Helmuth Cremer, Toulouse School of Economics (IDEI and Institut universitaire de

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1 Taxng sn goods and subsdzng health care Helmuth Cremer, Toulouse School of Economcs (IDEI and Insttut unverstare de France). Phlppe De Donder, Toulouse School of Economcs (GREMAQ-CNRS and IDEI). Daro Maldonado, Department of Economcs and CeBA-Complejdad, Unversdad del Rosaro, Bogota. Perre Pesteau, CREPP, HEC-Management School Unversty of Lège; CORE, Unversté Catholque de Louvan. Abstract: We study the taxaton of sn goods n a two-perod, three-good model. Indvduals can buy health care to compensate for the damages caused by ther earler sn good consumpton. Indvduals are myopc and underestmate the e ect of the snful consumpton on health; n ther second perod, they may acknowledge ther mstake or persst n ther error. We characterze and compare the rst-best and the (lnear) second-best taxes n these d erent settngs. In partcular, we examne how the results are a ected by the way sn good consumpton and health care nteract n the health producton technology. Keywords: paternalsm, behavoral economcs, dual self vs sngle self JEL class caton: H21, I18 We thank partcpants to Queen s Unversty s 2007 Publc Economcs Workshop for helpful suggestons. We are also grateful to the referees for ther comments and remarks. Part of ths paper was wrtten whle the second author was vstng Yale Unversty. He thanks Yale Economcs Department for ts hosptalty. Ths paper has been nanced by the french Agence Natonale de la Recherche s POLNUTRITION grant (ANR-05-PNRA-012, Poltques Nutrtonnelles, Régulaton des Flères Almentares et Consommaton). We thank ANR for ts support.

2 I Introducton In our everyday lfe we consume a number of goods that all brng us utlty. For most of them, that s all. For some, today s consumpton can also have some e ects on tomorrow s health. For example, smokng leads to shorter lves or excess sugar to dabetes. To the extent that we mpose costs on ourselves, there s no need for government acton except f, out of gnorance or myopa, we do not take nto account the delayed damage done to our health. 1 If ths s the case, then there s a paternalstc mandate for publc acton, assumng that the government has a correct percepton of the health damage generated by our snful consumpton. Optmal sn taxes have been studed by O Donoghue and Rabn (2003, 2006). They model an economy where ndvduals have hyperbolc preferences and d er n both ther taste for the sn good and n ther degree of tme-nconsstency. They show how (heterogenety n) tme nconsstency a ects the optmal (Ramsey) consumpton tax polcy. Ther mansght s that, although taxes create consumpton dstorton for fully selfcontrolled people, such dstortons are second-order relatve to the bene ts from reducng over-consumpton by people wth self-control problems (O Donoghue and Rabn, 2006, p. 1827). Gruber and Koszeg (2001) study a Pgouvan tax used to counteract overconsumpton due to self-control problems, and apply ther model to the determnaton of optmal cgarette taxes. Gruber and Koszeg (2004) also study cgarette taxaton wth self-control problems, but ther focus s the tax ncdence for d erent ncome groups rather than optmal taxes. O Donoghue and Rabn (2003, 2006) are representatve of the lterature studyng present-based preferences (such as Labson (1997)) n two respects. Frst, they assume 1 We are not concerned here by two mportant ssues: addcton and externaltes assocated wth snful actvtes. 1

3 that all based ndvduals dsapprove or regret ther past consumpton decsons and, second, there s nothng that agents can do to mtgate the current mpact of past consumpton decsons. Our paper lfts these two assumptons and studes ther consequences on optmal sn taxes. We model a two-perod settng where ndvduals consume a sn good, wth postve mmedate grat caton but negatve mpact on second perod health status. In the second stage, ndvduals may nvest n health care servces that have a postve mpact on ther health status. Indvduals d er ncome and n ther awareness of the lnk between sn good consumpton and health care on the one hand, and health status on the other hand. We contrast two possbltes. In the rst one, ndvduals n ther second perod realze the mstake they commtted prevously. They regret ther past hgh sn good consumpton, and nvest n health care understandng ts correct mpact on ther health and utlty. In other words, ndvduals su er from myopa n the rst perod, but use ther true or correct preferences later on when they choose health expendtures. They thus exhbt dual selves, usng a term coned n the behavoral economcs lterature. A second case, referred to as persstent error, occurs when all decsons, ncludng the determnaton of health expendtures, are made accordng to the mstaken preferences. In other words, whendvduals realze ther error (n the mddle of the second perod) t s too late to correct for t. We use the concept of snful consumpton a narrower sense than other authors. We are not nterested by externaltes such as those related to passve smokng. We are only concerned by the damage the snful consumpton exerts on the consumer s own health. Snful consumptos part of the lfestyles regarded as rsk factors, the so-called Holy Four, namely tobacco smokng, drnkng of alcoholc beverage, eatng an unhealthy det 2

