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Journal of Financial Economics

Journal of Financial Economics 109 (2013) 466–492

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Collateral and capital structure$

Adriano A. Rampini n, S. ViswanathanDuke University, Fuqua School of Business, 100 Fuqua Drive, Durham, NC 27708, USA

a r t i c l e i n f o

Article history:Received 13 February 2012Received in revised form15 January 2013Accepted 16 January 2013Available online 15 March 2013

JEL classification:D24D92E22G31G32G35

Keywords:CollateralCapital structureRisk managementLeasingTangible assets

5X/$ - see front matter & 2013 Elsevier B.V.x.doi.org/10.1016/j.jfineco.2013.03.002

thank Michael Brennan, Francesca CorneFishman, Piero Gottardi, Dmitry Livdan, Ell

, Ilya Strebulaev, Tan Wang, Stan Zin, an anoparticipants at Duke University, the Federale Toulouse School of Economics, the Universirk University, Boston University, MIT, Virgington University, McGill, Indiana, Koc, Lausanthe 2009 Finance Summit, the 2009 NBEMeeting, the 2009 SED Annual Meetin

n Summer Symposium in Financial MarMeeting, the 2010 UBC Winter Financea Center Conference, the 2010 FIRS ConferenSociety World Congress, and the 2011 AFAcomments and Sophia Zhengzi Li for researcesponding author. Tel.: þ1 919 660 7797; faail address: rampini@duke.edu (A.A. Rampin

a b s t r a c t

We develop a dynamic model of investment, capital structure, leasing, and risk manage-ment based on firms' need to collateralize promises to pay with tangible assets. Bothfinancing and risk management involve promises to pay subject to collateral constraints.Leasing is strongly collateralized costly financing and permits greater leverage. Moreconstrained firms hedge less and lease more, both cross-sectionally and dynamically.Mature firms suffering adverse cash flow shocks may cut risk management and sell andlease back assets. Persistence of productivity reduces the benefits to hedging low cashflows and can lead firms not to hedge at all.

& 2013 Elsevier B.V. All rights reserved.

1. Introduction

We argue that collateral determines the capital struc-ture and develop a dynamic agency-based model of firm

All rights reserved.

lli, Andrea Eisfeldt,en McGrattan, Lukasnymous referee, andReserve Bank of Newty of Texas at Austin,ia, UCLA, Michigan,

ne, Yale, Amsterdam,R Corporate Financeg, the 2009 CEPR

kets, the 2010 AEAConference, the IDCce, the 2010 Econo-Annual Meeting forh assistance.x: þ1 919 660 8038.i).

financing based on the need to collateralize promises topay with tangible assets. We maintain that the enforce-ment of payments is a critical determinant of both firmfinancing and whether asset ownership resides with theuser or the financier, i.e., whether firms purchase or leaseassets. We study a dynamic neoclassical model of the firmin which financing is subject to collateral constraintsderived from limited enforcement and firms choosebetween purchasing and renting assets. Our theory ofoptimal investment, capital structure, leasing, and riskmanagement enables the first dynamic analysis of thefinancing vs. risk management trade-off and of firmfinancing when firms can rent capital.

In the frictionless neoclassical model, asset ownershipis indeterminate and firms are assumed to rent all capital.The recent dynamic agency models of firm financingignore the possibility that firms rent capital. Of course, africtionless rental market for capital would obviate finan-cial constraints. We explicitly consider firms' dynamic

A.A. Rampini, S. Viswanathan / Journal of Financial Economics 109 (2013) 466–492 467

lease vs. buy decision, modeling leasing as highly collater-alized albeit costly financing. When capital is leased, thefinancier retains ownership which facilitates repossessionand strengthens the collateralization of the financier'sclaim. Leasing is costly since the lessor incurs monitoringcosts to avoid agency problems due to the separation ofownership and control.

We provide a definition of the user cost of capital in ourmodel of investment with financial constraints that issimilar in spirit to Jorgenson's (1963) definition in thefrictionless neoclassical model. Our user cost of capital isthe sum of Jorgenson's user cost and a term whichcaptures the additional cost due to the scarcity of internalfunds. In our model, firms require both tangible andintangible capital, but the enforcement constraints implythat only tangible capital can be pledged as collateral andborrowed against, resulting in a premium on internalfunds; tangibility restricts external financing and henceleverage.

There is a fundamental connection between the opti-mal financing and risk management policy that has notbeen previously recognized. Both financing and risk man-agement involve promises to pay by the firm, leading to atrade-off when such promises are limited by collateralconstraints. Indeed, firms with sufficiently low net worthdo not engage in risk management at all because the needto finance investment overrides the hedging concerns. Thisresult is in contrast to the extant theory, such as Froot,Scharfstein, and Stein (1993), and is consistent with theevidence that more constrained firms hedge less providedby Rampini, Sufi, and Viswanathan (2012), and the litera-ture cited therein, and with the strong positive relationbetween hedging and firm size in the data.

With constant investment opportunities, risk manage-ment depends only on firms' net worth and incompletehedging is optimal, i.e., firms do not hedge to the pointwhere the marginal value of net worth is equated acrossall states. In fact, firms abstain from risk management withpositive probability under the stationary distribution.Thus, even mature firms that suffer a sequence of adversecash flow shocks may see their net worth decline to thepoint where they find it optimal to discontinue riskmanagement. We moreover characterize the comparativestatics of firms' investment, financing, risk management,and dividend policy with respect to other key parametersof the model. Firms subject to higher risk can choose tohedge more and reduce investment due to the financingneeds for risk management. Firms with more collateraliz-able or tangible assets can lever more and increaseinvestment, while at the same time raising corporate riskmanagement to counterbalance the increase in the volati-lity of net worth that higher leverage would otherwiseimply. Firms with more curvature in their productionfunction operate at smaller scale and may hence hedgeless, not more as one might expect. Thus, our model hasinteresting empirical implications for firm financing andrisk management both in the cross section and the timeseries.

With stochastic investment opportunities, risk manage-ment depends not only on firms' net worth but also ontheir productivity. If productivity is persistent, the overall

level of risk management is reduced, because cash flowsand investment opportunities are positively correlated dueto the positive correlation between current productivityand future expected productivity. There is less benefit tohedging low cash flow states. Moreover, risk managementis lower when current productivity is high, as higherexpected productivity implies higher investment andraises the opportunity cost of risk management. Withsufficient but empirically plausible levels of persistence,the firm abstains from risk management altogether, pro-viding an additional reason why risk management is solimited in practice. Furthermore, when the persistence ofproductivity is strong, firms hedge investment opportu-nities, i.e., states with high productivity, as the financingneeds for increased investment rise more than cash flows.

Leasing tangible assets requires less net worth per unitof capital and hence allows firms to borrow more. Finan-cially constrained firms, i.e., firms with low net worth,lease capital because they value the higher debt capacity;indeed, severely constrained firms lease all their tangiblecapital. Over time, as firms accumulate net worth, theygrow in size and start to buy capital. Thus, the modelpredicts that small firms and young firms lease capital. Weshow that the ability to lease capital enables firms to growfaster. Dynamically, mature firms that are hit by asequence of low cash flows may sell assets and lease themback, i.e., sale-leaseback transactions may occur under thestationary distribution. Moreover, leasing has interestingimplications for risk management: leasing enables highimplicit leverage; this may lead firms to engage in riskmanagement to reduce the volatility of net worth thatsuch high leverage would otherwise imply.

In the data, we show that tangible assets are a keydeterminant of firm leverage. Leverage varies by a factor 3from the lowest to the highest tangibility quartile forCompustat firms. Moreover, tangible assets are an impor-tant explanation for the “low leverage puzzle” in the sensethat firms with low leverage are largely firms with fewtangible assets. We also take firms' ability to deploytangible assets by renting or leasing such assets intoaccount. We show that accounting for leased assets inthe measurement of leverage and tangibility reduces thefraction of low leverage firms drastically and that firmswith low lease adjusted leverage are firms with low leaseadjusted tangible assets. Finally, we show that accountingfor leased capital has a striking effect on the relationbetween leverage and size in the cross section of Compu-stat firms. This relation is essentially flat when leasedcapital is taken into account. In contrast, when leasedcapital is ignored, as is done in the literature, leverageincreases in size, i.e., small firms seem less levered thanlarge firms. Thus, basic stylized facts about the capitalstructure need to be revisited. Importantly, the leaseadjustments to the capital structure we propose basedon our theory are common in practice, and accounting rulechanges are currently being considered by the US andinternational accounting boards that would result in theimplementation of lease adjustments similar to oursthroughout financial accounting. Our model and empiricalevidence together suggest a collateral view of the capitalstructure.

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Our paper is part of a recent and growing literaturewhich considers dynamic incentive problems as the maindeterminant of the capital structure. The incentive pro-blem in our model is limited enforcement of claims. Mostclosely related to our work are Albuquerque andHopenhayn (2004), Lorenzoni and Walentin (2007), andRampini and Viswanathan (2010). Albuquerque andHopenhayn (2004) study dynamic firm financing withlimited enforcement. In their setting, the value of defaultis exogenous, albeit with fairly general properties, whereasin our model the value of default is endogenous as firmsare not excluded from markets following default. More-over, they do not consider the standard neoclassicalinvestment problem.1 Lorenzoni and Walentin (2007)consider limits on enforcement similar to ours in a modelwith constant returns to scale and adjustment costs onaggregate investment which implies that all firms areequally constrained at any given time. However, theyassume that all enforcement constraints always bind,which is not the case in our model, and focus on therelation between investment and Tobin's q rather than thecapital structure. Rampini and Viswanathan (2010) con-sider a two-period model in a similar setting with hetero-geneity in firm productivities and focus on thedistributional implications of limited risk management.While they consider the comparative statics with respectto exogenously given initial net worth, net worth isendogenously determined in our fully dynamic model,and our model moreover allows the analysis of thedynamics of risk management, risk management bymature firms in the long run, and the effect of persistenceon the extent of risk management. Rampini, Sufi, andViswanathan (2012) extend the model in this paper toanalyze the hedging of a stochastic input price, e.g.,airlines' risk management of fuel prices.

The rationale for risk management in our model isrelated to the one in Froot, Scharfstein, and Stein (1993)who show that firms subject to financial constraints areeffectively risk averse and hence engage in risk manage-ment. Holmström and Tirole (2000) recognize that finan-cial constraints may limit ex ante risk management. Melloand Parsons (2000) also argue that financial constraintscan restrict hedging. None of these papers provide adynamic analysis of the financing vs. risk managementtrade-off.2 A literature, e.g., Leland (1998), argues that“[h]edging permits greater leverage” by allowing firms toavoid bankruptcy costs when they are financed withnoncontingent debt. To the extent that low net worthfirms have high leverage and a high probability of bank-ruptcy, this implies a negative relation between net worth

1 The aggregate implications of firm financing with limited enforce-ment are studied by Cooley, Marimon, and Quadrini (2004) and Jermannand Quadrini (2007). Schmid (2008) considers the quantitative implica-tions for the dynamics of firm financing.

2 Bolton, Chen, and Wang (2011) study risk management in aneoclassical model of corporate cash management; in contrast to ourtheory, they find that the hedge ratio decreases in firms' cash-to-capitalratio, and low cash firms hedge as much as possible while high cash firmsdo not hedge at all. The cost of hedging they consider is an inconvenienceyield of posting cash to meet an exogenous margin requirement ratherthan the financing risk management trade-off we emphasize.

and risk management, which is the opposite of ourprediction and the relation in the data.

Capital structure and investment dynamics determinedby incentive problems due to private information aboutcash flows or moral hazard are studied by Quadrini (2004),Clementi and Hopenhayn (2006), DeMarzo and Fishman(2007a), Biais, Mariotti, Rochet, and Villeneuve (2010), andDeMarzo, Fishman, He, and Wang (2012). Capital structuredynamics subject to similar incentive problems butabstracting from investment decisions are analyzed byDeMarzo and Fishman (2007b), DeMarzo and Sannikov(2006), and Biais, Mariotti, Plantin, and Rochet (2007).3 Inthese models, collateral plays no role.

The role of secured debt is also considered in aliterature which takes the form of debt and equity claimsas given. Stulz and Johnson (1985) argue that secured debtcan facilitate follow-on investment and thus ameliorate aMyers (1977) type underinvestment problem in the pre-sence of debt overhang.4 Morellec (2001) shows thatsecured debt prevents equityholders from liquidatingassets to appropriate value from debtholders. Using anincomplete contracting approach, Bolton and Oehmke(2012) analyze the optimal priority structure betweenderivatives used for risk management and debt.

Several papers study the capital structure implicationsof agency conflicts due to managers' private benefits.Zwiebel (1996) argues that managers voluntarily choosedebt to credibly constrain their own future empire-building in a model with incomplete contracts. Morellec,Nikolov, and Schürhoff (2012) study agency conflicts in aLeland (1998) type model in which managers divert a partof cash flows as private benefits leading them to lever less.

Moreover, none of these models consider intangiblecapital or the option to lease capital. An exception isEisfeldt and Rampini (2009) who argue that leasingamounts to a particularly strong form of collateralizationdue to the relative ease with which leased capital can berepossessed, albeit in a static model. We are the first, tothe best of our knowledge, to consider the role of leasedcapital in a dynamic model of firm financing and provide adynamic theory of sale-and-leaseback transactions.

In Section 2 we report some stylized empirical factsabout collateralized financing, tangibility, and leverage,taking leased capital into account. Section 3 describesthe model, defines the user cost of tangible, intangible,and leased capital, and characterizes the payout policy.Section 4 analyzes optimal risk management and providescomparative statics with respect to key parameters.Section 5 characterizes the optimal leasing and capitalstructure policy and Section 6 concludes. All proofs are inAppendix A.

3 Relatedly, Gromb (1995) analyzes a multi-period version of Boltonand Scharfstein's (1990) two-period dynamic firm financing problemwith privately observed cash flows. Gertler (1992) considers the aggre-gate implications of a multi-period firm financing problemwith privatelyobserved cash flows. Atkeson and Cole (2008) consider a two-period firmfinancing problem with costly monitoring of cash flows.

4 Hackbarth and Mauer (2012) provide a Leland (1994) type modelwhere priority rules mitigate this underinvestment problem.

Table 1Tangible assets and liabilities.

This table reports balance sheet data on tangible assets and liabilities fromthe Flow of Funds Accounts of the United States for the ten years from 1999to 2008 [Federal Reserve Statistical Release Z.1], Tables B.100, B.102, B.103,and L.229. Panel A measures liabilities two ways. Debt is Credit MarketInstruments which for (nonfinancial) businesses are primarily corporatebonds, other loans, and mortgages and for households are primarily homemortgages and consumer credit. Total liabilities are Liabilities, which, inaddition to debt as defined before, include for (nonfinancial) businessesprimarily miscellaneous liabilities and trade payables and for householdsprimarily the trade payables (of nonprofit organizations) and security credit.For (nonfinancial) businesses, we subtract Foreign Direct Investment in theU.S. from Table L.229 from reported miscellaneous liabilities as Table F.229suggests that these claims are largely equity. For households, real estate ismortgage debt divided by the value of real estate, and consumer durables isconsumer credit divided by the value of consumer durables. Panel B reportsthe total tangible assets of households and noncorporate and corporatebusinesses relative to the total net worth of households. The main types oftangible assets, real estate, consumer durables, equipment and software, andinventories are also separately aggregated across the three sectors.

Panel A: Liabilities (% of tangible assets)

Sector Debt Total liabilities

(Nonfinancial) corporate businesses 48.5% 83.0%

(Nonfinancial) noncorporate businesses 37.8% 54.9%

Households and nonprofit organizationsTotal tangible assets 45.2% 47.1%Real estate 41.2%Consumer durables 56.1%

Panel B: Tangible assets (% of household net worth)

Assets by type Tangible assets

Total tangible assets 79.2%Real estate 60.2%Equipment and software 8.3%Consumer durables 7.6%Inventories 3.1%

A.A. Rampini, S. Viswanathan / Journal of Financial Economics 109 (2013) 466–492 469

2. Stylized facts

This section provides some aggregate and cross-sectional evidence that highlights the first-order impor-tance of tangible assets as a determinant of the capitalstructure in the data. We first take an aggregate perspec-tive and then document the relation between tangibleassets and leverage across firms. We take leased capitalinto account explicitly and show that it has quantitativelyand qualitatively large effects on basic stylized facts aboutthe capital structure, such as the relation between leverageand size. Tangibility also turns out to be one of the fewrobust factors explaining firm leverage in the extensiveempirical literature on capital structure, but we do notattempt to summarize this literature here.

2.1. Collateralized financing: the aggregate perspective

From the aggregate point of view, the importance oftangible assets is striking. Consider the balance sheet datafrom the Flow of Funds Accounts of the U.S. for (non-financial) corporate businesses, (nonfinancial) noncorpo-rate businesses, and households reported in Table 1 for theyears 1999–2008 (detailed definitions of variables are in

the caption of the table). For businesses, tangible assetsinclude real estate, equipment and software, and inven-tories, and for households mainly real estate and consumerdurables.

Panel A documents that from an aggregate perspective,the liabilities of corporate and noncorporate businessesand households are less than their tangible assets andindeed typically considerably less, and in this sense allliabilities are collateralized. For corporate businesses, debtin terms of credit market instruments is 48.5% of tangibleassets. Even total liabilities, which include also miscella-neous liabilities and trade payables, are only 83.0% oftangible assets. For noncorporate businesses and house-holds, liabilities vary between 37.8% and 54.9% of tangibleassets and are remarkably similar for the two sectors.

