Galpe i, Simone
A icle
A heo y o pe sonal budge ing
Theo e ical Economics
P o ided in Coope a ion wi h:
The Econome ic Socie y
Sugges ed Ci a ion: Galpe i, Simone (2019) : A heo y o pe sonal budge ing, Theo e ical Economics,
ISSN 1555-7561, The Econome ic Socie y, New Ha en, CT, Vol. 14, Iss. 1, pp. 173-210,
h ps://doi.o g/10.3982/TE2881
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Theo e ical Economics 14 (2019), 173–210 1555-7561/20190173
A heo y o pe sonal budge ing
Simone Galpe i
Depa men o Economics, Uni e si y Cali o nia, San Diego
P ominen esea ch a gues ha consume s o en use pe sonal budge s o manage
sel -con ol p oblems. This pape analyzes he link be ween budge ing and sel -
con ol p oblems in consump ion–sa ing decisions. I shows ha he use o good-
speci ic budge s depends on he combina ion o a demand o commi men and
he demand o lexibili y esul ing om unce ain y abou in a empo al ade-
o s be ween goods. I explains he sub le mechanism ha ende s budge s use-
ul commi men s, hei in e ac ion wi h minimum-sa ings ules (ano he widely
s udied o m o commi men ), and how budge ing depends on he in ensi y o
sel -con ol p oblems. This heo y ma ches se e al empi ical indings on pe sonal
budge ing.
Keywo ds. Budge , minimum-sa ings ule, commi men , lexibili y, in a empo-
al ade-o , unce ain y, p esen bias.
JEL classi ica ion. D23, D82, D86, D91, E62, G31.
1. In oduc ion
Many s udies a gue ha pe sonal budge ing is a pe asi e pa o consume beha io .1
This p ac ice in ol es g ouping expenses in o ca ego ies and cons aining each wi h an
implici o explici cap applied o a speci ied ime pe iod (a week, a mon h, e c.).2While
his p ac ice canno be explained by he classic li e-cycle heo y o he consume , i has
impo an consequences. I can accoun o “mys e ious” la ge di e ences in weal h ac-
cumula ion be ween consume s ha ime o isk p e e ences canno explain (Ame iks
Simone Galpe i: [email p o ec ed]
I hank S. Nageeb Ali, Eddie Dekel, Alexande F ankel, Ma k Machina, Meg Meye , Alessand o Pa an, Ca lo
P a o, Ron Siegel, Joel Sobel, Ran Spiegle , Cha les Sp enge , B uno S ulo ici, Balazs Szen es, Alexis Aki a
Toda, Joel Wa son, he co-edi o , and wo anonymous e e ees o use ul commen s and sugges ions. I also
hank semina pa icipan s a Bos on College, Columbia Uni e si y, London School o Economics, Ox o d
Uni e si y, Penn S a e Uni e si y, Uni e si y o Pennsyl ania, Uni e si y College London, Unu e si y o
Cali o nia–Ri e side, Uni e si y o Wa wick, and Tel A i Uni e si y. This pape supe sedes a p e ious
pape i led “Delega ing Resou ce Alloca ions in a Mul idimensional Wo ld.” John N. Rehbeck p o ided
excellen esea ch assis ance. All emaining e o s a e mine.
1See Bakke (1940), Lee e al. (1962), Thale and She in (1981), Thale (1985,1999), Hende son and Pe-
e son (1992), Baumeis e e al. (1994), Hea h and Soll (1996), Zelize (1997), We enb och (2002), Ame iks
e al. (2003), Bénabou and Ti ole (2004), An onides e al. (2011), and Beshea s e al. (2016).
2This pape uses he e m “pe sonal budge ing” a he han “men al accoun ing” because he la e has
he much b oade meaning o a gene al p ocess whe eby people ame e en s, ou comes, and decisions.
This also includes choice b acke ing, na ow aming, and gain–loss u ili y, which di e om budge ing.
©2019 The Au ho . Licensed unde he C ea i e Commons A ibu ion-NonComme cial License 4.0.
A ailable a h p://econ heo y.o g.h ps://doi.o g/10.3982/TE2881
174 Simone Galpe i Theo e ical Economics 14 (2019)
e al. 2003). By iola ing he p inciple o ungibili y o money, i shapes demand di e -
en ly om sa ia ion and income e ec s (Hea h and Soll 1996). I a ec s how i ms p o-
mo e hei p oduc s so as o a oid compe ing o he same budge (We enb och 2002).
I is a he ounda ion o he economics o commi men de ices (B yan e al. 2010). Al-
mos all exis ing s udies in o mally sugges ha consume s use budge s o manage sel -
con ol p oblems, o en caused by p esen bias, which in e e e wi h hei sa ing goals
(Thale 1999,Ame iks e al. 2003,An onides e al. 2011).
Despi e his consensus, a o mal in es iga ion o he link be ween budge ing and
sel -con ol p oblems seems o be missing. The pape o e s such a ounda ion using a
b oadly s udied aspec o ime p e e ences: p esen bias. I shows, howe e , ha p esen
bias alone canno explain budge ing. P esen -biased consume s alue cons ain s on
u u e choices. Bu o budge s o eme ge, his p e e ence o commi men has o be
combined wi h a p e e ence o lexibili y o a p ecise bu plausible kind, namely ha
caused by unce ain y abou in a empo al ade-o s due, o ins ance, o good-speci ic
as e shocks. The pape also unco e s a ension be ween good-speci ic budge s and
minimum-sa ings ules, an o en s udied o m o commi men . This leads o a nega i e
ela ionship be ween he le el o p esen bias and he use o budge s. These p edic ions
help o ganize he e idence on budge ing and can guide u u e empi ical s udies.
Conside an agen , Ann, who has wo sel es: a ime-consis en sel -0and a p esen -
biased sel -1.3Bo h sel es ha e he same pe -pe iod consump ion u ili y. In each pe-
iod, sel -1chooses consump ion and sa ings subjec o he usual income cons ain .
Suppose ha (i) consump ion in ol es mul iple goods (no a single uni o m commod-
i y) and (ii) bo h sel es’ p e e ences depend on a s a e o he wo ld (cap u ing as e
shocks) ha a ec s no only he a e o subs i u ion be ween p esen and u u e u ili y,
bu also he a es o subs i u ion be ween goods wi hin pe iods. In each pe iod, be o e
he s a e ealizes, Ann’s sel -0can adop a commi men plan ha dic a es which income
alloca ions sel -1is allowed o choose. This c ea es a ade-o be ween commi men
and lexibili y. The pape ocusses on plans ha can eely combine good-speci ic bud-
ge s and an o e all limi on consump ion expenses ia a sa ings loo . This is in line
wi h i s mo i a ion and o e s an in e es ing lowe bound on sel -0’s payo . O cou se,
one would wan o allow o gene al o ms o commi men , which is, howe e , much
ha de in he p esence o he o egoing ea u es (i) and (ii) (see Sec ion 4), and is beyond
he scope o his pape .4
Fea u es (i) and (ii) a e he key di e ences be ween his pape and Amado e al.
(2006), whe e consump ion in ol es a single commodi y and as e shocks a ec only
he in e empo al u ili y ade-o . Tha pape shows ha , unde e y gene al condi-
ions on he shock dis ibu ion, he op imal ule is o impose a sa ings loo and g an
sel -1 lexibili y o he wise, e en i sel -0can choose among a bi a ily gene al o ms o
3Dual-sel models appea in Thale and She in (1981), Bénabou and Pycia (2002), Be nheim and Rangel
(2004), Benhabib and Bisin (2005), Fudenbe g and Le ine (2006,2012), Loewens ein and O’Donoghue
(2007), B ocas and Ca illo (2008), Cha e jee and K ishna (2009), and Nageeb (2011).
4This pape akes he p ocess o no icing an expense and epo ing i o i s budge as a de ining aspec o
budge ing i sel . To ocus on he issues o in e es he e, i also assumes ha people s ick o hei commi -
men plans, as jus i ied in Sec ion 2.
Theo e ical Economics 14 (2019) A heo y o pe sonal budge ing 175
commi men . To es ablish a benchma k, Sec ion 3.2 shows ha hei esul ca ies o e
o a wo ld wi h mul iple goods i he e is no unce ain y abou in a empo al ade-
o s be ween goods. In ui i ely, in his case binding good-speci ic budge s o ces sel -
1 o choose ine icien consump ion bundles, which is akin o was ing esou ces (i.e.,
“money bu ning”). Amado e al. (2006) al eady showed ha money bu ning is gene ally
subop imal.
Unce ain in a empo al ade-o s change hings subs an ially, as summa ized by
he main esul s o he pape . Fi s , i he goods sa is y app op ia e subs i u abili y and
no mali y condi ions, op imal commi men plans always in ol e good-speci ic budge s
when p esen bias is su icien ly weak, bu only a sa ings loo when p esen bias is su -
icien ly s ong. Second, ixing a weak bias, o some ange o pa ame e s he op imal
plans combine budge s wi h a sa ings loo , bu o ano he ange hey ely only on he
budge s. By con as , in Amado e al. (2006) op imal plans always in ol e a sa ings
loo .5The subs i u abili y and no mali y condi ions ensu e ha he consump ion dis-
o ions caused by budge s cu ail how much sel -1gains in e ms o p esen u ili y by
unde sa ing, he eby esul ing in highe sa ings. This imp o emen ma e s mo e han
hose dis o ions o he ime-consis en sel -0.
