Financial Capital and the Macroeconomy: Policy Considerations

Finance and Economics Discussion Series Divisions of Research & Statistics and Monetary Affairs Federal Reserve Board, Washington, D.C. Financial Capital and the Macroeconomy: Policy Considerations Michael T. Kiley and Jae W. Sim 2011-28 NOTE: Staff working papers in the Finance and Economics Discussion Series (FEDS) are preliminary materials circulated to stimulate discussion and critical comment. The analysis and conclusions set forth are those of the authors and do not indicate concurrence by other members of the research staff or the Board of Governors. References in publications to the Finance and Economics Discussion Series (other than acknowledgement) should be cleared with the author(s) to protect the tentative character of these papers.

Financial Capital and the Macroeconomy: Policy Considerations (cid:3) Michael T. Kiley Jae W. Sim y z May 5, 2011 Abstract We develop a macroeconomic model in which the balance sheet/liquidity condition of (cid:133)nancial institutions plays an important role in the determination of asset prices and economic activity. The (cid:133)nancial intermediaries inourmodelarerequiredtomakeinvestmentcommitmentsbeforeacomplete resolution of idiosyncratic funding risk that can be addressed only bycostlyre(cid:133)nancing,forcingthemtobehaveinarisk-aversemanner. The model shows that the balance sheet condition of intermediaries can drive asset values away from their fundamentals, causing aggregate investment and output to respond to shocks to intermediaries. We use this model to evaluate several public policies designed to address balance sheet problems at (cid:133)nancial institutions. With regard to short-run policies, we (cid:133)nd thatcapitalinjectionsconditioneduponvoluntaryrecapitalizationcanbe a more e⁄ective tool than direct lending/asset purchases. With regard tolong-runpolicies,wedemonstratethathighercapitalrequirementscan have sizable short-run e⁄ects on economic activity if not implemented carefully, and that a long transition period helps avoid such e⁄ects. 1 Introduction To understand the links between (cid:133)nancial intermediation and the real economy and to assess related public policies, it is essential to have a model that captures key aspects of the dynamic frictions that cause (at least short-run) deviations from the Modigliani-Miller theorem and hence make the capital structure of the banking sector important for credit provision. These links are thin to non-existent within the workhorse framework for macroeconomic analysis, althoughresearchhasbegun(forinstance,AdrianandShin(2010),Brunnermeier We would like to thank seminar participants at the 2011 ASSA meetings, the Board of (cid:3) Governors of the Federal Reserve System, the Bank of Canada, and the Bank of Japan for commentsonanearlierdraft(circulatedunderthetitle(cid:147)TheDynamicE⁄ectsofBankCapital in General Equilibrium(cid:148)). The views expressed herein are those of the authors, and do not re(cid:135)ect the views ofthe FederalReserve Board orits sta⁄. Board ofGovernors ofthe FederalReserve System y Board ofGovernors ofthe FederalReserve System z 1

and Pedersen (2009), Gertler and Kiyotaki (2010) and He and Krishnamurthy (2008)). Moreover, banking, (cid:133)nance, and macroeconomics are typically not integratedinthemodelsusedinpolicycircles(e.g.,thediscussioninBoivinetal. (2010)). Our goal in this paper is twofold: First, we develop a dynamic model in which the balance sheet/liquidity condition of (cid:133)nancial institutions plays an important role in the determination of asset prices and economic activity. Second, using this model, we evaluate the macroeconomic e⁄ects of short-term creditpoliciesaimedatstabilizingthebalancesheet/liquidityconditionoftroubled (cid:133)nancial institutions, and assess the long-term and transitional e⁄ects of implementing higher capital standards. These policies are stylized examples of the types considered in recent discussions.1 The (cid:133)nancial intermediaries in our model are required to make investment commitmentsbeforeacompleteresolutionofidiosyncraticfundingriskthatcan be addressed only by costly re(cid:133)nancing (of the type emphasized by, for example,MyersandMajluf(1984)andBoltonandFreixas(2000)). Thecommitment structurecauses(cid:133)nancialintermediariestobehaveinarisk-aversemanner. The resultingcautionagainsttakingalargeunhedgedpositiongivenshort-runfunding uncertainty creates an intermediary speci(cid:133)c pricing kernel that can deviate fromthestochasticdiscountfactorofarepresentativehouseholdevenwhenthe intermediaryisfullyownedbythehousehold,pushingequilibriumassetreturns awayfromtheircounterpartintheabsenceofsuchintermediationfriction,causing aggregate investment and output to respond to shocks to intermediaries. Our framework yields a highly tractable quantitative framework with which we can assess implications for asset markets and real economic activity. We (cid:133)rst show that our model has plausible long-run properties. Shifts in the mix of debt and equity in the capital structure of (cid:133)nancial intermediaries are essentially neutral in the long run, as, for example, a shift toward (cid:147)more expensive(cid:148) equity is o⁄set by a decline in the borrowing rate (echoing the reasoning in Admati et al. (2010) and Hanson et al. (2011)). In addition, our framework is a business-cyclemodelthatgeneralizestheLAPM(Liquidity-BasedAssetPricing Model) of Holmstr(cid:246)m and Tirole (2001), and, as a result, implies a substantial equity premium. The magnitude of this premium can explain almost a half of the measured equity premiums under our baseline calibration, suggesting that incorporation of frictions such as those we consider has important implications beyond our speci(cid:133)c focus on the links between the level of capitalization at intermediaries, lending and lending spreads, and real activity.2 In an earlier analysis we illustrated how this model provides realistic re- 1For a discussion of capital requirements, see BIS (2010a) and BIS (2010b). For a recent summaryoftheTroubledAssetReliefProgram(TARP),whichincludedmeasurestostabilize the balance sheet position of(cid:133)nancialintermediaries,see CBO (2011). 2The intermediary speci(cid:133)c pricing kernel in our framework provides a structural justi(cid:133)cation of the time-varying discount factor of Jermann and Quadrini (2009), who derive a super(cid:133)cially-similar pricing kernel from a reduced-form quadratic adjustment cost for dividends. Other work developing intermediary-speci(cid:133)c discount factors includes He and Krishnamurthy (2008), who derive their intermediary-speci(cid:133)c pricing kernel by assuming risk aversion for(cid:133)nancialintermediaries. 2

sponses of macroeconomic variables to (cid:133)nancial shocks (Kiley and Sim (2011)), and herein we use the model to assess the e¢ cacy of public policies designed to address balance sheet problems at (cid:133)nancial institutions. We consider two types of stabilization policy: direct lending/asset purchase by a public authority and a capital injection conditioned on voluntary recapitalization. Our results indicatethatthecapitalinjectionpolicycanbemuchmorepowerfulthanthedirect lending/asset purchase policy in stabilizing output (cid:135)uctuations. In our baseline simulation, the former turns out to be 6 times more e⁄ective than the latter in terms of output stabilization e⁄ects from interventions of the same size. The key mechanism behind this di⁄erence is that an asset purchase policy su⁄ers from a classic case of crowding out: while aggregate investment is lifted by the increase in government demand, higher government holdings lowers the supply available for private investment. This decrease in supply for private investment boosts asset prices, which causes private demand to decline along the downward sloping demand curve. Overall, the improvement in liquidity conditions and the business environment boosts aggregate demand, but this boost is not enough to overcome the crowding out e⁄ect and the size of the stimulativee⁄ectdiesoutratherquickly. Incontrast,thecapitalinjectionpolicy increases private demand for capital assets by improving the capital position directly, which boosts the risk appetite for risky assets.3 Finally, we use our model to analyze the transition costs associated with a substantial increase in the minimum capital ratio for banking institutions, a subject that has been the focus of recent debate, as discussed by Admati et al. (2010) and Hanson et al. (2011), but for which a general-equilibrium quantitative assessment has been wanting. Our model (cid:133)lls this gap in quantitative assessments. Though we (cid:133)nd the capital structure of the (cid:133)nancial sector to be neutral in the long run, shifts in capital requirments can have sizable short-run e⁄ects because of the (cid:133)nancial frictions facing intermediaries. Holding the asset side of the balance sheet constant, raising the regulatory capital ratio increases the probability of costly equity (cid:133)nance since the cash in(cid:135)ow from borrowing has to be reduced to comply with the higher capital constraint. To avoid a sti⁄rise in the required return on capital, the intermediaries choose to cut back on the asset side of their balance sheet (lending). However, this is costly because the bank earns strictly positive intermediation margins owing to the low cost of funds associated with deposits. Thus, a transition to a higher minimum capital ratio leads (cid:133)nancial institutions to balance the marginal cost of issuing equity (reducing dividends) with the marginal cost of cutting back on lending, bringing about a mix of some degree of (cid:133)nancial dis-intermediation and bank recapitalization in conjunction with rising lending spreads. As a consequence, the higher capital requirements can have sizable short-run e⁄ects on economic activity if not implemented carefully, and a long transition period helps avoid such e⁄ects. 3HeandKrishnamurthy(2008)alsosuggestthatcapitalinjectionsaremoree⁄ectivethan asset purchases, but their focus is on the asset market recovery. Our framework provides a more comprehensive assessment of impacts of the stabilization policies on employment, investment and output in a production-based business-cycle framework. 3

2 Model The model consists of a representative household, a continuum of (cid:133)nancial intermediaries, a continuum of competitive (cid:133)nal-goods producers, a continuum of competitive investment-goods producers, and a government. The model extends that of Kiley and Sim (2011) through inclusion of a government sector. The following assumptions are central: First, the complexity of the (cid:133)nancial markets create prohibitively large transaction costs for households. For this reason, households participate in the (cid:133)nancial markets only through (cid:133)nancial intermediaries, either in the form of deposits, or in the form of ownership of the intermediaries. Second, households value liquidity services from deposits at (cid:133)nancial intermediaries, which implies that households accept returns on intermediary deposits below the risk free rate, creating a strictly positive intermediation margin. Third, the (cid:133)nancial intermediaries face capital (margin) constraint in their capital structure choice. The constraint creates an environment where (cid:133)nancial intermediaries may not be able to fully arbitrage away all pro(cid:133)t opportunities because of the constraint on leverage, and thus leaves a room for policy intervention. We start with the (cid:133)nancial intermediaries. 2.1 Financial Intermediaries 2.1.1 Return Structure A (cid:133)nancial intermediary i [0;1] purchases capital asset KB (i) at a market 2 t+1 price Q . The intermediary rents out this capital to (cid:133)nal-goods (cid:133)rms for net t rental incomes de(cid:133)ned as RK =R~K UB (i) (cid:24)(UB (i))P t+1 t+1 t+1 (cid:0) t+1 t+1 where R~K is the nominal rental rate per utilization unit of capital asset t+1 (KB (i)UB (i)), UB (i) is the utilization rate, (cid:24)(UB (i)) is the real cost of t+1 t+1 t+1 t+1 utilization and P is the price level of the (cid:133)nal goods. Equivalently, the t+1 rental income can be thought of as dividends from the (cid:133)nal goods (cid:133)rms. In this case, KB (i) should be interpreted as the number of shares. The total ret+1 turnfromtheinvestmentiscomposedofrents/dividends(RK KB (i))andthe t+1 t+1 capital gains associated with the changes in the price of capital assets/shares ((1 (cid:14))Q KB (i)=Q ) where (cid:14) is the depreciation rate of capital.4 The su- (cid:0) t+1 t+1 t perscripts B associated with K and U indicates that the capital stock is t+1 t+1 owned by a (cid:133)nancial intermediary; as discussed below, we also allow the government to own capital assets, denoted by KG . The capital market clearing t+1 condition then requires 0=KG + KB (i)di K for all t. (1) t+1 t+1 (cid:0) t+1 Z 4Inbroadterms,thereturnstructureofourintermediariesshareaspectsofthoseanalyzed by Gertlerand Kiyotaki(2010). 4

