Consumption Patterns over Pay Periods. Clare Kelly 1 Departments of Economics, University College Dublin and Keele University,

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1 Consumpion Paerns over Pay Periods Clare Kelly Deparmens of Economics, Universiy College Dublin and Keele Universiy, and Gauhier Lano Deparmen of Economics, Keele Universiy. January 00 VERY PRELIMINARY DRAFT DO NOT QUOTE JEL Classificaion: D; D; D9. Key Words: Consumpion; liquidiy consrains; uncerainy; credi cards Corresponding auhor: Deparmen of Economics, Universiy College Dublin, Belfield, Dublin 4, Ireland. Tel: , Fax: ,

2 Absrac How do individuals smooh consumpion when hey are paid monhly bu make consumpion decisions a leas every week wihin ha monh? Are consumers credi consrained in he shorrun wih respec o non-durable consumpion and can we deermine he impac of his? This paper examines behaviour of individuals who receive income periodically bu make consumpion decisions on a more frequen basis when hey are faced wih an imperfec capial marke wih respec o he price and quaniy of credi, and wih price uncerainy. We presen for he firs ime a dynamic model o characerise opimal behaviour and prove he exisence of he saionary soluion for each micro consumpion period wihin he paymen period. We simulae he numerical soluion o he model and here is a clear u-shape over he paymen period. A a given level of wealh, consumpion is highes in he firs week afer income receip, decreases in he second and hird weeks and increases again in he fourh. A pseudo maximum likelihood esimaor is applied o daa from he Family Expendiure Survey o esimae he coefficien of relaive risk aversion. However here is evidence of subsanial measuremen error in he daa and we esimae ha his accouns for abou 50% of he oal variaion in he daa.. Inroducion

3 In he recen pas a new problem has arisen for consumers wih respec o consumpion decisions. Currenly (998/99), close o 58% of working persons in he UK receive heir income on a monhly basis bu hey make consumpion decisions more frequenly. How does such an individual choose o allocae heir spending over he monh? Do individuals spend he same proporion of income in each week or are expendiures significanly higher on receip of heir income and hen reduced unil he nex payday? An addiional issue is how his shor-run consumpion decision fis ino a life-cycle consumpion plan i.e. combining food and oher non durable consumpion wih morgage and oher credi repaymens and pension plan conribuions. Prima facia evidence in Figure shows he paern of non-durable expendiure over he paymen cycle for nine differen income groups and here is a clear paern. For mos groups consumpion is relaively high in he firs week when income is received, hen decreases for he second and hird and is hen relaively high in he fourh week, in anicipaion of paymen being received a he end of ha week. However, does such a paern reflec opimal behaviour? Tradiional consumpion heory suggess ha consumpion should be smooh given ha income is known wih cerainy each monh. However, here is sill uncerainy wih regard o expendiure levels which arises from many sources: repairs o household durables, school rips for children, sponaneous purchases, and variaion in prices. Inuiion suggess ha individuals may pospone unnecessary consumpion unil all informaion is known so ha hey have resources o deal wih any unexpeced negaive shocks during he monh: his in effec is he precauionary moive for saving widely discussed in he lieraure (see Carroll and Kimball 00). This implies posiive consumpion growh during he monh as uncerainy is gradually resolved, paricularly beween he second las and las week of he paymen cycle, when all informaion is known and income is received in he nex week. A a given level of wealh, he opimal consumpion is likely o be differen in he second week of he paymen cycle han in he fourh week because in he fourh

4 week all uncerainy is resolved and income is received in he nex period whereas in he second here are sill wo weeks of uncerain expendiure remaining. The paern shown in Figure is consisen wih consumers showing some degree of cauion during he earlier weeks: alhough expendiure is high when monhly income is received, here is a subsanial drop in weeks wo and hree, bu a sharp increase in he final week when uncerainy is resolved and income is expeced. Figure Average Non-durable expendiure by income group over paymen cycle 98/ However, an addiional facor in he consumpion decision may be a limi on unsecured borrowing for non-durable consumpion for a shor period of ime. In his case he oucome will differ from when we assume individuals hold liquid asses or have unlimied credi o use as a buffer agains negaive shocks and who herefore have an expendiure paern which in independen of income. In ha case he liquidiy consrains will reinforce he cauious behaviour of individuals as a response o uncerainy as poined ou in Deaon 99, page. Therefore he decisions on consumpion earlier in he paymen cycle will ake accoun of he fac ha here is limied credi wih which o finance consumpion unil income is received again 3

5 when posiive wealh is exhaused. This also implies ha an individual will no be a heir credi limi before week four because consumpion would hen be zero unil he sar of he nex cycle. In addiion, he Marginal Propensiy o Consume (MPC) ou of wealh and borrowings will differ during he monh; early in he monh he MPC ou of deb should be low, while boh should increase as he nex receip of income ges closer. If here is an ineres rae differenial beween borrowing and saving, hen borrowing will no occur unil all posiive wealh is depleed. Secion of his paper presens some illusraive empirical evidence showing he exisence of a paern in expendiure decisions over a paymen cycle. Secion 3 presens a simple model of simulaneous consumpion and income decisions incorporaing a limi on borrowing and an ineres rae differenial. Secion 4 describes he exension and soluion o his model which allows for periodic receip of income relaive o consumpion decisions. Secion 5 describes he esimaion procedure and presens some Mone Carlo resuls. Secion 6 concludes.. Empirical descripion of Shor-Run Consumpion decisions Wih a given a level of uncerainy and he possibiliy of liquidiy consrains, he consumpion level consisen wih any level of wealh will depend on he poin a which he individual is in he paymen cycle. However, as already noed above, his is no due o preferences or discoun raes ha vary over ime: only he variaion in he amoun of uncerainy which makes he marginal uiliy of any given level of expendiure a funcion on he poin in he paymen cycle a which he expendiure occurs. Thus he poin in he paymen cycle acs as a shif facor in he uiliy funcion which is similar o incorporaing demographic facors in he long-run consumpion models. In he weeks in which here is more uncerainy, he marginal uiliy of consumpion for given expendiure is shifed downwards relaive o weeks when here is less uncerainy and by implicaion consumpion mus be lower in he weeks wih more uncerainy. 4

