Pricing Systemic Crises: Monetary and Fiscal Policy When Savers are Uncertain

PRICING SYSTEMIC CRISES: Monetary and Fiscal Policy When Savers Are Uncertain (cid:3) Andreas Lehnert Wayne Passmore BoardofGovernors ofthe BoardofGovernorsof the FederalReserveSystem FederalReserveSystem MailStop93 Mail Stop93 Washington DC,20551 WashingtonDC,20551 (202)452-3325 (202)452-6432 alehnert@frb.gov wpassmore@frb.gov Versioninformation: Revision: 6.8 LastRevised: August 12,1999 (cid:3) Theviewsexpressedinthispaperareoursaloneanddonotnecessarilyreflectthose oftheBoardofGovernorsoftheFederalReserveSystemoritsstaff.WethankWen-Fang Liu,DarrelCohen,JohnCurran,EricEngen,StephenOliner,andseminarparticipantsat theFederalReserveBoardforhelpfuldiscussions. Anyremainingerrorsareourown.

PRICING SYSTEMIC CRISES: Monetary and Fiscal Policy When Savers Are Uncertain Abstract In our model, the return on assets depends on the joint behavior of all savers; if all savers sell the asset simultaneously then there will be a financial “Armageddon” and the return will be quite low. We assume that individualsavers arerisk-neutral but uncertainty-averse andcannot form precise estimates of the behavior of the market (all other savers). We determineequilibriuminvestment usingthemodern theory ofKnightian uncertainty, that is, decision-making with multiple subjective prior distributions,whichshowsthatsaverswillacttomaximizetheirminimumpayoff in the presence of uncertainty. Savers divide their portfolios between two assets: A risky asset, representing a fully-diversified share in economywide production, and a riskless asset, representing government bonds. Government bonds are backed by the tax authority of the state, and so always pay off. In times of high uncertainty, savers' demand for government bonds will increase, and spreads between the return on bonds and the expected return on risky assets will widen. As a result, investment in the risky asset will decrease. If the government responds with a purely monetarypolicyofreducingtherisk-freerate,itwillmakethebondlessattractiveandforcesaverstoholdmoreoftheriskyasset. Weshowthatsuch monetaryexpansionsarePareto-improving, but thattheyfailto recapture the optimal allocation. To restore investment andsavings to their optimal levels,thegovernmentmustalsouseafiscalpolicyofdistortionarytaxesto discourage current consumption andleisure. Journal ofEconomicLiterature classification numbers: G2,E5,E44,D8. Keywords: Knightian uncertainty, financial crisis, monetary and fiscal policy

1 Introduction During times of market turmoil, savers are unable to form precise estimates of the distribution of returns of their portfolios. Such times arealso oftenmarkedbyunusuallyhighinterest-ratespreadsbetweenrisklessand risky assets, unusually high levels of asset return volatility, and an attendant difficulty in obtaining project financing. The distribution of an asset's returns is difficult to forecast, at least in part, because it depends on the likely actions of other market participants. Ifworried savers liquidate partoftheirportfolios,thevalueofallriskyassetscanfalldramatically. To combatsuchcrisesofconfidence,governmentsareurged,ontheonehand, to take dramatic action to restore confidence in financial markets with a mix of an expansionary monetary policy and explicit or implicit guarantees to producers or to financial institutions. On the other hand, governmentsarecautioned not toprovoke inflationorpromote moralhazard. In this paper, we characterize the government's optimal policy responses to suchanepisode. Financialassetstypicallycarryapromisedrateofreturn,butthisreturnis subjecttoahostofso-called“risks.” Theserangefromthewellunderstood and predictable (such as actuarial risk) to the exotic (such as “fat-finger” risk) or unpredictable (such as political risk).1 Somewhere in the middle 1“Fat-finger”riskistheriskofmis-keyinganorderintoanelectronicterminal. Fora representative“taxonomyofrisks”seeRahl(1998). 1

lies so-called model risk, the risk of having mis-modelled the underlying stochastic process of interest. This “risk” is largely what we have in mind when we discuss “uncertainty,” although, of course, all risks carry some uncertaintywiththem. Uncertaintyoccurswhentheinformationavailable about a particular event is too sparse or too vague to form a precise estimate of its probability; the sparser or vaguer the information, the greater theuncertainty. Uncertainty has a qualitatively different effect on financial markets than risk. Forexample,onNovember3,1998theChicagoMercantileExchange (CME) listed a futures contract based on the Quarterly Bankruptcy Index (QBI).2 If savers were truly risk-neutral, there should have been a brisk tradeinthecontract, giventhedisparityinestimatesofthedistribution of bankruptcies,andthefactthat theCMEwaiveditscontract fee. However, as of mid-April, 1999, not a single contract had been traded. While there are several possible explanations for this, market participants often cite high levels of model risk for staying away from contracts based on the CME-QBI,suggesting that uncertainty indeedplaysarole.3 Similarly, in the late summer and fall of 1998 spreads between risky and riskless assets widened dramatically, while asset return volatility also in- 2TheCME-QBIisacount,measuredinthousands,ofallnewbankruptcycasefilingsin U.S.bankruptcycourts. Formoreinformation,seeChicagoMercantileExchange(1999). 3Asafurtherexampleofthiskindofuncertainty,Christofferson,Diebold,andSchuermann (1998)analyze the typical volatility forecast underpinning the popular “Value at Risk” risk-management model and find that, beyond a ten-day trading horizon, the volatilityforecastisofverypoorquality. 2

creased. Market participants reported that they were dissatisfied with theirrisk-managementmodels. Therewasawidespreadsense that savers were less willing to accept risk, which reduced liquidity in financial marketsand,asaresult,jeopardizedthecontinuedexpansionofrealeconomic output. The Federal Reserve responded by cutting the target Fed Funds rate by 75 basis points. The return on risky assets also declined, although spreadsremainedelevated,andtherealconsequencesofthefinancialturmoil never materialized. We argue that the turmoil was a direct consequence of uncertainty: savers began to doubt their estimates of the distribution of portfolio returns, in part because these returns depend on the behavior of other savers. In particular, savers' estimates of the probabilities of extreme financial events became more imprecise, causing them to fleerisky assets infavor of riskless assets.4 In our model savers are able to invest in two assets: A risky asset, representing a share in economy-wide production and a riskless asset, representing a government bond. The risky asset carries only aggregate risk because we assume that the saver has diversified away all of the idiosyncratic, project-specific, risk. We model financial crises by assuming that if total investment in the risky asset falls below a critical level, its per-share returnwillplunge. Abovethecriticallevel,thereturnissmoothlydecreasingintotal investment. 4Formally, wemodeluncertainty following Liu(1999)andEpsteinandWang(1994), whereuncertaintyanduncertainty-aversionaresummarizedbyascalarparameter. 3

Government bonds, in contrast, pay a return that varies neither with the state of nature nor with the level of aggregate investment. They are ultimately backed by the tax authority of the state and so will be paid off evenifabadstateofnatureisrealizedorifinvestmentfallsbelowthecritical level [see Burnham (1989)]. Uncertainty-averse savers will demand an uncertainty premium over the return on these bonds to hold a share in the productive, though risky, asset. In periods of greater uncertainty thesespreadswillwiden,investmentwilldecline,andtotaloutputwillbe lower. We show that, in times of increased uncertainty, savers will “flee to quality” and demand too much of the government's safe bond relative to the risky asset. Wefurther show that apurelymonetarypolicy of reducingthe risk-freerateisParetoimproving,becauseitmakesthebondlessattractive andinduces saversto hold moreof therisky asset,although it alsocauses savers to save less in total. We also discuss the consequences of a policy that holds the stock of riskless bonds constant, so that uncertainty drives downtheequilibriumrisk-freerate. Whenthereisabadtechnologyshock (so that labor's product is low), we show that the scope for monetary expansions will be sharply limited; and we further show that if uncertainty is very high, monetary policy may be wholly ineffective. To recapture the optimum–that is, to restore total savings, labor effort and consumption to theiroptimallevels–thegovernmentmustuseanadditionalfiscalpolicyof 4

distortionary taxes to discourage current consumption andleisure. Thepaperisorganizedasfollows: Insection2wepresentthemodelwithout uncertainty, in section 3 we present a condensed discussion of the modern theory of uncertainty and show how it affects outcomes in our model andin section 4 we deriveoptimal monetary and fiscal policies. In section5weprovideconclusionsandabriefdiscussionoftheimplications ofthisworkfortheconductofmonetarypolicy. Proofsofpropositions are relegatedto anappendix. 2 The Model 2.1 Savers Manyindividual savers, whosepopulation is normalizedto one,are born in each period t and live for two periods. Savers consume an amount c t 0 while young and c t 1 while old. (We will generally denote generations with superscripts and age with subscripts.) In addition, young savers are endowed with asingle unit of time which they may split between leisure, ‘ t 0 ,andlaboreffort, 1 (cid:0) ‘ t 0 . Savers havepreferences given by: u ( c t 0 ; ‘ t 0 ) + c t 1 ; where u c ; u ‘ > 0 , u c c ; u ‘ ‘ < 0 and u 2 c ‘ < u c c u ‘ ‘ (1) . 5

