Advances in MATHEMATICAL - Nguyễn Thành An · PDF fileAdvances in MATHEMATICAL ECONOMICS Managing Editors Shigeo Kusuoka University of Tokyo Tokyo, JAPAN Akira Yamazaki Meisei University

Advances in

MATHEMATICAL ECONOMICS

Managing Editors

Shigeo Kusuoka University of Tokyo Tokyo, JAPAN

Akira Yamazaki Meisei University

Tokyo, JAPAN

Editors

Robert Anderson University of California, Berkeley

Berkeley, U.S.A. Charles Castaing

Universite Montpellier II

Montpellier, FRANCE Frank H. Clarke

Universite de Lyon I

Villeurbanne, FRANCE Egbert Dierker

University of Vienna Vienna, AUSTRIA

Darrell Duffie Stanford University Stanford, U.S.A.

Lawrence C. Evans University of California, Berkeley

Berkeley, U.S.A. Takao Fujimoto

Fukuoka University Fukuoka, JAPAN

Jean-Michel Grandmont CREST-CNRS

Malakoff, FRANCE

Norimichi Hirano Yokohama National University Yokohama, JAPAN

Leonid Hurwicz

University of Minnesota

MinneapoUs, U.S.A.

Tatsuro Ichiishi Hitotsubashi University Tokyo, JAPAN

Alexander loffe Israel Institute of Technology Haifa, ISRAEL

Seiichi Iwamoto Kyushu University

Fukuoka, JAPAN

Kazuya Kamiya University of Tokyo

Tokyo, JAPAN

Kunio Kawamata Keio University

Tokyo, JAPAN

Norio Kikuchi Keio University

Yokohama, JAPAN

Tom M aniyama Keio University

Tokyo, JAPAN

Hiroshi M atano University of Tokyo

Tokyo, JAPAN

Kazuo Nishimura Kyoto University

Kyoto, JAPAN

Marcel K. Richter University of Minnesota

Minneapolis, U.S.A.

Yoichiro Takahashi Kyoto University

Kyoto, JAPAN

Michel Valadier Universite Montpellier II

Montpellier, FRANCE

Makoto Yano Keio University

Tokyo, JAPAN

Aims and Scope. The project is to publish Advances in Mathematical Economics once a year under the auspices of the Research Center for Mathematical Economics. It is designed to bring together those mathematicians who are seriously interested in obtaining new challenging stimuli from economic theories and those economists who are seeking effective mathematical tools for their research.

The scope of Advances in Mathematical Economics includes, but is not limited to, the following fields:

- Economic theories in various fields based on rigorous mathematical reasoning.

- Mathematical methods (e.g., analysis, algebra, geometry, probability) motivated by economic theories.

- Mathematical results of potential relevance to economic theory. - Historical study of mathematical economics.

Authors are asked to develop their original results as fully as possible and also to give a clear-cut expository overview of the problem under discussion. Consequently, we will also invite articles which might be considered too long for publication in journals.

S. Kusuoka, A. Yamazaki (Eds.)

Advances in Mathematical Economics Volume 10

Springer

Shigeo Kusuoka Professor Graduate School of Mathematical Sciences University of Tokyo 3-8-1 Komaba, Meguro-ku Tokyo, 153-0041 Japan

Akira Yamazaki Professor Department of Economics Meisei University Hino Tokyo, 191-8506 Japan

ISBN 978-4-431-72733-0 Springer Tokyo Berlin Heidelberg New York

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Table of Contents

Research Articles

C. Castaing, M. Saadoune Komlos type convergence for random variables and random sets with applications to minimization problems 1

J.-P. Gamier, K. Nishimura, A. Venditti Capital-labor substitution and indeterminacy in continuous-time two-sector models 31

T. Ibaraki, W. Takahashi Weak and strong convergence theorems for new resolvents of maximal monotone operators in Banach spaces 51

S. Iwamoto Golden optimal policy in calculus of variation and dynamic programming 65

S. Kusuoka A remark on law invariant convex risk measures 91

A. Rubinchik, S. Weber Existence and uniqueness of an equilibrium in a model of spatial electoral competition with entry 101

H. Hata, J. Sekine Publisher's Errata: Solving long term optimal investment problems with Cox-IngersoU-Ross interest rates 121

Subject Index 123

Instructions for Authors 125

Adv. Math. Econ. 10, 1-29 (2007) Advances in


©Springer 2007

Komlos type convergence for random variables and random sets with applications to minimization problems

C. Castaing^ and M. Saadoune^

^ Departement de Mathematiques, Universite Montpellier II, 34095 Montpellier Cedex 5, France (e-mail: [email protected])

^ Departement de Mathematiques, Universite Ibnou Zohr, Lot. Addalha, B.P. 8106, Agadir, Maroc (e-mail: [email protected])

Received: August 10, 2006 Revised: October 16, 2006

JEL classification: C61

Mathematics Subject Classification (2000): 49J40, 49J45,46N10

Abstract. Let £ be a separable super reflexive Banach space and let (^, T, \x) be a complete probability space. We state some Komlos type theorems in the space C?^ iSl, T, \x) of ^-valued random variables and a version of Komlos slice theorem in the space £^ , ,^x(^, T, /x) of convex weakly compact random sets. Weak Komlos type theorems for some unbounded sequences in £jr(^ , T, ^x) and £j. ,[F](^, T, /i) when F is a separable Banach space are also stated. A Fatou type lemma in Mathematical Economics and minimization problems on convex and closed in measure subsets of £ ^ (^, T, /JL) are presented. Further Minimization problems and Min-Max type results involving saddle-points and Young measures are also investigated.

Key words: Biting Lemma, Komlos convergence, minimization, Min-Max, saddle points, young measures

1. Introduction and preliminaries

Throughout £" is a separable Banach space, {Q,J^, /x) is a complete probability space, >C^(^, T, jx) is the space of all ^-measurable £-valued functions defined on Q. Let cwk(E) be the set of all nonempty convex weakly compact subsets of E. Let us denote by >C^^^^^^(^, ^ , /x) the space of all scalarly ^-measurable cw;/:(£')-valued mapping defined on Q (alias convex weakly

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2 C. Castaing and M. Saadoune

compact random sets). Recall that a cu;/:(£')-valued mapping X : Q -^ cwk(E) is scalarly -measurable if the support functions 5* (jc*, X (.)) are -measurable for all X* G E\ Given a convex weakly compact random set X, we denote by |X| the ^-measurable real valued function |X| : Q, -^ \X((JO)\ := sup{|5*(x*,X(ft>))| : 11 *11 < 1}. We refer to [18] for details concerning Convex Analysis and Measurable Multifunctions. Also we will use the following Um-iting notions. Let (Cn)neNu{oo} be a sequence of nonempty closed convex subsets of E, {Cn)nef^ Mosco Converges to Coo if the two following inclusions are satisfied:

Coo C s-liCn := {x e E : \\x - XnW -> 0; Xn e Cn}

w-lsCn :={x e E \Xn^^ X weakly; x„ e C„ J c Coo-

Given two nonempty subsets B and C in £", the gap between B and C is defined by

D{B, C) = M{\\x -y\\:x eB,y eC}.

The slice topology Xs on cc{E) (nonempty closed convex subsets of E) is the weakest topology r on cc(E) such that for each nonempty bounded closed convex subset B of E, the function C i-> D{B, C) is r-continuous. (Cn)neN slice converges to Coo if one has

lim D(5, Cn) = D{B, Coo)

for all nonempty bounded closed convex subset 5 of £". It is well-known that the sUce convergence and the Mosco convergence coincide on cc{E) when E is reflexive. We refer to [6] for the topologies on closed convex subsets in Banach spaces. If F is a reflexive Banach space, any bounded sequence (/«) in the space £}^(Q, !F, ix) has the Mazur property, namely, there exist a subsequence (g^) of ifn) and /oo e C\{Q., T, \x) and a sequence i}in) of convex combinations of {gm). i-e. hn G co{gm : m > n},Wn E N, such that (hn) converges a.e. to /oo, with respect to the norm topology of F. Further if F is super reflexive, then any bounded sequence (fn) in £^(Q, T, ji) has the Komlosproperty, namely, there exist a subsequence (fa(n)) in C\^(^, T, /i) and /oo e £]^(Q, T, fx), such that

lim -Eyî/yô) = /oo

a.e., with respect to the norm topology, for every subsequence (/^(n)) of (/«(«))• Let (/„)«eNu{oo} in C\(Q., T, \x) , the notation /x-lim«_ôo fn = /oo means that (fn) converges to /oo in measure. For more on Komlos theorem [20] in L } ^ ( ^ , T, jx) where F is super reflexive (or B-convex) space, we refer to [7, 19].

Komlos type convergence with applications 3

In § 2 we present several versions of Komlos theorem [20]. The first ones concerns with a version of Komlos theorem in £ ^ ( ^ , J", /x) and a version of Komlos sHce type theorem on the space >Cû; (£)( , - ^ M) when £" is a super reflexive. The second ones deals with weak Komlos type theorems for some unbounded sequences in C\^{Q, T, /x) and £^,[F](^, T, \i), respectively, here fi\,\F\(Sl, T, \i) denotes the space of all F^-valued functions f : Q. -^ F' such that for all jc e F, the scalar function {x, f) is integrable and such that the function | / | : ^ ^ R given by \f\{(jo) := H / M H F S (JO e Q is integrable, when F is a separable Banach space, here the notation weak Komlos means that the associated Cesaro sums converges a.e to the functions under consideration, with respect to the weak topology of F and the weak* topology of F\ respectively. Our main purpose is to introduce a new type of tightness condition for sequences in these spaces. Namely, a sequence (/„) in £)r(^, T, /x), is Mazur tight if it satisfies the condition (*): for every subsequence (/„^) of (/„) there exists a sequence (r„) in £^ (^ , T, JJL) with r„ e co{\\fni(.)\\ '• i > n} such that Hm sup„ r„ e £j j(^, ^ , /i) and similarly for (gn) in £^,[F](Q, !F, /x). It is worthy to mention that the above Mazur tightness condition does not imply that (fn) is bounded in C]^(Q, T, JJL). Indeed, it suffices to consider the space £jj(^, ^ , M) where Q = [0, 1] endowed with the Lebesgue measure and fn is given by fnico) := «^l[o,i/n](^), < e ^ . Then, (/„) is not bounded in C^^îQ, T, IJL) but satisfies (*), because it converges a.e. to 0. These considerations led to several tightness conditions in the study of Fatou lemma in Mathematical Economics [15,16] and allow to give a new light in the problem under consideration. In § 3 a series of new apphcations illustrating the results obtained in the previous sections is given. We present a characterization of convex closed sets for the convergence in measure by means of Komlos theorem in C^^(Q, T, /x). In particular we show that for every convex and closed in measure subset W of .C^C^, T, /x) and for every convex weakly compact random set r , the set <Sr H- 7i where <Sr is the set of all ^-measurable selections of F, is convex and closed in measure. A Fatou-type lemma in Mathematical Economics and some Minimization problems are also given. We provide also two min-max type results involving Young measures and saddle points for a class of convex integral functionals.

We refer to [7,14,19,20,23] for Komlos theorem for vector-valued random variables and to [4,12,17,21] for random sets.

2. Komlos theorem for random variables and random sets with applications

The following version of Komlos theorem in £ ^ ( ^ , T, /x) is crucial for our purpose.


Theorem 2.1. Let E be a separable super reflexive Banach space and let (fn) be a sequence in C^^(Q^, T, [x) satisfying the following condition: Of or every subsequence (fm^) of (fn) there exists a sequence (rn) in C^(Q,T, p) with rn e co{||/„.(.)|| '. i > n] such that limsup^ rnico) is finite for each CD in Q. Then there exist a subsequence (gn) of{fn) and an E-valued random vector f such that

1 «

Z hi{a))- f{a))\\ = 0 , a.e.coe Q, liml " " , = 1

for every subsequence {hn) of(gn). Furthermore, if we suppose in this condition lim sup„ r„ e Cl^(Q,J^, p), then fô e C\{Q,, T, p).

Proof By hypothesis, there exists a sequence (r„) of the form

rn^Y.^1\\fi^n\\ iein

with X^ >0 and ^.^j k^ = I such that Umsup„ rn{(o) is finite (equivalently sup„ rn(co) is finite) for each a; in ^ . For p e N, define

Ap := {(o e Q : supr„(a;) < p}. n

As Unip- oo l^(^p) = 1. we can choose pi € N such that p(Ap^) > 1 — ^. By integrating we get

S U P X ^ ? / \\fi+n(co)\\dp<pi

Hence, there exists a subsequence (f^) such that sup„ /^ 11 / ^ (co) \ \dp < p\. In view of the Biting lemma (see e.g. [13], Theorem 6.1.4) there exists an increasing sequence (5^) in T with lim^ p{B^) = 1 and a subsequence {g\) of (/ ) such that {g\) is uniformly integrable on each Ap^ H B^. Now, by virtue of Komlos theorem [19], there exist a subsequence of {g\) still denoted by {g\), f^ e £ ^ ( ^ , T, p) and cp]^ e C\^{Q, T, p) such that the following hold

lim ||-Efî/z/(a;) - f^((o)\\ = 0, a.e. co e Ap, n-ôo n

and

hm -Efîl | / i ,Mll = (plc^ a.e. co e Ap, nôo n

for every subsequence Qin) of {g\). As {g\) is uniformly integrable on each B^ n Ap,, by Lebesgue-VitaU theorem, we get


V ^ , V A G ^ , lim / -J:^^^\\hi\\dn= f (pldfji

for every subsequence (hn) of (gl). This is equivalent to

V ^ , V A e ^ , lim / \\gl\\diJi=[ cp^dfji.

Next, applying condition (*) to (gl) instead of (/„) and using again the preceding arguments we find a measurable set Ap^ with iJiiAp^) > 1 — | , an increasing sequence (B^) in T with Hm^ M(5^) = 1, a subsequence (g^) of (gl), f^ € £ ^ ( ^ , JF, /x) and(^^ e /:]^(^, JT, /x) such that the following hold

lim ll-Sfî/z/M - / ^ M l l = 0, a.e. co e Ap,

for every subsequence Qin) of {g^) and

V ^ , V A G ^ , hm / \\gl\\dfi= [ cpl^dii. ""^"^JADAry^nB} JAOAn^nBi

Repeating the preceding arguments provides a measurable set A^ with /x (^^^) > 1 — ^ , an increasing sequence (BJ^) in^withlim^ M(^^ ) = 1, a subsequence (8n) of (g^^), / 4 e £ ^ ( ^ , ^ , M) and ^ ^ G 4 ( ^ , , /x) such that the following hold

lim | | -Efî /z ,M - / 4 M I I = 0, a.e. co e Ap,

for every subsequence (hn) of (g^) and

V^ ,VA€^ , l i m / \\gl\\dn=f cpldfjL.

Finally, define

^pi •= ^Pi and A' ^ := Ap, \ Ap,_, for/: > 1 k=oo k=oo

gn '•= gl f^ := X ^^pj^ ^ ^ ~ •= S ^ W ' /:=1 k=\

Then

( k=oo \ /k=oo \


because /x(Ap^) > 1 — ^ for all k eN. Further, it is not difficult to see that

lim l l -Efî / i /M - foc((o)\\ = 0, a.e. co e Q (2.1.1) n-ôo n

for every subsequence {hn) of {gn) and

V^, V/:, VA 6 ^ , lim / WgnWdî = f (pocdfi.(2A.2)

Now, let us prove the second part of Theorem 2.1. Applying the new condition to the sequence (gn) provides a sequence (r„) of the form r„ = Y.iein^'i^^î+nW with X^ > 0 and ZieinK = 1 such that limsup^r^ e £ ^ ( ^ , J^, /i). From (2.1.2) and Fatou lemma it follows that

/ (Pood 11= lim X ^ ? / Wgi^nWdî

- lim / Tx'lWgi^nWdn -^^Pk^^q ieIn

L limsupr^JjU. (2.1.3) A'p.C^B^, nôc

Since the sequence (^Sf=i^n+/) is uniformly integrable on each A^ fl 5^, it follows from (2.1.1) and Lebesgue-VitaU theorem that

lim / |hEfî^,+, | I^M= / WfocWdfi. (2.1.4)

Therefore, by (2.1.4), (2.1.2) and (2.1.3) we deduce that

/ l l /ooll^M<lim-I],t i / Wgn+iWdn

= lim / WgnWdfi

= / (Poodfi

J A' nfii ^Pk <i

'-L Hmsupr jLC. p.^B^g «^00


Then it follows that

/ ||/ocll^M = S^'T 1™ / WfooWdli

< E ^ Hm / Urn snip Kfidfi

= / XmiswpVndî < +00.

The proof is therefore complete. •

Before going further let us mention a useful convergence result.

Proposition 2.1. Let H be a subset of C^^(Q, T, /x) with € = 0, 1. Then the following are equivalent: (1) Given any sequence (fn) in 7i, there exist a subsequence (gn) of(fn), cpoo i^ £^^(Q, T, /JL) and an increasing sequence (Q) in J-with limjt oD l^{Ck) = 1 such that for every k eN, Ickôo ^ >^^+(^, ^ , M) and

VAeJT, lim / \gn\dfi= (PoodfJi.

(2) Given any sequence (fn) in H there exists a sequence (rn) in C^(^, T, \x) with rn e co{\fi\ : i > n} such that (rn) pointwisely converges a.e to a measurable function roo ^ >Cj ( , J^, ji). (3) Given any sequence (fn) in H, there exists a sequence (rn) in C^(Q, T, /x) with rn € co{\fi I : / > «} such that hm sup„ r„ € £ ^ ( ^ , T, /JL).

Proof Suppose (1) holds. Let (/„) be a sequence in H. There exist a subsequence (gn) of (fn), (foo e C^-^ (^, T, /JL) and an increasing sequence (Ck) in J^ with lim^-^00 M ( Q ) = 1 such that

V/:, VA G ^ , lim / \gn\d/ji = / (foodfi < 00. "^^"^JADCk JAnCk

For each/: choose n^ such that, for every n > nk, J^ Ignldf^ < +oc and define the following sequence of integrable functions:

gn,k = ^Ckgn ifw >nk,

= 0 otherwise.

It is clear that, for each k, the sequence (\gn,k\)n or(^\ L^)-converges to 1Q<ÔO. By Lemma 3.1 in [17], there exists a sequence (rn,k)n,k with r„,^ ==


ll'iLnK\8i,k\ where X^ > 0 and Xtn = 1 such that, for every k, {rn,k)n converges a.e. to Ickôo- Since

Wk, WcoeCk, lim y XI \gi,kI = lim Y A \gi | = lim V X |g,-1 /=n i=n i—n

and limits00 /xCQ) = 1 we deduce that the sequence (^{L^ A |g/1) converges a.etO(;ôo.Thisproves (2). The implication (2) =^ (3) is trivial. Now, to prove the implication (3) ^ (1) let (/„) be a sequence in H and let (g„), (poo, Apj^ andB^, (k, q > 1), be defined as in the proof of Theorem 2.1. Since lim^ M(5^) = 1, then there exists qk > 1 such that M(^^^) > 1 - p Taking Q := u|=^(Ap. 0 B^.) it is clear that (gn), (Poo and (Ck) have the required properties. D

There is a simple version of Theorem 2.1.

Corollary 2.1. L r Ebea separable super reflexive Banach space and let (fn) be a sequence in C^^(Q,J^, fi) such that Um sup„ 11 /«(<^) 11 is finite for each co in ^ . Then there exist a subsequence (gn) of (fn) and an E-valued random vector f^ such that

1 " \im\\-y^hi((o) - foo{(o)\\ = 0 , a.e. co eQ, n n ^^^

i—\

for every subsequence (hn) of (gn). /jf Umsup„ ||/n(.)|| e £ ^ ( ^ , ^ , /x), then

Proof Corollary 2.1 is a direct consequence of Theorem 2.1, because for every subsequence (/„^) and for every sequence (r„) in £^ with r„ € co{| |/„. 11 : / > n] we have

lim sup rn (co) < lim sup 11 fn (co) 11, co e Q, n n

see Lemma 4.1 in [15] or more generally Lenmia 1.1 in [5]. D

Remark. Theorem 2.1 is not true for unbounded sequences, it suffices to take

Now we proceed with some significant variants of Theorem 2.1 when the Banach space E is not super reflexive. We recall some tightness notions.

Let F be a separable Banach space. Let cwk(F) (respectively TZwc(F)) be the set of all convex weakly compact (respectively, weakly closed, ball-weakly compact) subsets in F; a weakly closed subset in F is ball-weakly compact if its intersection with any closed ball of F is weakly compact. If F is reflexive, any weakly closed subset of F is ball-weakly compact. Similarly, any weak*


closed subset of the weak* dual of a Banach space F, is ball-weakly* compact. A sequence (/„) in C]^{^, T, /x) is 'JZwc(F)-tight, if, for every e > 0 there exists a 7?.u;c(F)-valued measurable multifunction Vg : Q ^ F such that

sup fi{{(jo eQ: fn{o)) ^ ^^(0))}) < £ n

Theorem 2.2. Let Fbea separable Banach space and D := {e'p)p>\bea dense sequence in F' for the Mackey topology. Let (fn) be a IZwciF)-tight sequence in C\ {Q, T, jJi) satisfying the condition C):for every subsequence {fn,^) of(fn) there exists a sequence (rn) in £|^(^, T, IJL) with rn e co{||/n,(-)ll ^ ' ^ «} such that lim sup„ r„ {co) is finite for each co in Q. Then there exist a subsequence (gn) of(fn) and an F-valued random vector fô such that

1 '^ V/7>1, -îêp,hi(co)^->{ê'p,foc(co)y a.e.coeQ,

i = \

1 "" Vw e C%[F]{Q, T, /x), /x- lim - V(w, hi) = (M, /OO>

i=\

for every subsequence (hn) of(gn). Furthermore, if we suppose in the condition (*), Umsup„ rn(.) eC^^(Q, T, /x), then foo ^ C.\(Sl, T, /x).

Proof Let us prove the first part of Theorem 2.2. By Proposition 2.1, there exist a subsequence {gn) of (/„), (poo e >C^+(^, T, /x) and an increasing sequence (Ck) in T with lim t /x(Q) = 1 such that for every /: e N

lim / \\gn\\dii= / ipoodpi. (2.2.1)

for all A e ^ . It is clear that, for each ^ > 1, the sequence ( 1 Q ^„) is uniformly integrable and 7û;c(F)-tight in C],(Q, T, /x). Using (2.2.1) and Corollary 2.1 in [23] via an appropriate diagonal procedure, we find a subsequence of {gn) still denoted {gn) and / 4 G il\{Q.,T, /x) such that

1 "" ^k > 1, V/7 > 1, lim - Tie' IcM^)) = iC fU^))^ a.e. oyeQ

1 "" V/: > 1, Vi/ e C%[F]{Q, T. M), /X - lim - Y d / , l^/i/) = («, / 4 )

for every subsequence {hn) of (g„). Putting

C; — Ci and C := Q \ U ^' ^ r i > 1


and

foo •= 2Ld ^C[foo^ k=\

it is not difficult to verify that

k=OQ

1 " Vp>l , ^J^^-Y^{e'p,hi{w)) = (e'p,fo^{co)), a.e. c; 6 J2 (2.2.2)

f= i

1 " VM e 4 , [F ] ( f i , T, fi), M- lim - Y{u, hi) = (u, fô) (2.2.3)

i=\

for every subsequence (/z„) of (^„). Now, let us prove the second part of Theorem 2.2. Applying the new

condition to the sequence {gn) provides a sequence (r„) of the form r„ = Hiein KWî^nW with X > 0 and ^.iein K = 1 such that limsup^ r„(.) e C\^(Q, T, fji). From (2.2.1) and Fatou lemma it follows that

< / limsupr„d/x. (2.2.4)

On the other hand, since the sequence {\ckSn) is uniformly integrable, from (2.2.3) and Lebesgue-VitaU it follows that (2.2.5)

^k > 1,VM e L^,[F](^,J^,/x), Um / (M, -S,tig/+itWM= / {u, foo)dii.

Since the norm ||.||î is weakly lower semi-continuous, (2.2.5) impHes

V/: > 1, / WfocWdfJi < Uminf / ||-Efî^,+^||J/x

< Uminf / -Sfîllgz+itll^/x

= lim / WgnWdii

Jci


where the two last inequalities follow from (2.2.1). Thus the above estimate and (2.2.4) imply that

VA: > 1, / ll/ooll^M < / limsuprnd/ji. Jc: Jci n

Hence

/ | |/OO||JM = 5]+!? / WfocWdfi k

+CX) < ^kS / limsupr„J/i6 Jq n

lim sup rndfi < +oc. / JQ

As a special case of Theorem 2.2 we give the following result.

Corollary 2.2. Let F be a separable Banach space and let (fn) be a sequence in CîO,, T, /x) satisfying the condition (*) of Theorem 2.2 and the condition (**).• there exists a measurable IZwc(F)-valued multifunction F \ Q =^ F such that fn (CL>) e T{a})for alln e N and all co e Q.. Then there exist a subsequence (gn) of(fn) and an E-valued random vector /o© such that

1 " Vx* 6 F\ lim - V ( x * , hi((o)) -> (JC*, /oo(< )>, a.e. co e Q

/=!

for every subsequence Qin) of (gn), here the negligible set depend only the subsequence under consideration. Furthermore, if we suppose in the condition (*) limsup^ rn{.) € £j j(^, J^, /x), then foo e C\{a, T, /x).

Proof It clear that (/„) is 7^M;c(F)-tight. Applying Theorems 2.1 and 2.2, respectively, to the sequences (H/nlD^ and {{e[, fn))n (k > 1) via a diagonal process provides a subsequence (gn) of (fn) and /o© e i 2 ^ ( ^ , ^ , î), (or e £jr(^, •^,M)»if we suppose in condition (*),lim sup„r„(.) e i2}^(Q,^,/x)) such that

lim -i:^_.\\hi(co)\\ exists, a.e. coeQ, (2.2.5)

1 " Vk>h lim - y ( 4 , / z / ( a ; ) ) = (4,/oo(^)), a.e.a;€Q, (2.2.6)

for every subsequence (hn) of (gn). To complete the proof take a subsequence (/i^)of (/;,)andputr(.) := /?(.)nr(.):^£, wherer(.) :^ sup„ ii:fî||/z,(.)||.


Then, by (2.2.5), r{co) is weakly compact a.e and Vn e N, ^DJLi/i/M e coF (co) a.e. Hence, it follows from (2.2.6), by using a routine density argument, that

1 "" VJC* 6 F', Um - Y(jc*, /z/(a;)) = (x\ /oo(^)), a.e. a; e Q.