6 The framework here closely follows Zeldes (989), using a CRRA uiliy in order o derive an approximaion o he Euler equaion. We assume a CRRA uiliy funcion of he form ρ c z uc (, z) = exp( ), ρ where ρ is he coefficien of risk aversion, c is consumpion and z represens he effec of differen poins in he paymen cycle. If here is no limi on he amoun of borrowing, hen in each period he individual maximises he sum of uiliy in all fuure periods in his lifeime i.e. T ρ c β = 0 max E exp( z) ρ subjec only o a budge consrain governing he evoluion of wealh ( )( ) w = + r w + + y c, where β is he discoun rae, y is regular income, r is he ineres rae. The firs order condiions for his problem leads o an Euler equaion which implies ha discouned marginal uiliy should be consan beween ime periods ρ β ρ c+ z+ c exp( z ) ( + r) exp( ) = + e, + where boh r and β are assumed o be consan over ime and across households. The lef hand side of his equaion should differ from one only by he expecaion error e +, which has mean zero and is independen of informaion known a period including monhly income. Taking logs of boh sides and rearranging gives he equaion for consumpion growh as ln( c+ ) = ( z+ z) ln( + r) ln( ) + ln( + e+ ) ρ [ β ] (.) Given ha changes in regular monhly are likely o be known in advance, we assume ha he level of income is known wih cerainy. However significance of income in explaining consumpion growh would indicae he imporance of liquidiy consrains because he 5

7 individual canno simply borrow as necessary o achieve heir opimal pah and so ha pah canno be independen of he iming of receip of income. The economeric specificaion of (.) herefore also includes he level of monhly income and is given by ln e = α ln y + γ d + ε (.) iw i w, w w, w iw where ln e iw = growh in expendiure in week w, for individual i, y i = level of pay received by individual i a he sar of monh, d ww, = dummies for he ransiion beween weeks w, w-, ε iw = sochasic error erm which has mean zero and is iid. If d represens a dummy variable which akes he value equal o one in w, (d w - d w- ) becomes a zero sum dummy over he paymen cycle for each individual. The dummy variables capure he effec of moving beween weeks and hus heir significance indicaes he imporance of he changing level of uncerainy. We esimae his Euler equaion using daa from he Family Expendiure Survey (FES) in he UK which records expendiure for wo consecuive weeks for each individual in he survey, as well as he daes over which expendiure is recorded, and he dae, amoun, and period covered by he las received income. Thus for all monhly paid individuals, he weeks of he paymen cycle over which hey are observed is clear. The FES for 996/97, 997/98 and 998/99 are used bu he sample is resriced o hose individuals who are in full ime employmen, are usually paid monhly and whose las income receip is heir usual income. We separae he sample ino individuals who may be consrained wih respec o shor-erm credi based on wheher hey are recorded as holding a credi card in he daase. However his depends on he individual paying an annual charge for heir card and herefore we supplemen his by also recording wheher individuals made any purchases on credi during heir diary period. While his is no a perfec indicaor of liquidiy, close o 55% of our sample The esimaions exclude ouliers which are defined as persons whose non-durable consumpion in any one week is greaer han heir monhly income. These observaions (less han 3%) biased he resuls owards he insignificance of income. Monhly effecs are also conrolled for bu he coefficiens are no repored. 6

8 is classified as unconsrained which seems realisic alhough conservaive. 3 This may be explained by he fac ha we record individuals who did no use heir credi card during he fornigh or pay an annual charge during he previous year as consrained. The resuls of he esimaion of equaion are presened in Table : monhly dummies were included alhough he coefficiens are no repored. The effec of moving beween differen weeks in he paymen cycle is measured relaive o he effec of going from week four when uncerainy is leas o week one when i is greaes. The significan coefficien on d 43 for all years indicaes individuals respond in a posiive way o he resoluion of all uncerainy. The negaive sign on d 3 is consisen wih he paern in Figure for he income groups where consumpion falls hroughou he monh and increases a he end. Because uncerainy is less in week hree han week wo, we would expec his coefficien o be posiive however, alhough he negaive values are insignifican in wo hirds of he resuls. The resuls are also quie clear when he sample is spli ino groups which we consider o be liquidiy consrained (wihou a credi card) and hose who are no. Again for all years, he effec of paymen cycle is significan for he consrained group, while here is no effec on he unconsrained group lending weigh o he argumen ha hose wih access o credi have less need o ac in a cauious manner. However hese individuals also have higher income we canno deermine wheher hese individuals are dissaving or borrowing. In 97/98 and 98/99 he level of monhly income is significan for he whole sample even when he posiion in he paymen cycle is conrolled for. This poins owards he possibiliy of liquidiy consrains given ha he level of income is cerain when consumpion decisions are made alhough he coefficien on income is posiive which is a odds wih he resuls of virually all previous ess for excess sensiiviy (see Table 5., Browning and Luscardi, 996). 4 3 However, following he findings of Japelli e al (998), his may sill provide a more accurae classificaion ha wealh o income raio. 4 In his shor-run framework, higher income relieves shor-run liquidiy consrains and leads o a higher level of consumpion and hence increased consumpion growh. In he long-run, lagged movemen of 7

9 Table Esimaion of Euler Equaion All observaions Unconsrained Sample Consrained Sample 98/99 97/98 96/97 98/99 97/98 96/97 98/99 97/98 96/97 Dependen variable: dlndexp N llaspay 0.060* * * * d * * * d * * * d * * * * * * consan -0.58* * * * * 0.70 When he wo groups are considered separaely however, income is always insignifican for he group we consider o be consrained, bu his may sill be due o misclassificaion of hose unconsrained individuals. More difficul o explain is he significance of income for he unconsrained group in 97/98 and 98/99 which is direcly a odds wih he presence of liquidiy consrains. However, income is generally he greaes deerminan of access o credi and herefore i is possible ha we are capuring his relaionship here. In resuls no repored here, we correc for seleciviy in credi card ownership by using a Heckman selecion esimaor and in ha case, he level of income and he paymen cycle are insignifican for he unconsrained group. Also including squared erms of log income, which simply relaxes he assumpion ha expendiure growh is a linear funcion of log income, also resuls in insignifican income erms for he unconsrained group. income owards is permanen income level leads o a lower long-run consumpion growh rae as he 8

