A Neoclassical Theory of Liquidity Traps

Transcription

1 A Neoclassical Theory of Liquidiy Traps Sebasian Di Tella Sanford Universiy November 2017 Absrac This paper provides an equilibrium heory of liquidiy raps and he real effecs of money. Money provides a safe sore of value ha prevens ineres raes from falling enough during downurns, and he economy eners a persisen slump wih depressed invesmen. This is an equilibrium oucome prices are flexible, markes clear, and inflaion is on arge bu i s no efficien. Invesmen is oo high during booms and oo low during liquidiy raps. Alhough money has large real effecs, moneary policy is ineffecive he zero lower bound is no binding, money is superneural, and Ricardian equivalence holds. The opimal allocaion requires he Friedman rule and a ax/subsidy on capial. 1 Inroducion Liquidiy raps occur when money prevens ineres raes from falling enough during downurns, and he economy eners a persisen slump wih depressed invesmen. They are associaed wih some of he deepes and mos persisen slumps in hisory. Japan has arguably been experiencing one for almos 20 years, and he US and Europe since he 2008 financial crisis. Bu how can money have such large and persisen real effecs? This paper provides an equilibrium heory of liquidiy raps and he real effecs of money. I show ha liquidiy raps arise naurally in a moneary economy wih incomplee idiosyncraic risk sharing, even if prices are compleely flexible. To fix ideas, consider a simple AK growh model wih log uiliy over consumpion and money, and incomplee idiosyncraic risk sharing. During downurns idiosyncraic risk goes up and makes risky capial less aracive. Wihou money, real ineres raes would fall and keep invesmen a he firs-bes level. Bu money prevens equilibrium ineres raes from falling enough, depressing invesmen. The liquidiy rap is an equilibrium oucome prices are flexible, markes clear, and inflaion is on arge. In conras o convenional models of liquidiy raps, he zero lower bound is no binding, money is superneural, and Ricardian equivalence holds. The real effecs of money become I d like o hank Manuel Amador, Narayana Kocherlakoa, Pablo Kurla, Yuliy Sannikov, Chris Tonei, Chad Jones, Adrien Aucler, Arvind Krishnamurhy, Bob Hall, and John Taylor. sdiella@sanford.edu. 1

2 gradually larger as ineres raes fall and he value of liquidiy rises. The compeiive equilibrium is inefficien invesmen is oo high during booms and oo low during liquidiy raps. However, while money can have large real effecs, moneary policy is ineffecive. The opimal allocaion requires he Friedman rule and a ax or subsidy on capial. When invesmen is oo low, subsidize i. When i s oo high, ax i. How can money affec real ineres raes and invesmen? The key insigh is ha money provides a safe sore of value ha improves idiosyncraic risk sharing, and is value endogenously rises during liquidiy raps. Beer risk sharing weakens agens precauionary saving moive, which keeps real ineres raes high and invesmen depressed relaive o he economy wihou money. The value of money is he presen value of expendiures on liquidiy services, and i becomes very large when ineres raes fall. In paricular, if risk is high enough he real ineres rae can be very negaive wihou money, bu mus remain above he growh rae if here is money. The value of liquidiy endogenously grows, raising he equilibrium real ineres rae and depressing invesmen unil his condiion is saisfied. The resul is a liquidiy rap. Wha makes money special? Is his abou money or is i really abou safe asses? Agens can rade risk-free deb, bu i doesn produce a liquidiy rap. Neiher does a diversified (safe) equiy index. I also allow for safe governmen deb and deposis. They produce a liquidiy rap only o he exen ha hey have a liquidiy premium. To see why, noice ha safe asses wihou a liquidiy premium mus be backed by paymens wih equal presen value. Agens own he asses bu also he liabiliies, so he ne value is zero. Bu he value of liquid asses, ne of he value of he paymens backing hem, is equal o he presen value of heir liquidiy premium. This is wha allows hem o serve as a sore of value and improve risk sharing in general equilibrium. Agens wih a bad shock can sell par of heir liquid asses o agens wih a good shock in order o reduce he volailiy of heir consumpion. Money, and more generally safe asses wih a liquidiy premium, are special because hey are simulaneously safe (no idiosyncraic risk) and have a posiive ne value. There are many asses ha have posiive ne value, such as capial, housing, or land. Bu he saring poin of his paper is ha real invesmens are risky and idiosyncraic risk sharing is incomplee. For example, agens mus buy a paricular house or plo of land, whose value has significan idiosyncraic risk ha can be fully shared. There are also safe financial asses, such as AAA corporae deb, bu heir ne value is zero (someone owes he asse). Safe asses wih a liquidiy premium have he rare combinaion of safey and posiive ne value ha allow hem o funcion as a safe sore of value. How quaniaively imporan is he role of liquid asses as a safe sore of value? Quie small during normal imes, bu very large during liquidiy raps. The ne value of liquid asses is equal o he presen value of expendiures on liquidiy services. During normal imes when ineres raes are high heir value is relaively small, close o he expendiure share on liquidiy services (around 1.7% by my calculaions), and can be safely ignored. Bu during hose relaively rare occasions when he real ineres rae becomes persisenly very low relaive o he growh rae of he economy, heir value can become very large. And hese are precisely he evens we are ineresed in sudying. In 2

