NBER WORKING PAPER SERIES RARE DISASTERS, ASSET PRICES, AND WELFARE COSTS. Robert J. Barro. Working Paper

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1 NBER WORKING PAPER SERIES RARE DISASTERS, ASSET PRICES, AND WELFARE COSTS Rober J. Barro Working Paper hp:// NATIONAL BUREAU OF ECONOMIC RESEARCH 1050 Massachuses Avenue Cambridge, MA December 2007 This research is suppored by he Naional Science Foundaion. I appreciae commens from Fernando Alvarez, Marios Angeleos, John Campbell, Raj Chey, John Cochrane, Xavier Gabaix, Francois Gourio, Lars Hansen, David Laibson, Jong-Wha Lee, Greg Mankiw, Ian Marin, Casey Mulligan, and Ivan Werning. The views expressed herein are hose of he auhor(s) and do no necessarily reflec he views of he Naional Bureau of Economic Research by Rober J. Barro. All righs reserved. Shor secions of ex, no o exceed wo paragraphs, may be quoed wihou explici permission provided ha full credi, including noice, is given o he source.

2 Rare Disasers, Asse Prices, and Welfare Coss Rober J. Barro NBER Working Paper No December 2007 JEL No. E2,G12,O4 ABSTRACT A represenaive-consumer model wih Epsein-Zin-Weil preferences and i.i.d. shocks, including rare disasers, accords wih key asse-pricing observaions. If he coefficien of relaive risk aversion equals 3-4, he model accords wih observed equiy premia and risk-free real ineres raes. If he ineremporal elasiciy of subsiuion is greaer han one, an increase in uncerainy lowers he price-dividend raio for equiy, whereas a rise in he expeced growh rae raises his raio. In a model wih endogenous saving, more uncerainy lowers he saving raio (because subsiuion effecs dominae). The mach wih major feaures of asse pricing suggess ha he model is a reasonable candidae for assessing he welfare cos of aggregae consumpion uncerainy. In he baseline simulaion, he welfare cos of disaser risk is large -- sociey would be willing o lower real GDP by as much as 20% each year o eliminae he small chance of major economic collapses. The welfare cos from usual economic flucuaions is much smaller, hough sill imporan, corresponding o lowering GDP by around 1.5% each year. Rober J. Barro Deparmen of Economics Liauer Cener 218 Harvard Universiy Cambridge, MA and NBER rbarro@harvard.edu

3 In a previous sudy, Barro (2006), I used he Riez (1988) idea of rare economic disasers o explain he equiy premium and relaed asse-pricing puzzles. My quaniaive examinaion of large macroeconomic conracions in 35 counries during he 20 h cenury suggesed a disaser probabiliy of roughly 2% per year. The size disribuion of GDP conracions during hese evens ranged beween 15% (he arbirary lower bound) and over 60%. A simple represenaive-agen economy, calibraed o accord wih his disaser experience, can explain an equiy premium of around 4-6% and a risk-free real ineres rae of abou 1-2%. Wih power-uiliy preferences, hese resuls require a coefficien of relaive risk aversion of 3-4. The analysis applies in a Lucas-ree economy wih i.i.d. producion shocks or o an AK model wih endogenous saving and sochasic depreciaion. The presen analysis exends he framework o consider addiional aspecs of asse pricing and o assess he welfare cos of consumpion uncerainy. As observed by Bansal and Yaron (2004), power-uiliy preferences wih a coefficien of relaive risk aversion above one generae wo implausible predicions. Firs, an increase in uncerainy raises he price-dividend raio for equiies, and second, a rise in he mean growh rae lowers he price-dividend raio. More reasonable predicions require an ineremporal elasiciy of subsiuion (IES) above one. However, in he power-uiliy framework, his propery conflics wih a coefficien of relaive risk aversion greaer han one a condiion needed o mach observed equiy premia and risk-free raes. Therefore, o fi a broad se of asse-pricing facs, i is essenial o use a preference specificaion, such as ha of Epsein and Zin (1989) and Weil (1990), ha de-links he IES from he

4 coefficien of relaive risk aversion. Power uiliy, alhough aracive for is simpliciy, canno work. The framework is sill a represenaive-consumer model wih i.i.d. shocks o producion. In his seing, he key asse-pricing condiions under Epsein-Zin-Weil (henceforh, EZW) preferences resemble hose wih power uiliy. However, wo key differences emerge. Firs, under EZW preferences, consumpion eners ino asse-pricing formulas wih an exponen ha involves he coefficien of relaive risk aversion, no he IES. Second, he formulas involve an effecive rae of ime preference, denoed ρ*, ha deviaes from he usual rae of ime preference, ρ, when he coefficien of relaive risk aversion is unequal o he reciprocal of he IES. The value of ρ* depends on ρ, he IES, he coefficien of relaive risk aversion, and he oher parameers of he model including parameers ha describe expeced growh and uncerainy. Wih i.i.d. shocks, he EZW framework ends up as simple as he power-uiliy seing, and i accords wih a broader se of asse-pricing facs. Firs, when calibraed o he observed frequency and size disribuion of macroeconomic disasers, he model can explain he equiy premium and risk-free rae, sill wih a coefficien of relaive risk aversion of 3-4. Second, wih an IES above one, he model predics ha an increase in uncerainy lowers he dividend-price raio, whereas a rise in he expeced growh rae raises his raio. Third, in an AK model ha allows for endogenous saving, he IES above one implies ha more uncerainy lowers he saving raio (because subsiuion effecs dominae when he IES exceeds one). Lucas (1987, Ch. 3; 2003, secion II) argued ha he welfare gain from eliminaing aggregae consumpion uncerainy is rivial. One problem wih his 2

5 calculaion, apparen from Mehra and Presco (1985), is ha simulaions wih Lucas s model do no ge ino he righ ballpark for explaining he high equiy premium and low risk-free rae. These failures wih respec o asse reurns sugges, as observed by Akeson and Phelan (1994), ha he model misses imporan aspecs of consumpion uncerainy. Hence, he model s esimaes of welfare effecs from aggregae consumpion uncerainy are unlikely o be accurae. A reasonable principle is ha analyses of he impacs of consumpion uncerainy should be carried ou wihin models ha a leas roughly replicae he way ha asse markes price his uncerainy. This Akeson-Phelan principle was followed by Alvarez and Jermann (2004) and is also adoped in he presen paper. In my case, he prospecs of rare economic disasers, as in Riez (1988), are criical for maching asse-pricing facs. Wihin his seing, changes in consumpion uncerainy ha reflec shifs in he probabiliy of disaser have major implicaions for welfare. Individuals would willingly relinquish as much as 20% of GDP each year in exchange for eliminaing all chances for macroeconomic disaser. The welfare cos from usual economic flucuaions is much smaller, hough sill imporan corresponding o lowering GDP by around 1.5% each year. Secion I works ou he Lucas-ree model wih rare disasers. The key assepricing formulas under EZW preferences are derived here. Secion II compues welfare coss wihin his model; firs for marginal changes in uncerainy and hen for large changes. Secion III discusses he sensiiviy of he welfare-cos calculaions o he choices of he wo key preference parameers: he coefficien of relaive risk aversion and he ineremporal elasiciy of subsiuion. Secion IV allows for endogenous labor 3