4 (fat, salt, sweets) and lack of physcal actvty. 2 In our settng, sn goods have the followng features: they have detrmental e ects on health, ther consumers do not fully antcpate these e ects, they brng regret wth some lag and they can be partally o set by costly treatment. The ssue of regret s documented by a number of recent surveys. For example, Fong et al. (2004) show on the bass of telephone surveys n Canada, the US, the UK and Australa that about 90% of smokers agree wth the statement: If you had to do t agan, you would not have started smokng. Fnally there s the ssue of ex post compensatory treatment. In some nstances there exsts no treatment, or there s no choce as n the case of emphysema that requres oxygen therapy. In many other nstances, however, there are treatments that can partally allevate the problems brought about by one of the Holy Four. Note that the formal structure of our model can be appled to account for crcumstances that go beyond snful consumpton. It can, for example, apply to stuatons where habt leads to unforeseen consequences. For nstance, people can face needs for whch they are not prepared and whch can force them to work longer than expected or even to unretre. 3 Whch preferences should the socal planner use when assessng optmal taxes/subsdes on sn good consumpton and health care expendtures? The recent lterature on paternalsm has studed the mpact of behavoral consderatons on the socal objectve. Thaler and Sunsten (2003) make a strong case for lbertaran paternalsm, whch apples when no coercos nvolved, such as when the planner chooses the default opton (for example, automatc enrollment n 401(k) employee savngs plans n the US). Other papers 2 See Harrs (1999). There may be other rsk factors but there s less research nto ther prevalence or nto ther mportance for the burden of dsease. 3 See Cremer et al. (2008). 3

5 go further and envson the possblty of coercng ndvduals by taxng certan goods or even prohbtng ther consumpton. The lterature has focused upon the case where people d er n ther degree of non-ratonalty. These contrbutons advocate the use by the planner of cautous (O Donoghue and Rabn (1999)) or asymmetrc (Camerer et al. (2003)) paternalsm, whch trades-o the bene ts of paternalstc nterventons for people makng mstakes aganst the costs for fully ratonal ndvduals. Ths lterature shows that ths knd of paternalsm usually leads to some nterventon, because devatons from lassez-fare mpose second-order costs on ratonal ndvduals, but mply rst-order gans for non-ratonal persons. Moreover, t s shown (O Donoghue and Rabn (2003, 2006)) that even a small probablty (or proporton) of people makng mstakes can have dramatc e ects for optmal polcy. We depart from ths lterature n two ways. Frst, n addton to studyng the dual self settng, we consder a paternalstc objectve where ndvduals are adamant n ther mstakes.e., where they ether never realze (for nstance because of gnorance or cogntve dssonance) or realze too late that they base ther decsons on wrong premses. 4 Second, rather than mxng ratonal and non-ratonal ndvduals, we contrast the results obtaned when all ndvduals are repentant n the second perod or when none s. Obvously, paternalsm s easer to defend whendvduals have dual selves. Smlarly, t does not appear to be problematc n the case of persstent errors f ndvduals eventually realze they made a mstake (albet too late to take any correctve measures). When mstakes are truly persstent and ndvduals never realze ther mstakes we return to an 4 Aronsson and Thunström (2008) and Aronsson and Sjögren (2009) also analyze settngs where ndvduals consume a snful good and canvest n health captal. Ther analyss d ers from ours n several ways. Both papers ntroduce myopa or gnorance n a d erent way than we do and whch does not allow for the ntroducton of the dstncton of persstent errors and dual self. Addtonally, the rst paper does not consder the ssue of the complementarty/substutablty of sn good consumpton and health expendture on health stock and they only consder rst best settngs. The second paper ntroduces non-lnear taxatonstead of lnear taxaton. 4

6 analyss smlar to the older lterature on mert goods (Musgrave (1959), more recently Besley (1988)), where we add that the reason for the d erence between the planner s and the ndvduals preferences resdes n the (unrecognzed) mstakes made by ndvduals. To keep the analyss tractable, we make a number of smplfyng assumptons. Frst, we assume that ndvduals lve two perods. Consequently, we cannot have hyperbolc preferences whch requre a three perod spec caton. However, ther essental feature, namely the regrets for yeldng to the short term concerns, s represented n our settng. 5 Second, we study lnear tax nstruments. Non-lnear nstruments are clearly more general but at the same tme they constran the analyss to focus on a lmted number of ndvduals. 6 We obtan the followng man results. We show that the rst-best outcome can be decentralzed wth ndvdualzed lnear taxes and subsdes n the two scenaros (persstent error and dual self). In the rst one, t s necessary to tax the sn good consumpton whle subsdzng health care expendtures. There s no need to n uence savng. The second scenaro s more complex, because the socal planner faces a problem wth changng preferences. The planner has to ntervene n the rst perod by taxng the sn good whle subsdzng savngs. There s no need to n uence health care expendtures, whch are optmally chosen provded that rst perod choces are optmal. Comparng the sn tax n the two scenaros, we obtan that t s smaller n the dual self case f and only f the margnal e ect of health care on health status ncreases wth sn good consumpton. We also show that under ths same condton the possblty of compensatng health expendture makes the sn tax smaller n the dual-self case settng. 5 Addtonally n our settng the dstncton between sophstcated and non sophstcated ndvduals s rrelevant as we do not have any commtment devce. For example, n poltcal economy models, sophstcated ndvduals can constran ther short term self by ther vote (see Cremer et al., 2007 and Haavo and Kotakorp, 2009). 6 See Cremer et al. (2009) and Blomqust and Mcheletto (2006). 5

