Optimal strategy for corporate international investment and consumption problem with stochastic hyperbolic discounting

Long, Jun; Zeng, Sanyun; Gup a, B ij; Zhang, Jindan; Nedjah, Nadia
A icle
Op imal s a egy o co po a e in e na ional in es men
and consump ion p oblem wi h s ochas ic hype bolic
discoun ing
Jou nal o Inno a ion & Knowledge (JIK)
P o ided in Coope a ion wi h:
Else ie
Sugges ed Ci a ion: Long, Jun; Zeng, Sanyun; Gup a, B ij; Zhang, Jindan; Nedjah, Nadia (2024) :
Op imal s a egy o co po a e in e na ional in es men and consump ion p oblem wi h s ochas ic
hype bolic discoun ing, Jou nal o Inno a ion & Knowledge (JIK), ISSN 2444-569X, Else ie ,
Ams e dam, Vol. 9, Iss. 1, pp. 1-12,
h ps://doi.o g/10.1016/j.jik.2023.100459
This Ve sion is a ailable a :
h ps://hdl.handle.ne /10419/327365
S anda d-Nu zungsbedingungen:
Die Dokumen e au EconS o dü en zu eigenen wissenscha lichen
Zwecken und zum P i a geb auch gespeiche und kopie we den.
Sie dü en die Dokumen e nich ü ö en liche ode komme zielle
Zwecke e iel äl igen, ö en lich auss ellen, ö en lich zugänglich
machen, e eiben ode ande wei ig nu zen.
So e n die Ve asse die Dokumen e un e Open-Con en -Lizenzen
(insbesonde e CC-Lizenzen) zu Ve ügung ges ell haben soll en,
gel en abweichend on diesen Nu zungsbedingungen die in de do
genann en Lizenz gewäh en Nu zungs ech e.
Te ms o use:
Documen s in EconS o may be sa ed and copied o you pe sonal
and schola ly pu poses.
You a e no o copy documen s o public o comme cial pu poses, o
exhibi he documen s publicly, o make hem publicly a ailable on he
in e ne , o o dis ibu e o o he wise use he documen s in public.
I he documen s ha e been made a ailable unde an Open Con en
Licence (especially C ea i e Commons Licences), you may exe cise
u he usage igh s as speci ied in he indica ed licence.
h ps://c ea i ecommons.o g/licenses/by-nc-nd/4.0/
Op imal S a egy o Co po a e In e na ional In es men and
Consump ion P oblem wi h S ochas ic Hype bolic Discoun ing
Jun Long
a
, Sanyun Zeng
b,
*, B ij B. Gup a
c,d,e, ,g
, Jindan Zhang
h
, Nadia Nedjah
i
a
School o In o ma ion Technology & Enginee ing, Guangzhou College o Comme ce, Guangzhou 511363, PR China
b
School o Accoun ing, Guangdong Uni e si y o Fo eign S udies, Guangzhou 510006, PR China
c
Depa men o Compu e Science and In o ma ion Enginee ing, Asia Uni e si y, Taichung 413, Taiwan
d
Kyung Hee Uni e si y, 26 Kyungheedae- o, Dongdaemun-gu, Seoul 02447, Ko ea
e
Symbiosis Cen e o In o ma ion Technology (SCIT), Symbiosis In e na ional Uni e si y, Pune, India
Depa men o Elec ical and Compu e Enginee ing, Lebanese Ame ican Uni e si y, Bei u 1102, Lebanon
g
Cen e o In e disciplina y Resea ch, Uni e si y o Pe oleum and Ene gy S udies (UPES), Deh adun, India
h
Xianyang Voca ional Technical College, Xianyang, Shanxi, China
i
Depa men o Elec onics Enginee ing and Telecommunica ions o he Enginee ing Facul y, S a e Uni e si y o Rio de Janei o, Rio de Janei o, B azil
ARTICLE INFO
A icle His o y:
Recei ed 10 Ma ch 2023
Accep ed 30 Decembe 2023
A ailable online 12 Janua y 2024
ABSTRACT
This s udy in es iga es he dynamics o a s ochas ic hype bolic discoun ing model in a con inuous- ime
amewo k o add ess he complexi ies associa ed wi h he co po a e in e na ional in es men consump ion
p oblem (CIICP). By o mula ing a dynamic p og amming equa ion based on he p inciples o dynamic p o-
g amming, we seek o un angle he in icacies o CIICP by inco po a ing log u ili y. Ou esea ch p o ides
aluable insigh s in o he economic amifica ions o he p oposed model and p esen s a comp ehensi e anal-
ysis o nume ical sensi i i ies. This in es iga ion no only con ibu es o a deepe unde s anding o he CIICP
phenomenon bu also con ibu es o mo e in o med decision-making wi hin he field.
© 2024 The Au ho s. Published by Else ie España, S.L.U. on behal o Jou nal o Inno a ion & Knowledge. This
is an open access a icle unde he CC BY-NC-ND license
(h p://c ea i ecommons.o g/licenses/by-nc-nd/4.0/)
Keywo ds:
Co po a e in e na ional in es men and con-
sump ion p oblem (CIICP)
S ochas ic hype bolic discoun ing
ime-inconsis en
Dynamic p og amming equa ion (DPE)
JEL Classifica ion:
C610
G150
In oduc ion
Hype bolic discoun ing e e s o a phenomenon whe ein indi id-
uals end o place g ea e alue on immedia e benefi s compa ed
wi h u u e benefi s, esul ing in a ime-inconsis en p e e ence
s uc u e ha can be ca ego ized in o nai e o sophis ica ed. Nai e
indi iduals a e unawa e o hei changing p e e ences o e ime,
leading o inconsis en planning ha is no execu ed p ac ically. In
con as , sophis ica ed indi iduals a e awa e o hei ime-inconsis-
en p e e ences and choose an op imal ime-consis en plan. In he
con ex o in e na ional in es men and consump ion o a co po a-
ion, his s udy hypo hesizes a con inuous- ime model wi h
s ochas ic hype bolic discoun ing. Specifically, we p opose ha co -
po a e manage s p e e sho - e m benefi s o e u u e benefi s, and
his p e e ence s uc u e may ha e implica ions o he co po a ion’s
in es men and consump ion decisions, wa an ing u he
in es iga ion.
The majo i y o economic decisions in ol e a adeo be ween
p esen and u u e e u ns, which essen ially gi es hem a ime-span-
ning na u e. Discoun ed u ili y heo y has become he main ame-
wo k o e alua ing in e empo al choices in economics (see, Hajdini,
2012;Mankiw, 2020;Po ney & Weyan , 2013). The discoun unc-
ion, which is an essen ial componen o he heo y o decision mak-
ing, alues delayed e u ns in he p esen . In p oblems wi h fini e
ho izons, indi iduals ypically discoun sho - and long- e m u ili y
unc ions applying he same cons an discoun a e o hei p e e en-
ces o e ime. Howe e , expe imen al s udies on ime p e e ences
ha e demons a ed ha he ime-consis en hypo hesis is un ealis ic
(Ainslie, 1992;Loewens ein & P elec, 1992;Thale , 1981) since indi-
iduals beha e mo e pa ien ly when bo h ewa ds a e a emo ed
om he p esen and mo e impa ien ly when bo h ewa ds app oach
This esea ch is suppo ed by g an s om: The Key P ojec o Na ional Na u al Sci-
ence Funds (No. 71231008), Guangdong P o ince Philosophy and Social Sciences Plan-
ning P ojec (No. GD23CGL28), and Guangdong P o ince Gene al Uni e si y
Humani ies and Social Sciences Resea ch Cha ac e is ic Inno a ion P ojec (No.
2018WTSCX035).
