Advances in Mathematical Economics

201 Pages·2003·6.254 MB·English

by  Guillaume Carlier (auth.), Shigeo Kusuoka, Toru Maruyama (eds.)

Advances in
MATHEMATICAL
ECONOMICS
Managing Editors
Shigeo Kusuoka Toru Maruyama
University ofTokyo Keio Univeristy
Tokyo,JAPAN Tokyo, JAPAN
Editors
Robert Anderson Jean-Michel Grandmont Norio Kikuchi
University of California, CREST-CNRS Keio Univeristy
Berkeley Malakoff, FRANCE Yokohama, JAPAN
Berkeley, U.S.A. Norimichi Hirano Hiroshi Matano
Charles Castaing
Yokohama National Universityof Tokyo
Universire Montpellier II University
Tokyo, JAPAN
Montpellier, FRANCE Yokohama,JAPAN
Frank H. Clarke Leonid Hurwicz Kazuo Nishimura
Universite de Lyon I University of Minnesota Kyoto University
Villeurbanne, FRANCE Minneapolis, U.S.A. Kyoto, JAPAN
Gerard Debreu Tatsuro Ichiishi Marcel K. Richter
University of California, Ohio State University University of Minnesota
Berkeley
Ohio, U.S.A. Minneapolis, U.S.A.
Berkeley, U.S.A.
Alexander loffe Yoichiro Takahashi
Egbert Dierker
Israel Instituteof Kyoto University
University of Vienna
Technology
Kyoto, JAPAN
Vienna, AUSTRIA
Haifa, ISRAEL
Darrell Duffie Michel Valadier
Seiichi Iwamoto
Stanford University Universite Montpellier II
Kyushu University
Stanford, U.S.A. Montpellier, FRANCE
Fukuoka, JAPAN
Lawrence C. Evans
Kazuya Kamiya Akira Yamazaki
University of California,
University of Tokyo Hitotsubashi University
Berkeley
Tokyo,JAPAN
Tokyo, JAPAN
Berkeley, U.S.A.
Takao Fujimoto Kunio Kawamata Makoto Yano
Kagawa University Keio Univeristy Keio Univeristy
Kagawa, JAPAN Tokyo, JAPAN Tokyo, JAPAN
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Springer
Tokyo
Berlin
Heide/berg
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London
Milan
Palis
s.
Kusuoka, T. Maruyama
(Eds.)
Advances in
Mathematical Economics
Volume 5
i
Springer
Shigeo Kusuoka
Professor
Graduate School of Mathematical Seiences
University ofTokyo
3-8-1 Komaba, Meguro-ku
Tokyo, 153-0041 Japan
Toru Maruyama
Professor
Department of Economics
Keio University
2-15-45 Mita, Minato-ku
Tokyo, 108-8345 Japan
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Table of Contents
Research Articles
G. Carlier
Duality and existence for a dass of mass
transportation problems and economic applications 1
C. Castaing, A. G. Ibrahim
Functional evolution equations governed by
m-accretive operators 23
T. Fujimoto, J. A. Silva, A. Villar
Nonlinear generalizations oftheorems on
inverse-positive matrices 55
1. Hurwicz, M. K. Richter
Implicit functions and diffeomorphisms without Cl 65
1. Hurwicz, M. K. Richter
Optimization and Lagrange multipliers: non-C'!
constraints and "minimal" constraint qualifications 97
S. Kusuoka
Monte Car10method for pricing of Bermuda type
derivatives 153
Historical Perspective
I. Mutoh
Mathematical economics in Vienna between the
VVars 167
Subject Index 197
Adv.Math. Econ.5, 1-21(2003) Advancesin
MATHEMATICAL
ECONOMICS
©Springer-Verlag2003
Duality and existence for a dass of mass
transportation problems and economic
applications
Guillaume Carlier
Universite Bordeaux I, MAB, UMR CNRS 5466 and Universire Bordeaux IV,
GRAPE,UMRCNRS5113,AvenueLeon Duguit,33608, Pessac,FRANCE
(e-mail:[email protected])
Received:April 15,2002
Revised:May 20,2002
JELcIassification:C61,C82
MathematicsSubjectClassification (2000): 90C08, 90C46, 91B40
Abstract We establish duality, existence and uniqueness results for a class of mass
transportations problems. We extend a technique of W.Gangbo [9] using the Euler
Equationofthedualproblem.Thisisdone byintroducingthe h-FenchelTransformand
usingitsbasicproperties.The costfunctionsweconsidersatisfyageneralizationofthe
so-calledSpence-Mirrleesconditionwhichiswell-known byeconomistsindimension
I.Wethereforeend thisarticle byasomehow unexpectedapplication tothe economic
theoryofincentives.
