Tax, Credit Constraints, and the Big Costs of Small Inflation. Andrew Coleman Motu Working Paper Motu Economic and Public Policy Research

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1 Tax, Credi Consrains, and he Big Coss of Small Inflaion Andrew Coleman Mou Working Paper Mou Economic and Public Policy Research

2 Auhor conac deails Andrew Coleman Mou Economic and Public Policy Research Acknowledgemens I would like o hank Ma Benge, Arhur Grimes, Suzi Kerr, Dave Maré, Gran Scobie, Dominic Sephens and seminar paricipans a he Reserve Bank of New Zealand for commens on his paper. Financial suppor from he Marsden fund is graefully acknowledged. Mou Economic and Public Policy Research PO Box Wellingon New Zealand info@mou.org.nz Telephone Websie Mou Economic and Public Policy Research Trus and he auhors. Shor exracs, no exceeding wo paragraphs, may be quoed provided clear aribuion is given. Mou Working Papers are research maerials circulaed by heir auhors for purposes of informaion and discussion. They have no necessarily undergone formal peer review or ediorial reamen. ISSN (Prin), ISSN (Online). i

3 Absrac This paper develops an overlapping generaions model incorporaing credi consrains, owneroccupier and renal secors, and deailed ax regulaions o examine how he ineracion of he inflaion and he ax sysem affec he housing marke. I shows ha even modes raes of inflaion can have very large effecs on he home-ownership raes of young households, paricularly a low real ineres raes. This occurs even if here is a large supply response in he quaniy of housing. The model suggess ha he welfare coss of inflaion could be amelioraed by exemping he inflaion componen of ineres paymens from income ax. This version (published Augus 2009) refers o addiional resuls from an exended model developed for Mou Working Paper 09-13, The Long Term Effecs of Capial Gains Taxes in New Zealand. JEL codes E40, E58 Keywords Inflaion, credi consrains, capial income axes, housing markes, home-ownership raes, moneary policy. ii

4 Conens 1. Inroducion

5 1. Inroducion This paper develops a model ha analyses he long run effecs of inflaion on housing markes. I has been sparked by concern among New Zealand policy makers ha he homeownership raes of young, low-income households are declining. The decline in home-ownership raes does no appear o be volunary, for i has no been welcomed by younger, poorer, households. Raher, i has been accompanied by complains ha landlords have ou-bid poenial owner-occupiers because of ax laws ha favour invesmen in propery raher han ineres earning asses when here is inflaion. The argumen ha he ineracion of inflaion wih he ax sysem can cause subsanial economic disorions is well esablished (for example Viner (1923); Aaron (1976); Fischer and Summers (1989) or Feldsein (1996, 1997)). The primary issue concerns he axaion of capial income. Since mos counries ax nominal ineres earnings raher han real ineres earnings, and since nominal ineres raes ypically rise in response o an increase in long-erm inflaion expecaions, he real afer-ax reurn on ineres earning asses declines as inflaion rises. This creaes an incenive for owners of capial o seek ou ax-shelered asses as he inflaion rae increases. A favoured invesmen class is residenial housing, for in he long run prices end o rise a he rae of inflaion bu he capial gains are axed lighly, if a all. By iself, he ineracion of inflaion wih he ax sysem is no enough o reduce home ownership raes, for residenial propery acs as a ax sheler for owner-occupiers as well as landlords. Indeed, owner-occupiers wih significan equiy in heir houses gain a larger ax advanage from housing han landlords, for impued ren is ypically no axed. Bu inflaion has a second effec on capial markes: i exacerbaes he credi consrains facing borrowers. This is because bank imposed resricions on he amoun households can borrow are rarely adjused for inflaion, even hough nominal ineres raes increase when he inflaion rae rises. If banks do no increase he amoun credi-consrained households can borrow when nominal deb servicing paymens increase, i becomes more difficul for hese households o purchase houses (Modigliani (1976); Kearl (1979)). The heoreical model developed in his paper aemps o unravel hese compeing effecs. The model analyses how he ineracion of households who differ by age and income deermines house prices, and how hese prices affec housing allocaions. Because he ax and credi marke effecs of inflaion depend on he wealh and income of households, and because hese depend in urn on household age, a version of he Modigliani-Brumberg syle overlapping generaions model developed by Oralo-Magné and Rady (1998, 2006) is used. 1

6 In he model, here is a plehora of households ha differ by age, income, and marginal ax rae. These households choose o consume goods and services, o live in large or small houses, and o ren or buy. They save for reiremen, pay ax, borrow and lend, face realisic borrowing consrains, and choose wheher or no o inves in housing. House prices are deermined endogenously by maching he supply of houses wih he collecive demand for housing by owner-occupiers and landlords. The model is used o idenify some key economic facors ha deermine when he ineracion of inflaion and he ax sysem are likely o have large effecs on he housing marke. One of he mos imporan facors is he long run supply elasiciy of housing. This paper examines he cases when he housing supply is compleely inelasic and when i has (approximaely) uni elasiciy. A second facor is he exen o which he oal demand for housing changes wih house prices and rens, as people choose o live ogeher raher han alone. In his paper, he elasiciies of he oal demand for housing wih respec o house prices and house rens are implicily derived from he consumer maximisaion problem. Following Oralo- Magné and Rady, he main resuls concern he case when variaion in he number of households occurs because of variaion in he age a which adul children choose o leave he parenal home. However, he model has also been solved for he siuaion ha people can share rened housing, which increases hese elasiciies. The laer case is discussed a lengh in a companion paper ha analyses how capial gains axes may affec he housing marke (Coleman 2009). The model suggess ha small changes in he inflaion rae can have very large effecs on home ownership raes even when here is no uncerainy and inflaion is perfecly anicipaed. In he main parameerisaion sudied, in which real ineres raes are 5 percen, a one percenage poin increase in inflaion leads o an 8-11 percen age poin decline in home ownership raes among young households. This occurs wheher he housing supply is elasic or inelasic. When inflaion rises, young households find morgage repaymens more onerous because nominal ineres raes increase. If he housing supply is inelasic, landlords araced by ax-free capial gains bid up propery prices and young households are squeezed ou of he propery marke. If he housing supply is elasic, landlords bid down rens, and credi consrained households are induced o delay purchasing a house and ren insead. While he size of he effec of inflaion on home-ownership raes seems large, i should be noed ha if real ineres raes are 5 percen a one percenage poin increase in he inflaion rae increases nominal ineres paymens by approximaely 20 percen. I is no implausible ha a price change of his magniude could induce a large number of young households o delay he purchase of a home and ren insead. In he model, he change in he number of households 2

