Existence and uniqueness of solutions to the Bellman equation in stochastic dynamic programming

Rincón-Zapa e o, Juan Pablo
A icle
Exis ence and uniqueness o solu ions o he Bellman
equa ion in s ochas ic dynamic p og amming
Theo e ical Economics
P o ided in Coope a ion wi h:
The Econome ic Socie y
Sugges ed Ci a ion: Rincón-Zapa e o, Juan Pablo (2024) : Exis ence and uniqueness o solu ions o he
Bellman equa ion in s ochas ic dynamic p og amming, Theo e ical Economics, ISSN 1555-7561, The
Econome ic Socie y, New Ha en, CT, Vol. 19, Iss. 3, pp. 1223-1260,
h ps://doi.o g/10.3982/TE5161
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Theo e ical Economics 19 (2024), 1223–1260 1555-7561/20241223
Exis ence and uniqueness o solu ions o he Bellman equa ion
in s ochas ic dynamic p og amming
Juan Pablo Rincón-Zapa e o
Depa amen o de Economía, Uni e sidad Ca los III de Mad id
In his pape , we de elop a amewo k o analyze s ochas ic dynamic op imiza-
ion p oblems in disc e e ime. We ob ain new esul s abou he exis ence and
uniqueness o solu ions o he Bellman equa ion h ough a no ion o Banach
con ac ions ha gene alizes known esul s o Banach and local con ac ions.
We apply he esul s ob ained o an endogenous g ow h model and compa e
ou app oach wi h o he well-known me hods, such as he weigh ed con ac ion
me hod, coun able local con ac ions, and he Q- ans o m.
Keywo ds. S ochas ic dynamic p og amming, Bellman equa ion, con ac ion
mapping, weigh ed con ac ion, local con ac ion, Q- ans o m, endogenous
g ow h.
JEL classi ica ion. C61, E21.
1. In oduc ion
S ochas ic dynamic p og amming inco po a es unce ain e en s in o a sui able ame-
wo k o ind op imal policies. A use ul app oach o showing he exis ence o op imal
s a iona y plans is o p o e ha he dynamic p og amming equa ion admi s a unique
solu ion— he alue unc ion—in a sui able space o unc ions. See Blackwell (1965),
Mai a (1968), Fu ukawa (1972), Be sekas and Sh e e (1978), S okey, Lucas, and P esco
(1989), He nández-Le ma and Lasse e (1999), o Bäue le and Riede (2011), whe e his
p oblem is analyzed in de ail. Also, he e is a la ge amoun o li e a u e ha applies
s ochas ic dynamic p og amming o economics. B ock and Mi man (1972), Mi man
and Zilcha (1975), Donaldson and Meh a (1983), Dan hine and Donaldson (1981), Ma-
jumda , Mi a, and Nya ko (1989), Hopenhayn and P esco (1992), o Mi a (1998)a e
only a ew o he many ele an pape s ha ha e con ibu ed o de eloping his ield
o esea ch. Olson and Roy (2006) make a e iew o he con ibu ions o he s ochas ic
op imal g ow h model.
Many dynamic p og ams ha e bo h unbounded ewa ds and unbounded shocks,
which canno be handled by he heo y ini ia ed by Blackwell (1965), based on he p op-
e ies o mono onici y and discoun o he dynamic p og amming ope a o . In gene al,
Juan Pablo Rincón-Zapa e o: [email p o ec ed]
This pape is based on my wo king pape s Rincón-Zapa e o (2019,2022). I acknowledge o i e e e -
ees o his and o ano he jou nal o hei insigh ul commen s, which signi ican ly imp o ed he expo-
si ion, leading o he cu en e sion. Suppo om he Spanish Minis e io de Economía y Compe i i idad,
g an s PID2020-117354GB-I00, CEX2021-001181-M, MICIN/AEI/10.13039/501100011033, and Comunidad
de Mad id (Spain), g an EPUC3M11 (V PRICIT), a e g a e ully acknowledged.
©2024 The Au ho . Licensed unde he C ea i e Commons A ibu ion-NonComme cial License 4.0.
A ailable a h ps://econ heo y.o g.h ps://doi.o g/10.3982/TE5161
1224 Juan Pablo Rincón-Zapa e o Theo e ical Economics 19 (2024)
a p oblem wi h unbounded u ili y canno be ans o med in o an equi alen bounded
p oblem, since he op imal policies o bo h models will di e due o he dynamic s uc-
u e. Also, imposing a i icial bounds o deal wi h a compac shock space may be incom-
pa ible wi h modeling unce ain y by means o a i s -o de s ochas ic p ocess. Think,
o ins ance, o he simple andom walk. I akes e e y in ege wi h posi i e p obabili y.
Weigh ed con ac ions, (Wessels (1977), S okey, Lucas, and P esco (1989), Boyd
(1990), He nández-Le ma and Lasse e (1999), o Bäue le and Riede (2011)), coun able
local con ac ions, (Ma kowski and Nowak (2011), Ja´
skiewicz and Nowak (2011), Balbus,
Re e , and Wozny (2018)) and he ecen Q- ans o m due o Ma, S achu ski, and Toda
(2022) a e use ul app oaches o deal wi h s ochas ic p og ams wi h unbounded ewa ds
and shocks.
The weigh ed no m app oach needs o iden i y a sui able bounding unc ion. This
is no immedia e in some models. Ou pape p o ides su icien condi ions, which do
no need a bounding unc ion. In ac , we show a one-sec o op imal g ow h model wi h
linea echnology and s ic ly inc easing and conca e u ili y unc ion, which does no
admi a bounding unc ion, bu ou app oach applies.
Coun able local con ac ions1impose, oughly speaking, ha he condi ional p ob-
abili y measu es de ined by he ansi ion ke nel ha e bounded suppo . This is due
o he need o cons uc ing a coun able amily o inc easing compac se s co e ing he
s a e space. We dispense wi h his assump ion. The same g ow h model desc ibed abo e
se es o show ha his me hod gi es a mo e es ic ed condi ion o he discoun ac o .
The Q- ans o m consis s in aking condi ional expec a ions a bo h sides o he Bell-
man equa ion ha , in some models, con e s an unbounded dynamic p og am in o a
bounded one. This is a simila idea o wha we do o ob ain he companion ope a o
pa ame e L(see P oposi ion 3below) bu he pu pose is di e en , as we wo k wi h
he o iginal Bellman ope a o . The Q- ans o m deals wi h a ans o med ope a o . Fo
unbounded om abo e ewa ds, he Q- ans o m is he same o ha weigh ed con-
ac ion, bu o unbounded om below ewa ds i may ake ad an age o he a e aging
ope a ion o ob ain a bounded p og am. We p esen a quad a ic example whe e i is
no possible o apply he Q- ans o m, no he coun able con ac ion app oach, bu ou
esul s show ha he Bellman equa ion de ines a con ac ion mapping.2
Ou aim is o de elop a new amewo k o s udy p og ams wi h unbounded e-
wa ds and/o unbounded shocks by ex ending he local con ac ion me hod de eloped
in Rincón-Zapa e o and Rod íguez-Palme o (2003,2009)andMa ins da Rocha and
Vailakis (2010) o de e minis ic p og ams o he s ochas ic se ing, while p ese ing he
1The local con ac ion app oach gene alizes he Banach con ac ion p inciple o unc ion spaces whose
opology is de ined by a amily o semino ms. Had˘
zi´
c and S anko i´
c(1970) is one o he i s pape s deal-
ing wi h his ex ension. Rincón-Zapa e o and Rod íguez-Palme o (2003,2007,2009), independen ly, in o-
duced di e en hypo heses and applied he esul s o he de e minis ic Bellman and Koopman’s equa ions.
Ma ins da Rocha and Vailakis (2010) ex ended he heo y o he case o an uncoun able amily o semi-
no ms.
2I is wo h no ing ha he esul s in Ma, S achu ski, and Toda (2022) and Ja´
skiewicz and Nowak (2011)
may be applied o unbounded om below p og ams whe e he u ili y unc ion may ake he alue −∞ on
he s a e space; his is beyond he scope o his pape .
Theo e ical Economics 19 (2024) Exis ence and uniqueness o solu ions 1225
mono onici y o he Bellman ope a o . To his end, we de ine a sui able space o unc-
ions and a sui able amily o semino ms. The semino ms combine he usual sup emum
no m in he endogenous a iables wi h an L1no m in he exogenous a iables, and de-
ine a comple e space o unc ions—a Ca a héodo y unc ion space.
To wo k wi hin his amewo k, we need o ex end he no ion o con ac ion map-
ping by conside ing he con ac ion pa ame e (s) in he local con ac ion de ini ion as
an ope a o ac ing on he amily o semino ms. This ope a o is wha we call he com-
panion ope a o associa ed wi h he con ac ion mapping.3The heo ywede elopis
qui e gene al and could be applied o o he equilib ium p oblems in economics beyond
s ochas ic dynamic p og amming.
Ou amewo k allows us o elax con inui y o he pe iod u ili y unc ion wi h e-
spec o he exogenous a iable. This is impo an since Felle con inui y o he Ma ko
chain is no enough o p ese e con inui y when he space o shocks is no compac .
Also, he L1 ype no m de ined on shocks makes i possible o ob ain less es ic i e
bounds on he discoun ac o han wi h o he known me hods, as i is demons a ed in
he pape .
Ou app oach is designed o p oblems whe e u ili y unc ions may be unbounded
om below, bu no aking he alue −∞ on he s a e space. This is es ic i e, as i akes
ou o conside a ion impo an p oblems in economics. Ne e heless, we s ill ge new
insigh s in be e -beha ed models.4
The pape is o ganized as ollows. Sec ion 2de elops a heo y o con ac ion map-
pings on opological spaces whose opology is gi en by a amily o pseudome ics ha
makes i Hausdo and sequen ially comple e. The con ac ion pa ame e is gi en by
an ope a o ac ing on pseudome ics. Sec ion 3applies he esul s o Sec ion 2 o he
s ochas ic dynamic p og amming equa ion o models wi h shocks d i en by an exoge-
nous Ma ko chain. The main assump ion used o ob ain ou esul s s a es ha oday’s
condi ional expec a ion o he u ili y unc ion is bounded by he p esen alue o o-
mo ow’s condi ional expec a ion, in such a way ha he esul ing in ini e sum o all
expec ed alues is ini e. In Sec ion 4, we s udy a model o endogenous g ow h, allow-
ing o co ela ed and unbounded shocks. Sec ion 5makes a compa ison o ou esul s
wi h hose ob ained wi h weigh ed con ac ions, coun able local con ac ions, and he
Q- ans o m add essed abo e. Sec ion 6concludes. Appendices Aand Bcon ain he
p oo s no in he main ex o Sec ions 2and 3, espec i ely. Appendix Cp o ides a pu e
cu ency model whe e he alue unc ion is discon inuous wi h espec o he shock a i-
able, showing in a simple economic model he well-known ac ha Felle con inui y
3This idea is no new. Kozlo , Thim, and Tu esson (2010) de eloped a ixed-poin heo em in locally
con ex spaces whose opology is gi en by a amily o semino ms. Howe e , he esul s ob ained depend
on he companion con ac ion pa ame e ope a o being linea , p ecluding applica ion o he dynamic
p og amming equa ion, since i genuinely demands a nonlinea companion con ac ion pa ame e , due o
he p esence o a maximiza ion ope a ion in he de ini ion o he Bellman equa ion.
4To adap ou esul s o his class o models, i may be p omising o wo k wi h pseudome ics, ins ead o
semino ms, as in Rincón-Zapa e o and Rod íguez-Palme o (2003); his pape does no explo e his issue.
Also, he heo y we de elop applies only o p oblems whe e unce ain y is exogenous. Howe e , o ex end
he app oach o models whe e ac ions a ec unce ain y is possible, and in ac Rincón-Zapa e o (2022)
d a s how i could be done.
1226 Juan Pablo Rincón-Zapa e o Theo e ical Economics 19 (2024)
o he Ma ko chain is no enough o p ese e con inui y when he shock space is no
compac .
2. A gene al class o Banach con ac ions
Le (E,D)be a opological space, whe e Eis a se whose opology is gene a ed by a
sa u a ed amily o pseudome ics D={da}a∈A,wi hAan a bi a y index se . Since he
amily Dis sa u a ed, he opology i gene a es is Hausdo .5We suppose ha (E,D)
is sequen ially comple e: i {xn}is a sequence in E, which is Cauchy wi h espec o all
da∈D, ha is,i da(xn,xm)→0asn,m→∞, hen he e is x∈Esuch ha da(xn,x)→0
as n→∞ o all a∈A.
Gi en a sequen ially comple e subse F⊆E, we s udy he exis ence and uniqueness
o a ixed poin o a mapping T:F→E.
Le RAbe he se o unc ions d:A→R+and le RA
+be he nonnega i e cone o RA.
On his se , we conside he o de i gene a es, ha is, o wo elemen s d,d∈RA
+,we
say ha d≤di and only i d(a)≤d(a) o all a∈A. The amily Dcan be embedded
in o RA
+, since ha , o x,y∈Egi en, he mapping a→ da(x,y)de ines a unc ion in
RA
+, ha wedeno edx,y(a):=da(x,y). In gene al, o a gi en subse F⊆E,wele D(F)
be he se o unc ions in RA
+, which a e gene a ed by pai s x,y∈F, ha is,
D(F):=d:A→R+:d=dx,y o some x,y∈F.
De ini ion 1. Le F⊆E. The mapping T:F→Eis an L-local con ac ion on Fwi h
con ac ion ope a o pa ame e (COP) L, i he e a e a se C⊆RA
+such ha D(F)⊆C,
and an ope a o L:C→RA
+,such ha
da(Tx,Ty)≤Ldx,y(a),
o all x,y∈Fand o all a∈A.
No e ha he inequali y abo e can be ew i en dTx,Ty ≤Ldx,y, ha is, as an o de
ela ion in he space RA
+. The de ini ion o L-con ac ions o mappings T:F−→ E,
no imposing T:F−→ F, will acili a e he de ini ion o he COP pa ame e Lo he
Bellman ope a o in Sec ion 3. O cou se, he p ope y T:F−→ Fis undamen al o
Theo em 2below, and will be checked ca e ully in Sec ion 3.
The ollowing wo examples show ha he ope a o Lis a gene aliza ion o he con-
cep o con ac ion pa ame e o a (local) con ac ion mapping.
5Apseudome icd:E×E→R+is a unc ion sa is ying d(x,y)≥0, d(x,x)=0, d(x,y)=d(y,x), and
d(x,z)≤d(x,y)+d(y,z) o any x,y,z∈E,bu d(x,y)=0 does no imply x=y. The amily Do pseudo-
me icsissa u a edi da(x,y)=0 o alla∈Aimplies x=y. Some imes, he pseudome ics a e de ined
h ough semino ms pa,a∈A,byda(x,y)=pa(x−y),whe enowEis a eal ec o space. A semino m is
a unc ionp:E→R+ ha sa is ies all he axioms o be a no m, excep ha p(x)=0 does no imply ha x
is he null ec o o E. I he amily o semino ms is sa u a ed, hen he opology de ined by he amily is
Hausdo and he space Eis a locally con ex space. See Willa d (1970) o u he de ails.

