Optimal Investment With Fixed Financing Costs

Optimal Investment With Fixed Financing Costs ∗ JasonG.Cummins DivisionofResearchandStatistics FederalReserveBoard jason.g.cummins@frb.gov IngmarNyman HunterCollegeandtheGraduateSchool TheCityUniversityof NewYork ingmar.nyman@hunter.cuny.edu September8,2004 Abstract Modelswithapremiumonexternalfinanceproducecounterfactualpredictionsabout liquidity management. We address this shortcoming by introducing a fixed cost of increasingexternalfinanceintoanotherwisestandardinvestment/financingproblem. This additional financial friction is well motivated by case studies and our analysis shows that it generates more realistic predictions about liquidity management: firms holdexternalfinanceandidlecashsimultaneously,andmayinvestanadditionaldollar ofcashflowinliquidityratherthanrepayingexternalfundsorinvestinginproductive capital. Inadditiontobetterfittingthestylizedfactsaboutthetime-seriesandcrosssectionalpatternofliquidityholding,theseresultsmayhelpshedlightonthefragility ofestimatesofinvestment-cashflowsensitivities. JELClassification: D21,E22,G31. Keywords: financialadjustmentcost;liquidityconstraints;corporatecashholdings. Wethankananonymousreferee,R.Gençay(theeditor),SteveBond,andseminarparticipantsattheCUNY ∗ GraduateCenter,theFederalReserveBoard,HunterCollege,andNYUforhelpfulcommentsandsuggestions. Nyman thanks DELTA and the Stockholm School of Economics for their generous hospitality. The views presentedaresolelythoseoftheauthorsanddonotnecessarilyrepresentthoseoftheBoardofGovernorsof theFederalReserveSystemoritsstaffmembers.

1 Introduction Conflicts of interest and asymmetric information make internal and external funds imperfectsubstitutes. Inmostmodels,theresultingfinancialfrictiontakestheformofanexternal finance premium. This setup has some intuitive implications, one of which is that the optimal capital stock is decreasing in the amount of external finance (see, for example, Hubbard 1998). But, at the same time, this setup also produces counterfactual predictions about liquidity management: a firm that has outstanding financial obligations cannot hold liquidresourcesorinvestadditionalcashflowinliquidity. We address this shortcoming by introducing a fixed cost of acquiring more external financeintoanotherwisestandardinvestment/financingproblem. Thisadditionalfinancial friction is motivated by evidence that corporate managers feel that the very act of dealing with external financiers is costly. For example, Donaldson’s (1984) case studies suggest that corporate managers prefer internal funds because they perceive that external finance carriessignificantnon-pecuniaryfixedcosts.1 Ouranalysisshowsthatafixedfinancingcostgeneratesmorerealisticpredictionsabout liquidity management. To avoid paying the fixed cost in the future, firms may hold external finance and idle cash even though the monetary marginal cost of external finance is greater than the marginal return on liquidity. Moreover, firms may invest an additional dollar of cash flow in liquidity rather than repaying external funds or investing in productive capital. In addition to better fitting the stylized facts about the time-series and cross-sectionalpatternofliquidityholding,theseresultsmayhelpshedlightonthefragility ofestimatesofinvestment-cashflowsensitivities(see,forexample,Cummins,Hassett,and Oliner2003,EricksonandWhited2000). 1AccordingtoDonaldson(1984),“Managersmistrustexternalsources[offinancing]thattheycouldneither predictnorcontrol.Inresponse,theymadefinancialself-sufficiencythecentraltenetintheirfinancialplanning sothatthetiming,magnitudeandformofstrategicallycriticalinvestmentremainedtheirstodecide.Inpractical terms,thismeansthattheyreliedoninternallygeneratedfunds(orretainedearnings).”(p.12) 1

2 Model Physical capital, K, is the sole factor of production. The price of investment is p while thepriceofoutputisnormalizedtounity. Revenuefromproductioninperiodt, Π(K ), is t strictly increasing and strictly concave and satisfies standard boundary conditions. There is a stochastic revenue shock, z , which is additive and independently and identically dist tributed with a density function f(z ). As a result, cash flow from operations at time t t is the sum of the shock z and revenue from production Π(K ). The additive uncertainty t t makes capital a risk-free asset and allows us to contrast the features of our model with a benchmarkinwhichtheshockhasnoeffectontheoptimalpolicy. Capitalevolvesaccording to the accounting identity: K = (1 δ)K + I , where δ is the constant rate of t+1 t t − economicdepreciation,andI isinvestment. Apartfromdepreciation,investmentincapital is completely reversible. Risk and irreversibility can generate inertia in investment, so to focus on the effect of financial frictions, we keep the investment technology free of those complications.2 The firm can invest its liquid financial resources, L, across periods and earn a rate of return of i. The discount factor is therefore β = 1/(1+i). Debt, D, can be dynamically managed,buttostreamlinethepresentationweassumethatequityistrappedinsidethefirm —specifically, dividendsareset equal tozerountilthefinal periodwhentheshareholders are paid the equity value of the firm. We adopt this simplified treatment of dividends for two reasons. First, our focus is on the trade-off between internal and external sources of financing rather than the on trade-off between different sources of external funds such as debt and equity. Second, corporate payout policy is a poorly understood area of finance. Intheabsenceofarobustandwidely-acceptedtheoreticalexplanationfordividendpayout decisions,wechoosetoimposeanassumptionthatfitswiththekeystylizedempiricalfact about dividends, namely that they are remarkably stable over time. To be sure, we force 2Worthnoting,however,isthatcapitalirreversibilitydistortsonlytheinvestmentpolicy;itcannotbyitself generateliquidityholdingsincerepayingexpensivedebtremainsastrictlypreferableuseofresources. 2