Note that we do not consider whether liabilities areexplicitly collateralized or only implicitly in the sense thatfirms have tangible assets exceeding their liabilities. Ourreasoning is that even if liabilities are not explicitlycollateralized, they are implicitly collateralized sincerestrictions on further investment, asset sales, and addi-tional borrowing through covenants and the ability not torefinance debt allow lenders to effectively limit borrowingto the value of collateral in the form of tangible assets.That said, households' liabilities are largely explicitlycollateralized. Households' mortgages, which make upthe bulk of households' liabilities, account for 41.2% ofthe value of real estate, while consumer credit amounts to56.1% of the value of households' consumer durables.

Finally, aggregating across all balance sheets and ignor-ing the rest of the world implies that tangible assets makeup 79.2% of the net worth of U.S. households, with realestate making up 60.2%, equipment and software 8.3%, andconsumer durables 7.6% (see Panel B). While this providesa coarse picture of collateral, it highlights the quantitativeimportance of tangible assets as well as the relationbetween tangible assets and liabilities in the aggregate.

2.2. Tangibility and leverage

To document the relation between tangibility andleverage, we analyze data for a cross section of Compustatfirms. We sort firms into quartiles by tangibility measuredas the value of property, plant, and equipment divided bythe market value of assets. The results are reported inPanel A of Table 2, which also provides a detailed descrip-tion of the construction of the variables. We measureleverage as long-term debt to the market value of assets.

The first observation that we want to stress is thatacross tangibility quartiles, (median) leverage varies from7.4% for low tangibility firms (i.e., firms in the lowestquartile) to 22.6% for high tangibility firms (i.e., firms inthe highest quartile), i.e., by a factor 3. Tangibility alsovaries substantially across quartiles; the cut-off value forthe first quartile is 6.3% and for the fourth quartile is 32.2%.

To assess the role of tangibility as an explanation forthe observation that some firms have very low leverage(the so-called “low leverage puzzle”), we compute thefraction of firms in each tangibility quartile which havelow leverage, specifically leverage less than 10%. Thefraction of firms with low leverage decreases from 58.3%

Table 2Tangible assets and debt, rental, and lease adjusted leverage.

Panel A displays the relation between tangibility and (debt) leverage and Panel B displays the relation between tangibility and leverage adjusted forrented assets. Annual firm-level Compustat data for 2007 are used excluding financial firms. Panel A: Tangibility: Property, Plant, and Equipment-Total (Net)(Item #8) divided by Assets; Assets: Assets-Total (Item #6) plus Price-Close (Item #24) times Common Shares Outstanding (Item #25) minus Common Equity-Total (Item #60) minus Deferred Taxes (Item #74); Leverage: Long-Term Debt-Total (Item #9) divided by Assets. Panel B: Lease Adjusted Tangibility: Property,Plant, and Equipment-Total (Net) plus 10 times Rental Expense (Item #47) divided by Lease Adjusted Assets; Lease Adjusted Assets: Assets (as above) plus 10times Rental Expense; Debt Leverage: Long-Term Debt-Total divided by Lease Adjusted Assets; Rental Leverage: 10 times Rental Expense divided by LeaseAdjusted Assets; Lease Adjusted Leverage: Debt Leverage plus Rental Leverage.

Panel A: Tangible assets and debt leverage

Tangibility Quartile Leverage (%) Low leverage firms (%)

quartile cutoff Med. Mean (leverage ≤10%)(%)

1 6.3 7.4 10.8 58.32 14.3 9.8 14.0 50.43 32.2 12.4 15.5 40.64 n.a. 22.6 24.2 14.9

Panel B: Tangible assets and debt, rental, and lease adjusted leverage

Lease Quartile Leverage (%) Low leverage firms (%)adjusted cutoff (leverage ≤10%)

Tangibility (%) Debt Rental Lease adj. Debt Rental Lease

Quartile Med. Mean Med. Mean Med. Mean adj.

1 13.2 6.5 10.4 3.7 4.2 11.4 14.6 61.7 97.7 46.02 24.1 9.8 12.9 6.9 8.1 18.4 21.0 50.1 68.2 16.13 40.1 13.1 14.8 8.0 10.5 24.2 25.3 41.7 60.6 12.04 n.a. 18.4 20.4 7.2 13.8 32.3 34.2 24.4 57.3 3.7

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in the low tangibility quartile to 14.9% in the hightangibility quartile. Thus, low leverage firms are largelyfirms with relatively few tangible assets.

2.3. Leased capital and leverage

Thus far, we have ignored leased capital which is theconventional approach in the literature. To account forleased (or rented) capital, we simply capitalize the rentalexpense (Compustat item #47). This allows us to imputecapital deployed via operating leases, which are the bulk ofleasing in practice.5 To capitalize the rental expense, recallthat Jorgenson's (1963) user cost of capital is u≡rþδ, i.e.,the user cost is the sum of the interest cost and thedepreciation rate. Thus, the frictionless rental expense foran amount of capital k is

Rent¼ ðrþδÞk:

Given data on rental payments, we can hence infer theamount of capital rented by capitalizing the rental expenseusing the factor 1=ðrþδÞ. For simplicity, we capitalize therental expense by a factor 10. We adjust firms' assets,tangible assets, and liabilities by adding 10 times rental

5 Note that capital leases are already accounted for as they arecapitalized on the balance sheet for accounting purposes. For a descrip-tion of the specifics of leasing in terms of the law, accounting, andtaxation, see Eisfeldt and Rampini (2009) and the references citedtherein.

expense to obtain measures of lease adjusted assets, leaseadjusted tangible assets, and lease adjusted leverage.6

We proceed as before and sort firms into quartiles bylease adjusted tangibility. The results are reported in PanelB of Table 2. Lease adjusted debt leverage is somewhatlower as we divide by lease adjusted assets here. There is astrong relation between lease adjusted tangibility andlease adjusted leverage (as before), with the median leaseadjusted debt leverage varying again by a factor of about 3.Rental leverage also increases with lease adjusted tangi-bility by about a factor 2 for the median and more than 3for the mean. Similarly, lease adjusted leverage, which wedefine as the sum of debt leverage and rental leverage, alsoincreases with tangibility by a factor 3.

Taking rental leverage into account reduces the fractionof firms with low leverage drastically, in particular, forfirms outside the low tangibility quartile. Lease adjustedtangibility is an even more important explanation for the“low leverage puzzle.” Indeed, less than 4% of firms withhigh tangibility have low lease adjusted leverage.

It is also worth noting that the median rental leverageis on the order of half of debt leverage or more, and ishence quantitatively important. Overall, we conclude thattangibility, when adjusted for leased capital, emerges as a

6 In accounting, this approach to capitalization is known as con-structive capitalization and is frequently used in practice, with “8 � rent”being the most commonly used. E.g., Moody's rating methodology usesmultiples of 5� , 6� , 8� , and 10� current rent expense, depending onthe industry. We discuss the calibration of the capitalization factor we usein footnote 12.

Table 3Leverage and size revisited.

This table displays debt and rental leverage across size deciles (measured by lease adjusted book assets). Lease Adjusted Book Assets: Assets-Total plus 10times Rental Expense; Debt Leverage: Long-Term Debt-Total divided by Lease Adjusted Book Assets; Rental Leverage: 10 times Rental Expense divided byLease Adjusted Book Assets; Lease Adjusted Leverage: Debt Leverage plus Rental Leverage. For details of data and variables used, see caption of Table 2.

Size deciles

Median leverage 1 2 3 4 5 6 7 8 9 10

Debt 6.0 7.3 7.4 14.1 19.5 22.6 20.6 20.2 21.6 17.8Rental 21.8 14.6 10.8 11.1 11.2 9.1 9.7 9.1 7.8 7.3

Lease adjusted 30.6 24.2 21.0 28.8 36.4 37.7 33.4 36.6 31.7 26.3

1 2 3 4 5 6 7 8 9 100

5

10

15

20

25

30

35

40

(Lease adjusted) size deciles

Leas

e ad

just

ed, d

ebt,

and

rent

al le

vera

ge (%

)

Fig. 1. Leverage versus size revisited. Lease adjusted leverage (solid), debtleverage (dashed), and rental leverage (dash-dotted) across size decilesfor Compustat firms. For details see caption of Table 3.

8 We are implicitly using the Leontief aggregator of tangible andintangible capital minfðkpþklÞ=φ,ki=ð1−φÞg which yields ki ¼ ð1−φÞk,kpþkl ¼ φk, and k¼ kiþkpþkl as above, simplifying the firm's investmentproblem to the choice of capital k and leased capital kl only. If tangible

A.A. Rampini, S. Viswanathan / Journal of Financial Economics 109 (2013) 466–492 471

key determinant of leverage and the fraction of firms withlow leverage.

2.4. Leverage and size revisited

Considering leased capital changes basic cross-sectionalproperties of the capital structure. Here we document therelationship between firm size and leverage (see Table 3 andFig. 1). We sort Compustat firms into deciles by size. Wemeasure size by lease adjusted assets here, although usingunadjusted assets makes our results even more stark. Debtleverage is increasing in size, in particular for small firms,when leased capital is ignored. Rental leverage, by contrast,decreases in size, in particular for small firms.7 Indeed, rentalleverage is substantially larger than debt leverage for smallfirms. Lease adjusted leverage, i.e., the sum of debt and rentalleverage, is roughly constant across Compustat size deciles. Inour view, this evidence provides a strong case that leasedcapital cannot be ignored if one wants to understand thecapital structure.

3. Model

This section provides a dynamic agency-based model tounderstand the first-order importance of tangible assetsand rented assets for firm financing and the capitalstructure documented above. Dynamic financing is subjectto collateral constraints due to limited enforcement. Weconsider both tangible and intangible capital as well asfirms' ability to lease capital. Moreover, we define the usercost of tangible, intangible, and leased capital. Finally, wecharacterize the dividend policy and show how tangibilityand collateralizability of assets affect the capital structurein the special case without leasing.

3.1. Environment

A risk neutral firm subject to limited liability discountsthe future at rate β∈ð0,1Þ and requires financing for invest-ment. The firm's problem has an infinite horizon and wewrite the problem recursively. The firm starts the periodwith net worth w and has access to a standard neoclassicalproduction function with decreasing returns to scale.

7 Eisfeldt and Rampini (2009) show that this is even more dramati-cally the case in Census data, which includes firms that are not inCompustat and hence much smaller, and argue that for such firms,renting capital may be the most important source of external finance.

There are two types of capital, tangible capital andintangible capital. Tangible capital can be either purchased(kp) or leased (kl), while intangible capital (ki) can only bepurchased. The total amount of capital is k≡kiþkpþkl andwe refer to total capital k often simply as capital. Forsimplicity, we assume that tangible and intangible capitalare required in fixed proportions and denote the fractionof tangible capital required by φ.8 Both tangible andintangible capital can be purchased at a price normalizedto 1 and depreciate at the same rate δ. There are noadjustment costs. An amount of invested capital k yieldsstochastic cash flow Aðs′Þf ðkÞ next period, where Aðs′Þ is therealized total factor productivity of the technology in states′, which we assume follows a Markov process describedby the transition function Πðs,s′Þ on s′∈S.

Tangible capital which the firm owns can be used ascollateral for state-contingent one-period debt up to afraction θ∈ð0,1Þ of its resale value. These collateral con-straints are motivated by limited enforcement. We assumethat enforcement is limited in that firms can abscond withall cash flows, all intangible capital, and 1−θ of purchasedtangible capital kp. Further, we assume that firms cannotabscond with leased capital kl, i.e., leased capital enjoys a

and intangible capital were not used in fixed proportions and had aconstant elasticity of substitution γ > −∞, i.e., the aggregator of tangibleand intangible capital was ½sðkpþklÞγþð1−sÞkγi �1=γ with γ≤1, then thecomposition of capital would vary with firms' financial condition, withmore financially constrained firms using a lower fraction of intangiblecapital.

11 In practice, there may be a link between the lessor's monitoringand the repossession advantage of leasing. In order to monitor the useand maintenance of the asset, the lessor needs to keep track of the assetwhich makes it harder for the lessee to abscond with it.

12 To impute the amount of capital rented from rental payments, weshould hence capitalize rental payments by 1=ðR−1ðrþδþmÞÞ. In docu-

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repossession advantage. It is easier for a lessor, who retainsownership of the asset, to repossess it, than for a securedlender, who only has a security interest, to recover thecollateral backing the loan. Leasing enjoys such a reposses-sion advantage under U.S. law and, we believe, in mostlegal systems. Importantly, we assume that firms whoabscond cannot be excluded from any market: the marketfor intangible capital, tangible capital, loans, and rentedcapital. As we show in Appendix B, these dynamic enfor-cement constraints imply the above collateral constraints,which are similar to the ones in Kiyotaki and Moore(1997), albeit state contingent, and are described in moredetail below.9 We emphasize that any long-term contractthat satisfies the enforcement constraints can be imple-mented with such one-period ahead state-contingent debtsubject to the above collateral constraints and hence long-term contracts are not ruled out. The motivation for ourassumption about the lack of exclusion is twofold. First, itallows us to provide a tractable model of dynamic collater-alized firm financing. The equivalence of the two problems(with limited enforcement and collateral constraints,respectively) enables us to work directly with the problemwith collateral constraints and use net worth as the statevariable. In contrast, the outside options considered in theliterature result in continuation utility being the appro-priate state variable, which typically makes the dualproblem easier to work with (see, e.g., Albuquerque andHopenhayn, 2004). Second, a model based on this assump-tion has implications which are empirically plausible, inparticular by putting the focus squarely on tangibility.

We assume that intangible capital can neither becollateralized nor leased. The idea is that intangible capitalcannot be repossessed due to its lack of tangibility and canbe deployed in production only by the owner, since theagency problems involved in separating ownership fromcontrol are too severe.10

Our model considers the role of leased capital in adynamic model of firm financing subject to limited enfor-cement. The assumption that firms cannot abscond withleased capital captures the fact that leased capital can berepossessed more easily. This repossession advantagemeans that by leasing, the firm can effectively borrowagainst the full resale value of tangible assets, whereassecured lending allows the firm to borrow only against afraction θ of the resale value. The benefit of leasing is its

9 These collateral constraints are derived from an explicitly dynamicmodel of limited enforcement similar to the one considered by Kehoe andLevine (1993). The main difference to their limits on enforcement is thatwe assume that firms who abscond cannot be excluded from futureborrowing whereas they assume that borrowers are in fact excluded fromintertemporal trade after default. Similar constraints have been consid-ered by Lustig (2007) in an endowment economy, by Lorenzoni andWalentin (2007) in a production economy with constant returns to scale,and by Rampini and Viswanathan (2010) in a production economy with afinite horizon. Krueger and Uhlig (2006) find that similar limits onenforcement in an endowment economy without collateral implyshort-sale constraints, which would be true in our model in the specialcase where θ¼ 0.

10 The assumption that intangible capital cannot be collateralized orleased at all simplifies the analysis, but is not required for the mainresults. Assuming that intangible capital is less collateralizable and morecostly to lease would suffice.

higher debt capacity. However, leasing also has a cost:leased capital involves monitoring costs m per unit ofcapital incurred by the lessor at the end of the period,which are reflected in the user cost of leased capital ul.Leasing separates ownership and control and the lessormust pay the cost m to ensure that the lessee uses andmaintains the asset appropriately.11 A competitive lessorwith a cost of capital R≡1þr charges a user cost of

ul≡rþδþm

per unit of capital. Equivalently, we could assume thatleased capital depreciates faster due to the agency problemat rate δl > δ and set m¼ δl−δ. Due to the constraints onenforcement, the user cost of leased capital is charged atthe beginning of the period and hence the firm pays R−1ul

per unit of leased capital up front. Specifically, since thelessor recovers the leased capital at the end of the period,no additional payments at the end of the period can beenforced and the leasing fee must hence be charged upfront. Recall that in the frictionless neoclassical model, therental cost of capital is Jorgenson's (1963) user cost u≡rþδ.Thus, the only difference to the rental cost in our model isthe positive monitoring cost m (or, equivalently, the costsdue to faster depreciation δl−δ). Note that as in Jorgenson'sdefinition, we define the user cost of capital in terms ofvalue at the end of the period.12

We assume that the firm has access to lenders whohave deep pockets in all dates and states and discount thefuture at rate R−1∈ðβ,1Þ. These lenders are thus willing tolend in a state-contingent way at an expected return R. Theassumption that firms are less patient than lenders, whichis quite common,13 implies that firms' financing policymatters even in the long run, i.e., even for mature firms,and that the financing policy is uniquely determined.Moreover, firms are never completely unconstrainedand firms which currently pay dividends that are hit bya sequence of low cash flow shocks may eventuallystop dividend payments, cut risk management, and switch

menting the stylized facts, we assumed that this factor takes a value of 10.This calibrated value is based on an approximate (unlevered) real cost ofcapital for tangible assets of 4%, an approximate depreciation rate forleased tangible assets of 5% (using a depreciation rate of 3% for structuresand 15% for equipment, which are based on U.S. Bureau of EconomicAnalysis (BEA) data, and a weight of 80% (20%) on structures (equipment),since leased tangible assets are predominantly structures and structuresare 66% of nonresidential fixed assets overall), and an assumed monitor-ing cost m of 1%. The implicit debt associated with rented capital isR−1ð1−δÞ times the amount of capital rented, so in adjusting liabilities, weshould adjust by R−1ð1−δÞ times 10 to be precise. In documenting thestylized facts, we ignored the correction R−1ð1−δÞ, implicitly assumingthat it is approximately equal to one.

13 E.g., this assumption is made by DeMarzo and Sannikov (2006),Lorenzoni and Walentin (2007), Biais, Mariotti, Plantin, and Rochet(2007), Biais, Mariotti, Rochet, and Villeneuve (2010), and DeMarzo,Fishman, He, and Wang (2012); DeMarzo and Fishman (2007a, 2007b)consider β≤R−1; in contrast, firms and lenders are assumed to be equallypatient in Albuquerque and Hopenhayn (2004), Quadrini (2004),Clementi and Hopenhayn (2006), and Rampini and Viswanathan (2010).

