The Banking Industry and the Safety Net Subsidy

THE BANKING INDUSTRY AND THE SAFETY NET SUBSIDY (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: 2.2 LastRevised: August 11,1999 (cid:3) The views areexpressed in this paper areours alone and do not necessarily reflect thoseoftheBoardofGovernorsoftheFederalReserveoritsstaff.WethankDarrelCohen andStephenOlinerforhelpfulcomments. Anyremainingerrorsareourown.

THE BANKING INDUSTRY AND THE SAFETY NET SUBSIDY Abstract Governments use monetary policies to counteract the effects of financial crises. In this paper we examine the subsidy that such “safety net” policiesprovide tothebankingindustry. Usingamodelofuncertainty-driven financial crises, we show that any monetary policy designed to maintain riskyinvestmentinthefaceofinvestoruncertainty(andthuspromoteeconomic growth and stability) will subsidize the banking industry. In addition,weshowthatthemerepresenceofamonetaryauthoritywillingtosupportafailingbankingsysteminbadtimessubsidizesthebankingindustry, evenifthosebadtimesdonotoccur. Aconditionalbailoutpolicythatdoes notextendequallytoallfinancialinstitutions createsagreatersubsidyfor those institutions perceived as being “close” to the central bank, possibly giving these institutions a competitive advantage. Economic profits, in this model, are required to cover fixed costs of entry into the banking system. Journal of Economic Literature classification numbers: G2, E5, E44, D8. Keywords: Knightianuncertainty,safety-netsubsidy, monetaryandfiscalpolicy

1 Introduction In a companion paper [Lehnert and Passmore (1999)], we argued that Knightian uncertainty–that is, an imprecise estimate of the true probabilitydistribution ofevents(especiallyof extremeevents)–can causeafinancial crisis with many of the hallmarks of the market turmoil of the fall of 1998. We showed that an expansionary monetary policy in the face of uncertainty improved social welfare, but that a combination of fiscal and monetary policies was required to recapture the first-best allocations. In this paper, we augment our model by relaxing the assumption of a competitive,zero-cost financialintermediary. Inits place,weconsiderabanking industry made up of many firms, each of which must pay some fixed costs associated with intermediation. We model such a banking industry with a single firm. This “representative firm” realizes profits sufficient to cover the average fixed cost of entry. We then use this model to analyze thesubsidyprovided to thebanking industry bythegovernment's monetary policy, andhow that subsidy isdistributed across intermediaries and savers. The typical analysis of the safety-net subsidy concentrates narrowly on the explicit government guarantee of insured deposits. Thus certain financial institutions are seen as privileged because they can raise funds from depositors at the riskless rate of return. Broader definitions of the 1

safety-net includes access to payments systems and the discount window [see Whalen (1997)]. Beyond that, however, is the subsidy provided by thegovernment'sdesiretoavoidfinancialcrises[seeKwastandPassmore (1999)]. Financial institutions generally suffer during financial crises (for example,duringthefall1998marketturmoil,theequityvaluesoftheNew York money center banks fell precipitously), and governments generally implementmonetary policiestoassuagefinancialcrises. Thusamonetary policydesignedtoundotheilleffectsofafinancialcrisishastheadditional effectof increasing profits at financialintermediaries. Thisbroaderdefinitionofasubsidyencompassestheuseanddistribution of public goods–where, in this paper, the public good is financial stability supported with a monetary policy. By analogy, consider the example of the government's purchase of streetlights for a dark street. Because such lightslowertheoddsofacarcrash–andcarcrashescreatecoststhatarepotentially borne by everyone–everyone benefits. But those who drive cars benefit the most, whereas those who walk benefit least. Because tax paymentsarenottiedtothebenefitthetaxpayerreceivesfromthestreetlights, thosetaxpayerswhowalksubsidizethosetaxpayerswhodrive. Similarly, hereweargue thatmonetary policy, when usedto offset theeffects of afinancialcrisis,potentiallybenefitseveryonebut some–such asthebanking industry–benefit more thanothers. As in our companion paper,if the government must respond to an uncer- 2

taintyledfinancialcrisiswithonlyamonetarypolicy(perhapsbecausefiscalpoliciesare too difficult toadjust quickly), its bestpolicy isto decrease the risk-free rate. This policy provides a direct subsidy to the banking industry, although it may or may not make up for the loss in profits caused by the financial crisis. We further show that such an unconditional policy maybeinterpretedasaconditionalpolicyinwhichthegovernmentundertakesamonetaryexpansiononlyifabadstateisrealized. Themerepresence ofamonetary authority thought to befollowing suchaconditional policy provides asafety-net subsidy. The paper is organized as follows. In section 2 we briefly analyze the investor's problem and in section 3 we analyze the bank's problem (see the companion paper for a more detailed analysis). We characterize the government's optimal unconditional monetary policy in section 4. In section 5 we consider how conditional monetary policies can subsidize intermediaries, and in section 6 we show how an explicit or implicit government guarantee policy—e.g. a “too big to fail” policy—can complement monetary policy. Section 7concludes thepaper. 2 A Model of Investors Investors live for two periods, t = 0 ; 1 and are born with a single unit of the consumption good that they may either consume immediately or 3

