Taxation, transfer income and stock market participation

Transcription

1 Taxatio, trasfer icome ad stock market participatio Bjare Astrup Jese Marcel Marekwica This versio: December 211 We are grateful for helpful commets ad suggestios from Marc Griblatt, Cria Pugulescu, Paolo Sodii, ad Matti Suomie. The authors gratefully ackowledge acial support from the Daish Ceter for Accoutig ad Fiace. Copehage Busiess School, Departmet of Fiace, Solbjerg Plads 3, DK-2 Frederiksberg, Demark. Bjare Astrup Jese, phoe: , Marcel Marekwica, phoe: ,

2 Abstract Taxatio, trasfer icome ad stock market participatio We study a redistributive tax system that taxes ivestmet prots ad redistributes them i such a way that relatively rich agets are et cotributors to relatively poor agets. Our closed form solutio allows us to draw two mai coclusios. First, eve though the redistributio mechaism seeks to reduce the disparity i the distributio of wealth amog agets, wealth levels are ot harmoized despite ogoig trasfers from richer to poorer agets. Specically, whe the level of poorer agets' trasfer icome is iversely related to their wealth levels, poorer agets have a icetive to reduce their wealth levels to icrease the level of future trasfer icome. Secod, sice trasfer icome is subject to stock market risk poorer agets optimally reduce their exposure to equity. I particular, a redistributive tax system ca thus cotribute to explaiig the low empirically documeted equity exposures ad stock market participatio rates of poorer agets. JEL Classicatio Codes: G11, E21, H24 Key Words: redistributive taxatio, portfolio choice, asset pricig, stock market participatio

3 1 Itroductio Trasfer of icome from relatively rich to relatively poor idividuals through a redistributive tax system is a widespread pheomeo throughout may coutries i the world. For example, the U.S. govermet has spet more tha 1.2 trillio U.S. dollars o wealth trasfers i 29 via icome security ad social security. 1 Eve though ivestmet prots are subject to taxatio i most coutries aroud the world, ad despite the empirically documeted evidece Dai et al. 28), Sialm 29)), surprisigly little is kow about the geeral equilibrium impact of a redistributive tax system o the distributio of wealth, cosumptio, asset prices ad optimal ivestmet strategies. A otable exceptio is the work of Sialm 26) who studies geeral equilibrium eects of stochastic tax rates assumig a represetative aget. I our work, we explicitly allow for aget heterogeeity. I particular, we take ito accout that agets may dier by their acial edowmets. We develop a stylized model of a exchage ecoomy i the traditio of Lucas 1978) where agets ca trade a risk-free asset that comes i zero et supply ad a risky stock that represets a claim o aggregate cosumptio i the ecoomy. The govermet taxes ivestmet prots ad immediately redistributes them i a way that seeks to reduce the disparity i the distributio of wealth amog agets. I.e., i cotrast to partial equilibrium models with taxatio goig back to Domar ad Musgrave 1944), the overall risk i the ecoomy is ot reduced. Istead it is redistributed together with the trasfer icome. The closed form solutio of our model shows that a redistributive tax system aects optimal cosumptio ad ivestmet strategies of poorer agets that are et recipiets of trasfer paymets ad richer agets, the et payers, i dieret ways. Specically, we show that the depedecy of tax reveues o the evolutio of the stock market implies that trasfer icome is subject to stock market risk. As a cosequece, poorer agets that are the et recipiets of trasfer icome optimally reduce their exposure to stocks. Furthermore, we demostrate that eve though the redistributive tax system is implemeted i a attempt to reduce the disparity i the distributio of wealth amog the agets, wealth levels are ot ecessarily harmoized despite ogoig trasfers from richer to poorer agets. O the cotrary, the spread betwee wealth levels of richer ad poorer agets ca actually wide whe relatively poor agets have a icetive to cosume ad thereby remai relatively poor i order to icrease the level of future trasfer icome. Our work cotributes to two strads of literature. First, we cotribute to the asset pricig literature by showig how a redistributive tax system aects asset prices ad optimal cosumptioivestmet strategies i geeral equilibrium. It has log bee kow Brea ad Kraus 1978), Merto 1971), Rubistei 1974)) that whe CRRA ivestors have idetical prefereces it is straightforward to get from idividual prefereces to a represetative ivestor, the pricig kerel ad the market equilibrium. I particular, a Pareto 1 This correspods to 34.57% of total federal outlays. 1

4 optimal sharig rule is liear; it requires the amout of risk bore by ay idividual ivestor to be proportioal to his wealth level. I.e., agets do ot trade risk-free bods with each other. We show that with a redistributive tax system agets still seek to establish a liear sharig rule. However, this requires richer agets to icrease their exposure to risky stocks ad poorer agets to decrease it. Simultaeously, the bod market plays a active role i establishig the liear sharig rule. Our work also cotributes to the growig literature that seeks to explai the low empirically observed household equity exposures ad stock market participatio rates. It has log bee observed that stock market participatio historically has bee far from uiversal Blume ad Fried 1974), Makiw ad Zeldes 1991), Kig ad Leape 1998)). Accordig to the 27 Survey of Cosumer Fiaces SCF), oly 51.1% of U.S. families have stock holdigs i direct or idirect form, eve though theoretical research cocludes that households should usually hold stocks to ear the equity premium ad to diversify risks. The same fractio whe oly direct stock holdig is accouted for is 17.9%. At the same time, 91.% of the 1% of households with the highest icome have stock holdigs i direct or idirect form, whereas stock market participatio drops to oly 13.6% for the 2% of households with the lowest icome. same umbers for direct stock holdigs show a drop from 47.5% to 5.5%. Although empirical research documets that participatio is icreasig with wealth, age ad educatio Makiw ad Zeldes 1991), Haliassos ad Bertaut 1995), Bertaut 1998), Guiso et al. 23), Calvet et al. 27), Christiase et al. 28), Calvet et al. 29a,b)), varies greatly across coutries ad has icreased recetly Guiso et al. 23), Giaetti ad Yrjö 21)), the overall impressio is that participatio is still low ad at odds with the predictios from commoly applied asset allocatio ad asset pricig models Campbell 26)). There are two mai strads of literature tryig to explai the low empirically observed equity exposure ad stock market participatio rates. A rst strad of literature bases its argumets o aget-specic characteristics, icludig loss aversio Ag et al. 25)), lack of trust i acial markets Guiso et al. 28)), low experieced stock returs Malmedier ad Nagel 211)), arrow framig Barberis et al. 26)), itelligece Griblatt et al. 211)), acial literacy va Rooij et al. 211)), marital status ad childre Love 21)), historical portfolio decisios Alessie et al. 24)), iteret access Boga 28)), ad political prefereces Kaustia ad Torstila 211)). Aother strad bases its argumets o market frictios, icludig diereces betwee risk-free retur ad borrowig rate Davis et al. 26), Becker ad Shabai 21)), liquidity costraits Haliassos ad Michaelides 23)), lackig diversicatio of labor icome risk ad its correlatio with stock returs Haliassos ad Bertaut 1995), Makiw ad Zeldes 1991), Vissig-Jørgese 22), Guo 24)), ad stock market etry costs Abel 21), Alle ad Gale 1994), Gomes ad Michaelides 25, 28)). 2 2 Approaches that do ot fall ito oe of these two categories iclude model ucertaity Dow ad Werlag 1992), Epstei ad Scheider 27)), backgroud risk correlated with the stock market Heato ad Lucas The 2

