Risk-taking over the Life Cycle: Aggregate and Distributive Implications of Entrepreneurial Risk

Transcription

1 Risk-aking over he Life Cycle: Aggregae and Disribuive Implicaions of Enrepreneurial Risk Dejanir H. Silva UIUC Rober M. Townsend MIT April 018 Absrac We sudy he risk-aking behavior of enrepreneurs in an environmen wih wo main ingrediens: finie lives and uninsurable idiosyncraic risk on he business. We show ha he fracion of wealh invesed in he business depends on he idiosyncraic risk premium and ha i declines subsanially over he life cycle. The consumpion-wealh raio is U-shaped over he life cycle. We solve for he wealh disribuion boh across and wihin age groups. We show ha he variance of wealh condiional on age has an invered-u shape, iniially increasing wih age and evenually declining. We find suppor for hese predicions in he daa using a survey of enrepreneurial aciviy in Thailand. We also consider he impac of financial developmen and demographic ransiions on asse prices, economic aciviy, and inequaliy. We show ha an increase in he fracion of idiosyncraic risk enrepreneurs can insure or a decline in populaion growh will lead o a reducion in he idiosyncraic risk premium, an increase in he capial sock of he economy, and a decline in inequaliy. Universiy of Illinois a Urbana-Champaign, College of Business. dejanir@illinois.edu 1

2 1 Inroducion Enrepreneurship is inherenly a risky aciviy, even more so in he conex of developing counries. The decision o become an enrepreneur as well as he choice of he appropriae scale o operae will depend on how risky a given aciviy is, he expeced reurn on he business, and he capaciy or willingness of he enrepreneur o bear such risks. Hence, in order o undersand he behavior of enrepreneurs and he impac of enrepreneurship on he economy, i is imporan o undersand he implicaions of enrepreneurial risk. In his paper, we propose a framework o analyze he aggregae and disribuive implicaions of such risks. We sar by documening a subsanial variaion in he enrepreneurs exposure o heir businesses. Using daa on enrepreneurs for rural and semi-urban villages in Thailand, we show ha he share of financial wealh invesed in he business declines sharply over he life cycle, wih young enrepreneurs having a share of wealh invesed in he business up o 60% higher han old enrepreneurs. Moreover, he level of financial wealh varies significanly boh wihin and across age groups. In order o capure such heerogeneiy, we consider a life cycle model where enrepreneurs face a porfolio choice of how much o inves in he business and how much o inves in a safe low reurn alernaive. The heerogeneiy in risk-aking behavior over he life cycle is by no means accidenal. We show ha he effecive risk aversion of an enrepreneur, i.e. he curvaure of her value funcion, will vary over he life cycle even if he insaneneous uiliy funcion is of he CRRA ype. In paricular, he effecive risk aversion will be decreasing in he human-financial-wealh raio of an enrepreneur. Differences in human wealh, he presen discouned value of non-business income, will hen induce differences in risk-aking. 1 In order o es wheher his mechanism can generae he amoun of heerogeneiy we observe in he daa, we consruc an empirical measure of human wealh. An imporan aspec of his calculaion is he choice of he discoun rae. Given ha labor income is risky, boh in he daa and in he model, he risk free ineres rae is no he appropriae discoun rae. We show how o use daa on reurns on a porfolio of business aciviies o find he correc risk-adjused discoun rae. Given our measure of human wealh, we show ha he human-financial wealh raio declines over he life cycle. Moreover, he resuling increase in effecive risk aversion is enough o quaniaively explain he decline in he exposure o he business over he life cycle. The human-financial wealh raio, hrough is impac on he effecive risk aversion, deermines how an enrepreneur evaluaes a risk-reurn rade-off. Given he effecive risk aversion, he decision of how much o be exposed o he business will depend on he acual expeced reurn and volailiies. Samphanharak and Townsend 018 documens ha reurns have an imporan idiosyncraic componen. More han 90% of he variance is accouned for idiosyncraic risk. Ineresingly, idiosyncraic risk accouns for only half of he expeced reurns, so he Sharpe raio of aggregae risk is hree imes he one for idiosyncraic risk. We inroduce hese feaures ino he model by assuming ha reurns are subjec o aggregae and idiosyncraic risk. Given he differences in volailiies, we are able o generae 1 This resul is reminiscen of he work on porfolio choice wih labor income of Bodie e al. 199 and Viceira 001. See also Heaon and Lucas 1997 and Koo 1998 See Hugge and Kaplan 016 for a similar approach on valuing human wealh.

3 endogenously he differences in reurns and Sharpe raios. Inroducing idiosyncraic risk ino he model opens he door o a poenial moral hazard problem, as idiosyncraic shocks are ypically hard o monior. We assume ha aggregae shocks are public informaion, bu idiosyncraic shocks are privae informaion o he enrepreneur. This will limi he amoun of idiosyncraic insurance an enrepreneur can conrac. Enrepreneurs will be subjec o a skin-in-he-game consrain and will be able o insure a mos a fracion 0 φ 1 of idiosyncraic risk, where φ 1 amouns o full insurance and φ 0 financial auarky. In conras, enrepreneurs are allowed o buy aggregae insurance freely. The Lagrange muliplier on his skin-in-he-game consrain plays an imporan role in he enrepreneurs risk-aking decision. I urns ou ha he risk-aking decision of he enrepreneur depends only on he effecive risk aversion, he level of idiosyncraic risk, and his muliplier, which we refer o as he shadow price of idiosyncraic insurance. In order o es he heory, i is hen crucial o come up wih a measure of such shadow price. We show ha, given he skin-in-he-game parameer φ, one can idenify he shadow price of idiosyncraic insurance from he inercep of he ime-series regression of reurns of a given enrepreneur on he reurn of a porfolio of all businesses of a given region. The slope of his regression gives informaion abou he aggregae risk premium. This is analogous o he α and β of a CAPM regression of reurns on he marke porfolio. 3 The level of aggregae and idiosyncraic reurns, ogeher wih he human-financial wealh raio, are also imporan o undersand enrepreneur s savings behavior. We documen ha he consumpionwealh raio has a U-shaped paern over he life cycle, iniially decreasing and evenually increasing by he end of he cycle. The model is able o replicae his paern by a combinaion of wo forces. Firs, he decline in he human-financial wealh raio end o reduce he rae of consumpion. On he oher hand, he marginal propensiy o consume MPC increases wih age, which ends o increase he consumpion-wealh raio. The firs force dominaes ealy in life, while he second dominaes as he end of life approaches. The level of consumpion-wealh raio will depend on preferences and he reurn on he porfolio. Having deermined he risk-aking and consumpion decisions of enrepreneurs, we can solve for he whole disribuion of wealh in he economy. In paricular, we can analyze how he presence of uninsurable idiosyncraic risk can affec wealh inequaliy and evenually asse prices and he level of economic aciviy. Firs, we consider he deerminaion of beween-age-group inequaliy, i.e. differences on average wealh of enrepreneurs of differen ages. Average wealh has an invered-u shape over he life cycle, as i iniially increases wih age and evenually falls a older ages. This is a resul of having more ime o accumulae wealh, and he higher he level of aggregae and idiosyncraic risk premium, he faser he enrepreneur accumulaes, combined wih an increase in he MPC wih age, especially by he end of he life cycle. We es hese predicions in he daa and find similar life cycle paern and overall level of beween-group inequaliy. We also consider he wihin-group inequaliy, i.e., differences in he level of wealh for enrepreneurs of a given age. The evoluion of he disribuion wih age follows a parial differenial equaion, he so-called Kolmogorov Forward Equaion. We are able o solve for he disribuion of wealh condiional 3 Samphanharak and Townsend 018 sudied exensively hese regression and characerized he empirical behavior of α, bu wihou giving an explici srucural inerpreaion o he regression. 3

