Idiosyncratic investment risk and business cycles

Finance and Economics Discussion Series Divisions of Research & Statistics and Monetary Affairs Federal Reserve Board, Washington, D.C. Idiosyncratic investment risk and business cycles Jonathan E. Goldberg 2014-05 NOTE: Staff working papers in the Finance and Economics Discussion Series (FEDS) are preliminary materials circulated to stimulate discussion and critical comment. The analysis and conclusions set forth are those of the authors and do not indicate concurrence by other members of the research staff or the Board of Governors. References in publications to the Finance and Economics Discussion Series (other than acknowledgement) should be cleared with the author(s) to protect the tentative character of these papers.

Idiosyncratic investment risk and business cycles Jonathan Goldberg∗ January 10, 2014 Abstract I show that, due to imperfect risk sharing, aggregate shocks to uncertainty about idiosyncratic return on investment generate economic contractions with elevated risk premia and a decrease in the risk-free rate. I present a tractable real business cycle model in which firms experience idiosyncratic shocks, to which managers are at least partially exposed; the distribution of these shocks is time-varying and stochastic. I show that the path for aggregate quantities, the price of physical capital, and the equity premium are the same as in a model without idiosyncratic risk, but with timepreference shocks. That is, in response to an increase in idiosyncratic uncertainty, the responseofthesevariablesisthesameasiftherewerenoidiosyncraticuncertaintybut managers were suddenly reluctant to invest. However, time-preference and idiosyncratic uncertainty shocks are not isomorphic: an increase in idiosyncratic uncertainty leadstogreaterdemandforprecautionarysavingandhenceadecreaseintherisk-free rate; in contrast, an increase in impatience has the opposite effect. In addition, with anidiosyncraticuncertaintyshock,investmentinphysicalcapitalcanremainloweven after the stock market and firm profitability recover, because managers cannot fully transferidiosyncraticrisktodiversifiedinvestors. Thus,shockstoidiosyncraticinvestment risk can explain, qualitatively, the aftermath of financial panics – elevated risk premia, a sharp and persistent decrease in investment, and a decrease in the risk-free rate. Inacalibration,anincreaseinidiosyncraticinvestmentrisksimilartothatexperiencedduringtheGreatRecessionleadsfirmstoinvestasiftheircostofcapitalwere 10 percentage points higher than the cost of capital implied by financial markets, and to a large decrease in the real risk-free rate. Keywords: Incompletemarkets,idiosyncraticrisk,businesscycles,equitypremium, risk-free rate JEL codes: D52, E44, G11 ∗Federal Reserve Board. I thank Anna Orlik, Vasia Panousi, Skander Van den Heuvel, and Francisco Vazquez-Grande for helpful comments. Any views expressed here are those of the author and need not representtheviewsoftheFederalReserveBoardoritsstaff. Contact: jonathan.goldberg@frb.gov 1

For a given firm, uncertainty about idiosyncratic returns varies over time. Moreover, across firms, this time variation in idiosyncratic uncertainty has an aggregate component. And because a firm’s managers are at least partially exposed to the firm’s idiosyncratic risks, a shock to idiosyncratic uncertainty can affect a firm’s investment decisions and its managers’ consumption and savings behavior. In this paper, I study how aggregate shocks to idiosyncratic investment risk affect business cycles and risk premia. To do so, I develop a tractable real business cycle model in which managers face moral hazard and hence are exposed to firm-specific shocks. Each period, firms experience an idiosyncratic shock to their return on investment. In a given period, firm-level return shocks are independent and identically distributed across firms. However, the distribution of these idiosyncratic shocks is itself an aggregate shock. Thus, the model has two aggregate shocks: a standard labor-augmenting productivity shock that is common across firms; and an aggregate shock to idiosyncratic uncertainty. When there is no aggregate uncertainty, comparing across steady-states, I show that an increase in idiosyncratic uncertainty leads to a decrease in aggregate capital, consumption, employment,andtherisk-freerate;also,thewedgebetweenthetheexpectedreturntophysical capital and the risk-free rate increases. More generally, when idiosyncratic uncertainty and aggregate productivity are stochastic, I show that the path for aggregate quantities, the price of physical capital, and the equity premium are the same as in a model without idiosyncratic risk, but with a time-varying discount factor. That is, in response to an increase in idiosyncratic uncertainty, the response of these variables is the same as if there were no idiosyncratic uncertainty but managers were suddenly impatient. Intuitively, when idiosyncratic risk increases, the risk-adjusted return to investing in physical capital falls, because managers are risk averse and investing in physical capital requires bearing idiosyncratic risk. Thus, aggregate quantities and the equity premium behave as if there were no investment risk, but managers had become impatient and thus reluctant to invest. Theseresultsareimportantbecauseanumberofrecentpapershaveusedtime-preference shocks to explain asset-pricing puzzles (Albuquerque, Eichenbaum and Rebelo (2012)) and business-cycledynamics(Hall(2013),Christiano,EichenbaumandRebelo(2011),Smetsand Wouters (2003)). My results imply that if time-preference shocks in a relatively standard real business cycle model can explain the dynamics of aggregate quantities and the equity premium, then, in a model where managers are unable to pledge a fraction of firm profits, there exists a stochastic process for uncertainty about idiosyncratic return on investment that also explains these dynamics. However,shockstoidiosyncraticuncertaintyarenotisomorphictotime-preferenceshocks: anincreaseinidiosyncraticuncertaintyleadstoadecreaseintherisk-freerate, whereasthe time-preference shock that generates the same path for aggregate quantities leads to an increase in the risk-free rate. The reason is that an increase in idiosyncratic uncertainty leadstoagreaterdemandforprecautionarysaving,andhencelowerreturnsontherisk-free asset, whereas an impatience shock implies higher expected returns on all assets, including risk-free bonds. Thus, in several papers that use time-preference shocks, a time-preference shockthatcausesadropinthestockmarketoraneconomiccontractionleadstoanincrease in the (real) risk-free rate, which limits the ability of time-preference shocks to explain the joint behavior of the risk-free rate, aggregate quantities and risk premia. For example, Albuquerque, Eichenbaum and Rebelo (2012) show that a simple Lucas treeeconomywithtime-preferenceshockscanexplainseveralasset-pricingpuzzles. Intheir model,a“bad” shock–onethatleadstoanincreaseintheequitypremium–isanincreasein impatience, which leads to higher expected returns for financial assets and correspondingly 2