/ = 1

Remark, Theorem 2.2 is valid in the space £^(Q, T, fi) under obvious modifications.

Now we present some convergence properties for a class of unbounded sequences in the space L J ^ , [ F ] ( ^ , J", /x) of all F'-valued mappings / : ^ -> F^ such that co h-> {f((o),x) is integrable, VJC € F, and such that | / | ( .) := ||/(.)||/r/ belongs to L^(^ , T, /x), when F is a separable Banach space. We summarize some properties of this space. For more details, we refer to [5,18] and the references therein. We will endow L\^,[F](Q, T, \i) with the norm

Aî(/) = Aî(ll/ll), / G L ) , , [ F ] ( Q , J ^ , / X )

1 here A i denotes the norm in L^ (^, ^ , /x). By Theorem 4.1 in [5] L ^ (^, ^ , /x) 1 is included in the topological dual ( L L [ F ] ( ^ , T, [X))' of L L [ F ] ( / X ) and we

have

Aî(/)= sup f {h,f)dix heTii J^

here H\ denotes the set of all simple mappings from ^ into the closed unit ball Bp of F, so that the mapping / i-^ N\(f) is lower semicontinuous on L } „ [ F ] ( Q , JT, /X) for the topology or(L},,[F](/x), Lf(fi)).

The following result deals with some convergence properties for a class of unbounded sequences in (L^p,[F](Q, T, /x), A i) and leads to interesting apphcations in several problems of convergence of F'-valued scalarly integrable random variables, in particular, Fatou type Lemma in L\,\F\(Q.,T, IJL).

Theorem 2.3. Let F be a separable Banach space. Let (fn) be a sequence in ( L J ^ , [ F ] ( ^ , T, /X), N\) which satisfies the condition {""): for every subsequence ifrik) ofifn) there exists a sequence (rn) in L^(^ , ^ , /x) with rn G co{\frn \ : / > n} such that Hm sup„ r„ 6 L]^(Q, !F, fi). Then there exist a subsequence (gn) ofifn), foQ G L^p,[F](Sl, T, /x) and an increasing sequence (Q) in T with limjt /x(Q) = 1 ^^^^ that

V/:> l,Vi;eL^(^,JP^,/x), lim / {v,gn)d[i= f (u,/oo)^M


and such that

fooico) colf^ w*-cl{fm((o) : m > n} I, a.e.

Furthermore, we have

1 " VJC e F, lim - ^{x, hi) = (jc, /o©), a.e,

nôo n ^-^ i = \

for every subsequence (hn) of(gn), here the negligible set depends only on the subsequence under consideration.

Proof. Step 1 Applying Proposition 2.1 to the sequence {\fn\) provides a subsequence {gn) of (/„), (poo e L^+(^, T, ii) and an increasing sequence (Q) in !F with limjt M ( Q ) = 1 such that for every k e N

lim / \\gn\\d^l= I (pocdfJi. (2.3.1)

for all A e T. Then for every k e N, (lck8n)neN is uniformly integrable in L]^f[F](Ck, Ck n T, /X|Q). In view of Theorem 6.5.9 in [13] and a diagonal procedure, we produce a subsequence (not relabelled) of {gn) and a sequence ( /4) with / 4 e LJ„[F](Q, Q n T. /x |cj such that

I p|u;*-d{/^(a;):m>/7} I, / 4 ( c ^ ) € c o ( [ ]u;*-d{/^(a;):m>/7} | , a.e. a; E Q (2.3.2)

and such that

V/: € N, Vu € Lf{n, T. M), lim / (i;, gjJ /x = / (u, /4)^/x.

Define

C; := Ci and C^ := Q \ | J Q for i > 1

and

/oo := Z lq/< it


It follows that

V/:6N,VMeL^[F](^, j r , /x) , lim / {u,gn)dfjL= [ (u, foc)dfi.

(2.3.3)

On the other hand, let D := {ep)p>\ be a dense sequence in F for the norm topology. Applying Theorem 2.1, respectively, to the sequences ((^p, gn{')))n. (p € N), and (| |^„ (.)11) via a standard diagonal extraction procedure provides a subsequence (not relabelled) of (gn) such that

1 " V/7 e N, lim -^{ep,hi{a))) exists a.e. (2.3.4)

n-ôo n ^^

and

1 " lim - V| | /Z/ (CD) | | exists a.e. (2.3.5)

n-ôo n ^~^ i—\

for every subsequence (/z„) of (^„). By (2.3.5) the sequence (^ ^YH^x ll^«ll) i pointwise bounded almost everywhere. By (2.3.3) and (2.3.4) it is immediate that

1 "" Vp G N, Hm - X^^P ' ^^•(^)) = <^P' / ^ ) -

Using the separability of F and the pointwise boundedness of (^ X/^=i I l n 11) we get, by a routine argument

1 " lim -y]{e,hi) -> (^,/oc

for all ^ € F and almost everywhere. Step 2 /oo e LUF]{Q,J=', /X). AS

Aî ( / )=sup [ {Kf)dn= f \\fi.)\\dfi, V / e L^,[F](/x)

A i is cr(Lj.,[F], L^) lower semicontinuous, (2.3.3) implies

V/: > 1, / ll/ooll^M < liminf / ||gn(.)II^M

= lim / WgnWdfi = / ( oo /x


where the two last inequahties follow from (2.3.1). Hence

/ | | / o c | | ^ M = S+!? / WfocWdfJi

Jn Jcl

^ ^ j £ T / ôodfJi = / (Poodfji < + 0 0 .

Finally by (2,3.2) we have

fooico) ^colf] w''-cl{fm{co) : m > n} I, a.e.

Remarks. If (| /„ |) is bounded, condition (*) is satified. See for instance Proposition 2.1.

Now we proceed to a new version of Komlos sUce theorem in the space C^cwk(E)^^' J", ix). Compare with ([17], Theorem 4.1 and Corollary 4.2). See [4,12] for other related results. Let us recall the following result.

Proposition 2.2. Hess [9] Let F be a separable Banach space, and D = (^k^k>l be a dense sequence in F' for the Mackey topology. Let (Xn) be a sequence in ^^yjj^(f)i^^ ^)- Assume that the two following conditions are satisfied:

(i) There is a convex weakly compact valued random set C such that, Vn > l,â)eQ,Xn{(o)cC(oj).

(ii) Wk > 1, lim„-ôo ^*(^^' Xnico)) exists a.e.

Then there exists a convex weakly compact valued random set X^ and a negligible set N c Q such that

lim 8*(e\ Xn(oj)) = 8*(e\ Xôicv)) /i->oo

for all e' e F' and for allco e Q\N.

Theorem 2.4. Let E be a separable super reflexive Banach space. Let (Z„) be a sequence of convex weakly compact random sets in E such that for every subsequence (X„^) of (Xn) there exists a sequence (rn) in £ ^ ( ^ , T, jx) with rn e co{\Xn. \ '. i >n} such that lim sup„ ^n(^) is finite for each co in Q. Then there exist a subsequence (Xp(n)) of{Xn) and a convex weakly compact random set Xoo such that, for all bounded closed convex subset B in E and for every subsequence (Xy^n)) of(X^(n))> the following hold:

lim D (B, -'^'j=iXy^j)(co)) = D(B, Xoc(co)) (*)

for almost every co e ^ .


Proof. Let D[ := ( pA:>i be a dense sequence in BE* for the norm topology. For each k, we pick a maximum ^-measurable selection aj^ of Z„ associated to el, that is, (^|, a^) = <5*( , X„). By virtue of Theorem 2.1, there are a subsequence (Xy3(„)), a subsequence (<T^(„)) and a^ G £ ^ ( ^ , T, /x) and areal valued ^-measurable function cp^ such that for each k the following hold

Um 5* ( 4 , -E^îX^(y)(a;)) = (pîco) a.e. (2.4.1)

for every subsequence (Xj/(„)) of (X^(„)), and

1 •^n ^k lim - £ " = ! < ( . ) M = aîco) a.e. (2.4.2)

for every subsequence (o^y(„)) of (o^L ))- For simphcity we set

n -^

for all n e N and for all tD e Q. Again by Theorem 2.1, we may assume that for each 6t) G ^ ,

sup 15nMl < +00 (2.4.3) n

a.e. By (2.4.1), (2.4.3) and Proposition 2.2, there is a convex weakly random set Xoo and a negligible set N such that

^lim^r L \ \i:^%xXyU){cJ\ = 5* {e\ Xooico)) (2.4.4)

for all (oj, e*) e (Q\N) x BE*- NOW by obvious properties of (or;f), (2.4.1), (2.4.2) and the definition of s-li ^ D^^j Xy(j) (co) we have

5* « , Xooico)) = [el a^(co)) < 5* I el s-li^Y.^yU)(co) j (2.4.5)

for all k and almost every a; G ^ . By (2.4.4), (2.4.5) we conclude that (Sn) Mosco-converges to Xoo- We will prove (*) that is a formulation of slice convergence in C^cwkiE)^^^ •^' /^)- ^y (2-4.4) we have


liminf D(B, Sn(a))) = liminf sup {-5*(JC*, Sn(o))) - 5*(-JC*, B)]

> sup liminf{-(5*(jc*,5^(a;))-r(-Jc*,B)} x*eBE'

• sup x*eBE'

-limsup(5*(x*, Snico)) - 8*(-x\ B)

= sup {-(5*(jc*,Xoo(ft>))-5*(-;c*,5)}

= D(B,Xoo(co))

for any bounded closed convex subset B inE and for almost every co e Q, This proves the liminf part.

Let us prove now the limsuppart. Let (gk) be a Castaing's representation of 5-//Êy^jXy(j)(.). Then we have

Vo; G ^, Vx e £:, VA: E N, limsupJ Ix, -'E''^^Xy(j)((jo)] < d(x,gkia)))

which impHes that

limsup J Ix, -l^^-iXyn)(co) I < inf d(x, gk((o)) n-ôo \ n •' ) keN

= dUs-iiî:^^,Xyu)((^)Y

Now let 5 be a bounded closed convex subset of E, then

limsupD I 5, -J:''-^Xy(j)(a))] = limsup inf d(x, -T^''-^Xyn)(a))) n->oo V ^ / n-ôo xeB n •'

< inf limsup J (x, '-Ti^^-^Xy(j)(a)) I xeB nôo \ ^ /

< inf dlx^s-li-TÎ.Xyaîco)) xeB \ n J~ J

^Di^B.s4i^-i:)^,Xy^j^{co)^

<D(B,Xoc{(o))

for any bounded closed convex subset B in E and for almost every co e ^. D

We give some apphcations of the preceding results.

Proposition 2.3. Let E be a separable super reflexive Banach space. Let H be a convex subset ofC^^(Q, T, /x). Then the following are equivalent:


(a) H is closed for the topology of convergence in measure, (b) For any almost everywhere pointwise bounded sequence (un) in H, that is,

SUp||M;^(6t>)|| < + 0 0 n

for almost every CO e Q, there is a subsequence (uy^n)) andu^ e £ ^ ( ^ , T, p) such that

/x - lim -E'J_iMy(/) = Moo

with respect to the norm topology.

Proof (a) =^ (b). In view of Theorem 2.1, any almost everywhere pointwise bounded sequence (un) in H has the Komlos property. Hence there exist a subsequence {uy(n)) and MQO € >C^(^, ^ , p) such that

lim - H'J-i My (/)= Moo n-ôo n • •~

a.e., with respect to the norm topology. In particular, we have

IX- lim - E'Jî My (;)= Moo

with respect to the norm topology. Since H is convex and closed for the convergence in measure, it is immediate that MOO (b) =^ (a). Let (M„) be a sequence in H which converges in measure to u e C^^(Q,T, p). There is a subsequence of (u„) of (M„) which converges to M for almost every co e Q. Thus the sequence (i;„) is pointwise bounded for almost every co e Q. Using (b) and Theorem 2.1, there is a subsequence (ua(n)) and Voo ^H such that

lim -Ey^jiJaQ-) = Uoo

a.e., with respect to the norm topology. So we have u = vô e Hfoi almost every o) eO.. •

The following result is a combined effort of Theorem 2.1 and Theorem 2.4. So we omit the proof.

Proposition 2.4. Let Ebea separable super reflexive Banach space. Let (Xn)

be a sequence in lyjÊ)^^' ^ ^^^^ ^^^^

SUP|XJG4(^,J^,M).


Then there exist a subsequence {X^(n)) of{Xn) and a convex weakly compact random set Zoo such that, for all bounded closed convex subset B in E and for every subsequence {Xy(n)) of(Xp(n)), the following hold:

lim D (B, -^^-^Xy^jîco)) = D(B, XocM) (*)

for almost every 6D G ^.

For the sake of transparency, let us examine first the particular case where Xn G L^(^ , ^ , /x). Since E is reflexive, by Dunford-Pettis theorem, (X„) is relatively sequentially weakly compact in L^(J^, ^ , /x). See [11] for further references of weak compactness in Bochner and Pettis integration. So there is a subsequence (X^) of (Xn) which converges weakly to an element Y^Q^ 6 L\(SI,T ,{i) with I Fool Sg and, by Mazur's lemma, there is a sequence (Z„) of convex combinations of (Xn) which converges strongly to Foo for almost every o) ^Q.. Using Theorem 2.1 or Proposition 2.3, we get more. There is a subsequence (Jn) of (Xn) and a random integrable E-valued vector Zoo such that

l i m | | - 5 ] f ^ i Z / M - X o c M | | = : 0 , a .e .a ;G^, (*) n n

for every subsequence (Z„) of (F„). Let W be the unit ball of L ^ ( ^ , J^, JJL). Then (*) imphes

lim -Efî {W(a)), Ziioj)) = {W{a)), X^(co)), a.e. a)eQ. (**) n n

Integrating equahty (**) gives

lim-Ef_i f (W(aj),Zi(co))i^(dcv)= f {W{cv), Xoc(o)))fi(da)) ( )

for every subsequence (Z„) of (F„). (***) shows that (Yn) weakly converges to Xoo in the Banach space L^(^ , T, fi). Similarly, in the multivalued case. Proposition 2.4 provides a shce type convergence for the sequence (Xn). This result is sharper thanjhe Mazur's TL convergence initiated in [17], namely, there is a sequence (Xn) with Xn e co{Xni : m > n] (i.e. X„_has the form Xn = ^ZnKî with 0 < Af < 1, i:^^^k^ = 1) such that (Xn) converges in the linear topology xi [6] to a convex weakly compact random set Zoo ^ ^\wk{E)^^' ^ ) for almost everywhere a; G ^ , that is

VJC* G E\ lim r (x*, Xn) = r (jc*. Zoo) n->oo

Vx G E, lim d(x, Xn) = d(x, Zoo) M—>00


for a.e. co e Q. The weak convergence in the space ^cu;/:(£:)*^ ' •^' ^^ ^^^^ £• is a separable Banach space, has been extensively developed in [17], see, for instance, ([17], Theorem 3.4). At this point, let us mention that the weak convergence of (Xn) towards Xô in ^ lui cf)*^ ' •^' ^ ) ' ^ ^ ^ '

V/iGL??,(^,jr,;x), lim I 8\h,Xn)dî= I b\h,Xoo)dî

does not follow trivially from the following one

VA 6 jr, VJC' e E\ Hm / 8*(x\ Xn) t//x = / ^*(x^ Xcx)) ^/x

because the support function is subhnear, by constrast to the L^(^ , ^ , /x) case, one may use trivially a density argument, meanwhile in the space ^cwk(E)^^'^' /x), this needs a subtile argument [17] which shows the difference between the spaces ^lyjk(E)(^^ ^^ M) and L^(^ , T, /i). In short the routine density argument involving L^, (^, T, \i) cannot be appUed to the case ^^^\wk{E)^^' ^ , /x) space. See [17], Theorem5.3) for the weak convergence in ^cwk(E) ^' •^' ^^ ^ ^ Komlos type arguments. So the preceding considerations and the characterization of convex closed sets for the convergence in measure (Proposition 2.3) focus the differences when deaUng with super reflexive Banach spaces.

In this vein, we present a Fatou type lemma in Mathematical Economics.

Proposition 2.5. Let E be a separable super reflexive Banach space and let (Ufi) be a sequence in C^^{^, T, /x) satisfying condition (*) of Theorem 2.1. Let (p : Q X E ^^^ [0, +oo[ be an T <S> B(E)-measurable integrand such that, Vco e Q,(p((i>,.) is convex lower semi-continuous on E (alias convex normal integrand) and such that I(p(u) := J^(p(co, u{cD))dii{o)) is finite for every u G £ ^ ( ^ , ^ , /x). Assume further that there exists b e R" such that b := lim„_>oo Icpit^n)- Then there exist a subsequence (uy^n)) of(un) and u e C\{Q,T, II) such that

lim -ll^:Ûy(i\{(o) = u{o)) a.e. n-ôo n

and such that

b= lim / (p{o),Un{o))) ii{da)) > / (p{(o,u{co)) ii{d(jo).

Proof In view of Theorem 2.1, there exist a subsequence (uy(n)) of (un) and u e£^(Q, : r , /x ) such that


lim -Ti^-Ûy(j)(a)) = u(co) n-ôo n •'~

for almost everywhere co eQ. By convexity we have

Hence

lim sup (p \(JO, -Ti^jîUy(j)((jt)) I fjiidco) < b. n JQ \ n ^ )

By the lower semicontinuity of (^(ft;,.) and by Fatou lemma, we get

liminf / {p\o},-YA^^Ûyn\{ci>)\\x{do))^ \ (p((jo,u{co)) iiidco)

which impHes

b= lim / (p(a),Un{(jo)) iji(da)) > / (p((jo,u(co)) iiidco).

We finish this section with the following closure property for the convergence in measure (compare with Proposition 5.2 in [14]).

Proposition 2.6. Let E be a separable super reflexive Banach space, let F be a convex weakly compact random set and let Sr be the set of all T-measurable selections ofV. Let H be a convex closed in measure subset of C^^(Q, T, //). Then the set Sr -\-H is convex and closed in measure.

Proof We need to prove that if (un) is a sequence inSr +H which converges almost everywhere to a function u e C^^{Q, J^, /JL), then u e Sr+H. There are fn 6 Sr and gn ^H such that Un = fn + gn for all n. Repeating the arguments given in the proof of Proposition 2.2, there exist a subsequence (fy(n)) of (/„) and f e Sr such that

1 " lim -y]fyij){o)) = f{o))

;= i

for almost every ct> 6 ^ , with respect to the norm topology. Now it is obvious that the sequence (^ X}=i Syij)) converges almost everywhere tog :=u- f. Since His convex closed in measure, we have e W.ThusM = f-\-g e Sr+H.

D


3. Some Minimization problems and min-max type results

We give some applications of the preceding results to some Minimization problems.

Proposition 3.1. Let E be a separable super reflexive Banach space, let T be a convex weakly compact random set and let SY be the set of all !F-measurable selections ofY. Then SY is convex and closed for the convergence in measure. Let J : SY -^ [0, oo[be a convex and lower semicontinuous for the convergence in measure. Then J reaches its mimimum on SY-

Proof. It is obvious that SY is convex and closed for the convergence in measure. Let (M^) be a minimizing sequence in 5r , that is, lim,t J{un) = m := inf{y(M) : u e «Sr}. Notice that sup„ l|wn(<w)|| < \^(o))\ < +oo for each ct> e Q. In view of Theorem 2.1, there exists a subsequence {uy(n)) of (M„) and M e £ ^ ( ^ , ^ , M) such that

1 "" lim \\-y]uy(j)(o))-u(a))\\=0

7 = 1

for almost every co eO^.By convexity we have

^ 1 ^ \ „ 1 =i--^("yO))-

Since J is Isc for the convergence in measure and (^ Xj=i ^YU)^n converges in measure to w 6 <Sr, we have

> J{u) > m. \ J=^ /

But is obvious that

" 1 limy^ -J(uy(j)) = m. n ^-^ n

It follows that

y=i

J{u) =m := inf{J(M) : u e SY)-

D

There is a direct application which arises from evolution problems. See ([18], Theorem VII-18).


Proposition 3.2. Let H be a separable Hilbert space, let dt be the Lebesgue measure on [0, 1]. Let T : [0, 1] =^ H be a convex weakly compact random set such that \T\ e L^([0, l],dt)andS^thesetofallL]^{[0, I], dt)-selections ofT. For each u e S^, let Pu be the primitive ofu e L^([0, 1], dt)

û{t) := u Jo

{s)ds, Vr € [0, 1].

Let j be a nonnegative convex normal integrand defined on[0, l]x H such that the associated convex integral fiinctional

J(u) := I j(t,u(t))dt, u e L^([0, II dt) Jo

is finite on S^ and the conjugate convex integral fiinctional

r{u) := [ fit, u(t))dt, Vw € L^([0, 1], dt) Jo

is proper on L^([0, 1], dt). Let ^ be the integral functional

VJ/(M) := / {Pu{t), u{t))dt + J{u), VM e L^([0, 1], dt). Jo

Then the functional ^(u) reaches its minimum on S^.

Proof (a) First proof. By a remark in ([18], Theorem VII-18), it is not difficult to check that

1/2 / {Pu(t),u(t))dt= f \\u{t)\\^dt. Jo Jo

Using the duahty of convex integral functional in ([18], Theorem VII-7), J is proper convex lower semicontinuous on L^([0, 1], dt) with respect to the O{L\, L\) topology, and is finite on 5p, here the nonnegativity of j is unnecessary, so is ^ . As it is obvious that S^ is nonempty convex and weakly compact in L^([0, 1], dt), it is immediate that ^ reaches its minimum on 5p. (b) Second proof. It is obvious that S^ is nonempty convex and closed for the convergence in measure and the convex integral functional ^ is finite, nonnegative, convex and lower semicontinuous on Sy for the convergence in measure. Then Proposition 3.1 shows that ^ reaches its minimum on 5p. D

Now we proceed to further results of Minimization with other techniques.

24 C, Castaing and M. Saadoune

Proposition 3.3. Let H be a separable Hilbert space, F be a nonempty weakly closed subset o/L^([0, 1], dt) and let A : F -^ ^^([0, 1], dt) be a compact mapping, (that is, A transforms any weakly convergent sequence in F into strongly convergent sequence in L^([0, 1], dt)), satisfying

{Au,u).j2 j l , >Qf | |M| |^2

for all u e F, for some positive constant a. For each u e L^([0, 1], dt), let Pu be its primitive

Jo is)ds, Vf6[0, 1].

Then the functional integral ^(u) := (Au,u),^2 2 v + jQ{Pu{t),u(t))dt reaches its minimum on F.

Proof It is immediate that

^(U) = {AU,U)^^2^^^2^^ + ~\\U\\12^ 1.. ,2

verifies

îu)>(a + ^\\\u\\l2^

for all u e F. Let (un) be a minimizing sequence for ^ , that is Um„ ^(un) = infueF ^(w) with Un e F for all n. We may assume that

^(M/I) < 1 + inf ^{u) ueF

for all n, so that, using the preceding estimate, we get

foralln. Hence the sequence (un) isrelatively weakly compact in L?,([0, 1], dt). So we may extract a subsequence (M„) not relabelled which a(Lj^, L^) converges to M G F because F is weakly closed. As J(u) := i||M||^2 is lower

semicontinuous on L^([0,1], dt) with respect to the topology (7(L^, L^), it follows that

inf ^(u) = \im^{un) = Hm {Aun,Un),ii ji x + J{un)\ ueF n n L ^^H'^H' J

> {Au,u)^^2^j^2^^-\-J(u).

The following is a variant of the preceding ones.


Proposition 3.4. Let F be a separable Banach space. Let G be a nonempty weakly compact subset of L\^{[0, 1], dt) and let W be the closed unit ball of L^,([0, 1], dt). Let A '. G ^^ W a mapping transforming weakly convergent sequences in G into sequences in W converging in measure with respect to the norm topology of F\ Let j be a convex normal integrand defined on[0, I] x F satisfying

0<j(t,x)<p\\x\\-\-y

for all (t,x) G [0, 1] X F for some positive constant p and y. Assume that there is v e L^,([0, l],dt) such that the conjugate integral functional 7*(i;) := JQ j*{t,v{t))dt is finite. Then the functional integral 4>(M) := {Au, u),ôo i\ X + J{u) reaches its minimum on G.

Proof By our assumption it is immediate that

4>(M) := {Au, u)iîôoî^\ ^ -h J(u)

verifies

\^{u)\<{\ + mu\\ii^y F

for all u e G. Let (M„) be a minimizing sequence for 4>, that is lim„ 0(M„) = infêG ^(w) with Un e G for all n. We may assume that (w„) converges a(Lj^, L^f) (alias weakly) in L]^([0, l],dt) to u e G. By our assumption, the sequence (Aun) is bounded in L^([0, 1], dt) and converges in measure to Au with respect to the norm topology of F\ By virtue of Castaing [10], see also Grothendieck [8] for the one dimensional case, we conclude that

l im(^M„, Un)ijoo / I \ = {Au, u),joo j \ \.

As J(u) := /Q j(t, u(t))dt is convex lower semicontinuous on L\^([0, 1], dt), when L) , ( [0 , 1], dt) is endowed with the topology a (L) , , Lf,), see ([18], Theorem VII-7), we have

lim<l>(w„) = inf 4)(M) = lim[(^M„, M„),,OO riv -h J{Un)] n ueG n \^f'^^Fi

> {Au, w) / oo 1 ^ + J(u).

As an application of Proposition 3.1 we provide the following min-max type result.


Proposition 3.5. Let E be a separable super reflexive Banach space, T be a convex weakly compact random set, SY the set of all measurable selections of r and K be a compact space. Assume that 0 : 5 r x /C -> R" is such that for every fixed u € 5 r , ^ (w, . ) is upper semicontinuous on JC and for every fixed V e /C, <!>(., u) is convex and lower semicontinuous for the convergence in measure on Sr. Then there exists a pair (M, £>) e iSp x /C such that

max min <I>(M, V) < <!>(«, v) < min max 0 ( M , V). VGK, ueSr u^Sr veJC

Proof Let us set

p{u) \= max <I>(M, U), VM G Sr veK,

q{v) := min 4>(M, i;), Wv e IC ueSr

Then p(.)is convex lower semicontinuous on Sr for the convergence in measure and q(.) is upper semicontinuous on /C. By virtue of Proposition 3.1, there is u e Sr such that

p{u) = min p{u). ueSr

As q{.) is upper semicontinuous on /C, there isveIC such that

q(v) = max^(i;). veJC

So we get

q(v) < ^(u,v) < p{u).