4 on age in closed-form for he case where he enrepreneur leave no bequess. In paricular, we are able o show ha he variance of wealh also has an invered U-shape paern, i iniially increases wih age and hen evenually decreases. We documen ha he variance of wealh follows a similar life cycle paern in he Thai daa. Finally, we discuss he deerminaion of equilibrium prices and consider a few counerfacuals exercises. Firs, we consider he impac of changes in he skin-in-he-game parameer φ. This is mean o capure he process of financial developmen as he consrains in idiosyncraic insurance ges relaxed. We show ha as enrepreneurs have access o a greaer exen of idiosyncraic insurance, he idiosyncraic risk premium falls. The reducion in required reurns simulaes enrepreneurs o expand heir business, so he aggregae capial sock of he economy increases. Financial developmen goes ogeher wih economic developmen. Moreover, he reducion in he shadow price of idiosyncraic risk will lead o a reducion in wihin-group inequaliy. Second, we consider he effec of demographic ransiions, where he populaion growh of he economy g is reduced. We show ha, in consras o parial equilibrium analysis ha emphasize he role of he difference r g, a reducion in he populaion growh reduces wealh inequaliy. The reason is he endogenous response of he idiosyncraic risk premium, which falls in response o a drop in g, leading o a reducion in inequaliy. Moreover, he capial sock of he economy increases, so a reducion in populaion growh has more benign effecs han in environmens where he role of idiosyncraic risk is ignored. Lieraure: This paper is relaed o several srands of lieraure. Firs, he developmen lieraure on he risk and reurn of producion aciviies: see, for insance, Rosenzweig and Binswanger 199, Morduch 1995, Udry and Anagol 006, De Mel e al Karlan e al. 014, Beaman e al The lieraure on financial fricions and misallocaion, as in Buera and Shin 013, Midrigan and Xu 014, Moll 014. However, in conras o mos of his work, we focus on insurance consrains insead of borrowing consrains. The paper is also relaed o he lieraure on porfolio choice and asse pricing over he life cycle: Poerba and Samwick 1997, Soresleen e al. 004, Viceira 001, Consaninides e al. 00, and Cocco e al The res of he paper is organized as follows. Secion discuss he enrepreneur s problem and he risk-aking and savings decisions of enrepreneurs over he life cycle. Secion 3 discuss he deerminaion of beween-group and wihin-group inequaliy. Secion 4 closes he model in general equilibrium and discuss he equilibrium pricing condiions. Secion 5 presens he financial developmen and demographic ransiion counerfacuals and secion 6 concludes. Risk-aking and savings over he life cycle In his secion, we will focus on he problem of an enrepreneur in isolaion. We will focus on how consumpion and risk-aking of enrepreneurs evolve over he life cycle, for a given sequence of prices. In paricular, we will focus on prices consisen wih a saionary equilibrium, where properly scaled prices are consans. In he nex secion, we will embed he enrepreneur s problem in a general equilibrium seing, and consider he deerminaion of a saionary equilibrium. 4

5 .1 Enrepreneur s problem Consider he problem of an enrepreneur who lives for T periods and mus choose a each insan how much o consume and how o divide her savings beween a riskless financial invesmen and a risky producion aciviy. The producion echnology is subjec o boh aggregae and idiosyncraic shocks. Enrepreneurs can purchase any amoun of aggregae insurance, bu hey are limied on he amoun of idiosyncraic insurance hey can buy. We show in he appendix ha his paricular marke srucure implemens he allocaion under an opimal conrac of a dynamic moral hazard problem. Producion Technology Enrepreneur i combines capial k i, and labor l i, o produce a final good y i, using he echnology: y i, A k α i, l1 α i, 1 The enrepreneur is subjec o wo ypes of shocks. Firs, an aggregae produciviy shock which follows a geomeric Brownian moion: 4 where Z is a sandard Brownian moion. da A µ A d + σ A dz Capial accumulaion is subjec o idiosyncraic shocks: dk i, k i, ι i, δ d + σ id dz i, 3 where ι i, represens he ne invesmen rae and is a choice variable o he enrepreneur. Enrepreneurs can hire labor a he wage rae w and buy capial a he price q. We will focus on a saionary equilibrium, such ha w w A and q q A. The relaive price of capial hen evolves according o dq /q µ A d + σ A dz. Invesmen is subjec o quadraic adjusmen coss Φι i, A k i,, where Φι Φ 0 ι + Φ 1 ι, Φ 1 > 0. Adjusmen coss depend on he amoun of capial in efficiency unis. This will be imporan o guaranee he economy has a balanced growh pah. The reurn of invesing in he projec can be wrien as dr i, y i, w l i, Φι i, A k i, q k i, d + dq k i, q k i, µ R i, d + σ AdZ + σ id dz i, µ A σ A 4 The level of produciviy can be wrien as A A 0 e dae is disribued as a log-normal random variable. +σ A Z, where Z N 0,, so produciviy a any given 5