anincreaseintherisk-freerate. Thisisconsistentwiththepositivecorrelationintheirdata between equity returns and their measure of the risk-free rate. However, this also suggests thatastandardtime-preferenceshock,atleastinaLucastreeeconomyorrealbusinesscycle model, cannot do a good job of explaining the qualitative behavior of financial variables during and after the 2008-2009 financial crisis – characterized by elevated risk premia and a decrease in the risk-free rate. To explain financial and macroeconomic dynamics during and after the financial crisis, other papers, including Christiano, Eichenbaum and Rebelo (2011), have used a timepreference shock with the opposite sign: a decrease in impatience. In a model without nominal rigidities, following such a shock, the risk-free rate falls and investment increases. Incontrast,withnominalrigiditiesandazero-lower-boundonnominalrates,therealinterest ratecanincreasesharply,leadingtoadecreaseininvestment,asinChristiano,Eichenbaum and Rebelo (2011). However, this is again at odds with the decrease in the risk-free rate during and after the financial crisis. Instead,theuncertaintyshockinmypapercanexplainthediscountrateshockthatHall (2013)usestoexplainemploymentdynamicsafterthefinancialcrisis: Hall(2013)considers a discount-rate shock that increases the required return on risky investments even as the risk-freeratedecreases. Thus,mypaperaddressesthedifficultyofstandardmacroeconomic models to match counter-cyclical equity premia and the decrease in risk-free rates associated with financial panics, by demonstrating that a shock to idiosyncratic investment risk decreases the risk-free rate and affects aggregate quantities, the equity premium, and the price of capital as if there were an impatience shock. Another difference between a time-preference shock and a shock to idiosyncratic uncertaintyisthat,withidiosyncraticuncertainty,thereisabreakdowninthestandardQ-theory result that the return on firm assets is equal to the return on a financial claim on firm assets. Instead, the return on investment is greater than, rather than equal to, the return on financial claims on firms. This wedge is required to compensate managers for bearing idiosyncratic risk. I show that this wedge increases in response to an idiosyncratic uncertaintyshock. Thus,withashocktoidiosyncraticuncertainty,investmentinphysicalcapital mayappear“toolow” givenstock-marketvaluations. PanousiandPapanikolaou(2012)find that when idiosyncratic risk rises, firm investment falls, and more so when managers own a larger fraction of the firm. Panousi and Papanikolaou (2012) also find that, during the financial crisis, firms with higher fractions of managerial ownership reduced investment (as a share of existing capital stock) by 6 percentage points more than firms with a more diversified shareholder base. Buera and Moll (2012) provide a simple measure of the ratio of the return on physical capital to the risk-free rate, and show that it increased during the financial crisis. Smets and Wouters (2007) and Galí, Smets and Wouters (2011) consider a “riskpremium” shockthatgeneratesawedgebetweenthereturnonphysicalcapitalandthe return on government bonds, and find that this shock is important for explaining short-run movements in output, employment and the Federal Funds rate. Galí, Smets and Wouters (2012) show that output and the labor market recovered more quickly after pre-1990s recessions than after the three most recent recessions, and attribute much of the difference to this “risk premium” shock. This paper is closely related to the literature on disaster risk, especially Gourio (2012) and Gourio (2013). These papers model disasters as a series of shocks to aggregate productivity and capital quality. In Gourio (2012), there is a representative firm and financial markets are complete. In this setting, if disasters last exactly one period and involve equal-sizedandpermanentreductionsinproductivityandcapital,theresponseofaggregate 3

quantities to an increase in the risk of a disaster is the same as if there was no disaster risk, but agents became suddenly impatient; also, the risk-free rate falls. Thus, the response of aggregate quantities and the risk-free rate to an increased risk of disaster in his model is qualitativelysimilartotheresponseofthesevariablestoashocktoidiosyncraticuncertainty in my model. However, our papers differ in a number of ways. First, disaster-risk shocks are different than shocks to idiosyncratic uncertainty; disaster risk concerns the level of (or uncertaintyabout)acoupledshocktoaggregateproductivityandaggregatecapitalquality, whereas idiosyncratic investment risk concerns only uncertainty about firm-level return on investment. Second, in Gourio (2012), markets are complete and Q-theory holds, whereas in my model, financial contracting is limited by moral hazard and without this limitation, the uncertainty shock would not matter. In addition, moral hazard gives rise to a wedge between the returns on physical capital and a financial claim on those returns; this wedge can be measured, as in Panousi and Papanikolaou (2012) and Buera and Moll (2012), to potentially distinguish between the two models. Third, Gourio’s disaster shocks are motivated by catastrophic wars and natural disasters; these disasters are rarely observed and hence difficult to learn about; and, even over a fairly long sample, observed business cycles patterns and risk premia will be driven not only by changes in the probability of disaster, butalsobythenumberofraredisastersthatactuallyoccurintheperiod. Incontrast,panel data on stock returns, private business income and consumption provide information about the types of idiosyncratic risk studied here. This paper is also closely related to the literature on idiosyncratic investment risk, includingAngeletos(2007),AngeletosandPanousi(2009),AngeletosandPanousi(2011)and Panousi (2012). One key difference between my paper and this earlier literature on idiosyncratic investment risk is that these papers assume that idiosyncratic uncertainty and all other aggregate exogenous variables are deterministic, even if they are time-varying. In contrast, I allow idiosyncratic uncertainty and aggregate productivity to follow a stochastic process, permitting a characterization of risk premia and business cycle dynamics. I calibrate the model to be consistent with estimates of the volatility of idiosyncratic consumption and idiosyncratic returns of public and private firms. In the calibration, followinga50percentincreaseinthestandarddeviationofidiosyncraticreturnoninvestment, aggregate quantities and the equity premium respond as if there were a negative discountfactor shock (e.g., an “impatience” shock) of 70 basis points. However, the risk-free rate declines by 5 percentage points, relative to the change in the risk-free rate caused by such a time-preference shock. In addition, the investment wedge – the spread between the return on a firm’s physical capital and the return on financial claims on the firm – increases from 5 percentage points to 10 percentage points. If there were no aggregate uncertainty, the increase in idiosyncratic investment risk would lead to a decrease in the steady-state risk-free rate from 1 percent to -3 percent. These results show that, due to imperfect risk sharing, shocks to idiosyncratic uncertainty of the size contemplated in Bloom (2009) and Gilchrist, Sim and Zakrajšek (2013) can lead to very large shocks to the real risk-free rate, of similar magnitude to the exogenous “shock to the natural rate of interest” studied in Eggertsson and Woodford (2003) and Christiano, Eichenbaum and Rebelo (2011). 1 Model Overview. Time is discrete, indexed by t ∈ {0,...,∞}. There is a continuum of infinitelylived firms, indexed by i, that produce consumption goods using capital goods. Each firm 4

is run by a manager. Atthebeginningofperiodt,managerihasanexpectedcapitalstockke,i . Themanager t−1 thenexperiencesanidiosyncraticcapital-qualityshocksi,whichisdrawnaccordingtocumut lative distribution function P . The manager also learns the aggregate technology shock t−1 z and the uncertainty shock P . Next, the manager hires labor in a competitive market, t t produces, and pays di to creditors. Finally, the manager chooses how much to consume, t ci, an expected next-period capital stock, ke,i, and a portfolio of state-contingent debts, t t di . The extent to which managers can offload idiosyncratic risk through the portfolio of t+1 state-contingent debts is limited by moral hazard. Denote manager i’s history of idiosyncratic shocks by si,t = {si,si,...,si}, the history 0 1 t of aggregate shocks by ht = (Pt,zt) and the history of idiosyncratic and aggregate shocks by hi,t = (cid:8) si,t,ht(cid:9) . Technology. Eachperiod,firmsexperienceidiosyncraticcapital-qualityshocks: although manager i chooses expected capital ke,i in period t, the manager’s actual capital stock in t period t+1 is ki =si ke,i (1) t+1 t+1 t where E [si ] = 1 and si ∈ S = (cid:2) smin,smax(cid:3) has a cumulative distribution function t t+1 t+1 P ∈P. The corresponding density function p is continuous and positive over S. t t The manager’s output in period t+1 is yi =F(ki ,z li ) t+1 t+1 t+1 t+1 where F is a neoclassical production technology and li is labor hired by manager i. The t+1 price of capital goods in period t+1 is pK . The firm’s period-t+1 assets are: t+1 ai =yi −ω li +(1−δ)pK ki . (2) t+1 t+1 t+1 t+1 t+1 t+1 The shock si is independent across firms and across time. P and z follow a Markov t+1 t t process. Financial markets. Inperiod t, themanager-ownercantradeafullsetofArrow-Debreu securitieswithperiod-t+1payoutsthatdependonthehistoryofaggregateandidiosyncratic shocks, hi,t+1. Specifically, at history hi,t, the manager sells a portfolio of Arrow-Debreu securities that represent a promise to pay di next period. The proceeds from this sale t+1 areE (cid:2) q di (cid:3) ,whereq isastate-pricedensity. Becausethecapital-qualityshocksare t t+1 t+1 t+1 idiosyncratic, q will depend only on the history of aggregate shocks, ht+1. t+1 Although the promises are state-contingent, markets are incomplete because of moral hazard. In particular, at history hi,t+1, manager i can abscond with (1−θ) share of the firm’s period-t+1 assets. Thus, di ≤θai . t+1 t+1 Iassumethatθ ∈(0,1− 1 ). Ifθweregreaterthan1− 1 ,thentherewouldbecomplete smax smax risk sharing in general equilibrium and the model would reduce to a standard real-business cycle model. If θ were equal to zero, then in general equilibrium, no risk sharing would be possible. Preferences. I assume that managers have Epstein-Zin preferences with constant elasticity of intertemporal substitution and constant relative risk aversion. That is, associated with a stochastic consumption stream {ci}∞ is a stochastic utility stream {vi}∞ that t t=0 t t=0 satisfies the following recursion: vi =U−1(cid:2) U(ci)+βU (cid:0)CE (cid:2) vi (cid:3)(cid:1)(cid:3) (3) t t t t+1 5