invest. They have preferences over consumption today and consumption tomorrow of: U ( C 0 ; C 1 ) = (cid:30) ( C 0 ) + C 1 ; where (cid:30) 0 > 0 and (cid:30) 00 < 0 (1) . For simplicity, assume that there are two states of nature (cid:10) which will be realized in period t = 1 : A good state ! , and bad state ! . The true probability of the good state is p . If we imagine this model to represent a single period in a dynamic setting (as we do in the companion paper) then the bad state is expected to occur only once every 1 = ( 1 (cid:0) p ) periods. If p = 0 : 9 9 , for example, and a period is taken to be a year, then the bad staterepresents aonce-per-century outcome. There are two assets: A government bond, b , that pays a gross return r , and an uninsured deposit at the monopolist bank (which we take to represent the banking industry) x , that pays a gross return R ( ! ) in each state ! . The second asset represents the return to intermediated loans to risky borrowers, andhencewereferto itasthe“risky” asset. The portfolio choice of investors will be affected by Knightian uncertainty: investors will have an imprecise estimate of the probability distribution over states. Inaseries of axiomatic derivations, Gilboa (1987),Schmeidler (1989), and Gilboa and Schmeidler (1989) show that uncertainty-averse agents withimprecise probability estimates behavelike pessimists, acting 4

to maximize their minimum payoff in the “maxmin” form of the utility function. We follow the (cid:15) -contamination form of Liu (1999) and Epstein and Wang (1994), by describing investors as having a non-unique set of prior distributions overstates P (cid:15) : P (cid:15) (cid:17) 8 > < > : ( 1 (cid:0) (cid:15) ) 0 B @ 1 p (cid:0) p 1 C A + (cid:15) 0 B @ 1 m (cid:0) m 1 C A : 0 (cid:20) m (cid:20) 1 9 > = > ; : (2) This set of subjective probability distributions implies a set of estimates for the probability of thegood state: q ( ! ) = ( 1 (cid:0) (cid:15) ) p + (cid:15) m ; 0 (cid:20) m (cid:20) 1 : Thus the representative investor has many different subjective prior distributions over the two states, ranging from optimistic, when m = 1 , to pessimistic,when m = 0 . However,thesedistributionsarecenteredonthe true distribution f p ; 1 (cid:0) p g . If (cid:15) is small, uncertainty is low and the distributions are tightly clustered around the true distribution. If (cid:15) is large, uncertainty ishigh anddistributions are spreadout. Assumefor nowthat thegovernment bondpays areturn of r inallstates, so that it is truly riskless. Note that we can characterize the investor as choosingaleveloftotalsavings, S ,andanamountofthatsavingstoinvest in the risky asset, X (cid:20) S . The remaining S (cid:0) X is placed in government 5

bonds. With this in mind, and using the “maxmin” form of expected utility,wecanthenwritetherepresentativeinvestor'sproblem,contaminated byKnightian uncertainty, as: m S a ;X x (cid:30) ( 1 (cid:0) S ) + r ( S + (cid:0) f q X ! ( ) ) m ;1 (cid:0) i q n ! ( ) g 2 P (cid:15) n q ( ! ) R ( ! ) X + [ 1 (cid:0) q ( ! ) ] R ( ! ) X o : The minimization embedded in the preferences arises from the axiom of uncertainty-aversion. Investorsarepessimistic: theyacttomaximizetheir payoff assuming that the distribution is the worst one from their set of priors, P (cid:15) . However, because the set of priors P (cid:15) is centered on the true distribution f p ; 1 (cid:0) p g , it is not the case that investors believe that the bad state will occur with certainty. Pessimistic investors will assume that the badstateoccurs withprobability ( 1 (cid:0) (cid:15) ) ( 1 (cid:0) p ) + (cid:15) ,which,forsmallvalues of (cid:15) ,is notmuch largerthan 1 (cid:0) p . Althoughwewilldeferacomplete discussion ofthebanking industry we assume for now that banks will not pay a higher return in the bad state thaninthegoodstate,so R ( ! ) (cid:21) R ( ! ) . Withthisassumptioninmind,the solution to the minimization problem in the investor's problem above is thentheprobabilitydistributionin P (cid:15) thatweightsthebadstatewithhighest probability. Thus we can write the representative investor's problem 6

as: m S a ;X x (cid:30) ( 1 (cid:0) S ) + r ( S (cid:0) X ) + ( 1 (cid:0) (cid:15) ) [ p R ( ! ) + ( 1 (cid:0) p ) R ( ! ) ] X + (cid:15) R ( ! ) X : Because investors are risk-neutral with respect to consumption in period t = 1 , they hold both assets in positive quantities only if an uncertaintyadjusted rateof return equalitycondition holds: ( 1 (cid:0) (cid:15) ) R + (cid:15) R ( ! ) = r : (3) Here R denotes the true expected rate of return on risky assets, which is simply p R ( ! ) + ( 1 (cid:0) p ) R ( ! ) : Sotheinvestorbehavesasthoughhebelieves the true probability model, expecting R from the risky asset, holds with only 1 (cid:0) (cid:15) probability. With the remaining (cid:15) probability, the investor believes that the bad outcome is assured. If the Knightian uncertainty contaminatedrateofreturnequalityrelation(3)holds,theinvestor'sproblem maybewritten: m a S x (cid:30) ( 1 (cid:0) S ) + r S : The investor is indifferent to any division of total savings, S ( r ) , between the government bond and the risky asset when (3) holds. This implies a 7