To see he in ui ion o hese esul s, suppose Ann consumes wo goods and is un-
ce ain whe he he ma ginal u ili y o each good will be high o low. An icipa ing he
endency o unde sa e, she i s conside s se ing a sa ings loo . This limi s o e spend-
ing i bo h ma ginal u ili ies a e high, which makes he wan o consume a lo o bo h
goods. I only one ma ginal u ili y u ns ou o be high, howe e , he loo may no bind;
his is especially likely i p esen bias is weak. In his case, Ann ealizes ha she will s ill
o e spend and his will be mos ly d i en by he good wi h high ma ginal u ili y. She can
hen also cap his good wi h a a ge ed budge , which aises he sa ings because now
she can o e spend only on he good wi h low ma ginal u ili y. By con as , when p esen
bias is s onge , o e spending becomes mo e se e e e en o he good wi h low ma ginal
u ili y, and he budge leads o a small (i any) ise in sa ings a he cos o a ioning a
good wi h high ma ginal u ili y. As a esul , Ann p e e s o adop only a sa ings loo ,
because i cu bs unde sa ing wi hou dis o ing consump ion.
The esul s in ol e some no ewo hy sub le ies. An agen may adop budge s ha
dis o consump ion spending e en hough he sel es always ag ee on how o di ide
e e y dolla be ween goods wi hin a pe iod. By con as , a binding loo dis o s only
he income di ision be ween spending and sa ing. Pe haps coun e in ui i ely, i is no
he case ha i a p esen -biased agen adds budge s o a loo , hen a mo e biased agen
should do he same. In addi ion, agen s who use budge s may also se igh e loo s, as
he budge s’ dis o ions lowe he alue o lea ing mo e income o consump ion. Once
budge s a e allowed, agen s wi h a s onge p esen bias may adop a slacke loo (in
con as o P oposi ion 5 in Amado e al. 2006). Sec ion 3 u he discusses he esul s
ela i e o he e idence on budge ing, highligh ing indings ha o he heo ies s uggle
o explain.
This pape expands ou unde s anding o consump ion–sa ings beha io unde
sel -con ol p oblems and he esul ing demand o commi men . Since Thale and
5These p ope ies con inues o hold o pa ially nai e agen s who inco ec ly an icipa e hei bias.
176 Simone Galpe i Theo e ical Economics 14 (2019)
She in’s (1981)andLaibson’s (1997) seminal wo k, he li e a u e has almos always
assumed a single, pe -pe iod commodi y (“money”).6As his pape shows, ha as-
sump ion is no innocuous wi h p esen -biased consume s (in con as o he case
o ime-consis en consume s) and his c ucially depends on unce ain in a empo al
ade-o s. The li e a u e has ocussed on he p oblem o cu bing unde sa ing and he
use ulness o de ices like illiquid asse s and sa ings accoun s. This pape shows ha
consume s can do s ic ly be e by (also) adop ing good-speci ic budge s, which opens
he doo o o he commi men de ices, such as pe sonal budge ing se ices.7I also sug-
ges s which ype o consume s will demand which ype o de ices, which can be used by
hi d-pa y p o ide s.8
To de i e he esul s, he pape uses echniques di e en om he s anda d mecha-
nism-design app oach. The idea is o exploi he in o ma ion in he Lag ange mul iplie s
o he cons ain s ha budge s and sa ings loo s add o sel -1’s op imiza ion p oblem.
Relying on sensi i i y-analysis echniques (Luenbe ge 1969), we can use his in o ma-
ion o quan i y, a e app op ia ely adjus ing o sel -1’s bias, he ma ginal bene i o
sel -0o modi ying a budge o a loo .
Rela ed li e a u e
Exis ing explana ions o pe sonal budge ing a e based on Thale (1985). Using he no-
ions o “ ansac ion u ili y” and gain–loss u ili y, he a gues ha agen s ea he conse-
quences o each ansac ion in isola ion. Gi en his, hey can sol e hei consump ion–
sa ings p oblems by means o ansac ion-speci ic budge s, a esul ha echoes S o z
(1957). In eali y, people se budge s o su icien ly long pe iods so ha each co e s
many ansac ions. Also, in Thale ’s de e minis ic model, he agen s can achie e he
same u ili y wi h and wi hou budge s, bu wi h unce ain y, hey would ne e se bind-
ing budge s. The e o e, hey do no exhibi a s ic demand o budge s as commi men
de ices. Finally, ansac ion and gain–loss u ili y di e concep ually om sel -con ol
p oblems, which he li e a u e iews as he main cause o budge ing. Gain–loss u ili y
can explain o he phenomena o men al accoun ing, such as choice b acke ing (Koch
and Na zige 2016), which, howe e , di e s om budge ing.
O he pape s in he mechanism-design li e a u e s udy he ade-o s be ween com-
mi men and lexibili y, usually imposing no es ic ion on he easible mechanisms. Pa-
pe s by Amado e al. (2006)andHalac and Ya ed (2014) a e he closes o he p esen
pape .9We bo ow hei baseline model, bu adds mul iple consump ion goods and
unce ain y abou in a empo al ade-o s. In so doing, we show how his unce ain y
a ec s he commi men – lexibili y ade-o and i s solu ions. Ano he di e ence is ha
6B ocas and Ca illo (2008) discuss a model wi h wo goods, one o which has ex an e unce ain u ili y,
and sel -1is ully myopic. In his case, he op imal commi men s a egy consis s o a nonlinea plan ha
punishes spending on one good by cu ing spending on he o he , which is no a budge ing plan. E en i
one ocusses on hese plans, sel -0ne e se s budge s wi h a ully myopic sel -1(see P oposi ion 3).
7This kind o se ice is cu en ly o e ed by i ms like Min , Quicken, and S ickK.
8In eali y, i may be ha d o obse e each consume ’s deg ee o p esen bias and o e de ices acco d-
ingly. Some o he issues ha a ise in his case a e analyzed by Galpe i (2015).
9See also A hey e al. (2005), Amb us and Ego o (2013), and Amado and Bagwell (2013).

Theo e ical Economics 14 (2019) A heo y o pe sonal budge ing 177
Halac and Ya ed (2014) ocus on he ole o in o ma ion pe sis ency. In hei se ing, an
op imal commi men plan can dis o u u e choices, e en hough hey cause no con-
lic be ween he agen ’s sel es gi en oday’s choice. Pe sis ence links sel -1’s cu en
in o ma ion and expec ed u ili y om u u e choices, which can be used o elax oday’s
incen i e cons ain s, as in o he dynamic mechanism-design p oblems.10 Co ela ion
among sel -1’s pieces o in o ma ion is no he d i e o he p esen pape ’s esul s.
An olde li e a u e examined how a ioning a ec s consume beha io (Howa d
1977,Ellis and Naugh on 1990,Madden 1991). By se ing a sa ings loo o good-speci ic
budge s, an agen essen ially a ions his u u e sel es jus as he go e nmen may a ion
consume s. In con as o ha li e a u e, he e a ioning assumes he ole o a commi -
men de ice. Tha li e a u e shows ha p edic ing he budge s’ e ec s is a om i ial.
I s insigh s a e use ul o iden i y condi ions unde which budge s can help he agen .
2. The model
Conside an agen , Ann, who li es o wo pe iods. In he i s , she chooses a consump-
ion bundle c=(c1c2)∈R2
+and a le el o sa ings s∈R+. In he second pe iod, con-
sump ion in ol es a single good and, hence, equals s. Ann ecei es he income, no mal-
ized o 1, in he i s pe iod.
Ann has sel -con ol p oblems caused by a con lic be ween a long- un sel -0and
a sho - un sel -1. Thei p e e ences depend on some as e shocks, ep esen ed by he
s a e (θ 1 2),whe eθ>0and =( 1 2)∈R2. In each pe iod bo h sel es ha e he
same (conca e) consump ion u ili y: u(c; )in pe iod 1 and (s) in pe iod 2. In pe iod 1,
howe e , sel -0and sel -1e alua e s eams (cs)using, espec i ely, he u ili y unc ions
θu(c; )+ (s) and θu(c; )+β (s)
Fo cla i y and ac abili y, o now assume ha
u(c; )=u1(c1; 1)+u2(c2; 2)wi h ∂2ui(ci; i)
∂ci∂ i
=ui
c (ci; i)>0 o i=12
Sel -1’s p esen bias is cap u ed by β∈(01).Sel -0knows β(sophis ica ion); we discuss
nai e é la e .
A key no el y o his model is ha he s a e a ec s bo h in e -andin a empo al
ade-o s. While θa ec s only he subs i u ion a e be ween p esen and u u e u ili y,
also a ec s he subs i u ion a es be ween goods wi hin pe iod 1. He ea e , le ω=
(θ ),le Gbe i s dis ibu ion, and le be he s a e space. Dis ibu ion Gis allowed o
ha e ich o ms o dependence as well as ull independence ac oss θ, 1,and 2.
Sel -0delega es he consump ion–sa ings choice o sel -1by designing a commi -
men plan ha dic a es which choices sel -1is allowed o implemen . In he case o
budge ing, such a plan in ol es spending limi s on speci ic consump ion ca ego ies, de-
no ed by bi( o budge ), o an o e all limi on consump ion expendi u es implemen ed
10See, o example, Cou y and Hao (2000), Ba aglini (2005), and Pa ane al.(2014).