6 supply one poin here is ha any wage elasiciy of labor supply is compaible wih a given coefficien of relaive risk aversion. Secion V includes endogenous saving and invesmen and shows how adjusmens of saving affec welfare coss. Secion VI concludes by emphasizing he effecs of policies and insiuions on disaser probabiliies and sizes. I. A Lucas Frui-Tree Model The iniial model is a version of Lucas s (1978) represenaive-agen, frui-ree economy wih exogenous, sochasic producion. Oupu of frui in period equals real GDP, Y. Populaion is consan. The number of rees is fixed; ha is, here is neiher invesmen nor depreciaion. (The model in secion IV allows for invesmen.) Governmen purchases are nil. Since he economy is closed and all oupu is consumed, consumpion, C, equals Y. The log of oupu evolves as a random walk wih drif: (1) log(y +1 ) = log(y ) + g + u +1 + v +1. The random erm u +1 is i.i.d. normal wih mean 0 and variance σ 2. This erm reflecs normal economic flucuaions. The parameer g 0 is a consan ha reflecs exogenous produciviy growh. The random erm v +1 in Eq. (1) picks up low-probabiliy disasers, as in Riez (1988) and Barro (2006). In hese rare evens, oupu and consumpion jump down sharply. The probabiliy of a disaser is he consan p 0 per uni of ime. The probabiliy of more han one disaser in a period is assumed o be small enough o 4

7 neglec; laer, he arbirary period lengh shrinks o zero. In a disaser, oupu conracs by he fracion b, where 0<b<1. The disribuion of v +1 is given by probabiliy 1-p: v +1 = 0, probabiliy p: v +1 = log(1-b). The disaser size, b, follows some probabiliy disribuion (gauged subsequenly by he empirical disribuion of disaser sizes). Unlike Lucas (1987, Ch. 3), bu in line wih Obsfeld (1994), he shocks u +1 and v +1 in Eq. (1) represen permanen effecs on he level of oupu, raher han ransiory disurbances o he level. Tha is, he economy has no endency o rever o a deerminisic rend line. Cochrane (1988, Table 1) used variance-raio saisics for k-year differences o assess he exen of reversion o a deerminisic rend in he log of U.S. real per capia GNP for He found evidence for reversion in ha he raio of he k-year variance (divided by k) o he 1-year variance was beween 0.30 and 0.36 for k beween 20 and 30 years. Therefore, a large k, he empirical variance raio was much less han he value 1.0 prediced by Eq. (1). However, Cogley (1990, Table 2) showed ha he Cochrane finding was paricular o he Unied Saes. For 9 OECD counries, including he Unied Saes, from 1871 o 1985, he mean of he variance raio a 20 years was 1.1; hence, close o he value 1.0 prediced by Eq. (1). Cogley s resuls hold up for a broader sample comprising 19 OECD counries. The daa on per capia GDP are for from Maddison (2003), updaed from World Bank, World Developmen Indicaors (and using U.S. daa from Balke and Gordon [1989] before 1929). For k=20, he mean of he variance raios for he 19 counries is 5

8 1.22 and he median is 1.00, while for k=30, he corresponding values are 1.30 and These values accord wih Eq. (1). The Unied Saes wih variance raios of 0.42 when k=20 and 0.38 when k=30 has he lowes raios a hese values of k among he 19 counries. 1 The criical facor for he Unied Saes is ha he urbulence of he Grea Depression and World War II happened o be followed by he log of per capia GDP revering roughly o he pre-1930 and pre-1914 rend lines. Mos oher counries do no look like his. My inference from he long-erm GDP daa for he OECD counries is ha he evidence conflics srongly wih reversion o a fixed, deerminisic rend. The key, couner-facual predicion from his model is he comparaively low uncerainy abou he disan fuure. In conras, he variance-raio resuls are consisen wih he sochasicrend specificaion in Eq. (1). Therefore, I use his model for he presen analysis. Richer models of GDP and consumpion ha I am currenly sudying (in join work wih Emi Nakamura, Jon Seinsson, and Jose Ursua) allow for rend breaks (analyzed saring from Banerjee, Lumsdaine, and Sock [1992]) and for gradual reversion o pas levels afer major disasers, such as desrucive wars and financial crises. Previous research (Barro [2006, Table 1 and Figure 1]) gauged he probabiliy and size disribuion of disaser evens from ime series on per capia real GDP for 35 counries for he full 20 h cenury. 2 In he main empirical analysis, ha sudy defined a 1 The nex smalles values for k=20 are 0.55 for New Zealand, 0.68 for Germany, and 0.77 for Swizerland. A k=30, he nex smalles values are 0.40 for New Zealand, 0.53 for Germany, and 0.54 for Canada. For smaller values of k, he mean and median of he variance raios are, respecively, 1.16 and 1.18 a k=2, 1.23 and 1.31 a k=5, and 1.13 and 1.06 a k=10. The U.S. raios a hese values of k are, respecively, 1.30, 1.34, and The GDP daa were from Maddison (2003). In he frui-ree model, GDP and consumpion coincide. More generally, consumpion would be more appropriae han GDP for analyses of asse pricing and welfare coss. However, long-erm daa on real consumer expendiure are no repored by Maddison and are no readily available for many counries. An ongoing research projec, described in Barro and Ursua 6