5 We cotribute to this lie of research by showig that redistributive features of may tax systems foud aroud the world ca help explaiig the low empirical equity exposures ad stock market participatio rates of poorer agets. Overall, our cotributio to the literature is three-fold. First, to the best of our kowledge, we are the rst to solve a geeral equilibrium model with a redistributive tax system ad heterogeeous agets. Secod, we show that whe tax reveues deped o the evolutio of the ecoomy ad the stock market, poorer agets optimally reduce their exposure to stocks. Third, we show that eve whe the govermet implemets the trasfer mechaism attemptig to reduce the disparity i the distributio of wealth amog the agets, this objective is ot ecessarily attaied despite ogoig trasfers from richer to poorer agets. I.e., eve i the log ru poorer agets with iitial wealth levels below average might optimally keep their wealth levels below their iitial edowmets i order to icrease the level of future trasfer icome. The paper is orgaized as follows. Sectio 2 itroduces our model. I sectio 3 we preset its closed form solutio, ad sectio 4 shows umerical examples ad demostrates how the optimal solutio is quatitatively aected by the values of the chose iput variables. Sectio 5 shows both aalytical ad umerical results from a taxatio system that allows for the separatio of wealth trasfer from the trasfer of stock market risk. Sectio 6 geeralizes some of the results i a settig with agets that are heterogeous with respect to their levels of risk aversio; sectio 7 cocludes. 2 The model We cosider a model for a exchage ecoomy with agets ad a acial market o which two assets ca be traded. First, agets ca trade a locally risk-free asset payig a pre-tax retur of r t from time t to t+1. This asset comes i zero et supply. I.e., if a subset of agets wat to hold a positive fractio of their wealth i the risk-free asset, the market equilibrium has to brig about a iterest rate that makes the remaiig agets willig to issue such a risk-free asset. Secod, agets ca trade a risky stock that represets the owership to aggregate cosumptio. Risk is modeled by assumig that aggregate cosumptio is the fruit/divided from a biomial Lucas tree Lucas 1978)). The iitial divided at time t = is ormalized to D = 1. The divided growth G from time t to t + 1 depeds o the evolutio of the ecoomy. We assume there is a 5% probability each for a boom ad a bust i the ecoomy. The correspodig divided growths are deoted by G + ad G. Hece, the market is dyamically complete. The risky stock comes i et supply ormalized to oe uit. The iitial edowmets of the agets are deoted by α,j >, j =1, 2,...,, with j=1 α,j =1. I the followig, we assume that the agets are idexed i ascedig order relative to their iitial shares of wealth α,j ; i.e., aget 1 has the lowest iitial edowmet, ad aget has the highest. 2), Bezoi et al. 27)) ad chages i correlatio of equity, icome ad cosumptio over the life cycle Costatiides et al. 22)). 3

6 2.1 The redistributive tax system We cosider a tax system where ivestmet returs are subject to taxatio at the commo rate τ [, 1). 3 By taxig ivestmet returs the tax system collects tax reveues of τ P t + D t P t 1 ), t = 1, 2,..., N 1) where P t is the price of the risky stock at time t ad D t is its divided paymet at time t. Sice the et supply of the risk-free asset is zero, the tax reveues oly deped o the evolutio of the price ad the divided of the risky stock. We assume that the model is parameterized i such a way that P t +D t >P t 1, i.e. that tax reveues are o-egative. We further assume that the govermet redistributes the tax reveue amog the agets i a attempt to reduce ecoomic disparities amog the agets. Assumig that oly part of the tax reveue is used to ace trasfer icome ad the remaiig part is used to ace a supply of public good, as, e.g., i Sialm 26), aects our results quatitatively, but ot qualitatively. We therefore disregard this possible dual role of the govermet. We require that the redistributive tax mechaism fullls three coditios. First, it has to fully distribute the collected tax reveues, i.e. the govermet either builds up wealth or debt. This is the govermet budget costrait i this paper. Give that the agets are the same i all periods, govermet debt wealth) would ever be cosidered et wealth debt) for the idividuals i the ecoomy. Ay outstadig positio for the govermet must be settled by the same idividuals withi the time horizo of the model, cf. the reasoig i Barro's semial work Barro 1974)). Secod, the redistributio of tax reveues must icrease the wealth level for the relatively poor agets ad decrease the wealth level for relatively rich agets. 4 Third, the ascedig orderig of the agets with respect to their wealth levels before takig trasfers ito accout must remai uchaged after takig trasfers ito accout. A simple rule that fullls these coditios is to redistribute the total tax reveue equally amog the agets. We therefore use this rule i the followig. 2.2 The optimizatio problem We assume that our agets are expected preset discouted utility maximizers with time-additive CRRA prefereces, ad that they oly dier by their iitial edowmets. 5 I order to study dyamic eects, we allow for multiple periods ad assume that the ivestmet horizo of our agets is N periods, such that the agets dyamically have to make cosumptio ad ivestmet 3 I sectio 5 we allow for a more geeral taxatio mechaism. We also cosidered a cosumptio tax as, e.g., used i Sialm 26). However, a cosumptio tax oly leads to ecoomic eects equivalet to a upfrot redistributio of wealth. These results are therefore ot preseted i the paper, but they are available from the authors upo request. 4 To be precise, the relatively poorer agets are those with a iitial owership of aggregate wealth less tha 1/. 5 I sectio 6 we also allow for heterogeeity i the agets' degree of risk aversio. 4

