Financial Markets and Firm Dynamics

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Financial Markets and Firm Dynamics

By THOMAS F. COOLEY AND VINCENZO QUADRINI*

Recent studies have shown that the dynamics of firms (growth, job reallocation, andexit) are negatively correlated with the initial size of the firm and its age. In thispaper we analyze whether financial factors, in addition to technological differences,are important in generating these dynamics. We introduce financial-market frictionsin a basic model of industry dynamics with persistent shocks and show that thecombination of persistent shocks and financial frictions can account for the simul-taneous dependence of firm dynamics on size (once we control for age) and on age(once we control for size). (JEL D21, G3, L2)

Recent studies of the relationship betweenfirm size and growth have overturned the con-clusion of Gibrats Law, which holds that firmsize and growth are independent. Studies byDavid S. Evans (1987) and Bronwyn H. Hall(1987) show that the growth rate of manufac-turing firms and the volatility of growth is neg-atively associated with firm size and age. Firmsize and age also play an important role incharacterizing the dynamics of job reallocation.Steven J. Davis et al. (1996) show that the ratesof job creation and job destruction in U.S. man-

ufacturing firms are decreasing in firm age andsize and that, conditional on the initial size,small firms grow faster than large firms. Theempirical regularities of firm dynamics are:1

Conditional on age, the dynamics of firms(growth, volatility of growth, job creation, job destruction, and exit) are negatively re-lated to the size of firms.

Conditional on size, the dynamics of firms

(growth, volatility of growth, job creation, job destruction, and exit) are negatively re-lated to the age of firms.

We will refer to the first regularity as thesize dependence and to the second regularityas the age dependence.

Existing models of the growth of firms ex-plain these features as arising from learningabout the technology or from persistent shocksto the technology. Examples are the modelsstudied by Boyan Jovanovic (1982); Hugo A.

Hopenhayn (1992); Hopenhayn and RichardRogerson (1993); Jeffrey R. Campbell (1998);Campbell and Jonas D. M. Fisher (2000). Thesemodels capture some of the empirical regulari-ties mentioned above but they are unable tosimultaneously account for both the size depen-dence (once we control for the age of the firm)and the age dependence (once we control for thesize of the firm).2

* Department of Economics, Stern School of Business,New York University, 44 West Fourth Street, New York, NY10012; Quadrini: also Centre for Economic Policy Research(CEPR). We have received helpful comments from Jeff Camp-bell, David Chapman, Harold Cole, Tom Cosimano, JoaoGomes, Hugo Hopenhayn, Jose-Vctor Ros-Rull, HaraldUhlig, and Lu Zhang. We would also like to thank two anon-ymous referees whose suggestions helped us to improve thepaper significantly. This research is supported in part by Na-tional Science Foundation Grant No. SBR-9617396.

1 Some of these empirical facts are shown using estab-lishment data while others are shown using firm-level data.However, many of the empirical facts based on establish-ment data also hold for single-unit establishments (i.e.,establishments that are firms).

2 These models can generate an unconditional depen-dence of the firm dynamics on size and age. In other words,without conditioning on age, the firm dynamics are nega-tively related to its size, and without conditioning on size,the firm dynamics are negatively related to the firms age.But, they cannot account simultaneously for the conditionaldependence on both size and age. Consider, for example, themodels developed in Jovanovic (1982) and Hopenhayn(1992). They are the primary models of industry dynamicsthat emphasize learning and persistent productivity shocks.In the learning model of Jovanovic, age is the main dimen-sion of heterogeneity: for a certain distribution of the shock,(lognormal) firms of the same age experience the samegrowth rate and variability of growth, independently of theirsize. For a more general distribution of the shock, it is notclear whether size has a positive, negative, or nonmonotone

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In this paper, we ask whether the introductionof financial frictions in an otherwise standardmodel of industry dynamics can account for thesimultaneous dependence of firm dynamics onboth size and age. It seems natural to try to link

patterns of firm growth with their financial de-cisions because there are also important regu-larities in the financial characteristics of firmsthat are related to their size. Empirically, thefinancial behavior of firms is characterized bythe following facts3:

Small and younger firms pay fewer divi-dends, take on more debt, and invest more.

Small firms have higher values of Tobins q. The investment of small firms is more sensi-

tive to cash flows, even after controlling fortheir future profitability.

The results of the paper can be summarizedas follows. First, we show that a model withfinancial frictions can capture the features of thefinancial behavior of firms cited above. Weshow that smaller and younger firms pay fewerdividends, take on more debt, and invest more.Moreover, their investment is more sensitive tocash flows and they have higher values of To-bins q. Second, we show that the combination

of persistent shocks and financial frictions cangenerate the simultaneous dependence of indus-try dynamics on size (once we condition on age)and age (once we condition on size). In contrast,a model with only persistent shocks or onlyfinancial frictions cannot account for both thesize and age dependence.

It is important to emphasize that this paperdoes not provide a microfoundation for marketincompleteness. Rather, we evaluate how mar-ket incompleteness affects the dynamics of firm

growth. For this reason, we take a simple ap-proach in modeling financial frictions. We as-

sume that firms can finance investment in twoways; with equitywhich can be increased byissuing new shares or by reinvesting prof-itsand with one-period debt. The financialfrictions arise because of the following assump-

tions: (a) there is a cost or premium associatedwith increasing equity by issuing new shares,compared to reinvesting profits; (b) defaultingon the debt is costly. Although the cost ofraising equity is exogenous in the model, itcaptures the fact that firms prefer to increaseequity with internally generated funds and theyissue new shares only occasionally [see, forexample, Smith (1977) and Ross et al. (1995)].The assumption of a cost to issuing new sharesis also made in Joao F. Gomes (2001). The

default cost can be justified as a verification costwhich is paid in the event of default (costly stateverification).

Firms have access to a decreasing returns toscale technology with inputs of capital and la-bor. The firms productivity changes accordingto a persistent stochastic process. In the absenceof financial frictions, the efficiency level of thefirm fully determines its size. With financialfrictions, however, the size of the firm alsodepends on its assets (equity). Because there isa cost to issuing new shares and a default cost,

equity and debt are not perfect substitutes andthe investment choice of the firm depends on theamount of equity it owns. New firms are createdwith an initial amount of equity which changesendogenously as firms issues new shares andretain earnings. Because the profitability of thefirm is stochastic, at each point in time therewill be a stationary distribution of heteroge-neous firms.

The debt contract is a standard one-perioddebt contract signed with a financial intermedi-

ary. The financial intermediary lends funds atthe end of the period and the firm commits toreturn the borrowed funds plus the interest at theend of the next period. If at that time the firmdoes not repay the debt, the firm faces a bank-ruptcy problem. In this case the financial inter-mediary incurs the cost to verify the financialcondition of the firm. The financial intermediaryanticipates the possibility that the firm may notrepay the debt and the interest rate chargeddepends on the probability of default.

Firms face a trade-off in deciding the optimalamount of debt: on the one hand, more debt

impact on the firm dynamics. (See Jovanovic, 1982 p. 656).In Hopenhayns model, size is the only dimension of het-erogeneity: firms of the same size experience the samedynamics independently of their age. (See Hopenhayn, 1992p. 1141). The models developed in Hopenhayn and Roger-son (1993), Campbell (1998), and Campbell and Fisher(2000) are similar to Hopenhayns model.

3 See Clifford W. Smith, Jr. (1977); Stephen M. Fazzariet al. (1988); Simon Gilchrist and Charles P. Himmelberg(1995, 1999); Stephen A. Ross et al. (1995).