where β <1 and CE (vi )=Υ−1(cid:0) E (cid:2) Υ(vi ) (cid:3)(cid:1) is the certainty-equivalent of vi condit t+1 t t+1 t+1 tional on hi,t. Υ and U are given by: Υ(c)=c1−γ and U(c)=c1−1 (cid:15). (4) Note that γ > 0 denotes the coefficient of relative risk aversion and (cid:15) > 0 denotes the elasticity of intertemporal substitution (EIS).1 Budgets. Manager i’s budget constraint at state hi,t is: ci+pKke,i ≤wi+E (cid:2) q di (cid:3) t t t t t t+1 t+1 where wi =ai−di. t t t Consumption and expected capital cannot be negative: ci >0 and ke,i >0. t t Capital goods. Capital-goods firms participate in a perfectly competitive capital-goods market. In period t, capital-goods firm j purchases φ( It )Ij consumption goods and Kt−1 t transforms them into Ij capital goods, where φ is continuous and increasing and φ(δ)=1. t Aggregate capital satisfies: K =(1−δ)K +I t t−1 t−1 and, in equilibrium, the price of capital satisfies pK =φ( It ). t Kt−1 In order to have expected capital goods of ke,i, manager i must purchase ke,i capital t t goods in period t. Workers and limited participation. Thereisarepresentativeworkerthatdoesnotparticipate in financial markets. The worker’s preferences over consumption and labor are given by: (cid:20) (cid:21) ζ u =U−1 U(c − l1+v)+βU(CE [u ]) t t 1+v t t t+1 where Υ and U are given by (4). The mass of workers is normalized to L=1. Equilibrium. An equilibrium is: 1. amappingoftheaggregatehistoryht intoastate-pricedensityq ,wagesω ,theprice t t of capital goods pK, aggregate capital K , aggregate consumption C , aggregate labor t t t L , and aggregate debt D ; t t 2. a mapping for each manager i from hi,t into consumption ci, capital ki, repayment di t t t and labor-demand li; and t 3. a mapping for the representative worker from ω into labor supply l t t such that: (cid:110) (cid:111)∞ 1. theplans ci,ke,i,di ,li maximizetheutilityofeachentrepreneur,takingprices t t t+1 t t=0 (cid:8) q ,ω ,pK(cid:9)∞ as given; t t t t=0 2. the plan l maximizes the utility of the representative worker, taking ω as given, and t t aggregate capital K is consistent with profit maximization by capital goods firms; t 1Ingeneral,IassumethattheEISisgreaterthanone,butIalsocharacterizethedynamicsifthisisnot thecase. 6

3. financial, labor, capital goods and consumption goods markets clear; and 4. aggregate quantities´are determined by individual polic´ies (i.e., aggregate managerial consumption CM = cidi and aggregate capital K = ke,idi). t t t t The initial condition of the economy is given by the distribution of capital goods ki and 0 debt di across firms and an initial aggregate state h0 =(P ,z ). 0 0 0 The risk-free rate, the return on equity, and the public-company model of financial markets. Because there exist a full set of Arrow-Debreu securities, it is possible to price any financial claim. The risk-free rate between period t and t+1 is given by Rrf =E [q ]−1 t t t+1 I define the return on equity as the return between period t and t+1 on a financial claim on aggregate firm assets. That is, Yt+1−ωt+1Lt+1 +(1−δ)pK Requity = Kt t+1 . (5) t+1 E [q (cid:16) Yt+1−ωt+1Lt+1 +(1−δ)pK (cid:17) ] t t+1 Kt t+1 I define the aggregate return to investing as: Yt+1−ωt+1Lt+1 +(1−δ)pK R = Kt t+1 (6) t+1 pK t Note that, due to idiosyncratic risk, the standard Q-theory result that R =Requity will t+1 t+1 not hold. Thus, (5) differs from the return on equity in production-based asset pricing paperssuchasBoldrin,ChristianoandFisher(2001),inwhichthepriceofafinancialclaim to aggregate firm assets is pKK and hence (5) is equal to (6). One contribution of the t t paper is to characterize how the investment wedge, Rt+1 =E [q R ], and the equity Requity t t+1 t+1 t+1 premium, Et [R t e + qu 1 ity] , are related to idiosyncratic risk, parametrized by P . Rrf t t Remark 1. An alternative way to model financial contracting is to envision the creation of publicly traded, limited-liability equity claims on all future cash flows of a firm, when contractingbetweenthemanagersandtheequityholdersissubjecttothesamemoralhazard problem as in the sequential-trading “entrepreneurial” setup above. Suppose that, in period 0,managersandinvestorsmeetandcreatealimited-liabilitypubliclytradedcompany. Each contributeswealthtocreatethecompany. Thecompanyisownedbytheinvestorsandsigns a contract with the manager to provide a stream of consumption, given by ci, and to make t a stream of investments in physical capital, given by ke,i, where ci and ke,i depend on t t t the history of aggregate and idiosyncratic shocks. In this model, ci resembles managerial t compensation. Themanagerialmoral-hazardconstraintisthatthemanager’slifetimeutility vi mustbegreaterthanorequaltotheoutsideoptionofabscondingwith(1−θ)shareof t+1 thefirm’sassets, ai , andre-contractingwithanewsetofequityholders. Thisalternative t+1 model results in the same equilibrium policies, aggregate quantities and state price density asinthesequential-trading“entrepreneurial” setupabove. Moreover,ingeneralequilibrium, the return on the aggregate limited-liability equity claims is given by (5). Remark2. Thereturnonequityisthereturnonanunleveredfinancialclaimtoaggregate firm assets. However, it is possible, as in Boldrin, Christiano and Fisher (2001), to study 7

the return on a levered claim to aggregate firm assets: (cid:16) (cid:17) Rlevered =Requity+λ Requity−Rrf t+1 t+1 t+1 t where λ measures leverage. The expected excess return on the levered claim is given by: E [Rlevered] E [Requity] t t+1 =(1+λ) t t+1 −λ Rrf Rrf t t 2 Equilibrium characterization 2.1 Partial equilibrium In the model, the firm’s end-of-period assets and labor demand are linear in the manager’s capital,duetoconstantreturnstoscaleintechnologyandtheabilitytoadjustlabordemand according to the realization of the idiosyncratic shock: ai =R pKki and li =l ki t+1 t+1 t t+1 t+1 t+1 t+1 wherel =argmax (F(1,z l)−ω l)andR = 1 (cid:0) F(1,z l )−ω l +(1−δ)pK (cid:1) . t+1 l t+1 t+1 t+1 pK t+1 t+1 t+1 t+1 t+1 t Thus, the firm’s problem can be written recursively as: V(wi;t)= max U−1(cid:2) U(ci)+βU (cid:0)CE (cid:2) V(wi ;t+1) (cid:3)(cid:1)(cid:3) (7) t t t t+1 ke,i,ci,{wi } t t t+1 subject to the budget constraint E (cid:2) q wi (cid:3) ≤wi−ci+E [q R −1]pKke,i t t+1 t+1 t t t t+1 t+1 t t and, for each si , the limited-enforcement constraints t+1 wi ≥(1−θ)R pKki (8) t+1 t+1 t t+1 Below, I guess and verify that V is linear in the manager’s wealth. Thus, conditional on aggregate history ht+1 and idiosyncratic history hi,t, manager i’s wealth wi will equal t+1 some constant amount following every idiosyncratic shock si below a threshold si∗, and t+1 t for every state si greater than this threshold, wealth will be determined by (8). That is, t+1 wi will be equal to the greater of a fixed amount or the minimum wealth level consistent t+1 with repayment. I denote the fixed amount by ni , so that t+1 wi =max (cid:8) ni ,(1−θ)R pKki (cid:9) . (9) t+1 t+1 t+1 t t+1 Using this intuition, firms’ optimal decisions for given prices can be characterized. Lemma1. Givenprices,optimalconsumptionci,expectedcapitalke,i,andminimumpayoff t t ni are linear in wealth wi t+1 t ci t = c˜ wi t t pKke,i t t = (1−c˜)κ (10) wi t t t ni t+1 = (1−c˜)η wi t t+1 t 8