6 where µ R i, y i, w l i, Φι i, A k i, q k i, } {{ } dividend yield + µ A + ι i, δ }{{} expeced capial gain 4 Preferences and labor supply Enrepreneurs have CRRA preferences wih parameer, hey live for T periods and derive uiliy of leaving bequess U i,si E si ˆ si +T s i e ρ s i c 1 i, 1 d + e ρt s i 1 ψ V n 1 i,s i +T 1 where s i is he dae enrepreneur i was born and n i, denoes financial wealh or ne worh. The parameer ψ measures he srengh of he beques moive. If ψ 1, he enrepreneur gives no weigh o fuure generaions and here are no inergeneraional linkages. If ψ 0 he behavior of he enrepreneur will coincide wih he one of an agen wih infinie horizon. 5 The case 0 < ψ < 1 hen capures a form of imperfec alruism. 5 In addiion o business income, enrepreneurs are allowed o receive labor income. Hence, as in he daa, households have muliple sources of income. Labor is supplied inelasically, i is denoed by l i,, and can vary deerminisically over he life cycle. 6 We will show in secion.3 ha here is significan variaion in he imporance of labor income over he life cycle. The profile of labor supply over he life cycle will follow a flexible funcional form in order o capure he corresponding empirical paern: l i, Γ 1 e ϕ 1 s i + Γ e ϕ s i 6 The sum of exponenials is analyically convenien and, as we are going o see, i fis well he empirical labor income life-cycle profile. The maximizaion problem Enrepreneurs have access o a riskless savings insrumen wih reurn r. The enrepreneur can also buy aggregae and idiosyncraic insurance. The enrepreneur pays p ag θ ag i, o reduce aggregae volailiy by θ ag i,. There is no resricion in how much he enrepreneur can buy of aggregae insurance. In conras, he cos of idiosyncraic insurance will be zero in equilibrium, as providers of insurance can perfecly diversify across enrepreneurs, bu enrepreneurs can insure a mos a fracion 1 φ of he idiosyncraic volailiy: θ id i, 1 φq k i, σ id 7 5 The coefficien V equals he value funcion coefficien for an infinie horizon agen. 6 Noice ha l i, denoes he amoun of labor supplied by an enrepreneur born a s, while l i, is he amoun of labor demanded by an enrepreneur born a s o run his projec. 6

7 where θi, id is he amoun of idiosyncraic insurance. This skin-in-he-game consrain can be moivaed by a moral hazard problem. If idiosyncraic shocks are privae informaion and enrepreneurs can diver a leas par of he capial, hen enrepreneurs will have o bear some of he risk in order o suppor some risk-sharing. In he appendix, we derive he skin-in-he-game consrain following he lieraure on dynamic moral hazard problems. Enrepreneurs in his economy are insurance-consrained, as hey are forced o hold idiosyncraic risk if hey decide o operae heir producion echnology. In consras, we absrac from ad-hoc borrowing consrains. Firs, his will allow us o highligh he implicaions of he less explored insurance consrain. Second, Samphanharak and Townsend 018 shows ha he negaive correlaion beween wealh and reurns, an indicaive of borrowing consrain, disappears afer we conrol for he appropriae risk premium. This is suggesive ha, a leas in he conex of Thailand during his period, imperfec risk sharing may be a more prominen fricion. Enrepreneurs can hen borrow freely agains heir human wealh, i.e., he value of heir fuure labor income: where h i, denoes human wealh. 7 n i, h i, 8 The enrepreneur s problem is o choose a vecor of processes c i, k i, l i, ι i, θ ag i, θi id, aking he processes for prices q, w, r, p ag as given, o solve he following program: V i,si max c i,k i,l i,ι i,θ ag i ˆ si +T E si,θi id s i e ρ s i c 1 i, 1 d + e ρt s i 1 ψ V n subjec o 7, 8, non-negaiviy consrains c i,, k i, 0, and he law of moion of a i, dn i, n i, q k i, r + q k i, µ R i, pag given n i,si > h i,si. θ ag i, + w l i, c i, d + q k i, σ A θ ag i, 1 i,s i +T 1 dz + q k i, σ id θ id i, 9 dz i, The erm in brackes in he expression above represens he expeced growh rae of financial wealh. The enrepreneur invess n i, q k i, in he riskless asse, wih reurn r, and he amoun q k i, in he risky echnology, wih expeced reurn µ i, R. The cos of aggregae insurance is pag θ ag i,. The enrepreneur receives labor income w l i, and consumes c i,. The las wo erms represen he exposure o aggregae and idiosyncraic risk. The risk exposure depends on he scale on which he business is operaed, q k i,, volailiies, σ A and σ id, and aggregae and idiosyncraic insurance, θ ag i, and θid i,. 7 Human wealh is given by he discouned value of fuure labor income h i, s+t π E z π w z l i,z dz, where π z π denoes he sochasic discoun facor for his economy. 7

8 . Soluion in a saionary equilibrium We will focus on a saionary equilibrium where, despie he aggregae shocks, scaled variables are consan. This is he analogous of a balanced growh pah o a sochasic economy. The economy is in a saionary equilibrium if prices saisfy w wa, q qa, r r, p ag p ag and he aggregae capiallabor raio is consan and denoed by k. We will mainain his assumpion hrough his secion. Maximizing expeced reurns Noice ha l i,, ι i, only ener he maximizaion problem hrough µ i, R. Hence, he enrepreneur will choose hese variables o maximize he expeced reurn on he business. Labor demand assumes he usual form: w 1 α ki, l i, α 10 Hence, he capial-labor raio is equalized across enrepreneurs and i will equal he aggregae capial-labor raio: k i, /l i, k. The ne invesmen rae ι i, is given by using he quadraic cos specificaion Φι Φ 0 ι + 0.5Φ 1 ι. Φ ι i, q ι i, q Φ 0 Φ 1 ι q 11 Since he capial-labor raio and he ne invesmen rae are all equalized across enrepreneurs, he expeced reurn on he business is also equalized. We will hen denoe he expeced reurn by µ R i, µr. Plugging he previous wo equaions ino 4, we obain Financial and human wealh µ R αkα 1 Φιq q + µ A + ιq δ 1 Lemma 1 indicaes ha he relevan noion of wealh for he enrepreneur is oal wealh, ω i, n i, + h i,, he sum of financial and human wealh. Lemma 1. Suppose he economy is in a saionary equilibrium. a The value funcion is given by V i, n i, Vs n i, + h i, 1 i, 1 where Vs i, is deerminisic and given in he appendix. The effecive risk aversion of enrepreneur i is given by 13 V i,nnn i, V i,n 1 + h i, n i, 14 8