dividends to be zero, but our results would be qualitatively unaffected if we instead fixed thematastrictlypositivelevel. Themarginalcostofdebt,r(D ),isstrictlyincreasingandstrictlyconvexwithr(0) = t i.3 Thisfunctionisareducedformthatcapturesthepremiumcreditorschargethefirmdue to moral hazard or adverse selection problems and could be derived endogenously if we were to assume that shareholders enjoy limited liability. However, we assume no limited liability to economize on space and because little is sacrificed by taking such a shortcut. Inparticular,limitedliabilitywouldmodifyourconclusionsintwoways(apartfromintroducing standard credit risk due to default that would be priced in equilibrium through the r(D )function). First,ifcapitalwereriskythenlimitedliabilitywouldcreateincentivesfor t excessiverisk-takingandtheoptimalcapitalstockwouldbelargerinallcircumstances. But capitalisrisklessinourmodel,sothispointisoflittlerelevance. Second,ifdebtcouldbe differentiated in terms of its treatment in bankruptcy, then debt-overhang problems would make refinancing in future periods problematic. This situation mimics a very high fixed costofincreasingdebtfinancingandshouldthereforetendtoreinforceourconclusions. Inadditiontotheexternalfinancepremium,weassumethatthefirmfacesafixedcost, µ 0,whenitacquires,butnotwhenitretires,debt: ≥ 0 ifD D t+1 t m(D ,D )= ≤ t t+1    µ ifD t+1 >D t .   We think of the fixed cost as a simple way of capturing the idea that seeking external finance can exacerbate the conflict of interest between managers and outside financiers, as suggested by Donaldson’s (1984) evidence. But there are certainly other sources of fixed costs,suchasthesizableone-timefeeschargedbyinvestmentbanksfordebtorequityofferings(see,forexample,Smith1977,CalomirisandHimmelberg2002). Inthefinalanalysis, 3Astochasticmarginalcostofdebtwouldnot, byitself, generatecashholding. Aslongastheexpected costofexternalfinanceexceedsthereturntoidleliquidity(andthefirmfacesnootherfinancialfrictions),itis strictlypreferableforarisk-neutralfirmtorepayinthecurrentperiodasmuchdebtaspossibleand,ifneedbe, refinancetomorrow. 3

the key ingredient for our results is that additional external finance carries an adjustment costthatcannotberecoupedbyrepayment.4 Indeed,thistypeofasymmetrygeneratespredictions similar to ours in a model in which borrowing incurs a proportional adjustment cost(Kim,Mauer,andSherman1998). Atthebeginningofeachperiod,thefirminheritsthestatevariablesD ,K ,andL . The t t t firmservicesitsdebt,r(D )D ,andreceivesinterestonitsliquidity,iL . Productionandthe t t t stochasticshockgeneratecashflowfromoperations,Π(K )+z . Thesefinancialandreal t t activities plus the market value of the completely liquid surviving capital stock define the firm’snetstockofinternalfunds(cash),C (1+i)L r(D )D +Π(K )+z +p(1 δ)K . t t t t t t t ≡ − − Finally, at the end of the period the firm chooses next period’s stocks of debt and capital, subjecttothefixedcost,ifapplicable. Outgoingliquidityisaresidual,definedasthenetstockofinternalfundsplusdebtless thecostofinvestment, L =C +D D pK . (1) t+1 t t+1 t t+1 − − In period three, the firm is liquidated and shareholders receive the payoff, L . Otherwise, 3 thefirm’spayoffineachperiodisequaltooutgoingliquidity,L —theaccumulatedcash t+1 flowfromthefirm’sproduction,investment,andfinancingactivity—netofthefixedcost, m(D ,D ): t t+1 V(D ,K ,L )= (1+i)L +Π(K )+z p[K (1 δ)K ] t t t t t t t+1 t − − − (2) [1+r(D )]D +D m(D ,D ). t t t+1 t t+1 − − 4Aconstraintontheamountofexternalfundsthatthefirmcanraisewouldalsoencourageself-financing. Alongtheselines,Gross(1995)andAlmeida,Campello,andWeisbach(2004)showthatfirmshoardresources whentheyfacebankruptcy,astateinwhichtheycannotborrow. Quantityconstraintsofonetypeoranother surelyaffectsomefirms,andcan,infact,beseenasthespecialcaseofourframeworkinwhichthefixedcost isinfinitelylarge.Butthewidespreadavailabilityofdistressedfinancingforriskyprospectssuggeststousthat themoresalientfinancialfrictionsarecapturedinprices. 4

The firm’s objective is to maximize the present discounted value of current and expected future payoffs defined by equation (2). Using backward induction to solve this problem givesthefollowingper-perioddecisionproblem: max V(D ,K ,L )+βE V(D ,K ,L ) t t t t t+1 t+1 t+1 Dt+1,Kt+1 (3) { } subjectto L 0 and D 0. t+1 t+1 ≥ ≥ The first constraint forces investment to be less than or equal to the amount of available resources, which could include new debt. The second constraint forces debt to be nonnegative. 2.1 SolutionWithout aFixedCost Toestablishabenchmark, wefirst characterizetheoptimalfinancingandinvestment decisions in the absence of a fixed cost. To conserve on space, we sketch the intuition for our findingsandrefertheinterestedreadertoourworkingpaperforthecompleteproofsofthe propositions. Proposition 1: For any given history, D , K , and L , the financing and investment decit t t sionsofthefirmcanbecharacterizedintermsoftworegionsofitsinternalfunds,C . t 1. Wheninternalfunds,C ,aresufficientlysmall,thefirmreliesonexternalfinanceand t holds no liquidity (L = 0). The optimal capital stock and external finance are ∗t+1 givenjointlybythefirstconstraintinequation(3),whichholdswithequality,andthe followingmarginalcondition: Π(K )=p[r(D )+r (D )D +δ]. 0 t∗+1 t∗+1 0 t∗+1 t∗+1 2. Wheninternalfunds,C ,aresufficientlylarge,thefirmreliesexclusivelyoninternal t financingandhasnoexternalfinancialobligations(D = 0). Theoptimalcapital t∗+1 stockisimplicitlydefinedbyΠ(K ) =p(i+δ). Outgoingliquidity,L ,isgiven 0 t∗+1 ∗t+1 asaresidualbyequation(1). The threshold level of internal funds above which the firm can hold liquidity is strictly increasingintheamountofdebtthatitcarries,D. t Since the marginal cost of debt is greater than the return on internal funds, the firm always uses an additional dollar to retire debt rather than holding it as liquidity. Thus, the 5

