Utah State Univesity DigitalCommons@USU All Gaduate Plan B and othe Repots Gaduate Studies 8-2013 The Effects of Financing Unfunded Social Secuity with Consumption Taxation when Consumes ae Shotsighted Michael P. Clagg Utah State Univesity Follow this and additional woks at: https://digitalcommons.usu.edu/gadepots Pat of the Economics Commons Recommended Citation Clagg, Michael P., "The Effects of Financing Unfunded Social Secuity with Consumption Taxation when Consumes ae Shotsighted" 2013). All Gaduate Plan B and othe Repots. 339. https://digitalcommons.usu.edu/gadepots/339 This Thesis is bought to you fo fee and open access by the Gaduate Studies at DigitalCommons@USU. It has been accepted fo inclusion in All Gaduate Plan B and othe Repots by an authoized administato of DigitalCommons@USU. Fo moe infomation, please contact dylan.buns@usu.edu.
The Effects of Financing Unfunded Social Secuity with Consumption Taxation when Consumes ae Shotsighted Michael P. Clagg Depatment of Economics and Finance Utah State Univesity August 5, 2013 Abstact Using a epesentative-agent life-cycle model with consume shotsightedness, I study an unfunded social secuity pogam financed via consumption taxation. Compaed to financing an unfunded pogam with payoll taxation, I find that thee is only a slight incease in well-being acoss planning hoizons that is geneated by a pogam with a consumption tax. I acknowledge and thank T. Scott Findley fo his guidance and seving as my thesis committee chaiman, James Feigenbaum and Ryan Boswoth fo seving as thesis committee membes. 1
We suggest that both ou data and the available time-seies evidence ae consistent with Milton Fiedman s view that people save to smooth consumption ove seveal yeas but, because of liquidity constaints, caution, o shotsightedness do not seek to smooth consumption ove longe hoizons.... Indeed, Milton Fiedman explicitly ejected the idea that consumes had hoizons as long as a lifetime in discussing the pemanent income hypothesis Caoll and Summes 1991 pp. 307, 355). 1 Intoduction The unfunded social secuity pogam in the United States is the lagest such pogam in the wold. It has been justified by many as: insuance against disability o pematue death; edistibution fom the wealthy eldely to the poo eldely; as a eplacement fo failed annuity makets; to compensate fo unde saving behavio. The most common justification used is the unde saving fo etiement Kotlikoff 1982; Feldstein 1985; Docquie 2002; İmohooğlu et al. 2003; and many othes). Feldstein 1985) stated moe specifically, the pincipal ationale fo such mandatoy [social secuity pogams is that some individuals lack the foesight to save fo thei etiement yeas p.303). It is well known that the well-being of a life-cycle pemanent-income consume is educed by the pesence of a social secuity pogam, if the pogam has a negative net pesent value. This is due to the fact that the lifetime budget constaint is deceased. The existence of myopic agents in a model can esult in a social secuity pogam that has an optimal tax ate geate than zeo Docquie 2002; Caliendo and Gahamanov 2009; Findley and Caliendo 2009). The tax ate in such studies is usually set to maximize a patenalistic social welfae function following behavioal economic pactices Akelof 2002; Kanbu et al. 2006). The fist study that uses a mixed economy of life-cycles and myopic agents to estimate an optimal social secuity pogam was Feldstein 1985). Feldstein used an agent who exponentially discounts utility and one who does not discount in ode to show that thee is an oppotunity fo a welfae impovement. İmohooğlu et al. 2003) model quasi-hypebolic agents following Laibson 1998) and find that in a patial-equilibium and geneal-equilibium setting it is not welfae-impoving to have a social secuity pogam. Kumu and Thanopoulos 2008) models consumes with temptation pefeences and show that some of the welfae loss is mitigated by a social secuity pogam. The payoll-tax 2
financed pogam can have an effect on labo supply, causing less labo to be supplied as demonstated in OLG geneal equilibium models Auebach and Kotlikoff 1987; and Hugget and Ventua 1999). As an altenative to some of ou poposals fo benefit eductions o evenue inceases, policy makes could dedicate evenue fom anothe specific souce to Social Secuity Diamond and Oszag 2005, p. 5). As foeshadowed in the quotation above, the idea of moving towads a consumption tax fo geneal govenment expenditues has eceived some consideation by policy makes. Majo changes to the U.S. social secuity pogam have not been enacted since it is viewed as politically contovesial by many. To date, the majoity of the consumption-tax liteatue is focused along two lines: using detailed taxpaye infomation to estimate liabilities unde cuent and poposed egimes; and the study of theoetical economies and the effects of policy changes inside this famewok. Two studies in which a flat tax was consideed as a eplacement fo the cuent U.S. tax egime wee conducted by Feenbeg, Mitusi and Poteba 1997) and Genty and Hubbad 