policy fortotal savings S ( r ) thatsatisfies: (cid:30) 0 ( 1 (cid:0) S ) = r : (4) Now relax the assumption that the government bond pays a return of r inallstates, andassumeinstead that it paysstate-contingent returns r ( ! ) . The representative investor now chooses a level of total savings S , bank deposits X ,andbondholdings S (cid:0) X to solve: (5) m S a ;X x (cid:30) ( 1 (cid:0) + S q f ) ! ( ) m ;1 (cid:0) i n q ! ( ) g 2 P (cid:15) n q ( ! + ) [ r [ ( 1 ! (cid:0) ) ( S q ( ! (cid:0) ) X ] [ r ) ( ! + ) ( R S ( ! (cid:0) ) X X ] ) + R ( ! ) X ] o : 3 The Banking Industry's Problem Thetotalinvestment X madebytherepresentativeinvestorinthebanking industry iscompletelyintermediatedtoborrowers, whopayanaggregate gross return of (cid:26) ( ! ; X ) ineachstate ! . Define (cid:26) ( X ) tobethetrueexpected aggregate gross repaymentamount whentotal investment is X : (cid:26) ( X ) = p (cid:1) (cid:26) ( ! ; X ) + ( 1 (cid:0) p ) (cid:1) (cid:26) ( ! ; X ) : 8

Thesereturns, whicharepaidback tothe representative bank,satisfy: (cid:26) ( ! ; X ) (cid:21) (cid:26) 0 ; all X (cid:21) 0 ,and (cid:26) ( ! ; X ) = (cid:26) 0 ; all X (cid:21) 0 . Thus returns in the good state always (weakly) exceed returns in the bad state. Further, (cid:26) representsanunderlyingtechnologythatexhibitsdecreasingreturns to investment andfinite returns, so: @ (cid:26) ( @ ! X ; X ) < 0 ; and: (cid:26) ( ! ; 0 ) = M ; M < 1 : Finally,tosimplifytheanalysis,wewilloftenassumethat (cid:26) 0 = 0 ,although no results of importance depend on this assumption. We take the good state to stand for normal times, including relatively bad times like recessions. Thebadstatestandsfor aggregaterealizationsthat arewelloutside therangeexperiencedeveninconventionallybadtimes–inotherwords,a severefinancialcrisis or economicdepression. The representative bank pays a fixed cost to enter the intermediation industry,buthaszeromarginalcostsofintermediation. Ifinvestorshavedeposited an amount X and the bank pays gross returns of R ( ! ) and R ( ! ) , 9

ithasexpectedprofits of: X n p [ (cid:26) ( ! ; X ) (cid:0) R ( ! ) ] + ( 1 (cid:0) p ) [ (cid:26) 0 (cid:0) R ( ! ) ] o : (6) Note that investment X depends on the returns paid by the bank, R ( ! ) and R ( ! ) , because investors base their portfolio allocation decisions on thesereturns. Thebankingindustry'soptimizationproblemisthatofmaximizingitsexpectedprofits(6),bychoiceofstate-dependentreturns R ( ! ) ,whereinvestment X solves the general investor's problem (5). However, the solution tothisproblemisintimatelytiedupwiththegovernment'schoiceofbond returns, r ( ! ) . Because the banking industry and the representative investor work with different subjective probability distributions over states of nature (the investor acts like a pessimist because of uncertainty aversion), the banking industry finds it profitable to provide a completely riskless asset; that is, to set the returns in both states thesame, R ( ! ) = R ( ! ) = R c . Thebanking industry takes the government's choice of monetary policy, the risk-free interest rate r , as given. It can raise any amount of deposits X less than S ( r ) so long as there is rate-of-return equality between (uninsured) bank deposits andthe government bond: R c = r . The banking industry's prob- 10

lemis thus: m a X x p X (cid:26) ( ! ; X ) + ( 1 (cid:0) p ) X (cid:26) 0 (cid:0) R c X ; (7) subject to: X (cid:20) S ( r ) and R c = r . Notice that in the bad state of the world the banking industry will make expostprofits of X ( (cid:26) 0 (cid:0) R c ) or X ( (cid:26) 0 (cid:0) r ) ,which,forlowvaluesof (cid:26) 0 ,will benegative. Assumingthatthereispositivedemandforgovernment bondsinequilibrium,so X (cid:20) S ( r ) ,thesolution tothebankingindustry's problem(7)isto raisedeposits of X 0 ( r ) : X 0 ( r ) : (cid:26) ( ! ; X 0 ) + X 0 @ (cid:26) ( ! @ ; X X 0 ) = r p (8) (wherewehaveassumedthat (cid:26) 0 = 0 ). Therepresentativeinvestor holdsa portfoliowith X 0 ( r ) heldinbankdepositsandtheremainder S ( r ) (cid:0) X 0 ( r ) held in the government bond, where the function X 0 ( r ) is implicitly defined by equation (8). From this analysis, a decrease in the risk-free rate r increases both the amount of investment, X 0 ( r ) ,andthe banking industry's profits. The banking industry's problem is shown graphically in figure1. ThelinemarkedMR(X)isthebank'smarginalrevenuefromraising anextraunitofdepositsandloaningthemout. Becausethebankcanoffer a deposit contract that pays a certain rate of return of R c = r , investors' 11