firm never holds liquidity and debt simultaneously. In fact, liquidity arises as a residual manifestationof goodfortunerather thanasadeliberatedecision. Interms of its financial policy, the firm uses a pecking-order: relatively cheap internal funds are used until they areexhaustedandonlythenisrelativelyexpensive debt tapped. Intermsof itsinvestment policy, the firm acquires capital until its marginal revenue product (net of depreciation) is equal to the marginal cost of funds. Since debt is increasingly more expensive than internal funds, the optimal capital stock decreases with the amount of borrowing. Finally, in terms of investment-cash flow sensitivities, investment is unconstrained when the firm has liquidity and no debt. When the firm has debt, investment is financially constrained, displayingexcesssensitivitytoanextradollarofcashflowwiththatdollardividedbetween investmentandpayingdowndebt. 2.2 SolutionWithaFixedCost Financingchoicesinthebaselinemodelarestraightforwardbecauseinternalfundsarejust negative external finance with a lower marginal cost (and return). This changes when we introduceafixedcostofaddingdebt. In the third period, the firm makes no decisions because it is exogenously forced into liquidation. There is no limited liability, so the firm’s payoff in period three, whether it is positive or negative, is equal to the net stock of cash, C , minus debt, D . In the second 3 3 period, thefirmcan get newfinancingonlybyincurringthediscreteadjustment cost, µ > 0. After the firm has incurred the fixed cost in period two, however, there are no future financing decisions. Hence, if the firm acquires additional debt (i.e., if D > D ), then 3 2 the first part of proposition 1 describes its external financing and investment decisions. In particular,additionaldebtwouldonlybeusedtobuycapital. Whetherthefirmrefinancesinperiodtwoisdeterminedbythenetbenefitfromacquiring debt and investing it in physical capital (in which case D , K , and L = 0, where 3∗ 3∗ ∗3 optimal choices are denoted by an asterisk), rather than keeping debt at its current level 6

and settling for the capital stock that can be internally financed (in which case D = D , 3 2 K˜ = p 1C , and L = 0, where financially-constrained choices are denoted by a tilde). 3 − 2 3 e Thisnetbenefitcanbeexpressedasthedifferencebetweenthevaluefunctionevaluatedat e theoptimalchoicesandthefinancially-constrainedchoices: V (D ,K ,L )+βE [V (D ,K ,L )] 2 2 2 2 2 3∗ 3∗ ∗3 H ≡ V (D ,K ,L ) βE V D ,K ,L . 2 2 2 2 3 3 3 − − h ³ ´i e e e AssumingthatC p(1 δ)K ,whichmeansthatthefirmhassufficientinternalfunds 2 2 ≥ − − toallowit to leavetheperiod with non-negative liquidity(lest thefirmhaveno choicebut torefinance),thisconditioncanbeexpressedas =β [Π(K ) Π(K˜ )] p[K K˜ ](i+δ) 2 3∗ 3 3∗ 3 H { − − − }− β [r(D ) i][D D ]+[r(D ) r(D )]D +(1+i)µ . 3∗ 3∗ 2 3∗ 2 2 − { − − − } The two bracketed expressions capture the real andfinancial effects, respectively, of additional financing. Inthereal sector, thefirmenjoysrevenue fromalarger capital stock, but also pays the user cost for this investment. In the financial sector, the increased stock of debtispaidfor,thecostofcarryingtheexistingdebtincreases,andthe(compounded)fixed costisincurred. Inproposition2,weusethenetbenefitfunction toderiveathresholdlevelofinternal 2 H fundsthattriggersrefinancing. Proposition 2: Forany givenhistory, D , K , andL , thefinancing andinvestment deci- 2 2 2 sionsofthefirminperiodtwocanbecharacterizedintermsoftworegionsofitsperiod-two internalfunds,C ,definedbythethresholdCˆ . 2 2 1. WhenC <Cˆ ,thefirmincreasesexternalfinance(D >D )andinvestsoptimally 2 2 3 2 asdescribedinthefirstpartofproposition1. 2. WhenC Cˆ ,thefirmdoesnotincreaseexternalfinance(D =D ). IfC islarge 2 2 3 2 2 ≥ enough to internally finance the capital stock that is optimal with debt unchanged 7

at D , then the firm invests optimally as described in the first and second part of 2 proposition 1. Otherwise, there is under-investment relative to the optimal capital stockdescribedinthefirstpartofproposition1. In the first period, the firm’s decision problem is complicated by the specter of future costsofadditionalfinancing. However, theriskofincurringthesecostscanbereducedby accumulating resources inside the firm. Such behavior amounts to precautionary saving, whichcandistortbothfinancialandrealdecisions. In period one, the risk of facing the fixed cost in period two can be derived from the optimalfinancingpolicydescribedinproposition2: thecostwillbeincurredifandonlyif C <Cˆ . Wemapthisconditionintothespaceofrealizationsoftherevenueshock,calling 2 2 zˆ (D ,K ) the critical value of the shock that triggers refinancing. For all realizations of 2 2 2 z <zˆ (D ,K ),thefirmmustincurthefixedcost,sotheprobabilityofthateventisequal 2 2 2 2 to zˆ2(D2,K2) f(z )dz. 2 −∞ RInlemma1,wederivethemarginaleffectsoftheperiodonedecisionvariables,D and 2 K ,ontheexpectedfuturecostofadditionalfinancing.5 2 Lemma1: Defineψ Π(K˜ ) Π(K ),whereK˜ andK areevaluatedatz =zˆ . 0 3 0 3∗ 3 3∗ 2 2 ≡ − ∂E[m(D2,D 3∗ )] =µf(zˆ ) dzˆ2 = µf(zˆ ) (1+i) [r(D )+r (D )D ] pr 0 (D2)D2 ∂D2 2 dD2 − 2 ( − 2 0 2 2 − ψ ) · ¸ ∂E[m(D2,D 3∗ )] =µf(zˆ ) dzˆ2 = µf(zˆ ) Π(K ) p(1+i)+p(1 δ) . ∂K2 2 dK2 − 2 0 2 − − · ¸ ½ ¾ Anincreaseindebtinperiodone,D ,hasthreeeffectsonzˆ . First,theadditionaldebt 2 2 increasesthestockofcashinperiodtwobythefacevalueoftheaddedfinanceplusthereturnthatitearns. Thisenablesthefirmtoabsorbalessfavorableshock,thusdecreasingthe thresholdthat triggersrefinancing. Second, theadditional debt must beserviced, although not paid-in-full, which decreases the stock of cash and therefore increaseszˆ . Finally, the 2 5Byassumption, the firm’s decisions cannot affect the magnitude ofthefixed cost, µ, ortheprobability distributionoftherevenueshocks,f(z). 8