Note that savers are risk-neutral with respect to consumption while old. This allows us to isolate the pure effect of uncertainty aversion. Finally, becauseallsaversareidentical,wewilldefineequilibriausingarepresentativesaver. Laboreffortearnsacertainrealwageof W t ,whichwilldependonthestate (seebelow). Thusattheendofthefirstperiodoflife,agentshaveavailable to them an amount s t = ( 1 (cid:0) ‘ t 0 ) W t (cid:0) c t 0 to save for consumption while old. Theirsavingsportfolioisdividedbetweeninvestmentinaneconomywide risky asset, x t and holdings of the safe asset, b t . The safe asset will pay a gross return in the following period of r t . The risky asset will pay a state-contingent gross return in the following period, t + 1 , conditional on the total amount invested (see below). For now, assume that the risky assetpaysagross per-sharereturn of R ( ! t+ 1 ; X t ) . Aggregatestates ! ,whichrepresentatechnologyshock,mustlieintheset of allowed states (cid:10) . There is a known, true, distribution over these states (cid:25) ? ,whichdoes not changeover time. The wage earned by young agents will depend on the state in the current period, W t = W ( ! t ) . Assumethat thewagefunction W ( ! ) satisfies: W ( ! ; X ) > 0 ; all ! (A1) ; This assumption guarantees that young workers will always earn a posi- 6

tive wage. Savers must pay an excisetax rate of (cid:28) c on consumption whileyoung and a labor income tax rate of (cid:28) ‘ . In addition, the government makes lumpsum transfers (taxes if negative) in each period of life of H t 0 and H t 1 . The saver'sbudget constraints ineach periodoflifeare: ( 1 + (cid:28) c ) c t 0 + ( 1 (cid:0) (cid:28) ‘ ) W t ‘ t 0 + x t + b t (cid:20) ( 1 (cid:0) (cid:28) ‘ ) W t + H t 0 ; (2) c t 1 ( ! t+ 1 ) = R ( ! t+ 1 ; X t ) x t + r b t + H t 1 : (3) Savers take asgiventhereturn to the risky asset ineachstate, R ( ! t ) . 2.2 Producers The risky asset stands for a fully diversified investment share of a large number of risky projects. These projects are owned and operated by twoperiod-livedentrepreneurswhoconsumeonlyinthesecondperiodoflife, arerisk-neutral andhavenootherproductiveassets. Asaresult,theywill be willing to accept a loan to finance their projects if the expected value of output net of loan costs is greater than or equal to zero. Projects' ex ante risk characteristics and ex post outputs are costlessly observed by all parties in the economy. There is a continuum of producers and their associatedprojects;thesearenamed n andaredistributeduniformlyinthe 7

range [ 0 ; I 1 ] . Projects of type n potentiallyproduce areturn to their owners (the producers) of (cid:26) ( n ; ! ) in state ! , where (cid:26) ( n ; ! ) is twice continuously differentiable in n for each ! . Assume that these projects are ordered so that: @ (cid:26) ( @ n n ; ! ) (cid:17) (cid:26) n ( n ; ! ) < 0 ; all ! in (cid:10) . Inotherwords,themostproductiveprojects havethelowestname. Inaddition, assume that (cid:26) ( 0 ; ! ) < 1 , so that even the most productive project in the best possible state has a finite return. Finally, define the potential expectedreturn to projects of type n as (cid:26) ( n ) : (cid:26) ( n ) (cid:17) Z (cid:10) (cid:26) ( n ; ! ) d (cid:25) ? : When a project type is funded by savers, it generates an external social benefit. Ifnotenoughprojectsarefunded,thentheprojectsthatarefunded do not achieve their potential, and produce a very low return, (cid:26) 0 . Let this critical level of investment be denoted as X L . Thus if some total amount X is invested in the risky projects (i.e. the project type named X and all better project types n < X are financed), total expected economy-wide 8

production is: f ( X ) = 8 > < > : (cid:26) 0 X if X (cid:20) X L , R X 0 (cid:26) ( n ) d n if X > X L . Finally, assume that the return when aggregate investment is too low, (cid:26) 0 , issmallerthanthereturn earnedbyanyproject inany state: (cid:26) 0 < (cid:26) ( n ; ! ) ; all n 2 [ 0 ; I 1 ] , ! in (cid:10) . Anexpectedreturnfunctionsatisfyingthesecharacteristicsisdisplayedin figure 1. X) (r ,nruter detcepxE Critical investment level, X L r Expected return on 0 risky asset X Investment, X L Figure 1: Expected returns to the risky asset as a function of total investment, (cid:26) ( X ) . 9

Thisformofproductioncapturesthesenseofrecentresearchintofinancial crises. As total investment decreases, the projects that are financed may suffer because they do not have access to the products of other projects, as in the “race for the exit” model of Subrahmanyam and Titman (1999). Further, the projects themselves may be illiquid, so that a decrease in investment from one period to the next requires the premature (and costly) liquidation of fundamentally illiquid assets, as in Allen and Gale (1998) and Diamond and Dybvig (1983). Finally, failure to invest in one project maycausethe default ofanother, asinLagunoff andSchreft (1998,1999). Note that all of our results are unaffected if the critical aggregate investment level is zero, X L = 0 . All that we require is that the returns realized bytheveryfirstprojecttobefundedarequitelowifthatprojectistheonly one funded. As a result, a saver who believes that no other savers intend to financeprojects will not findit profitable to fundaproject. Savers place their investments x t with a zero-cost, zero-profit economywidecompetitiveintermediarythatallowsthemtoownafully-diversified shareoftheprojectsthatarefunded. Bothsaversandproducerswillactas price takers with respect to the interest rate charged on individual loans. All projects will be charged the same rate in each state. If a total quantity X > X L is committed to the intermediary, the owners of the last project typethatisfinancedmustbejustindifferentbetweenacceptingtheirloans and undertaking production and not. Thus savers realize per-share re- 10

turnsof (cid:26) ( X ; ! ) . Ifanamountbelow thecriticallevelisinvested, X < X L , weassumethatallproduction issurrenderedtothesaversbytheproducers,so that theper-share return is (cid:26) 0 . Thus if an amount X > X L is invested, total expected producer's surplus (the sumof allexpectedutilities) is givenby: PS ( X j X (cid:21) X L ) = Z 0 X [ (cid:26) ( n ) (cid:0) (cid:26) ( X ) ] d n : (4) If less than X L is invested, we assume that producers make zero surplus and all output is divided among savers. Although we do not explicitly consider them here, we also assume that the government has available to it a complete array of lump-sum taxes and transfers that it can levy on producers. 2.3 The Government Bond The government receives goods in period t from the young of that period in exchange for tokens which may be redeemed in period t + 1 for some known amount. In this paper, we assume that the government has a monopoly on the safe asset, which is backed by a storage technology earning a unit return. Thus by manipulating the return to the only truly safeassetintheeconomy,thegovernmentwillbeabletodirectlyaffectthe risk-free rateof return. 11