decisions are not affected by their uncertainty. Notice the importance of the assumption that S ( r ) > X 0 ( r ) : the banking industry faces a perfectly elasticsupply curvefor funds. We now relax the assumption that banks can earn negative profits in the bad state. Instead, we assume that output in the bad state is so low as to preclude any organization, even the government, from smoothing across aggregate realizations. As a result, banks will be unable to pay a return on deposits that is constant across all states. In the bad state, banks will beabletopayatmostareturnof (cid:26) 0 . Becausethepessimistic investor puts undue weight on the bad state, the banking industry will find it optimal topayashighareturnaspossibleinthatstate, R ( ! ) = (cid:26) 0 ,andthusrealize zero profits in that state. As a result, Knightian uncertainty adjusted rate of return equality between the risky and the riskless asset [equation (3)] requires that, for investors to hold both deposits at the bank and governmentbonds,thereturn on depositsin thegood state, R ( ! ) ,must satisfy: R ( ! ) = 1 1 (cid:0) (cid:15) r p : (9) Thebanking industry must therefore payamarkup over therisk-free rate to attract investors. Here, because we are assuming that 0 R ( ! ) = (cid:26) 0 = , the markup is exactly 1 = ( 1 (cid:0) (cid:15) ) . Because the banking industry only realizespositiveprofitsinthegoodstateoftheworld,itsproblem(7)now 12

becomes: m a X x p X [ (cid:26) ( ! ; X ) (cid:0) R ( ! ) ] ; subject to: X < S ( r ) (10) . Here R ( ! ) isasdefinedinequation(9)above. Thesolutiontothisproblem isalevelof investment X ? thatsatisfies: (cid:26) ( ! ; X ? ) + X ? @ (cid:26) ( ! @ ; X X ? ) = R ( ! ) ; (11) or,from (9): = 1 1 (cid:0) (cid:15) r p : This equation implicitly defines the amount invested as a function of the risk-free rate, X ? ( r ) . The effect of uncertainty on the banking industry is displayed in figure 2. Notice immediately that when investors are uncertain, so that (cid:15) > 0 , the amount invested falls below the optimum: X ? ( r ) < X 0 ( r ) forthesamelevelof r . Also,noticethatprofits (theshaded area in each diagram) fall as uncertainty rises. In essence, uncertainty increases the cost of funds of banks. By lowering the risk-free rate, the government can force investors to reshuffle their portfolios to contain more of the risky asset. Such a policy will lower the cost of funds and result in increasedprofits tothebanking industry. 13

Expected return r r (X) X,S nruteR Profits S(r) MR(X) X0 Investment, Saving Figure1: Bankingindustry'sproblemwhenthereisnouncertainty. Expected return Uncertainty premium r r (X) X,S nruteR Profits S(r) MR(X) X* Investment, Saving Figure2: Bankingindustry'sproblemwithanuncertaintypremium. 14

4 Unconditional Government Monetary Policy Weassumethatthegovernmentwillacttomaximizeasocialwelfarefunctionthatsumsthebankingindustry'sprofitsandtheinvestor'sutilitywith equal weights, netting out the surplus-shifting effects of monetary policy. Changesin r changethe surplus claimedbybanks relativeto that heldby the investor, as wellas changing the investor's optimal portfolio. It is this latter effectthat affects total economy-wide surplus. Thegovernment sells S (cid:0) X bonds inthe first period andpays out atotal of r ( S (cid:0) X ) in the second period. The government has a riskless storage technology that pays a unit return in all states. The government can then payagreaterorlowerreturnthanthis“natural”rateonitsbonds. Ifitpays ahigherreturn, itmustlevytaxestopayforthereturn on its bondsabove the technologically-determined rate. If it pays a lower return, it realizes revenue, which we assume is refunded lump-sum. We refer to a policy of depressing the risk-free rate below the natural rate as an expansionary monetary policy. Thegovernmentcanmakelump-sumtransfers(taxesifnegative)of H 1 ( ! ) inthesecondperiod. Thusthegovernment's budgetconstraint is: ( 1 (cid:0) r ) ( S (cid:0) X ) (cid:0) H 1 (cid:21) 0 : (12) 15

If the gross return on government bonds is less than unity, r < 1 , then government realizes seigniorage revenue. By setting H 1 = ( 1 (cid:0) r ) ( S (cid:0) X ) itcanlump-sumrefund anysuchrevenue. Proposition 1(OptimalMonetaryPolicy) When there is no uncertainty, so (cid:15) = 0 , the optimal monetary policy will be to set the risk-free rate to unity: r = 1 . When there is uncertainty, so (cid:15) > 0 , the optimal monetary policy will be to decrease the risk-free rate, so that r < 1 . The optimal risk-free rate will be greater than 1 (cid:0) (cid:15) . Such a policy will increase the total surplus divided between the investor and the bank industry, and will also directlyincreasethebanking industry'sprofits. Notice that proposition 1 merely states that an expansionary monetary policy increases aggregate social welfare. However, not all parties benefit equally, and some might even be made worse off (although the other partiesthenwouldallbemadesomuchbetteroffthattheywouldbewilling to transfer resources to those who were made worse off). Now we consider the question directly of whether or not savers are made directly better offby amonetary expansion. Proposition 2(Distribution ofBenefits toMonetaryPolicy) Savers will be made directly better off (that is, without requiring a lump-sum transfer of the banking industry's profits) by a monetary expansion, beginning from a neutral policy stance of r = 1 , if and only if the elasticity of investment 16