3 fac, he liquidiy rap survives even in he cashless limi where expendiures on liquidiy services vanish, and is robus o differen specificaions of money demand. The inefficiency in his economy comes from hidden rade. 1 I microfound he incomplee idiosyncraic risk sharing wih a fund diversion problem wih hidden rade. The compeiive equilibrium is he oucome of allowing agens o wrie privaely opimal long-erm conracs in a compeiive marke. I hen characerize he opimal allocaion by a planner who faces he same environmen, and show how i can be implemened wih a ax or subsidy on capial. The compeiive equilibrium is inefficien because privae conracs don inernalize ha when hey make heir consumpion and invesmen decisions, hey are affecing prices, such as he ineres rae, and herefore he hidden-rade incenive-compaibiliy consrains of oher conracs. The model is driven by counercyclical idiosyncraic risk shocks for he sake of concreeness. Bu an increase in risk aversion is mahemaically equivalen; i will also raise he risk premium and precauionary moives. Higher risk aversion can represen wealh redisribuion from risk oleran o risk averse agens afer bad shocks (see Longsaff and Wang (2012)) or weak balance shees of specialized agens who carry ou risky invesmens (see He and Krishnamurhy (2013) and He e al. (2015)). I can also capure habis (Campbell and Cochrane (1999)) or higher ambiguiy aversion afer shocks ha upend agens undersanding of he economy (see Barillas e al. (2009)). Here I focus on simple counercyclical risk shocks wih homogenous agens, bu hese are poenial avenues for fuure research. I firs sudy a simple saionary model and do comparaive saics across balanced growh pahs for differen levels of idiosyncraic risk in Secion 2. I characerize he opimal allocaion in Secion 3. This simple environmen capures mos of he economic inuiion and can be solved wih pencil and paper. I hen inroduce aggregae risk shocks in a dynamic model in Secion 4 and characerize he compeiive equilibrium as he soluion o a simple ODE. Secion 5 discusses he link o a bubble heory of money, and he relaionship beween he mechanism in his paper and sicky-price models of he zero lower bound. In he Appendix I consider a cash-in-advance version of he model and a general CES demand for money, and I also solve he model wih Epsein-Zin preferences. The Online Appendix has he echnical deails of he conracual environmen. New Keynesian models of he zero lower bound. The mainsream view of liquidiy raps focuses on he role of he zero lower bound on nominal ineres raes in New Keynesian models wih nominal rigidiies, saring wih he seminal work of Krugman e al. (1998). 2 If here is money in he economy he nominal ineres rae canno be negaive. So if he naural ineres rae (he real ineres rae wih flexible prices) is very negaive, he cenral bank mus eiher abandon is inflaion arge or allow he economy o operae wih an oupu gap (or boh). In conras, his paper argues ha liquidiy raps are no essenially abou nominal rigidiies and he zero lower bound. Money makes he naural rae posiive, and he depressed invesmen does 1 See Farhi e al. (2009), Kehoe and Levine (1993), Di Tella (2016). 2 Eggersson e al. (2003), Werning (2011), Eggersson and Woodford (2004), Eggersson and Krugman (2012), Svensson (2000), Caballero and Simsek (2017). 3

4 no reflec a negaive oupu gap, bu raher he real equilibrium effecs of money. An aracive feaure of he mechanism here is ha he liquidiy rap is a gradual phenomenon. The real effecs of money become larger as he value of liquidiy endogenously rises. While here is a zero lower bound on he nominal ineres rae, i s no binding and i doesn play any imporan role in he liquidiy rap. In fac, since money is superneural, changing he inflaion arge will always fix he zero lower bound problem, bu i will have no real effecs. And under he Friedman rule, he zero lower bound is never binding. The wo approaches are complemenary. In he shor-run prices may very well be sicky, markes segmened, and informaion imperfec. I absrac from hese issues o focus on he underlying fricionless aspecs of liquidiy raps. The model in his paper can be regarded as he fricionless version of a richer model wih shor-run fricions. For example, in he model consumpion is counercyclical in he very shor-run, for well undersood reasons. Imporanly, his is no a model of shor-run flucuaions, bu raher of persisen slumps associaed wih liquidiy raps. The resuls in his paper have an imporan ake-away for New Keynesian models of he zero lower bound. Inroducing money ino an economy doesn jus place a lower bound on ineres raes i also raises he naural ineres rae. To he exen ha he naural rae is posiive he cenral bank will be able o reproduce he flexible-price equilibrium, hiing he inflaion arge and zero oupu gap, even if he economy is in a liquidiy rap. Bu i may no be opimal. Invesmen is oo depressed during a liquidiy rap, so unless an invesmen subsidy is used o obain he opimal allocaion, reproducing he flexible-price allocaion is subopimal. I conjecure ha simulaing invesmen wih low ineres raes may be he opimal moneary policy in ha siuaion, bu his is beyond he scope of his paper. Oher lieraure review. Buera and Nicolini (2014) provide a flexible-price model of he zero lower bound, based on borrowing consrains and lack of Ricardian equivalence. Aiyagari and McGraan (1998) sudy he role of governmen deb in a model wih uninsurable labor income and binding borrowing consrains. In conras, here Ricardian equivalence holds (agens have he naural borrowing limi), he zero lower bound is no binding, and money is superneural. Changing he amoun of governmen deb can only affec he liquidiy premium on governmen deb and oher asses, bu no he real side of he economy. Changing he inflaion arge raises nominal ineres raes and fixes he zero lower bound problem, bu doesn have any real effecs. I is easy o break Ricardian equivalence and superneuraliy, bu hey are useful heoreical benchmarks ha highligh ha he real effecs of money don hinge on a fiscal side. There is a large lieraure on risk or uncerainy shocks boh in macro and finance. 3 The seing here is closes o Di Tella (2017), who shows ha risk shocks ha increase idiosyncraic risk can help explain he concenraion of aggregae risk on he balance shees of financial inermediaries and creae financial crises. Here I remove inermediaries and inroduce money, and show ha hese 3 Bloom (2009), Bloom e al. (2012), Campbell e al. (2001), Bansal and Yaron (2004), Bansal e al. (2014), Campbell e al. (2012), Chrisiano e al. (2014). 4

