NELLCO NELLCO Legal Sholarship Repository Harvard Law Shool John M. Olin Center for Law, Eonomis and Business Disussion Paper Series Harvard Law Shool 8-5-006 Myopia and the Effets of Soial Seurity and Capital Taxation on Labor Supply Louis Kaplow Harvard Law Shool Follow this and additional works at: http://lsr.nello.org/harvard_olin Part of the Law and Eonomis Commons Reommended Citation Kaplow, Louis, "Myopia and the Effets of Soial Seurity and Capital Taxation on Labor Supply" (006). Harvard Law Shool John M. Olin Center for Law, Eonomis and Business Disussion Paper Series. Paper 555. http://lsr.nello.org/harvard_olin/555 This Artile is brought to you for free and open aess by the Harvard Law Shool at NELLCO Legal Sholarship Repository. It has been aepted for inlusion in Harvard Law Shool John M. Olin Center for Law, Eonomis and Business Disussion Paper Series by an authorized administrator of NELLCO Legal Sholarship Repository. For more information, please ontat tray.thompson@nello.org.
HARVARD JOHN M. OLIN CENTER FOR LAW, ECONOMICS, AND BUSINESS ISSN 045-6333 MYOPIA AND THE EFFECTS OF SOCIAL SECURITY AND CAPITAL TAXATION ON LABOR SUPPLY Louis Kaplow Disussion Paper No. 555 08/006 Harvard Law Shool Cambridge, MA 038 This paper an be downloaded without harge from: The Harvard John M. Olin Disussion Paper Series: http://www.law.harvard.edu/programs/olin_enter/ The Soial Siene Researh Network Eletroni Paper Colletion: http://papers.ssrn.om/abstrat_id######
JEL Classes: D, D9, H, H4, H55, J Myopia and the Effets of Soial Seurity and Capital Taxation on Labor Supply Louis Kaplow * Abstrat Myopia is inreasingly believed to be a signifiant determinant of behavior and also plays a entral role in justifiations for soial seurity and poliies toward the taxation of apital. It is important, however, to aount for labor supply effets, partiularly in light of the preexisting distortion due to labor inome taxation. For example, might even atuarially fair soial seurity have the highly distortionary effet of a tax on top of an existing tax (the inome tax) beause myopi individuals give exessive weight to present levies on earnings that finane distant future benefits? Similarly, might greater reliane on apital rather than labor inome taxation be attrative beause olletions are in the future rather than when earnings are reeived? To answer these and other questions, this artile analyzes the effet of suh poliies on labor supply in a model that expliitly inorporates myopi deision-making. Many of the results may seem ounterintuitive. In most respets, even with myopia, soial seurity has qualitatively different effets than those of a tax levied on top of an existing tax. Both soial seurity and apital taxation may ause labor supply to rise or fall when individuals are myopi, depending on the urvature of individuals utility as a funtion of onsumption. Moreover, whatever is the sign of these effets under one assumption about how myopia relates to labor supply deisions, the sign is reversed under the other assumption that is onsidered. Additionally, some interventions have a first-order effet on labor supply from the outset but others do not, and some labor supply effets rise with the magnitude of the intervention whereas others fall. * Harvard University and National Bureau of Eonomi Researh. I am grateful to Steven Shavell and NBER workshop partiipants for omments, Stephanie Gabor and James Kvaal for researh assistane, and the John M. Olin Center for Law, Eonomis, and Business at Harvard University for finanial support.
Myopia and the Effets of Soial Seurity and Capital Taxation on Labor Supply Louis Kaplow Louis Kaplow. All rights reserved.. Introdution Eonomists have long been interested in the possibility that individuals may behave myopially. See, for example, Strotz (956). Inreasingly in the past deade, researhers have emphasized myopia as an explanation for savings behavior, inluding notably those features that have been diffiult to reonile with idealized models of lifeyle maximization. See, for example, Laibson (996, 997) and the surveys by Bernheim (00) and Bernheim and Rangel (005). Empirial work has foused on suh subjets as whether individuals retirement savings are adequate and, relatedly, the existene of and explanation for a signifiant drop in onsumption upon retirement. It is also appreiated that myopi savings behavior may have normative impliations. Laibson (996) estimates that orretives for inadequate savings due to myopia ould raise individuals lifetime welfare to an extent equivalent to almost an additional year s worth of inome. Soial insurane and tax provisions for retirement savings seem motivated in part by the onern that shortsighted individuals may inadequately provide for themselves. Aordingly, eonomists reently inreased attention to soial seurity and longstanding interest in optimal apital taxation should be extended to inorporate myopi behavior. The present study pursues suh an inquiry with an emphasis on the impliations of myopi savings behavior for labor supply. The subjet of labor supply is signifiant in itself and is espeially important in light of the preexisting distortion due to labor inome taxation. Although labor supply in general and how it may be affeted by soial seurity and apital See, for example, Kotlikoff, Spivak, and Summers (98), Banks, Blundell, and Tanner (998), Engen, Gale, and Uello (999), Moore and Mithell (000), Bernheim, Skinner, and Weinberg (00), Sholz, Seshadri, and Khitatrakun (004), and Aguiar and Hurst (005). No attempt is made here to undertake a omplete welfare analysis of any partiular poliy. In order to do that, one ould first examine the welfare effets of a poliy ignoring hanges in labor supply and then supplement the analysis aordingly. Importantly, labor supply affets government revenue beause the existene of an inome tax; in standard optimization problems, this welfare effet would be determined, at the margin, by the produt of the rate of hange in revenue (equal to the rate of hange in labor supply times the pertinent marginal tax rate) and the shadow prie on the government s revenue onstraint. The present artile s fous on labor supply effets is motivated in signifiant part by this formulation of the soial maximization problem. - -