9 macroeconomic disaser as a decline in per capia real GDP by a leas 15% over consecuive years (such as for France and for he Unied Saes). These kinds of evens are rare only 60 cases were found in he long-erm experiences of he 35 counries; ha is, less han 2 per counry. 3 Therefore, o use hisory o gauge he probabiliy and size disribuion of macroeconomic disasers, i is hopeless o rely on he experience of a single counry, such as he Unied Saes, even if we are willing o assume ha he U.S. economic srucure remained fixed for 100 years or more. 4 In conras, in long ime series for a broad inernaional sample, enough disaser realizaions are available o allow for reasonably accurae inferences abou disaser probabiliies and size disribuions. Underlying his calculaion, of course, is he assumpion ha he underlying probabiliy disribuions are reasonably similar across counries, as well as roughly sable over ime. For he 35 counries, he main global disasers were World War II (18 counries wih large GDP conracions), he Grea Depression (16 counries), World War I (13 counries), and pos-world War II depressions in Lain America and Asia (11 counryevens). The empirical frequency 60 evens for 35 counries over 100 years corresponds o a disaser probabiliy, p, of 1.7% per year. (The disasers need no be (2008), involves he assembly of a daa se on long-erm real personal consumer expendiure for as many counries as possible. 3 The 60 cases exclude 5 pos-war GDP conracions ha did no involve large declines in real personal consumer expendiure. The lower limi of 15% is arbirary. Exending o 10% brings in anoher 21 conracions for he 35 counries. However, he inclusion of hese smaller conracions has a minor effec on he resuls. 4 Chaerjee and Corbae (2007) use he U.S. hisory of he unemploymen rae o noe (p. 1534) ha here is only one depression episode in he sample. From hese daa and an assumpion of unchanged economic srucure since 1900 hey infer a probabiliy of moving from normalcy o depression of once every 83 years. This probabiliy and he size disribuion of depressions canno be gauged accuraely from his one ime series. Moreover, hey assume wihou discussion ha real GDP always revers o a deerminisic rend line, alhough, as already noed, Cogley s (1990) and oher analyses indicae ha he daa for mos counries srongly rejec his hypohesis. Salyer s (2007) analysis is similar in spiri o Chaerjee and Corbae s. 7

10 independen across counries; in fac, hey end o congregae ino evens such as world wars, he Grea Depression, he Asian financial crisis, and he Lain American deb crisis.) The conracion proporion b for he observed 20 h cenury disasers ranged from 15% o 64%, wih a mean of 29%. 5 However, wih subsanial risk aversion, he effecive average value of b is subsanially above he mean. For example, wih a coefficien of relaive risk aversion of 4, a consan b of around 40% generaes abou he same equiy premium and welfare effecs as he empirically observed frequency disribuion of b. The formulaion neglecs rare bonanzas. Wih subsanial risk aversion, bonanzas do no coun nearly as much as disasers for he pricing of asses and for welfare effecs. Moreover, long-erm daa on annual growh raes of per capia GDP end o exhibi negaive skewness. For 19 OECD counries from 1880 o 2005, 14 exhibi negaive skewness, and he only subsanially posiive values are for France, he Neherlands, and Swizerland. The expeced growh rae of real GDP depends no only on he growh-rae parameer, g, bu also on he uncerainy parameers. As he lengh of he period approaches zero, he specificaion in Eq. (1) implies ha he expeced growh rae of GDP and consumpion, denoed by g*, is given by (2) g* = g + (1/2)σ 2 p Eb, where Eb is he expeced value of b 0.29 in he sample of 60 observed crises. In pracice, he erm (1/2)σ 2 ends o be negligible in he calibraions considered laer, for which σ=0.02. However, he erm p Eb is no rivial when p= The 29% figure refers o raw levels of per capia GDP. Wih an adjusmen for rend growh, he mean conracion size was 35%. 8

11 and Eb=0.29. In his case, g=0.025 corresponds o g*=0.020, he value used in he main calibraions. I sar wih he familiar formulaion where he represenaive consumer maximizes a ime-addiive uiliy funcion wih iso-elasic preferences: 1 1 γ (3) U = E [( C ) 1]/(1 γ ) i= 0 i + i, (1 + ρ) where ρ 0 is he rae of ime preference. As is well known, his power-uiliy specificaion implies ha he key parameer γ>0 represens boh he coefficien of relaive risk aversion and he reciprocal of he ineremporal elasiciy of subsiuion, henceforh labeled IES. This resricion maers for welfare-cos calculaions, as observed by Obsfeld (1994), and also generaes predicions abou asse prices ha are probably couner-facual, as argued by Bansal and Yaron (2004). Therefore, I soon generalize he preference formulaion o he seing of Epsein and Zin (1989) and Weil (1990), in which he coefficien of relaive risk aversion is de-linked from he IES. Asse prices and raes of reurn can be deermined in he usual way from he firsorder condiions for consumpion over ime. Wih he power-uiliy formulaion from Eq. (3), he familiar firs-order condiions are γ 1 γ (4) C = ( ) E ( R C+ 1), 1+ ρ where R is he gross reurn on any asse from ime o ime +1. A key variable is he marke value, V, of a ree ha iniially produces one uni of frui. One way o calculae his value is o sum he prices of equiy claims on fuure dividends, C +i =Y +i. (In order o correspond o he summaion in Eq. [3], i is convenien o rea C, raher han C +1, as he firs payou on ree equiy.) These prices 9

12 can be deermined readily from Eq. (4). As he arbirary period lengh approaches zero, he reciprocal of V urns ou o be (5) 1/V = ρ + (γ-1) g* (1/2) γ (γ-1) σ 2 p [E(1-b) 1-γ 1 (γ-1) Eb], where g* is he expeced growh rae (of dividends) from Eq. (2), E(1-b) 1-γ is he expecaion of (1-b) 1-γ, and Eb is he expecaion of b. The variable V corresponds o he price-dividend raio for an unlevered equiy claim on a ree. Given he pricing formula in Eq. (5), he expeced rae of reurn on unlevered equiy can be deermined (when he period lengh approaches zero) o be (6) r e = ρ + γg* - (1/2) γ (γ-1) σ 2 - p [E(1-b) 1-γ (γ-1) Eb]. Therefore, he righ-hand side of Eq. (5) is he difference beween r e and g*. The ransversaliy condiion, which guaranees ha he marke value of a ree is posiive and finie, is ha his righ-hand side be posiive; ha is, r e >g*. The risk-free ineres rae, r f, can also be deermined from Eq. (4). The resul (when he period lengh approaches zero) is (7) r f = ρ + γg* - (1/2) γ (γ+1) σ 2 - p [E(1-b) -γ γ Eb]. (Since he model has i.i.d. shocks, he erm srucure of risk-free raes is fla; ha is, r f is he shor-erm and long-erm risk-free rae.) Depending on he uncerainy parameers paricularly p and he disribuion of b r f can be very low. In fac, r f can be less han g* and even less han zero. The equiy premium is (8) r e - r f = γσ 2 + p [ E(1-b) -γ - E(1-b) 1-γ - Eb]. Columns 1 and 2 of Table 1 show average real raes of reurn on socks and governmen bills from 1880 o 2005 for 11 OECD counries ha have he necessary longerm daa. The equiy premium, in he sense of he difference beween he wo average 10