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allows them to increase their expected profits byexpanding the production scale; on the other,the expansion of the production scale implies ahigher volatility of profits. Given that the firmsobjective is a concave function of profits, its

volatility has a negative impact on the firmsvalue. This is reinforced by the higher interestrate associated with a higher probability of de-fault (to compensate for the default cost). As theequity of the firm increases, the firm becomesless concerned about the fluctuation of profits(in absolute value) and expands the productionscale. But, because of the decreasing returns toscale, the increase in production scale is notproportional to the increase in equity and thefirm will reduce its leverage as it grows. This

financial behavior has important consequencesfor the dynamics of the firm.The paper is organized as follows. In Section

I, we describe the basic model of firm dynamicswithout financial frictions and analyze its prop-erties. In Section II we extend the model byintroducing financial frictions. After describingthe firms problem and deriving some analyticalresults, we analyze the properties of the modelnumerically. We look first, in subsection C, atthe case in which the shocks are not persistent.This allows us to illustrate the impact of finan-

cial factors on the dynamics of the firm, sepa-rately from the impact of persistent shocks.Then in subsection D, we study the generalmodel with both persistent shocks and financialfrictions. As we will see, the model with onlypersistent shocks and the model with only fi-nancial frictions replicate some of the size de-pendence facts, but are unable to account for theage dependence. In both models, size is the onlydimension of heterogeneity. However, the com-bination of financial frictions and persistent

shocks allows the model to account for both thesize and age dependence. We then conclude thatboth financial factors and persistent shocks areimportant for the properties of the growth offirms.

I. The Basic Firm Dynamics Model in the

Frictionless Economy

The basic model of firm dynamics that westart with is a simplified version of the modeldeveloped in Hopenhayn (1992). Assume thereis a continuum of firms that maximize the ex-

pected discounted value of dividends, that is,E0{t0

tdt}, where dt is the dividend dis-tributed at time t, the discount factor for thefirm, and E0 the expectation operator at timezero.

In each period, firms have access to a pro-duction technology. We will work directly withthe revenue function implied by this technologyy (z )G(k, l ) where y is the revenue, kis the input of capital that depreciates at rate ,l is the labor input, and the variables z and areidiosyncratic shocks that determine the effi-ciency of the firm. The inputs of capital andlabor are decided one period in advance.

Capital and labor are perfect complements,which implies that the capital-labor ratio em-

ployed by the firm is always constant.

4

Giventhis assumption, we write the revenue functionas (z ) F(k). Similarly, we will denote byk [ w(l/k)]k, the depreciation of cap-ital plus the cost of labor, where w is the wagerate. Because the wage rate is constant in themodel, will also be constant.

We make the following assumptions aboutthe revenue function and the shocks.

ASSUMPTION 1: The function F : R 3Ris strictly increasing, strictly concave, and con-

tinuously differentiable.

ASSUMPTION 2: The shock z takes values inthe finite set Z {z0, z1, ... , zN} and followsa first-order Markov process with transitionprobability (z/z). The shock is indepen-dently and identically distributed (i.i.d.) in theset of real numbers R, with zero mean. Thedensity function f : R 3 [0, 1] is continuous,differentiable, and f() 0, @ R.

The concavity of F implies that the revenuefunction displays decreasing returns to scale.The decreasing returns to scale could be ratio-nalized by assuming limited managerial or or-ganizational resources as in Robert E. Lucas, Jr.(1978). Alternatively, we could assume thatthese properties derive from the monopolisticnature of the competitive environment where

4 This is without loss of generality. Given that the wagerate is constant, the capital-labor ratio would be constanteven if the two inputs were substitutable.

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the firm faces a downward-sloping demandfunction.

The structure of the shocks allows us to sep-arate the persistent component, the z, from thenonpersistent component, the . This in turn

allows us to separate the properties of firmgrowth induced by technological differences(due to persistent shocks) from pure accidentalevents (due to i.i.d. shocks). The persistentshock, z, is revealed one period in advance,while the i.i.d. shock, , is revealed in the cur-rent period. Therefore, at the moment of decid-ing the production plan, the firm knows z butdoes not know .

There are different ways to induce the exit ofa firm. As in Jovanovic (1982), for example, we

could assume that the firm has some outsideopportunity. Alternatively, as in Hopenhayn(1992), we could assume that in addition to thecost for the inputs of capital and labor, there isa fixed cost of production. In both cases, thefirm will exit when its value, net of the oppor-tunity or fixed cost, is negative. At this stage,however, it is convenient to assume that thefirms exit is exogenous. Accordingly, we sim-ply assume that with a certain probability thefirm becomes unproductive and exits. This iscaptured in the model by assuming that z0 0

and (z0/z0) 1. In Section III, we will discussthe properties of the model with endogenous exitwhen there is a fixed cost of production.

For the frictionless economy we assume that1/ 1 r, where r is the market interest rate.This condition can be interpreted as a general-equilibrium property of the model. In this econ-omy the Modigliani-Miller theorem applies, andwhether capital is financed with debt or equity isirrelevant. The problem of the firm is then staticand, conditional on surviving, it consists of the

maximization of expected profits, that is:

(1) maxk

z Fkfd r k max

k

zFk r k.

The solution is given by the optimal input ofcapital which depends on the ex ante produc-tivity of the firm z. We denote it by k*(z).

Given this solution we can then determine thevalue of an active firm denoted by V(z).

The last feature of the model that needs to bespecified is the entrance of new firms. In eachperiod there is a large number of projects,

drawn from the invariant distribution of z. Theimplementation of a project requires a fixed cost. In this partial-equilibrium analysis, withfixed prices, the mass of new entrant firms isnondegenerate only if the surplus from creatingnew firms is nonpositive, that is, V(z) 0. Because the surplus from creating new firmswith high z is larger than for firms with small z,this condition must be satisfied with equalityonly for z zN. In equilibrium it must be thecase that V(zN) , and all new firms will be

of the highest efficiency. Although we do notconduct a general-equilibrium analysis, this ar-bitrage condition will be guaranteed by general-equilibrium forces: the entrance of new firmswould induce changes in the prices (particularlywages) and in the value of firms until there areno gains from creating new firms. For the fric-tionless economy, this feature of the model im-plies that younger firms are larger than olderones. As we will see, however, this is not thecase in the model with financial frictions.

A. Industry Dynamics Properties of theFrictionless Economy

The fact that z is a sufficient statistic for thesize of the firm implies that, once we conditionon the firms size, age becomes irrelevant for itsdynamics (growth, volatility of growth, and jobreallocation). This is formally stated in the fol-lowing proposition.

PROPOSITION 1: In the frictionless economy,

firms of the same size experience the same dy-namics independently of their age.

This result is intuitive and the formal proof isomitted. What about the dependence of thefirms dynamics on size? To simplify the anal-ysis, lets consider the case in which the vari-able z takes only three values (z0, z1, z2) wherez0 is the absorbing shock and the transitionalprobability matrix for z {z1, z2} (conditionalon not being z0) is symmetric. Denote the con-ditional transitional probability by c(z/z).We then have the following proposition.



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PROPOSITION 2: Assume that c(z1/z1) c(z2/z2). Then small firms grow faster thanlarge firms and the rate of job reallocation isindependent of the firm size.

The proof of the proposition is trivial. In thiseconomy, surviving firms are only of two types:small firms with current shock z1 and largefirms with current shockz2. Neglecting the pos-sibility of exit, small firms will only grow whilelarge firms will only shrink. Moreover, smallfirms will never destroy jobs (except in the caseof exogenous exit) while large firms will nevercreate jobs. Job reallocation is defined as thesum of job creation and job destruction. Be-cause the number of jobs destroyed by a large

firm when z switches from z2 to z1 is equal tothe number of job created by a small firm whenz switches from z1 to z2, we have that thevolume (and rate) of job reallocation is the samefor small and large firms. Looking at the indi-vidual components of job reallocation, thismodel is consistent with the observation that job creation is decreasing in the firms size,but it is inconsistent with the empirical fact that job destruction is also decreasing in the firmssize.