taxation has been studied extensively, it is important to onsider how the analysis hanges if it is assumed that savings deisions are myopi. Regarding soial seurity, if mandatory ontributions, typially a funtion of labor inome, are viewed by individuals as an additional tax, labor supply effets should be highly signifiant beause payroll taxes that fund soial insurane are of a similar order of magnitude to inome taxes. Following the rule of thumb that distortion is proportional to the square of the tax rate, the total distortion would quadruple. Furthermore, even an atuarially fair soial seurity retirement sheme, not in itself a tax, might be imagined to have similar effets to a urrent tax on labor if individuals are signifiantly myopi, for the ontributions are extrated from earnings in the present whereas the benefits they finane are in the distant future. Although prior theoretial literature, suh as Diamond and Mirrlees (978, 986, 007), has examined a number of respets in whih soial seurity may influene retirement deisions, labor supply during working years has reeived more limited srutiny. See, for example, Diamond (00) and the survey by Feldstein and Liebman (00). 3 In partiular, it has not been systematially analyzed for the ase in whih individuals are myopi. Yet this ase is a natural one to onsider in light of the fat that myopia is thought to provide one of the most important justifiations for the existene of soial seurity. Regarding the taxation of apital, it is understood that suh taxation serves indiretly as a tax on labor and thus ontributes to the distortion of labor supply. To hold the overall level of taxation on labor supply onstant, one an examine apital taxation as a form of differential ommodity taxation, where different ommodities orrespond to onsumption in different time periods. Atkinson and Stiglitz (976) showed that, in the presene of an optimal inome tax, no differential taxation is optimal when ommodities as a whole are weakly separable from labor, with the impliation that apital should not be taxed. However, taxes on apital or, equivalently, differential taxes on future onsumption are ordinarily levied in the future, raising the possibility that they may have less of an effet on the urrent labor supply of myopi individuals. Perhaps deferred taxation generally, whether substituting a ash-flow onsumption tax for a tax on wages as earned or shifting from the taxation of earnings to the taxation of apital, would have a favorable effet on labor supply when individuals are myopi. Relatedly, perhaps the subsidization of apital inome, whih may otherwise appear attrative as a ounterweight to myopi savings deisions, will have an offsetting ost of inreasing labor supply distortion. To determine whether these and other onjetures are valid, it is neessary to revisit prior analysis of apital taxation, inorporating myopia expliitly. This artile analyzes the effets of soial seurity and apital taxation (or subsidization) on labor supply in a setting in whih individuals onsumption alloation deisions are myopi. Beause this is a preliminary investigation of a omplex problem, it will fous on a fairly simple 3 Prior work that examines aspets of the interation between soial seurity and labor supply muh based on the absene of perfet tax-benefit linkage inludes Auerbah and Kotlikoff (987), Browning (985), Burkhauser and Turner (978, 985), Gordon (983), and Moffitt (987). See also Diamond and Köszegi (003) on myopia and retirement and Feldstein (985) on myopia, soial seurity, and savings. - -
model that highlights entral issues but abstrats from a number of ompliations. Aordingly, the analysis will employ a standard two-period model with separable utility, wherein individuals make only two deisions: how muh labor to supply in the first period and how to alloate disposable inome between the two periods. Individuals will be subjet to a linear inome tax on their earnings and also to a soial seurity regime in the first variation and to a tax or subsidy on apital inome (modeled as a differential tax on onsumption in the two periods) in the seond variation. Consumption deisions are assumed to exhibit a simple form of myopia wherein exessive weight is plaed on first-period onsumption. For labor supply, two ases are onsidered: when labor supply deisions exhibit the same myopia as onsumption deisions, and when labor supply deisions are nonmyopi but are made with an awareness that onsumption will be alloated myopially. Setion examines soial seurity. To isolate the effets of myopia, atuarially fair shemes are onsidered. Given individuals myopi onsumption deisions, the diret effet of soial seurity s fored-savings feature is to raise welfare. When labor supply deisions are myopi, soial seurity may raise or redue labor supply, depending on the urvature of individuals utility funtions: Labor supply falls on aount of soial seurity reduing myopi individuals pereived value of disposable inome, but it rises beause of the relatively greater effet of soial seurity on the pereived marginal utility of onsumption in the first period. Importantly, as the fored-savings onstraint just begins to bind, there is no first-order effet on labor supply, even though there is a first-order welfare gain in improved onsumption alloation. In this respet, despite the existene of even signifiant myopia, the soial seurity tax does not behave as a tax imposed on top of a preexisting tax (the linear labor inome tax in this model): A small pure additional tax, unlike the soial seurity tax here, would have a first-order effet on labor supply and thus on distortion. Moving to the ase in whih individuals labor supply deisions are nonmyopi, the sign of the effet of soial seurity on labor supply reverses. That is, in ases in whih the effet on labor supply is negative (positive) with myopi labor supply, it is positive (negative) with nonmyopi labor supply. This differene results beause nonmyopi individuals view soial seurity s fored savings positively, whih reverses both the diret effet of soial seurity on the value of onsumption and the indiret effet on marginal utilities of onsumption in the two periods, as will be explained. Additionally, in this ase there is a first-order effet of soial seurity on labor supply as the fored-savings onstraint just begins to bind, but one that falls as the onstraint beomes tighter. One again, the effet of soial seurity is qualitatively different from that of a tax upon a tax. Setion 3 analyzes the taxation of apital. As noted, apital taxation or subsidization is modeled as a differential tax on onsumption in the two periods. Furthermore, to abstrat from hanges in the level of taxation, it is assumed that, when the tax on seond-period onsumption is inreased (redued), the tax on first-period onsumption is redued (inreased) so that the same amount of revenue is raised for a given amount of labor inome. It is first shown that, as one would expet, apital subsidization opposes the effet of myopia on onsumption alloations, whih raises welfare. Regarding labor supply, the effets are analogous to those of soial seurity. For myopi labor supply, there is no effet of apital taxation or subsidization on labor supply at the point at whih a differential is first introdued. But with nonmyopi labor - 3 -