additional debt drives up the marginal cost of funds in period one. Because debt taken on earlier softens the blow from future refinancing costs, investment in period two is more attractiveandthisleadstoahigherzˆ . 2 An increase in the outgoing capital stock, K , has three effects on zˆ . First, an addi- 2 2 tional unit of capital generatesrevenue ofΠ(K )inperiodtwo which booststhestock of 0 2 cashanddecreaseszˆ . Second,buyingcapitalinperiodonedecreasesthestockofcashin 2 periodtwobyp(1+i), whichincreaseszˆ . Finally, an additional unit ofcapital inperiod 2 oneincreasestheliquidation valueof theexisting capital stockinperiod twoby p(1 δ). − Thismeansthat a larger capital stockcanbe internallyfinanced, whichmakes refinancing lessurgentanddecreaseszˆ . Combiningthelattertwoeffectsyieldsthestandardexpression 2 fortheusercostofinternally-financedcapital,p(i+δ). The heart of the firm’s decision problem in period one is the trade-off between direct effects in the current period and expected future financing costs. This comparison is captured by φ = µf(zˆ2) , which measures the relative importance of the future refinancing 1+µf(zˆ2) cost in the firm’s objective function. The parameter φ lies between 0 and 1 and is strictly increasinginµandinf(zˆ ). 2 Proposition 3 and its corollary summarize the firm’s unconstrained policy in period one. We can close the model like we did in proposition 2 by defining the net benefit from refinancing in period one — a function analogous to — and then show that there is a 2 H uniquethresholdofinternalfundsthattriggersrefinancing. Proposition 3: Suppose that the fixed cost in period one is irrelevant either because it is sunkorbecauseitwasnotincurred. Then,foranygivenhistory,D ,K ,andL ,financing 1 1 1 and investment decisions of the firm in period one can be characterized in terms of two regionsofitsperiod-oneinternalfunds,C ,definedbythethresholdC¯ . 1 1 1. WhenC C¯ ,thefirmreliesonexternalfinanceandholdsnoliquidity.Theoptimal 1 1 ≤ capital stock and debt are given jointly by thefirst constraint in equation (3), which holdswithequality,andthefollowingmarginalcondition: pr (D )D Π(K )=p r(D )+r (D )D +δ φ 1 0 2∗ 2∗ . 0 2∗ 2∗ 0 2∗ 2∗ − − ψ ( ) · ¸ 9

2. WhenC >C¯ ,thefirmholdsbothliquidityandexternalfinance.Theoptimalcapital 1 1 stockanddebtaregivenjointlybythefollowingtwomarginalconditions: pr(D )D Π(K ) =p(i+δ)and r(D )+r (D )D φ 1 0 2∗ 2∗ =i. 0 2∗ 2∗ 0 2∗ 2∗ − − ψ " # Outgoingliquidityisgivenasaresidualbyequation(1). Corollary1: C¯ isgivenbythefirstconstraintinequation(3)whenitholdswithequality. 1 C¯ isstrictlydecreasinginφandapproachesinfinityasφapproacheszero. WhenC C¯ , 1 1 1 ≤ the firm divides an additional dollar between capital investment and retirement of debt. When C > C¯ , neither investment nor financing is responsive to additional cash flow 1 1 becauseitischanneledinitsentiretyintoidleliquidity. Thekeyfeatureofthemodelisthatthefixedcostmodifiesthemarginalcostofdebtby φ 1 pr 0 (D2)D2 . Twoforcesareat workinsidethebracketsofthisterm: adollarthat is − ψ h i notusedtorepaydebtcanbeusedinthefuturetostaveoffrefinancingcosts;however,not repaying debt drives up its marginal cost in the current period. On balance, the benefit of retaining the dollar dominates, so that the marginal cost of debt falls by an amount that is scaledbytherelativeimportanceoftherefinancingcosts,φ. Giventheaddedbenefitfrom debt, internal finance is not always less expensive than external finance; instead, internal financeismoreexpensiveforlowlevelsofdebt. Asaconsequence,thefirmsholdsliquidity anddebt simultaneously whenC > C¯ . Thismodificationalsoupsetstheusual pecking- 1 1 order of funds: the firm uses debt until its marginal cost is equal to the marginal cost of internal finance, i, then relies on internal finance until it isexhausted and, finally, borrows more. The added benefit of debt affects the firm’s investment policy, too. When productive capitalisexternallyfinanced(C C¯ ),thefixedcostreducestheusercostofcapitaland 1 1 ≤ raises the optimal capital stock compared with the benchmark solution in the first part of proposition 1. Thus, the fixed-cost financial friction partly offsets the negative effect that the external finance financial friction has on the optimal capital stock. When productive 10

capital is internally financed (C > C¯ ), the optimal capital stock remains the same as in 1 1 thebenchmarkcaseinthesecondpartofproposition1becausethetrade-offbetweencapital andliquidityisunaffectedbytheintroductionofthefixedcost. Theempiricalpredictionsofourmodellineupnicelywiththestylizedfactsaboutcorporate cash holding and investment (see, for example, Almeida, Campello, and Weisbach 2004,Kim, Mauer, andSherman1998,Opler, Pinkowitz, Stulz, andWilliamson1999). In thecross-sectionaldimension,liquidassetstendtobeheldbyfirmswithstronggrowthopportunities, ascapturedbyahighmarket-to-bookratio. Likewiseinourmodel,firmswith relativelystronginvestmentopportunitieshavealargerdistortionterm(becauseψisbigger) and,therefore,areapttoholdidlecash.6 Intuitively,growthfirmsholdmorecashbecause it is especially costly for them to have their future investment hampered by a short-fall in financial resources. In the time-series dimension, studies have found that firms with cash holdings continue to accumulate cash flow. Likewise in our model, firms with idle cash add to their stockpile because it provides a buffer against adverse shocks that may occur. Lastly, a fixed-cost financial friction makes the relationship between investment and cash flowamisleadingindicatoroffinancialfrictions. Inparticular,investment(aswellasdebt) iscompletelyunresponsivetocashflowwhenC >C¯ ;instead,thefirmchannelsallofan 1 1 additionaldollarintoliquidity, aresult thatmayhelpexplainthefragilityof reduced-form estimates of investment-cash flow sensitivities. Our results also suggest that researchers should use caution when screening for liquidity constraints using firm characteristics. For example, one might think that a cash-rich firm would not face financial constraints; but in our model a large stock of cash may indicate that the firm anticipates facing financing constraintsinthefuture. 6Achangeininvestmentopportunitiescanbemodeledbymultiplyingtheproductionfunctionbyascaling parameter:αΠ(K).Anincreaseinαwidensthedifferencebetweenthemarginalrevenueproductsofanytwo levelsofcapital;italsoincreasestheoptimalcapitalstock. Bothoftheseeffectsincreaseψand,therefore,the distortionterm. 11