Other variants of Proposition 3.5 are available. Compare with Proposition 8.3.3 in [13] and the results stated below.

The following is a min-max result involving the use of Sion's theorem.

Proposition 3.6. Let E be a separable reflexive Banach space, and Z be a compact metric space. Let j : [0, l]xExZ be anon-negative normal integrand

satisfying the following conditions (i) for each (t, z) e [0, I] x Z, j (t,., z) is convex lower semicontinuous on E, (ii) for each (t, x) e [0, 1] x E, j(t, Jc,.) is continuous on Z, (Hi) there is a constant c > 0 such that 0 < j{t,x,z) < c( l + ||jc||)/(9r all

(t,x,z) G [0, 1] x £ X Z. Let K := 3^(10, 1], M\{Z)) be the space of Young measures associated with Z, that is, the set of all Lebesgue measurable mappings X : [0, 1] -^ M\.(Z);


M\.(Z) being the compact metrizable space of all probability Radon measures defined on Z, endowed with the vague topology and let Hbe a convex weakly compact subset 6>/L^([0, 1], dt). Let us consider the integral fiinctional

J{u, k) := \ j(t, u(t), z)Xt(dz) dt; V(M, k) eHxIZ. Jo Uz J

Then the following hold: (a) there is a pair (u,X) eH x7Z such that

(b)

maxmin/(M, X) < J(u, X) < vmnm3xJ(u,X). XeTlueH ueH veU

max min J(u, k) = min max 7(M, k).

Proof (a) follows the same line of the proof of the preceding result. Nevertheless this need a careful look. Let us set

and

p(u) := max 7(M, A,), Wu eH ken

q(k) := min J(u,X), VA eU. ueH

For each A, 6 7^ the convex integral functional

J(u, X):= \ j(t, u(t), z)kt{dz) dt

is convex lower semicontinuous on 7Y, indeed, it is easy to see that

jx(t,x) := / j(t,x,z)kt(dz)

is a nonnegative convex normal integrand on [0,1 x E satisfying (iii), so that by ([13], Theorem 8.1.4 or Theorem 8.1.6) the associated integral functional

J(u,k)= / jx(t,u{t))dt Jo

is (sequentially) lower semicontinuous on H with respect to the a(L|,, L^,) topology and so it reaches a minimum on the weakly compact set H. Now we prove that ^ (.) is upper semicontinuous on 71. Indeed, for each u eH.we have the estimate


0<j{t,u{t),z)<c(l-{-\\u(t)\\)

Hence (t,z) -^ j(t, u(t), z) is a Caratheodory integrable integrand. Consequently, for each u eH, the functional

f \f jit,u(t),z)K(dz)\dt

is affine and continuous on (the convex compact) 7?., taking account into the definition of stable convergence for Young measures [2,13]. Then q is concave upper semicontinuous on TZ, recalUng that it is compact for the a{L^.^y, L\..^.)-

topology (aUas stable topology). See [2,13,18] for details. As H is weakly compact, and p is weakly lower semicontinuous on H, there is an element u eH such that

p(u) = min P(M).

As q(.) is upper semicontinuous on TZ, there isX elZ such that

q{X) = max^(A).

So we get

(A.) < J{u,X) < p{u)

which proves (a). (b) follows by applying Sion's theorem [24] and the above considerations to the convex Isc-concave use function J on the product 7i xlZ. D

Remarks. Variants of Proposition 3.6 are available by combining new techniques for Young measures ([13], Theorem 8.1.6) and a general version of inf - sup theorem due to Moreau [22].

Acknowlegments. We wish to thank the referee for his careful reading and useful comments.

References

1. Amrani, A., Castaing, C, Valadier, M.: Methodes de troncature appliquees a des problemes de convergence faible ou forte dans L^. Arch. Rational Mech. Anal. 117, 167-191 (1992)


2. Balder, EJ.: New fundamentals of Young measure convergence. In: Calculus of variations and optimal control, Haifa (1998), Chapman Hall, Boca Raton 24-48 (2000)

3. Balder, EJ.: A general approach to lower semicontinuity result and lower closure in optimal control theory. SIAM J. Control Optim. 22, 570-598 (1984)

4. Balder, E.J., Hess, Ch.: Two generalizations of Komlos theorem with lower closure-type aplications. J. Convex Anal. 3, 25-44 (1996)

5. Benabdellah, H., Castaing, C: Weak compactness and convergences in L^,[£']. Adv. Math. Econ. 3, 1-44 (2001)

6. Beer, G.: Topologies on closed and closed convex sets. Kluwer, Dordrecht 1993 7. Bourgain, J.: The Komlos theorem for vector valued functions. Wrije Universiteit,

Brussels, 1979/12 8. Grothendieck, A.: Espaces vectoriels topologiques. Publicacao da Sociedade de

Matematica de Sao Paulo (1964) 9. Castaing, C: Quelques resultats de convergence des suites adaptees. Sem. Anal.

Convexe Montp. 17, 1-24 (1989) 10. Castaing, C: Topologie de la convergence uniforme sur les parties uniformement

integrables de L l et theoremes de compacite faible das certains espaces du type Kothe-Orlicz. Seminaire d' Analyse Convexe 10, 1-27 (1980)

11. Castaing, C.: Weak compactness and convergences in Bochner and Pettis integration. Vietnam J. Math. 24(3), 241-286 (1996)

12. Castaing, C , Ezzaki, K: Convergences for convex weakly compact random sets in B-convex reflexive Banach spaces. Atti Sem. Mat. Fis. University of Modena, XLVI, 123-149(1998)

13. Castaing, C, Raynaud de Fitte, P., Valadier, M.: Young Measures on Topological Spaces. With applications in control theory and probability theory. Kluwer Dordrecht 2004

14. Castaing, C, Guessous, M.: Convergences in L]^(/X). Adv. Math. Econ. 1, 17-37 (1999)

15. Castaing, C, Hess, Ch., Saadoune, M.: Tightness conditions and Integrability of the sequential weak upper limit of a sequence of Multifunctions. Working paper 2005

16. Castaing, C, Hess, Ch., Saadoune, M.: On various versions of Fatou lemma. Working paper 2006

17. Castaing, C, Saadoune, M.: Dunford-Pettis-types theorem and convergences in set-valued integration, J. Nonlinear Convex Anal. 1(1), 37-71 (2000)

18. Castaing, C , Valadier, M.: Convex Analysis and Measurable Multifunctions, Lectures Notes in Mathematics, Springer, Berlin, 580, 1977

19. Garling, D.J.H.: Subsequence principles for vector-valued random variables. Math. Proc. Cambridge Philos. Soc. 86, 301-311 (1979)

20. Komlos, J.: A generalization of a problem of Steinhaus. Acta Math. Acad. Sci. Hungar 18, 217-229 (1967)

21. Krupa, G.: Komlos theorem for unbounded random sets. Set-Valued Anal. 8(3), 237-251(2000)

22. Moreau, J.J.: Theoremes "inf-sup". C. R. Acad. Sci. Paris, 258, 2720-2722 (1964) 23. Saadoune, M.: A new extension of Komlos theorem in infinite dimensions. Appli

cation: weak compactness in LJ^. Portugaliae Math. 55, 113-128 (1998) 24. Sion, M.: On general minimax theorems. Pacific J. Math. 8, 171-176 (1958)

Adv. Math. Econ. 10, 31^9 (2007) Advances in


©Springer 2007

Capital-labor substitution and indeterminacy in continuous-time two-sector models*

Jean-Philippe Garnier^, Kazuo Nishimura^ and Alain Venditti^

^ Universite de la Mediterranee, GREQAM, Marseille 13002, France (e-mail: [email protected])

^ Institute of Economic Research, Kyoto University, Kyoto 606-8501, Japan (e-mail: [email protected])

^ CNRS, GREQAM, Marseille 13002, France (e-mail: venditti @ ehess.univ-mrs.fr)

Received: August 8, 2006 Revised: November 24, 2006

JEL classification: C62, E32, 041

Mathematics Subject Classification (2000): 37C70, 37N40, 37C75, 39A11

Abstract. The aim of this paper is to discuss the role of the elasticity of capital-labor substitution on the local determinacy properties of the steady state in a two-sector economy with CES technologies and sector-specific externalities.

Key words: sector-specific externalities, constant returns, capital-labor substitution, indeterminacy.

1. Introduction

Recently, a large number of papers have established the fact that locally indeterminate equilibria and sunspots fluctuations easily arise within two-sector infinite-horizon growth models with sector specific external effects in production and linear preferences. For instance, considering Cobb-Douglas technologies, Benhabib and Nishimura [2] prove within a continuous-time model that the existence of local indeterminacy is obtained if and only if there is a reversal of factor intensities between the private and social levels. The consumption

* We are grateful to T. Seegmuller, K. Shimomura and an anonymous referee for useful comments and suggestions.

^ See Benhabib and Farmer [1].



http://ehess.univ-mrs.fr

32 J.-P. Gamier et al.

good has indeed to be capital intensive from the private perspective but labor intensive from the social perspective.

When CES technologies are introduced, a much larger set of configurations is compatible with local indeterminacy. In particular, Leontief or Unear technologies for the consumption good sector may be considered. Therefore, we will consider CES technologies in a continuous-time model and provide a complete analysis of the local determinacy properties of equilibria. We will show that, even with asynmietric factor substitutabiUty properties, local indeterminacy is still based on a reversal of factor intensities between the private and social levels, what we call quasi factor intensity. We will also show that local indeterminacy is compatible with a Leontief technology in the consumption good sector, and that this result is preserved if the elasticity of capital-labor substitution in that sector is not too large.

The paper is organized as follows: § 2 presents the basic model with the production structure, the intertemporal equihbrium and the steady state. In § 3 we provide all the results on the existence of local indeterminacy depending on different configurations for the CES coefficients at the private and social levels. § 4 finally contains some concluding comments. All the proofs are gathered in a final Appendix.

2. The model

2.1. The production structure

We consider an economy producing a pure consumption good yo and a pure capital good yi. Each good is assumed to be produced by capital XQJ and labor xij, j = 0,1, through a CES technology which contains sector specific ex-temahties. The representative firm in each industry indeed faces the following function:

yj = (i oyôT + î^-îT + 'J^^^J^ ^^J^y'^''' ' = 0,1, (1)

with îj > 0, py > - 1 and cxj = 1/(1 -h />;) > 0 the elasticity of capital-labor substitution. The positive extemahties are equal to

ej(Xoj, Xij) = bojX^j ^ + b\jX^j\

with bij > 0 and Xij denoting the average use of input / in sector j . We assume that these economy-wide averages are taken as given by each individual firm. At the equilibrium, since all firms of sector j are identical, we have Xij = Xij and we may define the social production functions as follows:

yj

Capital-labor substitution and indeterminacy 33

= {ôjx-p + ^rjx;py'"' (2)

with Pij = Pij + bij. The returns to scale are therefore constant at the social level, and decreasing at the private level. We assume that in each sector 7 = 0 , 1 , Ôj -^ Pij = 1 so that the production functions collapse to Cobb-Douglas in the particular case pj = 0. Labor is normalized to one, i.e. JCQO + - oi = 1» and the total stock of capital is given byxi =jcio + xii.

Choosing the consumption good as the numeraire, i.e. /?o = 1, a firm in each industry maximizes its profit given the output price pi, the rental rate of capital wi and the wage rate WQ. Its profit is

^j = Pjyj - ^0X0j - w\x\j.

The first-order conditions subject to the private technologies (1) are

pjPij {yj/îjY^'' = ^ ^ ^ J = 0 ,1. (3)

From (3) we have

1

Xij/yj = [pjPij/wi)^^ = aijiwi, Pj), /, j = 0, 1. (4)

We call Uij the input coefficients from the private viewpoint. If the agents take account of externalities as endogenous variables in profit maximization, the first-order conditions subject to the social technologies (2) are

Pjîj [yj/îj] '^^' =î^ /, 7 = 0, 1.

and the input coefficients become

atjiwi, Pj) = \^pjPij/wij , /, j = 0, 1. (5)

We call aij the input coefficients from the social viewpoint. We also define

aij (wi ,pj) = (Pij /Pij)aij (Wi ,pj), (6)

as the quasi-input coefficients from the social viewpoint, and it is easy to derive that

aijiwi, Pj) = aijiwi, Pj) yPij/Pijj \Pj/il+Pj)

Notice that aij = dtj if there is no extemahty coming from input / in sector j , i.e. bij = 0, or if the production function is Cobb-Douglas, i.e. pj = 0. As we will show below, the factor-price frontier, which gives a relationship between


input prices and output prices, is not expressed with the input coefficients from the social viewpoint but with quasi-input coefficients from the social viewpoint.

These quasi-input coefficients at the social level correspond to input coefficients evaluated at the equihbrium, i.e., taking into account that the sector-specific extemahties are equal to the actual equilibrium amounts of input used in each sector. They differ from the input coefficients at the social level since the producers consider the external effects as given when they maximize profit.

Based on these input coefficients straightforward computations allow to establish various lemmas. We first show that the factor-price frontier is determined by the quasi-input coefficients from the social viewpoint.

Lemma 1. Denote p = (1, pxY, w = {WQ, wiY and A{W, p) = [aij{wi, pj)]. Then p = A\w, p)w.

The factor market clearing equation depends on the input coefficients from the private perspective.

Lemma 2. Denote x = {l,x\)\ y = (yo, yiY and A(w, p) = [aij(wi, pj)]. Then A(w, p)y = x.

The basic intuition for these two results is the following: since the producers consider the external effects as given when they maximize profit, the input demand functions do not directly depend on the externalities so that the factor market clearing equation is only determined from the input coefficients at the private level. On the contrary, since external effects actually modify the equilibrium amount of output in both sectors, the prices of the final goods are affected so that the factor-price frontier is determined from the quasi-input coefficients at the social level.

Note that at the equihbrium the rental rate is a function of the output price only, w\ = w\(p\), while outputs are functions of the capital stock and the output price, yj = yj(x\, p\), j =0,1.

We now examine some comparative statics. The factor-price frontier satisfies the Stolper-Samuelson theorem:

Lemma 3. dw\/dp\ = -— —7—• auaoo - a\oao\

The factor market clearing equation finally satisfies the Rybczynski theorem:

Lemma 4. dy\/dx\ = . «11«00 -«10«01

In an optimal growth model without external effects, i.e. bij = 0, we have A{w, p) = A(w, p). The Rybczynski and Stolper-Samuelson theorems are equivalent since [dy/dx] = [dw/dpY. However, in presence of externalities, the Rybczynski effects depend on the input coefficients from the private perspective while the Stolper-Samuelson effects depend on the quasi-input coefficients from


the social perspective. The duality between these two effects is thus destroyed. This property is again explained by the same intuition as the one given above. Local indeterminacy of equihbria will be a consequence of this broken duaUty.

2.2. Intertemporal equilibrium and steady state

A representative agent optimizes a hnear additively separable utility function with discount rate 8 > 0. This problem can be described as:

+00

max / [Pooxooity + iîoîoCO"^ + ô(Xoo(0, Xio(O))"^ e-^'dt {Xijit)] J ^ ^

0

s.t. yi(t) = {Poixoiit)-'' + PuxniO-f^' êi(Xoi{t),Xu(t)))~^ x\{t) = y\(t) - gxiit) 1 =ôo(0 + -ôi(0 xi(t) = xioit) + xn(t) xiiO) and {ej(Xoj(t), Xiy(r))},>o, j = 0, 1, given

where g > 0 is the depreciation rate of the capital stock. We can write the modified Hamiltonian in current value as

n = [Pooxooity + Pioxioity -êoiXooit), XioW))'^ +woit) (1 - xoo(t) - xoi(0) + mit) (xi(0 - xioit) - xu(t))

+P\{t)({Poixoi(tr^' +Puxnity êiiXoiiO^Xuit)))"^ -gxiit)).

The static first-order conditions are given by equation (3). The necessary conditions which describe the solution to the optimization problem are given by the following equations of motion:

x\it) = y\{xi(t), piit)) - gxiit)

piit) = (8^g)p,(t)~w^(pi(t)). ^'^

Any solution {x\ (0, P\ (t)}t>o that also satisfies the transversality condition

lim ^-^';7i(r)jci(0=0 r^+oo

is called an equiUbrium path. A steady state is defined by a pair (jCj*, pj) solution of

y\(xup\) = gxi mipx) = (S^g)pi. ^""^

We introduce the following restriction on parameters' values which will ensure the existence of an unique non trivial steady state:

36 J.-R Gamier et al.

Assumption 1. fiu > 8 -\- g and p\ e (pi, +oo) with

Pi = InPu

ln{^)-lnPu € ( - 1 , 0 ) (9)

Considering the fact that, within continuous-time models, the discount rate 8 and the capital depreciation rate g are quite small, the restriction fi\\ > 8+g does not appear to be too demanding. Assumption 1 precisely guarantees positiveness and interiority of all the steady state values for input demand functions jc/y. Moreover, it allows to prove the following result:

Proposition 1. Under Assumption 1, there exists a unique steady state (jCp p\) > 0 such that

(tt) l+PO

Po\

(&i) 1

1+pl (a) 1 1+PO

(fe)A,^„^'» )Soi

P\ = 8-\-g 3 plO/Ôl\ 1+PO m r+pT •Pn

PQ(1+PI) \ Pl(l+P0)

Po\ + P\0

l+PO

3. Main results

We start by hnearizing the dynamical system (7) around (jc*, /?*):

(

t(^hPV-8 0 )

Any solution from (7) that converges to the steady state (jc*, /?*) satisfies the transversaUty condition and is an equilibrium. Therefore, given :ci (0), if there is more than one initial price pi(0) in the stable manifold of (Xj*, pj), the equilibrium path from jci (0) will not be unique. In particular, if J has two roots with negative real parts, there will be a continuum of converging paths and thus a continuum of equilibria.

Definition 1. If the locally stable manifold of the steady state (x*, p*)is two-dimensional, then (x*, pj) is said to be locally indeterminate.


The roots of J are given by the diagonal terms. We know from Lemmas 3 and 4 that dy\/dx\ corresponds to the factor intensity difference from the private viewpoint and dw\/dp\ corresponds to the quasi-factor intensity difference from the social viewpoint. Using the definitions of input coefficients given in § 2, we may indeed interpret the elements of dy\/dx\ and dwi/dpi as follows:

Definition 2. The consumption good is said to be:

(i) capital intensive at the private level if and only ifa\\aoo — ^loôi < 0. (ii) quasi capital intensive at the social level if and only if a\\aoo —

a\oao\ < 0, (iii) capital intensive at the social level if and only if a\\aoo — «ioôi < 0-

We may thus relate the input coefficients to the CES parameters:

Proposition 2. Let Assumption 1 hold. At the steady state:

(i) the consumption good is capital (labor) intensive from the private perspective if and only if

^00^11/

j _ +P0

Pl(l+PO)

^01

(ii) the consumption good is quasi capital (labor) intensive from the social perspective if and only if

\ Pl(l+P0)

< ( > ) 0

J (iii) the consumption good is capital (labor) intensive from the social per

spective if and only if

p\ P\-PO

V 00^11/ I h\ < (>)0.

/

The following proposition estabhshes that local indeterminacy requires a capital intensity reversal from the private input coefficients to the quasi-input coefficients.


Proposition 3. Let Assumption 1 hold. The steady state is locally indeterminate if and only if the consumption good is capital intensive from the private perspective, but quasi labor intensive from the social perspective.

To get indeterminacy in a framework with constant returns to scale at the social level, we need a mechanism that nuUify the duahty between the Rybczynski and Stolper-Samuelson effects. As shown in § 2.1, the Rybczynski effect is given by the input coefficients from the private perspective while the Stolper-Samuelson effect is given by the quasi-input coefficients from the social perspective. In the presence of external effects, the duahty between these coefficients is broken and local indeterminacy may appear.

This mechanism is very similar to the one exhibited in the contribution of Benhabib and Nishimura [2]. However in the current paper, it follows that depending on the value of the elasticities of capital-labor substitution, the capital intensity reversal from the private input coefficients to the quasi-input coefficients does not necessarily requires a capital intensity reversal from the private to the social level.

Propositions 2 and 3 show that the stabihty properties of the steady state depend, among all the parameters, on whether the ratios jîoiôi/iîiiôo and ^xofimlûÔQ are lower or greater than 1. Around the steady state, it can be easily shown that if the elasticities of capital-labor substitution are identical across sectors, the consumption good is capital intensive at the private level if and only if fimPox/fin^Qo > 1 while it is capital intensive at the social level if and only if ^ lOi oi/Aii oo > 1- With asymmetric elasticities, as it is shown in Proposition 2, the capital intensity differences between sectors also depend on the parameters po and p\.

Notice also from Proposition 2 that the capital intensity differences at the private and quasi social level are linked as follows:

yîito

We know from Proposition 3 that local indeterminacy requires the consumption good to be capital intensive at the private level, i.e. Z? < 0, but quasi labor intensive at the social level, i.e. > 0. It follows from the above expression that when Z? < 0, a necessary condition for b to be positive is given by the following Assumption:

Assumption 2.

Let us first consider the configuration ^xoPm/^n^m < 1 which is known in the case with symmetric elasticities of capital-labor substitution to imply


local deteraiinacy of the steady state.^ The following Proposition shows on the contrary that as soon as the elasticities are sufficiently asymmetric, there is room for local indeterminacy.

Proposition 4. Under Assumptions 1-2, let

1 > and 1 > -—7— > —n 7—. (10)

Then there are p G (—1, 0), po > 0 such that the following results hold:

(i) for any given po e (p , po), there exist p > max{0, po} and p\ > p such that the steady state is locally indeterminate when p\ 6 (p , p\);

(ii) for any given po > Po, there exists p > po such that the steady state is locally indeterminate when p\ > Py

Before giving interpretations of our results, we need to justify the part of condition (10) which concerns the CES coefficients at the social level: this inequahty ensures the existence of the bounds po and p^, and the possible occurrence of local indeterminacy when both elasticities of capital labor substitution are close to zero.

All the various conditions for the existence of indeterminacy in Proposition 4 are based on the restriction p\ > po which implies a lower elasticity of capital-labor substitution in the investment good sector than in the consumption good sector. Since under Assumption 1 the elasticity of capital-labor substitution in the investment good sector is necessarily finite, it clearly appears that the same condition has to be considered for the investment good sector. In case (i), if the elasticity of capital-labor substitution in the consumption good sector is large enough, the extreme configuration of a Leontief technology in the investment good sector is also ruled out. On the contrary, if the factor sub-stitutabihty of the consumption good sector is low enough, local indeterminacy becomes compatible with a Leontief technology in the investment good sector.

Let us now consider the configuration with ^loiôi/îi/ôo > 1 and P\QPQ\/PuPm < 1 which is known to imply the existence of local indeterminacy in the case with Cobb-Douglas technologies in both sectors.^ Notice that Assumption 2 is then necessarily satisfied.

^ See Benhabib and Nishimura [2] for Cobb-Douglas economies and Nishimura and Venditti [3] for CES economies with symmetric elasticities of capital-labor substitution.

^ See Benhabib and Nishimura [2].

40 J.-R Gamier et al.

Proposition 5. Under Assumption 1, let

——— >\and\> . . > ^ - ^ . (11)

r/i^n r/z r are p_ e (—1,0) and po > 0 such that the following results hold:

(i) for any given po e {p^^, po), there exist p € (pi, 0) and p\ > 0 such that the steady state is locally indeterminate when p\ E (p , pi);

(ii) for any given po > po, there exists p e (0, po) such that the steady state is locally indeterminate when p\ > p..

Proposition 5 covers the formulation with symmetric elasticities of capital-labor substitution across sectors previously analyzed in Benhabib and Nishimura [2] with Cobb-Douglas technologies and Nishimura and Venditti [3] with CES technologies having symmetric elasticities of capital-labor substitution. The part of condition (11) which concerns the CES coefficients at the social level is introduced to ensure the possibiUty of local indeterminacy when the consumption good technology is Cobb-Douglas while the investment good technology is Leontief [see case (ii)]. This explains why such a condition does not occur under the assumption of symmetric elasticities. However, since the plausible values of 8 and g are close to zero, this condition does not imply a strong restriction on the CES coefficients.

Contrary to the previous Proposition, the existence of local indeterminacy does not require the elasticity of capital-labor substitution in the consumption good sector to be larger than the one in the investment good sector. The extreme configuration of an infinite factor substitutability in the consumption good sector is again ruled out. Moreover, as shown in case (ii), the occurrence of local indeterminacy with factors complementarity in the investment good sector is not compatible with a Cobb-Douglas technology in the consumption good sector and requires an elasticity of capital-labor substitution significantly lower than 1.

Let us finally consider the configuration with P\o^mlP\\^{yo > 1 which is known in the case with symmetric elasticities of capital-labor substitution to be compatible with the existence of local indeterminacy only when the common elasticity is lower than one, or po = pi = p > 0. In the next proposition, we show that in the asymmetric case, po has to be positive but under Assumption 1, pi can be negative, equal to 1 or positive.

Proposition 6. Under Assumptions 1-2, let

^10^01 , ,^10^01 . ..^. > 1 and -—:r- > 1. (12)

^11^00 i^lliÔO

^ See Nishimura and Venditti [3].


Then there are Pf > 0 and Po > Pr^ such that the following results hold:

(i) for any given po ^ (PQ. PQ), there exist p e (p\, po) ^^^ Pi > 0 such that the steady state is locally indeterminate when p\ e {p , p\);

(ii) for any given po > po, there exists p > 0 such that the steady state is locally indeterminate when p\ > p .

Proposition 6 confirms only part of the conclusions obtained by Nishimura and Venditti [3] under symmetric elasticities of capital-labor substitution across sectors: local indeterminacy requires some elasticity lower than one but only in the consumption good sector. Indeed, the existence of multiple equilibria does not rely on particular restrictions for the factor substitutability in the investment good sector. In particular, some elasticities greater than 1 are compatible with local indeterminacy in case (i). However, as in the previous Propositions, a Leontief technology for the investment good can be reached only if the elasticity in the consumption good sector is low enough.

4. Concluding comments

Within the framework of CES technologies, we precisely show how much the elasticities of capital-labor substitution can differ across the industries while leading to local indeterminacy of equilibria. For instance, given a value for the elasticity of substitution in the consumption good sector, the elasticity of substitution in the investment good sector can be arbitrarily large.

5. Appendix

5.1. Proof of Lemma 1

Substituting (4) into the social production functions (2) gives

It follows that

P

-VPj

Multiplying both sides by p- ^^ then gives

Pj = aojWo-\-a\jW\.

