- Email: [email protected]
Financial constraints and firm post-entry performance
International Journal of International Journal of Industrial Organization 13 (1995) 543-565
ELSEVIER
Industrial Organization
Financial constraints and firm post-entry performance Paulo Brito a'b'*, A n t 6 n i o S. Mello a'c'd aBanco de Portugal, DEE, Av. Almirante Reis, 71, 1100 Lisbon, Portugal bUniversidade T~cnica de Lisboa- ISEG, 1200 Lisbon, Portugal CUniversidade Cat6lica Portuguesa, Camino Palma de Lima, 1600 Lisbon, Portugal dCEPR, 25-28 Old Burlington Street, London, WIX 1LB, UK
Abstract Firms finance production by internally generated funds and external loans. The benefits of leverage, however, come with a cost. This cost is related to the uncertainty banks face about the firm's quality and output price. As time evolves banks learn about the firm and adjust the terms of the loan contract. Because of this, firms do not have equal access to credit: small, young firms face greater binding debt constraints than more mature firms with well-known prospects. The firm survival rate, as well as the firm rate of growth, are, therefore, important issues in analyzing firm post-entry performance. Keywords: Financial constraints; Firm growth; Demography J E L classification: G3; L1
1. Introduction
Recent models that analyze the problem of financing firm's production and investment opportunities consider that different sources of capital are not perfect substitutes for one another. With imperfect information, invest-
* Corresponding author at: Banco de Portugal, DEE, Av. Almirante Reis, 71, 1100 Lisbon, Portugal. 0167-7187/95/$09.50 O 1995 Elsevier Science B.V. All rights reserved S S D I 0167-7187(95)00504-8
544
P. Brito, A.S. Mello / Int. J. Ind. Organ. 13 (1995) 543-565
ment spending is constrained by a financial hierarchy, in which internally retained funds are cheaper than external capital provided by investors who are unable to evaluate precisely future prospects of the firm and the quality of its management team. In this paper we analyze the problem of financing firms' production opportunities when firms cannot secure sufficient internal funds and need additional external finance. There is asymmetric information between those that own and control the assets of the firm and outside investors. However, as time evolves, these can learn more about the quality of the firm's m a n a g e m e n t and accordingly adjust the terms of the financing contract. By doing so we are able to capture the effects of capital constraints on output decisions as well as on the probability of firm survival and firm growth. Firms do not have equal access to capital markets, because small, young and relatively unknown firms face greater liquidity restrictions and higher cost of capital than more mature firms with well-known prospects. Currently there are two main rival approaches in the literature analyzing the demography of firms. The first, known as Gibrat's law, states that the growth rates of firms are independent of firm size. The second relies on the work by Jovanovic (1982), which establishes a relationship between firm age and firm growth. Firms can start small and suffer from scale disadvantages. T h e successful firms grow and become more efficient, not just because costs are reduced but also because firms learn from their own experience; unsuccessful firms, on the contrary, remain small and eventually may exit the industry. Our model belongs to the recent literature highlighting the interaction between finance and production decisions. One earlier paper on this topic is that by Greenwald and Stiglitz (1990), who show the importance of financial constraints on firms' investment decisions and discuss how the strength of firms' financial position can explain the propagation of macroeconomic disturbances and the persistance of their effects. In our paper we study how growth and survival of firms is affected by the evolution of the c r e d i t o r borrower relationship, when there exist multiple sources of asymmetrical information. We show that the firm's age and size are not independent of the duration of the relationship between those who control the assets and those who finance the company. Although we think that learning by those who manage the assets is important, it is not the complete story or even the most crucial factor during the first stages of a firm. We think that learning by outside financiers about the firm is perhaps of the same degree of importance or even more important in determining survival and growth. Our model, just as in Jovanovic's model, implies that the exit rates are a decreasing function of firm size and age, consistent with what Dunne et al. (1989) have shown. Furthermore, we show that the growth rate of successful
P. Brito, A.S. Mello / Int. J. Ind. Organ. 13 (1995) 543-565
545
firms increases at a decreasing rate with age, consistent with the findings in several empirical studies about post-entry performance. The importance of taking financial and production decisions jointly is key to a better understanding of entry and exit decisions of firms, as well as to the study of post-entry performance. In most cases, firms simply do not cease to exist or have any sort of limitations self-imposed. They live, are forced out of the industry or suffer major restructuring from the actions taken by the different claim-holders. Models that ignore that firms are governed by combinations of contracts established between and monitored by parties with different interests and information are too simplistic to provide accurate explanations of reality. Of the various external sources we concentrate on debt finance from banks. Information on the financing practice of firms in different countries shows that banks are the dominant source of external finance (see Taggart, 1985; Fazzari et al., 1988; Mayer, 1990). Mayer, for instance, observes that banks are the dominant source of external finance in the G-7 countries and that small and medium-sized firms are considerably more dependent on external debt than large firms. Also, usually banks establish long-term relationships with borrowers, which allow them to learn progressively more about companies, an issue in which we are particularly interested in this paper. Specialization in monitoring, as well as the reduced severity of holdout problems, give a clear cost advantage to bank debt relative to diffusely held public debt, as James (1987), Gilson et al. (1990) and Hoshi et al. (1990) have noted. Although firms could alternatively issue additional equity, we exclude this possibility on the assumption that equity issues would be subject to adverse selection and therefore too costly to firms (see Greenwald et al., 1984; Myers and Majluf, 1984). The model focuses on the behavior of firms that maximize expected profits net of debt repayments owed to risk neutral banks, that have temporarily inferior information about the type of the owner-manager. In case the firm does not repay the promised amount to the bank the firm defaults. The dead-weight costs of financial distress are paid by the equityholders of the firm, who therefore make output decisions taking into account the possibility of bankruptcy. As in the case of Greenwald et al. (1990), because output is partly financed with debt, when the firm increases production the probability of bankruptcy also increases. The interesting feature of our model is that by explicitly considering a time-varying degree of asymmetrical information, in order to capture the learning process, we can analyze firm output and capital structure decisions and bank contractual s c h e d u l e s - interest rate charged and amount lent to finance production b e y o n d the level which the firm is able to finance out of its own cash-flows as the b o r r o w e r / l e n d e r relationship evolves through time.
546
P. Brim, A.S. Mello / Int. J. Ind. Organ. 13 (1995) 543-565
2. The model
In this section we model the behavior of a firm, with some of the features found in Greenwald et al. (1990), although there are significant differences in the postulated a s y m m e t r y of information, as well as in content. The firm m a k e s decisions at discrete intervals of time t = 1 . . . . . T. At the beginning of period t the firm has a notional amount of fixed-rate debt outstanding, B,, which has b e e n contracted with a bank at the nominal rate R,. The firm needs cash in advance from the bank to pay for the use of inputs for production at time t. The firms asks for m o n e y from the bank when its own accumulated funds, A,, are insufficient. Therefore, there is a preference for the use of retained funds before the firm looks for external sources, because internal funds have a cost advantage over new debt. The higher cost of debt capital results from agency problems that might arise because the bank extending the loan has a rough idea of the output price and does not observe with complete precision the type of entrepreneur(s) that manages the firm and is assumed to maximize the value of his (their) equity position. We assume that the debt contract is short-term and lasts for a single period. The loan can be renegotiated for the next period or not. The short-term maturity of the debt contract is somehow equivalent to a threat to cut off credit at the b a n k ' s discretion and therefore provides an incentive for the firm to take actions that reduce moral hazard (see Stiglitz and Weiss, 1983). Also, the firm does not pay out dividends to equity-holders. These two assumptions are quite c o m m o n in practice in the case of small and medium-size companies and can be interpreted as debt covenants imposed by the lender. T h e firm hires the amount of labor 1, to produce q, units of output. The labor requirement function, 1, = g(q,), is increasing and convex in q,: g ' > 0 and g" > 0. Total production costs at t are in nominal t e r m s e t w t l t , where w t is the real wage assumed to be determined in a competitive labor market, and P, is the aggregate price level. Therefore, firms borrow an amount equal to
B,= P,w,I,- A, .
(1)
The m a n a g e m e n t of the firm contributes to the value added of the firm by a factor e. This p a r a m e t e r , defining the type of the firm, can be seen as the skills affecting the organization of production or marketing or both. The firm knows its type, e, but outside financiers do not observe it. We assume that for the b a n k providing credit to the firm the type is a stochastic variable with a variance decreasing with time:
et=e+Ot(t*-t ) t~t*, where t* is the date when the bank learns the firm's type with complete precision and 0 is drawn from a uniform distribution over the interval (0, _0),
13 Brito, A.S. Mello / lnt. J. Ind. Organ. 13 (1995) 543-565
547
where 0 is positive and _0 is negative. Then e, has also a uniform distribution over the interval (~, e) = (e + O(t* - t), e + -0(t* - t)), where we assume that e>0foranyt. The output produced at time t, eqt, is sold at time t + 1 and is assumed to be perishable. The price at which the firm sells its output at time t + 1 is unknown at the time of production. This is because firms face a stochastic relative price as regards the overall price level, ff~+l, such as P[+I = tT~+,P,+,, where u f follows a uniform distribution over the interval (t/f,__uf), where t/f, u-f > O. The accuracy of the forecasts of the relative price by the banks, t ~ , is, however, inferior. As it also follows a uniform distribution with support (u-tub,t/b), then formally t/b _ _Ub i> t/f _ / f . That is, when anticipating the firm's selling prices as P~÷I : U~bt + l e t + l , banks are assumed to know less than the firm about the markets where the firms sell their products. We leave imprecise the relative level of the bank's ignorance, which can vary with the firm and the industry. Even if the length of time of the bank-firm relationship can help to reduce the gap, it is realistic to assume that banks do not specialize in knowing the idiosyncratic details of each client's business. When the output is sold, the firm either repays in full its debt with accrued interest (1 + R t ) B ,, or it is declared bankrupt. In that case, the bank takes possession of what is left, P~+leqt. It can be seen that bankruptcy occurs when P~+leqt <~ (1 + R t ) B t , which implies that A t + 1 ~ O. In the event of bankruptcy shareholders pay costs c f = c(qt), and c(.) is an increasing and convex function of q,. Thus, the costs increase with the size of the firm. The costs borne by shareholders could result from increasing concerns of suppliers, customers and employees (see Shapiro and Titman, 1986). Alternatively, these costs can represent the reputation lost by managers in the event of the firm's financial difficulties. Banks maximize the expected value of profits by determining the amount of credit, Bt, and the rate charged R,, simultaneously. The opportunity cost of lending is the market real interest rate, p,, which is set exogenously. It is assumed that the amount lent to each firm is small compared to the large and diversified portfolio of loans hold by the bank. It is reasonable to assume that u and e are independent random variables. We then define a new variable for the bank
Zt+l := Ubt+le' with a density function depending upon the supports of the price and management quality distributions, fz(x; t/b, U_b, £ e),
548
P. Brito, A.S. Mello / Int. J. Ind. Organ. 13 (1995) 543-565
1
~ f ~l (x; . ) ,
ifubg~aU_e '
f~(z;a u,_uu , g , e ) : - ( z i b _ u b ) ( y _ _ e ) x ( f ~ ( x ; . ) , where
and f z ( x ; . ) : = I¢a.abe)(x) log ~
+ I(ab~,ub~)(X) log
U°
+ l(ub~.s)(X ) log
,
where z := ube and ~? : = / i b e and Ic)(x ) are the indicator functions for z = x. T h e distribution fz is a non-linear function with truncated support. Given its shape and moments, we can easily see that the results found in the paper are robust for a wide class of distributions of z, requiring only the realistic assumption that z is bounded. T h e r e f o r e from the bank's viewpoint the equity of the firm next period can be written as At+l = P,+I(Z,+Iq,)- (1 + R,)B,. If we deflate by Pt+l, which is perfectly anticipated, and denote the real interest rate charged to the firm by r t, and note that 1 + r, = (1 + R,)P,/(Pt+t), then, the real value of the firm's equity, according to the bank, is
a,+, = (zt+,q,) - (1 + r,)b,. We assume that banks observe z but are unable to separate u from e. In other words, banks observe the firm's revenues but do not know whether these depend on the quality of the management or on the realization of the price at which the products are marketed. Banks' real rate of return for a particular loan is a stochastic variable which depends upon the solvency of the firm: ~(1 + r,)
(l+Ft)=t~t+iq,/b ,
ifa,+l ~>0 ifa,+, < 0 .