X ? ( r ) exceeds 1 = (cid:15) : (cid:0) @ X @ ? r ( r ) X r ? ( r ) (cid:12) (cid:12) (cid:12) (cid:12) r = 1 (cid:21) 1 (cid:15) : Noticethatmonetaryexpansionsarethusmorelikelytodirectlybenefitssaversat largervaluesoftheuncertaintyparameter (cid:15) . Finally, we consider directly the expected profits of the banking industry asawhole beforethe uncertainty parameterhasbeenrealized. Proposition 3(BankingIndustryProfits) Considertwoeconomieswithidenticaldistributionsoftheuncertaintyparameter, a ( (cid:15) ) . In the firsteconomy the government sets the risk-free rate to the technological rate of return to storage, r = 1 , under all realizations of the uncertainty parameter. In the second economy, the government sets the risk-freerate only afterobservingthe uncertainty parameter,followingan interest-rate policyof r ( (cid:15) ) . The policy satisfies 1 (cid:0) (cid:15) (cid:20) r ( (cid:15) ) (cid:20) 1 and also S ( r ( (cid:15) ) ) (cid:21) X ? ( r ( (cid:15) ) ) , all (cid:15) . The banking industry will make greater expected profits in the second economy than in the first. Total expected social surplus will be greater in the second economy thaninthefirst,althoughwithoutlump-sumtransfersbetweenbanksandsavers, saversmaybeworseoff. 17

5 Conditional Monetary Policies Inproposition1aboveweestablishedthatanexpansionarymonetarypolicyincreases thetotal surplusdividedbetweentheinvestor andthebanking industry (modeled as a representative monopolist); furthermore, a monetary expansion directly increases the bank's profits. In this section weexpandtherangeofpossiblemonetarypoliciestoincludestatecontingent valuesfor the return on bonds. Thus thegovernment may,for example, commit to an expansion only in the bad state of the world. We show thatthisclassofpolicies,ifchosencorrectly,willalsobePareto-improving. Therefore, the merepresenceofamonetary authority with the right kind of state-contingent monetary policy will haveallof the beneficial effects discussed in the previous section. However, unless the bad state is realized, themonetaryauthoritywillnotmovetherisk-freerateawayfromunity. If thebadstateisveryunlikely,themonetaryauthoritywillalmostcertainly be,expost, passive. Acontingent monetary policy isachoice ofrates-of-return on thegovernment bond in each state, r ( ! ) . We consider only the class of policies in which the government sets the return on the government bond equal to unity in the good state, and to some lower rate in the bad state. That is, the government does nothing in the good state (since a return of unity is thetechnologically determinedreturnonthestorage technology), anden- 18

gineersamonetaryexpansioninthebadstate. Weshowthatsuchapolicy isequivalent(withanappropriatechangeofvariables)toanunconditional policy ofthetype studiedintheprevious section. Monetary policy (in the restricted class that we are studying here) boils down to a choice of rate of return on the government bond in the bad state: r ( ! ) . The return in the good state is assumed to be unity: r ( ! ) = 1 . Thustheinvestor's problem(5)becomes: (13) m S a ;X x (cid:30) ( 1 (cid:0) + S f ) q ( ! ) m ;1 (cid:0) i q n ! ( ) g 2 P (cid:15) n q ( ! + ) [ [ 1 ( S (cid:0) (cid:0) q ( X ! ) ) ] + [ r ( R ! ( ) ! ( S ) X (cid:0) ] X ) + H 1 ( ! ) ] o : Here,asbefore, weareassuming that (cid:26) 0 = 0 andso R ( ! ) = 0 ,for simplicity. Inthebadstatethegovernmentwillrealizesomeseignioragerevenue, which is then refunded lump-sum to the investor via H 1 ( ! ) , determined bythegovernment'sbudgetconstraint (12). Inthegoodstate,thegovernment does not manipulate the return on bonds, so realizes no seigniorage revenue. For both assets tobeheld in positive amounts, thebanking industry must 19

payareturninthegoodstate, R ( ! ) ,thatsatisfiesarateofreturnequation: q ? ( ! ) R ( ! ) = q ? ( ! ) + [ 1 (cid:0) q ? ( ! ) ] r ( ! ) : (14) Here q ? ( ! ) is the solution to the minimization problem in (13) above; it is the most pessimistic distribution in P (cid:15) , given the agent's choices. As we saw above, the most pessimistic distribution in P (cid:15) is the one that weights thatbadstate themost, andthegood state theleast: q ? ( ! ) = ( 1 (cid:0) (cid:15) ) p : Compare condition (14) to (3), the rate-of-return equality condition when thegovernment pursuesan unconditional monetary policy. Wewillshow that, within a feasible range, a conditional policy r ( ! ) is equivalent to an unconditional policy r . Let r u denote an unconditional return on bonds that produces a desired equilibrium. The allocations associated with this desired equilibrium, f S ( r u ) ; X ? ( r u ) g , may be mimicked by a conditional policy r ( ! ) that,inturn,leadsbankstopaythesamereturns f R ( ! ) ; R ( ! ) g . Ifwecontinuetoassume,forsimplicity,that (cid:26) 0 = 0 ,andso R ( ! ) = 0 ,then, from equation(3): R ( ! ) = ( 1 r (cid:0) u (cid:15) ) p : Given the conditional policy r ( ! ) , from equation (14), we can calculate 20