7 choosing o ren when he inflaion rae increases is an increasing funcion of he discoun rae and a decreasing funcion of real ineres raes. When real ineres raes are above 6 percen (as hey were in New Zealand beween 1985 and 1998) moderae inflaion has relaively lile effec on he fracion of households choosing o ren. The effec of inflaion depends on he ineracion of axes and credi consrains, and largely disappears if here is only one disorion. When households can borrow as much as hey like (bu sill repay heir deb), moderae inflaion raes have almos no effec on homeownership raes irrespecive of ax raes. Households simply borrow enough early in life o make he addiional morgage paymens wihou reducing real consumpion, knowing hey will be able o repay he deb laer on. If banks impose credi consrains bu capial income is no axed, a one percenage poin rise in inflaion leads o a 2 3 percenage poin reducion in homeownership raes among young households, raher han an 8-11 percenage poin reducion. Bu axaion of capial income is no he primary reason why inflaion has such a big effec on homeownership raes when here are credi consrains. Raher, i is he axaion of he inflaion componen of ineres paymens. If real ineres income is axed a normal income ax raes bu he inflaion componen of ineres earnings is exemped from ax, a one percenage poin increase in inflaion again only leads o a 2 3 percenage poin reducion in home ownership raes. This reinforces he radiional view of economiss ha axing he inflaion componen of ineres earnings is highly disorionary, even a wha has been considered low inflaion raes (Aaron 1976; Feldsein 1997). The model is used o calculae how he inflaion rae affecs welfare. Under he assumpions ha households ha he supply of housing is inelasic, ha agens have a log-linear uiliy funcion, and ha ineres income is axed, an increase in he long erm inflaion rae from 0 o 3 percen lowers lifeime welfare for all agens. In he main simulaions examined, he welfare loss is equivalen o beween 2 percen and 4 percen of lifeime consumpion. This loss occurs for wo reasons. Firs, inflaion leads o higher nominal ineres raes and higher house prices, forcing young people o ren for longer or o buy a small sarer house raher han a large house. Secondly, i reduces real ineres raes, disoring iner-emporal consumpion and providing less incenive o accumulae capial o spend in reiremen. However, inflaion is no necessarily bad when here is an elasic supply of housing. In his case, inflaion will raise he lifeime welfare of low income households by reducing he ren hey pay early in life, enabling hem o increase consumpion when young. Inflaion sill causes welfare losses for middle and high income households, however, as he higher nominal ineres raes make house purchase more difficul while young, and he lowers afer ax real ineres raes disor he iming of 3

8 consumpion. Noneheless, when here is an elasic supply of houses, he overall welfare consequences of inflaion will depend on he relaive numbers of low income and high income households. The resul ha inflaion is good for young people who ren has no been prominen in he lieraure (for an excepion, see Pozdena 1988). While his resul may seem perverse, i occurs because of he exisence of wo disorions: he resricions imposed on young people ha preven hem borrowing agains fuure income; and he excessive axaion of real ineres earnings when here is inflaion. In his world of second bes, inflaion can improve he welfare of credi consrained young people who ren by inducing unconsrained landlords o offer hem low rens. The model highlighs wo issues relevan o curren policy in New Zealand. Firs, given he srucure of axes in New Zealand, in which capial gains are exemp from ax, he recen deerioraion in housing affordabiliy may have been parly caused by he increase in he inflaion arge in 2001 and he subsequen increase in he average inflaion rae. The analysis in his paper does no suppor he view ha i is largely inconsequenial wheher inflaion is in he op end or he boom end of he 1 3 percen inflaion arge range. Secondly, if i is oo difficul for he cenral bank o achieve very low raes of inflaion in he medium erm - say 0-2 percen - he adverse effecs on homeownership can be amelioraed hrough ax reform. The model is relaed o several earlier papers. I is concepually similar o Feldsein (1997) as i examines he welfare effecs of low inflaion, bu i clarifies some of his insighs by incorporaing a renal housing secor and by allowing he afer-ax ineres rae o depend on wheher an agen is a borrower or lender. I builds on he model used by Slemrod (1982) and Hayashi, Io, and Slemrod (1988) o sudy individual opimisaion over a lifecycle when here are axes and credi consrains, bu exends i by finding equilibrium house prices. I exends he equilibrium lifecycle model of housing markes analysed by Oralo-Magné and Rady (1998) and Coleman (2007) by incorporaing axes and by allowing he supply of housing o be elasic. The paper is organised as follows. In secion 2, he deails of he model are presened and a skech of he soluion echnique is offered. In secion 3, key resuls of he model are shown for four differen scenarios: when he housing supply is elasic or inelasic; and when he inflaion componen of ineres is axed or ax exemp. This is followed by an ouline of he way he resuls of he model depend on key parameers such as ineres raes and credi consrains in secion 4. Finally, he resuls are discussed and conclusions are offered in secion 5. 4