7 decisios at time t =, 1,... N 1 ad cosume their remaiig wealth at time t = N. The followig otatio is used: ρ deotes the agets' commo utility discout factor. α t,j deotes the umber of uits of the risky stock held by aget j from time t to t+1. β t,j is the umber of uits of the risk-free asset held by aget j from time t to t+1. C t,j is aget j's cosumptio at time t. r t R t 1 + r t ) is the et gross) risk-free rate from time t to t+1. r t R t 1 + r t ) is the et gross) risk-free rate after tax from time t to t+1. W t,j is aget j's wealth level at time t before cosumptio. Havig itroduced the otatio we ca ow state aget j's optimizatio problem, where we have omitted the subscript j for otatioal simplicity: with α N =β N. max {{C t} t=n t=,{αt,βt}t=n 1 t= } s.t. C 1 γ N [ ] 1 γ + ρ t C 1 γ t E 1 γ t=1 C t = W t α t P t β t t =, 1,..., N 3) [ W t = α t 1 1 τ) + τ ] P t + D t ) + β t 1 Rt 1 + τ α t 1 1 ) P t 1, t = 1, 2,..., N 4) W = P + D ) α 5) Equatio 3) is the ivestor's budget costrait ad equatio 4) describes the dyamic evolutio of wealth. They are equalities betwee radom variables reectig the biomial ltratio. 2) 3 Cosumptio ad ivestmet i geeral equilibrium Havig itroduced our model, we ext tur to a demostratio of how a redistributive tax system aects the agets' optimal cosumptio ad portfolio policies. If the tax rate is zero, there is o redistributio of wealth ad ay aget's optimal exposure to the risky stock will be equal to his eterig exposure to the risky asset at ay poit i time. Similarly, there will be o holdig of risk-free assets by ayoe. After the itroductio of taxatio 5

8 ad redistributio of tax reveues, this is o loger true. The redistributio mechaism implies a dyamic trasfer of wealth with a edogeously determied wealth distributio, icludig the capitalized values of future trasfers. Furthermore, the redistributio mechaism also implies a trasfer of stock market risk. This causes the relatively poor agets the et recipiets of wealth trasfers to ivest less i the stock market, because their trasfer icome is already subject to imputed stock market risk. The geeral equilibrium model ca be solved i closed form. We state the solutio as Theorem 1. Theorem 1. The geeral equilibrium solutio to the optimizatio problems for agets dierig oly i their iitial edowmets, as stated i equatios 2)-5), is give as follows: 1. The allocatio of market risk is i accordace with a liear sharig rule relative to the wealth distributio after tax. 2. The martigale measure is uiquely determied ad idepedet of the iitial distributio of wealth as well as the tax rate τ. probability for a boom i the ecoomy, are give as q = The risk eutral probabilities, with q deotig the G + ) γ G + ) γ + G ) γ, 1 q = G ) γ G + ) γ + G ) γ 6) 3. The discout factor after tax, R, is costat ad idepedet of the tax rate τ: R = 2 1 ρ G + ) γ + G ) γ 7) The risk-free rate of iterest before tax, r, is also costat. However, it depeds o the tax rate τ: 4. The asset prices are give by r = R 1 1 τ = r 1 τ P N 1 = 1 + r) 1 E Q N 1 [D N] = 1 + r) 1 D N 1 E Q [ N 1 G ± ] 9) P t = 1 + r) 1 E Q t [P t+1 + D t+1 ] 1) 8) The risk premium is costat for a give tax rate τ ad takes o the value: [ 1 G + + G ] R 2 qg q)g 1 = R EP [G] EQ [G] E Q 1 [G] 11) 5. The equity exposure is give by the followig dyamic relatios with X t beig a sequece 6

9 of time-depedet but state-idepedet coeciets X t : α t = 1 + X t α t 1 1 ), t = 1, 2,..., N 1 12) RP t + D t ) X t = [ N 1 ], t = 1, 2,..., N 1, 13) RP t + j=t+1 X j RDt α = P + D ) R 1 τ RP + RD [ N 1 ] α 1 ) j=1 X j Provided that the risk-free rate of iterest as well as the tax rate are both positive it is the case that X t, 1) t. Furthermore, the deviace elarged, i.e. α 1/ > α 1/, whe the choice of the iitial exposure α is made. 6. The cosumptio policy is give by a costat share of aggregate output: 14) C t = α N 1 1 τ) + τ D t = α N 1 1 ) 1 τ) + 1, t =, 1,..., N 15) 7. The risk-free asset plays a role i order to establish a liear sharig rule. The positio i the risk-free asset is determied by ) β t = R 1 1 τp t α t, t =, 1,..., N 1 16) 8. The share of wealth ivested, α t + βt P t, ca be expressed as α t + β [ t = α t 1 τ R ] P t + τ R 1 17) For relatively poor rich) ivestors the share of wealth ivested is icreasig decreasig) over time. 9. The level of the received et trasfer paymet for each idividual at time t is τ ) ) 1 α R t 1 P t + D t P t 1 R 18) There is a xed relatio, idepedet of time ad state, betwee the et trasfer paymets received i the boom ad the bust states, respectively. The ratio is give by τ 1 α ) ) t 1 P t + + D t + R P t 1 R τ 1 α ) RG + E Q t 1 ) = [G] t 1 Pt + Dt R P t 1 RG E Q t 1 [G] = RG + E Q [G] RG E Q [G] 19) R Proof The details of the derivatios are foud i Appedix A. 7