7 We then turn to the second best settng where the planner observes nether ncome, preferences nor savngs and uses unform lump sum transfer, and taxes/subsdes on the sn good and on health expendture. In the sngle self case, optmal lnear sn taxes and health expendture subsdes depend upon two terms: a (classcal) covarance term re ectng dstrbutve consderatons and a Pgouvan term that re ects the nternaltes andvdual mposes on hmself. In the dual self settng the optmal tax formulas also contan a thrd term, whch s lnked to the nablty to control savngs. Ths addtonal term would call for hgher tax on sn good/subsdy on health care provded that ths tax/subsdy encourages savngs. The rest of the paper s organzed as follows. In secton II, the model, the rstbest soluton and the decentralzaton condtons are presented for the two spec catons. Then secton III we turn to the second-best problem whendvduals persst n ther gnorance. In secton IV, we study the alternatve second-best problem, that s when ndvduals realze havng made a mstake. A nal secton concludes. II Frst-best and decentralzaton Model We consder a socety consstng of I types of ndvduals ndexed by subscrpt. Each type of ndvdual s characterzed by a wealth endowment w and subjectve and objectve health parameters and. The proporton of type ndvduals n the populatos gven by ( P I = 1). Each ndvdual s lfe spans two perods. In the rst one, he consumes a numerare good c and a sn good x. He also saves s for future expenses. In the second perod, he consumes an amount d of the numerare and he nvests e n health mprovement. In ths second perod, he enjoys a qualty of health h (x ; e ), 6

8 on whch x has a negatve e ect and e a postve e ect. For reasons of gnorance or myopa, the ndvdual has a percepton of ths functon that underestmates the mpact of both arguments. In other words, he perceves a health functon equal to h (x ; e ) wth < : We assume for the moment that ndvduals underestmate the functon h(:) n both perods (persstent error). Hs two-perod utlty functon can be wrtten as: U = u (c ) + ' (x ) + u (d ) + h (x ; e ) ; (1) wth budget constrants: w = (1 + ) s + (1 + ) x + c a ; d = s (1 + ) e ; where u and ' are strctly concave functons, ; ; are tax rates and a s a lump sum transfer. For smplcty, we assume a zero tme dscount rate and a zero rate of nterest. Here the myopa parameter concerns the health functon. An alternatve spec caton could be h ( x ; e ), n whch case myopa only concerns the sn good and not health care. Ths would not change the qualtatve nature of our results. A thrd approach, often used n papers on socal securty and savng, s to assume that myopa concerns the whole second perod, namely the utlty of health but also the utlty for second perod consumpton. In that case, the utlty functon would be gven by u (c ) + ' (x ) + [u (d ) + h (x ; e )] : Frst-best We assume that the government s paternalstc utltaran. In other words, ts objectve conssts of the sum of utltes (1) n whch replaces. As a benchmark, we derve 7

9 the rst-best (FB) condtons by maxmzng the followng Lagrangan expresson L 1 = [u (c ) + ' (x ) + u (d ) + h (x ; e ) (c + x + d + e w )] ; where s the multpler assocated wth the resource constrants. The FOCs yeld: u 0 (c ) = u 0 (d ) = ' 0 (x ) + h x (x ; e ) = h e (x ; e ) = ; (2) wth h x < 0 and h e > 0: Denote the rst-best soluton by c ; x ; d and e. We can also de ne s = d + e, the (mplct) ndvdual savngs at the rst-best soluton. The utltaran planner equalzes margnal utlty of consumng the numerare good n both perods. Snce preferences for ths good are the same for all ndvduals, ths calls for c and d to be equal and the same for all. Margnal utlty for numerare and sn goods are also equalzed, wth the latter composed of the mmedate margnal grat caton and of the (true) delayed margnal mpact on health. Fnally, the planner also equalzes second perod margnal utlty from consumng the numerare good and from consumng health care. If the margnal mpact of sn good consumpton on health s the same for all ndvduals ( = ), then x and e are also dentcal for all. We now contrast the rst-best soluton wth the lassez-fare allocaton obtaned when ndvduals maxmze ther own utlty. We show that the rst-best allocaton can be decentralzed wth ndvdualzed (redstrbutve) lump sum taxes and Pgouvan taxes or subsdes. We have to dstngush two settngs: persstng errors on the one hand, and dual self on the other hand. 8