9 2. An inergeneraional model of housing demand 2.1. The basic framework The paper calculaes equilibrium prices, saving paerns, consumpion and housing allocaions in an overlapping generaions model of an economy ha comprises a muliude of households ha differ by age and income. The model is an exension of he model used by Coleman (2007) o analyse he effec of inflaion and credi consrains on he housing marke. In urn, i is based on he housing model of Oralo-Magné and Rady (1998). In he model, here are four cohors, each of which lives four periods and hen dies. Agens have exogenously deermined labour income and consume a single non-sorable good. They also gain uiliy from rening or purchasing a single uni of housing. These housing unis come in wo sizes, small flas or large houses. Agens choose among differen paerns of housing and consumpion o maximise heir uiliy. Agens can borrow or lend a exogenously deermined ineres raes, alhough young agens face credi consrains. Agens can also become landlords. They pay income ax on any ineres earnings or on renal income. In he las period of life agens consume all wealh excep heir house, which is inheried by a younger generaion. The model is solved under wo differen assumpions abou he supply of housing. In he firs case, he supply of flas and houses is deermined exogenously. In he second case, separae upward sloping supply curves for he quaniy of flas and houses are specified, so ha he equilibrium number and mix of properies is deermined endogenously Agens The N agens in each cohor live for four periods labelled i = {0,1,2,3}. A period is T years long. Agens differ by income and while any paern of income is possible, agens are assumed o have a consan place in he wihin-cohor income disribuion. Agen 1 has he lowes income. In period, agen j born in period -i has real labour income Y = ω gy (1) i, j 0 j i i where ω j = idiosyncraic facor affecing agen j relaive o average cohor earnings; g i = facor reflecing he life-cycle earnings of he cohor in is i h period; and 0 Y i= average income of cohor a ime of birh. 5

10 Nominal income is PY i, j, where P is he pre-ax price of he good. An indirec goods g and services ax is applied o goods oher han housing a rae τ, so he pos ax price of he g good is (1 + τ ). Incomes and he prices of goods boh increase a a consan inflaion rae π, P where 1+ π = P+ 1 P. consumpion Agens obain uiliy from he consumpion of goods and housing. An agen chooses real i j c, i, jh, i, jr, i, jf, i, jh, I ={,, }, and has housing choices described by a vecor of hree indicaor variables I I I ha equal one if he agen has housing enure h in period i of his or her life a ime, and zero oherwise. There are hree possible housing enures: an agen can ren a fla (R), purchase a fla (F), or purchase a house (H). Age zero agens can live wih heir parens a zero cos, alhough hey gain zero uiliy from doing so. 1 In period agens obain uiliy uc (, I ) ln( c ) vi (2) = + i, j i, j, h i, j h i, j, h h H F F R I is assumed v > v as houses are bigger han flas, and v > v, as agens can shape an owned fla in heir own image, whereas hey canno modify a rened fla. Agens can only live in one housing uni in any period. Agens born a ime choose consumpion and housing pahs o maximise discouned lifeime uiliy: U 3 i i, j i, j, h β uc+ i, + i i= 0 = ( I ) (3) Households are assumed o receive heir income, purchase, ren, or sell propery, borrow or lend, and consume a he sar of each period, alhough hey gain uiliy from housing by living in i hroughou he period. In he las period, agens are assumed o sell or realise all asses excep heir las owned housing uni, repay any debs, and consume all of heir wealh. They die a he end of period 3, a which poin heir housing uni is disribued o younger cohors. A ime a fracion κ i is lef o he cohor born a -i for i=0,1,2; by assumpion i is eiher equally disribued across all agens of ha cohor, or lef o he jh agen, hus preserving he inergeneraional income ranking 2. The weighs are chosen so ha agens do no receive an κi 1 Coleman (2009) alers his srucure and allows households o ren half a house. If hey do so, hey pay half he full ren, bu he uiliy hey ge is a free parameer ha can be allowed o be smaller han, equal o, or greaer han half he uiliy from rening a whole house. The resuls are similar, bu here are some suble differences in he properies of he model. The main difference is ha more low income people ren when hey can ren half a house as his allows hem o consume more when young. See Coleman (2009) for a longer discussion. 2 Coleman (2009) has a hird opion: every odd numbered person receives no inheriance, while every even numbered person inheris wo properies. The resuls are similar, indicaing ha he resuls are no dependen on he assumpion ha agens receive some inheriance. 6

11 inheriance unil relaively lae in life. This proves o be an imporan choice, for if agens inheri early, hey have much less need o save for a housing uni. In his paper, κ 2 = 1. In he uiliy maximisaion equaion below, Inheri is he average value of he inheriance lef by he generaion dying a he end of ime -1 and inheried a ime Taxes and he housing marke Because he focus of he paper is he way capial income ax affecs housing and consumpion choices, he ways ha he New Zealand ax sysem affecs housing have been carefully modelled. Five feaures of he ax sysem have been modelled. Firs, ineres and ren income is axed a an agen s marginal ax rae. There are wo marginal ax raes: τ 1 for agens wih real income in period less han τ * ; and τ 2 τ 1 for agens wih real income greaer han or equal o τ *. I is assumed ha he ax hreshold is auomaically adjused for inflaion and hus consan in real erms. Secondly, he capial gains ax rae is zero. No propery appreciaion, eiher for an owner-occupied house or for a leased fla, is axed. Thirdly, impued ren is ax exemp. Fourhly, a landlord can deduc ineres paymens associaed wih a morgage when calculaing axable income. Thus a landlord pays ax on ren ne of ineres paymens, bu no ax on any capial appreciaion. Fifhly, here is a goods and services ax ha is applied o consumpion bu no o ren or propery. In he model, he goods and service ax rae is se endogenously a a rae ha makes he oal ax ake (ax on capial income plus ax on goods and g* services) equal o a se fracion of labour income, in his case τ = 10 percen. This ensures ha any changes in he srucure of capial incomes axes do no have revenue implicaions for he Governmen. Agens do no receive uiliy from governmen expendiure. Alhough he deails of he ax sysem are closely based on he New Zealand ax regime, i is sraighforward o change heses deails. In he paper he effecs of an alernaive ax regime in which real ineres earnings raher han nominal ineres earnings are axed are also examined. Coleman (2009) invesigaes he effecs of inroducing differen ypes of capial gains axes: many of he resuls are qualiaively similar, so long as he capial gains ax rae is lower han he op marginal ax rae. Flas and houses cos F P and H P o purchase. Flas can also be leased, a price R P ha is paid in advance a he beginning of he lease. The ren is paid o a landlord, who, for convenience, is resriced o be an agen in period 2 of heir lives. The number of landlords is,, * endogenous; an indicaor variable I i jr indicaes wheher or no he j h agen owns a renal 7