We set Z,+l as the level of z such that a,+~ = 0: i.e. d,+,
(1 + rt)b , q,
The expected real return from lending to the firm is
(2)
P. Brito, A.S. Mello / Int. J. Ind. Organ. 13 (1995) 543-565
549
qt f zt+l g[(l+~t)]=(l+rt)(l--f(zt+l)+-~t zdF(z) = (1 + r,)(1 - F(~t+1) + G ( ~ , + , ) ) ,
(3)
where Prob[z ~< z~t+l] = F(z~t+z) = ff,+l dF(z) is the probability of bankruptcy calculated by the bank, and G(~,+~)= (1/(~t+l)) f~t+l dF(z) is the average liquidation income when bankruptcy occurs. Since the credit risk on the loans is diversifiable, banks determine the optimal contractual interest rate from El(1 + f',)l = 1 + p,.
(4)
The resulting bank's supply schedule, according to the following results, is positively sloped and displays credit rationing.
Lemma 1. The bank's supply of loans schedule is positively sloped in the (r,, q,) space. The real volume of credit extended is positively related to the amount of equity available, a,, and to the limits of the perceived distributions of the quality of the management and of the relative price faced by the firm. Proof. See the appendix. Intuitively, the bank offers the firm a schedule of the pair (r, b), to partially finance production, from which the firm chooses. Interestingly, the bank may decide not to extend any credit to the firm. That is the case when the prospects of the firm are so low that the bank expects to have a return on capital below the risk-free rate, p, as the following lemma shows:
Lemma 2. If ~ >I ~ then the bank rations credit at the level of output E(z)q/b <- (1 + p) for q >1qb.
qb, since
Proof. See the appendix. At the margin, when the bank does not expect to have a return at least equal to the opportunity cost of risk-free capital, no further money is lent to the firm and the credit is rationed. Fig. 1 shows the effect of a change in E(e), the bank's evaluation of the firm's management. The supply curve moves to the right as the bank improves its knowledge of the firm. Thus, for two firms with management of the same type, the older firm gets more funds than the newer one, for the same interest rate charged. Alternatively, other things being equal, two firms that have been dealing with the bank for the same amount of time are charged rates that depend on the bank's assessment of the firm's quality.
550
P. Brito, A.S. Mello / Int. J. Ind. Organ. 13 (1995) 543-565
1+r t
t
s
/,
s/
J+p i-
qb
qb'
qt
Fig. 1. Bank supply of funds and assessment of firm's management quality.
As the perceived prospects of the firm deteriorate, the bank's supply curve shifts to the left. In some cases the assessment of the risk is so high that the situation is practically equivalent to the firm being excluded from the pool of the bank's clients. The firm chooses the level of output by looking at the conditions of the debt contract, (r,, b,). Management maximizes the expected value of the equity of the levered firm. Because the firm knows its own type, the only source of uncertainty comes from the shock to the relative price of its output next period, u~+ 1. Financial distress imposes a cost on shareholders of the firm. This cost may reflect the loss of reputation faced by the management, or represent the more stringent conditions imposed by suppliers, customers and employees in future contracts with the owners of the firm. The model could easily incorporate the situation where the bankruptcy costs were shared by banks. In this case the profit function of the firm - and that of the b a n k - w o u l d be modified, since the interest-cost paid would include an additional component due to the expected portion of the liquidation cost borne by banks. Without loss of generality, we only consider the costs to shareholders from bankruptcy. The probability of bankruptcy is given by F(t~t+ 1)
=
f
dF(ut+l)=
,
f
(tif-_u ') d u t + , -
Ut+l
--
fff__uf ,
P. Brito, A.S. Mello / Int. J. Ind. Organ. 13 (1995) 543-565 where
/~,+1 = ( l q--
551
rt)bt/eqr
Real profits are stochastic and the firm's problem is max E(Ht+ 1) - c(q,)F(gtt+ 1), qt
for b/> 0, and where Ylt+l are real profits to equity-holders after payments to debt-holders. The first order condition for a maximum is then given by eE(u~) - (1 + rt)wtg'(qt) - 7rt + ~t(wtg(q) - at) = 0,
(5)
where {: is the Kuhn-Tucker multiplier, zr, is the premium related to bankruptcy, which induces the firm to reduce its output: 7r,:=
Oc(q,)F(•,+l) Oc Oq, - Oq, F(/J'+I) + 1
-- /~f - u f
c -f
U
--
Uf
Oa,+ 1 Oq,
(C,(tit+ __u__f)+C~t+l [ ( g) 1 ~ W, g' --
at] } +-~ > 0.
(61
The fact that firms act to avoid bankruptcy generates a kind of risk-averse behavior. Indeed, if there are incentives for managers to build their reputations, Hirshleifer and Thakor (1992) have shown that firms' investment policies tend to be more conservative. Given Eqs. (5) and (6), the demand for funds from the firm, shown in Fig. 2, is given in the following lemma: L e m m a 3. The demand for external funds f r o m the bank is inversely related to the rate charged by the bank. However, the firm may decide to operate with no debt when the bank charges a rate above some specified level, r f. Proof. See the appendix. Normally the firm would ask for money from the bank, but there are instances when the firm chooses to operate as a purely equity-financed firm, even knowing the power given by leverage to increase profits. This occurs when the bank charges a rate above some level, r e. L e m m a 4. The higher the quality o f the firm's management, e, the higher the expected relative output price, E(uf), the lower the operating costs, w, and the lower the costs associated with financial distress, the higher the debt capacity and the desired leverage ratio. Proof. See the appendix.
552
P. Brito, A.S. Mello / Int. J. Ind. Organ. 13 (1995) 543-565
!+r t
D l+rf
qf
1,
qt
Fig. 2. Firm's demand for funds.
3. Equilibrium In equilibrium the firm's output level and the bank contractual c o n d i t i o n s - interest rate and amount of debt - are functions of the market real interest rate, the real wage rate, the level of equity capital available, the type of the firm's management, the precision error and the distributions of the firm's relative price for both the firm and the bank: r, = r t ( l + p,, w,, at; g, e, t/f, _uf, ~/b, L b)
(7)
qt = q,(1 + Pt, w,, at; Y, _c, t/f, u f, t/b, u b) ,
(8)
Consider the ratio a t / w t l , , representing the amount of labor that firms can hire without having to resort to external financing. The increase in supply from hiring more labor services is only possible through additional external funds. Given the contractual conditions offered by the bank, which depend upon the bank's expectations about the firm's behavior, there is an equilibrium interest rate and amount of debt, and a corresponding level of firm's output. T h r e e possible situations may occur, corresponding to equilibria that differ with respect to the parameter values characterizing the borrowing firm (see Fig. 3). First, if the contractual conditions are too stringent, firms may
P. Brito, A.S. Mello / Int. J. Ind. Organ. 13 (1995)543-565
553
i +r t D
S
l+rf
l+r~
iiii
1+/3 I I
I I
I I
b
qf
q~
qb
qt
Fig. 3. Equilibrium. decide not to borrow from the bank and the level of production is determined by the amount of internal finance available. Second, if firms may decide to borrow more than the bank is willing to lend, given the bank's expectation about the firms' type and the relative output prices, there is credit rationing by the bank, resulting in an output constrained by the amount and not by the price of credit. This would be the case when the interest rate which the firm is willing to pay may act as a screening device for the bank, since those firms that accept paying high interest rates are, with greater probability, the worse risks (see Stiglitz and Weiss, 1981). Third is an intermediate case, where firms operate partially financed with debt, although with a leverage ratio smaller than that under the first best case. The following proposition summarizes the scale of production and the capital structure of the firm in equilibrium.
Proposition I. Under the prevailing labor and output market conditions, the firm's production scale is bounded by the capital structure. The lower bound is determined by the amount of internal funds available and the upper bound is determined by the maximum level of indebtedness that the bank is prepared to accept. Proof. See the appendix.
554
P. Brito, A.S. Mello / Int. J. Ind. Organ. 13 (1995) 543-565
In the following section we analyze how the differential information between the parties, which is assumed to decrease over time, affects firm performance, growth and probability of survival.
4. The effect of liquidity constraints on post-entry performance
The model in the last section is consistent with the view that information and incentive problems in debt markets affect corporate production decisions. Recent empirical evidence (see Whited, 1992) seems to indicate that the effect of a liquidity constraint appears to be stronger for firms that rely more on bank debt and are not able to participate in the corporate debt market. On the other hand, firms with long-term relationships with the lender usually get loans more easily. It is often the case that relationship factors that favor large well-established firms and discriminate against startups may outweigh the profitability of investments in determining which firms get loans. In this section we explore the importance of the long-term relationship between lender and borrower and show that with imperfect information, small unknown firms, even with better prospects, may face worse contractual terms. A bank that lends to a firm can obtain information about the client in the course of lending. It takes time to learn the firm's type, and only by date t* the bank knows it with complete precision. The learning period may be firm as well as industry specific and may depend on various factors such as the technology, the degree of ownership concentration, the stability and formality of procedures regarding the internal organization of information reporting, size, etc. When the bank assesses on average the quality of the firm below its true value, that is, when E(O,)= E ( e , ) - e < 0 , during the learning period, the following proposition holds. Proposition 2. As the bank's error in observing the true type of the firm decreases with time, the equilibrium interest rate charged by the bank will decrease, the binding debt constraint will decrease and, as a consequence, the firm's output increases. Proof. See the appendix.