178 Simone Galpe i Theo e ical Economics 14 (2019)
h ough a minimum-sa ings ule ( o loo ). Fo mally, le
F=(cs)∈R3
+:c1+c2+s≤1
Think o ciand sas he sha e o income alloca ed o good iand sa ings. A budge ing
plan, B, can hen be exp essed as
B=(cs)∈F:s≥  c1≤b1c2≤b2
whe e ∈[01]and bi∈[01] o i=12.Le Bbe he se o all budge ing plans. F om he
ex an e iewpoin , we call and bibinding i hey bind wi h s ic ly posi i e p obabili y
unde G.No e ha
Bde ines a speci ic subclass o commi men plans ha — hough
in ui i e and ac able— ule ou many o he possible ways o es ic sel -1’s choices.
Sec ion 4 discusses some in icacies o allowing o mo e gene al plans.
In eali y, agen s commi o hei plans p io o obse ing all he necessa y in o ma-
ion o making a decision. This c ea es a ade-o be ween commi men and lexibil-
i y. In he model, i s sel -0commi s o a plan B, and hen only sel -1obse es ωand
chooses some (cs) om B.Sel -0designs B o maximize he expec ed payo om sel -
1’s choices. No e ha i sel -0knew ω, he p oblem would be unin e es ing: Se ing
a he le el o sa ings ha sel -0 inds op imal gi en ωalways induces sel -1 o choose c
and s ha maximize sel -0’s u ili y.
The goal o he pape is o unde s and whe he and how sel -0se s minimum-
sa ings ules and goods-speci ic budge s. The p oblem can be s a ed as
max
B∈BU(B) =θuc(ω); + s(ω)dG(ω) (1)
s. . c(ω)s(ω)∈a gmax
(cs)∈B
θu(c; )+β (s) ω ∈ (2)
A solu ion is called an op imal plan.
Technical assump ions
In o ma ion dis ibu ions Le =[θθ]×[ 1 1]×[ 2 2],whe e0<θ<θ<+∞ and
0<
i< i<+∞ o i=12. We only assume ha he join p obabili y dis ibu ion G
o (θ 1 2)has ull suppo ( ha is, G(O) > 0 o e e y open O⊂). The condi ions
on θand θ ule ou he implausible si ua ion whe e Ann does no ca e a all abou he
p esen o he u u e.11 The condi ions on iand iha e bi e only when combined wi h
he p ope ies o ulis ed nex .
Di e en iabili y, mono onici y, and conca i y The e m is wice con inuously di e -
en iable wi h >0and  <0.Fo i=12and i∈[ i i],ui(·; i):R+→Ris wice
di e en iable wi h ui
c(·; i)>0and ui
cc(·; i)<0;also,ui
cand ui
cc a e con inuous on
(01]×[ i i]. This implies ha ui
cis bounded below and away om ze o; his non-
sa ia ion p ope y seems plausible o he ex en ha i e e s o ood, housing, o en e -
ainmen and a pe iod co esponds o a week o a mon h.
11A simila assump ion appea s in Amado e al. (2006), who poin ou ha wi h unbounded suppo i
may be op imal o g an sel -1 ull lexibili y.
Theo e ical Economics 14 (2019) A heo y o pe sonal budge ing 179
Bounda y condi ions We ha e lims→0 (s) =+∞and limc→0ui
c(c; i)=+∞ o i∈
[ i i]and i=12. This allows us o ocus on in e io solu ions.
Discussion o he model
No hing signi ican changes i Ann ecei es income in bo h pe iods and can bo ow in
pe iod 1 o i he consump ion bundle cin ol es mo e han wo goods. In ac , he
p oo s conside he gene al case o n≥2goods. I is s aigh o wa d o allow o mul iple
goods in he second pe iod also.
The wo sel es’ p e e ences a e consis en wi h he quasi-hype bolic discoun ing
model o Laibson (1997) and wi h iewing he agen as a household agg ega ing i s
membe s’ p e e ences, which a e ime consis en bu he e ogeneous (Jackson and Ya i
2015). A public inance in e p e a ion o he model is also possible along he lines o Ha-
lac and Ya ed (2014). In each pe iod, a go e nmen chooses spending on a lis o public
goods and se ices, c, and sa ing o bo owing, s, subjec o he cons ain gi en by he
ax e enues. The go e nmen may exhibi p esen bias as a consequence o agg ega ing
he p e e ences o he e ogeneous ci izens (Jackson and Ya i 2015) o unce ain y in he
poli ical u no e (Aguia and Amado 2011).
The assumed in o ma ion s uc u e has some edundancy, as bo h an inc ease in
θand an inc ease in all componen s o ende pe iod-1 consump ion mo e aluable.
None heless, i is con enien o di e en ia ing unce ain y abou in a- and in e em-
po al ade-o s and o showing ha he o me is c ucial o budge s o a ise (Sec-
ion 3.2). In a nu shell, his is because i allows o si ua ions whe e o e spending is
d i en by all goods and si ua ions whe e i is mos ly d i en by only some good.
To ocus on he issues o in e es o his pape , i is assumed ha Ann s icks o
he plans. This is no a mino assump ion, o cou se, bu he li e a u e has p oposed
se e al mechanisms ha can jus i y i . These include a desi e o in e nal consis ency
(Fes inge 1962), he plans’ wo king as e e ence poin s (Hea h e al. 1999,Hsiaw 2013),
sel - epu a ion mechanisms (Bénabou and Ti ole 2004), in e nal con ol p ocesses ha
p e en impulsi e p ocesses om b eaking ex an e ules (Benhabib and Bisin 2005), and
sel -en o cemen sus ained by h ea s o swi ching o less desi able equilib ia (Be nheim
e al. 2015). Pe haps in eali y people a e able o ca y ou hei plans p o ided ha hey
a e no oo s ingen o cos ly ex pos . E en in his case, i is wo h unde s anding which
o ces lead people o ind budge s and loo s use ul despi e hei ex pos ine iciency.
Fo ins ance, some p esen -biased agen s may no use budge s no because hey canno
s ick o hem, bu simply because hey do no ind hem use ul. This can also be alu-
able o hi d pa ies ha design commi men de ices o help people s ick o hei plans
(such as i ms like Min , Quicken, and S ick).
3. Op imal budge ing plans
3.1 P elimina ies
Fi s o all, ea ing sel -0’s payo as a unc ion o ,b1,andb2, one can easily es ablish
exis ence o an op imal plan using he maximum heo em.12
12See Lemma 2 in he Appendix.
180 Simone Galpe i Theo e ical Economics 14 (2019)
I is wo h de ining wo benchma k alloca ions. Fo each ω,le (cd(ω) sd(ω)) be sel -
1’s choice i g an ed ull disc e ion, namely, he solu ion o max(cs)∈F{θu(c; )+β (s)}.
Also, le (cp(ω)sp(ω)) ep esen wha sel -0would like sel -1 o choose in ω,which
is he solu ion o max(cs)∈F{θu(c; )+ (s)}.Call(cdsd) he ull-disc e ion alloca ion
and call (cpsp) he i s -bes (o planned) alloca ion. They sa is y he ollowing use ul
p ope ies.
Rema k 1. (i) The mechanisms (cpsp)and (cdsd)a e con inuous in ω.
(ii) Each componen o (cpsp)and (cdsd) akes alues in a closed in e al and is
bounded away om ze o.
(iii) Fo i=12,cp
iand cd
ia e s ic ly inc easing in iand θ, and dec easing in j o
j= i.
(i ) Sa ings spand sda e s ic ly dec easing in θ, 1,and 2.
( ) Fo ω∈,sd(ω) < sp(ω) and sd(ω) is con inuous and s ic ly inc easing in β.
( i) Fo ω∈and i=12,cd
i(ω) is con inuous and s ic ly dec easing in β.
Ano he p ope y wo h no ing is ha all consump ion goods a e no mal o bo h
sel es.13
Fo illus a ion, conside a ully symme ic model wi h espec o goods 1and 2.
Since sel -1sa es wha e e he does no consume ( >0), we can ocus on his choices
o c ep esen ed in Figu e 1. No e ha cs on nega i e 45° lines close o he o igin co -
espond o a highe s. To unde s and he shape o cp, suppose o he momen ha θ
akes only one alue. S a om (θ 1 2), which leads o he highes s.I we aise 1
up o 1,cp
1inc eases while cp
2and spdec ease, which means ha we mo e along he
sou h pa o he dashed line. I we now s a om (θ 1 2)and aise 2up o 2,cp
2in-
c eases while cp
1and spdec ease; ha is, we mo e along he eas pa o he dashed line.
P oceeding in his way, we can map he en i e dashed line; con inui y o cpimplies ha
Figu e 1. Fi s -bes and ull-disc e ion alloca ions.
13This p ope y ollows, o ins ance, om P oposi ion 1 in Quah (2007).
Theo e ical Economics 14 (2019) A heo y o pe sonal budge ing 187
A simple change o his h ee-s a e se ing su ices o explain why op imal plans can
in ol e only good-speci ic budge s. Fix g>g
∗and all he o he pa ame e s excep θ.
I we inc ease θ, bo h sel es wan o consume mo e in ω0. This e en ually leads o a
si ua ion as in Figu e 3(b), whe e cp
1(ω0)>c
p
1(ω1)and cp
2(ω0)>c
p
2(ω2).In hiscase,
Ann wan s o keep b1and b2,bu d op . In ac , he op imal Bsa is ies b1=b2and
cp
i(ωi)<b
i<c
p
i(ω0) o e e y i=12,bu =0.20
Figu e 3(b) helps wi h he in ui ion.