10 The model without taxatio i.e. τ = ) is a bechmark model i asset pricig theory with a umber of well kow properties, such as the liear sharig rule for cosumptio, liear asset demad fuctios ad straightforward aggregatio of agets ito a represetative aget with prefereces correspodig to those of the agets i the ecoomy. I this bechmark model we kow, cf., e.g., Brea ad Kraus 1978), Merto 1971) ad Aase 22), that o aget takes ay positio i the bod market ad that the allocatio α t is costat ad equal to the iitial distributio α. I other words, o dyamic tradig takes place. Additioally, the martigale measure is determied by the margial utilities of the represetative aget. I our model with a redistributive tax system some of these key properties are reproduced: The liear sharig rule for cosumptio items 1 ad 6), the martigale measure item 2) ad the asset pricig equatios ad the risk premium item 4). The equity premium takes the same value at all poits i time ad all states cosidered, reectig that the aggregate risk i the divided growth is idetical at each ad every poit i time ad i each ad every state. Hece, the P - expected values of the growth rates are idetical at each ad every poit i time ad i each ad every state. The premium to be paid for bearig risk is idepedet of the wealth distributio. This is reected i the Q-expected values of the growth rates, which are also idetical at each ad every poit i time ad i each ad every state. The risk-free rate of iterest is also beig reproduced, but i a after tax settig item 3). 6 However, the redistributive tax system also implies importat chages. First, the rst term i equatio 4) cotais the liear exposure to the market portfolio, icludig the eect from the trasfer paymet. This term shows that the tax mechaism reduces the volatility i the aget's exposure to market risk stemmig from his direct ivestmet i the market portfolio α t 1 1 τ)p t +D t )). It also shows that the trasfer mechaism ivolves a simultaeous trasfer of wealth ad market risk τ/)p t +D t )). This trasfer sigicatly aects cosumptio levels ad portfolio decisios. Secod, the relatively poor agets with iitial edowmet below 1/ are et recipiets of trasfers ad reach a costat cosumptio share C t /D t ) i excess of their iitial share of wealth α, which correspods to their share of wealth after takig their edogeously determied future trasfers ito accout. As time goes by they also kow that the capitalized value of their future et receipts from trasfers decreases, which requires them to save ad icrease their share of wealth ivested, α t + βt P t Theorem 1, item 8). To keep their cosumptio share at the desired level over the etire ivestmet horizo they eed to keep their savigs o a suciet level. O the other had, by savig they also dimiish their future et receipts savig is costly i this respect, but ecessary i order to smooth out the relative cosumptio patter over time. Eve though X t, 1) implies a covergece of wealth towards a equal distributio, this covergece is rather slow for reasoable parameter choices. I.e., after the iitial adjustmet of the stock market positio from α to α, accordig to equatio 14), the future levels of α t oly 6 This is because the taxatio mechaism is eutral, which implies that the discoutig mechaism is aected, but the martigale measure remais uchaged whe taxes are itroduced. For further details, see Jese 29). 8

11 move very slowly towards 1/. Hece, i additio to aectig cosumptio levels via wealth trasfers, the redistributive tax system also sigicatly aects the asset demad fuctios Theorem 1, items 5 ad 7). Theorem 1, item 6 shows that the Pareto optimal solutio requires cosumptio shares to be costat over time, which requires a liear exposure of the agets to stock market risk. Equatio 4) shows that such a liear exposure to stock market risk is ot attaied if agets were to disregard the bod market ad choose a exposure to the risky asset correspodig to their shares of wealth. Specically, the secod ad third summad i equatio 4) brig agets away from the Pareto optimal liear sharig rule. However, both of these terms are predictable. That is, agets ca eutralize the risk trasfer by actively usig the bod market i such a way that these two terms vaish Theorem 1, item 7). I.e., with our redistributive tax system asset demad fuctios are o loger liear but become ae fuctios with time-varyig coeciets istead of liear fuctios with costat coeciets. The optimal portfolio strategies ow ivolve dyamic tradig i both the bod ad the stock market, reectig that the agets take ito accout the redistributio of stock market risk as well as the capitalized value of future trasfer paymets whe solvig for optimal dyamic cosumptio-ivestmet strategies. Although the coeciet X t i frot of α t 1 1/) i equatio 12) is less tha oe, these X t values ca be very close to oe whe the remaiig ivestmet horizo is log. Eve though the redistributio mechaism implies a covergece of equity holdigs ad wealth towards a equal distributio, it is a iterestig feature of our model that this equal distributio of wealth is ot attaied. I the followig subsectio we show a umerical example i graphical form ad discuss this issue i more detail. 4 Numerical examples After presetig the closed form solutio to our model, we ext tur to illustratig the impact of a redistributive tax system o optimal cosumptio-ivestmet strategies, as well as the evolutio of wealth ad trasfer icome over the ivestmet horizo. Throughout our umerical examples, we restrict ourselves to a settig with = 2 agets, which allows us to depict our results i graphical form. Because of the well-kow aggregatio properties of the CRRA utility fuctio Merto 1971), Rubistei 1974), Brea ad Kraus 1978)), this setup ca be iterpreted as a settig with two groups of agets. 4.1 Base case We choose a ivestmet horizo of N = 6 periods which eables us to demostrate log-ru eects of the redistributive tax mechaism. The growth of aggregate cosumptio is calibrated to the real empirical aual estimates for U.S. cosumptio i Lettau ad Ludvigso 25), i.e. G + =1.315 ad G =1.87. I.e., throughout our umerical examples, oe period correspods to oe year. The level of risk aversio, the subjective utility discout factor ad the tax rate are 9

12 set to γ = 5, ρ =.96, ad τ = 2%. 7 We assume that the poor aget 1 is iitially edowed with a claim o α,1 = 1% of aggregate cosumptio; this does ot iclude the preset discouted value of future trasfer paymets. We summarize this set of parameters i Table 1 ad refer to it as our base case parameter choice i the followig. Parameter Descriptio Value Number of agets 2 N Ivestmet horizo 6 α,1 Aget 1's iitial edowmet 1% γ Degree of risk aversio 5 ρ Utility discout factor.96 τ Tax rate 2% G + Divided growth boom G Divided growth bust 1.87 Table 1: Parameter values for the base case I Figure 1 we depict the impact of the legth of the remaiig ivestmet horizo o aget 1's exposure to equity α t,1, upper left graph), his share of wealth ivested α t,1 + β t,1 P t, upper right graph), his cosumptio share lower left graph), ad his et trasfer icome i percet of the preset value of the divided lower right graph). The upper left pael i Figure 1 shows that the relatively poor aget 1 with a iitial claim of α,1 =1% of aggregate cosumptio actually starts out by ot participatig i the stock market whe the remaiig ivestmet horizo is the full 6 periods. This is because his relatively low wealth levels implies that he will receive a sigicat amout of trasfer icome i the followig periods. Sice this trasfer icome is subject to stock market risk ad already meets the aget's risk-appetite, it is optimal for him ot to participate i the stock market at all. Agai, our umerical example shows that a redistributive tax system ca help explai the low empirically observed equity exposures ad stock market rates of poorer agets. As the remaiig ivestmet horizo decreases, aget 1 slowly starts icreasig his wealth level as ca be see from the upper right graph. Simultaeously, he icreases his exposure to the risky asset as a higher wealth level implies a lower future level of trasfer icome ad thereby a lower level of imputed stock market risk. However, whe the remaiig ivestmet horizo is log, both the stock market positio ad aget 1's wealth remai at a low level. I fact, aget 1's wealth level is below his iitial wealth level whe the remaiig ivestmet horizo is sucietly log. This is heavily drive by aget 1's icetive to smooth his cosumptio share over time. The lower left graph shows that aget 1's cosumptio share is costat at about 18% of aggregate cosumptio. I.e., the aget sells o from his iitial edowmet of stocks i order to ace his high cosumptio rate. However, the aget does ot oly sell stocks for acig cosumptio, but also for purchasig the log positio i bods required by Theorem 1, item 7, 7 We allow for other tax rates i sectio 4.3 ad a dieret tax mechaism i sectio 5. 1