1997). Thee is no publicly available data set containing all necessay infomation, hence both studies use a combination of diffeent public souces. Feenbeg, Mitusi and Poteba 1997) use infomation on income, tax liabilities and consumption while Genty and Hubbad 1997) us data on use data on household potfolio choice. Feenbeg, Mitusi and Poteba 1997) find those lowe income households bea a dispopotionate shae of the tax buden in compaison to the high income households. This is in contast to the findings of Genty and Hubbad 1997) who find that the tax liabilities could be pogessive in natue, the moe affl uent in the economy bea a lage pecentage of the tax buden. Impotant assumptions undelying these computational studies ae that aggegate quantities and facto picing emains constant unde both egimes being consideed. The line of liteatue which uses calibated geneal-equilibium models to examine the diffeences of tax liabilities unde tax egime changes also exhibit mixed esults with espect to the pogessivity of tax budens. Ventua 1999) uses an OLG model with age and labo effi ciency shocks to simulate heteogeneity, and he finds that the change fom the cuent U.S. tax egime to a flat tax esults in inceased concentations of wealth. Altig et al. 2001) compute the tansition dynamics in moving fom the cuent U.S. tax code 3
to a flat tax and finds that the poo ae wose off duing the tansition and in the new steady state as compaed to befoe the tansition. Coeia 2010) uses an infinite-hoizon model with heteogeneity in initial wealth and income levels. She studies the efinancing of govenment expenditues fom an income-tax egime simila to that in the U.S. to one of a flat consumption tax. She finds that well-being inceases acoss all initial wealth levels. Fo an in depth discussion of consumption taxation and optimal taxation levels, see Coleman 2000). The idea of financing the social secuity pogam fom altenative souces is not new. Gahamanov and Tang 2013) use a geneal-equilibium OLG model with endogenous labo decisions and motality isk to investigate an optimal tax policy ove capital taxes, payoll taxes, income taxes, and consumption taxes. They find that the optimal policy to maintain benefits at cuent levels is to eliminate the payoll tax and incease a consumption tax above baseline. This leads to welfae gains acoss the economy. They then investigate the welfae dynamics duing the tansitional peiod, whee they find that the newly etied and nealy etied face the lagest welfae cost of this estuctuing. They advocate fo an additional tansfe payment to be made duing the tansition to maintain utility levels fo these individuals. Findley and Caliendo 2009) study the shot-tem planning model of Caliendo and Aaland 2007), supplemented with an unfunded social secuity pogam that is financed with payoll taxation. They demonstate that the pogam can be welfae impoving fo some planning hoizons in geneal equilibium. An open question emains as to whethe o not a payoll tax is the best instument to finance an unfunded pogam. Indeed, I evisit the ability fo social secuity to povide adequate etiement esouces in the shot-hoizon famewok used by Findley and Caliendo 2009). My contibution is the addition of an unfunded social secuity pogam financed with taxation on consumption. I find that a payoll-tax financed pogam can be eplaced with a consumption-tax financed pogam. Such a move geneates welfae gains in patial equilibium, although the welfae gains ae small. 4
2 Model I model a epesentative individual who optimizes consumption and saving behavio ove a shot planning hoizon. Time is continuous and indexed by t. The individual entes the wokfoce at t = 0. The individual eties at t = T, and dies at age t = T. Duing the woking peiod t [0, T the individual eceives wages at ate w, and duing the etiement peiod t [ T, T the individual eceives social secuity benefits b = Rwν w + Cν c / ) T T. The individual supplies one unit of labo inelastically while woking. R = T is the T T woke to etiee atio. ν w is the payoll tax ate. ν c is the consumption tax ate. C is aggegate consumption in the economy. Consumption at each instant is ct) and is the contol vaiable. Any income not consumed at each instant is placed in the individual s asset account, kt), which gows at ate. Thee ae no boowing constaints placed on the individual, and k0) = k T ) = 0 is assumed. The planning hoizon length x is the amount of time ove which an individual optimizes. I impose the estiction x T T fo ease of modeling, as is customay in this model. It allows fo a simple compatmentalization of the life cycle: Phase 1 [0, T x Phase 2 [T x, T Phase 3 [ T, T x Phase 4 [ T x, T Phase 1 is the peiod of the life cycle duing which the individual is in the wokfoce and does not foesee etiement. Phase 2 is that peiod of the life cycle when the individual is still in the wokfoce, but can see the futue date of etiement. Phase 3 is afte the individual is etied fom the wokfoce, but does not foesee the date of death. Phase 4 is that pat of etiement when the individual can see the date of death. Inside the shot planning hoizon model, the individual s behavio is time-inconsistent in Phases 1-3. This is due to the sliding planning window which moves though time with the individual. I model a naive individual, meaning that the individual does not anticipate 5