* Co esponding au ho .
E-mail add ess: [email p o ec ed] (S. Zeng).
h ps://doi.o g/10.1016/j.jik.2023.100459
2444-569X/© 2024 The Au ho s. Published by Else ie España, S.L.U. on behal o Jou nal o Inno a ion & Knowledge. This is an open access a icle unde he CC BY-NC-ND license
(h p://c ea i ecommons.o g/licenses/by-nc-nd/4.0/)
Jou nal o Inno a ion & Knowledge 9 (2024) 100459
Jou nal o Inno a ion
&Knowledge
h ps://www.jou nals.else ie .com/jou nal-o -inno a ion-and-knowledge
he cu en ime. Laibson (1997) p oposed a quasi-hype bolic dis-
coun unc ion in which he discoun a e dec eases o e ime o cap-
u e he phenomenon o ime- a ying impa ience. This ype o
p e e ence is e e ed o as p esen -biased p e e ences by Palacios-
Hue a and P
e ez-Kakabadse (2013). Time-inconsis en p e e ences
ha e been ex ensi ely in es iga ed by economis s, wi h a la ge body
o li e a u e explo ing hype bolic discoun ing models (Ainslie & Has-
lam, 1992;Ainslie & He ns ein, 1981;DellaVigna & Malmendie ,
2006;Ki by & He ns ein, 1995;Loewens ein & P elec, 1992;McClu e
e al., 2004;Mye son & G een, 1995;Thale & She in, 1981). These
models ha e been applied o a wide ange o economic p oblems,
including Ba o (1999),DellaVigna and Malmendie (2004),G ena-
die and Wang (2007),O’Donoghue and Rabin (1999), and Palacios-
Hue a and P
e ez-Kakabadse (2013), among o he s. Howe e ,
hese models ha e inhe en limi a ions, as wi h he assump ion o
consis en p e e ences o e ime and he di ficul y o es ima ing dis-
coun a es accu a ely. Fu he esea ch is needed o add ess hese
limi a ions and imp o e ou unde s anding o ime-inconsis en p e -
e ences.
A su ge in esea ch ac i i ies ega ding co po a e in e na ional
in es men has occu ed o e he yea s, explo ing a ious aspec s o
his phenomenon om di e en pe spec i es. Choi (1989) examined
he e olu ion o he heo y o co po a e in e na ional in es men ,
ocusing on i s financial dimension. Bellalah and Wu (2009) p oposed
a model o co po a e in es men among in e na ional conce ns ha
inco po a es s ochas ic and de e minis ic elemen s. The model con-
side s he impac o domes ic and o eign p ices on fi ms’ou pu s
and inpu s, along wi h exchange a e isk and compe i i eness
be ween fi ms. Zhang and Zhang (2012) de eloped an op imal co po-
a e in es men model ha accoun s o day ime and nigh ime di -
e ences. O e all, hese s udies ha e con ibu ed o ou
unde s anding o he ac o s ha influence in e na ional in es men
in co po a ions; howe e , some limi a ions emain. Fo ins ance,
Choi’s (1989) analysis p ima ily ocused on he financial aspec s o
in e na ional in es men , neglec ing o he significan ac o s such as
poli ical and cul u al conside a ions. Bellalah and Wu’s (2009) model
assumed ha fi ms a e a ional and p ofi -maximizing, which migh
no always be he case in eali y. Zhang and Zhang’s (2012) model is
based on he assump ion ha day ime and nigh ime di e ences
ha e a significan impac on in es men decisions, which migh no
be ue in all con ex s. The e o e, u u e esea ch mus add ess hese
limi a ions and p o ide a mo e comp ehensi e unde s anding o co -
po a ions’in e na ional in es men .
The li e a u e on fi ms’in e na ional in es men and consump-
ion has included se e al s udies. Wu and Zhang (2006) examined
he op imal co po a e po olio and consump ion choice p oblem, in
which in es o s could in es hei weal h in bonds (bank accoun s)
and eal p ojec s owned by p oduc ion. Huang and Wu (2011)
ocused on he op imal in e na ional po olio and consump ion
choices o co po a ions, in which in es o s could in es hei weal h
in domes ic bonds (bank accoun s) o physical p ojec s ab oad.
Huang and Zhang (2013) p oposed a model o fi ms’op imal in es -
men and consump ion ha conside ed he e ec s o infla ion and
di e ences in ma ke opening and closing. Long and Zeng (2016)
in es iga ed an in e na ional in es men p oblem o co po a ions,
p oposing ime-consis en s a egies wi h a mean- a iance c i e ion.
Bellalah e al. (2016) conduc ed a s udy on co po a e in e na ional
in es men and analyzed he impac o in o ma ion cos , sho sell-
ing, and axa ion on managemen decision making, finding ha hese
ac o s ha e a significan impac on co po a e in es men decisions.
Bellalah and Zhang (2017) in es iga ed he impac o fi ms’in e na-
ional in es men decisions a ound ma ke closu e on exchange a e
isk, incomple e in o ma ion, and sho selling es ic ions, de e min-
ing ha hese ac o s had a significan impac on in es men . Choi e
al. (2018) examined he p edic ions o eal op ion heo y, confi ming
ha i accu a ely p edic s in es men decisions. Daszkiewicz (2019)
explo ed he in e na ionaliza ion pa e ns o amily high- ech fi ms
and ound ha such fi ms end o in e na ionalize slowly and cau-
iously. Glodowska e al. (2019) in es iga ed mul ina ional en e -
p ises’ isk mi iga ion s a egies in eme ging ma ke s, finding ha
hese s a egies a e essen ial o success ul in e na ionaliza ion. Zhu
and Sa dana (2020) in es iga ed he mode a ing e ec o in e na-
ionaliza ion mo i es on Polish fi ms’mul i- na ionaliza ion−pe o -
mance ela ionship, de e mining ha such mo i es had a significan
impac on he ela ionship. Ba lozewski and T apczyzski (2021)
examined he impac o o eign exchange a e isk on he expec ed
e u ns gene a ed by Sou h A ican in es o s’po olios, demons a -
ing ha his isk has a significan impac on e u ns. Djemo e al.
(2021) examined how small amily fi ms esponded o he COVID-19
c isis, finding ha such fi ms adap ed and esponded o he c isis in
a ious ways. Ma janski & Sulkowski, 2021 examined how new mul-
ina ional fi ms s uc u ed hei o eign di ec in es men loca ion
po olios and ound ha such decisions ha e a significan impac on
pe o mance ou comes. G ieco (2021) p o ided a comp ehensi e
o e iew o he heo ies and e idence on o eign in es men and
de elopmen , while Song (2021) examined he condi ions unde
which mul ina ional co po a ions adjus p oduc ion olume among
subsidia ies. Ko en and Vodopyano a (2022) ocused on me hods o
inc ease ansna ional co po a ions’in es men a ac i eness, and Li
e al. (2022) cons uc ed a p oposed ma hema ical model o s udy
he op imal in es men p oblem. Finally, Nguyen (2023) in es iga ed
he ela ionship be ween financial cons ain s and fi m p oduc i i y
in Vie namese manu ac u ing indus ies.
