The Debt Tax Shield, Economic Growth and Inequality

Transcription

1 Arbeitskreis Quatitative Steuerlehre Quatitative Research i Taxatio Discussio Papers Marcel Fischer / Bjare Astrup Jese The Debt Tax Shield, Ecoomic Growth ad Iequality arqus Discussio Paper No. 219 May ISSN

2 The Debt Tax Shield, Ecoomic Growth ad Iequality Marcel Fischer Bjare Astrup Jese This versio: May 2017 Prelimiary ad icomplete, please do ot distribute We are grateful for helpful commets ad suggestios from Raphael Flore, Olaf Kor, Joh Neuma, Malte Schumacher, Robert Schwager, Lauret Weill ad semiar participats at the Uiversity of Göttige, the DGF Coferece, Bo 2016, the Kostaz-Strasbourg Workshop o Ivestmets, Portfolio Choice ad Asset Pricig, the World Fiace Coferece, New York 2016 ad the SGF Coferece, Zürich Copehage Busiess School, Departmet of Fiace, Solbjerg Plads 3, DK-2000 Frederiksberg, Demark ad Uiversity of Kostaz, Departmet of Ecoomics, Postbox 147, D Kostaz, Germay, phoe: , marcel.fischer@ui.k. Copehage Busiess School, Departmet of Fiace, Solbjerg Plads 3, DK-2000 Frederiksberg, Demark, Bjare Astrup Jese, phoe: , ba.fi@cbs.dk.

3 Abstract The Debt Tax Shield, Ecoomic Growth ad Iequality We study the implicatios of the corporate debt tax shield i a growth ecoomy that taxes household icome ad firm profits ad redistributes tax reveues i a attempt to harmoize the lifetime cosumptio opportuities of households that differ i their edowmets. Our model predicts that the debt tax shield 1 icreases the risk-free rate, 2 leads to a higher growth rate of the ecoomy, ad 3 icreases the degree of disparity i households lifetime cosumptio opportuities. We further show that the debt tax shield affects the tradeoff betwee the goals of achievig a high growth rate of the ecoomy ad a low degree of iequality ad quatify this tradeoff. Key Words: debt tax shield, macroecoomic growth, redistributive tax system JEL Classificatio Codes: E21, E23, G11, H23, H31, H32

4 1 Itroductio Departig from the pioeerig work of Modigliai ad Miller 1958, a huge literature ivestigates corporate capital structure decisios. Especially the tax-deductibility of corporate iterest expeses the corporate debt tax shield has caught a lot of attetio i both theoretical ad empirical work. This literature demostrates that the debt tax shield heavily affects corporate fiacial structure decisios. However, the macroecoomic implicatios of the debt tax shield, which our work focuses o, have bee largely overlooked so far. We set up a geeral-equilibrium model with a represetative firm, households that differ by their iitial edowmets, ad a govermet that taxes household icome as well as firm profits ad redistributes tax reveues i a attempt to reduce disparities i lifetime cosumptio opportuities amog households. Households ear icome by ivestig ito risky corporate equity, risk-free corporate debt, ad risk-free bods traded amog the households. Our model makes two mai predictios. First, it predicts the debt tax shield to icrease the risk-free rate, reflectig that debt holders wat their share of the tax advatage from the debt tax shield. Secod, the higher risk-free rate decreases the price of future over preset cosumptio, which leads to higher savigs ad thus, i the ed, to a higher growth rate of the ecoomy. O the other had, a partial effect of the debt tax shield is to lower tax reveues, thereby lowerig trasfers paid out to poorer households which, ultimately, leads to a higher degree of iequality i households lifetime cosumptio opportuities. I this maer, a high growth rate of the ecoomy ad a reductio of the disparities i lifetime cosumptio opportuities amog households trade off agaist each other. For example, a reductio i the gross growth rate of cosumptio by 0.7% ca icrease the cosumptio share of a poorer household with a iitial edowmet of 20% of the average iitial edowmet from 25.2% of the average cosumptio level to 29.8%. A edogeous determiatio of corporate capital structure is importat for a better uderstadig of the macroecoomic implicatios of the debt tax shield. I our model, we operate with a represetative firm for which corporate leverage is chose such that the payout to shareholders is o-egative i all states of the world. Although ay sigle firm may be subject to default risk, this is reasoably assumed ot to be the case for the represetative firm, which is assumed to maximize shareholder value by choosig a maximal degree of leverig subject to this costrait. That is, if the tax burde o firm profits paid out to households as iterest is lower tha that paid out as divided, there is a tax advatage to debt fiacig, ad the firm operates with leverage; Otherwise the firm remais ulevered. Whether such a tax advatage exists depeds, 1

5 amog other thigs, o whether the debt tax shield applies. Hece, the decisio of the firm o whether to lever up or ot is affected by whether a debt tax shield exists or ot. Our work cotributes to two importat lies of literature. It complemets the literature dealig with the macroecoomic implicatios of taxes. It is a well-kow fact that it is geerally ot optimal to tax accumulatig productio factors, because this discourages savigs, slows dow factor accumulatio ad thus, ultimately, iovatio Mukherjee, Sigh, ad Zaldokas, 2016 ad ecoomic growth e.g., Diamod, 1975; Eato ad Rose, 1980; Varia, 1980; Judd, 1985; Chamley, 1986; Joes, Mauelli, ad Rossi, Simultaeously, Hackbarth, Miao, ad Morellec 2006 ad Che 2010, amog others, demostrate that macroecoomic coditios affect corporate capital structure decisios. However, the reverse chael, i.e., how the debt tax shield ad its effect o corporate capital structure decisios affect macroecoomic coditios, such as the risk-free rate, the growth rate of the ecoomy, or the allocatio of resources amog households, has received surprisigly little attetio so far. Simultaeously, our work exteds this lie of literature by allowig for o-uiform taxatio of capital icome. Our work further cotributes to a growig literature o the implicatios of the debt tax shield. The debt tax shield has recetly caught reewed iterest i both theoretical ad empirical work. Empirical work estimates that the debt tax shield accouts for about 10% of corporate values e.g., Graham, 2000; Kemsley ad Nissim, 2002; vabisberge, Graham, ad Yag, 2010, depicts the evolutio of corporate leverage ratios over time DeAgelo ad Roll, 2015; Graham, Leary, ad Roberts, 2015 as well as over the busiess cycle Korajczyk ad Levy, 2003; Hallig, Yu, ad Zecher, 2016, ad documets that taxes i geeral, ad the debt tax shield i particular, sigificatly affect corporate capital structure decisios e.g., MacKie-Maso, 1990; Graham, 1996, 1999; Gordo ad Lee, 2001; Hovakimia, Opler, ad Sherida, 2001; Bell ad Jekiso, 2002; Graham ad Lucker, 2006; Becker, Jacob, ad Jacob, 2013; Logstaff ad Strebulaev, 2014; Devereux, Maffii, ad Xig, 2015; Doidge ad Dyck, 2015; Faccio ad Xu, 2015; Heider ad Ljugqvist, 2015; Faulkeder ad Smith, 2016; Ljugqvist, Zhag, ad Zuo, Schepes 2016 argues that this makes tax shields a valuable tool for policy makers. However, all these papers focus o the impact of the debt tax shield for corporate valuatio ad capital structure decisios, but do ot ivestigate the broader macroecoomic implicatios of the debt tax shield, which are the focus of our work. Theoretical work, icludig Miles ad Ezzell 1980 ad Cooper ad Nyborg 2006, has so far primarily focused o the valuatio of the debt tax shield. A otable exemptio is the 1 Boski 1978, Blachard ad Perotti 2002, Romer ad Romer 2010, ad Cloye 2013 provide supportig empirical evidece. Optimal redistributio is studied, amog others, i Golosov, Troshki, ad Tsyviski