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back to leasing capital, implications that are empiricallyplausible.14

Firms in our model thus have access to two sources ofexternal financing, state-contingent secured debt andleasing. In Section 4 we provide an alternative interpreta-tion, which is equivalent, in which firms have access tononcontingent secured debt, leasing, and risk manage-ment using one-period ahead Arrow securities subject toshort-sale constraints.

3.2. Firm's problem

The firm's problem can be written recursively as theproblem of maximizing the discounted expected value offuture dividends by choosing the current dividend d,capital k, leased capital kl, net worth wðs′Þ in state s′ nextperiod, and state-contingent debt bðs′Þ given current networth w and state s:

Vðw,sÞ≡ maxfd,k,kl ,wðs′Þ,bðs′Þg∈R3þ S

þ �RSdþβ∑

s′∈SΠðs,s′ÞVðwðs′Þ,s′Þ ð1Þ

subject to the budget constraints for the current and nextperiod

wþ ∑s′∈S

Πðs,s′Þbðs′Þ≥dþðk−klÞþR−1ulkl, ð2Þ

Aðs′Þf ðkÞþðk−klÞð1−δÞ≥wðs′ÞþRbðs′Þ, ∀s′∈S, ð3Þ

the collateral constraints

θðφk−klÞð1−δÞ≥Rbðs′Þ, ∀s′∈S, ð4Þ

and the constraint that only tangible capital can be leased

φk≥kl: ð5Þ

The program in (1)–(5) requires that dividends d and networth wðs′Þ are non-negative which is due to limitedliability. Furthermore, capital k and leased capital kl haveto be non-negative as well. We write the budget con-straints as inequality constraints, despite the fact that theybind at an optimal contract, since this makes the con-straint set convex as shown below. There are only twostate variables in this recursive formulation, net worth wand the state of productivity s. This is due to our assump-tion that there are no adjustment costs of any kind andgreatly simplifies the analysis. Net worth in state s′ nextperiod wðs′Þ ¼ Aðs′Þf ðkÞþðk−klÞð1−δÞ−Rbðs′Þ, i.e., equals cashflow plus the depreciated resale value of owned capitalminus the amount to be repaid on state s′ contingent debt.Borrowing against state s′ next period by issuing state s′contingent debt bðs′Þ reduces net worth wðs′Þ in that state.In other words, borrowing less than the maximum amountwhich satisfies the collateral constraint (4) against state s′amounts to conserving net worth for that state and allowsthe firm to hedge the available net worth in that state.

14 While we do not explicitly consider taxes here, our assumptionabout discount rates can also be interpreted as a reduced-form way oftaking into account the tax-deductibility of interest, which effectivelylowers the cost of debt finance.

We make the following assumptions about the stochasticprocess describing productivity and the production function:

Assumption 1. For all s,s∈S such that s > s, (i) AðsÞ > AðsÞand (ii) AðsÞ > 0.

Assumption 2. f is strictly increasing, strictly concave,f ð0Þ ¼ 0, limk-0 f kðkÞ ¼ þ∞, and limk-þ∞ f kðkÞ ¼ 0.

We first show that the firm financing problem is a well-behaved dynamic programming problem and that thereexists a unique value function V which solves the problem.To simplify notation, we introduce the shorthand for thechoice variables x, where x≡½d,k,kl,wðs′Þ,bðs′Þ�′, and theshorthand for the constraint set Γðw,sÞ given the statevariables w and s, defined as the set of x∈R3þ S

þ � RS suchthat (2)–(5) are satisfied. Define operator T as

ðTgÞðw,sÞ ¼ maxx∈Γðw,sÞ

dþβ∑s′∈S

Πðs,s′Þgðwðs′Þ,s′Þ:

We prove the following result about the firm financingproblem in (1)–(5):

Proposition 1. (i) Γðw,sÞ is convex, given (w,s), and convex inw and monotone in the sense that w≤w implies Γðw,sÞDΓðw,sÞ. (ii) The operator T satisfies Blackwell's sufficientconditions for a contraction and has a unique fixed point V.(iii) V is continuous, strictly increasing, and concave in w. (iv)Without leasing, Vðw,sÞ is strictly concave in w forw∈intfw : dðw,sÞ ¼ 0g. (v) Assuming that for all s,s∈S suchthat s > s, Πðs,s′Þ strictly first-order stochastically dominatesΠðs,s′Þ, V is strictly increasing in s.

The proofs of parts (i)–(iii) of the proposition arerelatively standard. Part (iii), however, only states thatthe value function is concave, not strictly concave. Thevalue function is linear in net worth when dividends arepaid. The value function may also be linear in net worth onsome intervals where no dividends are paid, due to thelinearity of the substitution between leased and ownedcapital. All our proofs below hence rely on weak concavityonly. Nevertheless, we can show that without leasing, thevalue function is strictly concave where no dividends arepaid (see part (iv) of the proposition).15

Denote the multipliers on the constraints (2)–(5) by μ,Πðs,s′Þβμðs′Þ, Πðs,s′Þβλðs′Þ, and ν l. Let νd and ν l bethe multipliers on the constraint d≥0 and kl≥0. Thefirst-order conditions of the firm financing problem inEqs. (1)–(5) are

μ¼ 1þνd, ð6Þ

μ¼ ∑s′∈S

Πðs,s′Þβfμðs′Þ½Aðs′Þf kðkÞþð1−δÞ�þλðs′Þθφð1−δÞgþνlφ, ð7Þ

ð1−R−1ulÞμ¼ ∑s′∈S

Πðs,s′Þβfμðs′Þð1−δÞþλðs′Þθð1−δÞgþν l−νl, ð8Þ

μðs′Þ ¼ Vwðwðs′Þ,s′Þ, ∀s′∈S, ð9Þ

15 Section 5 discusses how the linearity of the substitution betweenleased and owned capital may result in intervals on which the valuefunction is linear and shows that when m is sufficiently high, the valuefunction is strictly concave in the non-dividend paying region even withleasing.

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μ¼ βμðs′ÞRþβλðs′ÞR, ∀s′∈S, ð10Þwhere we use the fact that the constraints k≥0 andwðs′Þ≥0, ∀s′∈S, are slack as Lemma 6 in Appendix Ashows.16 The envelope condition is Vwðw,sÞ ¼ μ; the mar-ginal value of (current) net worth is μ. Similarly, themarginal value of net worth in state s′ next period is μðs′Þ.

Using Eqs. (7) and (10), we obtain the investment Eulerequation,

1≥∑s′∈S

Πðs,s′Þβ μðs′Þμ

Aðs′Þf kðkÞþð1−θφÞð1−δÞ1−R−1θφð1−δÞ

, ð11Þ

which holds with equality if the firm does not lease all itstangible assets (i.e., ν l ¼ 0Þ. Notice that βμðs′Þ=μ is the firm'sstochastic discount factor; collateral constraints render thefirm as if risk averse and hence provide a rationale for riskmanagement.

3.3. User cost of capital

This section defines the user cost of (purchased) tangi-ble and intangible capital in the presence of collateralconstraints extending Jorgenson's (1963) definition. Thedefinitions clarify the main economic intuition and allow asimple characterization of the leasing decision as we showin Section 5.

Denote the premium on internal funds for a firm withnet worth w in state s by ρðw,sÞ and define it implicitlyusing the firm's stochastic discount factor as1=ð1þrþρðw,sÞÞ≡∑s′∈SΠðs,s′Þβμðs′Þ=μ; internal funds com-mand a premium as long as at least one of the collateralconstraints is binding. Define the user cost of tangiblecapital which is purchased upðw,sÞ for a firm with networth w in state s as

upðw,sÞ≡rþδþ ρðw,sÞRþρðw,sÞ ð1−θÞð1−δÞ, ð12Þ

where ρðw,sÞ=ðRþρðw,sÞÞ ¼∑s′∈SΠðs,s′ÞRβλðs′Þ=μ. The usercost of purchased tangible capital is the sum of theJorgensonian user cost of capital and a second term, whichcaptures the additional cost of internal funds for thefraction ð1−θÞð1−δÞ of capital that has to be financedinternally. Analogously, the user cost of intangible capitalis uiðw,sÞ≡rþδþρðw,sÞ=ðRþρðw,sÞÞð1−δÞ.

Using our definitions, we can rewrite the first-ordercondition for capital, Eq. (7), as

φ minfupðw,sÞ,ulgþð1−φÞuiðw,sÞ ¼ ∑s′∈S

Πðs,s′ÞRβ μðs′Þμ

Aðs′Þf kðkÞ:

Optimal investment equates the weighted average of theuser cost of tangible and intangible capital with theexpected marginal product of capital, where the applicableuser cost of tangible capital is the minimum of the usercost of purchased and leased capital.17

16 Since the marginal product of capital is unbounded as capital goesto zero by Assumption 2, capital is strictly positive. Because the firm'sability to issue promises against capital is limited, this in turn implies thatthe firm's net worth is positive in all states next period.

17 We can rewrite Eq. (12) in a weighted average (user) cost of capitalform as upðw,sÞ¼R=ðRþρðw,sÞÞððrþρðw,sÞÞ½1−R−1θð1−δÞ�þr½R−1θð1−δÞ�þδÞ,where the fraction of capital that can be financed externally, R−1θð1−δÞ, is

3.4. Dividend payout policy

We start by characterizing the firm's payout policy. Thefirm's dividend policy is very intuitive: there is a state-contingent cutoff level of net worth wðsÞ, ∀s∈S, abovewhich the firm pays dividends.18 Moreover, wheneverthe firm has net worth w exceeding the cutoff wðsÞ, payingdividends in the amount w−wðsÞ is optimal. All firms withnet worth w exceeding the cutoff wðsÞ in a given state s,choose the same level of capital. Finally, the investmentpolicy is unique in terms of the choice of capital k. Thefollowing proposition summarizes the characterization offirms' payout policy:

Proposition 2 (Dividend policy). There is a state-contingentcutoff level of net worth, above which the marginal value ofnet worth is one and the firm pays dividends: (i) ∀s∈S, ∃wðsÞsuch that, ∀w≥wðsÞ, μðw,sÞ ¼ 1. (ii) For ∀w≥wðsÞ,½doðw,sÞ,koðw,sÞ,kl,oðw,sÞ,woðs′Þ,boðs′Þ� ¼

½w−wðsÞ,koðsÞ,kl,oðsÞ,woðs′Þ,boðs′Þ�where xo≡½0,koðsÞ,kl,oðsÞ,woðs′Þ,boðs′Þ� attains VðwðsÞ,sÞ.Indeed, koðw,sÞ is unique for all w and s. (iii) Without leasing,the optimal policy xo is unique.

3.5. Effect of tangibility and collateralizability withoutleasing

We distinguish between the fraction of tangible assetsrequired for production, φ, and the fraction of tangibleassets θ that the borrower cannot abscond with and that ishence collateralizable. This distinction is important tounderstand differences in the capital structure acrossindustries, as the fraction of tangible assets required forproduction varies considerably at the industry levelwhereas the fraction of tangible assets that is collateraliz-able primarily depends on the type of capital, such asstructures versus equipment (which we do not distinguishhere). Thus, industry variation in φ needs to be taken intoaccount in empirical work. That said, in the special casewithout leasing, higher tangibility φ and higher collater-alizability θ are equivalent in our model. This result isimmediate as without leasing, φ and θ affect only (4) andonly the product of the two matters. Thus, firms thatoperate in industries that require more intangible capitalare more constrained, all else equal. Furthermore, the factthat firms can only borrow against a fraction φθ of totalcapital is quantitatively relevant as the model predictsmuch lower, and empirically plausible, leverage ratios.

4. Risk management and the capital structure

Our model allows an explicit analysis of dynamic riskmanagement since firms have access to complete markets

(footnote continued)charged a cost of capital r, while the fraction that has to be financed internally,1−R−1θð1−δÞ, is charged a cost of capital rþρðw,sÞ.

18 Consistent with this prediction, DeAngelo, DeAngelo, and Stulz(2006) find a strong positive relation between the probability that firmspay dividends and their retained earnings.

19 In our model, we do not take a stand on whether the productivityshocks are firm specific or aggregate. Since all states are observable, asthe only friction considered is limited enforcement, our analysis applieseither way. Hedging can hence be interpreted either as using loancommitments, e.g., to hedge idiosyncratic shocks to firms' net worth oras using traded assets to hedge aggregate shocks which affect firms'cash flows.

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subject to the collateral constraints. We first show how tointerpret the state-contingent debt in our model in termsof risk management and provide a general result about theoptimal absence of risk management for firms withsufficiently low net worth. Next, we prove the optimalityof incomplete hedging with constant investment opportu-nities, i.e., when productivity shocks are independentlyand identically distributed; indeed, firms abstain from riskmanagement with positive probability under the station-ary distribution, implying that even mature firms thatexperience a sequence of low cash flows eliminate riskmanagement. Moreover, with stochastic investmentopportunities, persistent shocks further reduce risk man-agement and may result in a complete absence of riskmanagement for empirically plausible levels of persis-tence. Strong persistence of productivity may result infirms hedging higher productivity states, because finan-cing needs for increased investment rise more than cashflows. Finally, we analyze the comparative statics of firms'investment, financing, risk management, and dividendpolicy with respect to key parameters of the model.

4.1. Optimal absence of risk management

Our model with state-contingent debt bðs′Þ is equiva-lent to a model in which firms borrow as much as they canagainst each unit of tangible capital which they purchase,i.e., borrow R−1θφð1−δÞ per unit of capital, and keepadditional net worth in a state-contingent way by pur-chasing Arrow securities with a payoff of hðs′Þ for state s′.This formulation allows us to characterize the corporatehedging policy. Specifically, we define risk management interms of purchases of Arrow securities for state s′ as

hðs′Þ≡θðφk−klÞð1−δÞ−Rbðs′Þ: ð13ÞWe say a firm does not engage in risk management whenthe firm does not buy Arrow securities for any state nextperiod. Under this interpretation, firms' debt is not state-contingent and hence risk-free, as are lease contracts, sincewe assume that the price of capital is constant for allstates. We denote the amount firms pay down per unit ofcapital by ℘ðφÞ≡1−R−1θφð1−δÞ and the amount firms paydown per unit of tangible capital by ℘≡℘ð1Þ ¼ 1−R−1θð1−δÞ.Using this notation, we can write the budget constraints forthe current and next period (2) and (3) for this implementa-tion as

w≥dþ ∑s′∈S

Πðs,s′ÞR−1hðs′Þþ℘ðφÞk−ð℘−R−1ulÞkl, ð14Þ

Aðs′Þf ðkÞþ½ð1−φÞkþð1−θÞðφk−klÞ�ð1−δÞþhðs′Þ≥wðs′Þ, ð15Þand the collateral constraints (4) reduce to short-sale con-straints

hðs′Þ≥0, ∀s′∈S, ð16Þimplying that holdings of Arrow securities have to be non-negative. The budget constraint (14) makes the trade-offbetween financing and risk management particularly appar-ent; the firm can spend its net worth w on purchases ofArrow securities, i.e., hedging, or to buy fully levered capital;leasing frees up net worth as long as ℘−R−1ul > 0 which weassume (see Assumption 3 in Section 5) as leasing is

otherwise dominated. Eq. (15) states that net worth wðs′Þin state s′ next period is the sum of cash flows, the value ofintangible capital and owned tangible capital not pledged tothe lenders, and the payoffs of the Arrow securities, if any.Our model with state-contingent borrowing is hence amodel of financing and risk management.

In the implementation considered in this section, firmspledge as much as they can against their tangible assets tolenders leaving no collateral to pledge to derivativescounterparties. The holdings of Arrow securities are hencesubject to short-sale constraints and the cost of riskmanagement is the net worth required in the currentperiod to purchase them. That said, since firms in ourmodel have access to complete markets, subject to collat-eral constraints, firms can replicate any type of derivativescontract, including forward contracts or futures that do notinvolve payments up front, implying that our results holdfor forward contracts and futures, too. Such contractsinvolve promises to pay in some states next period whichcount against the collateral constraints in those states.There is an opportunity cost for such promises, becausepromises against these states could alternatively be usedto finance current investment. Thus, there are no con-straints on the type of hedging instruments firms can usein our model, and the only constraint on risk managementis that promised payments need to be collateralized, whichis identical to the constraint imposed on financing.

The next proposition states that for severely con-strained firms, all collateral constraints bind, which meansthat such firms do not purchase any Arrow securities at all,and, in this sense, do not engage in risk management. Thenumerical examples in Sections 4.2 and 4.4 show that theextent to which firms hedge low states is in fact increasingin net worth.19

Severely constrained firms optimally abstain from riskmanagement altogether:

Proposition 3 (Optimal absence of risk management). Firmswith sufficiently low net worth do not engage in riskmanagement, i.e., ∀s∈S, ∃whðsÞ > 0, such that ∀w≤whðsÞ, allcollateral constraints bind, λðs′Þ > 0, ∀s′∈S.

Collateral constraints imply that there is an opportunitycost to issuing promises to pay in high net worth statesnext period to hedge low net worth states, as suchpromises can also be used to finance current investment.The proposition shows that when net worth is sufficientlylow, the opportunity cost of risk management due to thefinancing needs must exceed the benefit. Hence, firmsoptimally do not hedge at all. Notice that the result obtainsfor a general Markov process for productivity. The result isconsistent with the evidence that firms with low networth hedge less, and is in contrast to the conclusionsfrom static models in the extant literature, such as Froot,

A.A. Rampini, S. Viswanathan / Journal of Financial Economics 109 (2013) 466–492476

Scharfstein, and Stein (1993). The key difference is that ourmodel explicitly considers dynamic financing needs forinvestment as well as the limits on firms' ability to promiseto pay.