Proposition 2 also holds when the shock z

takes more than two values and the transitionprobability matrix is symmetric with decreas-ing probability of changing the current z tomore distant values. Without some restrictionon the transition probability matrix, we can-not derive a general pattern for the dynamicsof firms and it is possible to have a nonmono-tone relation.

To summarize, the basic model of firm dy-namics captures some of the size dependence offirm growth but is unable to capture the age

dependence once we control for the size of thefirm. This result is not affected by the initial sizeof new entrant firms (determined by the initialvalue ofz). The initial size of new firms affectsthe size distribution of firms but not their size-and age-conditional dynamics.5

II. The Economy with Financial Frictions

We now extend the basic model of firm dynam-ics by introducing financial-market frictions. Ateach point in time, firms are heterogeneous in the

amount of assets they own as well as their tech-nology level z. Henceforth, the assets of the firms,denoted by e, are referred to as equity and itcorresponds to the firms net worth.

The input of capital is financed with equityand by borrowing from a financial intermediary.If we denote the firms debt by b, the input ofcapital is k e b. Financial frictions derivefrom two assumptions:

(a) There is a cost per unit of funds raised by

issuing new shares.(b) The firm can default and the default proce-dure implies a cost .

The cost of raising funds implies that thefirm will prefer to increase equity by reinvestingprofits, rather than issuing new shares. The de-fault cost increases the cost of borrowing. Aswe will see, for a given value of equity, theprobability of default increases when the firmborrows more because it is more vulnerable toidiosyncratic shocks. This increases the ex-

pected default cost and the financial intermedi-ary will demand a higher interest rate.

It is important to note that both assumptions arenecessary to have effective financial frictions. If itis costly for the firm to issue new shares, but it canborrow at the market interest rate without limit,then the firm will implement the desired scale ofproduction by borrowing more, whatever thevalue of its equity. On the other hand, the defaultcost becomes irrelevant if the firm can increase itsequity by issuing new shares without cost.

The particular structure of the debt contractcan be justified by asymmetry in informationand costly monitoring as in a standard costlystate verification model. Assume that at the endof the period, if the firm defaults on the debt, thefinancial intermediary has the ability to liqui-date the firm by paying the cost . Under theseconditions, the firm will default only if theend-of-period resources (net worth) are smallerthan a certain threshold and the financial inter-mediary will verify the firm only in the event ofdefault. The threshold point is such that thevalue of the firm at that level of net worth is

5 It will affect the unconditional dependence on age.When new entrants are small, the growth rate of firms isnegatively related to their age (if we do not control forsize). If they are large, the growth rate is positivelyrelated to the age of the firm (again, if we do not controlfor its size).



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exactly zero, so that the firm is indifferent be-tween repaying the debt or defaulting.

In this economy we make a slightly differentassumption about the relation between the in-terest and the discount rates:

ASSUMPTION 3: The risk-free interest rate ris such that 1/ 1 r 0.

This condition can be interpreted as a general-equilibrium property of economies with these fea-tures. In the frictionless economy, the condition1/ 1 r implies that firms are indifferentbetween accumulating equity or distributing divi-dends. In the economy with financial frictions,however, if the interest rate is equal to the discount

rate, firms would strictly prefer to reinvest profits,no matter what the value of the equity is. Eventu-ally, after reaching a certain size, part of the equitywill be kept in the form of risk-free investmentsearning the interest rate r. In this case the debt bwould be negative, meaning that the firm lendsmoney rather than borrowing. This would gener-ate an excessive supply of loans and the sub-sequent reduction in the lending rate r. WithAssumption 3, however, there is some upperbound emax which bounds the equity chosen by thefirm. At the same time, because the value of the

firm is obviously bounded, if the debt of the firmis very large, relative to its equity, the firm willrenegotiate the debt. This implies that there issome emin below which the equity of the firm (networth) will never fall. Therefore, we can restrictthe state space for equity to the compact andconvex set [emin, emax].

A. The Firms Problem

At the end of each period, after the realization

of the revenues and the observation of the newz, but before issuing new shares or paying div-idends, the firm decides whether to default onits debt. Given the initial equity and the debt,the end-of-period net worth of the firm is:

(2) e, b, z 1 e b

z Fe b 1 rb

where r is the interest rate charged by the in-termediary. The firm will default if its net worth

is such that the value of continuing the firm isless than zero. Denote by e(z) the value of networth below which the firm defaults. Figure1 shows how e(z) is determined. The figure

plots a possible shape for the value of the firm,which is increasing in the value of net worth(e, b, z ). The firms value is plotted fora particular z. For very low , the value of thefirm is negative. In this case, the firm will de-fault and its liabilities are renegotiated to bringits end-of-period net worth to e(z).

Associated with the default threshold, there isa value of the shock for which the end-of-periodnet worth is equal to e(z). The thresholdshock, denote by (z, e, b, z), is definedimplicitly by:

(3) e, b, z

1 e b

z

Fe b 1 rb e

z .

The interest rate charged by the intermediary ris implicitly defined by:

(4) 1 rb 1 rb

fd

1 e b

z Fe b ] fd.

This simply says that the expected repaymentfrom the loan (expression on the right-handside) is equal to the repayment of a riskless loan(expression on the left-hand side). Therefore,

FIGURE 1. VALUE OF THE FIRM AS A FUNCTION OFEND-OF-PERIOD NET WORTH (EQUITY)



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the expected return from the loan is equal to themarket interest rate r.

Using (4) to eliminate r in (3), the thresholdshock(z, e, b, z) is determined by the condition:

(5) 1 rb e

z

fd

fd 1 e b

Fe b

where (

) z

f(d

)

f(d

).Note that default does not lead to the exit ofthe firm. It simply leads to the renegotiation ofthe debt to the point where the firm would notdefault. This is because the liquidation of thefirm is not in the interest of the financial inter-mediary. Of course, the firm defaults only if thenet worth is negative, (e, b, z ) 0. Thisimplies that, in case of default, the intermediarywould not get the full repayment of its debt ifthe firm is liquidated. On the other hand, byrenegotiating the loan and giving the firm the

incentive to continue operating, the firm willrepay a larger fraction of the debt, either byissuing new shares and/or by contracting a newloan. More specifically, assume that (e, b,z ) e(z) 0. If the firm is liquidated,the intermediary loses (e, b, z ). If instead the intermediary renegotiates the loan, itloses only (e, b, z ) e(z).

Using (2) and (3) and taking into accountthat the debt is renegotiated when (e, b,z ) falls below e(z), the end-of-period

resources of the firm or net worth can beexpressed as:

(6) qe, b, z , z

e

z

z, e, b, z Fe b,if

z, e, b, ze

z ,if

z, e, b, z .

After the default decision, the firm will decidewhether to issue new shares or pay dividendsand will choose the next period debt. Although

the default choice, the dividend policy, and thechoice of the next period debt are all decided atthe same time, it is convenient to think of themas decided at different stages. Define (z, x)to be the end-of-period value of the firm after

renegotiating the debt but before issuing newshares or paying dividends. The variable x de-notes the corresponding equity (again, after therenegotiation of the debt but before issuing newshares or paying dividends). Also, define (z,e) to be the value of the firm at the end of theperiod after issuing new shares or paying divi-dends, but before choosing the next period debt.The variable e is the end-of-period equity afterraising funds with new shares or distributingdividends. The firms problem can be decom-

posed as follows:

6

(7) z, e

maxb

z

z,e,b,z

z, qe, b, z , zz/zfd subject to

(8) qe, b, z , z

ez Fe b, if e

z , if

(9) 1 rb e

z

fd

fd

1 e b Fe b

(10) z, e 0

(11) z, x maxe

dx, e z, e

subject to

6 Notice that the exogenous probability of exit is implic-itly accounted in the formulation of the problem by assum-ing that z

0

is an absorbing shock, that is, z0

0 and(z0/z0) 1.