6 work of Fischer ad Jese 2016 that ivestigates how the debt tax shield affects households cosumptio-ivestmet strategies via the govermet s budget costrait. However, their work builds o a edowmet ecoomy model. I their framework, the growth rate of the ecoomy is thus exogeously give, which reders a ivestigatio of the broader macroecoomic implicatios impossible. Our work istead builds o a productio ecoomy ad predicts that the debt tax shield sigificatly affects macroecoomic variables, such as the risk-free rate, iequality, ad the growth rate of the ecoomy. This paper cotributes to a lie of research that Fama 2011 calls oe of the big ope challeges i fiacial ecoomics: uderstadig the implicatios of corporate taxatio. Our work exteds the existig literature i several importat dimesios. It shows that the debt tax shield ot oly affects corporate leverage, but also has broader macroecoomic implicatios. The debt tax shield has a icreasig effect o the risk-free rate, which i tur decreases the relative price of future versus preset cosumptio. As a result, savigs ad ivestmets icrease, resultig i a higher growth rate of the ecoomy. The debt tax shield also leads to lower tax reveues, thereby to lower trasfers from richer to poorer households ad thus, ultimately, to a higher degree of iequality i households lifetime cosumptio opportuities. That is, the debt tax shield cotributes to a higher growth rate of the ecoomy at the expese of a higher degree of iequality. More geerally, the objectives of attaiig a high growth rate of the ecoomy ad a reductio of the degree of disparity i households lifetime cosumptio opportuities trade off agaist each other. Ituitively, the govermet could try to icrease the growth rate of the ecoomy by ivestig ito the productio process itself. However, households ca ad will udo ay effect that such govermet policy may be iteded to have. Hece, there is o scope for such fiscal itervetios i our model. This paper proceeds as follows. Sectio 2 outlies our model. I sectio 3, we preset its aalytical solutio ad discuss our model s predictios. I sectio 4, we illustrate the quatitative implicatios of the debt tax shield for the risk-free rate, ecoomic growth, ad iequality. Sectio 5 cocludes. The appedix provides proofs of our theorems. 2 The debt tax shield i a productio ecoomy 2.1 The ecoomy We cosider a ecoomy populated by households ad a represetative firm, that makes up the productio sector. The firm has a risky oe-period productio techology. That is, it oly geerates a output i the ext period. This output ca either be cosumed or be 3

7 reivested i preparatio for cosumptio i the subsequet periods. The output produced ad available at time t depeds o the evolutio of the ecoomy. It is give by G t It 1, a where G t is the gross growth factor per uit of ivestmet made at time t 1 ad It 1 a deotes the aggregate ivestmet i the productio techology. For simplicity, we assume that the growth rates are idepedet ad idetically distributed copies of a discrete radom variable G with M possible realizatios G m, where G 1 > G 2 > > G M. I the sequel we assume that these M realizatios have equal probabilities 1/M. 2 Our productio techology model is a discrete time versio of the classical Cox-Igersoll- Ross model Cox, Igersoll, ad Ross, 1985, without itertemporal ucertaity about the productio techology. The productio techology ca be thought of as a farmer growig a perishable output, such as cor, with a idetical distributio of the outcome from year to year. The ext year s harvest depeds o how much of this year s harvest is used for replatig ad o the exogeously give realizatio of the growth rate of the output process, that may, e.g., reflect differet weather coditios. 2.2 Corporate leverage To ru the productio techology, the firm issues equity ad corporate debt that the households ca ivest ito. The aggregate ivestmet, I a t, made at time t is fiaced by the aggregate amout of equity ivested, E a t, ad the aggregate amout of corporate debt, δ a t, outstadig from time t to t + 1. A edogeous determiatio of corporate leverage i respose to tax icetives is a ceterpiece i uderstadig the macroecoomic implicatios of the debt tax shield. We follow oe of the stadard assumptios i the literature ad assume that the firm s CEO chooses a costat leverage ratio L. Apart from this costrait o the relatio betwee Et a ad δt a, the supply of aggregate ivestmet opportuities is perfectly elastic. The use of a costat leverage ratio is ofte referred to as the Miles-Ezzell assumptio Miles ad Ezzell, It is used, amog others, i the work of Cooper ad Nyborg 2006, If the total tax burde o firm profits paid out to households as iterest o corporate debt is higher tha that paid out as divided to equity holders, i.e., there is a tax advatage of equity fiacig, the CEO decides to remai ulevered. Otherwise, the CEO chooses the maximum possible degree of leverage that esures a o-egative retur to shareholders i all states. 3 The o-egativity of et returs simultaeously esures a positive tax basis from 2 The assumptio of equal probabilities is solely made to ease otatio. Our results ca be geeralized to allow for uequal probabilities with similar qualitative coclusios, although with a sigificatly blow-up amout of otatio. 3 I the proof of Theorem 1, we show that this is tatamout to maximizig the expected gross growth o ivestmets ito firm equity uder the risk-eutral measure. 4