To model the balance sheet/liquidity risk that (cid:133)nancial intermediaries face, weassumethattherateofreturnfrominvestmentissubjecttoamultiplicative idiosyncratic shock such that the total rate of return can be decomposed into two components, idiosyncratic and aggregate, RF (i) = (cid:15) (i)RF (2) t+1 t+1 t+1 RK +(1 (cid:14))Q = (cid:15) (i) t+1 (cid:0) t+1 t+1 Q (cid:20) t (cid:21) where (cid:15) (i) is the idiosyncratic component of the return and RF is the t+1 t+1 aggregate component. We assume that the idiosyncratic shock follows an iid lognormal distribution, log(cid:15) (i) N( 0:5(cid:27)2;(cid:27)2).5 t (cid:24) (cid:0) 2.1.2 Capital (Margin) Constraint To (cid:133)nance the investment described above, the intermediaries mix debt (deposits) and equity. In doing so, they face a capital (margin) constraint, which requires that every dollar of investment asset should be backed by at least m t centsofcapital. DenotingtheamountofborrowedfundsbyB (i),thecapital t+1 constraint can be stated as B (i) 1 t+1 m : (3) (cid:0) Q KB (i) (cid:21) t t t+1 A constraint such as the above can arise in various contexts. In Kiley and Sim(2011),weshowhowsuchaconstraintcanbederivedfromaValue-at-Risk (VaR)constraintsuchasinBrunnermeierandPedersen(2009)andAdrianand Shin (2008). In particular, we show that the constraint can arise as a limit point of VaR constraint, where (cid:133)nancial intermediaries are never allowed to default, i.e., the intermediaries are always required to raise enough amount of equitycapitaltostaya(cid:135)oat,apointoriginallymadebyAdrianandShin(2008).6 Becauseourgoalistoshowtheimplicationsofsuchconstraintinanenvironment where the recapitalization of an intermediary is costly owing to equity market frictions, we avoid the complications associated with endogenously motivating the capital constraint. 5Janicki and Prescott (2006) show that a lognormal distribution is a good approximation of the size distribution of the banks in U.S. In Kiley and Sim (2011), we consider a timevaryinguncertaintyprocess,wheretheparameter(cid:27)followsaMarkovprocess,andexplorethe implication ofan increase in uncertainty forthe cost ofcapitaland economic activity. 6Suchaconstraintalsocanexistasaconsequenceofimperfectcontractenforceabilityasin GertlerandKiyotaki(2010). Aswillbemadeclearbelow,therisk-neutralityofintermediary, thelinearity ofinvestmenttechnology and thelinearity ofequity (cid:135)oating costimply thatthe intermediary problem is linearly homogeneous such that the intermediary value function can beexpressedasVt=(cid:23)tQt 1Kt &tBtasinGertlerandKiyotaki(2010)despitethedi⁄erence in capital structure choice(cid:0)s of t (cid:0) he two approaches . It is then straightforward to derive the capitalconstraintbyrequiringthatthevalueof(cid:133)rmshouldbegreaterthanalowerboundfor the (cid:133)rm value, V(cid:22) t. One structural interpretation of the lower bound could be that of Gertler and Kiyotaki(2010): assuming that(cid:24) portion ofbalance sheetassetisdivertable,one could t d th e i r s iv c e as t e h , e t c h o e n m st i r n a i i m nt u f m ro c m ap a i n ta i l n r c a e t n io tiv c e an co b m e p s a e t e i n bi a li s ty m ,i t .e = ., 1 (cid:23)t (cid:0) Q ( t (cid:23)(cid:0) t 1 (cid:0) Kt (cid:24) t (cid:0) )= & & t t B . t (cid:21) (cid:24) t Qt (cid:0) 1Kt. In 5

Inequilibrium,thecapitalconstraintisalwaysbindingfortworeasons: First, as mentioned before, the household is willing to pay a liquidity premium for its deposits since the intermediary deposits create non-pecuniary returns for the household. Second, even without the liquidity premium, (cid:133)nancial intermediaries prefer to issue debt rather than to issue equity owing to the dilution cost associated with equity issuance, which will be explained shortly. As a consequence, the (cid:133)nancial intermediaries follow a (cid:147)pecking order(cid:148)in their capital structure choice. 2.1.3 Modeling Liquidity Risk The primary function of (cid:133)nancial intermediary is the transformation of shortterm liquid assets into long-term illiquid capital assets. Such a transformation exposes the (cid:133)nancial intermediaries to liquidity risk. To model this liquidity risk, we adopt the following timing convention for a given period of time t: 1. Atthebeginningofeachperiod,theaggregatecomponentofreturns(RF) t becomes known. 2. After observing the aggregate shocks, the intermediary makes investment (Q KB (i)) and borrowing (B (i)) decisions. t t+1 t+1 3. After the investment/borrowing decisions are made, the level of the idiosyncratic shock ((cid:15) (i)) becomes known to the intermediary and dividend t payout /equity issuance decisions (D t (i)T0) are made. Thetimingconventionimpliesthatthe(cid:133)nancialintermediarieshavetomake investment commitments before they know their (random) realization of internal funds. It also implies that the revenue shock becomes known only after the borrowing markets for intermediaries are closed. While this precise timing is somewhat arbitrary, it captures important features of reality. In particular, the timingconventionrepresentsparsimoniouslyshort-runfundingrisks. Forexample, (cid:133)nancial intermediaries always face uncertainty about the balance between their short-run loanable funds and/or the cost of such funds in retail/wholesale borrowingmarketsandtheuseofoutstandingloancommitments; alternatively, realized income can fall short of the funding needs associated with their precommitments due to credit losses or (cid:135)uctuations in asset values. Under such conditions and when outside equity is more expensive than borrowing, funding uncertaintycanmaketheintermediariesadoptaprecautionary stanceinmaking investment/deposit decisions even when all intermediaries are risk-neutral.7 7A similar timing convention has been used by Wen (2009) in the context of bu⁄er stock savingofrisk-aversehouseholdsandbyGertlerandKiyotaki(2010)inthecontextofinterbank marketborrowingdecisionofriskneutralbanks. Notethatinterbankborrowing,apossibility from which we abstract,does not ameliorate the potentialfunding distortions under realistic assumptions. In particular, borrowing more through the interbank market to cope with cash (cid:135)owshortfallssimplyworsensanyfundingshortfallbecauseitincreasesleverage. Ane¢ cient secondarymarketforbalancesheetassetscouldhelpinthissituation. However,itisnatural to assume that the same information problem that makes equity (cid:133)nance costly also makes 6

2.1.4 Costly Recapitalization Tocapturetheroleof(cid:133)nancialmarketfrictionsfortheintermediaries,weadopt a costly equity (cid:133)nance framework. Owing to the information asymmetry between the intermediaries and the potential owners, equity issuance involves a dilution e⁄ect, a phenomenon that a dollar amount of equity issuance reduces the value of existing shares more than a dollar. We operationalize this e⁄ect by assuming that the actual cash (cid:135)ow related with equity is given by a function ’(D (i)) de(cid:133)ned as, t D (i) if D (i) 0 ’(D t (i)) = (1 ’(cid:22) t )D (i) if D t (i) (cid:21) <0 t t (cid:26) (cid:0) = D (i) ’(cid:22) min D (i);0 : t t (cid:0) (cid:1) f g In words, when the intermediary pays out a positive amount of dividends, the cash out(cid:135)ow associated with equity is simply given by the dividends payout, D (i). Howeverwhentheintermediaryissuesnewequities(D (i)<0),thecash t t in(cid:135)ow associated with the notional value D (i) is reduced to (1 ’(cid:22))D (i). t t (cid:0) (cid:0) (cid:0) Following Bolton and Freixas (2000), we call the foregone cash (cid:135)ow ’(cid:22)D (i) t (cid:0) dilution cost.8 In each period, (cid:133)nancial intermediaries face the following (cid:135)ow of funds constraint, 0 = (cid:15) (i)RFQ KB(i)+B (i) (4) t t t 1 t t+1 (cid:0) Cash In(cid:135)ow [RBB (i)+Q KB (i)+’(D (i))]: (cid:0) | t t {zt t+1 } t Cash Out(cid:135)ow The cash in(cid:135)ow is compos|ed of revenue fr{ozm last period(cid:146)s}investment (lending) (cid:15) (i)RFQ KB(i) and new borrowing from the household B (i). The t t t 1 t t+1 cash out(cid:135)ow co(cid:0)nsists of repayment to the household for last period(cid:146)s borrowing RBB (i), where RB is the borrowing rate of the intermediary, and new t t t investment Q KB (i). The last item in (4) can be cash in(cid:135)ow or cash out(cid:135)ow t t+1 depending on the sign of D (i). When it is negative, the actual cash in(cid:135)ow t interbank transfer of balance sheet assets di¢ cult (as was apparent in the (cid:133)nancial crisis of 2008,where secondary markets forbank loans became severely distressed). 8Inreality,thecostofissuingequitycouldstemfrommanysources. Forexample,outsiders whoinvestinnewsharesoftheintermediarymaynotbeabletodistinguishanegativeincome shock from diversion or ine¢ ciency of management. In such an environment, outsiders need to investigate the balance sheet of the intermediary before they invest to verify that the intermediary complies with the rule of truthful reporting. Furthermore, as shown by Ross (1977)andMyersandMajluf(1984),outsiders,notknowingthetrueinvestmentopportunities of the intermediary, require initialdiscounts to protect themselves from (cid:147)lemons(cid:148). This type of friction is evident in market data, where, for example, equity issuance costs take the form of underwriting fees for investment banks and initial discounts of seasoned equity o⁄erings (SEOs). 7

is reduced by a constant factor, ’(cid:22).9 By rearranging the terms and using the de(cid:133)nition of capital, the (cid:135)ow of funds constraint can be interpreted as the law of motion for equity capital, i.e., E (i)= N (i) + ’(D (i)) t t t (cid:0) Net-Worth Cash Flow for Equity where the net-worth of the int|erm{zed}iary is g|iven{zby } N (i) = (cid:15) (i)RFQ KB(i) RBB (i) t t t t (cid:0) 1 t (cid:0) t t = E (i)+[(cid:15) (i)RF 1]Q KB(i) (RB 1)B (i): t (cid:0) 1 t t (cid:0) t (cid:0) 1 t (cid:0) t (cid:0) t 2.1.5 Value Maximization Problem Tode(cid:133)netheoptimizationproblemofanintermediaryunderthespeci(cid:133)ctiming convention discussed above, it is useful to introduce an expectation operator that accounts for idiosyncratic uncertainty, E i t ( (cid:1) ) (cid:17) E( (cid:1)j sA t ). The conditioning set of the operator includes all aggregate information up to time t (denoted by sA) except the current realization of the idiosyncratic shock (cid:15) (i). We can then t t formally state the value maximization problem of the intermediary as follows. The intermediary optimizes over Q KB (i) , B (i) and D (i) to maximize s s+1 s+1 s V t B(i)=max 1 (cid:12)s (cid:0) t Et (cid:3) P s E i t [D s (i)] (5) s=t (cid:20) s (cid:21) X + 1 (cid:12)s (cid:0) t Et (cid:3) P s(cid:22) s (i) (1 (cid:0) m s )Q s K s B +1 (i) (cid:0) B s B +1 (i) s=t (cid:26) s (cid:20) (cid:21)(cid:27) X + s 1 =t (cid:12)s (cid:0) t Et (cid:26) (cid:3) P s s E i t (cid:20) (cid:21) s (i)[(cid:15) s (i)R s FQ s (cid:0) 1 K s B(i)+B s+1 (i) X RBB (i) Q KB (i) ’(D (i))] (cid:0) s s (cid:0) s s+1 (cid:0) s (cid:21)(cid:27) where(cid:3) isthemarginalutilityoftherepresentativehousehold,(cid:22) (i)and(cid:21) (i) s s s are the Lagrangian multipliers associated with the capital constraint and the (cid:135)ow of funds constraint, respectively. Notethattheintermediaryisrisk-neutralanddiscountsthefuturedividends bythemarginalutilityofrepresentativehousehold,theowneroftheinstitution. 9Gomes (2001) points out that the per unit cost of equity issuance is either constant or declining, exhibiting an increasing returns to scale. An alternative approach considered in JermannandQuadrini(2009)assumesaquadraticadjustmentcostindividendpayouts/equity issuance. Such an assumption is motivated by empirical evidence that dividend payouts are smooth. In contrast to dividend payouts, equity (cid:133)nancing and/or share repurchases are better described as lumpy, discrete event. In reality, modeling the mix of smooth dividend streamsandlumpyequityissuance/sharerepurchasesjointlywouldrequireconsideringavery complicated corporate (cid:133)nancing problem, which lies well outside our interest in focusing on key factors driving the links between bank capitalization and realeconomic activity. 8