Now Ann is willing o spend e en mo e in ω0. The e o e, he budge s she would se o
cu b splu ging in ω1and ω2s a o bind also in ω0. As a esul , she wan s o elax hem.
She ealizes, howe e , ha he i s -bes spending on clo hes and ood in ω0is jus abo e
hose budge s. Relaxing hem a bi will allow he o cu b splu ging in ω1and ω2as well
as ω0. Since hese budge s al eady push sa ings abo e he i s bes in ω0, Ann canno
bene i by adding a binding .
In sho , a weakly p esen -biased agen may use only good-speci ic budge s o he
ollowing eason. To cu b unde sa ing in s a es wi h la ge asymme y in consump ion
ma ginal u ili ies, she may p e e o use he budge s a he han a sa ings loo , which
would ha e o be oo s ingen . Toge he he budge s hen impose a cap on o al spend-
ing. I his al eady ensu es su icien ly high sa ings in s a es whe e p esen consump ion
is e y aluable o e all, hen any binding loo will ha e o cause addi ional o e sa ing
and his ine iciency can exceed he loo ’s commi men bene i s.
4. Discussion
Theo y and e idence
How does his heo y ela e o he e idence on budge ing and o he explana ions
he eo ? One inding is ha people may se budge s on “unobjec ionable goods like
spo s icke s and blue jeans” (Hea h and Soll 1996) o housing, ood, and e en cha i a-
ble gi ing (Thale 1985,1999). This is di icul o explain wi h an al e na i e heo y a -
guing ha people se budge s o he goods hey ind emp ing (“ ice goods”), al hough
his can be ue in some cases. By con as , p esen bias combined wi h unce ain in-
a empo al ade-o s can lead people o se budge s on “unobjec ionable goods,” as
doing so helps hem manage hei o e all endency o o e spend be e han wi h jus a
sa ings loo . Thus, i is no he emp ing na u e o a good ha ma e s.
Ano he plausible heo y is ha budge ing is a echnique o simpli y he complex
ma e o household inance (Simon 1997,Johnson 1984). This heo y is complemen a y
o he one in his pape , bu again s uggles o explain some e idence. Fo ins ance,
i is no clea why compu a ional complexi y would lead people o sys ema ically se
budge s ha seem oo s ic and o cause unde consump ion, as ound by Hea h and Soll
(1996). By con as , p esen -biased people do op imally se budge s ha sys ema ically
bind and hus exhibi hose p ope ies.
20Again, his claim is shown as pa o he cons uc i e p oo o P oposi ion 4. No e ha , al hough his
h ee-s a e example is in ui i e, i akes some wo k o ule ou he possibili y o mul iple, pe haps asym-
me ic, op imal plans ea u ing di e en p ope ies om hose in he p oposi ion.

188 Simone Galpe i Theo e ical Economics 14 (2019)
The p edic ion ha only weakly biased agen s should use good-speci ic budge s is
consis en wi h some indings in An onides e al. (2011). In hei sample, people who ex-
hibi a “sho - e m ime o ien a ion” (which acco ding o hei desc ip ion is consis en
wi h s ong p esen bias) a e less likely o use budge s han people who exhibi a “long-
e m ime o ien a ion” (a weak bias). Un o una ely, An onides e al. (2011) do no mea-
su e how s ingen budge s o loo s a e in ela ion o p esen bias. Fo ha ma e , we
saw ha his ela ion need no be mono onic. As an al e na i e explana ion, s ongly bi-
ased agen s may no use budge s because hey a e less sophis ica ed o able o commi .
The an icipa ed bias (no he ue one) is wha ma e s o sel -0’s p oblem, howe e .
The e o e, by P oposi ion 2 unde es ima ing ha bias—no en i ely, o cou se—may ac-
ually ende i mo e likely ha sel -0 inds budge s bene icial. I his same agen we e
ins ead sophis ica ed, he migh no adop any budge by P oposi ion 3. We also saw ha
once a s ongly biased agen can use a sa ings loo , he eason why budge s do no wo k
o him is no ha he canno hono hem: E en i he could, hey would s ic ly lowe his
u ili y.
Relaxing sepa abili y
The message o he pape gene alizes o se ings whe e u ili y is no sepa able ac oss
goods. Con inue o assume ha u(c; )is s ic ly conca e in cand wice di e en iable
wi h con inuous uci(c; )>0and ucicj(c; )in bo h a gumen s o all iand j.Wesaw
ha budge s help cu b sel -1’s unde sa ing i (a) hey inc ease sa ings and (b) he e ex-
is s a es ha call o high consump ion o some good, bu no o all goods. P ope y (b)
holds i some good is a su icien ly s ong subs i u e o all o he goods. P ope y (a) holds
i he capped good is a Hicks subs i u e o sa ings (Howa d 1977); in gene al, such a good
always exis s (Madden 1991, Theo em 2). As no ed, howe e , a budge has o cu b un-
de sa ing as e han i exace ba es o e spending on o he goods o i o bene i sel -0.
Gi en space cons ain s, hese p ope ies a e s a ed di ec ly in e ms o alloca ions.
Condi ion 1. Bo h (cpsp)and (cdsd)a e in e io o e e y ω.Bo hspand sda e
s ic ly dec easing in θand i o i=12. The e exis s some good j ha sa is ies he
ollowing s a emen s: (i) cp
jand cd
ja e s ic ly inc easing in θand j, and dec easing in
i o i= j; (ii) he e exis s ε>0such ha , o e e y bj<maxωcd
j(ω),sel -1’s op imal
(c∗s∗)subjec o plans in ol ing only bjsa is ies s∗(ω) −sd(ω) ≥ε[cd
j(ω) −c∗
j(ω)] o
all ω∈.
Appendix A.6 p esen s an example ha sa is ies Condi ion 1.
To s a e he esul , conside a mo e gene al class o budge ing plans, deno ed by B,
which allow us o also se good-speci ic loo s and a sa ings cap,
B=(cs)∈F: 0≤s≤b0 1≤c1≤b1 2≤c2≤b2
whe e ibi∈[01]sa is y i≤bi o i=012and 0+ 1+ 2≤1.
P oposi ion 5. Unde Condi ion 1, he eexis sβ∗∈(01)such ha , i β∗<β<1.Then
e e y op imal B∈Bmus use dis o ing good-speci ic es ic ions.
Theo e ical Economics 14 (2019) A heo y o pe sonal budge ing 189
The p oo is omi ed, because using Condi ion 1, one can adap he p oo o P opo-
si ion 2 o show ha plans using only 0a e s ic ly domina ed o su icien ly high β.
Since se ing a binding b0is ne e op imal, he esul ollows.
Do op imal good-speci ic es ic ions always ake he o m o budge s? The answe
depends on he subs i u abili y and complemen a i y be ween goods and be ween each
good and sa ings, which can be a ec ed by he es ic ions hemsel es. A su icien con-
di ion o op imal plans o ne e use 1and 2is ha all goods a e Hicks subs i u es and
collec i ely su icien ly no mal (see Ellis and Naugh on 1990 o a o mal s a emen o
his p ope y). Gi en his, by Theo ems 3 and 4 o Madden (1991) wo goods emain
subs i u es independen ly o which goods a e es ic ed, and Ellis and Naugh on’s (1990)
analysis implies ha , gi en any 1and 2, elaxing hem aises s. Hence, since 1and 2
dis o consump ion, hey s ic ly ha m sel -0. Op imal plans use only b1and b2 o he
example in Appendix A.6.
Gene al mechanisms
One may wonde wha he bes among all concei able plans (no jus hose in B) looks
like and whe he i belongs o B. These a e impo an ques ions, bu also ha d in
he p esence o mul idimensional consump ion and unce ain y. The main challenges
come om he income cons ain and he complexi y o he incen i e cons ain s, which
as usual canno be educed o only he local ones. He e one can y o apply he insigh s
om mul idimensional sc eening (Roche and S ole 2003), bu subs an i e di e ences
emain. Fi s , sc eening p oblems allow o ans e s. He e one can iew he u ili y om
sa ings as a ans e and use Roche and Choné’s (1998) app oach o simpli y he incen-
i e cons ain s and sel -0’s objec i e. Bu he s a e-wise income cons ain ( he second
di e ence om sc eening) canno be simpli ied. Gene al echniques exis o handling
such cons ain s (Luenbe ge 1969), bu unlike in he case o unidimensional consump-
ion, he e hey do no go a .
5. Concluding ema ks
This pape p o ides a heo e ical analysis o he link be ween sel -con ol p oblems and
pe sonal budge ing using a pa simonious consump ion–sa ings model wi h a p esen -
biased agen . Unlike minimum-sa ings ules, good-speci ic spending caps help o
cu ail o e spending because hey cause ine iciencies in consump ion ha lowe he
e u n om unde sa ing, he eby coun e ac ing p esen bias. Consequen ly, good-
speci ic budge s a e no ee lunch and a e used only by agen s who a e weakly biased and
unce ain abou hei in a empo al ade-o s be ween goods. Those who a e s ongly
biased o do no ace such unce ain y p e e o ely exclusi ely on a minimum-sa ings
ule.
