Income Inequality and Stock Market Returns

Transcription

1 Income Inequaliy and Sock Marke Reurns Agnieszka Markiewicz Erasmus Universiy Roerdam Tinbergen Insiue Rafal Raciborski European Commission Visula School of Economics July, 2018 Absrac In his paper, we sudy he relaionship beween income inequaliy and sock marke reurns. We develop a quaniaive general equilibrium model ha links shifs in boh labour and capial income inequaliy o sock marke variables. An increase of he share of capial owners income from risky capial leads o higher equiy premium and a rise in heir non-risky, labor share of income reduces i. When we calibrae our model o mach he empirical size of shifs in he las five decades, we find ha he negaive impac of he higher labour share of income of capial owners dominaes and brings he equiy premium below he hisorical value by 0.79 percenage poins, in line wih he daa. If boh capial and oal income shares of op decile would coninue growing a he hisorical rae beween 1970 and 2014, he equiy premium would coninue decreasing o 6.11% in 2030, 0.92 percenage poin lower han hisorical equiy premium of 7.03%. If insead only he capial share of income coninues o grow, he equiy premium would be higher han he hisorical average by 0.57 percenage poin. If he labour income dispersion remains consan, he hisorical equiy premium of 7.03% would be reached by 2030 if he capial share of income was growing by 1.4% each year. Keywords: Asse Pricing, Risk Premium Dynamics, Income Inequaliy, Compuaional Macroeconomics. JEL Classificaion: D31, E32, E44, H21, O33. For helpful commens and suggesions we hank Bünyamin Erkan, Kevin Lansing, Andreas Pick, Vincen Rebeyrol and paricipans of he 25h Sociey of Nonlinear Dynamics and Economerics meeing in Paris, Fifh Inernaional Symposium in Compuaional Economics and Finance in Paris, Sociey of Economic Dynamics meeing in Edinburgh, Fifh Inernaional Symposium in Compuaional Economics and Finance in Paris and Naional Bank of Poland workshop in Warsaw. corresponding auhor: Erasmus Universiy Roerdam, P.O. Box 1738, 3000 DR Roerdam, he Neherlands, markiewicz@ese.eur.nl The conen of his sudy does no reflec he offi cial opinion of he European Union. Responsibiliy for he informaion and views expressed in he paper lies enirely wih he auhor.

2 1 Inroducion Equiies are concenraed in he hands of households a he op of he U.S. income disribuion. 1 One migh herefore expec ha changes in heir labor and capial incomes affec he movemens in asse prices. Boh labor and capial shares of income of he U.S. riches households increased during he las five decades, which was accompanied by large shifs in reurns from risky asses. The equiy premium, which was as high as 9% during he firs 25 years afer he war, dropped o below 6%, on average, since he early 1970s. In his paper, we sudy he relaionship beween income inequaliy and sock marke reurns. We develop a general equilibrium model ha links shifs in boh labour and capial income inequaliy o sock marke variables and invesigae in how far hese shifs could explain he fall in equiy reurns. Our model includes wo groups of agens. The op income group (capial owners) owns 100% of he economy s financial wealh a seup ha roughly approximaes he highly-skewed disribuion of U.S. financial wealh. 2 The res of he economy is populaed by workers who consume heir labor income and income from risk-free governmen and corporae bonds. This se-up is similar o Greenwald e al. (2016). The key difference of our model is ha, in addiion o he capial income, capial owners earn labor income, in line wih empirical observaion ha he op wealh holders of oday are also he op earners. Saez and Zucman (2016) show ha in 2012 he op 0.1% of wealh holders earned 31 imes he average labor income and heir pre-ax income share almos ripled beween 1960 and Our model provides a simple inuiion on how shifs in income inequaliy, as measured by changes in income shares, affec asse pricing variables and, in paricular, he equiy premium. An increase in he share of risky capial in income and, hence, in consumpion of capial owners leads o a higher equiy premium. The opposie is rue when he share of non-risky labor income in he oal income of capial owners rises. However, according o he daa, capial owners benefied from increase in boh incomes over he las four decades. As a resul, he oal quaniaive impac of income inequaliy shifs on he sock marke reurn is unclear. 3 We build a quaniaive model ha allows us o assess he impac of he join 1 Chen and Safford (2016) argue ha even fewer han 20 percen of households own sock direcly. 2 The share of oal financial wealh owned by he op 10% of households is around 80% in he sample period (Wolf, 2010). 3 The oal share of income of capial owners or, equivalenly in our model, op decile earners, in he U.S. increased from 32% o 47% beween 1970 and 2014 as shown in Figure 1, op panel. Boh, op decile labor and capial shares increased during his period as shown in Figure 2. 1

3 increases in capial and labor income shares of he op decile of he U.S. income disribuion on he sock marke. The model exends he sandard RBC seing used in he producion-based (CCAPM) lieraure allowing for heerogeneiy of agens, which differ by heir abiliy o hold financial asses and having differen oupu elasiciies of labor. This se-up urns ou suffi cien o broadly mach boh, sock marke and real economy saisics. The source of macroeconomic risk in he model is a sandard echnology shock. The model delivers high mean equiy premium and realisic Sharpe raio via a combinaion of hree facors. Firs, o generae a high price for risk, following Greenwald e al. (2016), we inroduce high and ime-varying coeffi cien of risk-aversion in capial owners uiliy funcion. Second, capial is subjec o adjusmen coss in he spiri of Uhlig (2007) and Jermann and Quadrini (2012). Finally, financial leverage of firms increases he quaniy of risk borne by capial owners hus helping o generae a realisic Sharpe raio. Careful calibraion of hese channels allows for precise maching of he mean and sandard deviaion of equiy premium (and hence he Sharpe raio), which has proven a challenge in models ha use oher mechanisms. Addiionally, inroducing financial leverage pins down he non-zero rae of risk-free savings on he side of consumers, which increases he realism of he model. We calibrae he model o he U.S. pos-war economy when he equiy premium of S&P500 shares reached 7.03%. In order o gauge he quaniaive impac of changes in income inequaliy on he equiy premium, we carry ou hree counerfacual scenarios. Firs, we consider an increase in capial income inequaliy and change he value of capial share of income, from is baseline value of 0.28 o 0.34, is empirical value in This increase leads o he mean equiy premium ha is higher by 0.43 percenage poins han he hisorical average. Second, we raise he value of he labour share of income of op decile from 3.5% o 13%, is 2014 value. We show ha he seady increase of he capial owners labour share beween 1970 and 2014 should have exercised a srong downward pressure on he equiy premium. In fac, he model predics ha if here was no increase in capial income inequaliy, bu only in labour income, he equiy premium would have been lower by 1.64 percenage poins. Finally, considering he wo changes ogeher, we show ha he ne effec on he equiy premium is negaive, in line wih wha has been observed in he S&P500 daa. The reason is ha, capial owners benefied from a faser growh in heir non-risky labour income share relaive o he risky capial income share bringing he equiy 2