Furthermore, the probability of survival of the firm will improve as time goes by, because by improving the terms of the loan the firm needs a lower fi,+~ = (1 + r,)bt/eq, to avoid bankruptcy, which is directly related to the terms of the loan contract. This result is formally stated in the following proposition.
P. Brito, A.S. Mello / Int. J. Ind. Organ. 13 (1995) 543-565
555
Proposition 3. I f the labor requirement function is sufficiently convex, i.e. if w,g'q, - b, - g"q,bt/g' < O, then the probability o f bankruptcy will decrease over time. Proof. See the appendix.
Both propositions establish conditions discriminating against start-ups and against more opaque firms. The bank's inference of the quality of the firm depends on 0 and on the length of the relationship with the client firm, t. In comparing two different firms, or the same firm when new and mature, the proposition says that, ceteris paribus, when 0 is smaller or the firm is more mature, that is, when the firm is better known, the binding debt constraint becomes weaker. Furthermore, new firms tend to be more liquidity constrained and therefore to be potentially smaller than better-known and established businesses. Depending on the degree of asymmetrical information, there may be instances when firms either do not apply for credit and work only with internal sources, or if they ask for a loan the probability of survival will depend on having either a high type or a good state that realizes for u. In any case, the slower is the process of learning the more likely it is that the firm will not survive. Notice that because the production function displays decreasing returns to scale, the incentives to expand capacity decrease as output goes up. Therefore, expanding production financed by additional loans will be limited by the greater probability of bankruptcy. Fig. 4 shows how learning by banks will shift the supply of loanable funds to the right. Two implications result from the increase in precision: (i) the bank will lower the interest rate charged for a given level of debt, and (ii) the bank will increase the level of credit available to the firm as well as the level at which credit rationing occurs. Some firms initially rationed may later get funds and increase the level of output produced. Implication (i) could be alternatively stated in terms of requested collateral, and Berger and Udell (1992) find that the bank is less likely to demand collateral if the relationship with a firm is long term. Petersen and Rajah (1994), however, find that, according to implication (ii), the relationship has value but works more through quantity than price, because banks, either because of market power or because of internal control and agency problems along the managerial hierarchy, prefer instead to increase the availability of credit or to relax the credit-rationing constraint. The results above provide interesting explanations for often observed facts. Now we want to analyze what are the implications for firm growth of informationally generated liquidity constraints. As a simplifying assumption, assume that the bankruptcy cost function is now linear. Then, Fig. 5
•q l t a o a ~ tu.lt3 p u e uop, e n l e A a s,,,luuq j o s a ! t u e u , { o
' g "~!zl
,1
lb
' u o ! s ! a o a d SplUeq u! a s e o a o u I 17 "~!al
lb 1
,qb t I i
qb
7b
i I l
I
(I
ff9K-fPS- (ff66I) f I "uv8.tO "pul T "lUl / °llalN "S'V '°l~..t~t "d
9gg
P. Brito, A.S. Mello / Int. J. Ind. Organ. 13 (1995) 543-565
557
illustrates the dynamics of firm's growth, measured by output, which, in this case, is equivalent to asset value, from the moment the firm enters the industry until it is perfectly identified by the bank t*. After t*, if everything else is kept constant the same level of production will replicate period after period. Notice that the situation after t* is still a second best, because the firm and the bank have different uncertainty regarding the firm's relative price next period. The following proposition states the post-entry growth rate of the firm. Proposition 4. Assume that the firm's bankruptcy cost function is linear and that wtg'/2 - bt/q, > O. Then as time passes and the bank learns more about the firm's type, output increases at a positive and decreasing rate. Proof. See the appendix.
Even when the rate of learning by the bank is constant, the marginally improving conditions on the debt contract are such that the firm performance, measured by the growth rate of output, or by the return on assets, decreases over time. This result depends on the hazard rate, measured by the probability of default, declining with time at a decreasing rate, suggesting that there are decreasing returns to learning by outsiders (first impressions are more i m p o r t a n t ) - r e s u l t i n g in a non-linear (r, b) s c h e d u l e - and also on the assumption of a production function displaying sufficiently high decreasing returns to scale (see Fig. 5). From Lemma 1 and Propositions 3 and 4 taken together it follows immediately that the probability of a firm's failure is decreasing in the initial level of e. Given that the initial size of the firm is directly proportional to the initial level of e, our model has the additional implication that the probability of failure decreases with the initial size of the firm, an empirical fact found in Audretsch (1991). In terms of empirical implications our model seems to be more in line with work of Jovanovic than with Gibrat's law. But our version is completely different from that of Jovanovic and is a new approach that combines financial and production decisions in analyzing the post-entry performance of firms. With imperfect capital markets the role played by the different claim-holders in deciding the future of firms seem to lend support to this view in dealing with the issue of firm demography. Although we have represented t* to be the date when the bank finally discovers the true type of the borrower, one can think of the period before t* as the period during which the firm builds its reputation by taking on costly bank debt. Then, in the spirit of Diamond (1991), firms that have established a good reputation move on to the bond market to save on the monitoring costs, or any rents extracted from a bank that has acquired
558
P. Brito, A.S. MeUo / Int. J. Ind. Organ. 13 (1995) 543-565
bargaining power over the firm's profits (see Rajan, 1992). Before t*, the costs associated with switching to a new alternative lender and the negative signals that potentially may result from the move may be significant and outweigh any benefits (see Stiglitz and Weiss, 1981). Indeed, according to the findings of James and Wier (1990), banking relationship renewal provides a positive signal to other investors in the economy.
5. Conclusions
In a simple model we analyze the effects of financial market imperfections on productive decisions of firms. The fact that banks initially cannot observe the true quality of the firm's potential imposes on small, less-known firms an additional burden in the form of a high required premium and greater capital constraints due to agency problems. For more mature companies, with an established relationship with banks and with sufficient reputation to issue traded debt in the capital markets, interest rates are lower and loan sizes correspondingly less limited. Maturity is therefore a factor of access differentiation among firms in the same industry. Known firms have a cost advantage over newly established but unknown firms. So long as firms need external financing the relationship gap affects output decisions and the rate of firm growth, that is, the performance of firms. Firm size and age are positively correlated with growth of entrants. In some cases only very efficient small firms can survive the initial stages of their relationship with banks, and firm growth can be significantly affected during infancy, implying that the height of entry barriers can be high (see Geroski, 1995). Moreover, the slower the process of learning by outside investors the more likely it is that firms will not survive. Thus, financial market imperfections can have a negative impact on economic growth and on development stimulated by the appearance of new firms that replace older and established businesses. In addition, and similar to Greenwald and Stiglitz (1990), the model has important implications for business cycle behavior. The effect of a binding liquidity constraint can have significant effects on the level of aggregate output when firm's prospects and balance sheets deteriorate, especially in countries where most firms are small, under-capitalized and the ability of financial intermediaries to obtain reliable information is reduced. Such circumstances could also explain why often firms require venture capital during the first years of their existence, and why debt to equity ratios, within the same industry, are often a function of the firm's maturity.
P. Brito, A.S. Mello / Int. J. Ind. Organ. 13 (1995) 543-565
559
Acknowledgements We are grateful to Pedro D. Neves, three anonymous referees and seminar participants at CEMFI (Madrid) for helpful suggestions. The usual disclaimer applies.
Appendix: Proofs
Proof of Lemma 1. Eq. (4) can be written in implicit form as (deleting the time subscripts from now on) ~bb(1 +r, q; 1 +p, w,a, 5,_u, g, e) = E(1 + r ) - (1 + p ) = 0 . In Eq. (3), after performing the necessary integrations, we have the following expressions, for the case where ~bg~/tb_e (deleting the time subscripts), z - d(1 - log(d/z))
F(d;ttb,-ub, e,e) :-- (fib +
ttb)(~_e)
I(,.ub~)(d)
Z -- Ubg + ~ log(g/e)
- -
w
[ +
1
£ - d(1 + log(:?/d)) ] (-~--_ub)(g_ e ) I(~%,e)(d),
and b
G(d;ffb, u , e , e ) : -
_Z2 -- ~2(1 --log(d/Z)) 4 d ( t i b _ u b ) ( ~ _ e ) l(z,ub~)(d) +-
Z2 -- (__ubg)2 + 2dE log(g/e) 4~07b __ub)(g__e) I(ub~,"be)(~)
+ [ E(Z)z= £2-4d(5 dz(1b +ub)(g2 log(Z/~))_ _e) ] I(a%'',(~) Making the suitable substitutions we get d = [(1 + r)(wg(q)- a)/q]. For -b any support of the distributions (i.e. for any relation between _UbYand u e) we get the following partial derivatives, a4) b 0(1 + r) - 1 - F(.) E (0, 1) ; (1 + r)G(.)(wg'(q)q - b)
a~b b
0~-
-
qb
<0,
560
P. Brim, A.S. Mello / Int. J. Ind. Organ. 13 (1995) 543-565
from the convexity of g(.); O~bb (1 + r)G(.) >0, Oa b O~bb (1 + r)G(.)l <0, Ow b O4,b --1, 0(1 + p) O~bb
1
(l+r)(2--U._bY)Zl(ubo,~)(5)]
[
0g - ( O - e )
r-p-
04~b 1 [ 3e - (g-_e) [ I + P +
2g(aU_u~)2 (1 + r)E(ub)e_ 2
(1 + 0 ( 2 - t~be) z ~Klu
/>0,
]
_u),,~
0~bb 1 [ 0rib -- (tib__.ub) r - p
(1 + r ) ( 2 -- / i b e ) 2
Oth b
(1 + r)E(e)u_ b
2ffb(g__e) 2
] I(au,.z)(2 ) />0,
and 1
[-
ae --(~b--_Ub) [ l + p - +
J
(1 + r)(2 - ubg) 2 i(,,ub~)(2)] 2ub(g -- _e)2 ~> 0
The signs for the partial derivatives involving the support for Y and tT, though they may seem to be uncertain, were evaluated for 2, from z until 2: they are zero for 2 = z, and positive for z < 2 ~< 2. Therefore, the supply of loanable funds schedule in the (r, q) space has a positive slope. It shifts downwards for increases in a and upwards for increases in w and p, and shifts to the right for increases in u ~, ti b, e and g. The change in the amount of credit supplied by the bank, for a given contractual interest rate, when a given parameter x changes, is a monotonous function of its effect on production:
db dx
_
Ob dq Oq dx
_
-
-
Ob Oq .