10 3. A Model of Shor-Run Consumpion Before looking a he behaviour of agens wih periodic income relaive o heir consumpion decisions, a simpler model is presened o analyse he case of simulaneous paymen and consumpion decisions, when accoun is aken of he ineres rae differenial beween borrowing and saving eviden in capial markes. We also incorporae an upper bound on he level of borrowing given ha he amoun of unsecured shor-erm borrowing is very likely o be limied for a leas some of he populaion, paricularly hose wih low income. Uncerainy is assumed o arise from variaion in prices which also capures he choices individuals can make over expendiure when similar producs are differeniaed wih respec o price. Alhough no assumpions regarding preferences are made a his sage, once he marginal uiliy funcion is no linear, he opimal pah of he individual will be changed because expecaions are affeced by uncerainy. The problem of uiliy maximisaion in his framework is one of dynamic programming wih he consumer in each period choosing a consumpion level which maximises curren uiliy plus heir preference for fuure wealh, given consrains and expecaions. Consumpion will be deermined by he real value of wealh which depends on he realisaion of curren price p and herefore he value funcion has wo sae variables, nominal wealh and price. The realisaion of fuure price is denoed by π and we assume boh p and π are drawn from he disribuion F wih suppor P over which expecaions are aken. We also assume ha price does no exhibi any persisence: if his was no he case, he curren price level would conain informaion abou he price disribuion in he nex period over which expecaions are aken. The budge income level sabilizes. I is his long-run effec ha previous sudies using panel daa have capured. 9

11 consrain for each ime period is pc w+ y+ d, and he evoluion of wealh follows he w = w + y pc ( + r) + + d( r δ ), process 5 : ( ) where pc = expendiure on consumpion, w = wealh, y = regular income, d = deb in period, subjec o an upper limi of d, r = reurn on savings, δ = cos of borrowing. Wealh a he sar of each period before income is received is denoed by w. We assume ha y, d, r and δ are exogenous and known wih cerainy and ha he differenial beween borrowing and saving raes is consan. The Bellman equaion for his problem is { } (, ) = max ( ) + β (( + )( + ) + ( δ), π) V w p u c E V r w y pc d r P (3.) pc w+ d + y 0 d d where he value funcion V(.), and he feliciy funcion uc () are assumed o be increasing, monoonic and concave. I is assumed ha he discoun facor β is such ha β(+δ) < i.e. he consumer is impaien and hus has a preference for consumpion in he presen. For a given wealh and price realisaion, he conrol variable consumpion is chosen o maximise he righ hand side, subjec o a lower limi on negaive wealh. The problem fulfils he condiions for coninuiy and differeniabiliy of he value funcion as laid down by Benvenise and Scheinkman (979) and so he value funcion in 3. will hold for all levels of wealh and price. The Lagrangean equaion for he problem gives 5 I would no be unreasonable o assume ha over a sufficienly shor period of ime, he reurn on savings equals zero. 0

12 (, ; γ, µ, µ ) = ( ) + β (( + )( + ) + ( δ), π) L c d u c E V r w y pc d r ( pc ( w y d) ) ( d ) ( d d ) γ µ + µ giving rise o he firs order condiions (, ; γ, µ, µ ) L c d c L c d (, ; γ, µ, µ ) d L c d (, ; γ, µ, µ ) (, ; γ, µ, µ ) L c d (, ; γ, µ, µ ) L c d γ µ µ * * * * * * EV = u' ( c ) β( + r) p (. ) γ p = 0, w EV = β( r δ) (. ) + γ + µ µ = 0, w = pc ( w y + d ) 0, = d 0, * = d d 0. (3.) The soluion o his problem can be characerised by examining differen values of he mulipliers. Firsly if γ =0, his implies ha he agen does no consume all cash in hand, which can only occur if he consumer is a saver. In his siuaion borrowing mus be zero 6, µ >0, µ =0, and here is posiive wealh carried over o he nex period. In all oher cases γ >0, i.e. consumpion is greaer han or equal o he sum of wealh and income, and asses carried beween periods and less han or equal o zero. Obviously µ >0 implies µ =0, and vice versa, bu µ = µ = 0 can occur if he level of opimal borrowing is posiive bu below he limi. Thus, when he individual consumes heir cash in hand, γ >0, bu has zero borrowing, µ >0, consumpion exacly equals wealh in ha period, and here are no asses or liabiliies carried beween periods. Then wealh equals income in he nex period. Alernaively, if γ >0 and here is some level of posiive borrowing, µ = µ =0, consumpion is equal o he opimal level of deb d*, plus cash on hand and a negaive wealh of ( δ ) + d * is carried ino he nex period. The final possibiliy is when γ >0, µ >0 and µ =0, a which poin he individual is a

13 his credi limi, consuming he maximum amoun available and has maximum deb a he sar of nex period ( δ ) + d bu consumpion is sill below he opimal. Therefore we have only four possible regimes for he individual Regime : pc* < w+ y, d * =0, γ =0, µ >0, µ =0, Regime : pc* = w+ y, d * =0, γ >0, µ >0, µ =0, Regime 3: pc* = w+ y+ d*, 0< d* < d, γ >0, µ = µ =0, (3.3) Regime 4: pc = w+ y+ d, d* = d, γ >0, µ =0, µ >0. 3. Regime boundaries The regime wihin which (3.) is maximised depends on he level of opimal consumpion relaive o he real value of wealh and will deermine how much borrowing or saving occurs. The boundaries beween regimes can be idenified from he firs order condiions, (3.). Following Deaon (99) denoe cash on hand as he sum of wealh and income. Considering firs when all cash on hand is consumed, γ >0, bu opimal deb is zero so µ >0, µ =0, i.e. here is zero wealh carried beween periods, so he firs order condiions from equaion 3. are ( ) EV ( 0, π ) u c c β( + r) p γ p= 0 w and ( 0, π ) EV β( r δ) + γ + µ = 0. w Solving for γ gives ( / ) ( 0, π ) u w + y p EV > β ( + r) p c w, (3.4) because γ >0 and hen subsiuing his ino he expression for µ, leads o 6 A consumer will never borrow and no consume ha exra cash when he cos of borrowing is greaer han he reurn on savings which rules ou γ=0, µ =µ =0 and γ=0, µ >0 µ =0.