Appendix Theappendixcontainstheproofsofthepropositions. ProofofProposition1: Whenthereisnofixed-costfinancialfriction(µ=0)themaximizationproblemandthetwoconstraintsin(3)yieldthefollowingfirst-orderconditions: p+β[Π(K )+p(1 δ)] λ =0 (4) 0 t∗+1 1 − − − 1 β[1+r(D )+r (D )D ]+λ p 1+λ =0. (5) t∗+1 0 t∗+1 t∗+1 1 − 2 − When D > 0, λ = 0. In this case, we can solve for λ using the first-order t+1 2 1 conditionfordebt(equation(5)): λ =pβ[1+r(D )+r (D )D (1+i)] 1 t∗+1 0 t∗+1 t∗+1 − =pβ[r(D )+r (D )D i] >0. (6) t∗+1 0 t∗+1 t∗+1 − Sinceλ > 0,thefirstconstraintinequation(3)holdswithequality,whichmeansthatthe 1 firm holds no liquidity. K and D are determined jointly by the first constraint and t∗+1 t∗+1 the following marginal optimality condition, which is derived by substituting the solution forλ (equation(6))intoequation(4): 1 p+β[Π(K )+p(1 δ)] =pβ[r(D )+r (D )D i] 0 t∗+1 t∗+1 0 t∗+1 t∗+1 − − − ⇔ Π(K )=p[r(D )+r (D )D +δ]. 0 t∗+1 t∗+1 0 t∗+1 t∗+1 When D = 0, r(D ) = i and r (D )D = 0. In this case, we can solve for t+1 t+1 0 t+1 t+1 λ andλ usingthefirst-orderconditionfordebt(equation(5)): 1 2 λ p 1+λ =pβ[1+r(D )+r (D )D (1+i)] 1 − 2 t∗+1 0 t∗+1 t∗+1 − =pβ[(1+i) (1+i)] =0. − 12

Themultipliersmustbenon-negativesoλ = λ = 0. Sinceλ = 0,wecansolveforthe 1 2 1 optimalcapitalstockusingequation(4): p+β[Π(K )+p(1 δ)] =0 0 t∗+1 − − ⇔ Π(K ) =p(i+δ). 0 t∗+1 The firm’s liquidity holding is determined as a residual using the first constraint in equation(3). Finally, the firm is better off when it holds liquidity and no debt than the other way around. This follows from the fact that in the former case L and K are larger and ∗t+1 t∗+1 D issmaller,andfromV ()beingstrictlyincreasinginL andK andstrictlydecreasing t∗+1 t t · inD t . ¥ Proof of Proposition 2: Notice that the firm has a non-trivial refinancing decision to make only if it has enough internal funds to leave the period with non-negative liquidity. Otherwise, it is exogenously forced to refinance in order to escape financial distress, i.e., the first constraint in the maximization problem (3) is violated if K < 0 and D = D . 3 3 2 Therefore,wecanrestrictourselvestothecaseofC p(1 δ)K . Usingthefactthat 2 ≥− − e 2 allnewlyacquiredfundsareinvestedinphysicalcapital,i.e.,thatD D =p K K , 3∗ 2 3∗ 3 − − h i thenetbenefitcanberewrittenas e =β [Π(K ) Π(K˜ )] p[K K˜ ][r(D )+δ] 2 3∗ 3 3∗ 3 2 H { − − − } β [r(D ) r(D )]D +(1+i)µ . 3∗ 2 3∗ − { − } 13

DifferentiationwithrespecttoC yields. 2 ∂ ∂K ∂K˜ ∂K ∂K˜ ∂ H C 2 = β Π 0 (K 3∗ ) ∂C 3∗ − Π 0 (K˜ 3 ) ∂C 3 − p ∂C 3∗ − ∂C 3 [r(D 2 )+δ] 2 ( 2 2 2 2 · ¸ ∂D ∂D [r(D ) r(D )] 3∗ r (D )D 3∗ − 3∗ − 2 ∂C − 0 3∗ 3∗ ∂C 2 2) ∂K ∂K˜ = β Π(K ) p[r(D )+δ] 3∗ Π(K˜ ) p[r(D )+δ] 3 0 3∗ − 2 ∂C − 0 3 − 2 ∂C ( 2 2 ½ ¾ ½ ¾ ∂D [r(D )+r (D )D r(D )] 3∗ − 3∗ 0 3∗ 3∗ − 2 ∂C 2) ∂K ∂K˜ = β Π(K ) p[r(D )+δ] 3∗ Π(K˜ ) p[r(D )+δ] 3 0 3∗ − 2 ∂C − 0 3 − 2 ∂C ( 2 2 ½ ¾ ½ ¾ ∂K [r(D )+r (D )D r(D )] p 3∗ 1 − 3∗ 0 3∗ 3∗ − 2 ∂C − 2 ) µ ¶ ∂K = β Π(K ) p[r(D )+r (D )D +δ] 3∗ 0 3∗ − 3∗ 0 3∗ 3∗ ∂C ( 2 ½ ¾ p 1 Π(K˜ ) p[r(D )+r (D )D +δ] − 0 3 3∗ 0 3∗ 3∗ − − ) ½ ¾ = βp 1 Π(K˜ ) p[r(D )+r (D )D +δ] − 0 3 3∗ 0 3∗ 3∗ − − ) ½ = βp 1[Π(K˜ ) Π(K )] <0. − 0 3 0 3∗ − − NoticethatasK˜ approacheszero,Π(K˜ )approachesinfinityand ∂ 2 approachesnega- 3 0 3 ∂HC2 tiveinfinity. ThisimpliesthatasC approachesitslowerboundof p(1 δ)K , (C ) 2 2 2 2 − − H approachesinfinity. Moreover,forasufficientlylargeC ,K˜ = K andD = D ,which 2 3 3∗ 3∗ 2 impliesthat (C ) = µ < 0. ItnowfollowsfromtheIntermediateValueTheoremthat 2 2 H − thereexistsauniqueCˆ suchthat (Cˆ ) =0. 2 2 2 H Therearetwocasestoconsider: 1. C < Cˆ . In this case, the firm refinances. The fixed cost of refinancing is sunk so 2 2 thefirstpartofproposition1describestheoptimalbehaviorwhenD >0. 3∗ 14