4 premium below he hisorical value by 0.79 percenage poins. Nex, we consruc scenarios for he size of he equiy premium in 15 years from now, assuming ha he U.S. income inequaliy coninues increasing. We consider wo differen scenarios for fuure rends in income inequaliy and inroduce hem ino he model o derive he equiy premium response. The firs scenario assumes ha boh capial and oal income shares of he op decile will grow a he same rae as beween 1970 and 2014 unil 2030, namely 0.5% and 0.92%, respecively. In his case, we find ha he equiy premium would coninue decreasing o 6.11% in 2030, 0.92 percenage poin lower han hisorical average. In he second, more likely scenario, we assume ha he labour share of income of capial owners sops growing and only capial share of income coninues o increase. 4 capial share of income coninues o grow a he annual rae of 0.5% and he labour share of income of op decile remains unchanged, capial would represen 37% of oal income in he U.S. in Toal income of op decile would amoun o half of he income in he U.S. economy. In his scenario, he equiy premium would be higher han he hisorical average by 0.57 percenage poin. Finally, we use he model o perform a reverse engineering exercise, in which we compue he income inequaliy required o reach he pos-war average equiy premia of 7.03% and 9.07%, saring from oday s values. We make predicions abou he behavior of one of he shares of income and compue a change in he oher share required o mach hisorical equiy premium. If he labour income dispersion sops increasing, he hisorical equiy premium of 7.03% would be reached by 2030 if capial share of income was growing by 1.4% each year, consisen wih Pikey s (2014) predicions. If he equiy premium was insead, o reach is value from immediaely afer he war of 9.07%, he capial share of income would need o grow a he annual 4.31% unil 2030, an unlikely scenario. The model developed here builds on he lieraure embedding risky asse markes ino RBC models, as proposed by Jermann (1998), Boldrin e al (2001) and Danhine and Donaldson (2002). Guvenen (2009) and Guvenen and Kuruscu (2006) inroduce ino an oherwise sandard RBC framework wo ypes of consumers, which differ by heir elasiciy of ineremporal subsiuion. The laer elemen differeniaes hese 4 This assumpion is moivaed by observaion ha since 2000, he labour income inequaliy has no been increasing. In conras, capial share of income sared rising since These rends are in line wih he lieraure on skill-biased echnological change which is commonly hough o have iniially driven income inequaliy and generaed higher dispersion in savings from labour income and higher capial income inequaliy. See also Saez and Zucman (2016) for empirical suppor of his hypohesis. If 3

5 papers from our approach, in which we assume ha he preferences of our wo ypes of consumers are idenical. Our paper is probably mos closely relaed o Lansing (2015) who, in an RBC seing, sudies he consequences of shor-erm flucuaions in he capial income disribuion. However, similar o he oher papers menioned above, he goal of Lansing (2015) is o explain he high pos-war equiy premium. In conras, in his paper we focus on he longer-erm, srucural shifs in income shares and we link hem o long-erm rends observed in he equiy premium and in paricular is recen decline. Finally, we share he ineres in he longer-erm rends in income shares and heir consequences for equiy markes wih Greenwald e al. (2016). Our modelling sraegy largely benefied from heir se-up, namely, we use heir risk aversion shock specificaion o deliver a sizeable average equiy premium. Unlike in Greenwald e al. (2016), however, in our model capial owners are also high labour income earners. The laer elemen urns ou o be crucial for explaining a par of he observed downward rend in he average equiy premium. By sudying he link beween income shares and asse reurns, we also conribue o he fas-growing lieraure emphasizing he imporance of wealh dispersion and resuling capial income inequaliy in he U.S. and oher developed economies. Kacperczyk e al. (2016) show ha capial income inequaliy is large and growing fas, accouning for a considerable porion of oal income inequaliy in he U.S. In addiion, Saez and Zucman (2016) demonsrae an increased correlaion beween op labor and op capial incomes in he U.S. daa. Our framework is moivaed by hese resen empirical observaions and herefore models capial owners also as high labor income earners. The paper is organized as follows. In Secion 2, we describe a se of sylized facs on changes in income inequaliy and equiy premium. In Secion 3, we describe he model and is main inuiion. Specifically, we explain he mechanisms generaing high equiy premium and Sharpe raio and how hey respond o shifs in income shares. In his secion, we also describe our calibraion sraegy. Secion 4 evaluaes he quaniaive performance of he model in boh macroeconomic and financial dimensions. In Secion 5, we carry ou a se of counerfacual exercises which allow us o assess he quaniaive impac of he recen increase in income inequaliy on he equiy premium. In his secion, we also implemen a number of scenarios for fuure behavior of equiy premium based on he income inequaliy predicions. Secion 6 concludes. 4