Oq .
Ox / Then the supplied b rises with a, _ub, li b, e and g and decreases with w and l+p. []
561
P. Brito, A.S. Mello / Int. J. Ind. Organ. 13 (1995) 543-565
P r o o f o f L e m m a 2. The credit supply function has a slope given by b b b b --~)q/~)l+r" ~)q is negative and finite and ~bl+r is positive and lime~e ( ~ lb+ r = 0. Therefore, there is an asymptote for the level of production qb such that z~(qb) = zT. After some derivation, it is easy to see that (1 - F(F.) + G ( Z ) ) = E(z)/Y.. Then q b : = ( q : E ( z ) = ( l + p ) ( b / q ) } . If ~ > ~ , that is, for q > q b , the equilibrium condition for the bank will not hold: (1 + p) > E ( z ) ( q / b ) . [] P r o o f o f L e m m a 3. The Lagrangian for the firm's problem is Aq = eE(uf)q (1 + r)(wg( q) - a) - c( q)F( (¢ ) + ~(wg( q) - a). The Kuhn-Tucker f.o.c, for a maximum are: (1) eE(u f) - (1 + r)wg' - c ' F - c(OF/Off)/(O~/aq) - ~wg' = 0, (2) wg - a >i 0 and (3) ~(wg - a) = O. Assume that sc 1> 0. Then, a minimum level of optimal production, qf, is defined such as w g ( q f) = a. Then the f.o.c. (1) determines an equation in two variables, sc and 1 + r. Assuming that ~: = 0, then wg(q f) > a and the f.o.c. (1) determines an equation in q and 1 + r, 6 f ( l + r, q , a , w , a f _uf)=0
The ~pair (1 + r f, qf) is determined by the intersection of the curves q = qf and ~b = 0. At this point ~ = 0 and 1 + r e : : - {1 + r:eE(u) - (1 + r)wg' c ' F - c ( O F / O t i ) / ( O t ~ / O q ) = 0}. Two issues are left to be shown: the slope of the demand for funds schedule for the firm, and that the kink involves interest rates higher than r I. If q i> qf (and ~: = 0) the partial derivatives for the function ~bf(.) = 0 are O(1 + r) - - w g ' = -wg' 04~Oq -
( c )
0a
eq(a f __uf)
w(1 + r)g"
(ae--uf)b
c'-
c' - q
< 0,
( 1 + eq(- ~-c- _uf))
{c"QJ - _uf) + 2 ti
04/ Ow = - ( 1 + r)g' =-(l+r)g'
.
1 + e q ( a f__uf)
(1 + r)wg . . .Oq. . 1
0---a--
0(1 + r)
b
>0,
01r Ow l+eq(af_u_f)
(af~uf) b
C'-q
<0,
P. Brito, A.S. Mello / Int. J. Ind. Organ. 13 (1995) 543-565
562
Oe
~=E(U)+e(/i
0~b f Oa f
e 2
0zr Ot~f
04/ 0u f
e 2
o37r
e
0b/f
2
e 2
f__uf) b
b c'-
+wcg'
>0,
"+ ( l i f - - b/f) • 0 , "/7" - - C'
(/if-b/f) <
.
T h e r e f o r e , if q > qf, the demand function has a negative slope in the (r, q) diagram, its intercept increases with a, e, and /if, decreases with w and is ambiguous with b/f. Additionally, qf increases with a and decreases with w. As Ob/Oq = wg' > O, the slope of the demand for funds curve is also negative in the (r, b) space. Given the negative slope of the curve, the kinked portion of the curve is above ~bf = O, given that the firm maximizes output for any given level of the interest rate• []
Proof of Lemma 4. The proof results from the last part of the proof of L e m m a 3, from the signs of the arguments of the function 4~f and from the monotonic relation between b and q. Note also that O~r
O~ f
OE(uf) -
OE(uf) = e > O,
which also shifts upwards the demand function for funds•
[]
Proof of Proposition 1. For a proof that there are two bounds arising from the optimal solutions of the bank, qb, and the firm, qf, see Lemmas 2 and 3. It is easy to see that q b > qf, because qf is determined from the assumption that b = 0 and because qb involves E(z)> 0. Then b > 0.
Proof of Proposition 2. Assume that b > 0 and that t < t*. Then there are two possible types of equilibria depending on the values of the firm's parameters: (i) the equilibrium is interior; or (ii) the demand schedule will intercept the supply schedule a t qb, which represents an equilibrium with credit rationing. Let us make the comparative statics when t changes. (i) For the interior solutions the multipliers are f b-+ •
dq
I~ " = ~
(l+r),
I~l+r(l~O
"~- ¢fibr/~ f
I~b~0 )
- - t-fi b
rfi f
O(l+;)q'e'l+r ~q i+.r'*'q Ot
4~fl+~ q "
P. Brito, A.S. Mello / Int. J. Ind. Organ. 13 (1995) 543-565
563
In Lemmas 1 and 3, we proved that &l+r b is positive and that 4~, ~bfl+rand &fq are all negative. Then, the denominator of q is positive and it has the opposite sign of ~bbo+~bb_0. Since this is a cumbersome expression, involving indicator functions, we determine its sign by evaluating it at the extreme values for ~: lim (&b~+ &b0 ) = 0, ~g
lira (~bb0 + oh0) -- (1 + r)E(O) ~ ~(g-e) <0, as a result of the assumption E(O) 0 and (1 ~ - r ) < 0 for (ii) For the equilibrium with rationing qb := {q:E(z)q- (1 + p)b = 0} also increases with time, as (1 + r)E(O)b 2 dt = - ZE(z)(g- e)(wg'q - b) > O.
dq b
[]
Proof of Proposition 3. We already know that the probability of bankruptcy is given by F ( 6 ) = 0 1 - uf)/(d f - _uf). Differentiating in relation to time we have F(ti)
U = (/~f_/~f)
Z
-
e(t/f_ u f )
.
But
~-
r(1) ~+
qZ
(
g"
\
wg'q-b--grbq)
if the second term is negative. A sufficient condition for this is that g" should be high enough in absolute terms to dominate wg'q- b, which is positive. []
Proof of Proposition 4. If bankruptcy costs are linear c( q)= cq then simplifies to q = N/D, where N : = - g ' (y(l+r)__e)( ¢~ E(ub)e02 -
2e(u b _ub) ~ I(u~,~) + ~ and
(1-F+G)(O-_0) - a
e)
\
--_U~ I(z'abe)) '
564
P. Brito, A,S. Mello / Int. J. Ind. Organ. 13 (1995) 543-565
D :=
g' G(wg' q - b) b2
+ g"(1 + r)(1 - F ) .
Now, the limits for ~ in its support give the limits for q as lira t~ = 0 and as we assume that E(O)< O,
E(O )qb E(e)(wg'q - b) > O .
z~ztim-q =
The curvature of the learning function is formally determined b y / j := 0q/
at. After some tedious algebra, we have lim/j = 0 , ~z
and, as ~ < 0 and assuming that (b/q)- wg'/2
2(g'qE(ub)E(O))2[q+ lim/j =
b£
g" g'(1 + r)
(b
wg')] <0 2 " []
References Audretsch, D., 1991, New-firm survival and the technological regime, Review of Economics and Statistics 72, 441-450. Berger, A. and G. Udell, 1992, Small firms, commercial lines of credit and collateral, Mimeo. (Board of Governors of the Federal Reserve System). Diamond, D., 1991, Monitoring and reputation: The choice between bank loans and directly placed debt, Journal of Political Economy 99, 688-721. Duune, T., M.J. Roberts and L. Samuelson, 1989, The growth and failure of US manufacturing plants, Quarterly Journal of Economics 104, 671-698. Fazzari, S.M., R. Glenn Hubbard and B. Petersen, 1988, Financing constraints and corporate investment, Brookings Papers on Economic Activity 1, 141-206. Geroski, P.A., 1995, What do we know about entry? International Journal of Industrial Organisation, 13, 421-440. Gilson, S.C., K. John and L.H. Lang, 1990, Troubled debt restructurings, Journal of Financial Economics 27, 315-353. Greenwald, B. and J. Stiglitz, 1990, Asymmetric information and the new theory of the firm: Financial constraints and risk behavior, American Economic Review 80, 160-165. Greenwald, B., J. Stiglitz and A. Weiss, 1984, Information imperfections in the capital market and macroeconomic fluctuations, American Economic Review, 74, 194-199. Greenwald, B., M. Kohn and J. Stiglitz, 1990, Financial market imperfections and productivity growth, Journal of Economic Behavior and Organization 13, 321-345. Hirshleifer, D. and A. Thakor, 1992, Managerial conservatism, project choice and debt, Review of Financial Studies 5, 437-470.
P. Brito, A.S. Mello / Int. J. Ind. Organ. 13 (1995) 543-565
565
Hoshi, T., A. Kashyap and D. Scharfstein, 1990, The role of banks in reducing the costs of financial distress, Journal of Financial Economics 27, 67-88. James, C., 1987, Some evidence on the uniqueness of bank loans, Journal of Financial Economics 19, 217-235. James, C. and P. Wier, 1990, Borrowing relationships, intermediation and the cost of issuing public securities, Journal of Financial Economics 28, 149-171. Jovanovic, B., 1982, Selection and evolution in industry, Econometrica 50, 649-670. Mayer, C., 1990, Financial systems, corporate finance and economic development, in: R. Glenn Hubbard, ed., Asymmetric information, corporate finance, and investment (University of Chicago Press, Chicago). Myers, S. 1977, Determinants of corporate borrowing, Journal of Financial Economics 5, 146-175. Myers, S. and N. Majluf, 1984, Corporate financing and investments decisions when firms have information that investors do not have, Journal of Financial Economics 13, 187-222. Petersen, M.A. and R.G. Rajan, 1994, The benefits of lending relationships: Evidence from small business data, Journal of Finance 49, 3-37. Rajan, R., 1992, Insiders and outsiders: The choice between informed and arm's-length debt, Journal of Finance 47, 1367-1400. Shapiro, A. and S. Titman, 1986, An integrated approach to corporate risk management, in: J. Stern and D. Chew, eds., The revolution in corporate finance (Blackwell, New York). Stiglitz, J. and A. Weiss, 1981, Credit rationing in markets with imperfect information, American Economic Review 71, 393-410. Stiglitz, J. and A. Weiss, 1983, Incentive effects of terminations: Applications to credit and labor markets, American Economic Review 73, 912-927. Taggart, R.A. 1985, Secular patterns in the financing of U.S. corporations, in: B.M. Friedman, ed., Corporate capital structure in the United States (University of Chicago Press, Chicago). Whited, T.M. 1992, Debt, liquidity constraints and corporate investment: Evidence from panel data, Journal of Finance 47, 1425-1460.