The result follows considering that po = 1-



By definition xij = atjjj and thus

Xi =« /oyoH-f l / i> ' i .

The resuh follows considering that JCQ = 1. D


The result immediately follows from the fact that the function A{w, p) is homogeneous of degree zero in w and p. u


The result follows from a direct differentiation of JC, = a/ojo + «/iJi under the assumption that prices are constant. D

5.5. Proof of Proposition 1

From equation (3) considered at the steady state with y\ = gxi and w\ =

Using now the social production function (2) for the investment good we derive

• 01 = (fe) - "n

J_

\ Po\

(llL.) 1

^^1

and thus

^01

xu

/(g,)A -A,^

V Po\

J_ ' p\

Finally, we easily obtain from (3):

i+Pi

/SoojSii \xnj V- oo/

\/3oo/5ii/

1+Po

i+Pi

^00

(13)

(14)


Considering (13), (14) and the fact that XQQ + XQI = l,x\ = x\o + x\\, we get the final expression of JCJ*. Equation (3) for / == 1 and 7 = 0 gives

u ^ i ^ i ^ i o ^ A i ^ ^ ^ + ^ ^

The final expression of /7j is then derived from (14) and the fact that w\ =


(i) From (4) we derive

/an aio\ xu (, ^lOôA aiiaoo - aioôi = ôofloi I I = ^ oo oi — I 1 I

\«oi am/ - 01 \ ^00- 11 /

and the result follows after substitution of (13) and (14) into the previous expression,

(ii) From (6) we derive

< ii«oo - «io«oi = oo oi ô\ P\\ aox PooPioaooJ

^n Po\ - 11 /< i ioii oi i ooi ii - 10 01 \

Po\ p\\ - 01 \ PooPu Piopoi ^00^11 y

and the result follows after substitution of (13) and (14) into the previous expression,

(iii) The result is derived as in (i) but considering instead the input coefficients at the social level (5). n


From Lemma 2, xi = fliojo + îiji- Moreover, at the steady state, y\ = gx\, and it follows

aioyo + gaux\ = xi ^ aioyo = [1 - gau]x\ > 0

Therefore,

dy\ «oo «oo[l - gau] + a\oao\g g = g = < 0

dx\ auaoo-aioaoi «ii«oo - «io«oi if and only if anaoo - «io«oi < 0. From Lemma 1, pi = ao\wo -i- a\iwi. Moreover, at the steady state, (8 + g)pi = w\, and it follows


{8 -f g)aoiWo + (5 + g)auw\ = wi ^ (5 + g)ao\wo

= [l-(8 + g)an]wi > 0

Therefore,

9/?i auaoo-d\oao\

_ [1 - (6 + g)aii]flQQ + (3 + g)aioaoi

auaoo -^10^01

if and only if auaoo — aioaoi > 0.

< 0


Proposition 3 shows that local indeterminacy occurs if and only if ^ < 0 and ^ > 0. Under Assumption 1, jSn > 8-{- g and pi e {p\, -hoo), while po > — 1 without particular restriction. We will then study the sign of Z? and b for different values of po and p\ over these intervals. Our strategy consists in considering a fixed value for po and varying p\ in order to find intervals of values in which local indeterminacy occurs. To simplify the analysis, we have to impose some restrictions on the parameters po and p\ such that the following function

g{p\) = '(fc)*-^\^

/3oi (15)

/

IS monotone mcreasmg.

Lemma 5. Under Assumption 1, there is a po € (—1,0) such that for any given Po > Po there exists p\ e [pi, 0) such that g(p\) is a monotone increasing function for all pi > p\.

Proof Notice first that

PO

a p i \ i + p i / (i + P i ) ^ ' ' ' ap i \ P i ( i + Po)/ Pi(l + Po)/ Pf(l + Po)

It follows that for any given po > 0, g(p\) is a monotone increasing function when p\ > 0. Consider now the case po e (~1,0). We can write g{p\) as follows:

g(p\) = exp Pi - Po , -In

^^^p[Tfk^-(fe)]-^ i Pi(l + Po) I Po\

= exp{/(pi)}.


If / (p i ) is monotone increasing, so is g(p\). We easily compute

/O0(l+Pl) {§^)'^-Pu ^01

f(pl) =

mim-Po){^)'în{i^)

Pf ( l+P l )2 ( l+P0) (li^)^-^n

We have now to compute the sign of Um^jô f'iPx)- A Taylor expansion of order two allows to show that Hmpi_>o f iPx) > 0 for any po > po with

Po = - l4-/n € ( - l , P i )

Notice finally that Umpj_^pj /^(/Oi)= —ooandlimpj_^^j /(pi)=+ooifpo<Pi, while limp J _^pj f'(p\) = +ooandlimpj_^pj f(p\) = -ooifpo > Pi.Wethen conclude from all this that for any given po > Po, there exists p\ e [p\, 0) such that g(p\) is a monotone increasing function for all p\ > p\. u

It clearly appears from Proposition 2 that under the conditions of Lemma 5, h and b are monotone decreasing functions of p\.

We may now prove Proposition 4: in the first part of the proof we consider positive values of p\ only. Notice that UHopital's rule gives

]im g{pi) V^ + ^y

-_p^

(l+P0)^01 (16)

Consider the capital intensity difference at the private level b. The previous result imphes

\im b = \ Pl->0 \ôoP\\) m _PQ_

O+P0)^0l (17)

Notice that the right-hand side of (17) is a monotone function of po. Under Assumption 2 and ^XQPQX/Pnfioo < 1, we derive Hm 1-0 Û>0 ^ - t follows that there exists PQ G [—1, 0) such that limpj_ô^ > Oforanypo > PQ-Similarly, we get

lim Pl-»+CXD

1

(18)

Notice again that the right-hand side of (18) is a monotone function of po. If the following condition holds


POOPll )Soi

then limpi_^+oo^|pQ^_^^ = 0_ and limpi^+oo^|pQ<o < 0- It follows that under (19), linipi^+oo ^ < 0 for any po > -I. Notice finally that if po = p\ we have > 0. Considering Lemma 5 under (19), we then conclude that for any given Po > Po' ^ ^ ^ exists p\ > max{0, po} such that Z? < 0 for all p\ > pi.

Consider now the quasi capital intensity difference at the social level b. equation (16) implies

yîo/ôi 1 PO

li,n ^ = 1 - | l l |2^ ( ^ ^ "^ (^-±1) " ' ' •" . (20) PI 1 POO

Notice that the right-hand side of (20) is a monotone function of po. Under

Assumption 2 and ^\oô\/P\\^m < 1, we derive limp^ô^l >o ^ ^ ^^ that

there exists PQ G [—1,0) such that limpjô^ > 0 for any po > PQ- Since

b > b,we get when po = PQ, b\ ^Q ~ ^ while b\ ^Q ^ ^^ ^ ^ o ^ ' o-Similarly, we have

nm ^ = i _ | i i ^ f ^ l o ^ ) ^ ( i ± i ^ ) . (21)

Notice again that the right-hand side of (21) is a monotone function of po. If the following condition holds

we get with hmpi_^+oo ^p^ô < ^ and hmpi_^+oo |po_^+oo ^ - Therefore, under (22), there exists PQ > 0 such that hmpj^+oo ^ > 0 for any po > PQ-Considering Lemma 5 under (22), we then conclude the following results:

- for any given po e (p^, PQ), there exists p^ > 0 such that ^ > 0 for all

Pi € [0, p^). Notice that since b > b, p\ > p\; - for any given p{)> p^.b > ^ for all p\ > po-

Notice that under Assumption 2, (22) imphes (19). Therefore, under (22), we conclude that local indeterminacy occurs in the following cases:

(i) for any given po e (max{p(J, p^}, p^) and p\ e {p_^,p\) with p^ =

p\ > max{0, Po} and px = p^ > p_^\

(ii) for any given po > PQ and p\ > p with p = p^ > po. n



We use the same kind of arguments as in Proposition 4. Consider first the capital intensity difference at the private level b. Under Assumption 2 and i iO) oi// i 1) 00 > 1, we derive from (17) that limp^ôb\^^Q < 0 while lim^jô 1 _L > 0- Therefore, there exists PQ > 0 such that limpj_ô ^ < 0 for any po e ( - 1 , /OQ). Similarly, we derive from (18) that lim^j^+cx) ^ < 0 for any po > — 1. Notice also that

Considering Lemma 5, we then conclude the following results:

(a) for any given po ^ (— 1. Po)» < 0 for all p\ > 0; (b) for any given po > PQ' ^ ^ ^ exists p | e (0, po) such that Z? < 0 for all

P\ > PI

Consider now the quasi capital intensity difference at the social level b. Under Assumption 2 and PioPox/^wPm < 1, we derive from (20) that hm^jô 1 >o ^ ^ while hmp^ô ^| _^_i < 0- Therefore, there exists PQ € (-1,0) such that hmp^ô ^ > 0 for any po > PQ. Similarly, we derive from (21) that limpi^+oo ^l^^ô ^ ^ ^^^^^^ ^^^^ ' ^^^^^ hmp,-^+oo ^|po^+oo ^ • Therefore, under (22), there exists PQ > 0 such that Ump - +oo > 0 for any Po > PQ- Notice also that

1 _ ^ 1 _ ^ Q Q /^00^1 A ^ 0 ^24)

is an increasing function of po since ^QQPU/P\QPQ\ < 1. Considering that PI e (-1,0) is defined as hmp,_ô ^L_^2 = 0, we derive from (20):

«U=„=.. = ' - (^)^^>0. (25)

Therefore, ^|pi=po ^ ^ ^ ^ ^y Po ^ PQ-Considering Lemma 5 under (22), we then conclude the following results:

(c) for any given po e (PQ , PQ), there exists p^ > max{0, po} such that > 0 for all PI G[0,p2);

(c) for any given po > Pg, ^ > 0 for all p\ > 0.

Therefore, under (22), we conclude from points (a)-(e) that local indeterminacy occurs in the following cases:


(i) for any given po e (p^, max{p^, p^}) and p\ e (Pj, pi) with p^ e

(pi, 0) and pi= p\> max{0, po}; (ii) for any given po > max{pQ, PQ} and p\ > p with p = pj > 0. n


We use the same kind of arguments as in Proposition 5. Consider first the capital intensity difference at the private level b. Under Assumption 2 and ^xoPoxIPnPm > 1, we have already shown in the proof of Proposition 5 that there exists PQ > 0 such that:

(a) for any given po e (~1, PQ), Z? < 0 for all p\ > 0; (b) for any given po > Pg, there exists p | e (0, po) such that Z? < 0 for all

Pi > PI

Consider now the quasi capital intensity difference at the social level b.

Under Assumption 2 and ^XQ^QX/PWÔO > 1, we derive from (20) that

Umpjô^l ==0 " ^ while limpi_>o^| _ _L.OO ^ ^' Therefore, there exists

PQ > 0 such that Umpjô^ > 0 for any po > Pg- Since b > b, we

get PQ < PQ. Similarly, we derive from (21) that limpj^+oo^| ^Q < 0

while Umpi^+oo^l _>+oo ^ ^' Therefore, there exists PQ > 0 such that

hmpi^+oo ^ > 0 for any po > PQ. Now notice from (20) and (21) the following property:

1

-^+^ ^po=Po y ^Qi J \8 + gj

Therefore, we get PQ < PQ. Consider finally equations (24) and (26). Since PQ > 0 is such that ]imp^ôb\ 2 = 0, we derive from (20): b\^^^^^^2 < 0. On the contrary,

since Pn > 0 is such that Umî^+cx) 1 3 = 0, we derive from (21):

Therefore, we get b\p^=pQ > 0 for any po > PQ-Considering Lemma 5, we then conclude the following results:

(c) for any given po € (PQ, PQ), there exists p^ e (0, po] such that Z? > 0 for all PI €[0,p?);

(c) for any given po > PQ, ^ > 0 for all p\ > 0.

Notice that since ^ > Z?, we get p^ > p}.


Therefore, we conclude from points (a)-(d) that local indeterminacy occurs in the following cases:

(i) for any given po ^ (PQ' ^^^{Po^ Po^) and pi e (Pj, pi) with p^ e

(pi, po) and px= p\ e (0, po]; (ii) for any given po > max{p(J, p^} and pi > p^ with p^ = p} e (0, po).

n

References

1. Benhabib, J., Farmer, R.: Indeterminacy and sunspots in macroeconomics. In: Taylor, J.B., Woodford, M. (eds) Handbook of Macroeconomics. Amsterdam North-Holland pp. 387^48 (1999)

2. Benhabib, J., Nishimura, K.: Indeterminacy and sunspots with constant returns. J. Econ. Theory 81, 58-96 (1998)

3. Nishimura, K., Venditti, A.: Indeterminacy and the role of factor substitutability. Macroecon. Dyn. 8, 436-465 (2004)



©Springer 2007

Weak and strong convergence theorems for new resolvents of maximal monotone operators in Banach spaces Takanori Ibaraki^ and Wataru Takahashi^

^ Information Security Promotion Agency, Nagoya University, Furo-cho, Chikusa-ku, Nagoya, Aichi 464-8601, Japan (e-mail: [email protected])

^ Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, Oh-okayama, Meguro-ku, Tokyo 152-8552, Japan (e-mail: [email protected])

Received: September 11, 2006 Revised: November 2, 2006

JEL classification: C61, C63, C02

Mathematics Subject Classification (2000): 49M05,47H05,47H09

Abstract. In this paper, we prove weak and strong convergence theorems for new resolvents of a maximal monotone operator in a Banach space which are connected with the proximal point algorithm of Rockafellar (SIAM J. Control. Optim. 14:877-898, 1976). Using these results, we consider the problem of finding minimizers of convex functions defined on Banach spaces.

Key words: Banach space, generalized nonexpansive retraction, proximal point algorithm, convex minimization problem

1. Introduction

Let / / be a Hilbert space and let T be a maximal monotone operator from H to / / . It is well known that many problems in nonlinear analysis and optimization can be formulated as follows: Find

u eH such that O G TU. (1.1)

Such a M 6 / / is called a zero point (or a zero) of 7 . A well-known method for solving (1.1) in a Hilbert space H is the proximal point algorithm: x\ e H and



52 T. Ibaraki and W. Takahashi

Xn+\=Jrn^n. « = 1, 2, . . . , (1.2)

where {r„} c (0, oo)andyr = (Z+rJ)"^ forallr > 0. This algorithm was first introduced by Martinet [5]. In [19], Rockafellar proved that if hm inf^ôo n > 0 and r~ ^ 0 ^ 0, then the sequence {xn} defined by (1.2) converges weakly to a solution of (1.1). Motivated by Rockafellar's result, Kamimura and Takahashi [6] proved the following two convergence theorems.

Theorem 1.1 ([6]). Let H be a Hilbert space and letT c H x H be a maximal monotone operator. Let Jr — {I -\- rT)~^ for all r > 0 and let {xn} be a sequence defined as follows: x\ = x e H and

Xn-\-\ = OinXn + (1 - an)^r„Âi, « = 1, 2 , . . . ,

where {an\ C [0, 1] and {r } C (0, oo) satisfy

limsupa„ < 1 and liminf r > 0.

If T~^0 ^ 0, then the sequence {xn'S converges weakly to an element v of r~^0, where v — Um„_>oo ^r-io^n ^^^ ^T-Ô ^ ^^^ metric projection of H onto I 'Ô.

Theorem 1.2 ([6]). Let H be a Hilbert space and let T c H x H be a maximal monotone operator Let Jr = (I + rT)~^ for all r > 0 and let {JC„} be a sequence defined as follows: x\ = x e H and

Xn-^\ = anX + (1 - an)Jr„Xn, « = 1, 2 , . . . ,

where {an} C [0, 1] and {r„} C (0, oo) satisfy hm„_ôoa« = 0, X ^ i <^n = cx), and liminfn_>oo rn = oo.IfT~^0 ^ 0, then the sequence {jc„} converges strongly to Pj-ô^^^^ where PJ-IQ is the metric projection of H onto T~^0.

These results were extended to more general Banach spaces; see Kamimura and Takahashi [7-9], Ohsawa and Takahashi [13], Kohsaka and Takahashi [11] and Kamimura et al [10]. Recently, Ibaraki and Takahashi [4] found new resolvents in a Banach space which are connected with a maximal monotone operator.

Our purpose in this paper is to extend Kamimura-Takahashi's theorems to Banach spaces by using new resolvents of a maximal monotone operator. Such problems were posed in [22,23].

In § 3, we prove a weak convergence theorem which generaUzes Theorem 1.1. In § 4, we obtain a strong convergence theorem which generaUzes Theorem 1.2. Using these theorems, we consider the problem of finding mini-mizers of convex functions defined on Banach spaces.

Weak and strong convergence theorems for new resolvents 53

2. Preliminaries

Let £" be a real Banach space with its dual £"*. We write Xn -^ XQ to indicate that the sequence {x„} converges weakly to XQ. Similarly, x„ -^ XQ will symbolize the strong convergence. A Banach space E is said to be strictly convex if

\\x\\ = \\y\\ = \, x ^ j ^ P ^ < 1.

Also, E is said to be uniformly convex if for each e e (0, 2], there exists 8 > 0 such that

\\x\\ = \\y\\ = h \\x-y\\>6=>\\^^ < 1 - 5 .

The normahzed duahty mapping J from E into £"* is defined by

J(x) := jjc* eE* : (x,x*) = \\xf = \\x*f], Vjc e E.

An operator T c E x E"^ with domain D(T) = {x e E : Tx ^ 0} and range R(T) = U{Tx : X e D(T)} is said to be monotone if (jc - j , jc* - j*) > 0 for any (JC, jc*), (y, j * ) e T. An operator T is said to be strictly monotone if {x — y,x* — y*) > 0 for any (x, x*), (y, j* ) e T {x ^ y). A monotone operator T is said to be maximal if its graph G(T) = {(JC, JC*) : JC* € TJC} is not properly contained in the graph of any other monotone operator. If T is maximal monotone, then the set r~^0 = {u e E : 0 e Tu] is closed and convex. If E is reflexive and strictly convex, then a monotone operator T is maximal if and only if R(J + XT) = £* for each A > 0 (see [18,21] for more details).

A Banach space E is said to be smooth if

lim (2.1) t-^0 t

exists for each x,y e {z e E : \\z\\ = 1}(=: S(E)). In this case, the norm of E is said to be Gateaux differentiable. The space E is said to have a uniformly Gateaux differentiable norm if for each y e S{E), the hmit (2.1) is attained uniformly for x e S(E). The norm of E is said to be Frechet differentiable if for each X e S(E), the hmit (2.1) is attained uniformly for y e S(E). The norm of E is said to be uniformly Frechet differentiable (and E is said to be uniformly smooth) if the limit (2.1) is attained uniformly for JC, j G S(E).

We also know the following properties (see [20] for details):

1. Jx ^ 0 for each x e E. 2. y is a monotone operator. 3. If £• is strictly convex, then J is one to one, that is,

X y/^y =^ Jxr\Jy = &.


4. If E is reflexive, then 7 is a mapping of E onto £"*. 5. If £" is smooth, then the duality mapping J is single valued. 6. If E has a Frechet differentiable norm, then J is norm to norm continuous. 7. If £ has a uniformly Gateaux differentiable norm, then J is norm to weak*

uniformly continuous on each bounded subset of E. 8. E is strictly convex if and only if 7 is a strictly monotone operator. 9. E is uniformly convex if and only if £"* is uniformly smooth.

Let £; be a smooth Banach space and consider the following function studied in Alber [1] and Kamimura and Takahashi [9]:

V(x,y) = \\xf-2{x,Jy) + \\y\\\

for each x,y e EAiis obvious from the definition of V that

( l k l | - | | j l l ) ^ < V ( x , ^ ) < ( | | x | | + ||3;||)^ (2.2)

for each x,y e E. We also know that

V(x, y) = V{x, z) + V(z, y) + 2(jc - z, 7z - Jy), (2.3)

for each x,y,z e £'(see [9]). The following lemma is well known.

Lemma 2.1 ([9]). Let E be a smooth and uniformly convex Banach space and let [Xn] and {yn} he sequences in E such that either [Xn] or {yn} is bounded. If Um^ôo y{^n. yn) = 0, then lim^ôo \\Xn - Jnll = 0.

Let £• be a smooth Banach space and let Z) be a nonempty closed convex subset of E. A mapping R : D -^ D is called generalized nonexpansive if F(R) ^ 0 and V(Rx, y) < V(x, y) for each x e D md y e F{R), where F{R) is the set of fixed points of R. Let C be a nonempty closed subset of E. A mapping R : E -> C is said to be sunny if

R{Rx -\-t(x- Rx)) = Rx, VJC e E, Wt > 0.

A mapping R : E -^ C is said to be a retraction if Rx = x, VJC e C. If £ is smooth and strictly convex, then a sunny generaUzed nonexpansive retraction of E onto C is uniquely decided (see [3,4,14]). Then, if £ be a smooth and strictly convex, a sunny generaUzed nonexpansive retraction of E onto C is denoted by Re- Let C be a nonempty closed subset of a Banach space E. Then C is said to be a sunny generaUzed nonexpansive retract (resp. a generaUzed nonexpansive retract) of E if there exists a sunny generaUzed nonexpansive retraction (resp. a generaUzed nonexpansive retraction) of E onto C (see [3,4, 14] for more details). The set of fixed points of such a generaUzed nonexpansive retraction is C.


The following result was obtained in [4].

Lemma 2.2 ([4]). Let C he a nonempty closed subset of a smooth and strictly convex Banach space E. Let Re be a retraction ofE onto C. Then Re is sunny and generalized nonexpansive if and only if

{x-Rcx,JRcx-Jy) > 0,

for each x e E and y e C, where J is the duality mapping of E.

Let £ be a reflexive, strictly convex, and smooth Banach space with its dual £*. If a monotone operator B C E* x E is maximal, then E = R(I -\- rBJ) for all r > 0 (see Proposition 4.1 in [4]). So, for each r > 0 and x e £, we can consider the set JrX = {z e E : x e z + rBJz}. From [4], JrX consists of one point. We denote such ^ Jr by (I -\- rB J)~^. 7 is called a generahzed resolvent of B (see [4] for more details).

The following two results were obtained in [4].

Lemma 2.3 ([4]). Let E be a reflexive and strictly convex Banach space with a Frechet differentiable norm and let B c E* x E be a maximal monotone operator with B~^0 ^ 0. Then the following hold:

1. D{Jr) = E for each r > 0. 2. (BJ)~'^0 = F{Jr) for each r > 0, where F{Jr) is the set of fixed points

of Jr. 3. (BJ)-Ô is closed 4. Jr is generalized nonexpansive for each r > 0. 5. Forr > Oandx e E, j{x — Jrx) e BJJrX.

Theorem 2.4 ([4]). Let E be a uniformly convex Banach space with a Frechet differentiable norm and let B C E* x E be a maximal monotone operator with B~^0 7 0. Then the following hold:

L For each x e E, lim^-ôo JrX exists and belongs to (BJ)~^0. 2. If Rx := lim^ôo JrX for each x e E, then R is a sunny generalized

nonexpansive retraction of E onto (BJ)~^0.

3. Weak convergence theorem

In this section, we first start with the following lemma. Compare this lemma with the results in Kamimura and Takahashi [9], and Kohsaka and Takahashi [11].


Lemma 3.1. Let E be a reflexive, strictly convex, and smooth Banach space, let B C E* X E be a maximal monotone operator with B~^0 ^ 0, and J, = (I-\- rBJ)-^ for all r > 0. Then

V(JC, JrX) + VUrX, U) < V(x, w),

for all r > 0, u e (BjyÔ, andx e E,

Proof Let r > 0,u e (BJ)~^0, and x e Ehc given. By the monotonicity of B, (2.3) and Lemma 2.3 (5), we have

V(X, U) = V{X, JrX) + V{JrX, u) + 2{x — JrX, J JrX — J u)

= V(X, JrX) + VUrX, U) + 2r T ~ '^ - 0, J JrX - JU

> V(X, JrX)-\- V{JrX,u).

Next we can prove the following weak convergence theorem, which is a generalization of Kamimura-Takahashi's weak convergence theorem (Theorem 1.1).

Theorem 3.2. Let E be a smooth and uniformly convex Banach space whose duality mapping J is weakly sequentially continuous. Let B C E* x E be a maximal monotone operator, let Jr = {I -\- rBJ)~^ for all r > 0 and let {Xn} be a sequence defined as follows: x\ = x e E and

Xn+l = OlnXn + (1 " Q^n)^r„^n, n = 1, 2, . . . ,

where {Q?„} C [0, 1] and {r„} C (0, oo) satisfy

lim sup otn <\ and lim inf r„ > 0.

If B~^0 7 0, then the sequence {xn} converges weakly to an element of {BJ)-^0.

Proof Note that ^"^0 y^ 0 implies (BJ)-Ô i^ 0. In fact, if w* e ^"^0, we obtain 0 G 5M* and hence 0 e BJJ-û\ So, we have / " ^ M * 6 {BJ)-^0.

Put yn = Jr^Xn for all w € N and let z e (BJ)~^0 be given. We first prove that {xn} is bounded. From Lemma 3.1 and the convexity of || • ||^, we have

V(Xn-^\,Z) = V(anXn-\-il-an)yn,z)

< ar,V(xn.z) + {l-an)V{yr,,z)

< «n V(;c„, z) + (1 - an){V(xr,, z) - V(jcn, yn)]

< anV(Xn, Z) + (l- an)V(Xn, z)

= V{Xr,,z),


for all n eN. Hence, lim„_ôo Vi^n^z) exists. So, we have from (2.2) that the sequence {jc„} is bounded and

(1 - an)V{Xn, yn) < V(x„, z) - ViXn+uz),

for all n e N. Then it follows from Hm sup„_^^ otn < ^ that

Urn V(xn.yn) = 0. (3.1)

We also know that the sequence {jc„} is bounded. From (3.1) and Lemma 2.1, we have that

lim | | x , - > ; , | | = 0 . (3.2)

Since {x„} is bounded, we have a subsequence {x„.} of {x„} such that x^ -^ f e £ as / -> oo. Then it follows from (3.2) that y^ -^ v e E ^si -> oo.On the other hand, from (3.2) and hm inf „_»oo n > 0. we have

lim = 0.

If (z*, z) € J5, then it holds from monotonicity of B that

^rii ~ yrii * J \ ^ r\

for all i e N. Since J is weakly sequentially continuous, letting / -^ oo, we get {z, z* — Jv) > 0. Then, the maximality of B imphes Jv e B~^0, That is, i; 6 {BJ)-^0.