that: R ( ! ) = 1 + (cid:18) q ? 1 ( ! ) (cid:0) 1 (cid:19) r ( ! ) : Thereturnontheriskyasset(thatis,uninsureddepositsatthebank)inthe good state, R ( ! ) ,is thesameiftheconditional monetary policy satisfies: r ( ! ) = r u 1 (cid:0) (cid:0) p p ( 1 ( 1 (cid:0) (cid:0) (cid:15) ) (cid:15) ) : (15) Because negative gross interest rates are not allowed, the smallest unconditionalinterest rate thatmaybemimicked is: r u = ( 1 (cid:0) (cid:15) ) p : Thisisbelowthelowestpossibleoptimalchoiceoftheunconditionalriskfree rate, 1 (cid:0) (cid:15) . (See proposition 1 above.) Thus the allocation associated withtheoptimalchoice ofunconditional monetary policyisalways availablewithaconditionalmonetarypolicyofthekindwehaveoutlinedhere. Thus, in reacting to a level of uncertainty (cid:15) > 0 , the government may use either an unconditional monetary policy or a conditional monetary policy. From equation (15), it is clear that if the government chooses to use a conditional monetary policy, and the bad state is realized, the monetary expansion will be greater than if the government had chosen an uncondi- 21

tional monetary policy. 6 Implicit Government Guarantees of the Banking Industry Governmentsoften,eitherexplicitlyorimplicitly,guaranteethesafetyand soundness of their national banking industries. Investors feel confident thatcertainclassesofassets(e.g. depositsatbanks)arebackedbythegovernment. Investors may also believe that certain ostensibly risky assets arealso, inreality, backed by thegovernment, e.g. equity in an institution consideredtobe“toobigtofail”or“tooembarrassingtofail.” Thedegree towhichanyparticularinstitutionisthoughttobeprotected,expressedas a probability of being bailed out in the bad state, will also affect its profitability. Institutions that are perceived as being more likely to be bailed out will pay a lower cost of funds than those perceived to be less likely to be bailed out. Finally, investors may believe that the government will undertake a wholesale support of many different types of risky assets if theyarethreatened by “systemic risk” or “contagion.” Investors, inshort, may trust that many classes of risky assets are implicitly protected by the government from aggregate shocks. Wecanmodelsuchanimplicitguaranteeherebyallowingthegovernment 22

to fund the banking industry directly if the badstate of nature is realized. The banking industry as a whole will continue to make zero profits in the bad state, but investors will realize more than the pure liquidation valueoftheinstitutions. Thegovernment raisesseigniorage revenuewith a monetary expansion in the bad state, and then instead of refunding this revenuelump-sum,it differentially rewards holdersof therisky asset. Weshall,itturnsout,beabletorestrictourattentiontoaslightgeneralization of the class of conditional monetary policies considered in the previous section. As before, the government does not distort the rate-of-return tobondsawayfromunityinthegoodstate,andengineersamonetaryexpansion only if the bad state is realized. Now however, the government will no longer refund the resulting seigniorage revenue lump-sum, but will useitto fundthebankingindustry's claimantsdirectly. Recallfromthegovernment'sbudgetconstraint(12),thatitstotalseignioragerevenuefrom amonetaryexpansion,thatis,setting astate-contingent rateof return ongovernment bonds lessthanunity,in state ! is: F ( ! ) = [ 1 (cid:0) r ( ! ) ] ( S (cid:0) X ) : Here S and X represent the representative household's choices of total savings and holdings of the risky asset. These choices will be affected by the conditional monetary policy that the government chooses. Assume 23

that seigniorage revenue F is completely distributed, pro rata, to holders of the risky asset. Thus, in the bad state, the risky asset no longer earns zero,but rather: R ( ! ) = F ( X ! ) ; or: = [ 1 (cid:0) r ( ! ) X ] ( S (cid:0) X ) ; if S > X (16) . Assumethat,asalways,returnsinthegoodstateexceedreturnsinthebad state,so that R ( ! ) = R ( ! ) ,despitethegovernment's implicit guarantee. A monetary policy that combines monetary expansions with direct fundingoftherisky assetinbadtimeswillact likeastronger monetary expansion without direct funding (that is, without an implicit guarantee). Ifthe government chooses a policy of setting the return on bonds to r ? < 1 in the bad state without a guarantee policy, it would be able to achieve the same results with a policy of setting the return on bonds to r ? ? > r ? with a guarantee policy. To see this, note that the banking industry must pay a return inthe good stateof: q ? R ( ! ) + ( 1 (cid:0) q ? ) R ( ! ) = q ? (cid:1) 1 + ( 1 (cid:0) q ? ) r ( ! ) ; (17) where: q ? = ( 1 (cid:0) (cid:15) ) p ; and: R ( ! ) = [ 1 (cid:0) r ( ! ) X ] ( S (cid:0) X ) ; if S > X . As the guarantee amount R ( ! ) increases, the uncertainty premium that 24