If the government pays agross return r t < 1 on its bonds from period t to t + 1 ,itrealizesnetrevenuesinperiod t + 1 ,whileifitpaysagross return r t > 1 ,itmustmakeanetoutlayinperiod t + 1 . Inthefirstcase,weimagine that this “seigniorage” revenue is refunded lump-sum to the generation thatpaysit,whileinthelattercase,weimaginethatthegovernmentlevies taxes on either generation to meet its obligation. Thus government bonds arespecialbecausetheyarebackedbythetaxauthorityofthegovernment and because the return on bonds is totally independent to the quantity of bonds sold. Thegovernment's periodt budget constraint is thus: (cid:0) H t 0 (cid:0) H (cid:0)t 1 1 + (cid:28) ‘ ( 1 (cid:0) ‘ t 0 ) W t + (cid:28) c c t 0 + ( 1 (cid:0) r (cid:0)t 1 ) B (cid:0)t 1 + B t (cid:21) 0 : (5) Recall that superscripts denote generations and subscripts denote age. In period t the government removes from storage the bond purchases of the current old generation, B (cid:0)t 1 . The government promised a gross return r (cid:0)t 1 on its bonds, and so realizes revenue of ( 1 (cid:0) r (cid:0)t 1 ) B (cid:0)t 1 on its bonds. Noticethatthegovernmentcantransferresourcesbetweenthetwogenerations alive in each period with lump-sum transfers (or taxes) on the current young ( H t 0 ) and the current old ( H (cid:0)t 1 1 ). We assume (see below) that thegovernmenttreats eachgenerationasessentiallyindependentfromits neighbors, and so does not favor one over the other. Because we assume thatthegovernmenthasnoexpensesofitsowntheinequalityin(5)holds 12

withequality. 2.4 Equilibrium Without Knightian Uncertainty In the case without Knightian uncertainty we look for an equilibrium in each period t , conditional on a realization of the shock term ! t (and thus the wage rate W t ), in which the government does nothing but provide a zero-cost storage asset paying agross return r t > 0 . Assume that the risk-free rate lies between the maximum and minimum expectedreturns ontherisky asset: m X i (cid:21) n 0 (cid:26) ( X ) (cid:20) r t (cid:20) m X a (cid:21) x 0 (cid:26) ( X ) : Because savers are risk neutral, they are not concerned with the distribution of returns, and will demand a common expected return of r t on elements of their savings portfolio (if both assets are held in positive quantities). As a consequence of this, we can characterize a saver's optimal policy, the solution to maximizing (1) subject to the constraints (2) and (3), as a savings function s [ W ( ! t ) ; r t ] or s ( ! t ; r t ) . Note that a saver's total savings isincreasing inthe risk-free rate r t . Definition1(EquilibriumWithout KnightianUncertainty) Given (a) An announced choice for the risk-free rate r t ; (b) A realization of the aggregate state ! t ; and (c) A wage rate W t = W ( ! t ) , an additive probability 13

measure Q t (withsupportontheinterval [ 0 ; s ( W t ; r t ) ] )overtheinvestmentlevels x of other saversand an aggregate levelof investment X t are a Nash equilibrium if: 1. The representative saver is indifferent, under Q t , among all levels of risky investment, 0 (cid:20) x t (cid:20) s ( ! t ; r t ) . 2. Thelevelofaggregateinvestmentintheriskyassetisgivenby: X t = Z X x d Q t ( x ) : Further,theaggregatequantityofsavings, S t ,isgivenbytherepresentative saver'schoiceofsavings, S t = s ( ! t ; r t ) ;andtheaggregatequantityofbond holdings, B t ,isgivenby S t (cid:0) X t . Here, Q t is the probability distribution over levels of investment in the riskyassetthattherepresentativesaverascribestoallothersaversexisting attime t . Iftherepresentative saveristo beindifferent among alllevelsof investment, x t ,thenitmustbethecasethatthereisexpectedrate-of-return equalitybetweenassets: (cid:26) ( E Q f X t g ) = r t ; where E Q f (cid:1) g isthemathematicalexpectationsoperatorunderdistribution Q . There are many possible equilibria to the investment game in each 14

period, associated with many possible distributions Q , including one in which the saver expects no-one else to invest, and so does not himself. For positive quantities ofinvestment, aggregate investment willbe X ? ( r ) , definedas: (cid:26) [ X ? ( r ) ] = r : (6) The representative saver born each period has some probability distribution Q t inmindfor theinvestment behaviorof hisfellows,where Q t must only satisfy equation (6). We interpret Q t as a purely subjective distribution,anddeterminethelevelof aggregate investment from equation(6). As shown by figure 2, when the risk-free rate is r t = 1 and there is no uncertainty, total savings is s ( W t ; 1 ) and equilibrium investment is X ? ( 1 ) (we assume that for all realizations of the technology shock ! t , total savings exceed X ? ( 1 ) when r t = 1 ). The remaining S t (cid:0) X ? ( 1 ) of aggregate savingsisallocatedtotherisklessasset. Clearly,whenlabor'sproduct W t , and hence savings, is low, bond sales will also be low. As we shall see in the next section, this will limit the government's ability to use monetary policy. 15

X) (r ,nruter detcepxE r( )X S(r) Expected return risk-free to risky asset rate r 0 X L 1=r Total savings schedule B X*(1) S(1) Investment, Savings: X,S Figure 2: Equilibriumwithout uncertainty. 3 Analysis With Knightian Uncertainty 3.1 Description of Knightian Uncertainty Knight(1921)madetheoriginaldistinctionineconomicsbetweenriskand uncertainty that has since become conventional. For this reason, uncertainty in the sense of multiple prior distributions has become known as Knightian uncertainty. Keynes [see Glahe (1991)] famously argued that uncertainty opened the door for “animal spirits” to affect financial markets. However, the modern economic analysis of choice under randomness, due to Savage (1954), explicitly requires a unique subjective prior 16

distribution. The formal analysis of choice when there are multiplepriors begins with Gilboa (1987), Schmeidler (1989) and Gilboa and Schmeidler (1989). The Schmeidler-Gilboa analysis assumes, as does the classic Savage analysis, a series of axioms of choice. To accommodate the presence of multiple priors, they weaken the axiom of independence and add an axiom of uncertainty aversion. With these additions, preferences can be represented using a “maxmin” functional form—agents will act to maximizetheir payoff underthe most pessimisticprobability distribution. The theoretical advances of Schmeidler-Gilboa over the standard Savage economictheoryarenecessarilycomplexandbeyondthescopeofthisarticle. GilboaandSchmeidlerprovethatexpectedutilitymayberepresented by either a single but non-additive prior or a set of additive priors. (Nonadditive distributions are subjective probability distributions that do not sumtoone,andarealsoknownascapacities.) Insimplecases,liketheones consideredhere,thetwoapproachesyieldidenticalresults. [Seeespecially Gilboa and Schmeidler (1993) and Wakker (1989) for a description of the connection between the two approaches.] Because information about extreme events is relatively vague, Knightian uncertainty is particularly apposite for modeling such events [see Epstein and Wang (1995) and Dow andWerlang(1992a)]. A classic example of uncertainty is one of appraising the value of a (purported) Ming dynasty vase [see Shafer (1976) and Dempster (1968)]. The 17

vase in question is worth $1000 if it really is from the Ming dynasty, but zero if it is a fake. A risk-neutral saver who is not an expert in Chinese pottery examines the vase and concludes that it is as likely real as fake. A strictly Bayesian approach would then require us to ascribe a subjective probability distribution of f 0.5,0.5 g to the states f real, fake g . The agent would thenbuy the vase for any pricebelow $500andsellit for anyprice above $500. (Imagine that if the agent sells the vase, he receives the sale price with certainty and then loses the revealed true value of the vase. The agent is “shorting” the vase asset.) This description seems to miss something crucial about how agents, even risk-neutral agents, react to incomplete information. If the agent's behavior isinstead described by thenon-additive subjective probability distribution f 0.4,0.4 g the two events are still equiprobable (so that the agent still believes that the vase is as likely real as fake), but their sumislessthanone. Anamount0.2ofthetotalprobabilityoverstateshas been “lost” to uncertainty. Expected values of random variables under non-additive probabilities may be calculated using the Choquet integral [see Dow and Werlang (1992b) and Gilboa (1987)]. If the agent buys the vasefor some price p , andit turns out to be fake, his realized utility is (cid:0) p . If it turns out to be real, the agent's utility is 1 0 0 0 (cid:0) p , an improvement of $1000. Thus his expected utility from buying the vase is (cid:0) p + 0 : 4 ( 1 0 0 0 ) or 4 0 0 (cid:0) p . The agent would buy the vase for any price below $400. If 18

instead the agent sells the vase for some price p , and it turns out to be real, his utility is p (cid:0) 1 0 0 0 . If it turns out to be fake, his utility is p , an improvement of $1000. Thus his expected utility from selling the vase is p (cid:0) 1 0 0 0 + 0 : 4 ( 1 0 0 0 ) or p (cid:0) 6 0 0 . Theagentwouldsellthevaseforanyprice above$600. Atpricesbetween$400and$600,theagentwouldneitherbuy norsellthe vase. More generally, if agents have a subjective probability distribution over events of P (cid:15) = ( 1 (cid:0) (cid:15) ) Q , where Q is an additive probability distribution overeventsinthe (cid:27) -algebra S constructedfromthediscretesetofstates (cid:10) (so that P (cid:15) is a “uniform squeeze”of Q ), we say that agents have constant uncertainty aversion of degree (cid:15) . If an agent has a utility in each state of u ( ! ) ,then“expectedutility” isformed as: E P (cid:15) f u ( ! ) g = (cid:15) m i ! n u ( ! ) + ( 1 (cid:0) (cid:15) ) E Q f u ( ! ) g : (7) Thusuncertainsaversbehaveasif theywerepessimistic,inthesensethat theyascribe allof themissing probability to theworst-case scenario. Inrelatedwork,GilboaandSchmeidler(1989)andWakker(1989)demonstrate formally that, under certain assumptions, agents with many additive prior distributions behave as if they had a single but non-additive probability distribution. Such agents will also have the maxmin form of the utility function. Assume that an agent has the collection of additive 19