5 risk shocks may also be responsible for liquidiy raps. This may explain why liquidiy raps and financial crises ofen appear ogeher. Cochrane (2011) highlighs he role of ime-varying risk premia in asse prices, and herefore invesmen. The driving force behind he liquidiy rap in his paper is a ime-varying idiosyncraic risk premium. Bu a high risk premium is no enough o depress invesmen. The real ineres rae could drop enough o absorb he hi, leaving asse prices and invesmen unaffeced. This paper provides a heory of why he equilibrium real ineres rae will no drop enough, so ha a high risk premium will be refleced in lower invesmen. The liquidiy premium is he focus of a large lieraure ha micro-founds he role of money as a means of exchange in a search-heoreic framework. 4 Here I use money in he uiliy funcion as a simple and ransparen way o inroduce money ino he economy (I also solve a cash-in-advance version in he Appendix). The purpose of his paper is no o provide a new explanaion for why people hold money in equilibrium, bu raher o undersand how money can produce liquidiy raps and have real effecs. However, a more micro-founded accoun of he liquidiy premium can help undersand how i is affeced by aggregae shocks and policy inervenions. There is also a large lieraure modeling money as a bubble in he conex of OLG or incomplee risk sharing models. 5 The closes paper is Brunnermeier and Sannikov (2016b), who use a similar environmen wih incomplee idiosyncraic risk sharing o sudy he opimal inflaion rae. 6 imporan conribuion of ha paper is o develop a version of he Bewley (1980) model of bubble money ha is racable and yields closed-form soluions. They find ha counries wih high risk should have a higher inflaion rae. In conras, here bubbles are ruled ou, money is superneural, and he focus is on how money can produce liquidiy raps. 7 An I sudy some of he differences and similariies of he bubble and liquidiy views of money and how hey relae o he issue of liquidiy raps in Secion 5. The conracual environmen micro-founding he incomplee idiosyncraic risk sharing wih a fund diversion problem wih hidden rade is based on Di Tella and Sannikov (2016), who sudy a more general environmen. 8 Di Tella (2016) uses a similar conracual environmen o sudy opimal financial regulaion, bu does no allow hidden savings or invesmen. Insead, i focuses on he exernaliy produced by hidden rade in capial asses by financial inermediaries. Tha exernaliy is absen in his paper because he price of capial is always one (capial and consumpion goods can be ransformed one-o-one). 4 See Kiyoaki and Wrigh (1993), Lagos and Wrigh (2005), Aiyagari and Wallace (1991), Shi (1997). 5 See Samuelson (1958), Bewley (1980), Diamond (1965), Tirole (1985), Asriyan e al. (2016), Sanos and Woodford (1997) 6 Brunnermeier and Sannikov (2016a) use a similar environmen bu focus on he role of financial inermediaries. 7 In heir model money is a bubble and is inroduced proporionally o wealh, so higher inflaion acs as a subsidy o saving. Here bubbles are explicily ruled ou and money is inroduced in a lump-sum, non-disorionary way. 8 Cole and Kocherlakoa (2001) sudy an environmen wih hidden savings and risky exogenous income, and find ha he opimal conrac is risk-free deb. Here we also have risky invesmen. 5

6 2 Baseline model In his secion I inroduce he baseline saionary model. I s a simple AK growh model wih money in he uiliy funcion and incomplee idiosyncraic risk sharing. The equilibrium is always a balanced growh pah, and o keep hings simple I will consider compleely unexpeced and permanen risk shocks ha increase idiosyncraic risk (comparaive saics across balanced growh pahs). In Secion 4 I will inroduce he fully dynamic model wih aggregae risk shocks. 2.1 Seing The economy is populaed by a coninuum of agens wih log preferences over consumpion c and real money m M/p [ˆ U(c, m) = E e ρ( ] ) (1 β) log c + β log m d 0 Money and consumpion ener separaely, so money will be superneural. Money in he uiliy funcion is a simple and ransparen way of inroducing money in he economy. 9 As we ll see, wha maers is ha money has a liquidiy premium. Agens can coninuously rade capial and use i o produce consumpion y = ak, bu i is exposed o idiosyncraic qualiy of capial shocks. The change in an agen s capial over a small period of ime is d k i, = k i, σdw i, where k i, is he agen s capial (a choice variable) and W i, an idiosyncraic Brownian moion. Idiosyncraic risk σ is a consan here, bu we will look a comparaive saics of he equilibrium wih respec o changes in σ. This is mean o capure a shock ha makes capial less aracive and drives up is risk premium. Laer we will inroduce a sochasic process for σ and allow for aggregae shocks o σ. Idiosyncraic risk washes away in he aggregae, so he aggregae capial sock k evolves dk = (x δk )d (1) where x is invesmen. The aggregae resource consrain is c + x = ak (2) where c is aggregae consumpion. Money is prined by he governmen and ransferred lump-sum o agens. In order o eliminae any fiscal policy, here are no axes, governmen expendiures, or governmen deb; laer I will inroduce safe governmen deb and axes. For now money is only currency, bu laer I will add 9 In he Appendix I also solve he model wih a cash-in-advance consrain and a more general CES uiliy funcion. 6

7 deposis and liquid governmen bonds. The oal money sock M evolves dm M = µ M d The cenral bank chooses µ M endogenously o deliver a arge inflaion rae π. This means ha in a balanced growh pah µ M = π + growh rae. Markes are incomplee in he sense ha idiosyncraic risk canno be shared. They are oherwise complee. Agens can coninuously rade capial a equilibrium price q = 1 (consumpion goods can be ransformed one-o-one ino capial goods, and he oher way around) and deb wih real ineres rae r = i π, where i is he nominal ineres rae. There are no aggregae shocks for now; I will add hem laer and assume ha markes are complee for aggregae shocks. Toal wealh is w = k + m + h, which includes he capialized real value of fuure money ransfers h = ˆ The dynamic budge consrain for an agen is 10 e s rudu dm s p s (3) dw = (r w + k α c m i )d + k σdw (4) wih solvency consrain w 0, where α a δ r is he excess reurn on capial. Each agen chooses a plan (c, m, k) o maximize uiliy U(c, m) subjec o he budge consrain (4). Remark. As in Brunnermeier and Sannikov (2016b) and Angeleos (2006), his seing has several feaures ha make i very racable and easy o solve in closed-form wih pencil and paper. Uninsurable idiosyncraic risk comes from radable capial, raher han non-radable labor income. Togeher wih homoheic preferences, his produces policy funcions linear in wealh, which eliminae he need o keep rack of he whole wealh disribuion and yields closed-form expressions Balanced Growh Pah Equilibrium A BGP equilibrium will be scale invarian o aggregae capial k, so we can normalize all variables by k ; e.g. ˆm = m /k. A Balanced Growh Pah Equilibrium consiss of a real ineres rae r, invesmen ˆx, and real money ˆm saisfying r = ρ + (ˆx δ) σ 2 c Euler equaion (5) r = a δ σ c σ Asse Pricing (6) 10 This is equivalen o defining financial wealh w = k + m + d, where d is risk-free deb (in zero ne supply), and using he dynamic budge consrain d w = (d r + k (a δ) m π c )d + k σdw, and he naural deb limi is w = h, so ha w w. This is equivalen o (4) wih w = w + h Angeleos (2006) does no have money. Brunnermeier and Sannikov (2016b) develop a racable version of he Bewley (1980) model of bubble money. They inroduce money proporionally o wealh. Here bubbles are explicily ruled ou and money is inroduced lump-sum. I discuss he similariies and differences beween he bubble and liquidiy views of money in Secion 5. 7