his time-inconsistant behavio. Theefoe, the individual s actual behavio is the envelope of the initial moments solved fo duing the optimization. The deivations of the following solutions can be found in Appendix A. 2.1 Phase 1 At any [0, T x the individual solves max ct) : t0 +x e ρt ) ct)1 φ 1) 1 φ subject to dk t) = k t) + w 1 ν w ) 1 + ν c ) c t) 2) k ) given 3) k + x) = 0, 4) whee ρ is the pesonal discount ate and φ is the invese elasticity of intetempoal substitution IEIS). The solution to 1)-4) is the optimal planned path fom the pespective of [0, T x, [ k t0 ĉt) = e gt ) e + +x w 1 ν w ) e j dj 1 + ν c ) +x, 5) e g )j dj fo t [, + x whee g = ρ φ. Following Caliendo and Aaland 2007), the actual consumption pofile can be deived by eplacing with t ct) = e gt k t) e t + w 1 ν w) e t e t e x) 1 + ν c e g g )t+x) e g )t), 6) fo t [0, T x. This can be moe simply expessed as ct) = kt)z 1 + wz 2, 7) 6
whee z 1 g ) 1 + ν c ) e g )x 1 ) 8) and the asset account follows the path z 2 1 ν w) g ) 1 e x ) 1 + ν c ) e g )x 1 ), 9) kt) = e Ωt 1 ) w [1 ν w ) 1 + ν c ) z 2, 10) Ω whee Ω = [ 1 + ν c ) z 1. 2.2 Phase 2 At any [T x, T the individual solves max ct) : t0 +x e ρt ) ct)1 φ 11) 1 φ subject to dk t) = k t) + w 1 ν w ) 1 + ν c ) c t) 12) dk t) = k t) + b 1 + ν c ) c t) 13) k ) given 14) k + x) = 0. 15) The planned consumption path is the solution to equations 11)-15), ĉ t) = k ) e + T w 1 ν w ) e j dj + +x T be j dj 1 + ν c ) +x e gt, 16) e gt e j dj fo t [, + x. The actual path is c t) = k t) e t + T t w 1 ν w ) e j dj + t+x T 1 + ν c ) t+x t e g )j dj be j dj e gt 17) 7
fo t [T x, T. Using z 1 fom above, it can be ewitten as c t) = kt)z 1 + w 1 ν w) z 1 e t e t e T ) + b z 1e t e T e t+x)) 18) whee k t) = e Ωt e ΩT x) k T x) + w 1 ν w) Ω [ e T + w 1 ν w) 1 + ν c ) z 1 + b 1 + ν [ c) z 1 e T Ω e ΩT x) e Ωt) e Ω)t e Ω)T x) 1 Ω Ω e Ω)T x) e Ω)t) + e x Ω [e ΩT x) e Ωt) e ΩT x) e Ωt)).19) 2.3 Phase 3 At any [ T, T x the individual solves max ct) : t0 +x e ρt ) ct)1 φ 20) 1 φ subject to dk t) = k t) + b 1 + ν c ) c t) 21) k ) given 22) k + x) = 0. 23) The solution is the planned consumption path fo t [, + x, The actual path fo t [ T, T x is [ k t0 ĉt) = e gt ) e + +x be j dj 1 + ν c ) +x. 24) e g )j dj ct) = e gt k t) e t + b e t e t+x)) 1 + ν c e g g )t e g )t+x)), 25) 8
which is the envelope of initial planned consumption allocations given kt) = kt )e ΩT t) + bν cz 3 Ω 1 e ΩT t)). 26) 2.4 Phase 4 Since the individual can see the date of death in this phase, behavio is time-consistent. The planned consumption path fom the pespective of = T x will be the actual consumption path, ct) = e gt z 4, 27) whee z 4 = g 1 + ν c k T x ) e x b 1 ex ) e gt 1 e xg )). 28) This chaacteizes the asset path duing Phase 4 with dk t) = k t) + b 1 + ν c ) c t) 29) and k T x ) known. 2.5 Social secuity in the model I will examine two options fo social secuity financing in this model. A tax on consumption, ν c, will be levied against all consumption in the model. I will also examine a payoll tax, ν w, as done in Findley and Caliendo 2009). I will compae the two altenate tax egimes. The unfunded pogam has a balanced budget and the individual does not take into account the effects that his consumption level has on the level of benefits, such that b = ν c [ T x 0 ct) + T T x ct) + T x T T T ct) + T T x ct) + Rwν w 30) fo t [T, T. The use of a consumption tax to finance benefits ceates an implicit function, since ct) 9
is a function of b while b is a function of ct). Yet, it is possible to numeically appoximate the level of benefits. Due to the inelasticity of labo supply in this model the payoll-tax potion of benefits is easily demonstated analytically. 3 Simulation and numeical execises 3.1 Baseline model paametes The baseline paametes ae summaized in Table1). I set T = 40 and T = 55 which epesents an individual who entes the wok foce at age 25, eties at age 65, and dies at age 80. I set the eal ate of etun,, to 0.035. The woke to etiee atio is appoximately 2.667. I set = ρ following convention. I set φ = 1, making utility logaithmic. 3.2 Optimal tax ates I will allow the model to detemine the optimal payoll tax ate, ν w, and the optimal consumption tax ate, ν c, fo each planning hoizon length, x. The optimal ate fo both pogams is the ate that patenalistically maximizes lifetime utility fo the individual, ν c ag max ν c [0,1;ν w=0 { T x 0 e µt ct)1 φ 1 φ + T ct)1 φ T x e µt 1 φ + T x T e µt ct)1 φ 1 φ + T T x e µt ct)1 φ 1 φ } 31) ν w ag max ν w [0,1;ν c=0 { T x 0 e µt ct)1 φ 1 φ + T ct)1 φ T x e µt 1 φ + T x T e µt ct)1 φ 1 φ + T T x e µt ct)1 φ 1 φ }, 32) whee µ is the social discount ate. 3.3 Individual life-cycle consumption pofiles Simulated consumption pofiles using the baseline paametes in Table1) can be seen in Figue1) fo the case of no tansfe pogam ν c = 0 and fo the case of a pogam with ν c = 0.10. The individual consumes less duing Phase 1 and pat of Phase 2, but has inceased consumption duing pat of Phase 2 and all of Phases 3 and 4. The consumption tax does 10