Theo e ical Economics 19 (2024) Exis ence and uniqueness o solu ions 1227
Example (Banach Con ac ions). In he classical Banach’s heo em, Eis endowed wi h
a comple e me ic d, so he index se Ais a single on, D={d},andTis a con ac ion o
cons an pa ame e β,wi h0<β<1: d(Tx,Ty)≤βd(x,y), o anyx,y∈E.TheCOPis
L=βI,whe eIis he iden i y map in R+.
A gene aliza ion o he Banach con ac ion concep is p o ided in Wong (1968),
whe e i is conside ed T:E−→ E o which he e is a unc ion L:R+−→ R+sa is ying
d(Tx,Ty)≤Ld(x,y),(1)
o all x,y∈E. No e ha ou de ini ion is an ex ension o his concep o opological
spaces whose opology is gi en by a amily o pseudome ics. ♦
Example (k-Local Con ac ions). Suppose ha A=Nis coun able. In Rincón-Zapa e o
and Rod íguez-Palme o (2003,2007), we in oduced he concep o k-local con ac ion
in he s udy o he de e minis ic Bellman and Koopmans equa ions, espec i ely. A k-
local con ac ion on F,k=0, 1, 2, is a mapping T:F⊆E−→ Esa is ying
dj(Tx,Ty)≤βjdj+k(x,y)
o some ixedsequenceo numbe s{βj}j∈Nwi h 0 <β
j<1, and o all x,y∈F.I wele
s=RNbe he se o eal sequences and s+be he subse o so nonnega i e sequences,
hen he COP associa ed wi h Tis he linea ope a o L:s+−→ s+ac ing on sequences
gi en by
L(d1,d2,,dj,)=(β1d1+k,β2d2+k,,βjdj+k,),
whe e k≥0is ixed.
Suppose ha Ais uncoun able and le a mapping α:A−→ A.Ma ins da Rocha and
Vailakis (2010) wo ked wi h he ollowing gene aliza ion o he coun able class abo e:
T:E−→ Eis an α-local con ac ion i he e exis s a unc ion β:A−→ [0, 1)such ha
da(Tx,Ty)≤β(a)dα(a)(x,y).
The COP Lac s on unc ions d:A−→ R+by ansla ion in he independen a iable by
α, and a mul iplica ion by β, ha is,(Ld)(a)=β(a)d(α(a)).I u nsou ha Lis also a
linea mapping, as in he coun able case abo e. ♦
In wha ollows, we use he s anda d no a ion o successi e i e a ions o he ope -
a o s Tand L. Fo ins ance, L0is he iden i y ope a o on C,L1=L, and o ≥2,
L =L◦L −1.Weimpose oC,L,andT he Assump ions (I) o (VI) lis ed below. The
Assump ions (I) o (V) conce n he beha io o Lon he se C. Assump ion (VI) links
di ec ly he ope a o s Tand L.
(I) D(F)⊆C(hence he null unc ion 0∈C). Fo all d,d∈C, he sum d+d∈C,and
any bounded subse o Cis coun able chain comple e.6Mo eo e , i d∈C,d∈RA
+,
and d≤d, hend∈C.
6Asubse S⊆Cis bounded wi h espec o he o de inhe i ed om RAi he e is d∈Csuch ha d≤d
o all d∈S. The bounded subse Sis coun ably chain comple e i o any coun ably chain d1≤d2≤···d ≤
··· in S,sup ∈Nd ∈S.
1228 Juan Pablo Rincón-Zapa e o Theo e ical Economics 19 (2024)
(II) L(C)⊆C;L0=0.
(III) Lis mono one: o all d,d∈Cwi h d≤d,Ld ≤Ld.
(IV) Lis subaddi i e: o any d,d∈C,
Ld+d≤Ld +Ld.
(V) Lis uppe semicon inuous sup-p ese ing:7 o any bounded coun able chain in
C,d1≤d2≤···≤d ≤···,
Lsup
d ≤sup
Ld .
(VI) The e a e x0∈Fand 0∈Cwi h da(x0,Tx0)≤ 0(a)and
R0(a):=
∞