ELSEVIER
Industrial Organization
Financial constraints and firm post-entry performance Paulo Brito a'b'*, A n t 6 n i o S. Mello a'c'd aBanco de Portugal, DEE, Av. Almirante Reis, 71, 1100 Lisbon, Portugal bUniversidade T~cnica de Lisboa- ISEG, 1200 Lisbon, Portugal CUniversidade Cat6lica Portuguesa, Camino Palma de Lima, 1600 Lisbon, Portugal dCEPR, 25-28 Old Burlington Street, London, WIX 1LB, UK
Abstract Firms finance production by internally generated funds and external loans. The benefits of leverage, however, come with a cost. This cost is related to the uncertainty banks face about the firm's quality and output price. As time evolves banks learn about the firm and adjust the terms of the loan contract. Because of this, firms do not have equal access to credit: small, young firms face greater binding debt constraints than more mature firms with well-known prospects. The firm survival rate, as well as the firm rate of growth, are, therefore, important issues in analyzing firm post-entry performance. Keywords: Financial constraints; Firm growth; Demography J E L classification: G3; L1
1. Introduction
Recent models that analyze the problem of financing firm's production and investment opportunities consider that different sources of capital are not perfect substitutes for one another. With imperfect information, invest-
* Corresponding author at: Banco de Portugal, DEE, Av. Almirante Reis, 71, 1100 Lisbon, Portugal. 0167-7187/95/$09.50 O 1995 Elsevier Science B.V. All rights reserved S S D I 0167-7187(95)00504-8
544
P. Brito, A.S. Mello / Int. J. Ind. Organ. 13 (1995) 543-565
ment spending is constrained by a financial hierarchy, in which internally retained funds are cheaper than external capital provided by investors who are unable to evaluate precisely future prospects of the firm and the quality of its management team. In this paper we analyze the problem of financing firms' production opportunities when firms cannot secure sufficient internal funds and need additional external finance. There is asymmetric information between those that own and control the assets of the firm and outside investors. However, as time evolves, these can learn more about the quality of the firm's m a n a g e m e n t and accordingly adjust the terms of the financing contract. By doing so we are able to capture the effects of capital constraints on output decisions as well as on the probability of firm survival and firm growth. Firms do not have equal access to capital markets, because small, young and relatively unknown firms face greater liquidity restrictions and higher cost of capital than more mature firms with well-known prospects. Currently there are two main rival approaches in the literature analyzing the demography of firms. The first, known as Gibrat's law, states that the growth rates of firms are independent of firm size. The second relies on the work by Jovanovic (1982), which establishes a relationship between firm age and firm growth. Firms can start small and suffer from scale disadvantages. T h e successful firms grow and become more efficient, not just because costs are reduced but also because firms learn from their own experience; unsuccessful firms, on the contrary, remain small and eventually may exit the industry. Our model belongs to the recent literature highlighting the interaction between finance and production decisions. One earlier paper on this topic is that by Greenwald and Stiglitz (1990), who show the importance of financial constraints on firms' investment decisions and discuss how the strength of firms' financial position can explain the propagation of macroeconomic disturbances and the persistance of their effects. In our paper we study how growth and survival of firms is affected by the evolution of the c r e d i t o r borrower relationship, when there exist multiple sources of asymmetrical information. We show that the firm's age and size are not independent of the duration of the relationship between those who control the assets and those who finance the company. Although we think that learning by those who manage the assets is important, it is not the complete story or even the most crucial factor during the first stages of a firm. We think that learning by outside financiers about the firm is perhaps of the same degree of importance or even more important in determining survival and growth. Our model, just as in Jovanovic's model, implies that the exit rates are a decreasing function of firm size and age, consistent with what Dunne et al. (1989) have shown. Furthermore, we show that the growth rate of successful
P. Brito, A.S. Mello / Int. J. Ind. Organ. 13 (1995) 543-565
545
firms increases at a decreasing rate with age, consistent with the findings in several empirical studies about post-entry performance. The importance of taking financial and production decisions jointly is key to a better understanding of entry and exit decisions of firms, as well as to the study of post-entry performance. In most cases, firms simply do not cease to exist or have any sort of limitations self-imposed. They live, are forced out of the industry or suffer major restructuring from the actions taken by the different claim-holders. Models that ignore that firms are governed by combinations of contracts established between and monitored by parties with different interests and information are too simplistic to provide accurate explanations of reality. Of the various external sources we concentrate on debt finance from banks. Information on the financing practice of firms in different countries shows that banks are the dominant source of external finance (see Taggart, 1985; Fazzari et al., 1988; Mayer, 1990). Mayer, for instance, observes that banks are the dominant source of external finance in the G-7 countries and that small and medium-sized firms are considerably more dependent on external debt than large firms. Also, usually banks establish long-term relationships with borrowers, which allow them to learn progressively more about companies, an issue in which we are particularly interested in this paper. Specialization in monitoring, as well as the reduced severity of holdout problems, give a clear cost advantage to bank debt relative to diffusely held public debt, as James (1987), Gilson et al. (1990) and Hoshi et al. (1990) have noted. Although firms could alternatively issue additional equity, we exclude this possibility on the assumption that equity issues would be subject to adverse selection and therefore too costly to firms (see Greenwald et al., 1984; Myers and Majluf, 1984). The model focuses on the behavior of firms that maximize expected profits net of debt repayments owed to risk neutral banks, that have temporarily inferior information about the type of the owner-manager. In case the firm does not repay the promised amount to the bank the firm defaults. The dead-weight costs of financial distress are paid by the equityholders of the firm, who therefore make output decisions taking into account the possibility of bankruptcy. As in the case of Greenwald et al. (1990), because output is partly financed with debt, when the firm increases production the probability of bankruptcy also increases. The interesting feature of our model is that by explicitly considering a time-varying degree of asymmetrical information, in order to capture the learning process, we can analyze firm output and capital structure decisions and bank contractual s c h e d u l e s - interest rate charged and amount lent to finance production b e y o n d the level which the firm is able to finance out of its own cash-flows as the b o r r o w e r / l e n d e r relationship evolves through time.
546
P. Brim, A.S. Mello / Int. J. Ind. Organ. 13 (1995) 543-565
2. The model
In this section we model the behavior of a firm, with some of the features found in Greenwald et al. (1990), although there are significant differences in the postulated a s y m m e t r y of information, as well as in content. The firm m a k e s decisions at discrete intervals of time t = 1 . . . . . T. At the beginning of period t the firm has a notional amount of fixed-rate debt outstanding, B,, which has b e e n contracted with a bank at the nominal rate R,. The firm needs cash in advance from the bank to pay for the use of inputs for production at time t. The firms asks for m o n e y from the bank when its own accumulated funds, A,, are insufficient. Therefore, there is a preference for the use of retained funds before the firm looks for external sources, because internal funds have a cost advantage over new debt. The higher cost of debt capital results from agency problems that might arise because the bank extending the loan has a rough idea of the output price and does not observe with complete precision the type of entrepreneur(s) that manages the firm and is assumed to maximize the value of his (their) equity position. We assume that the debt contract is short-term and lasts for a single period. The loan can be renegotiated for the next period or not. The short-term maturity of the debt contract is somehow equivalent to a threat to cut off credit at the b a n k ' s discretion and therefore provides an incentive for the firm to take actions that reduce moral hazard (see Stiglitz and Weiss, 1983). Also, the firm does not pay out dividends to equity-holders. These two assumptions are quite c o m m o n in practice in the case of small and medium-size companies and can be interpreted as debt covenants imposed by the lender. T h e firm hires the amount of labor 1, to produce q, units of output. The labor requirement function, 1, = g(q,), is increasing and convex in q,: g ' > 0 and g" > 0. Total production costs at t are in nominal t e r m s e t w t l t , where w t is the real wage assumed to be determined in a competitive labor market, and P, is the aggregate price level. Therefore, firms borrow an amount equal to
B,= P,w,I,- A, .
(1)
The m a n a g e m e n t of the firm contributes to the value added of the firm by a factor e. This p a r a m e t e r , defining the type of the firm, can be seen as the skills affecting the organization of production or marketing or both. The firm knows its type, e, but outside financiers do not observe it. We assume that for the b a n k providing credit to the firm the type is a stochastic variable with a variance decreasing with time:
et=e+Ot(t*-t ) t~t*, where t* is the date when the bank learns the firm's type with complete precision and 0 is drawn from a uniform distribution over the interval (0, _0),
13 Brito, A.S. Mello / lnt. J. Ind. Organ. 13 (1995) 543-565
547
where 0 is positive and _0 is negative. Then e, has also a uniform distribution over the interval (~, e) = (e + O(t* - t), e + -0(t* - t)), where we assume that e>0foranyt. The output produced at time t, eqt, is sold at time t + 1 and is assumed to be perishable. The price at which the firm sells its output at time t + 1 is unknown at the time of production. This is because firms face a stochastic relative price as regards the overall price level, ff~+l, such as P[+I = tT~+,P,+,, where u f follows a uniform distribution over the interval (t/f,__uf), where t/f, u-f > O. The accuracy of the forecasts of the relative price by the banks, t ~ , is, however, inferior. As it also follows a uniform distribution with support (u-tub,t/b), then formally t/b _ _Ub i> t/f _ / f . That is, when anticipating the firm's selling prices as P~÷I : U~bt + l e t + l , banks are assumed to know less than the firm about the markets where the firms sell their products. We leave imprecise the relative level of the bank's ignorance, which can vary with the firm and the industry. Even if the length of time of the bank-firm relationship can help to reduce the gap, it is realistic to assume that banks do not specialize in knowing the idiosyncratic details of each client's business. When the output is sold, the firm either repays in full its debt with accrued interest (1 + R t ) B ,, or it is declared bankrupt. In that case, the bank takes possession of what is left, P~+leqt. It can be seen that bankruptcy occurs when P~+leqt <~ (1 + R t ) B t , which implies that A t + 1 ~ O. In the event of bankruptcy shareholders pay costs c f = c(qt), and c(.) is an increasing and convex function of q,. Thus, the costs increase with the size of the firm. The costs borne by shareholders could result from increasing concerns of suppliers, customers and employees (see Shapiro and Titman, 1986). Alternatively, these costs can represent the reputation lost by managers in the event of the firm's financial difficulties. Banks maximize the expected value of profits by determining the amount of credit, Bt, and the rate charged R,, simultaneously. The opportunity cost of lending is the market real interest rate, p,, which is set exogenously. It is assumed that the amount lent to each firm is small compared to the large and diversified portfolio of loans hold by the bank. It is reasonable to assume that u and e are independent random variables. We then define a new variable for the bank
Zt+l := Ubt+le' with a density function depending upon the supports of the price and management quality distributions, fz(x; t/b, U_b, £ e),
548
P. Brito, A.S. Mello / Int. J. Ind. Organ. 13 (1995) 543-565
1
~ f ~l (x; . ) ,
ifubg~
f~(z;a u,_uu , g , e ) : - ( z i b _ u b ) ( y _ _ e ) x ( f ~ ( x ; . ) , where
and f z ( x ; . ) : = I¢a.abe)(x) log ~
+ I(ab~,ub~)(X) log
U°
+ l(ub~.s)(X ) log
,
where z := ube and ~? : = / i b e and Ic)(x ) are the indicator functions for z = x. T h e distribution fz is a non-linear function with truncated support. Given its shape and moments, we can easily see that the results found in the paper are robust for a wide class of distributions of z, requiring only the realistic assumption that z is bounded. T h e r e f o r e from the bank's viewpoint the equity of the firm next period can be written as At+l = P,+I(Z,+Iq,)- (1 + R,)B,. If we deflate by Pt+l, which is perfectly anticipated, and denote the real interest rate charged to the firm by r t, and note that 1 + r, = (1 + R,)P,/(Pt+t), then, the real value of the firm's equity, according to the bank, is
a,+, = (zt+,q,) - (1 + r,)b,. We assume that banks observe z but are unable to separate u from e. In other words, banks observe the firm's revenues but do not know whether these depend on the quality of the management or on the realization of the price at which the products are marketed. Banks' real rate of return for a particular loan is a stochastic variable which depends upon the solvency of the firm: ~(1 + r,)
(l+Ft)=t~t+iq,/b ,
ifa,+l ~>0 ifa,+, < 0 .
