. Canova
Seasonalities in Foreign Exchange Markets
8918
F. van der Ploeg
Monetary Disinflation, Fiscal Expansion and
the Current Account in an Interdependent
World
8919
W. Bossert and
F. Stehling
On the Uniqueness of Cardinally Interpreted
Utility Functions
8920
F. van der Plceg
Monetary Interdependence under Alternative
Exchange-Rate Regimes
8921
D. Canning
Bottlenecks end Persistent Unemployment:
Why Do Booms End?
8922
C. Ferahtman and
A. Fishman
Price Cycles end Booms: Dynamic Search
Equilibrium
8923
M.B. Canzoneri and
C.A. Rogers
Is the European Community en Optimal Currency
Area? Optimal Tax Smoothing versus the Cost
of Multiple Currencies
8924
F. Groot, C. Withagen
and A. de Zeeuw
Theory of Natural Exhaustible Resources:
The Cartel-Versus-Fringe Model Reconsidered
8925
O.P. Attanasio and
G. Weber
Consumption, Productivity Growth and the
Interest Rate
No.
Author(s)
Title
8926
N. Rankin
Monetary and Fiscal Policy in a'Hartian'
Model of Imperfect Competition
8927
Th. van de Klundert
Reducing External Debt i n a World with
Imperfect Asset and Imperfect Commodity
Substitution
8928
c. Dang
The D1-Triangulation of Rr~ for Simpliciel
Algorithms for Computing Solutiona of
Nonlinear Equations
8929
M.F.J. Steel and
J.F. Richard
Bayesien Multivariate Exogeneity Analysis:
An Application to a UK Money Demand Equation
8930
F. van der Ploeg
Fiscal Aspects of Monetary Integration in
Europe
8931
H.A. Keuzenkamp
The Prehistory of Rational Expectations
8932
E. van Damme, R. Selten
and E. Winter
Alternating Bid Bargaining with a Smallest
Money Unit
8933
H.
E.
Global Payoff Uncertainty and Riak Dominance
8934
H. Huizinga
National Tax Policies towards ProductInnovating Multinational Enterprises
8935
c. Dang and
D. Talman
A New Triangulation of the Unit Simplex for
Computing Economic Equilibria
8936
Th. Nijman and
M. Verbeek
The Nonresponse Bias in the Analysis of the
Determinants of Total Annual Expenditures
of Households Based on Panel Data
8937
A.P. Barten
The Estimation of Mixed Demand Systems
8938
G. Marini
Monetary Shocks and the Nominal Interest Rate
8939
w. Guth and
E. van Demme
Equilibrium Selection in the Spence Signaling
Game
8940
G. Marini and
P. Scaramozzino
Monopolistic Competition, Expected Inflation
and Contract Length
8941
J.K. Dagsvik
The Generalized Extreme Value Random Utility
Model for Continuous Choice
Carlsson and
van Damme
P.O. BOX 90153, 5000 LE TILBURG. THE NETHERLAN
Bibliotheek K. U. Brabant
~i ~r ~~u~~w~~~~M i~m