This heo y o e s insigh s in o he sub le o ces unde lying a widely obse ed phe-
nomenon, which has a - eaching consequences o consume beha io and wel a e by
a ec ing demand di e en ly om sa ia ion and income e ec s, and by signi ican ly con-
ibu ing o households’ weal h accumula ion. The heo y ma ches exis ing empi ical
190 Simone Galpe i Theo e ical Economics 14 (2019)
indings, such as ha people o en se budge s o goods no mally no iewed as emp -
ing and only hose who exhibi weak p esen bias seem o use budge s. The heo y also
sugges s new di ec ions o en iching he spa se e idence on budge ing by demons a -
ing i s dependence on unce ain in a empo al ade-o s, and o designing commi -
men de ices whose unc ions a e a ge ed o he igh ype o p esen -biased agen s.
Appendix
A.1 Technical lemmas
Lemma 2. The e exis s B ha maximizes U(B) o e B.
P oo .EachB∈Bcan be iewed as an elemen ( b)o he compac se [01]n+1.Thus,
we can hink ha sel -0chooses ( b)∈[01]n+1.
Gi en any such ( b),le (c(ω|  b)s(ω|  b)) be sel -1’s op imal alloca ion in s a e
ω om he compac se B bde ined by ( b). Since B bis con ex (Theo em 2.1 in
Rocka ella 1997), (c(ω|  b)s(ω|  b)) is unique o e e y ω∈by s ic conca i y o
sel -1’s u ili y unc ion. Clea ly, he co espondence ha o each ( b)∈[01]n+1maps
o B bis nonemp y, compac alued, and con inuous. I ollows om he maximum
heo em ha (c(ω|··) s(ω|··)) is con inuous o e e y ω∈.
We can now show ha sel -0’s payo is con inuous in ( b). Fo each ( b)∈
[01]n+1,le
U( b)=θuc(ω|  b); + (s(ω|  b)dG(ω)
Since he in eg and is con inuous in ( b) o e e y ω∈and is uni o mly bounded
o e B( b), Lebesgue’s domina ed con e gence heo em implies he claimed p ope y
o U(··).
A second applica ion o he maximum heo em gi es he esul .
Lemma 3. Fix i∈{1n}and conside B∈Bwi h bj=1 o all j= i. Fo any ω,i
bi<c
d
i(ω), sel -1chooses s>s
d(ω) and cj>c
d
j(ω) o all j= i.
P oo .Le i=1and b1∈(0cd
1(ω)).Conside sel -1’s p oblem in s a e ω o maximize
θu(c )+β (s) o (cs) ∈Fsubjec o c1≤b1. The i s -o de condi ions o i s La-
g angian a e β (s(ω)) =μ(ω),θu1
c(c1(ω); 1)=μ(ω) +λ1(ω),andθui
c(ci(ω); i)=μ(ω)
o all i= 1,whe eμ(ω) ≥0and λ1(ω) ≥0a e he Lag ange mul iplie s o n
i=1ci≤1
and c1≤b1.
Suppose s(ω) ≤sd(ω). Since c1(ω) =b1<c
d
1(ω) and s(ω) +jcj(ω) =sd(ω) +
jcd
j(ω) =1by s ong mono onici y o p e e ences, cj(ω) > cd
j(ω) o some j= 01.
By s ic conca i y o ujand ,θuj
c(cj(ω); j)<θu
j
c(cd
j(ω); j)=β (sd(ω)) ≤β (s(ω)).
This iola es he i s -o de condi ions o c(ω). Sowemus ha es(ω) > sd(ω).This
in u n implies ha θuj
c(cj(ω); j)=β (s(ω)) < β (sd(ω)) =θuj
c(cd
j(ω); j) o j= i.By
conca i y, cj(ω) > cd
j(ω) o j= 1.
Theo e ical Economics 14 (2019) A heo y o pe sonal budge ing 191
Fo k=p d,le sk=minωsk(ω) and sk(ω) =maxωsk(ω). Focussing on ∈[sdsp],21
deno e by B he co esponding policy in B.
Lemma 4. De ine ( ) ={ω∈:sd(ω) ≤ }and le c (ω) be he maximize o u(c; )
subjec o n
i=1ci≤1− . Then U(B )is di e en iable in o e [sd sp]wi h
d
d U(B )=( ) ( ) −θucic (ω); dG o any i=1n
P oo .Gi en and any ω, de ine
˜
u( ;ω) ≡uc (ω); =max
{c∈Rn
+:n
i=1ci≤1− }
u(c; )
and ˜
U( ;ω) =θ˜
u( ;ω) + ( ). Since u(·; )is s ic ly conca e in c,sois ˜
u(·; )in by
s anda d a gumen s. Hence, ˜
U(·;ω) is also s ic ly conca e in . Now conside ˜
U( ;ω).
Whene e i is de ined, ˜
U( ;ω) =θ˜
u( ;ω) + ( ). By i s -o de condi ions o he La-
g angian de ining ˜
u( ; ),weha euci(c (ω); )=λ(ω; ) o i=1n,whe eλ(ω; )
is he Lag ange mul iplie o n
i=1ci≤1− . Since c (ω) is con inuous in o e -
e y ω,soisλ(ω; ).ByTheo em1o Luenbe ge 1969, p. 222) λ(ω; )(  − )≤
˜
u( ; )−˜
u( ; )≤λ(ω; )(  − ) o e e y   ∈(01). Con inui y o λ(ω;·)im-
plies ha ˜
u( ; )exis s o e e y ∈(01)and ˜
u( ; )=−λ(ω; ) =−uci(c (ω); ).
The e o e,
˜
U( ;ω) = ( ) −θucic (ω); ω∈ (3)
Fo any ,deno eby(c s )sel -1’s beha io as a unc ion o ωunde B .By he
maximum heo em, (c (ω)s (ω)) is con inuous in bo h and ω. Since, ixing any s,
bo h sel es would choose he same cin e e y ω, by de ini ion
( ) ≡U(B )=
˜
Us (ω);ωdG
Conside any > ˆ
and ecall ha ( ) ={ω:sd(ω) ≤ }.Then
( ) −( ˆ
)=( )˜
U( ;ω) −˜
Usˆ
(ω);ωdG
=( )∩(( ˆ
))c˜
U( ;ω) −˜
Usˆ
(ω);ωdG +( ˆ
)˜
U( ;ω) −˜
U( ˆ
;ω)dG;
he i s equali y holds because s (ω) =sˆ
(ω) o ω/∈( ) and s (ω) = o ω∈( ).
Di ide bo h sides by −ˆ
and conside he limi as ↓ˆ
.Fi s , o allω,weha e
lim
↓ˆ
˜
U( ;ω) −˜
U( ˆ
;ω)
−ˆ
=˜
U(ˆ
;ω)
21Any o he is domina ed by one in his ange.
192 Simone Galpe i Theo e ical Economics 14 (2019)
Since ˜
U(·;ω) is conca e,

˜
U( ;ω) −˜
U( ˆ
;ω)
−ˆ

≤max˜
U( ;ω)˜
U(ˆ
;ω)
Since ˜
U( ;ω) is con inuous in ωand as illus a ed by (3), |˜
U( ;ω)|is bounded by
some M<+∞ o ( ω) ∈[sdsp]×. The e o e, by Lebesgue’s bounded con e gence
heo em,
lim
↓ˆ
( ˆ
)
˜
U( ;ω) −˜
U( ˆ
;ω)
−ˆ
dG =( ˆ
)
˜
U(ˆ
;ω)dG
Conside now he second pa o he limi . Again, by conca i y o ˜
U(·;ω) and since
s (ω) ∈[sd sp] o ∈[sd sp],weha e

˜
U( ;ω) −˜
Usˆ
(ω);ω
−sˆ
(ω) 
≤M
The e o e,
( )∩(( ˆ
))c
˜
U( ;ω) −˜
Usˆ
(ω);ω
−ˆ
dG
≤( )∩(( ˆ
))c
˜
U( ;ω) −˜
Usˆ
(ω);ω
−ˆ

dG
≤( )∩(( ˆ
))c
˜
U( ;ω) −˜
Usˆ
(ω);ω
−sˆ
(ω) 
dG
≤M( )∩(( ˆ
))c
dG
Obse e ha ( ) ∩(( ˆ
))c={ω:ˆ
<s
ˆ
(ω) ≤ }, which con e ges o an emp y se
as ↓ˆ
. Since hen he second pa o he limi con e ges o ze o as ↓ˆ
, o e e y
ˆ
∈[sd sp),
(ˆ
+)=( ˆ
)
˜
U(ˆ
;ω)dG
A simila a gumen implies ha (ˆ
−)=( ˆ
) ˜
U(ˆ
;ω)dG o e e y ˆ
∈(sd sp].Hence,
( ) is di e en iable o e [sd sp].