2. C Cˆ . Inthiscase, theoptimalbehavior dependsontherealizationofC . IfC 2 2 2 2 ≥ is insufficient to finance the capital stock that is optimal when debt is equal to D , 2 then the firm invests as much as possible, holds no liquidity, and leaves debt at D . 2 Otherwise,asC increases,theoptimalbehaviorisdescribedinitiallybythefirstand 2 subsequentlybythesecondpartofproposition1. ¥ Proof of Lemma 1: Since both µ and the density function are independent of the decision variables in period one, Leibnitz’s rule implies that ∂E[m(D2,D 3∗ )] = µf(zˆ ) dzˆ2 . ∂D2 2 dD2 · ¸ ThesameargumentappliestothederivativewithrespecttoK . 2 Next, recall the derivation of ∂ 2 and that because all new financing is used to buy ∂HC2 capital,D D =p K K˜ . Differentiate withrespecttoz ,K ,andD . 3∗ 2 3∗ 3 2 2 2 2 − − H h i d ∂ ∂C ∂ d H z 2 = ∂ H C 2 ∂z 2 = ∂ H C 2 = − βp − 1[Π 0 (K˜ 3 ) − Π 0 (K 3∗ )] = − βp − 1ψ. 2 2 2 2 d ∂K ∂C ∂K˜ ∂C ∂K ∂C ∂K˜ ∂C d H D 2 = β Π 0 (K 3∗ ) ∂C 3∗ ∂D 2 − Π 0 (K˜ 3 ) ∂C 3 ∂D 2 − p ∂C 3∗ ∂D 2 − ∂C 3 ∂D 2 [r(D 2 )+δ] 2 ( 2 2 2 2 2 2 2 2 · ¸ ∂D ∂C p[K K˜ ]r (D ) [r(D ) r(D )] 3∗ 2 − 3∗ − 3 0 2 − 3∗ − 2 ∂C ∂D 2 2 ∂D ∂C r (D )D 3∗ 2 +r (D )D − 0 3∗ 3∗ ∂C ∂D 0 2 3∗ 2 2 ) ∂K ∂K˜ ∂K ∂K˜ = β Π(K ) 3∗ Π(K˜ ) 3 p 3∗ 3 [r(D )+δ] 0 3∗ ∂C − 0 3 ∂C − ∂C − ∂C 2 ( 2 2 2 2 · ¸ ∂D ∂D ∂C [r(D ) r(D )] 3∗ r (D )D 3∗ 2 − 3∗ − 2 ∂C − 0 3∗ 3∗ ∂C ∂D 2 2) 2 +βr (D ) D p[K K˜ ] 0 2 3∗ 3∗ 3 − − ½ ¾ ∂ ∂C 2 2 = H +βr 0 (D 2 )D 2 . ∂C ∂D 2 2 15

d ∂K ∂C ∂K˜ ∂C ∂D ∂C d H K 2 = β Π 0 (K 3∗ ) ∂C 3∗ ∂K 2 − Π 0 (K˜ 3 ) ∂C 3 ∂K 2 − [r(D 3∗ ) − r(D 2 )] ∂C 3∗ ∂K 2 2 ( 2 2 2 2 2 2 · ¸ ∂K ∂C ∂K˜ ∂C ∂D ∂C = p 3∗ 2 3 2 [r(D )+δ] r (D )D 3∗ 2 − ∂C ∂K − ∂C ∂K 2 − 0 3∗ 3∗ ∂C ∂K 2 2 2 2 2 2) ½ · ¸¾ ∂K ∂K˜ ∂D = β Π(K ) 3∗ Π(K˜ ) 3 [r(D ) r(D )] 3∗ 0 3∗ ∂C − 0 3 ∂C − 3∗ − 2 ∂C ( 2 2 2 · ¸ ∂K ∂K˜ ∂D ∂C p 3∗ 3 [r(D )+δ] r (D )D 3∗ 2 − ∂C − ∂C 2 − 0 3∗ 3∗ ∂C ∂K 2 2 2) 2 ½ · ¸¾ ∂ ∂C ∂ 2 2 2 = H = H Π 0 (K 2 ) p(1+i)+p(1 δ) ∂C ∂K ∂C − − 2 2 2 · ¸ ∂ 2 = H Π 0 (K 2 ) p(i+δ) . ∂C − 2 · ¸ WeusetheImplicitFunctionTheoremtocompletetheproof: dzˆ2 = d dHD2 2 = ∂ ∂HC2 2 ½ (1+i) − [r(D2)+r 0 (D2)D2] ¾ +βr 0 (D2)D2 dD2 −d dHz2 2 ¯z2=zˆ2 − ∂ ∂ H C2 2 ¯ ¯ = (1+i) [r(D )+r (D )D ] βr 0 (D2)D2 ¯ − ½ − 2 0 2 2 ¾ − − βp − 1[Π 0 (K˜ 3) − Π 0 (K 3∗ )] = (1+i) [r(D )+r (D )D ] pr 0 (D2)D2 . − − 2 0 2 2 − ψ ( ) dzˆ2 = d dHK2 2 = ∂ ∂HC2 2 ½ Π 0 (K2) − p(i+δ) ¾ dK2 −d dHz2 2 ¯z2=zˆ2 − ∂ ∂HC2 2 ¯ ¯ ¯ = Π(K ) p(i+δ) . 0 2 ¥ − − ½ ¾ 16