8 k σ c σ = (1 λ)σ k + m + h Risk Sharing (7) λ m + h ρβ = k + m + h ρ ((1 λ)σ) 2 Value of Liquidiy (8) ˆm = β a ˆx 1 β r + π Money (9) As well as i = r + π > 0 and r > (ˆx δ). These las condiions make sure money demand is well defined and rule ou bubbles. Equaion (5) is he usual Euler equaion. ˆx δ is he growh rae of he economy and herefore consumpion, and σ 2 c is he precauionary saving moive. The more risky consumpion is, he more agens prefer o pospone consumpion and save. Equaion (6) is an asse pricing equaion for capial. Agens can choose o inves heir savings in a risk-free bond (in zero ne supply) and earn r, or in capial and earn he marginal produc ne of depreciaion a δ. The las erm α = σ c σ is he risk premium on capial. Because he idiosyncraic risk in capial canno be shared, agens will only inves in capial if i yields a premium o compensae hem. Equaion (7) is agens exposure o idiosyncraic risk. Because of homoheic preferences each agen consumes proporionally o his wealh, and his exposure o idiosyncraic risk comes from his invesmen in capial. In equilibrium, he porfolio weigh on capial is k /w = k /(k + m + h ) = (1 λ) where we define λ (m + h )/w as he share of wealh in money (presen and fuure). λ capures he value of liquidiy in he economy, and (8) gives us an equaion for λ in erms of parameers. Finally, (9) is an expression for real money balances. Because of he log preferences agens devoe a fracion β of expendiures o liquidiy and 1 β o consumpion. Using i = r + π and he resource consrain (2), we obain (9). The BGP has a simple srucure. We can solve (8) for λ, plug ino (7) o obain σ c, hen plug ino (6) o obain r, and plug ino (5) o obain ˆx. equilibrium, we use (9) o obain ˆm. Finally, once we have he real par of he The share of wealh in money λ capures he value of liquidiy in he economy, and plays a cenral role. Money provides a safe sore of value ha improves risk sharing, and i is worh he presen value of expendiures on liquidiy services. From he definiion of h we obain afer some algebra and using he No-Ponzi condiions, 12 m + h = m + ˆ e r(s ) dm s p s = ˆ e r(s ) m s ids = m i r (ˆx δ) (10) Because of log preferences, we ge m i = ρβ(k + m + h ) which yields λ m + h k + m + h = ρβ r (ˆx δ) (11) Finally, use he Euler equaion (5) and he definiion of σ c in (7) o obain (8). 12 Wrie m + r(s ) dms e p s = m + e r(s ) dm s + e r(s ) m sπ sds = lim T e r(t ) m T + e r(s ) m s(r s + π s)ds, and use he No-Ponzi condiion o eliminae he limi. 8

9 1.0 λ σ Figure 1: The value of liquidiy λ as a funcion of σ. Parameers: a = 1/10, ρ = 4%, π = 2%, δ = 1%, β = 1.7%. β How big is he value of liquidiy λ? In normal imes when he real ineres rae is high relaive o he growh rae of he economy, r ˆx δ, he value of liquidiy λ is small, close o he expendiure share on liquidiy services β. To fix ideas, use a conservaive esimae of β = 1.7%. 13 Bu when he real ineres rae r is small relaive o he growh rae of economy ˆx δ, he value of liquidiy can be very large (in he limi λ 1). This happens when idiosyncraic risk σ is large while capial is discouned wih a large risk premium, liquidiy is discouned only wih he risk-free rae, which mus fall when idiosyncraic risk σ is large. Figure 1 shows he non-linear behavior of λ as a funcion of σ. This is an imporan insigh he value of liquidiy may be small in normal imes, bu can become quie large during periods of low ineres raes such as liquidiy raps. 14 Proposiion 1. For any β > 0, he value of liquidiy λ is increasing in idiosyncraic risk σ, and ranges from β when σ = 0 o 1 as σ. Furhermore, idiosyncraic consumpion risk σ c = (1 λ)σ is also increasing in σ, and ranges from 0 when σ = 0 o ρ(1 β) when σ. For β = 0, λ = Non-moneary economy As a benchmark, consider a non-moneary economy where β = 0. In his case, ˆm = ĥ = 0 and herefore λ = 0. The BGP equaions simplify o r = a δ σ 2 and ˆx = a ρ. Higher idiosyncraic risk σ, which makes invesmen less aracive, is fully absorbed by a lower real ineres rae r (and herefore lower nominal ineres rae i = r + π), bu a consan invesmen 13 As Secion 2.5 shows, β is he expendiure on liquidiy premium across all asses, including deposis and reasuries. Say checking and savings accouns make up 50% of gdp and have an average liquidiy premium of 2%. Krishnamurhy and Vissing-Jorgensen (2012) repor expendiure on liquidiy provided by reasuries of 0.25% of gdp. Consumpion is 70% of gdp. This yields β = 1.7%. 14 I s worh sressing ha he value of liquidiy includes no only he value of money m, bu also fuure money h. As Figure 2 shows, mos of he value of liquidiy is in he fuure, h. 9