Despi e he significan p og ess made in he field o co po a e
in es men and consump ion decisions, some limi a ions emain in
he p e ious li e a u e. Specifically, he majo i y o s udies ha e p i-
ma ily ocused on co po a ions’op imal in es men and consump-
ion choices, while neglec ing he po en ial impac o ex e nal ac o s
such as poli ical and economic isks. Fu he mo e, some s udies ha e
solely concen a ed on a pa icula ype o in es men , such as bonds
o physical p ojec s, and ha e no explo ed o he in es men op ions.
To add ess hese limi a ions, u u e esea ch should expand on he
app oaches o p e ious esea ch by conside ing a b oade ange o
in es men op ions and ex e nal ac o s o p o ide mo e comp ehen-
si e insigh s in o fi ms’in e na ional in es men and consump ion
decisions. As no ed, some s udies ha e only ocused on specific ac-
o s and did no conside o he impo an ac o s ha may a ec
in es men decisions. Addi ionally, some s udies ha e only ocused
on specific ypes o fi ms o indus ies, which may limi he gene -
alizabili y o he findings. Finally, some s udies ha e elied on sel -
epo ed da a, which could be subjec o bias and may no accu a ely
eflec co po a ions’ac ual in es men decisions. The e o e, u u e
esea ch mus inco po a e a mo e comp ehensi e se o ac o s and a
b oade ange o fi ms and indus ies o es ablish a mo e comple e
unde s anding o co po a e in e na ional in es men decision mak-
ing. In doing so, schola s can gain a deepe unde s anding o he com-
plex decision making p ocesses ha fi ms engage in when making
in e na ional in es men and consump ion decisions and p o ide
mo e accu a e and eliable ecommenda ions o p ac i ione s and
policymake s.
As he scope and complexi y o p oblem-sol ing expands, demand
o cen al p ocessing uni (CPU) esou ces also ises. To op imize
CPU esou ces, Ja a weh e al. (2019) p oposed he use o pa alleliza-
ion and ec o iza ion echniques o enhance he pe o mance o he
Needleman−Wunsch algo i hm; a widely used sequence alignmen
algo i hm in bioin o ma ics wi h high compu a ional complexi y. The
au ho s sugges ed dis ibu ing he wo kload o mul iple p ocesso s
using pa alleliza ion echniques and op imizing he use o CPU
esou ces using ec o iza ion echniques. They implemen ed he pa -
allelized and ec o ized Needleman−Wunsch algo i hm, es ing i on
a se o DNA sequence da a. The esul s demons a ed ha hei algo-
i hm significan ly imp o ed pe o mance compa ed wi h adi ional
J. Long, S. Zeng, B.B. Gup a e al. Jou nal o Inno a ion & Knowledge 9 (2024) 100459
2
algo i hms. This echnology has significan implica ions o u u e
esea ch in his field.
We combine s ochas ic hype bolic discoun ing (SHD) and a co po-
a e in e na ional in es men and consump ion p oblem (CIICP) o
in es iga e hei implica ions o unde s anding and sol ing con empo-
a y economic and financial issues, p o iding aluable insigh s o co -
po a e in es men and consump ion decisions. Ou CIICP is cons uc ed
e e encing Bellalah and Wu (2009), and he SHD me hod was p oposed
by Zou e al. (2014). We ob ain some in e es ing esul s by combining
CIICP and SHD. This s udy makes se e al no able con ibu ions. Fi s ,
dynamic p og amming is applied o o mula e a dynamic p og amming
equa ion (DPE) ha desc ibes he op imal co po a e in e na ional
in es men s a egy and consump ion choice o sophis ica ed indi id-
uals. Second, we p o ide explici solu ions o loga i hmic u ili y. Thi d,
we conduc a sensi i i y analysis o examine he ola ili y pa ame e o
he op imal solu ion. The esul s o his s udy p o ide aluable insigh s
o CIICP ha a e o conside able significance o co po a e in es men
and consump ion decisions.
The emainde o he pape is o ganized as ollows. Sec ion 2
p esen s he CIICP ma ke se ing and in oduces he concep o SHD.
Sec ion 3 de ails he DPE o he p oblems o in e na ional in es -
men and consump ion wi h complex indi idual co po a ions. In Sec-
ion 4, we ob ain an explici solu ion o log u ili y. Sec ion 5 p o ides
simula ion esul s o illus a e he impac o ola ili y pa ame e s on
o eign p oduc ion ma ke s and exchange a es. We p esen some
conclusi e and sugges ed ex ensions in Sec ion 6. Finally, he Appen-
dix p esen s he model’s echnical de ails.
Model hypo heses
In his sec ion, we p esen he basic model o he CIICP ame-
wo k and desc ibe SHD e e encing Zou e al. (2014).
Le ðV;F;F;PÞdeno e a fil e ed p obabili y space, whe e Vis a
p obabili y space, P is a measu able p obabili y, F ep esen s he
in o ma ion a ailable up o ime , and he fil a ion F¼
F g; 2½0;T
sa isfies he usual condi ions (i.e., F g; 2½0;Tis igh -con inuous
and P-comple e). He e, T ep esen s he e minal ho izon. We assume
ha all s ochas ic p ocesses and andom a iables a e defined in he
fil e ed p obabili y space ðV;F;F;PÞ. Addi ionally, le B1
;B2
, and B3
be h ee mu ually dependen , one-dimensional B ownian mo ions
wi h co ela ion coe ficien s 12; 13, and 23 2½1;1, espec i ely,
which ep esen unce ain ex e nal sou ces in he ma ke .
The indus ial ma ke model
In his sec ion, we cons uc a ma ke e e encing Bellalah and Wu
(2009), conside ing a “ wo-coun y”fi m o which domes ic ðR Þand
o eign ðR
Þs a ic cash flows a e deno ed as ollows:
R ¼ð1kÞðP C ÞQ ;Q ¼Pb
;
R
¼ð1kÞðP
C
ÞQ
;Q
¼ðP
Þb;ð1Þ
whe e P ;C (domes ic), P
and C
(ab oad, ma ked wi h *) a e he ou -
pu and inpu p oduc ion p ices. Q and Q
ep esen he amoun o
p oduc ion ou pu , band bdeno e posi i e cons an s, e ep esen s
he exchange a e, which is unce ain, and kand ka e he ax a es
in home and o eign coun ies. Choi (1989) assumed he quan i y o
ou pu (Q ) o be ce ain and did no conside he ax.
Simila o Choi (1989) and Bellalah and Wu (2009), we assume
ha he dynamic inpu and ou pu p ices o p oduc s and he
exchange a e can be exp essed in he ollowing o m:
dP
P
¼mpd þsdB1
;dC
C
¼mcd þsdB1
;
dP
P
¼mpd þsdB2
;dC
C
¼mcd þsdB1
;
ð2Þ
and
de
e
¼med þsedB3
;ð3Þ
whe e he e ms mp;mp;mc;mc, and mea e cons an -bounded and
ep esen he a es ha a e expec ed ins an aneously o a ious a -
iables. The e ms s;sand se ep esen he coe ficien s o ins an a-
neous fluc ua ions. The ini ial alues o he abo e a iables a e P0;C0;
P
0;C
0, and e0, espec i ely.
The cash flow gene a ed om he domes ic ma ke ðR Þis
exp essed as ollows:
dR
R
¼ ð Þd þðbþ1ÞsdB1
;ð4Þ
wi h
ð Þ¼1
2ðbþ1Þ2s2þðbþ1Þmp1
2s2

