Loan financing, bankruptcy, and optimal supply

www.elsevier.com/locate/econbase

International Review of Economics and Finance

13 (2004) 115–140

Loan financing, bankruptcy, and optimal supply

Nils Hauenschild*, Peter Stahlecker

Institute for Statistics and Econometrics, University of Hamburg, Von-Melle-Park 5, 20146 Hamburg, Germany

Received 31 January 2002; received in revised form 1 September 2003; accepted 25 September 2003

Abstract

We consider a model of an economy consisting of heterogeneous firms that are faced with uncertainty in future

prices when deciding upon production and financing. It is shown that the firms’ supply behaviour is significantly

affected by their attitude towards a possible bankruptcy in case of loan financing. In particular, both the individual

and the aggregate supply curve depend on the price uncertainty, the current real cash flow, and the current price

level if at least some firms choosing loan financing are not protected by limited liability but take the bankruptcy

risk resulting from the uncertain prices into account.

D 2004 Elsevier Inc. All rights reserved.

JEL classification: D21; E23; G33

Keywords: Price uncertainty; Loan financing; Bankruptcy risks; Limited liability

1. Introduction

For quite a long time, real and financial decisions of a firm have been treated as two strictly separate

issues in the economic literature. Only recently, it has been recognized that such an approach ignores

several important aspects of the firm’s decision problem and that both real and financial variables should

be considered simultaneously in a single model framework. The main point lies in the fact that the

possibility of bankruptcy constitutes a linkage between the financial and the real ‘‘sector’’ of a firm. In

case of debt financing the firm is (legally) obliged to repay this debt at a specific point of time or

otherwise goes bankrupt. When making its real decisions (investment and output, say) it thus has to

ensure that those obligations can be met out of the associated operating profits. Hence, its behaviour

differs from that of a self-financed firm and this in turn also influences the determination of the capital

structure.

1059-0560/$ - see front matter D 2004 Elsevier Inc. All rights reserved.

doi:10.1016/j.iref.2003.09.002

* Corresponding author. Tel.: +49-40-42838-3539; fax: +49-40-42838-6326.

E-mail address: [email protected] (N. Hauenschild).

N. Hauenschild, P. Stahlecker / International Review of Economics and Finance 13 (2004) 115–140116

Building on this insight, an important line of research has developed that aims at explaining the real

decisions of individual firms under external financing and the resulting consequences for macroeco-

nomic variables. Perhaps surprisingly, however, most of the papers in this field emphasize the role of

asymmetric information between lenders and borrowers in this respect and almost exclusively

concentrate on the firms’ investment behaviour.1 The mere existence of bankruptcy risks that was

identified as the key linkage between real and financial decisions and that would also be relevant under

symmetric information only plays a minor role. Moreover, the impact of external financing on the firms’

supply behaviour has largely been neglected.2

In this paper we are concerned with precisely the latter aspect; that is, we analyze how bankruptcy

risks affect the output decisions of individual firms as well as aggregate supply. Building on previous

work by Greenwald and Stiglitz (1988, 1993), we consider competitive firms that have to prefinance

their production costs and resort to (partial) loan financing if their current cash flow falls short of these

costs. The future selling price of the product is assumed to be uncertain, such that a loan financed firm

may not earn sufficient revenues to repay the loan and is thus confronted with the risk to go bankrupt. It

is shown that a firm’s supply behaviour depends crucially on how this risk enters the relevant decision

problem. A firm that is protected by limited liability in case of a bankruptcy can essentially ignore the

possibility to go bankrupt because all negative consequences (i.e., the loan loss) have to be borne by the

bank. Therefore, such a firm behaves exactly like a completely self-financed firm and derives its optimal

output from the ‘‘standard’’ parameters, that is, the real interest rate, the real factor prices, the relative

expected price and the production technology. If the owners of the firm are at least partially liable for the

losses in case of a bankruptcy, however, the firm’s supply behaviour changes significantly. In this case,

the firm will no longer ignore a possible bankruptcy but intends to avoid it instead. Hence, the

bankruptcy risk is taken into consideration, and output is chosen to keep the risk as small as possible. As

a consequence, output declines compared to the case of limited liability and, in addition to the

aforementioned parameters, now also depends on the expectations about future prices, the volatility

in prices, the current real cash flow, and the current price level. In particular, output is an increasing

function of the latter two variables and (under additional assumptions) of the expected price as well as

decreasing in the volatility of prices. The reason is that higher expected prices, a higher real cash flow,

and a higher price level reduce the risk to go bankrupt and thus allow for a larger output that is closer to

the one under self-financing or limited liability, whereas a higher volatility in prices increases the

bankruptcy risk and leads to a smaller output.

1 The analysis regarding the consequences of asymmetric information can be traced back to the seminal work of Jensen and

Meckling (1976) and Myers and Majluf (1984). Its impact on the investment behaviour is surveyed by Hubbard (1998); see also

Fazzari and Athey (1987) and Fazzari, Hubbard, and Petersen (1988). Broadly, it is shown that agency costs associated with

asymmetric information result in higher interest rates and credit rationing, which leads to a decline in the overall investment

activity. In this context, one can also establish the existence of a credit channel of monetary policy transmission; see Bernanke

and Gertler (1995), Gertler and Gilchrist (1993), and Hubbard (1995).2 Among the notable exceptions that are both concerned with bankruptcy risks and output decisions are Bernanke, Gertler,

and Gilchrist (1996), Greenwald and Stiglitz (1988, 1993), and Jefferson (1994). The fact that the supply behaviour is not

considered is the more surprising as there is a closely related line of research in industrial economics, where it is analyzed how

debt financing affects a firm’s output decisions and the resulting (Nash) equilibria on oligopolistic product markets; see, for

example, Brander and Lewis (1986, 1988), Glazer (1994), and Wanzenried (2003). Furthermore, much more attention is

devoted to bankruptcy risks than to asymmetric information in this line of research.

N. Hauenschild, P. Stahlecker / International Review of Economics and Finance 13 (2004) 115–140 117

It is likely that all three types of firms indicated above are present in an economy. Hence, aggregate

supply depends on all of the aforementioned parameters, where the relative size of groups consisting of

firms in different financing ‘‘regimes’’ (i.e., self-financing, loan financing under limited liability, loan

financing without limited liability) determines the sensitivity of aggregate supply to the respective

parameters. If a large fraction of firms resorts to loan financing and considers the risk of bankruptcy, this

finding has two important implications. Firstly, aggregate supply is significantly below the level it could

achieve under complete self-financing. Even worse, an ‘‘unfavourable’’ economic environment (a

recession, say) being characterized by low cash flows, pessimistic price expectations, and highly volatile

prices has a further contractive impact on aggregate supply, thereby enhancing the already ‘‘bad’’

situation. On the other hand, a ‘‘favourable’’ environment (a boom) with high cash flows, optimistic

price expectations, and low volatility in prices supports higher levels of aggregate supply, thus mitigating

the problem of too low aggregate supply under bankruptcy risks. Secondly, the aggregate supply curve is

upward sloping in the price–quantity graph, which implies some potential for monetary policy to

stipulate aggregate supply by expansionary measures. This is even enhanced by the fact that the

established results are not of temporary character but also hold in the long run.

The paper is organized as follows. In Section 2 we present the basic setup of the general model that is

able to capture different attitudes towards bankruptcy on part of the firms. In Section 3 we discuss the

optimal supply of a firm with limited liability and a firm that considers the risk of bankruptcy separately.

The impact of several important variables on the optimal supply of both types of firms is analyzed in

Section 4 by means of a comparative static analysis. In Section 5 we then show how far the results

obtained for individual firms extend to aggregate supply. Some concluding remarks are presented in

Section 6. Finally, Appendix A contains the proofs of all propositions.