not distot the asset account duing Phase 1 as seen in Figue2). The tax popotionally deceases consumption duing Phase 1. This non-distotion of the asset account holds fo a wide aay of paametes as shown in Figues2,4,6). Duing Phase 2-4 the tax ate does change saving ates and consumption levels. It is impotant to note that the asset account with a pogam in place is always less than o equal to the asset account when a pogam is not pesent. This is simila to the esult fo the LCPI consume in which the pesence of an unfunded pogam causes the individual to save less fo etiement. 3.4 Welfae analysis: social secuity vs. no pogam Hee, I study an unfunded pogam financed by a consumption tax compaed to the countefactual of no pogam at all. In doing this, I define a compensating vaiation CV ) as the pecentage incease in peiod consumption that is needed to equalize lifetime utility without a pogam to the lifetime utility with an optimally paameteized social secuity pogam. In Table 2) I display the compensating vaiation. Fo all planning hoizons which have a non-zeo optimal tax, an unfunded pogam aises well-being. I also epot in Table 3) that an optimally paametized payoll-tax financed pogam is welfae impoving, compaed to no pogam at all. 3.5 Welfae analysis: consumption-tax financing vs. payoll-tax financing The welfae metic that I use is that of a patenalistic social planne, whee the social discount ate of µ evaluates utils ove the entie life span, even though the individual is optimizing ove a shot-hoizon. This is consistent with the majoity of the behavioal economics liteatue. I now define a uniquely diffeent compensating vaiation,, to measue the pecentage incease in ct) unde a paticula tax egime in ode to appoximate the value of paticipating in an optimally paameteized social secuity pogam. With φ = 1 the utility 11
function becomes logaithmic, and solves the following equation, { T x 0 e µt ln [1 + ) c νw t) + T T x e µt ln [1 + ) c νw t) + T x e µt ln [1 + ) c νw t) + } T T x e µt ln [1 + ) c νw t) T = { T x 0 e µt ln [c νc t) + T T x e µt ln [c νc t) + T x T e µt ln [c νc t) + T T x e µt ln [c νc t) }. 33) Solving fo gives whee [ U c U w = exp T 1 34) 0 e µt U w = { T x + 0 T x T T e µt ln [c νw t) + e µt ln [c νw t) + T x T T x e µt ln [c νw t) } e µt ln [c νw t) 35) U c = { T x + 0 T x T T e µt ln [c νc t) + e µt ln [c νc t) + T x T T x e µt ln [c νc t) } e µt ln [c νc t). 36) The optimal tax ates ae epoted in Tables4-9) fo a ange of paamete values. I compae the utility of two identical individuals unde the diffeent tax egimes using the metic. As epoted in Tables4-9), I find that the consumption-tax financed pogam has a highe total welfae, but only maginally. When using the Ramsey citeia fo measuing welfae such that the social planne does not discount utility, µ = 0), I find that thee ae lage gains in well-being fom a consumption-tax financed pogam compaed to a payoll-tax financed pogam. In this patial-equilibium model the diffeences between the consumption and saving pofiles ae elatively small, with the paths almost laying on top of each othe. But the cumulative utility gains fom the consumption-tax financed pogam ae sizable. 12
3.6 Robustness check of computational code To check fo potential computational eos in the simulation envionment, I calculate the pesent value of taxes collected ove a given planning hoizon. If the pesent value of taxes is equal acoss tax egimes, then the behavio should be the same egadless of which tax egime is in place. I analytically solve fo when each of the two egimes have the same pesent value of tax evenues fo a given planning hoizon. The pesent value of taxes in a payoll-tax financed egime is I w = t0 +x e t wν w, 37) and the pesent value of taxes in a consumption-tax financed pogam is I c = t0 +x e t ν c ĉt), 38) whee ĉt) is the planned consumption path fom the pespective of the planning instant. Setting 37) equal to 38) yields ν w = ν c 1 + ν c 39) whee the deivation is found in Appendix B. I use this equation to estimate the diffeence in the pesent values of the tax egimes within the simulations envionment. I found the two calculations to be almost identical. 4 Summay and possible extensions fo futue wok The pesence of an unfunded secuity pogam can impove well-being. A consumption-tax financed pogam leads to slightly highe levels of well-being as compaed to a pogam using payoll taxation. Due to the smoothing of consumption ove the life-cycle, thee is an incease in lifetime utility. Thee is an oppotunity to extend this eseach by allowing facto pices to adjust given behavio in the model. A geneal-equilibium setting would likely lean to diffeent quantitative esults. This meits futhe investigation. Anothe in- 13