priors P (cid:15) ,definedas: P (cid:15) (cid:17) f ( 1 (cid:0) (cid:15) ) Q + (cid:15) m : m 2 M g ; (8) where the constant, 0 (cid:20) (cid:15) (cid:20) 1 indexes uncertainty, Q is the reference or true additive probability distribution and M is the set of all probability measures defined on the support of Q . Then expected utility is again formedusing themaxminformulation, equation(7),above. Notice the interesting parallel with certain types of difficult-to-quantify financial risks, e.g. model risk. Savers have many different competing hypothesesabouttheprobabilitydistributionofthereturnstosomefinancialasset. Thesecorrespond with the manydistributions lying in P (cid:15) . Such savers, according to the theory of uncertainty, will act to maximizeutility, equation (7). They will, as a result, be disproportionately concerned with the worst-case scenario. The more the competing hypotheses differ, i.e. themoreuncertain agents are,thegreater the weighton theworst-case. 3.2 Nash Equilibrium With Knightian Uncertainty In this section, we extend the definition of a Nash equilibrium in the investment game, played by the representative saver against the market, to thecasewhenthesaverisuncertainaboutthemarket'sbehavior. Indoing so, we use the extension of Nash equilibrium due to Dow and Werlang 20

(1994). Notice that uncertainty willnot beover the aggregate state, which follows the known i.i.d. process (cid:25) ? , but rather over the behavior of other savers. rotsevnI l h Market L H 0,0 0,a -b,0 a,a Figure3: Asimplifiedversion of theinvestment game. To motivate our definition of a symmetric Nash equilibrium with Knightian uncertainty, we provide the following example. Figure 3 displays a stylizedversionoftheinvestmentgamethattherepresentativesaverplays against “the market” (all other savers taken together) in which both sides are allowed only two simple actions. The individual saver can play either a high (“h”) or a low (“ ‘ ”) level of investment in the risky asset. The market can do the same, playing either “H” or “L”. The behavior of the individual (small) saver will not affect the payoff of all other savers taken together. Ifthemarketinveststhelowamount,itrealizestherisk-freerate, herenormalizedtozero,andifitinveststhehighamount,itrealizessome increment a > 0 over that. If the individual follows the market, he real- 21

izes the same return as the market. However, if the individual invests the high amount when the market invests the low amount, he suffers a loss of (cid:0) b < 0 . This corresponds to the case of being the last saver out of a “rush to the exit” model, or a depositor at the back of the line in a bank runmodel. AlthoughthesavercaneliminateLasastrategy ofthemarket (becauseitisstrictlydominatedbyH),ifthemarket(forwhateverreason) did play L, and b was large, then the saver would be exposed to a large loss. We use the augmented definition of a Nash equilibrium due to Dow and Werlang(1994). Assumethatthesaver'sbeliefsabouttheplayofthemarket can be described by the set of probabilities f p L ; p H g . Assume further that the saver ascribes probability zero of the market playing L , but, because ofKnightian uncertainty, aprobability lessthanunity of themarket playing H , p H < 1 . Wecanformexpectedutilitiesasdescribedinequation (7)above,andseethattheindividualsaverinveststhehighamountifand only if p H > b = ( a + b ) . Thus if the loss exposure b is small, the individual saver will invest even if p H is quite small, while if b is large, p H must be relatively close to unity. This is equivalent to assuming that the saver has anuncertainty parameterof (cid:15) = 1 (cid:0) p H . With this example in mind, we generalize the definition of a Nash equilibrium in the periodt investment game to include the case when agents have Knightian uncertainty. Because he is uncertain about the behavior 22

of the market, the representative saver will have beliefs about aggregate investmentthatcanberepresentedbyanon-additiveprobabilitydistribution. As a result, the saver will demand a premium for holding the risky asset. He is guarding against the possibility of a financial Armageddon– whenaggregate investmentfalls belowthecritical level, X L . Definition2(EquilibriumWith Knightian Uncertainty) Given (a) An announced choice for the risk-free rate r t ; (b) A realization of the aggregatestate ! t ; (c)A wagerate W t = W ( ! t ) ; and (d) A realization ofthe uncertainty parameter (cid:15) t , an additive probability measure Q t (with support on the interval [ 0 ; s ( W t ; r t ) ] ) over the investment levels x of other savers and an aggregate level of investment X t are a Nash equilibrium with Knightian uncertainty if: 1. The representative saver is indifferent, under the non-additive probability measure P (cid:15) t ( x ) = ( 1 (cid:0) (cid:15) t ) Q t ( x ) , amongall levelsof riskyinvestment, x t (cid:20) s ( ! t ; r t ) 0 (cid:20) . 2. Thelevelofaggregateinvestmentintheriskyasset isgivenby: X t = Z X x d Q t ( x ) : The level of aggregate savings S t and bond holdings, B t are given by the representativesaver'schoices: S t = s ( ! t ; r t ) and B t = S t (cid:0) X t : We further assume that the uncertainty parameter (cid:15) t is drawn i.i.d. each 23

period from a known distribution on the interval [ 0 ; 1 ] , and that it is costlessly observed by all agents before they make their consumption, effort andinvestment decisionsinaperiod. Wemakenoassumptions aboutthe covariance (if any) between the preference shock (cid:15) t and the technology shock ! t . 3.3 Investment Under Knightian Uncertainty For a saver with uncertainty parameter (cid:15) t to be indifferent between all portfoliodivisions,therateofreturnequalitycondition(6)mustbealtered to reflect uncertainty. Thus aggregate investment will be given by X ? (cid:15) t ( r t ) , definedimplicitly by: ( 1 (cid:0) (cid:15) t ) Z (cid:10) (cid:26) [ ! ; X ? (cid:15) t ( r t ) ] d (cid:25) ? + (cid:15) t (cid:26) 0 = r t : (9) Because (cid:26) ( X ) isdecreasingin X ,theuncertainty-contaminated levelofinvestmentintherisky assetinperiod t , X ? (cid:15) t ( r t ) ,isbelow theno-uncertainty level of investment X ? ( r t ) , as shown in figure 4. Recall from section 2.2 that the return (cid:26) ( X ) is finite; as a result, if the uncertainty parameter (cid:15)t grows large enough, investment in the risky asset falls to zero because even the project with the highest return, when corrected for the saver's fear of a financial Armageddon, does not yield more than the risk-free rate. 24

X) (r ,nruter detcepxE r( )X Expected return to risky asset without Knightian uncertainty Expected return to risky asset with Knightian uncertainty risk-free rate r 0 X L 1=r X1 X0 Investment, X Figure 4: Effect of uncertainty. Notice that aggregate investment in the risky assetfalls from X 0 to X 1 . 25

The following proposition formalizes the effect of increases in the uncertainty parameter (cid:15) t . It is quite close in spirit to proposition 1 from Liu (1998). Proposition 1(EffectofUncertainty) If no government action is taken, a generation j born with an uncertainty parameter (cid:15) j will invest less than all generations i born with smaller uncertainty parameters, (cid:15) i < (cid:15) j . In addition, in times of high uncertainty, spreads between riskyassetsand therisklessassetwillwiden. Proposition 2(FinancialArmageddon) Ineachperiod,givenanannouncedrisk-freerate r t ,if r t > (cid:26) 0 thereissomelevelof uncertainty (cid:15) ? < 1 such that, if (cid:15) t (cid:21) (cid:15) ? , no equilibriumwithpositiveinvestment exists. 4 Optimal Monetary and Fiscal Policy In periods of certainty (when generations are born with uncertainty parameters (cid:15) t = 0 ) there is no need for government intervention. In such periods, the government sets the interest rate on bonds to the storage rate r t = 1 and does not levy any taxes on the young. In periods of uncertainty(when (cid:15) t > 0 )thegovernmentmaychooseamixoffiscalandmonetary policies to undothedistortions–decreased investment andpossibly a complete financial Armageddon–described in section 3.3 above. We take 26