14 ( / ) ( 0, π ) u w + y p EV < β( + δ) p c w. (3.5) Thus, when he marginal uiliy of real cash on hand falls beween he discouned value of having no asses or deb in he fuure and he agen consumes exacly cash on hand. If he curren real marginal uiliy of real cash, ( / ) u w+ y p p c, is less han he discouned fuure value of zero savings, i.e. he opposie of 3.4, opimal consumpion is below he level of real cash on hand and saving occurs, and he consumer is in regime. Similarly, if i is greaer han he discouned fuure value of zero borrowing, consumpion is financed ou of deb in order o reach opimal consumpion. 7 When he consumer is consrained i.e. consumpion is equal o w+ y+ d, deb is a he upper limi, γ > 0, µ >0, µ =0, so he firs order condiions in his case are (from equaion 3.) ( / ) ( ( δ), π) u w+ y+ d p EV + d β p( + r) γ p = 0 and c w ( ( δ) d, π) EV + βδ ( r) + γ µ = 0. w Again solving for γ and subsiuing ino µ gives ( / ) ( ( δ), π) u w+ y+ d p EV + d > β( + δ) p c w. (3.6) When his condiion holds he real marginal uiliy of maximum possible consumpion is greaer han he discouned fuure value of having maximum deb, he consumer uses all available credi up o he limi. In his case, wealh combined wih oal available credi is below he agens opimal consumpion, so he credi consrain is binding and if i were possible, marginal uiliy would be furher decreased. However, his is no possible and so consumpion remains below he opimal. If he opposie of 3.6 is rue bu he individual sill has a preference for consumpion greaer han cash on hand, hen posiive borrowing will lower curren marginal 3

15 uiliy of consumpion and increase he fuure value of wealh unil he wo erms equae a µ = µ =0 when here is some negaive wealh carried beween periods. These condiions allow he idenificaion of he regime ha an individual will fall ino, given a price realisaion, p a wealh level w, income y, fixed ineres raes, discoun rae and preferences. However, he effec of price realisaion is ambiguous: a low price will increase he value of real wealh, decreasing marginal uiliy bu will increase he real value of marginal uiliy. These firs order condiions and corresponding regimes can be wrien as an augmened Euler equaion where λ( c) denoes he marginal uiliy of consumpion, and he condiions relaing consumpion in any wo periods, and + are * β( + re ) λ( c+ ), * w + y * λ( c ) = max min λ, β(+ δ)ελ( c + ), p p p w + y + d λ p p (3.7) wih expecaions aken wih respec o price. Consider firs if he consumer did no have access o any credi, hen he maximum value ha consumpion could ake would be ( w + y)/ p and once * c is below ha, marginal uiliy of consumpion is above w y λ + p p and he Euler equaion is fulfilled in an unconsrained way. When he credi consrain binds, oal cash on hand ( w + y)/ p is he highes possible consumpion and his gives a lower bound on marginal uiliy. Therefore * * w + y λ( c ) = max β( + r) Eλ( c+ ), λ p p p 7 Obviously if r = δ, here are only hree regimes and every consumer is eiher a saver or a borrower. 4

16 as shown in Deaon (99). Wih access o credi markes limied o d, maximum consumpion is now ( w + y+ d )/ p. If c is below he consrained level, ( w + y+ d )/ p, * marginal uiliy will be higher han a he maximum possible consumpion level. Hence, he * individual chooses he maximum available marginal uiliy eiher β( re ) λ ( c + ) + when no consrained or w + y+ d λ p p when consrained. Wih he addiion of an ineres rae differenial, here may be consumers who have a wealh level and preferences such ha opimal consumpion is exacly cash in hand because hey are no impaien enough o borrow, or paien enough o save. i.e. boh equaion 3.4 and 3.5 hold. However for some individuals in his regime, δ may be sufficien reward for saving which would raise he discouned value of * savings and lower curren consumpion so ha ( c ) p λ w y > p λ +. However his is p no possible given capial marke imperfecions and so consumpion would insead equal cash in hand and he consumer mus be saisfied wih a lower marginal uiliy hence he need for he minimum condiion in 3.7. Assuming ha a saionary soluion exiss o equaion 3.7, his will define he opimal relaionship beween expendiure and wealh and prices, and can be wrien as pc f ( w, p) for all wealh and price. In ha case, he marginal uiliy of real wealh V (, ) denoed as qwp [, ] and assume he following properies of he funcions hold w =, w p can be (, ) (, ) f w p f w p 0; 0, and w p (3.8) λ ( c) qwp [, ] qwp [, ] p 0; ; w p p 5

17 i.e. he consumpion policy funcion is increasing in wealh bu decreasing in prices while he marginal uiliy of money is unambiguously decreasing in wealh bu he effec of price depends on he slope of real marginal uiliy of consumpion wih respec o price. Subsiuing ( ) c = f w, p / p for consumpion gives he marginal uiliy of money as λ p p qwp [, ] = f ( wp, ) Rewriing 3.7 in erms of qwp [, ], and he policy funcion becomes f ( wp, ) p. λ ( pqwp. [, ]) =. β( + r) q ( r) ( w y pλ ( pq[ w, p] )), π + + df( π), P w+ y qwp [, ] = max min λ, β(+ δ) q ( δ) ( w y pλ ( pqwp [, ])), π df( π), p p + + (3.9) P w+ y+ d λ p p he soluion o which will describe he marginal uiliy of money and herefore define he policy funcion. For ease of noaion denoe he following funcions ( [ ] ) = ( [ ]) P H qwp,, wp, β ( r) q( r)( w y pλ pqwp,, π df( π) λ w y = p λ + p ( [ ] ) = ( [ ]) P H qwp,, wp, β ( δ) q( δ)( w y pλ pqwp,, π df( π) λ w y d p λ + + =, p where H ( qwp [, ], wp, ) and ( [, ],, ) H qwp wp are he discouned fuure value of savings and borrowings, respecively, for curren wealh and price, w, p, and marginal uiliy of money qwp [, ]. Exending he work of Deaon and Laroque (99) and under a similar se of assumpions i is possible o show he following Theorem 6