6 2 Income Inequaliy and Asse Pricing Variables in he Daa 2.1 Capial and labor shares of income The recen increase of income inequaliy has been accompanied by rising capial share of income. Karabarbounis and Neiman (2014) show ha he labour share of income has significanly declined since he early 1980s in he large majoriy of counries, including he U.S. Figure 1: Top decile income share and capial share of income in he U.S. Top decile income share series comes from he World Inequaliy Daabase a hps://wid.world/. Capial share of income is compued from Bureau of Economic Analysis daabase as 1-compensaion of employees. Figure 1 plos increase in he op decile and capial shares of income in he U.S. There is a clear posiive rend in boh series. Capial share of income increased from 35% o 43% beween 1970 and 2014 and he op decile income share raised from 32% o 47% during he same period. Given ha he capial sock is concenraed in hands of a relaively small group of wealhy households, he observed increase in capial share of income direcly implied an increase in income inequaliy. While oal labor share of income declined in U.S. in he recen decades, he relaive labor share of op incomes increased as well. Figure 2 plos he composiion of he op decile income share in he U.S. beween 5

7 Figure 2: Top decile income share and is composiion beween 1970 and 2011 The solid area shows capial share of income and is composiion: capial gains, rens, ineres income, dividends and enrepreneurial income. Sriped area shows labor share of income which covers wages and salaries, bonuses, exercised sock-opions, and pensions. The figures are compued as shares of oal U.S. income. Income share daa is described in Akinson e al. (2011) and can be found a hps://wid.world/ 6

8 1947 and The sriped area of he figure corresponds o he labor income and demonsraes ha is share increased during he pas hiry years from 25 % o 33 % of oal income in he U.S. economy. The solid area shows how capial gains, rens, ineres raes, dividends, and business income evolved during his period. These componens are combined o form one caegory: capial share of income, which also increased during he las hree decades. I represened 8.9% of op decile income share in 1980 and 13% in Top incomes hus experienced an increase from boh sources: capial and labor and, here is in fac, a srong posiive correlaion beween op labor and capial incomes. The op wealh holders of oday are also he op earners Equiy premium While income inequaliy increased during he las five decades he equiy premium has decreased. In Table 1, we repor he figures for he equiy premium compued form he Shiller s websie: hp:// The firs column denoes he period over which he saisic was compued. In he second column, we show he values for he equiy premium (EP) and in he hird column for Sharpe raios (SR). The firs row shows ha he equiy premium reached 7.03% on average during he pos-war period while Sharpe raio was equal o 0.43 during he same period. Table 1: Equiy premium and Sharpe raio for SP 500 EP SR % % % 0.34 EP sands for equiy premium and SR for Sharpe raio. Boh saisics are compued using he annual daa from Shiller s websie: hp:// Equiy premium is compued as he difference beween he reurn including dividends on S&P 500 and he risk free rae is 6 monhs rolled commercial paper rae. The following wo rows repor he saisics for he period before 1970 and afer The equiy premium before 1970 was a hird higher han afer In case 5 As previously menioned, here is an increasing evidence ha op labor and op capial incomes are srongly correlaed. For insance, Saez and Zucman (2016) show ha in 2012 he op 0.1% of wealh holders earned 31 imes he mean labor income. 7

9 of he Sharpe raio saisics, his paern is even sronger as he value before 1970 is almos wice as high as afer Model As our analyical ool we build on he model by Lansing and Markiewicz (2018). The model exends he sandard RBC seing used in he producion-based Consumpion Capial Asse Pricing Model (CCAPM) lieraure allowing for wo ypes of agens differing by heir abiliy o hold financial asses and who face differen oupu elasiciies of he labor hey provide. This exension o he sandard framework les us race he impac of he shifs in he capial share and labor-income shares on he sock marke price behavior and, in paricular, on he mean equiy premium. There are several addiional elemens of he model worh emphasizing. The high mean equiy premium and realisic Sharpe Raio obained in our seing are due o a combinaion of hree facors: (i) ime-varying coeffi cien of risk aversion, as in Greenwald e al. (2016), which increases he price for risk in he economy; and (ii) capial adjusmen coss and (iii) financial leverage, which boh increase he quaniy of risk borne by he agens invesing in socks. Careful calibraion of hese channels allows for precise maching of he mean and sandard deviaion of equiy premium (and hence he Sharpe Raio), which has proven a challenge in models ha use oher mechanisms o generae high equiy premium. 6 Addiionally, inroducing financial leverage pins down he non-zero rae of risk-free savings on he side of consumers, which increases he realism of he model Workers Workers, of mass 1 η, maximize a discouned sum of uiliy over consumpion, c w : max c w,a f,a c E 0 =0 β w (c w ) 1 γw 1 γ w, (1) 6 This lieraure is oo broad o be discussed here. For an early accoun of he puzzle in a framework wih fully-fledged producion see, e.g., Jermann, I is worh emphasizing ha here is ypically a rade-off beween fiing he asse marke and real-economy saisics and ha i is no possible o fi exacly boh ses of saisics in our model. This observaion is of general naure, however, and holds for many modern producion-based CCAPMs (e.g. Guvenen, 2009). 7 As is well known, in his class of models, due o he Euler equaion ha does no conain quaniies, he prevailing risk-free rae of reurn is consisen wih an arbirary asse porfolio composiion. A zero-risk-free saving rae is, herefore, usually assumed in hese models. 8

10 where β w is an individual worker s discoun facor and γ w is her coeffi cien of risk aversion. The maximizaion is subjec o a budge consrain: c w + a f P f + a c,w P c = W w n w + a f 1 + ac,w 1, (2) wih W w he wage rae received by workers, P f and P c he prices of zero-supply risk-free bonds and (risk-free) corporae bonds, respecively, and a f and a c,w he respecive posiions aken by workers in hese asses. Workers are assumed o incur a ransacion cos for rading socks which prohibis heir paricipaion in sock exchange. 8 Finally, n w = n w is he consan supply of labor hours per worker. Assuming he usual ransversaliy condiion, firs order condiions for he worker s problem are sandard: (c w ) γw = λ w 1 = β w λ w +1 E λ w R f (3) 1 = β w λ w +1 E λ w R c (4) wih λ w he worker s marginal uiliy of consumpion and E represening he mahemaical expecaion operaor condiional on informaion a he end of period. By definiion, he reurns on risk-free asses saisfy: R f = 1 P f R c = 1 P c By consrucion we also have R c R f. Noe ha we implicily assumed ha firms do no defaul on heir deb. 3.2 Capial Owners Capial owners, of mass η, represen he op decile of income disribuion. Similarly o workers, hey maximize a discouned sum of uiliy over consumpion, c c, max c c,a c,c,a s E 0 =0 β c (c c ) 1 γc 1 γ c, (5) 8 In conras, hey are assumed o have full access o risk-free saving vehicules such as bank deposis (a f ) and corporae bonds (a c ). I is possible o assume a c 0 wihou undermining he resuls of he paper. 9