Alsonotethatthe(cid:135)owoffundsconstraintanditsshadowvalue(cid:21) (i)arewithin s the expectation operator E i t ( (cid:1) )(cid:150)under our timing assumption, the intermediary hastodecidehowmuchtoborrowandinvestbeforeitcomestoknowthevalue of idiosyncratic shock (cid:15) (i). This implies that the intermediary does not know s its own shadow value of internal funds until the idiosyncratic cash (cid:135)ow shock becomes known and the intermediary needs to form an expectation based on aggregateconditions. Wecansummarizethee¢ ciencyconditionsoftheproblem as follows, FOC for Q K (i): t t+1 (cid:15) E i t [(cid:21) t (i)] = (cid:22) t (i)(1 (cid:0) m t ) (6) (cid:3) RF + (cid:12)Et (cid:3) t+1 E i t+1 [(cid:21) t+1 (i)(cid:15) t+1 (i)] (cid:5) t+1 (cid:20) t t+1(cid:21) FOC for B (i): t+1 (cid:15) (cid:3) RB E i t [(cid:21) t (i)]=(cid:22) t (i)+(cid:12)Et (cid:3) t+1 E i t+1 [(cid:21) t+1 (i)] (cid:5) t+1 (7) (cid:20) t t+1(cid:21) FOC for D (i): t (cid:15) 1=(cid:21) (i)’(D (i)) (8) t 0 t where (cid:5) P =P . On the right side of the FOCs for investment and bort+1 t+1 t (cid:17) rowing, all macroeconomic variables at t+1 are taken out of the expectation operator E i t+1 ( (cid:1) ), since the conditioning set of E i t+1 ( (cid:1) ) includes those variables attimet+1. Incontrast,theFOCfordividendsisnotintegratedovertheidiosyncratic uncertainty. This is because the dividends/equity (cid:133)nancing decisions are made after the realization of the shock. To see that the capital constraint binds in the steady state, consider the version of (7) that arises in the absence of aggregate uncertainty, i.e., when (cid:3) t =(cid:3) t+1 , E i t [(cid:21) t (i)]=E i t+1 [(cid:21) t+1 (i)], and (cid:5) t+1 =1, (cid:22) 1 =(cid:12)RB (cid:0) Ei[(cid:21)(i)] Since the idiosyncratic uncertainty does not disappear in the steady state, the shadowvalueofthe(cid:135)owoffundsconstraintisstillintegratedoveridiosyncratic uncertainty. Bindingcapitalconstraint,i.e.,(cid:22)>0requires(cid:12)RB <1. Asshown below, this is indeed the case owing to the liquidity premium households place on deposits.10 By multiplying 1 m to both sides of (7) and subtracting the t (cid:0) 10Thereareotherwaystoensureabindingcapitalconstraint. Forexample,onecanassume that the intermediary is impatient or subject to a constant death probability. Second, one can introduce a tax shield. 9

resulting expression from (6), we can merge the two FOCs into (cid:3) RF m tE i t [(cid:21) t (i)]=(cid:12)Et (cid:3) t+1 E i t+1 [(cid:21) t+1 (i)(cid:15) t+1 (i)] (cid:5) t+1 (9) (cid:20) t t+1(cid:21) (cid:3) RB (cid:0) (cid:12)Et (cid:3) t+1(1 (cid:0) m t )E i t+1 [(cid:21) t+1 (i)] (cid:5) t+1 (cid:20) t t+1(cid:21) This is the version of the e¢ ciency condition that will be used extensively in our analysis that follows. To operationalize (9) for a sharper characterization of the equilibrium, we need to show how the intermediaries in the model form expectations regarding their liquidity condition, which is summarized by two measures, E i t [(cid:21) t (i)] and E i t [(cid:21) t (i)(cid:15) t (i)]. 2.1.6 Intermediary Asset Pricing Our model has a symmetric equilibrium for three reasons: (cid:133)nancial intermediaries are risk-neutral; the (cid:133)rst moment of the idiosyncratic shock is timeinvariant; and (cid:133)nally, the intermediaries decide how much to invest and to borrow before the realization of their idiosyncratic shocks. In this symmetric equilibrium: all(cid:133)nancialintermediarieschoosethesamelevelofinvestmentand borrowing,i.e.,KB (i)=KB (j)andB (i)=B (j)foralliandj [0;1]. t+1 t+1 t+1 t+1 2 This greatly facilitates aggregation. However, dividends/equity issuance decisions are conditioned upon the realization of the idiosyncratic shock. The same thing can be said about the shadow value of the (cid:135)ow of funds constraint, which is the summary measure of the liquidity condition of a particular intermediary. Afterimposingthebindingcapitalconstraintandthesymmetricequilibrium condition, we can express the (cid:135)ow of funds constraint as D (i) ’(cid:22) min D (i);0 =(cid:15) (i)RFQ KB t (cid:0) (cid:1) f t g t t t (cid:0) 1 t RB(1 m )Q KB m Q KB : (cid:0) t (cid:0) t (cid:0) 1 t (cid:0) 1 t (cid:0) t t t+1 At the time of dividend payout/equity issuance decision, all other quantities of the above expression are predetermined. Since the LHS is strictly increasing in D (i) everywhere, we can (cid:133)nd a unique level of the revenue shock that satis(cid:133)es t the (cid:135)ow of funds constraint with D (i)=0. If we let D (i)=0 and solve for (cid:15) , t t t we obtain an equity (cid:133)nancing threshold, RB 1 Q KB (cid:15) =(1 m ) t +m t t+1 : (cid:3)t (cid:0) t (cid:0) 1 R t F t R t F Q t 1 K t B (cid:0) If (cid:15) (i) (cid:15) , paying out a strictly positive amount of dividends is optimal t (cid:21) (cid:3)t while it is optimal to issue equities (D (i) < 0), incurring the dilution cost of t ’(cid:22) if (cid:15) (i) < (cid:15) . This and (8) imply that the shadow value of internal funds of t (cid:3)t the intermediaries depends on the realization of the idiosyncratic shock in the following way: 1 if (cid:15) (i) (cid:15) (cid:21) t (i)=1=’ 0 (D t (i))= 1=(1 ’(cid:22))>1 if (cid:15) t (i) (cid:21) <(cid:15) (cid:3)t : (10) (cid:26) (cid:0) t (cid:3)t 10

The discussion above regarding the equity (cid:133)nance threshold can be used to transform the e¢ ciency condition (9) into a form that is more convenient for a quantitative analysis of the model, which requires us to evaluate two measures of liquidity condition: E i t [(cid:21) t (i)] and E i t [(cid:21) t (i)(cid:15) t (i)]. To that end, let s t (i) be a standardization of (cid:15) (i) de(cid:133)ned as t s (i)=(cid:27) 1(log(cid:15) (i)+0:5(cid:27)2): (11) t (cid:0) t Sinces (i)isamonotonictransformationof(cid:15) (i)andfollowsastandardnormal t t distribution, we can integrate the shadow value over the idiosyncratic uncertainty as follows 1 E i t [(cid:21) t (i)] = 1 (cid:1) dF((cid:15))+ 1 ’(cid:22) (cid:1) dF((cid:15)) (12) Z(cid:15)t(cid:21) (cid:15)(cid:3)t Z(cid:15)t(cid:20) (cid:15)(cid:3)t (cid:0) (cid:8)(s ) ’(cid:22) = 1 (cid:8)(s )+ (cid:3)t =1+ (cid:8)(s )>1: (cid:0) (cid:3)t 1 ’(cid:22) 1 ’(cid:22) (cid:3)t (cid:0) (cid:0) (12) implies that the intermediary(cid:146)s ex ante valuation of a sure dollar is always greater than a dollar as long as the probability of costly recapitalization is strictly positive. What is uncertain here is not the dollar, but its valuation. While the realized shadow value takes only two values: it is either 1 or 1=(1 ’(cid:22)), the expected shadow value is time varying as aggregate conditions (cid:0) change. It is this expected value that matters for the commitment decisions for investment/borrowing. The more likely is costly equity (cid:133)nancing, the higher the expected shadow value of internal funds. To evaluate E i t [(cid:21) t (i)(cid:15) t (i)], the following properties of lognormal distribution is useful,11 1 (cid:15)f((cid:15))d(cid:15)=[1 (cid:8)(s (cid:27))] (cid:15)f((cid:15))d(cid:15); (cid:0) (cid:3)t (cid:0) Z(cid:15) (cid:21) (cid:15)(cid:3)t Z0 where f() is the pdf of the lognormal distribution conditioned upon the para- (cid:1) meter(cid:27) ands isde(cid:133)nedas(11). Usingpropertiesofthelognormaldistribution (cid:3)t and noting that 0 1(cid:15)f((cid:15) j (cid:27) t )d(cid:15) = 1 for all bounded positive parameter (cid:27) t , one can easily see that R (cid:15) E i t [(cid:21) t (i)(cid:15) t (i)]= (cid:15) t dF((cid:15))+ 1 t ’(cid:22) dF((cid:15)) (13) Z(cid:15)t(cid:21) (cid:15)(cid:3)t Z(cid:15)t(cid:20) (cid:15)(cid:3)t (cid:0) (cid:8)(s (cid:27)) ’(cid:22) =1 (cid:0) (cid:8)(s (cid:3)t (cid:0) (cid:27))+ 1 (cid:3)t (cid:0) ’(cid:22) =1+ 1 ’(cid:22) (cid:8)(s (cid:3)t (cid:0) (cid:27))>1: (cid:0) (cid:0) where(cid:8)(s (cid:27))comesfromthetruncatedlognormaldistribution.12 (13)implies (cid:3)t(cid:0) 11See Johnson et al.(1994). 12The following property of lognormal distribution is used to derive the expression in the main text (see Johnson et al.(1994) ): Z(cid:15) (cid:21) (cid:15)(cid:3)t (cid:15)f((cid:15) j (cid:27))d(cid:15)=[1 (cid:0) (cid:8)(s(cid:3)t (cid:0) (cid:27))] Z0 1(cid:15)f((cid:15) j (cid:27))d(cid:15); where f( (cid:27)t)is the pdfofthe lognormaldistribution conditioned upon the parameter(cid:27) and (cid:1)j s is de(cid:133)ned as (11). (cid:3)t 11