18 There is a unique saionary soluion qwp [, ], he soluion o he funcional equaion 3.9, which is non-increasing and coninuous in wealh. In addiion he soluion can be characerised as follows qwp [, ] = H ( qwp [, ], wp, ), when [ ] β ( + r) q 0, π df( π) λ ; P qwp [, ] = λ, when [ ] β( + r) q 0, π df( π) λ, and P β( + δ) q[ 0, π] df( π) λ; P qwp [, ] = ( [, ],, ) H qwp wp, when β( δ) q ( δ d, π + + ) df( π) λ, and P β( + δ) q[ 0, π] df( π) λ; P [, ] qwp = λ, when β ( + δ) q ( + δ) d, π df( π) λ; P Proof available upon reques. This heorem formalises he inuiion given above: behaviour is deermined by he value of wealh and opimal consumpion oday relaive o valuaion of fuure wealh, for any paricular level of curren wealh and prices, expecaions and a consan se of preferences over consumpion and wealh. When he wealh level is sufficienly high ha he real value of curren marginal uiliy from consuming all of ha wealh plus income is below he fuure marginal value of zero wealh, i is opimal given he individuals preferences o shif wealh oward he fuure, increasing curren marginal uiliy and decreasing fuure marginal uiliy of wealh unil a unique level of savings equaes he wo erms. A a lower level of wealh, he individual will consume all cash on hand when i is opimal o carry zero asses beween periods, given he cos of increasing curren consumpion and reducing fuure wealh δ, and decreasing curren consumpion while increasing fuure wealh, r. When he value of carrying 7

19 zero deb beween periods is below he value of consuming all cash on hand, he individual will incur borrowings in order o decrease curren marginal uiliy and increase fuure marginal uiliy bu hese borrowings will be below he maximum amoun as long as he opimal consumpion is below he poin of he credi limi. If wealh is sufficienly low ha opimal consumpion is above he sum of wealh, income and maximum credi, he individual will consume he maximum amoun possible and would increase consumpion if i were possible because he marginal value of fuure deb is below he marginal uiliy of consuming a he credi limi. The saionary equilibrium funcion qwp [, ] will be non-linear given he min condiion resuling from he ineres rae differenial and he max condiion resuling from he limi on borrowing. In addiion he policy funcion which is recoverable by using (, ). λ (. [, ]) f wp = p pqwp is non-differeniable a he hree criical values of wealh levels where he soluion swiches beween regimes: * w< w, where * q w y pc df : P β ( + δ) ( + δ) ( + ), π ( π) = λ * (, ) f wp = w+ y+ d w < w< w where * * * q w y pc df ; P β ( + δ) ( + δ) ( + ), π ( π) = λ (, ) w + y+ d < f w p < w + y * * w < w< w, where * * 3 * 3 P ; β ( + r) q ( + r)( w + y pc), π df( π) = λ (, ) w + y < f w p = w+ y < w + y * * 3 * w3 < w where * 3 P ; β ( + r) q ( + r)( w + y pc), π df( π) = λ ( ) w + y < f w p < w+ y * 3, A low levels of wealh, herefore, he soluion racks λ, follows an opimal pah for wealh levels beween * w and * w where some posiive borrowing incurs, follows λ beween w * and 8

20 * w 3, and above * w 3 again follows an opimal pah wih consumpion less han cash on hand. Given λ < λ, he soluion funcion qwp [, ] will be flaer han λ while above λ and beween λ and λ. 4. Muli consumpion periods wihin each paymen period 4. Model In his secion he focus shifs o a model wih a number of consumpion periods, bu where paymen is received periodically. There are hree possible siuaions for he consumer: paymen occurs in his period and no in he nex period, paymen does no occur his period or nex period, and paymen does no occur his period bu does occur nex period. Thus, a hree period model is sufficien o capure he characerisics of he problem bu any number of inerim periods can be included. I seems naural o consider individual periods as weeks and each hree period cycle as a monh. The value funcion for each week will reflec he proximiy o receip of income i.e. in week i will reflec ha income is received bu ha any deb will incur an ineres cos in each week unil he nex paymen, while behaviour in week 3 accouns for he receip of income in following period and he immediae repaymen of deb. Combining his wih he cos differenial beween savings and borrowings ensures ha deb will never be incurred unil all posiive wealh is depleed. We assume ha he posiive reurn on savings and he cos of borrowing are incurred each week, so he source of borrowing is more like an overdraf han a credi card. The evoluion of wealh beween weeks follows he process ( ) w = w + y pc ( + r) + d ( r δ ), where y=0 for i. i+ i i i i The maximisaion problem herefore wrien as a Bellman equaion for each week is: { P } (, ) = max ( ) + βε (( + )( + ) + ( δ), π) V w p u c V r w y pc r d + ) pc w+ y+ d d 0 d ( δ { P } (, ) = max ( ) + βε (( + )( ) + ( δ), π) V w p u c V r w pc r d + 3 pc w+ d d 0 d ( δ ) 9