Let {x„.} and {xnj} be two subsequences of {xn} such that x^ -^ v\ and Xfij -^ V2- As above, we have v\,V2 ^ (BJ)~^0. Put

a = lim {V(xn, vi) - V(xn, V2)).

Note that

V(Xn, Vi) - ViXn, V2) = 2(jC„, JV2 -Jvi) + \\Vi f - \\v2f, W = 1, 2, . . . .

From Xfij -^ vi and Xnj -^ f2, we have

fl = 2{i;i,7i;2-7i;i) + ||i;i||^-||i;2ll^ (3.3)

and a = 2(i;2, Jv2 - Jvx) + ||i;i \\^ - \\v2f^ (3.4)

respectively. Combining (3.3) and (3.4), we obtain

{V\ — V2, Jv\ — JV2) = 0.

Since J is strictly monotone, it follows that vi = V2. Therefore, {xn} converges weakly to an element of {BJ)~^0. n


We know that the duality mappings J oniP, \ < p < cx) and smooth finite dimensional Banach spaces are weakly sequentially continuous. However, we do not know whether Theorem 3.2 hold without assuming that J is weakly sequentially continuous.

4. Strong convergence theorem

Let £• be a reflexive, strictly convex, and smooth Banach space and let J be the duahty mapping from E into £"*. Then J~^ is also single valued, one-to-one, and surjective, and it is the duaUty mapping from E* into E. We make use of the following mapping Vi studied in Alber [1], and Kohsaka and Takahashi [11]:

yi(;c,x*) = | |x| |2-2(x,x*) + |U*||2, (4.1)

for 2i\lx e E and jc* e E*. In other words, Vi (jc, jc*) = V (jc, J~^ (jc*)) for all X e E and JC* G £"*. For each x* e £*, the mapping g(x) = V\{x, x*) for all JC € £" is a continuous and convex function from E into R. As in Kohsaka and Takahashi [11], we can prove the following lemma.

Lemma 4.1. Let E be a reflexive, strictly convex, and smooth Banach space and let V\ be as in (4.1). Then

Vx (JC, JC*) + 2 ( j , yjc - jc*> < Vi (JC + y, JC*)

for all x, y e E and x* G E*.

Proof. Let JC* G E* be given. Define ^(jc) = Vi(x, JC*) and f(x) = \\x\\^ for all JC G E. We have

dg(x) = a ( / - 2(-, JC*)) (JC) = 27JC - 2JC*

for all JC G £". By the definition of dg, we have

g{x) + 2{y,Jx-x*)<g(x + y),

that is,

yi(jc,jc*) + 2 ( j , y j c - j c * ) < Vi(jc + >;,x*)

for all JC,}' e E. •

Next we can prove the following strong convergence theorem, which is a generalization of Kamimura-Takahashi's strong convergence theorem (Theorem 1.2).


Theorem 4.2. Let Ebea uniformly convex and uniformly smooth Banach space and let B C E"^ X E be a maximal monotone operator Let Jr = (I -\-rBJ)~~^ for allr > 0 and let {Xn} be a sequence defined as follows: x\ = x e E and

Xn-{-\ = GlnX + (1 - Otn)Jrn^n. H = 1, 2 , . . . ,

where {Qf„} C [0, 1] and {r„} C (0, oo) satisfy \\mn-ôoOin = 0, X ^ i ^n = oo, and lim„_ôo^n = ^^- If B~^0 ^ 0, then the sequence {xn} converges strongly to R(BJ)-'Ô(^^' where R(BJ)-Ô ^^ ^ sunny generalized nonexpansive retraction of E onto (BJ)~^0.

Proof Put yn = Jrn^n for all n eN. We denote a sunny generalized nonexpansive retraction R(BJ)-Ô of ^ onto (BJ)~^0 by R. We first prove that {JC„} is bounded. It is obvious that V(x\, Rx) < V(x, Rx). Suppose that V(xn, Rx) < V(x, Rx) for some n eN. Then from Lemma 3.1 and the convexity of || • ||^, we have

V(xn-^\,Rx) = V{anX + {l-an)yn.Rx)

< anV(x,Rx) + (l-an)V(yn,Rx)

< anV(x, Rx) + (1 - an)V(xn, Rx)

< anV(x, Rx) -h (1 - an)V(x, Rx)

= V(x,Rx).

Hence, by induction, we have V(xn, Rx) < V(x, Rx) for all n e N. From (2.2), the sequence {xn} is bounded. From Lemma 3.1, we have that V(yn,Rx) = V(Jr^Xn,Rx) < V(xn,Rx) for all n e N. So, {yn] is also bounded. We next prove

limsup(x - Rx, Jxn - JRx) < 0. (4.2)

Put Zn = Xn+\ for all n G N. Since {Jzn} is bounded, without loss of generaUty, we have a subsequence {Jzm} of {Jzn) such that

lim {x - Rx, Jzm - JRx) = limsup(jc - Rx, Jzn - JRx)

and {Jzm} converges weakly to some f* e E*. From the definition of {x„}, we have

Zn -yn =Oin(x - yn),

for all n eN. Since {yn} is bounded and hm„_ôo ctn = 0, we have

lim Wzn -yn\\= lim an\\x - ynW = 0. (4.3) n—^00 n-ôo


Since E has a uniformly Gateaux differentiable norm, the duahty mapping J is norm to weak* uniformly continuous on each bounded subset of E. Therefore, we obtain from (4.3) that

Jzm - Jjrii - ^ 0 as / -> 00.

This implies Jy^ -^ i * as / -^ cx). On the other hand, from Um„- oo n = oo, we have

hm = 0.

If (z*, z) e B, then it holds from the monotonicity of B that

z • -, z - Jym > 0

for all / G N. Letting / -^ oo, we get (z, z* — f*) > 0. Then, the maximality of B implies v* e B~^0. Put v = J~^v*. Applying Lenuna 2.2, we obtain

limsup(jc — Rx, Jzn — JRx) = Hm {x — Rx, Jzm — JRx)

= (jc - Rx, V* - JRx)

= {x- Rx, Jv - JRx) < 0.

Finally, we prove that Um„_ cxD Xn = Rx. Let £ > 0 be given. From (4.2), we have m G N such that

{x - Rx, Jxn - JRx) < s, (4.4)

for all n > m. If n > m, then it holds from Lenrnia 4.1 and (4.4) that

V(Xn-^\,Rx) = V\(Xn+\,JRx)

= Vi(anx + (1 - an)yn, JRx)

< V\{anX + (1 - an)yn - oin(x - Rx), JRx)

— 2{—an(x — Rx), Jxn-\-\ — JRx)

= Vi(anRx + {l-an)yn,JRx)

~\-2an{x — Rx, JXn-\-\ — JRx)

< V(anRx + (1 - an)yn, Rx) + 2anS

< anV{Rx, Rx) + (1 - an)V(yn, Rx) + 2anS.

From V{Rx, Rx) = 0 and Lemma 3.1, we have

V(jc„+i, Rx) < (1 - an)V(Xn, Rx) + 2ane

= 26{l-(l-an)} + {l-an)V{xn,Rx).


Therefore, we have

V(Xn-ûRx)

< 2s{l-(l-an)}

+(1 - an)[26{l - (1 - an-i)} + (1 - an-i)V(xn-uRx)]

= 2s{l - (1 - an)(l - an-\)} + (1 - an)(l - an-i)V(xn-uRx)

<26 l-Yl(l-ai) -^l[(l-ai)V(xm.Rx)

for all n >m. Since YA^\ î — ^ ' ^^ ^^^^ D/Sm ( - a/) = 0 (see Takahashi [21]). Hence, we have

limsupy(x„, Rx)

= lim sup V(Xfn-\-i-\-\, Rx)

m-\-l

l - [ ] ( l - a / ) < lim sup 2e m+l

+ Yl{l-ai)V{x,n,Rx) = 2e.

This implies lim sup„^^ Vix„, Rx) < 0. Hence, we get

lim V

Applying Lemma 2.1, we obtain

lim \\x

Therefore, {JC„} converges strongly to R(BJ)-ÔM'

lim V{xn,Rx) = 0 .

lim ||jc„ - Rx\\ = 0 .

5. Applications

In this section, we study the problem of finding a minimizer of a proper lower semicontinuous convex function; see [2,16,21].

As a direct consequence of Theorem 3.2, we obtain the following result.

Corollary 5.1. Let E be a smooth and uniformly convex Banach space whose duality mapping J is weakly sequentially continuous. Let f* : E* -> (—cx), oo] be a proper lower semicontinuous convex function such that (3/*)~H0) ^ 0. Let {Xn} be a sequence defined as follows: x\ = x ^ E and

yn argmin y*eE*

/ * ( / ) + ; ^ l l / l l ' - - { ^ n , j * ) 2rn rn

(5.1)

= anXn + {\-an)J ^y^, « = 1,2,.. . ,


where {«„} C [0,1] and {r„} C (0, oo) satisfy

lim sup an < I and lim inf r„ > 0.

Then the sequence {jc„} converges weakly to an element of{df*J)~^0.

Proof. By Rockafellar's theorem [15,17], the subdifferential mapping 9/* c E* X Eis maximal monotone. Fix r > 0, z e E, and let Jr be the generahzed resolvent of 9/*. Then we have

zeJrZ + rdf'JJrZ

and hence,

0 € df'JJrZ + -J-^JJrZ - -Z = 9 f/* + -^ || • H - -(z, )) J U^ r r \ 2r r )

Thus, we have

J JrZ = argmin /*(/) +îl/ll'-;(^,/>}.

Therefore, J ^y^ = J ^JJr^Xn = Jr^Xn for all n e N. By Theorem 3.2, {xn} converges weakly to an element of (9/*7)~^0. D

As in the proof of Corollary 5.1, we get the following result from Theorem 4.2.

Corollary 5.2. Let E be a uniformly convex and uniformly smooth Banach space and let / * :£"* ^- (—oo, oo] be a proper lower semicontinuous convex function such that (9/*)~^ (0) ^ 0. Let {xn} be a sequence defined as follows: x\ = x e E and

y; = argnun y*eE* 2rn rn

(5.2)

Xn^\ = anX^-{\-an)J Vn ' « = 1, 2, . . . ,

where {«„} C [0, 1] and {r„} C (0, oo) satisfy ]imn-ôoOin = 0, X ^ i ^n = 00, and limn-ôo^n = oo. Then the sequence [xn] converges strongly to R(:df*j)-\{){x), where R(sf*j)-\o ^^ ^ sunny generalized nonexpansive retraction of E onto {df" J)-^^.


References

1. Alber, Y.I.: Metric and generalized projection operators in Banach spaces: properties and applications. Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, pp. 15-50. Dekker, New York 1996

2. Butnariu, D., lusem, A.N.: Totally Convex Functions for Fixed Points Computation and Infinite Dimensional Optimization. Kluwer Academic, Dordecht 2000

3. Ibaraki, T, Takahashi, W.: Convergence theorems for new projections in Banach spaces (in Japanese). RIMS Kokyuroku, vol. 1484, pp. 150-160, 2006

4. Ibaraki, T, Takahashi, W.: A new projection and convergence theorems for the projections in Banach spaces, (to appear)

5. Martinet, B.: Regularsation d'inequations variationnells par approximations succes-sives (in French). Rev. Francaise Informat. Rech. Oper. 4, 154-158 (1970)

6. Kamimura, S., Takahashi, W.: Approximating solutions of maximal monotone operators in Hilbert spaces. J. Approx. Theory 106, 226-240 (2000)

7. Kamimura, S., Takahashi, W.: Iterative schemes for approximating solutions of accritve operators in Banach spaces. Sci. Math. 3, 107-115 (2000)

8. Kamimura, S., Takahashi, W.: Weak and strong convergence of solutions to accretive operator inclusions and applications. Set- Valued Anal. 8, 361-374 (2000)

9. Kamimura, S., Takahashi, W: Strong convergence of a proximal-type algorithm in a Banach space. SIAM J. Optim. 13, 938-945 (2002)

10. Kamimura, S., Kohsaka, F., Takahashi, W: Weak and strong convergence theorems for maximal monotone operators in a Banach space. Set Valued Anal. 12, 417^29 (2004)

11. Kohsaka, F, Takahashi, W.: Strong convergence of an iterative sequence for maximal monotone operators in a Banach space. Abstr. Appl. Anal. 2004, 239-249 (2004)

12. Matsushita, S., Takahashi, W: Weak and strong convergence theorems for relatively nonexpansive mappings in Banach space. Fixed Point Theory Appl. 2004, 37-47 (2004)

13. Ohsawa, S., Takahashi, W: Strong convergence theorems for resolvents of maximal monotone operators in Banach spaces. Arch. Math. 81, 439-445 (2003)

14. Reich, S.: Strong convergence theorems for resolvents of accretive operators in Banach space. J. Math. Anal. Appl. 75, 287-292 (1980)

15. Rockafellar, R.T: Characterization of the subdifferentials of convex functions. Pacific J. Math. 17, 497-510 (1966)

16. Rockafellar, R.T: Extension of Fenchel's duality theorem for convex functions. Duke Math. 33, 81-89 (1966)

17. Rockafellar, R.T: On the maximal monotonicity of subdifferential mappings. Pacific J.Math. 33, 209-216(1970)

18. Rockafellar, R.T: On the maximality of sums of nonlinear monotone operators. Trans. Am. Math. Soc. 149, 75-88 (1970)

19. Rockafellar, R.T: Monotone operators and proximal point algorithm. SIAM J. Control. Optim. 14, 877-898 (1976)

20. Takahashi, W: Nonlinear Functional Analysis: Fixed Point Theory and Its Applications. Yokohama Publishers, Yokohama 2000

21. Takahashi, W.: Convex Analysis and Approximation of Fixed Points (in Japanese). Yokohama Publishers, Yokohama 2000


22. Takahashi, W.: Fixed point theorems and proximal point algorithms. In: Takahashi, W., Tanaka, T. (eds.) Nonlinear Analysis and Convex Analysis, pp. 471-481. Yokohama Publishers, Yokohama 2003

23. Takahashi, W.: Convergence theorems for nonlinear projections in Banach spaces (in Japanese). RMS Kokyuroku, vol. 1396, pp. 49-59 (2004)



©Springer 2007

Golden optimal policy in calculus of variation and dynamic programming

Seiichi Iwamoto

Department of Economic Engineering, Graduate School of Economics, Kyushu University, Fukuoka 812-8581, Japan (e-mail: [email protected])

Received: April 22, 2006 Revised: September 12, 2006

JEL classification: C61, D81

Mathematics Subject Classification (2000): 90C39, 90C40, 90A43

Abstract. This paper discusses four dynamic optimization problems on an infinite continuous time interval from a viewpoint of Golden optimality. The problem is whether an optimal policy is Golden or not. We solve two control processes with quadratic cost criterion and two allocation processes with discounted square-root reward criterion. Both processes have a linear dynamics. It is shown that one cotrol process does not admit a Golden optimal policy. The other three processes have a Golden optimal policy. Further we illustrate the Golden optimal trajectories through three approaches: (i) one-parametric method, (ii) Euler equation and (iii) Bellman equation.

Key words: golden policy, optimal policy, Euler equation. Bellman equation, golden optimality, control process, allocation process, golden section

1. Introduction

E. Phelps says in ''Golden Rule of Accumulation: A Fable for Growthmen'' [12] as follows: The King commended the task force for its informative and stimulating report. He invited all his subjects to join in search of an optimal investment poHcy. Solovian theorists considered dozen of fiscal devices for their efficiency, equity and effectiveness. Mathematicians, leading the quest for a growth strategy, grappled with extremals, functional and Hamiltonians. Yet nothing practical emerged. Then a poUcy-maker was heard to say, "Forget grand optimaUty. Solovians are a simple people. We need a simple pohcy. . ."

On the other hand, this paper discusses an optimahty. We are concerned with Golden optimality [8-11] for both control processes and allocation processes on


66 S. Iwamoto

continuous time. Here we also need a simple policy. However, it is the Golden optimal policy. A trajectory is called Golden if any current state moves to a next state constantly repeating a Golden section [4,13] in unit time. A policy is called Golden if it, together with a relevant dynamics, generates a Golden trajectory. We direct our attention to the Golden policy as well as optimal poUcy.

We consider four dynamic optimization problems on time interval [0, oo) through three approaches. Two control processes with a quadratic criterion and two allocation processes with a discounted square-root criterion are solved in § 3 and in § 4, respectively. Our three approaches are (i) one-variable optimization method, (ii) calculus of variation and (iii) dynamic programming method. While (i) and (ii) concern a Golden trajectory, (iii) concerns a Golden pohcy. We show that one control process does not admit any Golden optimal policy and that the other three processes have a Golden optimal pohcy.

2. Golden trajectories

A real number

, 1 + V5 1.618

2 is called Golden number [4, 5, 13]. It is the larger of the two solutions to quadratic equation

j c ^ - j c - l = 0. (1)

Sometimes (1) is called Fibonacci quadratic equation [5]. The Fibonacci quadratic equation has two real solutions: 0 and its conjugate 0 := 1 — 0. We note that

0-h0 = l, 0-0 = - l . Further we have

0^ = 1 + 0 , 0 = 2 - 0

0^ + 0^ = 3, 0^ .0^ = 1.

Definition 21. A differentiable function x : [0, oo) -> / ^ is called Golden if and only if either

4 T = >og(<A-l) or ^=log{2-4>). (2) x(t) xit)

Lemma 21. A Golden function x is either

;c(0=;c(0)e-"°g* or x(0 = A:(0)e-"°S<'+*'.

We remark that in Fig. 1

Golden optimal policy in calculus of variation 67

1 2 3 4 5

Fig. 1. Golden trajectories x = c{<t) — l)^ c = 1, 2, 3

where

log(/> « 0.481, 0 - 1 = 0 " ^ « 0.618

and that in Fig. 2

^-rlog(l+0) ^ ^rlog(2-0) ^ (1 ^ ^yt ^ (2 - 0)^

where

log(l + 0) « 0.962, 2 - 0 ^ ( 1 + 0)-^ « 0.382.

1 2 3 4 5

Fig. 2. Golden trajectories JC = c(2 - 0)^ c = 1, 2, 3

68 S. Iwamoto

Let us introduce a controlled linear dynamics with real parameter b as follows [2]:

X = bx -\-u on [0, oo), (3)

where u : R^ -> R^ is called control function or control. If u{x) = px (resp. px -\- q), the control is cdlledproportional (resp. linear), where p, q are real constants. A differentiable solution x to (3) is called trajectory.

Definition 22. A proportional control u on dynamics (3) is called Golden if and only if it generates a Golden trajectory.

Lemma 22. A proportional control u = px on (3) is Golden if and only if

/7 =-Z? + log(0 ~ 1) or p = -fc + log(2-0) .

3. Control processes

This section minimizes two quadratic cost functions

/ {x^ + uAdt and / ix^-û^\dt

under a common additive dynamics

X —bx -\-u

(see [2, 3]).

3.1. First quadratic criterion

Let C^ be the set of all continuously differentiable functions on the nonnegative half-hne [0, oo) :

C^ = \x = x(t) \ X : [0, oo) R^ continuously differentiable|.

First we take the quadratic criterion

I{x) = f \x^-^(bx-x)^]dt.

Now we consider a variational problem for a given real constant c:

MPi(c) minimize I(x) subject to (i) x e C \ (ii) x(0) = c.


Let us evaluate a few special trajectories:

1. A constant x{t) = c yields I(x) = oo. 2. A proportional JC(0 = ce""' (0 < a < oo) yields

Kx) = c^ [l + (a+.)^]^

2 1 + (fl + b)^

e-^'^'dt

la

3. Since \ + {a + b)

min 2

0<a<c» 2fl

is attained at 5 = v T T ^ , we have the minimum value b + Vl + ^^ • Thus, a proportional jc(0 = ce~^^^^ ^ yields the optimal value

I{x) = (/7 + y r T ^ ) c ^

in the class of proportional trajectories.

Since 1 > l o g ( l + 0 ) « 0.962 neither

V T + ^ = log ( l+0)

nor

yi+Z?2 = l o g 0

has solution. This implies that the proportional trajectory x{t) = ce~^^^^ ^ should not be Golden. Thus, we have no Golden optimal trajectory in MPi (c).

3.1.1. Euler equation

Let us consider an extremal problem

lize / Jo

EPi (c) extremize / f{t, x, x)dt subject to (i) x e C\ (ii) x(0) = c. Jo

Then an extremal curve x = JC (•) satisfies a variational equation, Euler equation,

J^fx-fx = 0 (4)

and a boundary condition at infinity oo, transvesality condition,

lim Mt,x{t),x{t)) = 0. (5)

70 S. Iwamoto

Formally both equations are shown as follows (see [6]). Let rj = rj(-) be any C^-function satisfying rj(0) = 0. Then y := x +€riis feasible for any e e R^. Let us define

/•oo rT

J(€):= f(t,y,y)dt = lim / f{t,y,y)dt.

Then 7 ( ) must take a minimum value at € = 0 for any such r]. This impUes J\0) = 0.

J(h) - 7(0) Let us now calculate 7 (0) = Um . First we note that

hÔ h

J{h) - 7(0) ,. [^ . _ = hm / g{t)dt,

where

f(t, X -hhrj.x + hi]) — f(t, jc, x) g(t):=g(t',h) =

h

From the mean value theorem, there exists ^ (0 < ^ < 1) satisfying

^(0 = fxrj + fxrj,

where

fx = Mt,x+Ohr],x + Ohrj)

fx = fx(t,x + ehr],x + ehrj).

Hence, from integration by parts and r}(0) = 0, it holds that

/ g(t)dt = [ (fxTj-^ fxr])dt Jo 7o

r]dt + fMT).

Then we have

im / T

lim / g(t)dt = Mm j iu-^f\r]dt+\im firj(T).


Thus

J (0) = lim ; hÔ h

= lim lim / ^(0^^ h-ÔTôoJo

= lim lim / I fjc - — fx) r]dt + Mm lim fxr](T)

= lim / lim \ fx - — fx] rjdt-\- lim Um fxr]{T).

Consequently, it holds that

/(O) - lim / (f^-:^f^r]dt+ lim firjiT),

where

fx = fx(t,X,x)

fx = fxit,X,x).

Since J\0) must vanish for any C^-function r] satisfying r]{0) = 0, we have

fx--fx = 0 on[0,c^) dt

lim fi{t,xit),x(t)) = 0.

Thus, both Euler equation and the transversaUty condition are derived. •

Now let us take the quadratic criterion

f(t,x,x) =

Then the Euler equation (4) imphes

f(t,x,x) = x^-\-(bx - i ) ^ .

x-a^x = 0, (6)

where

The transvesahty condition (5) imphes

lim [x(t)-bx(t)] = 0. (7)

By solving the system of three equations, initial condition (ii), (6), and (7), we have

x{t) = ce-^^

72 S. Iwamoto

3.1.2. Bellman equation

Let us now consider a control process with constant parameter b {—oo < b < oo):

minimize / Ix"^ -{-u^]dt f{,^W). ^, , subject to (i) X = bx -\- u C(c) ' : 1 0 < r < o o

(ii) u e C

(iii) x(0) = c.

Let v(c) be the minimum value. When c = 0, the feasible x{t) = u{t) = 0 yields null integral value. Hence D(0) = 0. The value functions = i;(jc) satisfies the Bellman equation [1-3, 7] :

min rjc^ + M + i;'(jc)(^Jc + M)1 = 0 , i;(0) = 0. (8) — 00<M<00 L J

Since — [• • • ] = 0 yields u = , we have du 2

min I"] = x^-\-bx' v\x) (v\x)f . -(X)<M<00 4

Thus equation (8) imphes

{v\x)Y -bx'v\x)-x^ = 0. 1 / /. .\2 , / ^ .2 4

Assuming a quadratic form v{x) = kx^ (k > 0), the coefficient k must satisfy a quadratic equation

k^-2bk-l= 0.

Thus, we have

Hence, we have the following result.

Lemma 31. A pair of proportional control and quadratic value function

u(x) = —kx, v(x) = kx^

satisfies the Bellman equation

min \x^ + M + v\x)(bx + M)1 = 0. — 00<M<00 L J


Then the dynamical system

X = -y/b^-\- IJC, JC(0) = c,

yields an trajectory

x(t) = ce~ -Vh^t

The proportional poUcy u generates an optimal trajectory x, which is not Golden. Thus, the control process C(c) does not admit a Golden optimal poUcy. However, when b = 1/2, the coefficient k reduces to the Golden ratio k = (p.

3.2. Second quadratic criterion

Second we take the quadratic criterion

J(x) = I \x^-^ (bx - xf]dt onC^

where 0 < /? < oo. Now we consider a mathematical programming problem for a given real

constant c:

MP2(c) minimize J(x) subject to (i) jc e C^ (ii) x(0) = c.


1. A constant x(t) = c yields J(x) = oo. 2. A proportional x(t) = ce~^^ yields

I(x) = c^ \a^ + (a + bf] I e-^'^'dt

2a^-\-(a-\-bf

3. Since

mm

2a

a^^{a-V bf

(0 < fl < oo).

0«2<oo 2a

is attained at a = —;=. we have the minimum value (1 + \[l)b. V2

Thus, a proportional x{t) = ce~^^^^^^ yields the optimal value

I(x) = (l + V2)bc^


74 S. Iwamoto

The proportional trajectory x(t) = ce~^^^^^^ is Golden if and only if either of the following two cases holds:

b • Case(i) —=r = l o g ( l + 0 )

V2 impHes

bi = x/2"log(l+(/>)« 1.361

Then we have a Golden trajectory

b • Case (ii) —•p=r = log 0

implies

Z?2 = \ /2"log0« 0.681.

Then we have a Golden trajectory

Thus, we have the two Golden optimal trajectories jci and Jc2.


We consider an extremal problem

EP2(c) extremize / f(x, x)dt subject to (i) jc € C^ (ii) JC(0) = c Jo

where

/ ( j c , i ) = i ^ + (Z7JC-i)^.

Then the Euler equation (4) implies

x-a^x = 0, (9)

where b

a = V2-

The transvesahty condition (5) impHes

\imi2x(t)-bx(t)) = 0. (10)

By solving the system of three equations, initial condition (ii), (9), and (10), we have

x(t) = c^-^^



Let us now consider a control process with constant parameter b {—oo < b < oo):

minimize dt ly^"')' _,, , subject to ii) X = bx -\- u C\c) .: 1 0 < r < o c

(u) u e C

(iii) x{0) = c.