2. The model

Consider an economy consisting of i = 1, . . ., n individual single-product firms that produce different

goods with j = 1, . . ., m variable input factors. Both the commodity and factor markets are assumed to be

competitive such that the nominal prices pit > 0, the price level pt > 0 as well as the vector wt = (w1t, . . .,wmt), wjt > 0, j = 1, . . ., m, of nominal factor prices are exogenous variables with respect to the firms’

decision problem in each period t. Here, the price level pt is defined by ptw Si = 1n gipit, where gi > 0 for all i

andSi = 1n gi = 1 for the exogenously given weights gi. Let xit denote the quantity produced by firm i and let

ci(wt, xit) denote the corresponding cost function. The firms’ factor demand functions can be obtained from

the cost function by using Shephard’s lemma. We assume that for every i = 1, . . ., n, ci is twice

differentiable and has the properties ci(wt,0) = (Bci/Bxit)(wt,0) = 0, (Bci/Bxit)(wt, xit)!lfor

xit!l and (B2ci/Bxit2)(wt, xit>0 for all xit>0 and all wtaRm

þþ . Let us indicate two

important special cases for which these requirements on the cost function are met. First, consider

the case that labour is the only input factor and that the firms’ technology is represented by a standard

neoclassical production function whose inverse hi has the properties hi(0) = hiV(0) = 0,

limx!lhiV(x)!l and hiW(x) > 0 for all x > 0.3 Here, wt denotes the nominal wage, hi(x) is

the ith firm’s labour demand, and the cost function simplifies to ci(wt, xit) =wthi(xit). Second,

3 This is the case studied by Greenwald and Stiglitz (1988, 1993) and Großl and Stahlecker (1998).


consider the case of several inputs and a production function that is homogeneous of degree s > 0. It

is well known that the cost function can then be written as ci(wt,xit) = xit1/sci(wt,1), where ci(wt,1)

denotes the unit cost function. If s < 1 the cost function possesses all the above properties.

The quantity xit supplied in any period t + 1 has to be produced in the previous period t. If the cash

flow qit > 0 earned in that period is not sufficient to finance the production of xit, the firm has to take

up a loan from a representative bank that charges the nominal interest rate qit.4 In the production

period, all variables with the time subscript t are known with certainty. For notational convenience, we

omit the corresponding subscripts in the subsequent analysis. When deciding on the output level xi,

however, the firm does not know the next period’s price pit + 1 with certainty. Instead, the future price is

given by a positive and continuous random variable pit + 1 with the corresponding density function fiand cumulated density function Fi as well as a compact support [0, pi], pi > 0.5 Owing to this price

uncertainty, the firm cannot rest assured that the revenues earned by selling a specific output xi will be

sufficient to pay back the loan in the next period. Of course, the precise amount due and the

consequences of a possible default on this obligation depend on the institutional arrangement of the

financial relationship between the firm and the bank. Here, we make the standard assumption that the

loan contract takes the form of what is usually referred to as a ‘‘standard debt contract.’’ In such a

contract a (nominal) interest rate is agreed upon and the firm is committed to pay back the loan plus

interests in one sum if it is ‘‘successful,’’ that is, if the revenues are sufficiently high. If total revenues

fall short of the firm’s debt obligation the firm goes bankrupt and all revenues are passed to the

creditor. It is well known that the existence (and optimality) of standard debt contracts may be justified

by asymmetric information with respect to the realized price and by costly state verification (see

Townsend, 1979; Gale and Hellwig, 1985). Thus, we essentially assume that informational asymmetries

of this kind are present in the economy. In our model framework, this institutional setup implies the

existence of some critical price pi for which total revenues pixi are just sufficient to pay back the debt.

We may express pi as a function of xi given by

piðxiÞ ¼ð1þ qiÞbiðxiÞ

xi; ð1Þ

where

biðxiÞwciðw; xiÞ � qi ð2Þ

denotes the amount of the loan demanded by the firm.6

Assuming a risk neutral bank, the nominal interest rate that is written into the loan contract has to be

determined such that the bank earns an expected yield that equals the nominal yield of a riskless asset.

4 We thus assume that there is a one-period lag between the production and the sale of the product. However, the factors

needed to produce the good must be purchased and paid in the first planning period. Furthermore, output cannot be stored and

must completely be sold in the second period.5 The probability distribution may depend on the time index t or on the information available at time t. This modification,

however, will be omitted since it is not essential within the above two-period setup, and an extension of the model to a

multiperiod framework is beyond the scope of this paper.6 If the cash flow is sufficient to finance the optimal output xi, both bi and the critical price pi are equal to zero.


The latter is composed of the safe real interest rate7 r and the expected rate of inflation. Hence, the

nominal loan yield Kt + 1x has to fulfil the relation8

1þ E½Kxitþ1� ¼

lpð1þ rÞ; ð3Þ

where lwE[Pt + 1] =E[SgiPit + 1] =Sgili is the expected price level and liwE[Pit + 1] = mpit + 1dFi(pit + 1)

is the expected future price of firm i. According to the definition of a standard debt contract, the nominal

loan yield is a discrete random variable with two possible realizations given by9

Kxitþ1 ¼

qi; if pitþ1 z piðxiÞ

pitþ1xi�biðxiÞbiðxiÞ ; else

8<: ð4Þ

The expected yield E[Kt + 1xi ] of the loan can thus be expressed as

E½Kxitþ1� ¼

Z piðxiÞ

�l

pitþ1xi � biðxiÞbiðxiÞ

� �fiðpitþ1Þdpitþ1 þ

Z l

piðxiÞqi fiðpitþ1Þdpitþ1

which, by partial integration as well as Fi(0) = 0 and Fi(pi) = 1, simplifies to

E½Kxitþ1� ¼ qi �

xi

biðxiÞ

Z piðxiÞ

�lFiðpitþ1Þdpitþ1: ð5Þ

Combining Eqs. (3) and (5) we immediately see that the nominal interest rate qi = qi(xi) is set by the

bank according to

1þ qi ¼lpð1þ rÞ þ xi

biðxiÞ

Z piðxiÞ

�lFiðpitþ1Þdpitþ1: ð6Þ

Eq. (6) shows that the nominal interest rate exceeds the riskless nominal yield by a markup reflecting

the firm’s default risk.10 At this interest rate, however, the bank is willing to meet the firm’s loan demand

8 Eq. (3) can be extended to include further components like market power or risk aversion of the bank, which for simplicity

will subsequently be ignored.9 Throughout the whole paper, we denote random variables by a capital letter and their realizations by the corresponding

small letter.10 Modelling loan contracts with (nominal) interest rates according to Eq. (6) is standard in the literature; see, for example,

Dotan and Ravid (1985) or Greenwald and Stiglitz (1993), where further arguments for the validity of Eq. (6) are provided.

7 The real interest rate is exclusively determined by real and exogenous parameters, e.g., the rate of time preference.


completely; that is, there is no credit rationing. Finally, inserting Eq. (6) into (1), the critical price pi isdetermined by

piðxiÞ ¼lð1þ rÞbiðxiÞ

pxiþZ piðxiÞ

�lFiðpitþ1Þdpitþ1; ð7Þ

which renders pi as an implicit function of xi.

Taking the above technological and financial conditions into account, each firm decides on optimal

production and optimal financing simultaneously. The firm’s objective is to maximize expected utility of

total nominal profits

Qxitþ1 ¼ Pitþ1xi � ð1þ qiÞbiðxiÞ ¼ Pitþ1xi � piðxiÞxi; ð8Þ

where the debt obligation is given by Eqs. (7) and (2). Moreover, we assume the particular utility

function

UiðQÞ ¼Q; if Qz 0

riQ; if Q < 0

;

8<: ð9Þ

where riz 0 denotes a parameter that reflects the firm’s attitude towards the risk of negative profits. For

ri = 1 the utility function is linear and the firm is risk neutral with respect to total profits. Analogously, U

is concave for ri >1 and convex for ri < 1, referring to a risk-averse and a risk-loving firm, respectively.