4.2. Risk management with constant investmentopportunities

With independent productivity shocks, risk manage-ment only depends on the firm's net worth, because theexpected productivity of capital is independent of thecurrent state s, i.e., investment opportunities are constant.More generally, both cash flows and investment opportu-nities vary, and the correlation between the two affects thedesirability of hedging, as we show in Section 4.4 below.

With constant investment opportunities, the marginalvalue of net worth is higher in states with low cash flowsand complete hedging is never optimal:

Proposition 4 (Optimality of incomplete hedging). Supposethat Πðs,s′Þ ¼Πðs′Þ, ∀s,s′∈S. (i) The marginal value of networth next period μðs′Þ ¼ Vwðwðs′ÞÞ is (weakly) decreasing inthe state s′, and the multipliers on the collateral constraintsare (weakly) increasing in the state s′, i.e., ∀s′,s′þ∈S such thats′þ > s′, μðs′þ Þ≤μðs′Þ and λðs′þ Þ≥λðs′Þ. (ii) Incomplete hedging isoptimal, i.e., ∃s′∈S, such that λðs′Þ > 0. Indeed, ∃s′,s′∈S, suchthat wðs′Þ≠wðs′Þ. Moreover, the firm never hedges the higheststate, i.e., is always borrowing constrained against the high-est state, λðs′Þ > 0 where s′¼maxfs′ : s′∈Sg. The firm hedgesa lower interval of states, ½s′,…,s′h�, where s′¼minfs′ : s′∈Sg,if at all.

Complete hedging would imply that all collateral con-straints are slack and consequently, the marginal value ofnet worth is equalized across all states next period. Buthedging involves conserving net worth in a state-contingent way at a return R. Given the firm's relativeimpatience, it can never be optimal to save in this state-contingent way for all states next period. Thus, incompletehedging is optimal. Further, since the marginal value of networth is higher in states with low cash flow realizations, itis optimal to hedge the net worth in these states, if it isoptimal to hedge at all. Firms' optimal hedging policyimplicitly ensures a minimum level of net worth in allstates next period.

We emphasize that our explicit dynamic model ofcollateral constraints due to limited enforcement is essen-tial for this result. If the firm's ability to pledge were notlimited, then the firm would always want to pledge moreagainst high net worth states next period to equate networth across all states. However, in our model the abilityto credibly pledge to pay is limited and there is anopportunity cost to pledging to pay in high net worthstates next period, since such pledges are also required forfinancing current investment. This opportunity costimplies that the firm never chooses to fully hedge networth shocks.

To illustrate the interaction between financing needsand risk management, we compute a numerical example.We assume that productivity is independent and, forsimplicity, that productivity takes on two values only,

Aðs1Þ < Aðs2Þ, and that there is no leasing. The results anddetails of the parameterization are reported in Fig. 2.

Investment as a function of net worth is shown in PanelA, which illustrates Proposition 2. Above a threshold w,firms pay dividends and investment is constant. Below thethreshold, investment is increasing in net worth anddividends are zero.

The dependence of risk management on net worth isillustrated in Panel B. With independent shocks, the firmnever hedges the high state, i.e., hðs′2Þ ¼ 0, where hðs′Þ isdefined as in Eq. (13) (see Proposition 4). Panel B thusdisplays only the payoff of the Arrow securities that thefirm purchases to hedge the low state, hðs′1Þ. Importantly,note that the hedging policy is increasing in firm networth, i.e., better capitalized firms hedge more. Thisillustrates the main conclusion from our model for riskmanagement. Above the threshold w, hedging is constant(as Proposition 2 shows). Below the threshold, hedging isincreasing, and for sufficiently low values of net worth w,the firm does not hedge at all, as Proposition 3 shows moregenerally. Indeed, hedging is zero for a sizable range ofvalues of net worth (up to a value of around 0.1 in theexample).

The values of net worth next period are displayed inPanel C and illustrate the endogenous dynamics of networth. Since optimal hedging is incomplete, net worthnext period is higher in state s′2 than in state s′1. The figuremoreover plots the 451 line (dotted), where w¼w′, tofacilitate the characterization of the dynamics of net worthand the stationary distribution. Net worth next period ishigher than current net worth, i.e., increases, when wðs′Þ isabove the 451 line and decreases otherwise. In the low(high) state next period, net worth decreases (increases)for all levels of net worth above (below) the intersection ofwðs′1Þ ðwðs′2ÞÞ and the 451 line, which we denotewðs1Þ ðwðs2ÞÞ. These transition dynamics of net worthtogether with the policy functions describe the dynamicsof investment, financing, and risk management fully; if thelow (high) state is realized, investment and risk manage-ment both decrease (increase) as long as current net worthis in the interval ½wðs1Þ,wðs2Þ�.

Since levels of net worth below wðs1Þ and above wðs2Þare transient, the ergodic set must be bounded below bywðs1Þ and above bywðs2Þ, and the support of the stationarydistribution is a subset of the interval ½wðs1Þ,wðs2Þ�. Indeed,firms with net worth abovewðs2Þwill pay out the extra networth and start the next period within the ergodic set.Moreover, evaluating the first-order condition (10) for bðs′Þat wðs1Þ and s′¼ s1, we have Vwðwðs1ÞÞ ¼ RβVwðwðs1ÞÞþRβλðwðs1ÞÞ and thus λðwðs1ÞÞ > 0. This means that thefirm abstains from risk management altogether at wðs1Þ.By continuity, the absence of risk management is optimalfor sufficiently low values of net worth in the stationarydistribution, a result that is more generally true (seeProposition 5 below).

The multipliers on the collateral constraints λðs′Þ areshown in Panel D. The first-order conditions (10) for bðs′Þimply that μðs1′Þþλðs1′Þ ¼ μðs2′Þþλðs2′Þ; firms do not simplyequate the marginal value of net worth across states, butthe sum of the marginal value of net worth and the multiplieron the collateral constraint. The figure shows that λðs2′Þ > 0 for

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Fig. 2. Investment and risk management. Panel A shows investment k; Panel B shows risk management for the low state hðs′1Þ; Panel C shows net worth inlow state next period wðs′1Þ (solid) and in high state next period wðs′2Þ (dashed); Panel D shows scaled multipliers on the collateral constraint for the lowstate next period βλðs′1Þ=μ (solid) and for the high state next period βλðs′2Þ=μ (dashed); all as a function of current net worth w. The parameter values are:β¼ 0:93, r¼0.05, δ¼ 0:10, m¼ þ∞, θ¼ 0:80, φ¼ 1, Aðs2Þ ¼ 0:6, Aðs1Þ ¼ 0:05, and Πðs,s′Þ ¼ 0:5, ∀s,s′∈S, and f ðkÞ ¼ kα with α¼ 0:333.

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all w, whereas the multiplier on the collateral constraint forthe low state λðs1′Þ is zero for levels of w at which the firmhedges and positive for lower levels at which the firm abstainsfrom risk management. Collateral constraints result in a trade-off between financing and risk management.

4.3. Risk management under the stationary distribution

We now show that firms do not hedge at the lowerbound of the stationary distribution as we observed in theexample above. Indeed, firms abstain from risk manage-ment with positive probability under the (unique) sta-tionary distribution.

Proposition 5 (Absence of risk management under the sta-tionary distribution). Suppose Πðs,s′Þ ¼Πðs′Þ, ∀s,s′∈S, andm¼ þ∞ (no leasing). (i) For the lowest state s′, the wealthlevel w for which wðs′Þ ¼w is unique and the firm abstainsfrom risk management at w. (ii) There exists a uniquestationary distribution of net worth and firms abstain fromrisk management with positive probability under the sta-tionary distribution.

Proposition 5 implies that even if a firm is currentlyrelatively well capitalized and paying dividends, a suffi-ciently long sequence of low cash flows will leave the firmso constrained that it chooses to stop hedging.20 Thus, it isnot only young firms with very low net worth that abstainfrom risk management, but also mature firms that sufferadverse cash flow shocks. Consistently, Rampini, Sufi, and

20 Eq. (10) implies that there is an upper bound on the extent to whichthe marginal value of net worth can increase as Vwðwðs′Þ,s′Þ=Vwðw,sÞ≤ðβRÞ−1;however, the marginal value of net worth could increase, e.g., when cashflows are sufficiently low.

Viswanathan (2012) show that airlines that hit financialdistress dramatically cut their fuel price risk management.

4.4. Risk management with stochastic investmentopportunities

With stochastic investment opportunities, risk manage-ment depends not only on net worth but also on the firm'sproductivity, since the conditional expectation of futureproductivity varies with current productivity when pro-ductivity is persistent. Positive autocorrelation reduces thebenefit to hedging and can eliminate the need to hedgecompletely.

We first show that incomplete hedging is optimal evenwhen investment opportunities are stochastic:

Proposition 6 (Optimality of incomplete hedging with persis-tence). Suppose m¼ þ∞ (no leasing). Optimal risk manage-ment is incomplete with positive probability, i.e., ∃s such thatfor all w, λðs′Þ > 0, for some s′.

Thus, firms engage in incomplete risk management forany Markov process of productivity, generalizing theresults from Proposition 4. As in the case without persis-tence, firms engage in risk management to transfer fundsinto states in which the marginal value of net worth ishighest (if at all). With persistence, however, the marginalvalue of net worth depends on both net worth andinvestment opportunities going forward. Positive autocor-relation, which is typical in practice, implies that invest-ment opportunities are worse (better) when productivityis low (high), decreasing (increasing) the marginal valueof net worth in states with low (high) realizations. Thisreduces the benefit to hedging and thus firms hedge lessor not at all. Indeed, if this effect is strong enough, firmsmay hedge states with high productivity, despite the fact

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that cash flows are high then, too. Current investmentopportunities also affect the benefits to investing and theopportunity cost of hedging, and thus the extent of riskmanagement depends on current productivity as well.

To study the effect of persistence, we reconsider theexample with a two-state symmetric Markov process forproductivity from the previous section. The results arereported in Fig. 3. We increase the transition probabilitiesΠðs1,s1Þ ¼Πðs2,s2Þ≡π, which are 0.5 when investmentopportunities are constant, progressively to 0.55, 0.60,0.75, and 0.90. Since the autocorrelation of a symmetrictwo-state Markov process is 2π−1, this corresponds toprogressively raising the autocorrelation from 0 to 0.1, 0.2,0.5, and 0.8. Given the symmetry, the stationary distribu-tion of the productivity process over the two states is 0.5and 0.5 across all our examples, and the unconditionalexpected productivity is hence the same as well.

Panels A and B display the investment and hedgingpolicies in the case without persistence from the previoussection (see also Fig. 2). Since these policy functions areindependent of the state of productivity s, there is only onefunction for each policy. In the panels with persistence, thesolid lines denote the policies when current productivity islow (s1) and the dashed lines the ones when currentproductivity is high (s2).

The left panels show that as persistence increases,investment is higher when current productivity is high,because the conditional expected productivity is higher.The right panels show the effect of persistence on thehedging policy. Panel D shows that hedging (for the lowstate) decreases relative to the case without persistence,but more so when current productivity is high. Mostnotably, for given net worth, the firm hedges the low stateless when current productivity is high than when it is low.The economic intuition has two aspects. First, persistentshocks reduce the marginal value of net worth whenproductivity is low (since the conditional expected pro-ductivity is low then too) while raising the marginal valueof net worth when productivity is high. Thus, there is lessreason to hedge. Second, high current productivity leads tohigher investment when productivity is persistent (andhigher opportunity cost of risk management), which raisesnet worth next period and further reduces the hedgingneed. This second effect goes the other way when currentproductivity is low, but the first effect dominates in ourexample even in this case.

When persistence is raised further, the benefit tohedging is still lower and the firm abstains from hedgingcompletely when current productivity is high and onlyhedges (the low state) when current productivity is low(Panel F); indeed, the firm stops hedging completely at anautocorrelation of productivity of 0.5 (see Panel H). Thissuggests that, for empirically plausible parameterizations(see, e.g., the calibrated autocorrelation of 0.62 in Gomes,2001), even firms with high net worth may engage in onlylimited risk management or none at all.

For the persistence levels considered thus far, firmshedge the low cash flow state, if at all. With severepersistence (see Panels I and J), the difference in invest-ment is very large across the two productivity states, andfirms have an incentive to hedge the high state due to its

substantially greater investment opportunities whencurrent productivity is low (see the dash-dotted line inPanel J). But notice that even in this case, hedging isincreasing in net worth.

This example illustrates the dynamic trade-off betweenfinancing current investment and risk management. First,current expected productivity affects the benefits to investingand hence the opportunity cost of risk management. Second,expected productivity next period affects the benefits tohedging and which states the firm hedges; for plausible levelsof persistence, firms abstain from riskmanagement altogether.However, whatever the persistence, firms do not hedge at allwhen they are severely constrained.

4.5. Effect of risk, tangibility, and collateralizability

We now study the comparative statics of the firm'sinvestment, financing, risk management, and dividend policywith respect to key parameters of the model. Specifically, weconsider how the firm's optimal policy varies with the risk ofthe productivity process Aðs′Þ, the tangibility φ and collater-alizability θ, and the curvature of the production function αwhen f ðkÞ ¼ kα. For simplicity, we consider the case withoutleasing, which implies that the effects of tangibility andcollateralizability are identical. Moreover, we assume thatinvestment opportunities are constant.

First, consider how the firm responds to an increase inrisk of the productivity process Aðs′Þ in the Rothschild andStiglitz (1970) sense. An increase in risk decreases firmvalue. Intuitively, an increase in productivity risk results inan increase in risk in net worth, given the optimal policy,which reduces value since the value function is concave innet worth. In contrast, in a frictionless world, firm valuewould be unaffected by such an increase in risk asexpected cash flows are unchanged.

The increase in risk also affects the firm's dividend andinvestment policy. Relative to the deterministic case, a firmsubject to risk pays dividends only at a higher level of networth since there is a precautionary motive to retainingnet worth. Such a firm also invests more in the dividendpaying region essentially because of a precautionarymotive for investment. That said, when the firm engagesin risk management, the financing needs for risk manage-ment can reduce investment given net worth. The follow-ing proposition summarizes these results:

Proposition 7 (Effect of risk). Suppose that Πðs,s′Þ ¼ πðs′Þ,∀s,s′∈S, and m¼ þ∞ (no leasing). (i) Valuation: SupposeAþ ðs′Þ is an increase in risk from Aðs′Þ in the Rothschild andStiglitz (1970) sense and let V ðV þ Þ be the value functionassociated with Aðs′Þ ðAþ ðs′ÞÞ; then VðwÞ≥V þ ðwÞ, ∀w, i.e., anincrease in risk reduces the value of the firm. (ii) Investmentand dividend policy: Suppose A0′ is a constant and Asðs′Þ isan increase in risk from A0′ and denote the associatedoptimal policy by x0 ðxsÞ; then ks≥k0 and ws≥w0, i.e., theinvestment of a dividend paying firm and the cutoff net worthat which the firm starts to pay dividends are higher in thestochastic case. Moreover, suppose that S¼ fs,sg; if a divi-dend paying firm is hedging, then ks is decreasing in risk.

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Fig. 3. Risk management with stochastic investment opportunities. Panels A through I: Investment (k) and risk management for the low state ðh1ðs′ÞÞ as afunction of current net worth w for low current productivity (s1) (solid) and high current productivity (s2) (dashed). Panel J: Risk management for the highstate ðh2ðs′ÞÞ as a function of current net worthw for low current productivity (s1) (dash-dotted). Persistence measured by Πðs1 ,s1Þ ¼Πðs2 ,s2Þ≡π is 0.50, 0.55,0.60, 0.75, and 0.90 in Panels A/B (no persistence), Panels C/D (some persistence), Panels E/F (more persistence), Panels G/H (high persistence), and PanelsI/J (severe persistence). For other parameter values, see the caption of Fig. 2.

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Fig. 4 illustrates the comparative statics with respect torisk. Panel A plots firms' investment policy for differentlevels of risk. The precautionary motive for risk manage-ment increases the optimal investment of dividend paying

firms. At the same time, the financing needs for riskmanagement reduce investment given net worth whenfirms do not pay dividends and also reduce investment fordividend paying firms that hedge. Since investment is

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Fig. 4. Investment and risk management: Effect of risk. Panel A shows investment k and Panel B shows risk management for the low state hðs′1Þ as afunction of current net worth w as risk varies. The parameter values are as in Fig. 2 except that ðAðs2Þ,Aðs1ÞÞ vary from (0.625,0.025) (dashed), (0.600,0.050)(solid), (0.575,0.075) (dash-dotted), to (0.550,0.100) (dotted).

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constant in the dividend paying region, Panel A also showsthat as risk increases, firms postpone paying dividendsuntil a higher cutoff of net worth is reached. Panel B showsthat as risk increases, firms start to hedge earlier and domore risk management given net worth.21

Second, consider how collateralizability θ and tangibi-lity φ affect the firm's optimal policy. We emphasize that inthe absence of leasing, the effects of collateralizability andtangibility are the same, and that collateralizability andtangibility are primarily determined by the nature of theassets used in a particular industry; tangibility φ isdetermined by the extent to which tangible assets (struc-tures and equipment) are used and collateralizability θ isdetermined by the extent to which structures, which arearguably more collateralizable, are used instead of equip-ment. We discuss the effects in terms of collateralizability.Firm value increases in collateralizability; intuitively, afirm with higher collateralizability could choose the samepolicy as a firm with lower collateralizability, but choosesnot to do so. A larger fraction of the capital can be financedwith debt. Moreover, and most directly, collateralizabilityand tangibility increase leverage. These results are statedformally in the following proposition:

Proposition 8 (Effect of tangibility and collateralizability).Suppose m¼ þ∞ (no leasing). (i) Valuation: Suppose θþ > θand let V ðV þ Þ be associated with θ ðθþ Þ; then V þ ðw,sÞ >Vðw,sÞ, for all w and s, i.e., an increase in collateralizability(or tangibility) strictly increases the value of the firm. (ii)Investment policy: Suppose that S¼ fs,sg and Πðs,s′Þ ¼ πðs′Þ,∀s,s′∈S; if a dividend paying firm is hedging, then k isincreasing in collateralizability (or tangibility).