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(12) dx, e

x e, if x ex e 1 , if x e.

Notice that the dynamic program can besolved backward. The second part of the prob-lem defines the new shares or dividend policy ofthe firm. The function d(x, z , e) isdefined in (12), where x is the end-of-periodequity of the firm before the new shares ordividend choices are made. If the firm issuesnew shares, d is negative. In this case the firmpays the cost per unit of equity raised. If thefirm pays dividends, d is positive.

Given the function

(z, x), equation (10)defines the value of the net worth below whichthe firm defaults, that is, the function e(z). Thefirm defaults if the value of repaying the debtand continuing the firm is negative. Then, givene(z), equation (9) determines the thresholdshock (z, e, b, z). Once e(z) and (z, e, b,z) are determined, the firms problem (7) iswell defined. The following proposition charac-terizes some features of the firms problem.

PROPOSITION 3: There exists a unique func-

tion *(z, e) that satisfies the functional equa-tion (7). In addition, if for1 and2 sufficientlysmall, f() 1 for all 2, then

(a) the firms solution is unique, and the policyrule b(z, e) is continuous in e;

(b) the input of capital k e b(z, e) isincreasing in e;

(c) there exist functions e(z) e (z) e(z),z Z, for which the firm renegotiates theloan if the end-of-period resources are

smaller than e(z), will issue new shares ifthey are smaller than e (z), and distributedividends if they are bigger than e(z);

(d) the value function *(z, e), is strictly in-creasing and strictly concave in [e, e].

PROOF:See Appendix A.

The restrictions on the second part of theproposition are motivated by the fact that theend-of-period resource function (net worth),q(e, b, z , z), is not concave in e and b for

all values of . In order to assure that the firmspolicy is unique, we have to impose some re-strictions on the stochastic process for theshock. After imposing these restrictions, theoptimal debt policy of the firm is unique asstated in point (a).

Point (c) characterizes the new shares anddividend policies of the firm. The intuition forthese policies is provided in Figure 2, whichplots the value of a firm as a function of equity,for low and high values of z. If the equity falls

below the threshold e (z), the firm issues newshares to bring its equity to the level e (z),despite the cost of issuing new shares. This isbecause the concavity of the firms value [point(d) of the proposition] implies that, when theequity is small, the marginal increase in thefirms value with respect to e is larger than 1 . In the range (e (z), e(z)), instead, the mar-ginal increase in the value of the firm is notsufficient to cover the cost of one unit of newequity. Therefore, no shares are issued. In this

range, the marginal increase in the value of thefirm is also larger than one and the firm prefersto reinvest all the profits. Finally, for values ofequity above e(z) the marginal increase in thevalue of the firm is smaller than one and the firmdistributes dividends.

The figure also shows another interesting fea-ture of the new shares policy. Consider a firmwith a low z and suppose that its equity is inthe range (e (z1), e (z2)). If the productivityswitches from z1 to z2, the firm will issue newshares. Therefore, in this economy firms issuenew shares in two cases: when they are making

FIGURE 2. FIRMS VALUE AS A FUNCTION OF EQUITY FORLOW AND HIGH VALUES OF z



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losses that dissipate their net worth and whentheir future prospects improve. In both cases, arecapitalization of the firm is the optimal policy.

The monotonicity of the investment function[point (b) in the above proposition] along with

the reinvestment of the firms profits imply thatthe investment of firms is sensitive to cash flowseven after controlling for the future profitabilityof the firm (controlling for z) and this sensi-tivity is greater for smaller firms. In this way themodel captures an important empirical regular-ity of the investment behavior of firms as shownin Fazzari et al. (1988) and Gilchrist and Him-melberg (1995, 1999).

B. Entrance of New Firms and Invariant

Distribution of Firms

New firms are created with an initial value ofequity raised by issuing new shares. The opti-mal equity of a new firm with initial productiv-ity z is the lower bound e (z), as determined inthe previous section. Therefore, the cost of cre-ating a new firm with initial productivity z is (1 )e (z) and the surplus generated bycreating the firm is (z, e (z)) (1 )e (z).

As in the frictionless model, many projects

are drawn in each period from the invariantdistribution of z, and they will be implementedonly if the surplus from creating a new firm isnonnegative. In equilibrium, the following arbi-trage condition must be satisfied:

(13) zN, e zN 1 e zN.

As emphasized earlier, this arbitrage conditioncan be interpreted as a general-equilibriumproperty: the entrance of new firms would in-

duce changes in the prices and in the value ofthe firm , until there are no gains from creat-ing new firms.7

This framework generates complex dynamicsand at each point in time the economy is char-acterized by a certain distribution of firms .

Technically, is a measure of firms over theproduct set i1

N [e (zi), e(zi)]. In the analysisof the next subsections, we will concentrate onthe invariant distribution of firms denoted by*. The existence of the invariant distribution

depends on the properties of the transition func-tion generated by the optimal decision rule b(z,e). The transition function gives rise to a map-ping which returns the next period measureas a function of the current one. The invariantdistribution is the fixed point of this mapping,that is, * (*). In this subsection we onlystate the main existence result. The proof re-quires the introduction of some formal defini-tions and the derivation of intermediate resultswhich are relatively technical. They are in Ap-

pendix A.

PROPOSITION 4: An invariant measure offirms * exists. Moreover, if the probability ofdefault is decreasing in e , then * is uniqueand the sequence of measures generated by thetransition function, {n(0)}n0

, convergesweakly to * from any arbitrary 0.

PROOF:See Appendix A.

The convergence result is especially impor-tant because it allows us to find this distributionnumerically through the repeated application ofthe mapping .

C. Properties of the Economy with FinancialFrictions: The Case of i.i.d. Shocks

In this section, we describe the financial be-havior and industry dynamics properties gen-erated by the model with financial-market

frictions starting with the special case in whichz takes only two values: the absorbing shockz0 0 and z1. We will refer to this as the i.i.d.case because, conditional on surviving, theshock is independently and identically distrib-uted. This simple case facilitates an understand-ing of how the financial mechanism affects thedynamics of firms, as opposed to the dynamicsinduced by changes in the productivity level.This will also facilitate an understanding ofhow the interaction between persistent shocksand financial frictionsstudied in subsectionDaffects the dynamics of the firm.

7 Condition (13) implies that all new firms are of the highproductivity type. As we will see, this feature of the modelhas important consequences for the dynamics of firms.Although we consider this to be the relevant case, we willalso discuss the alternative case in which new entrants are ofthe low productivity type.



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For the frictionless economy the dynamicproperties of the firm can be characterized an-alytically. In the economy with financial fric-tions we need to use numerical methods. Thesemethods are described in Appendix B.

Parameterization.We parameterize themodel assuming that a period is a year, and weset the risk-free interest rate r to 4 percent andthe depreciation rate to 0.07.

The revenue function is characterized by thefunction F, the value of z1, and by the stochas-tic properties of the shock . The function F isspecified as F(k) k, and the technologyshock is assumed to be normally distributedwith mean zero and standard deviation . The

parameter determines the degree of returns toscale. Studies of the manufacturing sector as inSusanto Basu and John G. Fernald (1997), findthat this parameter is close to one. We assign avalue of 0.975.

