Effects of a capital gains tax on asset pricing

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1 Busness Research (2018) 11: ORIGINAL RESEARCH Eects o a captal gans tax on asset prcng Marko Volker Krause 1 Receved: 21 Aprl 2016 / Accepted: 23 November 2017 / Publshed onlne: 4 December 2017 Ó The Author(s) Ths artcle s an open access publcaton Abstract I extend and generalze the work o Kruschwtz and Löler (BuR Busness Research 2(2): , 2009). I nd that, wth a zero rsk-ree rate, the mplct prce o captal gans tax payments s zero. I provde condtons n stochastc dscount actor language when a captal gans tax has no eect on asset prces or the case o a zero rsk-ree rate. A sucent condton or prce equalty wth a zero rsk-ee rate s that agents consume the same n any state wth and wthout taxes. Equlbra exst that guarantee equal consumptons, and they mply the same portolo rules that Kruschwtz and Löler (BuR Busness Research 2(2): , 2009) nd or the CAPM. Furthermore, or an exogenous non-zero rsk-ree rate, I show that exponental utlty wth multvarate normal payos, as well as lnear margnal utlty leave prces unchanged. Equlbrum prces are ndependent o captal gans taxes n those cases. However, total wealth o agents s derent between the tax and the no-tax economy. Keywords Captal gans tax Stochastc dscount actor Portolo theory Constant absolute rsk averson Lnear margnal utlty JEL Classcaton G11 G12 & Marko Volker Krause marko.krause@capco.com 1 Capco-The Captal Markets Company GmbH, Opernplatz 14, Frankurt am Man, Germany

2 116 Busness Research (2018) 11: Introducton I buld on the work o Kruschwtz and Löler (2009) who assumed a sngle-perod mean-varance captal asset prcng model (CAPM) wth a lat tax on captal gans and tax transers back to nvestors. They nd that prces n a world wth taxes on returns are the same as prces n a world wthout taxes the rsk-ree rate s zero or nvestors have constant absolute rsk averson mean-varance utlty. Instead o regardng a mean-varance CAPM as n Kruschwtz and Löler (2009), I construct a model wth agents that value expected utlty over consumpton,.e., a consumpton CAPM wth heterogeneous agents. The undamental results rom Kruschwtz and Löler (2009) also hold or such economes, but I nd mportant extensons. Frst, I look at economes wth consumpton at two tmes, and I examne the eect o the rsk-ree rate on asset prcng. I nd that a non-zero rsk-ree rate leads to non-zero prces o tax payments. Even though not traded, prces o tax payments can be constructed rom tradeable assets. For a zero rsk-ree rate captal gans taxes and the respectve transer payments have a zero (mplct) prce. I construct two economes that have agents wth equal endowments wth shares o nancal assets and consumpton goods, equal utlty unctons and payos. I mpose a tax on captal gans on one economy. I show that, or any tax economy, there s a no-tax economy wth equal prces. Ths holds or two economes n whch ndvdual consumpton o agents n one economy s the same as the consumpton n the other economy n every state. Then, the stochastc dscount actor n the no-tax economy o any agent s the same as n the tax economy. Snce taxes are not prced ths leads to the same asset prces n both economes. Furthermore, I obtan the same portolo rule as n Kruschwtz and Löler (2009). Ths rule makes consumpton proles o nvestors equal n both economes wth a zero rsk-ree rate. It ollows that ths rule s not just applcable to mean-varance CAPM economes but also to economes wth expected utlty maxmzng agents, and n whch a rsk-ree asset s traded and has a zero return. Wthout a zero rsk-ree rate prce equalty does not generally hold. For lnear margnal utlty t can be shown that t never holds. I also regard the case o economes wth consumpton only n the uture. In ths case the rsk-ree rate s exogenous. For a zero rsk-ree rate prce equalty can be obtaned agan. For a non-zero rsk-ree rate, I show that exponental utlty and multvarate normal payos lead to a par o economes wth equal prces. It s only necessary to pck equal prces o the rsk-ree assets n both economes. In contrast to Kruschwtz and Löler (2009), who use mean-varance utlty arguments, I use SDF arguments to derve ths result. Furthermore, I show that aggregate wealth ater ntal consumpton n the no-tax economy s derent to the one n the tax economes - even though prces are the same. In the tax economy aggregate wealth s derent rom wealth n the no-tax economy by the prce o aggregate transer payments, whch do not have zero prces as wth a zero rsk-ree rate. The portolo rule or rsky assets s agan the same as the one proposed n Kruschwtz and Löler (2009). However, the rule or the rsk-ree asset ders. Furthermore, I nd that utlty unctons that lead to margnal utlty lnear n consumpton also lead to prce equalty. The reason here s that ndvdual prcng equatons can easly be

3 Busness Research (2018) 11: aggregated to a prcng equaton that does not depend on the tax rate. Wth nonlnear margnal utlty prces cannot generally be obtaned snce aggregaton regularly does not lead to a prcng equaton that s ndependent rom the tax rate. I contrbute to the asset prcng lterature that s especally concerned wth tax eects on asset prcng. Much o the lterature s concerned wth the classc meanvarance CAPM such as Kruschwtz and Löler (2009) and Ekseth and Lndset (2009), who consder tax transers back to the nvestors. Salm (2006), n turn, uses a representatve agent model wth an uncertan tax on consumpton and tax transers. He nds that aggregate consumpton and thereore margnal utlty growth s not aected when all taxes are transerred back. Wth certan and constant taxes there would not be an eect on asset prces versus no taxes. Brennan (1970) sa classc paper that ncorporates varous personal tax rates nto the CAPM to arrve at pre-tax expected returns, but t does not consder transers. Wese (2007) bulds on Brennan s work to develop a model that relects the German tax code. I especally nclude SDF and consumpton arguments nto my analyss n the ashon o Cochrane (2014). In Sect. 2, I ntroduce the basc economy wthout taxes and the economy wth a lat and certan tax rate on captal gans. In the ollowng Sect. 3, I show that or every no-tax economy there s a tax economy wth equal asset prces n the cases when the rsk-ree rate s zero. When consumpton only takes place n the uture and thereore the rsk-ree rate s exogenous, equal prces are obtaned when agents have exponental utlty wth normal consumpton or when they have lnear margnal utlty. I contnue to dscuss the results and the lmtatons o the analyses n Sect. 4, where I also provde a smple numercal example. I conclude n Sect The two basc economes 2.1 The basc nance economy wthout taxes Payo space I model an endowment economy wth nancal assets. The economy exsts at dates t ¼ 0, when decsons are made and ntal consumpton takes place, and at t ¼ 1, when payos are pad out and consumed. I add to the model o Kruschwtz and Löler (2009) consumpton at t ¼ 0. I denote X r as an N S matrx o tradeable, rsky and elementary payos, n whch N s the number o payos and S the number o possble states at t ¼ 1. Wth elementary or basc payos, I mean non-redundant payos. Non-redundant, n turn, means that any sngle elementary payo cannot be constructed through lnear combnatons o other payos. Ths matrx s augmented by a rsk-ree payo X 0, whch s also non-redundant, so that, X ¼ðX 0 X r Þ 0 s an N þ 1 S matrx o non-redundant payos. Thus, the payo space s spanned by N elementary rsky asset payos and a rsk-ree payo. The number o states S can be greater than the number o assets so that an ncomplete market s possble. I use the subscrpt s or ndvdual states and the subscrpt j or the derent nancal assets so that the payo j pays X js n state s. To smply notaton, I put tme subscrpts only when necessary, such as or consumpton, whch