supply, there is a first-order effet of the same sign and analogous magnitude to that arising under soial seurity s fored savings. This setion also onsiders the impliations of heterogeneity in suseptibility to myopia. Capital subsidization improves the welfare of the myopi by reduing their intertemporal onsumption misalloation but redues the welfare of the nonmyopi by distorting theirs. Additionally, apital subsidization tends to redistribute from the myopi to the nonmyopi (at eah level of earnings), the desirability of whih depends on the form of the soial welfare funtion and on the urvature of individuals utility funtions. Setion 4 examines the hoie between soial seurity and apital subsidization as remedies for myopia and also onsiders the relevane of liquidity onstraints to the analysis. Setion 5 offers onluding remarks.. Soial Seurity.. Model Consider a two-period model in whih individuals supply labor, l, in the first period at the wage w and alloate disposable inome between onsumption, and, in periods and. Suppose that utility takes the following form: () u (,,) l + δ zl (), ρ ρ where * is the subjetive disount fator and z measures the disutility of labor effort, with zn > 0 and zo > 0. (To larify, * is taken here to be a real trait of individuals utility, for purposes of assessing soial welfare; myopia will be introdued separately below.) The positive onstant D is the oeffiient of relative risk aversion, utility from onsumption in eah period taking the onstant-relative-risk-aversion form (where it is understood that, when D, utility from onsumption i is instead given by ln i ). This funtional form further implies that there is a onstant elastiity of intertemporal substitution in onsumption. Individuals are assumed to be subjet to a linear inome tax with marginal rate t and grant g, and savings earn interest at the rate r, so their budget onstraint (before we onsider soial seurity or apital taxation) is ( ) + ( twl ) + g. + r The first-order ondition for onsumption alloation between the two periods by a nonmyopi individual (whih an be determined by solving the budget onstraint () for, substituting it into expression (), and differentiating) is u () 3 δ ( + r) 0 or δ ( + r) / ρ. - 4 -
Let * denote the optimal level of first-period onsumption by a nonmyopi individual (whih an be stated expliitly by simultaneously solving expressions () and (3) for and ). To introdue myopia in a simple manner, suppose that, in alloating disposable inome between and, individuals behave as if they are maximizing the following variant of the utility funtion given by expression (): ( 4) u (,, l) β + δ zl ( ), ρ ρ where the weight on first-period onsumption $ is taken to exeed. 4 This formulation ould be taken to represent quasi-hyperboli disounting, as in Laibson (996, 997), for the ase of a two-period model, or other forms of myopia. The important point is that individuals behavior is ditated by a different utility funtion (4) from that whih defines their atual well-being (), i.e., that whih is relevant for assessing soial welfare. 5 The first-order ondition for onsumption alloation by a myopi individual is u () 5 β δ( + r) 0 or β δ ( + r) / ρ. Let M denote the optimal level of first-period onsumption by a myopi individual. Comparing expressions (3) and (5), it is lear that $ > implies M > *. At this point, soial seurity an be introdued into the model. This ould be done fairly generally by letting T S (wl) denote the soial seurity tax on labor earnings that individuals are required to pay in period and B S (wl) as the level of soial seurity benefits they reeive in 4 It would also be natural to weight the disutility of labor, z, by $ beause labor is supplied in the first period. Inluding suh a weight would not materially affet the results below. (Speifially, eah of the zn and zo terms in expressions (7), (8), (9), (4), and (5), involving the ases with myopi labor supply, would be weighted by $, but the ultimate expressions of interest would be unhanged beause this added weight would impliitly be in d u/dl in the denominator of expressions (9), (0), (8), and (9).) Note also that, instead of weighting firstperiod soures of utility by $, one ould weight seond-period utility from onsumption by a fration less than one. The results would be nearly idential, the differene between the two formulations being in the ardinalization of utility as funtion of onsumption. (Reall that the disount fator * does not already reflet suh a downward weighting of seond-period onsumption beause * is taken to be the true subjetive disount rate, a feature of the normatively relevant utility funtion.) 5 As has beome familiar, myopia is being taken as a problem of self-ontrol rather than as a feature of true utility (the latter being aptured here by *). - 5 -