The case ri = 0 is of particular interest because Ui(Q) simply reduces to the firm’s cash flow then. The

utility function specified in Eq. (9) appears to be the most simple approach to capture different attitudes

towards the incident of bankruptcy on part of the firm, which will become evident momentarily.

Using Eqs. (8) and (9), the firm’s objective function Zir is given by

ZirðxiÞ ¼ E½UðQxitþ1Þ� ¼

Z l

piðxiÞq xitþ1 fiðPitþ1Þdpitþ1 þ ri

Z piðxiÞ

�lq xitþ1 fiðPitþ1Þdpitþ1

¼Z l

�lq xitþ1 fiðPitþ1Þdpitþ1 þ ðri � 1Þ

Z piðxiÞ

�lq xitþ1 fiðPitþ1Þdpitþ1: ð10Þ

Hence, a neutral firm simply maximizes its expected nominal profits, whereas a risk-loving firm with

ri = 0 maximizes its expected cash flow.11 In the latter case the firm is protected by limited liability

because it is only concerned with positive cash flows (profits) and essentially ignores the possibility of

bankruptcy or rather its consequences. In any case, however, the firm has to take the nominal interest rate

qi given by Eq. (6) into account. This interest rate (as well as the critical price) depends on the firm’s

11 In financial economics those two cases are usually referred to as firm value maximization (equity plus debt value) and

equity value maximization, respectively.


production decision, which will in turn have an impact on its objective function and its supply behaviour.

In view of Eqs. (7) and (8), the first term in Eq. (10) may be written as

Z l

�lq xitþ1 fiðPitþ1Þdpitþ1 ¼ lixi �

lpð1þ rÞðciðw; xiÞ � qiÞ � xi

Z piðxiÞ

�lFiðPitþ1Þdpitþ1;

while the second term in Eq. (10) simplifies to

ðri � 1ÞZ piðxiÞ

�lq xitþ1 fiðPitþ1Þdpitþ1 ¼ �ðri � 1Þxi

Z piðxiÞ

�lFiðPitþ1Þdpitþ1

by partial integration. Hence, the objective function takes the form

ZirðxiÞ ¼ lixi � lð1þ rÞðciðw�; xiÞ � qiÞ � rixi

Z piðxiÞ

�lFiðPitþ1Þdpitþ1; ð11Þ

where w and qi denote the vector of real factor prices and the real cash flow, respectively, and where we

have used the fact that the cost function is homogeneous of degree one in prices. Obviously, the specific

form of the debt contract in which the nominal interest rate depends on the quantity produced leads to

significant restrictions of the firm’s behaviour. It ensures that even a firm with limited liability (ri = 0)

cannot ignore the possibility of bankruptcy because this would negatively affect the expected loan yield.

In fact, Eq. (11) shows that a firm with limited liability behaves like a risk neutral firm that has to pay the

safe interest rate (plus inflation rate) for its loan.12 For all firms that consider the risk of bankruptcy

‘‘voluntarily’’ (ri > 0), the debt contract introduces an additional term into the objective function that

accounts for the potentially unfulfilled debt obligation. This term just equals the risk parameter ri

multiplied by the expected loan loss Ri(xi):

RiðxiÞ ¼Z piðxiÞ

�lðpiðxiÞ � pitþ1Þxifiðpitþ1Þdpitþ1 ¼ xi

Z piðxiÞ


where the second equality again follows by partial integration. It can be shown that the function Ri is

strictly increasing and strictly convex for all pi(xi)a(0, pi).13 Hence, a firm that is not protected by

limited liability but takes into account the risk of bankruptcy also minimizes the expected loan loss. In

what follows, we shall refer to Ri(xi) as the firm’s risk costs since, according to Eq. (11), they can be

interpreted as another (subjective) cost component entering the objective function. Perhaps surprisingly,

it is no longer important whether ri is larger, equal to, or smaller than one because the loan contract

ensures that all firms do not take actions at the expense of the bank, which implies that they behave

qualitatively alike. In the subsequent sections, however, we shall make evident that the behaviour of

12 The same result, though in a completely different context, was also obtained by Appelbaum (1992, p. 405).13 In fact, RiV(xi)>0 can be verified by Eqs. (17) and (18). Analogously, RiW(xi)>0 follows from the strict convexity of

ci(w, xi ) in xi.


firms with ri = 0 and of firms with ri > 0 differs substantially. It is thus only important whether firms are

protected by limited liability or not.

3. Individual supply behaviour

3.1. The limited liability case

Let us first consider a firm that is protected by limited liability (ri = 0) and maximizes its expected

cash flow.14 Since Zi0 is strictly concave (by the assumed properties of the cost function), the necessary

and sufficient optimality condition is given by

li

l¼ ð1þ rÞ Bci

Bxiðw�; xiÞ: ð13Þ

Hence, the optimal quantity supplied depends only on the real factor prices (e.g., the real wage), the

safe real interest rate, the production technology (which determines the cost function), and the relative

expected price. Neither the current cash flow, the critical price, nor the nominal loan rate has an impact

on the firm’s output decision.

The economic intuition behind this result becomes evident if the bank’s point of view is considered. If

the price in t + 1 exceeds the firm’s critical price, the firm redeems the entire loan and pays the interest

rate charged by the bank, which in this case earns a yield above the one of the alternative (riskless) asset.

If the price in t + 1 is below its critical value, the bank sustains a loss amounting to the difference

between the loan plus the due interest payment and the firm’s total revenues, which have completely

passed to the creditor. Since the bank is risk neutral, the nominal interest rate qi was fixed such that thosegains and losses balance out on average, and the riskless yield (plus inflation rate) is earned. From the

firm’s point of view, the opposite effect occurs: In case of a sufficiently high price, the interest payments

exceed those based on the risk free rate, whereas they fall short of the risk free rate if the price is too low,

thus ‘‘compensating’’ the firm for excessive payments in former periods.15

Of course, such an ‘‘averaging’’ of gains and losses is possible only if the production decision can be

repeated many times under identical conditions. In particular, it must be possible for the entrepreneur to

found an identical firm each time a bankruptcy has occurred. This assumption, however, appears to be

rather implausible, at least from the firm’s point of view. If the firm owners are privately liable for all

losses, they have to redeem the loan out of their private wealth or future incomes. Even if their liability is

completely limited, it has to be borne in mind that the owners also incur a loss amounting to that part of

personal wealth that had been invested in the firm, and it is by no means sure that they are able to found

another identical firm. It therefore appears plausible to assume that the owners will not be indifferent

with respect to bankruptcy. The same is true for managers acting on behalf of the firm owners. On the

14 The following analysis is carried out only for those production levels xi>0 being compatible with the existence of the

critical price pi(xi). It can be shown that the critical price exists for all xiaBi*wfxiaR : xi > 0; Zi0ðxiÞ > 0g, see Section A.2.1

in Großl and Stahlecker (1998), where a simpler predecessor of this model is considered.15 Note that a firm maximizing its expected nominal cash flow is also indifferent towards its financial structure if any

surplus can be invested at the riskless interest rate r in case of complete self-financing. We then have bi(xi)V 0 in all the above

equations.


one hand, their reputation will suffer once they have led a firm into bankruptcy preventing them from

being offered a new job at equally favourable conditions. Beyond that, it may happen that the owners

will call their managers into account by charging a penalty fee.16

To summarize the above discussion, there are good reasons to assume that the firm will take the risk

of going bankrupt into account when deciding upon optimal production and financing and will thus

choose some value ri > 0 in its utility function.17 Such an approach will be presented in the next section.