10 Decentralzaton wth persstng errors Wth persstng errors there s just one optmzaton problem at the start of the rst perod. It amounts to maxmzng: U = u (w (1 + ) s (1 + ) x + a ) +' (x ) + h (x ; e ) + u (s (1 + ) e ) : whch yelds the followng FOCs (1 + )u 0 (c ) u 0 (d ) = 0 (3) (1 + )u 0 (c ) + ' 0 (x ) + h x (x ; e ) = 0 (4) (1 + )u(d ) + h e (x ; e ) = 0: (5) In the lassez-fare (LF), = = = a = 0 and we have u 0 (c ) = u 0 (d ) = ' 0 (x ) + h x (x ; e ) = h e (x ; e ) : Margnal utlty of consumng the numerare and the sn goods are also equalzed (though they d er across agents f there s heterogenety n ), but not at the correct level snce ndvduals make a mstake when assessng the mpact of both sn good and health care consumpton on ther second perod utlty (health status). To decentralze the rst-best optmum, we need ndvdualzed redstrbutve lump sum taxes a and ndvdualzed correctve taxes or subsdes on the sn good and health expendture. Combnng (2) wth (3) (5) yelds the followng lemma. Lemma 1 In the case of persstent errors, t s possble to decentralze the rst-best allocaton by usng ndvdualzed lump sum taxes together wth a tax on sn good and a 9

11 subsdy on compensatory health gven by = ( ) h x (x ; e ) u 0 (c ) > 0; (6) = ( ) < 0: (7) No tax or subsdy on savngs s needed and we have = 0. The tax on sn good consumpton forces the ndvdual to nternalze the full mpact of hs sn good consumpton on hs health. It s proportonal to the share (gven by the d erence between and ) of the margnal mpact of sn good on health that he does not spontaneously nternalze. It s also necessary to subsdze health care, snce ndvduals underestmate ts mpact on health. Intutvely, the subsdy rate s equal to the percentage of underestmaton by the ndvdual ( )=. There s no need to n uence savng, snce ndvduals do not exhbt tme-nconsstent preferences. The smplcty of formula (7) s due to the spec caton adopted: addtve utltes and multplcatve myopa parameter. The taxes,, and subsdes,, are ndvdualzed as long as the parameter or vares across ndvduals. Naturally, wth = and =, taxes would be dentcal for all. Decentralzaton wth dual self In the prevous subsecton we have assumed that ndvduals stck to ther belefs n the second perod when they choose e. Let us now make the reasonable assumpton that n the second perod they realze that they have made a mstake out of gnorance or myopa and take ther decson concernng health care usng ther correct preferences. In behavoral economcs, one then speaks of dual self. 10

12 When the reasonable self prevals n the second perod, the choce of e s determned by the equalty (1 + )u 0 (s (1 + )e ) = h e (x ; e ) : (8) However, ths level of e s not the one that the ndvdual envsoned when he chose hs sn good consumpton and savng n the rst perod. The amount of health care that the ndvdual orgnally planned to buy, denoted by e P, s gven by (1 + )u 0 (s (1 + )e P ) = h e (x ; e P ): (9) The levels of s and x then satsfy the followng rst-order condtons: (1 + )u 0 (c ) + u 0 (s e P ) = 0; (10) (1 + )u 0 (c ) + ' 0 (x ) + h x x ; e P = 0: (11) Is t possble to decentralze the rst-best optmum n these condtons wth our lnear nstruments that are chosen the rst perod? Combnng (2) wth equatons (8) (11) shows that ths s possble usng and plus a. Wth these nstruments, and denotng optmal values wth a, one obtans x and s, whch themply e. De nng e P as the planned level of e when the tax nstruments are set to decentralze the rst-best, we obtan the levels of the tax and subsdy that decentralze the rst-best n ths case. These levels are shown the followng lemma. Lemma 2 In the case of dual self, t s possble to decentralze the rst-best allocaton by usng ndvdualzed lump sum taxes together wth a subsdy on savngs and a tax on the sn good gven by = u0 (s e P ) u 0 (c ) u 0 (c ) ; (12) = h x x ; e P h x (x ; e ) u 0 (c ) : (13) 11

13 No tax or subsdy on health expendtures s needed and we have = 0. Equaton (8) shows that the ndvdual wll take the optmal health care decson the second stage, provded that he chose the optmal values of x and s n the rst stage. In uencng the health care decsos then unnecessary, provded that tax nstruments on savng and sn good consumpton decentralze these two optmal choces. The sn tax s proportonal to the mstake made by the ndvdual. Ths mstake comes from two sources: under-estmaton of the mpact of sn good on health (snce < ) and msplannng of the future amount of health care consumed (e P as opposed to e ). Snce ndvduals msplan ther future health care need, t s also necessary to n uence ther savng decson, as shown by (12). 7 Comparson of sn taxes n the two spec catons It s nterestng to compare the sn taxes obtaned under the two spec catons. To make the comparson easer, we assume that = > 0 and = >. Note that, under ths assumpton, the use of a personalzed lump sum transfer wll make all ndvduals demand the same amounts of all goods (c = c, x = x, e = e, e P = e P ). We thus have (wth S for sngle self and D for dual self): S = ( ) h x (x ; e ) ; u 0 (c ) D = ( ) h x (x ; e ) + h x x ; e P h x (x ; e ) u 0 (c ) : (14) In the two cases, the sn tax s proportonal to the error made n the rst stage when 7 Assumng = and =, we obtan = and = : In words, a makes everyone dentcal and the Pgouvan tax and subsdy rates are dentcal. To llustrate ths pont, assume a sngle ndvdual wth = 0 < = 1: We then have e P = 0 so that the mplementng tax rates or subsdy are = u0 (s ) u 0 (c ) 1 < 0 and = h x (x ; e ) u 0 (c ) > 0: 12