4 118 Busness Research (2018) 11: s possble at t ¼ 0 and at t ¼ 1. I use all random varables as row vectors o dmenson 1 S. Constants such as prces o a sngle asset j, denoted p j, can also be wrtten as a 1 S vector o constant values. Characterzaton o the agents and ther maxmzaton problems There are ¼ 1;...; I agents n the economy. Agents are ratonal and have the same complete set o normaton,.e., they know the dstrbutons o the payos. They are characterzed through a tme separable utlty uncton u ðþ over consumpton and through ntal (pre-trade) portolo holdngs n. At date t ¼ 1 and n state s agent consumes c s unts o a composte consumpton good. One unt o a consumpton good has a prce o one at all tmes so that a payo o one can buy exactly one unt o the consumpton good. To address random varables such as agent s consumpton or the j th payo at t ¼ 1, I leave out the subscrpt s or states and wrte c 1 and X j, respectvely. I denote c 0 the endowment o agent wth consumpton goods at tme t ¼ 0. Agents maxmze expected utlty o consumpton, max E½b u ðc 1 ÞŠþu ðc 0 Þ; c 0 ;c 1 ð1þ subject to the budget constrants at t ¼ 0, n 0 p þ c 0 ¼ n 0 p þ c 0 ð2þ and at t ¼ 1 c s ¼ n 0 X s; ð3þ or s ¼ 1;...; S: I use all collectons o prces and asset weghts as column vectors. I denote E½Š the expected value at tme zero o ts argument, p ¼ ðp 0 p 1 p j p N Þ 0 s the prce vector o the N þ 1 assets, n ¼ ðn 0 n 1 n j n N Þ 0 s a vector o ater-trade portolo weghts (I use n or pretrade portolos.), b the subjectve tme dscount actor (or mpatence actor), and u ðþ the utlty uncton. The expected value operator wth a sngle random varable means a probablty nner product. Wth a random varable z that means E½zŠ ¼ P S s¼1 z s, n whch s the probablty o state s. For prces I mostly use the short notaton so that p j s the prce o a payo X j. When necessary, I also use prces as operators to make more clear what s prced, or example p j ¼ pðx j Þ s agan the prce o the payo j. Furthermore, I use the subscrpt r to reer only to the rsky assets p r ¼ðp 1 p j p N Þ 0 and n r ¼ðn 1 n j n N Þ 0, the subscrpt zero s related to the rsk-ree asset. 1 The utlty uncton s derentable and strctly monotonously ncreasng at a decreasng rate. Thereore, any addtonal unt o consumpton adds to utlty, and t s optmal to consume all o the payos, whch justes to wrte the budget constrants as equaltes (Lengwler 2004, p. 52). The equalty o the budget constrants allows to substtute out consumpton and to restate the maxmzaton problem wth respect to the portolo weghts and ntal consumpton. 1 Wthout too much abuse o notaton, I also use the subscrpt zero or consumpton at t ¼ 0.

5 Busness Research (2018) 11: The rsk-ree asset s n zero net supply P I ¼1 n 0 ¼ 0. Thereore, I dene a vector o aggregate asset holdngs n ¼ P I ¼1 n, whch s n 0 ¼ð Þ because the rsk-ree asset s n zero net supply. Equlbrum The equlbrum s gven through a vector o prces p, consumpton proles c 0 ; c 1 and portolos n or ¼ 1;...; I so that each agent maxmzes utlty subject to hs budget constrant, gven prces p. Furthermore, the market or the consumpton good clears: P I ¼1 c 0 ¼ P I ¼1 c 0 and P I ¼1 c s ¼ n 0 X s or s ¼ 1;...; S. Fnancal assets are n postve net supply and markets clear so that P I ¼1 n j ¼ P I ¼1 n j ¼ 1 or j ¼ 1;...; N and P I ¼1 n 0 ¼ 0 or the rsk-ree asset. I assume that at least one equlbrum exsts. Notce that equlbrum prces mply the absence o arbtrage opportuntes (Lengwler 2004, p.50). Prcng equatons I wrte the agent s optmzaton problem n terms o a Lagrangan: L ¼ E½b u ðc 1 ÞŠþu ðc 0 Þ k ðn 0 p þ c 0 n 0 p c 0Þ; ð4þ where k s a Lagrange multpler. I substtute n Eq. (3) and take the partal dervatves wth respect to portolo weghts and to ntal consumpton. Combnng the results I obtan, u 0 p ¼ E Xb ðc 1Þ u 0 ðc : ð5þ 0Þ I denote more compactly, u 0 m ¼ b ðc 1Þ u 0 ðc ð6þ 0Þ as agent s stochastc dscount actor (SDF). Usng ths, I can prce any sngle payo X j through: p j ¼ Em X j : ð7þ Here the expected value means that probabltes are nduced to the nner product o X j and m : E½m X j Š¼ P S s¼1 m X js. Through tradng, agents nd a prce vector on whch everyone agrees,.e., p ¼ E½m XŠ or ¼ 1;...; I, and whch maxmzes utlty. In complete markets, X s a square matrx wth ull rank,.e., there are as many basc assets as states. The equaton p ¼ E½m XŠ can be wrtten as: p ¼ Xj, n whch state prces are: j s ¼ m s or s ¼ 1;...; S and are objectve probabltes o states s. When X has ull rank, there s a unque soluton or j. Snce probabltes are objectve probabltes, there s a unque SDF,.e., every agent has the same SDF. It also ollows that the state prce vector can be expressed as a lnear combnaton o bass assets and thereore les n the payo space. The same s true or the SDF. 2 2 For propertes o the SDF under derent assumptons such as market ncompleteness, see (Cochrane 2005, pp ).

6 120 Busness Research (2018) 11: Wth ncomplete markets,.e., wth S [ N þ 1, X does not have ull rank. The system o equatons p ¼ Xj has less equatons than unknowns so that there s more than one soluton to the system. That means state prces and SDFs among agents may der. Prcng a rsk-ree payo o one, I dene the rsk-ree rate as R ¼ 1=E½m Š¼ 1=p 0 or ¼ 1;...; I. The term R s the gross rsk-ree rate: R ¼ 1 þ r. Thus, the prcng Eq. (7) can be restated as: p j ¼ EðX jþ þ Covðm ; X j Þ; ð8þ R n whch Covðm ; X j Þ s the covarance between the SDF and the payo. As stated n Cochrane (2014), n ncomplete markets the SDFs o agents m can der and do not have to be wthn the payo space. But there s one SDF m wthn the space o tradeable assets that prces all assets. Ths SDF s the probablty nduced projecton o all o the agents SDFs onto the payo space. The relaton between the unque SDF wthn the payo space and any ndvdual SDF s m ¼ m þ, where s an error term orthogonal to the (probablty nduced) payo space and thereore does not nluence prces: p j ¼ Eðm X j Þ¼Eððm þ ÞX j Þ¼EðmX j Þ, because Eð X j Þ¼0 holds or all payos o the payo space (Cochrane 2005, p. 66). The unque SDF wthn the payo space can be used to prce all payos but t wll not necessarly lead to a possble portolo rule or all agents,.e., to a consumpton prole that s wthn the payo space. In complete markets the SDF s the same or every agent. In the standard CAPM, whch does not requre complete markets, the SDF s a lnear combnaton o the market return: m ¼ a þ br M, where R M s the return on the market portolo and a and b are constants (Cochrane 2005, p. 152). In those two cases the sngle SDF leads straghtorwardly to consumpton rules wthn the payo space. 2.2 The nance economy wth taxes I ntroduce another economy that has, compared to the no-tax economy, equal utlty unctons o agents u ðþ, equal mpatence actors b, and an equal (pre-tax) dstrbuton o payos o nancal assets X. The ntal or pre-trade portolos o agents wth shares o assets are also the same, as well as the agents perect normaton about the payo dstrbutons. I ntroduce taxes on captal gans. To account or possble derences n prces, ater-trade portolos, and consumpton proles rom the ones n the no-tax economy, I add an astersk to them. Prces o taxed payos are denoted as, j ¼ p ðxj s Þ and prces o pre-tax payos are denoted as, p j ¼ p ðx j Þ. Taxes I dene the tax base as the derence between the payo and the prce o the payo: X js j, n whch j s the prce o the ater-tax payo,.e., o the payo Xjs s ¼ X js T js ¼ X js sðx js j Þ, n whch T js are taxes on the asset j ¼ 0; 1;...; N n states s ¼ 1;...; S. Any observed prces relect possble tax eects. Investors consder the taxes they have to pay on the payo when prcng the asset. I use s 2ð0; 1Þ as the tax rate and also as a superscrpt to denote ater-tax