and the consumption-wealth ratio c˜ satisfies: t 1 =(βU(ρ ))(cid:15)+1 (11) c˜ t t where (cid:20) (cid:21) 1 ρ =CE c˜1−(cid:15) max{η ,(1−θ)si R κ } (12) t t t+1 t+1 t+1 t+1 t and (cid:20) (cid:21) 1 {η ,κ }=argmaxCE c˜1−(cid:15) max{η,(1−θ)si R κ} (13) t+1 t t t+1 t+1 t+1 {η},κ subject to E (cid:2) q max{η,(1−θ)si R κ} (cid:3) ≤1+E [q R −1]κ (14) t t+1 t+1 t+1 t t+1 t+1 Lemma 1 simplifies the manager’s problem by transforming it into a canonical portfolio choice problem in which there is a single asset, with return Ri ≡max{η ,(1−θ)si R κ } (15) t+1 t+1 t+1 t+1 t where η and κ are given by (13). The return on the single asset reflects the optimal t+1 t mix of investments in physical capital and financial assets, given prices and the limited enforcement constraints. Correspondingly,asinacanonicalportfoliochoiceproblemwithEpsteinZinpreferences, the lifetime utility function V that solves the firm’s problem is given by: t 1 Vi =c˜1−(cid:15)wi t t t Thus, ρ is the period-t risk-adjusted return to saving, where return to saving is measured t dVi 1 in units of the marginal lifetime utility of wealth, t+1 =c˜1−(cid:15). That is, dwi t+1 t+1 (cid:20)dVi (cid:21) ρ =CE t+1Ri t t dwi t+1 t+1 Managers’consumptionchoiceisdescribedbytheEulerequation(11);ahigherrisk-adjusted return ρ is consistent with lower consumption and greater investment if the elasticity of t intertemporal substitution is greater than one. To understand the manager’s problem, consider manager i’s pricing kernel mi , where t+1 ∂vi/∂dci (cid:18)ci (cid:19)−1 (cid:15) (cid:32) vi (cid:33)1 (cid:15) −γ mi ≡ t t+1 =β t+1 t+1 . (16) t+1 ∂vi/∂dci ci CE (cid:2) vi (cid:3) t t t t t+1 (16) depends only on the assumption of Epstein Zin preferences. The Euler equation (11) can be stated as the familiar asset-pricing condition: E (cid:2) mi Ri (cid:3) =1 (17) t+1 t+1 If there are no limited-enforcement constraints (that is, θ =1), then equilibrium requires mi =q t+1 t+1 9

and the model becomes a standard real-business cycle model. However, if θ <1, then mi <q t+1 t+1 wheneverthelimitedenforcementconstraint(8)binds. Thatis,forsi >si∗,themarginal t+1 t lifetime utility from an additional unit of consumption is low, relative to the market price of consumption, but reducing wealth and consumption at state si without violating the t+1 limited-enforcement constraint would require reducing ke,i, thus affecting consumption at t other histories hi,t+1. 3 General equilibrium 3.1 Cross-sectional dynamics Define manager i’s share of total managerial consumption by ci t . The next result shows CM t that risk sharing across managers is imperfect. Lemma 2. The (gross) growth rate of manager i’s consumption share in period t+1 is given by: ci /CM gi ≡ t+1 t+1 =max{ψ ,(1−θ)si } (18) t+1 ci/CM t t+1 t t where ψ <1 is the unique solution to t E[max{ψ ,(1−θ)si }]=1. (19) t t+1 Lemma2impliesthathowidiosyncraticriskbetweenperiod-tandperiod-t+1isshared dependsonlyonthedistributionofperiod-t+1idiosyncraticshocksandtheextentofmoral hazard, 1−θ. This result follows from market clearing and the linearity of each manager’s investment and consumption in her wealth. Later,Iwillcharacterizehowmacroeconomicandfinancialvariablesrespondtoameanpreserving spread in gi . A paper from the options literature, Rasmusen (2007), provides t+1 a definition of increased risk in si that is useful here. t+1 Definition. The distribution P(cid:101)t is pointwise riskier than P t if si t+1 has the same mean under each distribution and if there exist s(cid:48) and s(cid:48)(cid:48) with smin <s(cid:48) <s(cid:48)(cid:48) <smax such that p˜ > p for all si ∈[smin,s(cid:48)] (cid:83) [s(cid:48)(cid:48),smax] t t t+1 and p˜ ≤p otherwise. t t Anincreaseinpointwiseriskinessshiftsprobabilitymassfromeachpointinthemiddleof the support to points at each extreme of the support. As the next result shows, an increase in pointwise riskiness in si is a sufficient condition for a mean-preserving spread in gi .2 t+1 t+1 Lemma 3. An increase in pointwise risk in P leads to a mean-preserving spread in the t growth rate of manager i’s consumption share, gi , and a decrease in risk-sharing, ψ . t+1 t 2An increase in pointwise riskiness is a mean-preserving spread, but a mean-preserving spread is not necessarilyanincreaseinpointwiseriskiness. Notethatamean-preservingspreadinsi isnotasufficient t+1 condition for a mean-preserving spread in gi : a mean-preserving spread in si that leaves the density t+1 t+1 functionunchangedforallsi >s∗ willhavenoimpactongi . t+1 t t+1 10

According to Lemma 3, the growth rate of each manager’s consumption share becomes riskier when idiosyncratic uncertainty increases. The deterioration of risk sharing has two components: the mechanical effect that P is riskier; as well as an endogenous, generalt equilibrium effect, in the form of a strict decrease in ψ , the worst-case growth of manager t i’sconsumptionshare. Intuitively, worst-caseconsumption-sharegrowthdecreasesbecause, with greater uncertainty about idiosyncratic depreciation, managers seek to hedge all of their increased downside risk, but can only sell θ share of their increased upside risk. 3.2 Aggregate dynamics Using the market-clearing condition κ = 1, one can write the idiosyncratic return to int vestment, defined in (15), as: Ri =gi R (20) t+1 t+1 t+1 The term gi reflects idiosyncratic risk, given by (18). The term R is the aggregate t+1 t+1 return to investment, given by (6). Profit maximization of consumption-goods firms and capital-goods firms, together with labor market clearing, implies (cid:16) (cid:17)1 F (K ,z ωt+1 v)+(1−δ)φ(It+1) R = K t t+1 ζ Kt t+1 (cid:16) (cid:17) φ It Kt−1 where ω is the unique solution to t+1 (cid:18) (cid:19)1 ω =F (K ,z ω t+1 v )z . (21) t+1 L t t+1 ζ t+1 These results permit further characterization of the dynamics of aggregate capital and consumption. Proposition 4. The path for aggregate capital, K , aggregate consumption, C , and the t t price of physical capital, pK, are the same as in a model without investment risk (θ = 1), t but with a different discount factor β¯ that follows a stochastic process given by: t β¯ =βU (cid:0)CE [gi ] (cid:1) (22) t t t+1 where gi is the growth rate of manager i’s share of aggregate managerial consumption, t+1 given by (18). A mean-preserving spread in gi is associated with a decrease in the equivt+1 alent discount factor β¯ if and only if (cid:15)>1. t Proof. Combining (11) and (12), we obtain the Euler equation: 1 c˜ = . t (cid:18) (cid:18) (cid:18) (cid:19)(cid:19)(cid:19)(cid:15) 1 βU CE c˜1−(cid:15)Ri +1 t t+1 t+1 where Ri is given by (20). Because the growth rate of the idiosyncratic consumption t+1 share, gi , is independent of h , we can re-write this as: t+1 t+1 11

