Competitive Equilibrium Theory of Bank Capital
Structure
Preliminary and Incomplete
Douglas Gale
New York University
Piero Gottardi
European University Institute
September 29, 2016
1
Introduction
The Financial Crisis of 2007-08 started a vigorous debate about the regulation of the banking
system, much of it focused on bank capital regulation. It is widely accepted that capital
adequacy regulation in the pre-crisis period did not ensure that banks had su¢ cient capital
to weather the storm. The risk weights used under Basel II underestimated the risk of certain
asset classes and regulatory arbitrage further reduced its e¤ectiveness. In response, regulators
at both the national and international level have sought to introduce more stringent capital
adequacy requirements.
Although new policies are already being put in place, our theoretical understanding of the
role of bank capital lags behind. What is the market failure that requires capital regulation?
What role do capital requirements play in the corporate governance of large and complex
banks? What are the costs of regulation, directly in the form of complicance costs and
indirectly in the form of distortions of economic decisions?
Capital regulation has its roots in bank supervision, which traditionally focuses on the
“safety and soundness”of individual banks. Not surprisingly, recent proposals for increasing
the stability of the …nancial system are simply stronger versions of the policies aimed at
insuring the resilience of individual banks. Similarly, much of the theoretical literature on
capital regulation focuses on the behavior of individual banks, rather than the …nancial
system. This is unfortunate, because macroprudential regulation is concerned with the
stability of the …nancial system as a whole. It requires an understanding of systemic risk,
as distinct from the risk of individual banks. In addition, capital regulation may have
macroeconomic e¤ects. For example, a global increase in bank capital will have an e¤ect
on the cost of capital. Finally, stability is not the only objective of …nancial regulation. Its
objectives should include an e¢ cient, innovative, and competitive …nancial system. This
requires an understanding of the welfare economics of regulation.
1
For all these reasons, we take the view that a general equilibrium approach is needed.
A useful starting point is to identify the conditions under which laisser-faire equilibrium is
e¢ cient, as a precursor to identifying the market failures that makes regulation necessary
and the distortions that may be introduced by regulation.
Admati and Hellwig (2013) have argued that the starting point for any discussion of
capital regulation should be the seminal Modigliani and Miller (1958)’s theory of capital
structure. We agree the work of Modigliani and Miller (1958) is the foundation of the
literature on corporate capital structure: any explanation of the determinants of the capital
structure of …rms must move away in some aspect from the environment considered by
Modigliani and Miller, introducing costs and bene…ts of using debt and equity. Moreover,
in analyzing banks’ capital structure it is important to explicitly recognize their role as
intermediaries, as well as the fact that bank deposits are not simply debt claims, but are also
valued by agents because they function as money. In this paper we assume that deposits can
be used for transactions and this function gives rise to a spread between the (risk adjusted)
returns on equity and deposits. At the same time, bank deposits are costly because the
higher banks’ debt the higher the probability that banks default and this is costly. More
precisely, we assume that when a bank defaults (because it is unable to meet the demand
for withdrawals) a fraction of the value of its assets is lost.
Unlike in the Modigliani-Miller model, in which capital structure is irrelevant, capital
structure matters in our model. In equilibrium, it is determined by a tradeo¤ between the
funding advantage of deposits and the possibility of costly default1 . In addition, the role of
banks as intermediaries generates an interdependence between the capital structure of banks
and that of …rms, which obtain funding from banks.
In this paper, we study a representative agent economy consisting of consumers, …rms
and banks. Consumers have an initial endowment of capital goods and want consumption
goods. Firms have access to a variety of risky technologies that use capital goods to produce
consumption goods. Firms issue equity to households and borrow from banks in order to
fund the purchase of capital goods. Banks lend to …rms and issue equity and deposits to
households so as to fund their loan portfolios. Households purchase equity in …rms and make
deposits in banks to fund their future consumption. Only banks can lend to …rms and only
households can invest in equity.2
Firms and banks are restricted in the securities they can issue. Firms are restricted to
issuing debt and equity and banks are restricted to taking deposits and issuing equity. In this
sense, markets are incomplete. Nonetheless, there are many types of securities issued, because
…rms and banks could di¤er in their risk characteristics (respectively of the technology and
the loan portfolio they choose) and in their capital structures. We assume markets for these
securities are competitive and complete, in the sense that there is a market and associated
price for each type of security which could be issued.
In this framework, we obtain analogues of the fundamental theorems of welfare eco1
Note that an analogous trade-o¤ arises in models proposed for …rms’ capital structure, which rely on
the trade-o¤ between the tax advantages of debt and the costs of default.
2
The assumption that banks cannot hold …rm equity is of little importance, but it simpli…es the analysis.
2
nomics. First, we show that a competitive equilibrium, where investment and …nancing
decisions of …rms and banks are taken so as to maximize their market value, is constrained
e¢ cient.3 Second, we show that any constrained e¢ cient allocation can be decentralized4 as
an equilibrium.5
We then turn to the critical question: What is the constrained e¢ cient - and hence also
the equilibrium - level of equity for banks and …rms? Our …rst result shows that, if the
technologies available to …rms satisfy a property known as co-monotonicity, which implies
that their productivities are positively correlated, the value of bank capital will be zero
in equilibrium. This is a striking result but one that has an intuitive explanation. The
fundamental source of uncertainty in the economy is the randomness of …rms’productivity.