14 evaluatng the damage of sn good consumpton on health, measured at the optmal sn good and health care consumptons. An addtonal term s present n the dual self case, whch s proportonal to the second mstake made by the ndvdual n that case. Snce ths ndvdual msestmates how much health care he wll buy at the optmum, he s also mstaken hs assessment of the margnal damage done by the optmal amount of sn good consumpton, as measured by the functon h(x; e). The sgn of ths mpact depends on the cross-dervatve of ths functon. Assume for nstance that t s postve. Snce the ndvdual under-estmates how much health care he wll buy, he then over-estmates how bad the margnal mpact of sn good wll be 8 (as measured by the functon h(:)). Ths calls for decreasng the tax on the sn good, compared to a sngle self ndvdual. Assumng that h xe has everywhere the same sgn, we then obtan the followng proposton Proposton 1 In the rst-best, f = > 0 then S S D () h xe S 0. Proposton 1 says that the sn tax s hgher (lower) n the case of dual self than the case of persstent error f the margnal productvty of health expendtures decreases (ncreases) wth the consumpton of the sn good. Consequently the comparson depends on the sgn of the cross dervatve. A postve sgn means that the margnal e ectveness (productvty) of health care expendtures ncreases wth the consumpton of the sn good; wth a negatve sgn, health care expendtures are less e ectve for hgher sn good consumpton levels. The sgn of ths cross dervatve depends upon the knd of sn good under consderaton. For nstance, t seems reasonable to assume that t s postve f the sn good s sugar: the more you eat, the more medcatons desgned to treat dabetes may be helpful to you. Wth ths assumpton the sn tax s smaller when the ndvdual 8 Recall that h x (:) < 0. 13

15 acknowledges hs mstake n the second perod of hs lfe. Gven the ncreased productvty of health care, n the dual case the ndvdual easly corrects for hs earler excess whle the ndvdual n the persstent errors case s unable or unwllng to do so. The opposte assumpton can be made for smokng: heavy smokers ncrease ther probablty of gettng lung cancer, for whch there s up to now no e cent cure n the majorty of cases. Put bluntly, there s not much utlty that you can get from consumng health care f you end up wth lung cancer followng heavy smokng. So far we have contrasted the rst-best polces wth dual self on the one hand and persstent error on the other hand. However, t s also nterestng to study the mpact of the avalablty of a compensatory treatment. To do ths, we compare the polcy wth dual self when a compensatory treatment exsts and whet does not; let D nc represent the optmal sn tax when there s no compensatory treatment n the dual self case. To make ths comparson we have to set the value of e to get a meanngful counterfactual. The most natural approach s to adopt the optmal value of e, that s e such as de ned n the rst-best secton, and to assume that t s exogenous. In that case, t s plan that the optmal sn tax s exactly the same as the one wth persstent error, spec ed by equaton (6),.e. D nc = ( ) h x (x ; e ) : (15) u 0 (c ) Ths tax rate, by ts very de nton, nduces the ndvdual to choose the optmal level of the sn good for gven health expendtures, e. Recall that n the case of persstent error, the planned and e ectve levels of e are equal and no correcton for the msplannng of e s needed n the rule for the sn tax. When there s no compensatory treatment, such a correctos of course not needed ether. In other words, n the case of persstent gnorance there s no wllngness to correct for the sn good nduced health deteroraton; 14

16 when there s no compensatory treatment there s no room for correcton. Ths leads to the followng proposton. Proposton 2 In the case of dual self, f = > 0 then D nc S D () h xe S 0. Proposton 2 says that the possblty of compensatng for the snful consumpton leads to a hgher (lower) rst-best sn tax (compared to a settng where such a compensaton s not avalable) f the margnal productvty of health expendtures decreases (ncreases) wth the consumpton of the sn good. III Second-best n the case of persstent errors We now turn to the second-best settng wth lnear tax nstruments and a unform lump sum transfer. It s the combnaton of these two assumptons whch explans the move to a second best settng. We assume n the remanng of ths paper that = > : In other words, the objectve e ect of both e and x on health s the same for all, but ndvduals d er n ther degree of myopa (as well as ncome). We also assume that taxes/subsdes on savng are not avalable anymore (ether because savng s not observable, or because elements not modeled, lke nternatonal moblty of captal, prevent savng from beng taxed or subsdzed). 9 Ths allows us to concentrate on the two nstruments whch are the most relevant from the perspectve of ths paper. We rst consder the case where the ndvduals never acknowledge that the true health parameter s (or do so after e s determned). In that case, restrctng the nstruments 9 Introducng ths extra nstrument would complcate the expressons consderably wthout a ectng the results n a sgn cant way. In partcular, even wth the three tax nstruments (but wthout non unform lump-sum transfers) the rst best cannot be mplemented except n very specal cases; see below for addtonal dscusson. 15