11 where β c is he capial owners discoun facor and γ c is heir coeffi cien of risk aversion. The capial owners coeffi cien of risk aversion is ime-varying, as in Greenwald e al. (2016). In combinaion wih echnology shocks, which are he sandard source of macroeconomic risk in our economy, shocks o he coeffi cien of risk aversion increase he price for risk ha is, he reurn on he risky asse, ne of risk-free reurn, per uni of risk, as radiionally measured by he excess reurn s sandard deviaion. In oher words, hey increase he Sharpe raio. The ime-varying coeffi cien of risk aversion is defined as γ c = γ c 1 + exp (x 1 ) wih γ c being he maximum degree of risk aversion and x an auoregressive process of order 1 wih mean µ x : x µ x = ρ x (x 1 µ x ) + ε x and ε x an iid shock. The maximizaion is subjec o a budge consrain: c c + a s P s + a c,c P c = W c n c + a s 1 (P s + d s ) + a c,c 1, (7) (6) wih W c he wage rae received by capial owners, a s and a c,c he number of socks and corporae bonds held by capial owners, respecively, P s he sock price and d s he (economic) dividend received by capial owners from holding socks. n c = n c is heir consan supply of labor. Assuming he usual ransversaliy condiion, firs order condiions for he capialis s problem are: (c c ) γc = λ c wih λ c he marginal uiliy of consumpion and R s +1 socks: 1 = β c λ c +1 E λ c R+1 s (8) 1 = β c λ c +1 E λ c R c (9) is he nex period reurn on R s +1 = P s +1 + ds +1 P s The form of he above firs order condiions is very similar o he sandard firs order condiions dirived in he CCAPM lieraure. However, since risk-free asses are now held by wo ypes of agens, whose consumpion is allowed o display differen dynamics, Euler equaions of capial owners and workers associaed wih hese asses 10

12 are, in mos general case, inconsisen wih each oher. To deal wih his problem, we apply he definiion of ime-varying discoun facors as in Greenewald e al. (2016): [ β c λ c ] 1 +1 β E λ c [ β w λ w ] 1 +1 β E λ w where 0 < β < 1. These definiions guaranee consisency of boh ses of Euler equaions, while a he same ime deal wih he problem of oo volaile risk-free rae, see Greenewald e al (2016), making he risk-free rae in he economy consan: 9 R c R c 1 β 3.3 Firms Idenical compeiive firms, of mass 1, maximize he presen value of heir fuure profis, D, discouned a he marginal rae of subsiuion, β λc +1 λ c, of he firms 0 owners: max K,N c,n w =0 β λ c +1 E λ c D (K 1, K, N c, N w ) 0 where K is he end of period capial sock and N c and N w indicae demand for he capial owners and workers labor, respecively. Profis are defined as: D (K 1, K, N c, N w ) D = Y W c N c W w N w I + P b µk µk 1 (10) Above, Y is he curren oupu, I is oal invesmen and µ measures he degree of financial leverage of firms. When µ = 0, he new capial of a firm is fully financed hrough reained earnings. I will be assumed ha profis are redisribued via dividends o capial owners: d s = D. 10 Oupu is produced wih Cobb-Douglas echnology: Y = AK θ 1 (exp (z ) (N c ) α (N w ) 1 α) 1 θ (11) 9 Anoher well-esablished soluion o he volailiy puzzle is disenangling he risk aversion and ineremporal elasiciy of subsiuion coeffi ciens, which are he reciprocals of each oher in our model, by using he Epsein-Zin-Weil uiliy funcion (Epsein and Zin, 1989 and 1991; Weil, 1990). While his would solve he risk-free rae volailiy problem, i would no, on is own, assure consisency beween he arbirage condiions. 10 Noe ha, as is common in he lieraure, e.g. Lansing (2015), we use he concep of macroeconomic dividends in place of financial dividends. 11

13 where θ is he capial income share, α (1 θ) is he capial owners share of labor income and z = z 1 + µ z + ε z is a TFP shock wih possibly non-zero growh rae, µ z, and ε z is zero-mean N.I.D. I is assumed ha ransforming invesmen I ino capial K is cosly, which permis he shadow price of insalled capial o diverge from he price of an addiional uni of capial. In specifying he capial adjusmen cos, we follow Uhlig (2007) and Jermann and Quadrini (2012) so ha he capial accumulaion equaion akes he form: wih funcion G ( ) such ha ( ) I K = (1 δ) K 1 + G K 1 (12) K 1 ( ) I G = a 1 K ξ ( I K 1 ) 1 1 ξ + a2 where a 1, a 2 are wo posiive consans. For ξ <, adjusmen coss become sricly posiive. Assuming ransversaliy condiion, firs order condiions are as follows: W w = (1 θ) (1 α) Y N w (13) W c = (1 θ) α Y N c (14) where reurn on capial, R k, is defined as: ( ) G I K 1 R k +1 = 1 G ( I K 1 ) µp b 1 = β c λ c +1 E λ c R+1 k (15) θ Y +1 K µ + ( ) 1 δ + G I+1 K G ( I+1 K ) I +1 K (16) which can also be rewrien as R k +1 = D +1 + ( ( 1 G I+1 ) )µp K b +1 ( G I+1 ) K +1 K ( ( ) 1 G I )µp K b 1 ( ) G I K K 1 (17) The reurn on capial has an inuiive inerpreaion, which will be presened below. 12