þðmpmcÞC0emc
P0emp C0emc :ð5Þ
Simila ly, he cash flow ðR Þin he o eign ma ke ha has no ye
been emi ed is deno ed as ollows:
dR
R
¼ ð Þd þðbþ1ÞsdB2
;ð6Þ
whe e
ð Þ¼1
2ðbþ1Þ2ðsÞ2þðbþ1Þmp1
2ðsÞ2

þðmpmcÞC
0emc
P
0emp C
0emc :ð7Þ
F om R¼e R and Eqs. (3) and (6), he a ia ion in cash flow om
ab oad ðR
Þis ob ained as ollows:
dR
R
¼½ ð Þþmed þðbþ1ÞsdB2
þsedB3
;ð8Þ
whe e
ð Þ¼ ð Þþðbþ1Þsse 23:ð9Þ
Howe e , a sunk cos is paid by in es o s who equi e ex a
e u ns o compensa e o he esou ces wi h which hey engage in
he sea ch o ob ain in o ma ion. In o ma ion cos has an inc emen al
e u n dimension, and i is included in he desc ip ions o he dynam-
ics o di e en a iables and added o he d i . Thus, he ac ual
change in domes ic cash flow should be indica ed as ollows:
dR
R
¼ ðÞþλR
½d þbþ1ðÞsdB1
;
R0¼1kðÞPb
0P0C0
ðÞ:
ð10Þ
whe e λR ep esen s he a e o in o ma ion cos in he domes ic ma -
ke .
The eal changes in he exchange ma ke a e exp essed as ollows:
de
e
¼½meþλed þsedB3
;
eð0Þ¼e0:
ð11Þ
whe e λe ep esen s he a e o in o ma ion cos in he exchange
ma ke .
Then, om R¼e R and Eqs. (8) and (11), he change in cash flow
a a specific ime coming om he o eign ma ke can be easily
ob ained as ollows:
dR
R
¼½ ð ÞþλRþmeþλed þðbþ1ÞsdB2
þsedB3
;
R
0¼ð1kÞPb
0ðP
0C
0Þ:
ð12Þ
J. Long, S. Zeng, B.B. Gup a e al. Jou nal o Inno a ion & Knowledge 9 (2024) 100459
3
Weal h p ocess
Le W ep esen co po a e weal h and p deno e he p opo ion
o co po a e in es men ab oad a ime 0. 1 p is he p opo ion
o home ma ke in es men . Re e encing he co po a e consump ion
in oduced by Sumne (2008), in addi ion o decision making on
in es men , he manage can also make consump ion decisions o
sa is y sha eholde demands.
Eqs. (10) and (12) indica e ha co po a e weal h can be exp essed
as ollows:
dW ¼p ð ÞþλRþmeþλe
ðÞW þð1p Þ ð ÞþλR
ðÞW c
½d
þð1p Þðbþ1ÞW sdB1
þp ðbþ1ÞW sdB2
þp W sedB3
;
ð13Þ
whe e c is co po a e consump ion. We assume a posi i e ini ial co -
po a e weal h (W0), and pð¢Þis a measu able p ocess ha is almos
su ely locally bounded.
SHD
We use he amewo k o Me on (1969) o in oduce he s ochas-
ic hype bolic discoun unc ion om Ha is and Laibson (2013) o
examine ime-inconsis en indi iduals’consump ion and in es men .
Re e encing Ha is and Laibson (2013), he discoun unc ion Dð ;sÞis
exp essed as ollows:
Dð ;sÞ¼ e ðs Þ;s2½ ; þ Þ;
he ðs Þ;s2½ þ ;þ1Þ;
ð14Þ
whe e ½ þ ;þ1Þis he o wa d in e al and ½ ; þ Þis he closes
in e al. We assume ha he closes in e al ð Þhas an exponen ially
dis ibu eddu a ionwi hpa ame e λ, and he s a iona i y assump ion
is sa isfied o he s ochas ic hype bolic discoun unc ion Dð ;sÞ(i.e., Dð
; þsÞ¼Dð0;sÞ). The expec ed leng h o he closes in e al is E½ ¼1
λ.
Alowe λindica es a longe expec ed du a ion. The du a ion o he clos-
es in e al is 1as λ¼0, and he discoun unc ion degene a es o a
cons an exponen ial discoun , and he p oblem ep esen s Me on’s
classical case; howe e , when λ!1, he closes in e al becomes
ze o, and he discoun unc ion is ans o med in o he ollowing:
Dð ;sÞ¼ 1;s¼ ;
he ðs Þ;s2½ ;þ1Þ;
ð15Þ
Ajumpoccu sa s¼ . The pa ame e hð0<h1Þshows he p es-
en de ia ion in he deg ee o p e e ence. A smalle hindica es a la ge
p esen de ia ion. When h¼1, he p esen and u u e in e als a e
no dis inguished and he discoun unc ion Dð ;sÞonce again degen-
e a es o an exponen ial discoun unc ion wi h a discoun a e o ,
and his implies ha he p oblem educes o Me on’s classical case.
CIICP op imiza ion model
This s udy simul aneously conside s fi ms’in es men and con-
sump ion o maximize he u ili y o weal h by de e mining he
in es men s a egy and consump ion choice. A ma u e co po a e
in es o wi h empo ally inconsis en p e e ences is equi ed o
allow o u u e sel es in he cu en decision. The ollowing op imi-
za ion p oblems mus be sol ed by complex indi idual fi ms o fini e
and infini e planning pe iods wi h SHD o ob ain op imal and ime-
consis en consump ion and po olio policies, espec i ely:
max c ;p gER þ
e Uðc Þd þhRT
þ e Uðc Þd þhe TBðT;WTÞ
hi
s: :dW ¼p ð ÞþλRþmeþλe
ðÞW þð1p Þ ð ÞþλR
ðÞW c
½d
þð1p Þðbþ1ÞW sdB1
þp ðbþ1ÞW sdB2
þp W sedB3
;
ð16Þ
and
max c ;p gER þ
e Uðc Þd þhR1
þ e Uðc Þd
hi
s: :dW ¼p ð ÞþλRþmeþλe
ðÞW þð1p Þ ð ÞþλR
ðÞW c
½d
þð1p Þðbþ1ÞW sdB1
þp ðbþ1ÞW sdB2
þp W sedB3
;
ð17Þ
whe e Uð¢Þis he co po a e consump ion u ili y in he in es men
pe iod, Bð¢Þis he co po a e e minal weal h u ili y, is he manag-
e ’s ime p e e ence in he ecen pe iod, h ep esen s he diminished
a e o u u e discoun ing, and ep esen s he leng h o ime o he
cu en phase.
In he nex subsec ion, we fi s de i e he p oblem o complex
indi iduals, and hen ex end he analysis o yield DPE ha a e alid
o infini e le el p oblems.
The DPE o he fini e ho izon p oblem
Ou analy ical app oach is based on Zou e al. (2014), and has also
been used by Ka p (2007). Specifically, a con inuous- ime p oblem is
con e ed in o a disc e e- ime p oblem, which we sol e using back-
wa d induc ion, ob aining he op imal ac ion o he u u e sel fi s ,
and hen we ob ain he DPE o a fini e ho izon p oblem o ma u e
indi iduals by i e a ing and passing he p oblem o he limi o con-
inuous ime.
Deno ed by Vð ;W Þ he alue unc ion o Eq. (16) is as ollows:
Vð ;W Þ¼ max c ;p g
EZ þ
e Uðc Þd þhZT
þ
e Uðc Þd þhe TBðT;WTÞ