Key words: mass transportation,duality,generalFencheltransform, economictheory
ofincentives, Spence-Mirrleescondition
1. Introduction and main statement
1.1 Assumptionsand notations
Let us first recall that, given a probability space (Ol,Al,lll), a measurable
space (02,A2) and a measurable map f :01 --> O2, the push-forward of 111
through f,denoted fUlll istheprobability measureon (02,A2) defined by:
forevery B EA2.
S. Kusuoka, et al. (eds.), Advances in Mathematical Economics
© Springer-Verlag Tokyo 2003
2 G.Carlier
In allthefollowing, nissome boundedconnectedopen subset ofIRn, and
J-l is some probability measure in n which is absolutely continuous with re
specttothen-dimensionalLebesguemeasure, withapositiveRadon-Nikodym
derivative with respect to the n-dimensional Lebesgue measure and such that
J-l(Bn) = O.
Weare also givenacompactPolish space Y,aRadonprobabilitymeasure
v on Y andafunction h :0 x Y --+ IRwhich satisfies:
hE 0°(0 x Y,IR), (I)
foreverywce nthere exists c(w)> 0such that forall (Xl,X2) Ew2
sup Ih(XI'y) - h(X2,y)1 :::; c(w)llxI - x211, (2)
yEY
forallYE Y, h(.,y)isdifferentiableinnandforall (YI'Y2,x) E y2Xn
Bh Bh
Bx(x,Yd = Bx(x,Y2)=> YI =Y2· (3)
Assumption (3) playsan importantrole intheproofs and we shall seethat
it maybe interpretedasageneralization ofthe well-known oneofSpence and
Mirrlees, thisassumption wasfirstintroducedbyLevin in[13].
Our aim istostudy the following Monge's mass transportationproblem:
r
(M) sup J(s) := h(x,s(x))dJ-l(x)
Jo
sEtl(J.!,v)
with:
D..(J-l,v) := {s isaBorel map :n--+ Y s.t.sttJ-l = v}.
The associated Monge-Kantorovich problem is the linear (relaxation of
(M)) program:
r
(MK) sup K('y):= h(x,y)d'Y(x,y)
, Er(J<,v) JOXY
with:
r(J-l,v) := bisaBorel probabiliymeasure onnxY S.t.1Tlh = J-l, 1T2h = v}
where1TI(x,y) = x,1T2(X,y) = Yforall (x,y) E nx Y.
Finally, wedefinethe (dual of (M)) problem:
with:
Eh := {(vJ,c/» ,real-valued measurables.t,vJ(x)+c/>(Y) ~ h(x,y),V(x,y) E nxY}.
Dualityandcxistcncc foradassofmasstransportationproblems 3
1.2 Main result
If 'l/J is a given real-valued function defined on n,we define the h-Fenchel
Transforrnof'l/J,'l/Jhby:
'l/Jh(y):= suph(x,y)- 'l/J(x), forall y E Y.
xEIl
In a similar way, if r/J is a given real-valued function defined on Y, we define
the h-FenchelTransforrn ofr/J,r/Jhby:
r/Jh(x):= sup h(x,y) - r/J(y), forallx E n.
yEY
Our mainresult can then bestatedasfolIows:
Theorem 1 Underassumptions(1),(2),(3)thefollowingassertionshold:
1)problems(M),(M K) and(V )admitatleastonesolution,
2) (V )isdualto(M) and(M K) inthesense:
inf'(D) = sup(M ) = sup(MK),
3) theminimumin(V) isattainedbyapair(~,([» suchthat:
- -h - - h
l/; = r/J ,r/J ='l/J
thereexistsmoreoversomeBorelmapsfromnto Y whichsatisfies:
~(x) +([>(s(x)) = h(x,s(x)),jorallx E n,
s E/:'(J-l,v)andisasolution of (M ),and(id,s)UJ-l isasolutionof(M K),
4) uniqueness also holds: if s is a solution of(M ) then s = s u-a.e.,
(id,s)UJ-l is the unique solution of(M K), and if ('l/J,r/J) is a solution of(V)
thenib- ~(respectivelyr/J - ([» isequaltosomeconstantu-a.e.(respectively
v-a.e.).
In Section 2, technical lemmas are established and basic properties of the
h-Fencheltransforrnareproved.InSection3,the mainresult isproved.Finally,
inSection4,weadressaquestion arising intheeconomictheory ofincentives
andshow how assumption(3) canbe interpreted as anaturalgeneralizationof
the Spence-Mirrlees condition. In this framework, our main result enables to
proveageneralre-allocation principle.