period, where benefits depend in some fashion on prior ontributions. In order to isolate the effets of the fored-savings feature of soial seurity from redistributive and revenue effets, it will be assumed that soial seurity is atuarially fair, i.e., T S (wl)(+r) B S (wl) for all wl. 6 In that ase, soial seurity would have no effet on the budget onstraint (), and thus on behavior, if individuals ould freely borrow against (period ) soial seurity benefits and, furthermore, if despite their myopia they fully appreiated the availability of suh benefits and had no ompuntion against borrowing. Thus, soial seurity is only interesting when suh borrowing is impossible or otherwise will not our, as will be assumed throughout. Aordingly, the (only) effet of soial seurity in the present model is to onstrain the amount individuals an onsume in period. For onveniene, this upper bound on will be stated as a fration of disposable inome, so soial seurity poliy may be desribed simply yet ompletely by P, the minimum required fration of savings. 7 There is now the additional onstraint [ ] () 6 ( χ )( t) wl + g. If the onstraint is binding, this expression is satisfied as an equality, whih in turn for any given level of l ditates the alloation of disposable inome between and. If labor supply was unaffeted by soial seurity, welfare analysis would be straightforward. Raising P has no effet until the point at whih the onstraint (6) begins to bind, i.e., the point at whih the level of that satisfies (6) as an equality just equals M. As P is raised further, will fall. This initially produes a first-order welfare gain beause M > *, and raising P ontinues to produe suh a gain until just equals *. (The value of Mu/M 6 In the presene of a tax on labor inome, the assumption that soial seurity is atuarially fair is primarily a matter of notational onveniene, for any differene between taxes and expeted benefits at a given inome level ould instead be understood as a omponent of the labor inome tax. 7 Other formulations of soial seurity ould be analyzed, inluding one in whih the bound is a fration of earnings, wl, and another in whih the bound is a fration of after-tax earnings, (!t)wl (whih differ from disposable inome on aount of g). For any given tax rate t, these two formulations are equivalent to eah other, and the effet of hanging the bound differs between the two ases only in magnitude beause raising the latter bound (on after-tax earnings) one unit has a smaller effet than raising the former bound (on before-tax earnings) when t > 0. Analysis of these two ases yields results qualitatively similar to those of the ase onsidered in the text. Note that inluding g in the quantity that is subjet to fored savings, as is done here, has the effet that fored savings are a onstant fration of disposable inome at all inome levels, whereas the other formulations would make fored savings a rising fration of disposable inome (beause the grant g, exempt from the fored-savings requirement in the alternative formulations, is a greater share of disposable inome for lower-inome individuals). Alternatively, if the grant payment was divided between the two periods (and borrowing against it was impossible), these two alternative models would be even loser to the present model. - 6 -
given by expression (3) is negative when > *.) In this respet, the fored savings that results from soial seurity an be viewed as a substitute for a ommitment tehnology that myopi individuals would employ to redue their level of if they had the ability to do so. 8 In subsequent disussion, attention will largely be onfined to levels of P suffiiently high that the onstraint binds (for lower values of P, sine there is no effet on onsumption, there an be no effet on labor supply) but not so high as to fore below *. The question to be explored is whether the effet of raising P on labor supply reinfores or detrats from (or has no effet on) the welfare gains from fored savings in realloating onsumption between the two periods. Two subases regarding the effets of soial seurity on labor supply should be distinguished: When individuals labor supply deisions are subjet to the same myopia that determines the alloation of onsumption between periods, and when these deisions are rational in the sense that individuals not only understand that they will alloate their earnings myopially but also appreiate what their realized utility atually will be (i.e., that suh an alloation involves too high a level of ). Both ases are of potential interest in light of the fat that myopi behavior is not that well understood and the reognition that myopia is ontext speifi. 9 Note speifially that the effet of myopia on labor supply may depend on the nature of the deision in question: Deisions about whether to pursue higher eduation or what job to hoose from among many that require different effort levels may perhaps be made nonmyopially, whereas the same individuals may forgo overtime opportunities beause of the immediate temptation to go out with friends or home to wath favorite television shows. For simpliity, analysis will fous on the two pure ases, beginning with that in whih the same behavioral utility funtion (4) determines both onsumption alloation and labor supply... Myopi Labor Supply As stated, the present assumption is that individuals labor supply deision is also myopi in the sense that it is determined by maximizing utility as defined in expression (4) rather than as defined in expression (). It should be emphasized that individuals are nevertheless assumed to appreiate that their onsumption alloations are myopi in the same sense. In some respets, this assumption seem natural; one might say that suh myopi individuals only know of one utility funtion in period, and that is the myopi one. In addition, it is the only ase of 8 For individuals who already an ommit to lower levels of, soial seurity would have no effet until the onstraint began to bind on them, i.e., until first-period onsumption is fored lower than the level to whih they an suessfully ommit themselves. For simpliity, it is assumed that no private ommitment tehnologies exist. 9 For example, some individuals employ ommitment devies (automati ontributions to retirement aounts, not purhasing types of food they know they will overeat), many fail to borrow (fully or at all) from inreased home equity despite their tendeny to onsume all of their payheks, and savings behavior may be influened by modest hanges in framing, suh as when individuals ontributions to 40(k) plans depend on what ontribution, if any, is speified as the default. On the latter, see Madrian and Shea (00) and Choi et al. (004). - 7 -