13 1 18 α t,1 i %) s share of wealth i %) Remaiig ivestmet horizo Remaiig ivestmet horizo s cosumptio share i %) Remaiig ivestmet horizo 1 s et trasfer icome i %) Boom Bust Remaiig ivestmet horizo Figure 1: Base case dyamics: This gure shows the base case behavior over time for aget 1's equity holdigs upper left graph), share of wealth ivested upper right graph), cosumptio share lower left graph), ad et trasfer icome i percet of aggregate cosumptio lower right graph) i our base case parameter settig. That is, we cosider a settig with =2 agets, a tax rate of τ = 2%, a ivestmet horizo of N = 6 periods ad assume that the relatively poor aget 1 iitially has a claim o α,1 =1% of aggregate cosumptio i the ecoomy. to smooth his cosumptio share. Reducig his iitial wealth level for acig a higher cosumptio level has the importat side-eect for aget 1 that it icreases the level of future trasfer icome. I.e., aget 1 has a strog icetive to choose a high cosumptio level as this simultaeously provides him with high utility from preset cosumptio ad also icreases his future trasfer paymets. The graph i the upper right had pael does ot iclude the capitalized value of future trasfers, which is decreasig over time. The lower right pael shows the level of these trasfers, expressed as fractios of aggregate cosumptio, i both the boom state ad the bust state. As show i Theorem 1, item 9, the level of trasfer paymets i case of a boom i the ecoomy is a xed multiple of the level of trasfer paymets i case of a bust. I our umerical example this 11

14 multiple is The lower right graph shows that aget 1's trasfer icome i percet of aggregate preset cosumptio is below his optimal cosumptio share irrespective of the legth of the remaiig ivestmet horizo ad irrespective of whether the ecoomy was experiecig a boom or a bust i the previous period. Cosequetly, aget 1 eeds savigs to attai his desired cosumptio share. Whe the remaiig ivestmet horizo is very log, the aget uses most of his trasfer icome for cosumptio ad oly uses a small fractio for savig to avoid reducig his future trasfer icome too much ad to keep his cosumptio rate at the desired level. As the legth of the ivestmet horizo decreases, the aget's wealth level grows at a faster rate sice aget 1 ears prots from his ivestmets ad thus ca use a higher share of his trasfer icome for savig. Eve though this icrease i wealth is dampeed by the decrease i aget 1's trasfer icome lower right graph), both his wealth level ad his exposure to equity icrease most whe the remaiig ivestmet horizo is short. However, his exposure to the risky asset always remais below his eterig exposure of α,1 =1%. Aother iterestig implicatio of our model relates to the evolutio of wealth. Eve though the redistributive tax system redistributes tax reveues i a attempt to reduce the disparity i the distributio of wealth, this goal is ever attaied. O the cotrary, there is a tedecy that poorer agets remai poor. Eve though Theorem 1, item 5 idicates that the poorer aget's exposure to the risky asset ad thereby also his wealth level icreases as the legth of the remaiig ivestmet horizo decreases, this does ot imply that a equal distributio of wealth is ultimately attaied. This is because the X t values are very close to 1 whe the legth of the ivestmet horizo is log. I.e., the relatively poor aget 1 i) maitais a low direct stock market positio sigicatly below his iitial edowmet α,1, ad ii) uses most of the received trasfer icome for cosumptio whe the legth of the remaiig ivestmet horizo is log. Especially, the secod eect is so strog that the distributio of wealth does ot, eve for a log ivestmet horizo, coverge to a equal distributio of wealth. 4.2 Legth of ivestmet horizo ad cosumptio Our results i Theorem 1, item 6, ad the lower left pael of Figure 1 show that cosumptio shares are time- ad state-idepedet for a give legth of the ivestmet horizo. I.e., give the legth of the ivestmet horizo N, the optimal cosumptio shares do ot deped o the legth of the remaiig ivestmet horizo. However, the cosumptio shares vary with the legth of the ivestmet horizo N. This is because the legth of the ivestmet horizo aects the preset value of the future trasfer icome. We depict the quatitative impact of the legth of the ivestmet horizo N o aget 1's optimal cosumptio share i Figure 2. The results i Figure 2 show that the poorer aget 1's cosumptio share icreases with the legth of the ivestmet horizo. However, it icreases at a decreasig rate. Both these eects 12