=0
L 0(a)<∞,
o all a∈A.
Since L 0∈C, o all =0, 1, , and he coun able chain { 0, 0+L 0,, 0+L 0+
···+L 0,}is bounded in Cby (VI), R0is in Cby Assump ion (I).
Fo F⊆E,x0∈F,andm∈RA
+, le he se
VF(x0,m)=x∈F:da(x0,x)≤m(a),∀a∈A.(2)
When Eis a me ic space, ha is, when Ais a single on, he pseudome ic is a me ic,
and VF(x0,m)is simply he in e sec ion wi h Fo heclosedballcen e eda x0and
adius m.
Le N0:={0}∪Nand se J:=A×N0. Conside nex he amily o pseudome ics
:=(δj)j∈Jon Fwhe e
δa, (x,y)=L dx,y
a.
Assump ions (I)–(IV) imply ha δjis a pseudome ic while a s aigh o wa d compu a-
ion shows ha
δa, (Tx,Ty)=L da(Tx,Ty)≤L +1da(x,y))=δa, +1(Tx,Ty).
Now le he map :J→Jbe de ined by (a, )=(a, +1).ThenTis a local con ac ion
wi h espec o (, ).8
7Fo ins ance, he sup-p ese ing p ope y, L(sup d )=sup Ld , plays a p ominen ole in he ixed-
poin heo em o Kan o o ich–Ta ski. In ou con ex , i can be weakened o a kind o uppe semicon inui y.
8This cons uc ion was shown o he au ho by a e e ee. I allows us o apply di ec ly Ma ins da Rocha
and Vailakis (2010, Theo em 2.1) o ob ain exis ence and uniqueness o he ixed poin . Howe e , in some
cases, he amily could be no sa u a ed o no de ining a sequen ially comple e opology. An example
wi hin he dynamic p og amming class is as ollows. Suppose a de e minis ic p oblem wi h X=R+and
(x)={x+1}. Then, o all compac se s Ko Xand all unc ions p=p ,wi h con inuous, Lp(K)=
βmaxx∈Kp((x))=βpK+1and L p(K)=β pK+ ,whe eK+ ={x+ :x∈K}, o all =1, 2, I is clea
Theo e ical Economics 19 (2024) Exis ence and uniqueness o solu ions 1229
Theo em 1. Le (E,D)be a Hausdo and sequen ially comple e opological space. Le
T:F→Fbe an L-local con ac ion on he sequen ially comple e subse F⊆Eand le
x0∈Fbe such ha (I)–(VI) hold ue. Suppose ha he amily o pseudome ics de ined
abo e is sa u a ed and he space (E,)is comple e. Then he e is a unique ixed poin
x∗∈VF(x0,R0)o T, which is he limi o any i e a ing sequence y +1=Ty , =0, 1, 2, ,
whe e y0=x∈VF(x0,R0)is a bi a y.
P oo . The esul ollows om Ma ins da Rocha and Vailakis (2010, Theo em 2.1), wi h
K=VF(x0,R0). See Lemma 2in Appendix A.
The nex esul is a co olla y o he abo e heo em ha p o ides condi ions o he
uniqueness o he ixed poin in Fand no only in VF(x0,R0).
When Tis indeed an L-local con ac ion on he whole E, his esul p o ides global
uniqueness o he ixed poin on E.
Co olla y 1. Le (E,D)be a Hausdo , sequen ially comple e space. Le T:F→Fbe
an L-local con ac ion on he sequen ially comple e subse F⊆Eand le x0∈Fbe such
ha (I)–(VI) hold ue. Suppose ha o all x∈F he e exis s 0∈Csa is ying (VI) such
ha x∈VF(x0,R0),whe eR0=∞
=0L 0. Then he e is a unique ixed poin o Tin F
and con e gence o he ixed poin o successi e i e a ions o Tis a ained om any x∈F.
Nex , we es ablish a use ul su icien condi ion o (VI). No e ha he Bellman ope -
a o sa is ies he ex a condi ion imposed on L.
P oposi ion 1. Le (E,D)be a Hausdo and sequen ially comple e opological space.
Le T:F−→ Fbe an L-local con ac ion on F⊆E,wi hCOPLsa is ying (I) o (V) and
L(αd)≤αLd, o all d∈C, o all α∈[0, 1].Le x0∈F, o which he e is 0∈{0, 1, 2, },
s∈C,andθ∈[0, 1)such ha
L 0d0≤sand Ls ≤θs,
whe e d0(a)=da(x0,Tx0).Then(VI)holdswi h 0=d0.
3. S ochas ic dynamic p og amming and Bellman equa ion
Conside a dynamic p og amming model (X,Z,,Q,U,β),whe eX×Zis he se
o possible s a es o he sys em, is a co espondence ha assigns a nonemp y se
(x,z)o easible ac ions o each s a e (x,z),andQis he ansi ion unc ion, which
associa es a condi ional p obabili y dis ibu ion Q(z,·)on Z o each z∈Z.Hence,
he law o mo ion is assumed o be a i s -o de Ma ko p ocess, which could be de-
gene a e, gi ing ise o a de e minis ic model. We will use indis inc ly he no a ion
ha {δK, }is no a sa u a ed amily o semino ms. Supposing ha and ga e con inuous unc ions on R+,
such ha = gin [0, 1]and =gin (1, ∞);ye ,δK, ( −g)=β pK+ ( −g)=0 o allKand all =1, 2, 
A di ec app oach, wi hou he use o ,isshowninRincón-Zapa e o (2022). The applica ions we s udy in
u he sec ions ha e bo h sa u a ed and comple e.
1230 Juan Pablo Rincón-Zapa e o Theo e ical Economics 19 (2024)
Qz(·)=Q(z,·); he unc ion Uis he one-pe iod e u n unc ion, de ined on he g aph
o ,={(x,y,z):(x,z)∈X×Z,y∈(x,z)},andβis a discoun ac o .
S a ing a some s a e (x0,z0), he agen chooses an ac ion x1∈(x0,z0), ob aining
a e u no U(x0,x1,z0)and he sys em mo es o he nex s a e (x1,z1),whichisd awn
acco ding o he p obabili y dis ibu ion Q(z0,·). I e a ion o his p ocess yields a an-
dom sequence (x0,z0,x1,z1,)and a o al discoun ed e u n ∞
=0β U(x ,x +1,z ).A
his o y o leng h is z =(z0,z1,,z ).Le Z be he se o all his o ies o leng h ,and
le Z =Z×···×Z( imes), whe e Zis he Bo el σ-algeb a o Z. A ( easible) plan πis a
cons an alue π0∈Xand a sequence o measu able unc ions π :Z −→ X,such ha
π (z )∈(π −1(z −1),z ), o all =1, 2, .Deno eby(x0,z0) he se o all easible
plans s a ing a he s a e (x0,z0). Any easible plan π∈(x0,z0), along wi h he an-
si ion unc ion Q, de ines a dis ibu ion Pπ,(x0,z0)on all possible u u es o he sys em
{(x ,z )}∞
=1, as well as he expec ed o al discoun ed u ili y
u(π,x0,z0)=Eπ,(x0,z0)∞