as monetary policy a choice for the return on government bonds, r t ; and asfiscalpolicy achoiceofdistortionary taxpolicy, f (cid:28) t c , (cid:28) t ‘ g . Anexpansionary monetary policy is one which pushes the gross return on government bondsbelowits naturallevelof one(thestorage return). The government will take as its problem that of maximizing a social welfare function formed by the equally-weighted sum of the expected utility of the representative saver plus the producers' surplus. Notice that because savers and producers are risk-neutral with respect to consumption while old, an increase in this social welfare function means that the government,byusingalump-sumtaxandtransferpolicyinthesecondperiod of life, could make both producers and savers (at least weakly) better off. Thus we identify increases in the social welfare function with potential Pareto improvements, with the caveat that the government may have to lump-sumtransfer resources from oneclassof agentsto another. Governments often use monetary policy to respond to (at least the initial stages of) a financial crisis, perhaps because fiscal policy is costly to change. Inthismodel,becausegovernmentbondsarebackedbyariskless storage technology, the optimal monetary policy when there is no uncertaintyistosetthegrossreturnonbondstothetechnologically-determined return of unity. Whenweintroduce Knightian uncertainty, asolelymonetary policy (that is, a monetary policy without an associated fiscal policy) will have two countervailing effects. First, it causes agents to work less 27

and consume more while young, pushing down total savings. Second, it will, by theportfolio balance equation (9), cause them to hold more of the risky asset, which has a true expected rate of return greater than the riskfreerate. Thesecond,good, effectwilldominatethefirst, bad,effect forat leastsmall decreasesintherisk-free rate,ifuncertainty isnot too large. Theeffect of amonetary expansion willdependon thecurrent realization level of labor productivity W ( ! t ) , and the uncertainty level (cid:15) t . Consider the case, displayed in figure 5, of “good times,” when labor productivity, W t , is high. In such periods, the savings schedule s ( W t ; r t ) lies relatively far to the right–for any given interest rate, savers will save in total more when W t is high. If the saver had no uncertainty, then the equilibrium would be the optimum presented in section 2.4 above. At the point marked“C”inthefigure,thesaversavesatotalamount s ( W t ; 1 ) ,ofwhich an amount X ? ( 1 ) , marked “A,” is invested in the risky asset. The difference, the interval marked “B,” is devoted to bonds. The uncertainty realization (cid:15) t shifts theapparent (to the saver) return on the risky asset down, fromthesolidlinetothedashedline. Iftherisk-freerateisstillheldatone, thenthesaver'sportfolionowcontainsanamount X a < X ? ( 1 ) oftherisky asset,wherethedashedlineandthesolidlineintersectatthepointmarked “D.” The government (in this figure) responds with a large monetary expansion, forcing down the risk-free rate from its initial level of one (the solid line) to r ? < 1 (the dashed line). Now the saver's portfolio contains 28

Expected return with Expected return without Knightian uncertainty Knightian uncertainty r 0 X X s(W ,r*) s(W ,1) L a t t nruteR 1=r Risk-free rate *r=r S(good,r) D B A C E F Saving, Investment X*(1) Figure5: Effectofmonetarypolicyinrelativelygoodtimes. Investmentin therisky asset returns to itsoptimal level,but total savings falls. just the right amount, X ? ( 1 ) , of the risky asset again, at the point marked “E.” However, the amount of total savings fall to s ( W t ; r ? ) < s ( W t ; 1 ) , at thepointmarked “F.” Incontrast,whenlabor'sproductivityislow,i.e. in“badtimes,”asshown infigure6,thescopeformonetaryexpansionsislimited. Ifthere isnouncertainty,aggregateinvestmentintheriskyassetisstill X ? ( 1 ) (atthepoint marked “A”) and aggregate bond holdings are still given by the interval marked “B.” Because the savings schedule is shifted quite far to the left, 29

the demand for government bonds when there is no uncertainty is quite small. Weassumed,insection 3.2above,that without uncertainty,thereis alwaysapositive demandfor bondsat r t = 1 . When uncertainty is present in addition to lower labor productivity, the apparent expected return to the risky asset is given by the dashed line in figure 6. As before, when the government decreases r t total savings fall and the portfolio level of risky assets rises, so the demand for bonds falls. Whentheinterest rateis r ,thedemandforbondsfallstozero(at the point marked “E”) and hence further decreases in r t do not fuel further increasesinholdingsoftheriskyasset. Themaximumlevelofinvestment intheriskyassetthatcanbegeneratedfrommonetarypolicyaloneis X ? ( 1 ) X c < . Now consider the effect of the uncertainty parameter (cid:15) t on the effectiveness of monetary policy. As uncertainty increases, the apparent (to the saver) rate of return on risky assets falls. If uncertainty is high enough, the highest possible apparent return to the risky asset (when aggregate investment is X L ) would not stimulate enough savings to cover the minimum investment level X L . Here monetary policy is completely ineffective and the saver will hold a portfolio made up entirely of government bonds. As shown on figure 7, if the uncertainty-adjusted apparent rate of return schedule on risky assets (the dashed line) at X L falls below the point marked S a , then savers would never be willing to save enough to 30

Expected return with Expected return without Knightian uncertainty Knightian uncertainty Lowest possible risk-free rate r r X c r 0 X X L a nruteR 1=r S(bad,r) B D A Risk-free rate C E Saving X*(1) Figure 6: Effect of monetary policy in bad times. Note the lower bound on the effective risk-free rate and the (suboptimally low) upper bound on aggregate investment intherisky asset. 31

Expected return without Knightian uncertainty Expected return with Knightian uncertainty S a r 0 X L nruteR 1=r S(bad,r) B Risk-free rate A C X*(1) Saving Figure 7: Effect of very high uncertainty. No monetary policy can stimulate investment; without fiscal policy, portfolios are made up entirely of thesafeasset. cover the critical investment level X L . In contrast, in figure 5, because the savingsscheduleisshiftedquitefartotheright,evenatverylowrisk-free rates the saver is willing save enough to cover the critical level X L . Thus thereisfurtherinteractionbetweenthetechnologyshockandtheeffectiveness of monetary policy: Whenlabor's product is low, there may besome levelof uncertainty so large that no monetary policy canavoid a financial Armageddon. Wenowformalize theseideas. 32

Proposition 3(Pareto ImprovingRoleofMonetaryExpansions) In timesofmoderateuncertainty,when 0 < (cid:15) t (cid:20) (cid:15) ? t ? ,where (cid:15) ? ? isdefinedas: s [ W t ; ( 1 (cid:0) (cid:15) ? t ? ) (cid:26) ( X L ) + (cid:15) ? t ? (cid:26) 0 ] = X L ; decreasing the risk-free rate r t from r t = 1 increases the social welfare function. Monetary expansions alone cannot restore the optimal allocation. If there is no uncertaintytheoptimalrisk-freerateisone. Proposition 4(LimitstoMonetaryExpansions) There is some minimum effective risk-free rate, r t ( W t ) , available to the government. The level of aggregate risky investment at this rate may be below the optimum. The minimum rate r t ( W t ) is higher when labor's productivity, W t , is low. To fix ideas, consider figure 8 below, which shows the relationship of ? (cid:15)t ? to labor productivity, W t . When labor is productive, it is more likely that at least a small decrease in the risk-free rate will have an effect. However, evenwithintheregionmarkedaseffective,asaneconomyapproachesthe border with ineffectiveness (from a combination of high uncertainty and low labor productivity), the minimum possible levelfor the risk-free rate, r t ,will beincreasing. So far we have imagined that the government has conducted monetary policy by choosing a risk-free rate and accommodating the resulting de- 33

mand for bonds. In this view, the demand for bonds is irrelevant, as long asitispositive. However,wemightimaginethatthegovernment'sability to vary bond sales was limited either because the government has some minimum financing needs, or because its monetary policy is simply to sell the same number of bonds each period, regardless of the level of labor's product or uncertainty. Figure 9 below plots the demand schedule forbondsunderdifferentrealizationsoflabor'sproduct W ( ! t ) anduncertainty (cid:15) t . The two solid schedules correspond to different realizations of labor's product W ( ! t ) whenuncertainty iszero, and the dashed schedule showstheeffectofuncertainty. Forthereasonsdiscussed above,bonddemand drops to zero at r t ( W t ) . This level is higher in bad times and lower whenuncertainty ishigh. Demandfor bonds rises smoothly with therate they pay, until the critical level at which bonds pay a higher return than the largest possible return, correcting for Knightian uncertainty, that the risky asset can pay. If bonds pay a return greater than this critical level, demand for bonds jumps up as savers adjust their portfolios to contain only bonds. Bonds have completely crowded out investment in the risky asset. Ifuncertaintyexceeds (cid:15) ? t ? asdefinedinproposition3above,thenthis critical level is at or below r t , so no equilibrium with positive investment ispossible. If the government followed a policy of always selling (if possible) some amount B ? > 0 of bonds, andadjusting the risk-free rate to accommodate 34