In the sample of firms analyzed by Evans(1987), the average probability of exit is about4.5 percent. Therefore, we assign a value of0.045 to the probability of the absorbing shock(z0/z1). The default cost is set to be 1percent of the value of the equity of the largestfirm and the premium for new shares is set to

0.3. These two parameters do not affect thequalitative properties of the model.

There are still four parameters to be pinneddown. Those are , , , and z1. The calibrationof these parameters is obtained by imposingfour conditions: (a) the equity of the largest firmis normalized to 100;8 (b) the average probabil-ity of default is 1 percent; (c) the value of debtof the largest firms is 25 percent of the totalvalue of its assets (b/k 0.25); (d) the capital-output ratio is 2.5. Condition (b) derives from

the estimates of Dun and Bradstreet Corpora-tion for the period 1984 1992. Condition (c)derives from balance sheet evidence of largefirms. For example, in the sample of firms an-alyzed by Hall and Robert E. Hall (1993), theratio of debt to total assets is about 0.25. Noticethat, once we fix , the wage rate w and thecapital-labor ratio can be determined to yield acertain capital income share and a desired range

of heterogeneity in employment. For example,we can choose a capital income share of 0.36and have the largest firm employing 2,000

workers. In this way the model generates heter-ogeneity that is comparable to empirical studiesas in Evans (1987). The full set of parametervalues is reported in Table 1.

Financial Behavior and Invariant Distribu-tion.Figure 3 shows the key properties of thefinancial behavior of firms. These properties canbe summarized as follows:

Small firms take on more debt (higherleverage).

Small firms face higher probability of default. Small firms have higher rates of profits. Small firms issue more shares and pay fewer

dividends.

The value of debt, plotted in panel (a) ofFigure 3, is increasing in the equity of thefirm. But, debt as a fraction of equity (leverage)is decreasing in the firms equity [see panel(b)]. To understand why debt is an increasingfunction of equity, we have to consider the

trade-off that firms face in deciding the optimalamount of debt. On the one hand, more debtallows them to expand the production scaleand increase their expected profits; on the other,the expansion of the production scale impliesa higher volatility of profits and a higher prob-ability of failure. Given that a large fractionof profits is reinvested, and the firms futurevalue is a concave function of equity [see panel(f)], the firms objective is a concave functionof profits. This implies that the volatility ofprofits (for a given expected value) has a neg-ative impact on the firms value. Therefore, in

8 Alternatively, we could fix z1

and the equity of thelargest firm would be determined endogenously.

TABLE 1CALIBRATION VALUES FOR THEMODEL PARAMETERS

Lending rate r 0.040Intertemporal discount rate 0.956Returns to scale parameter 0.975

Depreciation rate 0.070Standard deviation of the shock 0.280Productivity parameter z1 0.428Probability of exogenous exit (z0/z1) 0.045Default cost 1.000New shares premium 0.300



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FIGURE 3. FINANCIAL BEHAVIOR WITH i.i.d. SHOCKS



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deciding whether to expand the scale of produc-tion by borrowing more, the firm compares themarginal increase in the expected profits withthe marginal increase in its volatility (and there-fore, in the volatility of next period equity). Due

to diminishing returns, as the firm increases itsequity and implements larger production plans,the marginal expected profits from further in-creasing the production scale decrease. Conse-quently, the firm becomes more concernedabout the volatility of profits and borrows less inproportion to its equity. As a consequence ofhigher borrowing, small firms face a higherprobability of default as shown in panel (c).

Panels (d) and (e) plot the expected rates ofprofits, new shares, and dividends. The ex-

pected profit (as a fraction of equity) is decreas-ing in the size of the firm. This is a result of thefinancing policy outlined above and the de-creasing returns to scale property of the revenuefunction. The higher profitability of smallerfirms implies that they have a greater incentiveto reinvest profits and when the equity of thefirm falls below a certain threshold, the firmissues new shares even if this requires a pre-mium. In fact, the expected rate of issuance ofnew shares is decreasing in the equity of thefirm while the dividend rate is increasing in the

size of the firm. The higher profitability of smallfirms, associated with their lower dividends,implies that small firms invest more.

Panels (f) and (g) plot the firms value and thevalue of Tobins q. As can be seen from thesefigures, the firms value is an increasing andconcave function of equity and Tobins q isdecreasing in the firms size.

Panel (h) plots the invariant distribution offirms. If we exclude the largest size, the shapeof this distribution presents a degree of skew-

ness toward small firms which is also an empir-ical regularity of the data. There is aconcentration of firms at the bottom of the dis-tribution because the optimal size of new en-trants is small. The concentration of firms in thelargest class, instead, follows from the exis-tence, in the model, of an (endogenous) upperbound to the firms size. In the data, we havefirms that employ many more workers than thelargest firms in the model. Although the numberof these firms is relatively small, they accountfor a large fraction of aggregate production.Accordingly, the largest firms in the model must

be interpreted as representing the production offirms employing more than 2,000 workers: thelarge share in production of these big firms isaccounted for in the model by an increase in thenumber of firms rather than their size.

Finally, Figure 4 plots the joint distribution offirms over size (equity) and age. Because newfirms are small, the invariant distribution ischaracterized by a concentration of firms insmall and young classes.

Industry Dynamics.Figure 5 shows the keyproperties of the firms dynamics. These prop-erties can be summarized as follows:

Small firms grow faster and experience

higher volatility of growth. Small firms face higher probability of default. Small firms experience higher rates of job

reallocation (with some qualification for jobdestruction).

Without conditioning on size, young firmsexperience higher rates of growth, default,and job reallocation (with some qualificationfor job destruction).

Panel (a) of Figure 5 reports the expectedgrowth rate of equity as a function of the initial

size of the firm and panel (b) its standard devi-ation. The growth rate of the firm is decreasingin size. This derives from the higher rate ofprofits of small firms, and from their lowerdividend payments [see panels (d) and (e) ofFigure 3]. The standard deviation of growth isalso decreasing in the size of the firm except forvery small firms. This is because there is alower bound to the size of the firms. When theirequity falls below a certain threshold, they issuenew shares.

Panel (d) plots the rates of job creation andjob destruction. Following Davis et al. (1996), job creation is defined as the sum of employ-ment gains of expanding firms, and job destruc-tion is defined as the sum of employment lossesof contracting firms. Both creation and destruc-tion are decreasing in the firms size. The onlyexception is the job destruction of very smallfirms. This is because there is a lower bound eto the firms equity. Therefore, a firm with eq-uity e never destroys jobs, with the exception ofexit. But, for this model the probability of exit isthe same for all firms. As discussed in Section



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III, by making exit endogenous, the probabilityof exit decreases with the size of the firm. In thisway small firms will destroy more jobs due toexit, as opposed to the reduction of the produc-tion scale.

The model generates an unconditional agedependence of firm dynamics as shown in pan-els (e) through (h) of Figure 5. These figuresshow that growth, variability of growth, failure

rates, and job reallocation are decreasing in theage of the firm. This dependence, however, de-rives from the fact that young firms are small,which in turn derives from the small size of newentrants. However, if we control for the size ofthe firm, these dynamics are independent of age.This is because equity, which determines thesize of the firm, is the only dimension ofheterogeneity.

To summarize, the model with financial fric-tions and i.i.d. shocks is able to generate thedependence of the firm dynamics on size but itshares with the basic frictionless model the in-

ability to generate the age dependence, once wecontrol for size. This is because in both modelsthere is only one dimension of heterogeneity(identified by the variable e in the economywith financial friction and i.i.d. shocks, and bythe variable z in the basic model with persistentshocks). To account simultaneously for the sizeand age dependence, another dimension of het-erogeneity is needed. As we will see in the next

subsection, this is obtained by combining per-sistent shocks and financial frictions.