13 raes of reurn, is per year. Since he sock reurns reflec leverage, he premium for unlevered equiy would be smaller. For example, wih a deb-equiy raio of 0.5 (corresponding o recen U.S. values), he premium would be around For he model o ge ino he righ ballpark for explaining he equiy premium, he coefficien of relaive risk aversion, γ, has o be well above one. Barro (2006) showed, for plausible values of he uncerainy parameers, especially p and he disribuion of b, ha γ=4 was saisfacory. 6 In any even, γ could no be less han abou 3. One difficuly is ha, if γ>1, Eq. (5) delivers he likely couner-facual predicion ha an increase in uncerainy (higher σ or p or a shif in he disribuion of b oward larger values), for given g*, implies a higher price-dividend raio, V. Bansal and Yaron (2004, p. 1487) make an analogous observaion abou he connecion beween he volailiy of consumpion growh and he price-dividend raio in heir model. The predicion for a posiive relaionship beween he exen of uncerainy and he pricedividend raio conflics wih he usual view ha an increase in aggregae uncerainy ends o depress sock prices. The reason ha he model makes his couner-inuiive predicion is ha, wih power uiliy, he IES is consrained o equal he reciprocal of he coefficien of relaive risk aversion. Therefore, I now relax his resricion (as do Bansal and Yaron [2004]) by adoping he preference specificaion of Epsein and Zin (1989) and Weil (1990). Using a minor modificaion of he Weil (1990) formulaion, he exended uiliy formula is (10) 1 θ (1 θ ) /(1 γ ) {(1 β ) C + β[(1 β )(1 γ ) E U ] } (1 γ ) /(1 θ ) + 1 U =, (1 β )(1 γ ) 6 Tha analysis also ook accoun of parial defaul on bills, ypically due o high warime inflaion. 11

14 where he discoun facor β equals 1/(1+ρ), 1/θ > 0 is he IES, and γ>0 is he coefficien of relaive risk aversion. Equaion (3) is he special case of Eq. (10) when θ=γ. In general, EZW preferences do no allow for simple formulas for pricing asses. However, when he underlying shocks are i.i.d., as already assumed, he analysis simplifies dramaically. A key propery of he soluion under i.i.d. shocks is ha aained uiliy, U, ends up as a simple funcion of conemporaneous consumpion, C : (11) U = ΦC 1-γ, where he consan Φ depends on he parameers of he model. 7 The applicaion of a sandard perurbaion argumen o Eq. (11) implies ha he firs-order condiions for uiliy maximizaion can be expressed as γ 1 γ (12) C = ( ) E ( R C+ 1), 1+ ρ * where R is he gross, one-period reurn on any asse. As usual, hese firs-order condiions will be he basis for asse pricing. Thus, an imporan resul is ha, wih i.i.d. shocks, he condiions for asse pricing under EZW preferences look similar o hose in he power-uiliy model, described by Eq. (4). However, wo key feaures of he EZW resuls are worh sressing. Firs, he exponens on C and C +1 in Eq. (12) involve γ, he coefficien of relaive risk aversion, no θ, he reciprocal of he IES. Second, he effecive rae of ime preference, ρ*, differs from ρ when γ and θ diverge. The formula for ρ* is, if γ 1, 7 Giovannini and Weil (1989, appendix) show ha, wih he uiliy funcion in Eq. (10), aained uiliy, U, is proporional o wealh raised o he power 1-γ. The form in Eq. (11) follows because C is opimally chosen as a consan raio o wealh in he i.i.d. case. The formula for Φ is, if γ 1 and θ 1, ρ Φ = ( 1 γ ( θ γ ) /(1 θ ) ) ρ + ( θ 1) (1 γ ) /(1 θ ) 2 θ 1 1 γ g * (1/ 2) γ ( θ 1) σ ( ) p [ E(1 b) 1 ( γ 1) Eb]. γ 1 12

15 (13) 2 p ρ* = ρ ( γ θ ) g * (1/ 2) γσ ( ) [ E(1 b) γ 1 1 γ 1 ( γ 1) Eb]. In his and subsequen cases, resuls when γ (or, subsequenly, θ) approach one can be derived from sandard limi calculaions. Noe from Eq. (13) ha ρ* depends no only on preference parameers ρ, γ, and θ bu also on parameers ha describe expeced growh and uncerainy g*, σ, p, and he disribuion of b. The resuls imply ha, in he i.i.d. case, asse-pricing formulas derived under EZW preferences coincide wih formulas under power uiliy if ρ* replaces ρ. In paricular, he formulas for V, r e, and r f in Eqs. (5)-(7) remain valid wih he subsiuion of ρ* for ρ. Therefore, in he EZW case, he IES, 1/θ, affecs he price-dividend raio (Eq. [5]) and levels of raes of reurn (Eqs. [6] and [7]) hrough influences on ρ* bu no he equiy premium (Eq. [8]). The equiy premium depends on he coefficien of relaive risk aversion, γ, exacly in he way as in he power-uiliy case. Since he poweruiliy model accorded reasonably well wih observed equiy premia when γ=4, i follows ha he EZW specificaion fis he equiy premium jus as well when γ=4. Wih EZW preferences, he formula for he price-dividend raio, V, in Eq. (5) becomes, afer replacemen of ρ by ρ* from Eq. (13), 2 θ 1 1 (14) 1/ V = ρ + ( θ 1) g * (1/ 2) γ ( θ 1) σ p ( ) [ E(1 b) 1 ( γ 1) Eb] γ 1 γ, if γ 1. For any γ>0, he condiion θ<1 implies ha, wih g* held fixed, V is lower if uncerainy is greaer (higher σ or p or a shif of he b-disribuion oward higher values). 8 8 These resuls apply when he price-dividend raio, V, perains o unlevered equiy. We can insead consider levered equiy, as in Barro (2006, secion III). The relaion beween uncerainy and he price of levered equiy can be negaive even if θ>1. The condiion for increased σ o reduce he levered equiy price is θ < 1 + 2λ, where λ is he deb-equiy raio for claims on rees. For increased p, he condiion depends on he disribuion of disaser sizes, b, and he coefficien of relaive risk aversion, γ. For he baseline specificaion wih γ=4 and he hisorical disribuion of b, he condiion is θ < λ. 13