15 1 s cosumptio share i %) N Figure 2: Impact of legth of ivestmet horizo: This gure shows the impact of the ivestmet horizo N o aget 1's cosumptio share i our base case parameter settig. That is, we cosider a settig with =2 agets, a tax rate of τ =2%, ad assume that the relatively poor aget 1 iitially has a claim o α,1 =1% of aggregate cosumptio i the ecoomy. are drive by the impact of the legth of the ivestmet horizo o the preset discouted value of future trasfer icome at time t =. Theorem 1, item 6, shows that cosumptio shares are time- ad state-idepedet for a give legth of the ivestmet horizo. I.e., as already argued above, the agets agree about the asset prices ad the cosumptio-ivestmet strategies i the sese that these brig about a solutio that allows them to attai the cosumptio shares they have agreed upo at time t =. The loger the legth of the ivestmet horizo, the higher the preset value of future trasfer paymets. As a cosequece, the poorer aget's cosumptio share icreases as the legth of the ivestmet horizo does. However, the fact that future trasfer icome is discouted implies that the impact of extedig the legth of the ivestmet horizo o the preset value of future trasfer icome decreases as the legth of the ivestmet horizo icreases. As a cosequece, the poorer aget's cosumptio share icreases at a decreasig rate. Throughout the remaider of this sectio, we demostrate how chages i our assumptios impact our results. We demostrate how the level of the tax rate, the iitial distributio of wealth ad dieret tax mechaisms aect our results. I particular, we demostrate that our two key digs low equity holdigs of poor agets ad poorer agets remaiig poor are robust to various variatios of our assumptios., 4.3 Level of the tax rate I this sectio we study the impact of the level of the tax rate τ o aget 1's optimal cosumptioivestmet strategy over the ivestmet horizo. To visualize the eect of the tax rate τ o aget 1's optimal cosumptio-ivestmet strategy we vary the tax rate betwee τ =% ad τ =5% i our base case parameter settig. Similar to Figure 1, we show i Figure 3 aget 1's optimal 13

16 exposure to equity i the upper left graph, his share of wealth ivested i the upper right, his cosumptio level i the lower left ad his et trasfer icome as a fractio of aggregate cosumptio i the lower right graph. Give that aget 1's et trasfer icome i case of a boom i the ecoomy is a costat multiple of it i case of a ecoomic bust, cf. Theorem 1, item 8, we improve the readability of that graph by oly showig the level of aget 1's trasfer icome for a boom throughout the backdatig period. α t,1 i %) τ i %) Remaiig ivestmet horizo 1 s share of wealth i %) τ i %) Remaiig ivestmet horizo 1 s cosumptio share i %) τ i %) Remaiig ivestmet horizo 1 s et trasfer icome i %) τ i %) Remaiig ivestmet horizo Figure 3: Impact of tax rate: This gure shows the impact of the tax-rate τ for aget 1's equity holdigs upper left graph), his share of wealth ivested upper right graph), his cosumptio share lower left graph), ad his et trasfer icome i percet of aggregate cosumptio whe the ecoomy was boomig throughout the previous period lower right graph) i our base case parameter settig. That is, we cosider a settig with = 2 agets, a ivestmet horizo of N =6 periods ad assume that the relatively poor aget 1 iitially has a claim o α,1 = 1% of aggregate cosumptio i the ecoomy. I lie with the well kow key properties of the model without taxatio ad redistributio, Figure 3 shows that for a tax rate of τ = %, aget 1's exposure to the stock coicides with his eterig exposure of α,1 = 1%. Furthermore, his share of wealth ad his cosumptio share also remai at a costat level of 1%. The upper two graphs show that a icreased level of the tax rate icreases the extet to which aget 1 chages his exposure to stocks, reectig the icreased level of imputed stock market 14

17 risk. For a remaiig ivestmet horizo of 6 periods aget 1's optimal exposure to the stock drops from 1% at a tax rate of % to -3% at a tax rate of 5%. That is, aget 1 partly aces his preset cosumptio by shorteig the risky stock ad repurchasig it usig the aticipated future trasfer icome. Likewise, the evolutio of aget 1's share of wealth is most aected for high tax rates. Whereas it is costat at 1% for a tax rate of %, it icreases from 4.8% to 26.3% at at tax rate of 5%. This reects that the aget's implicit wealth from the capitalized value of future trasfers is sigicat ad icreasig i both the tax rate ad the legth of the horizo. However, this capitalized value is depreciatig as the legth of the horizo shortes, ad the aget eeds to save i lie with this depreciatio to maitai his high cosumptio share costat. As ca be see from Theorem 1, item 8, the level of the et trasfer paymet for each idividual is: τ ) ) 1 α R t 1 P t + D t P t 1 R This relatio is close to beig liear i the tax rate τ as show i the lower right graph. I additio to the obvious ad direct liear iuece of the tax rate τ as the rst term i 2), the eects o the remaiig two terms cotribute to makig the level of et trasfer paymets progressively icreasig i the tax rate τ. However, these eects are very small. 2) Accordig to Theorem 1, item 6, the cosumptio share is give by 15) as α N 1 1 ) 1 τ) ) This expressio depeds i a direct liear maer o the tax rate τ ad i a idirect maer through α N 1. Agai, the eect o α N 1 is very small as is also visible from the upper left graph. Overall, our results i this sectio demostrate that eve though variatios i the tax rates aect our results qualitatively, our two key digs that poorer agets choose lower equity exposures ad that poorer agets remai poor are robust to varyig the tax rate. 4.4 Iitial distributio of wealth The iitial distributio of wealth to the agets aects to which extet they are et recipiets or et payers to the redistributive tax system. I this sectio we vary aget 1's iitial exposure α,1 to the stock to illustrate its quatitative impact o his cosumptio-ivestmet strategies as well as the evolutio of his wealth ad trasfer icome over the life cycle. I Figure 4 we study the impact of aget 1's iitial share, α,1 of the risky stock o his exposure to the risky stock upper left graph), his share of wealth upper right graph), his cosumptio 15

18 α t,1 i %) α, Remaiig ivestmet horizo 1 s share of wealth i %) α, Remaiig ivestmet horizo 1 s cosumptio share i %) α, Remaiig ivestmet horizo 1 s et trasfer icome i %) α, Remaiig ivestmet horizo Figure 4: Impact of iitial wealth: This gure shows the impact of aget 1's iitial claim α,1 o aggregate cosumptio i the ecoomy o his equity holdig upper left graph), his share of wealth ivested upper right graph), his cosumptio share lower left graph), ad his et trasfer icome i percet of aggregate cosumptio whe the ecoomy was boomig throughout the previous period lower right graph) i our base case parameter settig. That is, we cosider a settig with =2 agets, a ivestmet horizo of N =6 periods, ad a tax rate of τ =2%. share lower left graph), ad his et trasfer icome expressed as a share of aggregate cosumptio i the ecoomy lower right graph) i our base case parameter settig. I.e., we stick to the settig with =2 agets, a ivestmet horizo of N =6 periods, ad a tax rate of τ =2%. I lie with ecoomic ituitio, our results i Figure 4 show that a icrease i aget 1's iitial edowmet α,1 results i a icrease of his equity holdigs α t,1, a icrease i his share of wealth ivested, a icrease i his cosumptio share ad a decrease i the trasfer icome he receives. However, it seems worth otig that α t,1 icreases at a faster rate tha α,1, reectig that with icreasig iitial edowmet the et trasfer icome received by aget 1 decreases. Sice this does ot oly decrease the et wealth trasfer to aget 1 but simultaeously decreases the imputed stock market risk, aget 1 optimally icreases his equity holdigs α t,1 at a faster rate tha the iitial edowmet α,1 does i order to be edowed with the same exposure to stock market risk after accoutig for trasfer icome. Likewise, aget 1's optimal cosumptio 16