=0
β U(x ,x +1,z ).
The expec a ion Eπ,(x0,z0)is aken wi h espec o he dis ibu ion Pπ,(x0,z0).Thep ob-
lem is hen o ind a plan π∈(x0,z0)such ha u(π,(x0,z0)) ≥u(π,(x0,z0)) o all
π∈(x0,z0), o all(x0,z0)∈X×Z. The alue unc ion o he p oblem is (x0,z0)=
supπ∈(x0,z0)u(π,(x0,z0)).
Conside he unc ional equa ion co esponding o he abo e dynamic p og am-
ming p oblem as s a ed in S okey, Lucas, and P esco (1989). Fo x∈X,z∈Z,
(x,z)=sup
y∈(x,z)U(x,y,z)+βZ
y,zQz,dz.(3)
A solu ion o he Bellman equa ion sa is ying addi ional assump ions is he alue
unc ion o he in ini e p og amming p oblem. This is he con en o Theo em 2below,
whose p oo needs he no ion o he p obabili y measu e μ de ined on he sequence
space o shocks (Z ,Z ) o ini e =1, 2, ,whe e
Z ,Z =(Z×···×Z,Z×···×Z)(
imes)
and whe e Zis de ined in (B1) below. Fo any ec angle B=A1×···×A ∈Z ,μ is
de ined by
μ (z0,B)=A1
···A −1A
Qz −1(dz )Qz −2(dz −1)···Qz0(dz1),
and by he Hahn ex ension heo ems, μ (z0,·)has a unique ex ension o a p obabili y
measu e on all o Z . We omi he de ails, which can be ound in S okey, Lucas, and
P esco (1989), Sec ion 8.2, whose p esen a ion we ollow closely.
De ining he Bellman ope a o in a sui able unc ion space E,such ha o ∈E,
(T )(x,z)=sup
y∈(x,z)U(x,y,z)+βZ
y,zQz,dz,
Theo e ical Economics 19 (2024) Exis ence and uniqueness o solu ions 1237
≤Z
β
∞

s=
lsπ0(z0),z1Qz0(dz1)
≤
∞

s= +1
ls(x0,z0),
again by he mono one con e gence heo em, and whe e we ha eused Fubini’s heo em
and he induc ion hypo hesis. This and (9) imply (iii), since he se ies w0con e ges.
Thus, ∗is he alue unc ion. The claims abou ∗a e immedia e om he heo em
o he maximum o Be gé and he measu able maximum heo em; see Alip an is and
Bo de (1999).
The ollowing esul p o ides a su icien condi ion o (B6).
P oposi ion 4. Le Assump ions (B1) o (B5) hold. Suppose ha he e is l0∈L1(Z;
C(X)) wi h |ψ|≤l,α≥0such ha αβ < 1,and
Z
max
y∈(x,z)l0y,zQzdz≤αl0(x,z),
o all x∈X,z∈Z. Then (B6) holds, wi h R0(K,z)=pK,z(l0)/(1−αβ).
P oo . Choose l =(αβ) l0, o =0, 1, Then
βZ
max
y∈(x,z)l y,zQzdz=β(αβ) Z
max
y∈(x,z)l0y,zQzdz
≤(αβ) +1l0(x,z)=l +1(x0,z0).
Hence, w(x0,z0)=l0(x0,z0)/(1−αβ )and R0(K,z)=pK,z(l0)/(1−αβ) o K∈Kand
z∈Z.
3.1 Sha pe es ima es
Many in e es ing p oblems ha e u ili y unc ions, which a e unbounded om below.11
In his case, he es ima es gi en in Assump ion (B6) a e no e icien . A gene aliza ion o
(B6) allows us o cons uc sha pe es ima es in he o m o an o de in e al o unc ions
ha is mapped in o i sel by he ope a o T.
(B6’) The e a e wo collec ions o unc ions k ,l :X×Z→R,wi hk ,l ∈L1(Z;
C(X)), o all =0, 1, , sa is ying ha o all x∈X,allz∈Z, he eexis s
y(x,z)∈(x,z)such ha
k0(x,z)≤minUx,y(x,z),z,0
;
11We e e he e o p oblems whe e he u ili y unc ion is no bounded om below, bu ne e akes he
alue −∞.

1238 Juan Pablo Rincón-Zapa e o Theo e ical Economics 19 (2024)
k +1(x,z)≤βZ
k y(x,z),zQzdz;
l0(x,z)≥maxψ(x,z),0
;
l +1(x,z)≥Z
max
y∈(x,z)l y,zQzdz,all =1, 2, ;
and he se ies
u (x,z):=
∞

s=
ks(x,z)and w (x,z):=
∞

s=
ls(x,z)
a e uncondi ionally con e gen , o all =0, 1, 2, 
Le Iu0,w0={ ∈L1(Z;C(X)) :u0≤ ≤w0}. The ollowing esul s a es Theo em 2
in he mo e es ic ed space o unc ions VIu0,w0, hus p o iding a me hod o ge he
con ac ion p ope y o he Bellman ope a o when Uis unbounded om below bu
ne e akes he alue −∞ on . In he ollowing heo em, we le 
l =max{|k |,|l |} o all
=0, 1, and 
R0(K,z)=∞
=0pK,z(
l ) o all K∈Kand z∈Z.
Theo em 3. Suppose ha (B1)–(B5) and (B6’) hold. Then Theo em 2holds wi h
VIu0,w0(0, 
R0), eplacing V(0, 
R0).
P oo . By Lemma 7in Appendix B,Tis a sel -map on Iu0,w0.No ice ha |ψ(x,z)|≤

l0(x,z). Hence, (B6) holds ue o 
l by he de ini ion o 
l and Theo em 2applies in
VIu0,w0(0, 
R0).
In wha ollows, o simpli y he exposi ion, we in oduce he ollowing no a ion: o
a unc ion ∈Ca(X×Z),

x,z,z=max
y∈(x,z) y,z. (10)
4. Applica ion o endogenous g ow h
Endogenous g ow h models ha e become undamen al o unde s and economic
g ow h. Se e al con ibu ions conside an unbounded shock space, as in S achu ski
(2002), Kamihigashi (2007), Ma kowski and Nowak (2011), o Bäue le and Ja´
skiewicz
(2018). I conside he e he s ochas ic endogenous g ow h model s udied in Jones,
Manuelli, Siu, and S acche i (2005), which is desc ibed as ollows. The p e e ences o
he agen o e andom consump ion sequences a e gi en by
max E
∞

=0
β c1−σ
υ( )
1−σ, (11)
subjec o
c +k +1+h +1≤z Akα
(n h )1−α+(1−δk)k +(1−δh)h , (12)
Theo e ical Economics 19 (2024) Exis ence and uniqueness o solu ions 1239
 +n ≤1, (13)
c ,k ,h , ,n ≥0 (14)
o all =0, 1, ,wi hk0and h0gi en. He e, {z }is a Ma ko s ochas ic p ocess wi h
ansi ion p obabili y Qz(·)and Z=(0, ∞);c is consump ion;  is leisu e; n is hou s
spen wo king; k and h a e he s ock o physical and human capi al, espec i ely; δk
and δha e he dep ecia ion a es on physical and human capi al, espec i ely; and υis
a con inuous unc ion on (0, 1], s ic ly inc easing. The usual nonnega i i y cons ain s
on consump ion, in es men , leisu e, and hou s wo ked apply. I we le k=k +1,h=
h +1,andk=k ,h=h ,c=c ,n=n ,and= , hen he easible co espondence is
(k,h,z)=k,h:The ea ec,n,such ha (12)–(14)hold
,
and he u ili y unc ion is U(c,)=c1−συ()/(1−σ). Rega ding he unc ion υ,wecon-
side υ()=ψ(1−σ). The endogenous s a e space is X=[0, ∞)×[0, ∞)and he amily
o compac se s Kis o med by compac se s in he p oduc space R+×R+.TheMa ko
chain is gi en by he log-log p ocess
lnz +1=ρln z +lnw +1, (15)
wi h ρ≥0andwhe e hew’s a e i.i.d., wi h suppo in W⊆(0, ∞).Le μbe he dis i-
bu ion measu e o he w’s. No e ha ρ=0 co esponds o shocks z ha a e i.i.d. Jones
e al. (2005) suppose ha z =exp(ζ −(1/2)σ2
/(1−ρ2)),whe eζ +1=ρζ + +1and
he ’s a e i.i.d., no mal wi h mean 0 and a iance σ2
. This co esponds o (15)wi h
w +1=exp( +1−(1/2)σ2
/(1+ρ)).Wedono es ic  o be no mally dis ibu ed.
To sho en no a ion, we will use he ollowing de ini ions in his subsec ion:
γ=αα(1−α)1−α,
δ=min{δk,δh},
ν=(1−δ)1−σ,
E=
∞