1 c˜ = t (cid:18) (cid:18) (cid:18) (cid:19)(cid:19)(cid:19)(cid:15) βU (cid:0)CE (cid:0) gi (cid:1)(cid:1) U CE c˜1− 1 (cid:15)R +1 t t+1 t t+1 t+1 That is, conditional on the aggregate return to investing R , the consumption share of t+1 wealth c˜ will be the same as in a model without investment risk but with a discount factor t given by (22). Therefore, the equilibrium path for aggregate capital and consumption will be the same in both models and hence the path for the aggregate return to investing will be the same as well. Ifthereisnoaggregateuncertaintyaboutidiosyncraticrisk(e.g.,ifP =P forallt),then t aggregate quantities are the same as in a model without investment risk (θ = 1), but with an adjusted discount factor. The direction of adjustment depends on the EIS: if managers are willing to substitute across time ((cid:15) > 1) , a mean-preserving spread in gi makes the t+1 managers more reluctant to invest. The intuition is that an increase in idiosyncratic risk leadstoadecreaseintherisk-adjustedreturntosaving;inresponse,managerswillconsume more and save less if they are willing to substitute across time. Similarly, if there is aggregate uncertainty about idiosyncratic risk (e.g., if P follows a t stochasticprocess),thenaggregatequantitiesarethesameasinamodelwithoutinvestment risk (θ =1), but with time-preference shocks.3 3.3 Asset pricing and the investment wedge It is not the case that shocks to idiosyncratic uncertainty are isomorphic to time-preference shocks. In particular, the prices of financial assets are different than in the model without investment risk, but with time-preference shocks given by (22), even though the paths for aggregate quantities and the price of physical capital are the same. One way to see this is to examine how the equilibrium state-price density q is related t+1 to the state-price density in a model without investment risk, but with time-preference shocks. Lemma5. Attimet,thestate-pricedensityq isthesame,uptoapre-determinedscalar, t+1 as in a model without investment risk (θ = 1) but with stochastic discount factor β¯ given t by (22). That is, if qβ is the state-price density in the model without investment risk and t+1 with time-preference shocks given by (22), then q t+1 = CE t [ g ψ t i + t 1]γ >1. (23) q t β +1 CE t [g t i +1 ] Proof. For any idiosyncratic history hi,t+1 for which the limited-enforcement constraint (8) is not binding, we have q =mi . Hence, substituting into (16), we have t+1 t+1  1−γ (cid:15) q t+1 =β (cid:18) c˜ t c˜ + t 1ψ t R t+1 (1−c˜ t ) (cid:19)−1 (cid:15)    CE c (cid:20) ˜ t c 1 ˜ + − 1 1 1 − (cid:15) 1 ψ (cid:15)g t R i t+ R 1 (cid:21)    t t+1 t+1 t+1 3Proposition4describeshowtheadjusteddiscountfactordefinedby(22)respondstoamean-preserving spreadingi ,anendogenousvariable. Onewaytoconnectthisresulttoanexogenousshockistorecall, t+1 fromLemma2,thatapointwiseincreaseinriskintheexogenousdistributionPt isasufficientconditionfor amean-preservingspreadingi . t+1 12

Because the growth rate of the idiosyncratic consumption share, gi , is independent of t+1 h , we can write t+1  1−γ (cid:15) q t+1 =β (cid:18) c˜ t c˜ + t 1ψ t R t+1 (1−c˜ t ) (cid:19)−1 (cid:15)    CE (cid:20) c˜1− c 1 ˜ t (cid:15) 1 + − R 1 1 (cid:15)ψ t R (cid:21) t C + E 1 (cid:0) gi (cid:1)    t t+1 t+1 t t+1 which, together with (22), implies (23). Finally, (19) and Lemma 2 imply qt+1 >1. qβ t+1 This result has immediate implications for the investment wedge, the risk-free rate and the equity premium. 3.3.1 Investment wedge In a model without idiosyncratic investment risk (θ = 1), the return on investment, (6), would equal the return on equity, (5). Correspondingly, we would have: E [q R ]=1. (24) t t+1 t+1 Thisno-arbitrageconditionisthestandardQ-theoryresultthatfinancial-marketpricesq t+1 can price the return to investing in physical capital. However, with idiosyncratic investment risk, (24) does not hold. Because investing in physical capital involves idiosyncratic risk, managers need to compensated to bear these risks. This will take the form of an investment wedge. Proposition 6. The investment wedge, Rt+1 =E [q R ], is greater than one. Requity t t+1 t+1 t+1 Proposition 6 is a corollary of Lemma 5. To see this, substitute from (23) to write the investment wedge as: q q E[q R ] = t+1E [qβ R ]= t+1 (25) t+1 t+1 qβ t t+1 t+1 qβ t+1 t+1 FromProposition4, wehavethattheequilibriumpathforR isthesameasinthemodel t+1 without investment-risk, but with time-preference shocks given by (22). Note that in the model without investment risk, the state-price density that prices financial assets qβ will t+1 price the return to investing in physical capital, R ; that is, E [qβ R ]=1. The result t+1 t t+1 t+1 then follows from Lemma 5. 3.3.2 The equity premium and the risk-free rate Additional corollaries of Lemma 5 concern the risk-free rate and the equity premium: Proposition 7. The equity premium, Et [R t e + qu 1 ity] , is the same as in the model without in- Rrf vestment risk, but with stochastic discount fa t ctor β¯ given by (22). t 13

Proof. From (5) and (6), the equity premium is given by: (cid:104) (cid:105) E Requity t t+1 = E t [R t+1 ] E [q ] R t rf E t [q t+1 R t+1 ] t t+1 E [R ] = t t+1 E [qβ ] (26) E (cid:104) qβ R (cid:105) t t+1 t t+1 t+1 wherethesecondequalityfollowsfrom(23). (26)istheequitypremiuminthemodelwithout investment risk, but with stochastic discount factor β¯. t Proposition 8. The risk-free rate, Rrf = E [q ]−1, will be lower than in the model t t t+1 without investment risk, but with stochastic discount factor β¯ given by (22). t Proof. Substitute from (23) to write the risk-free rate as: (cid:104) (cid:105)−1 qβ Rrf =E qβ t+1 (27) t t t+1 q t+1 In the model without investment risk, but with time-preference shocks, the risk-free rate is (cid:104) (cid:105)−1 given by E qβ . And, from Lemma 5, q >qβ . t t+1 t+1 t+1 Thus, although aggregate quantities and the equity premium are same as in a model without investment risk, but with time-preference shocks given by (22), the risk-free rate is lower. 3.3.3 Effects of an increase in idiosyncratic uncertainty Next, I will characterize how the investment wedge and the state-price density change in response to an increase in idiosyncratic uncertainty. Proposition 9. There exists a γ > 1 such that, if γ < γ, an increase in pointwise risk in P leads to a strict increase in the investment wedge Rt+1 = qt+1; and a strict decrease t Requity qβ t+1 t+1 in the ratio of the risk-free rate to the risk-free rate that would obtain in the model without investment risk, but with stochastic discount factor β¯ given by (22). t Proposition9highlightsthedifferentforcesaffectingassetpricinginageneralequilibrium environment with idiosyncratic risk. As is well known from Hadar and Seo (1990) and Gollier (1995), in a standard partial-equilibrium portfolio problem with one safe asset and one risky asset, a mean-preserving spread in the return on the risky asset does not, in general, implythatarisk-averseagentwillreduceherallocationtotheriskyasset. Instead, with constant relative risk aversion, Hadar and Seo (1990) show that a mean-preserving spread leads to a decrease in the allocation to that asset if γ < 1. Correspondingly, in my model, if γ > 1, it is possible to construct a mean-preserving spread in si such that the t+1 investment wedge decreases. In particular, any mean-preserving spread in si that leaves t+1 the density unchanged for all si < s∗ will result in a strict decrease in the investment t+1 t wedge. However, with an increase in pointwise risk in si , there is always an endogenous t+1 increase in “downside risk,” in the form of a strict decrease in ψ . This guarantees that if t 14