this,thenthereturnwouldvarydependingonlabor'sproduct, W ( ! t ) and uncertainty, (cid:15) t . In bad times, when labor's product was low, the equilibrium risk-free rate would be higher than in good times, when labor's product was high, even if there was no uncertainty. In times of increased uncertainty, as the demand schedule for bonds shifts out, the equilibrium risk-free ratewouldbelower. Thisimmediatelyraisesthepossibility that, if the government adopted a constant bond-sale policy, monetary policy would be self-stabilizing, with decreases in the risk-free rate in times of highuncertainty. However,this turns out notto bethecase: Corollary1 Thereisno levelofbonds, B ? (cid:21) 0 ,suchthat,ifthegovernmentmaintainedbond salesat B ? forallrealizationsof ! t and (cid:15) t ,alteringbondsaleswouldnot produce aParetoimprovement. This is a simple application of propositions 3 and 4. First, notice that if thegovernmentpursuesaconstantbond-salepolicy,therisk-freeratewill varyinresponsetoshockstolabor'sproduct W ( ! t ) . Weknowfromproposition 3 that the optimal risk-free rate is the same for all levels of W ( ! t ) . Second,fromproposition4weknowthatintimesofhighuncertaintyand a realization of labor's product, W ( ! t ) , the optimal monetary policy will betosellzerobonds. If B ? > 0 ,thenthisconditioncanneverbemet;while if B ? = 0 ,thentoo fewbonds willbesoldinperiodsof lowuncertainty. Next, we describe how a fiscal policy can encourage savings by discour- 35

aging current consumption and leisure, thus allowing the government to recapturetheoptimum. Byincreasingthetaxoncurrentconsumptionand decreasing the labor income tax (that is, actually subsidizing labor effort), the optimal tax system rewards saving. As a result, for any prevailing risk-free interest rate, savers facing such a tax system will save more than savers who faceno distortionary taxes. The government can then depress therisk-freerate,recoveringtheoptimallevelofriskyinvestmentthrough the portfolio-balance effect, without causing an over-all decrease in savings. The additional effect of a fiscal policy is illustrated in figure 10 below. A combined monetary and fiscal policy move the aggregate investmentleveltoA'andtotalsavingstoC'.Notethatthesearethesameasthe optimallevels,atpointsAandC.Suchapolicyisalwaysfeasible,because weassumedthat S ( ! t ; 1 ) > X ? ( 1 ) forall ! t . Proposition 5(OptimalFiscal andMonetaryPolicy) In times of uncertainty, when (cid:15) t > 0 , if the government sets the risk-free rate to r ? ( (cid:15) t ) < 1 satisfying X ? (cid:15) t ( r ? t ) = X ? ( 1 ) ;thetaxratesonconsumptionwhileyoung and labor income to: (cid:28) ? c = (cid:0) 1 + 1 = r ? ; and (cid:28) ? ‘ = 1 (cid:0) 1 = r ? ; and the lump-sum transfers to satisfy the government's budget constraint, then the representative saverwillconsume,work,saveandinvestexactlyasiftherewerenouncertainty. Inparticular,riskyinvestmentinperiod t willbeatitsoptimallevel: X t = X ? ( 1 ) . 36

e ,ytniatrecnU t e ** t W t 1 0 Ineffective Effective Labor Productivity Figure 8: Regioninwhichmonetary policy iseffective. 37

Bond demand schedules; various technology and uncertainty shock values (1-e)r( ) + er X L 0 Low tech. shock Increase in Knightian uncertainty r ,etaR ) W(r t r=1 High technology shock Bond sales, B Figure 9: Bond demand as a function of technology shock and Knightianuncertainty. Thedashedlineindicates an increase inKnightian uncertainty. 38

Expected return with Expected return without Knightian uncertainty Knightian uncertainty Savings schedule (no fiscal policy) Savings schedule (with fiscal policy) r 0 X L nruteR 1=r Risk-free rate X1 X0 S1 S0 Saving *r=r S A C A’ C’ Figure 10: Effect of fiscal and monetary policy in good or bad times. The appropriate fiscal policy manipulates aggregate savings so that monetary policy isalwayseffective. 39

5 Conclusion In this paper we used the relatively new theory of choice under Knightian uncertainty to study the equilibrium effects of saver uncertainty and the optimal government policy response. Because uncertain savers behave like pessimists, they will underinvest in the risky asset, starving the economy of productive capital, and overinvest in the risk-free asset, the government bond. The optimal monetary policy response to uncertainty is one that decreases the rate of return to bonds, inducing savers to hold risky assets. A strategy of combating uncertainty with a purely monetary policy is shown to be Pareto improving, but to be unable to recapture the optimal allocation. To recapture the optimum, monetary policy must be combined with a fiscal policy of taxing current consumption and subsidizingcurrent labor effort, sothat total savings doesnot fall. This analysis allows us to draw several conclusions about the market turmoil of the late summer and fall of 1998. First, it implies that portfolio adjustments may have been the primary reason that a monetary expansion undid some of the ill-effects of the financial turmoil. Second, it impliesthatinterestratespreadsbetweenrisklessandriskyassetsareinlarge part determined by the level of uncertainty in the economy. As a result, spreads remained elevated even after the monetary expansion, indicating thatsavers continuedtobeuncertain. Third,our analysisimpliesthat itis 40

easierforthegovernmenttocounteractuncertainty-drivenfinancialcrises in good times than in bad times. The fact that the U.S. economy was particularlyhealthyinthefallof1998allowedittousemonetarypolicyalone to calm financial markets. Finally, the model provides an explanation for the underlying cause of the sudden increase in uncertainty. If savers perceived that other savers were withdrawing from risky investments, they wouldhavehadno incentiveto maintain theirown risky investments. More generally, our analysis points to rules for conducting monetary policy in the face of shocks to labor income and uncertainty. Roughly speakingthere are three levelsof theuncertainty parameter: zero (or very low), moderate, and extremely high. In the same way, we can consider the two extremes of labor's product: low and high. If uncertainty is close to zero, then there is no role for monetary policy in our model (proposition 3). This result derives fundamentally from the fact that financial markets are assumed to be perfect; that is, they are zero-cost, fully transparent, and free of any underlying moral hazard problems. When there is moderate uncertainty,spreadsbetweenriskyandrisklessassetwillwiden,andthere will be a general flight to safety (proposition 1). An expansionary monetary policy will be Pareto improving, but will be limited by the current level of labor's product (propositions 3 and 4). If labor's product is high, demand for bonds will be relatively strong, and the government will be abletopushthe risk-freeratefairlylow. Iflabor'sproduct islow,demand 41

for bonds will be relatively weak, and the government will not be able to push the risk-free rate very far away from its natural level. The monetary expansion undoes, to a certain extent, the flight to quality. Even after the decrease in the risk-free rate, spreads will remain elevated, although returnstotheriskyassetwillfall. Thusalthoughspreadsareusefulinsignalling increases in uncertainty, monetary policy should not target them directly, insteadtargeting the returns on risky assets. Finally, if uncertainty is extremely high, investors may perceive that aggregate investment will fall below the critical level, and thus choose to prudently invest nothing in the risky asset, with disastrous consequences foroutput(proposition2). Further,atsomelevelsofuncertainty,monetary policy may be ineffective, and so monetary policy alone will be unable to stem a catastrophic decline in investment (proposition 4). In such cases, only a combination of monetary and fiscal policies will be able to avert thisfinancial Armageddon(proposition 5). 42

Appendix Proof of Proposition 1 Given that the government is not conditioning variables on the signal, we must show that investment in the risky asset is decreasing in (cid:15) for all values of the risk-freerate r > 0 andrealizationsoftheproductionshock, ! : X ? (cid:15) j ( r ) (cid:21) X ? (cid:15) i ( r ) ; all r > 0 , (cid:15) j > (cid:15) i . Here X ? (cid:15) ( r ) is determined by the portfolio balance equation (9). This equation impliesthattheexpectedreturntotheriskyassetisincreasingintheuncertainty parameter, (cid:15) , as savers demand an uncertainty premium for holding the risky asset. Becausetheproductionfunction (cid:26) ( ! ; X ) isdecreasinginaggregateinvestment X ,itmustbethecasethatequilibriuminvestmentintheriskyassetdeclines as uncertaintyincreases. Theeffect of theproductionshock ! is to shift thetotal savings curve s ( ! ; r ) . For any given level of total savings, higher values of (cid:15) are associatedwithadecreasedportfolioholdingoftheriskyasset. Tothinkaboutthespreadbetweenariskyandtherisklessasset,consideranasset that pays a state-contingentreturn of R ( ! ) . Assume for convenience's sake that the return can take on only two values: R 1 > 0 , when aggregate investment lies above X L , and zero, when aggregate investment falls below X L (imagine a AAA-rated corporate bond that repays in all states of the world, and defaults only if there is a financial “Armageddon”). Uncertainty-contaminated rate-ofreturnequalitythenrequiresthattheexpectedreturntothebond,formedunder thenon-additiveprobabilitymeasure P (cid:15) ,mustsatisfy: ( 1 (cid:0) (cid:15) ) R 1 = r ; where r is the prevailing risk-free rate. Thus for risk-neutral (but uncertaintyaverse)saverstoholdpositivequantitiesofthisasset,itsreturnmustsatisfy: R 1 (cid:21) 1 1 (cid:0) (cid:15) r : Thespreadisthen: R 1 (cid:0) r = r 1 (cid:15) (cid:0) (cid:15) : (A.1.1) 43