21 Variation in risk across firms thus induces a negative relationbetween value and risk management given net worth, all else equal, incontrast to the hypothesis of a positive relation typically explored inempirical work.

Fig. 5 illustrates these comparative statics. Panel Ashows that higher collateralizability θ increases invest-ment given net worth, because it allows higher leverageand thus a large fraction of capital can be financed withdebt, which is relatively cheap. Moreover, firms withhigher collateralizability start paying dividends at a lowerlevel of net worth since less net worth is required to run afirm of a given size. Panel B shows that collateralizabilityincreases risk management, because it allows higherleverage which makes the net worth of the firm morevolatile, all else equal.

Finally, consider the effect of the curvature of the produc-tion function α when f ðkÞ ¼ kα. One might expect that highercurvature of the production function, i.e., lower α, wouldinduce more curvature in the value function and result inincreased risk management. However, when the curvature ishigher, the marginal product of capital decreases faster,which reduces the optimal size of the firm. This in turnreduces the financing needs for investment and the cashflow risk and can result in reduced hedging. In fact, Fig. 6illustrates that both investment and risk managementdecrease as the curvature increases given our parameteriza-tion. The comparative statics of our model imply interestingempirical predictions for firm financing and risk manage-ment in both the cross section and the time series.

5. Leasing and the capital structure

This section analyzes the dynamic leasing decision.We first prove that leasing is optimal for firms withsufficiently low net worth. We then analyze the dynamicchoice between leasing and secured financing in thedeterministic case, which facilitates explicit characteriza-tion because the collateral constraint binds throughout,to highlight the economic intuition; leasing allows firmsto grow faster. In the stochastic case, leasing and riskmanagement are jointly determined and firms that leasemay engage in risk management because leasing enables

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Fig. 5. Investment and risk management: effect of tangibility and collateralizability. Panel A shows investment k and Panel B shows risk management forthe low state hðs′1Þ as a function of current net worth w as tangibility or collateralizability varies. The parameter values are as in Fig. 2 except that θ variesfrom 0.9 (dashed), 0.8 (solid), 0.7 (dash-dotted), to 0.6 (dotted).

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Fig. 6. Investment and risk management: Effect of curvature of production function. Panel A shows investment k and Panel B shows risk management forthe low state hðs′1Þ as a function of current net worth w as the curvature of the production function α varies. The parameter values are as in Fig. 2 exceptthat α varies from 0.366 (dashed), 0.333 (solid), 0.300 (dash-dotted), to 0.267 (dotted).

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higher leverage but reduces net worth in low cash flow states.Mature firms may engage in sale-and-leaseback transactionswhen they experience adverse cash flow shocks. Moreover,the model's implications for the capital structure are consis-tent with the empirical facts documented in Section 2.

5.1. Optimality of leasing

Using the definitions of the user cost of tangible capitaland (10), the first-order condition with respect to leasedcapital, (8), simplifies to

upðw,sÞ ¼ ulþRμðν l−ν lÞ: ð17Þ

The decision between purchasing capital and leasing reducesto a straight comparison of the user costs. If the user cost of

leasing exceeds the user cost of purchased capital, whichdepends on the firm's net worth w and state s, ν l > 0 and thefirm purchases all capital. If the reverse is true, ν l > 0 and allcapital is leased. When upðw,sÞ ¼ ul, the firm is indifferentbetween leasing and purchasing capital at the margin.

Using Eqs. (8) and (10) we moreover obtain an Eulerequation for leasing capital

1¼ ∑s′∈S

Πðs,s′Þβ μðs′Þμ

ð1−θÞð1−δÞ℘−R−1ul

þ 1μ

ν l−ν l℘−R−1ul

: ð18Þ

Leasing instead of purchasing a unit of tangible capital allowsthe firm to borrow the additional amount ℘−R−1ul, which isthe difference between the minimal down payment requiredto purchase a unit of capital and the leasing fee, in return forgiving up the incremental payoff of ð1−θÞð1−δÞ next period,which is the additional amount that the firm can pledge to

23 For a dividend paying firm, μ¼ 1, and, using (10), we haveRβλðs′Þ ¼ 1−Rβμðs′Þ≤1−Rβ, ∀s′, and thus upðw,sÞ ¼ rþδþ∑s′∈SΠðs,s′ÞRβλðs′Þð1−θÞð1−δÞ≤rþδþð1−RβÞð1−θÞð1−δÞ.

24 If one does not impose the second inequality in Assumption 3,then even dividend paying firms may lease capital. In practice, therearguably are different types of tangible assets which differ in theirmonitoring costs and so Assumption 3 may be satisfied for some types

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lessors. We define the implied interest rate paid on theadditional amount the firm can borrow by leasing as

Rl≡ð1−θÞð1−δÞ℘−R−1ul

¼ R1

1−m

ð1−θÞð1−δÞ> R:22

Effectively, leasing is a costly way to borrow more. Indeed,using the premium on internal funds ρðw,sÞ defined in Section3, the leasing Euler equation (18) simplifies to

1≷1

1þrþρðw,sÞRl,

and the firm leases some or all of its tangible capital if Rl≤1þrþρðw,sÞ, i.e., if the cost of leasing is less than or equal to thecost of internal finance, and does not lease otherwise. In asense there is a pecking order among the two modes ofexternal financing in our model: firms use secured financingand use lease financing only if their internal funds aresufficiently scarce.

The fact that leasing allows additional borrowing isevident when one writes the up-front user cost of leasedcapital using this notation as

R−1ul ¼ 1−R−1θð1−δÞ−R−1l ð1−θÞð1−δÞ:

By leasing, the firm implicitly borrows θð1−δÞ at an interestrate R and the additional amount ð1−θÞð1−δÞ at the impliedinterest rate Rl. The cost of leasing to implicitly borrow moreis constant, whereas the return on investment is decreasingas one can see from the investment Euler equation (11). Thebasic trade-off is that when the firm's net worth is low,investment is low and hence the marginal return on invest-ment is high whereas the cost of leasing is unchanged, andhence the firm leases all its tangible capital. In contrast,when the firm's net worth is high, investment is high and themarginal return on investment low, and the firm maypurchase some or all of its tangible capital as the cost ofborrowing more by leasing is too high.

The following assumption ensures that the monitoringcosts are such that leasing is neither dominated nor dominat-ing, which rules out the uninteresting special cases in whichfirms never lease or always lease tangible assets:

Assumption 3. Leasing is neither dominated nordominating, i.e.,

ð1−θÞð1−δÞ > m > ð1−RβÞð1−θÞð1−δÞ:

We maintain this assumption throughout. The condi-tion ð1−θÞð1−δÞ > m ensures that leasing is not dominated.There are two ways to derive this condition. First, thecondition is equivalent to assuming that ℘ > R−1ul, whichmeans that the minimal down payment required topurchase a unit of tangible capital is larger than the up-front payment required to lease the same capital, i.e., theadditional outlay for purchasing capital instead of leasingit is strictly positive. Otherwise, there would be no benefitto leasing at all. Second, the user cost of purchased capitalfor a severely constrained firm is limw-0 upðw,sÞ ¼ rþδþ

22 Assumption 3 below implies that the difference ℘−R−1ul ispositive and that Rl > β−1.

ð1−θÞð1−δÞ, which, given the assumption, strictly exceedsthe user cost of leased capital ul ¼ rþδþm.

The condition m > ð1−RβÞð1−θÞð1−δÞ ensures that divi-dend paying firms do not lease. This is because for a dividendpaying firm, the user cost of purchased capital is boundedabove by rþδþð1−RβÞð1−θÞð1−δÞ, which, given the assump-tion, is strictly less than the user cost of leased capital.23 Thus,leasing is not dominating secured financing either. In otherwords, the condition implies that Rl > β−1 making borrowingmore by leasing too costly for dividend paying firms.

We now prove that severely constrained firms lease alltheir tangible assets, while dividend paying firms, andfirms with sufficiently high net worth, do not lease:

Proposition 9 (Optimality of leasing). Suppose m satisfiesAssumption 3. (i) Firms with sufficiently low net worth leaseall tangible capital, i.e., ∀s∈S, ∃wlðsÞ > 0, such that ∀w≤wlðsÞ,kl ¼ φk. (ii) Firms with sufficiently high net worth do notlease any tangible capital, i.e., ∀s∈S, ∃wlðsÞ < wðsÞ, such that∀w≥wlðsÞ, kl¼0.

The proposition holds for any Markov process forproductivity, and hence cash flows, and does not requireany further assumptions. The intuition is that when networth is sufficiently low, the firm's investment mustbe very low and hence its marginal product very high.But then the firm's financing need must be so severe, thatit must find the higher debt capacity of leasing worth-while. Firms with sufficiently high net worth, in contrast,do not value the higher debt capacity enough and hencechoose to buy all their capital instead of leasing it.24

5.2. Dynamic deterministic choice between leasing andfinancing

We consider the capital structure dynamics in thedeterministic case next. To start, consider the determinis-tic dynamics of firm financing without leasing. As long asnet worth is below a cutoff w, firms pay no dividends andaccumulate net worth over time which allows them toincrease the amount of capital they deploy. Once networth reaches w, dividends are positive and firms nolonger grow.

When leasing is an option, firms have to choose aleasing policy in addition to the investment, financing, andpayout policy. In this case, the financing dynamics are asfollows: when firms have low net worth, they lease all thetangible capital and purchase only the intangible capital.Over time, firms accumulate net worth and increase theirtotal capital. When they reach a certain net worth

of assets but not others. This means that there are some types of assetsthat firms never lease, others that they only lease when severelyconstrained, and yet others that they lease even when their net worthis quite high or they pay dividends.

A.A. Rampini, S. Viswanathan / Journal of Financial Economics 109 (2013) 466–492 483

threshold, they start to substitute owned capital for leasedcapital, continuing to accumulate net worth. Once firmsown all their tangible and intangible capital, they furtheraccumulate net worth and increase the capital stock untilthey start to pay dividends. At that point, capital staysconstant. The following proposition summarizes the deter-ministic dynamics:

Proposition 10 (Deterministic capital structure dynamics).(i) Suppose m¼ þ∞ (no leasing). For w≤w, no dividends arepaid and capital is strictly increasing in w and over time. Forw > w, dividends are strictly positive and capital is constantat a level k. (ii) Suppose m satisfies Assumption 3. For w≤w,no dividends are paid and capital is increasing in w and overtime. For w > w, dividends are strictly positive and capital isconstant at a level k. There exist wl < wl < w, such that forw≤wl all tangible capital is leased and for w∈ðwl,wlÞ capitalis constant at k and some but not all tangible capital is leasedwith the fraction of capital leased linearly decreasing in wbetween wl and wl.

Leasing allows constrained firms to grow faster. To seethis, note that the minimum amount of internal fundsrequired to purchase one unit of capital is ℘ðφÞ ¼1−R−1θφð1−δÞ, since the firm can borrow against fractionθ of the resale value of tangible capital, which is fraction φof capital. The minimum amount of internal funds requiredwhen tangible capital is leased is 1−φþR−1ulφ¼℘ðφÞ−R−1

l ð1−θÞφð1−δÞ, since the firm has to finance allintangible capital with internal funds ð1−φÞ and has topay the leasing fee on tangible capital up front ðR−1ulφÞ.Per unit of internal funds, the firm can hence buy capital inthe amount of one over these minimum amounts ofinternal funds. Under Assumption 3, leasing allows higherleverage, i.e., 1=ð℘ðφÞ−R−1

l ð1−θÞφð1−δÞÞ > 1=℘ðφÞ. Thus,leasing allows firms to deploy more capital and hence togrow faster.

Corollary 1 (Leasing and firm growth). Leasing enables firmsto grow faster.

The same economic intuition carries over to the sto-chastic case analyzed below. However, the high leveragethat leasing entails can change firms' hedging policyconsiderably.

Finally, we briefly discuss the possibility of linearsegments of the value function on ½wl,wl�. Consideringthe limit as m-ð1−βRÞð1−θÞð1−δÞ, which implies thatβRl-1, we have k-k and wl-w; when leasing is notmuch more expensive than the firm's discount rate, thefirm stops leasing at a level of net worth wl which is closeto the level at which the firm starts to pay dividends, w.But then a firm with current net worth wl has networth next period w′ðwlÞ-wcum ¼wþd, and analogously,w′ðwÞ > w for w≤wl and close to wl. Thus, the firm paysdividends next period, implying that μ′¼ 1 and, using theEuler equation for leasing μ¼ βRl > 1. Therefore, there is aninterval ½w,wl�D ½wl,wl� on which the value function islinear with slope βRl > 1.

We can moreover show that large enough m is asufficient condition to ensure strict concavity of the valuefunction ∀w≤wðsÞ in the general stochastic case, by

considering the limit as m-ð1−θÞð1−δÞ, implying thatRl-þ∞. The leasing Euler equation (18) at an interiorsolution implies that μðs′Þ=μ-0, and hence λðs′Þ > 0, ∀s′∈S.The investment Euler equation (11) in turn implies thatk-0 when leasing is interior and wðs′Þ ¼ Aðs′Þf ðkÞþð1−θφÞkð1−δÞ-0 < wðs′Þ, i.e., the firm does not pay dividends inany state next period, and hence the firm's net worth is inthe strictly concave part of the value function next period.But then the value function is strictly concave this periodas well and is strictly concave for all w≤wðsÞ, ∀s∈S, for msufficiently high.

5.3. Leasing, leverage, and risk management

To study the leasing, financing, and risk managementpolicy jointly, we consider the stochastic example withoutpersistence analyzed in Section 4.2 and introduce leasing.The parameters are mostly as before, with details in thecaption of Fig. 7 which displays the results. Panel Adisplays the investment and leasing policy, which is verysimilar to the one in the deterministic case; more con-strained firms lease more, if not all, of their tangiblecapital; further, as net worth increases, firms substituteowning and borrowing for leasing and eventually stopleasing altogether.

Particularly noteworthy are the implications for riskmanagement in Panel B. For high values of net worth, thefigure shows the by-now familiar pattern for risk manage-ment, with risk management increasing in net worth untilthe dividend paying region is reached and constant fromthere on. However, for lower values of net worth at whichthe firm leases a substantial amount of capital, riskmanagement first increases and then drops quite drama-tically and in fact drops back to zero. To understand thisresult, recall that leasing allows higher leverage and hencefirms which are very constrained choose to lease to be ableto lever up more. But the high implicit leverage reducesfirms' net worth in the low state, and, if this effect issufficiently strong, firms undo some of it by purchasingsome Arrow securities for the low cash flow state nextperiod. Effectively, firms use leasing to borrow more fromthe high state next period while at least partially undoingthe effects of higher leverage for the low state next periodvia risk management. Panel D shows that the multiplier onthe collateral constraint for the low state next periodðλðs1ÞÞ is non-monotone. It is positive for low values ofnet worth, consistent with Proposition 3, zero for aninterval in which firms hedge when leasing a substantialamount of capital, again positive when firms purchaseenough of their capital, and finally goes back to zero forsufficiently well capitalized firms.

The transition function of net worth is displayed in PanelC and is reminiscent of Panel C of Fig. 2. Note that the solidline which denotes wðs′1Þ crosses the 451 line below thepoint where firms stop leasing. Recall that this intersectionis the lower bound of the support of the stationarydistribution. This implies that even mature firms that arehit by a sequence of low productivity shocks eventually willreturn to leasing capital, i.e., firms engage in sale-leasebacktransactions under the stationary distribution. Sale-leaseback transactions by financially constrained firms are

0 0.2 0.4 0.60

0.2

0.4

0.6In

vest

men

t and

leas

ing

Current net worth0 0.2 0.4 0.6

0

0.02

0.04

0.06

Ris

k m

anag

emen

t

Current net worth

0 0.2 0.4 0.60

0.2

0.4

0.6

Net

wor

th n

ext p

erio

d

Current net worth0 0.2 0.4 0.6

0

0.5

1

Mul

tiplie

rsCurrent net worth

0 0.2 0.4 0.60

0.5

1

Leve

rage

Current net worth

Fig. 7. Leasing, leverage, and risk management. Panel A shows investment (k) (solid) and leasing (kl) (dashed); Panel B shows risk management for the lowstate ðh1ðs′ÞÞ; Panel C shows net worth next period in the low state next period ðwðs′1ÞÞ (solid) and in the high state next period ðwðs′2ÞÞ (dashed); Panel Dshows the scaled multipliers on the collateral constraints for the low state ðβλðs′1Þ=mÞ (solid) and for the high state ðβλðs′2Þ=mÞ (dashed); and Panel E showstotal leverage ðθðφk−klÞþklÞ=k (solid), debt leverage θðφk−klÞ=k (dashed), and rental leverage kl=k (dash-dotted); all as a function of current net worthw. Forother parameter values, see the caption of Fig. 2 except that m¼0.01 and φ¼ 0:8.

A.A. Rampini, S. Viswanathan / Journal of Financial Economics 109 (2013) 466–492484

relatively common in practice. In our model, such transac-tions enable firms that are hit by adverse shocks to free upnet worth and cut investment by less than they would haveto otherwise.

Panel E illustrates the implication that rental leverage isdecreasing in net worth while debt leverage is increasingand total leverage is approximately constant, consistentwith the empirical leverage-size relation documented inSection 2. Thus, our model matches these basic cross-sectional facts on leasing and the capital structure.

6. Conclusion

We argue that collateral determines the capital struc-ture. We provide a dynamic agency-based model of thecapital structure of a firm with a standard neoclassicalproduction function subject to collateral constraints due tolimited enforcement. In the model firms require bothtangible and intangible capital, and the fraction of tangibleassets required is a key determinant of leverage and thedynamics of firm financing.