19 share ad his share of wealth icrease at a weaker rate tha α,1, reectig the decrease i his trasfer icome. Eve though variatios i aget 1's iitial edowmet α,1 aect our results quatitatively, our two key digs that poorer agets choose lower equity exposures ad that poorer agets remai poor are robust to varyig aget 1's iitial edowmet. 5 Dieret tax rates o risk-free rate ad equity premium I may tax codes aroud the world capital gais are subject to a lower tax rate tha eared iterest. I this sectio, we allow for dieret tax rates, τ r ad τ e, o the risk-free rate ad the equity premium, respectively. We restrict ourselves to a eutral tax system to avoid tax arbitrage ad to elimiate dispositios that are solely made i order to avoid tax paymets. I a eutral ad liear tax system, the imputed) risk-free retur, P t 1 r t 1, is taxed at the rate τ r, whereas the realized risk premium after correctio for the imputed risk-free retur, P t +D t 1+r t 1 )P t 1, is taxed at aother tax rate τ e. 8 is give by W t = Hece, the evolutio of a aget's wealth before cosumptio α t 1 1 τ e ) + τ e P t 1 α t 1 1 ) P t + D t ) + β t 1 Rt 1 + ) τ e + r t 1 τ e τ r )) 22) Specically, the key dierece compared to previous results is that the tax mechaism ow disetagles the trasfer of wealth ad the trasfer of imputed stock market risk. The latter is related solely to the tax rate τ e. The former is related to both of the tax parameters. I the special case with o trasfer of risk, but oly trasfer of wealth, i.e. τ e =, the wealth dyamics becomes W t = α t 1 P t + D t ) + β t 1 Rt 1 P t 1 α t 1 1 ) r t 1 τ r 23) Observe that the last term ow has the opposite sig of what was the case previously with a uiform tax rate τ for capital gais. Poorer agets will ow take short positios i the bod market. This is so because, as et recipiets of trasfer icome, they eed to lever up their positios to reach the desired exposure to stock market risk. The ecessary modicatios to Theorem 1 ca ow be stated for the case with dieret tax rates, τ r ad τ e, for the imputed) time value ad the risk premium, respectively. 8 A eutral tax system is the bechmark for a tax system that elimiates tax arbitrage opportuities i the tax code. Some key refereces to eutral taxatio systems i the public ecoomics literature are, e.g., Samuelso 1964) ad the retrospective tax system described i Auerbach 1991). For a short overview see, e.g., Harberger 28). The characteristics of such tax systems from a ace perspective have recetly bee described i detail i Jese 29), where the taxatio of imputed risk-free returs is called taxatio due to passage of time. 17

20 Theorem 2. The geeral equilibrium solutio to the optimizatio problems for agets that oly dier by their iitial edowmets ad who face dieret tax rates, τ r o risk-free returs ad τ e o risk premia, respectively, is idetical to the equilibrium i Theorem 1 for the martigale measure ad the iterest rate after tax. The followig modicatios apply: 1. The iterest rate before tax is r = R 1 1 τ r = r 1 τ r 24) 2. The equity exposure is give by α t = 1 + X t α t 1 1 ), t = 1, 2,..., N 1 25) RP t + D t ) X t = [ N 1 ], t = 1, 2,..., N 1 26) RP t + j=t+1 X j RDt α = P + D ) R 1 τ e RP + RD [ N 1 ] α 1 ) j=1 X j Provided that the risk-free rate of iterest as well as the tax rate τ r are both positive it is the case that X t, 1) t. Furthermore, provided that the tax rate τ e is sucietly large, the deviace α 1/ is elarged i compariso with α 1/, whe the choice of the iitial exposure α is made. A suciet coditio for this is that Rτ e rτ r >. 3. The cosumptio policy is give by a costat share of aggregate output: 27) C t = α N 1 1 τ e ) + τ e D t = α N 1 1 ) 1 τ e ) + 1, t =, 1,..., N 28) 4. The risk-free asset plays a role i order to establish the liear sharig rule. The positio i the risk-free asset is determied by ) 1 β t = R) 1 Rτe rτ r ) α t P t, t =, 1,..., N 1 29) 5. The level of the received et trasfer paymet for each idividual at time t is ) [ ] 1 α R t 1 τ e P t + D t ) P t 1 R Rτ e rτ r ) 3) There is a xed relatio, idepedet of time ad state, betwee the et trasfer paymets received i the boom ad the bust states, respectively. The ratio is give by RG + E Q t 1 [G]Rτ e rτ r ) RG E Q t 1 [G]Rτ e rτ r ) = RG + E Q [G]Rτ e rτ r ) RG E Q [G]Rτ e rτ r ) 31) 18