s=0
Ewρs(1−σ).
In his sec ion, con e gence means con e gence wi h espec o he semino ms
pK,z,K∈K,z∈Z. We only assume ha he expec a ion Ewis ini e.
P oposi ion 5. Conside he endogenous g ow h model desc ibed in (11)–(15)wi h0≤
σ<1and 0≤ρ<1.I
βAγ(Ew)1/(1−ρ)1−σ+ν<1, (16)
hen he associa ed Bellman equa ion admi s a unique solu ion, ∗, in he se V(0, R0)
de ined in (2)gi enby
V(0, R0)= ∈L1Z;C(X):pK,z( )≤R0(K,z),∀(K,z)∈K×Z,
1240 Juan Pablo Rincón-Zapa e o Theo e ical Economics 19 (2024)
whe e o K∈Kand z∈Z,R0(K,z)=pK,z(∞
=0l ), and he amily {l }∞
=0is gi en by (17)
and (21) in he p oo below. Mo eo e , ∗is he alue unc ion and Tn 0con e ge o
as n→∞ o all ini ial guesses 0∈V(0, R0). Finally, he op imal policies a e con inuous
unc ions o (k,h).
P oo . We check all he hypo heses o Theo em 2. I is clea ha (B1)–(B5) a e ul illed.
We ocus on (B6) and de ine g(k,h,z)=Azkαh1−α+(1−δ)(k+h). Since 0 ≤σ<1,
bo h Uand υa e bounded om below by ze o, and υis bounded abo e by 1. By he
de ini ion o δ,weha ezAkα(nh)1−α+(1−δk)k+(1−δh)h≤g(h,k,z).Then
ψ(k,h,z)=max
(k,h,c,n,)∈(k,h,z)u(c,)≤1
1−σg(k,h,z)1−σ≡l0(k,h,z). (17)
Acco ding o (10), le

l0k,h,z,z=max
(k,h,c,n,)∈(k,h,z)l0k,h,z.
Le us de e mine a bound o 
l0(k,h,z,z). To his end, conside he Lag ange p oblem
max l0k,h,z, (18)
such ha k+h≤g(k,h,z), (19)
k,h≥0, (20)
and no ice ha i s easible se is la ge han (k,h,z). The cons ain is binding a he
op imal solu ion, which is k=αg(k,h,z),h=(1−α)g(k,h,z). Subs i u ing his in o
he objec i e unc ion o (20), we ind i s op imal alue, which is
Azαα(1−α)1−α+(1−δ)1−σl0(k,h,z)≤A1−σz1−σγ1−σ+νl0(k,h,z),
whe e he inequali y is due o he unc ion c→ c1−σbeing subaddi i e, as i is conca e
and null a ze o. Compu ing he condi ional expec a ion o he igh -hand side o he
abo e inequali y, we ha e
Z
l0k,h,z,zQzdz≤A1−σzρ(1−σ)Ew1−σγ1−σ+νl0(k,h,z).
Thus, we de ine l1(k,h,z)=β(A1−σzρ(1−σ)Ew1−σγ1−σ+ν)l0(k,h,z).
To calcula e 
l1(k,h,z,z), we no e ha we ace he same Lag ange p oblem (20)
abo e, modulo he “cons an ” ac o β(A1−σ(z)ρ(1−σ)Ew1−σγ1−σ+ν). Thus, we will
ind

l1k,h,z,z≤A1−σzρ(1−σ)Ew1−σγ1−σ+νl1(k,h,z)
and hen
Z
l1k,h,z,zQzdz≤A1−σzρ2(1−σ)Ew1−σEwρ(1−σ)γ1−σ+νl1(k,h,z).
Theo e ical Economics 19 (2024) Exis ence and uniqueness o solu ions 1241
Now de ine l2(k,h,z)=β(A1−σzρ2(1−σ)Ew1−σEwρ(1−σ)γ1−σ+ν)l1(k,h,z). Using in-
duc ion,i canbep o edexac ly hesameas o hecases =1and =2, ha he amily
o unc ions {l }∞
=0gi en by
l +1(k,h,z)=βA1−σzρ +1(1−σ)Ew1−σEwρ(1−σ)···Ewρ (1−σ)γ1−σ+νl (k,h,z), (21)
o all =0, 1, , sa is ies he inequali ies demanded in (B6). Rega ding he se ies
w0(k,h,z)=∞
=0l (k,h,z), no e ha he a io
l +1(k,h,z)
l (k,h,z)=βA1−σzρ +1(1−σ)Ew1−σEwρ(1−σ)···Ewρ (1−σ)γ1−σ+ν,
con e ges o βA1−σEγ1−σ+νas →∞, which is smalle han one by he assump-
ion o he heo em, since by Jensen’s inequali y E≤(Ew)(1−σ)/(1−ρ).Thus,by he a-
io es , he se ies con e ges poin wise. I is easy o see ha inequali y (16) gua an ees
uncondi ional con e gence as well, hus (B6) is ul illed. To comple e he p oo , The-
o em 2shows ha he op imal policy co espondence is nonemp y and uppe hemi–
con inuous. The con ex maximum heo em o Be gé assu es ha he alue unc ion is
conca e and since he u ili y unc ion is s ic ly conca e, he op imal policies a e unique,
and so con inuous wi h espec o (k,h).
5. Compa ison wi h o he app oaches
In his sec ion, we compa e ou esul s wi h hose ob ained by o he me hods: he al-
eady classical weigh ed no m app oach, he one based on coun able local con ac ions,
and he ecen Q- ans o m. In each case, we gi e a b ie desc ip ion o he me hod, and
hen we wo k h ough model examples showing he di e ences wi h ou me hod. A
wo d o cau ion is needed he e. As said in he In oduc ion, we s udy dynamic p oblems
whe e he ac ions do no in luence he e olu ion o unce ain y; i is exogenous. The
p esen pape ’s aim is o ake ad an age o he special s uc u e o he s a e space o he
kinds o p oblems ha we analyze o ob ain u he insigh s.
5.1 The weigh ed no m app oach
In he weigh ed con ac ion app oach (see Boyd (1990), Becke and Boyd (1997) o he
de e minis ic case, and He nández-Le ma and Lasse e (1999), Ja´
skiewicz and Nowak
(2011)andBäue le and Riede (2011) o he s ochas ic case), i is pos ula ed ha he
exis ence o a con inuous unc ion ϕ:X×Z:−→ R++, called he bounding o weighing
unc ion, such ha he e exis nonnega i e cons an s Mand αsuch ha o all x∈X,
z∈Z,y∈(x,z),
(W1) |U(x,y,z)|≤Mϕ(x,z)and
(W2) Zϕ(y,z)Qz(dz)≤αϕ(x,z).
Gi en such a unc ion ϕ, he ollowing Banach space is conside ed:
Cϕ= :X×Z−→ Rcon inuous : sup
(x,z)∈X×Z (x,z)
ϕ(x,z)<∞,
whe e he no m is de ined by  ϕ=sup(x,z)∈X×Z(| (x,z)|/ϕ(x,z)).
1242 Juan Pablo Rincón-Zapa e o Theo e ical Economics 19 (2024)
Conside he ollowing esul ha can be ound in He nández-Le ma and Lasse e
(1999), Sec ion 8.3, o in Bäue le and Riede (2011), Theo em WSN, p. 208.
Theo em 4. Le ϕbe a con inuous unc ion sa is ying (W1)–(W2) abo e, such ha
αβ < 1. (22)
I
(WC1) is nonemp y and con inuous and (x,z)is compac alued o all (x,z)∈
X×Z;
(WC2) Uis con inuous;
(WC3) (x,z)−→ Zϕ(x,z)Qz(dz)is con inuous;
hen (a) he alue unc ion is he unique con inuous solu ion o he Bellman equa ion in
Cϕ, (b) alue unc ion i e a ion con e ges om any ini ial ∈Cϕ, (c) a leas one op imal
policy exis s, and (d) ha policy maximizes he igh -hand side o he Bellman equa ion.
The condi ions o Theo em 4a e s ic e han hose o Theo em 2. To see his, we can
ake l0=ϕin P oposi ion 4abo e o cons uc he amily {l }∞
=0such ha Theo em 2ap-
plies. The opposi e is no ue, ha is, Theo em 2is no con ained in Theo em 4.Fi s ,
Theo em 2does no impose con inui y in bo h a iables (x,z),as equi edin(WC3).
We show in Appendix Ca simple pu e cu ency model whe e he alue unc ion is dis-
con inuous wi h espec o he exogenous a iable and whe e no bounding unc ion ϕ
may sa is y (WC3). Second, e en i (WC3) is ul illed, he uniqueness o he solu ion o
he Bellman equa ion is gi en in a la ge space, since Ca(X×Z)con ains Cϕ(X×Z),
o any bounding unc ion ϕ. Thi d and las , mo e impo an ly beyond he issues o
con inui y jus discussed, he e a e bounded om below models o which a bounding
unc ion canno exis bu Theo em 2is applicable. Since ha , in p inciple, he e a e in-
ini ely many candida es o bounding unc ions, o ind a model wi h e u ns bounded
om below, o which he e is no a sui able bounding unc ion, ha is no an easy ask.
We will use he esul below, which s a es a necessa y condi ion o he exis ence o a
bounding unc ion ϕ.
Fo a unc ion depending on he a iables (x,z), emind he no a ion 
(x,z,z)=
maxy∈(x,z)∈X×Z (y,z). Also, de ine 0
0=1.
P oposi ion 6. Le he e be a dynamic p og amming p oblem (X,Z,,Q,U,β) o
which he e is a con inuous bounding unc ion ϕsa is ying he condi ions o Theo em 4.
De ine he amily o unc ions {l }∞
=0,wi hl0∈Cϕ(X×Z)and l +1=βZ
l Qz. Then he e
is a cons an Msuch ha
∞