8 taxig corporate profits. Sice the total tax burde o firm profits paid out to households as iterest depeds o whether the debt tax shield applies, corporate leverage is affected by whether the debt tax shield applies. 2.3 Traded assets Households ca trade three assets. First, households ca trade a locally risk-free oe-period bod payig a pre-tax retur of r t from time t to t + 1. We deote household j s positio i that asset by β t,j. This asset comes i zero et supply. That is, if some households wat to hold a log positio i that asset, the market equilibrium has to brig about a iterest rate that makes other households willig to issue such a asset. Secod, households ca ivest ito the firm s equity that etitles them to the firm s payout i proportio to their share of equity. We deote household j s ivestmet ito the firm s equity from time t to t+1 by E t,j ad its share of equity by α t,j = E t,j. Third, households ca ivest ito oe-period corporate Et a bods, issued by the firm. We deote household j s positio i corporate bods from time t to t + 1 by δ t,j. Sice the firm oly issues bods up to a limit where the et retur o equity is o-egative, corporate bods are default-free, therefore perfect substitutes for the risk-free bod traded amog households, ad thus bear the same yield. The households iitial edowmets are deoted by W 0,j > 0 ad their shares of the total iitial edowmet W0 a = j=1 W 0,j are deoted by α 0,j = W 0,j > 0. W0 a 2.4 The redistributive tax system Throughout the last ceturies, most idustrial atios aroud the world implemeted taxfiaced social isurace ad icome support programs for poorer households to reduce disparities i lifetime cosumptio opportuities. Romer ad Romer 2014 provide a overview over chages i social security beefits i the Uited States. We cosider a govermet that wats to reduce the disparity i lifetime cosumptio opportuities across households that differ by their iitial fiacial edowmets. To attai its goal, the govermet taxes corporate profits at rate τ C, households gais from ivestmets ito firm equity at rate τ E, ad households iterest icome at rate τ B. The govermet implemets a liear redistributive tax system from which each household receives a idetical share of tax reveues. That is, poorer households pay less i taxes tha they receive i trasfer icome. These households are therefore et recipiets of trasfer icome. Liear redistributive tax systems are commoly used i the public fiace literature. Their use rages back to the work of Romer 1975 ad Meltzer ad Richard 1981 ad has later bee used, amog others, i Alesia ad Ageletos 2005, Sialm 2006, Fischer ad Jese 5

9 2015, ad Pástor ad Veroesi 2016, The redistributio mechaism implies that the govermet either builds up wealth or debt. Withi the time horizo of our model, ay govermet debt must be settled through tax paymets by the households. 4 Cosequetly, govermet debt would ever be cosidered et wealth by the households, cf. also the reasoig i Barro s semial work Barro, We provide a more formal argumet that there is o room for a active fiscal policy i our model i sectio The debt tax shield Debt tax shields for corporate iterest expeses exist i may coutries to avoid a doubletaxatio of iterest at both the compay level ad the level of the fial recipiet of the iterest paymet. He ad Matvos 2016 eve suggest subsidizig debt i idustries with socially wasteful competitio to prepoe firm exits. Whether a debt tax shield exists or ot directly affects corporate capital structure decisios, because the debt tax shield reduces the after-tax cost of debt, thus makig debt-fiaced ivestmets more desirable. 5 The debt tax shield also directly affects the payout, P t, to equity holders: P t = It 1 a 1 + g t 1 τ C δt 1 a R t 1 1 at time t, i which g t = G t 1 is the et growth rate of ivestmets ito the firm s productio techology, R t 1 = 1+r t 1 1 τ C is the firm s gross after-tax risk-free rate after accoutig for whether the debt tax shield exists or ot, ad τ C with the debt tax shield τ C = 0 without the debt tax shield is the tax rate applicable to the firm s iterest paymets. 6 With the costat leverage ratio, L = δ a t 1/E a t 1, the payout to shareholders from Equatio 1 ca be rewritte as P t = E a t L 1 + g t 1 τ C E a t 1L R t We disregards the possibility that the govermet ca embark o a Pozi scheme ad ca igore its log-ru budget costrait. 5 We assume throughout that the represetative firm operates with corporate debt whe the debt tax shield applies. I the proof of Theorem 1, we show that this is the case, if the after tax retur o equity icome is lower tha the after tax retur o iterest icome. Without corporate debt, the debt tax shield obviously has o effect. 6 We do ot explicitly regard the case, where iterest expeses are deductible, but the tax compesatio for deductios is lower tha the tax paid o corporate profits 0 < τ C < τ C, throughout. Our model ca be readily applied to these cases. 6 2

10 Whe the debt tax shield applies, the firm faces lower debt servicig costs, implyig a higher amout remaiig for its shareholders. 2.6 The household optimizatio problem Each household maximizes its preset discouted utility from cosumptio over a N-period ivestmet horizo subject to its itertemporal budget costrait. Households have a commo utility discout factor ρ ad a time-additive costat relative risk aversio CRRA utility fuctio with risk aversio parameter γ 0. That is, the utility from a cosumptio of C is give by: C 1 γ if γ 1 1 γ U C = l C if γ = 1. The evolutio of household j s wealth after accoutig for taxes cosists of three compoets. First, the household receives the payout from its equity ivestmets. After accoutig for taxatio o the household level, this leaves the household with a icome of α t 1,j Pt E a t 1 1 τe + E a t Secod, the household receives icome from its holdigs of the risk-free asset ad corporate debt of β t 1,j + δ t 1,j R t 1, 6 where R t 1 = 1 + r t 1 1 τ B 7 is the gross risk-free rate from time t 1 to t after accoutig for taxatio o the household level. Third, the household receives trasfer icome, the level of which depeds o the govermet s tax reveues that i tur cosist of three compoets. First, the govermet geerates a tax reveue of τ E Pt Et 1 a by taxig gais from equity ivestmets. Secod, the taxatio of iterest o the household level provides a tax reveue of τ B r t 1 δt 1. a Fially, the govermet taxes the firm profit, Ω t : I t 1g a t r t 1 δt 1 a Ω t = I a t 1g t at the corporate tax rate τ C. Total tax reveues are thus give by with the debt tax shield 8 without the debt tax shield τ E Pt E a t 1 + τb r t 1 δ a t 1 + τ C Ω t, 9 7