that the intermediary(cid:146)s ex ante valuation of a random variable, whose mean is equal to a dollar, is always greater than a dollar. In contrast to the case of E i t [(cid:21) t (i)],whatisuncertainisboththecash-(cid:135)owanditsvaluation,whichmakes ’(cid:22) E i t [(cid:21) t (i)(cid:15) t (i)] = 1+ 1 ’(cid:22) (cid:8)(s (cid:3)t (cid:0) (cid:27)) (cid:0) ’(cid:22) < 1+ 1 ’(cid:22) (cid:8)(s (cid:3)t ) = E i t [(cid:21) t (i)] (cid:0) as long as (cid:27) > 0, re(cid:135)ecting a negative covariance between the shadow value andtheidiosyncraticshockin(10). Thisnegativecovarianceisintuitive(cid:150)(cid:133)rms with a large positive idiosyncratic shock do not need costly equity (cid:133)nancing, and hence have a lower shadow value of internal funds, than do (cid:133)rms with a large negative idiosyncratic shock. In summary, the caution created by the commitment structure imposed on the investmenttechnologyamid unresolved idiosyncratic fundingriskmanifests itselfintheconservativeexantevaluationofrandomandnon-randomcash(cid:135)ow. This sets a higher bar for the required return on investment as will be shown below. Using (12) and (13), we can eliminate all expressions involving the expectation operator E i t ( (cid:1) ) in (9). To that end, it is convenient to rewrite the FOC as m t =(cid:12)Et (cid:26) (cid:3) (cid:3) t+ t 1E i t+ E 1 i t [ [ (cid:21) (cid:21) t t + (i 1 ) ( ] i)] (cid:20) E i t+ E 1 [ i t (cid:21) + t 1 + [(cid:21) 1 ( t+ i) 1 (cid:15) ( t+ i) 1 ] (i)]R (cid:5) t t F + + 1 1 (cid:0) (1 (cid:0) m t ) R (cid:5) t t B + + 1 1 (cid:21)(cid:27) : Let(cid:17) ’(cid:22)=(1 ’(cid:22)). Afterdividingtheexpressionthroughbym andsubstituting t (cid:17) (cid:0) (12)and(13)intheabove,wecanderivetheintermediaryassetpricingformula, 1 R~F RB 1=Et ( M t B ;t+1 " m t (cid:5) t t + + 1 1 (cid:0) (1 (cid:0) m t ) (cid:5) t t + + 1 1 !#) (14) where the intermediary(cid:146)s pricing kernel is given by 1+(cid:17)(cid:8)(s ) (cid:3) 1+(cid:17)(cid:8)(s ) MB =MH (cid:3)t+1 =(cid:12) t+1 (cid:3)t+1 t;t+1 t;t+1 1+(cid:17)(cid:8)(s ) (cid:3) 1+(cid:17)(cid:8)(s ) (cid:20) (cid:3)t (cid:21) t (cid:20) (cid:3)t (cid:21) and the risk adjusted return is given as 1+(cid:17)(cid:8)(s (cid:27)) R~F =RF (cid:3)t+1(cid:0) <RF : t+1 t+1 1+(cid:17)(cid:8)(s ) t+1 (cid:20) (cid:3)t+1 (cid:21) The above asset pricing formula looks di⁄erent from a textbook version mainly for two reasons. First, the formula is a levered asset pricing formula. Unlike in the textbook version which assumes away leverage choice, the returns are levered up to the inverse of capital ratio. To see this point, assume m =1. t One can then see that the second term vanish and the formula looks closer to the conventional one, i.e., 1=Et [M t B ;t+1(cid:1) R~ t F +1 =(cid:5) t+1 ]: 12

Second, the intermediary speci(cid:133)c pricing kernel is a (cid:133)ltered version of the representative household(cid:146)s pricing kernel, where the (cid:133)lter is the ratio of the shadow value of internal funds today vs. tomorrow. The (cid:133)lter could potentially weaken the role of the representative household as a marginal investor even though all (cid:133)nancial intermediaries are owned by the households. Suppose that in the beginning of current period, a bad news about aggregate returns arrives. This, holding other things constant, increases the probability of costly recapitalization(cid:8)(s )sinceevenanormalrangeofidiosyncraticreturnmaynot (cid:3)t beenoughtomeetthefundingneedsassociatedwithtoday(cid:146)sinvestment. Ifthe aggregate shock is strong enough, the ratio of shadow values tomorrow vs. today substantially declines, making overall required return on capital (1=MB ) t;t+1 rise, which suppresses today(cid:146)s investment. Finally, we note that, when ’(cid:22) =0, the asset pricing formula collapses to 1 RF RB 1=Et M t H ;t+1 m (cid:5) t+1 (cid:0) (1 (cid:0) m t ) (cid:5) t+1 (cid:26) (cid:20) t (cid:18) t+1 t+1(cid:19)(cid:21)(cid:27) and idiosyncratic uncertainty plays no role in the determination of asset price. Anyarbitrarilylargeamountofuncertaintysimplydoesnotmatterforrealallocations. Inthissense, costlyequity(cid:133)nanceisthekeyfrictioninourframework. The form of the intermediary asset pricing formula is super(cid:133)cially similar to Jermann and Quadrini (2009), who derive a similar pricing kernel from a reduced-formconvexadjustmentcostofdividend;however,ourapproachderives from a speci(cid:133)c set of structural frictions. It is also super(cid:133)cially similar to the intermediary asset pricing formula of He and Krishnamurthy (2008); however, theyderivetheirintermediary-speci(cid:133)cpricingkernelfromtheassumptionofrisk averse intermediaries. The link to the LAPM (Liquidity-Based Asset Pricing Model)ofHolmstr(cid:246)mandTirole(2001)ismoredirect: Inourcase,theliquidity premium arises from costly recapitalization of (cid:133)nancial intermediaries, while the premium exists for non-(cid:133)nancial corporations with potential investment opportunity or working capital needs in Holmstr(cid:246)m and Tirole (2001). 2.1.7 Illiquidity of Balance Sheet Assets and Adjustment Cost In our timing convention, we assume that there exist factors that make the intraperiodadjustmentofbalancesheetassetsdi¢ cult,requiringthecommitment of participants. In reality, there are also reasons why interperiod as well as intraperiod adjustments of loan portfolio can be costly. As pointed out by many, for instance, Diamond and Rajan (2000), (cid:133)nancial assets of intermediaries are inherently illiquid: First, a substantial knowledge about the characteristics of borrowers is an indispensable prerequisite for successful selections of new borrowers and churning out ine¢ cient existing borrowers. Second, a substantial partofbalancesheetassetsiscomposedofitemsthatarenoteasilymarketable since the intermediaries cannot commit themselves to work for the second buyers after the sale of such (cid:133)nancial assets. Such an illiquidity of balance sheet assets may be the fundamental force behind the slow dynamics often found in balance sheet data. 13

To capture this aspect in a parsimonious way, we assume that there exists a constant return-to-scale convex adjustment cost associated with changing the nominal stock of (cid:133)nancial assets of the intermediaries: (cid:13)(cid:22) Q K 2 (cid:13)(Q K ;Q K )= t t+1 1 Q K , (cid:13)(cid:22) 0 t (cid:0) 1 t t t+1 2 (cid:18) Q t (cid:0) 1 K t (cid:0) (cid:19) t (cid:0) 1 t (cid:21) Withtheadjustmentfrictioninbalancesheet,itisstraightforwardtoshowthat the intermediary asset pricing formula is modi(cid:133)ed into 1 R~F RB 1=Et ( M t B ;t+1 " m t (cid:5) t t + + 1 1 (cid:0) (1 (cid:0) m t ) (cid:5) t t + + 1 1 !#) (15) (cid:13)(cid:22) Q K (cid:13)(cid:22) Q K 2 (cid:0) m t (cid:18) Q t t (cid:0) 1 t K +1 t (cid:0) 1 (cid:19) (cid:0) Et ( M t B ;t+12m t "(cid:18) Q t+ t K 1 t+ t+ 1 2 (cid:19) (cid:0) 1 #) : Though not explicit in (15), the (cid:135)ow of funds constraint and the equity (cid:133)nance threshold need to be modi(cid:133)ed accordingly as well. Thesedynamiccostsofadjustingthebalancesheetof(cid:133)nancialintermediaries are not important for the qualitative predictions of the model, but will help match the dynamics of adjustment apparent in the data. 2.2 Government Before we describe the other parts of private sector, we discuss the government sector. This makes it easier to explain the household(cid:146)s problem. Because we are only interested in credit policies, our model(cid:146)s government is solely focused on such activities. We consider two types of such credit policy: (i) direct lending/assetpurchasepolicy(ii)capitalinjectionpolicy. Tomakethepresentation as transparent as possible, we consider one policy at a time. 2.2.1 Direct Lending/Asset Purchase Policy We (cid:133)rst consider a policy under which the government purchases and holds a certain fraction of capital assets for a certain period of time. The policy is motivated by the recognition of the key problem that the (cid:133)nancial intermediaries do not have deep pockets under the capital constraint and the costly recapitalization. As a result, at a time when the prices of capital assets are unusually low, the intermediaries cannot su¢ ciently exploit opportunities for investment that are pro(cid:133)table absent (cid:133)nancial frictions, thus intensifying the depth and prolonging the duration of downturns (due to an inability to arbitrage inef- (cid:133)cient intermediation wedges, reminiscent of the limits of arbitrage noted by Shleifer and Vishny (1997)). If a third party which does not su⁄er from the deep pocket problem, say, the government, can intervene and purchase some fraction of capital assets, the undesirable downward spiral between the asset prices and aggregate investment can be mitigated. 14

LetSG;K andSB;K denotethesharesofcapitalassetsownedbythegovernt t mentandbytheintermediaries,respectively. Usingthesenotations,themarket clearing condition for capital assets (1) can be expressed as 1=SG;K +SB;K: t t Weassumethatthegovernmentdoesnotconsiderashort-salepolicy(cid:150)e.g.,governmentinvestmentisalwaysgreaterthanorequaltozero,restrictingthespace of SG;K to [0;1]. We also assume that the government maintains a balanced t budget and imposes a lump sum tax to meet the balanced budget constraint, T =Q SG;KK RFSG;KK (16) t t t+1 t+1 (cid:0) t t t Regardingthebudgetconstraint,threethingsareimportant: First,inouranalysis, all capital assets are homogeneous by construction. As a result, the issue of potential ine¢ ciency of the government in selecting investment projects is not addressed. Second, the policy can generate pro(cid:133)ts through the dividends and capital gains channel. Third, the tax policy turns into a transfer policy as the government starts implementing an exit strategy thatruns down its investment (SG;K <SG;K). t+1 t WeassumethefollowingAR(1)processwithaseriallycorrelatedshockterm for the law of motion of the government share KG, t SG;K = (cid:26) SG;K +uG;K (17) t+1 g t t uG;K = (cid:26) uG;K +(cid:15)G;K t u t 1 t (cid:0) The process is e⁄ectively an AR(2) process with AR(1) and AR(2) coe¢ cients given by (cid:26) +(cid:26) and (cid:26) (cid:26) , respectively. We choose this process to replicate a g u (cid:0) g u hump-shapedpolicyinterventioninaparsimoniousway: agradualphase-inand a gradual run-o⁄. Note that the government asset position has a zero steady stateand,asmentionedabove,weonlyconsiderpositivevaluesforSG;K (which t places restrictions on the parameters in 17 and the sequence of innovations (cid:15)G;K). t 2.2.2 Capital Injection Policy The key friction in our model is the funding risk facing the (cid:133)nancial intermediaries because raising outside equity is costly owing to the information problem in equity market. We also have shown that such cost at the aggregate level, denoted by (cid:4) , is given by t 1 (cid:4) = ’(cid:22)min D (i);0 di: t t (cid:0) f g Z0 Now consider a policy under which the government purchases the shares of the (cid:133)nancial intermediaries at market prices and refunds the cost of equity issuance only to the institutions that are raising equities and are owned by 15