D. Properties of the Economy with FinancialFrictions: The Case of Persistent Shocks

Parameterization.Relative to the case ofi.i.d. shocks, we only need to parameterize theprocess for the shock z. We assume that, con-ditional on surviving, z follows a symmetrictwo-state Markov process with c(z1/z1) c(z2/z2) 0.95. The two values of the shockare chosen so that in the invariant distribution,

FIGURE 4. AGE AND SIZE DISTRIBUTION OF FIRMS



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FIGURE 5. INDUSTRY DYNAMICS WITH i.i.d. SHOCKS



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the employment of the largest firm with currentz z1 is 50 percent the employment of thelargest firm with current z z2. The choice ofthese parameters and how they affect the dy-namic properties of the model are discussed

after the presentation of the results.

Financial Behavior and Invariant Distribu-tion.The key properties of the financial be-havior of firms are similar to the case of i.i.d.shocks. Now, however, firms differ over twodimensions: equity and productivity z. Panel (a)of Figure 6 plots the value of debt as a functionof its equity, for low (z z1) and high (z z2) productivity firms. For each equity size,high productivity firms borrow more and imple-

ment larger production scales in order to takeadvantage of their higher productivity. As dis-cussed for the model with i.i.d. shocks, in de-ciding the scale of production, firms face atrade-off. On the one hand, a larger productionscale allows higher expected profits. On theother, a larger production scale implies highervolatility of profits, to which the firm is averse(because the firms value is a concave functionof profits). For a high productivity level z, themarginal expected profit is higher for each pro-duction scale. Consequently, the firm is willing

to face higher risk by borrowing more, andexpands the scale of production. As shown inpanel (b), firms with a high value of z enjoyhigher profits. The higher profits allow thesefirms to grow faster [see panel (c)]. The increasein the volatility of profits induced by higherdebt, also implies that their growth rate ismore volatile. As a consequence of this, highproductivity firms experience higher failurerates [see panel (d)] and higher rates of jobreallocation [see panels (e) and (f)].

The last two panels of Figure 6 plot thefraction of low and high productivity firms inthe invariant distribution as a function of equity[panel (g)] and as a function of age [panel (h)].Of course, above a certain equity size, all firmsare of the high productivity type. This is be-cause the maximum size of the firm increaseswith z. Below this size, the fraction of lowproductivity firms tends to increase becausenew firms are small and have high productivity.After entering, some of them switch to lowproductivity but at a slow rate. In the interimthey grow so that, when they switch, they have

more equity. This process also implies that thefraction of low productivity firms increases withage as shown in panel (h).

Industry Dynamics.This heterogeneous be-

havior among firms of different productivityintroduces an age dependence in the dynamicsof firms. Figure 7 plots the average growth rate,the default rate, and the rates of job reallocation(creation and destruction), as a function of thefirms size and age. In order to separate the sizeeffect from the age effect, these variables areplotted for different age classes of firms (leftpanels) and for different size classes (right pan-els). These variables are all decreasing in thesize of the firm and in its age, even after con-

trolling, respectively, for age and size. The onlyexception is for the job destruction of very smallfirms. As observed previously, there is a lowerbound to the size of firms. Consequently, firmsthat are initially close to this bound destroy veryfew jobs. We have also observed previously thatthis feature could be corrected if we introduceendogenous exit. In that case small firms willdestroy more jobs because they face higher ratesof exit.

The size dependence in this analysis is drivenmainly by the same factors that generated this

dependence in the model with independentshocks. In contrast, the age dependence derivesfrom the technological composition of firms ineach age class. As shown in panel (h) of Figure6, the fraction of young firms with low produc-tivity is smaller than old firms. This is becausenew entrants have high productivity. Now be-cause firms with z z2 experience higher ratesof growth, failure, and job reallocation, we alsohave that for each size class, younger firmsgrow faster and face higher rates of failure and

job reallocation. Thus, in an economy with per-sistent shocks to technology, in order to have asignificant dependence of firm dynamics on age,there must be a heterogeneous composition offirm types in each age class of firms. In thisrespect the degree of persistence plays an im-portant role. If the shock is not highly persistent,the heterogeneous composition of firms be-comes insignificant after a few periods. With avery persistent shock, instead, the heterogeneityvanishes slowly and the age dependence ismaintained for a large range of ages.

Quantitatively, the age dependence is small,



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FIGURE 6. FINANCIAL BEHAVIOR AND INDUSTRY DYNAMICS FOR LOW AND HIGH PRODUCTIVITY FIRMS



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FIGURE 7. INDUSTRY DYNAMICS CONDITIONAL ON AGE AND SIZE



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but this may depend on the simple structurechosen for the stochastic process of z. Alsonotice that the age effect is more importantamong the class of small firms and it almostdisappears for very large firms.

The initial productivity of new firms plays animportant role. In the model, new entrant firmsare of the high productivity type. This is con-sistent with the view that new entrants possessbetter technologies and perform better than in-cumbent firms as in Jeremy Greenwood andJovanovic (1999). Alternatively we could as-sume that new firms initially enter with lowproductivity, that is, z z1. This assumptionwould be consistent with a learning-by-doinginterpretation of the technology shock z. How-

ever, if new entrant firms are of the low pro-ductivity type, then the age dependence wouldbe of the wrong sign, with old firms experienc-ing higher rates of growth, job reallocation, andfailure.

To summarize, the integration of a basicmodel of firms dynamics (frictionless economy)with a model of financial frictions (economywith frictions and i.i.d. shocks) is able to ac-count for most of the stylized facts about thefinancial behavior and the growth of firms.In particular, this more general model is able

to generate the simultaneous dependence ofindustry dynamics on size and age while theother two modelsthe frictionless modeland the model with financial frictions andi.i.d. shocks can only account for the sizedependence.

III. Endogenous Exit

In this section, we briefly discuss the exten-sion of the model that allows for endogenous

exit. Endogenous exit can be introduced by as-suming that, in each period, the firm faces afixed cost of production .

Consider first the frictionless economy. Inthis economy, the value of a firm is increasingin the value of z. If z is highly persistent, forsmall values of z the value of the firm will benegative. In this case the firm will exit. Supposethat the shockz can take N values. Then we canidentify z for which the firm will exit if z zand will continue operating ifz z. Now, theprobability that z z decreases with the cur-rent value of z. Because firms with a higher

value ofz are larger firms, we have that the exitrate decreases with the size of the firm. This isbasically the result of Hopenhayn (1992). Al-though the model is able to capture the sizedependence, if we control for the current size of

the firm (the current value of z), the survivalprobability of firms is independent of their age.Therefore, the model is not able to generate the(conditional) dependence of exit on age.

Now consider the model with financial fric-tions. We have seen that the value of the firm isincreasing in e and z. This is also the case whenthere is a fixed cost. Consequently, in thismodel, exit decreases with the size of the firm.The age dependence, however, cannot be estab-lished in general but will depend on the param-

eterization of the model. From the analysis ofthe previous section, we have seen that a largerproportion of young firms are of the high pro-ductivity type. On the one hand, this impliesthat younger firms borrow more and face higherprobabilities of falling to lower values of equityfor which the value of the firm is negative. Thismay imply that younger firms face a higherprobability of exit. This mechanism also ex-plains why younger firms face higher probabil-ity of default, as we have seen in the previoussection. On the other hand, because of their high

productivity, younger firms face a lower prob-ability of falling to lower values of z for whichthe value of the firm becomes negative. Thisdecreases the exit probability of young firms. Apriori, we cannot say which effect dominates.But in principle, for certain parameter values,the exit probability might be decreasing in theage of the firms, even after controlling for theirsize.