17 to lnear taxes and unform lump sum transfer we wrte the new Lagrangan as L 2 = [u(w s x (1 + ) + a) + ' (x ) + u (s (1 + ) e ) +h (x ; e ) (a x e )] ; where s, x and e are functons of a, and and are obtaned from the followng optmal condtons for ndvdual choces: u 0 (c ) + u 0 (d ) = 0; (16) u 0 (c ) (1 + ) + ' 0 (x ) + h x (x ; e ) = 0; (17) u 0 (d ) (1 + ) + h e (x ; e ) = 0: (18) The rst-order condtons for the maxmzaton of L 2 are provded n Appendx A. In compensated terms, these expressons can be wrtten as: 10 where ~ L 2 = cov (u0 (c ) ; x ) + ~ L 2 = cov (u0 (c ) ; e ) + ( ) H + ( ) H ~x + ~e = 0; (19) ~x + ~e = 0; (20) H = h x (x ; e ) ~x + h e(x ; e ) ~e and H = h x (x ; e ) ~x + h e(x ; e ) ~e : It s mportant to note that we here use the concept of average compensaton and not that of the standard Slutsky term. Usng a tlde (~) for our compensaton term and a 10 De nng L ~ 2 = L 2 + L 2 a x L and ~ 2 = L 2 + L 2 a e where x and e are the average values of x and e, respectvely. Addtonally, recall that for any two varables z and v de ned for our populaton cov (z ; v ) = z v z v : 16

18 hat (^) for the standard Slutsky term, we have ~x = x + x a e = ^x + x a (e e ) ; ~x = ^x + x a (x x ). Our approach s smple, but the sgns of the compensated terms have to be nterpreted wth cauton. For example, we know that ^x = < 0, but f x s much smaller than the average, x, ~x = could be postve. Interpretng the above FOCs, we assume that these own compensated dervatves are negatve. 11 Observe that wth dentcal ndvduals the rst-best optmum s obtaned wth just and. Equatons (19) and (20) jontly de ne the values of and. To obtan the tax formulas that de ne each of these taxes/subsdes we can use Cramer s rule to solve for and to obtan " = 1 cov (u 0 ~e (c ) ; x ) D pe ( ) H where " = 1 cov (u 0 (c ) ; e ) D pe D pe = ~x ( ) H " ~x cov (u 0 (c ) ; e ) ~e + ~e ~e ( ) H cov (u 0 (c ) ; x ) ~x + ~e ~x ~x ( ) H ~e ~x If we assume that the cross dervatves are neglgble, namely that ~x=! 0 and ~e=! 0, we obtan: # : # # ; ; = = P I ( ) h x (x ; e ) (~x =) cov (u 0 (c ) ; x ) P I n ; (21) (~x =) P I ( ) h e (x ; e ) (~e =) cov (u 0 (c ) ; e ) P I n : (22) (~e =) 11 That s (~x =) < 0 and (~e =) < 0. 17

19 The expressons for the optmal levels of the sn tax and the health subsdy nclude two terms: a Pgouvan term that corrects for the degree of myopa and an optmal tax term that measures the redstrbutve mpact of the sn tax or the health subsdy. The rst term of the numerator of (21) and (22) s the Pgouvan term found n (6) and (7) summed over all ndvduals wth weghts equal to the e ect of the tax ondvdual demands of ether x or e. Wth both dervatves of demand functons negatve, ths term calls for a tax on sn good and a subsdy on health care. The second term of the numerator of (21) and (22) re ects redstrbutve consderatons. It depends on the concavty of u, the ntal nequalty of earnngs and the correlaton between and w : Wth dentcal ndvduals, ths term dsappears and (21) and (22) reduce to (6) and (7). 12 Wth d erent ndvduals and no correlaton between and w, the covarance wll be negatve n both equatons, snce rcher people consume more of all goods (x, c and e ) than poorer people wth the same degree of myopa. Ths tends to ncrease the tax on the sn good and decrease the subsdy on health care, compared to the case wth dentcal ndvduals. A postve correlaton (between and w ) tends to reduce the consumpton of x of the hgh wage ndvduals. The sn good consumpton now ncreases less fast wth ncome and t may even decrease (when the myopa e ect domnates the ncome e ect). 13 Consequently, the cov(u 0 (c ) ; x ) wll ncrease and may even become postve whch leads to a smaller sn tax than wth zero correlaton. Intutvely, the redstrbutve bene ts of sn taxes are mtgated or even reversed when 12 We have a rst-best soluton wth = u 0 (c ). 13 There are many studes that look at the correlaton between wealth and other behavors or characterstcs such as smokng or obesty. They show that there s a strong negatve correlaton between ths characterstcs and wealth (e.g. Stunkard and Sorensen, 1993). Ths lterature provdes admttedly ndrect support for a postve correlaton between and w. The negatve correlaton between obesty or smokng and wealth can also be explaned by factors other than myopa; for example for the case of obesty a negatve correlaton wth wealth can also follow from d erences n food prces accordng to fat content. 18