the government. Let SG(i) denote the share of an intermediary owned by the t government. With this policy, the market clearing condition of a particular share is given by 1=SG(i)+SH(i); t t whereSH(i)istheshareownedbythehouseholds.13 Undertheproposedpolicy, t the cost of raising equity is reduced to 1 1 (cid:4) = ’(cid:22)min D (i);0 di+ ’(cid:22)min D (i);0 SG (i)di t (cid:0) f t g f t g t+1 Z0 Z0 1 = ’(cid:22)min D (i)[1 SG (i)];0 di (cid:0) f t (cid:0) t+1 g Z0 Assuming that the government purchases pro rata shares, i.e., SG (i) = SG , t+1 t+1 the cost can be simpli(cid:133)ed as 1 (cid:4) = (1 SG ) ’(cid:22)min D (i);0 di=’(cid:22)(1 SG )D t (cid:0) (cid:0) t+1 f t g (cid:0) t+1 t(cid:0) Z0 where 1 D min D (i);0 di= D (s )d(cid:8)(s ) t(cid:0) (cid:17) f t g t t t Z0 Zst(cid:20) s(cid:3)t Note that the policy can be seen as equivalent to a subsidy that is proportional to the amount of equity issuance with the subsidy rate given by the government share. Note the role of min operator in this expression: the (cid:133)nancial intermediary is eligible for the subsidy if and only if it is raising outside equity voluntarily. We assume that the policy is funded by a lump sum tax T (transfer when t negative) of households. Let PS(i) and PS (i) denote the ex-dividend value t t 1;t of equity at time t and time t value of e(cid:0)xisting shares outstanding at time t 1, respectively. Assumingabalancedbudgetineachperiod, thegovernment (cid:0) budget constraint is given by 1 T = 1(D (i) 0)PS(i)SG di t t (cid:20) t t+1 Z0 1 1(D (i) 0)[max D (i);0 +PS (i)]SGdi (cid:0) Z0 t (cid:0) 1 (cid:20) f t g t (cid:0) 1;t t To simplify the budget constraint into a form more convenient for a quantitative analysis, (cid:133)rst note that the ex-dividend value of equity PS(i) is the t same for all intermediaries owing to the assumption of iid idiosyncratic shock, i.e., PS(i) = PS 1 PS(i)di. To see this point more formally, let vB(i) t t (cid:17) 0 t t (cid:17) VB(i)=(cid:3) ,thevalueofanintermediaryinrealdollarunits. Therecursivenature t t R of (5) implies that the following Bellman equation holds, v t B(i)=d t (i)+Et [M t H ;t+1(cid:1) E i t+1 [v t B +1 (i)]] 13We do not considera short sale policy ofthe government. Hence,0 SG(i) 1. (cid:20) t (cid:20) 16

where d t (i) (cid:17) D t (i)=P t . Let v t B (cid:17) E i t [v t B(i)] = 0 1 v t B(i)di, the aggregate value of all intermediaries at time t. This is also the expected value of an individual R intermediary before the realization of the idiosyncratic shock. The ex-dividend value of equity at time t, P t S(i) is given by P tEt [M t H ;t+1(cid:1) v t B +1 ] and hence is the same for all intermediaries. Hence, the (cid:133)rst term on the RHS of the budget constraint is equivalent to 1 PSSG 1(D (i) 0)di=(cid:8)(s )PSSG : t t+1 t (cid:20) (cid:3)t t t+1 Z0 Alsonotethatbytheassumptionofiididiosyncraticshockandthelawoflarge number, we have 1 1(D (i) 0)[max D (i);0 +PS (i)]di Z0 t (cid:0) 1 (cid:20) f t g t (cid:0) 1;t 1 = (cid:8)(s ) [max D (i);0 +PS (i)]di t (cid:0) 1 Z0 f t g t (cid:0) 1;t Inwords,thepartialsumsofthedividendsandcurrentvaluesofexistingshares of the intermediaries that issued new shares at time t 1 are the same as the (cid:0) total sums multiplied by the measure of such intermediaries. We can then simplify the budget constraint as T =(cid:8)(s )PSSG (cid:8)(s )(D++PS )SG t t t t+1(cid:0) t (cid:0) 1 t t (cid:0) 1;t t where 1 D+ max D (i);0 di= D (s )d(cid:8)(s ): t (cid:17) f t g t t t Z0 Zst(cid:21) s(cid:3)t and PS PS (i)di. As mentioned above, the government purchases only th t (cid:0)e 1 s ;t ha (cid:17) res of t t(cid:0)h 1 e ;t institutions that are issuing new equities, which explains R the presence of (cid:8)(s ) in the (cid:133)rst term of the budget constraint. Under the t assumption of no persistence in the (cid:133)rst moment of the idiosyncratic shock and by the law of large numbers, the government portfolio held in time t 1 (cid:0) shares all the properties of the aggregates at time t. This explains why we can simplymultiply(cid:8)(s )tothesecondtermwithouthavingtokeeptrackofthe t 1 identities (or the dis(cid:0)tribution) of the intermediaries that issued new shares at time t 1. (cid:0) The last two remarks for the direct lending/asset purchase policy can be appliedhereaswell: thegovernmentcanearnpro(cid:133)tsduringtheimplementation ofthepolicyandthetaxpolicyturnsintoatransferpolicyoncetheexitstrategy (SG < SG) kicks in. For a straightforward comparison of this policy with t+1 t direct lending/asset purchase policy, we specify exactly the same process as in the latter case, SG;S = (cid:26) SG;S +uG;S (18) t+1 g t t uG;S = (cid:26) uG;S +(cid:15)G;S t u t 1 t (cid:0) 17

Again, to ensure that such policy does not a⁄ect the long run equilibrium of theeconomy, wesetthesteadystateofthegovernmentshareequaltozero. We then perform the same perfect foresight deterministic simulation to ensure that SG does not go outside the proper range, [0;1]. t 2.3 Household Therepresentativehouseholdconsumesthe(cid:133)nal-goodsandearnsmarketwages bysupplyinglaborinputsfortheproductionof(cid:133)nalgoods. Weassumethatthe householdlacksnecessaryskillstodirectlymanageinvestmentprojects. Forthis reason, the household invests its saving through (cid:133)nancial intermediaries. The household can either invest in the shares of the intermediaries or make deposits to the intermediaries. 2.3.1 Budget Constraint Undertheassumptionsmadeabove,thebudgetconstraintoftherepresentative household can be expressed as 1 0 = W H +RBB P C T PS(i)SH (i)di (19) t t t t (cid:0) t t (cid:0) t (cid:0) t t+1 Z0 1 B + [max D (i);0 +PS (i)]SH(i)di (cid:0) t+1 Z0 f t g t (cid:0) 1;t t where B = B (i)di, W is a nominal wage rate, H is labor hours, T is the t t t t t lump sum tax and SH(i) is the number of shares owned by the households R t outstanding at time t. Consider an accounting identity that relates the ex-dividend value of equity attime t (PS(i))tothe time t value ofexisting shareoutstandingattime t 1: t (cid:0) PS(i)=PS (i)+X (i) t t 1;t t (cid:0) where X (i) is the value of new shares. The dilution e⁄ect discussed in the t intermediary problem implies that the value of new shares, absent any public intervention,isgivenbytheamountofnegativedividendsreducedbyadilution factor ’(cid:22), X (i)= (1 ’(cid:22))min D (i);0 : t t (cid:0) (cid:0) f g The value of new shares under the capital injection policy is modi(cid:133)ed into X (i)= (1 ’(cid:22))min D (i);0 ’(cid:22)min D (i);0 SG t (cid:0) (cid:0) f t g(cid:0) f t g t+1 As a result, holding the value of outstanding share at time t 1 (PS (i)) constant, the capital injection policy increases the ex-dividend v (cid:0) alue o t f(cid:0)e 1 q ;t uity at time t exactly by ’(cid:22)min D (i);0 SG , i.e., (cid:0) f t g t+1 PS(i)=PS (i) [1 (1 SG )’(cid:22)]min D (i);0 : (20) t t (cid:0) 1;t (cid:0) (cid:0) (cid:0) t+1 f t g 18

Substituting the accounting identity (20) in (19), one can see that the budget constraint is equivalent to 1 0 = W H +RBB B P C T PS(i)SH (i)di (21) t t t t (cid:0) t+1 (cid:0) t t (cid:0) t (cid:0) t t+1 Z0 1 + [max D (i);0 +[1 (1 SG )’(cid:22)]min D (i);0 +PS(i)]SH(i)di f t g (cid:0) (cid:0) t+1 f t g t t Z0 2.3.2 Preferences Forthepreferencesoftherepresentativehousehold,weadoptthemoststandard speci(cid:133)cations for quantitative analyses in the literature. One such speci(cid:133)cation can be found in Smets and Wouters (2007). More speci(cid:133)cally, we adopt an internal habit in consumption and a labor disutility separable from the utility of consumption. To model the value households place on their deposits, we adopt the deposit-in-the utility speci(cid:133)cation originating from Sidrauski (1967), which captures the non-pecuniary bene(cid:133)ts provided by (cid:133)nancial institutions.14 Formally, the preferences are given by u(C ;C ;B =P ;H ) = log(C aC ) (22) t t 1 t+1 t t t t 1 (cid:0) (cid:0) (cid:0) (cid:16) B (i) (H )1+ + (cid:18)log t+1 di : t (cid:0) 1+ P (cid:18)Z t (cid:19) The household problem is straightforward: the household chooses C , H , t t f B (i), S (i) to maximize its value, t+1 t g V t H =max 1 (cid:12)s (cid:0) t Et u(C s ;C s 1 ;B s+1 =P s ;H s ) (cid:0) s=t X subject to the budget constraint (21). For later purpose, let (cid:3) denote the t Lagrangian multiplier associated with the budget constraint (21). 2.3.3 Pricing Financial Intermediaries We now show how the representative household prices the debts and equities of the (cid:133)nancial intermediaries. The FOCs for consumption, deposits and shares are given by FOC for C : t (cid:15) 1 a (cid:3) t = (cid:12)Et (23) C aC (cid:0) C aC t (cid:0) t (cid:0) 1 (cid:20) t+1 (cid:0) t(cid:21) FOC for B (i): t+1 (cid:15) (cid:18)=(cid:3) (cid:3) RB 1= t +(cid:12)Et t+1 t+1 (24) B (i)=P (cid:3) (cid:5) t+1 t (cid:20) t t+1(cid:21) 14Recent application also can be found in Van den Heuvel(2008). 19

FOC for SH (i): (cid:15) t+1 (cid:3) P t S(i) = (cid:12)Et (cid:3) t+1 E i t+1 [max f D t+1 (i);0 g (25) (cid:26) t +[1 (1 SG )’(cid:22)]min D (i);0 +PS (i)] (cid:0) (cid:0) t+2 f t+1 g t+1 (cid:27) The FOC for consumption is standard. The FOC for intermediary debt is di⁄erent from a standard asset pricing formula because of the non-pecuniary bene(cid:133)tofdeposit. Thiscreatesaliquiditypremiumthatthehouseholdiswilling to fore-go in making deposits at a rate lower than risk-free rate. Formally, the liquidity premium can be de(cid:133)ned as (cid:3) R RB (cid:18)=(cid:3) (cid:12)Et t+1 t+1 t+1 = t 0 (cid:3) (cid:5) (cid:0) (cid:5) B (i)=P (cid:21) (cid:20) t (cid:18) t+1 t+1(cid:19)(cid:21) t+1 t where R is a risk-free rate that satis(cid:133)es the (cid:133)ctitious asset pricing equation, t+1 1 = (cid:12)Et [((cid:3) t+1 =(cid:3) t )(R t+1 =(cid:5) t+1 )]. In the non-stochastic steady state, we have 1=(cid:12)R and (cid:18)=(cid:3) =1 (cid:12)RB; B=P (cid:0) which implies that (cid:12)RB 1 with the inequality strict if (cid:18) >0. This proves the (cid:20) statement that the capital constraint binds for the intermediaries in the steady state. 2.3.4 Cost of Capital From a theoretical perspective, the relevant cost of capital for a (cid:133)nancial intermediary is a marginal cost (or a weighted average of marginal costs), as can be seen directly by rewriting (9) as Et f M t H ;t+1E i t+1 [(cid:21) t+1 (i)(cid:15) t+1 (i)]R t F +1g MB of Investment RB |=m tE i t [(cid:21) t (i)]+{(z1 (cid:0) m t )Et M t H ;t+ } 1E i t+1 [(cid:21) t+1 (i)] (cid:5) t+1 : (26) (cid:26) t+1(cid:27) MC of Investment The above equ|ates the marginal bene(cid:133){tz(LHS) and the marginal}cost (RHS) of investment. Evidently it is an weighted average of two components, the one associated with the marginal cost of raising capital and the one associated with marginal borrowing cost. If the marginal cost of capital is constant, the distinction between marginal and average costs of capital is meaningless unless there is a (cid:133)xed cost component, which is absent in our environment. One might be tempted to take this conclusion given that the per-unit cost of issuing equity is constant and the 20