11 of which each household receives a equal share. From Equatios 8 ad 9, tax reveues are lower with the debt tax shield tha without. The evolutio of household j s wealth is give by XYZ Substitute for +τ C Ω t W t,j =α t 1,j Pt Et 1 a 1 τe + Et 1 a + βt 1,j + δ t 1,j R t τe Pt E a t 1 + τb τ C r t 1 δt 1 a + τ C It 1g a t. We deote the effective rate of double taxatio that the equity retur is subject to by τ: τ τ C + τ E 1 τ C = τ E + τ C 1 τ E 1 τ 1 τ E 1 τ C. 11 Household j s optimizatio problem is the give by: max {C t,j,e t,j,δ t,j,β t,j } t=n t=0 U C 0,j + N ρ t E 0 [U C t,j ] t=1 s.t. W t,j = C t,j + E t,j + β t,j + δ t,j, t = 0, 1, 2,..., N 12 E N,j = δ N,t = β N,j = Table 1 summarizes the otatio used i this paper. Havig preseted our model, we ext tur to its closed-form solutio ad show how the debt tax shield affects the ecoomy. 3 Implicatios of the debt tax shield I this sectio, we preset the geeral-equilibrium solutio to the model itroduced i sectio 2 i closed form. To esure a o-egative tax base, we impose a upper boud o the degree of corporate leverig: g M 1 τ B L ḡ g M 1 τ C 1 τ E, 14 where g M = G M 1 is the lowest possible et growth rate of the productio techology, ad ḡ is the expected value of the et growth rate g uder the risk-eutral measure. To esure a positive upper boud o the level of corporate leverage, we assume g M > 0 ad provide a formal derivatio i the proof of Theorem 1 that this costrait ot oly esures a positive upper boud o corporate leverage, but simultaeously also guaratees a o-egative tax base. 8

12 Table 1 Defiitio of variables Variable ρ γ α 0,j α t,j β t,j Descriptio The households commo utility discout factor The households commo degree of relative risk aversio Household j s iitial edowmet Household j s share of equity ivestmets i the productio process from time t to time t + 1 Number of risk-free assets, issued by households, that is held by household j from time t to t + 1 δ t,j Number corporate bods held by household j from time t to t + 1 E t,j Household j s equity ivestmet from time t to time t + 1 δt a Number of corporate bods outstadig from time t to t + 1 Et a Aggregate equity ivestmet from time t to time t + 1 It a Total ivestmet i productio process from time t to t + 1, It a = Et a + δt a L Firm s costat leverage ratio: L = δt a /Et a C t,j Household j s cosumptio at time t Ct a Aggregate cosumptio at time t τ E Tax rate applicable to household icome from equity τ B Tax rate applicable to household icome from bods τ C Corporate tax rate τ C Corporate tax rate, applicable to a firm s iterest paymets τ Total tax rate applicable to a household s equity icome: τ = τ C + τ E 1 τ C τ Tax rate measurig the loss i tax reveues from corporate ad equity taxatio per uit of equity replaced with debt: = τ C + τ E 1 τ C ξ, ψ The relative tax disadvatage of usig equity:=1 τ E 1 τ C /1 τ B R t Gross risk-free rate before taxes from time t to t + 1 r t Net risk-free rate before taxes from time t to t + 1: r t = R t 1 R t Gross risk-free rate after taxes o household level from time t to t + 1 R t Gross risk-free rate after taxes o corporate level from time t to t + 1 Output at time t O t Ω t P t G t Taxable corporate icome at time t Payout from the firm to equity holders at time t Gross growth factor of output O from time t 1 to t, G t = O t /O t 1 Versio of the idepedet stochastic gross growth factors G t G {G j } j=m j=1 Outcomes of G: G 1, G 2,..., G M g t Net growth factor of output O from time t 1 to t, g t = G t 1 W t,j N Household j s wealth level at time t, before cosumptio ad ivestmet Number of households i the ecoomy Legth of ivestmet horizo i periods 9

13 3.1 Macroecoomic effects We begi the presetatio of our results by turig to the implicatios of the debt tax shield for the risk-free rate ad growth of the ecoomy i Theorem 1: Theorem 1. For the risk-free rate ad the rate of ecoomic growth it holds that: 1. The risk-free rate r is costat ad give by r = ḡ 1 τ C 1 τ 1+L C = ḡξ 1 + L ψ, 15 1+L 1+L 1 1 τ B 1+L 1 τ E + L where ξ 1 τ E1 τ C 1 τ B = 1 τ 1 τ B, ψ 1 τ E1 τ C 1 τ B = 1 τ 1 τ B. 16 ξ is a measure of the tax burde o equity relative to debt fiacig whe the debt tax shield applies, 7 i which case ξ = ψ. If the tax shield does ot apply, ψ 1 τ E 1 τ B ad ψ is the similarly the measure of the tax burde o equity relative to debt fiacig. If there is a tax advatage of usig equity, the the firm remais ulevered L = 0, ξ > 1, ad the risk-free rate becomes: r = ḡξ. 17 If there exists a tax advatage to debt, the iterest rate is icreasig i the degree of leverage L. If the firm optimizes its leverage ratio by settig L equal to the right had side of Equatio 14 ad the iterest rate becomes: ḡξ + g M 1 ξ with debt tax shield r = 18 ḡξ + g M 1 ξ τ C without debt tax shield. 2. Aggregate cosumptio, C a t, ad total ivestmets, I a t ito the real ivestmet opportuity are give by C a t = 1 F t W a t, I a t = F t W a t, 19 where F t is the fractio of total output, Wt a = It 1G a t, that is ivested ito the real ivestmet opportuity as either equity or debt. F t is state idepedet, decreasig 7 Cf. Miller