IV. Conclusion

Existing models of industry dynamics thatabstract from financial-market frictions are un-able to account simultaneously for the depen-dence of the firm dynamics on size and age. Amodel with only persistent shocks to technol-ogy, as in Hopenhayn (1992), is able to accountfor the size dependence but does not capture the(conditional) age dependence. The learningmodel of Jovanovic (1982) is able to generatethe age dependence but does not capture the(conditional) size dependence. In this paper,we introduce financial-market frictions in an



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otherwise standard industry dynamics modeland we show that the integration of persistentshocks and financial-market frictions allowsthe model to account for the simultaneousdependence of the firm dynamics from size

(once we control for age) and age (once wecontrol for size). More importantly, the inte-gration of these two features helps to recon-cile the characteristics of firms growth withmany of the financial features related to sizethat are observed in the data.

This paper can be viewed as a first step to-ward the study of the importance of financial-market frictions for the dynamics of the firm. It

is a first step because we consider only one-period debt contracts. We leave for future re-search the goal of studying the dynamics of thefirm when financial contracts are not limited toone-period debt. Examples of this approach are

Rui Albuquerque and Hopenhayn (1997) andQuadrini (1999). The first studies optimal dy-namic contracts in an environment in whichinformation is symmetric and financial frictionsderive from the limited enforceability of thesecontracts; the second studies an environment inwhich financial frictions derive from informa-tion asymmetries which generate moral hazardproblems.

APPENDIX A: ANALYTICAL PROOFS

LEMMA 1: Let (e1, b1) and (e2, b2) be two arbitrary points in the feasible space, let (e, b) bea convex combination of these two points, and define the function () as:

(A1) z , z qe , b , z , z qe1 , b1 , z , z

1 qe2 , b2 , z , z

where q(e1, b1, z , z) is the end-of-period resource function defined in (6). Then, underthe conditions of Proposition 3, there exists (z, z) such that, for each z, z Z, (z , z) 0 if (z, z), and (z , z) 0 if (z, z). Moreover, (z, ,

z) f(d

) 0.

PROOF:The end-of-period resource function is equal to:

(A2) qe, b, z , z e z Fe b, if e

z , if

.

Therefore, q is a linear function of with slope F(e b). Because F is strictly concave, we havethat F(e

b

) F(e1 b1) (1 ) F(e2 b2). This implies that the slope of q(e, b,

z , z) is greater than the slope of q(e1, b1, z , z) (1 )q(e2, b2, z , z).

Therefore, if for small value of

the function q(e, b, z

, z

) is smaller than q(e1, b1, z

, z) (1 )q(e2, b2, z , z), as we increase the value of the first function gets closerto the second function until they cross. In the case in which the function q(e

, b

, z , z) is

greater than q(e1, b1, z , z) (1 )q(e2, b2, z , z), even for small values of , thenthe first function will be greater than the second for all values of .

Let us now prove the second part of the lemma. Simple algebra gives us:

(A3) qe, b, z , zfd e b1 1 rb zFe b

z,e,b,z

fd.

Because F is strictly concave, the first part of the above expression is strictly concave. The term

(z,e,b,z) f(d), however, is not necessarily concave. But, under the conditions of Proposition 3,



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the sensitivity of this term to changes in e and b are very small. For sufficiently small 1 , thesechanges are negligible and (z , z) f(d) 0.

PROOF OF PROPOSITION 3:Let us start the proof by assuming that z is a constant. The extension to the case of persistent

shocks is trivial. First we can restrict the feasible value of e to the set [emin, emax], where emin issufficiently small and emax sufficiently large so that the equity of the firm will never be outside thisinterval. We also observe that the choice of b is bounded for each value of e. Denote by k* theoptimal input of capital in the absence of financial frictions. Of course, b k* e (it is neveroptimal to expand the production scale beyond the optimal scale). At the same time e b 0(capital cannot be negative). Therefore, the correspondence that defines the feasible set for b iscontinuous, compact, and convex valued. We will denote this correspondence by B(e) {b e b k* e}.

Consider the firms problem as defined in (7):

(A4) Te maxbBe

e,b

qe, b, z fd

subject to(A5) qe, b, z e Fe b, if e

, if

(A6) 1 rb 1 e b z Fe b

fd e

fd

(A7) e

0

(A8) x maxe

dx, e e

subject to

(A9) dx, e x e, if x ex e1 , if x e.Notice that the mapping is solved backward. Problem (A8) defines the function . Even if the

function is decreasing for some values of e, the function is always strictly increasing. Thenequation (A7) determines the default value of equity e. Because is strictly increasing, e is unique.Given e, equation (A6) uniquely defines the default threshold for the shock and problem (A4) iswell defined.

We prove first that T maps bounded and continuous functions into itself and there is a uniquecontinuous and bounded function * that satisfies the functional equation * T(*). The factthat T maps continuous and bounded functions into itself is proved by verifying the conditions forthe theorem of the maximum. If is a continuous and bounded function, then the boundedness andcontinuity of q(e, b, z ) f(d) and (e) implies that the objective function is continuous andbounded. Because the correspondence B(e) is continuous, compact, and convex valued, and the set

[emin, emax] is compact and convex, the maximum exists and the function resulting from the mapping



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T()( e) is continuous and bounded. The fact that there is a unique fixed point of the mapping T isproved by showing that T is a contraction. This, in turn, is shown by verifying that T satisfies theBlackwell conditions of monotonicity and discounting.

We want to show now that, under certain conditions, * is strictly increasing and concave in theinterval [e, e], where e is the value of equity below which the firm defaults, and e is the value of

equity above which the firm distributes dividends. To show this, we study first a slightly differentmapping. Then by showing that for a certain range of equity the two mappings have the samesolution (fixed point), we can characterize the properties of the first mapping by studying the second.The modified mapping is obtained by replacing (A4) with the following:

T2 e

maxbB e

e ,b

q e , b, z fd , if e emax

bB e

e

,b

q e

, b, z fd

e

e, if e e

.

In (A10) we have added an extra term as if the firm, before borrowing, issues new shares anytimeits initial equity are below e. The new mapping (A10) is also a contraction and has a unique fixedpoint, denoted by **. We show now that *( e) **(e) for e e.

Consider problem (A10) where the function is derived from (A8) after substituting *. Ofcourse, for e e, T(*) (e) *( e). This implies that ** (e) *(e) for e e. For e e,the two fixed points are different. But we are interested only in values of e e.

After establishing the equivalence between the two fixed points for the relevant range of e, wenow characterize the properties of **. By doing so we also characterize the properties of * fore e. If is concave and satisfies (0) 0, then the function defined in (A8) is strictlyincreasing and concave. In addition, if q(e, b, z ) was strictly concave for each , then (q(e,b, z )) f(d) would be strictly concave in e and b and the maximizing value of b would beunique. This would also imply that T2() is strictly concave and satisfies T2()(0) 0 in [emin,emax]. Lemma 1, however, shows that q(e, b, z ) is strictly concave only for above a certainthreshold. The problem is that, in the neighbor of the failing shock, the end-of-period resourcefunction is not concave. However, if the density probability satisfies the restrictions of Proposition3, then T2 will map increasing and concave functions in strictly increasing and concave functions.

This point can be shown as follows: Take two points for equity e1, e2 and two points for debt b1,b2 and define e, b to be a convex combination of these two points with 0 1. Then, byLemma 1, if the density function of the shock satisfies the conditions of Proposition 3, there exists such that the term

(z ) q(e

, b

, z ) q(e1, b1, z ) (1 )q(e2, b2,

z ) is greater than zero if , and nonpositive if . The concavity of q(e, b, z

) f(d), however, implies that the concave points dominates the nonconcave ones and (z ) f(d) 0.

Now consider the term:

(A10) qe , b , z qe1 , b1 , z 1 qe2 , b2 , z .