retail borrowing market is competitive. However, the marginal cost of capital is increasing in the size of the balance sheet. To see this point, remember that the equity (cid:133)nancing threshold is given by RB 1 Q KB (cid:15) =(1 m ) t +m t t+1 : (cid:3)t (cid:0) t (cid:0) 1 R t F t R t F Q t 1 K t B (cid:0) Thethresholdisincreasinginthesizeofbalancesheet,Q KB : thegreaterthe t t+1 size of the balance sheet, the more likely to face the recapitalization problem. Also remember that the expected shadow value is increasing in the threshold, E i t [(cid:21) t (i)] = 1+(cid:17)(cid:8)[(log(cid:15) (cid:3)t +0:5(cid:27)2)=(cid:27)]. We can then consider a thought experiment in which the size of the balance sheet is permanently increased, boosting Q KB today and Q KB tomorrow in the same proportion. Holding t t+1 t+1 t+2 the asset return and borrowing rate constant, this increases the equity (cid:133)nance threshold (cid:15) and the probability of costly recapitalization today, and causes the (cid:3)t marginal cost of capital to rise, implying a convex cost of capital. While the marginal cost is the theoretically valid concept for capital budgeting, often policy debates have centered around a slightly di⁄erent concept of cost of capital: the weighted average cost of capital (WACC). This concept has played a signi(cid:133)cant role in assessments of the economic impact of changes in regulatory regime for capital standards, under the (potentially naive) view that equityismorecostlythandebt. Forexample,someassessmentsoftheeconomic impact of shifts in required capital starts with an estimate of the e⁄ect on the weighted average cost of capital for (cid:133)nancial institutions (see for instance BIS (2010a) and BIS (2010b)). The advantage of such a concept is its observability. In this subsection, we show how such an observable measure of cost of capital can be constructed. The weighted average cost of capital RW is de(cid:133)ned as the weighted sum of t+1 returns on equity and debt, i.e., Et R t W +1 =m tEt R t S +1 +(1 (cid:0) m t )R t B +1 : (27) Hence, the key issue in constructing the cost of capital is how to measure the returnonequity,especiallywhentheissuerfacescostlyequity(cid:133)nancingfriction. The household FOC for share can be used for this purpose. To that end, (cid:133)rst remember that P t S(i) = P t S for all i, and trivially, E i t+1 [P t S +1 (i)] = P t S +1 . Noting that E i t+1 [max f D t+1 (i);0 g ] = D t + +1 , E i t+1 [min f D t+1 (i);0 g ] = (cid:0) D t(cid:0)+1 and D =D+ D , we can rewrite the asset pricing formula (25) as t+1 t+1(cid:0) t(cid:0)+1 1 = (cid:12)Et ( (cid:3) (cid:3) t+ t 1 "(cid:18) D t+1 P + t S P t S +1 (cid:19) +’(cid:22)(1 (cid:0) S t G +2 ) D P t(cid:0) t + S 1 #) (cid:17) (cid:12)Et [M t H ;t+1(cid:1) R t S +1 ] (28) The (cid:133)rst component of return on equity is conventional: the return on equity is the sum of dividend/price ratio and the capital gain. The second component is the direct result of costly equity (cid:133)nancing assumption we adopt. From the 21

formula (28), one can easily see how the costly equity (cid:133)nance increases the cost of equity. One can also see how the capital injection policy directly lowers the cost of equity, and hence, the weighted cost of capital. 2.4 Technology To save space, our description of the rest of the model economy will be brief. Ourgoalinthisanalysisistoinvestigatetheroleoffunding-marketfrictionsfacing (cid:133)nancial intermediaries and to consider the e⁄ects of unconventional public policies designed to address balance-sheet strains at (cid:133)nancial institutions in an environment where other policy tools (such as traditional monetary policy) are not available. For this reason, we take the model as close as possible to a real businesscyclebenchmark. Whilewekeepdistinctionsbetweennominalandreal variables in our notation (thereby allowing easy integration of monetary policy questions at a later stage), price adjustment is frictionless in this analysis. 2.4.1 Final Goods A continuum of competitive (cid:133)rms produce (cid:133)nal goods using capital and labor in a constant return-to-scale (CRS) Cobb-Douglas technology. They solve the following static pro(cid:133)t maximization problem, max P Z (K (j)U (j))1 (cid:11)HH(j)(cid:11) W H (j) RK(K (j)U (j)) Kt(j)Ut(j);H t H(j)f t t t t (cid:0) t (cid:0) t t (cid:0) t t t g where Z is an aggregate technology shock. Since the scale of the problem is t indeterminate, one could assume a representative (cid:133)rm instead of a continuum. 2.4.2 Investment A continuum of competitive (cid:133)rms produce investment goods by combining an input of (cid:133)nal goods and a CRS adjustment technology. Following Christiano et al. (2003) and Smets and Wouters (2007), we specify a convex investment adjustment cost and model the investment problem as follows, V t I =m Is( a k x ) Et X s 1 =t (cid:12)t (cid:0) s (cid:3) P s s ( Q s I s (k) (cid:0) P s " I s (k)+ (cid:31)(cid:22) 2 (cid:18) I s I (cid:0) s ( 1 k (k ) ) (cid:0) 1 (cid:19) 2 I s (cid:0) 1 (k) #) : Again, the problem is scale-free and can be thought of as the one of a representative (cid:133)rm instead of a continuum. 2.4.3 Goods Market Clearing Condition The goods market clearing condition of the model economy is given by (cid:31)(cid:22) I I 2 (cid:28)(cid:22) Q K 2 Q K Y t =C t +I t + 2 (cid:18) t I (cid:0) t (cid:0) 1 t (cid:0) 1 (cid:19) I t (cid:0) 1 + 2 (cid:18) Q t t (cid:0) 1 t K +1 t (cid:5) t (cid:0) 1 (cid:19) t (cid:0) (cid:5) 1 t t: (29) 22

Note the absence of (cid:133)nancial (cid:135)ows related with equity issuance costs and government subsidy for equity issuance. This is due to our assumption that the dilution cost of equity issuance takes the form of discount sale, rather than an e¢ ciency loss for the economy (see the appendix in the working paper version for a detailed derivation for the goods market clearing condition). 3 Long Run E⁄ects of Capital Constraint In this section, we use some comparative statics and simulation exercises to analyze the e⁄ects of capital market frictions such as capital constraint and costlyequity(cid:133)nanceontheequilibriumreturnsandcapitalaccumulationinthe long run. In particular, considering the ongoing debate on the long-run e⁄ects of regulation on capital standards, we pay a special attention to the long-run e⁄ects of alternative levels of capital constraints. 3.1 Equilibrium Return Premium In this subsection, we show how the changes in model parameters related with the degree of funding-market frictions a⁄ect the return premium and capital accumulation. Tothatend, westartwiththesteady-stateversionoftheinvestmentEulerequationofthemodel. Inthesteadystate,theFOCforinvestment, equation (14), takes the form of 1+(cid:17)(cid:8)(s (cid:27)) m=(cid:12)+(1 m)RB = (cid:3) (cid:0) RF: (30) (cid:0) 1+(cid:17)(cid:8)(s ) (cid:1) (cid:20) (cid:3) (cid:21) To interpret the economic contents of the expression, it is useful to note that the stock market return of intermediary (28) is equalized to 1=(cid:12) in the steady state if and only if (cid:17) =’(cid:22)=(1 ’(cid:22))=0 since, with no aggregate uncertainty and (cid:0) ’(cid:22) = 0, RS = vB=(mH vB) = 1=(cid:12). Therefore, one can think of the left side of (cid:1) (30) as the frictionless weighted average cost of capital.15 Let us denote this by RW . We can then see immediately (cid:3) RW (cid:3) = 1+(cid:17)(cid:8)(s (cid:3) (cid:0) (cid:27)) = E i[(cid:21)(i)(cid:15)(i)] 1: (31) RF (cid:20) 1+(cid:17)(cid:8)(s (cid:3) ) (cid:21) Ei[(cid:21)(i)] (cid:20) From a mechanical standpoint, the inequality is due to the monotonicity of the standard normal distribution. From an economic standpoint, however, the inequality is the result of the negative correlation between the shadow value of internal funds and the idiosyncratic pro(cid:133)tability shock, which would not exist if the funding market were frictionless (’(cid:22) = 0). Facing costly recapitalization risk, the intermediaries adopt a cautionary stance before they enter commitments,reducingtheinvestmentlevelexante. Asaresult,giventhediminishing marginal productivity of capital, the return on assets RF =rK +1 (cid:14) cannot (cid:0) 15Note that in the frictionless economy, the marginal cost and the average cost of capital identicalsince the shadow value ofinternalfunds is always equalto one. 23

comedowntothelevelofthefrictionlesscostofcapitaldespitethecompetitive structure and free entry in the (cid:133)nancial industry. Obviously this is the direct resultofthecostlyequity(cid:133)nancingassumption. Wecallthebracketedtermthe intermediation wedge. The intermediation wedge plays an important role in the determination of excess returns of the risky asset. To see this, we can rewrite (30) as 1+(cid:17)(cid:8)(s ) RF R=RW (cid:3) R (32) (cid:3) (cid:0) (cid:1) 1+(cid:17)(cid:8)(s (cid:27)) (cid:0) (cid:20) (cid:3)(cid:0) (cid:21) Twothingsstandoutfromthisexpression. First, inourmodeleconomy, anequitypremiumcanexistinanon-stochasticsteadystate. Thepremiumarisesnot because of the covariance of asset return and the pricing kernel of the representativehousehold,butbecauseofthefrictionsinfundingmarketsforthe(cid:133)nancial intermediaries. In essence, the premium is closer to the liquidity premium in the LAPM (Holmstr(cid:246)m and Tirole (2001)) since it is the short-run funding risk associated with idiosyncratic return uncertainty that generates such a wedge. Second, in the special case of (cid:18) =0 (no deposits in the utility), RW =1=(cid:12) (cid:3) sinceRB =R. Inthisextremecase, theriskpremiumisentirelydeterminedby the intermediation wedge and is always strictly positive as long as idiosyncratic uncertainty exists. It is possible, though not plausible in a realistic calibration, that the premium can be negative. This happens when RW is too low relative (cid:3) to risk free rate R. For instance, if the non-pecuniary bene(cid:133)t of deposit is pathologically large, it is possible that the household is willing to make deposit at a negative net interest rate, i.e., RB < 1. In this case, the product of RW (cid:3) and the inverse of the intermediation wedge can be smaller than the rate of time preference, implying a negative premium. The same situation can also happen when the idiosyncratic uncertainty is su¢ ciently low. In this case, the intermediation wedge is close to 1 and the right side of (32) can be negative since RW 1=(cid:12). However, as will be shown below, such extreme cases do not (cid:3) (cid:20) happen with realistic calibrations. Anempiricallyrelevantquestionisifthemodelcangenerateasizableequity premium with a realistic calibration through the liquidity channel. Two crucial parametersarethedilutioncostparameter’(cid:22),alsoknownas(cid:135)oatationcost,and the amount of idiosyncratic uncertainty (cid:27). Figure 1 depicts the relationship between the idiosyncratic uncertainty and the equity premium for a range of parameter values for equity issuance cost. Evident are the positive relationship between the uncertainty and the return premium on one hand, and the positive relationship between the equity issuance cost and the return premium on the other hand. These are not surprising given the theoretical structure we adopt. What is interesting is the magnitude of return premium created by the capital market frictions. There exists a wide range of the dilution cost parameter in the literature. For instance, Gomes (2001) reports 0:08 of per-unit equity issuance cost. Recently Hennessy and Whited (2007), using simulated methods of moments, providesthestructuralestimatesofissuancecostfunction;theirestimatesofthe 24