14 over time, ad ca be expressed i explicit form as: 1 H N t for H 1 1 H F t = N t+1 N t for H = 1, N t+1 20 where ρ H = M M m=1 G γ m 1 γ R 1 γ. 21 F t is higher whe the debt tax shield applies. For N it holds that: lim F t = N 1 for H for H > 1. H Whe the debt tax shield applies, the share of wealth ivested, F t, as well as the the utility from aggregate cosumptio is higher. 3. The growth rate of cosumptio is the same for all households ad follows the i.i.d. process with distributio C a t+1 C a t = C t+1,j = 1 G. 23 C t,j H 4. I explicit form, the total tax reveue, T T R t = τ E Pt E a t 1 + τb τ C r t 1 δ a t 1 + τ C I a t 1g t, ca be writte as τg t g M 1 τ B 1 ξ T T R t = It 1 a τg t g M 1 τ B 1 ξ τ C τg t where I a t 1 ca also be expressed as I a t 1 = W a 0 t 1 G i i=1 t 1 F i = W0 a Proof The proof of Theorem 1 is provided i Appedix A. i=0 leverage ad tax shield leverage ad o tax shield o leverage, t 1 G i i= HN+1 t HN+1 Theorem 1 reveals how tax rates ad the debt tax shield affect macroecoomic variables, such as the risk-free rate, aggregate ivestmets ad ecoomic growth, as well as tax reveues. Eve though the debt tax shield oly directly affects the corporate tax basis of a levered firm, item 1 relevals that i geeral equilibrium, it also affects the risk-free rate. From 11

15 Equatios 16 ad 17, it holds that r 1 τ B = ḡ 1 τ whe the firm does ot lever up. I equilibrium, the retur o the risk-free asset after taxes the correspods to the expected after-tax retur o equity uder the risk-eutral measure. Whe there is a tax advatage to corporate debt, the firm levers up. From Equatio 18, the risk-free rate the cotais a additioal term, which takes a positive value. Corporate leverig icreases the risk-free rate, because the tax advatage to corporate debt icreases the desirability of ivestig ito corporate equity. To raise the desired amout of corporate debt, the firm has to offer a higher iterest rate. The effect of corporate leverig o the iterest rate is stroger whe the debt tax shield applies. I that case, the firm faces a lower corporate tax burde. The lower tax burde icreases the after-tax profit ad makes equity ivestmets eve more desirable. To make households willig to evertheless purchase the amout of corporate bods, the firm wats to issue, it has to offer a higher risk-free rate tha i the absece of the debt tax shield. Item 2 reveals two importat properties about the fractio F t of aggregate wealth ivested. First, F t is state idepedet, reflectig the i.i.d. growth rates of the productio process. Secod, F t is positively related to the iterest rate. The oly edogeous effect o H, ad thereby also o F t, is through the iterest rate. From Equatios 20 ad 21, a icrease i the iterest rate decreases the parameter H, which i tur icreases F t. Ecoomic growth thus icreases i the risk-free rate, reflectig that a higher risk-free rate decreases the price of future relative to preset cosumptio. Third, F t is higher whe the debt tax shield applies. Agai, the chael drivig this result is the impact of the debt tax shield o the risk-free rate. I the presece of the debt tax shield, the aggregate share of wealth cosumed is lower ad the share ivested is higher. The higher share of wealth ivested ultimately leads to more factor accumulatio ad thus, a higher growth rate of the ecoomy. The higher growth rate of the ecoomy leads to higher welfare level from aggregate cosumptio, i.e., to a higher welfare for a represetative ivestor. 8 I additio to the well-documeted effects of the debt tax shield o corporate fiacial structure, our model shows that i equilibrium, the debt tax shield also has importat macroecoomic effects. I particular, the debt tax shield icreases the risk-free rate, the growth rate of the ecoomy, ad aggregate welfare. Our model predicts a positive relatioship betwee the risk-free rate ad ecoomic growth. This result is i direct cotrast to the popular view that ivestmet icreases whe the risk-free rate decreases. I our model, ecoomic growth icreases with the risk-free rate, because a high risk-free rate decreases the 8 The expressio for utility from aggregate cosumptio ad the proof of this statemet is give i Appedix A. 12

16 price of future versus preset cosumptio ad thus icreases household savigs, which i tur has a positive impact o ecoomic growth. From item 4, tax reveues at time t deped o aggregate ivestmets, It 1, a i the previous period, as well as o whether the firm operates with leverage ad o whether the debt tax shield applies. Whe the firm operates without leverage, tax reveues at time t per uit of aggregate ivestmet at time t 1 stem from the taxatio of firm profits o the corporate ad household level. I particular, tax reveues are idepedet of whether the debt tax shield applies, because without corporate debt, there are o corporate iterest expeses that would be subject to the debt tax shield. Whe the firm operates with leverage, it does so because of a tax advatage to corporate debt. The secod term i the upper two cases i Equatio 24 measures the implied loss i tax reveues. It is higher whe the debt tax shield applies. Hece, the share of wealth redistributed is lower whe the debt tax shield applies. The absolute amout of tax reveues ad redistributios, however, may be higher, reflectig that aggregate ivestmets are higher whe the debt tax shield applies. From items 1, 2, ad 4, we ca coclude that the two objectives of attaiig a high rate of growth of the ecoomy ad a reductio i the disparities i lifetime cosumptio opportuities amog households trade off agaist each other. To reduce disparities i lifetime cosumptio opportuities, the govermet has to icrease trasfer paymets to poorer households. That is, it has to icrease tax reveues by icreasig tax rates or remove a existig debt tax shield. From Equatio 15, these policies lead to a reductio i the riskfree rate. The reductio i the risk-free rate i tur reduces F t, the share of wealth ivested, by icreasig the parameter H from Equatio 21. A reductio i disparities i lifetime cosumptio opportuities amog households thus comes at the cost of reducig ecoomic growth. I sectio 4.2 we ivestigate i more detail how reductios i lifetime cosumptio opportuities trade off agaist macroecoomic growth. I particular, we documet how this tradeoff is quatitatively affected by the existece or absece of the debt tax shield. From item 3, the growth rate of cosumptio is idetical amog households. Households establish a liear risk sharig rule via their tradig of fiacial assets. The attempt to establish such a liear sharig rule has importat implicatios for households cosumptioivestmet strategies ad the effectiveess of fiscal policy that we tur to i sectios 3.2 ad Households cosumptio-ivestmet policies Havig derived closed-form solutios for the risk-free rate ad ecoomic growth, we ext show how the debt tax shield affects idividual households cosumptio ad ivestmet 13