10 r σ c σ σ x m,m +h σ σ Figure 2: Real ineres rae r, invesmen ˆx, idiosyncraic consumpion risk σ c, and moneary variables ˆm (solid) and ˆm + ĥ (dashed) as funcions of idiosyncraic risk σ, in he non-moneary economy (dashed orange) and he moneary economy (solid blue). The lower bound on he real ineres rae π is dashed in black. Parameers: a = 1/10, ρ = 4%, π = 2%, δ = 1%, β = 1.7% rae ˆx and growh ˆx δ. economy for differen σ (dashed line). Figure 2 shows he equilibrium values of r and ˆx in a non-moneary Proposiion 2. Wihou money (β = 0), afer an increase in idiosyncraic risk σ he real ineres rae r falls bu invesmen ˆx remains a he firs-bes level. We can undersand he response of he non-moneary economy o higher risk σ in erms of he risk premium and he precauionary moive. Use he Euler equaion (5) and asse pricing equaion (6) o wrie r = a δ σ c σ }{{} risk pr. (12) ˆx = a ρ }{{} + σc 2 }{{} σ c σ }{{} = a ρ (13) firs bes prec. mo. risk pr. Larger risk σ makes capial less aracive, so he risk premium α = σ c σ goes up. Oher hings equal his depresses invesmen. Bu wih higher risk he precauionary saving moive σ 2 c also becomes larger. Agens face more risk and herefore wan o save more. Oher hings equal, his lowers he real ineres rae and simulaes invesmen. Wihou money σ c = (1 λ)σ = σ, so he precauionary moive and he risk premium cancel each oher ou and we ge he firs bes level of invesmen ˆx = a ρ for any level of idiosyncraic risk σ (his doesn mean ha his level of invesmen is opimal wih σ > 0). 10

11 This is a well known feaure of preferences wih ineremporal elasiciy of one (in he Appendix I solve he model wih general Epsein-Zin preferences). 15 For our purposes, i provides a clean and quaniaively relevan benchmark where higher idiosyncraic risk ha makes invesmen less aracive is compleely absorbed by lower real ineres raes which compleely sabilize invesmen. Bu noice in Figure 2 ha he real ineres rae r could become very negaive; in paricular, we may need r ˆx δ. This is no a problem wihou money because capial is risky, bu i will be once we inroduce money, which is safe, because is value would blow up if r ˆx δ. 2.4 Moneary economy Now consider he moneary economy wih β > 0, also shown in Figure 2. Money has large real effecs. When σ goes up, money prevens he real ineres rae r from falling as much as in he non-moneary economy. Insead, invesmen ˆx falls and he economy eners a persisen slump. 16 In paricular, wihou money he real ineres rae could be very negaive for high σ, bu wih money i mus remain above he growh rae of he economy. The value of liquidiy λ capures he real effecs of money. There are wo seps: (i) money serves as a safe sore of value and improves risk sharing, so a large value of liquidiy λ keeps he real ineres high relaive o he non-moneary economy and depresses invesmen; and (ii) he value of liquidiy λ endogenously rises during downurns when σ is high. The resul is a liquidiy rap he real ineres rae doesn fall as much as i would wihou money, and invesmen falls insead. To undersand sep (i) use he Euler equaion (5), he asse pricing equaion (6), and he risk sharing equaion (7) o obain an expression for r and ˆx in erms of σ and λ: r = a δ (1 λ)σ 2 }{{} risk pr. (14) ˆx = a ρ }{{} firs bes + (1 λ) 2 σ 2 }{{} prec. mo. (1 λ)σ 2 = a ρ ρ λ β }{{} 1 λ risk pr. Expressions (14) and (15) show ha a larger value of liquidiy λ raises he real ineres rae and depresses invesmen. Wha is going on is ha a large value of liquidiy λ improves idiosyncraic risk sharing, σ c = (1 λ)σ. Essenially, agens wih bad shocks sell par of heir money holdings o buy more capial and consumpion goods from agens wih good shocks. As a resul, he volailiy in heir consumpion and capial is smaller. Beer risk sharing dampens boh he risk premium σ c σ (raising r) and he precauionary saving moive σ 2 c bu crucially, i dampens he precauionary moive more. Inuiively, he risk premium comes from he risk of a marginal increase in capial 15 Alhough he real ineres rae r always falls wih higher risk σ, wihou money invesmen ˆx may go up or down depending on wheher ineremporal elasiciy is lower or higher han one. Bu for relevan parameer values he role of money is he same as in he baseline model wih log preferences: i prevens ineres raes from falling during downurns and depresses invesmen relaive o he non-moneary economy, producing a liquidiy rap. 16 Since oupu is fixed in he shor-run, lower invesmen implies higher consumpion. This is a well undersood feaure of his simple environmen. This is no a model of high-frequency business cycles; i s a model of persisen slumps produced by liquidiy raps. (15) 11

12 holdings, while he precauionary moive comes from he average risk in an agen s porfolio, ha now includes safe money. Money creaes a wedge beween he marginal and average risk ha weakens he precauionary moive relaive o he risk premium. Since he risk premium reduces invesmen and he precauionary moive increases i, a large value of liquidiy λ depresses invesmen. To undersand sep (ii), noice ha he value of liquidiy λ grows during downurns wih high σ, as shown in Figure 1. The value of liquidiy λ is equal o he presen value of expendiures on liquidiy services, as expression (11) indicaes. When idiosyncraic risk σ rises, he real ineres rae falls relaive o he growh rae of he economy because he precauionary moive rises (see he Euler equaion (5)), so his presen value becomes very large. I s imporan o sress ha he value of liquidiy λ includes no only curren real money balances m bu also fuure money h. As Figure 2 shows, mos of he value of liquidiy is in he fuure, h. Incomplee idiosyncraic risk sharing is essenial o he mechanism. If risk sharing is perfec or if here is no idiosyncraic risk, σ = 0, he moneary economy behaves exacly like he non-moneary one (classical dichoomy). During normal imes when idiosyncraic risk σ is small, he role of money is small and can be safely ignored. Bu i can become very large during periods of high idiosyncraic risk. Proposiion 3. Wih money (β > 0) afer an increase in idiosyncraic risk σ he real ineres rae r falls less han in he economy wihou money (β = 0), and invesmen ˆx falls insead, while he value of liquidiy λ and real money balances ˆm increase wih σ. (Classical Dichoomy) If σ = 0, he real ineres rae r and invesmen ˆx are he same in he moneary and non-moneary economies, even hough λ = β > 0. I is emping o inerpre he depressed invesmen as subsiuion from risky capial o safe money as a savings device; i.e., when capial becomes more risky, i is more aracive o inves in he safe asse. Bu his is misleading because he economy canno really inves in money. Goods can be eiher consumed or accumulaed as capial money is no a subsiue for invesmen in risky capial. Wha money does is improve how he idiosyncraic risk in capial is shared. Agens wih bad shocks use par of heir money holdings o buy more capial from hose wih good shocks. 17 As a resul of his risk sharing, he economy subsiues along he consumpion-invesmen margin. To drive home his poin, noice ha in a model wih risky and safe capial (bu no money), an increase in risk will ypically reduce risky invesmen bu increase he safe one. Money depresses all invesmen, which is an imporan feaure of liquidiy raps. 18 I s worh poining ou ha because oupu is fixed in he very shor run by he AK echnology, consumpion is negaively correlaed wih invesmen in he shor run. Over ime as he economy shifs o a BGP wih lower invesmen and growh, he correlaion becomes posiive. This is no a model of shor-run flucuaions, bu raher of persisen slumps. In he shor-run nominal rigidiies, segmened markes, or informaional fricions can play an imporan par. 17 They are no self-insuring in auarky by holding a less risky form of capial. They are sharing idiosyncraic risk. 18 The liquidiy rap is essenially abou an ineremporal wedge. From he poin of view of a fricionless model, invesmen is oo low. 12