relative risk aversion is not too large, an increase in pointwise risk in si will lead to an t+1 increase in the investment wedge.4 It also possible to define a stricter notion of an increase in risk such that the investment wedge increases with risk, for every γ. In particular, recall that a pointwise increase in risk transfers probability mass from the middle of the distribution, with si ∈ (s(cid:48),s(cid:48)(cid:48)), to the t+1 lower and upper parts of the distribution, with si ∈[smin,s(cid:48)] (cid:83) [s(cid:48)(cid:48),smax]. Any pointwise t+1 increase in risk with s(cid:48)(cid:48) <s∗ leads to a strict increase in the investment wedge.5 t 3.4 Deterministic steady state If there is no aggregate uncertainty, we can characterize the steady state of the economy: Proposition 10. Suppose that (cid:34) (cid:35)−1 logβ (cid:15)>(cid:15)≡ 1+ . (28) log (cid:0)CE [gi ] (cid:1) t t+1 Then there exist unique steady-state values K∗,C∗,L∗,I∗,Y∗. A mean-preserving spread in gi is associated with a decrease in K∗,C∗,L∗,I∗,Y∗ if and only if (cid:15)>1. t+1 Proposition 10 follows from the result that aggregate quantities are the same as in an economywithoutinvestmentrisk,butwithadifferentdiscountfactorgivenby(22). If(cid:15)>(cid:15), the equivalent discount factor is less than one and a unique steady state exists. Moreover, steady-state aggregate capital, consumption and employment decrease to a lower steady state following an increase in idiosyncratic risk if and only if (cid:15) > 1. To see this, note that aggregate quantities respond to an increase in idiosyncratic risk as if managers were more impatient, if and only if (cid:15)>1. Wecanalsofurthercharacterizetherisk-freerate. If(cid:15)>(cid:15),thereisauniquesteady-state value for the risk-free rate, given by: 1 1 Rrf∗ = β ψ t γCE t [g t i +1 ]1 (cid:15) −γ < β (29) Proposition 8 implies the inequality in (29). The analogous result to Proposition 9 is that if relative risk aversion γ is not too high, an increase in pointwise risk in P leads to a strict t decrease in the risk-free rate. However, whereas the threshold in Proposition 9 was γ > 1, here it is γ > 1. Similarly, any pointwise increase in risk with s(cid:48)(cid:48) < s∗ leads to a strict (cid:15) t decrease in the risk-free rate. 4 Quantitative implications of idiosyncratic investment risk Thecalibrationapproachhereistocharacterizethefinancialandmacroeconomicresponseto ashocktoidiosyncraticuncertainty,whilemakingasfewaspossibleparametricassumptions. 4Ofcourse,forthesamereason,anincreaseinpointwiseriskinsi willleadtoanincreaseintheratio t+1 oftherisk-freeratetotherisk-freeratethatwouldobtaininthemodelwithtime-preferenceshocksandthe samepathforaggregatequantities. 5To see this, note that such a pointwise increase in risk leads to an increase in CE t[γ t −1g t i +1 ]γ, the numeratoroftheinvestmentwedge(23),andadecreaseinthedenominator,CE t[g t i +1 ]. 15

Specifically, for several P ∈ P, I calculate the distribution of idiosyncratic consumption t growth, gi , and the investment wedge, Rt+1 . I also calculate, for several P ∈ P, the t+1 Requity t t+1 discount factor β¯ such that aggregate quantities and the equity premium in the model are t the same as in a model without idiosyncratic investment risk, but with discount factor β¯. t In doing so, I do not need additional assumptions about the technologies for producing consumption and capital goods. Also, I will not have to specify the Markov process for the technology shock z and the idiosyncratic risk shock P . That is, the results presented t t are such that, for any given Markov process for P ∈ P, one can calculate the Markov t processforβ¯. Althoughthisapproachoffersonlyapartialcharacterizationofthedynamics t of the economy, the results will be consistent with, for example: an AR(1) process for the (log) standard deviation of idiosyncratic shocks (as in Gilchrist, Sim and Zakrajšek (2013)); or an AR(1) process for the growth rate of the standard deviation of idiosyncratic shocks (which would give rise to “quantity-equivalent” time-preference shocks similar to those in Albuquerque, Eichenbaum and Rebelo (2012)). Also, previous theoretical results will be informative about how the calibration results would vary with different parametric assumptions: for example, the investment wedge does not depend on the EIS, and the “quantity-equivalent” discount factor is decreasing in the EIS. Inaddition, Icalculatethemarginalproductofcapitalandtherisk-freeratethatwould prevail in the steady state of the model when there is no aggregate uncertainty. 4.1 Parameter choice I assume that si follows a Pareto distribution with standard deviation σ ∈ (0,∞). As t+1 t before, I assume E [si ]=1. Thus, the tail parameter α satisfies t t+1 t (cid:113) α =1+ 1+σ−2 >2 t t and the cumulative distribution function P is given by t P (si )=1− (cid:0) 1−α−1(cid:1)αt(cid:0) si (cid:1)−αt t t+1 t t+1 for any si ≥1−α−1. Below, I repeat the analysis under the alternative assumption that t+1 t si follows a log-normal distribution. t+1 In the literature on idiosyncratic investment risk, Panousi (2012) uses a constant value of σ = 0.3, while Roussanov (2010) sets σ = 0.45. Using panel data on private-business income, DeBacker et al. (2012) also suggest σ = 0.45. One can also use idiosyncratic stock-market returns to calibrate σ , consistent with the public-company implementation t of financial markets discussed in Section 2. Goyal and Santa-Clara (2003) find that the average volatility of individual stock returns is 16 percent per month, suggestive of annual volatility of individual stock returns between 50 and 60 percent. Thus, below I consider σ ∈[0.15,0.6]. The corresponding range for the Pareto tail parameter is α ∈[2.9,7.7]. t t Tocalibratetherisk-sharingtechnology(parametrizedbyθ inthismodel),theliterature on idiosyncratic investment risk typically looks to data on the volatility of idiosyncratic consumption growth. Of course, disaggregated consumption data is subject to a variety of limitations: thereissignificantmeasurementerror;householdsintheConsumerExpenditure Survey (CEX) are followed for only a short period of time; and idiosyncratic consumption maybedrivenbydeterministicfactorssuchasageandaggregateconsumption. Todealwith these issues, Blundell, Pistaferri and Preston (2008) impute consumption using the Panel 16