7 Busness Research (2018) 11: gures when necessary. The tax rate s certan, constant, and the same or all agents. Ths s a smplcaton snce tax rates can be observed to have an uncertan element and they oten depend on certan characterstcs o agents such s ther ncome. 3 Introducng an uncertan tax rate may ntroduce an addtonal covarance as well as an expectaton nto the prcng equaton. An agent pays captal gans taxes at the amount T s ¼ s P N j¼0 n j ðx js j Þ¼sn 0 ðx s Þ, and they receve transer payments Q s ¼ sx n 0 ðx s Þ or ¼ 1;...; I, n whch x s the share o total tax revenues that s transerred to agent wth P I ¼1 x ¼ 1. Transer payments are predetermned amounts,.e., they cannot be nluenced by the agents. Postve and negatve captal gans are taxed the same way. I dscuss ssues o ths smpled tax system versus more realstc tax systems n Sect. 4. Aggregate tax payments are T s ¼ P I ¼1 T s. They must be equal to aggregate transer payments: T s ¼ Q s. Indvdual transer payments can also be wrtten as Q s ¼ x T s. 4 The ntroducton o taxes and transers does not ntroduce any new basc asset so that the payo space s the same as n the no-tax economy. Any tax payment T js ¼ X js sðx js j Þ¼X js ð1 sþþs j s just a lnear combnaton o the pretax payo X j and a rsk-ree payo. Characterzaton o the agents and ther maxmzaton problems Any agent maxmzes expected utlty o ater-tax (and transers) consumpton, max E½u ðc c 1 ÞŠ þ u ðc 0 Þ; ð9þ 0 ;c 1 subject to the budget constrants at t ¼ 0 n 0 ps þ c 0 ¼ n 0 ps þ c 0 and at t ¼ 1 ð10þ c s ¼ n0 ðx s sðx s ÞÞ þ Q s ; ð11þ or s ¼ 1;...; S. The varable n s a vector o ater-trade portolo weghts. I denote nancal wealth that s let ater ntal consumpton as W Fs ¼ n 0 and total nancal wealth ater ntal consumpton,.e., nancal wealth ncludng transers as W F ¼ W Fs þ p ðq Þ. Equlbrum The equlbrum s gven through a vector o prces, consumpton proles c 0 ; c 1 and portolos n or ¼ 1;...; I so that each agent maxmzes utlty subject to hs budget constrant, gven prces. Furthermore, the market or the P consumpton good clears: I ¼1 c 0 ¼ P I P ¼1 c 0 and I ¼1 c s ¼ n0 X s or s ¼ 1;...; S. That ths holds comes rom the act that taxes are just redstrbutons 3 See or example Salm (2006) or a theoretcal treatment o tax rate uncertanty on asset prces and Salm (2009) or an econometrc treatment. 4 A case when agents receve transers exactly at the amount they pay taxes s when T s ¼ Q s or sn 0 ðx s Þ¼sx n 0 ðx s Þ. Ths mples ðn 0 x n 0 ÞðX s Þ¼0. Snce ðx s Þ ncludes rsky assets, t cannot be a zero matrx. The vector n 0 x n 0 s a vector o zeros or n 0 ¼ x n 0, whch s a very specal case. Wth the rsk-ree asset n zero net supply ths requres the rst element o n be zero and all o the remanng elements be equal to the constant x. I the rsk-ree asset s n postve net supply, all elements o n must be equal to x.

8 122 Busness Research (2018) 11: and do not change aggregate values. Fnancal assets are n postve net supply and clear so that P I ¼1 n j ¼ P I ¼1 n j ¼ 1 or j ¼ 1;...; N, and P I ¼1 n 0 ¼ 0 or the rskree asset. Prcng equatons The rst-order condtons lead to a smlar prcng equaton as or the no-tax economy, except that ater-tax payos are prced: h j ¼ Em Xs j : ð12þ The ater-tax rsk-ree payo s X0 s ¼ X 0 sðx 0 0 Þ¼1 sð1 ps 0 Þ, and the ater-tax rsk-ree rate s: R s ¼ 1 sð1 p s 0 Þ ð1 sð1 ps 0 ÞÞ ¼ 1 1 sð1 ps 0 E½m ¼ Þ Š : ð13þ 0 Em The second equalty ollows rom the act that 1 sð1 0 Þ s a constant, whch can be taken out o the expectatons n the denomnator and thereore cancels wth the term n the numerator. The thrd equalty just restates that the denomnator s actually the prce o the cash low X0 s ¼ 1 sð1 ps 0 Þ. The pre-tax rsk-ree rate s then, R ¼ 1 Em ð1 sð1 ps 0 ÞÞ ¼ 1 : ð14þ 0 Usng R ¼ 1=ps 0 the ater-tax return can also be wrtten as Rs ¼ 1 þ r ð1 sþ. I the rsk-ree rate s not taxed, t s R ¼ 1=E½m Š. Notce that snce the rsk-ree asset s traded, every agent agrees upon the rsk-ree rate. It ollows that the expected ndvdual SDFs must be equal, whch, n turn, are equal to the expected SDF wthn the payo space: E½m Š¼E½m Š or ¼ 1;...; I. In an economy wth captal gans taxes, the expectatons o the SDFs E½m Š play a specal role. Ths s summarzed n the ollowng proposton: Proposton 1 Assume an asset j wth a pre-tax payo X j, and wth an ater-tax payo Xj s wth postve prces. Captal gans are taxed at a certan tax rate s 2ð0; 1Þ. Assume urther that 1=E½m Š [ s. The prces o the pre-tax payo p j and o the ater-tax payo j are only equal as long as E½m Š¼1 or ¼ 1;...; I. Wth E½m Š greater (less) than one the prce o the ater-tax payo ps j s greater (less) than the prce o the pre-tax payo p j. Proo Ater tax payos are dened as Xj s ¼ð1 sþx j þ s j. The respectve prce o ths payo s: j Ths can be rewrtten as: ¼ E½m Xs j Š¼E½m ðð1 sþx j þ s j ÞŠ ¼ð1 sþe½m X jšþs j E½m Š: ð15þ