There is a fundamental connection between firms'financing and risk management policy, since both involvepromises to pay by firms, and financing needs can overridehedging concerns. In fact, poorly capitalized firms opti-mally do not engage in risk management and firms abstainfrom risk management with positive probability under thestationary distribution. Our dynamic model which allowsexplicit analysis of the financing needs for investmentand the limits on firms' ability to promise to pay is criticalfor this result. Moreover, we prove the optimality ofincomplete hedging. It is not optimal for the firm to hedgeto the point that the marginal value of internal funds isequal across all states. Our dynamic analysis of riskmanagement shows that for plausible levels of the auto-correlation of productivity, firms may not hedge at all andthat even dividend-paying firms that are hit by a sequenceof adverse shocks eventually become so constrained thatthey cut risk management. An increase in risk can raise theamount of risk management and reduce investment due tothe financing needs for risk management. An increase incollateralizability or tangibility allows firms to lever more

A.A. Rampini, S. Viswanathan / Journal of Financial Economics 109 (2013) 466–492 485

and increase investment, and firms raise corporate riskmanagement to counterbalance the increase in the volati-lity of net worth that higher leverage would otherwiseimply. Thus, the comparative statics of our model implyinteresting empirical predictions for firm financing andrisk management both in the cross section and the timeseries.

Firms' ability to lease capital is explicitly taken intoaccount in contrast to previous dynamic models of firmfinancing and investment with financial constraints. Theextent to which firms lease is determined by firms'financial condition, and more constrained firms leasemore. Indeed, severely constrained firms find it optimalto lease all their tangible capital. Sale-leaseback transac-tions free up net worth and can be an optimal response toadverse cash flow shocks. Moreover, leasing enables firmsto grow faster. Leased capital is an important mode offinancing, in particular for constrained firms, and shouldbe taken into account not only in corporate finance, butalso in studies of the effect of financing on developmentand growth. Indeed, changes to financial accounting stan-dards are currently being considered by accounting boardsthat would result in adjustments to firms' capital structuresimilar to the ones suggested here.

We also provide stylized empirical facts that highlightthe importance of tangibility as a determinant of thecapital structure in the data. Firm leverage changes sub-stantially with the fraction of assets which is tangible.Moreover, the lack of tangible assets largely explains whysome firms have low leverage, and hence addresses the“low leverage puzzle.” Leased capital is quantitativelyimportant and further reduces the fraction of firms withlow leverage.

We conclude that the tangibility of assets and firms'ability to lease capital are critical determinants of thecapital structure. The simple form of the optimal contractin our dynamic agency-based capital structure modelshould facilitate the calibration and empirical implemen-tation, which has remained a challenge for other suchagency-based models. Moreover, due to its simplicity, ourmodel may also prove to be a useful framework to addressother theoretical questions in dynamic corporate financeand financial macroeconomics.

Appendix A. Proofs

Proof of Proposition 1. The proposition is proved in Lemmas1–5 below.

Lemma 1. Γðw,sÞ is convex, given (w,s), and convex in w andmonotone in the sense that w≤w implies Γðw,sÞDΓðw,sÞ.

Proof of Lemma 1. Suppose x,x∈Γðw,sÞ. For ϕ∈ð0,1Þ, letxϕ≡ϕxþð1−ϕÞx. Then xϕ∈Γðw,sÞ since Eqs. (2), (4), and(5), as well as the right-hand side of Eq. (3) are linear and,since f is concave,

Aðs′Þf ðkϕÞþðkϕ−kl,ϕÞð1−δÞ≥ϕ½Aðs′Þf ðkÞþðk−klÞð1−δÞ�þð1−ϕÞ½Aðs′Þf ðkÞþðk−klÞð1−δÞ�:

Let x∈Γðw,sÞ and x∈Γðw,sÞ. For ϕ∈ð0,1Þ, let xϕ≡ϕxþð1−ϕÞx. Since Eqs. (3), (4), and (5) do not involve w andw, respectively, and Γðw,sÞ is convex given w, xϕ satisfiesthese equations. Moreover, since x and x satisfy Eq. (2) at wand w, respectively, and Eq. (2) is linear in x and w, xϕsatisfies the equation at wϕ. Thus, xϕ∈Γðϕwþð1−ϕÞw,sÞ. Inthis sense, Γðw,sÞ is convex in w.If w≤w, then Γðw,sÞDΓðw,sÞ. □

Lemma 2. The operator T satisfies Blackwell's sufficientconditions for a contraction and has a unique fixed point V.

Proof. Suppose gðw,sÞ≥gðw,sÞ, for all ðw,sÞ∈Rþ � S. Then,for any x∈Γðw,sÞ,ðTgÞðw,sÞ≥dþβ∑

s′∈SΠðs,s′Þgðwðs′Þ,s′Þ≥dþβ∑

s′∈SΠðs,s′Þgðwðs′Þ,s′Þ:

Hence,

ðTgÞðw,sÞ≥ maxx∈Γðw,sÞ

dþβ∑s′∈S

Πðs,s′Þgðwðs′Þ,s′Þ ¼ ðTgÞðw,sÞ

for all ðw,sÞ∈Rþ � S. Thus, T satisfies monotonicity.Operator T satisfies discounting since

TðgþaÞðw,sÞ≤ maxx∈Γðw,sÞ

dþβ∑s′∈S

Πðs,s′ÞðgþaÞðwðs′Þ,s′Þ ¼ ðTgÞðw,sÞþβa:

Therefore, operator T is a contraction and, by the contrac-tion mapping theorem, has a unique fixed point V. □

Lemma 3. V is strictly increasing and concave in w.

Proof. Let xo∈Γðw,sÞ and xo∈Γðw,sÞ attain ðTgÞðw,sÞ andðTgÞðw,sÞ, respectively. Suppose g is increasing in w andsuppose w≤w. Then,

ðTgÞðw,sÞ ¼ doþβ∑s′∈S

Πðs,s′Þgðwoðs′Þ,s′Þ≥dþβ∑s′∈S

Πðs,s′Þgðwðs′Þ,s′Þ:

Hence,

ðTgÞðw,sÞ≥ maxx∈Γðw,sÞ

dþβ∑s′∈S

Πðs,s′Þgðwðs′Þ,s′Þ ¼ ðTgÞðw,sÞ,

i.e., Tg is increasing in w. Moreover, suppose w < w, then

ðTgÞðw,sÞ≥ðw−wÞþdoþβ∑s′∈S

Πðs,s′Þgðwoðs′Þ,s′Þ > ðTgÞðw,sÞ,

implying that Tg is strictly increasing. Hence, T mapsincreasing functions into strictly increasing functions,which implies that V is strictly increasing.Suppose g is concave. Then, for ϕ∈ð0,1Þ, let xo,ϕ≡ϕxoþ

ð1−ϕÞxo and wϕ≡ϕwþð1−ϕÞw, we have

ðTgÞðwϕ,sÞ≥do,ϕþβ∑s′∈S

Πðs,s′Þgðwo,ϕðs′Þ,s′Þ≥ϕ doþβ∑s′∈S

Πðs,s′Þgðwoðs′Þ,s′Þ" #

þð1−ϕÞ doþβ∑s′∈S

Πðs,s′Þgðwoðs′Þ,s′Þ" #

¼ ϕðTgÞðw,sÞþð1−ϕÞðTgÞðw,sÞ:

Thus, Tg is concave, and T maps concave functions intoconcave functions, which implies that V is concave. Since Vis increasing and concave in w, it must be continuousin w. □

Lemma 4. Without leasing, Vðw,sÞ is strictly concave in w forw∈intfw : dðw,sÞ ¼ 0g.

Proof. Without leasing, kl is set to zero throughout andall the prior results continue to hold. Supposew,w∈intfw : dðw,sÞ ¼ 0g, w > w. There must exist somestate st

n, where st ¼ fs0,s1,…,stg, which has strictly positive

A.A. Rampini, S. Viswanathan / Journal of Financial Economics 109 (2013) 466–492486

probability and in which the capital stock choice at w isdifferent from the choice at w, i.e., kðst

nÞ≠kðst

nÞ. Suppose

instead that kðstÞ ¼ kðstÞ, ∀st∈St ,t ¼ 0,1,… . Then there mustexist some state st

nnwith strictly positive probability in

which doðstnnÞ > doðstnnÞ and for which borrowing is notconstrained along the path of st

nn. Reducing doðstnnÞ by η

and paying out the present value at time 0 instead changesthe objective by ðR−t−βtÞðdoðstÞ−doðstÞÞ > 0, contradictingthe optimality of dðw,sÞ ¼ 0.Assume, without loss of generality, that koðs′nÞ≠koðs′nÞ, for

some s′n∈S. Rewrite the Bellman equation as

Vðw,sÞ ¼ maxx∈Γðw,sÞ,

xðs′Þ∈Γðwðs′Þ,s′Þ,∀s′∈S

dþβ∑s′∈S

Πðs,s′Þ dðs′Þþβ ∑s″∈S

Πðs′,s″ÞVðwðs″Þ,s″Þ( )

and note the convexity of the constraint set. Using the factthat koðs′nÞ≠koðs′nÞ, that V is concave and strictly increasing,and that f(k) is strictly concave, we have, for ϕ∈ð0,1Þ, anddenoting xo,ϕ ¼ ϕxoþð1−ϕÞxo and analogously for othervariables,

Vðwϕ,sÞ > do,ϕþβ∑s′∈S

Πðs,s′Þfdo,ϕðs′Þþβ ∑s″∈S

Πðs′,s″ÞVðwo,ϕðs″Þ,s″Þg

≥ϕVðw,sÞþð1−ϕÞVðw,sÞ:

The first (strict) inequality is due to the fact that for s″following s′n, Eq. (3) is slack and hence a net worth wðs″Þ >wo,ϕðs″Þ is feasible. The second inequality is due to con-cavity of V. □

Lemma 5. Assuming that for all s,s∈S such that s > s, Πðs,s′Þstrictly first-order stochastically dominates Πðs,s′Þ, V isstrictly increasing in s.

Proof. Let S¼ fs1,…,sng, with si−1 < si, ∀i¼ 2,…,n andN¼ f1,…,ng. Define the step function on the unit intervalb : ½0,1�-R as bðωÞ ¼∑n

i ¼ 1bðs′i Þ1Bi ðωÞ, ∀ω∈½0,1�, where 1 isthe indicator function, B1 ¼ ½0,Πðs,s′1Þ�, and

Bi ¼ ∑i−1

j ¼ 1Πðs,s′j Þ, ∑

i

j ¼ 1Πðs,s′j Þ

#, i¼ 2,…,n:

For s, define Bi, ∀i∈N, analogously. Let Bij ¼ Bi∩Bj, ∀i,j∈N, ofwhich at most 2n−1 are non-empty. Then, we can definethe step function b : ½0,1�-R as

bðωÞ ¼ ∑n

j ¼ 1∑n

i ¼ 1bðs′i Þ1Bij ðωÞ, ∀ω∈½0,1�:

We can then define the stochastic debt policy for Bj, ∀j∈N,with positive Lebesgue measure ðλðBjÞ > 0Þ, as bðs′i js′j Þ ¼ bðsi′Þwith conditional probability πðs′i js′j Þ ¼ λðBijÞ=λðBjÞ. Moreover,this implies a stochastic net worth

wðs′i js′j Þ ¼ Aðs′j Þf ðkÞþðk−klÞð1−δÞ−Rbðs′i js′j Þ

≥Aðs′i Þf ðkÞþðk−klÞð1−δÞ−Rbðs′i Þ ¼wðs′i Þ, a:e:,

with strict inequality when i < j, since under the assumptionin the statement of the lemma, λðBijÞ ¼ 0 whenever i > j.Moreover, wðs′i js′j Þ > wðs′i Þ with positive probability given thatassumption.

Now suppose s > s and gðw,sÞ≥gðw,sÞ, ∀w∈Rþ . Let xoattain ðTgÞðw,sÞ. ThenðTgÞðw,sÞ≥doþβ∑

s′∈SΠðs,s′Þ∑

s′∈Sπðs′js′Þgðwoðs′js′Þ,s′Þ

> doþβ∑s′∈S

Πðs,s′Þgðwoðs′Þ,s′Þ ¼ ðTgÞðw,sÞ:

Thus, T maps increasing functions into strictly increasingfunctions, implying that V is strictly increasing in s. □

This completes the proof of Proposition 1. □

Lemma 6. Under Assumption 2, capital and net worth in allstates are strictly positive, i.e., k > 0 and wðs′Þ > 0, ∀s′∈S.

Proof. We first show that if k > 0, then wðs′Þ > 0, ∀s′∈S.Note that (3) holds with equality. Using (4) we conclude

wðs′Þ ¼ Aðs′Þf ðkÞþðk−klÞð1−δÞ−Rbðs′Þ≥Aðs′Þf ðkÞþððk−klÞ−θðφk−klÞÞð1−δÞ > 0:

To show that k > 0, note that (7) and (10) imply that

μð1−R−1θφð1−δÞÞ≥∑s′∈S

Πðs,s′Þβμðs′Þ½Aðs′Þf kðkÞþð1−θφÞð1−δÞ�:

ð19ÞSuppose that μ¼ 1. Then k > 0 since μðs′Þ ¼ Vwðwðs′Þ,s′Þ≥1and hence the right-hand side goes to þ∞ as k-0, acontradiction. Suppose instead that μ > 1 and hence d¼0.For k sufficiently small, ∃s′∈S, such that μðs′Þ ¼ ðRβÞ−1μ. Butthen

0≥ ∑s′∈S\s

Πðs,s′Þβμðs′Þ½Aðs′Þf kðkÞþð1−θφÞð1−δÞ�

þfΠðs,s′ÞR−1½Aðs′Þf kðkÞþð1−θφÞð1−δÞ�−ð1−R−1θφð1−δÞÞgμ,where the first term is positive and the second term goesto þ∞ as k-0, a contradiction. □

Proof of Proposition 2. Part (i): By the envelope condition,μðw,sÞ ¼ Vwðw,sÞ. By Lemma 3, V is concave in w and henceμðw,sÞ is decreasing in w. The first-order condition (6)implies that μðw,sÞ≥1. If dðw,sÞ > 0, then μðw,sÞ ¼ 1 andμðw,sÞ ¼ 1 for all w≥w. Let wðsÞ ¼ inffw : dðw,sÞ > 0g.Part (ii): Suppose w > w≥wðsÞ and let xo attain Vðw,sÞ.

Since Vwðw,sÞ ¼ 1 for w≥wðsÞ, Vðw,sÞ ¼ Vðw,sÞþ Rww dv. Thechoice xo ¼ ½w−wþ do,ko,kl,o,woðs′Þ,boðs′Þ� attains Vðw,sÞand thus is weakly optimal.The optimal choice xo is unique in terms of the capital

stock ko. To see this, suppose instead that xo and ~xo bothattain Vðw,sÞ, but ko≠ ~ko. Recalling that Γðw,sÞ is convexand noting that

Aðs′Þf ðko,ϕÞþðko,ϕ−kl,o,ϕÞð1−δÞ > ϕ½Aðs′Þf ðkoÞþðko−kl,oÞð1−δÞ�

þð1−ϕÞ½Aðs′Þf ð ~koÞþð ~ko− ~kl,oÞð1−δÞ�,where xo,ϕ is defined as usual, we conclude that at xo,ϕ, (3)is slack, and hence there exists a feasible choice thatattains a strictly higher value, a contradiction. Indeed,xoðw,sÞ is unique in terms of koðw,sÞ, for all w and s.Now take w > w and let xo attain Vðw,sÞ. By part (i) of

Proposition 1, xo,ϕ∈Γðwϕ,sÞ. Moreover, if ko≠ko, then thereexists a feasible choice such that VðwϕÞ > ϕVðw,sÞþð1−ϕÞVðw,sÞ contradicting the linearity of V. Thus,koðw,sÞ ¼ koðsÞ, ∀w≥wðsÞ.