Let v{c) be the minimum value. Then the value function v = v(x) satisfies the Bellman equation with an initial condition:

min \{bx + uf + u^ + v\x)ibx + u)\=0, i;(0) = 0. (11) —oo<u<oo L J

Since — [•••]= 0 yields u = —— ilbx + v\x)), we have du 4 ^ ^

min [ . . ] = — -00<M<00 2

{bxf^bxv\x)--{v'{x)f

Thus, equation (11) implies

[v\x)Y -bxv\x)-b^x'^ = ^. 1 {,/(^S^^ u^.Jr^^ u2^2

Assuming a quadratic form v{x) = kx'^ {k > Q), the coefficient k satisfies a quadratic equation

k^ - Ibk -b^ = 0.

Hence, it follows that ^ = (1 + V2)b.

We have

Lemma 32. A pair of proportional control and quadratic value function

where

ii(x) = px, v(x) = kx^

p = --{1^4l)b, k = (l + ^/2)b

satisfies the Bellman equation

min \(bx + uf + w + v\x)(bx + u)] = 0. D<M<00 L J mm

-oo<

76 S. Iwamoto

Then the dynamical system

b X = —X, jc(0) = c,

yields an optimal trajectory

• Case(i) WhenZ? = >/2^1og(l + 0) « 1.361 the optimal proportional policy u(x) = - ( 1 + V2")log(l + 0)jc « -2.323 jc is Golden. It generates the Golden trajectory

x(t) = ce -rlog(l+0)

Case (ii) When b = \/2^1og0 « 0.681 the optimal proportional poHcy u(x) = —(I -{• \[l) log0;<c « —1.162JC is Golden. It generates the Golden trajectory

x(t) — ce -t\0g(f)

4. Allocation processes

This section maximizes two discounted square-root reward functions

/*oo poo / e~^^{y/x'-\-y/u)dt and / e~^^^/udt

Jo Jo

under a common subtractive dynamics

X = bx — u.

4.1. First square-root criterion

Let two nonnegative constants p and b be given inp — b> \[b. Then we have an inequahty with equaUty condition as follows.

Lemma 41. It holds that

l-\-VbTx 1 z < — on [—b, 00) .

p + ix " ^l + 2p-b-l

The sign of equality holds if and only if x = 2(1 + p — b — ^J\-\-2p — b) (see Fig. 3).


Fig. 3. Allocation ratio y = z , ir denotes maximum point

Let C be the set of all continuously differentiable functions satisfying x(t) < bx(t), jc(0 > 0 on the nonnegative half-Une:

C' = {x = x(t)\x : [0, oo) -^ R^ continuously differentiable, x < bx, JC > 0}.

First we take the first square-root criterion

K(x) = / e-^\V7-^y/bx-x)dt onC\ Jo


MP3(c) Maximize K(x) subject to (i) x e C\ (ii) jc(0) = c.


/— \/c 1. A constant x(t) = c yields K{x) = (1 + \/b)-^^—.

P 2. A proportional x{t) = ce~^^ {—b < a < oo) yields

/•oo J

K(X) = V^(l+^/bT^) / ^-^^+5^>^Jf Jo

.l-{-y/b-\-a = V^-p-hâ

78 S. Iwamoto

3. Since l + vF+a

Max -b<a<oo p -\- â

is attained a t a * = 2 ( l + p - Z ? - y/l+2p -b), we have the maximum value

1

./l+2p-b-l

Thus a proportional jc*(r) = ce~^ ^ yields the optimal value

y/l + 2p-b-l


4.1.1. Case Z» = 1

Let us consider the case b = I. Then, under the condition p > 2, we have the proportional trajectory x*(t) = QQ-'îp—J^M j ^ - QQi jg if and only if either of the following two cases holds.

• Case (i) 2(p - V2p) = log(l + 0) imphes

PI = - ( l + Vl + l o g ( l + 0 ) ) ^ « 2.882

Then we have a Golden trajectory jci (0 = ce~^ iog(i+' ), • Case (ii) 2(p - V^p") = log0

imphes

P2 = ^(1 + Vl + Iog0)^« 2.458

Then we have a Golden trajectory JC2(0 = ce~^^^^^.

Thus, we have the two Golden optimal trajectories Jci and X2.

4.1.2. Case b = 0

Let us consider the case b = 0. Since 2(1 + p - VI + 2p) = (^/^+Jp'- Ir, we have the proportional trajectory x*(r) = c^-(VH-2p -i) t ^nder the condition p > 0. It is Golden if and only if either of the following two cases holds.


• Case (i) ( V l + 2 p - if = log(l + 0) implies

PI = yiogd + 0) + y logd 4- 0) « 1.462

Then we have a Golden trajectory x\ (t) = (^^-t\ogi\-\-(f>) • Case (ii) ( V l + 2 p - 1)^ == log0

imphes

P2 = V l o g 0 + - l o g 0 « 0.934

Then we have a Golden trajectory JC2(0 = ce~^^^^^.

Thus, we have the two Golden optimal trajectories x\ and JC2.

4.2. Euler equation

Let us consider an extremal problem

EP3(c) extremize / f(t, x, x)dt subject to (i) x e C^ (ii) x(0) = c. Jo

Let us take the square-root criterion

f(t,x,x) = e-^'{y/x + >/bx-x).

Then we have

2 VV- ^x-x)

2 s/bx — X

dt^' 2 X^fbT^ 2{bx-x)^l^)-

The Euler equation (4) implies

p — b bx — X 1 + :;7 rwn = ^ ' (12) y/bx-x 2(jc-i)3/2 ^ '

The transversahty condition (5) imphes

hm ^ = 0. (13) tôc ^bx(t)-x(t)

80 S. Iwamoto

Let us now find a solution of a form x(t) = ce~^K Substituting x = -cae~^^, X = caê~^^ into (12), the coefficient a must satisfy

p ~b ba-\-a^

yfhT^ lib + afl^

namely,

p-b-^ = ^/bVa. (14)

Solving (14) yields

a = 2{\+p-b- y/\+2p-b).

Thus, we have a solution to (12), (13):

x{t) = ce-'''.

4.3. Bellman equation

Let us now consider a control process with constant parameter b (—oo < b < oo):

oo /•oo

Maximize / e~^^ f(x,u)dt Jo

subject to (i)x = g(xu) o<t<oo ^^ (ii) X € C\ ueU(x) -

(iii)jc(O) = c.

Let v(c) be the maximum value. Then the value function v = v(x) satisfies Bellman equation:

pv{x) = Max \f(x,u)-{-v\x)g{x,u)]. (15) ueUix)

Formally this is derived as follows (see [1-3]). Let us take any small A > 0. We define for any feasible paired (JC, M)-process a new paired (y, u;)-process as follows:

y(t) := jc(r + A), wit) := u(t + A), t e [0, oo).

Then the process y — {j()}[o,oo) satisfies

(i)' : = / C . « ) 0 < r < 00


Conversely, concatenating the paired process (jc (•), w (•)) on time interval [0, A] for any paired process(y(), w(-)) satisfying (i)^ (ii)^ (iii)^ we can construct a paired (jc, M)-process on the total time interval [0, oo) satisfying conditions (i), (ii), (iii). Thus, there is a one-to-one correspondence between (x, M)-pro-cess and (y, u;)-process.

First we note that /•oo rA POO

/ e-^'f{x,u)dt = / e-^'f(x,u)dt + e-^^ e-^'f(y,w)dt. Jo Jo Jo

From the mean value theorem, there exists 6> (0 < ^ < 1) satisfying

h(t)dt=h(6A)A, Jo

where

Thus, we have

v(c) > / e-^'f(x,u)dt Jo

POO

Jo From the preceding construction of the paired (x, M)-process on [0, oo) for any paired process(>'(), w;()) satisfying (i)^ (ii)^ (iii)\ we have

v{c) > /i(6>A)A + e-^'î;(jc(A)).

Applying the mean value theorem two times, there exist r ; , § ( 0 < ? ] , f < l ) satisfying

h(OA)A + e-P^v{x{A))

l-pA^le-^P^{pA)^ = /2(Â)A-f-

X [v{c) + v\x{r)A))x{r)A)A\ . (16)

Thus, we have

v(c) > h(eA)A + pA + ê-^^^(pA)%i;(c) + v\x{r]A))x(r]A)Al

Subtracting v{c), dividing A and taking a hmit as A J. 0, we have for any feasible u

pv{c) > f(c,u)-{-v\c)g{x,u).

82 S. Iwamoto

Thus, we have

pv{c) > Max [f{c,u) + v\c)g(c,u)]. ueUic)

Conversely, let us assume that a feasible (hence, optimal) paired (jc, u)-process attains the maximum value function v. Then we have from (16) once again

poo v(c) = / e-P^f{x,u)dt

JO

= h{e^)^-{•e-f'^v{x{^))

= h(0A)A + I 1 - pA + i^-^^^(pA)^ I [v(c) + v\x(r]A))x(r]A)A].

This in turn yields

pv(c) = f(c,u)-\-v\c)g(x,u).

Thus, we have the converse inequality

pv(c) < Max [fie, u) + v\c)g{c, u)]. ueU{c)

Thus, the Bellman equation (15) is vahd. •

Let us now take a square-root criterion and a subtractive dynamics as follows:

f(x,u) = y/x-\-^/u, g(x,u) = bx — U.

We solve an allocation process with a discounted total square-root criterion:

poo Maximize / e~^\'sfx-\-\fu)dt

^ , , subject to (i) i = Z?;f — M A c .' ^ ^ 1 ^ n 0 < r < o o

(n) JC € C , w > 0

(iii)jc(0) = c.

Let us solve the Bellman equation:

pv(x) = Max \^/x -h VM + v\x){bx - u)}. (17) M>0 "-


d Since — [• • • ] = 0 yields

du

V i.e., M = —r- (18)

we have

Thus, we have

1^ ' " 4i;'2

r- , 1 Max[---] = sl^-\-hxv H . M>0 ^V'

pv{x) — yfx + hxv'{x) -\-Av'{x)

Let us find a square-root solution f (jc) = k^/x (k > 0) to (17). Then the coefficient k must satisfy

( p - ^ ) ^ = l + ^ . (19)

Equation (19) has a solution

1 k =

^l + 2p-b- 1

This together with (18) and (iii) yields

X = —ax, x(0) = c,

where

a = 2{l-\-p-b- ^l + 2p-b).

Thus, we have the optimal trajectory

x*(t) = ce-'''

and an optimal control

u*(x) = (a -]-b)x.

4.3.1. Case b = l

Let us consider the case b = I. Then, under the condition p > 2, we have the proportional pohcy u*(x) = (I -\-2p - 2y/2p)x. It generates an optimal trajectory x*(t) = ce~^\ where a = 2(p — y/lp),

1 Case (i) When p = -(I-\- ^ 1 + l o g ( l + 0))^ « 2.882, the optimal trajectory X* is Golden jc*(0 = c^-Hog(i+0) j ^ ^ ^ - generated by a Golden u'lix) = (1 + log(l + 0))x « 1.962x. Thus, we have the Golden optimal policy w*.

84 S. Iwamoto

• Case (ii) When p = - (1 + ^\ +log</>)^ w 2.458, the optimal trajectory

jc* is Golden jc|(0 = c^"^^°^^. This is generated by a Golden M|(X) = (1 + log(/))jc « 1.481 JC. Thus ^2 is Golden optimal.

4.3.2. Case* = 0

Let us consider the case = 0. Then, under the condition p > 0, we have the proportional policy M*(x) = ax. It generates an optimal trajectory jc*(r) = ce~^\ where a = (VI + 2p - 1)^.

1 • Case (i) When p = yiog(l + </>) + - log( l + 0) « 1.462, the optimal

trajectory x* is Golden x\{t) = c^-îog(i+0). This is generated by a Golden u\{x) =:;tlog(l+0) « 0.962 JC.Thus, we have theGolden optimal poUcyM*.

• Case (ii) When p = vTog^ H—log(/> « 0.934, the optimal tra

jectory JC* is Golden jc|(r) = ce~^^^^^. This is generated by a Golden

w*(x) = JC log0 « 0.481 JC. Thus u^ is Golden optimal.

4.4. Second square-root criterion

Let p and b be given i n O < Z 7 < p < o o . Then we have an inequaUty with equality condition as follows.

Lemma 42. It holds that

-.— < ——=:z on [~b, 00). p + ix Îp-h

The sign of equality holds if and only ifx — 2{p — b) {see Fig. 4).

Let us take the second square-root criterion

POO

L(jc) = / e'^Wbx-xdt on C'. Jo


MP4(c) Maximize L(JC) subject to (i) JC € C\ (ii) x(0) = c.


Vbc 1. A constant jc(0 = c yields L{x) — .


/ I I y

Fig. 4. Allocation ratio y = —, i^ denotes maximum point

2. A proportional J (r) = ce ^^{—b < a < oo) yields

L(x) = v ^ V T T ^ / ^-^^+i^>^Jr Jo

-y/b + a ^c-

p + â

3. Since

Max VFTa 1 -b<a<oo p -\- â

is attained at (2* = 2(p — Z?), we have the maximum value

1

y/2p-b

Thus a proportional x*(0 = c "" ^ " ^ yields the optimal value

L(x*) = y/2p-b'


The proportional trajectory x*(r) = c ~^ ^~^ ^ is Golden if and only if either of the following two cases holds:

86 S. Iwamoto

• Case (i) 2(p - b) = log(l + 0). implies

pi = Z7 4-log0

Then we have a Golden trajectory jci (t) = ce~^ og(i+0) • Case(ii) 2{p-b) = l o g 0

impHes

P2 = ^ + - l o g 0 .

Then we have a Golden trajectory X2it) = ce~^^^^^.

Thus, we have the two Golden optimal trajectories x\ and X2-


Now we consider an extremal problem

roo

EP4(c) extremize / e~^Wbx — xdt subject to (i) jc € C\ (ii) x(0) = c. Jo

Then the second square-root criterion

f{t,x,x) = e~^Wbx - X

yields

/. = i.-"^ 2 y/bx — X

2 >Jbx — X d .f. = \e-pt( P bx-x \ ^^' 2 KVb^^ 2{bx-x)y^)

The Euler equation (4) impUes

bx — X P~^^^ ^ = ^'

2{bx — X) namely,

JC - (3^ - 2p)x + 2b(b -p) = 0. (20)

The transversahty condition (5) impHes

e -pt

Um , = 0. (21) tôo ^hx - X


Let us now find a solution to (20), (ii), (21). Equation (20) has a general solution

x(t) = Ae-^^^-^^' + Be^',

where A, 5 are real constants. Thus, we have a desired solution


Let us take a square-root criterion

f(x, u) = \[u

We solve an allocation process with a discouted total square-root criterion:

Maximize Jo

., , subject to (i) X = bx — u A (c) ^ ) ' ^1 ^ 0 < r < 00

(iii) jc(0) = c.

Let us solve the Bellman equation:

pv{x) = Max [Vw + v\x){bx - u)]. (22)

Since — [ • • ] = 0 yields du

we have

Max[---] = bxv\x)-\ .

Thus, equation (22) reduces to

pv{x) = bxv\x) + ——. (24) 4v^(x)

We will find a square-root solution v(x) = k^/x {k > 0) to (24). Thus, the coefficient k must satisfy

bk 1 M = y + ^ . (25)

88 S. Iwamoto

Equation (25) has a solution

1 k =

This together with (23) and (iii) yields

X = —ax, jc(0) = c

where a — 2{p — b). Thus, we have the optimal trajectory

jc*(0 = c^~^'

and an optimal control

M*(jc) = (b-\-a)x.

4.4.3. Two cases

Let us specify two cases where the allocation process A' (c) has a Golden optimal pohcy for any given b >0.

• Case (i) When p = Z? + log 0, the optimal trajectory jc* is Golden x^(t) =

^^-riog(i+0) jYns is generated by a Golden u\(x) = (b + 21og0)jc. Thus we have the Golden optimal poUcy M*.

• Case (ii) When p = b -\—log0, the optimal trajectory JC* is Golden

jc|(r) = ce~^^^^^. This is generated by a Golden u\{x) = {b -\- \og<p)x. Thus M2 is Golden optimal.

References

1. Bellman, R.E.: Dynamic Programming. Princeton University Press, NJ 1957 2. Bellman, R.E.: Introduction of the Mathematical Theory of Control Processes, vol. 1,

Linear Equations and Quadratic Criteria; vol. 2, Nonlinear Processes. Academic Press, New York 1967; 1971

3. Bellman, R.E.: Methods of Nonlinear Analysis, vol. 1, vol. 2, Academic Press, New York 1969, 1972

4. Beutelspacher, A. Petri, B.: Der Goldene Schnitt 2., uberarbeitete und erweiterte Auflange. Elsevier GmbH, Spectrum Akademischer, Heidelberg 1996

5. Dunlap, R.A.: The Golden Ratio and Fibonacci Numbers (Original). World Scientific, Singapore 1977

6. Gelfand, I.M., Fomin, S.V.: Calculus of Variations. Prentice-Hall, Englewood Cliffs 1963

7. Iwamoto, S.: Theory of Dynamic Program: Japanese. Kyushu University Press, Fukuoka 1987


8. Iwamoto, S.: Cross dual on the Golden optimum solutions, Mathematical Economics, Kyoto University RIMS Koukyuroku, vol. 1443, pp 27^3 , July 2005

9. Iwamoto, S.: The Golden optimum solution in quadratic programming. In: Takah-ashi, W., Tanaka, T. (eds) Proceedings of the International Conference on Nonlinear Analysis and Convex Analysis (Okinawa, 2005). Yokohama Publishers, Yokohama, pp.199-205, 2007

10. Iwamoto, S.: The Golden trinity - optimility, inequality, identity. Mathematical Economics, Kyoto University RIMS Koukyuroku, vol. 1488, pp 1-14, May 2006

11. Iwamoto, S., Yasuda, M.: Dynamic programming creates the Golden Ratio, too. In: Proceedings of the 6th International Conference on Optimization: Techniques and Applications (ICOTA 2004), Ballarat, Australia, December 2004

12. Phelps, E.S.: The Golden rule of accumulation: A fable for growthmen. Am. Econ. Rev. 51, 638-643 (1961)

13. Walser, H.: Der Goldene Schnitt. B.G. Teubner, Leibzig 1996



©Springer 2007

A remark on law invariant convex risk measures

Shigeo Kusuoka*

Graduate School of Mathematical Sciences, The University of Tokyo, Komaba 3-8-1, Meguro-ku, Tokyo 153-8914, Japan (e-mail: [email protected])

Received: September 11, 2006 Revised: November 22, 2006

JEL classification: C65, G19

Mathematics Subject Classification (2000): 60B05

Abstract. The author gives a simple proof of the representation theorem for law invariant convex risk measures which was obtained by Kusuoka (Adv.Math.Econ. 3:83-95,2001), Frittelli and Rossaza Gianin (Adv.Math.Econ. 7:33-46,2005) and Jouini et al (Adv.Math.Econ. 9:49-71,2006).

Key words: risk measure, law invariant.

1. Introduction

The idea of coherent risk measures has been introduced by Artzner et al[\\. Then FoUmer and Scheid [3] extended this notion to convex risk measures. Let me introduce the definition of convex risk measures first.

Let ( ^ , JT, P) be a probability space. We denote L ^ ( Q , JT, P) by L ^ .

Definition 1. We say that a map p : L^ ^^ R is a convex risk measure if the following are satisfied.

L p(0) = 0.

2. For any c G R and X E L ^ , we have

p(X + c) = p(X)-c.

3. IfX ^Y,X,Ye L ^ , then p(X) ^ p(Y).

Partly supported by the 21st century COE program at Graduate School of Mathematical Sciences, The University of Tokyo.


92 S. Kusuoka

4. For any k e [0, 1], and X,Y e L ^ ,

p{kX + (1 - k)Y) S kp{X) + (1 - k)p{Y).

Remark 2. From the conditions (2) and (3) in Definition 1, we see that a convex risk measure p : L ^ -^ R satisfies

\p{X)-p{Y)\^\\X-Y\U X,YeL^,

and so is continuous.

Also, we introduce the following notion.

Definition 3. We say that a convex risk measure p : L^ -^ Ris law invariant, ifp(X) = p(Y)forany X,Y e L^ with the same probability laws.

Let T> be the set of probabihty distribution functions of bounded random variables, i.e., T> is the set of non-decreasing right-continuous functions F on R such that there are zo, zi eR for which F(z) = 0, z < zo, and F{z) = 1, z ^ zi. Let us define Z : [0, 1) x P -> R by

Z(jc, F) = mf{z; F(z) > x], x e [0,1), F eV.

Z(jc, F) is a version of F '^jc) . Z(-, F) : [0,1) ^- R is a non-decreasing and right continuous function, and the probabihty distribution function of Z(JC, F) under the Lebesgue measure dx on [0,1) is F. We denote by Fx the probabihty distribution function of a random variable X.

For each a e (0,1], let pa : L^ ^- R be given by

p^{X) = - a " ^ / Z(jc, Fx)dx, X e L^. Jo

Also, we define po : L^ -^ Rhy

po(X) = -Z(0 , Fx) = -ess. inf X X e L^ .

Then it is easy to see that p(X) : [0, 1] -> R is a non-increasing continuous function for any X e L^.

Let M[o,\] be the set of probability measures on [0, 1]. Then combining the results by [5], Frittelh and Rossaza Gianin et al. [2]

and Jouini et al. [4], we have the following.

Theorem 4. Assume that (Q, J^, P) is a standard atomless probability space. Let p : L^ -^ R. Then the following conditions are equivalent

A remark on law invariant convex risk measures 93

1. There is a subset A of the set A^[o,i] x R such that

sup{b; (m, b) e A} = 0

and

p{X) = sup / Pet ^[0,1]

(X)mida)-\-b; (m,b) eA X e L ^ .

2. p is a law invariant convex risk measure.

Our purpose of the present paper is to give a simple and direct proof for this theorem.

Remark 5. One can easily prove that

Pa(X) = -m{\ E[gXl geL^Ô^g^-, E[g] = l a

, X e L °

foranyof e (0, 1]. Here we do not have to assume that (^, ^ , P) is atomless. So we can easily check that Pa, a e [0,1], are law invariant convex risk measures. Therefore, it is easy to prove that the condition (1) impUes the condition (2) in Theorem 4.

2. Preparations

Let iV ^ 2. In this section, we consider a probabiHty space (QN,GN, PN) such that A = {!,..., A }, GN be the set of subsets of Â^, and PNHO)}) = j ^ , CO e Qj\f.

Our aim in this section is to prove the following.

Theorem 6. Let p : L^ -^ R. Then the following conditions are equivalent.

1. There is a subset Ao of the set A^[o,i] x R such that

sup{b', (m, b) 6 AQ} = 0

and

/ Pa(X)m(da) -\-b, (m.b) e AQ p(X) = sup

2. p is a law invariant convex risk measure.

Z G L ^ .

94 S. Kusuoka

By Remark 4, it is sufficient to prove that the condition (2) impHes the condition (1). So we prove the converse. Let p is a law invariant convex risk measure and let C be a subset of L ^ x R given by

N

( « , Z ? ) 6 L ^ x R ; p(X)^-â{i)X(i) + b, for all X 6 L° i=\

Since p is a convex function defined in L ^ and L ^ is finite dimensional, we see that

p(X) = sup -â(i)X{i)-\-b; {a,b)eC i=l

X e L^ . (1)

Moreover, we have the following.

Proposition 7. For any (a,b) eC,we have the following.

1. a ( 0 ^ 0 , i = l,..,,iV. 2. Zf=i«(0 = l. 3. sup{b;(a,b) eC} = 0.

Proof. Let et e L^ , / = 1 , . . . , A , such that ei{i) = 1, and etij) = 0, 7 7 '• Then we have for any c > 0

0 ^ -c'^picet) ^ a{i) - c'^b.

Letting c -^ 00, we have the assertion (1). Note that for any c € R, we have

0 = —p{c) — c ^ c

So we have for any c > 0

c~^b<0 and

(Z«(o-ij-^.

(|„„-.). ( ! • < " - ' ) -\-c-^bÔ.

Letting c ^- 00, we have the assertion (2). The assertion (3) is obvious, since p(0) = 0 . D

Let SM be the set of permutations on QM- Then for any a e L^, there is a Ga e SN such that

aiaa(N)) ^ a(aa(N - 1)) ^ • • • a{aa{l))

Then we have the following.


Proposition 8. 1. For any (a,b) eC, and a e Sjsj, (a oa,b) eC. 2. Let (a,b) e C and let Ma be a measure on [0, 1] be given by

m, n ^ n = {a{aa{j))-a{Ga{j^l)))j. 7 - 1 , . . . , A - 1,

^^.({1}) = a{Ga{N))N and m , ^[0, 1] \ H , | , . . . , i H = 0.

Then nta G M[o,\] and

N

max - ^ ( « o cr){i)X(i); a e S^

i=\

= Pa ^[0,1]

(X)ma(da), X e L^.

Proof. Let X e L^. Then it is obvious that random variables X and X o a ^ has the same probability law . Therefore, we have

N N

p(X) = p{X o or-i) ^ -Y,^(i)X{cr-\i)) -\-b = - ^ f l ( a ( / ) ) Z ( / ) + b.

i=\ i=\

This imphes the assertion (1). Now we will prove the assertion (2). Let X e L^. Then there is xx € SM

such that

X( rx ( l ) ) ^ X(rx(2)) ^ • • • ^ X(TX(N)).

It is easy to see that

rk/N X(Tx(k)) f '

= N J(k-

\)/N Z(x;Fx)dx, k = l,...,N,

and so

Y,X(rx(j)) = -kpk/N(X), k = h...,N.

Then we have

i=l

A

= Y,(a(cra(N))-âiaa(i))-a(cra(N)))X(aa(i)) i=\

96 S. Kusuoka

a{cTa(N))l^X{aaii))j

N-\ / N

= a(aa{N))lY^X(aa(i))j

N j - \

7=2 \ / = l

^ «(^.(^))(X^(^^('')))

7=2 V/=1

«/[0,l] {X)ma{da).

Note that

A

yâ{aaOx-^{i))X(i) = Ya(i)X{Txoa-Hi)) = - [ pAX)ma(da).

So letting Z = 1, we see that ma([0, 1]) = 1. These also imply the assertion (2). •

Now let

AQ = {{ma, b) e M[o,i] x R; (a, b) e C}

Then we see from equation (1) and Proposition 7, that the condition (1) is satisfied for this AQ. This completes the proof of Theorem 6.

3. Proof of Theorem 4

By Remark 5, it is sufficient to prove that the condition (2) imphes the condition (1).


(m, b) e M[o i] x R ; p{X) ^ / Pa(X)m{da) + b, J [0,1]

Let p be a law invariant convex risk measure, and let

^ =

for all X 6 L ^ ( ^ ) | .