9 Busness Research (2018) 11: j ¼ p j ð1 sþþsps j E½m Š; ð16þ whch can be rearranged to: j ¼ p j ð1 sþ 1 se½m Š : ð17þ Thus, when E½m Š¼1, the tax terms cancel and prces o the pre-tax payo and the one o the ater-tax payo are the same. In any other case the prces are not the same. Equaton (17) shows urther that or E½m Š [ 1, t ollows that ð1 sþ=ð1 se½m ŠÞ [ 1 so that ps j [ p j and vce versa. Equaton (17) also shows that, gven E½m Š,.e., the prce o a payo o one n every state, one can derve prces o pretax rom ater-tax payos and vce versa. The condton 1=E½m Š [ s ensures that the denomnator o Equaton (17) s postve. h I assume that 1=E½m Š [ s holds throughout the paper. Notce that those pre-tax-ater-tax prce relatons use an SDF o the tax economy m. Any relatons to the SDFs o the no-tax economy,.e., to m, are stll to be obtaned. h Notce also that E½m Š¼1 mples that E u0 ðc 1 Þ u 0 ðc 0 Þ ¼ 1=b. Expected growth o margnal utlty o consumpton s exactly equal to the nverse o the mpatence actor. Hgher growth mples a lower rsk-ree rate and lower growth a hgher one. A smple log-normal model such as n (Cochrane 2005, pp ) allows or more nterpretatons o the rsk-ree rate n terms o consumpton growth. In ths case the rsk-ree rate s low when expected consumpton growth s low or mpatence s low,.e., when beta s hgh. The pror proposton has several mplcatons. Corollary 1 When the rsk-ree rate s not taxed, then, accordng to Eq. (13), R ¼ 1=E½m Š, and t ollows that E½m Š¼1 and r ¼ 0 are equvalent or all ¼ 1;...; I. Furthermore, E½m Š greater (less) than one s equvalent wth the rskree rate r beng less (greater) than zero. Corollary 2 When the rsk-ree rate s taxed, then, accordng to Eq. (13), R s ¼ 1=E½m Š, and E½m Š¼1 and rs ¼ 0 are equvalent or all ¼ 1;...; I. Furthermore, E½m Š greater (less) than 1 s equvalent wth the ater-tax rsk-ree rate r s beng less (greater) than zero. Corollary 3 In the case o a zero rsk-ree rate, the tax on captal gans has a zero prce. From the above proposton ollows that pre- and ater-tax prces are the same,.e., j ¼ p j p ðt j Þ¼p j, so that p ðt j Þ¼0. Furthermore, whether the rsk-ree rate o return s taxed as well does not matter when t s zero because taxes on that asset would also be zero. In the ollowng secton, I contnue to analyze equlbrum eects,.e., how taxes aect prces and quanttes n the no-tax and the tax economy.

10 124 Busness Research (2018) 11: Asset prces and portolos n the no-tax and the tax economy 3.1 General condtons or prce equalty I use the endowment economes, the one wthout and one wth a tax on captal gans, that I have outlned n the pror secton. I explore the general condtons under whch prces are the same n the two economes. 5 I contnue to state the general condtons or asset prces be equal. I start wth ndvdual prcng equatons and then contnue wth aggregate prcng equatons and projectons Indvdual prcng equatons Proposton 2 Asset prces n the no-tax and n the tax economy are equal,.e., p ¼, and only : E½m XŠ¼E m X Rs ; ð18þ or ¼ 1;...; I. Proo I start wth the vector o ater-tax prces. Smlar to the dervaton o Eq. (17) or a sngle prce, the prce vector s gven by: ¼ E½m Xs Š¼E½m ðð1 sþxs þ s ÞŠ ¼ð1 sþe½m XŠþsps E½m Š: ð19þ Ths can be rearranged to, ¼ From Eq. (13) we know that R s I obtan: R 1 s 1 se½m Š E½m XŠ: ð20þ ¼ 1=E½m Š. Substtutng that nto the pror equaton ¼ 1 s 1 s=r s E½m XŠ: ð21þ I multply the numerator and the denomnator by R s, whch yelds: ¼ Rs ð1 sþ R s s E½m XŠ: ð22þ The denomnator s R s s ¼ 1 þ r ð1 sþ s ¼ð1 sþþr ð1 sþ ¼ ð1 sþ, so that the 1 s terms cancel. Ths leads to: R 5 Notce that prce equalty concerns the tradeable nancal assets. Transer payments are not tradeable and do not belong to nancal assets.

11 Busness Research (2018) 11: ¼ E m X Rs ; ð23þ whch I set equal to p ¼ E½m XŠ to obtan the condton n the proposton. h Corollary 4 From Eq. (14),.e., rom the act that R ¼ 1=ps 0, and rom R ¼ 1=p 0 as well as rom prce equalty o the rsk-ree assets ollows that the rsk-ree rate n the no-tax economy s equal to the pre-tax rsk-ree rate n the tax economy: R ¼ R. Furthermore, R ¼ R mples E½m Š¼E½m Š and vce versa, whch ollows rom the denton o the rsk-ree rates. For example rom a pre-tax rsk-ree gross rate o return greater one,.e., R [ 1, ollows that the ater-tax rate s less than the pre-tax rate: R s \R ¼ R. Ater accountng or taxes, agents would requre less return than they would n the no-tax economy. They value a unt payo more than n the no-tax economy. For a zero rsk-ree rate pre- and ater-tax rates are the same so that the valuaton o a unt payo would not change. Corollary 5 Equaton (18) can also be rewrtten n terms o covarances: E½XŠ R R þ Covðm ; XÞ ¼ E½XŠ R þ Covðm ; XÞ Rs R : ð24þ Wth R ¼ R rom Corollary 4, I smply to obtan: Covðm ; XÞ ¼Covðm Rs ; XÞ R ; ð25þ or ¼ 1;...; I. Furthermore, Proposton 2 mples a condton that guarantees that the proposton holds. Corollary 6 The relaton o the ndvdual SDFs m R s R ¼ m ð26þ or ¼ 1;...; I s sucent to obtan prce equalty or all assets. Ths relaton consttutes a strong assumpton n that the SDF o any agent n the tax economy s proportonal to the SDF o an equal agent n the no-tax economy n every state. I assume the agents preerences to be the same n both economes so that a comparson makes sense. That means the agents ndvdual mpatence actors and the parameters and unctonal orm o ther utlty unctons are the same. That also means t s consumpton at t ¼ 0 and consumpton n the derent states at t ¼ 1 that determne the SDFs and possble derences n the SDFs o the two economes. Equaton (26) can be restated as: u 0 ðc 1Þ ¼u 0 ðc 1 Þ; ð27þ n whch ¼ u 0 ðc 0Þ=u 0 ðc 0 ÞRs =R s a constant that collects the rato o the rskree rates and the rst dervatves o the utlty unctons o consumpton at t ¼ 0.

