A theory of personal budgeting

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 This Ve sion is a ailable a : h ps://hdl.handle.ne /10419/217079 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/4.0/ 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. Re e ences Aguia , Ma k and Manuel Amado (2011), “G ow h in he shadow o exp op ia ion.” Qua e ly Jou nal o Economics, 126, 651–697. 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