3.2. Taking the risk of bankruptcy into account

Let us now turn to a firm with ri > 0 that does not ignore its bankruptcy risk. Such a firm maximizes

the objective function (11) with respect to all positive production levels yielding a positive expected cash

flow. If the optimal production plan xi can be financed by the current cash flow qi, the firm will only use

self-financing, which implies bi(xi)V 0, pi(xi)V 0, and Ri(xi) = 0. Hence, the case of self-financing is

equivalent to maximizing the expected cash flow. If complete self-financing is impossible, a loan has to

be taken up and the risk costs Ri(xi) > 0 become relevant. For simplicity, we focus only on the special

case ri = 1 since all results remain qualitatively unchanged as long as ri > 0. We then have to consider

the problem to maximize the objective function Zi given by

ZiðxiÞwZi0ðxiÞ � RiðxiÞ ð14Þ

subject to the constraint

xiaBi*wfxiaR : xi > 0; Zi0ðxiÞ > 0g: ð15Þ

The first-order necessary condition for an optimal solution xi* is given by

Zi0Vðxi*Þ ¼ RiVðxi*Þ; ð16Þ

where

RiVðxi*Þ ¼Z piðxiÞ

�lFiðPitþ1Þdpitþ1 þ xipiVðxiÞFiðpiðxiÞÞ ð17Þ

with

xipiVðxiÞ ¼1

1� FiðpiðxiÞÞlð1þ rÞ Bci

Bxiðw; xiÞ �

ciðw; xiÞxi

þ qi

xi

� �> 0: ð18Þ

The sign of xipiV(xi) follows from the strict convexity of ci with respect to xi and the assumption of

positive parameters qi, l, and r.

16 See Greenwald and Stiglitz (1988, 1993) for a similar argumentation.17 Of course, it is also questionable whether the assumption of a risk neutral bank is reasonable. If the existing opportunities

to diversify potential risks are not perfect, the bank might not be willing to completely ignore the risk associated with each

individual loan. We leave this extension of the model for future research, however.


To show how taking the risk of bankruptcy into account alters the firm’s supply behaviour we have to

compare the production level xi* to the optimal solution xiaBi* resulting from the maximization of the

expected cash flow, which is determined by Eq. (13).

Proposition 1 . There is a unique solution xi*aBi* of Eq. (16), and the following assertions hold

(i) If ci(w, xi )V qi, then xi*= xi.

(ii) If ci(w, xi ) > qi, then xi* < xi.

We see that firms that do not ignore the possibility of bankruptcy have the same production plan as

firms with limited liability if loan financing is not optimal; that is, production is exclusively financed by

internal resources. If, on the other hand, bi(xi) > 0, we have xi* < xi; that is, a firm taking the risk of

bankruptcy into account produces less than a firm with limited liability if partial loan financing is

optimal. In that case, Eqs. (16)–(18) show that in contrast to expected cash flow maximization, the

current cash flow qi and the expected price level l (not only the relative expected price li/l) are furtherdeterminants of the optimal production xi*.

4. Determinants of individual supply behaviour under loan financing

In this section we are going to have a closer look at the impact of different variables on the supply

behaviour of a firm maximizing its expected cash flow or choosing self-financing only, and a firm with

partial loan financing that takes its bankruptcy risk into account.

4.1. Price uncertainty

Let us first consider the impact of uncertain prices on a firm’s optimal production xi. Any change in

the firm’s expectations about the future price can be represented by a corresponding transformation of

the random variable Pit + 1 or rather its underlying probability distribution. Since the first-order

conditions (13) and (16) explicitly depend on the expected price li and the expected price level l[cf. Eqs. (17) and (18)], it seems only natural to begin the analysis with a change of those parameters.

For this purpose, suppose that some exogenous event leads to a shift of all random variables Pit + 1 by

some nonzero value ai, that is

Paiitþ1 ¼ Pitþ1 þ ai for all i ¼ 1; . . . ; n: ð19Þ

The cumulated density function Fii

a belonging to Pit + 1ai is thus given by

Faii ðpitþ1ÞwPrðPai

itþ1 V pitþ1Þ ¼ Fiðpitþ1 � aiÞ: ð20Þ

For simplicity, let us assume that the values ai are given by ai = api, where pi is the current nominal

price of good i and aaR is some constant. This implies that an exogenous event leads to a change in all


expected prices by some portion of each firm’s current nominal price, thus ensuring that the change in

expected prices is proportional to the price ‘‘levels’’ of the respective sectors (or commodity markets).18

Hence, we have

liðaÞwE½Paitþ1� ¼ li þ api ð21Þ

as well as

liVðaÞ ¼ pi: ð22Þ

Moreover, the definition of the price level Pt + 1 implies

lðaÞwE½Patþ1� ¼ l þ ap; lVðaÞ ¼ p: ð23Þ

If a > 0, the transformation (19) thus expresses more optimistic price expectations in the sense that all

expected nominal prices and the price level are higher, while a< 0 refers to the situation where an

exogenous event leads to more pessimistic price expectations. In view of the transformation (19), the

expected cash flow, the risk costs, and the critical price now all depend on the parameter a as well.

Hence, the objective function of a firm with limited liability takes the form

Zi0ðxi; aÞ ¼ liðaÞxi � lðaÞð1þ rÞðciðw�; xiÞ � qiÞ; ð24Þ

and the optimal production xi(a) is given as a solution of

liðaÞlðaÞ ¼ ð1þ rÞ Bci

Bxiðw�; xiÞ: ð25Þ

Proposition 2. Consider a transformation of the random variables Pit + 1 as in Eq. (19). Then

Axi

Aa> 0 ð ¼; < Þ ð26Þ

if and only if pi /p >li(a)/l(a) (=, < ).

The result stated in Proposition 2 follows from the fact that a transformation of all probability

distributions Fi with the additive term api increases the relative expected price li(a)/l(a) of all firms

18 It is also possible to consider the case where all firms are confronted with a shift of their relevant probability distributions

by the same parameter a; that is, Pit + 1a =Pit + 1 + a for all i= 1, . . ., n, but this approach is somewhat implausible because an

absolute change of all expected prices by the same a has a different meaning for firms with low and high nominal prices,

respectively. However, the qualitative results remain unchanged (see footnote 21).


with pi /p > li(a)/l(a) and reduces the relative expected price of all firms with pi /p < li(a) /l(a). While

the former firms will extend their production accordingly, the latter ones will definitely produce less.

The corresponding results for a firm without limited liability are somewhat different. Here, the

objective function is given by [cf. Eq. (11)]

Ziðxi; aÞ ¼ liðaÞxi � lðaÞð1þ rÞðciðw�; xiÞ � qiÞ � xi

Z piðxi;aÞ�api

�lFiðPitþ1Þdpitþ1; ð27Þ

where [cf. Eq. (7)]

piðxi; aÞ ¼lðaÞð1þ rÞðciðw�; xiÞ � qiÞ

xiþZ piðxi;aÞ�api

�lFiðPitþ1Þdpitþ1 ð28Þ

and we have made use of Eq. (20) and the substitution formula to simplify the integrals in Eqs. (27) and

(28).19 The optimal production level xi is given by the first-order condition20

ZixiVðxi; aÞ ¼ liðaÞ � lðaÞð1þ rÞ BciBxi

ðw�; xiÞ � RixiVðxi; aÞ ¼ 0; ð29Þ

where

RixVðxi; aÞ ¼Z piðxi;aÞ�api

�lFiðPitþ1Þdpitþ1 þ xipixiVðxi; aÞFiðpiðxi; aÞ � apiÞ ð30Þ

and

xipixiVðxi; aÞ ¼lðaÞð1þ rÞ

1� Fiðpiðxi; aÞ � apiÞBci

Bxiðw; xiÞ �

ciðw; xiÞxi

þ qi

xi

� �> 0: ð31Þ

Eq. (29) gives the optimal production xi = xi*(a) as an implicit function of the exogenous parameter a.As the case of complete self-financing coincides with expected cash flow maximization (limited

liability), we assume that partial loan financing of xi*(a) is optimal; that is, xi*(a) < xi (see Section 3).

19 In anology to Bi* , all functions containing the critical price pi are now defined over the set D

i*wfðxi; aÞaR2 : xi> 0; liðaÞ > 0; Zi0ðxi; aÞ > 0g.

20 For simplicity, we denote partial derivatives with respect to any variable by the corresponding subscript; that is, ZixiVwBZi/

Bxi and so on.