9 b Human wealh evolves according o dh i, r + p ag σ A h i, w l i, d + h i, σ A dz 15 Human wealh is hen given by h i, ˆ si +T e r+pag σ A z E w +z l i,z dz 16 c Demand for capial, aggregae, and idiosyncraic insurance solve he mean-variance problem: q k i, max µ R r + h i, p ag σ k i,θ ag i,θi id n i, n A p ag θ ag i, 1 q k i, i, n i, 1 + h σ i, n A + h i, σ i, n A θ ag i, q k + i, σ i, n i, n id θid i, i, n i, n i, 17 subjec o 7. The firs par of lemma 1 gives he value funcion, an age-dependen CRRA funcion of oal wealh. Imporanly, he effecive value funcion risk aversion of an enrepreneur will vary depending on he human-o-financial wealh raio. As discussed in secion.3, his raio varies significanly over he life cycle in he Thai daa, and i will play an imporan role explaining he differences in risk-aking over he life cycle. The second par of he lemma gives he law moion of human wealh. Human wealh is he presen discouned value of fuure labor income. Implicily, his expression gives he appropriae discoun rae o compue human wealh. Since he wage will move wih aggregae produciviy, labor income is risky, and he discoun rae for human wealh incorporaes he risk premium p ag σ A. The final par of he lemma shows ha he porfolio choice of an enrepreneur reduces o a simple mean-variance problem wih risk aversion /1 + h i, n i,. This is he resul of he coninuous-ime formulaion, as he mean-variance objecive comes from a direc rearrangemen of he HJB equaion. The firs erm in 17 capures he expeced excess reurn on oal wealh, which comes from he reurn on he business and from human wealh. The second erm is he produc of he effecive risk aversion and he aggregae and idiosyncraic variance. The maximizaion problem is subjec o he skin-in-he-game consrain. The Lagrange muliplier o his consrain, which we refer o as he shadow price of idiosyncraic insurance, will play an imporan role in he characerizaion of he enrepreneur s risk-aking decision. Savings behavior and risk-aking over he life cycle The nex proposiion characerizes he savings and risk-aking decisions of enrepreneurs Proposiion 1. Suppose he economy is in a saionary equilibrium. a The shadow price of idiosyncraic insurance is p id µr r p ag σ A φσ id 18 9

10 b Demand for capial is given by q k i, 1 + n i, h i, n i, p id φσ id 19 c The demand for aggregae insurance is θ ag i, 1 + h i, n i, n i, p id φσ id + h i, n i, σ A 1 + h i, n i, p ag 0 d The consumpion-wealh raio is given by c i, n i, r 1 + h i, 1 ψe rt s i n i, 1 where r 1 ρ r MV and r MV r + pag +p id. The firs par of he proposiion says he shadow price of idiosyncraic insurance, he Lagrange muliplier on he skin-in-he-game consrain, is equalized across enrepreneurs. Morever, i equals he reurn per uni of risk Sharpe raio of an invesor who fully insures he projec agains aggregae risk, so i is exposed only o idiosyncraic risk. In equilibrium, his erm will be posiive and he enrepreneur will buy as much idiosyncraic insurance as possible, i.e., θ id i, 1 φσ id. Demand for capial depends on he effecive risk aversion and he price and quaniy of idiosyncraic risk. Cross-secional differences in risk-aking are capured by differences in he effecive risk aversion, in paricular, he human-o-financial wealh raio. Since his raio has an imporan life-cycle componen, risk-aking will also vary over he life-cyle. The average scale of he business will depend on he raio of he shadow price of idiosyncraic risk and he amoun of idiosyncraic volailiy he enrepreneur canno insure. Hence, he decision abou he scale of he business does no depend on aggregae risk. The reason is he possibiliy of sharing risk hrough he insurance conrac. An enrepreneur ha is relaively less risk averse can increase he scale of he business and ge rid of he addiional aggregae risk by buying insurance. The decision of how much capial o have in he business is essenially a decision of how much idiosyncraic risk o hold. In paricular, if here is no idiosyncraic risk or if he enrepreneur can insure all of i, φ 0, hen he decision of how much o inves in he business is indeerminae: he enrepreneur is indifferen beween invesing in he safe asse or in he business, even if aggregae risk is sill presen. Equaion 0 gives he demand for aggregae insurance. The demand is linear in he price of aggregae insurance wih a slope given by he inverse of he effecive risk aversion and inercep given by he oal exposure o aggregae risk, coming from he business and from human wealh. Finally, he expression for he consumpion-financial-wealh raio is given in 1. The firs erm r/1 ψe rt s i is he marginal propensiy o consume MPC. I is increasing in age, as i is ypical in finie horizon problems, and he beques moive parameer ψ conrols he srengh of his effec. If ψ 0, he MPC will be consan, recovering he resul of he infinie horizon problem. If ψ 1, hen he MPC will ge arbirarily large as he enrepreneur approaches he end of life, so all he sock of wealh 10