Because is increasing and concave, this term is greater than zero if

(z ) 0, but may besmaller than zero if

(z ) 0. Although the points for which

(z ) 0 dominates the

points for which

(z ) 0 and

(z ) f(d) 0, this is not necessarily the case forthe function defined in (A10). For this to be true, we need to restrict the quantitative importance of for those values for which the firm defaults. The firm will never default if 0. Therefore, we

have to impose that f() is relatively small for 0 as assumed in Proposition 3. We then have:



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qe b , z qe1 b1 , z 1 qe2 b2 , z fd

qe b , z qe1 b1 , z 1 qe2 b2 , z fd 0.The first inequality comes from the imposed restriction on f, while the second inequality comes fromthe strict concavity of the expected value of q. Also notice that the slope of is always between 1and 1 . Therefore, there is a limit to the possible amplification of nonconcave points. Given thisresult, it is easy to prove that the mapping T2 maps concave functions into strictly concave functionsand the fixed point ** (e) is strictly concave in e. Moreover, the optimal solution for b is obviouslyunique (this is simply a problem of maximizing a continuous and concave function over a compactand convex set). The theorem of the Maximum will then guarantee that the solution is continuousin e. We can also show that the optimal value of b is such that k e b is nondecreasing in e.

Simply observe that, due to the concavity of the expected value of , the marginal return from k isdecreasing in k. Because a larger e relaxes the constraint on the feasible k, the reduction in k is notoptimal.

Given the strict concavity of *, the dividend policy assumes a simple form. More specifically,there exists a lower and upper bound e and e, with e e, for which dividends are negative (the firmissues new shares) when the end-of-period resources are smaller than e , and positive when theend-of-period resources are larger than e.

With persistent shocks, the proof follows exactly the same steps. The only difference is that thelower and upper bounds for the next period equity and the failure value of equity depend on the nextperiod z, that is, e(z) e (z) e(z).

PROOF OF PROPOSITION 4:Let us start the proof by assuming that, conditional on surviving, z is constant. The extensionof the proof to the case of persistent shocks is trivial. Let Q(e, A) : [e , e] A3 [0, 1] bethe transition function, where A is the collection of all Borel sets that are subsets of [e , e], andA is one of its elements. Note that we can define the transition function over the measurablespace ([e , e], A) given the optimal policy of the firm characterized in Proposition 3. Thefunction Q delivers the following distribution function for the next period equity e, given thecurrent value of e:

(A11) e

x

Qe, de

1

fd if x e

1

x e

/Fe b

fd if e x e

1 if x e

where is the mass of new entrant firms which is equal to the mass of exiting firms. In this way,the total mass of firms is constant. By normalizing the total mass of firms to one, the distribution offirms is represented by a probability measure. Given the current probability measure t, the functionQ delivers a new probability measure t1 through the mapping : M([e , e], A) 3M([e , e],A), where M([ e , e], A) is the space of probability measures on ([e , e], A). The mapping isdefined as:



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(A12) t 1 A t A e

Qe, Ad.

An invariant probability measure * is the fixed point of , i.e., * (*). In the followinglemma, we prove that Q has the Feller property. This property turns out to be useful in thesubsequent proof of the existence of an invariant probability measure.

LEMMA 2: The transition function Q has the Feller property.

PROOF:The transition function Q has the Feller property if the function T(Q)(e) defined as:

(A13) TQe e

e

veQe, de

is continuous for any continuous and bounded function v. Conditional on being productive (whichhappens with probability 1 ), the next period equity is given by:

(A14) e e if

e

Fe b if

e if .

Therefore, the function T(Q)( e) can be written as:

(A15) TQ e ve 1 ve e

f e, b e e

Fe b Fe bde

e

e

vef

e, b e e

Fe bFe bde

ve e

f e, b e eFe bFe bde .Because b(e) is a continuous function, then F(e b(e)) is also continuous. If in addition (e, b(e))is continuous in e, then the continuity off implies that T(Q) is continuous. So we only need to provethat is continuous.

The function (e, b) is implicitly defined by:

(A16) 1 rb e

z

fd

fd 1 e b

Fe b

where () z

f(d) f(d), is strictly increasing and continuous in under

Assumption 2. This implies that is invertible and the inverse function is continuous. Given this,



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it can be easily verified that for the relevant range of e and b, (e, b) is a continuous (singleton)function of e and b.

Because the probability measure of firms has support in the compact set [ e , e] and, as proved inLemma 2 the transition function Q has the Feller property, then Theorem 12.10 in Nancy L. Stokey

et al. (1989) guarantees that there exists an invariant distribution *. In order to prove that * isunique we need extra conditions. Theorem 12.12 in Stokey et al. (1989) establishes that, if Q ismonotone and satisfies a mixing condition, then the invariant probability measure * is unique. Wewant to show then that under the assumption made in the second part of the proposition, Q ismonotone and it satisfies the mixing condition.

(Monotonicity) To prove that Q is monotone, we have to show that Q(e1, ) is dominated byQ(e2, ) @e1, e2 [0 , e], with e1 e2. The dominance means that for all bounded increasingfunctions v, v(e)Q(e2, de) v(e)Q(e1, de). When the state space is defined in R

1, thenQ(e2, ) dominates Q(e1, ) if and only ife

x Q(e2, de ) ex Q(e1, de), @x [e , e]. Given

the monotonicity of k(e) e b(e) stated in Proposition 3, if the default probability isdecreasing in e[(e, b(e)) is decreasing in e], it can be verified that Q, defined in (A11), ismonotone.

(Mixing condition) We have to prove that there exists e [e , e], 0, and N 1 such thatN(e , [e, e]) and N(e, [e , e]) . Because we are assuming that f() 0 @ R,then the mixing condition is obviously satisfied.

APPENDIX B: COMPUTATIONAL PROCEDURE

The computational procedure is based on value function iteration, after the discretization of thestate space e. Following is the description of the individual steps.

1. Guess default thresholds e(z) and equity bounds e (z) and e(z). Then for each z

Z, choose agrid in the space of firms equity, that is, e E(z) {e1(z), ... , eN(z)}. In this grid, e1(z) e and eN(z) e(z).

2. Guess initial steady-state values of debt b*i(z), for i 1, ... , N and z Z.3. Guess initial steady-state values of firms value i(z), for i 1, ... , N and z Z.4. Approximate with a second-order Taylor expansion the function i(z, b), for i 1, ... , N,

around the guessed points for the steady-state values b*i(z). The value function (z) isapproximated with peacewise linear functions joining the grid points in which the value functionis evaluated. The definition of i(z, b) takes as given the dividend policy of the firm consistingin retaining issuing new shares in the interval [ e(z), e (z)] and retaining all profits until the firmreaches the size eN(z).

5. Solve for the firms policy bi by differentiating the functioni(z, b) with respect to b and update

the default threshold using the condition ( e (z) e(z))(1 ) (z, e ).6. Eliminate b from i(z, b) using the policy rules found in the previous step. The found values are

the new guesses for i(z, b). The procedure is then restarted from step 4 until converged.7. After value-function convergence, check whether the firm policies found in step 5 reproduce the

guesses for the steady-state values of debt b*i(z). If not, update these guesses and restart theprocedure from step 3 until convergence.

8. Check the optimality of the lower and upper bounds e (z) and e(z) by verifying

z, e

e

e e z

1 and z, e

e

e ez

1. To check for these conditions,

compute the numerical derivative of (z, e) at e1(z) and eN(z), taking as given the value ofb*1(z) and b*N(z) found previously. If the condition is not satisfied, update the initial guesses for

e (z) and e(z) and restart the procedure from step 1 until convergence.



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9. Given the decision rules, the transition function for the distribution of firms is well defined andthe invariant distribution of firms is determined by iterating on the law of motion for thisdistribution.

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Financial Markets and Firm Dynamics