analytial interest beause otherwise individuals would not expet soial seurity to be binding on themselves and thus there would be no effet on labor supply (keeping in mind that attention is foused on the ase in whih P is not set so high as to fore below *). 0 When the fored-savings onstraint is binding, expression (6), holding as an equality, determines as a funtion of l. The value of as a funtion of l an be determined by substituting for in the budget onstraint (), yielding P[(!t)wl+g](+r). The first-order ondition for labor supply in this ase is du ( 7) ρ β ( χ)( t) w+ δ( + r) χ( t) w z 0. dl (Throughout, primes denote derivatives.) Differentiating this expression with respet to P and rearranging terms yields () 8 l ( t) w χ ρ { [ ]} [ ] { } β ρ ( χ) ( t) wl + g δ( + r) ρ χ ( t) wl + g ( + r) ρβ ( χ) ( t) w ρδ( + r) χ ( t) w z where l P denotes dl/dp. Observe first that eah term in braes in the numerator redues to!d beause i! is followed by the expression for the respetive i. When this ommon term is fatored, what remains is equal to (Mu/M ) M, the partial derivative of utility with respet to for myopi individuals, as indiated by expression (5). Finally, the denominator an be shown to be equal to d u/dl (from differentiating expression (7), i.e., for the present ase of myopi labor supply). Aordingly, expression (8) simplifies to, ( 9) l χ ( ρ)( tw ) ( u/ ) d u/ dl M. To interpret this expression, note first that the denominator must be negative at the individual s optimum (and it an readily be shown to be stritly negative for all l in any event). Seond, observe that the presene of the term (Mu/M ) M in the numerator indiates that, as the onstraint just begins to bind, the effet on labor supply (in whihever diretion it may be) will 0 To state that there is no analytial interest in the alternative assumption that onsumption myopia is unantiipated does not mean that it ould not be of pratial interest. If individuals did behave in this alternative manner, soial seurity would raise welfare through improved onsumption alloation, and no further analysis would be required to take into aount effets on labor supply. - 8 -
be negligible. This result indiates that an atuarially fair tax on present disposable inome to finane fored savings does not effet labor supply in a manner qualitatively or (in general) quantitatively similar to that of further raising the marginal tax rate on urrent earnings, even though individuals are assumed to be myopi. As the onstraint beomes tighter (as P inreases one the onstraint binds), (Mu/M ) M beomes (more) positive. The reason is that this derivative reflets myopi individuals pereived marginal utility from raising rather than their atual marginal utility. When the soial seurity fored-savings onstraint is binding, is less than what myopi individuals would hoose, so the pereived marginal utility of raising further would be positive. Aordingly, l P is negative (positive) i.e., tightening the fored-savings onstraint redues (inreases) labor supply if D < (D > ). The intuition for this result an be understood by deomposing two effets (indiated by the and the!d in the leading term!d). A diret effet arises from more fored savings. When labor supply deisions reflet the same myopia as do individuals first-period onsumption alloation deisions, foring an inremental realloation of onsumption toward period is viewed as undesirable. Hene, the pereived return to labor effort falls. An indiret effet is due to hanges in relative marginal utilities of onsumption in the two periods. As P is inreased, the onsumption realloation toward period hanges the marginal utility of onsumption in eah period, making it higher in period and lower in period. Beause first-period rather than seond-period onsumption is pereived as too low, the realloation hanges the (pereived) marginal utility of onsumption more in the first period than in the seond. (Stated preisely, the third derivative of utility as a funtion of onsumption is positive, so the magnitude of the seond derivative is greater when onsumption is (pereived to be) low, as it is here in the first period, than when it is high, as in the seond period.) When relative risk aversion is low, speifially, when D <, this latter effet is less than the diret effet due to onsumption being pereived to be less well alloated between the two periods, so the overall pereived marginal benefit of inreasing labor effort falls. However, when risk aversion is high, D >, the latter effet dominates, so labor effort rises. In other words, when D >, the fat that soial seurity makes earnings seem less attrative is outweighed by the fat that the fored redution in greatly inreases the pereived marginal value of first-period onsumption, whih an only be raised, partially restoring it to its unonstrained level, by working more. When D <, this latter effet is present but insuffiient to outweigh the diret redution in the value of onsumption. Formally, this ondition and analysis are lose to what would be appliable if one were addressing the question of how generous should be alloations of soial resoures to a twoperson family when the two family members do not share the resoures equally. This oinidene should not be surprising. After all, with myopia, it is often stated that individuals behave as if there are two selves, see, for example, Thaler and Shefrin (98); in the ase of unequal sharing in the family, there literally are two persons, one of whih is given less weight than the other. - 9 -
These results may be summarized as follows: Proposition : When individuals onsumption alloation and labor supply deisions are subjet to myopia, the effet on labor supply of tightening the soial seurity fored-savings onstraint is: (a) zero as the onstraint just begins to bind, but thereafter is nonzero (unless D ) and (b) negative, if D <, and () positive, if D >. Further illumination regarding the differene between the soial seurity tax (i.e., the amount paid in period, as a funtion of wl, to fund onsumption in period ) and the inome tax (also a positive funtion of wl) an be gleaned by omparing l P to l t. For the latter, one an differentiate expression (7) with respet to t, rearrange terms, and make some substitutions to yield ( 0) l t [ ] [ ] ρ ρ ρ ρ ( ρ) w ( χ) β + χδ( + r) + ρwg ( χ) β + χ δ( + r) d u/ dl To ompare l t to l P, fous initially (and primarily) on how the first term in the numerator on the right side of expression (0) ompares to the numerator of the right side of expression (9). The term in brakets in the first term of (0) is positive, as is the value of (Mu/M ) M in (9) (when onsidering the range in whih the onstraint is binding). Likewise, the leading