17 strategies. Our key fidigs are summarized i Theorem 2: Theorem 2. For household j s cosumptio ad ivestmet policies, it holds that: 1. The allocatio of macroecoomic risk is i accordace with a liear sharig rule relative to the distributio of wealth after taxes. Household j s positio i the risk-free asset from time t to t + 1 is proportioal to the aggregate ivestmet, I a t, ad give by: β t,j + δ t,j I a t L = α t,j 1 + L + 1 R α t,j 1 L 1 + L r τ B τ τ. 26 For N, the positio i the risk-free asset is give by the expressio i Equatio 26 with α t,j substituted by the the limitig value of the equity positio, α j, from Equatio Household j s cosumptio share, ω j C 0,j /C a 0, is costat over time ad fulfills: ω j 1 = D α 0,j 1, 27 where D = H Y H N+1 1 H N+1 Y N+1 H 1 is a measure of the degree of disparity i lifetime cosumptio opportuities ad 28 Y = R τ + L r τ 1+L B τ 1 τ R. 29 It holds that lim D = H Y N H Household j s equity share, α t,j, is give by α t,j = 1 + ωj 1 Z t 31 1 τ Z t = HN t Y N t H 1 H Y H N t Poorer households equity shares icrease over time ad richer households decrease. It holds that D = 1/Z 1. For N, the equity share is a costat: α j = 1 + α 0,j τ

18 Proof A detailed proof for all items of Theorem 2 is give i Appedix B. Theorem 2 shows that the debt tax shield ot oly affects the risk-free rate ad ecoomic growth, but also the degree of harmoizatio i lifetime cosumptio opportuities as well as idividual households cosumptio ad ivestmet strategies. Theorem 2, item 1 reveals how households choose their exposure to the risk-free asset. From Equatio 26, household j s positio i the risk-free asset cosists of two terms. The L first term, α t,j is zero, whe the firm operates without debt ad is proportioal to the 1+L share of corporate debt of the firm s total capital otherwise. Whe the firm operates with corporate debt, its shareholders have a implicit short positio i the risk-free asset. For a household j that holds a share of α t,j of firm equity, the implicit short positio i the L risk-free asset is α t,j 1+L Ia L t. Hece, household j eeds a positio of α t,j 1+L Ia t i the risk-free asset to udo this implicit short positio. This hedgig demad simultaeously esures that aggregate demad for the risk-free asset meets aggregate supply. 1 I the secod term, αt,j 1 L R r τ 1+L B τ τ, the factor α t,j 1 is egative for poorer households with below-average equity holdigs, α t,j < 1. Whe the firm operates without leverage, the secod term is proportioal to τ, the effective tax rate applicable to returs from ivestmets ito firm equity. From Equatio 24, trasfer icome is the proportioal to the tax reveues from taxig firm profits at rate τ. Give that returs to equity are subject to macroecoomic risk, the trasfer icome of poorer households with below-average equity holdigs is subject to macroecoomic risk; i.e., these households have implicit log positios i firm equity, which they react to by decreasig their ivestmets ito firm equity ad icreasig their ivestmets ito the risk-free asset. Whe the firm operates with corporate leverage, the term L r τ 1+L B τ may become ozero. It measures how the govermet s tax reveues are affected by corporate debt. For every uit of corporate debt that replaces firm equity, the govermet collects a additioal tax reveue of rτ B from the taxatio of the retur o the risk-free asset, but loses a tax reveue of r τ from the taxatio of the replaced firm equity. Whe τ B < τ, tax reveues decrease with corporate leverig. Poorer households with below-average equity exposures react to this implied reductio i their risk-free trasfer icome by icreasig their exposure to the risk-free asset. With tax-eutrality betwee corporate debt ad equity, i.e., whe the Miller 1977 coditios hold ad τ B = τ, tax reveues are idepedet of the level of corporate leverig ad households do ot have to adjust their portfolio positios to chages i their trasfer icome. Because τ τ, there exists a tax advatage to equity for τ B > τ. The firm the operates without corporate leverage, ad L = 0. I sum, the term is oly ozero whe τ B < τ. L r τ 1+L B τ Theorem 2, item 2 reveals that the liear sharig rule from item 1 implies households at- 15

19 tai time- ad state-idepedet costat cosumptio shares. From Equatio 27, household j s deviatio from a equal cosumptio share is proportioal to its iitial deviatio from the average iitial edowmet. The proportioality factor D ca be iterpreted as a iequality measure ad take values betwee 0 ad 1. A value of D = 1 represets o reductio i the disparity i lifetime cosumptio opportuities where each households cosumptio share correspods to its iitial edowmet. It correspods to a world with o taxatio at ay level ad, cosequetly, o redistributio. D = 0 implies the largest possible degree i the reductio of disparity i lifetime cosumptio opportuities where all households, irrespective of their iitial edowmets, attai the same cosumptio share. Because such a perfect harmoizatio of lifetime cosumptio opportuities is oly achievable with a ifiite ivestmet horizo ad tax rates of 100%, a zero value of D is oly a theoretical lower boud. From Theorem 1, we kow that the two goals of achievig a high level of macroecoomic growth ad a reductio i the disparities i lifetime cosumptio opportuities amog households trade off agaist each other. High levels of the growth scalig parameter 1/H ted to occur simultaeously with high levels of the disparity measure D. We quatify this trade-off i more detail i sectio 4.2. Theorem 2, item 3 shows that with a fiite ivestmet horizo, households equity exposures are covergig towards each other over time. Poorer households equity exposures icrease ad richer households decrease, reflectig that poorer households eed to build up savigs to fiace their cosumptio share. 3.3 Fiscal policy Our results i Theorem 1 show that removig a existig debt tax shield decreases aggregate productio ad thus the growth rate of the ecoomy. I this sectio, we show why the govermet caot reestablish the same level of aggregate productio as i the presece of the debt tax shield via a active fiscal policy. Ituitively, the govermet could try to compesate for the decrease i aggregate productio by ivestig i the productio techology itself, thus fiacig this ivestmet by issuig govermet debt. Govermet bods ad privately issued bods are both risk-free assets ad are therefore perfect substitutes carryig the same iterest rate. To avoid legthy otatio, we assume that govermet bods are sigle-period bods. Hece, we do ot eed to itroduce further variables ito the model. Istead, we require that j=1 β t,j = β a t, where β a t is the total amout of govermet bods outstadig from time t to t + 1. If the govermet ivests ito firm equity, household j s evolutio of 16