;ð18Þ
wi h VðT;WTÞ¼BðT;WTÞ. We assume ha Vð ;W Þis o classC1.
Simila o Zou e al. (2014), we deduce he DPE o eplace he
Hamil on-Jacobi-Bellman (HJB) equa ion as ollows:
VV K¼max c ;p gUVWcþVWWp ð ÞþλRþmeþλe
ðÞ½
þð1p Þ ð ÞþλR
ðÞ
þ1
2VWW W2½ð1p Þ2ðbþ1Þ2s2þp2
ðbþ1Þ2ðsÞ2
þp2
ðseÞ2þ2ð1p Þp ðbþ1Þðbþ1Þss 12
þ2ð1p Þp ðbþ1Þsse 13 þ2p2
ðbþ1Þsse 23g;
ð19Þ
whe e
Kð ;W Þ¼ZT
eðλþ Þðs ÞUðc
sÞd ;ð20Þ
and c
sa isfies Eq. (19).
The e o e, he DPE o Eq. (16) is Eqs. (19) and (20), o which
he bounda y condi ions o complica ed indi iduals a e as ol-
lows:
VðT;WTÞ¼BðT;WTÞ:ð21Þ
To simpli y, we deno e he ollowing:
D1
u¼ ð ÞþλR;
D2
u¼ ð ÞþλRþmeþλe;
D1
d¼ðbþ1Þ2s2;
D2
d¼2ðbþ1Þ2s2þ2ðbþ1Þðbþ1Þss 12 þ2ðbþ1Þsse 13;
D3
d¼ðbþ1Þ2s2þðbþ1Þ2ðsÞ2þðseÞ2
2ðbþ1Þðbþ1Þss 12 2ðbþ1Þsse 13 2ðbþ1Þsse 23;
ð22Þ
J. Long, S. Zeng, B.B. Gup a e al. Jou nal o Inno a ion & Knowledge 9 (2024) 100459
4