Proposition 3. Consider a transformation of the random variables Pit + 1 as in Eq. (19).

1. If pi /pz li(a)/l(a) then

Axi*

Aa> 0: ð32Þ

2. If pi /p < li(a)/l(a), the optimal output xi* is not necessarily decreasing in a. In fact, we have @xi*/Ba >0 for all i = 1, . . ., n if

pi

p> ð 1 þ rÞ Aci

Axiðw�; xiÞ ð33Þ

for all i = 1, . . ., n.

The economic intuition behind Proposition 3 becomes evident when the result is compared to the

case of expected cash flow maximization (or complete self-financing). The first assertion in

Proposition 3 obviously establishes a similar result as in the limited liability case for those firms

resorting to partial loan financing and having a relative expected price below the current relative price.

Note, however, that the possibility of a higher production will generally be stronger in case of partial

loan financing because there is an additional effect resulting from the risk costs. If the expected price

(level) increases, the risk of bankruptcy (i.e., being confronted with a nominal price below the critical

price) declines, thereby reducing marginal risk costs as well [cf. Eq. (62) in Appendix A]. This effect,

which does not depend on a change in relative expected prices, is also responsible for the second

assertion of Proposition 3. Just as in the case of expected cash flow maximization, all firms with a

relative expected price larger than the current relative price have an incentive to reduce their

production because they are confronted with a lower relative expected price if some exogenous

event leads to a transformation of all random variables Pit + 1 as in Eq. (19). In contrast, the higher

expected nominal price still implies a lower risk of bankruptcy and falling marginal risk costs such

that there is an opposite effect in support of a higher production. Under the sufficient condition (33)

stated in Proposition 3, the latter effect will outweigh the former and even induce a higher production

level for firms with pi/p < li(a)/l(a).21

Proposition 3 gains some special importance if the case of identical firms is considered. In

such a situation, all current relative prices and all relative expected prices are equal to one, and

21 For the case indicated in footnote 18, that is, a = ai for all i, Propositions 2 and 3 also hold, the only difference being

that the relevant conditions pi/pz li(a)/l(a) and pi/p < li(a)/l(a) have to be substituted by 1z li(a)/l(a) and 1 < li(a)/l(a),respectively. Here, a transformation as in Eq. (19) increases relative expected prices and hence the production of all firms

with li(a) < l(a) both in case of ri = 0 and of partial loan finance with ri > 0, while relative expected prices decline for all

firms with li(a) > l(a). The latter case implies a lower production in case of ri = 0 and an ambiguous effect in case of loan

financing with ri > 0. In fact, both the proof and the economic interpretation of Propositions 2 and 3 can literally be applied

setting pi = p = 1.


Proposition 2 shows that optimal production is independent of the expected price (level) if the

firms maximize their expected cash flow or use self-financing only. Hence, a transformation as in

Eq. (19) will leave production unaffected. By Eq. (34), on the other hand, a firm using partial

loan financing and taking the risk of bankruptcy into account will increase its production if price

expectations become more optimistic (i.e., a > 0), whereas a more pessimistic view (a < 0) results

in a reduction of production and supply. In the latter situation, all factor demands will be

reduced as well.

So far we have only been concerned with implications from changing expected prices.

According to Eqs. (16)–(18), however, the entire distribution of the random variable Pit + 1 is

relevant for the optimal decision xi* in case of ri > 0. It is thus interesting to see in which way a

firm’s supply behaviour responds to other changes in the probability distribution and in particular

to the variance, which is usually viewed as a good measure of the ‘‘riskiness’’ of a random

variable and an important determinant of behaviour under uncertainty.22 We therefore consider a

transformation of the random variables Pit + 1 that leaves their expectations constant but increases

their variances, that is, a mean-preserving spread. Setting

Pbitþ1wbPitþ1 þ ð1� bÞli for all i ¼ 1; . . . ; n ð34Þ

for b > 0 we obtain

Fbi ðpitþ1ÞwPrðPb

itþ1 V pitþ1Þ ¼ Fi

pitþ1 � ð1� bÞli

b

� �ð35Þ

for the cumulated density function of the transformed random variables as well as

liðbÞwE½Pbitþ1� ¼ li; VarðPb

itþ1Þ ¼ b2VarðPitþ1Þ ð36Þ

and

liVðbÞ ¼ 0: ð37Þ

Furthermore, the definition of the price level Pt + 1 implies

lðbÞwE½Pbtþ1� ¼ l; lVðbÞ ¼ 0: ð38Þ

22 This approach parallels the one in Greenwald and Stiglitz (1993), see Proposition 2 in that paper.


By analogy to Eqs. (27) and (28), both the objective function and the critical price now depend on the

additional parameter b, that is,23

Zi0ðxi;bÞ ¼ liðbÞ � lðbÞð1þ rÞðciðw; xiÞ � qiÞ; ð39Þ

ZiðxibÞ ¼ liðbÞ � lðbÞð1þ rÞðciðw; xiÞ � qiÞ � xi

Z piðxi;bÞ

�lFi

pitþ1 � ð1� bÞli

b

� �dpitþ1

¼ liðbÞ � lðbÞð1þ rÞðciðw; xiÞ � qiÞ � xibZ piðxi ;bÞ�ð1�bÞli

b

�lFiðpitþ1Þdpitþ1

¼ liðbÞ � lðbÞð1þ rÞðciðw; xiÞ � qiÞ � xibZ ziðpiðxi;bÞÞ

�lFiðpitþ1Þdpitþ1 ð40Þ

and

piðxi; bÞ ¼lðbÞð1þ rÞðciðw; xiÞ � qiÞ

xiþ b

Z ziðpiðxi;bÞÞ


where we have set

ziðyÞwy� ð1� bÞli

b

to simplify notations. Hence, optimal supply xi = xi(b) in the limited liability case is given as an implicit

function of b by

Zi0xV ðxi;bÞ ¼ liðbÞ � lðbÞð1þ rÞ BciBxi

ðw; xiÞ ¼ 0; ð42Þ

and we obtain

Proposition 4. Consider a transformation of the random variable as in Eq. (34). Then

Axi

Ab¼ 0 for all i ¼ 1; . . . ; n: ð43Þ

Not surprisingly, a mean-preserving spread has no impact on the production decision of firms with

limited liability. Since marginal risk costs play no role for optimal production, a change in the riskiness

of future prices has no consequences. Matters are quite different for firms considering the risk of

23 Analogously to the case of a transformation as in Eq. (19), all functions containing the critical price pi are now defined

over the set Gi*wfðxi; bÞaR� Rþ : xi > 0; liðbÞ > 0; Zi0ðxi; bÞ > 0g, see footnote 19.


bankruptcy. Here, the objective function is given by Eq. (40), and optimal supply xi = xi*(b) is obtainedas a solution of

ZixVðxi; bÞ ¼ liðbÞ � lðbÞð1þ rÞ BciBxi

ðw; xiÞ � RixiVðxi; bÞ ¼ 0; ð44Þ

where

RixiVðxi;bÞ ¼ bZ ziðpiðxi;bÞÞ

�lFiðpitþ1Þdpitþ1 þ xipixiVðxi;bÞFiðziðpiðxi; bÞÞÞ ð45Þ

and

xipixiVðxi; bÞ ¼lðbÞð1þ rÞ

1� Fðziðpiðxi; bÞÞÞBci

Bxiðw; xiÞ �

ciðw; xiÞxi

þ qi

xi

� �> 0: ð46Þ

Again, we assume that partial loan financing of xi*(b) is necessary.

Proposition 5. Consider a transformation of the random variable as in Eq. (34). Then

Axi*

Ab< 0 for all i; . . . ; n: ð47Þ

Proposition 5 shows that a mean-preserving spread will lead to a reduction of production and supply

for all firms. Unlike the case of a changing expectation, such a transformation of the random variable

causes only one single effect, viz., a growing risk of bankruptcy due to a larger volatility in future prices.