teesting extension could be heteogeneity in the length of planning hoizons acoss diffeent individuals in the model population. Appendix A: deivations of consumption and savings pofiles Phase 1 [0, T x The individual solves subject to dk t) max ct) : t0 +x e ρt ) ct)1 φ 40) 1 φ = k t) + w 1 ν w ) 1 + ν c ) c t) 41) k ) given 42) k + x) = 0. 43) Using the Maximum Pinciple fo a one-stage poblem esults in the following Hamiltonian equation and optimality conditions, H = e ρt ) ct)1 φ 1 φ + λ t) k t) + w 1 ν w) 1 + ν c ) c t)) 44) H c = e ρt ) ct) φ λ t) 1 + ν c ) = 0 45) H k Solving the maximum condition fo ct) = λ t) = dλ H λ = k t) + w 1 ν w) 1 + ν c ) c t) = ct) = 46) dk t). 47) ) 1 e ρt ) 1 φ. 48) λ t) 1 + ν c ) Solving the costate equation dλ = λ t) λ t) = ae t. 49) 14
The constant of integation can be definitized such that λ ) = ae 50) a = λ ) e 51) λ t) = λ ) e t). 52) Substituting equation 52) into 48) gives ct) = whee g = ρ. This can be simplified as φ ) 1 e ρt ) 1 φ λ ) e t0 t) 1 + ν c ) = e gt 1 λ ) 1 + ν c ) e ρ) ) 1 φ 53) 54) 1 whee A = λ ) 1 + ν c ) e ρ) the state equation yields ) 1 φ ct) = e gt A 55) is a tansfomation of the unknown constant. Solving t k t) = e [q t + w 1 ν w ) 1 + ν c ) c j)) e j dj. 56) Using the bounday condition, k ) given, pins down the constant of integation k ) = e [ t0 q + w 1 νw ) 1 + ν c ) c j)) e j dj 57) k ) e = q + t0 w 1 νw ) 1 + ν c ) c j)) e j dj 58) q = k ) e t0 w 1 νw ) 1 + ν c ) c j)) e j dj. 59) 15
The paticula solution is t k t) = e [k t ) e + w 1 ν w ) 1 + ν c ) c j)) e j dj. 60) Using the othe bounday condition, k + x) = 0, [ e +x k ) e + t0 +x w 1 ν w ) 1 + ν c ) c j)) e j dj = 0 61) w 1 ν w ) 1 + ν c ) c j)) e j dj = 0 62) w 1 ν ) e dj = 1 + ν ) c j) e dj. 63) t0 +x k ) e + t0 +x t0 j +x k ) e + w c Substituting equation 55) into 63) gives t0 k ) e t +x t0 0 + w 1 ν w ) e j +x dj = 1 + ν c ) e gj Ae j dj 64) t0 +x t0 1 + ν c ) A e g )j dj = k ) e t +x 0 + w 1 ν w ) e j dj, 65) which allows us to solve fo the tansfomation of the unknown constant Theefoe, planned consumption is A = k ) e + +x w 1 ν w ) e j dj 1 + ν c ) +x. 66) e g )j dj [ k t0 ĉt) = e gt ) e + +x w j) 1 ν w ) e j dj 1 + ν c ) +x e g )j dj 67) in closed-fom. Since actual behavio will be decided fom eoptimization at evey instant, the actual paths can be mapped by eplacing with t in 67). This gives the actual consumption path [ k t) e ct) = e gt t + t+x t w 1 ν w ) e j dj 1 + ν c ) t+x t e g )j dj 68) 16
= e gt k t) e t + w 1 ν w) e t e t e x) 1 + ν c ) e g ) g )t e g )x e g )t) 69) = kt)z 1 + wz 2 70) with algebaic simplification, whee z 1 = g ) 1 + ν c ) e g )x 1 ) 71) z 2 = 1 ν w) g ) 1 e x ) 1 + ν c ) e g )x 1 ). 72) Substituting equation 70) into the law of motion that govens the actual evolution of the asset account, dk t) = k t) + w 1 ν w ) 1 + ν c ) kt)z 1 + wz 2 ) 73) = k t) z 1 1 + ν c )) + w 1 ν w ) 1 + ν c ) wz 2. 74) Solving this diffeential equation gives [ t kt) = e z 11+ν c))t q + w 1 ν w ) 1 + ν c ) wz 2 ) e z 11+ν c))j dj. 75) With the initial condition, k0) = 0, and with Ω = 1 + ν c ) z 1, the constant can be identified 0 0 = e [q Ω0 + w 1 ν w ) 1 + ν c ) wz 2 ) e Ωj dj 76) q = 0 w 1 ν w ) 1 + ν c ) wz 2 ) e Ωj dj 77) which povides a closed-fom solution fo the asset path duing Phase 1 t kt) = e Ωt w 1 ν w ) 1 + ν c ) wz 2 ) e Ωj dj 78) 0 = e Ωt 1 ) w 1 ν w ) 1 + ν c ) z 2 ). 79) Ω 17
Phase 2 [T x, T The individual can see both wok income and the social secuity benefits flow, but he is still woking. The individual solves max ct) : t0 +x e ρt ) ct)1 φ 80) 1 φ subject to dk t) = k t) + w 1 ν w ) 1 + ν c ) c t) 81) fo t = [, T and fo t = [T, + x, whee dk t) = k t) + b 1 + ν c ) c t) 82) k + x) = 0 83) k ) given. 84) Using the Maximum Pinciple fo two-stage poblems esults in the following Hamiltonians and optimality conditions, H 1 = e ρt ) ct)1 φ 1 φ + λ 1 t) k t) + w 1 ν w ) 1 + ν c ) c t)) 85) H 2 = e ρt ) ct)1 φ 1 φ + λ 2 t) k t) + b 1 + ν c ) c t)) 86) H 1 c = e ρt ) ct) φ λ 1 t) 1 + ν c ) = 0 87) H 1 k = λ 1 t) = dλ 1 88) H 1 λ 1 = k t) + w 1 ν w ) 1 + ν c ) c t) = dk t) 89) H 2 c = e ρt ) ct) φ λ 2 t) 1 + ν c ) = 0 90) H 2 k = λ 2 t) = dλ 2 91) 18
H 2 λ 2 = k t) + b 1 + ν c ) c t) = dk t). 92) The two multiplies ae defined as λ 1 fo t = [, T and λ 2 fo t = [T, + x and obey the costate equations, 88) and 91), ewitten as dλ 1 = λ 1 t) 93) fo t = [, T dλ 2 = λ 2 t) 94) fo t = [T, + x. Two-stage poblems equie a condition, λ 1 T ) = λ 2 T ), 95) which links the multiplies at the switch point. Solving equations 93) and 94) while definitizing the constants of integation yields λ 1 t) = a 1 e t a 1 = λ 1 T ) e T 96) λ 2 t) = a 2 e t a 2 = λ 2 T ) e T 97) Invoking the matching condition gives λ 1 T ) = λ 2 T ) a 2 = a 1, such that continuity exists acoss the switchpoint such that subscipts can be dopped. λ ) = ae 98) a = λ ) e 99) λ t) = λ ) e t) 100) Solving the fist maximum condition fo ct) gives ct) = ) 1 e ρt ) 1 φ. 101) λ t) 1 + ν c ) 19