16 Equaion (14) also implies, if θ<1, ha V is higher if he mean growh rae, g*, is higher (for given uncerainy parameers). This condiion is imporan in Bansal and Yaron (2004), who propose o explain he equiy premium no by disaser risk bu raher by shocks o heir counerpar of g*. They also allow for a ime-varying variance of hese shocks. One limiaion of heir approach is ha quaniaive success depends on very high risk aversion. The coefficien γ has o be around 10 o accoun for observed equiy premia in heir model. Thus, my inference is ha flucuaing long-run growh raes, g*, may usefully supplemen analyses ha include disaser risk bu probably canno be he main basis for explaining he equiy premium. Given he GDP process in Eq. (1), daa on raes of reurn, such as r e and r f, and he price-dividend raio, V, pin down γ and he effecive rae of ime preference, ρ*. Since ρ* depends on a combinaion of ρ and θ (in Eq. [13]), he daa would no allow separae idenificaion of ρ and θ, a finding ha relaes o he observaional-equivalence poin of Kocherlakoa (1990). However, he parameers ρ and θ could be separaely idenified from oher informaion; for example, if we know how V responds o changes in he uncerainy parameers σ, p, and he disribuion of b or he expeced growh rae, g*, in Eq. (14). Alernaively, in he model wih endogenous saving considered in secion IV, idenificaion would follow from informaion abou how he saving raio reacs o changes in σ, p, and he disribuion of b. To summarize, he model wih Epsein-Zin-Weil preferences, disaser risk, and i.i.d. shocks can accord wih some cenral asse-pricing facs. Firs, if he coefficien of relaive risk aversion, γ, is around 4, he equiy premium and risk-free rae can be roughly correc. Second, if θ<1, so ha he IES is greaer han 1, he price-dividend raio, V, 14

17 relaes o aggregae uncerainy and expeced growh in he righ direcions more uncerainy goes along wih lower V and higher expeced growh goes along wih higher V. The simpliciy of he underlying srucure (i.i.d. shocks, represenaive consumer, closed economy wih no invesmen) allows for a closed-form soluion for aained uiliy, U, as a funcion of he underlying parameers of preferences and he oupu process. Obsfeld (1994) derived analogous closed forms in a model wihou disaser risk. A convenien represenaion uses Eqs. (5) and (11) o express U as a funcion of he pricedividend raio, V. The formula, when γ 1 and θ 1, is, up o an inconsequenial addiive consan: 9 (15) ( θ γ ) /(1 θ ) ρ (1 γ ) /(1 θ ) 1 γ U = ( ) V Y. 1 γ Equaion (15), in conjuncion wih Eq. (14), allows for assessmens of he welfare effecs of uncerainy. II. Calculaion of Welfare Effecs Equaions (14) and (15) deermine he dependence of aained uiliy, U, on he expeced growh rae, g*, and he parameers ha govern consumpion risk: σ, p, and he disribuion of b. These effecs can be compared wih hose from proporionae shifs in he iniial level of GDP and consumpion, Y. 9 The form of Eq. (15) does no depend on he paricular sochasic process for oupu in Eq. (1). However, he consancy of he price-dividend raio, V =V, depends on he i.i.d. form of he shocks, u and v. A consan V conflics wih he observed volailiy of price-dividend raios for sock-marke claims. The model can mach his volailiy if he parameers of uncerainy, such as he disaser probabiliy, p, move around. Gabaix (2006) shows ha he main implicaions of he model for asse pricing go hrough if p evolves exogenously in random-walk-like fashion. 15

18 Eq. (15) by a. Local effecs on welfare The marginal effec on uiliy from a proporionae change in Y is given from (16) U ( θ γ ) /(1 θ ) (1 γ ) /(1 θ ) γ Y = ρ ( ) 1 V Y Y. The marginal effec from a change in g* follows from Eqs. (14) and (15) as (17) U ( θ γ ) /(1 θ ) [1+(1 γ ) /(1 θ )] γ = ρ V ( Y ) 1 g *. Therefore, he uiliy rae of ransformaion beween proporionae changes in Y and changes in g* is given by U / g * (18) = V. ( U / Y ) Y This resul gives he proporionae decrease in Y ha compensaes, a he margin, for an increase in g* in he sense of preserving aained uiliy. Equaion (18) shows ha his compensaing oupu change depends only on he combinaion of parameers ha ener ino he price-dividend raio, V, deermined in Eq. (14). To pin down a reasonable magniude for V, sar wih he already menioned specificaion p=0.017 per year. This and subsequen calibraion parameers are colleced in Table 1. The probabiliy disribuion for b is he hisorical one menioned before, for which Eb=0.29. Some oher baseline parameers are he same as hose used in he main calibraion exercise in Barro (2006, Table 5). The coefficien of relaive risk aversion is γ=4, he sandard deviaion of he u shocks is σ=0.020 per year, he growh-rae parameer is g=0.025 per year, and he expeced growh rae is g*=0.020 per year (from 16

19 Eq. [2]). 10 Since his earlier exercise assumed power-uiliy preferences, where θ=γ, he IES, 1/θ, was consrained o be 0.25 as already menioned, an IES his low produces implausible resuls concerning he link beween uncerainy parameers and he pricedividend raio. The EZW case now being considered requires a separae calibraion for he IES. Macroeconomic esimaes of he IES, 1/θ, represened by Hall (1988), come from regressions of consumpion growh raes on real raes of reurn, for example, on shorerm real ineres raes. The resuling esimaes of 1/θ cover a broad range and are ypically well below one. However, as observed by Bansal and Yaron (2004, p. 1501) and Barro (2005, secion VIII), hese coefficien esimaes end o be biased sharply oward zero because sample flucuaions in real ineres raes likely reflec, o a considerable exen, variaions in uncerainy parameers. This regression approach wih macroeconomic daa yields saisfacory esimaes of 1/θ only if he flucuaions in real ineres raes sem mainly from movemens in he expeced growh rae, g*, for given uncerainy parameers. Because of he shorcomings of macroeconomic esimaes of he IES, i is worhwhile o consider microeconomic evidence. The Gruber (2006) analysis is paricularly aracive because i uses cross-individual differences in afer-ax real ineres raes ha derive from arguably exogenous differences in ax raes on capial income. For 10 The values for g and σ come from daa on real personal consumer expendiure for 21 OECD counries for , a ranquil period wih no disaser evens for hese counries. The larges conracion was 14% for per capia real consumer expendiure (12% for per capia GDP) for Finland in For , he median of he growh raes of real per capia personal consumer expendiure for he 21 counries was per year, and he median sandard deviaion of he growh raes was The U.S. values were and 0.018, respecively. Wih γ=4, he expecaions associaed wih he hisorical disribuion of disaser sizes, b, are Eb=0.29, E(1-b) -γ = 7.69, and E(1-b) 1-γ =