hen Eq. (19) is equi alen o he ollowing:
VV K¼max c ;p gUVWcþVWWD1
uþ1
2VWW W2D1
d

þpVWWðD2
uD1
uÞþ1
2VWW W2D2
d

þ1
2p2VWW W2D3
dg:ð23Þ
By he fi s -o de condi ions, we ob ain he op imal solu ion o
Eq. (16) as ollows:
Uðc
Þ¼VW;
p
¼ VW
VWW W
D2
uD1
u
D3
d
1
2
D2
d
D3
d
:
8
>
<
>
:ð24Þ
Rema k 1.
(1) The e m VWW W
VWis he coun e pa o he P a −A ow measu e
(P a , 1976) wi h espec o isk a e sion. This measu e eflec s
how isk-a e se a company is owa d domes ic and o eign
in es men s.
(2) The e m D3
d e e s o he a iabili y o in es men s a egy.
(3) The fi s e m, VW
VWW W
D2
uD1
u
D3
d
in p
, is e e ed o as “specula i e
demand”o agg essi e demand and is con ingen on isk a e -
sion measu es.
(4) The second e m, 1
2
D2
d
D3
d
,inp
is e e ed o as hedging demand,
which co e s he demand o o eign in es men s on a isk
hedging basis and is no ela ed o he manage ’s isk a i ude.
Rema k 2.
Eq. (22) shows ha D2
uD1
u¼ ð Þ ð ÞþmeþðλRλeλRÞ.
In o ma ion cos s a e included in he in es men s a egy, and his
influence is ep esen ed by λRλeλR. This exp ession indica es
he in o ma ion cos s in ol ed in cos ad an age seeking, whe ein
manage s will no ag ee o in es ab oad un il hey ha e ob ained
some use ul in o ma ion ega ding he o eign and exchange ma -
ke s. Wi hou his in o ma ion, manage s a e mo e inclined o
make in es men s in he local ma ke .
The DPE o he infini e ho izon p oblem
The alue unc ion defines Vð ;W Þas in Eq. (17) as ollows:
Vð ;W Þ¼ max c ;p gEZ þ
e Uðc Þd þhZ1
þ
e Uðc Þd

:ð25Þ
Following Zou e al. (2014), we he DPE can be ob ained as ol-
lows:
dV K¼max c ;p gUVWcþVWWD1
uþ1
2VWW W2D1
d

þpVWWðD2
uD1
uÞþ1
2VWW W2D2
d

þ1
2p2VWW W2D3
dg:ð26Þ
whe e
K ;W
ðÞ¼λ1hðÞ
R1
eλþ ðÞsUc

s

ds:ð27Þ
We u he assume limT!1he ðT ÞBðT;WTÞ¼0. As a conse-
quence, a ans e sal condi ion wi h he bounda y condi ion Eq. (21)
becomes he ollowing:
lim !1he TVð ;W Þ¼0:ð28Þ
Fo his eason, he HJB o Eq. (17) is subjec o Eq. (13), which is
Eq. (26) o sophis ica ed indi iduals, and he ho izon al condi ion is
Eq. (28).
The explici solu ion o CIICP wi h log u ili y
In his sec ion, we conside he CIICP wi h log u ili y wi h he ol-
lowing:
Uðc Þ¼lnc ;BðT;WTÞ¼&lnWT:ð29Þ
We p esen he ollowing lemma o sol e he op imal p oblem
wi h log u ili y.
Lemma 1. The solu ion o he ollowing ODE:
_
a  a ¼P ;ð30Þ
can be cha ac e ized as ollows:
a ¼aTe ðT ÞZT
e s ðÞ
Psds;ð31Þ
whe e aTis a gi en scala , and P is a gi en unc ion.
The p oo is p esen ed in Appendix A.
F om Eq. (23), we ha e he ollowing:
Theo em 1. Eq. (16) is op imally sol ed as ollows:
c
¼W
a
;
p
¼1
D3
d
 ðÞ ðÞþλRþmeþλeλR1
2D2
d

;
ð32Þ
and he sophis ica ed -agen ’s alue unc ion is as ollows:
Vð ;W Þ¼a lnW þb ;ð33Þ
whe e a ;b sa isfies he ollowing:
a ¼e ðT Þ&h

1h
λþ eðλþ ÞðT Þþλhþ
ðλþ Þ;
b ¼RT
e ðs ÞPsds;
ð34Þ
wi h
P ¼lna D1
u1
2D1
dþ1
2D3
d
D2
uD1
u1
2D2
d