21 { P } (, ) = max ( ) + βε (( + )( ) + ( δ), π) V w p u c V r w pc r d 3 pc w+ d 0 d d Income eners he budge consrain only in he firs week and he maximum borrowing in each week is defined as oal limi on monhly borrowing d, discouned by he cos of borrowing, δ, which is incurred weekly. Therefore + ), d d ( δ ) d d/( δ /+, d3 d, while he minimum wealh possible a he sar of each week is given by w ( δ ) + d. The firs + order condiions in 3. arise for each value funcion and so he real marginal uiliy of consumpion for each period can be wrien as follows for each week * β( re ) λ( c ), + i+ * w+ y * λ( ci ) = max min λ, β(+ δ)ελ( ci+ ), p p p w+ y+ d i λ p p (4.) for d = d /( + δ ), d d/( δ = + ) and d3 = d and where y=0 if i. The inerpreaion of he Euler equaion is similar o he simpler one period case from he previous secion and again encompasses he four possible posiions ha he individual can be in for each week. The individual will save and earn r when wealh is sufficienly high ha he opimal consumpion is below cash on hand, maximum uiliy involves subsiuing consumpion owards fuure periods, and he Euler equaion is fulfilled in an unconsrained way. Consumpion will exacly equal cash on hand when he individual is neiher sufficienly paien enough o save a r or impaien enough o borrow a δ. An opimal consumpion which involves saving a δ i.e. decreasing consumpion below cash in hand is unaainable, so he individual will insead accep a marginal uiliy level equal o λ w+ y p p when he opimal would be above ha; hence he need for he minimum condiion. The Euler equaion is also 0

22 fulfilled wihou consrain in he hird regime which involves an opimal level of borrowing a δ which is below he maximum level. Finally for any given level of wealh, he minimum aainable marginal uiliy is a he poin of maximum borrowing is in he fourh regime and he soluion will be a his poin when all oher consumpion levels imply a lower unaainable marginal uiliy; hence he soluion mus be w y d λ + + p p i, he marginal uiliy from he consrained level of consumpion. Considering he behaviour over he paymen cycle, an individual who consumes in regime in he firs week i.e. consumes all available wealh plus income, (or repays all deb ou of income and consumes he remainder) mus consume ou of borrowing in he remaining weeks, i.e. mus be in regime 3 or 4. Similarly, an individual whose opimal consumpion is very high relaive o he real value of wealh and herefore chooses o be in regime 4 wih maximum possible borrowing, will have zero or negaive consumpion in he following weeks unil he nex income is received. However his is unlikely o be a soluion once here are resricions placed on preferences o allow only sricly posiive consumpion levels. The saionary soluion o equaion 4., will define he consumpion funcion relaionship beween wealh, prices and expendiure in each week. Using he same noaion as Secion 3, denoe pc f ( w, p) i real wealh as q [, ] =,for i =,,3, for all wealh and prices. Denoe he marginal uiliy of i i w p and we assume he opimal policy funcion is increasing in wealh bu decreasing in prices while he marginal uiliy of money is decreasing in wealh bu he price effecs depend on he slope of real marginal uiliy of consumpion (as in 3.8). Subsiuing ( ) c = f w, p / p for consumpion gives he marginal uiliy of money as i q [ w, p ] = f ( wp, ) i i λ p p (, ). λ (. i[, ]) i, and he policy funcion becomes f wp = p pq wp. Wriing he Euler equaion 4. in erms of he marginal uiliy

23 of money, gives he following funcional equaion o which here is a unique soluion q [ w, p ] for i=,,3 for each equaion given by: i q i [ w, p] = β( + r) q i+ ( + r)( w + y pλ ( pqi[ w, p] ), π df( π), P w+ y max min λ, β( + δ) q i+ ( δ)( w y pλ ( pqi[ w, p] ), π df( π), p p + + P w+ y+ d i λ p p (4.) The funcions H ( qi [ w, p], w, p ), ( i [, ],, ) H q w p w p, λ, λ, have analogous definiions o Secion 3 i.e. where H ( qi [ w, p], w, p ) and ( i [, ],, ) H q w p w p are he discouned fuure value of savings and borrowings, respecively, for wealh oday w, price oday p, and marginal uiliy of money [, ] q w p in week i. Following he analysis in Bailey and Chambers (996), i who grouped ime spells ogeher ino epochs in he analysis of season harves prices, we can develop a similar proof in order o sae he following Theorem : There is a unique se of funcions [, ] q w p, he saionary soluions o he funcional equaions i (4.) above, which are non-increasing and coninuous in wealh. In addiion he soluions can be characerised as follows qi [ w, p ] = H( qi+ [ w, p], w, p), when i+ [ ] β ( + r) q 0, π df( π) λ ; P i [, ] q w p = λ, when [ ] β( + r) qi+ 0, π df( π) λ, and P β( + δ) qi+ [ 0, π] df( π) λ; P

24 qi [ w, p ] = H( qi [ wp, ], wp, ) +, when β( δ) q i ( δ di, π ) df( π) λ, and P β( + δ) qi + [ 0, π] df( π) λ; P [, ] qi w p = λ when i β ( + δ) q + ( + δ ) di, π df( π) λ. P Proof available upon reques. The heorem ses ou he condiions for each regime: posiive saving is opimal when he wealh level is sufficienly high ha he real marginal uiliy from consuming all cash on hand, λ, is below he discouned fuure real value of he marginal uiliy of money from zero savings. In his case, consumpion is shifed owards he fuure so ha curren marginal uiliy increases and fuure marginal uiliy of money decreases unil a unique level of saving is reached ha resuls in equaliy beween he wo erms. When he wealh level is such ha he real marginal uiliy from consuming cash on hand is above he discouned marginal uiliy of zero savings in he following week discouned a r, he individual has an incenive o increase consumpion and carry negaive wealh ino he nex week. However, he cos δ of borrowing may bring he discouned fuure marginal uiliy of zero borrowing above he marginal uiliy of cash on hand in which case he preference for higher consumpion is no srong enough ha he individual is willing o borrow a δ. When wealh is lower however, i is opimal for he individual o have a posiive level of deb below he limi which decreases curren marginal uiliy of consumpion while increasing he fuure marginal uiliy of wealh. Borrowing is below he limi provided he marginal uiliy from maximum negaive wealh in he following week is above he marginal uiliy of maximum consumpion oday. Finally an individual consumes a maximum deb when opimal consumpion is so high relaive o he real value of wealh, ha even a maximum deb he individual would furher decrease curren marginal uiliy of consumpion by borrowing if i were possible. We assume CRRA preferences wih uiliy funcion of he form uc [] = c ρ for ρ >, ρ where ρ is he coefficien of relaive risk aversion and he simulaed numerical soluion is shown in Figure for a CRRA uiliy funcion where we assume β = 0.95, and ρ =.5. The cos of borrowing δ, is se equal o 0.4% giving an annualised rae of 5% which is 3