11 will be consumed a he final age T. While ψ is imporan o deermine how he MPC varies over he life cycle, r is imporan o deermine he average MPC. If 1, hen r ρ, he enrepreneur s discoun rae. In general, r is a linear combinaion of ρ and r MV, he variance adjused expeced reurn on oal wealh ω i,. The erm r MV can be wrien as a mean-variance objecive r MV 1 d E dω i, ω i, V dω i, ω i,. Afer some simplificaion, we obain r MV r + pag +p id. Hence, if > 1, an increase in riskadjused reurns will increase he average MPC. 8 Finally, he consumpion-wealh raio depends on he human-financial wealh raio, which poenially vary over he life cycle. Enrepreneurs wih more human wealh will consume more ou of heir curren asses..3 Tesing he life cycle predicions The human-financial wealh raio plays an imporan role on how risk-aking and consumpion decisions vary over he life cycle. In paricular, differences in risk-aking over he life cycle are enirely deermined by differences in he human-financial wealh raio. Hence, he abiliy of he heory o make esable predicions rely on he abiliy o discipline h i, /n i, empirically. From equaion 16, his requires a measure of expeced reurns, r + p ag σ, and a measure of how expeced labor income varies over he life cycle, E w +z l i,z. We urn nex o he descripion of he daa and he measuremen of hese variables. Daa We use daa from he Townsend Thai Monhly Survey, an ongoing inensive monhly survey iniiaed in 1998 in four provinces of Thailand. Two provinces, Chachoengsao and Lopburi, are semi-urban in a more developed cenral region near he capial, Bangkok. The oher wo provinces are rural, Buriram and Srisake, and are locaed in he less developed norheasern region by he border of Cambodia. In each of he four provinces, he survey is conduced in four villages, chosen a random wihin a given ownship. 9 Our sample covers 716 households and 14 years of monhly daa, saring in January During his ime, hese village economies were subjec o all sors of aggregae and idiosyncraic shocks. Rice culivaion is affeced by seasonal variaion in rainfall and emperaure. Resricions on expors o he EU affeced shrimp ponds. The produciviy of milk cows varies subsanially boh over ime for a given animal and over he heard. The daa colleced in he Townsend Thai Monhly Survey includes informaion on he ne income generaed by he business as well as oal asses and liabiliies of households. I also includes informaion on he household s labor income and consumpion. The consrucion of hese variables is described in deail in Samphanharak and Townsend 018. Figure 1 shows he life cycle profile for he share of financial wealh invesed in he business and he consumpion-wealh raio. The share of wealh invesed in he business falls sharply over he life cycle. From around 30% of financial wealh 8 This is an exension of he usual resul ha ineres raes have income and subsiuion effec on savings decisions, where he income effec dominaes for > 1. In an environmen wih risky reurns, he relevan noion is r MV insead of he riskless ineres rae. 9 For deails on he Townsend Thai Monhly Survey, see Samphanharak and Townsend

12 a age 5, o less han 0% a he end of cycle, a drop of more han 40%. The consumpion-wealh raio is roughly U-shaped over he life cycle. I declines unil he sixies, hen i reurns back up. Again, here is large variaion over he life cycle. From peak o rough, he consumpion-wealh raio falls by more han 40%. Figure 1: Risk-aking and Savings - Life Cycle Profiles 0.40 Share of wealh invesed in he business daa model 0.14 Consumpion-wealh raio age age Measuremen of reurns and volailiies The reurn on he business is measured as he ne income divided by oal asses ne of liabiliies, i.e., he reurn on asses ROA, a convenional financial accouning measure of performance of producive asses. Given our measure of reurns, we can compue he expeced reurn and volailiy for aggregae and idiosyncraic risk as follows. Firs, define he marke reurn" as he reurn on a diversified porfolio of individual businesses: dr M, µ M d + σ A dz Noice ha he marke porolio is no exposed o idiosyncraic risk, only aggregae risk. When he marke porfolio is publicly raded, hen no-arbirage imply µ M r + p ag σ A. We can now use he marke reurn o obain an esimae of he shadow price of idiosyncraic risk. Taking he difference beween he reurn on individual business i and he marke porfolio and rearranging dr i, µ R r p ag σ A d + dr M, + σ id dz i, α i d + β i dr M, + σ id dz i, where α i p id φσ id and β i 1. The erms α i and β i would be he inercep and slope, respecively, of a ime-series regression of individual reurns on he marke average reurn. 10 If he marke porfolio is no direcly raded, hen 10 Under our simplifying assumpions of no differences in produciviy or exposure o aggregae risk, he coefficiens of he regression would be he same across enrepreneurs. In general, differences in produciviy or risk would lead o differences 1

13 µ M would include an idiosyncraic risk premium and hen idenificaion would require an addiional sep: regressing expeced reurns on beas, as in Fama and MacBeh The coefficien on he bea would recover he price of aggregae risk and he slope would give he price of idiosyncraic risk. The idiosyncraic volailiy can be idenified from he volailiy of he residuals of his regression. Since he variance of individual reurns is σa + σ id, we can derive σ A from he volailiy of reurns and σ id. Finally, he price of aggregae insurance can be idenified from he expeced reurns on he marke porfolio, which can be idenified by a ime-series average of he marke reurn. Given he parameer φ ha conrols he exen of idiosyncraic insurance, he shadow price of idiosyncraic insurance can be obained from α i. We will ake φ as given for now and discuss is idenificaion in a laer secion afer covering he deerminaion of equilibrium prices. These reurn regressions, and several variaions of hem, were exensively sudied in Samphanharak and Townsend 018. We can hen obain esimaes for he aggregae and idiosyncraic volailiies and reurns direcly from heir work. 11 Measuremen of human wealh From 16, human wealh can be compued as h i, ˆ si +T e r+pag σ A µ A z w l e Γ 1 e ϕ 1z s i + Γ e ϕ z s i dz using he fac ha E w z e µ Az w and l i,z Γ 1 e ϕ 1z s i + Γ e ϕ z s i. Figure : Labor Income - Life Cycle Profile 1.50 daa model relaive o avg. life-ime income age The erm r + p ag σ A was obained from he reurn on he marke porfolio and µ A is simply he aggregae per capia growh rae of he economy. Noice ha if T and l i,z was consan, hen he in α i and β i. 11 Samphanharak and Townsend 018 consider he exremes of full risk sharing, φ 0 in our noaion, and auarky, φ 1, bu hey do no consider explicily he inermediary cases presened by 0 < φ < 1 13