20 wealth is give by: W t,j = E t 1,j Ea t 1 τ E E t 1,j Ea t 1 1 τ E + Ea t 1 Et 1 a Lr t 1 τ B τ C + 1 βa t L 1 + g t 1 τ C L R t τ C Ea t L g t + β t 1,j + δ t 1,j R t 1 + G R t Compared to the evolutio of wealth without govermet debt from Equatio 10, Equatio 34 cotais the additioal term 1 βa t 1 G R t 1 that, ceteris paribus, leads to a higher effective exposure to the real ivestmet ad a lower exposure to the risk-free asset for every household j. I our model, where markets are complete ad households have ratioal expectatios, households ca ad will udo ay effect fiscal policy might be iteded to have. Irrespective of whether the govermet implemets a active fiscal policy or ot, households strive for a liear risk sharig rule. Recallig that the leverage ratio L from Equatio 14 is give i terms of the iput parameters of the model, ay deviatio from its optimal level ca ad will be corrected by the supply ad demad decisios of the firm ad the households. Whe the govermet purchases firm equity, from Equatio 34, every household j s exposure to firm equity icreases by βa t 1, implyig a icrease i every household s exposure to macroecoomic risk. Households react to this icrease by reducig their equity holdigs by β a t 1 uits each. As a cosequece, the aggregate demad for firm equity is uchaged, ad the corporate leverage ratio is ot altered. The govermet s itervetio simultaeously implies a reductio i every household j s effective exposure to the risk-free asset by βa t 1 uits. To udo this govermet-itervetio-implied reductio, every household icreases its exposure to the risk-free asset by βa t 1 uits by purchasig govermet bods. As a result of the households adjustmets to their tradig strategies, the budget equatio is fulfilled, households do ot alter their cosumptio plas, they reestablish the liear sharig rule, markets clear, corporate leverage does ot chage, ad the level of real ivestmet remais uchaged. Hece, there is o room for a active fiscal policy. If the govermet decides to ivest ito the firm via corporate debt, household j s budget costrait from Equatio 10 is ot affected. For every uit of corporate iterest icome the govermet ears, it has to pay exactly the same amout to the govermet debt holders. The firm reacts to the deviatio from its optimal leverage ratio by decreasig its amout of corporate bouds outstadig to households by β a t 1 uits. I sum, each household reduces its corporate bod holdigs by βa t 1 uits ad icreases its govermet bod holdigs by βa t 1 uits. As a result of the firm s ad the households adjustmets to their tradig strategies, the 17

21 budget equatio is fulfilled, households do ot alter their cosumptio plas, they reestablish the liear sharig rule, markets clear, corporate leverage does ot chage, ad the level of real ivestmet remais uchaged. I other words, there is perfect crowdig out of the govermet itervetio ad o room for a active fiscal policy i our model. This model predictio is i lie with the empirical evidece i Graham, Leary, ad Roberts 2014 ad Demirici, Huag, ad Sialm 2017 that govermet debt crowds out corporate debt. Agai, there is o room for a active fiscal policy i our model. 4 Quatitative effects I this sectio we illustrate the quatitative implicatios of the debt tax shield for the riskfree rate, the growth rate of the ecoomy, ad household cosumptio. We wat to illustrate both immediate ad log-term cosequeces of the debt tax shield. We therefore choose a ivestmet horizo of N = 100 periods ad assume oe period to correspod to oe year. The degree of risk aversio ad the households time preferece parameter are set to γ = 1 ad ρ = 0.98, which is i the rage of values typically cosidered i the literature. 9 The tax rates are set to τ E = 20%, τ B = 39.6%, ad τ C = 35%, the curret top tax rates for U.S. households ad corporatios. For simplicity, we focus o a settig with M = 2 possible realizatios throughout our umerical aalysis. We set the mea of the growth rate of our real ivestmet opportuity to 3.1%, correspodig to the average real post-war GDP growth i the US. The stadard deviatio of the real ivestmet opportuity s growth rate is chose to attai a level of corporate leverig that is i lie with the historical empirical evidece. More specifically, we set the stadard deviatio to 1.8%, implyig a debt-to-capital ratio of 46%, which is i the rage of historical ratios reported by Graham, Leary, ad Roberts Risk-free rate, macroecoomic growth, ad cosumptio From Theorem 1, the debt tax shield icreases the risk-free rate, icreases ecoomic growth, ad alters the itertemporal allocatio of cosumptio. I this sectio we quatify the order of magitude of these effects. We begi the discussio of our results with the riskfree rate that drives both ecoomic growth Equatios 20 ad 21 ad the stregth of 9 We also explored the robustess of our results to other choices of γ ad ρ. These chages oly affect our results quatitatively, but ot qualitatively, ad are therefore ot preseted here. They are, however, available from the authors upo request. 10 We also explored other choices of the distributio of the growth rate of the real ivestmet opportuity. Give that these chages oly affect our results quatitatively, but ot qualitatively, they are ot reported here, but available from the authors upo request. 18

22 Relative cosumptio Figure 1 Impact of debt tax shield o cosumptio Time This figure depicts the evolutio of cosumptio over time i a settig with the debt tax shield relative to a settig without the debt tax shield. the harmoizatio of lifetime cosumptio opportuities Equatio 27. I our base case parameter settig, the risk-free rate is 2.8% whe the debt tax shield applies ad 2.6% whe it does ot apply. That is, the debt tax shield icreases the risk-free rate by seve percet or 20 basis poits. From Theorem 1, we kow that the higher risk-free rate icreases savigs, reflectig the decrease i the price of future relative to preset cosumptio. The higher savigs rate alters the itertemporal allocatio of cosumptio ad icreases the growth rate of the ecoomy. We quatify these effects i Figure 1. The figure depicts the evolutio of cosumptio i a settig with the debt tax shield relative to a settig without over time. I our base case parameter settig aggregate cosumptio at time t = 0 is 2.7% lower whe the debt tax shield applies Figure 1. That is, the debt tax shield sigificatly reduces immediate cosumptio. I the log ru, however, the higher savigs rate causes a wealth effect that results i the cosumptio level beig higher from time t = 20 owards. At time t = 100, cosumptio i the settig with the debt tax shield is 8.3% higher tha i the settig without. Havig show that whether the debt tax shield applies or ot affects the risk-free rate ad alters the itertemporal allocatio of cosumptio, we ext tur to a demostratio of how the level of the tax burde quatitatively affects our results. For that purpose, we vary the corporate tax rate betwee τ C = 0% ad τ C = 99%. 11 We first tur to showig how the 11 We also explored varyig τ E ad τ B, which resulted i similar results. The results maily chael them- 19