12 126 Busness Research (2018) 11: Ths relaton shows that Eq. (26) mples that margnal utlty at t ¼ 1 be proportonal. Wth a zero rsk-ree rate the condton n Eq. (18) smples to, E½m XŠ¼Em X ; ð28þ and m ¼ m s sucent to ulll ths condton, whch s the same as condton (26) or a zero rsk-ree rate. Notce that the above condtons are derved rom the prce equatons, whch, n turn, are the rearranged rst order condtons,.e., the optmalty condtons, o the agents. Thus makng those equatons hold guarantees optmalty. Together they orm an aggregate prcng equaton Aggregate prcng equaton To obtan an aggregate demand uncton, I sum the ndvdual equatons o the orm u 0 ðc 0 Þ¼E½b u 0 ðc 1 ÞXs rš over all agents. Rearrangng or prces I obtan: " P # I ¼1 ¼ E b u 0 ðc 1 ÞXs r P I ¼1 u0 ðc 0 Þ : ð29þ The aggregate SDF s then: m a ¼ P I ¼1 b u 0 ðc 1 Þ P I ¼1 u0 ðc 0 Þ : ð30þ Ths aggregate SDF prces all assets just as good as the ndvdual SDFs. Gven utlty unctons, t may help to nd an aggregate prcng uncton. Consumpton, be t ndvdual or aggregate, must le wthn the payo space. Even wth taxes, when there are non-tradeable transer payments, those payments can be replcated by tradeable payments because they are lnear unctons o tradeable payments. I margnal utlty s lnear n consumpton, the quadratc utlty case, all ndvdual SDFs must le wthn the payo space. Snce there can only be one SDF wthn the payo space, all ndvdual SDFs must be the same. Furthermore, t s well-known that ths SDF can be wrtten as a lnear uncton n terms o aggregate consumpton c 0 and c 1, when all agents have the same tme dscount actor. Appendx B shows a dervaton. I come back to ths mportant specal case later Projectons o SDFs As ponted out n Sect. 2.1, there s a unque SDF wthn the payo space that prces all assets, and whch s related to the ndvdual SDFs through m ¼ m þ, wth beng an error term orthogonal to the probablty nduced payo space (or the tax economy wth an astersk, respectvely). Proposton 3 Asset prces n the no-tax and n the tax economy are equal,.e., p ¼, and only :

13 Busness Research (2018) 11: E½mXŠ ¼E m X Rs ; ð31þ n whch m and m are the SDFs n the payo space n the no-tax and the tax economy, respectvely. Proo I use the relatons m ¼ m þ and m ¼ m þ wth errors orthogonal to the probablty nduced payo space,.e., E½ XŠ¼0 and E½ Xs Š¼0 or all. The prce vector o the tax economy s: ¼ E½m Xs Š¼E½ðm ÞXs Š ¼ E½m X s Š E½ Xs Š ¼ E½m X s Š: The term E½ Xs Š s zero snce the error term s orthogonal to the payo space. For the no-tax economy the dervaton s smlar. The remander s smlar to the proo o Proposton 2. h Snce the error terms do not aect the prcng o the assets, the corollares ollow just as beore. Corollary 7 Corollares 4 6 also ollow or the SDF wthn the payo space,.e., or m and m. R Budget constrants and market clearng So ar I have ound a necessary and sucent condton or prce equalty n Proposton 2 and a sucent condton n Corollary 6. For an actual equlbrum allocaton, budget constrants have to be met and markets need to clear as well. In the ollowng, a tax and an equvalent no-tax economy wll compared, whch are n equlbrum. Thus, apart rom meetng condtons o prce equalty the budget constrants and market clearng need to hold, so that ths step s also ncluded n the ollowng analyses. I wll contnue as ollows: under the assumptons that the no-tax economy s n equlbrum, I wll derve sucent condtons or the exstence o a tax equlbrum wth prces equal to the ones n the no-tax economy. To do that, I wll draw on the condtons establshed heren. 3.2 Economes wth consumpton at t ¼ 0 and t ¼ A zero rsk-ree rate and equal consumptons n both economes In the ollowng I wll show that, wth a zero rsk-ree rate, or an equlbrum n the no-tax economy there exsts an equlbrum n the tax economy n whch agents crcumvent redstrbuton through the captal gans tax and through the transer payments usng the same portolo rule as n Kruschwtz and Löler (2009). As

14 128 Busness Research (2018) 11: Kruschwtz and Löler (2009) pont out, equlbra need not be unque so that other equlbra may exst that are not consstent wth such an allocaton. From the pror secton t s obvous that wth a zero rsk-ree rate R s =R ¼ 1. Then, the equalty m ¼ m or ¼ 1;...; I s sucent to obtan prce equalty, snce Corollary 6 s met. Snce consumpton at the derent dates s the only varable argument n the SDFs o the agents, t s clear that equal consumpton o agents n both economes leads to equal SDFs. Ths ollows rom observaton o Eq. (27). It remans to show that there s a portolo rule that makes equal consumpton possble. Budget constrants have to hold and markets have to clear. I show that the portolo rule that ensures equal consumpton s the same as the one n Kruschwtz and Löler (2009). Beore I turn to the portolo rule, I wll make some remarks. Wth equal ndvdual SDFs,.e., wth m ¼ m or ¼ 1;...; I, and wth a zero rsk-ree rate, whch makes prces o taxes and transers zero, asset prces n both economes must be the same and pre-tax prces are equal to ater-tax prces: p ¼ ¼ p. Asset prces n the tax economy are ¼ E½m Xs Š¼E½m ðx TÞŠ. Usng ths and notng that E½m TŠ¼0, t ollows that ¼ E½m Xs Š¼E½m ðx TÞŠ ¼ E½m XŠ¼p. Wth equal ntal portolos equal prces mply that agents have the same nancal wealth ater ntal consumpton n both economes: W F ¼ W Fs ¼ W F. Agents receve the same utlty as n the no-tax economy. Wth a zero prce o taxes the ntal budget constrants o the agents are also equal to the ones o the notax economy. Thus, agents maxmze utlty and obey ther budget constrants. I contnue to construct the portolo rule so that consumpton s equal n both economes and that markets clear. Wth equal ntal portolos and wealth,.e., agents have the same ntal characterstcs n both economes, equal consumpton means that an optmum n the no-tax economy s equvalent to an optmum n the tax economy. Intal consumpton s just a constant, whch s set equal or any agent n both economes. Consumpton at t ¼ 1 needs more attenton. Proposton 4 Gven equal prces n the tax and the no tax economy, consumptons at t ¼ 1 o all agents ¼ 1;...; I are the same n both economes and only rsky portolos o all agents or the no-tax and the tax economy are related through n r ¼ n r ð1 sþþx sn r ; ð32þ and weghts on the rsk-ree assets are related through n 0 ¼ n 0 ð1 sþþsðwf x W F Þ: ð33þ Proo Consumpton o any agent at t ¼ 1 n the no-tax economy s smply c 1 ¼ n 0 X. Consumpton n the tax economy s;