ProofofProposition3: Whenthereisafixed-costfinancialfriction(µ = 0),themaximizationproblemandthetwoconstraintsin(3)yieldthefollowingfirst-orderconditions: ∂E[m(D ,D )] p+β Π(K )+p(1 δ) 2∗ 3∗ λ =0 (7) − 0 2∗ − − ∂K − 1 ( 2∗ ) ∂E[m(D ,D )] 1 β 1+r(D )+r (D )D + 2∗ 3∗ +λ p 1+λ =0. (8) − 2∗ 0 2∗ 2∗ ∂D 1 − 2 ( 2∗ ) Therearethreecasestoconsider. WhenD = 0,λ 0andλ 0. Inthiscase, wecansolveforλ andλ usingthe 2 1 2 1 2 ≥ ≥ first-orderconditionfordebt(equation(8)): ∂E[m(D ,D )] p 1λ +λ =β r(D )+r (D )D i+ 2∗ 3∗ − 1 2 2∗ 0 2∗ 2∗ − ∂D ( 2∗ ) ∂E[m(D ,D )] =β 2∗ 3∗ ∂D 2∗ = βµf(zˆ ) <0. 2 − Thisisacontradiction,whichrulesoutthepossibilitythattheoptimalstockofdebtisequal tozero. When L > 0 and D > 0, λ = 0 and λ = 0. The optimal capital stock and debt 2 2 1 2 aregivenjointly bythetwo first-order conditions. Start with thefirst-ordercondition with respecttocapital(equation(7)): ∂E[m(D ,D )] β Π(K ) p(i+δ) 2∗ 3∗ =0 0 2∗ − − ∂K ⇔ ( 2∗ ) Π(K ) =p(i+δ) µf(zˆ ) Π(K ) p(i+δ) 0 2∗ 2 0 2∗ − − ⇔ ½ ¾ Π(K ) =p(i+δ) (9) 0 2∗ 17

Next,considerthefirstorderconditionwithrespecttodebt(equation(8)): ∂E[m(D ,D )] r(D )+r (D )D i+ 2∗ 3∗ =0 2∗ 0 2∗ 2∗ − ∂D 2∗ pr (D )D r(D )+r (D )D i µf(zˆ ) (1+i) [r(D )+r (D )D ] 0 2∗ 2∗ =0 2∗ 0 2∗ 2∗ − − 2 − 2∗ 0 2∗ 2∗ − ψ ( ) pr (D )D r(D )+r (D )D φ 1 0 2∗ 2∗ =i. 2∗ 0 2∗ 2∗ − − ψ " # The firm’s liquidity holding is determined as a residual using the first constraint in equation(3). WhenL = 0andD > 0,λ 0andλ = 0. K andD aredeterminedjointlyby 2 2 1 2 2∗ 2∗ ≥ the first constraint in equation (3) and the following marginal optimality condition, which isderivedbysubstitutingthesolutionforλ (equation(8))intoequation(7): 1 ∂E[m(D ,D )] β Π(K ) p(i+δ) 2∗ 3∗ 0 2∗ − − ∂K ( 2∗ ) ∂E[m(D ,D )] pβ r(D )+r (D )D i+ 2∗ 3∗ =0 − 2∗ 0 2∗ 2∗ − ∂D ( 2∗ ) ∂E[m(D ,D )] ∂E[m(D ,D )] Π(K )= p[r(D )+r (D )D +δ]+p 2∗ 3∗ + 2∗ 3∗ 0 2∗ 2∗ 0 2∗ 2∗ ∂D ∂K 2∗ 2∗ Π(K )= p[r(D )+r (D )D +δ] pµf(zˆ ) (1+i) [r(D )+r (D )D ] 0 2∗ 2∗ 0 2∗ 2∗ 2 2∗ 0 2∗ 2∗ − − ( pr (D )D 0 2∗ 2∗ +Π(K )p 1 (i+δ) − ψ 0 2∗ − − ) pr (D )D Π(K )= p[r(D )+r (D )D +δ] pφ 1 0 2∗ 2∗ . 0 2∗ 2∗ 0 2∗ 2∗ − − ψ " # ¥ 18

ProofofCorollary1: (1)C¯ isimplicitlydefinedbythefollowingequation: 1 pr (D (C¯ ))D (C¯ ) =r(D (C¯ ))+r (D (C¯ ))D (C¯ )+φ 0 2∗ 1 2∗ 1 (i+φ)=0. G 1 2∗ 1 0 2∗ 1 2∗ 1 ψ − " # Differentiate withrespecttoC andwithrespecttoφ: 1 1 G d ∂D ∂D φ ∂D ∂D dC G 1 =2r 0 (D 2∗ ) ∂C 2∗ +r 00 (D 2∗ )D 2∗ ∂C 2∗ +p ψ r 0 (D 2∗ ) ∂C 2∗ +r 00 (D 2∗ )D 2∗ ∂C 2∗ 1 1 1 ( 1 1) µ ¶ φ ∂D = r (D )+ 1+p r (D )+r (D )D 2∗ <0. 0 2∗ ψ 0 2∗ 00 2∗ 2∗ ∂C ( ) 1 · µ ¶¸· ¸ d pr (D )D G 1 = 0 2∗ 2∗ 1. dφ ψ − WeusetheImplicitFunctionTheoremtocompletetheproof: dC¯ 1 d dGφ 1 ∂ 1 ∂ 1 − 1 = = G G . dφ − d d CG 1 1 ¯ ¯C1=C¯ 1 ∂φ " − µ ∂C 1 ¶ # ¯ ¯ ¯ Thisderivativetakesthesamesignas ∂ 1 whichisnegativeatC¯ : ∂Gφ 1 pr (D )D p(i+δ)=Π(K (C¯ ))=p[r(D )+r (D )D +δ] pφ 1 0 2∗ 2∗ 0 1∗ 1 2∗ 0 2∗ 2∗ − − ψ ⇔ · ¸ pr (D )D φ 1 0 2∗ 2∗ =r(D )+r (D )D i>0 − ψ 2∗ 0 2∗ 2∗ − ⇒ · ¸ pr (D )D 0 2∗ 2∗ 1 <0. ψ − (2)Thisstatementfollowsdirectlyfromtheproofofproposition3. (3) This statement follows from the fact that optimal solutions for debt and capital are independentofC 1 whenC 1 >C¯ 1 . ¥ Closing themodel: To closethemodel, definethenet benefit of refinancing inperiod oneasthedifferenceinpayoffbetweentheunconstraineddecisiondescribedinproposition 3(denotedbyanasterisk)andthedecisionconstrainedbytheavailabilityofinternalfunds 19