20 presen purposes, he key poin is ha he Gruber esimae of he IES is around 2.0. Thus, I use θ=0.5 in my baseline calibraion. The final parameer needed is he rae of ime preference, ρ. The main calibraions in Barro (2006) used ρ=0.030 per year. However, he pure rae of ime preference is no direcly observable. Typically, a reasonable value for ρ is inferred by considering is connecion o levels of raes of reurn, including he risk-free real ineres rae. Thus, a firs poin is ha, in he EZW conex, he link o raes of reurn involves he effecive rae of ime preference, ρ*, given in Eq. (13), no ρ, per se. Hence, I proceed by assuming ha ρ akes on a value ha, given he oher baseline parameers, generaes a ρ* ha yields a plausible risk-free real ineres rae. Table 2 shows ha he real rae of reurn on governmen bills (or analogous shorerm claims) for 11 OECD counries from 1880 o 2005 averaged per year. These bill reurns are no risk-free and include some very low realizaions due o war-relaed inflaions (such as in Germany around World War I). Therefore, risk-free raes (no direcly observed) would likely be somewha lower han he average real rae of reurn on bills. However, I ake as an approximaion o he risk-free rae. Given he oher baseline parameers, i urns ou ha a value ρ*=0.027 is required o generae r f =0.010 in he model (from Eq. [7] wih ρ* subsiued for ρ). Equaion (13) hen implies ha he required value of ρ is The full se of baseline parameers, shown in Table 1, generaes a price-dividend raio, V, in Eq. (14) of This value for V implies ha a small rise in he expeced growh rae, g* for example, by 0.1% per year has o be compensaed by a fall in he iniial level of GDP, Y, by 2.1%. Despie differences in specificaion, his resul accords 18

21 wih he one found by Lucas (1987, Ch. 3, p. 24). An economy should be willing o give up a lo in is iniial level of GDP o obain a small increase in is long-erm growh rae. The Lucas calculaions abou consumpion uncerainy relae in he presen model o he parameer σ. The marginal effec on aained uiliy, U, from a change in σ is given from Eqs. (14) and (15) by (19) U = σ 1 γ ( θ γ ) /(1 θ ) 2 ( Y ) ( ρv ) γσv. Therefore, Eq. (16) implies ha he uiliy rae of ransformaion beween proporionae changes in Y and changes in σ is given by ( U / σ ) (20) = γσv ( U / Y ) Y. This expression gives he proporionae increase in iniial GDP required o compensae, a he margin, for a rise in σ. The parameers specified before imply γσv = Therefore, o mainain aained uiliy, an increase in σ by, say, 10% (from o 0.022) requires a rise in he iniial level of GDP by approximaely 0.33%. Since he expeced growh rae, g*, is held fixed, his proporionae rise in GDP level should be viewed as applying each year. 11 These calculaions apply for small changes in σ. Large changes, considered in he nex secion, recognize ha he uiliy rae of ransformaion rises wih σ on he righhand side of Eq. (20). This consideraion means ha he welfare gain from reducing σ from o zero is smaller in magniude han he amoun 3.3% ha would be calculaed from Eq. (20) if he uiliy rae of ransformaion were consan. 11 Obsfeld (1994) observes ha Lucas (1987, Ch. 3) ges far smaller esimaes for he welfare cos of consumpion uncerainy because he reas he shock, analogous o u in he presen model, as a ransiory disurbance o he level of oupu. 19

22 Consider now he welfare consequences from a change in he disaser probabiliy, p, for a given disribuion of disaser sizes, b. Equaions (14) and (15) imply U 1 γ ( θ γ ) /(1 θ ) 2 1 γ (21) = ( Y ) ( ρv ) V [ E(1 b) 1 Eb ( γ 1)]/( γ 1). p This formula applies while holding fixed he expeced growh rae, g*; ha is, i does no allow for he negaive effec of p on g*, for given g, in Eq. (2). The uiliy rae of ransformaion beween proporionae changes in Y and changes in p is given by ( U / p) 1 γ (22) = V [ E(1 b) 1 Eb ( γ 1)]/( γ 1). ( U / Y ) Y Wih he parameer values used before, he righ-hand side equals As before, he resul applies o small changes. An increase in p by 10% (from o ) maches up approximaely wih a proporionae rise in iniial GDP by 2.6%. Again, his change in GDP level applies each year. We can modify he calculaions o allow for a growh effec from a change in p; ha is, for given g, g* falls wih p in Eq. (2). 12 The resul modifies Eq. (22) o ( U / p) 1 γ (23) ( incl. growh effec) = V [ E(1 b) 1]/( γ 1). ( U / Y ) Y Wih he same parameer values as before, he righ-hand side equals Therefore, a rise in p by 10% now maches up wih a proporionae increase in GDP by 3.6% larger han before because of he decline in g*. 12 See Barlevy (2004) for a discussion of models in which uncerainy affecs he expeced growh rae of GDP. 20

23 b. Welfare effecs from large changes Equaions (18), (20), (22), and (23) assess welfare effecs from small changes in Y, g*, σ, and p. We can insead use Eqs. (14) and (15) o assess he effecs on aained uiliy from large changes. Le V and Y be he values ha apply for he baseline specificaion of parameers in Table 1. Le V* and (Y )* be values ha apply in an alernaive siuaion ha delivers he same aained uiliy, U. Then he formula for U in Eq. (15) implies 13 (24) (Y )*/Y = (V/V*) 1/(1-θ). The resul in Eq. (24) relaes o Alvarez and Jermann (2004), who ry o go as far as possible o gauge he welfare coss of consumpion uncerainy by observing or esimaing various asse prices. 14 Equaion (24) provides insigh for he presen model on he exen o which welfare coss can be assessed from observaions of asse prices relaed o equiy shares. The price V may be observable in he Lucas-ree economy, V is he price-dividend raio for an unlevered equiy claim on a ree. However, he price V* is unlikely o be observable: V* is he price-dividend raio for unlevered ree equiy in a hypoheical economy, such as one wih zero uncerainy. If he hypoheical price-dividend raio, V*, could be observed or esimaed, Eq. (24) shows ha he welfare gain, measured by he compensaing oupu change (Y )*/Y, depends on he parameer θ, for given V and V*. The baseline specificaion in 13 Equaion (24) deermines he compensaing income change in he sense of Hicks (1946, pp ) for a shif in a parameer, such as g*, σ, or p. 14 Par of he Alvarez-Jermann analysis depends on he pricing of a claim o a consumpion rend. The price of such a claim is finie only if he risk-free rae, r f, exceeds he expeced growh rae, g*. This condiion need no hold in he model; ha is r f <g* can apply in Eq. (7). Moreover, he daa in Table 2 indicae ha he average real rae of reurn on governmen bills, 0.010, was below he long-erm average growh rae for OECD counries. These growh raes averaged around for per capia GDP and consumpion and for levels of GDP and consumpion. Under hese circumsances, he price of an Alvarez-Jermann consumpion-rend claim is infiniy. 21