fators both have the term!d, so they also have the same sign. However, the terms in brakets themselves are quite different. Raising t redues onsumption in both periods (by!p and P in periods and, respetively), whereas raising P redues onsumption in the first period while raising it in the seond (see expression (5) for (Mu/M ) M ). As noted when interpreting expression (9), the effet of the latter is nil at the point at whih the onstraint just begins to bind, rising thereafter, so the effet of the onstraint is in this respet qualitatively different from that of a tax levied on top of a tax. Observe that this differene persists even when myopia is signifiant. The reason is that myopi individuals do not have a higher marginal utility in period one they have optimally from their perspetive adjusted their levels of onsumption. At that point, they value inrements to and to savings equally. (Another distintion between l t and l P is due to the seond term in the numerator of expression (0), whih has no orresponding omponent in expression (9): Raising t pertains to only earned inome and not the grant omponent g of the. - 0 -
linear inome tax, so the inome effet is adjusted aordingly. ) In onlusion: Remark : When individuals onsumption alloation and labor supply deisions are subjet to myopia, the effet on labor supply of tightening the soial seurity fored-savings onstraint is qualitatively different from that of inreasing the existing inome tax rate..3. Nonmyopi Labor Supply Assume instead that individuals labor supply deisions are nonmyopi even though their onsumption deisions are myopi. Speifially, suppose that individuals hoose labor supply to maximize utility as defined by expression () rather than expression (4), but these labor supply deisions take into aount that, when it omes time to deide upon onsumption, the alloation will be given by expression (5), exept to the extent onstrained by soial seurity, expression (6). 3 The analysis of this ase is straightforward from the above derivation, although the onlusions differ. The first-order ondition for labor supply is now du ( ) ( χ)( t) w+ δ( + r) χ( t) w z 0. dl This is idential to ondition (7) for the ase of myopi labor supply exept that $. Differentiating this expression with respet to P, rearranging terms, and making analogous substitutions to those used previously yields ( ) l χ ( ρ )( tw ) ( u/ ). d u/ dl This ondition is deeptively similar to expression (9) for the ase of myopi labor supply, but it is importantly different beause, in the numerator, (Mu/M ) M from expression (5) is replaed by Mu/M from expression (3), i.e., the effet on nonmyopi utility from raising. When onsumption deisions are antiipated to be myopi, and assuming that the soial seurity parameter P is in the range in whih there is still overonsumption in period, the value of Mu/M is negative (whereas (Mu/M ) M was positive) beause, on aount of myopia, has been As a result, there will be a range of D somewhat in exess of one for whih l t is negative even though l P for the myopi ase is positive. See Chetty (forthoming) on how unearned inome influenes the values of D for whih labor supply is upward sloping. Note that, in the present problem, despite myopia, it an be shown that labor supply is affeted analogously by hanging t and by hanging w, just as in the standard problem that is, l t!wl w /(!t). 3 As mentioned in the preeding ase, one ould also onsider the possibility that myopia is not antiipated when hoosing labor supply, but then soial seurity would have no effet on labor supply beause the onstraint would not be expeted to bind. - -
raised past the point at whih the nonmyopi first-order ondition for onsumption is satisfied. An immediate impliation is that l P now has the same sign as!d, rather than the opposite sign. That is, as more savings are fored, labor supply will rise (fall) if D is less (greater) than. As before, the phenomenon has two omponents. First, when P is inreased, earnings are better alloated, whih enourages labor effort. Beause the labor supply deision is taken to be rational, the value of soial seurity as a substitute devie making it possible for individuals de fato to ommit to onsume less is positive in fat and is pereived as suh. On the other hand, as P is inreased, the redution in onsumption misalloation hanges the marginal utility of onsumption in eah period, making it higher in period and lower in period. Beause of the urvature of utility as a funtion of eah period s onsumption, the latter effet is greater. When D <, this latter effet is less than the diret effet due to onsumption being better alloated between the two periods, so the overall marginal benefit of inreasing labor effort rises. However, when D >, the latter effet dominates, so labor effort falls. Another differene between the present ase and that with myopi labor supply is that, there, l P 0 as the fored-savings onstraint just began to bind, whereas here this is not true: At that point, the marginal gain from onsumption realloation toward the future, whih is taken into aount in individuals labor supply deisions, is at its greatest, and thus fored savings affets labor supply nontrivially (exept when D or $ is lose to ) from the moment the onstraint begins to bind. This fator and, aordingly, the effet of fored savings on labor supply will equal zero not when the onstraint just begins to bind but rather when P reahes the point at whih the magnitude of fored savings just equals its optimal (nonmyopi) level, i.e., when *. If P were inreased further, Mu/M would reverse sign, beoming positive, and the sign of l P would reverse from whatever it had been when P was lower. In this respet also, foring additional savings has a qualitatively different effet from that of raising the tax rate. (One an also, as above, ompare l P to l t expliitly; here, l t is as given by expression (0) exept that $. The omparison is qualitatively different beause l P is qualitatively different.) These results may be summarized as follows: Proposition : When individuals onsumption alloation deisions are subjet to myopia but their labor supply deisions are not, the effet on labor supply of tightening the soial seurity fored-savings onstraint is: (a) nonzero (unless D ) as the onstraint just begins to bind and thereafter until onsumption equals the nonmyopi optimum, at whih point the effet is zero, after whih it reverses from what is desribed in (b) and () and in this initial range is (b) positive, if D <, and () negative, if D >. In addition, juxtaposing the results for the ases of myopi and nonmyopi labor supply further reinfores the sense in whih the effet of soial seurity on labor supply differs from that of raising the marginal tax rate t on labor inome. The fat that the sign of the effet on labor supply from tightening the fored-savings onstraint (whatever it might be) is opposite for myopi and nonmyopi labor supply deisions most learly demonstrates that soial seurity is - -