12 propery. 3 Because here is no uncerainy, he afer-ax reurn from purchasing a fla in period, leasing i, and selling i in period +1 is equal o he afer-ax reurn from lending money. As such, he relaionship beween ren, ax raes, fla prices, and ineres raes is or P (1 τ )(1 + r(1 τ )) + P = P (1 + r(1 τ )) (4) R T F F T (1 + r (1 τ )) (1 + π ) = T F R F 2 P T (1 τ2)(1 + r (1 τ2)) P (5) F where π is he rae of price appreciaion for flas. The righ hand side of equaion 4 is he afer-ax reurn in period +1 from invesing P in ineres earning bonds. The lef hand side is he afer-ax reurn a +1 from using he same sum o purchase a renal fla a ime. I F comprises he afer-ax ren paid a ime and reinvesed a ineres, plus he unaxed proceeds from selling he renal uni a ime +1. Since ineres paymens by landlords are fully ax deducible, he reurn o a landlord is independen of heir level of gearing. I is assumed ha he landlords are high income agens in period 2 of heir lives, so aferax reurns are calculaed using he op marginal ax rae τ 2. 4 In each period, agens choose beween one of he hree housing opions, or no having housing. Consequenly, here are poenially 256 differen housing paerns possible hrough an agen s lifeime. Raher han calculae he uiliy of each of hese paerns, I only le agens choose from a much smaller se of paerns, Η. To reduce he number of possible paerns, I impose a series of resricions on he lifeime housing opions available o an agen. The hree resricions are: (i) only 0 period agens may choose no housing; (ii) only period 0 and period 1 agens may choose o ren; and (iii) excep in he las period, agens housing choices mus no worsen hrough ime 5. By his means, he se H is reduced o 23, H ={0RFF, 0RHF, 0RHH, 0FFF, 0FHF, 0FHH, 0HHF, 0HHH, RRFF, RRHF, RRHH, RFFF, RFHF, RFHH, RHHF, RHHH, FFFF, FFHF, FFHH, FHHF, FHHH, HHHF, HHHH}. An agen s opimal discouned uiliy is calculaed for each of hese paerns, and he agen is assumed o choose he paern ha provides he greaes discouned uiliy. The model is solved using wo differen 3 If here is demand for f flas, he f highes income individuals are assumed o own one fla each. 4 If here is a high demand o ren propery, i is possible ha he las landlord in he model is on he low marginal ax rae. Noneheless, i is assumed ha compeiion beween high income landlords deermine rens, so he op marginal rae is used. 5 Coleman (2009) relaxes hese resricions furher, allowing agens ren in he hird and fourh periods of heir lives if hey ren hroughou heir lives. The resuls are qualiaively similar, alhough he fracion of he populaion rening increases. 8

13 assumpions abou he sock of housing. In he firs, he number of houses and flas is se exogenously and is compleely inelasic. There are n H houses and n F flas. I is assumed ha H F n + n 4N. This means here are no vacan houses, so propery prices and rens are posiive in equilibrium. In he second, here is an elasic supply of flas and houses, and he quaniy of each is deermined in equilibrium along wih rens and prices. Linear supply funcions are specified: ( ) P = α + α Q + Q F F F F H 0 1 P = P + α + α Q H F H H H 0 1 In his specificaion he price of flas is an increasing funcion of he oal number of properies (o reflec he possible scarciy of land), while he price of houses is deermined as a variable premium over he price of flas (o reflec he addiional building coss). In mos of he simulaions presened below, parameers are chosen so ha a 1 percen increase in he number of properies leads o abou a 1 percen increase in he price of flas The lending marke There is a non-profi financial inermediary ha acceps deposis and issues morgages a an ineres rae r. Agens can lend or borrow as much as he bank allows hem a he one period ineres rae r, subjec only o he resricion ha hey have a zero deb posiion a he end of heir life. The economy can eiher be closed, in which case he ineres rae is deermined endogenously and aggregae deposis equal aggregae loans, or open, in which case real ineres raes are deermined exogenously and he ne foreign asse posiion can be non-zero. There are no resricions on he deposi conrac, and ineres on a deposi made a ime is paid a ime +1. Agens pay ax on his ineres a heir marginal ax rae, bu do no ge a ax deducion for ineres paid on borrowed funds unless hey borrow o fund a renal propery. 6 An agen s posiive funds are labelled. i, B j The morgage conrac is subjec o hree resricions. 7 i) The loan o value resricion. (6) 6 To reduce compuaional complexiy, he marginal ax rae is calculaed on he basis of labour income, no oal income. Oherwise he marginal ax rae is deermined endogenously. 7 Noe ha banks impose hese resricions even hough here is no uncerainy in he model 9

14 The morgage may no exceed a cerain fracion of he value of he propery. In paricular, he gross amoun borrowed he loan o value raio θ: ha is D canno exceed he value of propery muliplied by i, j D θ PI (7) i, j h i, j, h h F, H (Noe D > 0 if he agen borrows.) This resricion means ha agens who ren canno i, j borrow o smooh consumpion, alhough hey can save. ii) The regular cash paymen resricion. Banks only issue η-year able morgages, and require a cash paymen in he period he morgage is issued. This resricion is imposed o mimic a sandard condiion of a able morgage, namely ha a cusomer is required o make regular cash repaymens CP of equal size hroughou he life of he morgage raher han a large repaymen a is erminal dae. The paymen size CP is chosen o ensure he morgage is reired a he end of he erm: if D 0- is iniially borrowed, he annual paymen is η 0 (1 + r) CP = D r η (1 + r) 1 (8) η is assumed o be 25 years. 8 I is no possible o exacly replicae his feaure of a sandard morgage conrac in he model. However, a close approximaion is achieved by requiring he cusomer o make a paymen ha pays off some of he ineres and principal in any period he or she has deb. In paricular, a cusomer wih gross deb of bank and make a deposi of size i, D j is required o open up a separae accoun wih he D ( 1+ r ) η / ( r ) η / T r = D i, j* i, j T r (9) ino his accoun. This deposi earns (unaxed) ineres a rae r. This means he ne borrowing posiion of a borrowing agen, D = D D, is less han he gross borrowing posiion. i, j i, j i, j* Wihou his cash paymen feaure, many agens would prefer o purchase raher han ren simply because he ineres paymen occurs a period laer han he renal paymen. When he 8 Unil recenly, his has been he sandard erm for a able morgage in Ausralia and New Zealand. 10