Study of Income Dynamics and the CEX, and regress imputed log annual consumption on year and year-of-birth dummies and a range of family characteristics. The first difference of these residuals corresponds, in my model, to the log of the consumption-share growth rate, log(gi ). Using the results in Blundell, Pistaferri and Preston (2008), a reasonable t+1 estimate of the cross-sectional standard deviation of this first difference is 13 percent per annum.6 Blundell,PistaferriandPreston(2008)studyconsumptiondynamicsfortheentire population. In this paper, the focus is on investors and managers. Jacobs and Wang (2004) estimate the standard deviation of idiosyncratic consumption growth across all households and also limiting the sample only to asset holders. Averaging across time, they find that the mean standard deviation across all households is roughly similar to the mean standard deviationacrossassetholders. However,controllingforageandeducation,JacobsandWang find that the standard deviation of idiosyncratic consumption growth across asset holders is about 50 percent larger than the standard deviation across all households. A number of calibration exercises make explicit or implicit choices about the volatility of the idiosyncratic consumption share. De Santis (2007) assumes that the cross-sectional standard deviation of log(gi ) has a mean of 10 percent. In Panousi (2012), the standard t+1 deviation of idiosyncratic consumption growth, in the steady state, is 7 percent. As a benchmark,Ichooseavalueofθ =0.3,whichresultsinacross-sectionalstandarddeviation of the log idiosyncratic consumption share growth rate equal to 8 percent if σ =0.3. t Fortheelasticityofintertemporalsubstitution,Ichoose(cid:15)=2. Gourio(2012)choosesthe samevalue. AnEISgreaterthanoneisrequiredforanincreaseinidiosyncraticuncertainty to generate an economic contraction. I also consider how a lower EIS effects the results. I setthediscountfactorβ equalto0.95andthecoefficientofrelativeriskaversionγ equalto 4. 4.2 Effects of idiosyncratic uncertainty on financial and macroeconomic variables Table 1 shows how different levels of idiosyncratic uncertainty affect risk sharing, macroeconomic dynamics and risk premia. The first column corresponds to the benchmark case, with σ =0.3. The second and third columns correspond to a 50 percent and a 100 percent t increase in the standard deviation of idiosyncratic shocks, as in Bloom (2009). The final column represents a 50 percent decrease in the standard deviation of idiosyncratic shocks. Relative to the benchmark case, a 50 percent increase in σ leads to a roughly prot portional increase in the volatility of idiosyncratic consumption. In the benchmark case, the lower bound of idiosyncratic consumption-share growth, ψ , is fairly high: following t the worst possible shock, which corresponds to a decrease in capital of about 20 percent, a manager’s consumption share falls only 2 percent.7 With a 50 percent increase in σ , a t manager’s consumption share falls twice as much (that is, about 4 percent) following a bad shock. The macroeconomic and financial effects of idiosyncratic risk are summarized in the next rows. With a 50 percent increase in σ , the response of aggregate quantities and the t 6Blundell,PistaferriandPreston(2008)usedatafrom1980to1992. InTable4oftheirpaper,theyshow varianceandfirst-orderautocovarianceoftheresidualsfromregressiondescribedhere. Tofindafirst-order measureofthevariancethatisnotcontaminatedbyimputationandothererrors,theysuggestsubtracting twice the absolute value of the first-order auto-covariance from the variance. Due to data availability, this ispossibleonlyfor8years. Theaverageofthisadjustedvariancemeasureoverthese8yearsis0.132. 7Theworstpossibleshocksi correspondsto1−α−1. t+1 t 17

equity premium are the same as in a model without investment risk, but with a discountfactor shock of approximately 70 basis points. That is, aggregate quantities and the equity premiumrespondasiftherewerenoinvestmentrisk,butthediscountfactordecreasedfrom 0.945 to 0.938. At the same time, the investment wedge – the spread between the return on a firm’s physical capital and the return on financial claims on the firm – increases from 5 percentage points to 10 percentage points. That is, in the benchmark case, firms invest as if their cost of capital were 5 percentage points higher than it actually is; with a 50 percent increase in idiosyncratic uncertainty, that wedge increases to 10 percentage points. Equivalently, the risk-free rate following the uncertainty shock will be about 5 percentage pointslowerthaninamodelwithoutinvestmentrisk, butwithadiscount-factorshockthat generates the same response for aggregate quantities. Thefinalrowshowsthesteady-staterisk-freeratethatwouldobtainiftherewerenoaggregate uncertainty. In the benchmark case, the risk-free rate would be about 1 percentage point per year. With a 50 percent increase in σ , the risk-free rate would be -3 percentage t points per year. These results show that, due to imperfect risk sharing, shocks to idiosyncratic uncertainty of the size contemplated in Bloom (2009) can lead to very large shocks to the real risk-free rate, of similar magnitude to the exogenous “shock to the natural rate of interest” studied in Eggertsson and Woodford (2003) and Christiano, Eichenbaum and Rebelo (2011). If the EIS were different, the distribution of idiosyncratic consumption-share growth, gi , and the investment wedge would not change. Thus, the only variables reported in t+1 Table1thatwouldchangearethe“quantity-equivalent” discountfactor,β¯,andthesteadyt state risk-free rate that would obtain if there were no aggregate uncertainty. From (22) and (27), one observes that, all else equal, β¯ is decreasing in the EIS and that Rrf∗ is t increasing in the EIS. As shown in Table 2, the size of the quantity-equivalent discountfactor shock corresponding to a 50 percent increase in idiosyncratic uncertainty would be somewhat smaller if the EIS were 1.5, at 50 basis points, rather than 70 basis points. The steady-staterisk-freerateatthebaselinelevelofidiosyncraticuncertaintywouldbe20basis points lower, and a 50 percent increase in idiosyncratic risk would result in a slightly larger drop in the steady-state risk-free rate. UnlikeachangeintheEIS,achangeinrelativeriskaversionwouldaffecttheinvestment wedge, as shown in Table 3. If relative risk aversion were halved, the investment wedge would be 3 percentage points at the baseline level of idiosyncratic uncertainty, rather 5 percentage points. Moreover, a 50 percent increase in idiosyncratic uncertainty would lead to an investment wedge of 7 percentage points, rather than 10 percentage points. Similarly, the effects of idiosyncratic uncertainty on the “quantity-equivalent” discount factor and the steady-state risk-free rate would also be somewhat smaller if relative risk aversion were halved. Finally, Table 4 shows how the results would be affected by assuming that P is logt normal, rather than Pareto.8 At the baseline level of idiosyncratic uncertainty, assuming that P is log-normal implies that idiosyncratic consumption volatility is somewhat lower, t and the lower-bound on consumption-share growth is a bit higher. Correspondingly, at the baseline level of idiosyncratic uncertainty, if P is log-normal, the quantity-equivalent t discount factor and the steady-state risk-free rate are slightly higher, and the investment wedge is slightly lower. However, at higher levels of idiosyncratic uncertainty, the effects of idiosyncratic uncer- 8Thatis,inTable4,Iassume: logsi t+1 isnormallydistributed;Var(si t+1 )=σ t 2;andEt[si t+1 ]=1. 18

taintyaremuchlargerifP islog-normalthanifP isPareto. Followingashiftfromσ =0.3 t t t to σ = 0.45, the volatility of idiosyncratic consumption-share growth increases 7 percentt age points, and the lower-bound on the consumption-share growth rate falls 3 percentage points. Correspondingly, the quantity-equivalent discount-factor shock and the increase in the investment wedge are larger in magnitude than if P is Pareto. t 5 Conclusion In this paper, I have argued that shocks to uncertainty about idiosyncratic return on investment can explain the aftermath of financial crisis – elevated risk premia, a sharp and persistent decrease in investment, and a decrease in the risk-free rate. More specifically, aggregate quantities and the equity premium respond to an increase in idiosyncratic uncertainty as if there were no idiosyncratic investment risk and instead managers experienced an“impatience” time-preferenceshock. However, unlikeanimpatienceshock, anincreasein idiosyncratic uncertainty leads to a decrease in the risk-free rate. Thus, a shock to idiosyncratic uncertainty is similar to the “risk premium” shock that Smets and Wouters (2007) , Galí, Smets and Wouters (2011) and Galí, Smets and Wouters (2012) find is important for explaining short-run movements in output, employment and interest rates, as well as the depth of and slow recovery from the Great Recession. In these papers, the “risk premium” shock makes it unattractive to hold physical capital. In order for the risk-free rate to fall meaningfully following such a shock, one needs the supply of financial assets to be somewhat inelastic. In Smets and Wouters (2003), this is the case because the only financial asset is a government bond, and government spending is exogenous. Inmypaper,followingashocktouncertaintyaboutidiosyncraticrateofreturn, thesupplyoffinancialassetsissomewhatinelasticbecausethewaytocreatefinancialassets is by deploying more physical capital. Thus, as each manager seeks to shift from investing in her own physical capital to investing in diversified claims on other managers’ physical capital, the other managers are trying to do the same. 19