Substituting in equation 100) gives ct) = whee g = ρ. This can be condensed φ ) 1 e ρt ) 1 φ λ ) e t0 t) 1 + ν c ) = e gt 1 λ ) 1 + ν c ) e ρ) ) 1 φ 102) 103) ct) = e gt A 104) ) 1 1 φ whee A = λ ) 1 + ν c ) e ρ) is a tansfomation of the unknown constant. Solving the second maximum condition fo ct) yields ct) = ) 1 e ρt ) 1 φ λ t) 1 + ν c ) 105) and substituting equation 100) into 105) yields ct) = ) 1 e ρt ) 1 φ λ ) e t0 t) 1 + ν c ) = e gt 1 λ ) 1 + ν c ) e ρ) whee g = ρ. This can also be simplified φ ) 1 φ 106) 107) ) 1 φ ct) = e gt A 108) 1 whee A = λ ) 1 + ν c ) e ρ) is also a tansfomation of the unknown constant. Note that 108) and 104) ae identical, theefoe no distinction will be made afte this. Solving the fist state equation gives t k t) = e [q t + w 1 ν w ) 1 + ν c ) c j)) e j dj 109) 20
fo t [, T. Using the initial condition k ) given definitizes the unknown constant, k ) = e [ t0 q + w 1 νw ) 1 + ν c ) c j)) e j dj 110) k ) e = q + t0 w 1 νw ) 1 + ν c ) c j)) e j dj 111) q = k ) e t0 w 1 νw ) 1 + ν c ) c j)) e j dj. 112) This gives the intended asset path fo t [, T, t k t) = e [k t ) e + w 1 ν w ) 1 + ν c ) c j)) e j dj 113) Evaluate 113) at t = T T k T ) = e [k T ) e + w 1 ν w ) 1 + ν c ) c j)) e j dj. 114) Solving the second state equation gives t k t) = e [q t + b 1 + ν c ) c j)) e j dj 115) fo t [T, + x. Using k + x) = 0 identifies the unknown constant, [ t0 k + x) = e t +x 0+x q + b 1 + ν c ) c j)) e j dj = 0 116) t0 +x q = b 1 + ν c ) c j)) e j dj. 117) Theefoe, the paticula solution is k t) = e t [ t +x b 1 + ν c ) c j)) e j dj, 118) which can be evaluated at t = T, k T ) = e T [ T +x b 1 + ν c ) c j)) e j dj. 119) 21
Set 114) equal to 119) e T [ T +x = e T [k ) e + b 1 + ν c ) c j)) e j dj T w 1 ν w ) 1 + ν c ) c j)) e j dj. 120) This can be eaanged = T k ) e + T 1 + ν c ) c j) e j dj T w 1 ν w ) e j dj T +x +x be j dj 1 + ν c ) c j) e j dj 121) and futhe simplified 1 + ν c ) t0 +x = k ) e + Substituting in fo c t), c j) e j dj T w 1 ν w ) e j dj + t0 +x T be j dj. 122) 1 + ν c ) A = k ) e + t0 +x T e gj e j dj w 1 ν w ) e j dj + the tansfomation of the unknown constant is identified, t0 +x T be j dj 123) A = k ) e + T w 1 ν w ) e j dj + +x T be j dj 1 + ν c ) +x. 124) e gj e j dj Inseting 124) into 108) yields planned consumption in closed-fom, ĉ t) = k ) e + T w z) 1 ν w ) e z dz + +x T b t) e z dz 1 + ν c ) +x e gt. 125) e gt e z dz 22
Replacing with t gives the actual consumption path c t) = k t) e t + T t w 1 ν w ) e j dj + t+x T 1 + ν c ) t+x t e g )j dj be j dj e gt 126) = k t) e t + w 1 ν w) e t e T ) + b e T e t+x)). 127) 1 + ν c ) e g ) t+x)g ) e g )t) Using z 1 fom above, this can be simplified c t) = kt)z 1 + w 1 ν w) z 1 e t e t e T ) + b z 1e t e T e t+x)). 128) Inseting 128) into dk t) = k t) + w 1 ν w ) 1 + ν c ) c t) gives dk t) = k t) + w 1 ν w ) 1 + ν c ) kt)z 1 + 1 + ν c) w 1 ν w ) z 1 e t e T e t) + 1 + ν c ) b z 1e t e t+x) e T ) 129) = k t) Ω + w 1 ν w ) + 1 + ν c) w 1 ν w ) z 1 e t e T e t) + 1 + ν c ) b z 1e t e t+x) e T ) 130) ewitten with Ω = 1 + ν c ) z 1 ). Solving this diffeential equation yields a geneal solution, t [ w 1 k t) = e q Ωt νw ) + w 1 ν w ) 1 + ν c ) z 1 e j e j e T ) + b [ z 1e j e T e j+x)) ) e Ωj dj. 131) Using the initial condition fo Phase 2 definitizes the unknown constant, q, such that T x [ w 1 k T x) = e q ΩT x) νw ) + w 1 ν w ) 1 + ν c ) z 1 e j e j e T ) + b [ z 1e j e T e j+x)) ) e Ωj dj 132) 23
T x [ w 1 q = k T x) e ΩT x) νw ) ) w 1 ν w ) 1 + ν c ) z 1 e j e f e T + b z 1e j [ e T e j+x)) e Ωj dj. 133) This yields the actual solution fo the asset path k t) = k T x) e ΩT x t) + e Ωt t T x [ w 1 νw ) 1 ν w ) w 1 + ν c ) z 1 e j e j e T ) + b z 1e j [ e T e j+x)) e Ωj dj 134) = e Ωt k T x) e ΩT x) + e Ωt t e Ωt t e Ωt t T x T x T x w 1 ν w ) e Ωj dj 1 + ν c ) w 1 ν w) z 1 e j e j e T ) e Ωj dj 1 + ν c ) b z 1e j e T e j+x)) e Ωj dj 135) [ = e Ωt e ΩT x) k T x) + w 1 ν w) e ΩT x) e Ωt) Ω w 1 ν w) 1 + ν c ) z t [ 1 1 e T j) e Ωj dj b 1 + ν c) z 1 t T x T x e j e T e j+x)) e Ωj dj 136) [ = e Ωt e ΩT x) k T x) + w 1 ν w) Ω w 1 ν [ w) 1 + ν c ) z t 1 b 1 + ν c) z t 1 e Ωj dj T x e T e Ω)j dj T x e ΩT x) e Ωt) t T x t T x e T j) e Ωj dj ) e x e Ωj dj 137) 24
= e Ωt e ΩT x) k T x) + w 1 ν w) Ω [ e T + w 1 ν w) 1 + ν c ) z 1 + b 1 + ν [ c) z 1 e T Ω Phase 3 [ T, T x e ΩT x) e Ωt) e Ω)t e Ω)T x) 1 Ω Ω e Ω)T x) e Ω)t) + e x Ω [e ΩT x) e Ωt) e ΩT x) e Ωt)). 138) The individual solves max ct) : t0 +x e ρt ) ct)1 φ 139) 1 φ subject to dk t) = k t) + b 1 + ν c ) c t) 140) k ) given 141) k + x) = 0. 142) Using the Maximum Pinciple, the Hamiltonian and optimality conditions ae H = e ρt ) ct)1 φ 1 φ + λ t) k t) + b 1 + ν c) c t)) 143) H c = e ρt ) ct) φ λ t) 1 + ν c ) = 0 144) H k Solving the maximum condition gives = λ t) = dλ H λ = k t) + b 1 + ν c) c t) = ct) = 145) dk t). 146) ) 1 e ρt ) 1 φ, 147) λ t) 1 + ν c ) and solving the costate equation yields dλ = λ t) λ t) = ae t, 148) 25