15 cash paymen requiremen is imposed, purchasing a house requires a larger paymen o he bank in period han he cos of rening a house. iii) The morgage-repaymen-o-income resricion. The maximum amoun an agen can borrow is resriced o ensure he morgage repaymen given by equaion 8 is smaller han a fracion δ of income: D r ( 1+ r ) η / ( r ) η / T δ PY i, j ij T 1+ r 1+ 1 (10) Noe ha his consrain is expressed in erms of nominal ineres raes. The morgage condiions are only imposed on agens in periods 0 and 1 of heir lives in order o simplify he soluion algorihm. In period 2 agens can borrow unresriced amouns. The absence of a resricion in period 2 has lile effec because agens are in heir peak earning years, receive heir inheriance a his ime, and are acively saving or reducing deb o finance heir reiremen Uiliy maximisaion An agen born a ime solves he following consrained maximisaion problem (he jh superscrip is omied): 3 i i ih, ih, = ( {, }, ) c β + ii (11) i + ii + + i i= 0 Max U u c λ PY B + D (1 + τ ) Pc P I 0 0 g 0 h 0, h 0 0 h (1 + π) PY + B (1 + r (1 τ )) D (1 + r ) B + D i i i i i i i + i + i 1 + i 1 + i 1 + i 1 + i + i 3 i g i h ih, h i 1, h λi (1 π) (1 τ ) Pc P+ ii+ i P+ ii+ i 1 κiinheri+ i i= 1 h h= FH, R i + P+ i(1 τ ) I i + ( P i P i (1 + r i (1 τ )) I i ir, * F F i i 1, R * i= 0 χ D θp I i, h ih, i + i + i + i h 11

16 1 τ / T i, + i + i i φi D+ i δy τ / T + i i= r+ i (1 + r+ i) 1 i i ( B + ) ν ( D + ) 3 3 ς i i i i i= 0 i= 0 r (1 + r ) Lines 2 and 3 of equaion (11) are he budge consrains facing he agen in he four periods. Lending and borrowing are enered separaely as here are differen afer ax ineres raes, and here are erms o reflec inheriance and renal income. Lending and borrowing in period 3 are i resriced o equal zero, and τ is he marginal ax rae applying in period i of he agen s life. The Kuhn-Tucker condiions in lines 4 and 5 reflec he loan-o-value raio consrains and he morgage-repaymen-o-income raio consrains respecively. The Kuhn-Tucker condiions in line 6 reflec he requiremen ha non-negaive amoun are len and borrowed. The agen solves he problem by calculaing he maximum uiliy for each housing paern in he se Η, and hen selecing he housing paern wih he highes uiliy. The use of log-linear uiliy funcions means i is relaively sraighforward o calculae an analyical soluion for he opimal consumpion pah given a paricular housing paern, even hough each soluion has 48 pars corresponding o he 48 possible combinaions of Kuhn-Tucker condiions Equilibrium condiions In he simulaions, he seady sae equilibrium is found for an open economy in which agens borrow or lend a he world ineres rae. In he seady sae, he following price relaionships hold: (1 + r) / (1 + π ) = 1+ r (12a) P P F + 1 F = 1+ π (12b) F P P H F H = ρ (12c) P P (1 + r(1 τ )) (1 + π ) R = = ρ R T F 2 F T (1 τ2)(1 + r(1 τ2)) (12d) Equaion (12a) saes ha real ineres raes are consan. In he open economy model, he rae r is he foreign real ineres rae. Equaion (12b) saes ha fla prices appreciae a a consan 9 In he periods 0 and 1, he financial asse posiion can be posiive, zero, negaive, or equal o he borrowing consrain; in period 2, he financial asse posiion can be posiive, zero or negaive; and in period 3 i is zero. 12

17 rae. 10 Equaion (12c) saes ha he raio of house prices o fla prices is consan. Equaion (12d) is a resaemen of equaion 5, linking rens o ineres raes and he fla price appreciaion rae. For a se of parameers and housing parameers eiher { F H F F H H n, n } or { 0, 1, 0, 1 } described by a se of prices {, F, H, R } 0 H F g* { NTY,,, ωj, gi, π, β, vh, κi, H, n, n, ηθδτ,,,, τ1, τ, τ * 2 } α α α α he seady sae equilibrium is r π ρ ρ, a GST raeτ g, a se of housing and consumpion s, j s, j, h demands { I } for each agen j in each cohor born in period -i, and a ne foreign c + i s, + i s s = 0,...,3 asse posiion B ne such ha all agens have maximal uiliy and r π 3 N 3 N i, j g* i, j ne c = (1 τ ) y B i= 0 j= 1 i= 0 j= 1 1+ π (13a) 3 N N i, j i, j F 2, j, R* ne ( B D ) P I = B i= 0 j= 1 j= 1 (13b) τ 3 N g* i, j y i= 0 j= 1 = 3 N 3 N N N g i, j i, j ij R 2, j 2, jr, * F 3, j 2, jr, * c + B 1r + P I P 1r I 1 i= 0 j= 1 i= 0 j= 1 j= 1 j= 1 τ τ τ τ (13c) and eiher 3 N i, jr, i, jf, F ( I + I ) = n i= 0 j= 1 (13d ) 3 N i, jh, H I = n (13e ) i= 0 j= 1 or 3 N i, jr, i, jf, F ( I + I ) = Q i= 0 j= 1 (13d ) 3 N i, jh, H I = Q (13e ) i= 0 j= 1 10 If he number of flas and houses is deermined exogenously, an equilibrium can be found in which incomes in he economy grow a a consan rae, and in his case he seady sae equilibrium will have propery prices growing a a faser rae han he rae of inflaion. If he number of properies is deermined endogenously and he income growh rae is posiive, he only possible seady saes occur when all people live in large houses, or when he qualiy of flas and houses seadily improves. This paper does no analyse hese cases alhough he model is se up o do so. 13