23 Risk-free rate i % Figure 2 Impact of corporate tax rate o risk-free rate TS TS Corporate tax rate i % This figure depicts the evolutio the risk-free rate i a settig with the debt tax shield solid lie, TS ad without the debt tax shield dashed lie, TS. corporate tax rate affects the risk-free rate. Figure 2 depicts the evolutio of the risk-free rate i a settig with the debt tax shield solid lie ad without dashed lie. Cofirmig our results from Theorem 1, it shows that the risk-free rate decreases i the level of the corporate tax rate. For corporate tax rates below 24.5% the effect is of the same order of magitude irrespective of whether the debt tax shield applies or ot. This is so, because for levels of the corporate tax rate below 24.5%, the firm operates without corporate leverage, irrespective of whether the debt tax shield applies or ot. Whe the firm operates without corporate leverage, it does ot matter whether the debt tax shield applies or ot, ad results do ot differ betwee the two settigs. For levels of the corporate tax rate exceedig 24.5%, it becomes optimal to operate with corporate debt whe the tax shield applies, ad it remais optimal to be ulevered without the debt tax shield, which results i the liear relatioship betwee the corporate tax rate ad the risk-free rate whe the debt tax shield does ot apply. With the debt tax shield, there is a kik at a corporate tax rate of 24.5%. The debt tax shield reduces the after-tax cost of corporate debt, which makes ivestmets ito equity more desirable. The order of magitude icreases with the level of the corporate tax rate. To evertheless fid ivestors that are willig to hold corporate debt, the firm has to offer a higher risk-free rate whe the debt tax shield applies. For example, whe corporate taxes do ot apply, i.e., for τ C = 0%, the risk-free rate is 4.1%. It decreases to 2.5% with the debt tax shield ad 2.1% without selves through the variatio of the parameters ξ ad ψ. 20

24 Relative cosumptio Figure 3 Impact of corporate tax rate o cosumptio 1.8 t=0 t= Corporate tax rate i % This figure depicts the evolutio of cosumptio i a settig with debt tax shield relative to a settig without debt tax shield as a fuctio of the corporate tax rate. The solid lies show results at time t = 0, the dashed lies at time t = 100. the debt tax shield for a corporate tax rate of τ C = 50%. Havig depicted how the corporate tax rate affects the risk-free rate, we ext ask how it affects aggregate cosumptio ad wealth i the ecoomy. Figure 3 depicts cosumptio i a settig with debt tax shield relative to a settig without as a fuctio of the corporate tax rate. The solid lies show results at time t = 0, the dashed lies at time t = 100. Cosistet with our results for the risk-free rate, cosumptio is idetical with ad without the debt tax shield for corporate tax rates below 24.5%. For rates exceedig 24.5%, the amout spet o cosumptio at time t = 0 is lower whe the debt tax shield applies ad the order of magitude of this effect amplifies i the level of the corporate tax rate, reflectig the decreasig price of future relative to preset cosumptio. For example, for a corporate tax rate of τ C = 35%, cosumptio at time t = 0 is 2.7% lower with the debt tax shield. The order of magitude of this effect icreases to 6.1% for a corporate tax rate of τ C = 50%. At time t = 100, however, relative cosumptio levels dramatically icrease i the level of the corporate tax rate, reflectig that the effect of more households savig i the presece of the debt tax shield amplifies whe the level of the corporate tax rate icreases. For example, for a corporate tax rate of τ C = 35%, cosumptio at time t = 100 is 8.3% higher with the debt tax shield. The order of magitude of this effect icreases to 22.0% for a corporate tax rate of τ C = 50%. 21

25 4.2 Tradeoff betwee growth ad iequality We kow from Theorem 1 that the growth rate of aggregate cosumptio ad the disparity i lifetime cosumptio opportuities amog households trade off agaist each other. I this sectio, we quatify this tradeoff. From Equatio 23 the growth rate of cosumptio is proportioal to 1/H. We therefore iterpret 1/H as a scalig factor ad a measure for ecoomic growth. From Equatio 27, the deviatio of each households cosumptio share from a equal cosumptio share is the household s deviatio from the average iitial edowmet times D. From Theorem 2, D ca therefore be iterpreted as a disparity measure. It ca take values betwee 0 ad 1. 1 represets the highest level of disparity, where poorer households cosumptio shares correspod to their iitial edowmets. This situatio occurs without taxatio ad redistributio, i.e., for tax rates of τ B = τ C = τ E = 0%. 0 represets the theoretically lowest possible level of iequality, where all households are edowed with equal cosumptio shares. I reality, a iequality measure of 0 should ot be a reasoable objective from a policy makers perspective, because it comes at the cost of removig households icetives to ivest. 12 Figure 4 depicts how the corporate tax rate affects ecoomic growth ormalized growth amplifier 1/H, left pael ad the disparity measure D, right pael. We ormalize the growth multiplier to 1 for a corporate tax rate of τ C = 0%, thus allowig us to easily measure reductios i the aual gross growth rate, G, of the ecoomy relative to a settig with a corporate tax rate of τ C = 0%. The solid lies show results whe the debt tax shield applies, the dashed lies, whe it does ot. Cosistet with Theorem 1, Figure 4 shows that higher levels of the corporate tax rate decrease both ecoomic growth ad iequality. These effects are amplified i the absece of a debt tax shield. That is, the debt tax shield sigificatly affects the tradeoff betwee ecoomic growth ad iequality. For example, i our base case parameter settig with the debt tax shield ad a corporate tax rate of τ C = 35%, the aual gross growth rate of the ecoomy is 0.7% lower tha with a corporate tax rate of τ C = 0%, ad the iequality measure is reduced from to 0.675, implyig that the cosumptio share of a poorer household with a iitial edowmet of 20% of the average iitial edowmet icreases from 25.2% to 29.8% whe the debt tax shield applies. Whe the debt tax shield does ot apply, the aual gross growth rate of cosumptio is 0.8% lower, ad the iequality measure is 0.657, implyig the cosumptio share of the poorer household icreases to 30.3%. For higher tax rates, these effects are further amplified. For example, for a corporate tax 12 I Fischer ad Jese 2016 we have ratioalized the liear taxatio ad redistributio scheme as the solutio to a optimizatio problem with a govermet objective fuctio for reducig disparity i cosumptio opportuities combied with frictio cost of collectig taxes. 22