thebankingindustryhastopayfalls. Comparetherate-of-returnequation with the guarantee to the one without it, that is, equation (17) to equation (14). By providing a guarantee the government gets some extra portfolio bangfor itsmonetary expansion buck. Althoughan(implicitorexplicit)guaranteepolicywillingeneralincrease the expected profits of the banking industry, the banking industry will continuetofareverypoorlyinthebadstate. Banksrealizetheextraprofits only if the good state is realized. Next, note that in our analysis there are no moral hazard considerations to degrade the benefits of a government guaranteepolicy. Here,byfundingthesysteminbadtimesattheexpense of bond holders, the government causes investors to readjust their portfolios more sharply than with a pure monetary expansion. If investors had some costly monitoring duties, then such implicit guarantees would weakentheirincentivetoproperlymonitorfinancialinstitutionsand,ultimately,borrowers. Finally, consider an intermediary institution that financial markets expect will be bailed out in the bad state only with probability (cid:11) . The greater (cid:11) , the “closer” the institution is to the government. For example, a large money-center bank might be seen as a high- (cid:11) institution, while a small finance company might be seen as a low- (cid:11) institution. Consider a single institution,toosmalltoindividuallyaffecttheequilibriumlevelsofaggregatesavingsandinvestment S and X . Thisinstitutionisassignedabailout 25

probability of (cid:11) i . Using the augmented rate-of-return equality condition, equation (17) above, we can determine what return the institution would haveto promise, inthehigh state, inorder to attract deposits, R i ( ! ) : R i ( ! ) = (cid:0) 1 (cid:0) q ? q ? R ( ! ) (cid:11) i + (cid:20) 1 + 1 (cid:0) q ? q ? r ( ! ) (cid:21) : As in equation (17) above, q ? is the Knightian-uncertainty adjusted probability of the good state, and R ( ! ) is the return on the risky asset (uninsured bank deposits), assuming that the institution is bailed out. Otherwise, investors anticipate salvaging nothing from their investment in the bad state. Notice immediately that the return institutions must promise, R i ( ! ) , is decreasing in the probability of a bailout, (cid:11) i . However, all institutions, eventhose for whom markets estimate no probability of abailout ( (cid:11) = 0 ), benefit from the government's policy of decreasing the risk-free rateinbadtimes;thatis, ofspecifying r ( ! ) < 1 . 7 Conclusion In this paper we augmented our model of uncertainty-driven financial crises to consider how optimal monetary responses would affect theprofits of the banking industry. We concluded that optimal monetary policy responsestouncertainty-ledfinancialcrises,whicharealwaysexpansions, 26

increasedexpectedprofits inthebanking industry. Wefurther considered monetarypoliciesthattooktheformofaconditionalmonetaryexpansion, inwhichthegovernmentreducestheratepaidonitsbondsonlyinclearly bad economic times. We showed that policies of this kind can always recapture the allocations associated with unconditional monetary policies (although in the companion paper we show that only a fiscal policy can recapture the first-best allocations). Finally, we showed that monetary policies combined with an implicit guarantee were more effective at altering investors' portfolio choice than monetary policies alone (in which the seigniorage revenue was refunded lump sum to investors), and that financial intermediaries that are perceived by investors as “close” to the government–thatis,morelikelytobebailedoutbythegovernmentinbad times–benefit more from monetary policy through a lower cost of funds. Such guarantee policies increase banking industry profits ex ante, and ex postifthe good state is realized. Thegovernment, inour paper,acts to maintaineconomic growth andstability by minimizing the number of worthwhile projects that are starved of capital. A monetary policy that maintains output in the face of uncertainty will, almost as a side-effect, subsidize the banking industry. Such a subsidy results in a lower cost of funds for banks, possibly giving them a competitive advantageover intermediaries that are viewed as less closely tiedto government policy. 27

Appendix Proof of Proposition 1 Thegovernmentactstomaximize thesocialwelfarefunction: m a r x V ( r ) + (cid:5) ( r ) ; subjectto: S ( r ) (cid:21) X ? ( r ) (A.1.1) . Here V ( r ) istheinvestor'svaluefunctionwhentherisk-freerateis r : S ( r ) V ( r ) = (cid:30) [1 (cid:0) S ( r ) ] + r [S ( r ) (cid:0) X ? ( r ) ] + X ? [p R ( ! ) + ( 1 (cid:0) p ) R ( ! ) ] + H 1 : is the investor's optimal savings policy. For the remainder of this section, assumethat,inthebadstate,thereisnooutput,sothat (cid:26) 0 = 0 . (Thisassumption merelysimplifiestheanalysis.) Thusthebank'spaymentsmustsatisfy: R ( ! ) = 0 ; and: R ( ! ) = r = [p ( 1 (cid:0) (cid:15) ) ]: Recall from the government's budget constraint (12) that the lump-sum transfer H 1 is H 1 = ( 1 (cid:0) r ) ( S (cid:0) X ) , because r = 1 is the technologically-determined naturalrateofreturn. Thuswecanrewritetheinvestor'svaluefunctionas: V ( r ) = (cid:30) [1 (cid:0) S ( r ) ] + S ( r ) (cid:0) X ? ( r ) + 1 r (cid:0) (cid:15) X ? ( r ) : (A.1.2) Theinvestor'soptimalsavingspolicy,fromequation(4),impliesthat (cid:30) 0 = r . The derivative of the investor's value function with respect to the monetary policy instrument,therisk-freerate r ,isthen: d V d ( r r ) = S 0 ( (cid:1)) [1 (cid:0) r ] + 1 X (cid:0) ? (cid:15) + @ X @ ? r ( r ) (cid:20) 1 r (cid:0) (cid:15) (cid:0) 1 (cid:21) : (A.1.3) Now consider the bank's profit function, assuming that R ( ! ) = 0 and r = [ p ( 1 (cid:0) (cid:15) ) ] R ( ! ) = : (cid:5) ( r ) = m a X x p (cid:26) ( X ) X (cid:0) 1 r (cid:0) (cid:15) X : (A.1.4) 28