A.A. Rampini, S. Viswanathan / Journal of Financial Economics 109 (2013) 466–492 487

Part (iii): We now show that without leasing, theoptimal policy is unique also in terms of state-contingentnet worth, state-contingent borrowing, and the dividend.Define S0 ¼ fs′ : woðs′Þ < wðs′Þg and Sþ ¼ S\S0. Then ∀s′∈S0,woðs′Þ is unique. To see this, suppose instead that there is a~xo with ~woðs′Þ≠woðs′Þ for some s′∈S0 that also attainsVðw,sÞ. Then a convex combination xo,ϕ is feasible andattains a strictly higher value due to strict concavity ofVðw,s′Þ for w < wðs′Þ (part (iv) of Proposition 1). For thealternative optimal policy ~xo, define ~S0 and ~Sþ analogouslyto S0 and Sþ . By above, ~S0⊇S0. For any s′∈Sþ , ~woðs′Þ≥wðs′Þ.For suppose instead that ~woðs′Þ < wðs′Þ, then by strictconcavity of V for w < wðs′Þ a convex combination wouldagain constitute a feasible improvement. Thus, ~Sþ⊇Sþ andas a consequence Sþ≡ ~Sþ ¼ Sþ and S0≡ ~S0 ¼ S0. For s′∈S0,boðs′Þ is uniquely determined by (3). For s′∈Sþ , Eq. (4)holds with equality and determines boðs′Þ uniquely, andwoðs′Þ is then uniquely determined by (3). Hence, theoptimal policy is unique. Moreover, the policy determinedby part (ii) (with kl,oðw,sÞ set to 0) is the unique optimalpolicy for w > wðsÞ. □

Proof of Proposition 3. Using the first-order conditions forinvestment (7) and substituting for λðs′Þ using Eq. (10), wehave

1≥∑s′∈S

Πðs,s′Þβ μðs′Þμ

½Aðs′Þf kðkÞþð1−θφÞð1−δÞ�℘ðφÞ

≥Πðs,s′Þβ μðs′Þμ

Aðs′Þf kðkÞ℘ðφÞ : ð20Þ

Using the budget constraint (2) and the collateral con-straints (4), we have

w≥ð1−φÞkþðφk−klÞ℘þR−1ulkl,

and thus as w-0, investment k-0. But then the marginalproduct of capital f kðkÞ-þ∞, which implies by (20) thatμðs′Þ=μ-0, and using (10), that λðs′Þ=μ¼ ðRβÞ−1−μðs′Þ=μ-ðRβÞ−1 > 0, ∀s′∈S. Therefore, by continuity, ∀s∈S, ∃whðsÞ > 0,such that ∀w≤whðsÞ, λðs′Þ > 0, ∀s′∈S. □

Proof of Proposition 4. Part (i): If wðs′Þ≤wðs′þ Þ, thenμðs′Þ≥μðs′þ Þ by concavity. Moreover, μðs′Þþλðs′Þ ¼ μðs′þ Þþλðs′þ Þ, so λðs′Þ≤λðs′þ Þ. Suppose instead that wðs′Þ > wðs′þ Þ.Then λðs′Þ ¼ 0 since otherwise net worth in state s′ couldnot be larger than in state s′þ . But then μðs′Þ ¼ μðs′þ Þþ λðs′þ Þ,implying μðs′þ Þ≤μðs′Þ. If μðs′þ Þ ¼ μðs′Þ, then λðs′þ Þ ¼ λðs′Þ andthe assertion is true. If instead μðs′þ Þ < μðs′Þ, then byconcavity, wðs′þ Þ≥wðs′Þ, a contradiction.Part (ii): Suppose that λðs′Þ ¼ 0, ∀s′∈S. Then (9),

(10), and the envelope condition imply thatVwðwÞ ¼ μ¼ βμðs′ÞR¼ RβVwðwðs′ÞÞ < Vwðwðs′ÞÞ and, by con-cavity, w > wðs′Þ, ∀s′∈S. If d¼0, then saving the entire networth w at R would imply net worth Rw > wðs′Þ in allstates next period and hence attain a higher value of theobjective, contradicting optimality. Suppose d > 0 andhence w > w as defined in Proposition 2. That proposi-tion also implies that V(w) can be attained by the sameoptimal policy as at w except that d¼w−w. SinceVwðwðs′ÞÞ > 1, we conclude that wðs′Þ < w. But then payingout d¼w−w as before and saving w at R raises net worth

in all states next period and hence improves the value ofthe objective, a contradiction.Hence, ∃s′∈S such that λðs′Þ > 0, and, since λðs′Þ is increasing

in s′ by part (i), λðs′Þ > 0 where s′¼maxfs′ : s′∈Sg. If λðs′Þ > 0,∀s′, then wðs′Þ ¼ Aðs′Þf ðkÞþkð1−θφÞð1−δÞ−klð1−θÞð1−δÞ andhence wðs′Þ≠wðs′Þ, s≠s′. If λðs′Þ ¼ 0 for some s′, thenμðs′Þ ¼ μðs′Þþλðs′Þ > μðs′Þ and wðs′Þ < wðs′Þ.Suppose λðs′Þ ¼ 0 for some s′∈S. For any s−′ < s′, μðs−′Þ≥ μðs′Þ

by part (i), and μðs−′ Þ≤μðs−′ Þþλðs−′ Þ ¼ μðs′Þ, implyingμðs−′Þ ¼ μðs′Þ. Thus, the firm hedges all states below sh′¼max fs′ : λðs′Þ ¼ 0g. Note that the set may be empty, i.e., thefirm may not hedge at all. □

Lemma 7 (Net worth transition dynamics). SupposeΠðs,s′Þ ¼ Πðs′Þ, ∀s,s′∈S, and m¼ þ∞ (no leasing). (i)∀s′,s′þ∈S, such that s′þ > s′, wðs′þ Þ≥wðs′Þ, with strict inequal-ity iff s′þ > s′h where s′h is defined in Proposition 4. (ii) wðs′Þ isincreasing in w, ∀s′∈S; for w sufficiently small, wðs′Þ > w,∀s′∈S; and for w sufficiently large, wðs′Þ < w, ∀s′∈S. (iii)∀s′∈S, ∃w dependent on s′ such that wðs′Þ ¼w.

Proof. Part (i): By part (iv) of Proposition 1, V(w) is strictlyconcave unless w > w. By Proposition 4, the firm hedges alower set of states ½s′,…,sh′ � if at all. If s′þ≤sh′ , thenμðs′Þ ¼ μðs′þ Þ > 1 and hence wðs′Þ ¼wðs′þ Þ < w. If s′þ > sh′ ,then either λðs′Þ ¼ 0 and μðs′Þ > μðs′þ Þ, implyingwðs′þ Þ > wðs′Þ, or λðs′Þ > 0, which together with (3) and (4)at equality implies wðs′þ Þ > wðs′Þ.Part (ii): If d > 0, then wðs′Þ is constant by Proposition 2

and hence (weakly) increasing. If d¼ 0, then wðs′Þ isstrictly increasing in w for fs′jλðs′Þ ¼ 0g using strict con-cavity of V and the fact that VwðwÞ ¼ RβVwðwðs′ÞÞ. Forfs′jλðs′Þ > 0g, (3) and (4) hold with equality and hencewðs′Þ is increasing in w if k is. We now show that k isstrictly increasing in w for w≤w. If λðs′Þ > 0, ∀s∈S, thenk¼w=℘ðφÞ and k is hence strictly increasing. Supposeλðs′Þ ¼ 0, some s∈S. Then using the first-order conditionsfor investment (7) and substituting for λðs′Þ using (10),we have

1¼ ∑s′∈S

Πðs′Þβ μðs′Þμ

½Aðs′Þf kðkÞþð1−θφÞð1−δÞ�℘ðφÞ

¼ ∑fs′jλðs′Þ>0g

Πðs′Þβ μðs′Þμ

½Aðs′Þf kðkÞþð1−θφÞð1−δÞ�℘ðφÞ

þ ∑fs′jλðs′Þ ¼ 0g

Πðs′Þβ μðs′Þμ

½Aðs′Þf kðkÞþð1−θφÞð1−δÞ�℘ðφÞ : ð21Þ

Take wþ > w and suppose that kþ≤k with the usual abuseof notation. Then f kðkþ Þ≥f kðkÞ. Moreover, for fs′jλðs′Þ ¼ 0g,μðs′Þ=μ¼ ðRβÞ−1. Since μþ < μ, (21) implies that ∃s′ suchthat μþ ðs′Þ < μðs′Þ. But kþ≤k implies that for fs′jλðs′Þ > 0g,wþ ðs′Þ≤wðs′Þ and hence μþ ðs′Þ≥μðs′Þ, a contradiction.Hence, k and wðs′Þ are strictly increasing in w for w≤w.To show that ∃w such that wðs′Þ > w, ∀s′∈S, note that

Proposition 3 implies that for w sufficiently small, λðs′Þ > 0,∀s′∈S, and thus w¼ k=℘ðφÞ and wðs′Þ ¼ Aðs′Þf ðkÞþkð1−θφÞð1−δÞ. But thendwðs′Þdw

¼ Aðs′Þf kðkÞþð1−θφÞð1−δÞ℘ðφÞ

¼ Aðs′Þf kðkÞkþkð1−θφÞð1−δÞ℘ðφÞk <

wðs′Þw

,

A.A. Rampini, S. Viswanathan / Journal of Financial Economics 109 (2013) 466–492488

where the inequality uses the strict concavity of f ð�Þ.Moreover, as w-0, f kðkÞ-þ∞ and thus dwðs′Þ=dw-þ∞and wðs′Þ=w > 1 for w sufficiently low.To show that ∃w such that wðs′Þ < w, ∀s′∈S, it is sufficient

to show that such a w exists for wðs′Þ given part (i). ByProposition 2, ∀w≥w, the optimal policy xo is independentof w (except for the current dividend), and thuswoðs′Þ ¼ Aðs′Þf ðkoÞþkoð1−δÞ−Rboðs′Þ < þ∞, and hence forw > woðs′Þ the assertion holds.Part (iii): By the theorem of the maximum wðs′Þ is

continuous in w, and the intermediate value theoremand part (ii) hence imply the result. □

Proof of Proposition 5. Part (i): Denoting the wealth level asdefined in part (iii) of Lemma 7 for the lowest state s′ by wand using wðs′Þ ¼w, (9), (10), and the envelope condition,we have VwðwÞ ¼ RβVwðwÞ þRβλðs′Þ and thus λðs′Þ > 0, andby Proposition 4 the firm abstains from risk managementaltogether at w. The level of net worth w is unique, sinceeither d > 0 at w, and hence wðs′Þ is constant, or d¼0 andthen using λðs′Þ > 0, ∀s∈S, and evaluating dwðs′Þ=dw as inpart (ii) at w

dwðs′Þdw

�����w¼w¼ Aðs′Þf kðkÞkþkð1−θφÞð1−δÞ

℘ðφÞk

�����w¼w<wðs′Þw

¼ 1:

Thus, the locus of wðs′Þ crosses the 451 line from above, i.e.,at most once. Moreover, wðs′Þ=w < 1,∀w > w.Part (ii): We adapt Theorem 12.12 from Stokey, Lucas,

and Prescott (1989). Let εw > 0 and wbnd ¼ Aðs′Þf ðkbndÞþkbndð1−δÞ where kbnd such that Aðs′Þf kðkbndÞ ¼ rþδ. Definethe induced state space W ¼ ½εw,wbnd�⊂R with its Borelsubsets W. Take P to be the induced transition function onðW ,WÞ, with the associated operator on bounded contin-uous functions T : BðW ,WÞ-BðW ,WÞ and the associatedoperator on probability measures Tn : PðW ,WÞ-PðW ,WÞ.We need to show that P is monotone (i.e., for any

bounded, increasing function g, the function Tg definedby ðTgÞðwÞ ¼ R

gðw′ÞPðw,dw′Þ, ∀w, is also increasing), hasthe Feller property (i.e., for any bounded, continuousfunction g, the function Tg is also continuous), and∃wo∈W , ε > 0, and N≥1, such that PNðεw,½wo,wbnd�Þ≥ε andPNðwbnd,½εw,wo�Þ≥ε.Take any bounded, increasing function g. Then

ðTgÞðwÞ ¼∑s′∈sΠðs′Þgðwðs′ÞðwÞÞ is increasing since wðs′ÞðwÞis increasing by part (ii) of Lemma 7. For any bounded,continuous function g, ðTgÞðwÞ is moreover continuous aswðs′ÞðwÞ is continuous by the theorem of the maximum.From Lemma 7 and part (i), we know that levels of net

worth below w and above woðs′Þ are transient. We nowprovide an explicit characterization of the stationary solu-tion when w≤w and then show that otherwise, wo can beset to wo ¼w, where w is the level of net worth abovewhich the firm pays dividends (see Proposition 2).Ifw≤w, then the stationary distribution is a subset of the

dividend paying set and the solution is quasi-determinis-tic, in the sense that capital ko is constant under thestationary distribution. In this case, ko solves

1¼ β∑s′∈SΠðs′ÞAðs′Þf kðkoÞþð1−θφÞð1−δÞ

℘ðφÞ :

Then w ¼℘ðφÞko and

w ¼wðs′Þ ¼ Aðs′Þf ðkoÞþkoð1−θφÞð1−δÞ

≥w ¼℘ðφÞko ¼ β ∑s′∈S

Πðs′ÞAðs′Þf kðkoÞkoþkoð1−θφÞð1−δÞ !

:

The condition for w≤w is thus

ð1−βÞð1−θφÞð1−δÞ≥β∑s′∈S

Πðs′ÞAðs′Þf kðkoÞ−Aðs′Þ f ðkoÞko

: ð22Þ

Concavity implies that f ðkoÞ=ko > f kðkoÞ and thus a suffi-cient condition is that Aðs′Þ≥β∑s′∈SΠðs′ÞAðs′Þ. If f ðkÞ ¼ kα, asufficient condition is Aðs′Þ≥αβ∑s′∈SΠðs′ÞAðs′Þ. Note that inthe quasi-deterministic case, the firm abstains from riskmanagement with probability 1 under the stationarydistribution.If (22) is violated, then w < w. Moreover, the firm cannot

be paying dividends in the lowest state next period if itcurrently is paying dividends, for if it paid dividends, itwould choose ko as determined above and

wðs′Þ ¼ Aðs′Þf ðkoÞþkoð1−θφÞð1−δÞ

< β ∑s′∈S

Πðs′ÞAðs′Þf kðkoÞkoþkoð1−θφÞð1−δÞ !

¼℘ðφÞko ¼w,

a contradiction. Thus, if wo ¼w, then Pðwbnd,½εw,w�Þ≥Πðs′Þand N1 ¼ 1 and ∃ε1 > 0 such that Πðs′Þ > ε1 > 0.Now given w¼ εw, and since the net worth could be paid

out, the objective has to exceed the value of net worth,

0 < εw ¼w≤E ∑∞

t ¼ 0βtdt

� �¼ E ∑

N

t ¼ 0βtdt

� �þE ∑

∞

t ¼ Nþ1βtdt

" #:

Note that dt≤doðs′Þ ¼woðs′Þ−w and the last expectationabove is thus bounded by βNþ1doðs′Þ=ð1−βÞ. For any εd > 0such that εw > εd, ∃N2 < ∞ such that the last expectation isless than εd. But then PN2 ðεw,½w,wbnd�Þ≥ðεw−εdÞ=doðs′Þ > 0.Let ε2≡ðεw−εdÞ=doðs′Þ, ε¼minfε1,ε2g, and N¼maxfN1,N2g.Finally, when w < w, dwðs′Þ=dw < 1 at w and wðs′Þ < w forall w≥w, and thus a sufficiently long sequence of thelowest productivity realization results in a net worth in aneighborhood of w and hence the firm abstains from riskmanagement with positive probability. □

Proof of Proposition 6. We first show that, for each s,k≤kfbðsÞ where kfb(s) solves rþδ¼∑s′∈SΠðs,s′ÞAðs′Þf kðkfbðsÞÞ, i.e., kfb(s) is the capital level in the frictionlesscase when cash flows are discounted at r. Using the first-order conditions (7) and (10), the investment Euler equa-tion implies

1¼ ∑s′∈S

Πðs,s′Þβ μðs′Þμ

½Aðs′Þf kðkÞþð1−θφÞð1−δÞ�℘ðφÞ

≤R−1∑s′∈S

Πðs,s′ÞAðs′Þf kðkÞþð1−θφÞð1−δÞ

℘ðφÞ : ð23Þ

Since the term on the right of the inequality equals 1 atkfb(s) and decreases in k, the result is immediate.Second, we show that, given s, k is weakly increasing in

w. We focus on the non-dividend paying region (as k isconstant otherwise). Let wþ > w. By the concavity of thevalue function (Proposition 1, Part (iii)), μþ≤μ. Suppose

A.A. Rampini, S. Viswanathan / Journal of Financial Economics 109 (2013) 466–492 489

kþ < k. If λðs′Þ ¼ 0, ∀s′∈S, at w and wþ , then k¼ kfbðsÞ atboth levels of wealth, a contradiction. Thus, suppose ∃s′∈S,such that λðs′Þ > 0 at w and assume that fs′ : λðs′Þ > 0g is thesame at w and wþ . Using (23), we have

1¼ ∑fs′:λðs′Þ>0g

Πðs,s′Þβ μðs′Þμ

½Aðs′Þf kðkÞþð1−θφÞð1−δÞ�℘ðφÞ

þ ∑fs′:λðs′Þ ¼ 0g

Πðs,s′ÞR−1 ½Aðs′Þf kðkÞþð1−θφÞð1−δÞ�℘ðφÞ :

The second summation must be higher at wþ and hence∃s′∈fs′ : λðs′Þ > 0g, such that μþ ðs′Þ < μðs′Þ. But for such an s′,wþ ðs′Þ < wðs′Þ and hence μþ ðs′Þ≥μðs′Þ, a contradiction.Therefore, kþ≥k, i.e., investment is weakly increasing in w.Finally, we show that ∃s, such that ∀w, there is an s′ for

which λðs′Þ > 0, i.e., incomplete hedging in state s isoptimal for all wealth levels. To see this, note that theremust be a current state (w,s) and state next period s′ suchthat d¼0 at (w,s) and dðs′Þ > 0, implying that λðs′Þ > 0 at(w,s). (Otherwise, the firm would either never pay adividend, which is not possible, or would always pay adividend in which case λðs′Þ > 0, ∀s′, in any state (w,s).)Since the firm pays a dividend in state s′ next period,wðs′Þ > wðs′Þ. Take wþ > w. Since k is weakly increasing inw, wþ ðs′Þ≥wðs′Þ > wðs′Þ, i.e., the firm still pays dividends ins′ and is hence constrained against s′ at ðwþ ,sÞ, too. Pickingwþ≥wðsÞ implies that λðs′Þ > 0 and thus kðsÞ < kfbðsÞarguing as in the first step of the proof. But then, given s,k≤kðsÞ < kfbðsÞ, which implies that ∃s′ such that λðs′Þ > 0, ∀w(using the argument in the firststep of the proof again).Note, however, that the state s′ against which the firm isconstrained in state s may not be the same for wealthlevels below the w we started with. Since state s haspositive probability, hedging is incomplete with positiveprobability. (We can moreover conclude that k is strictlyincreasing in w for w≤wðsÞ in state s. To see this, use thefact that the value function is strictly concave (Proposition1, Part (iv)) and the fact that fs′ : λðs′Þ > 0g≠∅ and argue asin the second step of the proof.) □

Proof of Proposition 7. Part (i): Let S be the support of s′ forAðs′Þ. Without loss of generality, consider Aþ ðs′Þ such thatAþ ðs′Þ ¼ Aðs′Þ, ∀s′∈S\s′. For s, let Aþ ðs′Þ ¼ Aðs′Þþ ~a whereE½ ~ajs� ¼ 0. Thus, Aþ ðs′Þ is obtained from Aðs′Þ by a mean-preserving spread. Note that given net worth w, the set offeasible choices coincide for the problem with Aðs′Þ andAþ ðs′Þ, and let x ðxþ Þ attain V(w) ðV þ ðwÞÞ, respectively. Byconcavity of the value function,

V þ ðwÞ ¼ dþ þβE½V þ ðw′þ Þ�≤dþ þβE½V þ ðE½w′þ js′�Þ�≤dþ þβfE½d′þ ðE½w′þ js′�Þ�þβE½Vþ ðE½w″þ ðE½w′þ js′�Þ�Þ�g ð24Þ

and so forth, where d′þ ðE½w′þ js′�Þ denotes the optimaldividend given net worth E½w′þ js′� and analogously forother variables. Now consider x such that d ¼ dþ , k ¼ kþ ,and bðs′Þ ¼ E½b′þ js′�, implying that ∀s′, wðs′Þ ¼ Aðs′Þf ðkÞþkð1−δÞ−Rbðs′Þ ¼ E½w′þ js′�. Therefore,VðwÞ≥dþβE½Vðw′Þ�≥dþβfE½d′ðw′Þ�þβE½Vðw″ðw′ÞÞ�g, ð25Þwhere d′ðw′Þ is chosen analogously to match the optimalpolicy for Aþ ðs′Þ given net worth wðs′Þ, and so forth. Theright-hand sides of (24) and (25) coincide in the limit.Therefore, VðwÞ≥V þ ðwÞ.