24 Table 1 uses he value θ=0.5, corresponding o an IES of 2. For his case, Eq. (24) implies an inverse-square law for he relaion beween he welfare effec and he raio of equiy prices: (25) (Y )*/Y = (V/V*) 2. Suppose, for example, ha a reducion in uncerainy (a decrease in σ or p or a shif in he b-disribuion oward smaller values) resuls in an increase in he price-dividend raio by 1% ha is, V* is 1% above V. In his case, he compensaing oupu change is abou 2% ha is, (Y )* is roughly 2% below Y. Lucas (1987, Ch. 3; 2003, secion II) focused on he consequences of eliminaing all consumpion uncerainy associaed wih usual business flucuaions in he presen conex, his exercise corresponds o seing σ=0. The formula for V in Eq. (14) implies for his case 1/V* = 1/V + (1/2) γ (θ-1) σ 2. Subsiuion ino Eq. (24) yields (26) (Y )*/ Y = [1 + (1/2) γ (θ-1) σ 2 V] 1/(1-θ). Suppose ha he magniude of (1/2) γ (θ-1) σ 2 V is much less han one a condiion likely o hold because (1/2) γ (θ-1) σ 2 V = in he baseline specificaion. In his case, he resul in Eq. (26) simplifies o (27) log[(y )*/Y ] -(1/2) γσ 2 V. Tha is, he welfare benefi from reducing σ o zero is approximaely one-half he effec ha would be calculaed from he local impac of a change in σ given by Eq. (20). Wih he parameer values assumed before, Eq. (26) implies ha (Y )* is 1.6% below Y. Tha is, sociey would be willing o give up 1.6% of oupu each year o 22

25 eliminae all of he cusomary economic flucuaions represened by σ. As noed before (n. 11), his effec is much larger han ha found by Lucas (1987) mainly because he impac of a shock, u, on he GDP level is permanen in he presen model. Seing he disaser probabiliy, p, o zero (or, equivalenly, he disaser size, b, o zero) has much greaer consequences for welfare. The formula, derived from Eqs. (14) and (24), is (28) Y * Y 1 θ 1 ( ) γ 1 1 γ = + pv E b Eb [ (1 ) 1 ( γ 1) ] 1/(1 θ ). Noe ha his formula holds fixed he expeced growh rae, g*; ha is, i does no allow for he inverse relaion beween p and g* in Eq. (2), for given g. Wih he same parameer values as before, (Y )* is 24.0% below Y. Hence, when gauged by he compensaing proporionae change in oupu, eliminaing disaser risk is worh 15 imes as much as eliminaing normal economic flucuaions. These large welfare coss of disasers arise even hough he presen analysis considers only he uiliy los from reduced consumpion. For wars, naural disasers, and epidemics, an allowance for he direc uiliy losses from deah, injury, and disease would raise he welfare effecs. See Hess (2003) for a discussion in he conex of conflics. We can again modify he calculaions o allow for a growh effec from a change in p; ha is, for given g, g* falls wih p in Eq. (2). The revised formula for he welfare gain is (29) Y * Y θ 1 = 1 + pv ( ) [ E(1 b) γ 1 1 γ 1] 1/(1 θ ) Wih he usual parameer values, (Y )* is 32.5% below Y. This resul is larger han before because he reducion in p raises g*. 23

26 We can also consider he eliminaion of all consumpion uncerainy by seing σ=0 and p=0 (or b=0) simulaneously. If g* is held fixed, (Y )* is 25.4% below Y. Allowing for he inverse relaion beween p and g*, he resul is 33.9%. These resuls correspond, as a good approximaion, o he sum of he effecs from seing σ=0 and p=0 separaely. Thus, he main effecs in each case come from seing p=0. III. Sensiiviy of he Welfare-Cos Esimaes The welfare esimaes, including he effecs from eliminaing all disaser risk, depend on he coefficien of relaive risk aversion, γ, and he IES, 1/θ. Table 3 shows how he compued welfare effecs depend on hese preference parameers. The line shown in bold, where γ=4 and θ=0.5, is he baseline specificaion already discussed. The firs four lines of Table 3 show he impac of raising θ, while holding fixed γ. One complicaion is ha, for given ρ, changes in θ influence he effecive rae of ime preference, ρ*, given in Eq. (13). The spiri of he calibraion exercise was o choose ρ o generae a ρ* ha produced reasonable levels of raes of reurn, including he risk-free rae. To accord wih his perspecive, ρ is varied in he able each ime θ or γ changes o mainain ρ* a is baseline value, For example, for γ=4, ρ=0.054 when θ=0.25, when θ=0.50, when θ=1, and when θ=4. Since γ and ρ* are held fixed, he raes of reurn, r e and r f, and he price-dividend raio, V, do no change as θ varies. For example, he equiy premium remains fixed a in hese cases. The general paern in Table 3 is ha an increase in θ implying a decrease in he IES lowers he welfare benefis from eliminaing uncerainy. However, for any given γ, since ρ* is held consan, an increase in θ say from 0.25 o 4 has only a minor effec 24

27 on he welfare gain from seing σ o zero. (This resul is apparen from Eq. [27] because, as an approximaion, he benefi does no depend on θ, for given γ and V.) For example, when γ=4, he welfare gain from seing σ=0 declines only slighly from 1.65% of oupu a θ=0.25 o 1.60% a θ=4. The negaive effec from raising θ on welfare is more pronounced for seing p=0. For example, when γ=4, he benefi decreases from 24.7% of oupu a θ=0.25 o 22.6% a θ=1 and 17.3% a θ=4. However, if we resric aenion o he range where θ<1, so ha he IES>1, he changes in θ have relaively small consequences for he welfare effecs. Table 3 shows, no surprisingly, ha decreases in he coefficien of relaive risk aversion, γ, reduce he welfare benefi from eliminaing uncerainy. These effecs are more imporan han hose from changing θ (given ha ρ* is mainained a in all cases). For example, if θ is fixed a 0.50, he welfare benefi from seing σ=0 declines from 1.65% of oupu when γ=4 o 1.30% a γ=3.5, 1.12% a γ=3, and 0.74% a γ=1. The corresponding gain from seing p=0 falls from 24.0% o 16.1%, 11.8%, and 4.6%. Thus, he large esimaed welfare gains from eliminaing disaser risk depend on agens having a subsanial degree of risk aversion. A problem wih he calculaions for low values of γ is ha he prediced equiy premium deviaes sharply from observed values of 4-6%. 15 Table 3 shows ha he model s prediced premium is 5.9% a γ=4, 3.9% a γ=3.5, 2.6% a γ=3, and only 0.3% a γ=1. Hence, even wih he presence of disaser risk, he predicions deviae sharply from observed equiy premia unless γ is a leas 3.5. The model s implicaions for welfare 15 Table 2 considers 11 counries wih daa on sock and bill reurns back o These daa show a mean equiy premium excess of he real rae of reurn on sock (7.4%) over ha on bills (1.0%) of 6.4%. However, he sock reurns refer o levered equiy. If he deb-equiy raio for corporaions is around 0.5, he equiy premium for unlevered equiy would be abou 4.3%. Since he risk-free rae is likely somewha lower han he average real bill reurn, he unlevered equiy premium would be somewha higher han 4.3%. 25