Hence, (marginal) risk costs will be higher inducing a lower supply, irrespective of the firm’s relative

expected price. Thus, being concerned with the risk to go bankrupt introduces some sensitivity of the

firm’s optimal production to possible deviations of future prices from their expectation.24

4.2. The current cash flow

The current real cash flow qi is another variable that has no impact on optimal supply behaviour under

expected cash flow maximization [cf. Eq. (13)] but that becomes relevant when the risk of bankruptcy is

taken into account. As in the previous section, we denote the risk costs, the critical price, and the

objective function by Ri(xi,qi), pi(xi,qi), and Zi(xi,qi), respectively, to emphasize their dependence on qi.

24 To complete the above analysis, it should be mentioned that a simple scale transformation of the form Pit + 1c wcPit + 1,

c > 0, implies Bxi*/Bc = 0; that is, a change of the unit in which prices are measured (e.g., euro instead of dollar) has no impact

on optimal supply.


According to Eq. (16), the optimal production of a firm with ri = 1 is given as an implicit function

xi = xi*(qi) of qi by

ZixiVðxi; qiÞ ¼ li � lð1þ rÞ BciBxi

ðw; xiÞ � RixiVðxi; qiÞ ¼ 0; ð48Þ

where [cf. Eqs. (17) and (18)]

RixiVðxi; qiÞZ piðxi;qiÞ

�lFiðpitþ1Þdpitþ1 þ xipixiVðxi; qiÞFiðpiðxi; qiÞÞ ð49Þ

with

xipixiVðxi; qiÞ ¼1

1� Fiðpiðxi; qiÞÞlð1þ rÞ Bci

Bxiðw; xiÞ �

ciðw; xiÞxi

þ qi

xi

� �> 0; ð50Þ

and we assume that partial loan financing (i.e., xi*(qi) < xi) is optimal.

Proposition 6. The optimal supply is an increasing function of the current real cash flow, that is,

Axi*

Aqi> 0 for all i ¼ 1; . . . ; n: ð51Þ

According to Proposition 6, a larger real cash flow implies falling marginal risk costs and hence a

larger production (and supply) level xi*. The economic intuition behind this result is that a higher

cash flow allows for a larger part of the production to be self-financed, which means a lower risk of

going bankrupt. Again, it has to be emphasized that in contrast to Proposition 3, Eq. (51) holds for

all firms expressing a phenomenon that is not present for a firm maximizing its expected cash flow;

see Eq. (13).

4.3. The current price level

In case of expected cash flow maximization (or complete self-financing), the firm’s supply xi depends

only on real variables [cf. Eq. (13)], which implies a vertical supply curve in the price–quantity graph.

If, on the other hand, partial loan financing is optimal and if the risk of bankruptcy is taken into account,

Eqs. (16)–(18) demonstrate that now the real cash flow is determined by the current price level, which

therefore is also relevant for the optimal production xi*. The reason is that all expenses in a period t are

given by the loan taken up in the previous period and are thus independent of the current price level p.25

25 Note that we implicitly impose the (unrealistic) assumption that a new and identical firm is founded by new owners if a

bankruptcy occurs in period t investing an amount which is equal to the ‘‘last’’ cash flow (i.e., of period t� 1) of its bankrupt

predecessor. A multiperiod sequential decision model that allows for entry and exit (due to bankruptcy) is beyond the scope of

this paper, however.


Here, of course, we have to assume that all current relative prices as well as the current real factor prices

remain unchanged; that is, nominal goods and factor prices adjust instantaneously to changes in the

current price level. This can be modelled by multiplying all current nominal goods and factor prices by

the same parameter kaR. By the definition of the price level, we then have pt(k) = kpt, and all relative

prices as well as all current factor prices remain unaltered. A change in the price level amounts to

(marginally) increasing or decreasing k. Of course, all variables now depend on k as well, and we

assume xit*(k) < xit; that is, partial loan financing is optimal.

Proposition 7. The optimal supply is an increasing function of the current price level, that is,

Axit*

Ak> 0 for all i ¼ 1; . . . ; n: ð52Þ

We have thus established that the firm’s supply curve has a positive slope in the price–quantity graph

if partial loan financing is optimal.26 Analogously to Proposition 3, the result formulated in Proposition 7

is particularly important in the special case where all firms are identical. There, all current relative prices

are equal to one and the assumption of constant relative prices trivially holds.27

5. Aggregate supply behaviour

5.1. Identical firms

In the preceding sections we have analyzed how loan financing and different degrees of risk aversion

with respect to bankruptcy as well as several important variables and parameters affect the supply

behaviour of each individual firm. When turning to the respective consequences for aggregate supply

behaviour, it appears appropriate to study the special case of identical firms as a benchmark.28 Here, all

(current) prices pit and all expected prices li are identical, and, according to the definition of the price

level, all relative current prices pit/pt and all relative expected prices li/l are equal to one. Aggregate

output xt in period t (supplied in period t + 1) is given by

xt ¼Xni¼1

xit; ð53Þ

since all goods are now identical.

First, consider the case that all firms are protected by limited liability and thus maximize their

expected cash flow (thereby ignoring the risk of bankruptcy) which is equivalent to the case that all firms

26 Note that the effect described here is different from the one mentioned in footnote 24. There, a scale transformation has

no impact on the firm’s supply behaviour since both the firm and the bank endogenously react to that change by adjusting the

loan demanded and the nomimal interest rate, respectively. In the scenario considered in Proposition 7, the nominal loan is

given from the previous period and does not respond to changes in current prices.27 The result established in Proposition 7 also holds in an extension of the present model where the probability distributions

underlying Pit + 1 are assumed to be time dependent. If there is a positive correlation between the current price pit and those

distributions, then the analysis in Section 4.1 establishes another reason for an increasing supply curve.28 In fact, that special case (a so-called symmetric equilibrium) is the dominant model framework in most of the New

Classical and New Keynesian literature.


exclusively choose self-financing. The optimality condition (13) as well as Eq. (53) then show that

individual and aggregate supply are determined only by the safe real interest rate, the real factor prices,

and the production technology. It is independent of the current real cash flow and the uncertainty about

future prices; that is, neither changes in expected prices nor changes in the riskiness of prices have an

impact on the quantity supplied (see Propositions 2 and 4). Moreover, the aggregate supply curve is

vertical in the price–quantity graph since optimal supply does not depend on the current price level.

If, however, all firms resort to partial loan financing and consider the risk of bankruptcy, aggregate

supply behaviour will change significantly. According to Proposition 1 and Eq. (53), aggregate output is

lower than for limited liability firms (or complete self-financing). Furthermore, the uncertainty about

future prices, the current real cash flow and the current price level become relevant determinants of

aggregate supply. In this context, Propositions 3 and 5 show that more pessimistic price expectations

(represented by a transformation of the random variable Pt + 1 as in Eq. (19) with a< 0) as well as more

risky prices (represented by a transformation of the random variable Pt + 1 as in Eq. (34)) lead to a lower

aggregate supply29, whereas more optimistic price expectations and less risky prices increase aggregate

supply. The same effects can be observed with respect to changes in the current real cash flow. By

Proposition 6, a larger real cash flow implies a higher production for all firms and hence a higher

aggregate supply, whereas a lower real cash flow leads to a reduction of aggregate output and supply.

These results suggest that a more ‘‘favourable’’ economic environment (i.e., optimistic price

expectations, a low volatility of prices, high real cash flows) supports high levels of output, aggregate

supply and factor demands even though they are still below the respective levels resulting from cash

flow maximization or complete self-financing. A more ‘‘unfavourable’’ scenario (pessimistic price

expectations, highly volatile prices, low real cash flows) results in a low level of output and factor

demands as compared to the case of limited liability.

Finally, Proposition 7 and Eq. (53) show that the aggregate supply curve has a positive slope in the

price–quantity graph, which establishes another sharp contrast to the case of expected cash flow

maximization. In particular, this result has implications for monetary policy, although one should not

jump to conclusions in this respect, given the partial equilibrium nature of the model.