Bytheenvelopetheorem,thishasslope: d (cid:5) d ( r r ) = (cid:0) 1 X (cid:0) ? (cid:15) : (A.1.5) Summing togethertheslopesfrom (A.1.3) and (A.1.5) givesus thenetchange in total surplus from a change in the risk-free rate, assuming that the constraint in (A.1.1)doesnotbind: d [V ( r ) d + r (cid:5) ( r ) ] = S 0 ( r ) ( 1 (cid:0) r ) + @ X @ ? r ( r ) (cid:20) 1 r (cid:0) (cid:15) (cid:0) 1 (cid:21) : (A.1.6) Noticefirstthatifthereisnouncertaintyandiftherisk-freerateisequaltounity, (cid:15) = 0 and r = 1 , then the derivative in (A.1.6) is zero. Thus the Pareto-optimal risk-free rate when there is no uncertainty is the technologically-determined return on storage, or just unity. In this model, there is no net gain to an activist monetarypolicyiftheuncertaintyparameteriszero. Next,notethatifthereisuncertainty( (cid:15) > 0 )andtherisk-freerateisequaltounity, thatthederivativeofthesocialwelfarefunction(A.1.6)isnegative. Theconstraint is not binding at r = 1 by assumption. So that a slight decrease in the risk-free rate from its natural level of r = 1 is always Pareto-improving. Decreasing the risk-freeratealsodecreasestotalsavings Note also that if r < 1 (cid:0) (cid:15) then the unconstrained derivative (A.1.6) is positive. Thuseveniftheconstraintthatbonddemandbenon-negativeisnotbinding,the optimalrisk-freeratewill lieintherange [1 (cid:0) (cid:15) ; 1 ] . Becausethesavingsschedule S ( r ) is upward-sloping, the constraint will require that the optimal constrained interestrateexceedtheoptimalunconstrainedinterestrate. Decreasing the risk-free rate in the face of uncertainty increases the net surplus in the economy because investors will shift their portfolios away from the lowreturn safe asset (the government bond) and towards the high-expected-return risky asset (an uninsured deposit in the bank). This surplus is shared between themonopolistbankandtheinvestors. Thebank'sprofitsincreasebyanamount: (cid:0) d (cid:5) d ( r r ) = X 1 ? (cid:0) ( r ) (cid:15) : The rest of the surplus is allocated to the investor. Thus bank profits increase as theresultofanoptimalmonetaryexpansion. 29

Proof of Proposition 2 Using the results from the proof of proposition 1 above, notice that the slope of therepresentativesaver'svaluefunctionis: d V d ( r r ) = S 0 ( (cid:1)) ( 1 (cid:0) r ) + 1 X (cid:0) ? (cid:15) + @ X @ r ? (cid:20) 1 r (cid:0) (cid:15) (cid:0) 1 (cid:21) : At r = 1 thisbecomes: d V d ( r r ) (cid:12) (cid:12) (cid:12) (cid:12) r = 1 ;(cid:15) > 0 = 1 X (cid:0) ? (cid:15) + @ X @ r ? (cid:20) 1 1 (cid:0) (cid:15) (cid:0) 1 (cid:21) : Thismayberewrittenas: d V d ( r r ) (cid:12) (cid:12) (cid:12) (cid:12) r = 1 ;(cid:15) > 0 = 1 X (cid:0) ? (cid:15) (cid:20) 1 + (cid:15) @ X @ r ? X 1 ? (cid:21) : Forthesavertobenefitdirectlyfromadecreaseintherisk-freerate,thisslopemust benegative. Thisisthecaseifandonlyif: (cid:0) @ X @ r ? X 1 ? (cid:12) (cid:12) (cid:12) (cid:12) r = 1 > 1 (cid:15) : Thisismorelikelytoholdif (cid:15) islarge. Proof of Proposition 3 Fromproposition1aboveweknowthat,alongeachrealizationoftheuncertainty parameter (cid:15) , the banking industry will make greater profits and the total social surplus will be greater if the government decreases the risk-free rate in the face ofuncertainty. Inparticular, if thegovernmentdoesnotaltertherisk-freerate,it risks an uncertainty-led financial crisis. Proposition 2 delivers the distributional componentsoftheproposition. 30

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