25 represenaive of he raes applicable on very shor-erm borrowing. Sandard numerical mehods of dynamic programming as oulined in Deaon (99,99) are used o reach he soluion. Figure : Soluion for four weeks The u-shape is eviden a all levels of wealh: consumpion is highes in he week when income is received, lowes in he nex week and he increasing over he remaining weeks. The model replicaes almos perfecly he paern in he daa in Figure. Hence he FES daa is he clearly he oucome of opimal behaviour once we allow for precauionary moives and limis on borrowing for he ype of expendiure considered here. This is despie he relaively simple uiliy funcion used in he simulaion of he model. The kinks a he poins where he soluion swiches beween differen regimes are clearly eviden in week four and so for any given wealh level is his week we can predic perfecly he regime wihin which he individual will be and hence heir borrowing/saving behaviour. The soluion also shows ha week four is he only week for which regime four can occur and again his is consisen wih he infinie cos of zero consumpion in he uiliy funcion ha we have assumed. 4

26 5. Empirical Esimaion The Family Expendiure Survey (FES) in he UK provides expendiure paerns for wo consecuive weeks for each individual in he survey and also records he daes over which his expendiure is recorded, and he level of and iming of he receip of regular labour income. The sample of ineres for he esimaion of our model is hose individuals who are in paid employmen and receive income on a monhly basis, and from he daa we can calculae a which poin in heir paymen cycle each individual is observed. The model however, is one of non-durable consumpion and so income should be inerpreed as a measure of disposable income, i.e. afer regular paymens such as housing and heaing coss, credi repaymens ec. are made. 8 The FES gives very deailed informaion regarding loans and higher purchase schemes ousanding for all individuals bu records housing and heaing coss a a household level only. Therefore we include only single individuals who are heads of households, abou hree fifhs of whom are homeowners and we can hen deduc regular coss from monhly labour income o give a measure of disposable income available for non-durable consumpion. This also allows us o use he household level indicaor of credi card ownership raher han he individual variable as an indicaor of possible liquidiy consrains 9 and absracs from he issue of resource pooling in households who receive income a differen poins in ime. The daa on hese individuals are pooled for hree years of he FES giving a oal of 77 observaions. The firs panel in Table provides some summary saisics according o ownership of a credi card. There is a clear income effec in credi card possession wih higher monhly income and living coss for card holders bu sill have higher disposable income. The second panel of Table shows he mean of weekly expendiure as a proporion of monhly disposable income for each week in he cycle. The range of expendiure share is very large and i would seem difficul for any model o reconcile individuals who spend less han 0.05% of heir disposable income in week one (weny observaions) wih hose who spend more han heir whole monhly disposable income (seven observaions) in week one. In addiion our model predics ha expendiure in he firs week should never be less han 0.0 of disposable income even a minimum wealh levels and for he wors price draw (see Figure ) alhough in fac here are 08 observaions in he daa for which his is no rue. Examining he means in each week 8 In he model, wealh should also be inerpreed as liquid cash on hand raher han (illiquid) saving or invesmens for long-erm purposes, and income refers o disposable income. 9 The recording of individual credi card ownership relies on he individual recalling paying an annual charge for heir card bu card possession a a household level is recorded irrespecive of charges. In his sample only 60% hold a credi card when using he individual level quesion, while 7% do using he household quesion, which seems more accurae given he choice of sample i.e. single salaried employees. 5

27 reveals ha he u-shape paern over he monh which is eviden for he whole daase and is prediced by he model is also clearly eviden for his sub sample of individuals alhough for non-credi card holders, he effec is less obvious. Table 3 Credi Card holders: N=56 Non Credi Card holders: N= Mean Sd Dev Mean Sd Dev Age Monhly Income Housing Cos Elecriciy Credi Repaymens Disposable Income Weekly expendiure as a share of monhly disposable income Credi Card holders Non Credi Card holders N Mean Sd Dev Min Max N Mean Sd Dev Min Max Week Week Week Week Esimaion procedure The simulaed policy funcion show in Figure is calculaed over an equally spaced grid of poins of W possible values for wealh and H possible values for price. Therefore he numerical soluion o he policy funcion is a W x H able of values each corresponding o a paricular combinaion of wealh and price. This soluion allows us o relae any level of wealh and price o he opimal consumpion by linear inerpolaion beween hose grid poins. Thus for any given wealh, price, and parameer vecor θ, we can calculae he opimal oucome as 6

28 (, ; ) pc = f w p θ. We can hen esimae he parameers which mos closely mach he observed expendiure level o ha implied by his non-linear policy funcion. However, in he daa neiher wealh nor price daa are available and so he esimaion insead relies on he relaionship beween expendiure in adjacen weeks. In shor, we ake expendiure in week and by invering he policy funcion for week we can find for any given price, he level of wealh combinaion consisen wih ha expendiure observaion, i.e. observed expendiure will give an implied level of cash on hand w f ( pc ; p, θ ) =. The success of his relies upon he monooniciy of he funcion if here is o be a unique associaion beween an expendiure observaion and wealh level. In addiion, he concaviy of he consumpion funcion means ha he soluion poins for expendiure for he wealh and price poins on he grid are no equally spaced. However, he use of linear inerpolaion ensures ha neiher of hese issues is a problem alhough i does require relaively large values for W and P. Once he implied wealh level is calculaed, he cash on hand carried ino he nex period + can be calculaed using he equaion for he evoluion of wealh: ζ ( θ) w + = ( + )( f pc; p, + y pc) where y=0 if, ζ = δ if ( w p c ) < 0, and ζ = r oherwise. Condiional on expendiure in and price oucomes in and +, p, π + respecively, expendiure in + can herefore be calculaed as ( ( ) ) ( ) ( ) ( ) p c ; pc, p, π, θ = f + ζ f pc ; p, θ + y pc, π ; π, θ, and by aking an expecaion over he price disribuion in he second period, we can arrive a an expecaion of expendiure for he second week of observaion condiional upon he observed level of expendiure in he firs week, he price oucome in he firs week and he parameer vecor which is given by ( ( ( ) ) ) π ( ) ( ) ( ) + π+ However he implied level of cash on hand w f ( pc ; p, θ ) E p c ; pc, p, θ = E f + ζ f pc; p, θ + y pc, π ; θ ; pc, p possible firs period prices w = mus be calculaed for all p. Therefore aking an expecaion again over he price disribuion gives he firs condiional momen of he second period expendiure as ( ; ) ( ;, θ = p ) π θ = (( )( ( ) ) π ζ θ π θ ) + m p c E E p c p c ( ;,, ; ; ) Ep E f f pc p y pc pc. The second momen consiss of he expecaion of variance of expendiure p c ; pc, p, π, θ around each condiional mean E ( p c pc p θ ) π ;,, + + +, his being 0 The calculaion for Week omis an observaion for which he repored expendiure is 6 imes income: his observaion disored he mean and sandard deviaion 7