5.2. Heterogeneous firms

We will now drop the assumption that all firms are identical and return to our original assumption of

heterogeneous firms. In this case, two additional, although minor, complications emerge. First, since all

firms produce different goods, one cannot simply add up all individual outputs to obtain aggregate

supply [cf. Eq. (53)] but rather has to consider a quantity index given by

xt ¼

Xni¼1

xitpit

pt¼

Xni¼1

xitpit

pt: ð54Þ

Second, the two cases studied in Section 5.1, viz., that all firms maximize their expected cash flow

(or use self-financing) or that all firms choose loan financing and take the risk of bankruptcy into

29 At the same time, all factor demands are reduced as well, which immediately follows from the monotonicity of the cost

function with respect to output and Shephard’s lemma.


account, represent only two polar cases. It can generally be expected that part of the firms maximize

their expected cash flows, some firms use only self-financing and some firms taking on loans take the

risk of bankruptcy into account. Nevertheless, it is intuitively convenient to begin with the special case

that all firms behave alike.

If every firm maximizes its expected cash flow or uses self-financing only, Eqs. (13) and (54)

show that almost all results established in Section 5.1 carry over to the case of heterogeneous firms.

In particular, aggregate supply does not respond to any changes in the riskiness of future prices or in

the current real cash flow. Moreover, as long as we assume that current relative prices and real

factor prices are determined in fully competitive markets, the aggregate supply curve is vertical in

the price–quantity graph, cf. Proposition 7. Only an exogenous event that leads to more optimistic

or more pessimistic price expectations represented by a transformation of the random variables Pit + 1

as in Eq. (19), aggregate supply behaviour differs from the one generated by identical firms. Even if

all firms maximize their expected cash flows (or use self-financing) their expected prices are not

necessarily identical, and Eq. (13) implies that the firms may both increase or reduce their output,

depending on whether their relative expected price is less or larger than their current relative price

(and whether a > 0 or a < 0). Hence, Eq. (54) implies that the net effect on aggregate supply is

indeterminate.

If all firms choose borrowing and are concerned with a possible bankruptcy, results similar to the

ones stated in Section 5.1 for the case of identical firms can be obtained as well: Aggregate supply is

below the corresponding level supplied by limited liability or self-financing firms, and aggregate

supply is reduced as future prices become more risky and the current real cash flow declines (and vice

versa). As long as current relative prices (as well as current factor prices) are determined in

competitive markets, Eq. (54) and Proposition 7 show that Bxt/Bk > 0; that is, the aggregate supply

curve is increasing in the price–quantity graph. An isolated change in expected prices [as in Eq. (19)]

has an ambiguous impact on aggregate supply, just as in the case of limited liability firms. As outlined

in Section 4.1, however, there is a much stronger tendency towards an increase (reduction) of

aggregate supply in case of more optimistic (pessimistic) price expectations. If pi/p > (1 + r)Bci(w, xi)/

Bxi for all i = 1, . . ., n, aggregate supply unambiguously increases if more optimistic price expectations

prevail.

Finally, consider the intermediate case that all kinds of firms are present in the economy. In view

of the above discussion it is then obvious from Eq. (54) that aggregate supply is always below the

corresponding level of the benchmark case with only limited liability firms, that the aggregate

supply curve is increasing in the price–quantity graph, and that aggregate supply is sensitive to

changes in the riskiness of prices, in expected prices, and in the current real cash flow as described

above. Both the slope and the location of the aggregate supply curve depend on the number of

firms that belong to the respective categories, where the curve is the further to the left of the

vertical supply curve and the flatter, the more firms choosing loan financing take the risk of

bankruptcy into account. Analogously, the stronger the response to changes in the current real cash

flow or future prices is, the more firms with ri > 0 and resorting to borrowing are present in the

economy.30

30 In any case, the weighting factors pit/pt in Eq. (54) are of course important as well, although their impact cannot be

characterized in more detail.


6. Concluding remarks

In this paper we have analyzed a model of an economy consisting of heterogeneous firms that

are faced with uncertainty in future prices when simultaneously deciding on optimal production

and financing. For all firms choosing loan financing this uncertainty constitutes a risk to go

bankrupt since the next period’s price may be too low for the firm to redeem the loan plus

interest payments. If the firms take this bankruptcy risk into account, their supply behaviour

significantly differs from the one of firms that are protected by limited liability and simply

maximize their expected cash flows and of firms using only self-financing. The output decisions

of the former firms are sensitive to price expectations, the riskiness in prices, the current real

cash flow, and the current price level, where output is positively correlated with the latter two

variables and, to some extent, with expected prices, as well as negatively correlated with the

volatility in prices. Notably, the results obtained hold both in the short and in the long run

provided that the long-run equilibrium price distribution does not reduce to a degenerated

distribution that assigns probability one to a single price.31 Furthermore, the results hold for both

individual and aggregate supply, and the respective effects are the stronger the more firms do not

ignore their risk to go bankrupt.

In most European countries, small and medium-sized firms contribute a considerable fraction to

aggregate output. Those firms regularly have to resort to loan financing and can also be considered

rather risk averse since a bankruptcy has very immediate consequences for the owners’ financial

position. This suggests that the above analysis is of some practical relevance.32 In a recession, for

example, the firms will only earn small revenues and hence small cash flows, which reduces their

ability to self-finance future projects and increases their need for bank loans. At the same time,

however, the recession will presumably be accompanied by more pessimistic price expectations and

more uncertainty about future prices (i.e., higher price volatility). This implies a significantly

increasing risk to go bankrupt and, according to our results, leads to a reduction of aggregate

supply. Thus, the economic downturn will even be enforced.33 Moreover, an increasing aggregate

supply curve seems to be reasonable for economies in which small and medium-sized firms play an

important role. This implies some potential for the effectiveness of monetary policy, particularly in

times of monetary contractions.

Acknowledgements

We thank an anonymous referee for helpful comments and suggestions. Of course, we are responsible

for all remaining errors.

31 One could proceed along the lines of Duffie, Geanakoplos, Mas-Colell, and McLennan (1994) to characterize long-run

equilibria of this kind in more detail.32 For some empirical evidence for the case of Germany, see Großl, Stahlecker, and Wohlers (2001) and Kirchesch,

Sommer, and Stahlecker (2001).33 Of course, there will be additional effects in the same direction resulting from a growing number of firms actually going

bankrupt.


Appendix A

Proof of Proposition 1. Since the objective function Zi is strictly concave, there is at most one

solution to Eq. (16). Let xi > 0 with Zi0V(xi) = 0 and Zi0V(xi) > 0 be given. It can easily be verified

that such a solution to Eq. (13) exists and that it is unique.

(i) Let ci(w, xi)� qiV 0. Then we have bi(xi)V 0, pi(xi)V 0 as well as RiV(xi) = 0, and hence

ZiV(xi) = Zi0V(xi) = 0; that is, xi*= xi.