We set Z,+l as the level of z such that a,+~ = 0: i.e. d,+,
(1 + rt)b , q,
The expected real return from lending to the firm is
(2)
P. Brito, A.S. Mello / Int. J. Ind. Organ. 13 (1995) 543-565
549
qt f zt+l g[(l+~t)]=(l+rt)(l--f(zt+l)+-~t zdF(z) = (1 + r,)(1 - F(~t+1) + G ( ~ , + , ) ) ,
(3)
where Prob[z ~< z~t+l] = F(z~t+z) = ff,+l dF(z) is the probability of bankruptcy calculated by the bank, and G(~,+~)= (1/(~t+l)) f~t+l dF(z) is the average liquidation income when bankruptcy occurs. Since the credit risk on the loans is diversifiable, banks determine the optimal contractual interest rate from El(1 + f',)l = 1 + p,.
(4)
The resulting bank's supply schedule, according to the following results, is positively sloped and displays credit rationing.
Lemma 1. The bank's supply of loans schedule is positively sloped in the (r,, q,) space. The real volume of credit extended is positively related to the amount of equity available, a,, and to the limits of the perceived distributions of the quality of the management and of the relative price faced by the firm. Proof. See the appendix. Intuitively, the bank offers the firm a schedule of the pair (r, b), to partially finance production, from which the firm chooses. Interestingly, the bank may decide not to extend any credit to the firm. That is the case when the prospects of the firm are so low that the bank expects to have a return on capital below the risk-free rate, p, as the following lemma shows:
Lemma 2. If ~ >I ~ then the bank rations credit at the level of output E(z)q/b <- (1 + p) for q >1qb.
qb, since
Proof. See the appendix. At the margin, when the bank does not expect to have a return at least equal to the opportunity cost of risk-free capital, no further money is lent to the firm and the credit is rationed. Fig. 1 shows the effect of a change in E(e), the bank's evaluation of the firm's management. The supply curve moves to the right as the bank improves its knowledge of the firm. Thus, for two firms with management of the same type, the older firm gets more funds than the newer one, for the same interest rate charged. Alternatively, other things being equal, two firms that have been dealing with the bank for the same amount of time are charged rates that depend on the bank's assessment of the firm's quality.
550
P. Brito, A.S. Mello / Int. J. Ind. Organ. 13 (1995) 543-565
1+r t
t
s
/,
s/
J+p i-
qb
qb'
qt
Fig. 1. Bank supply of funds and assessment of firm's management quality.
As the perceived prospects of the firm deteriorate, the bank's supply curve shifts to the left. In some cases the assessment of the risk is so high that the situation is practically equivalent to the firm being excluded from the pool of the bank's clients. The firm chooses the level of output by looking at the conditions of the debt contract, (r,, b,). Management maximizes the expected value of the equity of the levered firm. Because the firm knows its own type, the only source of uncertainty comes from the shock to the relative price of its output next period, u~+ 1. Financial distress imposes a cost on shareholders of the firm. This cost may reflect the loss of reputation faced by the management, or represent the more stringent conditions imposed by suppliers, customers and employees in future contracts with the owners of the firm. The model could easily incorporate the situation where the bankruptcy costs were shared by banks. In this case the profit function of the firm - and that of the b a n k - w o u l d be modified, since the interest-cost paid would include an additional component due to the expected portion of the liquidation cost borne by banks. Without loss of generality, we only consider the costs to shareholders from bankruptcy. The probability of bankruptcy is given by F(t~t+ 1)
=
f
dF(ut+l)=
,
f
(tif-_u ') d u t + , -
Ut+l
--
fff__uf ,
P. Brito, A.S. Mello / Int. J. Ind. Organ. 13 (1995) 543-565 where
/~,+1 = ( l q--
551
rt)bt/eqr
Real profits are stochastic and the firm's problem is max E(Ht+ 1) - c(q,)F(gtt+ 1), qt
for b/> 0, and where Ylt+l are real profits to equity-holders after payments to debt-holders. The first order condition for a maximum is then given by eE(u~) - (1 + rt)wtg'(qt) - 7rt + ~t(wtg(q) - at) = 0,
(5)
where {: is the Kuhn-Tucker multiplier, zr, is the premium related to bankruptcy, which induces the firm to reduce its output: 7r,:=
Oc(q,)F(•,+l) Oc Oq, - Oq, F(/J'+I) + 1
-- /~f - u f
c -f
U
--
Uf
Oa,+ 1 Oq,
(C,(tit+ __u__f)+C~t+l [ ( g) 1 ~ W, g' --
at] } +-~ > 0.
(61
The fact that firms act to avoid bankruptcy generates a kind of risk-averse behavior. Indeed, if there are incentives for managers to build their reputations, Hirshleifer and Thakor (1992) have shown that firms' investment policies tend to be more conservative. Given Eqs. (5) and (6), the demand for funds from the firm, shown in Fig. 2, is given in the following lemma: L e m m a 3. The demand for external funds f r o m the bank is inversely related to the rate charged by the bank. However, the firm may decide to operate with no debt when the bank charges a rate above some specified level, r f. Proof. See the appendix. Normally the firm would ask for money from the bank, but there are instances when the firm chooses to operate as a purely equity-financed firm, even knowing the power given by leverage to increase profits. This occurs when the bank charges a rate above some level, r e. L e m m a 4. The higher the quality o f the firm's management, e, the higher the expected relative output price, E(uf), the lower the operating costs, w, and the lower the costs associated with financial distress, the higher the debt capacity and the desired leverage ratio. Proof. See the appendix.
552
P. Brito, A.S. Mello / Int. J. Ind. Organ. 13 (1995) 543-565
!+r t
D l+rf
qf
1,
qt
Fig. 2. Firm's demand for funds.
3. Equilibrium In equilibrium the firm's output level and the bank contractual c o n d i t i o n s - interest rate and amount of debt - are functions of the market real interest rate, the real wage rate, the level of equity capital available, the type of the firm's management, the precision error and the distributions of the firm's relative price for both the firm and the bank: r, = r t ( l + p,, w,, at; g, e, t/f, _uf, ~/b, L b)
(7)
qt = q,(1 + Pt, w,, at; Y, _c, t/f, u f, t/b, u b) ,
(8)
Consider the ratio a t / w t l , , representing the amount of labor that firms can hire without having to resort to external financing. The increase in supply from hiring more labor services is only possible through additional external funds. Given the contractual conditions offered by the bank, which depend upon the bank's expectations about the firm's behavior, there is an equilibrium interest rate and amount of debt, and a corresponding level of firm's output. T h r e e possible situations may occur, corresponding to equilibria that differ with respect to the parameter values characterizing the borrowing firm (see Fig. 3). First, if the contractual conditions are too stringent, firms may
P. Brito, A.S. Mello / Int. J. Ind. Organ. 13 (1995)543-565
553
i +r t D
S
l+rf
l+r~
iiii
1+/3 I I
I I
I I
b
qf
q~
qb
qt
Fig. 3. Equilibrium. decide not to borrow from the bank and the level of production is determined by the amount of internal finance available. Second, if firms may decide to borrow more than the bank is willing to lend, given the bank's expectation about the firms' type and the relative output prices, there is credit rationing by the bank, resulting in an output constrained by the amount and not by the price of credit. This would be the case when the interest rate which the firm is willing to pay may act as a screening device for the bank, since those firms that accept paying high interest rates are, with greater probability, the worse risks (see Stiglitz and Weiss, 1981). Third is an intermediate case, where firms operate partially financed with debt, although with a leverage ratio smaller than that under the first best case. The following proposition summarizes the scale of production and the capital structure of the firm in equilibrium.
Proposition I. Under the prevailing labor and output market conditions, the firm's production scale is bounded by the capital structure. The lower bound is determined by the amount of internal funds available and the upper bound is determined by the maximum level of indebtedness that the bank is prepared to accept. Proof. See the appendix.
554
P. Brito, A.S. Mello / Int. J. Ind. Organ. 13 (1995) 543-565
In the following section we analyze how the differential information between the parties, which is assumed to decrease over time, affects firm performance, growth and probability of survival.
4. The effect of liquidity constraints on post-entry performance
The model in the last section is consistent with the view that information and incentive problems in debt markets affect corporate production decisions. Recent empirical evidence (see Whited, 1992) seems to indicate that the effect of a liquidity constraint appears to be stronger for firms that rely more on bank debt and are not able to participate in the corporate debt market. On the other hand, firms with long-term relationships with the lender usually get loans more easily. It is often the case that relationship factors that favor large well-established firms and discriminate against startups may outweigh the profitability of investments in determining which firms get loans. In this section we explore the importance of the long-term relationship between lender and borrower and show that with imperfect information, small unknown firms, even with better prospects, may face worse contractual terms. A bank that lends to a firm can obtain information about the client in the course of lending. It takes time to learn the firm's type, and only by date t* the bank knows it with complete precision. The learning period may be firm as well as industry specific and may depend on various factors such as the technology, the degree of ownership concentration, the stability and formality of procedures regarding the internal organization of information reporting, size, etc. When the bank assesses on average the quality of the firm below its true value, that is, when E(O,)= E ( e , ) - e < 0 , during the learning period, the following proposition holds. Proposition 2. As the bank's error in observing the true type of the firm decreases with time, the equilibrium interest rate charged by the bank will decrease, the binding debt constraint will decrease and, as a consequence, the firm's output increases. Proof. See the appendix.