20 the rch are less myopc. Ether way, the absolute value of the covarance wll tend to be larger the more concave s u and the more unequal s the wage dstrbuton. Ths wll ncrease the sn tax whet s progressve (.e., when cov(u 0 (c ) ; x ) < 0) and decrease the tax whet s regressve (postve covarance). The mpact of a postve correlaton on cov(u 0 (c ) ; e ) s less clear: wealther people buy less sn good, but they also better realze the mportance of health care, so that the net mpact on the amount of e consumed s not easy to determne. These results are summarzed n the followng proposton. Proposton 3 Consder the case of persstent errors where the polcy tools are restrcted to a lnear sn tax, a lnear subsdy on health and a lump sum transfer. If cross dervatves ~x= and ~e= are neglgble we have the followng results: a) The expressons for the optmal levels of and are gven by equatons (21) and (22). b) The level of the sn tax decreases as the correlaton between and w ncreases (.e., when wealther ndvduals tend to be less myopc). c) When cov(u 0 (c ) ; x ) < 0 (resp. cov(u 0 (c ) ; x ) > 0) the sn tax tends to be hgher (lower) the more unequal the wage dstrbuton and the more concave the utlty functon. IV Second-best wth dual self Now we assume that the ndvduals realze after one perod that they made a mstake and that the only correctve decson they can make s the choce of health expendture. As n secton II, we thus dstngush between the planned nvestment e P and the ex post choce e. The ndrect utlty functon used by the socal planner ts welfare maxmzaton has to take nto account these two values of e whch yeld two values of d (planned and e ectve). 19

21 In the rst perod, the functons x (; ; a), s (; ; a) and e P (; ; a) are obtaned as the soluton to u 0 (c ) + u 0 d P = 0; (23) u 0 (c ) (1 + ) + ' 0 (x ) + h x x ; e P = 0; (24) u 0 d P (1 + ) + he x ; e P = 0: (25) where d P = s e P (1 + ) > d = s e (1 + ). In the second perod the e ectve demand for e s de ned by (1 + )u 0 (d ) = h e (x ; e ); whch yelds e as a functon of x and s. Substtutng these varables by ther expressons n terms of (; ; a) yelds e = f s ; x ; e P = e (; ; a) : (26) The Lagrangas gven by L 3 = [u(w s x (1 + ) + a) + ' (x ) + u (s (1 + ) e ) +h (x ; e ) (a x e )] ; whch s smlar to L 2 except that ndvdual choces are now determned by (23), (24) and (26). The rst-order condtons for the maxmzaton of L 3 are gven Appendx B. These can be rewrtten as follows f we assume that the cross prce e ects are neglgble: = = " 1 P I n cov [u 0 (c ) ; x ] + u 0 (d ) u 0 d P ~s (~x =) # + hx (x ; e ) h x x ; e P ~x ; (27) " # 1 P I n cov [u 0 (c ) ; e ] + u 0 (d ) u 0 d P ~s : (28) (~e =) 20

22 The expresson for the optmal level of the sn tax ncludes three terms: an optmal tax term that measures the redstrbutve mpact of the sn tax, a savng compensaton term (whch accounts for the mpact of the sn tax on the, otherwse nsu cent, level of savng) and a Pgouvan term that corrects for the degree of myopa. The covarance term s the same as that n the sngle self scenaro and re ects the equty concern of publc polcy. The second term of the numerator n (27) has the same sgn as ~s=. The ntuton for ths term goes as follows: ndvduals over-estmate ther second perod consumpton (d P > d) snce they under-estmate ther health care needs. As a consequence, they do not save enough. To compensate for ths, the tax wll tend to be hgher f t has a postve e ect on savng. The thrd term of the numerator of (27) s the Pgouvan term found n equaton (13) summed over all ndvduals wth weghts equal to the e ect of the tax on ndvdual demands of x. Wth = 0 or h xe < 0, t s postve (as long as ~x= < 0). Equaton (28) has a smlar nterpretaton, except that only the rst two terms are present (redstrbuton and savngs compensaton). There s no Pgouvan term here. 14 In the case of dentcal ndvduals, (27) and (28) can be rewrtten as: = 1 u 0 d P u 0 (d) ~s= ~x= + 1 ( ) hx (x; e) + h x x; e P h x (x; e) ; = 1 u 0 d P u 0 (d) ~s= ~e= : Unlke n the case of persstent error, the rst-best can no longer be acheved wth the two consdered nstruments even whendvduals are dentcal. The second part of the sn tax s the famlar Pgouvan term, expressed as n the rst-best decentralzaton equaton (14). The other term n both equatons comes from our nablty to control savng drectly, whch would be necessary to decentralze the rst-best optmum. Savng 14 Recall that we are n the case where cross prce e ects are neglgble so that has no mpact on x. 21