13 Superneuraliy and he zero lower bound. While he presence of money has very large real effecs, money is sill neural and superneural. Doubling he amoun of money would jus double prices, leaving all real variables unaffeced. Demand for money ˆm grows during liquidiy raps as he nominal ineres rae i = r + π falls. A cenral bank ha arges inflaion mus increase he money supply endogenously o keep prices on pah. If i didn, prices would fall, bu he real allocaion wouldn change. The inflaion arge iself doesn affec any real variable excep real money holdings m. I simply does no appear in equaions (14), (15), and (8). As a resul, he opimal inflaion arge is given by he Friedman rule, i = r + π 0. I maximizes agens uiliy from money m wihou affecing any oher real variable. I is easy o break he superneuraliy, bu i is a useful heoreical benchmark ha highlighs ha he liquidiy rap does no hinge on violaing money neuraliy and superneuraliy. 19 Here super-neuraliy comes from log preferences, which imply a demand elasiciy of money of one. Recall ha he value of liquidiy is equal o he presen value of expendiures on liquidiy services m i. Wih log preferences a higher nominal ineres rae i reduces real money holdings m proporionally, so ha m i doesn change. As a resul, λ is no affeced and neiher is any real variable. 20 In conras o New Keynesian models wih nominal rigidiies, he zero lower bound on he nominal ineres rae, i = r + π 0, doesn really play any essenial role in he liquidiy rap. The liquidiy rap is a gradual phenomenon, and real effecs of money grow as he value of liquidiy rises. While he presence of money creaes his lower bound on ineres raes, i also raises he equilibrium ineres rae so ha he zero lower bound is no binding. As Figure 2 shows, he zero lower bound is no binding excep for very large levels of idiosyncraic risk σ. 21 When he zero lower bound is binding, he cenral bank is simply unable o deliver he promised inflaion arge. Bu he focus and conribuion of his paper is he wide parameer region where he zero lower bound is no binding, and ye we have a liquidiy rap. Wha s more, since money is superneural, changing he inflaion arge will always fix he zero lower bound problem, bu i will have no effecs on he real side of he liquidiy rap. In fac, under he opimal moneary policy, i 0, he zero lower bound is never a problem. 2.5 Undersanding he mechanism Is he liquidiy rap really abou money, or is i acually abou safe asses? Here I ll show ha i s abou safe asses wih a liquidiy premium. Agens can rade risk-free deb, bu i doesn produce a liquidiy rap. Neiher does a diversified (safe) equiy index. We can also add safe governmen deb and deposis. They only produce a liquidiy rap o he exen ha hey have a liquidiy premium. To undersand he role of he liquidiy premium, noice ha safe asses wihou a liquidiy 19 In he Appendix I solve he model wih a) a CES demand srucure for money and b) a cash-in-advance consrain. In boh cases inflaion arges have real effecs because he expendiure share on liquidiy services depends on he nominal ineres rae. 20 This may seem puzzling a firs. How can m fall bu λ remain consan? Recall ha λ = (m + h)/(k + m + h) includes no only curren real money balances m, bu also fuure money h. As we change he inflaion arge and i, m and h move in opposie direcions. 21 Of course, his depends on he inflaion arge π. If π is sufficienly negaive he ZLB will be binding for all σ. 13

14 premium mus be backed by paymens wih he same presen value. Agens may hold he safe asses, bu hey are also direcly or indirecly responsible for he paymens backing hem. The ne value is zero, so hey canno funcion as a safe sore of value. In conras, asses wih a liquidiy premium have a value greaer han he presen value of paymens backing hem. The difference is he presen value of he liquidiy premium. This is wha makes hem a sore of value ha can improve idiosyncraic risk sharing. Essenially, agens wih a bad shock can sell par of heir liquid asses o agens wih a good shock o reduce he volailiy of heir consumpion. And he ne value of hese safe liquid asses increases dramaically when he real ineres rae becomes very low. This is he origin of liquidiy raps. Money, and safe liquid asses more generally, are special because hey are boh i) safe, and ii) have posiive ne value because hey have a liquidiy premium. This allows hem o serve as a sore of value ha improves risk sharing and creaes a liquidiy rap. There are many asses ha have posiive ne value, such as capial, housing, or land. Bu he saring poin in his paper is ha real invesmens are risky, and risk sharing is incomplee. For example, an agen can buy a paricular plo of land, whose value has significan idiosyncraic risk ha can be fully shared. There are also many safe financial asses, such as AAA corporae deb. Bu since hey don have a liquidiy premium, heir ne value is zero and hey don produce liquidiy raps. I s worh sressing ha his is a general equilibrium mechanism. The only reason agens hold money is because i provides liquidiy services. From an agen s poin of view, risk-free bonds are jus as good as a sore of value for risk sharing purposes, and hey pay ineres on op. Bu agens can all hold risk-free bonds as a safe sore of value. Someone mus ake he oher side and issue risk-free deb. In general equilibrium he real ineres rae adjuss o ensure his. Money, and safe liquid asses, have posiive ne value, so hey can improve idiosyncraic risk sharing in general equilibruim. How does money improve risk sharing? To undersand how money improves risk sharing, inegrae an individual agen i s dynamic budge consrain (4) o obain 22 [ˆ E Q 0 e ] 0 rudu (c i + m i i )d w 0 = k 0 + ˆ 0 e 0 rudu m i d (16) Here for simpliciy I assume every agen owns an equal par of he aggregae endowmen of capial and money. On he lef hand side we have he presen value of his expendiures on consumpion goods and money services. On he righ hand side we have he aggregae wealh in he economy, k 0 +m 0 +h 0. The lef hand side is evaluaed wih an equivalen maringale measure Q ha capures he marke incompleeness; i.e. such ha W i + 0 (α u/σ)du is a maringale. A risky consumpion plan coss less because i can be dynamically suppored wih risky invesmen in capial ha yields an excess reurn α. The endowmen of money on he rhs is safe, however. Wih perfec risk sharing, σ = 0, we have α = 0, so marke clearing m i di = m means 22 The ineremporal budge consrain (16) is equivalen o he dynamic budge consrain (4) wih incomplee risk sharing if shoring capial k < 0 is allowed. This is no required in equilibrium of course. 14