total cost, including (cid:133)xed and variable costs, is somewhat higher than reported byGomes(2001). CooleyandQuadrini(2001)use0.30fortheiranalysis,which wetakeforourbaselinecase. Whilethiscalibrationisonthehighend,thislevel of dilution cost is appropriate for the analysis of the e⁄ects of liquidity policies designed to cope with crisis situation; moreover, the empirical analysis in Kiley andSim(2011)suggeststhisvaluehelps(cid:133)tthedataonmacroeconomic/(cid:133)nancial interactionsalongsomedimensions(seebelow). Forthevarianceofidiosyncratic shocks, or riskiness measure, of the (cid:133)nancial intermediaries, we choose (cid:27) =0:10 in annual frequency for our baseline calibration. Figure 1 shows that the model generates a sizable range of return premium at plausible levels of uncertainty as long as the dilution cost is greater 0.15. At ourbaselinecalibration, themodelcreatesareturnpremiumofabout300basis points. When the dilution cost is 0.20, the model generates a 200 basis points premium. Given our conservative calibration of the uncertainty level, one can seethatthemodelcanexplainasubstantialpartoftheequitypremiumthrough the capital market frictions facing (cid:133)nancial intermediaries. 3.2 Near Long Run Neutrality of Capital Constraint In policy circles, it is often emphasized that a higher minimum capital ratio may increase the weighted average cost of capital for (cid:133)nancial intermediaries in the long run. This is because the minimum capital regulation changes the mix of debt and equity so as to make (cid:133)nancial institutions more reliant on the costly equity funding (e.g., BIS (2010a) and BIS (2010b)). However, such a conclusion does not consider potential general equilibrium e⁄ects. With a higher level of capital, (cid:133)nancial intermediaries issue less amount of debts (or deposits)foragiven leveloflending. Forthis tohappen ingeneralequilibrium, the representative household should be discouraged from holding intermediary deposits,whichoccursthroughalowerreturnonsuchdepositsandhencealower borrowing rate for the intermediaries. This counteracts the partial equilibrium tendencyofthee⁄ectivecostofcapitalfortheintermediariestorisefromgreater reliance on equity.16 To analyze the general equilibrium e⁄ects in detail, we need to show how the deposit rate and return on assets are jointly determined in equilibrium. To that end, we derive a relationship between the deposit rate and return on asset that clears the goods market. We then combine this condition with the (cid:133)nancial market equilibrium condition given by (30) to (cid:133)nd the set of rates of returns that support equilibrium in both markets. By using the two FOCs of thehouseholdforconsumptionanddeposits,onecanderivealinearrelationship 16The general equilibrium e⁄ect in this analysis is reminiscent of, but economically quite di⁄erent from, the mechanism underlying the celebrated Miller-Modigliani theorem, where a shiftinthemixofdebtandequitychangestheriskassociatedwitheachtypeofliability. The intermediary deposits are default risk-free in ouranalysis. 25

between consumption and deposit, (1 a(cid:12))(1 (cid:12)RB) c = (cid:0) (cid:0) b (1 a)(cid:18) (cid:0) (1 a(cid:12))(1 (cid:12)RB)(1 m) = (cid:0) (cid:0) (cid:0) k (1 a)(cid:18) (cid:0) where the second equality is from the binding capital constraint. Here we use lower case letters for real quantities in the long run. From the rental market equilibrium condition, we have y RF (1 (cid:14)) = (cid:0) (cid:0) k 1 (cid:11) (cid:0) Substituting the two expressions in the resource constraint y=k = c=k+(cid:14), one can derive a linear relationship that RF and RB have to jointly satisfy, RF (1 (cid:14)) (1 a(cid:12))(1 (cid:12)RB)(1 m) (cid:0) (cid:0) = (cid:0) (cid:0) (cid:0) +(cid:14): (33) (1 (cid:11)) (1 a)(cid:18) (cid:0) (cid:0) We can then numerically solve the non-linear simultaneous equations system of (30) and (33) to determine RE and RB. Figure 2 shows the determination of two steady states, one with m = 0:10 and the other with m = 0:25. The red solid line shows the linear relationship between RF and RB that satisfy (33) (the real side of equilibrium) when m = 0:10.17 The blue solid line presents the locus of RF and RB that satisfy the (cid:133)nancial side of equilibrium, i.e., (30) when m = 0:10. Evident is that the locus has a steep upward slope: when the borrowing rate for the (cid:133)nancial intermediaries go up, the return on asset also has to go up to create an enough incentive for the intermediaries to invest. Point A is the intersection of the two loci, displaying the initial long run equilibrium when m=0:10. Inthe(cid:133)gure,weperformathoughtexperimentinwhichtheminimumcapital ratio is raised up to 0:25 and show how a new long-run equilibrium is determined. When the capital ratio goes up, the (cid:133)nancial locus shifts downward with a steeper slope between the borrowing rate and the asset return (the blue dotted line): with a much higher minimum capital ratio, the weighted average cost of capital goes up since the new regulation forces the (cid:133)nancial intermediaries to change a substantial portion of funding source from relatively cheap debts/deposits to more expensive equity. As a result, the investment in capital assets declines and the asset return goes up. The economy moves to point B. However, B cannot be an equilibrium because the real side of the equilibrium also changes. A new equilibrium requires the intermediaries to reduce the borrowing level substantially. The only way for the economy to achieve this is to discourage the households to hold intermediary debt/deposits: the borrowing 17The loci in the picture show the values of RB that satisfy the equilibrium conditions for 3,000 points RF between 1.040 and 1.065. We solve for these values using a nonlinear numericalroot (cid:133)nder. 26

rate for intermediaries (e.g., the deposit rate in our simple model) has to go down. The locus RF and RB that satisfy the real side of equilibrium shifts down to a new one (the red dotted line) and the new equilibrium is found at point C. Whetherthereturnonassetishigherorlowerthantheinitialvaluedepends on how sensitive this downward shift is. It turns out that the capital constraint in our model economy is near neutral: the shift in the red line almost perfectly o⁄sets the movement in the blue line. For instance when the economy moves from m = 0:10 to m = 0:25, the equilibrium asset returns drops less than 0:2 bps at an annual rate. This thought experiment shows how the capital constraint is nearly neutral for real outcomes in the long run; this echoes the conventional wisdom, due to "Modigliani-Miller" type e⁄ects, in related academic research (e.g. Admati etal.(2010)andHansonetal.(2011)). Thenear-neutralityofcapitalconstraint, however, does not mean that a transition from one equilibrium to another will be costless. The short-run transitional dynamics is a totally di⁄erent issue, to which we turn later in our analysis. 4 Policy Experiments Inthissection,weconsiderthee⁄ectsofvariousgovernmentpoliciestoimprove (cid:133)nancial stability. We start by assigning parameter values. We then consider the e¢ cacy of unconventional short-term credit policies designed to cope with extraordinary and exigent circumstances. Weclosethesectionbyanalyzingthe transitional dynamics of the economy moving from one steady state to another under the proposed higher capital standards. 4.1 Calibration There are three parameters that govern key aspects of the model(cid:146)s predictions for the macroeconomic e⁄ects of credit policies: the cost of equity issuance ’(cid:22), the standard deviation of return on asset (cid:27), and the weight on the deposit in the utility (cid:18). We try to adopt reasonable values for the (cid:133)rst two parameters by tying there values to data from (cid:133)nancial markets. As we mentioned earlier, we chose ’(cid:22) =0:30, following Cooley and Quadrini (2001) to replicate the harsh (cid:133)nancing environment seen during the recent (cid:133)nancial turmoil; this value also helps (cid:133)t the data on macro-(cid:133)nancial interactions, as shown below. Regarding the volatility, we set (cid:27) = 0:10 (in annual frequency) to match the standard deviationofreturnonasset(pro(cid:133)ts/totalasset)ofU.S.bankingsectorreported in Demirguc et al. (2003). With regard to the weight of deposits in the utility function ((cid:18)), we choose its value to match (roughly) the net interest margin of (cid:133)nancial intermediaries, RE RB. Saunders and Schumacher (2000) and (cid:0) Demirguc et al. (2003) provide an international comparison of such margins, which range from a low of 160 bps (Swiss) to a high of 500 bps (Spain and U.S.) on average during the period of 1988-1995. Conditioned upon ’(cid:22) = 0:30 27

and (cid:27) = 0:10, setting (cid:18) = 0:07 roughly matches the interest rate margin in the data. Note that the interest rate margin is a sum of two components, RE RB = RE R+R RB. With (cid:18) = 0:07, about half of the margin is (cid:0) (cid:0) (cid:0) explained by a return premium over risk free rate RE R and the rest of the (cid:0) margin is explained by the liquidity premium R RB in our framework. (cid:0) Withregardtootherparameters,wechoosetheinvestmentandbalancesheet adjustmentcostparametersandtheparametergoverninghabitpersistencesoas todeliverhump-shapedimpulsesresponsefunctiontotypicalshocks. Todeliver such slow dynamics for intermediaries(cid:146)balance sheet, we specify a small loan adjustmentcostbysetting(cid:13)(cid:22) equalto1. Thischoice,togetherwiththechoiceof investment adjustment cost parameter, helps us match the persistent response of lending. For the investment adjustment cost parameter, we set (cid:31)(cid:22) = 0:5, a moderate value similar to those reported in macroeconomic analyses (of other issues). We calibrate the habit persistence parameter as a = 0:75, a value in the typical range. For the parameters that can be considered traditional, we make standard choices whenever possible. The risk free rate in the steady state is set at R = 1=(cid:12) = 1:01 in quarterly frequency. The depreciation rate (cid:14) is set equal to 0:025. We assume a relatively elastic labor supply by setting the inverse of Frischelasticityparameter equalto0:1andwechoosetheweightofthelabor disutility as (cid:16) =1. We set (cid:11)=0:60, a fairly standard setting. As we have mentioned previously, this calibration helps (cid:133)t the data on macroeconomic/(cid:133)nancial interactions. Speci(cid:133)cally, Kiley and Sim (2011) show that this model (without the government sector developed in this research) can match the impulse responses of GDP, investment, and credit spreads following a shift in the level of uncertainty facing the (cid:133)nancial sector. For convenience, we reproduce these results in (cid:133)gure 3, which shows the impulse response to an uncertainty shock identi(cid:133)ed via a structural vector autoregression (along with 68-percent con(cid:133)dence intervals) and the model predictions (the dotted lines).18 Uncertainty increases by 10 percent (panel (a)).19. The borrowing spread rises notably (e.g., by about 20 basis points), indicating spillovers to (cid:133)nancial conditions more generally (panel (b)); Lending (panel (c)) jumps down. This shock has important macroeconomic e⁄ects: Hours, real investment, and real GDP decline notably (by about 1/3 percent, 1 1/4 percent, and 1/3 percent, respectively). Given this (cid:133)t to the data, the model is capable of quantitatively addressing the policy issues we consider. 4.2 Short-Run Stabilization Policies We now evaluate the e¢ cacy of the policies introduced above. To compare the two policies in an almost identical environment, we use the same parameteri- 18Kiley and Sim (2011) report more details and robustness checks. 19To modeltime-varying uncertainty,we assume the following process: log(cid:27)t=(1 (cid:0) (cid:26) (cid:27) )log(cid:27)(cid:22)+(cid:26) (cid:27) log(cid:27)t (cid:0) 1+ut, ut (cid:24) iid N(0;(cid:6)2): (34) We set (cid:26) =0:85;Kiley and Sim (2011) present more details (cid:27) 28