whee the unknown constant can be ewitten λ ) = ae 149) a = λ ) e 150) such that λ t) = λ ) e t). 151) Substituting 151) into 147), ct) = ) 1 e ρt ) 1 φ λ ) e t0 t) 1 + ν c ) 152) ) 1 = e gt 1 φ λ ) 1 + ν c ) e ρ), 153) whee g = ρ. The notation can be compessed fo simplicity, such that φ ct) = e gt A 154) ) 1 1 φ whee A = λ ) 1 + ν c ) e ρ) is again a tanfomation of the unknown constant of integation. Solving the state equation yields t k t) = e [q t + b 1 + ν c ) c j)) e j dj 155) fo t [, + x. Using the initial condition, k ) given, identifies q k ) = e [ t0 q + b 1 + νc ) c j)) e j dj 156) q = k ) e t0 b 1 + νc ) c j)) e j dj. 157) 26
The intended asset path is theefoe t k t) = e [k t ) e + b 1 + ν c ) c j)) e j dj. 158) Using the bounday condition, k + x) = 0, [ e +x) k ) e + t0 +x b 1 + ν c ) c j)) e j dj = 0, 159) which is simplified as t0 k ) e t +x t0 0 + be j +x dj = 1 + ν c ) c j) e j dj. 160) Substituting 154) in fo c t) gives t0 k ) e t +x t0 0 + be j +x dj = 1 + ν c ) e gj Ae j dj 161) t0 +x t0 1 + ν c ) A e g )j dj = k ) e t +x 0 + be j dj, 162) whee the tanfomation of the unknown constant is identified Theefoe, the planned consumption path is A = k ) e + +x be j dj 1 + ν c ) +x. 163) e g )j dj [ k t0 ĉt) = e gt ) e + +x be j dj 1 + ν c ) +x. 164) e g )j dj Replacing with t yields the actual consumption path, [ k t) e ct) = e gt t + t+x t be j dj 1 + ν c ) t+x t e g )j dj 165) = e gt k t) e t + b e t e t+x)) 1 + ν c ) e g ) g )t e g )t+x)). 166) 27
Using z 1 fom above and educing the faction, ct) = kt)z 1 + b g ) 1 e x ) 1 + ν c ) e g )x 1 ) 167) = kt)z 1 + bz 3 whee z 3 = g ) 1 e x ) 1 + ν c ) ). e g )x 168) 1 The actual law of motion fo Phase 3 is dk t) = k t) + b 1 + ν c ) c t) 169) = k t) + b 1 + ν c ) [kt)z 1 + bz 3 170) = k t) 1 + ν c ) z 1 ) + b 1 1 + ν c z 3 )). 171) Rewiting this with Ω = 1 + ν c ) z 1 and then solving gives a geneal solution t kt) = e [q Ωt + b 1 1 + ν c z 3 )) e Ωj dj. 172) Using the actual initial condition fo Phase 3 identifies the unknown constant T kt ) = e [q ΩT + b 1 1 + ν c z 3 )) e Ωj dj 173) q = kt )e ΩT T b 1 1 + ν c z 3 )) e Ωj dj. 174) The actual asset path is theefoe kt) = e Ωt [kt )e ΩT + t T b 1 1 + ν c z 3 )) e Ωj dj 175) [ = e Ωt kt )e ΩT + b 1 1 + ν cz 3 )) e ΩT e Ωt) 176) Ω 28
Phase 4 [ T x, T = kt )e ΩT t) + b 1 1 + ν cz 3 )) ) e ΩT t) 1. 177) Ω Thee is no time inconsistency in this phase. The path can be easily acquied by evaluating 164) at = T x, ct) = e gt k T x ) e T x) + T 1 + ν c ) T T x eg )j dj = e gt k T x ) e T x) b e T e 1 + ν c ) e g )T e g )T x)) T x be j dj T x)) 178) g ) 179) = e gt k T x ) e T x) be T 1 e x ) g ) 1 + ν c ) e g )T 1 e xg )) gt g ) = e 1 + ν c ) 180) k T x ) e x b 1 ex ) e gt 1 e xg )). 181) Defining z 4 = g ) 1 + ν c ) k T x ) e x b 1 ex ) e gt 1 e xg )), 182) actual consumption can be ewitten ct) = e gt z 4. 183) dk t) Coupled with = k t) + b 1 + ν c ) c t) and k T x ) given, 183) chaacteizes the asset path duing Phase 4. 29
Appendix B: deivation of the pesent value of taxes Assuming = 0 fo simplicity of demonstation, which coesponds to Phase 1, the pesent value of taxes paid ove a shot hoizon in the payoll-tax financed egime is I w = ν ww x 0 e t ν w w 184) 1 e x ). 185) The pesent value of taxes paid in the consumption-tax financed egime is I c x 0 e t ν c ĉt) 186) = ν cĉ0) g ) e g )x 1 187) whee ĉt) = ĉ0)e gt, 188) and whee ĉ0) = w1 ν w) x 0 e t 1 + ν c ) x 0 eg )t = 189) w1 ν w ) 1 e x ) 190) 1 + ν c ) ). e g ) g )x 1 This can be ewitten I c = ν ) w1 ν w) 1 e x ) c e g )x 1 g 1 + ν c ) e g ) g )x 1 ) 191) = ν cw1 ν w ) 1 + ν c ) Compaing only one tax egime at a time, ν w = 0 such that 1 e x ). 192) I c = ν cw 1 e x ) 193) 1 + ν c ) 30
Setting I c equal to I w with a scale inseted suggests I w Γ = I c, 194) o ewitten with substitution ν w w 1 e x ) Γ = ν cw 1 e x ). 195) 1 + ν c ) This can also be witten as Γ = ν c 1 + ν c )ν w, 196) o as with Γ = 1. ν w = ν c 1 + ν c ) 197) 31
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Kanbu, R., Pittilä J., & Tuomala, M. 2006). Non-Welfaist Optimal Taxation and Behavioual Public Economics. Jounal of Economic Suveys 205), 849-868. Kotlikoff, L. J., Spivak, A. & Summes, L. 1982). The Adequacy of Savings. Ameican Economic Review 725), 1056-1069. Kumu, Ç. S. & Thanopoulos, A. C., 2008. "Social secuity and self contol pefeences," Jounal of Economic Dynamics and Contol, Elsevie, vol. 323), pages 757-778, Mach. Laibson, D. 1998). Life-Cycle Consumption and Hypebolic Discount Functions. Euopean Economic Review 423-5), 861-871. Ventua, G. 1999). Flat Tax Refom: A Quantitative Exploation. Jounal of Economic Dynamics and Contol, 239 10): 1425 58. 33
Table 1 Baseline paametes Retiement date T 40 Date of death T 55 planning hoizon x 10 IEIS φ 1 Pesonal discount ate ρ 0.035 etiee to woke atio R 2.667 wage w 40000 eal ate o etun 0.035 Table 2 No pogam vs. consumption-tax financed pogam W 0 W w CV 1 236.3025252 252.6870847 34.70285% 2 246.2265692 252.7763546 12.64679% 3 249.5221449 252.8793832 6.29422% 4 251.1668483 252.9992367 3.38774% 5 252.152197 253.1384632 1.80939% 6 252.8081444 253.2981998 0.89499% 7 253.2759645 253.4785241 0.36897% 8 253.6262497 253.6790217 0.09600% 9 253.898201 253.899142 0.00171% 10 254.1153312 254.1153312 0 11 254.292599 254.292599 0 12 254.4399751 254.4399751 0 13 254.564361 254.564361 0 14 254.6706858 254.6706858 0 15 254.7625637 254.7625637 0 ρ = 0.035, µ =.035, = 0.035, T = 55, T = 40 34