15 Busness Research (2018) 11: c 1 ¼ n0 ðx sðx pþþ þ x sn 0 ðx pþ: ð34þ I use the no-tax prce notaton because, p ¼ must hold or the portolos that are mpled. Every nvestor consumes the same n both economes c 1 ¼ c 1 or n 0 X ¼ n0 ðx sðx pþþ þ x sn 0 ðx pþ: ð35þ Snce W F ¼ n 0 p and W F ¼ n 0 p, I restate the equaton as: n 0 X ¼ n0 ð1 sþx þ sw F þ x sn 0 X x sw F : ð36þ For complete markets gven n there s a unque soluton or n snce X s a square matrx o ull rank. For ncomplete markets the system o equatons s overdetermned,.e., a system wth more equatons (number o states) than unknowns (number o portolo weghts). Overdetermned systems need not have a perect soluton at all. 6 However n ths case there s a unque perect soluton, whch wll be vered below. I separate nto rsky and constant parts, whch leads to: n 0 þ n 0 r X r ¼ n 0 0 ð1 sþþswf Now, smple observaton shows that, x sw F þðn 0 r ð1 sþþx sn 0 r ÞX r: ð37þ n 0 r ¼ n0 r ð1 sþþsx n 0 r ; ð38þ n whch the vector n 0 r s the same as n0 wthout the rst element,.e., a vector o ones, and n 0 ¼ n 0 ð1 sþþsðwf x W F Þ ð39þ s a soluton to the system o equatons. Systems o lnear equatons can have zero, one or nntely many solutons. I ound that there s at least one soluton to ths system o lnear equatons. It s also exactly one snce the payos n the matrx X are lnearly ndependent, so that nntely many solutons are not possble. h Ths s the same relaton o shares o rsky assets that Kruschwtz and Löler (2009) propose or the mean-varance CAPM wth taxes on captal gans, wth transers, and wth a zero rsk-ree rate, to obtan equlbra at equal prces n a tax and a no-tax economy. Snce I do not assume any specc utlty uncton that would mply the mean-varance CAPM, I conclude that ther proposton or portolo weghts s not lmted to the mean-varance CAPM. In the CAPM, I can urther smply because every nvestor holds the market portolo 7 so that all elements wthn the vectors n r and n r are equal,.e., n 1 ¼ n 2 ¼¼n j ¼¼n N and n 1 ¼ n 2 ¼¼n j ¼¼n N. 6 One can stll obtan an approxmate soluton n the least squares sense (see also Wllams 1990). 7 Kruschwtz and Husmann (2012, pp ) present the Tobn Separaton Theorem together wth the Mutual Fund Theorem, whch state that every nvestor holds a share o the market portolo and o the rsk-ree asset.

16 130 Busness Research (2018) 11: The case o lnear margnal utlty A specal case s margnal utlty lnear n consumpton o all agents,.e., somethng lke u 0 ðc 1Þ ¼a þ b c 1, and equal tme dscount actors b ¼ b or all. Then, all ndvdual SDFs are equal and le wthn the payo space. Wth equal tme dscount actors, the SDF depends on aggregate consumpton n t ¼ 0 and t ¼ 1 and some constants (see Appendx B). In equlbrum agents consume all what they have snce t s optmal to do that. Aggregate consumpton must be the same n the no-tax and the tax economy, because agents are gven the same endowments, and pre-tax payos are the same. Thus, or lnear margnal utlty and equal tme dscount actors the SDF(s) are the same n the no-tax and the tax economy. Gven zero rskree rates, asset prces must be the same as well. An example o ths case or quadratc utlty s gven n Sect Furthermore, wth equal SDFs n both economes, the rsk-ree rate s not zero, there s no prce equalty, because Proposton 2 does not hold anymore Economes wth consumpton only at t ¼ General remarks I contnue to look at economes that have no tme zero consumpton. Kruschwtz and Löler (2009) lmt ther analyss to ths knd o economes. In ths case the rsk-ree rate s assumed to be exogenous to the economy. It s not the result o the trade-o o current and uture consumpton as n the model wth consumpton at t ¼ 0 and at t ¼ 1, because consumpton at t ¼ 0 does not take place. 9 I also smply to assume that all o the agents have a tme dscount actor o 1. An agent s maxmzaton problem s: max E½u ðc c 1 ÞŠ; ð40þ 1 subject to the budget constrants at t ¼ 0 and at t ¼ 1 n 0 0 þ n0 r ps r ¼ n 0 ps 0 þ n0 r ps r ð41þ c 1 ¼ n 0 ðx 0 sðx 0 0 ÞÞ þ n0 r ðx r sðx r r ÞÞ þ x sn 0 r ðx r r Þ: ð42þ As n Kruschwtz and Löler (2009), I rearrange the tme zero budget constrant or the quantty o the rsky asset to obtan: 8 Wth m ¼ m, n whch I leave out h the subscrpt snce all ndvdual SDFs are the same, the equaton n Proposton 2 turns nto E½mXŠ ¼EmX Rs R, a statement whch s not true or non-zero r. 9 The dentons n terms o prces o a pre- or ater tax cash low o one stll hold: R s R ¼ 1. 0 ¼ 1 sð1 ps 0 Þ p and s 0

17 Busness Research (2018) 11: n 0 ¼ 1=ps 0 n 0 0 þ n0 r ps r n 0 r ps r ; ð43þ and substtute ths expresson nto the one or consumpton and solve the maxmzaton problem to obtan: Eu 0 ðc ÞðX 0ð1 sþþs 0 Þps r =ps 0 ð 1ÞþX rð1 sþþs r ¼ 0: ð44þ The equaton can be restated as: Eu 0 ðc Þ Xs r ps r Rs ¼ 0 ð45þ so that rearrangement leads to, r ¼ E½u0 ðc ÞXs r Š R s Eu ½ 0 ðc Þ ð46þ Š; wth the SDFs m ¼ u0 ðc Þ R s Eu ½ An apparent queston s whether and n whch cases 0 ðc Þ Š. c ¼ c or all would lead to prce equalty. In ths case the SDFs can be rewrtten as m ¼ u0 ðc Þ. Multplyng by R s Eu ½ 0 ðc Rs ÞŠ =R gves m R s R ¼ u0 ðc Þ. SDFs n the notax economy are m ¼ u0 ðc Þ R Eu0 ½ ðc ÞŠ R Eu ½ 0 ðc ÞŠ. Snce the rsk-ree rate s exogenous, I set R ¼ R,.e., 0 ¼ p 0 as t s done n Kruschwtz and Löler (2009). Now the condton n Eq. (26) holds and prces must be equal. Notce that wth consumpton only at t ¼ 1, equal consumpton does not lead to equal SDFs, but rather to proportonal SDFs. However, n equlbrum, the budget constrants have to hold as well. I and only the portolo rules derved n the pror part hold, wll there be equal consumptons n both economes. It turns out that ths only holds or a zero rsk-ree rate. To show ths, I start wth the budget constrant n the no-tax economy denoted as n Eq. (43), and I substtute n Eq. (32). Ths leads to, n 0 ¼ n 0 þðn 0 r n0 r ð1 sþþx sn 0 r Þ p r ð47þ p 0 Now, I use the budget constrant n 0 ¼ n 0 þðn 0 r n0 r Þp r=p 0 rearranged to n 0 ¼ n 0 ðn0 r n0 r Þp r=p 0 and W F ¼ n 0 r p r and substtute both nto the pror equaton, whch leads to, n 0 ¼ n 0 þ s n0 r p r x W F 1 ð48þ p 0 ¼ n 0 ð1 sþþs WF x W F 1 : ð49þ p 0 Notce that ths s derent rom Eq. (33) when p 0 s not 1,.e., when the rsk-ree rate s not zero. Thus, portolo rules consstent wth equal consumptons o agents cannot be obtaned when prces are equal n both economes. Even though ths path s closed, there are some cases when prce equalty can be obtaned. However, consumptons are not equal anymore. Notce that or a zero