A.2 P oo o Lemma 1
Fix . Recall he de ini ion o U(D) in (1) o he main ex and ha (c(θ) s(θ)) ep esen s
sel -1’s op imal choice in s a e θ.The eexis sD⊂Fsuch ha U(D) ≥U(D) o all D⊂
Fi and only i he e exis unc ions χ:[θθ]→Rn
+and :[θθ]→R+ ha sa is y he
ollowing wo condi ions:
Condi ion 1. Fo all θ θ∈[θ θ],
θuχ(θ); +β  (θ)≥θuχθ; +β  θ

Theo e ical Economics 14 (2019) A heo y o pe sonal budge ing 193
and
n

i=1
χi(θ) + (θ)≤1
Condi ion 2. The pai (χ ) maximizes
θ
θθuχ(θ); +  (θ)g (θ) dθ
By con as , he e exis s D c ⊂F c such ha U(D c)≥U(ˆ
D c) o all ˆ
D c ⊂F c i and
only i he e exis unc ions ϕ:[θ θ]→R+and τ:[θ θ]→R+ ha sa is y he ollowing
wo condi ions:
Condi ion 1.Fo allθ θ∈[θθ],
θu∗ϕ(θ); +β τ(θ)≥θu∗ϕθ; +β τθ
whe e u∗(y; )=max{c∈Rn
+:n
i=1c
i≤y}u(c; )and
ϕ(θ) +τ(θ) ≤1
Condi ion 2.Thepai (ϕ τ) maximizes
θ
θθu∗ϕ(θ); + τ(θ)g (θ) dθ
Suppose (χ ) ha sa is ies Condi ions 1and 2. Then, by ou discussion on money
bu ning be o e he s a emen o Lemma 1, he e exis s a unc ion ϕ:[θ θ]→R+such
ha u∗(ϕ(θ); )=u(χ(θ); )and ϕ(θ) ≤n
i=1χi(θ) o all θ∈[θ θ]. Hence, le ing τ≡ ,
we ha e ha (ϕτ) sa is ies bo h Condi ions 1and 2.
Suppose (ϕτ) sa is y Condi ions 1and 2. Fo e e y θ∈[θ θ],le
χ(θ) =a g max
{c∈Rn
+:n
i=1ci≤ϕ(θ)}
u(c; )
Then, by de ini ion, u(χ(θ); )=u∗(ϕ(θ); ) o all θ∈[θ θ]. Le ing ≡τ,weha e ha
(χ ) sa is ies bo h Condi ions 1and 2.
A.3 P oo o P oposi ion 2
The p oo uses he ollowing h ee lemmas.
Lemma 5. When sel -0can se only , e e y op imal sa is ies sp< <sp.
P oo . We will show ha ( ) > 0 o all ∈(sdsp]and ( −)<0 o =sp. Recall
ha (c s )is con inuous in o e e y ωand, he e o e, so is ( ). These obse a ions
imply ha e e y op imal ∗is in (sp sp).
194 Simone Galpe i Theo e ical Economics 14 (2019)
Fo any ∈(sd sp], de ine +( ) ={ω:sp(ω) > }and −( ) ={ω:sp(ω) ≤ }.Fo
ω∈+( ), conside he ic i ious p oblem maximizing θu(c; )+ (s) o (cs)∈Rn+1
+
subjec o s+ici≤1and s≤ . Le ing μ(ω) and φ+(ω) be he co esponding La-
g ange mul iplie s, he i s -o de condi ions a e22 (s) =μ(ω)+φ+(ω) and θuci(c; )=
μ(ω) o all i. Clea ly, s= and φ+(ω) > 0 o ω∈+( ). Also, condi ional on s= ,
bo h sel es would choose he same cin s a e ω, which is, he e o e, c (ω). Using (3), i
ollows ha , o e e y i,
φ+(ω) = ( ) −θucic (ω);ω=˜
U( ;ω) ω ∈+( ) (4)
Fo ω∈−( ), conside he ic i ious p oblem o maximizing θu(c; )+ (s) o (cs)∈
Rn+1
+subjec o s+ici≤1and s≥ . Le ing μ(ω) and φ−(ω) be he co esponding La-
g ange mul iplie s, he i s -o de condi ions a e (s) =μ(ω) −φ−(ω) and θuci(c; )=
μ(ω) o all i. Clea ly, s= and φ−(ω) ≥0 o ω∈−( ). Also, condi ional on s= ,
bo h sel es would choose he same cin s a e ω, which is, he e o e, c (ω). Using (3), i
ollows ha , o e e y i,
φ−(ω) =θucic (ω); − ( ) =−˜
U( ;ω) ω ∈−( )
Conside any ∈(sdsp]. Recall ha ( ) ={ω:sd(ω) ≤ }. Using Lemma 4,we
ha e
( ) =( )∩+( )
˜
U( ;ω) dG +( )∩−( )
˜
U( ;ω) dG =( )∩+( )
φ+(ω)dG
whe e he las equali y ollows because ei he −( ) =∅o φ−(ω) =0 o ω∈−( ).
The unc ion φ+(ω) is s ic ly posi i e o e ( ) ∩+( ). We need o show ha
G(( ) ∩+( )) > 0, which implies ( ) > 0. This is immedia e i ∈(sdsp),be-
cause +( ) =.Conside =sp. Clea ly, (sp)∩+(sp)con ains he open se
◦(sp)∩+(sp)={ω:sd(ω) < sp<s
p(ω)}. I his se is nonemp y, we a e done
because Ghas ull suppo . Bo h ◦(sp)and +(sp)a e nonemp y. Suppose ha
+(sp)∩◦(sp)=∅.Then, o e e yω∈+(sp),weha esd(ω) ≥spand ha ◦(sp)⊂
−(sp)={ω:sp(ω) =sp}.Now,conside ˆω∈◦(sp)and any sequence {ωn}in +(sp)
con e ging o ˆω. Weha e ha limωn→ˆωin sd(ωn)≥sp>s
d(ˆω). Bu his iola es he
con inui y o sd—a con adic ion.
Now conside =sp. Again using Lemma 4,weha e
sp−=(sp)
˜
Usp;ωdG =
˜
Usp;ωdG =−
φ−(ω)dG
whe e φ−(ω) > 0 o all ωsuch ha sp(ω) < sp. The e o e, (sp−)<0.23
22He e—and in he o he p oo s— he complemen a y slackness condi ions a e omi ed o simplici y.
23I is easy o see ha he op imal sa is ies ≤sp. Suppose ∈(sp1). Then, o all ω,sel -1chooses
s(ω) = and c(ω) =c (ω). Take any ∈(sp ). Then, o e e y ω, =ζ(ω) +(1−ζ(ω))sp(ω) o
some ζ(ω) ∈(01). The e o e, o e e y ω,˜
U( ;ω) > ˜
U( ;ω) because ˜
U(sp(ω);ω) > ˜
U( ;ω) and ˜
U(·; ω)
is s ic ly conca e. I ollows ha sel -0’s payo is s ic ly la ge unde  han unde .
Theo e ical Economics 14 (2019) A heo y o pe sonal budge ing 195
Le E(β) be he se o op imal loo s and le u∗(y; )be he indi ec u ili y o spending
y∈[01].
Lemma 6. The se E(β) is dec easing in βin he s ong se o de .24 The la ges op imal
con e ges mono onically o spas β↑1.The eexis sβ > 0such ha E(β) ={ } o all
β≤β,whe e sa is ies <spand
U(B )=max
∈[spsp]θu∗(1− ; )+ ( )dG
P oo .Fix ∈[sdsp].These ( ) in Lemma 4 depends on β ia (cdsd).Bys anda d
a gumen s, i β<β
<1, hensd(ω;β) < sd(ω;β) o e e y ωand, hence, ( ;β)⊂
( ;β).Also,−( ) ⊂( ;β) o e e y β<1because sd(ω;β) < sp(ω) o e e y ω.So,
i β<β
<1,
( ;β) − ;β=(( ;β) ( ;β))∩+( )
φ+(ω)dG ≥0
whe e he inequali y uses (4). By s anda d esul s, E(β) dec eases in he s ong se o de .
De ine (β) =max{ : ∈E(β)}. Since (β) ≥sp o all βand (·)is dec eas-
ing, limβ↑1 (β) exis s; deno e i by (1−)≥sp. Clea ly, (1)=sp. Now suppose ha
(1−)> (1). By a simila a gumen , o any >s
p,limβ↑1( ;β) exis s and equals
−−( ) φ(ω)dG < 0. The e o e, o βclose enough o 1, (β) ≥ (1−)canno be
op imal—a con adic ion ha implies (1−)= (1).
No e ha sd(β) =maxsd(s;β) alls mono onically o 0as β↓0.Le β=max{β∈
[01]:sd(β) ≤sp}, which is s ic ly posi i e because sp>0.Then( ) = o all β≤β
and ∈[sp sp],and,hence,
( ;β) =θuc (ω); + ( )dG (5)
F om he p oo o Lemma 4,u(c (ω); )=˜
u( ;ω) is s ic ly conca e in o all ω.Thus,
he maximize o (5) is unique. F om he p oo o Lemma 5, he de i a i e o (5) is nega-
i e a spand, hence, <sp.
We now show ha sel -0bene i s om using only bi.
Lemma 7. Fix iand conside plans Bbiwi h bj=1 o all j= iand =0.The eexis s
bi<maxωcd
i(ω) ≡cd
isuch ha U(Bbi)>U(F).
P oo .Fixi=1and any b1∈(0 cd
1].Le (cb1sb1)desc ibe sel -1’s choices unde Bb1
and le
(b1)=θucb1(ω); + sb1(ω)dG
24Gi en wo se s Eand Ein R,E≥Ein he s ong se o de i , o e e y ∈Eand ∈E,min{  }∈E
and max{  }∈E(Milg om and Shannon 1994).