Table 3 No pogam vs. payoll-tax financed pogam W 0 W c CV 1 236.3025252 252.6862788 34.70088% 2 246.2265692 252.7732336 12.64040% 3 249.5221449 252.8726371 6.28118% 4 251.1668483 252.9881505 3.36690% 5 252.152197 253.123226 1.78118% 6 252.8081444 253.2802075 0.86199% 7 253.2759645 253.4606859 0.33642% 8 253.6262497 253.6661419 0.07256% 9 253.898201 253.898201 0 10 254.1153312 254.1153312 0 11 254.292599 254.292599 0 12 254.4399751 254.4399751 0 13 254.564361 254.564361 0 14 254.6706858 254.6706858 0 15 254.7625637 254.7625637 0 ρ = 0.035, µ =.035, = 0.035, T = 55, T = 40 Table 4 x ν c U c ν w U w 1.125 252.6871.112 252.6863 0.00330% 2.116 252.7764.104 252.7732 0.01279% 3.106 252.8794.096 252.8726 0.02765% 4.094 252.9992.086 252.9882 0.04544% 5.079 253.1385.073 253.1232 0.06246% 6.063 253.2982.059 253.2802 0.07376% 7.045 253.4785.042 253.4607 0.07312% 8.025 235.6790.022 253.6661 0.05279% 9.004 254.8991 0 253.8982 0.00386% 10 0 254.1153 0 254.1153 0 11 0 254.2926 0 254.2926 0 12 0 254.4400 0 254.4400 0 13 0 254.5644 0 254.5644 0 14 0 254.6707 0 254.6707 0 15 0 254.7626 0 254.7626 0 Note. ρ = 0.035, µ =.035, = 0.035, T = 55, T = 40 35
Table 5 x ν c U c ν w U w 1.174 308.4714.149 308.4695.00657% 2.165 308.5540.142 308.5463.02570% 3.155 308.6604.134 308.6437.05608% 4.142 308.7936.124 308.7654.09448% 5.126 308.9556.112 308.9151.13564% 6.109 309.1477.097 309.0962.17233% 7.089 309.3703.079 309.3119.19544% 8.066 309.6234.058 309.8617.19325% 9.042 309.9069.033 309.8617.15141% 10.016 310.2202.004 310.2047.05193% 11 0 310.5562 0 310.5562 0 12 0 310.8565 0 310.8565 0 13 0 311.1156 0 311.1156 0 14 0 311.3416 0 311.3416 0 15 0 311.5403 0 311.5403 0 Note. ρ = 0.025, µ = 0.025, = 0.035, T = 55, T = 40 Table 6 x ν c U c ν w U w 1.089 211.4763.082 211.4761.00123% 2.080 211.5726.074 211.5716.00477% 3.070 211.6777.066 211.6756.01039% 4.059 211.7964.056 211.7913.01668% 5.046 211.9263.044 211.9219.02148% 6.031 212.0735.030 212.0691.02193% 7.015 212.2356.013 212.2327.01427% 8 0 212.4104 0 212.4104 0 9 0 212.5603 0 212.5603 0 10 0 212.6770 0 212.6770 0 11 0 212.7700 0 212.7700 0 12 0 212.8452 0 212.8452 0 13 0 212.9070 0 212.9070 0 14 0 212.9585 0 212.9585 0 15 0 213.0018 0 213.0018 0 Note. ρ = 0.045, µ = 0.045, = 0.035, T = 55, T = 40 36
Table 7 x ν c U c ν w U w 1.374 565.3000.272 562.7388 4.76679% 2.371 565.3004.271 562.9414 4.38237% 3.366 565.3026.269 563.1766 3.94103% 4.359 565.3085.266 563.4393 3.45695% 5.349 565.3209.262 563.7199 2.95371% 6.337 565.3431.257 564.0081 2.45685% 7.323 565.3782.250 564.2955 1.98807% 8.306 565.4298.242 564.5765 1.56368% 9.288 565.5014.232 564.8478 1.19540% 10.267 565.5965.221 565.1082 0.89184% 11.245 565.7188.207 565.3572 0.65964% 12.220 565.8716.190 565.5949 0.50444% 13.194 566.0587.170 565.8220 0.43127% 14.166 566.2832.146 566.0400 0.44327% 15.137 566.5486.116 566.2936 0.46484% Note. ρ = 0.035, µ = 0, = 0.035, T = 55, T = 40 Table 8 x ν c U c ν w U w 1.374 565.2580.272 562.7530 4.65972% 2.371 565.2970.271 562.9989 4.26686% 3.365 565.3624.268 563.3064 3.80884% 4.356 565.4555.264 563.6687 3.30197% 5.344 565.5783.258 564.0726 2.77548% 6.329 565.7334.251 564.5032 2.26184% 7.311 565.9231.242 564.9476 1.78939% 8.291 566.1499.230 565.3959 1.38025% 9.268 566.4161.216 565.8412 1.0579% 10.243 566.7788.198 566.2785 0.91387% 11.215 567.1300.177 566.7046 0.77635% 12.186 567.5262.152 567.1175 0.74581% 13.154 567.9690.121 567.5568 0.75219% 14.121 568.4595.083 568.0839 0.68528% 15.087 568.9987.037 568.7190 0.5984% Note. ρ = 0.025, µ = 0, = 0.035, T = 55, T = 40 37
Table 9 x ν c U c ν w U w 1.374 565.2870.272 562.7246 4.76908% 2.372 565.2491.272 562.8844 4.39332% 3.368 565.1886.270 563.0481 3.96862% 4.362 565.1085.268 563.2125 3.50739% 5.354 565.0122.266 563.3713 3.02842% 6.345 564.9038.262 563.5181 2.55147% 7.334 564.7871.258 563.6479 2.09280% 8.321 564.6665.253 563.7588 1.66405% 9.306 564.5464.248 563.8506 1.27308% 10.290 564.4312.241 563.9246 0.92547% 11.272 564.3258.232 563.9828 0.62569% 12.253 564.235.222 564.0275 0.37794% 13.232 564.1636.211 564.0611 0.18659% 14.209 564.1167.196 564.0857 0.05648% 15.184 564.0995.179 564.1033 0.00702% Note. ρ = 0.045, µ = 0, = 0.035, T = 55, T = 40 Figue 1 Consumption pofiles with and without consumption-tax financed pogam Note. ρ = 0.035, = 0.035, T = 55, T = 40, x = 10 38
Figue 2 Asset accounts with and without consumption-tax financed pogam Note. ρ = 0.035, = 0.035, T = 55, T = 40, x = 10 Figue 3 Consumption pofiles with and without consumption-tax financed pogam Note. ρ = 0.025, = 0.035, T = 55, T = 40, x = 10 39
Figue 4 Asset accounts with and without consumption-tax financed pogam Note. ρ = 0.025, = 0.035, T = 55, T = 40, x = 10 Figue 5 Consumption pofiles with and without consumption-tax financed pogam Note. ρ = 0.045, = 0.035, T = 55, T = 40, x = 10 40
Figue 6 Asset accounts with and without consumption-tax financed pogam Note. ρ = 0.045, = 0.035, T = 55, T = 40, x = 10 41