(denotedbyatilde): =β Π(K ) Π(K˜ ) p[K K˜ ](i+δ) 1 2∗ 2 2∗ 2 H − − − ½ ¾ n o zˆ2(D1,K˜ 2) β [r(D ) i][D D ]+[r(D ) r(D )]D +(1+i)µ µ f(z)dz . 2∗ 2∗ 1 2∗ 1 1 − − − − − ( Z zˆ2(D 2∗ ,K 2∗ ) ) Again, keep in mind that because the firm is forced to avoid financial distress, refraining fromtakingonmoredebtisanoptiononlyifK˜ 0,sothedomainofthe isrestricted 2 1 ≥ H to C p(1 δ)K . Also, recall that all new financing must be used to buy capital, 1 1 ≥ − − makingD D =p K K˜ . 2∗ 1 2∗ 2 − − h i Differentiate withrespecttoC : 1 1 H ∂ ∂K ∂K˜ ∂K ∂K˜ ∂D ∂ H C 1 = β Π 0 (K 2∗ ) ∂C 2∗ − Π 0 (K˜ 2 ) ∂C 2 − p ∂C 2∗ − ∂C 2 (i+δ) − [r(D 2∗ ) − i] ∂C 2∗ 1 ( 1 1 1 1 1 · ¸ ∂D dzˆ ∂D dzˆ ∂K dzˆ ∂K˜ r (D )D 2∗ µf(zˆ ) 2 2∗ + 2 2∗ +µf(zˆ ) 2 2 − 0 2∗ 2∗ ∂C 1 − 2 · dD 2∗ ∂C 1 dK 2∗ ∂C 1 ¸ 2 · dK˜ 2 ∂C 1 ¸ ) ∂K ∂K˜ = β Π(K ) p(i+δ) 2∗ Π(K˜ ) p(i+δ) 2 0 2∗ − ∂C − 0 3 − ∂C ( 1 1 · ¸ · ¸ ∂K dzˆ ∂K [r(D )+r (D )D i] p 2∗ 1 µf(zˆ ) 2 p 2∗ 1 − 2∗ 0 2∗ 2∗ − ∂C − − 2 dD ∂C − µ 1 ¶ · 2∗µ 1 ¶¸ dzˆ ∂K dzˆ ∂K˜ µf(zˆ ) 2 2∗ +µf(zˆ ) 2 2 − 2 · dK 2∗ ∂C 1 ¸ 2 · dK˜ 2 ∂C 1 ¸ ) dzˆ dzˆ ∂K = β Π(K ) p[r(D )+r (D )D +δ] µf(zˆ ) 2 p+ 2 2∗ 0 2∗ − 2∗ 0 2∗ 2∗ − 2 dD dK ∂C ( ½ · 2∗ 2∗¸¾ 1 [Π(K˜ ) p(i+δ)]p 1 [r(D )+r (D )D i] 0 2 − 2∗ 0 2∗ 2∗ − − − − ½ dzˆ dzˆ µf(zˆ ) 2 + 2 p 1 . − 2 · dD 2∗ dK˜ 2 − ¸¾ ) 20

Proposition 3 establishes that the first bracketed expression following the last equality is equaltozero. Hence, ∂ dzˆ dzˆ ∂ H C 1 1 = − βp − 1 ( Π 0 (K˜ 2 ) − p[r(D 2∗ )+r 0 (D 2∗ )D 2∗ +δ] − µf(zˆ 2 ) · dD 2 2∗ p+ dK˜ 2 2¸ ) dzˆ dzˆ = βp 1 Π(K˜ ) p[r(D )+r (D )D +δ] µf(zˆ ) 2 p+ 2 − − 0 2 − 2∗ 0 2∗ 2∗ − 2 dD dK ( · 2∗ 2∗¸ ) dzˆ dzˆ βp 1 2 2 − − ( dK 2∗ − dK˜ 2) dzˆ dzˆ dzˆ dzˆ = βp 1[Π(K˜ ) Π(K )] βp 1 2 2 < βp 1 2 2 . − − 0 2 − 0 2∗ − − ( dK 2∗ − dK˜ 2) − − ( dK 2∗ − dK˜ 2) The bracketed expression on the right-hand side of the last inequality is strictly positive, making strictlydecreasinginC : 1 1 H dzˆ dzˆ 2 2 = Π(K ) p(i+δ) + Π(K˜ ) p(i+δ) dK − dK˜ − 0 2∗ − 0 2 − 2∗ 2 ½ ¾ ½ ¾ =Π(K˜ ) Π(K )>0. 0 2 0 2∗ − When C is sufficiently large, capital investment becomes unconstrained, i.e., K = K˜ , 1 2∗ 2 so that < 0. As K˜ approaches zero, ∂ 1 approaches negative infinity. This implies H 1 2 ∂HC1 that as C approaches its lower bound of p(1 δ)K , approaches infinity. It now 1 1 1 − − H follows that, just as in period two, there exists a unique threshold, Cˆ , that triggers refi- 1 nancing. When C < Cˆ , the firm increases external finance and invests as described in 1 1 thefirstpartofproposition3. WhenC Cˆ , thefirmdoesnotincreaseexternalfinance. 1 1 ≥ If C is large enough to internally finance the capital stock that is optimal with external 1 financeunchangedatD ,thenthefirminvestsasdescribedinthefirstandsecondpartsof 1 proposition 3. Otherwise, there is under-investment relative to the capital stock described inthefirstpartofproposition3. 21

Finally,Cˆ <C¯ . Toarguethatthisisthecase,noticethatifC C¯ ,thenK˜ =K . 1 1 1 1 2 2∗ ≥ This,inturn,impliesthat C¯ <0: 1 1 H ¡ ¢ zˆ2(D1,K˜ 2) = β [r(D ) i][D D ] [r(D ) r(D )]D (1+i)µ+µ f(z)dz 1 2∗ 2∗ 1 2∗ 1 1 H − − − − − − ( Z zˆ2(D 2∗ ,K 2∗ ) ) < β [r(D ) i][D D ]+[r(D ) r(D )]D +µi <0. 2∗ 2∗ 1 2∗ 1 1 − − − − ½ ¾ ¥ 22

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