196 Simone Galpe i Theo e ical Economics 14 (2019)
Le (b1)={ω:cd
1(ω) > b1}. Since cd
1is con inuous, (b1)is nonemp y and open i
b1< cd
1,and,hence,G((b1)) > 0.Weha e
(b1)−cd
1=(b1)θucb1(ω); + sb1(ω)−θucd(ω); + sd(ω)dG
=(1−β)(b1) sb1(ω)− sd(ω)dG
+(b1)˜
Vcb1
1(ω);ω−˜
Vcd
1(ω);ωdG
whe e
˜
V(ˆ
b1;ω) =max
{(cs)∈Rn+1
+:n
j=1cj≤1c1≤ˆ
b1}θu(c; )+β (s)
Clea ly, ˜
V(c
d
1(ω);ω) ≥˜
V(b
1;ω) o all ω. F om he i s -o de condi ions o he La-
g angian de ining ˜
V(ˆ
b1;ω),weha eλ1(ω;ˆ
b1)=θu1
c(c ˆ
b1
1(ω); 1)−β (s ˆ
b1(ω)),whe e
λ1(ω;ˆ
b1)is he Lag ange mul iplie o c1≤ˆ
b1. Since (cˆ
b1(ω) s ˆ
b1(ω)) is con inuous in
ˆ
b1and ω,soisλ1(ω;ˆ
b1).AgainbyTheo em1o Luenbe ge 1969, p. 222), ˜
V(ˆ
b1;ω)
exis s o all ˆ
b1and equals λ1(ω;ˆ
b1). I ollows ha ˜
V(cd
1(ω);ω) =0 o all ωby he
de ini ion o (cdsd). By he mean alue heo em (MVT), ˜
V(c
b1
1(ω);ω) −˜
V(c
d
1(ω);ω) =
˜
V(χ(ω);ω)(cb1
1(ω)−cd
1(ω)) and (sb1(ω))− (sd(ω)) = (ξ(ω))(sb1(ω)−sd(ω)),whe e
χ(ω) ∈[cb1
1(ω)cd
1(ω)]and ξ(ω) ∈[sd(ω)sb1(ω)].
Le bε
1=cd
1−ε o some ε>0.Fixω∈(bε
1)and, o now, d op he dependence
on ω. Recall ha sbε
1+icbε
1
i=sd+icd
i=1. Since sbε
1>s
d o ε>0(Lemma 3), we
can w i e
−cbε
1
1−cd
1
sbε
1−sd=1+
j=1
cbε
1
j−cd
j
sbε
1−sd(6)
Now, o any bε
1, he i s -o de condi ion β (s) −θuj
c(cj; j)=0mus hold o e e y
j= 1. The e o e, again by he MVT, o all j= 1,
cbε
1
j−cd
j=β sbε
1− sd
θuj
cc(ζj; j)
(7)
o some ζ∈[cd
jcbε
1
j]. Now, since  is con inuous, (y) − (ˆ
y) ≥ [y−ˆ
y] o e e y
y>ˆ
y≥sd,whe e  =minξ∈[sd1] (ξ) < 0. The e o e, using (6)and(7),
−cbε
1
1−cd
1
sbε
1−sd=1+1
sbε
1−sd
j=1
β
θuj
cc(ζj; j) sbε
1− sd
≤1+1
sbε
1−sd
j=1
β 
θuj
cc(ζj; j)sbε
1−sd≤1+β 
θ
j=1
1
uj
cc

Theo e ical Economics 14 (2019) A heo y o pe sonal budge ing 203
ha B(θ†)con ains he plan de ined by bi(θ†)=cp
i(ωi) o i=12and (θ†)=sp(ω0),
whe e (θ†)=1−b1(θ†)−b2(θ†)and, hence, is edundan . Thus, B(θ) con ains no
plan wi h (θ) > 1−b1(θ) −b2(θ), as such plans a e s ic ly domina ed o he same
a gumen ha ules hem ou in he p oo o Lemma 9. Since he la ges alue o (θ)
mus be close o (θ†) o θ∈(θ†θ†+ε), (θ) canno bind in ω1and ω2as well.
Fix θ∈(θ† θ†+ε).Claims6–9cha ac e ize he p ope ies o e e y B∈B(θ).
Claim 6. Fo e e y B∈B(θ),b1(θ) and b2(θ) mus bind in ω0; ha is,ci(ω0)=bi(θ) o
i=12. Gi en his, s(ω0)=1−b1(θ) −b2(θ) and, hence, can be emo ed.
P oo .No e ha sel -0’s objec i e in ωias a unc ion o biis s ic ly conca e and de-
c easing o bi>c
p
i(ωi)(see (10)). Thus, i , o example, b1(θ) is no binding o sel -1
in ω0( ha is, b1(θ) > c1(ω0)), sel -0can lowe b1wi hou a ec ing sel -1’s choice in ω0
and ω2, and can s ic ly imp o e he payo in ω1. Hence, he ini ial plan would no be
op imal.
Claim 7. We ha e b1(θ) =b2(θ) o e e y B∈B(θ).
P oo . Wi hou loss, suppose b1(θ) > b2(θ).No e ha b2(θ) < cd
2(ω0)because, o he -
wise, we would ha e b1(θ) > cd
1(ω0)=cd
2(ω0), con adic ing he p e ious claim. Con-
side he al e na i e wi h bε
1=b1(θ) −εand bε
2=b2(θ) +ε o ε>0.Fo εsmall, bε
1
and bε
2con inue o bind in ω0,so1−bε
1−bε
2=s(ω0).Inω0,sel -0’s payo is highe ,
because gi en s(ω0), he chosen cis close o being symme ic and, hence, o he bes
one acco ding o sel -0’s p e e ence. Due o symme y and he s ic conca i y in sel -
0’s payo induced by biin ωi o i=12(see (10)), he dec ease in he he payo in ω2
esul ing om he slacke b2is mo e han compensa ed by he inc ease in ω1 esul ing
om he igh e b1. Hence, o e all, sel -0’s payo is s ic ly la ge wi h (bε
1bε
2) han wi h
(b1(θ)b2(θ)), con adic ing he op imali y o he la e plan.
Claim 8. We ha e 1−b1(θ) −b2(θ) > sp(ω0) o e e y B∈B(θ).
P oo .I 1−b1(θ) −b2(θ) < sp(ω0),sel -0can se =sp(ω0)and ge a s ic ly highe
payo in ω0wi hou a ec ing sel -1’s choices in ω1and ω2.I 1−b1(θ) −b2(θ) =sp(ω0),
hen bi=cp
i(ω0) o i=12and so (c(ω0) s(ω0)) =(cp(ω0) sp(ω0)). The e o e, i would
be possible o lowe bo h b1(θ) and b2(θ) by he same small ε, so as o induce a i s -
o de gaininsel -0’s payo in ω1and ω2because bi(θ) > cp
i(ωi) o i=12, while caus-
ing only a second-o de loss in ω0.
Claim 9. All B∈B(θ) ha e he same b1and b2and sa is y he p ope ies in Lemma 10.
P oo .Le b1=b2=b.Sel -0’s payo in ω1and ω2is gi en by (10) up o a cons an :
θ[ ln(b)+ ln(1−b)]+ln(1−b).He payo inω0is gi en, up o a cons an , by 2θ ln(b)+
ln(1−2b). The e o e, sel -0’s expec ed payo om bis s ic ly conca e. To see ha
cp
i(ωi)>b
i>c
p
i(ω0) o i=12, conside he ollowing si ua ion. A bi>c
p
i(ωi)is s ic ly

204 Simone Galpe i Theo e ical Economics 14 (2019)
domina ed by bi=cp
i(ωi) o e e y i,as hisis heop imalle elo biin ωi.Consequen ly,
we mus ha e bi<c
p
i(ωi), because, by assump ion, 1−cp
1(ω1)−cp
2(ω2)>s
p(ω0) o
θ>θ†, and so educing bibelow cp
i(ωi)by he same small amoun o i=12causes a
i s -o de gain in ω0and a second-o de loss in ω1and ω2.
Pa III Le G b be a dis ibu ion o e (ω0ω1ω2) ha leads o Lemma 9 and le Gbe
he uni o m o e [θθ]×[  ]2.Le Gbbe a dis ibu ion ha leads o Lemma 10 and
le Gbe he uni o m o e [θ θ]×[  ]2,whe eθis as in Lemma 10.Fo α∈[01],le
G b
α=αG b +(1−α)Gand Gb
α=αGb+(1−α)G.P oposi ion 4 ollows om he nex
esul .
Co olla y 2. (i) The e exis s α∈(01)such ha , gi en G b
α, ,b1,andb2a e all binding
o e e y op imal B. (ii) The e exis s α∈(01)such ha , gi en Gb
α, o e e y op imal B,
bo h b1and b2bind, bu ne e binds.
P oo .Le
B ⊂Bcon ain all Bs ha canuseonly ,le Bbcon ain all Bs ha canuse
bo h b1and b2,andle Bbicon ain all Bs ha canuseonlybi o i=12. To indica e ha
sel -0’s expec ed payo is compu ed using some dis ibu ion ˆ
G,wewilluse heno a ion
U(B;ˆ
G).
Pa (i). Fo e e y Band α∈[01],U(B;G b
α)=αU(B;G b)+(1−α)U(B;G).Now
de ine
W b
(α) =max
B∈B
UB;G b
αand W b
b(α) =max
B∈Bb
UB;G b
αα∈[01]
Bo h W b
and W b
ba e well de ined by he same a gumen o Lemma 2 and con inuous
unc ions o αby he maximum heo em.26 Le B b deno e he op imal plan in Lemma 9.