=0 +
l ≤M
(1−αβ) +1ϕ, o all =0, 1,  (23)

Theo e ical Economics 19 (2024) Exis ence and uniqueness o solu ions 1243
P oo . We o en elimina e he a gumen s (x,z)in wha ollows o simpli y no a ion.
No e ha l1=βZ
l0Qz≤MβZϕQz≤M(αβ)ϕ o some cons an Mand αβ < 1, since
l0∈Cϕ(X×Z)and ϕis a sui able bounding unc ion. By induc ion, l ≤M(αβ) ϕ, o
all .Hence,w0=∞
=0l ≤Mϕ/(1−αβ)is ini e and
Z
w0Qz≤M
(1−αβ)ZϕQz≤Mα
(1−αβ)ϕ.
Le w1=∞
s=1ls. Then, as in he p oo o Theo em 2,weha e
Z
w0Qz=Z
∞

=0
l Qz=
∞

=0Z
l Qz=1
β
∞

=0
l +1=1
βw1.
Thus,weha eob ainedw1(x,z)≤Mϕ(x,z)αβ/(1−αβ). In gene al, he ollowing in-
equali y holds:
w (x,z)≤M(αβ)
1−αβ ϕ(x,z), o all =0, 1, , (24)
whe e w =∞
s= ls. To show his, we use he p inciple o induc ion. The cases =0, 1
ha e jus been p o ed. Suppose ha i is ue o .Then
Z
w Qz≤M(αβ)
1−αβ ZϕQz≤M(αβ)
1−αβ αϕ,
and on he o he hand, as in he compu a ion abo e,
Z
w Qz=Z
∞

s= 
lsQz=
∞

s= Z
lsQz=1
β
∞

s=
ls+1=1
βw +1,
hus we ge he inequali y sough . Adding (24) om =0 o =∞,weob ain
12
∞

=0
( +1) (x,z)≤M
(1−αβ)2ϕ(x,z)<∞. (25)
F om (25), eplacing xby xand zby zand in eg a ing wi h espec o Qzin bo h sides
o he inequali y and using he p ope ies o ϕ,weha e
Z
∞

=0
( +1)
l Qz=
∞

=0
( +1)Z
l Qz=1
β
∞

=0
( +1)l +1≤Mα
(1−αβ)2ϕ.
Thus, ∞
=0( +1)l +1≤Mϕαβ/(1−αβ)2,whichis(23)wi h =1. Repea ing he scheme
abo e ss eps, we ge ∞
=0( +1)l +s≤Mϕ(αβ)s/(1−αβ)2. Adding in sagain as in (25),
we ob ain
∞

=0 +2
2l =
∞

=0
( +2)( +1)
2l ≤M
(1−αβ)3ϕ.
12w0+w1+w2+···=(l0+l1+l2+···)+(l1+l2+···)+(l2+···)+···=l0+2l1+3l2+···.
1244 Juan Pablo Rincón-Zapa e o Theo e ical Economics 19 (2024)
By induc ion, and using he same a gumen s as o he cases =1and =2, i can be
p o ed ha o any ≥0,
∞

=0 +
l ≤M
(1−αβ) +1ϕ.
This esul will be used o show ha he weigh ed no m app oach canno be applied
o he ollowing simple g ow h model wi h a linea echnology, mul iplica i e shocks,
and a s ic ly inc easing, s ic ly conca e and bounded om below elici y unc ion (bu
ou app oach wo ks).
Example. Conside an op imal g ow h model, which he Bellman equa ion is
(k,z)=max
k∈[0,zk]Uk,k,z+βZ
k,zQzdz,
whe e k∈[0, ∞),Z={1, g},wi hg>1, Qz:=Qis gi en by Q(g)=p>0, Q(1)=q>0,
wi h p+q=1, and he u ili y unc ion is U(k,k,z)=u(zk −k),whe e
u(c)=1+c
3+ln2(1+c)−1
3.
The unc ion uis nonnega i e, con inuous, unbounded, s ic ly inc easing, and s ic ly
conca e on [0, ∞),wi hu(0)=0.13
The discoun ac o is aken o be β≤1/(gp +q)<1. Clea ly, any solu ion o he
Bellman equa ion has (0, z)=0, o all k,z.Le k>0andz∈{1, g}.Wi ha iew ouse
he necessa y condi ion in P oposi ion 6,le us ake0(k,z)=maxk∈[0,zk]U(k,k,z)=
u(zk). I is clea ha l0≤Mϕ o M=1, since l0=ψ≤ϕby de ini ion o he bounding
unc ion ϕ.Le
1(k,z)=βZ
max
k∈[0,zk]0k,zQdz=βpu(gzk)+qu(zk),
2(k,z)=βZ
max
k∈[0,zk]1k,zQdz=β2p2ug2zk
+pqu(gzk)+qpu(gzk)+q2u(zk)
.
.
.
In gene al,  (k,z)=β 
s=0
spsq −su(gszk), o all , which ollows by induc ion. We
ha e
∞

=0
l (k,z)=
∞

=0

s=0
s(βp )s(βq) −sugszk=
∞

s=0
(βp )sugszk∞

=s
s(βq) −s,
13Le ing c>0, deno e x=ln (1+c)>0. Then u(c)=(3+x2−2x)/(3+x2)2>0 and u(c)=−2e−x((x−
1)3+2)/(3+x2)3<0. Also, no e ha uis unbounded bu u(c)→0asc→∞. Fo ano he example whe e
he e is no sui able bounding unc ion ϕwha e e he alue o he discoun ac o β,seeRincón-Zapa e o
(2022).
Theo e ical Economics 19 (2024) Exis ence and uniqueness o solu ions 1245
whe e he change o o de summa ion is admissible since he double se ies is o posi i e
e ms. Now
∞

=s
s(βq) −s=1
s!
∞

=s
( −1)···( −s+1)(βq) −s.
The se ies a he igh -hand side is he alue o he s h de i a i e o he powe se ies
∞
=0x =1/(1−x), e alua ed a x=βq;hence,∞
=s ( −1)···( −s+1)(βq) −sequals
(ds/dxs)(1/(1−x))|x=βq =s!/(1−βq)s+1.Thus,
∞

=0
l (k,z)=1
(1−βq)
∞

s=0βp
1−βqs
ugszk. (26)
Le us see ha his se ies con e ges o all k>0andz∈Z.No e ha
ugszk=1
3+ln21+gszk+gszk
3+ln21+gszk−1
3
Thus, he se ies (26) decomposes in o he sum o h ee se ies. The i s and he hi d
se ies a e
1
(1−βq)
∞

s=0βp
1−βqs1
3+ln21+gszkand −1
3(1−βq)
∞

s=0βp
1−βqs
,
espec i ely, which a e con e gen , since βp < 1−βq (use he a io es ), and he se ies
in hemiddleis
1
(1−βq)
∞

s=0βpg
1−βqszk
3+ln21+gszk.
Ob iously, his se ies con e ges when β<1/(gp +q).Whenβ=1/(gp +q),(βpg )/(1−
qβ)=1, and he se ies educes o (1−qβ)−1zk∞
s=01/(3+ln2(1+gszk)), which is con-
e gen .14 A simila and s aigh o wa d a gumen shows ha he se ies ∞
=0pK,z(l )
con e ges o all compac se Ko [0, ∞)and z∈Z.15 Hence, all condi ions o Theo-
em 2a e ul illed and his g ow h model admi s a solu ion. Now we a gue by con a-
dic ion, assuming ha a bounding unc ion ϕsa is ying he assump ions o Theo em 4
14Since
lim
s→∞
3+ln21+gszk
3+ln2gszk=1,
by he limi compa ison es , he se ies has he same cha ac e han
∞

s=0
1
3+ln2gszk=
∞

s=0
1
3+slng+ln (zk)2,
which is con e gen o all k>0 and z∈Z.
15Since pK,z(l )=Zmaxk∈Kl (k,z)Q(dz)and uis inc easing, he p oblema ic pa in ∞
=0pK,z(l ),
which is (1−qβ)−1za∞
=01/(3+ln2(1+g za)), is also con e gen , whe e a=maxKand K= {0}a e a
compac se o [0, ∞).I hecompac se isK={0}, henpK,z(l )=0.
1246 Juan Pablo Rincón-Zapa e o Theo e ical Economics 19 (2024)
exis s when β=1/(gp +q). Then, by P oposi ion 6, he e a e cons an s Mand α≥0
such ha αβ < 1and∞
=0 +
l (k,z)≤αβM ϕ(k,z)/(1−αβ)2<∞, o allk>0, z∈Z,
o all ≥0. This is clea ly impossible, o ins ance, o =1, he le -hand se ies is
∞