Thisisincreasingin (cid:15) . Proof of Proposition 2 Thehighestpossibleexpectedreturntotheriskyassetoccurswhenaggregateinvestmentjustequals X L . Wecan thendefine (cid:15) ? from theportfolio balance equation(9) as thatlevel ofuncertaintyatwhich, evenif saversexpecttheriskytechnology to pay off at its highest possible level (that is, if aggregate investment is expectedtobejust X L ),theyarejustindifferentbetweentheriskyandtheriskless asset: ( 1 (cid:0) (cid:15) ? ) (cid:26) ( X L ) + (cid:15) ? (cid:26) 0 = r : Or,manipulating: (cid:15) ? = (cid:26) (cid:26) ( X ( X L L ) ) (cid:0) (cid:0) r (cid:26) 0 : Notice the importance of the assumption of r (cid:21) (cid:26) 0 . In the next proposition, we discusswhytherisk-freeratemightbeboundedfrombelow. Proof of Proposition 3 Inthissectionweconsidertheproblem ofa benevolentgovernmentconstrained tocombat uncertaintywith a purelymonetarypolicy. We constructa social welfare function and showthat, in times of uncertainty, it is increasing in an expansionary monetary policy (decreasing in the risk-free rate). Because increases in the risk-free rate can make savers better off by capturing some monopoly rents from the producers, we have to considerboth the welfare of savers and producers. The government will have available to it a lump-sum tax on (transfer to) producerswhichitusesasatransferto(taxon)savers. Note first that monetary policy will never influence portfolio decisions if uncertaintyisgreaterthanacritical level (cid:15) ? t ? ,definedimplicitly as: s [W t ; ( 1 (cid:0) (cid:15) ? t ? ) (cid:26) ( X L ) + (cid:15) ? t ? (cid:26) 0 ] = X L : If uncertainty (cid:15) t is greater (cid:15) ? t ? , the Knightian-uncertainty contaminated expected 44

rate of return schedule never intersectsthe savings schedule at any level of savings greater than the critical level, X L . Note that (cid:15) ? t ? is increasing in W t . If W t is largeenough,theremaybeno (cid:15) ? t ? lessthanunity,in whichcasemonetarypolicy iseffectivenomatterhowuncertainsaversare. Thuswhentheproductionshock, ! t ,takesonabadvaluemonetarypolicyismorelikelytobeineffective. Nowweturntothequestionoftheeffectofmonetarypolicywhenuncertaintyis nottoogreat: (cid:15) t < (cid:15) ? t ? . Fromequation(4)above, iftheprevailing risk-freerateis r t andsaversinvestan amount X ? (cid:15) t ( r t ) in theproductive(risky)technology,therepresentativeproducer hasanexpectedsurplusof: PS t ( r t ) = Z 0 X ? (cid:15) t ( r t ) f (cid:26) ( n ) (cid:0) (cid:26) [X ? (cid:15) t ( r t ) ] g d n : Note that the price paid for loans is not r t , but rather an uncertainty-premium over r t . Recall from equation (9) that the uncertainty adjusted expected rate of returnontheriskyassetsatisfies,inperiod t : (cid:26) [X ? (cid:15) t ( r t ) ] = r t 1 (cid:0) (cid:0) (cid:15) t (cid:15) (cid:26) t 0 : SubstitutingbackintotheexpressionforPS t ( r t ) produces: PS t ( r t ) = Z 0 X ? (cid:15) t ( r t ) (cid:26) ( n ) d n (cid:0) r t 1 (cid:0) (cid:0) (cid:15) t (cid:15) (cid:26) t 0 X ? (cid:15) t ( r t ) : (A.3.1) AppealingtoLeibnitz'srule,thederivativeis: PS 0 t ( r t ) = (cid:26) (cid:26) [X ? (cid:15) t ( r t ) ] (cid:0) r t 1 (cid:0) (cid:0) (cid:15) t (cid:15) (cid:26) t 0 (cid:27) @ X ? ( (cid:15) t @ r r t ) (cid:0) 1 1 (cid:0) (cid:15) t X ? (cid:15) t ( r t ) : Againsubstitutinginfortheuncertaintypremiumfromequation(9)produces: PS 0 t ( r t ) = (cid:0) 1 1 (cid:0) (cid:15) t X ? (cid:15) t ( r t ) : (A.3.2) To compute the saver's indirect utility, it will be convenient to abstract from his choice of consumption and leisure while young, c t 0 and ‘ t 0 . Consider the partial 45

valuefunction (cid:30) ( W t (cid:0) s t ) : (cid:30) ( W t (cid:0) s t ) (cid:17) m c t 0 a ;‘ x t 0 u ( c t 0 ; ‘ t 0 ) subjectto: c t 0 + W t ‘ t 0 (cid:0) H t 0 (cid:20) W t (cid:0) s t : Ifthesaverfacesarisk-freerateof r t ,hisproblemmaybeexpressedas: m a s t x (cid:30) ( W t (cid:0) s t ) + r t s t : As a direct consequence, notice that (cid:30) 0 ( (cid:1)) = r . This problem induces a savings relation s ( ! t ; r t ) in the usual way. Given that there is a continuum of savers of massunity,aggregatesavingsare S t = s ( ! t ; r t ) . Begin by considering the representative saver's expected utility, formed using maxminpreferences: V KU t ( r t ) = (cid:30) [W t (cid:0) S t ( r t ) ] + r t S t ( r t ) + H t 1 : The agent expectsa rate of return of r t on all parts of the total savings portfolio, including holdings of the risky asset (although he will earn, in reality, a higher return on the risky asset). Assume that the government refunds (taxes) lumpsum any seigniorage revenue (cost) derived from manipulating the rate paid on storage, r t : H t 1 = ( 1 (cid:0) r t ) [S t (cid:0) X ? (cid:15) t ( r t ) ]: Nowthesaver'sindirectutilityfunctionbecomes: V KU t ( r t ) = (cid:30) [W t (cid:0) S t ( r t ) ] + S t (cid:0) ( 1 (cid:0) r t ) X ? (cid:15) t ( r t ) : (A.3.3) Theslopewithrespectto r t is: d V KU t d r ( r t ) = S 0 ( t (cid:1)) [1 (cid:0) (cid:30) 0 ( (cid:1)) ] (cid:0) ( 1 (cid:0) r t ) @ X ? ( (cid:15) t @ r r t ) + X ? (cid:15) t ( r t ) : (A.3.4) Notice immediately that when the paid return on government bonds equals the technologicalrateofreturnunity,thisslopebecomes: d V KU t d r ( r t ) (cid:12) (cid:12) (cid:12) (cid:12) r t = 1 = X ? (cid:15) t ( r t ) : This term is positive because, if the risk-free rate increases, the saver captures somesurplusfromproducers. 46