14 expression above would boil down o he Gordon growh formula: h i, w l e /r + p ag σ A µ A. 1 The erm w l e represens he average labor income of enrepreneurs across all age groups. Hence, j1 Γ je ϕ jz s i represens he labor income of enrepreneurs wih age z s i relaive o he average. Using daa on he relaive labor income of each age group, we esimae he parameers Γ j, ϕ j by nonlinear leas squares. Figure shows ha he funcional form does a good job of approximaing he empirical labor income profile. Given he discoun rae and he labor income profile, we can compue he human-financial wealh, boh in he daa and in he model. Excep for early in life, he human-financial wealh raio end o decline over he life cycle, as repored in figure 3. Human wealh is quaniaively as imporan as financial wealh a he beginning of he cycle. By he age of 50, human wealh is only half of he financial wealh. The model does reasonably well in capuring he evoluion of he human-financial wealh over he life cycle, even hough i does no capure he iniial increase in human wealh. The parameer l e is calibraed o mach he average value of h i, /n i, across age groups, and he evoluion over he life is deermined by he discoun rae and he labor income profile esimaed above. Figure 3: Human-financial wealh raio - Life Cycle Profile 1.0 Human-financial-wealh raio daa model Implicaions for risk-aking and savings In order o solve for q k /n i, in he model, i remains o calibrae he risk aversion parameer. We will se 6.7, such ha he model implied aggregae risk premium p ag σ A coincides wih he esimae from he reurn regressions discussed above. Given he decline in he human-financial wealh raio, he effecive risk aversion of he enrepreneur increases over he life cycle, leading o a reducion in he share of wealh invesed in he risky business, as shown in figure 1. Noice he raio beween he share invesed in he business a he beginning and he end of life is enirely deermined by he human- 1 The Gordon growh formula is an expression ypically used o value socks which says he value of a company equals he dividend divided by he difference beween he invesor s discoun rae and he growh rae of dividends. 14

15 financial wealh raio, which was calibraed independenly of informaion on he cross-secional of risk-aking. To compue he consumpion-wealh raio, i remains o calibrae ρ and ψ. As previously discussed, he average consumpion-wealh raio is paricularly informaive abou r, and so ρ, and he raio of he consumpion-wealh raio a he beginning and he end of life is informaive abou ψ. We will use hese momens o calibrae ρ, ψ. The model is able o replicae he U-shaped paern in c i, /n i, over he life cycle, alhough wih less curvaure. This is he resul of wo opposing forces. On he one hand, he decline in he human-financial wealh raio ends o reduce c i, /n i,. On he oher hand, he increase in he MPC ends o increase he consumpion-wealh raio. This force is paricularly srong by he end of life, overurning he impac of he reducion in h i, /n i,. 3 Inequaliy beween and wihin age groups The presence of uninsurable idiosyncraic risk will disor he disribuion of wealh in he economy. Wih perfec insurance, here is no wealh inequaliy wihin age groups, and wealh across age groups will only vary for sandard life cycle consideraions. Imperfec insurance will creae dispersion in wealh wihin age groups, and wealh across age groups will depend on risk and risk premium. In his secion, we will characerize how he wealh disribuion is deermined by he ineracion of risk and demographics. Firs, we will characerize how wealh evolves on average over he life cycle. Second, we will show how we can solve for he wealh disribuion wihin age groups. In boh cases, he risk-aking and savings decisions discussed in secion will play an imporan role. Demographics and noaion: Firs, le s deail he demographic srucure of he economy and fix some noaion. Populaion grows a rae g. In paricular, he index i of an enrepreneur born a dae s i s belongs o he se 0, e gs. Hence, he mass of agens in cohor s is e gs and oal populaion a dae is e g 1 e gt /g. The child of enrepreneur i will have index ie gt and i will be born a dae s i + T. The child will inheri he financial wealh of he paren, so a ie gt,s i +T a i,s i +T. The age disribuion in he populaion is a runcaed expoenenial wih densiy f a ge ga 1 e gt 3 Le x s, e gs {i:s i s} x i,di and x e, T f sx s,ds denoe, respecively, he average value of variable x across enrepreneurs born a dae s and across all enrepreneurs, for any variable x i,. 3.1 Beween-group inequaliy Plugging in equaions 19 and 0 ino he law of moion of financial wealh, we obain he expression dn i, n i, E dni, n i, 1 + h i, n i, p ag h i, σ n A dz h i, n i, p id dz i, i, Noice ha an increase in h i, /n i, has wo opposing effecs on he exposure o aggregae risk. I reduces he effecive risk aversion, leading he enrepreneur o buy less aggregae insurance, bu i 15

16 increases background risk, leading he enrepreneur o buy more insurance. The ne effec will be zero if p ag σ A. This is a sandard formula in asse pricing heory for he price of aggregae risk and i will hold in he saionary equilibrium in his economy. Using his expression for p ag, we find ha enrepreneurs will have he same exposure o aggregae risk: dn i, n i, E dni, n i, σ A dz h i, n i, p id dz i, An implicaion of his common exposure o aggregae risk is ha financial and human wealh of enrepreneurs move in andem wih aggregae shocks. This propery is imporan o guaranee a saionary disribuion of wealh exiss, despie he presence of aggregae risk. Consider n s, and n e,, he average financial wealh of enrepreneurs born a dae s and he average across all enrepreneurs, respecively. Because we are averaging over a coninuum of enrepreneurs, he idiosyncraic risk is compleely diversified, bu n s, sill respond o aggregae risk. However, since n s, and n e, move wih aggregae shocks by he same proporion, hen he share of wealh held by enrepreneur wih age a, n a f an a, /n e,, is non-sochasic and independen of calendar ime. Similarly, define h a f ah a, /n e,. The nex proposiion provides a characerizaion of n a. Proposiion Beween-group inequaliy. Suppose he economy is in a saionary equilibrium. The share of wealh held by enrepreneurs of age a, n a, saisfies where log n a log n 0 + and mpc e 1 T T h 0 n log h a n }{{ a } human-financial wealh effec n 0 e + r + pag + pid g + µ A }{{} generalized "r-g" effec 1 e r+ pag + pid g+µ A mpc e T r+ pag + pid g+µ A mpc e T r 1 ψe rt a da is he average MPC across all enrepreneurs. ˆ a a r rt ã dã 0 1 ψe }{{} average MPC effec 4 h 0 5 Proposion decomposes he disribuion of wealh across differen age groups ino hree effecs. Firs, a human-financial wealh effec. As he enrepreneur ges older, he human wealh ges convered" ino financial wealh, i.e., he labor income received acceleraes he accumulaion of financial wealh. The second erm is a generalized r g effec. The firs componen is he reurn on oal wealh: r + p ag / + p id /. In he absence of risk, his erm simplifies o r. Hence, he correc noion of reurn in his conex is he reurn on oal wealh aking ino accoun he aggregae and idiosyncraic risk premium. The second componen is he growh rae of he economy, g + µ A, he sum of populaion and produciviy growh. The generalized r g effec implies ha if he reurn on oal wealh exceeds he growh rae of he economy, he share of wealh will end o increase wih age, as he enrepreneur has more ime o accumulae wealh relaive o he oher enrepreneurs. The hird erm is he average MPC effec. I capures he fac ha he amoun of wealh an enrepreneur accumulaes a age a depends on her previous consumpion decisions. In paricular, his erm ends o 16