(ii) Let ci(w, xi)� qi < 0. It follows that pi(xi) > 0 and hence ZiV(xi) = Zi0V(xi)�RiV(xi) =�RiV(xi) < 0since Ri is strictly increasing. On the other hand, since ci is strictly increasing in xi, ci(w, 0) and qi>0,

there exists an xiaBi* with 0 < xi < xi and ci(w, xi)� qi < 0. Hence, pi(x) < 0, RiV(xi) = 0 and ZiV(xi) =

Zi0V(xi). The strict concavity of Zi0 as well as the definition of xi imply Zi0V(xi) >Zi0V(xi) = 0. The

strict concavity of Zi thus yields the existence of a unique solution xi*aBi* of Eq. (16) with

xi < xi*< xi. 5

Proof of Proposition 2. Applying the implicit function theorem to Eq. (25) gives

Bxi

Ba¼ � Zi0xiaW ðxi; aÞ

Zi0xixiW ðxi; aÞ: ð55Þ

Since the objective function is strictly concave we only have to determine the sign of Zi0xiaW (xi, a). ByEqs. (22) and (23) we obtain

Zi0xiaW ðxi; aÞ ¼ pi � pð1þ rÞ BciBxi

ðw; xiÞ: ð56Þ

The assertions of the proposition now follow immediately from Eq. (56) and the optimality condition

(25). 5


Bxi*

Ba¼ � ZixiaW ðxi; aÞ

ZixixiW ðxi; aÞ: ð57Þ

Since the objective function is strictly concave we only have to determine the sign of ZixiaW(xi, a). By(Eqs. (22), (23), and (30) we obtain

ZixiaW ðxi; aÞ ¼ pi � pð1þ rÞ BciBxi

ðw; xiÞ � RixiaW ðxi; aÞ; ð58Þ

where

RixiaW ðxi; aÞ ¼ Fiðpiðxi; aÞ � apiÞðpiaVðxi; aÞ � piÞ þ xipixiVðxi; aÞfiðpiðxi; aÞ � apiÞðpiaVðxi; aÞ � piÞþ xipixiaW ðxi; aÞFiðpiðxi; aÞ � apiÞ: ð59Þ


Using

piaVðxi; aÞ � pi ¼1

1� Fiðpiðxi; aÞ � apiÞpð1þ rÞðciðw; xiÞ � qiÞ

xi� pi

� �ð60Þ

and

xipixiaW ðxi; aÞ¼1

1� Fiðpiðxi; aÞ � apiÞ

"xipixiV ðxi; aÞfiðpiðxi; aÞ � apiÞðpiaVðxi; aÞ � piÞ þ pð1þ rÞ

� Bci

Bxiðw; xiÞ �

Ciðw; xiÞxi

þ qi

xi

� �#; ð61Þ

Eq. (59) simplifies to

RixiaW ðxi; aÞ ¼1

1� Fiðpiðxi; aÞ � apiÞ

"Fiðpiðxi; aÞ � apiÞ pð1þ rÞ Bci

Bxiðw; xiÞ � pi

� �

þ xipixiVðxi; aÞfiðpiðxi; aÞ � apiÞðpiaVðxi; a � piÞ#: ð62Þ

Since RixiV(xi, a) > 0 if partial loan financing is optimal [see Eqs. (30) and (31)], Eq. (29) implies

liðaÞ > lðaÞð1þ rÞ BciBxi

ðw; xiÞ for xi ¼ xi*ðaÞ: ð63Þ

Furthermore, we have

liðaÞ > lðaÞ ð1þ rÞðciðw; xiÞ � qiÞxi

for xi ¼ xi*ðaÞ ð64Þ

because of Zi0(xi, a) > 0.34 It thus follows that RixiaW (xi, a) < 0 which yields ZixiaW (xi, a) > 0 for all

firms with pi/pz li(a)/l(a) by Eq. (63), such that Eq. (57) implies Eq. (32).

If pi/p < li(a)/l(a), the sign of ZixiaW (xi,a) is ambiguous. The last assertion directly follows from

Eqs. (58)–(62) because Eq. (33) also implies pi/p > (1 + r)(ci(w, xi)� qi)/Bxi by the strict convexity

of ci in xi and by ci(w, 0) = 0. 5

34 Remember that we restrict the analysis to all production levels that imply a positive expected cash flow; cf.

footnote 19.


Proof of Proposition 4. Since liV(b) = lV(b) = 0 the proposition immediately follows by application of the

implicit function theorem to Eq. (42). 5


Bxi*

Bb¼ � ZixibW ðxi; bÞ

ZixixiW ðxi; bÞ; ð65Þ

so we have to determine the sign of ZixibW (xi, b). Since liV(b) = lV(b) = 0 we obtain

ZixibW ðxi; bÞ ¼ �RixibW ðxi;bÞ; ð66Þ

where

RixibW ðxi; bÞ ¼Z ziðpiðxi;bÞÞ

�lFiðpitþ1Þdpitþ1 þ bFiðziðpiðxi;bÞÞÞ

bpibVðxi;bÞ � piðxi; bÞ þ li

b2

þ xipixibW ðxi; bÞFiðziðpiðxi; bÞÞÞ þ xipixiV ðxi;bÞfiðziðpiðxi; bÞÞÞ

� bpibVðxi;bÞ � piðxi; bÞ þ li

b2: ð67Þ

Using

xipixibW ðxi;bÞ ¼1

1� Fiðziðpiðxi; bÞÞÞxipixiV ðxi;bÞfiðziðpiðxi; bÞÞÞ

bpibVðxi; bÞ � piðxi;bÞ þ li

b2 ð68Þ

and

pibVðxi;bÞ ¼1

1� Fiðziðpiðxi;bÞÞÞ

Z ziðpiðxi;bÞÞ

�lFiðpitþ1Þdpitþ1 �

piðxi; bÞ � li

bFiðziðpiðxi; bÞÞÞ

" #;

ð69Þ

Eq. (67) simplifies to

RixibW ðxi;bÞ ¼1

1� Fiðziðpiðxi; bÞÞÞ

"Z ziðpiðxi;bÞÞ

�lFiðpitþ1Þdpitþ1 �

piðxi;bÞ � li

bFiðziðpiðxi; bÞÞÞ

þ xipixiV ðxi; bÞ1

bfiðziðpiðxi;bÞÞÞ pibVðxi; bÞ �

piðxi; bÞ � li

b

� �#: ð70Þ

Using Eq. (41) and applying Zi0(xi, b)>0 (cf. footnote 23) it immediately follows that pi(xi, b)� li < 0

and hence pibV(xi, b)>0 as well as RixibW (xi, b)>0. By Eq. (66) and the strict concavity of the objective

function, Eq. (47) thus follows from Eq. (65). 5



Bxi*

Bqi¼ � ZixiqiW ðxi; qiÞ

ZixixiW ðxi; qiÞ; ð71Þ

where by Eq. (48)

ZixiqiW ðxi; qiÞ ¼ �RixiqiW ðxi; qiÞ: ð72Þ

Using Eqs. (49), (50) and

piðxi; qiÞ ¼lð1þ rÞðciðw; xiÞ � qiÞ

xiþZ piðxi;qiÞ


straightforward calculations show that

RixiqiW ðxi; qiÞ ¼ � lð1þ rÞpixiVðxi; qiÞfiðpiðxi; qiÞÞð1� Fiðpiðxi; qiÞÞÞ2

< 0: ð74Þ

Inserting Eqs. (72) and (74) into (71) we finally obtain Eq. (51). 5

Proof of Proposition 7. By definition, the nominal cash flow is given by35

qitðkÞ ¼ kpitxit�1 � lð1þ rÞðcit�1ðwt�1; xit�1Þ � qit�1Þ � xit�1

Z pit�1ðxit�1Þ

�lFiðpitÞdpit; ð75Þ

where kpitxit � 1 denotes the nominal revenues earned by the sale of the output xit� 1 produced in the

previous period, and the last two terms are equal to (1 + qit � 1)bi(xit� 1), that is, the amount to be repaid

to the bank [cf. Eqs. (1) and (6)]. Dividing by pt(k) = kpt yields the real cash flow

qitðkÞ ¼qitðkÞkpt

¼ pit

ptxit�1 �

lkpt

ð1þ rÞðcit�1ðw t�1; xit�1Þ � qit�1Þ �xit�1

kpt


�lFiðpitÞdpit;

ð76Þwhich directly implies

Bqit

Bk¼ l

k2ptð1þ rÞðcit�1ðwt�1; xit�1Þ � qit�1Þ þ

xit�1

k2pt


�lFiðpitÞdpit > 0: ð77Þ

Since there is also a positive correlation between the real cash flow and the optimal supply xit (see

Proposition 6) and since the real factor prices do not change, we immediately obtain Eq. (52). 5

35 In a sequential model as indicated in footnote 25, both F and l as well as r could be modelled time dependent.


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Documents

Loan financing, bankruptcy, and optimal supply