28 coss of uncerainy likely should no be aken seriously in he range of values for γ where he model fails o ge ino he righ ballpark for explaining he equiy premium. Thus, i seems bes o focus on welfare effecs corresponding o a value for γ of a leas 3.5. For his case, when θ=0.5, he welfare gain from seing p=0 is 16.1% of oupu. I is possible o resore reasonable predicions for he equiy premium a lower values of γ if he disaser probabiliy, p, is raised subsanially above 1.7% per year. For example, a γ=3, p has o be 4.1% o generae he same equiy premium, 5.9%, as in he baseline case. Wih his unrealisically high p, he eliminaion of all disaser risk (seing p or b o zero) urns ou o balance agains a proporionae decline in oupu by 60%, well above he 24% calculaed originally. IV. Endogenous Labor Supply The model can be exended o encompass a simple model of producive labor and labor-leisure choice. Suppose ha he oupu of each ree is given by (30) Y = A L, α where A is exogenous produciviy, L is he quaniy of labor employed, and 0<α<1. The log of produciviy is generaed in he same way as oupu in he baseline Lucas-ree model; ha is, log(a +1 ) follows he sochasic process given by he form of Eq. (1). Thus, he underlying uncerainy in his model is he same as in he original seing. All labor is equally producive and earns he common real wage rae, w. Since he labor marke is compeiive, w equals he marginal produc of labor, deermined from Eq. (30). Each person is endowed wih one uni of ime, which can be allocaed beween leisure and marke work. Uiliy now depends on each period s consumpion, C, and 26

29 leisure, 1- L. One sraighforward way o model preferences is o use he Epsein-Zin- Weil formulaion of uiliy from Eq. (10) bu replace 1 θ C by λ 1 θ [ C (1 L ) ]. 16 The new parameer λ>0 is he consan elasiciy of subsiuion beween consumpion and leisure a a poin in ime. This form is consisen wih he King-Plosser-Rebelo (1988) prescripion ha preferences accord wih he propery ha work effor, L, be consan in he long run; ha is, when w and C advance a he same rae due o seady produciviy growh. In he presen seing, which lacks capial accumulaion, his propery also holds in he shor run, so ha L ends up consan in equilibrium. The new se of firs-order condiions involves subsiuion beween leisure and consumpion a each poin in ime: u / (1 L) (31) = w u C. / Given he assumed form of he uiliy funcion, hese condiions imply (32) L = 1 λ (C /w ), which can be viewed as a labor-supply funcion. The producion funcion in Eq. (30) and he condiion C = Y imply (33) C /w = (1/α) L. This resul, in conjuncion wih Eq. (32), implies (34) L = α/(α+λ). Hence, he fracion of ime worked is consan invarian wih shocks o produciviy, A. This resul applies because subsiuion effecs (from changing w ) exacly offse income effecs (associaed wih changing C ). 16 The basic resuls go hrough wih he more general specificaion saisfies ω(l )>0 and ω (L )<0. 1 θ [ C ω( L )], where he funcion ω 27

30 Since L is consan, oupu and consumpion, Y = C, and dividends paid on equiy claims are all proporional o A. Therefore, he pricing of equiy claims (and oher claims) is he same as in he iniial model. Chey (2006) shows, wihin an expeced-uiliy seing, ha labor-supply elasiciies and he exen of leisure-consumpion complemenariy imply resricions on he admissible range for he coefficien of relaive risk aversion he parameer γ in Eq. (10). In paricular, he argues ha empirical esimaes of income-compensaed laborsupply elasiciies sugges γ<2. Thus, he concludes ha he expeced-uiliy framework has o be abandoned o accommodae he higher values of γ needed o accord wih observed behavior in asse and insurance markes. The las finding fis wih my resuls in he EZW framework. The (consumpioncompensaed) wage elasiciy can be compued from Eq. (32) by aking a derivaive wih respec o w, while holding fixed C, o ge (35) compensaed wage elasiciy of labor supply = λ/α. Given he producion-funcion parameer α, he compensaed wage elasiciy can be anyhing, depending on λ, he elasiciy of subsiuion beween consumpion and leisure. Thus, in he EZW framework, labor-supply elasiciies place no resricion on he permissible range for γ. The EZW model, exended o incorporae labor-leisure choice, has hree independen parameers: one governing risk aversion (γ), anoher for he IES (θ), and a hird for consumpion-leisure subsiuion (λ). 28

31 V. Endogenous Saving and Invesmen In an endowmen economy, agens do no reac o changes in uncerainy by alering saving and invesmen. Generally, he poenial for such adjusmens affecs welfare coss no a he margin (by he envelope heorem) bu for large changes in parameers. This secion illusraes his process by using a version of he racable AK model of endogenous saving and invesmen developed in Barro (2006, secion VIII). The quaniy of rees is now variable and corresponds o he capial sock, K. Producion of frui is given by an AK producion funcion: (35) Y = AK. Unlike he original model, he produciviy level, A>0, is now consan. Oupu can be consumed as frui, C, or invesed as seed, I, so ha (36) C = Y - I = AK - I. The creaion of new rees hrough planing seeds (ha is, invesmen) is assumed o be rapid enough so ha, as in he convenional one-secor producion framework, he frui price of rees (capial) is pegged a a price normalized o one. This seing corresponds o Tobin s q always equaling one unlike in he previous model, where he marke price of rees was variable. The capial sock evolves because of gross invesmen and depreciaion, δ +1 K : (37) K +1 = K + I δ +1 K. The depreciaion rae is sochasic and equal o (38) δ +1 = δ + u +1 + v +1, where 0<δ<1. The u +1 shock, normally disribued wih mean 0 and variance σ 2, represens normal flucuaions, as in he previous seing. The v +1 shock represens rare 29