Furthermore, the probability of survival of the firm will improve as time goes by, because by improving the terms of the loan the firm needs a lower fi,+~ = (1 + r,)bt/eq, to avoid bankruptcy, which is directly related to the terms of the loan contract. This result is formally stated in the following proposition.
P. Brito, A.S. Mello / Int. J. Ind. Organ. 13 (1995) 543-565
555
Proposition 3. I f the labor requirement function is sufficiently convex, i.e. if w,g'q, - b, - g"q,bt/g' < O, then the probability o f bankruptcy will decrease over time. Proof. See the appendix.
Both propositions establish conditions discriminating against start-ups and against more opaque firms. The bank's inference of the quality of the firm depends on 0 and on the length of the relationship with the client firm, t. In comparing two different firms, or the same firm when new and mature, the proposition says that, ceteris paribus, when 0 is smaller or the firm is more mature, that is, when the firm is better known, the binding debt constraint becomes weaker. Furthermore, new firms tend to be more liquidity constrained and therefore to be potentially smaller than better-known and established businesses. Depending on the degree of asymmetrical information, there may be instances when firms either do not apply for credit and work only with internal sources, or if they ask for a loan the probability of survival will depend on having either a high type or a good state that realizes for u. In any case, the slower is the process of learning the more likely it is that the firm will not survive. Notice that because the production function displays decreasing returns to scale, the incentives to expand capacity decrease as output goes up. Therefore, expanding production financed by additional loans will be limited by the greater probability of bankruptcy. Fig. 4 shows how learning by banks will shift the supply of loanable funds to the right. Two implications result from the increase in precision: (i) the bank will lower the interest rate charged for a given level of debt, and (ii) the bank will increase the level of credit available to the firm as well as the level at which credit rationing occurs. Some firms initially rationed may later get funds and increase the level of output produced. Implication (i) could be alternatively stated in terms of requested collateral, and Berger and Udell (1992) find that the bank is less likely to demand collateral if the relationship with a firm is long term. Petersen and Rajah (1994), however, find that, according to implication (ii), the relationship has value but works more through quantity than price, because banks, either because of market power or because of internal control and agency problems along the managerial hierarchy, prefer instead to increase the availability of credit or to relax the credit-rationing constraint. The results above provide interesting explanations for often observed facts. Now we want to analyze what are the implications for firm growth of informationally generated liquidity constraints. As a simplifying assumption, assume that the bankruptcy cost function is now linear. Then, Fig. 5
•q l t a o a ~ tu.lt3 p u e uop, e n l e A a s,,,luuq j o s a ! t u e u , { o
' g "~!zl
,1
lb
' u o ! s ! a o a d SplUeq u! a s e o a o u I 17 "~!al
lb 1
,qb t I i
qb
7b
i I l
I
(I
ff9K-fPS- (ff66I) f I "uv8.tO "pul T "lUl / °llalN "S'V '°l~..t~t "d
9gg
P. Brito, A.S. Mello / Int. J. Ind. Organ. 13 (1995) 543-565
557
illustrates the dynamics of firm's growth, measured by output, which, in this case, is equivalent to asset value, from the moment the firm enters the industry until it is perfectly identified by the bank t*. After t*, if everything else is kept constant the same level of production will replicate period after period. Notice that the situation after t* is still a second best, because the firm and the bank have different uncertainty regarding the firm's relative price next period. The following proposition states the post-entry growth rate of the firm. Proposition 4. Assume that the firm's bankruptcy cost function is linear and that wtg'/2 - bt/q, > O. Then as time passes and the bank learns more about the firm's type, output increases at a positive and decreasing rate. Proof. See the appendix.
Even when the rate of learning by the bank is constant, the marginally improving conditions on the debt contract are such that the firm performance, measured by the growth rate of output, or by the return on assets, decreases over time. This result depends on the hazard rate, measured by the probability of default, declining with time at a decreasing rate, suggesting that there are decreasing returns to learning by outsiders (first impressions are more i m p o r t a n t ) - r e s u l t i n g in a non-linear (r, b) s c h e d u l e - and also on the assumption of a production function displaying sufficiently high decreasing returns to scale (see Fig. 5). From Lemma 1 and Propositions 3 and 4 taken together it follows immediately that the probability of a firm's failure is decreasing in the initial level of e. Given that the initial size of the firm is directly proportional to the initial level of e, our model has the additional implication that the probability of failure decreases with the initial size of the firm, an empirical fact found in Audretsch (1991). In terms of empirical implications our model seems to be more in line with work of Jovanovic than with Gibrat's law. But our version is completely different from that of Jovanovic and is a new approach that combines financial and production decisions in analyzing the post-entry performance of firms. With imperfect capital markets the role played by the different claim-holders in deciding the future of firms seem to lend support to this view in dealing with the issue of firm demography. Although we have represented t* to be the date when the bank finally discovers the true type of the borrower, one can think of the period before t* as the period during which the firm builds its reputation by taking on costly bank debt. Then, in the spirit of Diamond (1991), firms that have established a good reputation move on to the bond market to save on the monitoring costs, or any rents extracted from a bank that has acquired
558
P. Brito, A.S. MeUo / Int. J. Ind. Organ. 13 (1995) 543-565
bargaining power over the firm's profits (see Rajan, 1992). Before t*, the costs associated with switching to a new alternative lender and the negative signals that potentially may result from the move may be significant and outweigh any benefits (see Stiglitz and Weiss, 1981). Indeed, according to the findings of James and Wier (1990), banking relationship renewal provides a positive signal to other investors in the economy.
5. Conclusions
In a simple model we analyze the effects of financial market imperfections on productive decisions of firms. The fact that banks initially cannot observe the true quality of the firm's potential imposes on small, less-known firms an additional burden in the form of a high required premium and greater capital constraints due to agency problems. For more mature companies, with an established relationship with banks and with sufficient reputation to issue traded debt in the capital markets, interest rates are lower and loan sizes correspondingly less limited. Maturity is therefore a factor of access differentiation among firms in the same industry. Known firms have a cost advantage over newly established but unknown firms. So long as firms need external financing the relationship gap affects output decisions and the rate of firm growth, that is, the performance of firms. Firm size and age are positively correlated with growth of entrants. In some cases only very efficient small firms can survive the initial stages of their relationship with banks, and firm growth can be significantly affected during infancy, implying that the height of entry barriers can be high (see Geroski, 1995). Moreover, the slower the process of learning by outside investors the more likely it is that firms will not survive. Thus, financial market imperfections can have a negative impact on economic growth and on development stimulated by the appearance of new firms that replace older and established businesses. In addition, and similar to Greenwald and Stiglitz (1990), the model has important implications for business cycle behavior. The effect of a binding liquidity constraint can have significant effects on the level of aggregate output when firm's prospects and balance sheets deteriorate, especially in countries where most firms are small, under-capitalized and the ability of financial intermediaries to obtain reliable information is reduced. Such circumstances could also explain why often firms require venture capital during the first years of their existence, and why debt to equity ratios, within the same industry, are often a function of the firm's maturity.
P. Brito, A.S. Mello / Int. J. Ind. Organ. 13 (1995) 543-565
559
Acknowledgements We are grateful to Pedro D. Neves, three anonymous referees and seminar participants at CEMFI (Madrid) for helpful suggestions. The usual disclaimer applies.
Appendix: Proofs
Proof of Lemma 1. Eq. (4) can be written in implicit form as (deleting the time subscripts from now on) ~bb(1 +r, q; 1 +p, w,a, 5,_u, g, e) = E(1 + r ) - (1 + p ) = 0 . In Eq. (3), after performing the necessary integrations, we have the following expressions, for the case where ~bg~/tb_e (deleting the time subscripts), z - d(1 - log(d/z))
F(d;ttb,-ub, e,e) :-- (fib +
ttb)(~_e)
I(,.ub~)(d)
Z -- Ubg + ~ log(g/e)
- -
w
[ +
1
£ - d(1 + log(:?/d)) ] (-~--_ub)(g_ e ) I(~%,e)(d),
and b
G(d;ffb, u , e , e ) : -
_Z2 -- ~2(1 --log(d/Z)) 4 d ( t i b _ u b ) ( ~ _ e ) l(z,ub~)(d) +-
Z2 -- (__ubg)2 + 2dE log(g/e) 4~07b __ub)(g__e) I(ub~,"be)(~)
+ [ E(Z)z= £2-4d(5 dz(1b +ub)(g2 log(Z/~))_ _e) ] I(a%'',(~) Making the suitable substitutions we get d = [(1 + r)(wg(q)- a)/q]. For -b any support of the distributions (i.e. for any relation between _UbYand u e) we get the following partial derivatives, a4) b 0(1 + r) - 1 - F(.) E (0, 1) ; (1 + r)G(.)(wg'(q)q - b)
a~b b
0~-
-
qb
<0,
560
P. Brim, A.S. Mello / Int. J. Ind. Organ. 13 (1995) 543-565
from the convexity of g(.); O~bb (1 + r)G(.) >0, Oa b O~bb (1 + r)G(.)l <0, Ow b O4,b --1, 0(1 + p) O~bb
1
(l+r)(2--U._bY)Zl(ubo,~)(5)]
[
0g - ( O - e )
r-p-
04~b 1 [ 3e - (g-_e) [ I + P +
2g(aU_u~)2 (1 + r)E(ub)e_ 2
(1 + 0 ( 2 - t~be) z ~Klu
/>0,
]
_u),,~
0~bb 1 [ 0rib -- (tib__.ub) r - p
(1 + r ) ( 2 -- / i b e ) 2
Oth b
(1 + r)E(e)u_ b
2ffb(g__e) 2
] I(au,.z)(2 ) />0,
and 1
[-
ae --(~b--_Ub) [ l + p - +
J
(1 + r)(2 - ubg) 2 i(,,ub~)(2)] 2ub(g -- _e)2 ~> 0
The signs for the partial derivatives involving the support for Y and tT, though they may seem to be uncertain, were evaluated for 2, from z until 2: they are zero for 2 = z, and positive for z < 2 ~< 2. Therefore, the supply of loanable funds schedule in the (r, q) space has a positive slope. It shifts downwards for increases in a and upwards for increases in w and p, and shifts to the right for increases in u ~, ti b, e and g. The change in the amount of credit supplied by the bank, for a given contractual interest rate, when a given parameter x changes, is a monotonous function of its effect on production:
db dx
_
Ob dq Oq dx
_
-
-
Ob Oq .