=0
( +1)l (k,z)=zk
(1−qβ)
∞

=0
+1
3+ln21+g zk,
which di e ges, a aining a con adic ion.16 Thus, a unc ion ϕsa is ying he assump-
ions o Theo em 4canno exis .
This example can be ex ended o u(c)=(1+c)/(b+lna(1+c)) −1/b,wi ha>1
and b≥max(1+a,(1−a)(1−a)). These inequali ies gua an ee ha uis s ic ly inc easing
and s ic ly conca e. The se ies (26) con e ges since a>1 wi h he same condi ion o
β,β≤1/(gp +q),bu hese ies
∞

=0 +
l +1(k,z)=zk
!(1−qβ)
∞

=0
( + )( + −1)···( +1)
b+lna1+g zk
di e ges o all posi i e in ege s ≥a−1, since he nume a o is a polynomial o deg ee
and he se ies hen has he same cha ac e han ∞
=11/ a− , which is con e gen i
and only i a− >1. ♦
5.2 Coun able local con ac ions
Ma kowski and Nowak (2011)andJa´
skiewicz and Nowak (2011)ex end he(coun -
able) local con ac ion app oach de eloped in Rincón-Zapa e o and Rod íguez-Palme o
(2003) om he de e minis ic case o he s ochas ic case.17 In his sec ion, we desc ibe
he me hod and poin ou some o he ea u es ha may es ic i s applicabili y o
s ochas ic dynamic p og ams whe e he Ma ko chain is exogenous.
The ollowing a e he assump ions in Ja´
skiewicz and Nowak (2011,Sec ion3),and
adap ed o ou se ing:
(W) The echnological co espondence is uppe semicon inuous, (x,z)is com-
pac o each (x,z)∈X×Z,U:−→ R∪{−∞}is uppe semicon inuous and
16No e ha
+1
3+ln21+g zk
1
+1
=( +1)2
3+ lng+ln (zk)2
3+ lng+ln (zk)2
3+ln21+g zk→1
ln2g·1,
as →∞; hus, by he limi compa ison es , he se ies has he same cha ac e han he ha monic se ies
∞
=01/( +1).
17Ou app oach can be conside ed a way o ex end he (uncoun able) local con ac ion me hod in Ma -
ins da Rocha and Vailakis (2010) om he de e minis ic o a s ochas ic se ing. The wo ds “coun able”
and “uncoun able” e e o he ca dinali y o he amily o semino ms used o desc ibe he opology o he
unc ion space whe e a gi en ope a o is de ined.
Theo e ical Economics 19 (2024) Exis ence and uniqueness o solu ions 1253
(ii) Since pK,z( )<∞ o all K∈Kand all z∈Z,and(x,z)is a compac se o any
x∈X,z∈Z, henZ
| |(y,z)Qz(dz)=p(x,z),z( )<∞. Ob iously, he same is
ue o 
.
P oo o Lemma 1. We o ganize he p oo in se e al p e ious lemmas.
Lemma 4. Le Assump ions (B1) o (B6) o hold. Then:
(i) ∞
=0L pl0<∞;
(ii) R0[]∈L1(Z;C(X)) and pl0+LR0≤R0.
P oo .Gi enx∈Xand z∈Z,βpl [](x,z)=βZmaxy∈(x,z)l (y,z)Qz(dz)≤l +1(x,
z);hence,pl []∈L1(Z,C(X)) and hen Lpl (K,z)=βpK,z(pl []) ≤pK,z(l +1), o all
=0, 1, Thus, L pl0≤L −1pl1≤···≤pl .By(B6), hese ies∞
=0pl (K,z)con e ges
o all K∈Kand z∈Z, hus∞
=0L pl0con e ges. To conclude he p oo , by he iangle
inequali y
pK,zpl0[]+···+pl []≤pK,zpl0[]+···+pK,zpl0[]
≤(pK,z(l1)+···+pK,z(l +1).
Le ing →∞and adding pK,z(ψ) o bo h sides o he abo e inequali y, we ha e
pK,z(ψ)+pK,z(R0[]) ≤R0(K,z),showinga hesame ime ha R0[]∈L1(Z;C(X)).
Lemma 5. Le Assump ions (B1) o (B6) hold. Then ∈V(0, R0)implies T ∈L1(Z;
C(X)).
P oo .Le ∈L1(Z;C(X)).Weuse heno a ion xand zin oduced abo e a he
beginning o his sec ion. The unc ion xis Bo el measu able o all x∈Xand Qz-
in eg able o any z∈Z.Thus, xcan be w i en as he di e ence o wo posi i e,
Qz-in eg able unc ions, x= +
x− −
x,whe e +
x=max( x,0)and −
x=max(− x,0).
Applying Theo em 8.1 in S okey, Lucas, and P esco (1989), bo h M +
xand M −
xa e
Bo el measu able. Since (M )x=M( x)=M( +
x)−M( −
x),(M )xis measu able o any
x∈X. To see ha (M )zis con inuous, conside a sequence {xn}in X ha con e ges
o x∈X. Then he sequence and i s limi o m he compac se K={xn}∪{x}.Le
n:= xn, o n≥1. Fo all z∈Z, n(z)→ x(z)as n→∞, since is con inuous in x.
Mo eo e , | z|≤supx∈K| z(x)|,andz→ supx∈K| z(x)|is Qz-in eg able by de ini ion
o L1(Z;C(X)), hus by he Lebesgue domina ed con e gence heo em,
(M )(xn,z)=Z
nzQzdz→Z
xzQzdz=(M )(x,z),
and hus (M )zis con inuous. Hence, M is a Ca a héodo y unc ion, and hus
U(x,y,z)+βM (y,z)is con inuous in (x,y) o all z, and i is Bo el measu able in z
o all (x,y). By he Be gé maximum heo em, he unc ion T is hus con inuous in x