14 3.4 Equilibrium marke clearing condiions In he equilibrium, all markes clear: Consumpion goods marke: C w = (1 η) c w, C c = ηc c and C w + C c = C Goods marke: C + I = Y Labor marke: N w = (1 η) n w, N c = ηn c and N w + N c = N Asse markes: a s = η 1, a f B c,c = µk = 0, B c,w = (1 η) a w, B c,c = ηa c and B c,w + In he above, large-case-leers refer o aggregae (per capia) variables. Since, he share of corporae bonds held by capial owners (and workers) is no pinned down by he Euler equaions, we make an addiional assumpion ha B c,w or, alernaively, B c,c 0. Any alernaive assumpion, including B c,w µk, produces resuls similar o hose repored in he paper. B c,c µk 0 and The whole lis of equaions, afer having been saionarized (divided by exp (z 1 )) is shown in Appendix A). 3.5 Asse Pricing Variables We define he equiy premium (excess reurn on socks): R e +1 = R s +1 R f The uncondiional Sharpe raio is hen defined as: SR = E [Re ] σ [R e ] where E and σ sand for uncondiional expecaion and sandard deviaion, respecively. For he sake of compleeness, we also define price-dividend raio pd, he sandard sock marke diagnosic: pd = P s d s Finally, i is worh inerpreing he Euler equaion associaed wih capial. Comparing arbirage condiions (8) and (15), we see ha hey imply ( ( ) ) 1 G I P s K 1 µp b = ( ) K G I K 1 13

15 In a model wihou fricions, sock price, P s, would be simply equal o capial sock per capia, K. However, in our model, he price of insalled and uninsalled capial differ. The adjusmen erm G ( I K 1 ) ranslaes he value of insalled capial ino he uninsalled one. Moreover, sock owners own only a fracion 1 µ of capial sock, as hey have o borrow ( from workers ( he ) funds ) o cover he remaining par of capial. This is refleced in he 1 G I K 1 µp b adjusmen erm (again, wih a suiable adjusmen due o he price difference beween insalled and unisalled capial). 3.6 Inuiion of he model Mechanisms generaing high equiy premium and Sharpe raio Our model is able o easily generae realisic equiy premium and Sharpe raio, and produce a he same ime reasonable real-economy saisics. This is remarkable, given he well-known fac ha producion-based CCAPMs sruggle o achieve boh objecives a he same ime (e.g. Lansing 2015, Guvenen 2009). Several elemens of he model conribue o is success. The source of macroeconomic risk in he model is a echnology shock. To generae a high price for risk, following Greenwald e al. (2016), we inroduce high and imevarying coeffi cien of risk-aversion in capial owners uiliy funcion. In a sandard 1-ype of agens model, high and ime-varying risk aversion would, via he Euler equaion on capial, have a very srong impac on he real side of he economy. However, wih wo ypes of agens and a relaively small number of capial owners relaive o he oal number of consumers, he oal consumpion is no very srongly affeced by shocks o risk aversion. This helps us generae real economy saisics ha are similar o hose in a baseline RBC model and reasonably close o heir real-life counerpars. While high and volaile coeffi cien of risk aversion of capial owners makes i possible o produce almos arbirarily high equiy premium, i is no suffi cien on is own o generae a realisic Sharpe raio. Producion-based CCAPMs end o produce oo small he quaniy of risk, as measured by he uncondiional sandard deviaion of excess reurns. We inroduce wo mechanisms o deal wih his problem. Firs, we assume ha capial is subjec o adjusmen coss. This mechanism makes capial owners consumpion smoohing via adjusmen in he level of corporae invesmen coslier, making hem less keen on aking invesmen risks. Addiionally, capial adjusmen coss help us mach he volailiy of invesmen relaive o oupu. Second, we also inroduce a limied amoun of financial leverage of firms in he form of risk- 14

16 free corporae deb. This muliplies he risk aken up by capial owners per each sock held Expeced responses of equiy premium o shifs in income shares The objecive of he paper is o examine he impac of shifs in income shares on he sock marke variables and in paricular on he equiy premium. Using he model, we can make he following predicions. An increase in he capial income raio, as governed by he producion funcion parameer θ, will lead o an increase in he mean equiy premium. The reason is ha higher θ direcly increases he volailiy of reurn on capial, via (16), which raises he riskiness of invesmen in socks and hence increases he required excess reurn on socks. A shif in he labor income share of capial owners in oal income, α = α (1 θ), has a less direc effec. Combining he capial owners budge consrain (7) wih he definiion of dividends (10), we can wrie capial owners consumpion as: c c = α Y η + D η. (18) The firs erm on he RHS of (18) represens he oal labour income of capial owners. This is a par of capial owners income ha is relaively less risky. The second erm is simply capial income, equal o oal dividends income. This is he relaively more risky par of heir income. From he above, i is clear ha an upward shif in α will decrease consumpion risk faced by capial owners, by increasing he weigh of he less risky componen of capial owners income. As a resul, he volailiy of he associaed sochasic discoun facor, M c = β c λ c +1 λ c, and he mean sock excess reurn will fall. From he above reasoning, using definiion α = α (1 θ), i is also clear ha an increase in he capial share of income, θ, will have an addiional indirec effec on he mean equiy premium, by raising he relaive weigh of he riskier componen in capial owners consumpion. This effec reinforces he direc effec of a shif in θ discussed earlier. Since we observe an increase in boh shares of incomes in he daa, we use a se of simulaions of he model calibraed o he US economy o quanify he ne effec of hese wo simulaneous shifs An addiional rend ha could have been observed hrough he las decades was a seady increase in sock ownership among households. A seminal resul in he lieraure is ha limied sock marke paricipaion helps explain high excess reurns observed on sock markes (Vissing-Jorgensen, 2002). By his oken, he seady increase of sock ownership should have conribued o he falling equiy premium. Our model, in is baseline version presened above, is no able o capure his insigh. 15