2
"#
a þ1
þλð1hÞRT
eðλþ Þðs ÞRs
G0ð Þ 1
a

d lnas

ds;
G0ð Þ¼ D2
uD1
u
D3
d
1
2
D2
d
D3
d
!
ðD2
uD1
uÞþD1
u
1
2D1
dþD2
uD1
u
D3
d
1
2
D2
d
D3
d
!
aD2
dþ1
2
D2
uD1
u
D3
d
1
2
D2
d
D3
d
!
D3
d
"#
;
ð35Þ
and D1
u;D2
u;D1
d;D2
d;and D3
da e gi en in Eq. (23).
The p oo is p esen ed in Appendix B.
Rema k 3.
We find ha he op imal in es men s a egy does no de e mine he
weal h p ocess (Wp
), bu he op imal consump ion choice depends
on he weal h p ocess (Wp
).
Resul s o simula ion and pa ame e sensi i i y
In his sec ion, we examine he impac o pa ame e s wi h espec
o fi ms’op imal in es men s a egies and consump ion choices,
and supply se e al nume ical examples demons a ing he impac .
J. Long, S. Zeng, B.B. Gup a e al. Jou nal o Inno a ion & Knowledge 9 (2024) 100459
5
Fo he desc ip ion below, in he absence o any o he indica ion, he
undamen al pa ame e s a e p o ided as ollows:
b¼2;b¼2;mp¼0:08;mp¼0:06;mc¼0:02;mc¼0:03;
s¼0:2;s¼0:15;se¼0:05; 12 ¼0:8; 13 ¼0:6; 23 ¼0:4;
me¼0:004;λR¼0:001;λR¼0:002;λe¼0:003;P0¼6¼P
0;
C0¼2¼C
0:
(The da a a e ob ained om Bellalah & Wu, 2009).
(1) Based on Theo em 1, we can ob ain he po olio changing o e
ime.
In Fig. 1, (i) le ing λR¼0:001;λR¼0;and λe¼0, we ob ain he
cu e ha desc ibes he ela ionship be ween he po olio and in o -
ma ion cos in he home ma ke . (ii) Le ing λR¼0;λR¼0:002;and
λe¼0, o le ing λR¼0;λR¼0;and λe¼0:003;we ob ain a simila
cu e ha desc ibes he ela ionship be ween he po olio and in o -
ma ion cos in he home o exchange ma ke . (iii) Le ing λR¼0;λR
¼0;and λe¼0, we ob ain he cu e ha demons a es he po olio
wi hou any in o ma ion cos .
(2) In Theo em 1, we analyze he sensi i i y o in o ma ion cos
a es on op imal in es men s a egies.
Acco ding o Theo em 1,@p
@λR
<0;@p
@λR>0;@p
@λe
>0 and @p
@me
>0,
which means ha @p
@λR
<0, indica ing ha he manage should in es
less money in he o eign ma ke when he home ma ke ’s in o ma-
ion cos a e (λR) inc eases and o he pa ame e s a e fixed. @p
@λR>0
indica es ha mo e unds should be alloca ed o he o eign ma ke
when o he pa ame e s a e fixed, and as he in o ma ion cos a e in
he o eign ma ke ises, λRinc eases. I manage s do no wan o
spend much on in o ma ion cos in he o eign ma ke , hey choose
o in es in he home ma ke . The e o e, in o ma ion has a cen al
influence on o eign in es men and can explain he home bias in
in e na ional finance. @p
@λe
>0 means ha mo e in o ma ion cos in
he exchange ma ke indica es a highe p opo ion o he o al bud-
ge is alloca ed o o eign in es men . @p
@me
>0. This sugges s ha
when he exchange a e (me) inc eases and o he pa ame e s a e
fixed, he manage should alloca e mo e money o o eign ma ke
in es men .
I is c ucial o de e mine he ela ionship be ween he po o-
lio and he cos o a single piece o in o ma ion. We fi s se λR¼
0;λR¼0;λe¼0;me¼0 o e e y case. To no show he cu es
supe posed, we le ¼1;2;3;and 4. In Fig. 2, (i) le ing ¼1, we
ob ain he cu e ha demons a es he po olio changing wi h λR.
When λR¼0:009, he s a egy o achie ing he bes in he o eign
ma ke is p¼0:9735. In his case, he highes p opo ion o co po-
a e weal h is in es ed in he o eign ma ke . (ii) Le ing ¼2, we
ob ain he cu e ha demons a es he po olio changing wi h λe.
When λe¼0:006, he s a egy o achie e he bes in he o eign
ma ke is p¼0:9149. (iii) Le ing ¼3, we ob ain he cu e ha
demons a es he po olio changing wi h me.Whenme¼0:01, o
he CRRA case, he s a egy o achie ing he bes in he o eign
ma ke isp¼0:9344. (i ) Le ing ¼4, we ob ain he cu e ha
demons a es he po olio changing wi h λR.WhenλR¼0:001, o
he CRRA case, he s a egy o achie ing he bes in he o eign
ma ke is p¼0:729, indica ing ha 72.9 % o he co po a e weal h
is made a ailable o in es men in he domes ic ma ke .
To compa e wi h he esea ch esul s om Bellalah and Wu
(2009), his s udy uses he au ho s’in es men p opo ion in o -
eign ma ke s and ou own, espec i ely. When λR¼0:01, hei
esul is x= 0.9073, while ou esul is abou 0.8; when me¼0:01,
hei esul is x= 0.9508, while ou esul is abou 0.8; when
λe¼0:001, hei esul is x= 0.7769, while ou esul is abou 0.45;
and when λR¼0:0095, hei esul is x= 0.4943, while ou esul is
abou 0.25. Conside ing SHD, he esul s o his s udy a e gene ally
lowe han hose o Bellalah and Wu (2009), which indica es ha
SHD has a significan impac on in es men decision making, which
is consis en wi h he hypo hesis o ime-consis en p e e ences
being un ealis ic.
(3) F om Theo em 1, we nex examine he influence o λand Ton
he consump ion−weal h a io.
(i) The expec ed du a ion o he nea es in e al goes o
ze o as λ!1, and in he case o he s ochas ic
discoun unc ion, is changed o Eq. (15) as a de e minis ic
Fig. 1. Po olio changes o e ime.
J. Long, S. Zeng, B.B. Gup a e al. Jou nal o Inno a ion & Knowledge 9 (2024) 100459
6
jump unc ion ha jumps a .I T<1,byEq. (34), we ob ain
he ollowing:
a ¼e ðT Þ&h