29 due o he uncerainy of prices in +, plus he expeced variance around each mean condiional only on observed expendiure i.e. ( ; θ) = ( ;,, π, θ) [ ;,, θ] v pc E p c pc p E p c pc p πτ π [ ;, θ] ( ; θ) + E p E π p c pc Eπ Eπ p c The second exra erms is due o he fac ha firs period price is no observed o he economerician. Esimaion is carried ou using he pseudo maximum likelihood esimaor (PMLE) of Gourieroux e al, (984). The general mehodology used is herefore very similar o ha used by Deaon and Laroque (995,996) which esimaed a model of commodiy prices where he policy funcion is also dependan on wo unobservable variables. The log of he PMLE wrien as a funcion of he K-vecor of parameers θ and j=..n observaions in he daase ( p c m( p c ; θ )) v( pjcj ; θ ) N N N ( ( )) ln L = ln l = ln v p c ; N N j+ j+ j j j ( θ) j j θ (5.) j= j= j= The variance covariance marix for he PMLE is calculaed as ( ' ) V J G G J = (5.) where G is a (N-) x K marix where elemen G jk ln l j = and J is a K x K marix wih θ k elemen J ik ln L =. θ θ i k The relaionship beween wealh and expendiure has been shown o be non-linear wih kinks exising a he poins where he soluion swiches beween he four regimes, Therefore he likelihood will be non-differeniable wih respec o he parameers a hese poins, bu as shown in Laroque and Salanie (994), a PMLE will resul in consisen esimaes of he parameers despie he non-differeniabiliy. However, he non-differeniabiliy leads o subsanial difficulies in relying on numerical derivaives in he maximisaion procedure and so we use alernaive maximisaion mehods in he esimaion. Throughou he analysis we assume ha he price disribuion is he same in all periods and ha draws from ha disribuion are iid. To simplify he calculaions we consider H discree values from F and hese discree values are naurally he grid poins used in he numerical soluion. 8

30 Thus we avoid numerical inegraion in calculaing he expecaions over price. Equaion 6.3 shows he calculaion of he condiional expecaion and variance of second period expendiure under hese assumpions, for h=, H draws of price in period denoed by p, in +, denoed by π +. The expecaions are now calculaed as simply weighed averages of he oucome for each price draw. ( ; θ) = p ( ) ;, π pc θ H H f+ (( ζ )( f ( pc ; ph, θ) pc ), πh; θ) m pc E E p c = + H h= h= h= h= (( ) ( )) v( pc ; θ) = E p E ( ;, ) ; π p c pc θ m pc θ H H = ( f+ (( ζ )( f ( pc ; ph, θ) pc ), πh; θ) m( pc ; θ + )) H (5.3) These condiional momens are hen subsiued ino (5.) o give he PMLE. We carry ou a small Mone Carlo sudy o invesigae he properies of he PMLE in his framework using he same parameers values as previously: β = 0.95, ρ =.5, δ = 0.4% and r = 0. The parameer vecor in he esimaion procedure could include β, ρ, δ, rd, and possibly he parameers of he price disribuion also, bu we do no ry o idenify all hose parameers from he daa and insead resric he esimaion o he coefficien of risk aversion ρ. The replicaions are carried ou for hree values of ρ : (.5,.5, 5), and for wo differen price disribuions (normal and uniform). In he case of he uniform disribuion, prices can ake on any of six equally spaced discree values beween 0.75 and.5, for which he expecaion is one. We use a normal disribuion N(,0.7 ) so ha i has he same suppor as he uniform disribuion bu wice he variance. This is discreised by six poins, each poin being he midpoins of he inerval wihin which /6 of he densiy lies. (See Deaon and Laroque, 996). For each of hese scenarios we draw 00 daases of eiher 500 or 000 observaions. When esimaing jus one parameer, we use a simple golden-secion procedure o find he maximum and find his very fas and reliable. The resuls are presened below. The empirical disribuion of he esimaor is very close o he asympoic variance given by (5.) and in all cases he esimaed parameer is wihin wo sandard deviaions of he rue value. Therefore i is possible ha F is a discree disribuion or is a coninuous disribuion approximaed 9

31 These resuls sugges ha here are no obvious problems wihin he esimaion procedure and all he maxima are robus o changing he saring values. Table Mone Carlo resuls Uniform Disribuion of Prices ρ Sample Size Mean Esimae Sample SE Asympoic SE Normal Disribuion of Prices ρ Sample Size Mean Esimae Sample SE Asympoic SE The simulaed daase gives us an idea of wha we should expec o see in he acual daase if i is o be described by our model. Some problems were idenified earlier in his secion when we found very small and very large observaions and observaions below he minimum prediced by he model. However comparison of he simulaed and acual daases shows ha he change in expendiure beween weeks is very large in he acual daa relaive o wha he model predics in he simulaed daase. This is because for large ranges of wealh, he condiional expecaion given in 5.3 can be closely approximaed by a sraigh line wih a slope less han one. Figure 3a shows a hisogram of he proporional change in expendiures beween weeks wo and hree from a simulaed daase. The mean is close o 0.05% wih variance of 0.6. This is in sharp conras o he same hisogram for he acual daase from he FES where alhough he mean using a mehod of quadraure. 30