17 3.7 Compuaion For solving he model, we use a non-linear soluion mehod based on Coleman and Fenyes (1992) and following Davig (2004) and Lendvai and Raciborski (2014). 12 The deails of his procedure are described in Appendix A. In order o compue he saisical momens of macro and financial variables, for every model paramerizaion of ineres we simulae he economy for The momens repored are based on ime series consising of he las observaions Calibraion Our calibraion sraegy is as follows. We calibrae mos real economy parameers o reflec cerain empirical properies of he economy in he pos-war period, see Table 2. Parameer β is calibraed o mach he mean annual risk-free rae E ( R f ) = 2%. The share η of capial owners in he populaion is equal o he op income decile. The remaining hree real economy parameers (capial income share, θ, capial owners labor income share in oal labor income, α, and he raio of capial owners o workers employmen) are calibraed so ha o mach cerain economy-wide characerisics in 1970 (see Table 2). Finally, he financial leverage parameer (µ) and risk aversion shock parameers (γ c, µ x, ρ x, σ x ) are chosen o mach a se of sock marke saisics and are repored in Table 3. Saring from he real economy saisics, he value of σ z, is chosen such ha he volailiy of oupu growh in our economy maches is empirical counerpar of 2% annually. The value of capial adjusmen cos parameer, ξ, is chosen o come close o he raio of volailiy of invesmen growh o volailiy of oupu growh equal o The remaining parameers in he capial adjusmen cos funcion are specified so ha he seady sae invesmen o capial raio equals he depreciaion rae and he firs derivaive of his funcion in invesmen-capial raio is equal o 1, as in Uhlig (2007) and Jermann and Quadrini (2012). The value of depreciaion rae, δ, is se o 8%, annually. The following parameer values are se o heir 1970 empirical counerpars. The capial share of income was 35% in Because he op 10% he riches owns approximaely 80% of he oal capial sock in he U.S. economy, we compue heir share as = 0.28 and se θ 1970 = 0.28 in Similarly he value of θ 2014 is 12 Our codes are based on sample codes for he soluion of a bare-bone RBC model provided by Troy Davig. 13 The firs observaions are burn-in. 16

18 Table 2: Calibraion of he main parameer values Table 2: Baseline Parameer Values Parameer Value Descripion/Targe η 0.10 Capial owners = op income decile. θ Capial s share of income of op 10% in 1970: % = 0.28 α Top decile income share = 3.5% in N1970 c /N w Mean relaive wage W0 c/w 0 w = 2 in µ z 0.00 σ z e 4 Oupu volailiy of 2% in he pos-war period. ρ z 0 RW δ 0.08 Annual depreciaion rae of 8% ξ 0.33 Raio of volailiy of invesmen o volailiy of oupu σ I σ y a e 4 Seady sae invesmen o capial raio equal o depreciaion rae. a G ( I K 1 ) = 1 β 0.98 R f = 1.02 compued by muliplying oal capial share of income by 0.8. The value of parameer α 1970 is se so ha he op decile income share, α 1970 (1 θ 1970 ) = 3.5%. Furher, wihou a loss of generaliy, we se he demand for capial owners labor in he model equal o 1. Following Lansing (2015), we hen choose he value of demand for workers labor such ha he relaive wage of capial owners o workers, W c 0 /W w 0 in = 2, is value Table 3 repors baseline parameer values for sock marke. The financial leverage, µ, and maximum risk aversion parameer, γ c, are calibraed o reproduce as closely as possible wo fundamenal sock marke saisics, he mean equiy premium, ER e +1 = 7.03% and he Sharpe raio, SR = 0.43, in he pos-war S&P 500. We need a much lower maximum degree of capial owners risk aversion han Greenwald e al. (2016), γ c = 100. This also ranslaes ino a much lower mean degree of capial owners risk aversion, which in our baseline calibraion is E (γ c ) = 26. The main reason for his difference is he assumpion of a moderae degree of financial leverage, absen in Greenwald e al. (2016). The value of financial leverage, µ = 21%, is close o he value of 15% assumed by Guvenen s (2009) and is conservaive in comparison o values used in he lieraure (e.g. Boldrin e al., 2001, assume leverage of 50%). The mean of he shock process o he coeffi cien of risk aversion µ x = 1.1 is se o mach log price-dividend raio and is variance, σ x = 0.4, o replicae he empirical dividends growh volailiy. Finally, we se he AR(1) coeffi cien of he risk aversion shock, ρ x, 17

19 Table 3: Baseline parameer values for sock marke Parameer Value µ 0.21 γ c 100 min (γ c ) 1 µ x 1.1 σ x 0.4 ρ x 0.95 Table includes calibraed values of he coeffi cien of risk aversion, γ c, and is AR(1) shock. Risk aversion coeffi cien is defined as: γ c = γ c 1+exp(x ) wih γ c being maximum degree of risk aversion and x an AR(1) process wih mean µ x : x µ x = ρ x (x 1 µ x ) + ε x and ε x an iid shock wih variance (σ x ) 2. o mach he well documened long-horizon predicabiliy of equiy premium. 4 Quaniaive Performance of he Model We proceed as follows. We firs simulae our model for he parameer calibraion given in Tables 2 and 3, and in paricular, for he capial share of income θ = 0.28 and he share of capial owners labor income in oal labor income α = We assess is performance by comparing a se of model-based saisics wih he daa. In order o sudy he impac of changes in income inequaliy on he equiy premium, we hen run 3 addiional simulaions. We firs consider a rise in capial income inequaliy. We change he value of θ from is baseline value o 0.34 and sudy he impac of his increase on he mean equiy premium. Second, we increase he value of α o 13%, is 2014 value, and analyze he effec of his change. Finally, similar o he rends in he daa, we consider he join shifs in he capial and labour shares of income. To isolae he impac of income shares shifs on he equiy premium, we keep all he oher parameers of he model unchanged. 4.1 Baseline scenario We firs discuss he performance of he baseline version of he model calibraed o he pos-war U.S. real economy and S&P 500 index. In he op panel of Table 4, we repor sock marke saisics and in he lower panel real economy saisics. Boh 18