18 132 Busness Research (2018) 11: rsk-ree rate the equal consumpton approach stll leads to prce equalty the same way as n the model wth ntal consumpton Multvarate normal payos and exponental utlty I the rsk-ree rate s not zero, condton (18) has to hold to make prces n the notax and the tax economy equal. Kruschwtz and Löler (2009) dscover that or the CAPM wth constant absolute rsk averson (CARA), or every no-tax economy there s a tax economy wth equal prces. They use arguments rom a mean-varance utlty approach. I use exponental utlty, whch s a CARA utlty, and normal consumpton, whch lead to the CAPM (Cochrane 2005, pp ), and SDF arguments to derve the result that CARA utlty together wth multvarate normal payos works to obtan or every no-tax economy a tax economy wth the same prces. I wll keep the rsk-ree asset n zero net supply. Wth multvarate normal payos, I have to relax the assumpton o a nte and dscrete payo space. I use exponental utlty o the orm: u ðc Þ¼ expð a c Þ ; ð50þ a n whch a [ 0 s agent s coecent o absolute rsk averson. Consumpton s a lnear combnaton o multvarate normal payos so that consumpton s normal as well. Thereore, I rewrte the expected value n the maxmzaton condton as: E½u ðc 1 ÞŠ ¼ expð a E½c 1 Šþ0:5a2 Varðc 1 ÞÞ ; ð51þ a wth the budget constrants as n Eqs. (41) and (42). I maxmze wth respect to asset weghts to obtan the rst-order condtons. For rsky assets I obtan: ð1 sþe½x r Šþs r X 0 sðx 0 0 Þ r 0 a ð1 sþxðð1 sþn r þ x sn r Þ¼0; n whch X s the covarance matrx o the payos o rsky assets. Usng R s X 0 sðx 0 0 Þ 0 ð52þ and cancellng the 1 s terms leads to, E½X r Š r R a X ð1 sþn r þ x sn r ¼ 0; ð53þ whch can be rearranged or portolo weghts, n r ¼ 1 1 s 1 a X 1 ðe½x r Š r R Þ x sn r ¼ : ð54þ The equaton shows that ndvdual portolo weghts depend on the tax rate s, the coecent o absolute rsk averson a, and the share n transer payments x. Rearrangng and summng Eq. (53) over all agents leads to,

19 Busness Research (2018) 11: X I E½X r Š 1 r R ¼ Xn r : ð55þ a ¼1 Ths s the same as Equaton (26) n Kruschwtz and Löler (2009) when P I ¼1 2 a ¼ P I U E½cŠ ¼1 U VarðcÞ holds, n whch U E½cŠ s the dervatve o a mean-varance utlty uncton wth respect to the expected value o consumpton and U VarðcÞ s the rst dervatve o a mean-varance utlty uncton wth respect to the varance o consumpton. 10 Notce that Eq. (51) s a mean-varance utlty uncton. The dervatves wth respect to the expected value and the varance o consumpton are U E½cŠ ¼ expð a E½c 1 Šþ0:5a2 Varðc 1 ÞÞ and U VarðcÞ ¼ expð a E½c 1 Šþ 0:5a 2 Varðc 1 ÞÞ0:5a U. It ollows that E½cŠ U VarðcÞ ¼ 2 a. Summng ths expresson over agents shows that P I ¼1 a 2 ¼ P I U E½cŠ ¼1 U VarðcÞ holds. It turns out that the rato U E½cŠ U VarðcÞ only depends on the coecent o absolute rsk averson a. Notce that Kruschwtz and Löler (2009) state that ths rato depends on the ndvdual agents varances o consumpton and that ths s also stated n Meyer (1987) and Lajer-Chaherl and Nelsen (1993). However n those two sources the ratos presented are a bt derent n that the denomnator uses the dervatve o the mean-varance utlty uncton U wth respect to standard devaton,.e., E½cŠ U StdðcÞ, n whch StdðÞ stands or standard devaton. For the case at hand ths dervatve yelds U StdðcÞ ¼ expð a E½c 1 Šþ0:5a2 Varðc 1 ÞÞStdðc 1 Þa. It ollows that U E½cŠ U StdðcÞ ¼ 1 Stdðc 1 Þa, whch actually does depend on the standard devaton o consumpton. It turns out that or exponental utlty wth multvarate normal payos, captal gans taxes under the tax system descrbed heren do not nluence asset prces at all. As Eq. (55) shows, all o the tax terms and dependences on the tax rate dsappear n the aggregate prcng equaton. That leads to the ollowng proposton: Proposton 5 In the tax-economy set up above, n whch agents have exponental utlty and n whch consumpton only takes place at t ¼ 1, the product r R,.e., the rato, does not depend on the tax rate. Furthermore, the correspondng r =ps 0 no-tax economy wll have the same product as the tax-economy r R ¼ p rr,.e., the same rato r =ps 0 ¼ p r=p 0. Proo I rearrange Eq. (55) to r R ¼ E½X rš Xn r P I : ¼1 1 a ð56þ The rhs o ths equaton s exactly the same or the no-tax economy. The same rhs or both economes must lead to the same lhs. h 10 There s a mnor typo n Equatons (25) and (26) n Kruschwtz and Löler (2009). In Equaton (25) the mathematcal sgn n ront o the varance term should be postve as n Equatons (13) and (16). Equaton (26) has to be adjusted accordngly.

20 134 Busness Research (2018) 11: Notce that the relaton r R ¼ p rr just ollows rom the model. However, prce equalty s only there the R ¼ R. Otherwse, prces would only be proportonal but not equal. Snce consumpton takes place only at t ¼ 1 the rsk-ree rate does not say somethng about the trade-o o consumpton today versus consumpton tomorrow. It s exogenous to the economy and wll be chosen so that prces o rsk-ree assets are equal. Corollary 8 Asset prces are equal n the tax and the no-tax economy set up above, wth exponental utlty, multvarate normal payos and wth consumpton only at t ¼ 1, when R ¼ R,.e., when 0 ¼ p 0. Proposton 6 Gven equlbrum prces, rsky portolos o all agents or the notax and the tax economy are related through, n r ¼ n r ð1 sþþx sn r ; ð57þ and rsk-ree weghts are related through, n 0 ¼ n 0 ð1 sþþsðwf x W F Þ 1 p 0 : ð58þ Proo s, Equaton (54) or a zero tax rate shows that the equaton or the no-tax case n r ¼ 1 X 1 ðe½x r Š p a r R Þ: ð59þ Regardng the tax case, Eq. (54) can be rearranged to, n r ð1 sþþx sn r ¼ 1 X 1 E½X r Š r a R : ð60þ From Proposton 5 we know that r R ¼ p rr, and t ollows that the rhs o Eqs. (59) and (60) are equal. Thus, the lhs o the two equatons are equal as well. For the weght on the rsk-ree asset, I use the budget constrant o Eq. (43) but or the no-tax economy, and I substtute n Eq. (32). Ths leads to, n 0 ¼ n 0 þ n 0 r n0 r ð1 sþ x sn 0 r p r p 0 ð61þ I use the relatons n 0 þ n 0 r p r=p 0 ¼ W F =p 0 and W F ¼ n 0 r p r and substtute both nto the pror equaton, whch leads to: n 0 ¼ W F 1 n 0 r p ð1 sþ p r x sw F 1 ð62þ 0 p 0 p 0 Now, I add a constructve zero n the orm o n 0 ð1 sþp 0=p 0 n 0ð1 sþ and rearrange to obtan the weght on the rsk-ree asset rom the proposton, n 0 ¼ n 0 ð1 sþþsðwf x W F Þ 1 p 0 : ð63þ h