18 where Q F H and Q are he number of houses produced when he supply of properies is elasic, P P α P α Q = Q = Q. H F H F F H 0 F 0 H and H F α1 α1 Equaion (13a) requires ha oal consumpion plus ax plus real earnings on he ne bond posiion in each period equals oal producion. Equaion (13b) is he ne supply of foreign bonds, given ha landlords are assumed o borrow 100 percen of he price of a fla. This will change hrough ime if here is economic growh or inflaion. Equaion 13c says ha he oal ax ake is equal o oal GST revenue plus ax on ineres and ren minus he ax deducion for landlords. Noe ha while i has been assumed landlords borrow 100 percen of he value of he propery, ax revenue would no change if landlords had differen gearing as he ax rae on posiive balances is he same as he ax deducion hey ge when hey borrow. Equaions (13d ) and (13e ) require ha he oal demand for flas equals he supply of flas, and ha he oal demand for houses equals he supply of houses, when he supply of properies is deermined exogenously. Equaions (13d ) and (13e ) are he same equaions for he case when he supply of propery is elasic Parameerisaion 0 H F The se of baseline parameers { NTY,,, ω, g, π, β, v, κ, H, n, n, g* * 1 2 } j i h i ηθδτ,,,, τ, τ, τ and housing parameers { F H n, n } are nearly he same as hose used by Coleman (2007) and have been chosen o approximae feaures of he New Zealand economy. 11 These are lised in able 1. Excep for income disribuion, he income parameers approximaely mach he basic lifecycle and cohor income paerns of New Zealanders repored in census documens, , under he assumpion ha he basic agen is a household comprised of a male and female of he same age. For simpliciy, income is assumed o be uniformly disribued over he range $20000 o $ In he baseline model, he discoun rae is 3 percen, he real ineres rae is 5 percen (assumed equal o he world rae), and banks impose borrowing resricions ha limi households o borrow up o 90 percen of he value of a propery and o pay no more han 30 percen of heir income in deb servicing. The banking secor parameers are changed in some of he simulaions, bu hese reflec he condiions facing New Zealand borrowers since he year Coleman (2007) uses 5 cohors, no 4, and he parameers have been slighly modified. 14

19 The ax raes also reflec New Zealand ax seings in In he baseline model, he marginal ax is 20 percen for households wih incomes less han $50000, and 33 percen for households wih incomes above ha level. To demonsrae how axes and inflaion inerac, he model was solved under a variey of alernaive ax regimes: (τ 1,τ 2 ) = (0%,0%,), (10%, 10%), (20%, 20%), and (20%, 39%). 12 The model is also solved for a se of ax rules ha exclude he inflaion componen of ineres income from ax, and which only allow landlords o deduc real ineres paymens from heir axable income. 13 The GST rae was chosen o ensure ha capial income axes and consumpion axes oal o 10 percen of labour income. When he housing supply is inelasic, 57 percen of he properies are large houses, and 42 percen of he properies are small houses. The oal number of properies is 1 percen less han he number of agens, o ensure rens are posiive in equilibrium. The parameers R F H ( υ, υ, υ ) = (0.33,0.35,0.45) mean (approximaely) ha a he margin a household would be prepared o spend a hird of heir income on ren raher han have no accommodaion; he benefi from living in an owner-occupied fla raher han a rened fla is 2%, and he addiional benefi from living in a large house a furher 10 percen. Housing supply parameers were chosen so ha ha he quaniy of flas would increase by approximaely one percen for a one percen increase in prices, bu ha he number of houses and flas would be approximaely he same in he elasic and inelasic cases. The model was solved for inflaion raes ranging from 0 o 3 percen, reflecing he legal requiremen ha he Reserve Bank of New Zealand achieve sabiliy in he general level of prices. This requiremen currenly requires he Bank o keep inflaion beween 1 and 3 percen. The effec of higher inflaion raes were no invesigaed as he purpose of he paper was o ascerain wheher low raes of inflaion have an appreciable effec on economic welfare Soluion echnique The soluion is found numerically. The algorihm searches for a se of prices { τ g, P R, F, H P P } = 3,..0,...3 so ha when each agen j born in period -i, i= 0, 3 is consuming a s, j s, j, h sequence of goods and enure opions { I } c + i s, + i s s = 0,...,3 ha solves heir consrained uiliy 12 The op income ax rae was increased from 33 percen o 39 percen in However, many landlords can use propery russ o lower heir marginal ax rae o 33 percen. This is he reason why he main simulaions have been done using a op ax rae of 33 percen. 13 In his case he consrains in equaion 11 and he aggregaion condiion (13c) are modified accordingly. 15

20 problem given by equaion (11), he aggregaion condiions 13a 13e applied a ime are saisfied. In he seady sae, he vecor { τ g, P R, F, H P P } = 3,..0,...3 vecor P * = { τ, P, π, ρ } and he parameers { r, τ }. g F F H 0 The basic srucure of he algorihm is as follows. 2 can be calculaed from he Le he vecor P = { τ, P, π, ρ } be he k h esimae of he seady sae soluion *, g F F H k 0 * *,k P. Given P, calculae he opimal consumpion and housing enure pahs for each of he N households who are born a =0 by searching over he differen possible enure pahs in he se Η. a) Use hese resuls o calculae he demand for consumpion goods and housing a ime =0 for all households in he economy. b) Use hese resuls o calculae aggregae consumpion, he aggregae demand for flas, and he aggregae demand for houses a ime =0. Then calculae he excess demand funcions given by 13a 13e. c) If he excess demand funcions are no sufficienly close o zero, a new esimae of he * *, k 1 equilibrium prices P, P +, is calculaed. This is done using a discree approximaion o he Newon-Rhapson mehod. A se of quasi-derivaives is calculaed by g F F H recalculaing he se of excess demand funcions a he prices { τ + 1, P, π, ρ }, { τ g, P F + 2, π F, ρ H }, { g, F, F 3, H g F F H τ P π + ρ } and { τ, P, π, ρ + }. These 4 quasi derivaives are used o calculae he updaed price vecor using Broyden s mehod. The process is coninued unil he sequence of esimaes 2.9. Comparison o oher models *,k P converges. The model is relaed o earlier models by Slemrod (1982), Hayashi, Io and Slemrod (1988), Feldsein (1997), Oralo-Magné and Rady (1998, 2006), and Coleman (2007). Slemrod (1982) solves a similar model of he household o explore how axes and credi consrains affec iner-emporal housing and saving choices. His model has a similar asymmeric reamen of capial axes on posiive and negaive asse posiions, and imposes a maximum loan-o-value consrain. The price of housing is deermined exogenously, bu agens can choose he size of heir house. Hayashi, Io, and Slemrod solve a version of his model ha uses log-linear uiliy of he form used in his paper. The basic framework including he housing model is adaped from Oralo-Magné and Rady, alhough hey only use he version wih inelasic supply. Several of he modificaions were firs made in Coleman (2007) including he use of log-linear preferences 16