qualitatively different. (Note that the reversal in sign applies independently to eah of the two effets, the diret effet on the value of onsumption and the indiret effet due to the urvature of the utility funtion.) Additionally, in eah ase, the magnitude of the effet is determined differently. For myopi labor supply, the initial marginal effet is zero whereas adding a small ordinary tax on top of a preexisting tax has a first-order effet and it grows in magnitude as the onstraint tightens. For nonmyopi labor supply, the effet on labor supply (in whihever diretion) is initially first-order and tends to fall (rather than rising) as more savings is fored, reahing zero when fored savings equal the nonmyopi savings optimum (after whih the sign of the effet reverses from whatever it was). 4 As previously noted, an important part of the explanation for these differenes between fored savings and taxation is that, even with signifiant myopia that leads to substantial overalloation of disposable inome to, the result of the misalloation is an inrease in the atual and pereived marginal utility of, suffiiently so that, at the unonstrained myopi optimum, the individual is indifferent between realloations of onsumption between the two periods. Hene, foribly realloating some onsumption to period does not at at all like a tax, at least initially, in the myopi labor supply ase and is viewed as a benefit (rather than as a ost, as with a tax) in the nonmyopi labor supply ase. Although these features hange as the onstraint tightens, they do not immediately vanish. In onlusion: Remark : When individuals onsumption alloation deisions are subjet to myopia but their labor supply deisions are not, the effet on labor supply of tightening the soial seurity fored-savings onstraint is qualitatively different from that of inreasing the existing inome tax rate. 3. Taxation of Capital Inome 3.. Model This setion will analyze the same two-period model used in setion for purposes of analyzing soial seurity. The only differene is that, instead of introduing a onstraint (6) on first-period onsumption, differential ommodity taxation will be examined. As in standard models of ommodity taxation, suppose the government an impose a tax J i on eah of the ommodities i (orresponding to onsumption in eah period). For onveniene, let i +J i denote the full prie faing onsumers. Aordingly, the budget onstraint () an be rewritten as θ ( 3) θ + ( twl ) + g. + r 4 Of the two ases, soial seurity is more tax-like when labor supply is myopi, for individuals pereive its mandate to be undesirable, and inreasingly so as its magnitude rises (although, as noted previously, there are still substantial qualitative differenes). - 3 -
Although it is onventional to think of a tax on apital inome as applying to the interest rate r, the present formulation is equivalent and eases exposition and interpretation. (If the seond term on the left side of (3) was written instead as /[+r(!j r )], where J r is the tax rate on interest, one an let equal (+r)/[+r(!j r )], whih produes expression (3) as the budget onstraint.) Beause we wish to fous on the effet of differential ommodity taxation on onsumption, utility, and labor supply, abstrating from effets on the overall level of taxation (revenue), it will be useful to onsider a partiular poliy experiment: As is inreased, it is assumed that is dereased to keep the present value of revenue raised from ommodity taxation onstant, under the hypothetial assumption that labor supply is unhanged. (Instead of adjusting, one ould adjust t and derive analogous results, for it is familiar that hanging the overall level of ommodity taxes keeping onsumer prie ratios, here /, onstant is equivalent to hanging the level of inome taxation. Holding t onstant and making all the adjustments through the i s has the virtue of keeping lear both that the overall level of taxation is fixed and, relatedly, that the only hange involves relative taxation of onsumption in eah period.) Furthermore, it will be onvenient to follow the pratie in muh of the literature on optimal ommodity taxation by fousing on loal effets, in partiular, raising from a starting point at whih. 5 Under these assumptions, to determine the required adjustment to, one an differentiate an expression for (the present value of) ommodity tax revenue, whih is (!) + (!) /(+r), with respet to, set it equal to zero, and evaluate the result at, whih yields ( 4 ) θ ( ). + r (Throughout this setion, primes denote derivatives with respet to, exept for zn, where the derivative as before is with respet to l.) 3.. Constant Labor Supply The effet of soial seurity on onsumption in eah period and thus on the welfare of myopi individuals, ignoring effets on labor supply, was immediate. Beause the effet of differential ommodity taxation is not quite as straightforward and the details will prove useful for subsequent analysis, we begin by examining expliitly the ase in whih labor supply is imagined to be held onstant. To determine the first-order ondition for onsumption by a myopi individual, we an use the budget onstraint (3) to solve for, whih equals [(!t)wl+g! ](+r)/, substitute 5 In the present setting, analysis of further deviations would require realibration of the revenue-neutral tax adjustment ( N) to aount for the extent of the hange in. - 4 -