21 Busness Research (2018) 11: The portolo rule or rsky assets s the same as or the case wth the zero rskree rate and equal consumpton and t s the same as the one presented n Kruschwtz and Löler (2009) or the constant absolute rsk averson case. Derent rom Kruschwtz and Löler (2009) I nd that one only needs equal prces o the rsk-ree assets and prce equalty o rsky assets ollow. The weght on the rsk-ree assets s now derent rom the one presented beore (Eq. 33) n that the prce o the rsk-ree asset appears n the equaton. That means ndvdual consumptons o agents are not equal n the no-tax and the tax economy. The budget constrants are used n constructng portolo rules and the resultng portolo rules sum over agents to one or rsky assets and to zero or the rsk-ree assets. It ollows that budget constrants are met and markets clear. R s R From Eq. (26) we know that m ¼ m s a sucent condton to obtan prce equalty or the tax and the no-tax economy. However, the converse does not have to be true. There may be other relatons o SDFs that also lead to prce equalty. However, under the speccatons made n ths secton prce equalty also leads to m R s R ¼ m. Proposton 7 For the type o economy set up heren, gven equalty o prces o a tax and a no-tax economy, the condton n Eq. (26) holds. Proo To use the SDF language I use the budget constrant (43) n the consumpton part o Eq. (40) and take dervatves wth respect to n r. I obtan: 0 ¼ E½u 0 ðc 1 ÞðXs r Rs r ÞŠ. I rearrange the expresson to, u 0 r ¼ E ðc 1 Þ R s E½u 0 ðc 1 ÞŠ Xs r ; ð64þ n whch m ¼ m ¼ u0 ðc 1Þ R E½u 0 ðc 1ÞŠ u0 ðc 1 Þ R s E½u 0 ðc ÞŠ s the stochastc dscount actor. Wthout taxes the SDF s 1. I start wth the SDFs o the tax economy and rewrte them to obtan: m ¼ u0 ðc 1 Þ R s E½u 0 ðc 1 ÞŠ ¼ expð a c 1 Þ R s E½expð a c 1 ÞŠ expð a c 1 ¼ Þ R s expð a E½c 1 Šþ0:5a2 Varðc 1 ÞÞ : ð65þ The second equalty uses the rst dervatve o the utlty uncton (50) wth respect to consumpton. The thrd equalty uses the act that consumpton s normally dstrbuted. Consumpton rom Eq. (42) conssts o a rsky part c 1r and a rsk-ree part c 1 : c 1 ¼ c 1r þ c 1, n whch c 1r ¼ðn0 r ð1 sþþx sn 0 r ÞX r and c 1 ¼ n 0 ðx 0 sðx 0 0 ÞÞ þ sn0 r ps r x sn 0 r ps r. From Eq. (32) we know that the rsky part o consumpton n the tax and the no-tax economy are equal or any agent: c 1r ¼ c 1r. That also means that Varðc 1 Þ¼Varðc 1Þ. Usng that I rewrte the SDF to

22 136 Busness Research (2018) 11: m expð a c 1r Þ expð a c 1 ¼ Þ R s expð a E½c 1r ŠÞ expð a E½c 1 ŠÞ expð0:5a2 Varðc 1ÞÞ expð a c 1r Þ ¼ R s expð a E½c 1r ŠÞ expð0:5a 2 Varðc 1ÞÞ : In the second equalty the expð a E½c 1 ŠÞ terms cancel out. That leads to, R s R m expð a c 1r Þ ¼ R expð a E½c 1r ŠÞ expð0:5a 2 Varðc 1ÞÞ : ð66þ ð67þ The SDF o the no-tax economy can be wrtten as: m ¼ u0 ðc 1Þ R E½u 0 ðc 1ÞŠ ¼ expð a c 1 Þ R E½expð a c 1 ÞŠ expð a c 1 Þ ¼ R expð a E½c 1 Šþ0:5a 2 Varðc 1ÞÞ expð a c 1r Þ expð a c 1 Þ ¼ R expð a E½c 1r ŠÞ expð a E½c 1 ŠÞ expð0:5a 2 Varðc 1ÞÞ expð a c 1r Þ ¼ R expð a E½c 1r ŠÞ expð0:5a 2 Varðc 1ÞÞ : Wth prce equalty R ¼ R so that Eq. (68) s equal to Eq. (67). ð68þ Some remarks on wealth are approprate. It turns out that aggregate consumptons are the same n both economes because aggregate payos are the same. However, ndvdual consumptons der as was noted beore. Consumpton s also valued derently so that wealth ders between the economes. For example an agent would value aggregate wealth as ollows: W F ¼ Eðm c 1Þ¼Eðm c 1 Þ R R s ¼ W F R R s : ð69þ Equaton (69) shows that, wth a postve rsk-ree rate, wealth n the tax economy s greater than n the no-tax economy. Indvdually, equal ntal portolo holdngs and prce equalty mply that the values o the pre-trade portolos are the same n the tax-economy and the no-tax economy: n 0 ps ¼ n 0 p: h ð70þ The ncreased aggregate wealth n the tax economy s due to transer payments. I prce the sum o the ater-tax portolo payo n 0 Xs and transer payments Q ¼ x sn 0 r ðx r p r Þ, whch s an agent s total wealth,.e., the tradeable and the nontradeable part o wealth:

23 Busness Research (2018) 11: W F ¼ n 0 ps þ x sn 0 r r R R s ps r R s ¼ W Fs þ W Fs x sr R s ð71þ ¼ W Fs þ W F x sr R s ð72þ Notce that due to prce equalty n 0 r ps r ¼ W F ¼ W Fs. To obtan prce equalty agents must value the ater-tax payos n the tax economy equally to the untaxed payos n the no-tax economy. For ths reason the derent values o total wealth must result Margnal utlty lnear n consumpton I treat the case o margnal utlty lnear n consumpton,.e., margnal utlty o the orm u 0 ðc Þ¼a þ b c. That mples a quadratc utlty uncton. Integraton yelds u ðc Þ¼a c þ 0:5b c 2 þ d, n whch d s a constant. The constant d just shts the utlty uncton up or down and has no mpact on margnal utlty. To have rskaverse agents the second dervatve has to be negatve, whch leads to, u 00 ðc Þ¼b \0. 11 I addtonally assume that consumpton s nonnegatve (c 0) and less than or equal to blss pont consumpton at c c b ¼ a =b, whch s the extremum o the utlty uncton. Snce b \0, or a postve blss pont, a must be postve as well. Those condtons also ensure postve margnal utlty (a þ b c [ 0). For an mportant specal case o margnal utlty b ¼ 1, so that a ¼ c b s the blss pont consumpton. Agents have quadratc utlty o the orm: u ðc Þ¼ 0:5ðc c b Þ2 : Ths leads to margnal utlty lnear n consumpton o the orm, ð73þ u 0 ðc Þ¼c b c : ð74þ I use ths speccaton or the tax economy n Eq. (44) and smply to obtan: Summng over agents leads to, E½ðc b c ÞðX r E½ðc b c ÞðX r Ths leads to the ollowng proposton: r =ps 0 r =ps 0 ÞŠ ¼ 0: ð75þ ÞŠ ¼ 0: ð76þ Proposton 8 In the tax-economy set up above, n whch agents have quadratc utlty and n whch consumpton only takes place at t ¼ 1, the product r R,.e., the rato, does not depend on the tax rate. Furthermore, the correspondng r =ps 0 11 Wth rsk-lovng agents t s hard to ensure the exstence o equlbrum because o non-convexty o preerences. Araujo et al. (2014) show examples o equlbra o economes wth rsk-averse and rsklovng agents.