Then it is sufficient to prove the follow ing.

p{X) ^ sup / pa(X)m(da) 4- b; (m, b) e A U[0,1]

(2)

Since ( Q , ^ , P) is atomless standard probability space, we may think that Q = [0,1), T = B([0,1)), and P is a Lebesgue measure on [0, 1). For any w ^ 1, let

Tn =cr {l[(jt_i)2-«,it2-"); /c = 1, 2 , . . . , 2"} .

Then we see that

T\ cJîCTsC... and <Â T,

Let

<^n —

Then we have

(m, b) e M[o i] x R; p(X) ^ [ Pa{X)m(da) + b, J[0,\]

for all X eL^(^ , : r„ ,P)

AiD A2D A3D '"D A.

Note that A^[o,i] is a compact subset of the dual space of C([0, 1]; R) with weak * topology. Since p(X) : [0, 1] —> R is continuous for all X e L^ , A, ^ „ , z = 1, 2 , . . . , are closed in A^[o,i] x R.

Proposition 9. Let Aoo = f l ^ i A - Then Aoo = A.

Proof. Let (m, ^) e ôo- Let X e L ^ ( ^ , JT, P), and fix it. Let 7 6 L ^ be given by Y{(JO) = Z(co; Fx), CD e Q = [0, 1). Since random variables X and Y have the same probabiHty law, we see that p {X) = p(Y). Let y„, n = 1, 2 , . . . , be random variables given by

Ynico) = Z ( ^ ^ - ; Fx) , k — \ k l-±<co<~, k=l,2,...,2".

98 S. Kusuoka

Then we see that Yn(a)) i Y{a)), for ^nycoeQ. Since (m,b) e An.n > I, we have

P(y) ^ p(Yn) ^ [ Pa(Yn)m(da) -^b, ae [0, 1]. J [0,1]

It is easy to see that Pa(Yn) t Pa(Y), and so we have

p(X) = p(Y)^ [ Pa(Y)m{da)^b= f pAX)m(da) + b. J[OA] J[0,\]

This impHes that ôo C A. It is obvious that ôo D ^ » and so we have the assertion. n

Now let us prove Theorem 4. For each W e L^(^2", Gi", ^2"), let Un(W) : Q ^- R be given by

2"

[ik-\)2-^,k2-^)(^)' k=\

Then Un : L^(^2«, ^2", ^2«) ^ L^(Q, ^^, P) is bijective. Let p„ : L^(^2", &«, ^2") ^ R be defined by Pn(W) = p(Un(W)). Then it is easy to see that pn is law invariant, convex risk measure and that

/ Pa (W)m(da)-{-b, W 6 L^(Q2«, ^2",/'2«), Pn(W) ^

if and only if

^ / Pa( J[OM

for any (m, b) e M[o,\] x R. This observation and Theorem 6 show that

P(X)^ I Pa{X)m(da)-\-b, XeL'îQ.Tn.P). /[o,i]

p{X) = sup / Pa ^[0,1]

(X)m(da)-h b; {m,b) e An

(3) Let us take an arbitrary X e LîQ.T, P) and fix it. Let F and f„, n = 1, 2 , . . . , be random variables given by Y(a)) = Z(a), Fx), co e [0, 1), and

(^vc.,), k — ] k ''—l<o,<^, k = l,2,...,2".

2« - 2" Y„{w) = Z \ - ; ^ V 0; Fx

Then we see that

Z{x; Fy ) = Ynix) t Y{x~), x e (0,1)


and so we see that

PaiYn) i PaiY) = Pa(X). « -> (X) , « G (0, 1].

Also, we see that

Po(Yn) = -YniO) = -Y{0) = poiX)

So we see that Pa(Yn) converges to PaiX) uniformly in of G [0, 1]. Since Yn e L^(Q,Tn, P), we see from equation (3) that there exists

(m„, bn) e An, for each w > 1, such that

Note that

and that

p{Yn) ^ / Pa{Yn)mn{da) + 7 + - .

0 = P(0) ^ / Pa{())mn{da) + bn = bn

-WYnWoc = p(\\Yn\\oo)^P(Yn)^ [ Pa(Yn)mn(da) J [0,1]

+bn + -^\\Yn\\oo+bn + L

n So we have

0 ^ / 7 , ^ - 2 | | f , | | o c - l ^ - 2 | | X | | o c + l.

Since M[o,i] is compact, there are a subsequence {nk',k = 1,2,...} and (m, b) e M[o^\] X R such that

(fnni,,bnj^) -^ (m, b), n ^ oo, in M[o,i] x R.

It is obvious that (m, b) e Am,, /: = 1, 2 , . . . , and so we see that (m, b) e Aoo-Also we have

/ PaiYnk)jnnkida)-> Pa{Y)m{da). J[0,l] J[0,l]

On the other hand, we see that

p(Yn)^p(Y) = p(X).

So we see that

P(X)

This proves that

P(X) ^ sup

^ 1 Pa -'[0,1]

/ Pa

{X)m{da) + b.

(X)m(da)-\-b; {m,b) eA

This completes the proof of Theorem 4.

100 S. Kusuoka

References

1. Artzner, Ph., Delbaen, R, Eber, J.-M., Heath, D.: Coherent measures of risk. Math. Finance 9, 203-228 (1999)

2. Frittelli, M., Rossaza Gianin, E.: Law invariant cobvex risk measures. Adv. Math. Econ. 7, 33^6 (2005)

3. Follmer, H., Scheid, A.: Convex measures of risk and trading constraints. Finance stochastics 6, A29-AA1 (2002)

4. Jouini, E., Schachermayer, W., Touzi, N.: Law invariant risk measures have the Fatou property. Adv. Math. Econ. 9,49-71 (2006)

5. Kusuoka, S.: On law invariant coherent risk measures. Adv. Math. Econ. 3, 83-95 (2001)



©Springer 2007

Existence and uniqueness of an equilibrium in a model of spatial electoral competition with entry*

Anna Rubinchik^ and Shlomo Weber ' '*

^ Department of Economics, University of Colorado at Boulder, UCB 256, Boulder, CO 80309, USA (e-mail: [email protected]) SMU, Department of Economics, Dallas, TX, USA (e-mail: [email protected])

^ CORE, Catholic University of Louvain, Louvain, Belgium 4 CEPR, London, UK

2

Received: October 3, 2006 Revised: October 3, 2006

JEL classification: C62, C72, D72

Mathematics Subject Classification (2000): 91A10, 91B12

Abstract. Two incumbent parties choose their platforms in a unidimensional policy space while facing a credible threat of an entry by the third party. Relative electoral support is the predominant objective of each party, and the third party enters only if it can displace one of the incumbents by receiving at least the second highest support. We prove the existence and uniqueness of an equilibrium for a wide class of distributions of voters' ideal points, including, in particular, log-concave distribution functions. Moreover, in an equilibrium the incumbents prevent the entry and achieve the "balance of power" by choosing distinct positions in the policy space and equally splitting the electorate.

Key words: incumbent parties, threat of entry, entry-deterrence, rank concerns, balance of power

* We would like to thank Michel Le Breton and Jean-Frangois Mertens for their useful comments. A part of the paper was written when the first author was visiting CORE, the hospitality of which is greatly appreciated.



102 A. Rubinchik and S. Weber

1. Introduction

We consider a model of spatial competition with two incumbents and a potential entrant, where parties are pre-occupied by their rank (or relative performance).^ Competing parties choose their positions in the unidimensional poHcy space, and each voter supports the party that proposes a platform closest to her ideal point.- Similar to Palfrey [8], the incumbents in this model behave as Nash players with respect to one another, but as Stackelberg leaders with respect to the entrant. In other words, the incumbents choose their platforms simultaneously, but in full anticipation of the third party's location along the spectrum of political issues. For entry to be deemed successful, the third party must gamer more votes than at least one of the established parties; otherwise, the potential entrant stays out of the race altogether. Thus, becoming one of the top two is crucial for a party.

The focus of the paper is to show existence and to characterize an equiUb-rium of this game, referred to as 7?.-equilibrium.

We find that under rather mild restrictions on the distribution of voters' ideal points, a third party entry is not sustainable in an 7?.-equilibrium, i.e., the potential entrant can never become one of the top two. This immediately imphes that the set of 7?.-equiHbria is a subset of incumbent strategies which prevent entry by a third party, a notion introduced by Greenberg and Shepsle [7]. This is true even though in our setting the incumbents are forward looking; in particular, each of them might want to trigger an entry to displace the other incumbent from being among the top two, thereby improving her own rank or increasing the share of own supporters, while preserving its relative standing.

Greenberg and Shepsle [7] pointed out that a profile of entry-preventing incumbents' strategies, which we refer to as D-strategies, may fail to exist. We derive quite general sufficient conditions for existence and uniqueness of D-strategies. We go on to demonstrate that 7^-equilibrium exists under these conditions if, and only if incumbents achieve a balance of power whereby the electorate is shared equally among them.

The paper is organized as follows. The following section describes the spatial competition game. Section 3 demonstrates that an equilibrium of the game has to be entry-deterrent. Section 4 contains concluding remarks.

^ Although in what follows we refer to the competitors as political parties, the model is, clearly, applicable to other environments, e.g., an oligopolistic competition with differentiated products.

^ One could view this voting behavior as a desire to associate oneself with a certain platform, defining one's identity. This moUf might prevail in large elections, in which pivotalness of an individual voter is minuscule. See [6] for a related overview. Moreover, there are reasons to believe that sincere voting is a good "positive" assumpfion in the view of the recent empirical findings, see, for example, [1].

Existence and uniqueness of an equilibrium 103

2. The model

Consider two incumbent parties, 1 and 2, and one potential entrant, e. The competing parties choose their positions in the issue space / = [0, 1]. Having observed their choices, the entrant can either stay out (the option denoted by A ) or enter the race and choose a platform in /.

Payoffs to the parties are based on voters' support. Each voter has symmetric single-peaked preferences with the most preferred alternative, or ideal point, in / . Let F be the cumulative distribution of the voters' ideal points in the issue space / , with F(0) = 0 and F(l) = 1.

Given positions of the competing parties, each voter supports the party whose position is closest to her ideal point, and she randomly picks one of the closest by, in case there are many. No abstention is allowed — each voter identifies herself with a party. Let x = (xi, ^2) be a pair of positions chosen by the estabhshed parties 1 and 2 and assume that x\ < ^2, let X = ( j , x^) be the choices made by all three parties, where the entrant can either choose a platform or decide not to enter: Xe e / U {A }.

All three parties have lexicographic preferences. Each of them first considers the rank and then the fraction of votes they gamer. Given choice X of the parties, we denote by r\ (X), r2(X), re(X) their corresponding ranks. Rank r/ (X) can obtain one of the six values:

A1 the sole possession of the first place, A2 sharing the first place with one of the other parties, B1 the sole possession of the second place (with only one party being ahead), A3 sharing the first place with two other parties, B2 sharing the second place with one of the other parties, C1 the sole possession of the third place.

We assume the following natural preferences for all parties:

Al> A2^ Bl> A3> B2> CI.

If the entrant decides to stay out, we say that r^(X) = A , and her ranking reflects the desire to enter only if she can become at least the second (attaining rank Bl) in the electoral competition.^ That is, the preferences of the entrant are given by

Al > A2 > Bl ^ N > A3 > B2 > CI.

^ As mentioned in the introduction, in many situations "winning" is associated with being one of the top two and if entry is sufficiently costly, it will be prevented in case the entrant is not expecting to "win". An alternative specification in which the entrant still enters if she can assure sharing the first place with the incumbents will lead to the same results, as the outcome A3 can not be supported in 7 -equilibrium.


If two different outcomes generate the same rank, a party prefers the one that yields a higher vote share. This share is also determined by the positions of the three parties.

Suppose first that party e does not enter. If the incumbents choose the same position in the issue space, each party is supported by one half of the electorate. If their positions are different, jci < X2, then party 1 is supported by all those voters whose ideal points are to the left of the middle point ^ l i ^ , whereas party 2 is supported by voters whose ideal points are to the right of ^^^y^. That is, the support of party / = 1, 2 denoted by si (X) is determined by

52(X) = 1 - F

Suppose now that party e enters and chooses position Xe in the issue space. If all three parties choose the same position, then each is supported by one-third of the electorate, i.e., s\{X) — sîX) = Se{X) = ^. If the incumbents choose the same position but the entrant locates herself at a different point, say at jci = JC2 < Xe, then the support of each incumbent is given by FC^'^^')/2,

whereas party e is supported by 1 — FC^^^') voters. If all parties choose different positions, say jci < JC2 < JC , then the support of parties 1, 2 and e is given by F ( ^ ^ ) , F ( ^ ^ ) - F ( ^ ^ ) and 1 - F ( ^ ^ ) , respectively. That is, if party / has rivals both to the left and to the right,"* then its support covers the interval between two points: one equidistant from jc/ and its opponent from the left side, and the other equidistant from xi and its opponent from the right. If party has no rivals to its left, then its support covers the interval between 0 and the point which is equidistant from xt and its closest opponent from the right side. Similarly, if party / has no rivals to its right,^ then its support covers the interval between the point which is equidistant from jc/ and its closest opponent from the left side and the right endpoint of the issue space.

Unfortunately, subgame perfect equihbria of the game described above do not always exist. Let us first examine the reasons for the nonexistence and then modify the game to avoid the problem.

Recall party e enters only when it can win higher support than at least one of the incumbents. To describe these cases, fix the positions of the incumbent parties, jc = (jci, X2), and consider the following sets:

D\(x) = {Xe e I\Se(x,Xe) > max[s\{x, Xe), S2{x, Xe)]]

"* Eaton and Lipsey [4] call this location interior. ^ A position which has no rivals either to the left or to the right is called peripheral

in [4].


or the set of positions for the entrant that guarantee the entrant the sole possession of the first place;

D\2{x) = {Xe € I\Se{x,Xe) = m^x[s\{x, Xe), S2(x,Xe)]

> rmn[s\(x,Xe),S2(x,Xe)]}

or the set of positions that yield the entrant the share of the first place with one of the incumbents; and

D2{x) = {Xe e I\m3x[s\(x,Xe),S2(x,Xe)]

> Se(x,Xe) > min[si(x,Xe),S2{x,Xe)]}

or the set of positions where the entrant holds the second place. Also, let

D(x) = Dx (jc) U Di2(jc) U D2{x)

— {Xe e I\Se{x,Xe) > rmn[s\{x, Xe), S2{x, Xe)]\.

Given the preferences of party e, it will not enter, if the set D{x) is empty. Otherwise, it considers the sets D\ (jc), D\2{x), D2(x) (in this order) and makes its vote-maximizing choice over the first nonempty set in this sequence.

By using the arguments of Palfrey [8], it is easy to see that the best response of the entrant over the sets D\(x) or D2(x) may fail to exist. (We shall show that the set D\2(x) is either empty or consists of a unique element.) In this case, we adopt the procedure offered by Palfrey [8] and extended by Weber [10]. Namely, we consider the average of the incumbents' payoffs over the set of "e-best" responses of the entrant. Specifically, for each incumbent / and for each positive s we determine the average of player /'s payoffs over the set of "f-best" responses of the entrant over the set Di (x) if nonempty, and over D2, otherwise, and consider its limit when s approaches zero.

If the entrant chooses to enter, so that for a given pair of incumbents' positions, X, the set D (JC) is nonempty, define by E (x) the subset of D (x) that the entrant considers, i.e., let

E{x) = Di(x) if Di(x)^0 Di2(x) if Dx(x) = 0, Dn(x)9^0 D2{x) if Di(x)UDi2(x) = 0, D2(x) y^0.

For each pair of incumbent choices x e I^ and for £ > 0, denote by B^(x) the set of s-besi responses over E(x). That is,

B^(x) = {xe e E(x)\se(x, Xe) > Se{x, y) - 6 for all y e E(x)}.

Let a pair of incumbents' strategies x = (jci, X2) be such that the set D(x) is nonempty, and moreover, E{x) ^ D\2(x), While the set of best responses


BQ{X) might be empty, the set B^(x) is nonempty for every strictly positive £.^ Let

ix^{x)= I dz and Ui{x) = Urn ——-Si(x,z)dz

for each party /, / = 1,2. Palfrey [8] has shown that the functions ui are well defined. Roughly speaking, ut is a Umit of incumbent /'s electoral support, provided that the potential entrant is mixing across his £-best responses with equal probabiUty.

Now we can define the second component of the preferences for the incumbent parties. If the best response of the entrant exists - either the entrant does not enter, so that Xe = N, or her vote-maximizing position given is well defined^-this component is st, as defined before, and uniqueness of the best reply of the entrant in this case implies Xe is a function of the positions of the incumbents X = (jci, JC2). If the set BQ (X) is empty, then her payoff is set to be equal to the limit of the average support, M/, which is also fully determined by the positions of the incumbents. To sum up, an incumbent /, / = 1,2, derives payoff ndx) from her electoral support:

Ttiix) = Ui(x), ifD(x)^0, B^(x) = 0 Si (jc, jCe (x)), Otherwise.

We can now formally define game F between the incumbents, who foresee that the third party, e, enters only if she can displace one of the incumbents and to guarantee at least the sole position of the second place. Formally,

Definition 2.1. In the two-person game F the incumbents have strategy set /. Players' preferences are lexicographic in (1) rank, ri and (2) payoff, nt.Apure strategy equilibrium of the game F is called an IZ-equilibrium.

It is important to distinguish our equilibrium notion from that introduced by Greenberg Shepsle [7]. They refer to each pair of incumbent positions that prevent entry by a third party as 2-equilibrium. We simply call these strategies entry-deterrent:

Definition 2.2. A pair of positions of established parties x — {x\,X2)is called entry-deterrent fP-strategies), if the set D(x) is empty.

Note that an entry-deterrent pair of incumbent strategies is not necessarily consistent with 7?.-equilibrium. Indeed, the latter requires the incumbents'

^ It can be shown, using the argument in [11] that the set B^ (x) is the union of a finite set of intervals and that if the set of best responses B^ (x) is nonempty, it consists of a unique element (see claim 3.1 in [11]).

^ As is, for example, in case E{x) = Di2ix).


positions to be immune to unilateral deviations by the incumbents, while correctly anticipating the response of a potential third party. Thus, we allow an incumbent to induce the entry, if it is in her interest. The next section offers a characterization of 7?--equiHbria. The key result is that in 7?.-equihbria neither of the incumbents will want to induce the entry.

3. Balance of power in 7?.-equilibrium

In this section, we derive conditions for the existence and uniqueness of both 7^-equilibrium and D-strategies, and study the relationship between the two. Throughout the remainder of the paper we consider only pairs of strategies (jci, X2) where party 1 is located to the left of party 2, i.e., x\ < X2, so that the uniqueness of an equihbrium will be stated in terms of equiUbrium configurations up to a permutation of incumbents' strategies. The proofs of all results in this section are relegated to the Appendix.

We shall now introduce two assumptions, (A.l) and (A.2), which are maintained for the rest of the paper. The first is quite standard and requires the distribution of voters' ideal points to be unimodal and the density function / to be continuous:

Assumption (A.l). /(•) is continuous and strictly positive on [0, 1]. Moreover, there exists x e I, such that /(•) is strictly increasing on the interval [0, i ] and strictly decreasing on the interval [x, 1].

The second assumption assures that the ideal points of the voters are not too concentrated at any given interval. Following Haimanko, Le Breton and Weber (2005) we will refer to this assumption as gradually escalating median (GEM).^ Let / : [0, 1] -> [0, 1] be the median of [0, t] and r : [0, 1] -^ [0, 1] be the median of [t, 1] under F. Given the first assumption both functions are continuously differentiable.

Assumption (A.2). r (t) < l , r ' ( r) < 1.

Assumptions (A.l) and (A.2) will allow us to compare P-strategies and strategies of the incumbents under 7^-equilibria. The definition of P-strategies rules out a move by the entrant, while 7^-equilibrium requires an incumbent's position to be immune against a unilateral deviation of another incumbent that can "invite" an entry by the third party.

' This assumption is weaker than log-concavity, which is rather mild on its own, see [2] for the discussion and connection to other properties, including monotone hazard ratio. More precisely, a stronger assumption than (A.2) would require function F(t) to be log-concave on the interval [0,x] and the function 1 - F(l - /) to be log-concave on the interval [x, 1].


As defined, 7^-equilibrium does not preclude an entrance by the third party. It is important to estabhsh, therefore, whether there exists 7?,-equiHbrium in which the estabhshed parties allow for the entry of party e. Proposition 3.1 demonstrates that the answer is negative, implying that the notion of 7?,-equi-librium is no less restrictive than the notion of entry-deterrent strategies.

Proposition 3.1. Assume that (A.l) and (A.2) hold. Then in any TZ-equilib-rium party e does not enter. That is, every pair of incumbents' IZ-equilibrium strategies is also a pair of V-strategies.

Greenberg and Shepsle [7] concluded that, in general, the set of D-strategies might be empty. Providing sufficient conditions for existence of these strategies remained open. Cohen [3] has shown that P-strategies exists for the special case where the distribution of voters' ideal points is given by a normal density function. The following proposition demonstrates that the condition of normality, and even symmetry, of the distribution can be dropped. UnimodaUty and GEM yield existence and uniqueness of P-strategies.

Proposition 3.2. Under (A.l) and (A.2), there is a unique pair of V-strategies x^ = (jCp JC2). Moreover, x^ satisfies*^

F(î) = 2^\-^^r^\^ (2)

l - f ( - 2 ' ) = ^ ( l - ^ ( ^ ) ) . (3)

Our next proposition derives necessary and sufficient conditions for existence of 7^-equilibrium. Note that by Proposition 3.1, the set of 7^-equihbria is a subset of the set of P-strategies. We show that, in general, the converse is not true. Since, by Proposition 3.2, for a given distribution of ideal points a pair of P-strategies is unique, it follows that the set of 7?.-equilibria might be empty. That is, even though [under (A.l) and (A.2)], there is always a unique entry-deterring pair strategies for estabhshed parties, one of the incumbents could be better-off by deviating from it, thus allowing for entry of party e. Given that we impose an additional requirement of Nash behavior on incumbents, it is not surprising to find out that 7?.-equilibrium may fail to exist in circumstances which guarantee existence of P-strategies. Our result shows that 7?.-equihbrium exists only in the case where the estabhshed parties, while locating themselves

^ These equations were formulated in Greenberg and Shepsle [7] as necessary conditions for the existence of a 55-equilibrium.


at quartiles of the distribution, achieve a balance of power by equally splitting the total electoral vote.

Proposition 3.3. Assume that (A.l) and (A.2) hold and let the pair x^ = (jCpX2) be V-strategies. Then a pair of incumbents' strategies x^ is an IZ-equilibrium if and only if

Jxi+xi\^\ (5)

That is, if

(o--a)=--(0' then the set ofTZ-equilibria consists of the unique element, (F ^(•^), F ^4))-

Otherwise, the set ofTZ-equilibria is empty.

The intuition behind Proposition 3.3 is quite simple. When power is balanced between incumbents, in the sense that each gamers 50% of the vote, neither party can improve its standing by altering its position. Take, for example, the candidate located on the left. Moving further to the left will reduce her support relative to the other incumbent as well as make it possible for the entrant to locate "very close" on her right and displace her by garnering slightly more than 25% of the vote. On the other hand, should the left-most incumbent attempt to increase her support by moving closer to the current rival, the entrant will displace her by locating "very close" on her left and garnering shghtly more than 25% of the vote. By the same argument, the right-most candidate cannot improve her standing when the incumbents choose platforms in such a manner that the electorate is divided equally between the estabhshed parties.

Balance of power is essential for 7^-equilibrium to exist. If power is not shared equally between the two incumbents, the second place incumbent can improve its standing by moving slightly closer to the incumbent who ranks first. In doing so, the entrant can now gamer more votes than the top-ranked incumbent by entering "very close" on her outside. The incumbent formerly in second place will now win the election, the entrant will get the second place, and the incumbent formerly in first place will now be ranked third as it is "squeezed" between its old rival and the new third party.

Imphcitly, Proposition 3.3 characterizes societies (described by distributions of voters' ideal points) for which 7?.-equihbrium exists. In particular, any symmetric density function satisfies condition (6). Clearly, symmetry is not necessary for that condition to hold.


4. Conclusions

The model offers an analysis of electoral competition in the presence of rank concerns. If the distribution of voters' ideal points F is single-peaked and satisfies the gradually-escalating-median property, incumbents' strategies in an 7?.-equiUbrium are always entry-deterrent and in this case there exists a unique 7^-equilibrium if and only if the distribution F satisfies

-•(o--a)=--(o Appendix

Let a pair of incumbents' strategies x = (xi, JC2) be given. Let a = ^^^^, II ^ F(xi), I2 ^ F{a) - F{xi), I3 = F(X2) - F{a\ I4 ^ 1 - F f e ) , G(x) = 1 - F(x).

Assume, without loss of generahty, that

Ii > I4. (7)

Lemma 5.1. Ifl\ > I2 then the support of an entrant choosing a policy z G (xi, JC2) w/// not exceed that of the first party and if IT, < I4 the support of an entrant choosing a policy z G (xi, ^2) will not exceed that of the second party. If both inequalities Ii > I2 and I3 < L hold, then no entry will occur between x\ and X2, i.e., the set

D'îx) = {ze D(x)\xi <z<X2}

is empty.

Proof. Consider the entry of party e between jci and X2, say, at z. Let Ii > I2 hold. Then the support of the entrant is equal to F(z-\- r2) — F{z — r\) where r2 iz) = {X2 - z) /2, ri (z) = (z - x\) /2. This implies r[ (z) = -r^ (z) = \ and then by assumption (A.2),

l{z-\-r2)-l{ot) <z-\-r2-a

= z-r\-x\,

thus

/(z + r2) < z - r i ,

as / (a) < XI. Then F{z + ^2) < 2F(z - ri), so that the first party has higher support than the entrant. Similarly,if I3 < I4, G (1 - z + ri) < 2G (1 - z - r2)


and the second party has a higher support than the entrant. If both inequalities Ii > I2 and I3 < I4 hold,

F{z + r2) - F(z - n) < min[F(z - n ) , l-F(z + r2)l

Hence, the support of the entrant does not exceed that of any of the two estab-hshed parties. n

In order to prove Proposition 3.2 we shall use the following lemma:

Lemma 5.2. For each y,0 < y < 1, define the values ofa(y) and b(y) by

2F{aiy)) = F(y) and 2{l - F(b{y))) = 1 - F(y). (8)

Then there is a unique y^, satisfying

« ( / ) + ^ ( / ) = 2 / . (9)

Moreover, the value ofy^ is such that ^ < F(y^) < | .