þh
ðλ!1Þ;
which leads o
c
W
¼1
e ðT Þ&h

þh
:
(ii) Fo an infini e planning ho izon, i.e., Eq. (17),byEq. (34) and le -
ing T!1, we ha e a ¼λhþ
ðλþ Þ, and he consump ion−weal h
a io can be w i en as ollows:
c
W
¼ ðλþ Þ
λhþ :
(4) We analyze he sensi i i ies o pa ame e s λ;h, and ime on he
consump ion−weal h a io.
No e ha o any x>0;ex>1þx, and aking he pa ial de i a-
i es o a w. . pa ame e λ, we can ob ain he ollowing:
@a
@λ¼ 1h
ðλþ Þ2eðλþ ÞðT Þþ1h
λþ ðT Þeðλþ ÞðT Þ1h
ðλþ Þ2
¼1h
ðλþ Þ2eðλþ ÞðT Þ1þðλþ ÞðT Þeðλþ ÞðT Þ
hi
<0;
which means he ollowing:
@c =W
@λ¼@c =W
@a
@a
@λ¼ 1
a2
@a
@λ>0:
By aking he de i a i e o a in ela ion o he pa ame e h,we
ob ain he ollowing:
@a
@h¼1
eðT Þþ1
11
λþ

¼1
1eðT Þ
hi
þ1
λþ eðλþ ÞðT Þ1

:
Taking he pa ial de i a i es o he abo e equa ion in ela ion o
ime , we ob ain he ollowing:
Fig. 2. Consump ion−weal h a io c
/W
changes o e ime .
J. Long, S. Zeng, B.B. Gup a e al. Jou nal o Inno a ion & Knowledge 9 (2024) 100459
7
@
@
@a
@h

¼eðT Þ1eλðT Þ
hi
<0;
which shows ha @a
@his dec easing wi h espec o ime . Nex , we
will obse e whe he @a
@his nega i e by i s alue a =T. No ing ha
he pa ame e 2½0;1, we ob ain he ollowing:
@a
@h ¼T
¼1
λþ 11

<0:ð36Þ
Thus, we de e mine ha @a
@h<0, which means he ollowing:
@c =W
@λ¼@c =W
@a
@a
@λ¼ 1
a2
@a
@λ<0:
The e o e, he consump ion−weal h a io c
=W is dec easing
wi h espec oh.
Acco ding o Eq. (34), we ob ain he ollowing:
@a
@ ¼ &h

e ðT Þð1hÞeðλþ ÞðT Þ
¼1
1he ðT Þ &h
1heλðT Þ

:
I &h
1h1, hen @a
@ 0, which means @
@
c
W

¼@
@ 1
a
¼1
a2
@a
@ 0. I
&h
1h<eλT, hen @a
@ <0, which means @
@
c
W

>0. O he wise, i
eλT< &h
1h<1, hen when >Tþ1
λln &h
1h;@a
@ <0, o else @a
@ 0.
Fu he mo e, λ¼0o h¼1 leads o he ollowing:
a λ¼0
¼e ðT Þ&h

1h
e ðT Þþ1
¼1
1þð& 1Þe ðT Þ
hi
;
a h¼1
¼e ðT Þ&1

þλþ
ðλþ Þ
¼1
1þð& 1Þe ðT Þ
hi
;
ð37Þ
which means ha a jλ0o h¼1¼1
½1þð& 1Þe ðT Þ.
The e o e, he p oblem educes o Me on’s classical case yielding
he ollowing:
c
W
¼
1þð& 1Þe ðT Þ:
By Eq. (32) and o λ>0;0<h<1, we ob ain he ollowing:
a λ>0;0<h<1
;>a λ¼0
;a λ>0;0<h<1
>a h¼1
;
which indica es ha he consump ion−weal h a io in his s udy is
la ge han Me on’s.
In Fig. 2(i), le ing z1¼1 and h1¼0:3, we plo he cu es wi h
λ1¼0;0:5;1, and 10. The esul s show ha he consump ion−weal h
a io is inc easing wi h espec o pa ame e λ.
In Fig. 2(ii), le ing λ1¼1 and h1¼0:3, we plo he cu es o z1¼
1;2;3 and 4. The esul s show ha he influence o zon he consump-
ion−weal h a io is significan in he la e pe iod.
In Fig. 2(iii), le ing λ2¼1 and &1¼1, we plo he cu es wi h h1
¼0;0:05;0:1 and 1. The esul s show ha he influence o hon he
consump ion−weal h a io is significan in he ea ly s age.
(5) In Theo em 1,weob ain hefigu es demons a ing he op imal con-
sump ion−weal h a io ( c
W ) changes wi h he pa ame e s λand &.
In Fig. 3(i), using h ee di e en alues o &wi h 0.01, 0.1 and 0.8,
he esul s show ha he consump ion−weal h a io ( c
W ) is scaling up
ela i e o pa ame e λ.
In Fig. 3(ii), using h ee di e en alues o &wi h 1, 10 and 100,
he esul s show ha he consump ion−weal h a io ( c
W ) is also scal-
ing up ela i e o pa ame e λ.
In Fig. 4(i), aking h ee di e en alues o λwi h 0, 1 and 10, he
esul s show ha he consump ion−weal h a io ( c
W ) is educed in
ela ion o pa ame e &.
In Fig. 4(ii), aking h ee di e en alues o &wi h 1, 10 and 100,
he esul s show ha he consump ion−weal h a io ( c
W ) is also
educed in ela ion o pa ame e &.
Conclusion
This s udy builds upon he CIICP heo y and in oduces
he concep o SHD based on he p io esea ch conduc ed by Zou e
al. (2014) and Bellalah and Wu (2009). By aking in o accoun he
e ol ing p e e ences o sophis ica ed indi iduals, ou s udy enables
he o mula ion o consump ion and in es men s a egies in a con-
sis en manne o e ime. We employ a ecu si e app oach o de i e
explici solu ions o co po a e in e na ional in es men s a egy and
consump ion choices, conside ing log u ili y and encompassing bo h
infini e and fini e planning pe iods.
Fig. 3. Consump ion−weal h a io c
/W
changes wi h λ.
J. Long, S. Zeng, B.B. Gup a e al. Jou nal o Inno a ion & Knowledge 9 (2024) 100459
8