21 Proof The details of the derivatios are foud i Appedix A. Theorem 2 shows that allowig for dieret tax rates o the risk-free rate ad the risk premium aects our key digs relatively little. It remais the case that the optimal cosumptio shares are costat through time ad states Theorem 2, item 3). Likewise, the risk-free asset allows the agets to establish a liear sharig rule Theorem 2, item 4) ad the ratio of the et trasfer paymet i case of a ecoomic boom ad bust remais a time- ad state-idepedet costat. Dieret tax rates o the risk-free rate ad the risk premium, respectively, allows us to disetagle the wealth trasfer from the trasfer of stock market risk. We illustrate the impact of varyig the tax rate τ e o the risk premium betwee % ad 5% while keepig the tax rate o the risk-free rate as well as other parameters xed at our base case parameter choices of τ r =2%, =2, N =6, ad α,1 =1%. Figure 5 shows how the level of the tax rate τ e aects the relatively poor aget 1's exposure to equity upper left graph), his share of wealth ivested upper right graph), his cosumptio share lower left graph), ad the trasfer icome received i percet of aggregate cosumptio lower right graph). Compared to our results i Figure 3 where we varied a commo tax rate τ = τ r = τ e three key diereces become apparet. First, aget 1's share of wealth ivested does ot vary with τ e, oce the iitial adjustmet of α to α has bee made. Secod, his cosumptio share does ot vary with τ e, ad third, his et trasfer icome is less sesitive to chages i τ e tha i τ. Whereas the latter result is due to the fact that the tax basis that τ e applies to is smaller tha the basis τ applies to, the former two digs are closely related to the fact that the market value of the risk premium is zero. This implies that the agets are able to adjust their shares of the risk premium ad uwid the eect of its taxatio i order to obtai their desired liear exposure to market risk idepedet of the level of the tax rate τ e. If α t 1, β t 1 ) is the optimal ivestmet strategy at time t 1 for a give aget ad a give level of τ e, the same exposure ca be obtaied for ay dieret level τ e by choosig portfolio weights α, β) as follows: [ 1 τ e α t 1 = α t 1 + τ ] e τ e 1 τ e 1 τ e ) 32) β t 1 =β t 1 + P t 1 1 α t 1) τe τ e 1 τ e 33) Hece, the evolutio of wealth ad the cosumptio share does ot deped o τ e. I order to achieve the optimal allocatio of risk betwee the two agets show i Figure 5, aget 1 must decrease his exposure to equity as the level of imputed stock market risk icreases, i.e. as the tax rate τ e icreases. As ca be see from the upper left graph, for very high levels of τ e this might require holdig a short positio i the risky asset. That is, also i a tax system with dieret tax rates o the risk-free rate ad the equity risk premium poorer agets ted to hold lower equity exposures. Likewise, the relatively poor agets ted to remai relatively poor. 19

22 α t,1 i %) τ e i %) Remaiig ivestmet horizo 1 s share of wealth i %) τ e i %) Remaiig ivestmet horizo 1 s cosumptio share i %) τ e i %) Remaiig ivestmet horizo 1 s et trasfer icome i %) τ e i %) Remaiig ivestmet horizo Figure 5: Impact of dieret tax rates: This gure illustrates the impact of dieret tax rates o the risk-free rate ad the equity premium. It shows the impact of the tax rate τ e o the equity premium for aget 1's equity holdigs upper left graph), his share of wealth ivested upper right graph), his cosumptio share lower left graph), ad his et trasfer icome i percet of aggregate cosumptio whe the ecoomy was boomig throughout the previous period lower right graph) i our base case parameter settig. That is, we cosider a settig with =2 agets, ad a ivestmet horizo of N =6 periods. The tax rate o the risk-free rate is set to its base case parameter value of τ r = 2%. 6 Agets with dieret levels of risk aversio A aget's risk aversio is oe of the key determiats drivig the relatio betwee the demad for risky ad risk-free assets. However, it is a well-established fact that allowig for dieret levels of risk aversio greatly icreases the complexity of geeral equilibrium asset pricig models. There is a kow although yet limited literature o Pareto optimal sharig rules whe agets are heterogeous with respect to their degree of risk aversio. The semial paper ad modelig framework i this area is due to Dumas 1989). Other subsequet papers alog these lies are Beiga ad Mayshar 2), Bhamra ad Uppal 21), Cvitai ad Malamud 21), Cvitai et al. 211), Frake et al. 1998), Vasicek 25), Wag 1996) ad Weibaum 29). 2

23 The aalytical solutios i the previous sectios relied o some key relatios that o loger hold. First, whe agets dier i their attitude towards risk, they o loger agree ex ate about the correct level of the after-tax risk-free rate, i.e. equatio 7) o loger holds. Secod, agets o loger seek to smooth cosumptio shares through time ad dieret states. Istead, the relatively more risk averse agets seek to attai higher cosumptio shares i bust states ad i tur accept lower cosumptio shares i boom states ad vice versa for the relatively less risk averse agets. I other words the Pareto optimal sharig rules are o loger liear ad the role of the risk-free asset is o loger to establish such liear sharig rules; i.e., equatio 16) o loger holds. Istead, the after-tax risk-free rate as well as the martigale measure becomes time- ad state-depedet reectig the varyig wealth distributio across the agets. We kow that as a ecessary coditio for a optimal solutio the rst-order partial derivatives of the Lagragia with respect to the decisio variables α t, β t have to be equal to zero for each aget i geeral equilibrium. Sice the optimizatio problems are cocave ad the costraits are liear, a uique solutio exists, ad a solutio fulllig the ecessary coditio is guarateed to be optimal. For a problem with two agets ad N periods this leaves us with 4 2 N 1 ) rst-order coditios that have to be solved for the 4 2 N 1 ) decisio variables, asset prices, ad risk-free rates simultaeously. I this sectio, we therefore restrict ourselves to studyig a settig with N = 1 periods which is already umerically quite challegig as it requires us to solve a system of more tha 4, oliear equatios simultaeously for the more tha 4, ukows. 9 I Figure 6 we illustrate the impact of varyig aget 1's level of risk aversio betwee γ 1 =2 ad γ 1 =8, which is i the rage of values cosidered reasoable by Mehra ad Prescott 1985). The other parameters are set to our base case parameter values, i.e. = 2, τ = 2%, α,1 = 1%, ad γ 2 =5. I cotrast to the settigs studied i the previous sectios, the evolutio of wealth, cosumptio shares ad equity holdigs becomes state depedet. This state depedece also complicates the graphical illustratio of our results. I Figure 6 we therefore oly depict results for time t =, ad for time t = 9 i the best ad worst possible state of the ecoomy. Focusig o the latter two gives us a upper ad a lower boud o the attaiable cosumptio shares, shares of wealth ad equity exposures. I lie with ecoomic ituitio, aget 1's exposure to equity icreases as his level of risk aversio decreases. At time t = 9 his holdigs i the risky asset are larger whe he is the less risk averse ad lower whe he is the more risk averse aget. This is because whe aget 1 is the less risk averse aget he is bearig a higher share of risk, implyig that his wealth level is highest 9 Whe solvig the model umerically we make use of the fact that some of the variables ca be elimiated by simple substitutio. The optimal cosumptio strategy is ot a argumet i the umerical optimizatio as it directly follows from the budget equatio 3). This is also why the umber of variables i Appedix A, where we cout the umber of decisio variables icludig the optimal cosumptio decisios as well as the Lagrage multipliers, diers from the umber of argumets reported i this sectio. 21