17 be low a he beginning of life and i increases as he enrepreneur ges older. Enrepreneur will hen end o accumulae wealh a he beginning of life, bu evenually consumpion will increase unil he wealh achieves he desired beques level. The share of wealh held by he younges households depends on he beques of he older generaion, so i depends on he cumulaive reurn ne of consumpion over a lifeime. Hence, i is increasing in reurns and decreasing in he growh rae of he economy and he average MPC. Financial wealh of he young also depend on he iniial amoun of human wealh. Figure 4: Financial wealh disribuion across age groups Financial wealh daa model Even hough idiosyncraic risk is diversified by compuing group averages, he presence of he idiosyncraic risk premium will affec beween-group inequaliy hrough hese hree channels. Idiosyncraic risk premium increases he generalized r g erm and i will affec he level and dispersion of MPCs across age groups as r is a funcion of p id. Moreover, since he idiosyncraic risk premium affecs financial wealh accumulaion, i will affec he evoluion of he human-financial wealh raio. Figure 4 shows how financial wealh varies across age groups. Even hough his was no par of he calibraion arges, he model is able o capure he invered U paern of financial wealh. However, he variaion in he model is more modes han he one in he daa, poenially due o he smooher consumpion-wealh generaed by he model. 3. Wihin-group inequaliy We have considered so far he behavior of he average wealh of enrepreneurs by age. However, even afer condiioning on a given age group, wealh may vary subsanially due o presence of idiosyncraic risk. We now urn o he characerizaion of he whole wealh disribuion condiional on age. 17

18 In order o eliminae he impac of aggregae risk, consider he normalized wealh of agen i: ñ i, n i, /n e,. Similarly, define he normalized human wealh h i, h i, /n e,. Noice ha h i, is a deerminisic funcion of a i s i, he age of enrepreneur i. Normalized wealh evolves according o dñ i, r + pag + pid mpc ai µ A ñ i, + h i, r + pag w l i, pid µ h A h i, i, d + ñ i, + h i, }{{}}{{} µññ i,,a i σññ i,,a i r where mpc a and w l i, 1 ψe rt a h i, is a deerminisic funcion of a i. Noice ha he expeced change, µññ i,, a i, and he volailiy, σññ i,, a i, of normalized wealh depend only of he curren value of ñ i, and he age of he enrepreneur a i. We will denoe he join densiy of normalized financial wealh and age by f ñ, a and he condiional densiy by f ñ a, already imposing he fac ha he disribuion does no depend on calendar ime in a saionary equilibrium. Lemma Kolmogorov Forward Equaion. The condiional disribuion of normalized wealh f ñ a saisfies he parial differenial equaion dz i, f ñ a a f ñ aµ ññ, a ñ + 1 f ñ aσñ ñ, a ñ 6 and he boundary condiion f ñ 0 f ñ T. Despie he complexiy creaed by he age-dependen expeced change and volailiy of wealh, we are able o solve for he condiional disribuion of financial wealh in closed-form for he special case where enrepreneurs leave no bequess, i.e., ψ 1. This allow us o characerize analiically he evoluion of inequaliy over he life cycle. Proposiion 3 Wihin-group inequaliy: no bequess. Suppose ψ 1 and r + pag + pid µ A > 0. i. Shifed log-normal disribuion: The disribuion of normalized wealh ñ i, condiional on age a i a is given by a shifed log-normal disribuion wih suppor h a,, i.e., oal wealh ñ i, + h i, is lognormally disribued. ii. Mean and variance by age: The expeced value and variance of ñ i, condiional on age are given by Eñ i, a i a h 0 e Vñ i, a i a e r+ pag + pid µ A a e ra e rt p id a 1 h 0 e 1 e rt h a 7 a e ra e rt 1 e rt 8 r+ pag + pid µ A iii. Invered-U shape of inequaliy over he life cycle: here exiss 0 < â < T such ha Vñ i, a i a is increasing in a for a < â and decreasing for a > â. Proposiion 3 gives a complee characerizaion of he disribuion of wealh condiional on age. Wealh has a shifed log-normal disribuion, wih an age dependen shifer h a. Since enrepreneurs 18