No e ha U(B b;G) is ini e since sel -1’s esul ing choices a e bounded away om 0in
all dimensions. We ha e ha limα↑1U(B b;G b
α)−W b
j(α) > 0 o bo h j= and j=b.
The e o e, he e exis s ˆα∈(01)such ha B b s ic ly domina es e e y B∈B ∪Bbgi en
G b
ˆα.
Pa (ii). Fo e e y Band α∈[01],U(B;Gb
α)=αU(B;Gb)+(1−α)U(B;G).Le
Bbdeno e he op imal plan in Lemma 10. By he same logic in he p oo o pa (i),
he e exis s α ∈(01)such ha , o α∈(α1),Bbs ic ly domina es e e y B∈B ∪Bb∪
Bb1∪Bb2gi en Gb
α. I emains o show ha he e exis s α∈(α1)such ha Bbs ic ly
domina es e e y B∈Bgi en Gb
α. To his end, de ine B(α) =a g maxB∈BU(B;Gb
α).Se
B(·)is uppe hemicon inuous by he maximum heo em. No e ha B(1)is cha ac e ized
by ( ∗b∗
1b∗
2)such ha b∗
1and b∗
2a e unique and sa is y Lemma 10,and ∗∈[0 ],
whe e =1−b∗
1−b∗
2. Fo e e y η>0, he eexis sε>0such ha i α∈(1−ε 1],
hen ∈[0 +η],b1∈(b∗
1−ηb∗
1+η),andb2∈(b∗
2−ηb∗
2+η) o e e y ( b1b2)
co esponding o some B∈B(α). Choosing ηsmall enough ensu es ha o all B∈B(α),
we ha e ha (a) 1−b1−b2>s
p(ω0), (b) emo ing leads o a plan such ha b1and b2
26Recall oo no e 25.
Theo e ical Economics 14 (2019) A heo y o pe sonal budge ing 205
bind in ω0,and(c) canno bind in ω1and ω2,assel -1’s choice o ss ic ly exceeds
in ω1and ω2unde all B∈B(1).
Take any B∈B(α) and ix i s b1and b2.The comple ing Bmus be op imally chosen
gi en b1and b2. We claim ha i mus sa is y ≤1−b1−b2=k o αsu icien ly close
o 1. Suppose no . Conside sel -0’s expec ed gain om imposing >k. The gain in ω0
is
(1−β) ( ) − (k)+V ;ω0−Vk;ω0
and he expec ed gain unde he dis ibu ion Gis
( )(1−β) ( ) − ˆ
s(ω)+V( ;ω) −Vˆ
s(ω);ωdG(13)
whe e ( ) ⊂=[θ θ]×[  ]2is he se o s a es in which a ec s sel -1’s choices,
(ˆ
cˆ
s) is sel -1’s alloca ion unc ion unde he policy using only b1and b2,and
V(k;ω) =max
{(cs)∈B:c1≤b1c2≤b2s≥k}θu(c; )+β (s)k∈[
k1]ω∈
Since V( ;ω) ≤V(
ˆ
s(ω);ω) and ˆ
s(ω) ≥k o all ω∈, o all ≥k,(13)isbounded
abo e by
( )
(1−β) ( ) − ˆ
s(ω)dG≤(1−β) ( ) − (k)
No e ha he igh -hand side o his exp ession depends on αonly ia k.
Now ocus on V(k;ω0). Fo e e y >k,(a) always binds, because k>s
p(ω0)
and, hence, sel -1wan s o choose s< , (b) only one bican bind, because i bo h bind,
hen s(ω0)=k< , which is impossible, and (c) one bine e binds, because goods
a e no mal, so o all >k,sel -1’s chooses ci(ω0)<b
i o a leas one i=12.Wi h-
ou loss, suppose ha b2ne e binds. I we emo e b2,V(k;ω0)equals sel -1’s indi-
ec u ili y unde he plan de ined by k∈[
k1]and b1only, deno ed by V(k;ω0b1).
By he same a gumen in Lemma 4,V(k;ω0b1)is con inuously di e en iable in k o
k∈(01]and V(k;ω0b1)=−λ(ω0;k),whe eλ(ω0;k) is he Lag ange mul iplie as-
socia ed o s≥k. Using he Lag angian ha de ines V(k;ω0b1),weha eλ(ω0;k) =
θu2
c(c2(ω0;k); )−β (k).No e ha λ(ω0;k)>0 o all k∈[k 1], because such loo
le els mus always bind o sel -1.Mo eo e ,λ(ω0;k) is s ic ly inc easing in k∈[k 1]
because is s ic ly conca e, ui
cc <0,andc2(ω0;k) is noninc easing in kby no mali y
o goods. The e o e, V(k;ω0)=−λ(ω0;k) o all k∈(k 1]and V(k+; ω0)=−λ(ω0;k),
whe e he plus deno es he igh de i a i e.27 Mo eo e , V(k;ω0)is s ic ly dec easing
in k.
Obse e ha
(1−β) (k) +Vk;ω0= (k) −θu2
cc2ω0;k; (14)
27The unc ion V(k;ω0)is no di e en iable a k=k,asV(k;ω0)is cons an o k<kand V(k−;
ω0)=0.
206 Simone Galpe i Theo e ical Economics 14 (2019)
which is s ic ly nega i e. This is because b1<c
p
1(ω0)and b2<c
p
2(ω0)by Lemma 10
since αis close o 1, which implies ha b1and b2mus bind o sel -0;consequen ly,
=kand b1mus also bind o sel -0. The igh -hand side o (14) coincides wi h he
nega i e o he Lag ange mul iplie associa ed wi h s≥kin sel -0’s p oblem ha also
includes c1≤b1.
Recall ha kdepends on α—hence, deno e i by kα—and conside
gVkα;ω0+αg +(1−α)(1−β) (kα) (15)
This is s ic ly nega i e o α=1, in which case k1=1−b∗
1−b∗
2. By con inui y o (15)in
(α k) and uppe hemicon inui y o B(α), he eexis sε>0such ha (15) is s ic ly neg-
a i e o α∈(1−ε 1]. By he mono onici y o and V(·;ω0),(15) is s ic ly dec easing
o k≥kα.
Finally, o e e y α∈(1−ε 1]and >kα,
αg +(1−α)(1−β) ( ) − (kα)+gV ;ω0−Vkα;ω0
=
kααg +(1−α)(1−β) (k) +gV k;ω0dk
<αg +(1−α)(1−β) (kα)+gV kα;ω0( −kα)<0
We conclude ha sel -0is s ic ly wo se o by imposing a binding in addi ion o b1
and b2,and,hence,e e yop imalBmus use binding budge s o bo h goods, bu no
binding .
A.6 Example o non-addi i e u ili y
Suppose ha
u(c; )=1
1−γ 1c
e−1
e
1+ 2c
e−1
e
2e
e−1(1−γ) and (s) =s1−γ
1−γ
Assume ha e>1,0<γ<1,ande≤1/γ. By s anda d calcula ions, gi en o al expendi-
u es y∈[01]in a pe iod, he op imal alloca ion o good iis
ci( ;y)=y e
i
e
1+ e
2
(16)
We now show ha (cdsd)sa is ies Condi ion 1; simila s eps es ablish he desi ed
p ope ies o (cpsp). Fo e e y ω∈, maximizing
θτ( )
1−γ(1−s)1−γ+β
1−γs1−γ
yields
sd(ω) =β1
γ
θτ( )1
γ+β1
γ

Theo e ical Economics 14 (2019) A heo y o pe sonal budge ing 207
Clea ly, sd(s)is always in e io and s ic ly dec easing in θ, 1,and 2. Replacing ywi h
1−sd(ω) in o (16), we ge ha cd
i(ω) is s ic ly inc easing in θand i, and sa is ies
∂
∂ j
cd
i(ω) ∝sd(ω) 1−γ
γ(e −1)−1
Thus, o ∂cd
i(ω)/∂ j(and simila ly ∂cp
i(ω)/∂ j) o be s ic ly nega i e o all ω,asu i-
cien condi ion is ha
1−γ
γ(e −1)<1
sp
because sd(ω) < sd< sp<1.28
Now conside se ing only a budge on good 1(o , equi alen ly, on good 2). We will
show ha whene e b1binds, inc easing i leads o lowe sa ings and ha pa (ii) o
Condi ion 1 holds. Suppose b1binds in s a e ω.Thensel -1’s choice sa is ies c∗
2(ω) =
1−s∗(ω) −b1and he op imal s∗(ω) sol es he i s -o de condi ion
θu2b11−s∗(ω) −b1; =βs∗(ω)−γ
The e o e,
∂
∂b1
s∗(ω) =θu21(c; )−u22(c; )
θu22(c; )−γβ
s1+γ
(cs)=(c∗(ω)s∗(ω))

A su icien condi ion o his o be s ic ly nega i e is ha u21(c; )≥0, which holds un-
de ou assump ions since u21(c; )∝1−γe. Finally, since sel -1’s alloca ion o sand c2
is bounded away om ze o o e e y b1≤cd
1, and since u22(c; )and u21(c; )a e con in-
uous in bo h a gumen s, i ollows ha ∂s∗/∂b1is uni o mly bounded away om ze o.
The e o e, pa (ii) o Condi ion 1 holds. No e ha in he p e ious a gumen we can e-
place b1wi h 1. The e o e, in his example, binding good-speci ic loo s lead o lowe
sa ings, and, hence, hey a e ne e pa o op imal plans.
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