The problem ofoptimal measure preserving maps (M) has received a lot
of attention since related questions naturally arise in fluid mechanics [2], dif
ferential geometry (see [16]for relationwith aclassical resultof Aleksandrov
4 G.Carlier
[1]), shapeoptimization[4],functionalanalysis[11], [12],probability[19]and
economics.InthecaseY C IRn and h(x,y) = x'y,theproblemwassolvedby
Brenier [3] who proved the important Polar Factorization Theorem and exis
tenceand uniquenessofanoptimal map which isthe gradientofsomeconvex
potential.This resultwasthen extendedbyMc Cann and Gangbo[10] forcosts
of the form c(x - y)with cstrictlyconvex.The result stated inTheorem 1,is
very much inthat spiritsince itexpressesexistenceand uniquenessof anopti
mal allocation map which isameasurableselectionofthe h-subdifferential of
some h-convex potential. Similarcharacterization results were obtained by V.
Levin [13] using a different approach based on cyclical monotonicity and the
relaxed problem (MK).
2. Technical preliminaries andh-Fenchel transform
In what follows7/J will alwaysdenotesome function :0 -t IRU{+00}and </>
somefunction :Y -t IRU{+oo}.
Definition1 J) 7/J is h-convex ifandonlyifthereexistsanonemptysubset A
ofY x IRsuchthat:
7/J(x)= sup h(x,y)+t,foralt xE O.
(y,t)EA
2) </>ish-convexifandonlyifthereexistsanonemptysubsetB of O x IRsuch
that:
</>(y)= sup h(x,y)+t,foralty E Y.
(x,t)EB
Remark. If 7/J is h-convex then either 7/J is identically +00 or it is bounded.
Notealso that finite k-convexpotentialsare l.s.c, hence z--measurable. __
Definition2 I) The h-Fenchel Transformof7/J, 7/Jh, is the h-convexfunction
definedby:
7/Jh(y):= sup h(x,y)- 7/J(x),for alty E Y.
xE!!
2)Thek-FenchelTransformof </>, </>h istheh-convexfunctiondefinedby:
qi (x) := suph(x,y)- </>(y),for altx EO.
yEY
Obviously,Young'sinequalitieshold:
7/J(x)+7/Jh(y) ~ h(x,y), for all (x,y) E 0 x Y (4)
and:
</>h(x)+</>(y) ~ h(x,y), for all (x,y)E 0 x Y. (5)
Duality andexistence foradassofmasstransportation problems 5
Proposition 1
(7j;h)h(x) = sup{J(x):f :S 7j;. fis h-convex},foraltx Efl,
(qi)h(y) = sup{g(x) :g :S 4>. fis h-convex},foralty E Y.
Itfoltowsthat7j;(respectively4» ish-convex(respectivelyh-convex)ifandonly
if1j; = (7j;h)h (respectively4> = (4)h)h).
Proof.
First (7j;h)h is h-convex and Young's inequality yields (1j;h)h :S 7j; so that,
ifwedefine:
V(x) := sup{J(x) :f :S 7j;, f ish-convex },for allx Efl, (6)
then :
(7)
Since V ish-convex, there exists anonemptysubset Aof Y x IRsuch that:
V(x)= sup h(x,y)+t,foraII xEfl.
(y.t)EA
Let (Yo,to) E Aand ~ := h(.,Yo) + towe have:
J/J ~ ~ =? (7j;h)h ~ (~h)h (8)
ofcoursef ~ (~h)h andsincee(Yo) = - tothen (~h)h(x) ~ h(x,yo)+to =
~(x) for all x so (~h/' = ~, with (8) we get (7j;h)h ~ ~ and since (Yo,to)
is arbitrary in A taking the supremum yields (J/Jh/, ~ V so that V = (7j;h)h
using (7).The characterization of (4)h)h isproved inthesame way.
o
Definition3 I)Define,for altx E fl:
{)h7j;(X):= {y E Y:7j;(x')- J/J(x)~ h(x',y) - h(x,y), foralt x' E fl}
{)h7j;(x)iscaltedtheh-subdifferentialof7j; atz,and1j; is h-subdifferentiable
atx ifandonlyif{)hJ/J(x) -I- 0.
2)Define,for alty E Y:
{)h4>(Y):= {x En:4>(y')- 4>(y) ~ h(x,y') - h(x,y), foralty' E Y}.