15 [ ha money drops ou of he budge consrain in equilibrium; i.e. E Q 0 e ] 0 rudu m i i d = 0 e 0 rudu m i d. Money is worh more han he paymens backing i because i has a liquidiy premium (ha s why i appears on he rhs), bu agens spend on holding money exacly ha amoun, so i cancels ou of he budge consrain and has no effecs on he equilibrium. Bu if idiosyncraic risk sharing is imperfec, he excess reurn hen is posiive, α > 0. Then even if in equilibrium agens mus hold all he money, m i di = m, he presen value of expendiures on money services under Q is less han he value of he endowmen of money services (which is no [ risky), E Q 0 e ] 0 rudu m i i d < 0 e 0 rudu m i d. As a resul, money does no drop ou of he budge consrain, and hey can use he exra value o reduce he risk in heir consumpion c i. To make his clear, agens could choose safe money holdings m i = m if hey waned, in which case money would indeed drop ou. This corresponds o never rading any money; jus holding heir endowmen. Bu hey are beer off rading heir money coningen on he realizaion of heir idiosyncraic shocks. They ge a risky consumpion of money services m i, bu reduce he risk in heir consumpion c i. So an agen wih a bad idiosyncraic shock in his risky capial can sell par of his money o an agen wih a good idiosyncraic shock. Boh are beer off. The agen wih a bad shock ges more consumpion and capial han wihou rading, bu less money; he agen wih he good shock less consumpion and capial, bu more money. Governmen deb, deposis, and Ricardian equivalence. Now le s inroduce safe governmen deb and bank-issued deposis. Boh may have a liquidiy premium. 23 The boom line is ha governmen deb and deposis only produce a liquidiy rap if hey have a liquidiy premium. Le b be he real value of governmen deb, and dτ lump-sum axes. The governmen s budge consrain is 24 db = b (i b π)d dτ dm p dm = dm p πm d where i b is he nominal ineres rae on governmen bonds; I allow for he possibiliy ha i b < i so ha governmen deb also has a liquidiy premium. The governmen has a no-ponzi consrain lim T e T 0 rsds (b T + m T ) = 0. Inegraing boh equaions we obain m + b = ˆ e ( ) s rudu m s i s + b s (i s i b s) ds + ˆ e s rudu dτ s (17) The governmen s oal deb is b + m, and i mus cover i wih he presen value of fuure axes plus wha i will receive because is liabiliies b and m provide liquidiy services. When agens hold money, hey are effecively paying he governmen m i for is liquidiy services (he forgone ineres); when hey hold governmen deb hey are paying b (i i b ). In paricular, if governmen 23 Krishnamurhy and Vissing-Jorgensen (2012) show ha US Treasuries have a liquidiy or convenience yield over equally safe privae deb. 24 In he baseline model wihou governmen deb, we have b = 0 and dτ = dm /p. 15

16 deb is as liquid as money, i b = 0, he only hing ha maers is he sum (m + b )i. There are also banks ha can issue deposis d ha pay ineres i d < i. Banks are owned by households. The ne worh of a bank is n and follows he dynamic budge consrain dn = n r + d (i i d )d df where f are he cumulaive dividend paymens o shareholders. The bank earns a profi from he spread beween he ineres i pays on deposis i d and he ineres rae a which i can inves, i. Using he ransversaliy condiion lim T e rt n T = 0 we can price he bank a v : 25 v = n + ˆ e s rudu d s (i s i d s)ds The marke value of he bank includes is ne worh oday, plus he presen value of profis from he ineres rae spread on deposis, d (i i d ). Toal wealh is w = (k a ) + d + v + m + (b e s rudu dτ s ), where a is he bank s asses. Households own all he capial, money, and governmen deb (minus he presen value of axes), excep for whaever asses he bank holds. They also hold bank deb (deposis) d, and bank equiy v (so hey indirecly own he asses ha he bank owns). Since he bank s ne worh is n = a d, we have v + d a = v n = e s rudu d s (i s i d s)ds. And m + b e s rudu dτ s = e s rudu ( m s i s + b s (i s i b s) ) ds. So oal wealh is w = k + ˆ e ( ) s rudu m s i s + b s (i s i b s) + d s (i s i d s) ds (18) Toal wealh is capial plus he presen value of expendiures on liquidiy services, which now include money, liquid governmen bonds, and deposis (each weighed by is corresponding liquidiy premium). Governmen deb and deposis herefore only have an effec o he exen ha hey have a liquidiy premium. Safe governmen or privae deb wihou a liquidiy premium cancels ou and has no effecs. The corresponding expression for λ is λ = m i + b (i i b ) + d (i i d ) r (ˆx δ) 1 = w ρβ r (ˆx δ) where β should be inerpreed as he expendiure share on liquidiy services across all asses, β = (m i + b (i i b ) + d (i i d ))/oal expendiure. In he special case wihou deposis or governmen deb we recover expression (11). The easies way o inroduce governmen deb and deposis wih a liquidiy premium is o pu 25 Wrie n = obain v = n + e s r udu df s e s r udu d s(i s i d s)ds. e s r udu d s(i s i d s)ds + lim T e rt n T, and use v = e s r udu df s o 16