21 raher han linear preferences, he incorporaion of he morgage-repaymen-o-income consrain, and a wider range of housing opions. This model exends he earlier model by Coleman by incorporaing axes and an elasic housing supply. 3. Resuls The focus of he paper is he way ha he inflaion rae changes he effec of he ax sysem on he housing marke. The main resuls are esablished by finding he equilibrium prices and GST rae ha correspond o a se of parameers as he inflaion rae is varied from 0 o 3 percen. In ables 2 5, various oucomes of ineres are presened when he ax raes are ( τ, τ ) = {(0%, 0%), (20%, 20%), (20%, 33%)}. In ables 2 and 3, he housing supply is eiher 1 2 inelasic or elasic and all ineres earnings are axed; in ables 4 and 5 he housing supply is eiher inelasic or elasic and he inflaion componen of ineres earning are ax exemp. The ables show how rens, fla prices, house prices, he number of flas and houses, he number of people rening, and he GST rae vary wih he inflaion rae for each differen ax combinaion. Inelasic housing supply, all ineres earnings axed. Table 2 and figures 1 and 2 indicae how inflaion and ax affec he housing marke when he supply of housing is deermined exogenously. When he income ax rae is zero, fla prices, house prices and rens are lile affeced by he inflaion rae, falling by approximaely 1 percen as he inflaion rae increases from 0 o 3 percen. 14 However, he fracion of he younges cohor rening increases as he inflaion rae increases from 0 o 3 percen, from 21 percen o 31 percen. These resuls follows from he way rens are deermined. When he number of flas and houses is fixed, he price of a fla is deermined by is value o he marginal residen, which depends on he addiional benefi of living independenly raher han in he parenal home. For he assumed values of ν R and ν F, his is approximaely equal o 30 percen of he income of he marginal occupier; his sum varies only slighly wih he inflaion rae as lifeime uiliy and hus he amoun desired o be spen on housing changes only slighly. 15 As rens vary lile wih inflaion, fla prices vary lile. The housing enure choice is affeced by he inflaion rae, however, because more and more low income households find i preferable o ren raher han purchase as nominal ineres raes increase. 14 Coleman (2007) discusses he siuaion a lengh. 15 Assuming ha a low income young person spends all heir income, he amoun hey are willing o spend on ren is y 0, j R (1 e ν ). 17

22 When he ax raes are posiive, here are hree compeing endencies. When he inflaion rae is zero, he fracion of he young cohor rening is decreasing in he ax rae. This is because impued ren is ax exemp, providing households wih an incenive o own heir own home raher han o ren and accumulae financial asses. The effec is sufficienly large ha he fracion rening falls o zero when 1 2 ( τ, τ ) =(20%, 33%). As he inflaion rae increases, however, here are wo oher effecs. Firs, landlords are araced o he housing marke o ake advanage of ax free capial gains. Because hey are on a higher marginal ax rae han low income enans, hey value he ax concession more highly han enans and hus he amoun hey are prepared o pay for renal propery is a seeply increasing funcion of he inflaion rae (Lizenberg and Sosin (1979)). When boh he high and low marginal ax raes are 20 percen, he equilibrium fla price increases by 8 percen when he inflaion rae increases from 0 o 3 percen; when he op marginal ax rae is 33 percen, here is a 17 percen increase. Secondly, nominal ineres raes increase as he inflaion rae increases and low income, credi consrained households find i increasingly difficul o make morgage paymens. The resul is a squeeze ha sees landlords replace young households as he owners of flas. For 1 2 ( τ, τ ) =(20%, 20%), he fracion of he young cohor rening increases from 6 o 32 percen as he inflaion rae increases from 0 o 3 percen; for 1 2 ( τ, τ ) =(20%, 33%), he increase is larger, from 0 o 42 percen. Figure 3 shows he fracion of he young cohor ha rens as a funcion of ax raes and inflaion raes. When he inflaion rae is less han 1 percen, he fracion of he young cohor ha rens declines as ax raes increase, due o he rising value of he impued ren ax concession. When he inflaion rae is 2 or 3 percen, he effec of higher morgage raes and he increasing araciveness of propery o high marginal rae invesors means ha he fracion ha rens is an increasing funcion of he op marginal ax rae. Thus even hough landlords and owner-occupiers derive benefis from residenial housing ax concessions, he effec of axes on ownership paerns depends crucially on he inflaion rae. Elasic housing supply, all ineres earnings axed. Table 3 and figures 4 6 indicae how inflaion and ax affec he housing marke when he supply of housing is elasic. Again, fla prices, house prices, rens, and he oal number of houses are lile affeced by he inflaion rae when he income ax rae is zero. Moreover, as before, he fracion of he younges cohor rening increases as he inflaion rae increases from 0 o 3 percen, from 23 o 28 percen. There is a difference from he inelasic case, however: as he inflaion rae increases, he composiion of he housing sock changes, wih more flas and fewer large houses. This is because fewer young households can afford he nominal morgage 18