Sofar wehave concentratedontherepresentativesaver'sexpectedutilityunder Knightian uncertainty. We may also be interested in the true expected utility of thesaver,thusexplicitlyrecognizingthattheriskyassetwillpayoffmorethan r t inexpectedvalue. Therepresentativesaver'sindirectutilityfunctionisnow: V true t ( r t ) = (cid:30) [W t (cid:0) S t ( r t ) ] + S t ( r t ) (cid:0) X ? (cid:15) t ( r t ) + (cid:26) [X ? (cid:15) t ( r t ) ] X ? (cid:15) t ( r t ) : The representative saver divides his total savings S t into a portfolio of the risky asset, X ? (cid:15) t ( r t ) ,andthesafeasset, S t (cid:0) X ? (cid:15) t ( r t ) . Thesaverwillearnacertainreturn of unity on savings placed in the safe asset (because any difference between the prevailing risk-free rate and unitywill be refundedlump-sum in H t 1 ). However, thesaver will earn a true expectedreturnof (cid:26) [X ? (cid:15) t ( r t ) ] > 1 on investmentsin the risky asset. Thus by decreasing the risk-free rate, the government can stimulate greater investment in the high-return risky asset that is shunned by pessimistic savers. Substitutingoutfor theuncertainty premium, the representativesaver's true indirectutilityover r t becomes: V true t ( r t ) = (cid:30) [W t (cid:0) S t ( r ) ] + S t ( r t ) (cid:0) X ? (cid:15) t ( r t ) + r t 1 (cid:0) (cid:0) (cid:15) t (cid:15) (cid:26) t 0 X ? (cid:15) t ( r t ) : (A.3.5) Takingthederivativewithrespectto r produces: (A.3.6) V true t 0 ( r t ) = (cid:0) (cid:30) 0 [W t (cid:0) S (cid:0) t ( r @ ) t X 0 ]S ( r ) t t ? ( r ) t (cid:15) t @ r + (cid:20) 1 S (cid:0) 0 ( r ) t t (cid:15) (cid:0) t 1 (cid:0) r t (cid:15) + t (cid:15) t (cid:26) 0 (cid:21) + 1 1 (cid:0) (cid:15) t X ? (cid:15) t ( r t ) : Thefinaltermagainrepresentsthemonopolyrentscapturedbysaverswhenthe risk-freerateincreases. Itwillbeexactlyoffsetbyacorrespondingdecreaseinthe surplusoftheproducers. Thegovernment(initsroleasasocialplanner)choosesapurelymonetarypolicy, that is, a level of the risk-free interest rate, to solve the social welfare problem. HereweassumethatproducersandsaversareassignedequalParetoweights,so thesocialwelfarefunctionbecomes: m a r t x F t ( r t ) = V KU t ( r t ) + PS t ( r t ) ; subjectto: X ? (cid:15) t ( r t ) (cid:20) S t ( r t ) : The value functions PS t and V KU t are given by equations (A.3.1) and (A.3.4), respectively. Byassumption,theconstraintthat S t (cid:21) X ? (cid:15) t doesnotbindwhen r t = 1 , 47

foranyleveloftheuncertaintyparameter (cid:15) t (cid:21) 0 . Usingequations(A.3.2)and(A.3.4),theslopeofthesocialwelfarefunctionis: d F t d ( r r t ) = S 0 ( t r t ) [1 (cid:0) (cid:30) 0 t ( (cid:1)) ] (cid:0) ( 1 (cid:0) r t ) @ X ? ( (cid:15) t @ r r t ) (cid:0) 1 (cid:15) (cid:0) t (cid:15) t X ? (cid:15) t ( r t ) : Because (cid:30) 0 t ( (cid:1)) = r t ,when (cid:15) t = 0 thesolutiontothesocialwelfareproblemistoset r t = 1 : d F t d ( r r t ) (cid:12) (cid:12) (cid:12) (cid:12) r t = 1 ;(cid:15) t = 0 = 0 : If (cid:15) t > 0 and r t = 1 thentheslopeofthesocialwelfarefunctionisnegative: d F t d ( r r t ) (cid:12) (cid:12) (cid:12) (cid:12) r t = 1 ;(cid:15) t > 0 = 1 (cid:15) (cid:0) t (cid:15) t @ X ? ( (cid:15) t @ r r t ) : Thus,evenusingtheKnightianuncertaintycontaminated expectedutility of the representative saver, a decrease in the risk-free rate increases the total surplus available tobedividedbetweenproducersandsavers. ThereasonisthatKnightian uncertainty produces a wedge between what the producer must pay for a loanandwhatthesaverexpectstorealizeonit. Adecreaseintherisk-freeallows moreproduction(worthyprojectsarefunded),andthusanincreaseinproducer's surplus greater than the decrease in saver's surplus. With appropriate transfers betweenthetwoparties,saverscanbemadebetteroffwithouthurtingproducers. Inademocracy,allpartieswouldvoteforsuchasystem. This implies that an expansionary monetary policy, that is, a policy of depressing the risk-free rate below its natural rate of unity, increases the social welfare function,andisthuspotentiallyParetoimproving,when (cid:15) t > 0 . However,asthe risk-free rate decreases, the constraint that investment in the risky asset not exceedtotalsavingswillbegintobind,sothatwecannotderiveexactlytheoptimal pure monetary policy. Further, as we shall see, this constraint is more likely to bindinbadtimes(whenlabor'sproduct W t islow)thaningoodtimes. Withthe addition of the fiscal policy instruments of distortionary taxes, the government cancompletelymanipulatethesavingsscheduleandrestoretheoptimum. Finally,noticethatthetrueexpectedconsumptionofasaverwhileoldis: E f c t 1 ( r t ) g = S t ( r t ) (cid:0) X ? (cid:15) t ( r t ) + r t 1 (cid:0) (cid:0) (cid:15) t (cid:15) (cid:26) t 0 X ? (cid:15) t ( r t ) : 48

Takingthederivativewithrespectto r t : @ E f c @ t 1 r ( t r t ) g = S 0 ( t r t ) + 1 1 (cid:0) (cid:15) t X ? (cid:15) t ( r t ) + 1 (cid:15) (cid:0) t (cid:15) t ( 1 (cid:0) (cid:26) 0 ) @ X ? ( (cid:15) t @ r r t ) : Thismaybeeitherpositiveornegative,dependingonthesensitivityoftheportfoliolevelofinvestmenttotheprevailinginterestrate,whichinturndependson thesensitivityoftheproductionfunction (cid:26) ( (cid:1); X ) toinvestment. Proof of Proposition 4 Monetarypolicyfacesanotherlimit beyondthatimposedbythemaximum level of uncertainty which it can combat, (cid:15) ? t ? . Even if uncertainty is below this critical level,thereisalowerboundontherisk-freerate, r t ( W t ) . Whentherisk-freerate reachesthisbound,demandforbondswilldroptozeroandfurtherchangesin r t willnotaffectportfolioorsavingsdecisions. Thiscritical return r t ( W t ) isdefinedimplicitly from: s [ W t ; r t ( W t ) ] = X ? (cid:15) t [r t ( W t ) ] : If r t < r t ( W t ) thensavings S t fall belowinvestmentintheriskyasset, X ? (cid:15) t ( r t ) . In suchasituation,thenon-negativityconstraintonthestoragetechnologyisbinding: Savers would like to bring forward assets from the future to finance investmentinthepresent. Insection2wesawthatinthissituationsaverswouldinvest their portfolio entirely in the risky asset, so that further changes in r t would not affect portfolios. Notice that because thesavings schedule s ( W t ; r t ) is increasing in W t ,that r t isthereforealsoincreasingin W t . Inbadtimes,thegovernmentwill findthatmonetarypolicyislesseffective. Proof of Proposition 5 Assumingthatthesaver'ssavingsschedulecouldbemanipulatedtoanydesired level (see below), the right choice of risk-free rate is the one that induces a portfoliowiththeoptimalquantityofriskyinvestment: r ? : X ? (cid:15) t ( r ? ) = X ? ( 1 ) : 49

Evenif r ? isbelow r t ,it will stillbeachievable, becausethegovernmentwill use fiscal policies to manipulate the savings schedule. Further, by our assumption that S ( ! t ; 1 ) isalwaysgreaterthan X ? ( 1 ) ,thiswillbefeasible. Take as given for the moment the saver's choice of portfolio conditional on the risk-free rate r ? , and consider his optimization problem, that of maximizing (1) subject to the budget constraints (2) and (3), by choice of consumption while young, leisure and total savings. The first-order conditions from this problem are: u c = r t ( 1 + (cid:28) t c ) ; and: u ‘ = r t W t ( 1 (cid:0) (cid:28) t ‘ ) : Thus a decrease in r t will increase consumption and leisure while young. However, this decrease can be offset by an optimal choice of (cid:28) t c and (cid:28) t ‘ . For any 0 < r t (cid:20) 1 : (cid:28) ? c ;t = (cid:0) 1 + 1 = r ? t ; (A.5.1) and: (cid:28) ? ‘ ;t = 1 (cid:0) 1 = r ? t : (A.5.2) Notethatby(A.5.2),theoptimaltaxrateonlaborincomewillbenegative. Thetaxesandtransfersleviedontheyoung, (cid:28) ? c ;t and (cid:28) ? ‘ ;t ,maycausethegovernment to run a net surplus or deficit. The seigniorage revenue realized on bond holdings ( 1 (cid:0) r ? t ) will also leave the government a surplus in the second period of each generation's life. The differences are made up with lump-sum transfers (taxesifnegative)sothat: H 0 ? = (cid:28) ? c ;t c ? 0 + (cid:28) ? ‘ ;t ( 1 (cid:0) ‘ ? 0 ) ; (A.5.3) H 1 ? = ( 1 (cid:0) r ? t ) [1 (cid:0) X ? ( 1 ) ] : (A.5.4) Here c ? 0 and ‘ ? 0 denote the saver's optimal choices of consumption and leisure whileyoung,giventhegovernment'sfiscalandmonetarypolicychoices. 50

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