zation, (cid:26) = 0:85 and (cid:26) = 0:5. With this setting, a one time shock ((cid:15)G;K or g u t (cid:15)G;S) generates a peak in the government(cid:146)s asset market share after 3 quarters. t Thereafter,thegovernmentshareundergoesaveryslowrun-o⁄. Wesetthesizes of initial shocks so that the outlays equal about 2 percent of output, roughly matching13/4percentshareofGDPthatwasdevotedtobankrecapitalization under the Troubled Asset Relief Program (TARP). Figure 4 presents the economic e⁄ects of the two policies. The blue solid line shows the case of direct lending/asset purchase policy and black soliddotted line the case of capital injection. In panel (a), one can see that the twopolicyexperimentsarecalibratedsuchthatthesizesofgovernmentbalance sheet relative to aggregate output in the two cases are roughly the same: In both cases, the government balance sheet immediately jumps up to 2 percent level in the (cid:133)rst period, and continues to rise to reach the peak point of 2 3/4 percent in the third quarter. The government share starts running o⁄ very slowly thereafter and eventually reaches its steady state level, zero. Panel (b) shows the implied (period-by-period) outlays for each policy. In both cases, the maximum outlay of about 2 percent of output is reached in the (cid:133)rst period. While the government shares of assets continue to go up after the (cid:133)rst intervention period, a large chunk of public resources is required only for the (cid:133)rst period since the resources are being used to buy a stock of capital assets or intermediary shares. In fact, outlays drop signi(cid:133)cantly right after the (cid:133)rst period, and only the (cid:133)rst four periods are associated with positive outlays. Starting from the (cid:133)fth period, both policies generate a substantial amount of net pro(cid:133)ts and allow the government to disburse large amounts of money to the households as transfer. The potential payback to the households may be underestimatedinourexercisesbecauseweperformtheexperimentsaroundthe steadystatewhilesuchpoliciesarelikelytobeimplementedduringcriseswhen prices of asset are unusually low, which may not be well captured by a local approximation.20 In panel (c), one can see that the prices of capital assets immediately jump withthepoliciesinbothcases, althoughthepeaksizeandthedurationofprice boost is slightly greater for capital injection policy. Given our choice for a relativelysmalladjustmentfrictionininvestment,themagnitudeofassetprices responsesarenotbig. Nevertheless,suchanincreaseintheassetpricesimproves overall balance sheet conditions of the intermediaries in both cases. One can read this from the drop in the shadow value of internal funds, panel (d). However, panel (d) reveals a very di⁄erent picture about the e¢ cacy of the two policies in generating desired stabilization e⁄ect: The drop in the shadow value, which is the best summary measure of the liquidity/balance sheet conditions in our framework, is almost (cid:133)ve times greater for capital injection policy, suggestingthatthecapitalinjectionpolicymaybemuchmorepowerfulthanthe direct lending/asset purchase policy in handling liquidity/balance sheet crisis. 20The fact that the initial outlays of such policies overstate the long-run budget costs because of subsequent revenues from the purchased assets has been discussed in related policy discussions,e.g.,CBO (2011). 29

Panel (e) and (f) con(cid:133)rm this: In terms of peak response, the capital injection policyinduces5to6timesgreaterimpactsonaggregateinvestmentandoutput. What explains this di⁄erence? Panel (g) provides the answer: In contrast to capital injection policy, the directlending/assetpurchasepolicysu⁄ersfromaclassiccaseof(cid:145)crowdingout(cid:146) e⁄ect. To understand this point, it is useful to realize that the two policies act on asset markets di⁄erently. When the direct lending/asset purchase policy is executed, holding the market prices of assets constant, the policy shifts the supply of capital for private sector to the left, reducing the supply from K t+1 to (1 SS;G)K . As a consequence, asset prices go up while the private de- (cid:0) t+1 t+1 mand for assets decreases along the downward demand curve. While overall improvementinliquidityconditionandbusinessenvironmenthelpsthedemand recover, this is not enough to overcome the initial crowding out e⁄ect, as con- (cid:133)rmed by the large decline in private lending (investment) shown in panel (g). This explains why the size of the stimulative e⁄ect dies out so quickly. The capital injection policy works in a di⁄erent way. It improves the liquidity/balance sheet conditions of the intermediaries, which increases the risk appetiteforcapitalassetsassuggestedbythemassivedropintheshadowvalue in panel (d), making both the prices and the quantities of asset expand in the same direction. Roughly speaking, the vertical distance between the responses of private lending in panel (g) explains a large chunk of the di⁄erence in the e¢ cacy of two policies. More fundamentally, by tying the cash injection to the amount of equity (cid:133)nancing, the policy makes the (cid:133)rms reveal their liquidity conditions and allows the public resources to be directed to the right place (cid:150)directly at the location where the problem originates. In contrast, the direct asset purchase policy strengthens the balance sheets condition of all intermediaries, not only the cash strapped institutions, and cannot prevent the ones with large amount ofsurpluscash(cid:135)owfrompayingouttheextrapro(cid:133)tsasdividends(andperhaps bonuses in reality).21 Note that while the sizes of dividend payouts are similar inbothcases,thepayoutsinthecaseofassetpurchasepolicyarelesswarranted from the perspective of a policy maker given the lackluster fundamental of the economy (see panel (h)). It is also notable that the asset purchase policy is less successful in reducing the amount of costly equity (cid:133)nance (hence less dependenceonretainedearning)becauseitislesse⁄ectiveinimprovinginternalcash (cid:135)ow of intermediaries. 4.3 Transitional Dynamics to Higher Levels of Capitalization The current policy proposal from the Basel 3 process envisions a roughly 5 percentage point increase in the required ratio of common equity to risk-weighted 21This last aspect was not highlighted by Gertler and Kiyotaki (2010) because the intermediaries in their framework never pay out dividends (their problem is a terminal value maximization). 30

assets (from 2 percent to 7 percent) or in the ratio of tier 1 capital to riskweighted assets (from 4 to 8 percent). In the data, overall capital ratios have substantially exceeded these minima, both because regulators have de(cid:133)ned well capitalized as some notable margin above the minimum and because market pressures have led (cid:133)nancial institutions to maintain capital bu⁄ers. Consistent withvariouspolicyanalyses,weassumethattheincreaseintheregulatoryminimum is passed through to overall capital ratios and consider an increase in the overall ratio of capital to assets from 10 percent to 15 percent. We design two transition arrangements to compare two cases, fast vs slow transitions: one in which the capital ratio is raised by 5 percentage points approximately in 8 quarters and the other in which the capital ratio is raised by the same amount, but in 32 quarters. To achieve the transitional paths for theminimumcapitalratio, weassumethefollowingdatageneratingprocessfor the minimum capital ratio, logm =logm +(cid:14) ; t t 1 t (cid:0) (cid:14) =(cid:26)v +u : t t 1 t (cid:0) By setting an appropriate value (cid:26), one can control how fast the capital ratio reachesanewlongrunvalue. Inthecaseoffasttransition,weset(cid:26)=0:5while we (cid:26)=0:85 for the case of slow transition. Figure5displaysthetwotransitionalpathsforselectedendogenousvariables underthebaselinecalibration. Panel(a)showsthetwotransitionarrangements for the permanent increase in regulatory capital ratio. In both cases, (cid:133)nancial intermediariesfacesigni(cid:133)cantcapitalshortfalls,creatingfundingpressure,which leads to signi(cid:133)cantly higher costs of capital for the intermediaries as shown in panel(b). However,theslowertransitionisassociatedwithadisproportionately milderriseinthecostofcapital,roughlyonly1/4ofincreaseinthecostofcapital ascomparedwiththecaseinthefasttransition. Asaconsequence,thespillover e⁄ect on the general lending terms in other (cid:133)nancial markets, shown in panel (c), is much more mitigated: while the fast transition result in a maximum 300 bps increase in credit spreads, the credit spreads in the case of slow transition are maximized at around 20 bps. Inpanel(d),onecanseewhytheslowertransitionisassociatedwithsmaller (cid:133)nancial costs. The picture displays how much of the required increase in capital at each point in time is obtained by the costly equity issuance. In the faster transitioncase,theintermediarieshavetotapequitymarketmoreintensivelyas the funding needs far outstrip the available cash(cid:135)ows. Panel (f) highlights the same point from a di⁄erent angle: the faster transition is associated with much more aggressive contraction in lending. As both equity (cid:133)nance and decreasing lending are costly to the banks, intermediaries balance the two margins. The much higher funding costs, the loss of pro(cid:133)table lending opportunities, and further deterioration in cash (cid:135)ow owing to the ensuing overall economic downturn result in a massive drop in the price of intermediary shares in panel (e). In contrast,theslowertransitionallowsthebankstoearntheirwayoutbyrelying moreontheaccumulationofretainedearnings,allowingthemtoavoidthemuch 31

more costly (cid:133)nancing options and hence limiting the harmful e⁄ects on credit provision. Finally, aspredictedbythee⁄ectsonthecreditspreadsandlending, panel (e) (i) show that the faster implementation takes a much greater toll on (cid:24) economic activity: the declines in hours, lending, investment and output are about 2 to 3 times greater for the case of faster transition. 5 Conclusion Inthisresearch, weconsideratractablemacroeconomicmodelinwhichrealinvestment is intermediated through institutions that commit (cid:133)nancial resources amid idiosyncratic funding risk under a binding capital constraint. We show that the share of equity in the (cid:133)nancing base of intermediaries is neutral in the long run, but not in the short run, and that (cid:133)nancial frictions facing intermediaries imply a sizable equity premium for the aggregate economy. We then consider credit policies designed to address liquidity/balance sheet problems at intermediaries and show that a capital injection policy conditioned on voluntaryrecapitalizationisrelativelye¢ cientbecauseitdoesnotsu⁄erfroma (cid:147)crowding out(cid:148)e⁄ect on private investment. With regard to long-run policies, wedemonstratethatatransitiontohighercapitalrequirementscanhavesizable short-run e⁄ects on economic activity if not implemented carefully, and that a long transition period helps avoid such e⁄ects. 32

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1200 f =0.15 f =0.20 f =0.25 1000 f =0.30 f =0.35 800 s p b ,la u n 600 n A ,m u im e 400 rP n ru te R 200 0 200 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 volatility,s Figure 1: Uncertainty, Cost of Equity Issuance and Return Premium 36

1.035 1.03 1.025 )la u n n a ,B 1.02 A R B ( tis o p e D 1.015 n o C n ru te R 1.01 1.005 1 1.04 1.045 1.05 1.055 1.06 Return on Asset (RF, annual) Figure 2: Long-run Neutrality of Capital Constraint: m=0:10 vs m=0:25. 37

(a) uncertainty shock, % (c) lendding spreads, bps 15 30 10 20 5 10 0 0 5 10 0 5 10 15 0 5 10 15 (d) lending, % change at an annual rate (e) hours, % 0.5 0.2 0 0 0.5 0.2 1 0.4 1.5 0.6 2 0.8 0 5 10 15 0 5 10 15 (f) investment, % (g) output, % 0.5 0.2 0 0 0.5 1 0.2 1.5 0.4 2 2.5 0.6 3 0 5 10 15 0 5 10 15 Figure 3: Impact of Intermediation Shock in Model and Data (from identi(cid:133)ed VAR, see Kiley and Sim (2011)). 38

(a) government assets, % of GDP (b) tax(transfer), % of GDP (c) asset prices, % 3 3 0.15 2 2 0.1 1 1 0.05 0 0 0 1 1 0.05 0 10 20 30 40 0 10 20 30 40 0 10 20 30 40 (d) shadow value of internal funds, pp (e) investment, % (f) output, % 0.5 2 0.4 1.5 0.3 0 1 0.2 0.5 0.5 0.1 1 0 0 1.5 0.5 0.1 0 10 20 30 40 0 10 20 30 40 0 10 20 30 40 (g) lending, % (h) dividends payout, % (i) equity issuance, % 0.4 3 2 1 2 0.2 0 1 0 1 0 2 0.2 1 3 0.4 2 4 0 10 20 30 40 0 10 20 30 40 0 10 20 30 40 Figure4: E¢ cacyofPolicyIntervention: AssetPurchase(bluesolid)vsCapital Injection (black dash-dot). 39

(a) capital ratio, % (b) cost of capital, bps (c) spreads, bps 16 80 400 15 60 300 14 40 200 13 20 100 12 0 0 11 10 20 100 0 10 20 30 0 10 20 30 0 10 20 30 (d) recapitalization share, % (e) labor hours, % (f) lending, % 80 0.1 0.1 0 60 0 0.1 40 0.2 0.1 20 0.3 0.2 0.4 0 0.3 0.5 20 0.4 0 10 20 30 0 10 20 30 0 10 20 30 (g) consumption, % (h) investment, % (i) output, % 0.06 0.5 0.1 0 0.04 0 0.1 0.02 0.5 0.2 0 1 0.3 0.02 1.5 0.4 0.04 2 0.5 0 10 20 30 0 10 20 30 0 10 20 30 Figure5: TransitionalDynamicsofCapitalStandards: Fast(bluesolid)vsSlow (black dash-dot) Transition Arrangement. 40