19 are allowed o borrow, financial wealh clearly canno be log-normally disribued, as ñ i, can ake on negaive values. However, financial wealh canno go below he naural borrowing limi h i,, so oal wealh will assume only posiive values. Toal wealh follows a log-normal disribuion wih a growh rae which varies wih age. Expression 7 is essenially equaion 118 rearranged and specialized o he case ψ This can be seen by noing ha he MPCs cumulaed up o age a is given by a r 0 dã log e ra e rt. 1 e rt ã 1 e rt As we have seen, average wealh ends o increase wih age a he beginning of life and i goes down by he end of he life cycle, as he resul of he increase in MPC balancing ou he effec of wealh being accumulaed over ime. Expression 8 shows how he variance of wealh evolves over he life cycle. In he case wih no beques, he variance is zero a ages a 0 and a T. Since enrepreneurs leave no bequess, everyone sars wih zero financial wealh. By he end of life he MPC becomes arbirarily large, so he flow of consumpion exhauss he whole sock of wealh. The dispersion of wealh increases a he beginning of he life cycle. This is he resul of he fac ha some enrepreneurs will be lucky and receive a series of posiive shocks, while ohers will suffer a sequence of negaive shocks. This force is magnified wih he exposure o idiosyncraic risk p id /, bu also wih he magniude of porfolio reurns ne of he growh rae, i.e., a version of r g. The increase in MPC provides a counervailling force, as he impac of he proporional shocks is reduced as he level of wealh is brough down a he end of he life cycle. Figure 5 shows he evoluion of wihin group inequaliy over he life cycle in he Thai daa. I shows how he sandard deviaion of ñ i,, financial wealh normalized by he average wealh of enrepreneurs, increases sharply from ages 40 o age 55 and hen declines unil he end of he cycle. 4 Equilibrium In order o deermine he equilibrium prices, r, p ag, q, w and aggregae capial-labor raio k, we need o close he model by specifing he marke clearing condiions and he agens who will provide insurance and funds o enrepreneurs, which we will refer o as financiers. 4.1 Equilibrium definiion Financiers Financiers choose a pah of consumpion, c f, how much of aggregae insurance o provide, θ ag f,, given labor income w l f, and iniial value of ne asses n f,0. Labor endowmen of financiers l f, grows a rae g. Financiers maximize uiliy subjec o he law of moion of wealh dn f, r n f, + p ag θ ag f, + w l f c f, d + θ ag f, dz 9 13 One difference is he fac ha n a f aeñ i, a i a, i.e., i akes ino accoun no only he average wealh of enrepreneurs of age a, bu also he mass of such agens. 19

20 Figure 5: Sandard deviaion of financial wealh wihin age groups 10 sd. deviaion financial wealh age a naural borrowing consrain n f, h f, and given iniial ne worh n f,0 > h f,0. In order o keep he problem of financiers as simple as possible, we will assume hey are infiniely lived and have uni elasiciy of ineremporal subsiuion, so he consumpion funcion is given by he sandard formula c f, ρ f n f, + h f,, where ρ f, > 0 is he discoun rae of financiers and h f, is he human wealh. The risk aversion parameer is > 0. The expression for he opimal provision of insurance is analagous o he demand derived for enrepreneurs, excep for sign changes: θ ag f, n f, 1 + where h f, denoes he human wealh of financiers. h f, n f, p ag h f, n f, σ A 30 Compeiive Equilibrium A compeiive equilibrium is a sequence of allocaions c i, k i, l i, ι i, θ ag i, θi id, c f, θ ag f and prices r, p ag, q, w such ha a c i, k i, l i, ι i, θ ag i, θi id solves he enrepreneur s problem 9, given r, p ag, q, w b c f, θ ag f solves he financier s problem, given r, p ag c Marke clearing: 0

21 i. Goods marke ˆ ii. Financial markes T c s, + Φι s, A k s, ds + c f, ˆ T y s, ds ˆ T n s, q k s, ds + n f, 0; ˆ T θ ag s,ds θag f, iii. Facor markes ˆ T l s, dz ˆ T l s, ds + l f ; ˆ T k s, ds k The firs marke clearing condiion corresponds o he goods marke, implying ha he value of consumpion plus invesmen made by enrepreneurs and he consumpion of financiers mus equal oal oupu produced in he economy. The second se of condiions gives he marke clearing for he riskless asse and for aggregae insurance. Since bonds are in zero ne supply, he ne worh of enrepreneurs and financiers mus add o he asse in posiive ne supply, he capial sock. Finally, we have he marke clearing condiion for capial and labor. 4. Equilibrium characerizaion Le s now presen he equilibrium characerizaion. Firs, we will discuss he deerminaion of he price of aggregae insurance, ineres rae, and wages, he las one as funcion of he level of capial. Then, we will discuss how he capial sock and he idiosyncraic risk premium are simulaneously deermined. Proofs and deailed calculaions are provided in he appendix. Price of aggregae insurance The ne demand for insurance in he economy is given by ˆ T ˆ ˆ θ ag p s,ds θag f, ag q k s, + h s, ds + h f, σ A n s, + h s, ds + n f, + h f, T T From he marke clearing for bonds, he ne demand for insurance will be equal o zero if p ag σ A 31 Wages and he relaive price of capial In a saionary equilibrium, he capial-labor raio is consan, so capial grows a rae g. This requires an invesmen rae equal o g + δ. The price of capial mus hen saisfy ι i, q Φ 0 Φ 1 q Φ 0 + Φ 1 g + δ 3 1

22 Given he aggregae capial-labor raio, wages are deermined by he labor demand condiion w 1 αk α 33 Ineres rae In a saionary equilibrium, he growh rae of he financier s oal wealh mus be equal o g + µ A, he growh rae of he economy r + σ A ρ f g + µ A using he fac ha θ ag f, σ A and p ag σ A. Rearranging he expression above, we can solve for he ineres rae r ρ f + g + µ A σ A 34 Aggregae capial sock and he price of idiosyncraic risk From equaion 18, we obain he expression r + p ag σ A + p id φσ id αkα 1 Φg + δ q + g + µ A 35 The lef-hand side capures he required rae of reurn of invesing in he business, which includes a premium for holding aggregae and idiosyncraic risk. The righ-hand side gives he acual expeced reurn of invesing in he business, a funcion of he marginal produc of capial ne of adjusmen coss. Noice expression 35 generalizes he sandard relaion beween marginal produc of capial and ineres raes o an environmen wih growh, risk, and adjusmen coss. In he absence of risk, σ A σ id 0, growh, g µ A 0, and adjusmen coss, Φ 0 1, Φ 1 0, he expression above boils down o r αk α 1 δ. The usual resul ha higher ineres raes leads o a reducion in he capiallabor raio ranslaes here o a negaive relaionship beween reurns, inclusive of risk premium, and he capial-labor raio. The prices r, p ag, and q are enirely deermined by a small se of parameers, independen of he value of k. Hence, expression 35 essenially gives an inverse relaion beween he price of idiosyncraic risk and he sock of capial in he economy. The deerminaion of he idiosyncraic risk premium In order o deermine he idiosyncraic risk premium in his economy, we need anoher equaion relaing k and p id. Firs, aggregaing he demand for capial 19 across all enrepreneurs, we obain p id }{{} φσ id }{{} q k n e, + h e, }{{} risk aversion effecive risk leverage facor 36 where n e, and h e, denoe he average financial and human wealh of enrepreneurs.