20 panels include he momens observed in he U.S. daa and in he model. All he asse pricing saisics are based on he annual S&P 500 series coming from Rober Shiller s websie and covering pos-war period beween 1947 and The real economy saisics are based on he series compued by U.S. Bureau of Economic Analysis and rerieved from he FRED a S Louis Fed. The firs wo rows of Table 4 repor he average equiy premium and Sharpe raio. Our baseline model specificaion reproduces he hisorical equiy premium on he U.S. sock marke and produces sizeable Sharpe raio. The hird and fourh rows of he same able display log price-dividend raio, ln ( P s D ). While he model is able o mach he volailiy of he raio, σ ln( P D ), i somewha underpredics is log-level. Finally, he volailiy of dividend growh of 7% is well mached by he model. On he real side of he economy, he model replicaes he volailiy of oupu growh, σ y and comes close o maching he invesmen-oupu volailiy raio, σ I σ y. The model slighly overpredics he consumpion-oupu volailiy raio, σ c σ y. The reason is ha, in he model, relaively high volailiy of capial owners consumpion is required o mach he high equiy premium and Sharpe raio observed in he daa. Since workers consume less on average, capial owners consumpion consiues an imporan par of oal consumpion, despie he much lower number of hem in he oal populaion, so ha oal consumpion volailiy is srongly affeced by he volailiy of capial owners consumpion. In addiion o he uncondiional momens we compue he long-horizon predicabiliy of equiy premium based on he price-dividend raio. specificaion of he following form: where h j=0 h j=0 We esimae a R e +j+1= β P s D +ε +1,+h (19) R e +j+1 is a cumulaive excess reurn over h + 1 years wih h = 0, 1, 2, P s D is price-dividend raio a ime, and ε +1,+h is he error erm. The resuls of he esimaion of his specificaion for S&P 500 series are repored in he op panel of Table 5 while heir model-based counerpars are displayed in he lower panel of he same able. We find ha a drop in curren price-dividend raio predics increase in he fuure cumulaive excess reurns boh in he daa and in he model. The absolue value of esimaed coeffi cien β as well as R 2 increase in he horizon h. 19

21 Table 4: Uncondiional asse pricing and real economy momens in he model and he U.S. daa Variable Momen Daa Model Sock marke R+1 e Equiy premium E[R e] σ[r e] Sharpe raio ( ) P s ln D Log Price-Dividend raio σ ln( P D ) Volailiy of Log Price-Dividend raio σ D Volailiy of dividend growh Real economy σ y Volailiy of Oupu Growh 2% 2% σ I σ y Raio vol invesmen growh-vol oupu growh σ c σ y Raio vol consumpion growh-vol oupu growh Equiy premium, Sharpe raio and price-dividend raio are compued using he annual daa from Shiller s websie: hp:// Equiy premium is compued as he difference beween he reurn including dividends on S&P 500 and he risk free rae. Risk free rae is 6 monhs rolled commercial paper rae. Price-dividend raio is calculaed for S&P 500. Volailiy is measured by sandard deviaion. x sands for growh rae in variable x. The real economy daa is from FRED. Real oupu is proxied by real GDP, real invesmen by privae non-residenial fixed invesmen and consumpion by real personal consumpion expendiures. All he real economy variables are originally expressed in Billions of Chained 2009 Dollars and we compue heir annual growh raes. 20

22 Table 5: Long Run Reurn Predicabiliy Regressions Long Horizon Reurn Regressions h j=0 R e +j+1 = β P s D +ε +1,+h Daa P s D R 2 X : h (0.11) (0.18) (0.22) Model h (0.002) (0.002) (0.003) Table repors resuls of esimaion of he regression where h j=0 R e +j+1= β P s D +ε +1,+h h R+j+1 e is a cumulaive excess reurn over h+1 years wih h=0,1,2. j=0 is price-dividend raio a ime and ε +1,+h is he error erm. The specificaion is esimaed by OLS wih Newey-Wes correcion of he sandard errors. Sandard errors are repored in brackes. P s D 21

23 5 Increase in Income Inequaliy and Equiy Premium How did he rends in inequaliy affec sock marke and, in paricular, equiy premium? Beween 1970 and 2014 he capial income share of op decile increased by 6 percenage poins (from 28% o 34% of oal U.S. income), while he op decile labor income share increased by abou 9.5 percenage poins (from 3.5% o 13% of oal U.S. income). In order o gauge he impac of hese shifs on equiy premium, we simulae he model economy wih higher income inequaliy observed in Specifically, we se he income share parameers o heir 2014 seady sae values. Table 6: Income shares shifs and mean equiy premium Scenario Parameer values Equiy premium Change vs baseline Baseline θ = 0.28; α = % - θ θ = 0.34; α = % +0.43pp α θ = 0.28; α = % 1.64pp θ and α θ = 0.34; α = % 0.79pp θ denoes an increase in capial share of income and α an increase in he labour share of income of capial owners. The equiy premium in he model is compued over simulaed periods. All he parameers of he model excep for he ones included in he able are se o heir baseline calibraion values repored in Tables 2 and 3. As explained earlier (secion 3.6.2), we expec shifs in he capial income and op decile labor income shares o have opposie effecs on equiy premium. In order o es numerically his inuiion, we firs examine each income shif separaely. Firs, we keep α a is 1970 value bu increase θ o mach he 2014 value of he capial income share. Nex, we do he opposie: we keep θ a is 1970 value, bu increase α o is 2014 level. Finally, o assess he oal, quaniaive impac of he simulaneous rends in income shares observed in he daa, we inroduce boh shifs ogeher. The resuls of hese exercises are repored in Table 6. The firs row shows he baseline model simulaion oucomes where he equiy premium equals 7.04% and he second row displays he resuls of he simulaion wih a higher capial income inequaliy as measured by increased θ. Increase in he share of income derived from he risky source, requires a higher reurn on he risky asse and ranslaes ino a higher equiy premium. An increase in he capial share of income of 6 percenage poins generaes an increase in equiy premium of 0.43 percenage poins. 22