Proof. It is easy to verify that the functions a(') and Z?() are well defined, increasing and differentiable on the interval [0, 1], and so is h(y) = 2y — aiy) - b(y). Since a(0) = 0, b(0) > 0, a{l) < 1, (1) = 1, it follows that h{0) = -b(0) < 0 and h(l) = I - a(l) > 0. Hence, there exists a value y that solves (9).

For uniqueness rewrite the necessary conditions as a system of two equations,

2F(a)-F{^)=0 2F{b)-F{^) = 1

for (a, b) € [0, 1]^ . Its Jacobian is

. - i / ( ^ ) 2 / ( f e ) - i / ( ^ )

the principal minors of which are positive. Indeed, by (A.2),

/'W = ^ - 4 ^ < 1 (10) 2 / (/ (x)) 1 fix)

r (x) = < 1. 2 / ( r ( x ) )


This leads to the following conditions:

and, therefore,

(12) Conditions (11) and (12) assure the positiveness of the principal minors of the Jacobian and, thus, uniqueness by the Fundamental Global Univalence Theorem ([9], p. 20). 10

Finally, we need to verify the inequalities

- < F ( / ) < - . 3 ^^ ' 3

By the necessary conditions (8), we know / (y'') = a. By assumption (A.2) for any x €[>>'', fo], I' (x) < 1, or for r e[0,b- y'']

which implies

thus

and, moreover,

l[y'' + r^-r<l (y^) ,

1(b) < I (yA +b-y^ = y''

Fjbiy")) ^ F ( / ) _ ^ F{yd) " F{a(yd))

F ( / ) > Fibiy'')) - F ( / ) = ^^^ \

yielding Fiy") > j . Similarly,

1 - F ( f l ( / ) ) ^ 1 - F ( / ) ^ ^

1 - F ( / ) ^ 1 - Fibiyd)) '

^^ The initial formulation is due to [5].


which imphes

1 - F ( / ) > F ( / ) - F(f l ( / ) ) = 2 '

o r F ( / ) < f . D

Before proceeding to Proposition 3.1, we provide a proof of Proposition 3.2.

Proof of Proposition 3.2. Greenberg and Shepsle [7] have shown that conditions (2), (3), and (4) are necessary for a pair of strategies to be P-strategies. By Lemma 5.2, there is a unique pair of strategies which satisfies those conditions. Thus, it remains to show that this pair of strategies is indeed a pair of P-strategies under (A.l) and (A.2).

Lemma 5.2 yields the existence of a real number y € (0, 1) such that the pair w — {a(y), b{y)) is the unique pair of strategies which satisfies (2), (3) and (4). To demonstrate that set D(w) is empty, let us consider options available for party e, given the incumbents' locations at points a(y) and b{y), respectively. If party e enters to the left of a{y), it would not displace party L Moreover, since F{a(y)) < ^ < F{y), party 2 would not be displaced either. By using similar arguments one can show that neither of the established parties would be displaced if party e enters to the right of b{y). In the view of conditions (2), (3) lemma 5.1 guarantees that no entrant will choose a position in between a (y) Sind b(y). D

We shall turn now to the proof of Proposition 3.1. For each pair of positions of the estabhshed parties x = (x\,X2) denote by

^1 _ yi^x2-xi>^ ^^^ yi _ y2^x2-jci- ^^^ locations in the issue space which

satisfy

^(y ' + ^ ^ ^ ) = 2F(ji) ; (13)

G ( / - ^ ^ ^ ) = 2 G ( A (14)

Let

z\x) = ly^-xv, z^{x) = ly^-X2.

In the case of z (x) > x\, if party e enters at z (x) it would generate the support equal to that of party 1. Assumption (A.2) can be used to show that if party e enters between xi and z (x) it would displace party 1, and if it enters to the right of z (x) it would not displace party 1. Similarly, in case Z^(JC) < JC2,

if party e enters to the left of z^(x) it would not displace party 2, if it enters at z^(x) it would generate the same number of votes as party 2, and if it enters


between z^(x) and X2 it would displace party 2. In addition, for each pair of positions of the estabhshed parties x = {x\, JC2), consider the function s^(x, t) which determines the support of party e generated by entry at r E / . I t has been shown in [10] that assumptions (A.l) guarantee that s^(x, •) is continuous and strictly quasi-concave on the interval (jci, JC2). It allows us to continuously extend this function to the closed interval [x\, JC2]. Thus, there exists a unique value z*(x),x\ < z*(x) < X2 such that

z*(x) = 2iTg max s^(xj). X\<t<X2

z*(x) determines the vote-maximizing location^ of party e between the positions of the two estabhshed parties.

We will show now that if one of the sets Di(x), Dîx) or D2(JC) is nonempty then X is not an 7?.-equihbrium.

Lemma 5.3. Let a pair of strategies x = {x\,X2) be an IZ-equilihrium. Then set D{x)\D2 (x) = Di(x) U Duix) is empty.

Proof. Let x = {x\, X2) be an 7^-equihbrium. Suppose first that Di (x) is nonempty. Since by assumption (7), Ii > I4, party e may enter either to the left of x\ or between xi andjC2.

(i). Suppose that party e enters to the left of xi, which will happen, only if Ii > I4. The entrant gains the support of almost Ii and wins the race, if Ii > I4 -h I3. But then a move of player 2 to the left of X2 by a small £ would still hold party e entering to the left of jci. Thus, the payoff of party 2 will increase to 712(xi, jc2 — e) = 1 — F(a — f) > 7T2(X),

while the support of party 1 will drop to F(a — ^) — F{x\) < TTI (x), a contradiction to the fact that x is an 7^-equiHbrium.

(ii). Suppose now that party e enters between x\ and JC2, which will only happen if z\ (x) > Z2 (x). Then D\ (x) = (z\ (x), Z2 (x)), and party e maximizes its support over this interval. Thus, if any of the estabhshed parties makes a "slight" move towards its competitor, party e would still enter "in between" and, by Lemma 4.2 in [10], the party initiating the move, will increase its support, a contradiction. Thus, D\ (x) is empty,

(iii). Consider now the case, in which set D1 (JC ) is empty, whereas set D12 (x) consists of a unique element, xi. Indeed, the entrant cannot share the first place by entering either to the left of xi or to the right of X2. If zi (x) < Z2(x), then there is no position between x\ and X2 that guarantees the entrant the share of the first place. If z\ (x) > Z2{x), then, contrary to our assumption, set D\(x) is nonempty. Thus, the entrant

^ ^ When z* is equal either to jcj or to X2, it represents the limit of "almost" vote-maximizing positions of the entrant.


selects her position Xe at jci andwehave^F(Qf) > F(l—Qf)andxi = | . Then let party 2 leapfrog to the immediate left of xi, thus, allowing the entrant to move to the immediate right of xi. This move completely squeezes party 1, thus guaranteeing party 2 the sole possession of the second place, rather than being the last before the move took place. D

Proof of Proposition 3.1. Let a pair of strategies x = (x\, X2) be an 7?.-equi-Ubrium. In the view of Lemma 5.3, it remains to show that the set D2(x) is empty.

It suffices to demonstrate that Ii = I2 and 13=14 (Recall that Ii > I4.) Indeed, in this case, the same arguments as in the proof of Proposition 3.2 would yield the emptiness of the set D2(x).

Assume, in negation, that, at least, one of equalities Ii = I2 or I3 = I4 does not hold. There are several cases to consider.

(1) 11 > I2 and I3 < I4. Lemma 5.1 imphes that the entrant could not displace one of the estabhshed parties by entering between jci and X2. Depending on the relationship between Ii, I2,13 and I4, party e still may enter either to the left of xi or to the right of ^2.

(la) Ii > I4. Party e would enter to the left of jci, then jc is not an 7^-equilib-rium by the same argument as in case (i) in Lemma 5.3.

(lb) Ii = l4> I3. Party e would enter with equal probabihty to the left of x\ and to the right of X2, as both actions lead to being ranked as the second with equal electoral support. Hence

F{a)-F(xi) F(a) F(xi) n,(x) = ^—-^Fia)-^-,

recall, a — ^^^^. Then there exists £ > 0 such that a "slight" shift of party 1 to its left, to jci — e would force party e to enter to the right of X2, yielding 7i\ {x\ — e, X2) = F{a — | ) > TTI (X), again a contradicting X being an 7^-equilibrium.

(Ic) I i= 14= I3. Party ^wouldentertotheleftofxi, thusassuringthesecond placeandwithsupportof almost Ii. In this case 7ri(jc) = F{a) — F{x\). Then a move of party 1 to the left to x\ leads to an entry of party e to the left of X2, leaving the entrant in the second place with a higher support. Since the shift of party 1 can be chosen arbitrarily small, one can guarantee that party e enters "very close" to X2. Thus, there exists £ > 0 such that 7T\{X\ — e,X2) > TTI (x), a contradiction.

(2) Ii > I2,l3 < l4- Could be examined in the same manner as the case (1). (3) Ii < I2,l3£ l4- Coupled with assumption (7) it imphes l2> I4. Party e

would enter between x\ and JC2. Indeed, by entering to the right of jci, it could generate the support of "almost" I2 voters, thus assuring the second place. On the other hand, outside of this region it could attract the


support of no more than Ii < I2 voters, which gives her the second rank at best with less support. Moreover, since I4 > I3, Lemma 5.1 impUes that the entrant is unable to attract more votes than party 2, by entering between x\ and X2, so that D'îx) == (jci, z^ (x)). Thus, the entrant should maximize the electoral support on that interval. The payoffs of both players in game F will be determined by

7Tl(x) =

h if z* = XI

;r(a±|l(£)) if zHx) ^D'îx)

7t2{x) =

I3 + I4 if 1 _ F ( 2 + | ^ ) if l - F ( S ± | i W ) if

z*{x) =xi

z*ix) e D'"{x)

z*(x)^D'"{x)

Suppose now that party 2 moves its position to JC2, which is "sUghtly" to the left of X2. This would shrink the set of potential "in between" entry positions of party e, i.e., D^{x\,X2) C D^{x), as the aforementioned move of party 2 would shift y^ to the left by assumption (A.2) [combining (13) and (10)], and therefore z^ (î, ^2) < z^ {x). Thus, if z*(x) > z^{x), this move yields a higher payoff to party 2. If z*(x) < z^{x), then Lemma 4.2 in [10] implies that party 2 benefits from its move, a contradiction.

(4) Ii > I2,13> I4. Could be examined in the same way as case (3). The last case to be considered is:

(5) Ii < I2,13 > I4. Party e enters in the interval [x\, JC2], as D {x) c [jci, X2\ in this case. By using the previous arguments, we have D^{x) = (xi,z^(jc)) U(z^(jc), JC2), and, moreover, in the view of Lemma 5.3, we are left with the case z^(x) < z^{x). If zîx) is either less than zi (jc) or greater than Z2(x), then the consideration is the same as in the case (ii) of Lemma 5.3. Assume, therefore, that z\{x) < z^{x) < Z2(x)-

(5a) Let us first consider the case of 5- (x; z\(x)) ^ s^{x; Z2(x)) and, without loss of generality, s^{x; z\ix)) > s^{x; Z2{x)), Thus, party e enters to the left of z 1 (x). Then, by moving to the left of X2, party 2 would increase its support by forcing party e to shift its vote-maximizing position to the left by Lemma 4.2 in [10].

(5b) Let sîx; z\ix)) = s^(x', Z2(x)) and z*(x) = z\ (x) = Z2 (x), and so s'(x;zHx)) = l/3andD'"(jc) = (jci, z^jc)) UU^^),^2). The best the entrant can do in this case is to displace one of the incumbents by entering as close as possible to z*(x), and, being indifferent between


entering slightly to the left of z\ (x) or sUghtly to the right of Z2(x), she chooses both actions with equal probabihty.

Let us show that x is not an 7^-equilibrium. Take a "small" ^ > 0 and consider two alternative moves:

( 0 : Party 1 moves to the right to jc J = xi+e. Denote x^ = (-^1, 2) ^ ^^•

{r]): Party 2 moves to the left to jc2 = JC2 — e. Denote x^ = (x\, JC2) e M?.

Firstconsider^.By assumption (A.2), there are z =z^(x^) < z*(x) <

zl = z^(x^), such that the set of possible entry positions of party e

consists of two disjoint intervals: D^(x^) = (x\, zl)[j(zhx2) C M. In this case, again, only one of the estabhshed parties can be replaced, so provided party e can only guarantee to be the second, it maximizes voters' support. The best position ofparty sunder ^,z* = z*(x^), should by to the right of z*(x) by Lemma 4.2 in [10]. Therefore, zt > zl. If s^(x^; zl) < s^(x^; zl), the entrant will replace party 2, thus benefiting party 1, a contradiction.

It is left to consider the case, in which ^ (jc \z\) > s^{Xi^; zl) after the move of the first party. Note that the last inequality could be rewritten as

s'{x^',z\) = F(y\r-'-))>\-F ( / {r - 0 ) = s\x^',z]).

(15) The examination of the move r] is similar. Again assumption (A.2) yields the existence of zj = zUî, ^2) and z^ — zîxr^, such that D^(xr^) = (xi^zlj) U(^^' ^2). Again we assume that z^ < z^ and s^(xrj; z^) > s^(Xr^, zh. Thus, we have

/ (x , ; z^) = F[y\r-'-))>l-F ( / (r - '-)) = s\x,', z\).

(16) Combining (15) and(16), we conclude that party ^generates the support of the same number of voters, denoted by -Q, whenever it chooses one of the following positions z]. or z^ under ^ and zj or z^ under r]. Note also that since the functions y^ (•) and y^{-) depend only on the distance between the positions of the first two parties, the set of voters who pick z\ (z^, respectively) under ^ is the same as of those whose best choice zj (z^, respectively) under r]. Hence party 1 would be better off under ^ than under r], whereas the opposite is true for party 2, i.e.,

n\(x^) > n\(Xr^)\ TtliXrj) > 712(X^). (17)


As both moves generate support of SQ voters for party e, we also have

7Ti(x^)-\-7T2(x^) =7Ti{Xrj)-{-7T2(Xrj) = 1 - ^ 0 . (18)

Combining (17) and (18) we obtain 7ri(jc^)+7r2(jc^) > I - ^ Q . However, by Assumption (A.2), both moves ^ and rj lead to a dechne in support of party e, i.e., o < | = s^{x; Z*(JC)). Thus,

2 ni{x^)+7T2iXrj) > -

It follows, therefore, that at least one of the numbers 7t\ (x^) or 712(JC ) exceeds ^, yielding either 7ri(x^) > 7ri(jc) or 7t2{xrj) > 7t2{x). That is, at least one of the estabhshed parties would benefit by deviating, a contradiction.

(5c) Finally, let s^{x; z\(x)) = s^{x; Z2(x)) with zi(jc) < Z2(x), in which case party e enters with equal probabiUty to the left of zi (x) and to the right of Z2(x)- But then the similar arguments as in the consideration of the previous case show that x is not an 7?.-equihbrium. n

Proof of Proposition 3.3. By Proposition 3.2, there is a unique pair of D-strat-

egies (xf, x^). Then, by (2) and (3), F(xf) = ^ ^ ( ^ 4 ^ ) and 1 - F(x^) =

\(l- F ( ^ ^ ) ) . Assume that (4) holds, i.e., F ( ^ % ^ ) = i . Then the payoff of each player in game V is equal to ^. We shall show that if one of the players chooses a different strategy, her payoff will decUne.

Consider party 1. Suppose first that it moves to the right of ;cf by choosing jci > xf. Since this move of party 1 shrinks the mass of voters located between the incumbents' positions, party e could not displace an estabhshed party by entering between jci and JC2. Hence, the entrant should come either to the left of jci or to the right of x | . Since F(jci) > ^ = 1 — F(jc|), the optimal entrant's move would be to the left and "very close" to xi. Then the payoff of player 1 in game F would be less than F(x2) — F(xi) < \. Thus, player 1 would be worse off by moving to the right of xf.

Suppose now that party 1 moves to the left of xf by choosing jci < x^. Since the deviation of party 1 expands the mass of voters located between the incumbents' positions, party e could not displace an estabhshed party by entering to the right of JC2. Moreover, by assumption (A.2) (and by the argument analogous

to that in Lemma 5.1), F( 2 ) - ^ (^1) > i so that party e could not displace party one, and, clearly, not the second party whose support is over 1/2. Therefore, an entrant will not choose a platform to the left of xi, either. Since, by Proposition 3.1, D(jci, x^) is nonempty, the entrant could displace one of the estabhshed parties by entering between x\ and ^2. Then the payoff of player 1


would be less than F(—2"^) < ^ making her worse off. This completes the "if" part of the Proposition.

Suppose that (4) does not hold and assume, without loss of generality, that F(x^) < 1 - F(x^). Choose 8 > 0 such that F(jcf + 5) < 1 - F(jc^) and consider the move of party 1 to xi = x^ -\- 8. By entering between x\ and 2, as well as by entering to the left of x\ the entrant would get less than a quarter of the votes. However, by entering to the right and "very close" to ^2, party e would displace party 2 and would receive more than 1/4 of the votes. Moreover,

the support of the first party will increase, 7ri(xf + 5, x ^ = ^("^"^—~) >

F( ^ 2 ^) — ^1 (-^1' ^2 )• Thus, the party which gets less than 50% of the total vote in P-strategies would be better off by moving towards its rival incumbent and allowing entry of the third party.

To conclude the proof of the proposition, note that by (2), (3), (4), the pair (xi, JC2) constitutes an 7^-equilibrium if and only if F(xi) = ^, F(x2) = | and

References

1. Alvarez, R.M., Nagler, J.: A new approach for modeling strategic voting in multiparty elections. Br. J. Pol. Sci. 30, 57-75 (2000)

2. Bergstrom, T, Bagnoli, M.: Log-concave probability and its applications. Econ. Theory 26, 445-469 (2005)

3. Cohen, R.: Symmetric 2-equilibria of unimodal voter distribution curves. Mimeo, Harvard University 1985

4. Eaton, B., Lipsey, R.: The principle of minimum product differentiation reconsidered: Some new developments in the theory of spatial competition. Rev. Econ. Stud. 42,27-49(1975)

5. Gale, D., Nikaido, H.: The Jacobian matrix and global univalence of mappings. Math. Ann. 159, 81-93 (1965)

6. Green, D., Palmquist, B., Schickler, E.: Partisan Hearts and Minds: Pohtical Parties and the Social Identities of Voters. Yale University Press, New Haven 2002

7. Greenberg, J., Shepsle, K.: The effect of electoral rewards in multiparty competition with entry Am. Pol. Sci. Rev. 81, 525-537 (1987)

8. Palfrey, T: Spatial equilibrium with entry Rev. Econ. Stud. 51, 139-156 (1984) 9. Parthasarathy, T: On Global Univalence Theorems. Lecture Notes in Mathematics.

Springer, Berlin Heidelberg New York 1983 10. Weber, S.: On hierarchical spatial competition. Rev. Econ. Stud. 59,407^25 (1992) 11. Weber, S.: Entry deterrence in electoral spatial competition. Soc. Choice Welfare

15,31-56(1997)

Adv. Math. Econ. 10, 121-122 (2007) Advances in MATHEMATICAL

ECONOMICS

©Springer 2007

Publisher's Errata Solving long term optimal investment problems with Cox-IngersoU-Ross interest rates*

H. Hata and J. Sekine

^ Graduate School of Engineering Science, Osaka University, Toyonaka, Osaka 560-8531, Japan (e-mail: [email protected])

^ Institute of Economic Research, Kyoto University, Yoshida-Honmachi, Sakyo-ku, Kyoto 606-8501, Japan (e-mail: [email protected])

(c)Springer 2007

The following sentences were printed incorrectly in the above-mentioned article due to the publisher's error.

1. On p. 240, on line 2 from the bottom,

The effective domaineffective domain should be:

The effective domain

The original article was published in the Advances in Mathematical Economics volume 8, p.231-p.255. Its onhne version can be found at http://www.hiranolab.jks.ynu.ac.jp/ advances/4/current.html



http://www.hiranolab.jks.ynu.ac.jp/

122 Publisher's Errata

2. On p. 241, on line 2,

The difference A^ of growth rategrowth rates should be:

The difference A T of growth rates

On the same page, on the bottom line,

given by (1.3) is nearly optimalnearly optimal should be:

given by (1.3) is nearly optimal

3. On p.246, on line 14 from the bottom.

So from Feynman-Kac's formulaFeynman-Kac's formula should be:

So from Feynman-Kac's formula

Subject Index

additive dynamics 68 affine 28 allocation processes 76 allocation ratio 77

balance of power 109 ball-weakly compact 8 ball-weakly* compact 9 b-convex 29 Bellman equation 72 "e-best" response 105 biting lemma 4 Bochner 19

capital good 32 capital intensity differences 38 capital intensity reversal 37 capital intensive 37 CES technologies 32 closed in measure 3 closure property 21 Cobb-Douglas technologies 39 concatenating 81 conjugate convex integral functional 23 consumption good 32 control 29 control (function) 68 convergence in measure 18 convex function 61 convex Isc-concave use 28 convexity 21

depreciation rate of the capital stock 35 diagonal extraction 14 diagonal procedure 9 discount rate 35 Dunford-Pettis theorem 19

elasticities of capital-labor substitution 38 entry-deterrent {V-strategies) 106 equality condition 84 equilibrium path 35 Euler equation 69 evolution 22 extremal 69

Fatou lemma 6 Fibonacci quadratic equation 66 Frechet differentiable 53 function 45 fundamental global univalence

theorem 112

Gateaux differentiable 53 gap 2 generalized resolvent 55 golden number 66 golden optimal policy 83 golden optimality 65 golden policy 66 golden rule 65 golden section 66 golden trajectory 66 gradually escalating median 107 grand optimality 65

Hamiltonian 35

ideal point 103 inequality 84 inf - sup theorem 28 input coefficients 33 input coefficients at the private level 34 input coefficients from the social

viewpoint 34 integrand 20 integration by parts 70 issue space 103

Kamimura-Takahashi Komlos 1

52

labor intensive 38 Lebesgue-Vitali 4 Leontief technology 39 liminfpart 17 limsup part 17 linear additively separable

utility function 35 linear topology 19 local indeterminacy 31

124 Subject Index

Mackey topology 9 maximal momotone 53 Mazur property 2 Mazur tight 3 mean value theorem 70 measurable selection 16 minimization 22 minimizer 61 min-max 22 Mosco 2

negligible 11 normal integrand 20

optimal policy 66 optimal trajectory 73

Pettis 19 pointwise bounded 14 pointwisely converges 7 primitive 23 proportional 68 proximal point algorithm 51

quasi-input coefficients at the social level 34

radon measures 27 random 9 reflexive 8 /?-equilibrium 106 Rockafellar 51 Rwc(F)-iight 9 Rybczynski theorem 34

scalarly integrable 12 sector specific external effects sequentially 19 simple policy 65 slice 1 slice converges 2

31

slice topology 2 spatial competition 102 square-root 76 stable manifold 36 stable topology 28 steady state 35 Steinhaus 29 Stolper-Samuelson theorem 34 strictly convex 53 strong convergence theorem 58 sublinear 20 subsequence principles 29 subtractive dynamics 76 suites adaptees 29 sunny generalized nonexpansive

retract 54 sunny generalized nonexpansive

retraction 54 sunspots fluctuations 31 super reflexive 1 support function 20

tightness 3 transversality condition 35 transvesality condition 69 two-sector infinite-horizon growth

models 31

uniformly convex 53 uniformly integrable 4 upper semicontinuous 26

value function 72 variational 68

weak convergence theorem 55 weakly lower semi-continuous 10

Young measures 1

zero point 51

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Do not itahcize, underscore, or use boldface for headings and subheadings. Words that are to be italicized should be underscored in pencil.

Abbreviations must be spelled out at first mention in summary/abstract and main text. Abbreviations should also be spelled out at first mention in figure legends and table footnotes.

Short equations can be run in with the text. Equations that are displayed on a separate line should be numbered.

7. References: The list of references should be in alphabetical order and include the names and initials of all authors (see examples below). Whenever possible, please update all references to papers accepted for publication, preprints or technical reports, giving the exact name of the journal, as well as the volume, first and last page numbers and year, if the article has already been published or accepted for publication.

When styUng the references, the following examples should be observed:

Journal article: 1. or [F-M] Freed, D.S., Melrose, R.B.: A mod k index theorem. Invent. Math. 107, 283-299 (1992)

Complete book 2. or [C-S] Conway, J.H., Sloane, N.J.: Sphere packings, lattices, and groups (Grundlehren Math. Wiss. Bd. 290) Berlin Heidelberg New York: Springer 1988

Single contribution in a book: 3. or [B] Border, K.C.: Functional analytic tools for expected utility theory. In: Aliprantis, C D . et al. (eds.): Positive operators, Riesz spaces and economics. Berlin Heidelberg New York: Springer 1991, pp. 69-88

8. Citat ions in the t ex t should be either by numbers in square brack

ets, e.g. [1], referring to an alphabetically ordered and numbered list, or by the author's initials in square brackets, e.g. [F-M], or by author and year in parentheses, e.g. Freed and Melrose (1992). Any of these styles is acceptable if used consistently throughout the paper. In the third system, if a work with more than two authors is cited, only the first author's name plus "et al." need be given and if there is more than one reference by the same author or team of authors in the same year, then a, b, c should be added after the year both in the text and in the list of references.

9. Tables are to be numbered separately from the illustrations. Each table should have a brief and informative title. All abbreviations used in a table must be defined in a table footnote on first use, even if already defined in the text. In subsequent tables abbreviations need not be redefined. Individual footnotes should be indicated by superscript lowercase a, b, c, etc. Permission forms must be provided for any tables from previously published sources (same procedure as with figures; see below).

10. Figures: If you have access to suitable software, you can design your figures directly on a computer, but creation by other means is of course also possible. In any case the originals should be pasted into the camera-ready copy at the appropriate places and with the correct orientation. If necessary, the figures may be drawn overscale, but in this case suitably reduced copies should be pasted into the script.

If a figure has been published previously, acknowledge its source and submit written permission signed by author and publisher. The source must be included in the reference list. If a permission fee is required, it must be paid by the author. Responsibility for meeting this requirement lies entirely with the author.

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Advances in MATHEMATICAL - Nguyễn Thành An · PDF fileAdvances in MATHEMATICAL ECONOMICS Managing Editors Shigeo Kusuoka University of Tokyo Tokyo, JAPAN Akira Yamazaki Meisei University