Oq .
Ox / Then the supplied b rises with a, _ub, li b, e and g and decreases with w and l+p. []
561
P. Brito, A.S. Mello / Int. J. Ind. Organ. 13 (1995) 543-565
P r o o f o f L e m m a 2. The credit supply function has a slope given by b b b b --~)q/~)l+r" ~)q is negative and finite and ~bl+r is positive and lime~e ( ~ lb+ r = 0. Therefore, there is an asymptote for the level of production qb such that z~(qb) = zT. After some derivation, it is easy to see that (1 - F(F.) + G ( Z ) ) = E(z)/Y.. Then q b : = ( q : E ( z ) = ( l + p ) ( b / q ) } . If ~ > ~ , that is, for q > q b , the equilibrium condition for the bank will not hold: (1 + p) > E ( z ) ( q / b ) . [] P r o o f o f L e m m a 3. The Lagrangian for the firm's problem is Aq = eE(uf)q (1 + r)(wg( q) - a) - c( q)F( (¢ ) + ~(wg( q) - a). The Kuhn-Tucker f.o.c, for a maximum are: (1) eE(u f) - (1 + r)wg' - c ' F - c(OF/Off)/(O~/aq) - ~wg' = 0, (2) wg - a >i 0 and (3) ~(wg - a) = O. Assume that sc 1> 0. Then, a minimum level of optimal production, qf, is defined such as w g ( q f) = a. Then the f.o.c. (1) determines an equation in two variables, sc and 1 + r. Assuming that ~: = 0, then wg(q f) > a and the f.o.c. (1) determines an equation in q and 1 + r, 6 f ( l + r, q , a , w , a f _uf)=0
The ~pair (1 + r f, qf) is determined by the intersection of the curves q = qf and ~b = 0. At this point ~ = 0 and 1 + r e : : - {1 + r:eE(u) - (1 + r)wg' c ' F - c ( O F / O t i ) / ( O t ~ / O q ) = 0}. Two issues are left to be shown: the slope of the demand for funds schedule for the firm, and that the kink involves interest rates higher than r I. If q i> qf (and ~: = 0) the partial derivatives for the function ~bf(.) = 0 are O(1 + r) - - w g ' = -wg' 04~Oq -
( c )
0a
eq(a f __uf)
w(1 + r)g"
(ae--uf)b
c'-
c' - q
< 0,
( 1 + eq(- ~-c- _uf))
{c"QJ - _uf) + 2 ti
04/ Ow = - ( 1 + r)g' =-(l+r)g'
.
1 + e q ( a f__uf)
(1 + r)wg . . .Oq. . 1
0---a--
0(1 + r)
b
>0,
01r Ow l+eq(af_u_f)
(af~uf) b
C'-q
<0,
P. Brito, A.S. Mello / Int. J. Ind. Organ. 13 (1995) 543-565
562
Oe
~=E(U)+e(/i
0~b f Oa f
e 2
0zr Ot~f
04/ 0u f
e 2
o37r
e
0b/f
2
e 2
f__uf) b
b c'-
+wcg'
>0,
"+ ( l i f - - b/f) • 0 , "/7" - - C'
(/if-b/f) <
.
T h e r e f o r e , if q > qf, the demand function has a negative slope in the (r, q) diagram, its intercept increases with a, e, and /if, decreases with w and is ambiguous with b/f. Additionally, qf increases with a and decreases with w. As Ob/Oq = wg' > O, the slope of the demand for funds curve is also negative in the (r, b) space. Given the negative slope of the curve, the kinked portion of the curve is above ~bf = O, given that the firm maximizes output for any given level of the interest rate• []
Proof of Lemma 4. The proof results from the last part of the proof of L e m m a 3, from the signs of the arguments of the function 4~f and from the monotonic relation between b and q. Note also that O~r
O~ f
OE(uf) -
OE(uf) = e > O,
which also shifts upwards the demand function for funds•
[]
Proof of Proposition 1. For a proof that there are two bounds arising from the optimal solutions of the bank, qb, and the firm, qf, see Lemmas 2 and 3. It is easy to see that q b > qf, because qf is determined from the assumption that b = 0 and because qb involves E(z)> 0. Then b > 0.
Proof of Proposition 2. Assume that b > 0 and that t < t*. Then there are two possible types of equilibria depending on the values of the firm's parameters: (i) the equilibrium is interior; or (ii) the demand schedule will intercept the supply schedule a t qb, which represents an equilibrium with credit rationing. Let us make the comparative statics when t changes. (i) For the interior solutions the multipliers are f b-+ •
dq
I~ " = ~
(l+r),
I~l+r(l~O
"~- ¢fibr/~ f
I~b~0 )
- - t-fi b
rfi f
O(l+;)q'e'l+r ~q i+.r'*'q Ot
4~fl+~ q "
P. Brito, A.S. Mello / Int. J. Ind. Organ. 13 (1995) 543-565
563
In Lemmas 1 and 3, we proved that &l+r b is positive and that 4~, ~bfl+rand &fq are all negative. Then, the denominator of q is positive and it has the opposite sign of ~bbo+~bb_0. Since this is a cumbersome expression, involving indicator functions, we determine its sign by evaluating it at the extreme values for ~: lim (&b~+ &b0 ) = 0, ~g
lira (~bb0 + oh0) -- (1 + r)E(O) ~ ~(g-e) <0, as a result of the assumption E(O)
dq b
[]
Proof of Proposition 3. We already know that the probability of bankruptcy is given by F ( 6 ) = 0 1 - uf)/(d f - _uf). Differentiating in relation to time we have F(ti)
U = (/~f_/~f)
Z
-
e(t/f_ u f )
.
But
~-
r(1) ~+
qZ
(
g"
\
wg'q-b--grbq)
if the second term is negative. A sufficient condition for this is that g" should be high enough in absolute terms to dominate wg'q- b, which is positive. []
Proof of Proposition 4. If bankruptcy costs are linear c( q)= cq then simplifies to q = N/D, where N : = - g ' (y(l+r)__e)( ¢~ E(ub)e02 -
2e(u b _ub) ~ I(u~,~) + ~ and
(1-F+G)(O-_0) - a
e)
\
--_U~ I(z'abe)) '
564
P. Brito, A,S. Mello / Int. J. Ind. Organ. 13 (1995) 543-565
D :=
g' G(wg' q - b) b2
+ g"(1 + r)(1 - F ) .
Now, the limits for ~ in its support give the limits for q as lira t~ = 0 and as we assume that E(O)< O,
E(O )qb E(e)(wg'q - b) > O .
z~ztim-q =
The curvature of the learning function is formally determined b y / j := 0q/
at. After some tedious algebra, we have lim/j = 0 , ~z
and, as ~ < 0 and assuming that (b/q)- wg'/2
2(g'qE(ub)E(O))2[q+ lim/j =
b£
g" g'(1 + r)
(b
wg')] <0 2 " []
References Audretsch, D., 1991, New-firm survival and the technological regime, Review of Economics and Statistics 72, 441-450. Berger, A. and G. Udell, 1992, Small firms, commercial lines of credit and collateral, Mimeo. (Board of Governors of the Federal Reserve System). Diamond, D., 1991, Monitoring and reputation: The choice between bank loans and directly placed debt, Journal of Political Economy 99, 688-721. Duune, T., M.J. Roberts and L. Samuelson, 1989, The growth and failure of US manufacturing plants, Quarterly Journal of Economics 104, 671-698. Fazzari, S.M., R. Glenn Hubbard and B. Petersen, 1988, Financing constraints and corporate investment, Brookings Papers on Economic Activity 1, 141-206. Geroski, P.A., 1995, What do we know about entry? International Journal of Industrial Organisation, 13, 421-440. Gilson, S.C., K. John and L.H. Lang, 1990, Troubled debt restructurings, Journal of Financial Economics 27, 315-353. Greenwald, B. and J. Stiglitz, 1990, Asymmetric information and the new theory of the firm: Financial constraints and risk behavior, American Economic Review 80, 160-165. Greenwald, B., J. Stiglitz and A. Weiss, 1984, Information imperfections in the capital market and macroeconomic fluctuations, American Economic Review, 74, 194-199. Greenwald, B., M. Kohn and J. Stiglitz, 1990, Financial market imperfections and productivity growth, Journal of Economic Behavior and Organization 13, 321-345. Hirshleifer, D. and A. Thakor, 1992, Managerial conservatism, project choice and debt, Review of Financial Studies 5, 437-470.
P. Brito, A.S. Mello / Int. J. Ind. Organ. 13 (1995) 543-565
565
Hoshi, T., A. Kashyap and D. Scharfstein, 1990, The role of banks in reducing the costs of financial distress, Journal of Financial Economics 27, 67-88. James, C., 1987, Some evidence on the uniqueness of bank loans, Journal of Financial Economics 19, 217-235. James, C. and P. Wier, 1990, Borrowing relationships, intermediation and the cost of issuing public securities, Journal of Financial Economics 28, 149-171. Jovanovic, B., 1982, Selection and evolution in industry, Econometrica 50, 649-670. Mayer, C., 1990, Financial systems, corporate finance and economic development, in: R. Glenn Hubbard, ed., Asymmetric information, corporate finance, and investment (University of Chicago Press, Chicago). Myers, S. 1977, Determinants of corporate borrowing, Journal of Financial Economics 5, 146-175. Myers, S. and N. Majluf, 1984, Corporate financing and investments decisions when firms have information that investors do not have, Journal of Financial Economics 13, 187-222. Petersen, M.A. and R.G. Rajan, 1994, The benefits of lending relationships: Evidence from small business data, Journal of Finance 49, 3-37. Rajan, R., 1992, Insiders and outsiders: The choice between informed and arm's-length debt, Journal of Finance 47, 1367-1400. Shapiro, A. and S. Titman, 1986, An integrated approach to corporate risk management, in: J. Stern and D. Chew, eds., The revolution in corporate finance (Blackwell, New York). Stiglitz, J. and A. Weiss, 1981, Credit rationing in markets with imperfect information, American Economic Review 71, 393-410. Stiglitz, J. and A. Weiss, 1983, Incentive effects of terminations: Applications to credit and labor markets, American Economic Review 73, 912-927. Taggart, R.A. 1985, Secular patterns in the financing of U.S. corporations, in: B.M. Friedman, ed., Corporate capital structure in the United States (University of Chicago Press, Chicago). Whited, T.M. 1992, Debt, liquidity constraints and corporate investment: Evidence from panel data, Journal of Finance 47, 1425-1460.