into expression (4) for pereived utility of the myopi onsumer, and differentiate: u θ ( 5) β ( ) 0 θ δ θ + r or θ β δ ( + r) / ρ. This ondition, in turn, an be differentiated with respet to : u ρ ( 6) β δ( + r) θ θθ θ θ ( ) [( ) ] ρ θ ( ) θ δ θ θ + θ + t wl + g θ + r θ 0. To simplify, use the fat that we are evaluating this derivative at, that N is given by expression (4), that is determined by the budget onstraint (3), and make use of the firstorder ondition (5), allowing us to state ( 7) β [( t) wl + g] ρ ρβ + ρδ( + r). As expeted, N > 0, as the numerator and denominator of expression (7) are both positive (the denominator is the negative of Mu /M ). Noting the existene of partially ommon fators and making further use of the first-order ondition (5) and the budget onstraint (3), this an be further simplified to ( 8 ) ( ). ρ + r As expression (8) indiates, hanges more in response to an inrease in the smaller is the urvature of the utility funtion, indiated by D (whih, as noted, in this model is the oeffiient of onstant relative risk aversion and also /D indiates the onstant rate of intertemporal substitution). It is interesting to ompare this effet to what might be alled the mehanial effet of inreasing on, i.e., the rate at whih onsumption of would hange if savings were held onstant. The higher tax in period and lower tax in period would, in this respet, ause to fall and to inrease, the latter at the rate of /(+r). Aordingly, N simply equals the mehanial effet divided by D. When risk aversion (urvature) is low (below ), individuals raise by more than indiated by the mehanial effet i.e., they redue - 5 -
savings and when risk aversion is high (above ), they raise by less i.e., they inrease savings (despite that fat that / is higher). Consistent with intuition, when individuals are myopi, it is optimal to redue from its benhmark level, i.e., to employ a apital subsidy, in order to indue a derease in first-period onsumption, whih is too high relative to what would be ideal aording to the normative utility funtion (). When apital is subsidized, individuals have fewer resoures available in the first period and more in the seond, whih mehanially produes the desired effet i.e., it would result even if individuals were unaware of the hange and simply saved as they had previously. Under present assumptions, however, the inrease in savings ould be more or less, depending on the urvature of utility with respet to onsumption. Next we examine how the desirability of taxing or subsidizing apital in the present setting is influened by labor supply effets. 3.3. Myopi Labor Supply Following the approah used when analyzing soial seurity, it is now assumed that individuals, in hoosing labor supply, are maximizing myopi utility, expression (4), and that they are aware that their onsumption deisions are myopi as well. 6 The first-order ondition for labor supply in this ase is du d d ( 9) β + δ z 0. dl dl dl To determine the d i /dl s and for subsequent analysis, it is useful to introdue some additional notation. Let y (!t)wl+g (disposable inome, as given on the right side of the budget onstraint (3)), and let S equal the right side of the latter expression in (5), whih implies that S will also equal /. One an solve expressions (3) and (5) simultaneously, whih yields ( 0) y θ + θ / Ω ( + r), and ( ) y θ Ω + θ /( + r). 6 Unlike in the ase of soial seurity, it would not be analytially trivial to onsider the alternative assumption that individuals are myopi when hoosing labor supply but do not antiipate being myopi when hoosing onsumption. If one followed the subsequent derivations mutatis mutandis, it would be seen that the same result obtains (in sharp ontrast to the ase with soial seurity). - 6 -
Therefore, ( ) d dl ( tw ) θ + θ / Ω ( + r) ( tw ), y and ( 3) d dl θ Ω ( tw ) + θ /( + r) ( tw ). y Aordingly, expression (9) an be rewritten as ( 4) du dl [ β δ ] + ( ) tw y z 0. To find how labor supply is affeted by hanging, we an differentiate as follows: ( 5) d u dθ dl ρ ρ θ [ β δ ] + ( t) y w l + ( tw ) y ρ [ ρ β + ρ δ ] ( ) ( ) z l 0. θ (Reall that primes denote derivatives with respet to, exept for zo.) Differentiation of expressions (0) and () with respet to, making appropriate substitutions, allows one to determine that ( 6) ( tw ) y l θ + ρ( + r) θ, and ( 7) ( tw ) y l θ ρθ. Evaluating these expressions at, one an see that the seond terms in eah simply indiate the effet of raising when labor supply was held onstant. (For example, ompare expressions (6) and (8).) The first terms depit the effet of raising on labor supply multiplied by the orresponding expenditure shares. - 7 -
If one substitutes using expressions (6) and (7) in expression (5) and rearranges terms, the result is ( 8) l θ ρ ( tw ) ρ y( + r) + d u/ dl [ β δ( r) ]. The term in square brakets is simply the first-order ondition for myopi individuals onsumption alloation deisions, (Mu/M ) M, from expression (5). Therefore, ( 9) l θ ρ ( tw ) ρ y( + r) d u/ dl ( u/ ) M. Beause individuals first-order ondition for onsumption is satisfied, it follows that l θ 0. This result is as one would expet: As one just begins to raise (and lower as indiated by expression (4)), starting from, the resulting hange in the alloation of onsumption has no first-order effet on myopi individuals (pereived) utility; hene, the return to labor effort is unaffeted. This result is analogous to that for soial seurity when the onstraint just begins to bind. (Compare expressions (9) and (9).) Aordingly, we an state: Proposition 3: When individuals onsumption alloation and labor supply deisions are subjet to myopia, raising the relative tax rate on seond-period onsumption (i.e., in a onstantrevenue manner) has no effet on labor supply at the point of no differential taxation. 3.4. Nonmyopi Labor Supply As with soial seurity, we now onsider the ase in whih individuals labor supply deisions are nonmyopi they hoose labor effort to maximize expression () rather than expression (4) even though their onsumption hoies are myopi, a fat that they antiipate. 7 Analysis of this ase losely parallels that for myopi labor supply. Speifially, expressions (9) through (8) would be unaltered exept for the fat that the $ s in expressions (9), (4), (5) and (8) would be replaed by s. 8 Thus, expression (9) is replaed by 7 If their myopi onsumption hoies were not antiipated, they would not find that a small hange in apital taxation had other than a seond-order effet on their onsumption alloation, so labor supply would be unaffeted, just as was the ase with myopi labor supply. 8 There are also $ s impliit elsewhere, in partiular, in the definition of S; these remain unaffeted beause they indiate onsumption myopia. - 8 -