23 can be ndrectly controlled through the use of both and. If any of these nstruments stmulates savng, ths makes usng t more desrable. These results are summarzed n the followng proposton. Proposton 4 Consder the case of dual self where the polcy tools are restrcted to a lnear sn tax, a lnear subsdy on health and a lump sum transfer. If ~x=! 0 and ~e=! 0; we have the followng results: a) The optmal tax formulas are gven by equatons (27) and (28). b) Propertes b) and c) of Proposton 3, reman vald n the dual self case. c) The sn tax tends to be hgher (and the health subsdy lower) the hgher ts mpact on savngs. So far we have assumed that savngs could not be taxed. Introducng as an extra nstrument has a sgn cant mpact on the results only f all ndvduals are dentcal; n that case the lnear nstruments are su cent to mplement the rst-best. However, wth heterogeneous ndvduals ths s not true and the qualtatve results are not a ected by the avalablty of a tax (or subsdy) on savngs. 15 In partcular, the savng compensaton term does not dsappear from the expresson because does not provde a perfect control of ndvdual savngs. V Concluson In ths paper we have consdered the case of sn goods that have delayed negatve e ects that ndvduals gnore at the tme of consumpton but acknowledge later. Indvduals have then the possblty of partally compensatng those negatve e ects by nvestng n 15 Except that wth ths extra nstrument, the second-best soluton wll be close to the rst-best outcome when the degree of heterogenety s small. 22

24 health care. Assumng a paternalstc government, we show that the rst-best could be decentralzed wth a sn tax, a subsdy on savng and ndvdualzed lump sum transfers (or alternatvely, by assumng dentcal ndvduals). In the second-best, ndvdualzed lump sum transfers are not avalable and the only avalable nstruments are a lnear sn tax and a lnear subsdy on health care. We dscuss the optmal second-best tax subsdy polcy wheren dstrbutve and correctve Pgouvan consderatons are mxed. We also consder the case of what we call persstent error, namely the case where ndvduals acknowledge the negatve e ects of ther snful consumpton whet s too late to take any correctve acton (.e., after e has been chosen). From andvduals perspectve ths latter case s formally equvalent to yet another settng whch corresponds to what can be called persstng gnorance, where the ndvdual never acknowledges the negatve e ects of hs consumpton. The nteracton between sn goods and health spendng and ts ncdence on sn taxes s at the heart of ths paper. We have seen that n the rst-best the sn tax closely depends on the cross dervatve h xe. We have also shown that the mpact of the possblty of makng compensatory health expendtures on the sze of the sn taxes depends on ths same cross dervatve. In the second best the degree of substtutablty between e and x n uences the sn tax n a more complex way. All the e ects descrbed n the rst-best reman applcable (through the Pgouvan term whch depends on h xe ). However, there are now many more avenues through whch the nteracton a ects the optmal polcy. Spec cally, t a ects the compensated dervatves as well as the covarance terms. Our results provde some general qualtatve nsght nto these problems. To reach more spec c and quanttatve conclusons, emprcal studes or at least calbrated smulatons of our settng are necessary. 23

25 In ths paper we have focused on sn goods consumpton. Our method could be used for other problems. For example, lack of physcal exercses or hygene n the rst perod of lfe whch has delayed detrmental e ects. These e ects can be partally o set n the second perod. Another example s overtme or moonlghtng that lead to early dsablty. A fully ratonal ndvdual would understand the mportance of not abusng one s body when young to avod regretful consequences later on lfetme. The ngredents of these varous stuatons are: behavor wth delayed detrmental e ects, myopa and possblty of partal compensaton. Appendx A Frst-order condtons for the maxmzaton of L 2 Assumng nteror solutons and makng use of (16), (17) and (18), the FOCs of the socal problem are gven by: L 2 a = u 0 (c ) + L 2 = L 2 = x a u 0 (c ) x + h x (x ; e ) x a + h e (x ; e ) e ( ) a x + x + e u 0 (d ) e + e + x + e e = 0; a h x (x ; e ) x + h e (x ; e ) e ( ) = 0; h x (x ; e ) x + h e (x ; e ) e ( ) = 0: 24

26 B Frst-order condtons for the maxmzaton of L 3 The FOCs are gven by where L 3 a = u 0 (c ) + L 3 = L 3 = ~ H x a u 0 (d ) u 0 d P 1 x a s a e a = 0; u 0 (c ) x + u 0 (d ) u 0 d P s H ~ x + x + x + e = 0; u 0 (d ) e + u 0 (d ) u 0 d P s H ~ x + e + x + e = 0; ~H = h x (x ; e ) h x x ; e P : As n secton III we use L 3 =a to obtan the compensated expressons of L 3 = and L 3 = L ~ 3 = cov [u0 (c ) ; x ] + u 0 (d ) u 0 d P ~s + H ~ ~x + ~x + ~e = 0; L ~ 3 = cov [u0 (c ) ; e ] + u 0 (d ) u 0 d P ~s + H ~ ~x + ~x + ~e = 0: It s clear from the above that even wth dentcal ndvduals, one cannot acheve the 25

27 rst-best wth and as nstruments. Solvng for and, we obtan ( = 1 cov [u 0 (c ) ; x ] D ds H ~ ~x ~e ~e + u 0 (d ) u 0 d P + ( = 1 cov [u 0 (c ) ; e ] D ds cov [u 0 (c ) ; e ] ~s ~ H ~x u 0 (d ) u 0 d P ~ H ~x ~x ~x + u 0 (d ) u 0 d P + ~e ~s cov [u 0 (c ) ; x ] ~s ~ H ~x u 0 (d ) u 0 d P ~e ~x ~s ~e ~e ~x ~x ~x ) ) ; ; where D ds = " ~x ~e ~e ~x # : 26

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