17 hem ino he uiliy funcion (1 β) log(c ) + β log(a(m, b, d)) where A(m, b, d) is an homogenous aggregaor. Agens will devoe a fracion β of expendiures o he liquid aggregae, β = (m i + b (i i b ) + d (i i d ))/oal expendiures. As a resul, we don need o change anyhing in our baseline model. We jus need o reinerpre β as he fracion of expendiures on liquidiy services across all asses. 26 Ricardian equivalence holds in his economy. If governmen deb doesn have a liquidiy premium, changing b (and adjusing axes o service his deb) has no effecs on he economy. If governmen deb has a liquidiy premium, hen changing b can have an effec on he liquidiy premium of governmen deb and perhaps oher asses as well. Bu i will no have any effec on he real side of he economy. Proposiion 4. (Ricardian Equivalence) Wih log preferences for liquidiy, changes in governmen deb b have no effecs on he real ineres rae r, invesmen ˆx, or he value of liquidiy λ. Changes in b can only affec he liquidiy premiums of differen asses. To see his, noice ha he expendiure share on liquidiy services across all asses is a consan, β, and his is he only way ha liquid governmen deb can affec he economy. For example, if he liquidiy aggregaor is Cobb-Douglas, A(m, b, d) = m ɛm b ɛ bd ɛ d wih ɛ m + ɛ b + ɛ d = 1, hen he expendiure share on liquidiy services from each asse class is fixed; e.g. b (i i b )/expendiures = ɛ b β. Changing b only affecs he liquidiy premium on governmen bonds, bu no on deposis or money. As wih superneuraliy, Ricardian equivalence can be broken here if we move away from he log uiliy over liquidiy (see Appendix for CES and cash-in-advance formulaions). Bu i s a useful heoreical benchmark ha shows ha he liquidiy rap does no hinge on violaing Ricardian equivalence. Equiy markes. Bu wha abou equiy markes? The saring poin in his paper is ha capial is risky, and idiosyncraic risk sharing is incomplee. Bu if agens can hold a diversified (safe) marke index, can his funcion as a safe sore of value and produce a liquidiy rap? Here I ll show ha while issuing equiy improves risk sharing, i does no produce a liquidiy rap. In he baseline model agens canno issue any equiy. Le s say insead ha hey mus reain a fracion φ (0, 1) of he equiy, and can sell he res o ouside invesors. Issuing ouside equiy improves idiosyncraic risk sharing, of course. Ouside invesors can fully diversify across all agens equiy, creaing a safe marke index worh (1 φ)k. If agens could sell all he equiy, φ = 0, we would obain he firs bes wih perfec risk sharing; wih φ > 0 we have incomplee idiosyncraic risk sharing. 26 Wih i i d > 0 banks have incenives o supply as much deposis as possible. I m no providing a heory of wha limis hem (perhaps capial requiremens), bu i doesn maer. Regardless of how we fill in he deails of how banks operae, he expendiure share on liquidiy services across all asses will be β. 17

18 Since agens can finance an exra uni of capial parly wih ouside equiy, he effecive risk of capial for an agen is φσ. In fac, we can obain he compeiive equilibrium by replacing σ by φσ in (5)-(9). The dynamic budge consrain is now 27 dw = (r w + k α c m i )d + k φσdw The risk premium is α = σ c (φσ), and he volailiy of consumpion is σ c = k /(k +m +h ) (φσ) = ρβ (1 λ)(φσ). The value of liquidiy is given by λ =. ρ ((1 λ)φσ) 2 Bu while equiy improves risk sharing, i does no produce a liquidiy rap. In paricular, wihou money, β = 0, an increase in idiosyncraic risk σ is fully absorbed by lower real ineres raes r = a δ (φσ) 2, bu invesmen remains a he firs bes ˆx = a ρ. The reason is ha issuing equiy improves risk sharing in a way ha affecs he marginal risk from an exra uni of capial and he average risk in agen s porfolio equally. As a resul, i dampens he risk premium σ c φσ = (φσ) 2 and he precauionary moives σ 2 c = (φσ) 2 equally, canceling ou. And he value of equiy is backed by he firm s asses, so i s no a posiive ne value. The aggregae wealh in he economy is sill given by he righ hand side of (16), bu he oal value of capial is spli ino inside and ouside equiy k = φk + (1 φ)k. 28 In paricular, he value of he marke index does no blow up o infiniy as r approaches he growh rae ˆx δ, as he value of liquidiy does. 29 Cashless limi. The liquidiy rap does no hinge on a large expendiure share on liquidiy services β i survives even in he cashless limi β 0. As explained in Secion 2.2, he value of money λ is he presen value of expendiures on liquidiy discouned a he risk-free rae. When he real ineres rae is high relaive o he growh rae of he economy, λ is small; close o he expendiure share on liquidiy services β. Bu when he real ineres rae is very close o he growh rae of he economy, λ can become very large regardless of how small β is. This can be seen very clearly in equaion (11). I s all abou he denominaor. So if we ake he cashless limi, β 0, he compeiive equilibrium will no always converge o ha of he non-moneary economy wih β = 0. For σ such ha in he non-moneary economy he real ineres rae is above he growh rae, he moneary economy will indeed converge o he non-moneary one as β 0. Bu for σ such ha in he non-moneary economy he real ineres rae is equal or below he growh rae of he economy, his canno happen. As he real ineres rae drops and approaches he growh rae of he economy, he value of liquidiy λ blows up o keep r above ˆx δ, no maer how small β is. As a resul, we ge a liquidiy rap even in he cashless limi 27 Equiy can be diversified so is reurn mus be r. In equilibrium agens are holding w = n + m + h + e where n = φk is he inside equiy in heir firm ha hey reain, and e = (1 φ)k is he diversified ouside equiy in oher agens firms. Toal equiy n + e = k ; since here are no adjusmen coss, Tobin s q is 1 here. Boh inside and ouside equiy yield r, bu he inside equiy has idiosyncraic risk (ouside equiy also has id. risk bu i ges diversified). Agens herefore also ge a wage or bonus as CEO of heir firm o compensae hem for he undiversified idiosyncraic risk, k α. 28 More generally, if firms use deb, k = n + e + d, where n is inside equiy, e is ouside equiy, and d is deb. All he financial claims on firms add up o he value of heir asses. 29 Toal equiy is always worh oal capial, whose price akes ino accoun is uninsurable idiosyncraic risk. As σ grows and r drops, insider wages or bonuses αk increase o compensae for he idiosyncraic risk. 18