24 The hird row repors he resuls of he counerfacual wih θ kep unchanged a is 1970 value, and α se o is 2014 level. A 15 percenage poins increase in α ranslaes ino a large drop in equiy premium of 1.64 percenage poins. Unsurprisingly, considered simulaneously he shifs generae a drop in equiy premium. An economy wih higher shares of income, as observed in he U.S. daa, displays equiy premium of 6.25%. The reason is ha, during he las five decades he capial owners benefied from a higher average growh in heir non-risky labour income share relaive o he risky capial income share (2% versus 0.5% on per year). In addiion o he equiy premium, we look a he impac of changes in income shares on oher asse pricing variables. Table 7 shows he figures for Sharpe raio, log price-dividend raio, volailiy of log-price-dividend raio and volailiy of dividend growh. Table 7: Asse pricing momens in differen scenarios Scenario Parameer values SR log P/D sd log P/D sd. div. gr. Baseline θ = 0.28; α = θ θ = 0.34; α = α θ = 0.28; α = θ and α θ = 0.34; α = θ denoes an increase in capial share of income and α, an increase in he labour share of income of capial owners. The equiy premium in he model is compued over simulaed periods. All he perimeers of he model excep for he indicaed ones are se o he baseline calibraion values repored in Tables 2 and 3. The second row shows he saisics under counerfacual wih only θ shifing by 6 percenage poins. We see ha here is no impac on Sharpe raio because increase in capial share of income raises boh he reurn and he volailiy of reurn on capial. Because changes in labour income share of capial owners have an impac only on he sock reurn, we find ha an increase in α generaes a drop in Sharpe raio. 5.1 Alernaive scenarios and predicions How will he equiy premium change by 2030 if he U.S. income inequaliy coninues o increase? While i may be fuile o ry o predic income inequaliy changes, i is sill of ineres o sudy he mos likely scenarios. Recen research suggess ha, a leas o some exen, he curren rends will persis. Schwabish and Topoleski (2013) argue ha he hisorical paern of rising earnings inequaliy will coninue for he 23

25 nex wo decades. Pikey (2014) makes a srong claim ha he capial income share will coninue o grow as well. In line wih hese predicions, we exrapolae curren inequaliy rends ino he fuure, unil Specifically, we assume ha capial share of income, θ, coninues growing a a yearly 0.5%, as observed beween 1970 and 2014, on average. We also assume ha he oal op decile income share, (1 θ) α + θ, will grow a he same rae as on average beween 1970 and 2014, namely 0.92% per year. We hen compue he resuling labour share of income of op decile, (1 θ) α. These assumpions imply ha labour share of income of op decile increases a 2% per year. Nex, using our model, we calculae he resuling equiy premia, for boh of hese rends coninuing simulaneously (scenario 1) as well as for an alernaive scenario in which only capial share of income, θ, coninues increasing in he fuure (scenario 2). Figure 3: Top decile income shares in he U.S. daa and predicions (scenario 1) Doed grey line plos he labour share of income of capial owners: (1 θ) α. Grey solid line plos capial share of income: θ, and he dashed black line plos he sum of he wo: (1 θ) α + θ which corresponds o op decile income share. The black verical line indicaes 2014 when he available daa sample sops. Figures beyond 2014 correspond o he forecased values. I is assumed ha capial share of income, θ, coninues o grow a an annual 0.5% and he labour share of income of capial owners a an annual rae 2%. The evoluion of he exrapolaed income shares is ploed in Figure 3. figure plos he labour share of income of capial owners: (1 θ) α (grey doed line), capial share of income: θ (grey solid line) and oal share of income of op decile: (1 θ) α + θ (black dashed line). Beyond 2014, we plo linear forecass in income share rends. If boh labour and capial shares of income of op decile would coninue growing a heir rend raes unil 2030, he capial share of income, θ, would reach The 24

26 0.37, is labour share of income, (1 θ)α, 0.18, and he oal op decile income share Noe ha if hese rends would persis, op decile income share would double beween 1970 and Table 8: Mean equiy premium in differen scenarios Average growh raes Parameer values EP EP vs B1 EP vs B2 θ (1 θ)α θ α (1 θ)α θ + (1 θ)α Baseline Baseline 2 0.5% 2% pp - Scenario 1 0.5% 2% pp 0.14 Scenario 2 0.5% pp x sands for growh rae in variable x. θ is capial share of income. (1 θ)α is labour share of income of op decile and θ + (1 θ)α is heir oal income share. EP sands for equiy premium. EP vs B1 sands for equiy premium versus baseline 1 and EP vs B2 versus baseline 2. Saisics are compued using model simulaions of periods. All he parameers excep for he ones indicaed in he able are se o he baseline calibraion values repored in Table2. Table 8 repors he parameer values and resuling equiy premia in wo baseline scenarios and wo alernaive scenarios based on income shares rends exrapolaed in he fuure. The baseline scenarios assume ha income shares parameers remain fixed in he fuure. The firs baseline scenario (Baseline 1) corresponds o he iniial calibraion described in Table 2 where we mach he sock marke and real economy saisics in he pos-war U.S. daa. The second baseline scenario (Baseline 2) corresponds o he simulaion wih higher income inequaliy as observed in he daa in 2014, and described in Tables 6 and 7. The las hree columns of Table 8 repor he mean equiy premium, he difference relaive o he firs baseline and he difference relaive o he second baseline scenario. Unsurprisingly, under alernaive Scenario 1 (boh income share parameers coninue increasing a he raes depiced in Figure 3), he equiy premium would fall o 6.11 % in 2030, 0.92 percenage poin lower han he average pos-war equiy premium and 0.14 percenage poin lower han in The mechanism driving his resul is exacly he same as previously. An increasing share of risky asses in he porfolio of op decile requires higher reurn while a growing share of non-risky asses (labour income) implies lower reurn. Because he labour share of income grew faser han capial share of income beween 1970 and 2014 (2 % versus 0.5 %), he growh in he labour income share dominaes he one in he capial income share and generaes 25