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Appendix: Proofs omitted from the text Proof of Lemma 1 Define B(xi)= max CE (cid:2) V(wi ;t+1) (cid:3) (30) t t t+1 ke,i,{wi } t ,t+1 subject to E (cid:2) q wi (cid:3) ≤xi+E [q R −1]pKke,i t t+1 t+1 t t t+1 t+1 t t and (8). Using (30), we can re-write (7) as V(wi;t)=maxU−1(cid:2) U(wi−xi)+βU (cid:0) B(xi;t (cid:1)(cid:3) (31) t t t t xi t Conjecture the following solution: V(wi;t) = ξ wi (32) t t t ci t = c˜ (33) wi t t pKke,i t t = (1−c˜)κ wi t t t ni t+1 = (1−c˜)η wi t t+1 t where ξ is an endogenous random variable. t Substituting from (9) and (32) into (30), one obtains: (cid:104) (cid:110) (cid:111)(cid:105) B(xi)= max CE ξ max ni ,(1−θ)R si pKke,i (34) t t t+1 t+1 t+1 t+1 t t ke,i,{wi } t ,t+1 subject to (cid:104) (cid:110) (cid:111)(cid:105) E q max ni ,(1−θ)R si pKke,i ≤xi+E [q R −1]pKke,i (35) t t+1 t+1 t+1 t+1 t t t t t+1 t+1 t t Dividing the first-order condition with respect to ke,i by the first-order condition with t respect to ni , one obtains: t+1   1 ni (1−θ)E (cid:2) q R si 1{si >si∗ } (cid:3) −E [q R −1] ξ1−γ γ pK t t k + t e 1 ,i = (1− t θ) t 1 + − 1 γE t t + (cid:104) 1 (cid:0) ξ t t + + 1 1 R t+ t+ 1 s 1 i t+1 (cid:1) t 1 + − 1 γ 1{si t+1 t > t+ s 1 i t ∗ +1 t } + (cid:105) 1  q t t + + 1 1  where si∗ is defined by ni =(1−θ)R si∗ pKke,i. This, together with the linearity of t+1 t+1 t+1 t+1 t t (35), implies that the solutions ni and ki to (34)-(35) are linear in xi and that B(xi;t) is t t t t linear in xi. Then, from (31), one obtains that xi and ci are linear in wi, consistent with t t t t (33). Next, using the envelope condition of (31), one obtains: dVi (cid:18) ci (cid:19)−1 (cid:15) t = t =ξ dwi Vi t t t 22

Substituting from (32) and (33), we have 1 ξ =c˜1−(cid:15) t t and thus B(xi)=ρ xi. (36) t t t Substituting (36) into (31) and taking the first-order condition with respect to xi, one t obtains (11).♦ Proof of Lemma 2 From Lemma 1, each manager’s investment and consumption decisions are linear in her wealth. Thus, we have that gi = R t i +1. Market clearing requires that κ = 1: all cont+1 Rt+1 t sumptiongoodsthatarenotconsumedareusedasinputstocreatecapitalgoods. Financial market clearing implies that, for each ht+1, ˆ smax Ri dP (si )=R . t+1 t t+1 t+1 smin Dividing both sides by R and denoting ψ = ηt+1, one obtains (19). Note that ψ ∈ t+1 t Rt+1 t ((1−θ)smin,1) if smax > 1 and that ψ =1 otherwise. ♦ 1−θ t Proof of Lemma 3 ´ SupposethatP(cid:101)t ispointwiseriskierthanP t . Notethat s s m m i a n x max{ψ t ,(1−θ)si t+1 }dP(cid:101)t (si t+1 )> 1. Hence, if P(cid:101)t is the c.d.f. of si t+1 , then ψ t no longer solves (19). Instead, there is a unique ψ˜ <ψ that solves (19). t t Denotethecumulativedistributionfunctionofgi byG(gi ;P ),wheregi isdefined t+1 t+1 t t+1 by (18) and (19) and the c.d.f. of si is P . Then there exists a g¯∈[ψ ,(1−θ)smax) such t+1 t t thatG(g t i +1 ;P(cid:101)t )>G(g t i +1 ;P t )ifg t i +1 ∈[ψ˜ t ,g¯)andG(g t i +1 ;P(cid:101)t )≤G(g t i +1 ;P t )ifg t i +1 ≥g¯. In addition, G(g t i +1 ;P(cid:101)t )=G(g t i +1 ;P t )=0 if g t i +1 <ψ˜ t . ♦ Proof of Proposition 9 Let h(P ) denote CE [ g t i +1]γCE [gi ]−1, where the expectations are taken with respect t t ψt t t+1 to P and where gi and ψ are determined using (18) and (19) conditional on P . Thus, t t+1 t t from Lemma 5, h(P ) = qt+1. Suppose that P˜ is pointwise riskier than P . From Lemma t qβ t t t+1 3, a shift from P to P˜ induces a mean-preserving spread in gi and a strict decrease in t t t+1 ψ . Finally, note that lim h(P )−h(P˜)=ψ−1−ψ˜−1 <0 and that h(P )−h(P˜)<0 if t γ→1 t t t t t t γ <1.♦ 23

ytniatrecnu citarcnysoidi ot kcohs fo stceffE :1 elbaT 51.0= σ 6.0= σ 54.0= σ 3.0= σ t t t t 7.2 7.61 0.31 3.8 ) ig(gol fo .ved .dts noitpmusnoc citarcnysoidi fo ytilitaloV 1+t 3.0- 4.6- 2.4- 0.2- ) ψ(gol htworg erahs-noitpmusnoc no dnuob rewoL t 949.0 039.0 839.0 549.0 ¯β rotcaf tnuocsid ”tnelaviuqe-ytitnauQ“ t 0.1 9.31 7.9 1.5 1− 1+tR egdew tnemtsevnI ytiuqeR 1+t 3.4 7.5- 8.2- 7.0 1−∗frR ytniatrecnu etagergga on fi etar eerf-ksir etats-ydaetS .sliated rof txet eeS .rotcaf tnuocsid ”tnelaviuqe-ytitnauq“ eht rof tpecxe ,stniop egatnecrep ni detroper selbairav llA :etoN 24

5.1=(cid:15) fi ,ytniatrecnu citarcnysoidi ot kcohs fo stceffE :2 elbaT 51.0= σ 6.0= σ 54.0= σ 3.0= σ t t t t 059.0 739.0 249.0 749.0 ¯β rotcaf tnuocsid ”tnelaviuqe-ytitnauQ“ t 3.4 3.6- 2.3- 5.0 1−∗frR ytniatrecnu etagergga on fi etar eerf-ksir etats-ydaetS .sliated rof txet eeS .rotcaf tnuocsid ”tnelaviuqe-ytitnauq“ eht rof tpecxe ,stniop egatnecrep ni detroper selbairav llA :etoN 25

2= γ fi ,ytniatrecnu citarcnysoidi ot kcohs fo stceffE :3 elbaT 51.0= σ 6.0= σ 54.0= σ 3.0= σ t t t t 059.0 539.0 149.0 749.0 ¯β rotcaf tnuocsid ”tnelaviuqe-ytitnauQ“ t 6.0 2.01 9.6 4.3 1− 1+tR egdew tnemtsevnI ytiuqeR 1+t 7.4 0.3- 6.0- 2.2 1−∗frR ytniatrecnu etagergga on fi etar eerf-ksir etats-ydaetS .sliated rof txet eeS .rotcaf tnuocsid ”tnelaviuqe-ytitnauq“ eht rof tpecxe ,stniop egatnecrep ni detroper selbairav llA :etoN 26

lamron-gol si is fi ytniatrecnu citarcnysoidi ot kcohs fo stceffE :4 elbaT 1+t 51.0= σ 6.0= σ 54.0= σ 3.0= σ t t t t 5.0 5.81 0.21 5.5 ) ig(gol fo .ved .dts noitpmusnoc citarcnysoidi fo ytilitaloV 1+t 0.0- 4.9- 9.4- 5.1- ) ψ(gol htworg erahs-noitpmusnoc no dnuob rewoL t 059.0 629.0 939.0 849.0 ¯β rotcaf tnuocsid ”tnelaviuqe-ytitnauQ“ t 1.0 3.42 6.31 5.4 1− 1+tR egdew tnemtsevnI ytiuqeR 1+t 1.5 1.31- 3.6- 0.1 1−∗frR ytniatrecnu etagergga on fi etar eerf-ksir etats-ydaetS .sliated rof txet eeS .rotcaf tnuocsid ”tnelaviuqe-ytitnauq“ eht rof tpecxe ,stniop egatnecrep ni detroper selbairav llA :etoN 27