23 paymen on large houses when he inflaion rae increases, so hey purchase a small house as a emporary measure. When ax raes are posiive, he fracion of he young cohor ha rens increases wih he inflaion rae by a similar amoun as when he housing supply is inelasic: in he case ha 1 2 ( τ, τ ) =(20%, 33%), he fracion rening increases from 0 o 35 percen as inflaion increases from 0 o 3 percen. The mechanism is quie differen, however. In he inelasic case, fla prices increased as inflaion increased because landlords bid up prices o ake advanage of he ax concession. In he elasic case, prices of houses do no increase by very much, bu landlords bid rens down. The combinaion of slighly higher fla prices and significanly lower rens resuls in a big increase in he number of low income households choosing o ren, bu i also leads o a larger sock of flas and a reducion in he number of households living a home wih heir parens. Exemping he inflaion componen of ineres earnings from ax. I has long been argued ha many of he economic disorions caused by he ineracion of he inflaion and income ax could be avoided by changing he ax code so ha only real ineres income was axed (Aaron 1976; Fischer and Summers 1989; Feldsein 1996). In pracice, his would mean wo adjusmens: he inflaion componen of ineres earnings would be exemp from income ax; and borrowers would only be allowed o deduc he real ineres componen of ineres paymens from axable income. 16 The effecs of hese wo changes in he ax code on he housing marke are shown in ables 4 and 5, and in figures 1 2 and 4 6. The GST rae is adjused so ha oal nominal axes are he same. The resuls are dramaically differen. When he supply of housing is inelasic, changes in he inflaion rae only have small effecs on house prices and home ownership raes. Indeed he effecs are very similar o he case when axes are zero: as he inflaion rae increases from 0 o 3 percen, he fracion of he younges cohor rening increases because nominal morgage raes rise, bu by 8 percen raher han by 42 percen. The difference wih he normal ax case is ha wealhy households have lile incenive o become landlords as hey no longer pay ax on he Inflaion componen of ineres earnings if hey accumulae financial asses. 17 Consequenly, landlords do no bid up he price of flas as he inflaion rae increases. A similar resul holds 16 Economiss have also recommended changing depreciaion allowances and exemping he inflaion componen of capial gains from ax. 17 The ax on he inflaion componen of ineres earnings is ofen called he widows ax because mos lenders in an economy are over

24 when he supply of housing is elasic. In his case, wealhy households do no compee o become landlords, rens are no bid down as he inflaion rae increases, fewer houses are buil, and home-ownership raes amongs he young scarcely change. I is worh noing ha exemping he inflaion componen of ineres earnings from ax has lile effec on ax revenue. Even when he inflaion rae is 3 percen, he GST rae needs o be increased by less han 0.2% o raise he same amoun of revenue. This is because lile revenue is raised by axing he inflaion componen of ineres even when inflaion is moderae, for high income households rearrange heir finances o avoid i, in his case by becoming landlords. Even hough his generaes ax revenue on renal income, i is offse by a decrease in ax paid on ineres. This offse occurs parly because he morgage paymens made by landlords are ax deducible. Uiliy calculaions The maximum lifeime uiliy levels of each agen can be calculaed as a funcion of equilibrium prices. These can be used o calculae how welfare changes across he income disribuion as he inflaion rae changes. Figures 7 and 8 show how uiliy levels change when he inflaion rae increases from 0 o 3 percen. In each case he change in life-ime uiliy is shown as a funcion of income in he firs period. 18 Figure 7 shows how uiliy changes when he supply of houses is inelasic and he ax raes are ( τ1, τ 2) = (20%,33%). When nominal ineres earnings are axed, he increase in he inflaion rae reduces welfare across he board, by beween 2 and 4 percen. The effec is larges on middle income households. These households experience a reducion in welfare because of changes o heir housing consumpion paerns, as hey ren raher han purchase or delay heir purchase of large houses in response o higher nominal ineres raes. High income households, whose housing arrangemens are ypically unaffeced by he inflaion rae, experience a welfare reducion because hey change he iming of heir consumpion and face higher real capial income axes. The figure also shows how welfare changes when he inflaion componen of ineres earnings is exemp from ax. In his case here are few welfare changes. Middle income agens increase uiliy a lile; low income agens experience a small welfare reducion, as hey ren raher han own heir homes in response o higher nominal ineres raes; bu for boh groups he welfare changes are smaller han 1 percen, and ypically less han 0.5 percen. The middle 18 As he uiliy funcion is log linear, he change in uiliy has an approximae inerpreaion as he percenage change in consumpion ha would make an agen indifferen beween he wo inflaion raes. 20

25 income group experiences an improvemen in welfare because he GST rae falls when inflaion rises. This occurs because he Governmen derives revenue from he ax on ren. Figure 8 shows he change in uiliy ha occurs when he housing supply is elasic. In he case ha nominal ineres earnings are axed, he welfare changes are very differen han when he housing supply is inelasic. In paricular, inflaion improves he welfare of low income households, by up o 4 percen. There are wo reasons for his improvemen. Firs, rens fall, and his fall more han offses he decline in welfare ha comes from being a enan raher han an owner-occupier. Secondly, more flas are buil, and he lowes income households gain from he opporuniy o live independenly raher han in he parenal home. (The peak welfare improvemen accrues o he highes income person ha lived a home when he inflaion rae was zero.) These gains are offse by welfare losses of middle income and high income households, he laer of approximaely 2 percen, as hese households pay higher axes on heir ineres income. For he economy as a whole, average welfare is reduced, alhough his resul need no occur. 19 This resul warrans furher commen. Low income agens who ren when hey are young have low consumpion as hey have low curren income and canno borrow agains heir higher fuure incomes because of bank imposed credi consrains. Consequenly, anyhing ha reduces heir ren will lead o an increase in heir consumpion, and an improvemen in heir welfare. Landlords are prepared o accep lower rens when he inflaion rae increases since i reduces he afer ax real ineres rae, and capial gains are axed less han ineres paymens. In essence, he exisence of one economic disorion (he asymmeric axes on he inflaion componen of ineres earnings and capial gains) helps o reduce he effec of a second economic disorion (credi consrains prevening young people from borrowing agains fuure income) when here is inflaion. If young people were able o borrow agains fuure income for consumpion purposes, inflaion would sill reduce rens bu i would have minimal effecs on welfare as i would primarily affec he amoun young people borrowed, no he amoun hey consumed. When he inflaion componen of ineres is exemp from he ax, he welfare changes are almos he same irrespecive of wheher he housing supply is elasic or inelasic. This is because neiher ren nor he oal quaniy of housing change by much in he elasic case, and hus oucomes are similar o he siuaion when he housing supply is elasic. Noe ha his means ha if he inflaion componen of ineres earnings were ax exemp and he housing supply was sufficienly elasic, welfare would decline for some low income households. This 19 In paricular, he equally weighed average change in uiliy is negaive. 21