1254 Juan Pablo Rincón-Zapa e o Theo e ical Economics 19 (2024)
o all z, and by he measu able maximum heo em, i is Bo el measu able o any x.In
sho , he unc ion
(x,z)→ T (x,z)=max
y∈(x,z)U(x,y,z)+βM (y,z),
is a Ca a héodo y unc ion. Mo eo e , i ∈Fand x∈X,z∈Z,
T (x,z)≤max
y∈(x,z)U(x,y,z)+βmax
y∈(x,z)Z
max
y∈(x,z) y,zQzdz
≤l0(x,z)+βZ
max
y∈(x,z)wy,zQzdz
≤l0(x,z)+βp(x,z),z( ).
Since (x,z)∈K, o ∈V(0, R0),weha ep(x,z),z( )≤R0[](x,z). By Lemma 4,
pK,z(l0)+βpK,z(R0[]) ≤R0(K,z).Hence,pK,z(T )≤R0(K,z). Thisp o es ha
T ∈V(0, R0), and hence ha T ∈L1(Z,C(X)).
Lemma 6. Le Assump ions (B1) o (B6) hold. Then D(V(0, R0))⊆C.
P oo . Since ∈V(0, R0),p ≤R0, and hence we can ake c=1. Also, p ∈Ca(X×Z),
since p [](x,z)=Zmaxy∈(x,z)| (y,z)|Qz(dz)is con inuous in xand Bo el measu -
able in z, by Lemma 3.Mo eo e ,p []≤R0[]implies pK,z(p []) ≤pK,z(R0[]) ≤
1
βR0(K,z), by Lemma 4.Hence,p []∈L1(Z,C(X)).
Now we a e in posi ion o p o e Lemma 1. Fi s , le us see ha L:C−→ C.Le p∈C;
by he de ini ion o he ope a o Land Lemma 4,
Lp(K,z)=βpK,zp[]≤βpK,zcR0[]≤cR1(K,z)≤cR0(K,z),
and so, Lp[]≤cR0[]and Lp[]∈L1(Z,C(X)). Second, we p o e ha he Assump-
ions (I) o (VI) a e ul illed. Rega ding (I), no e ha p+q∈Ci p,q∈C, i ially, as
well i is also immedia e ha i p∈Cand p≤p, henp∈C. On he o he hand, i
a coun able chain o pa ial sums p0,p0+p1,p0+p1+p2, is bounded by an ele-
men Pin C, hen he in ini e sum, p:=∞
n=0pn, is well-de ined and p≤P≤cR0 o
some cons an c. Mo eo e , since p[]≤cR0[]and R0[]∈L1(Z;C(X)) by Lemma 4,
he mono one con e gence heo em implies ha p[](x,·)is Qz-in eg able o all z∈Z
and all x∈X. On he o he hand, each unc ion pi[](·,z)is con inuous in x, o all
i=1, 2, . By he Weie s ass M- es , he unc ion p[](·,z)is also con inuous in x
o all z∈Z. These wo obse a ions imply ha p[]∈L1(Z;C(X)). (II) is i ial and
(III) holds, since he in eg al is mono one, and ega ding (IV), i holds ue, since o
all p,q∈C,pK,z(p[]+q[]) ≤pK,z(p[]) +pK,z(q[]) by de ini ion o he semino ms
pK,z, and hence
L(p+q)(K,z)=pK,zp[]+q[]
≤pK,zp[]+pK,zq[]
=Lp(K,z)+Lq(K,z).
Theo e ical Economics 19 (2024) Exis ence and uniqueness o solu ions 1255
Lis clea ly sup-p ese ing in Cby he mono one con e gence heo em; hence, (V) also
holds. Finally, (VI) is implied by Lemma 4and Lemma 5.
Le he wo amilies o unc ions {u }∞
=0and {l }∞
=0de ined in (B6’). Le Iu0,w0={ ∈
L1(Z;C(X)):u0≤ ≤w0}.
Lemma 7. T:Iu0,w0−→ Iu0,w0.
P oo . We p o e ha o all x∈X,z∈Z,and =0, 1, 
βZ
u y(x,z),zQzdz≥u +1(x,z)
and
βZ
max
y∈(x,z)w y,zQzdz≤w +1(x,z).
In ac ,
βZ
u y(x,z),zQzdz=Z
s≥
ksy(x,z),zQzdz
=β
s≥ Z
ksy(x,z),zQzdz
=β
s≥
ks+1(x,z)
=u +1(x,z),
whe e he exchange o in eg al and summa o y is due o he mono one con e gence
heo em. In he same way,
βZ
max
y∈(x,z)w y,zQzdz≤βZ
max
y∈(x,z)
s≥
lsy,zQzdz
≤
s≥
βZ
max
y∈(x,z)lsy,zQzdz
≤
s≥
ls+1(x,z)
=
s≥ +1
ls(x,z)
=w +1(x,z).
Le ∈Iu0,w0.Tha T is in L1(Z,C(X)) is p o ed as in Lemma 5abo e. Le us show
i s ha T ≥u0.Fo x∈Xand z∈Z o simpli y no a ion, le y=y(x,z).Then
T (x,z)≥U(x,y0,z)|+βZ
| y,z|Qzdz
1256 Juan Pablo Rincón-Zapa e o Theo e ical Economics 19 (2024)
≥k0(x,z)+βZ
u0y,zQzdz
≥k0(x,z)+βu1(x,z)
=u0(x,z).
Also,
T (x,z)≤max
y∈(x,z)U(x,y,z)+βmax
y∈(x,z)Z
max
y∈(x,z) y,zQzdz
≤l0(x,z)+βZ
max
y∈(x,z) y,zQzdz
≤l0(x,z)+βZ
max
y∈(x,z)w0y,zQzdz
≤l0(x,z)+w1(x,z)
=w0(x,z).
Appendix C: Con inui y o he Ma ko ope a o
The issue o con inui y o he alue unc ion in he unbounded case (and unbounded
space o shocks) is no an easy one. The ansla ion o S okey, Lucas, and P esco (1989),
Lemma 12.14 o his case is no s aigh o wa d, e en i he Ma ko chain is s ong Felle
con inuous. Recall ha Qhas he weak (s ong) Felle p ope y i Mmaps bounded con-
inuous unc ions ( esp., bounded measu able unc ions) on Zin o bounded con inu-
ous unc ions. To see hese kinds o p oblems ha may eme ge o unbounded unc-
ions, conside he ollowing example, adap ed om S oyano (2013). Le Z=[0, ∞)
and le he ansi ion unc ion Q:Z×Z−→ Rbe de ined as ollows:
Q(z,B)=⎧
⎨
⎩
δ0(B),i z=0;
B
dFzz,i z>0, (28)
whe e δ0is he Di ac measu e a he poin 0, ha is, δ0(B)=1i 0∈Band δ0(B)=0
o he wise, and o 0 <z≤1,
Fzz=⎧
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎩
0, i z=0;
zz2+1−z,i 0<z
≤1
z;
1, i z>1
z.
Finally, o z≥1, Q(z,B)=λ(B∩[0, 1]),whe eλdeno es he Lebesgue measu e o R.
No e ha , o 0 <z<1, Fzis a dis ibu ion unc ion; i is nondec easing, con inuous
excep a 0, whe e he igh -sided limi exis s, 0 ≤Fz≤1, and
dFz=zz2+1−z−0z=0+1
z
0z2dz=1−z+z=1.
Theo e ical Economics 19 (2024) Exis ence and uniqueness o solu ions 1257
Mo eo e , i is clea ha Q(·,B)is Bo el measu able. Thus, Qis a ansi ion unc ion.
Le (y,z)= (z)be independen o yand con inuous in z.ThenM is well-de ined in
his pa icula example and depends only on z,wi h(M )(0)= (z)Q(0, dz)= (0).
Fo 0 <z<1, we ha e
(M )(z)= zQz,dz= zdFzz
= (0)zz2+1−z−0z=0+1
z
0 zz2dz= (0)(1−z)+z21
z
0 zdz.
Since is con inuous, M is con inuous o 0 <z<1. Fo z≥1, M is cons an and
gi en by
(M )(z)= zQz,dz=[0,1]
zdz.
Now, i is bounded, he e is k>0such ha −k≤ ≤k;hence,−kz ≤z21
z
0 (z)dz≤
kz,and husz21
z
0 (z)dz ends o 0 as z→0+.Hence,M (z)→ (0)=M (0),when
z→0+,and husM is con inuous a 0. On he o he hand, (0)(1−z)+z21
z
0 (z)dz
ends o 1
0 (z)dz=M (1)as z→1−,and husM is con inuous a 1. Thus, M is con-
inuous, and hence Qis s ong Felle con inuous. Howe e , conside ing he unbounded
unc ion g(z)=z,weha eMg(0)=g(0)=0andMg(z)=1/2 o z>0; hus, Mg is
discon inuous a 0.
Conside now he ollowing simple pu e cu ency model wi h linea u ili y, whe e
agen s’ p e e ences a e subjec o andom shocks; see S okey, Lucas, and P esco (1989)
o u he de ails abou his model. Le he u ili y he u ili y u(c,z)=(1+z)cdepend
on consump ion cand shock z,andle (m)=[0, m+y],whe em≥0, y>0isacon-
s an , X=R+,Z=[0, ∞], and le a discoun ac o βsuch ha β<2/3. The dynamic
p og amming equa ion is
(m,z)=max
m∈[0,m+y](1+z)m+y−m+β[0,∞)
m,zQzdz.
The andom shocks a e assumed o be go e ned by he Ma ko chain Qdesc ibed in
(28). We a e simply in e es ed in showing ha he alue unc ion is no join ly con inu-
ous in (m,z). I is easily checked ha
(m,z)=⎧
⎪
⎨
⎪
⎩
m+y+yβ
1−β,i z=0;
(1+z)(m+y)+3
2yβ
1−β,i z>0,
is a solu ion in he class Ca(R+×R+), which coincides wi h he alue unc ion, and i is
no con inuous in z.
To p o e ha (B6) is ul illed, ake l0(m,z)=ψ(m,z)=(1+z)(m+y).Now,no ic-
ing ha ZzQz(dz)=1/2 and ecalling ha Q0is he Di ac measu e a 0, i is easy o
1258 Juan Pablo Rincón-Zapa e o Theo e ical Economics 19 (2024)
compu e whe e m=m+y,
Z
0m,zQzdz=βm+2y,i z=0;
(3/2)(m+2y),i z>0.
Thus, l1(m,z)=β(m+2y),i z=0,andl1(m,z)=β(3/2)(m+2y), o z>0. In gene al,
l (m,z)=β (3/2)(m+( +1)y), o all ≥1. Clea ly, ∞
=0l is uncondi ionally con e -
gen o all β<1, hus Assump ion (B6) holds. Gi en β,ϕ(m,z)=(1+az)(m+y),wi h
a≥1isaboundo U(m,m,z). Howe e , Zϕ(m,z)Q0(dz)=ϕ(m,0)=m+yand
Zϕ(m,z)Qz(dz)=(1+(a/2))(m+by ),i z>0. Thus, (m,z)→ Zϕ(m,z)Qz(dz)is
discon inuous a z=0 and hen (WC3) in Theo em 4is no ul illed. Any o he majo an
unc ion o he selec ed ϕwill su e he same p oblem.
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