Part (ii): For a dividend paying firm, the investmentEuler equation (23) in the deterministic case determinesk0 as 1¼ βðA′0f kðkÞþð1−θÞð1−δÞÞ=℘, and in the stochasticcase implies that ks solves

1¼ ∑s′∈S

πðs′Þβμsðs′ÞAsðs′Þf kðkÞþð1−θÞð1−δÞ

℘: ð26Þ

Suppose μsðs′Þ ¼ 1, ∀s′∈S. Then k0 solves (26) and hencek0 ¼ ks. Suppose instead ∃s′∈S such that μsðs′Þ > 1, thenks > k0.In the deterministic case, investment is k¼w=℘ for

w≤w0 and k0 ¼w0 otherwise. Since k0≤ks, we have

w0=℘¼ k0≤ks≤ws=℘, implying that w0≤ws (with strict

inequality iff ks > k0). Using the fact that k is continuousand strictly increasing in w (see the proof of Proposition 6),we can provide a more explicit comparison: for w≤whs,ks ¼ k0, and for w∈½whs,w�, ks≤w=℘¼ k0; indeed, ifws > w0, ∃w∈½w0,ws� such that ks < k0, ∀w < w, andks≥k0, ∀w≤w.Suppose that S¼ fs,sg and that a dividend paying firm is

hedging, then 1¼ βRμðs′Þ, so the investment Euler equa-tion (26) can be written as

1¼ πðs′ÞR−1 Aðs′Þf kðkÞþð1−θÞð1−δÞ℘

þπðs′Þβ Aðs′Þf kðkÞþð1−θÞð1−δÞ℘

:

ð27ÞTotally differentiating given a mean-preserving spread inthe productivity process dAðs′Þ < 0 < dAðs′Þ with ∑s′∈Sπðs′ÞAðs′Þ ¼ 0, we have by continuity

0¼ ½πðs′Þ dAðs′Þþπðs′ÞβR dAðs′Þ�f kðkÞþ½πðs′ÞAðs′Þþπðs′ÞβRAðs′Þ�f kkðkÞ dk:

The first term in brackets is negative and since f kkðkÞ < 0,we must have dk < 0. □

Proof of Proposition 8. Part (i): Let x∈Γθðw,sÞ be the set ofx¼ fd,k,wðs′Þ,bðs′Þg∈R2þS

þ � RS such that for all s′∈S

wþ ∑s′∈S

Πðs,s′Þbðs′Þ≥dþk, ð28Þ

Aðs′Þf ðkÞþkð1−δÞ≥wðs′ÞþRbðs′Þ, ð29Þand

θkð1−δÞ≥Rbðs′Þ, ð30Þgiven θ. Let x ðxþ Þ be optimal at θ ðθþ Þ. Note that Γθðw,sÞ isincreasing in the sense that Γθðw,sÞDΓθþ ðw,sÞ and hencex∈Γθþ ðw,sÞ implying that V þ ðw,sÞ≥Vðw,sÞ. But at x, (30)given θþ is slack, so x cannot be optimal if s is the statefrom Proposition 6 for which λðs′Þ > 0 for some s′∈S.Otherwise, let the firm with θþ choose the optimal policyof the firm with θ until state s from Proposition 6 isreached, for which fs′ : λðs′Þ > 0g≠∅. In state s, (30) is slackand hence choosing x cannot be optimal. By uniqueness ofthe optimal policy, we must have V þ ðw,sÞ > Vðw,sÞ.Part (ii): If S¼ fs,sg and Πðs,s′Þ ¼ πðs′Þ, ∀s,s′∈S, the invest-

ment Euler equation of a dividend paying firm whichhedges (27) totally differentiated w.r.t. θ implies that

0¼ ∑s′∈S

πðs′Þβμðs′ÞAðs′Þf kkðkÞ℘

dkdθ

A.A. Rampini, S. Viswanathan / Journal of Financial Economics 109 (2013) 466–492490

þ ∑s′∈S

πðs′Þβμðs′ÞR−1ð1−δÞ℘

Aðs′Þf kðkÞþð1−θÞð1−δÞ℘

−R� �

:

Using the investment Euler equation (26) and the fact that1 > ∑s′∈Sπðs′Þβμðs′ÞR, since ∃s′∈S, such that λðs′Þ > 0, thesecond term must be strictly positive and thereforedk=dθ > 0. □

Proof of Proposition 9. Part (i): Proceeding as in the proof ofProposition 3, we conclude that as w-0, investment k-0and λðs′Þ=μ-ðRβÞ−1 > 0, ∀s′∈S. Therefore, given Assumption3, the user cost of owned tangible capital exceeds the usercost of leased capital in the limit asupðw,sÞ≡rþδþ∑s′∈SΠðs,s′ÞRβλðs′Þ=μð1−θÞð1−δÞ goes torþδþð1−θÞð1−δÞ > rþδþm¼ ul. Consequently, using (17)we obtain R=μðν l−ν lÞ ¼ upðw,sÞ−ul > 0, so ν l > 0 (andν l ¼ 0); all tangible capital is leased in the limit. Bycontinuity, ∀s∈S, ∃wlðsÞ > 0, such that ∀w≤wlðsÞ, upðw,sÞ >ul and all tangible capital is leased.Part (ii): Suppose the firm pays dividends and hence

μ¼ 1. Taking the first term on the right-hand side of theEuler equation for leasing capital (18) and using the factthat μðs′Þ≥1, the definition of ℘ and ul, and the secondinequality from Assumption 3, we have

∑s′∈S

Πðs,s′Þβ μðs′Þμ

ð1−θÞð1−δÞ℘−R−1ul

≥βð1−θÞð1−δÞ

R−1ðð1−θÞð1−δÞ−mÞ> βR

11−ð1−βRÞ ¼ 1,

which implies that ν l > 0. Therefore, firms with net worthw≥wðsÞ do not lease. Moreover, since ν l > 0 at wðsÞ, bycontinuity ∃ε > 0 such that for w∈ðwðsÞ−ε,wðsÞ� the multi-plier ν l > 0, and thus ∃wlðsÞ < wðsÞ such that kl¼0 forw≥wlðsÞ, ∀s∈S. □

Proof of Proposition 10. Part (i): Denote with a primevariables which in the stochastic case were a function ofthe state tomorrow, i.e., w′, b′, μ′, and λ′. We first char-acterize a steady state. From (9) and the envelope condi-tion, we have μ′¼ μ. Then (10) implies λ′¼ ððRβÞ−1−1Þμ > 0,i.e., the firm is constrained in the steady state, and (7) canbe written as 1−½R−1θφþβð1−θφÞ�ð1−δÞ ¼ βA′f kðkÞ, whichimplicitly defines the steady state value of capital k.Denoting steady state variables with a bar, using (4) and(3) at equality, we have b ¼ R−1θφkð1−δÞ and the cum-dividend net worth in the steady state wcum ¼ A′f ðkÞþkð1−θφÞð1−δÞ. Dividends in the steady state are

d ¼ A′f ðkÞ−kð1−½R−1θφþð1−θφÞ�ð1−δÞÞ> A′f ðkÞ−β−1kð1−½R−1θφþβð1−θφÞ�ð1−δÞÞ

¼Z k

0fA′f kðkÞ−β−1ð1−½R−1θφþβð1−θφÞ�ð1−δÞÞg dk > 0

and hence μ ¼ 1. The lowest level of net worth for which kis feasible is w≡wcum−d, and w is the ex-dividend networth in the steady state. Thus, for w < w, k < k. Using thefirst-order conditions and the envelope condition, we have

VwðwÞVwðw′Þ ¼

μ

μ′¼ β

A′f kðkÞþð1−θφÞð1−δÞ℘ðφÞ :

Note that the right-hand side equals 1 at k and isdecreasing in k. Thus, if k < ð>Þk, VwðwÞ > ð<ÞVwðw′Þ andw < ð>Þw′. Since k < k for w < w, w < w′ and w increasesover time. If w > w, then either d > 0 (and VwðwÞ ¼ 1) or

d¼0 and k > k. In the first case, concavity and the fact thatVwðw′Þ≥1 imply Vwðw′Þ ¼ 1 and hence k¼ k. In the secondcase, w > w′, but simply saving w at R would result inhigher net worth and hence k > k cannot be optimal.Part (ii): Consider the optimal policy without leasing

from part (i). The user cost of tangible capital at w isup ¼ rþδþð1−RβÞð1−θÞð1−δÞ < ul under Assumption 3.Thus, there is no leasing at w and the solution is as beforeas long as w is sufficiently high. Recall that as w decreases,μ′=μ decreases and hence λ′=μ increases. Note also thatunder Assumption 2, as w goes to zero, k and μ′=μ go tozero and hence λ′=μ goes to ðRβÞ−1 and up(w) goes torþδþð1−θÞð1−δÞ > ul given Assumption 3. Whenλ′=μ¼ ðRβÞ−1m=ðð1−θÞð1−δÞÞ, ul ¼ upðwÞ and (7) simplifies to

℘ðφÞ ¼ R−1 1−m

ð1−θÞð1−δÞ

� �½A′f kðkÞþð1−θφÞð1−δÞ�,

which defines k. At wl≡℘ðφÞk all the tangible capital isowned and at wl≡ð1−φ þR−1ulφÞk all the tangible capitalis leased. For w∈½wl,wl�, leased capital is

kl ¼℘ðφÞk−w℘−R−1ul

,

which is linear and decreasing in w. Moreover, w′ islinearly decreasing in kl and hence linearly increasing inw. For w < wl, k¼w=ð1−φþR−1ulφÞ and w′¼ A′f ðkÞþkð1−φÞð1−δÞ. □

Appendix B. Enforcement versus collateral constraints

In this appendix we prove the equivalence of enforce-ment constraints and collateral constraints. For simplicity,we abstract from leasing, but it is straightforward toextend the proof by recognizing that the firm cannotabscond with leased capital. The firm's problem withlimited enforcement at any time τ≥0, denoted PτðwðsτÞÞ,is to choose the sequence of dividends, capital levels, andnet payments to the lender fxðstÞgt≥τ , where xðstÞ ¼ fdðstÞ,kðstÞ,pðstÞg and st≡fs0,s1,…,stg, given current net worthwðsτÞ and history sτ , to maximize the discounted expectedvalue of future dividends

Eτ ∑∞

t ¼ τβðt−τÞdt

� �ð31Þ

subject to the budget constraints

wðsτÞ≥dðsτÞþkðsτÞþpðsτÞ, ð32Þ

AðstÞf ðkðst−1ÞÞþkðst−1Þð1−δÞ≥dðstÞþkðstÞþpðstÞ, ∀t> τ,

ð33Þthe lender's participation constraint,

Eτ ∑∞

t ¼ τR−ðt−τÞpt

� �≥0, ð34Þ

and the enforcement constraints

Eτ′ ∑∞

t ¼ τ′βðt−τ′Þdt

� �≥Eτ′ ∑

∞

t ¼ τ′βðt−τ′Þdt

� �, ∀τ′≥τ and ∀fdðstÞg∞t ¼ τ′,

ð35Þwhere fdðstÞg∞t ¼ τ′ together with fkðstÞg∞t ¼ τ′ and fpðstÞg∞t ¼ τ′solve Pτ′ðwðsτ′ÞÞ given net worth wðsτ′Þ ¼ Aðsτ′Þf ðkðsτ′−1ÞÞ

A.A. Rampini, S. Viswanathan / Journal of Financial Economics 109 (2013) 466–492 491

þð1−θφÞkðsτ′−1Þð1−δÞ, i.e., the same problem with a differ-ent level of net worth. We say a sequence of net paymentsis implementable if it satisfies the lender's participationconstraint and the enforcement constraints.

Proposition 11 (Equivalence of enforcement and collateralconstraints). (i) Any sequence of net payments fpðstÞg∞t ¼ τ isimplementable in problem PτðwðsτÞÞ iff

θφkðsτ′−1Þð1−δÞ≥Eτ′ ∑∞

t ¼ τ′R−ðt−τ′Þpt

� �, ∀τ′≥τ, ð36Þ

i.e., the present value of the remaining net payments neverexceeds the current collateral value. (ii) Moreover, the set ofsequences of net payments that satisfy (36) is equivalent tothe set of sequences of one-period state-contingent claimsfbðstÞg∞t ¼ τ which satisfy

θφkðst−1Þð1−δÞ≥RbðstÞ, ∀t > τ: ð37Þ

Proof. Part (i): Suppose the sequence fpðstÞg∞t ¼ τ is such that(36) is violated for some sτ′, τ′ > τ, i.e.,

θφkðsτ′−1Þð1−δÞ < Eτ′ ∑∞

t ¼ τ′R−ðt−τ′Þpt

� �:

Without loss of generality, assume τ′¼ τþ1. Supposethe firm defaults in state sτþ1 at time τþ1 and issues anew sequence fpðstÞg∞t ¼ τþ1 such that Eτþ1½∑∞

t ¼ τþ1R−ðt−ðτþ1ÞÞpt � ¼ 0 with pðstÞ ¼ pðstÞ, ∀t > τþ1, andpðsτþ1Þ ¼ −Eτþ1½∑∞

t ¼ τþ2R−ðt−ðτþ1ÞÞpt � (which has zero net

present value by construction), keeping the dividend andinvestment policies the same, except for the dividend instate sτþ1 at time τþ1. This dividend increases since thefirm makes payment pðsτþ1Þ instead of pðsτþ1Þ whilebuying back the tangible assets which have been seized,i.e., θφkðsτÞð1−δÞ, and thus

dðsτþ1Þ ¼ dðsτþ1Þþðpðsτþ1Þ−pðsτþ1Þ−θφkðsτÞð1−δÞÞ

¼ dðsτþ1Þþ Eτþ1 ∑∞

t ¼ τþ1R−ðt−ðτþ1ÞÞpt

" #−θφkðsτÞð1−δÞ

!> dðsτþ1Þ:

Such a deviation would hence constitute an improvement,a contradiction. Conversely, if (36) is satisfied, ∀τ′≥τ, thendefaulting cannot make the firm better off.Part (ii): Take any sequence of net payments fpðstÞg∞t ¼ τ

and define

Rbðsτ′Þ≡Eτ′ ∑∞

t ¼ τ′R−ðt−τ′Þpt

� �≤θφkðsτ′−1Þð1−δÞ, ∀τ′ > τ:

Then, Rbðsτ′Þ ¼ pðsτ′ÞþR−1Eτ′½Rbðsτ′þ1Þ� and thus pðsτ′Þ ¼Rbðsτ′Þ−Eτ′½bðsτ′þ1Þ� and Eq. (33) can be rewritten as

AðstÞf ðkðst−1ÞÞþkðst−1Þð1−δÞþEt ½bðstþ1Þ�≥dðstÞþkðstÞþRbðstÞ, ∀t > τ: ð38Þ

Similarly, setting bðsτÞ ¼ 0 yields pðsτÞ ¼−Eτ½bðsτþ1Þ�. Thus,any sequence of net payments satisfying (36) can beimplemented with a sequence of one-period contingentclaims satisfying (37).Conversely, take any sequence fbðstÞg∞t ¼ τ satisfying (37)

and define pðstÞ ¼ RbðstÞ−Et ½bðstþ1Þ�, ∀t≥τ. Then, ∀τ′ > τ,

Eτ′ ∑∞

t ¼ τ′R−ðt−τ′Þpt

� �¼ Eτ′ ∑

∞

t ¼ τ′R−ðt−τ′ÞðRbt−btþ1Þ

� �¼ Rbðsτ′Þ≤θφkðsτ′−1Þð1−δÞ,

i.e., the sequence of one-period contingent claims satisfy-ing (37) can be implemented with a sequence of netpayments satisfying (36). □

Thus, the sequence problem Pτ in Eqs. (31)–(35) isequivalent to maximizing (31) subject to wðsτÞ≥dðsτÞþkðsτÞ−Eτ½bðsτþ1Þ�, (37), and (38), which can be writtenrecursively as in Eqs. (1)–(4) by defining net worth afterrepayment of one-period claims issued previously aswðstÞ≡AðstÞf ðkðst−1ÞÞþkðst−1Þð1−δÞ−RbðstÞ, ∀t > τ.

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