When a …rm receives a negative shock to productivity, it may be forced to default on its
bank loans. This in turn makes bank loans risky and may trigger bank default as well.
Equity in …rms and banks represents a bu¤er that can absorb (small) losses and protect
them against costly default. The issue is whether it is e¢ cient to allocate a larger amount
of this bu¤er in …rms or in banks. In this regard, note that …rm equity plays a double
role, because by protecting the …rm against default it also shields the banks from the e¤ects
of the productivity shock and thus protects the banks from default. In the special case of
co-monotonic technologies this last consideration proves to be key: no matter what are the
relative costs of default in banks and …rms, it is always e¢ cient to put all the equity in
the corporate sector, at the top of the chain linking …rms and banks, where shocks …rst hit.
Equity in fact does “double duty”there, in the sense that it simultaneously provides a bu¤er
against default by the …rm and the bank.
But when the productivities of the di¤erent technologies available are negatively correlated, banks may reduce their probability of default by diversifying their portfolio of loans
among …rms who chose di¤erent technologies. In this case the probability that a large shock
hits any given technology is considerably smaller than the probability that such a shock hits
one of the technologies available in the economy. As a consequence, if the bank diversi…es
across …rms choosing all di¤erent types of technologies, it bene…ts in two ways. First, it
only needs to hold a relatively small amount of equity, since in a typical state only a small
fraction of its porfolio of loans defaults. Second, because there is often some type of …rm
that is defaulting, the equity bu¤er is being used most of the time, unlike equity held by
a …rm, which is only used a much smaller fraction of the time. Thus, there are now two
advantages in banks using equity compared to …rms and we show that in this case it may be
3
The …rst best or Pareto e¢ cient allocation cannot be attained because of the restriction to debt and
equity as the funding instruments available to …rms and banks. However, if we similarly restrict the planner
to the allocations attainable using debt and equity, using so a notion of constrained e¢ ciency, he would not
be able to improve on the laisser faire equilibrium.
4
Since there is a representative consumer lump sum taxes and transfers are not needed to decentralize
the constrained e¢ cient allocation.
5
In this paper we ignore asymmetric information in order to focus on the welfare properties of the choices
made by banks and …rms in a basic competitive environment. There is a large and well known literature on
moral hazard and risk shifting in banks. These considerations, as well as the possibility of bank bailouts in
the event of default may distort the choice of the capital structure and introduce ine¢ ciencies.
3
e¢ cient for banks, rather than …rms, to issue equity.
The rest of the paper is organized as follows. In Section 2 we present the economy and
the competitive equilibrium notion. In Section 3 we show the welfare properties of equilibria.
Section 4 examines then the properties of banks’capital structure in equilibrium. In particular, subsection 4.1 shows that when the returns on the technologies available to …rms are
co-monotonic, bank equity has zero value in equilibrium. Subsection 4.2 characterizes then
equilibrium capital structures in an environment where the yields of the technologies feature
both aggregate and idioncratic, fully diversi…able, shocks, where we show that under some
conditions bank capital structures feature equuity with a positive value. Section 5 concludes.
1.1
Related literature
This paper is closely related to Allen, Carletti and Marquez (2015), henceforth ACM.6 ACM
consider, like us, an environment where banks and …rms can only issue debt (deposits)
and equity, but where the markets for deposits and equity are segmented and the capital
structure is chosen so as to maximize the expected surplus for the representative bank…rm pair, subject to participation constraints for the equity holders and deposit holders.
This cooperative contracting approach guarantees the e¢ ciency of the equilibrium while, in
our framework, this is a property of competitive equilibrium when markets are open for all
possible types of debt and equity. The assumed segmentation of markets means that investors
in deposits have lower outside options than investors in equity. As a result, investors accept
a lower return on deposits than investors in equity and equity is “expensive.” ACM also
derive a result that bank equity has zero value in equilibrium, for the special case in which
…rms’ returns are perfectly correlated and uniformly distributed. In that case, the bank
is simply a pass-through for the …rms’ returns and the bank will default only if the …rms
default. ACM show that putting all the equity in the …rms minimizes the probability of
default for both banks and …rms.
ACM extend their zero-equity result to the case where …rms are divided into two sectors. Returns are uniformly distributed and perfectly correlated within each sector, but are
independently distributed across sectors. ACM show that for some (but not all) parameter
values, it is optimal for banks to lend only to …rms in a single sector. The contracting problem is therefore equivalent to the previous one-sector case and the no-equity result continues
to hold. The case of two independently-distributed sectors does not generalize, however.
One can see that, if the number of sectors were increased without limit, the law of large
numbers would eventually imply that a diversi…ed portfolio would dominate the risky portfolio invested in a single sector. When the risk of the diversi…ed portfolio is small enough, it
is optimal for the bank to include a positive value of equity in its capital structure.
As we already said, the key ingredients of the model of banks’capital structure consid6
The published version of ACM, Allen, Carletti and Marquez (2015), contains only the …rst part of the
working paper version and deals only with banks that invest directly in projects, rather than lending to
…rms. The results more closely related to the present paper are found in the working paper version and, in
what follows, we refer only to that version.
4
ered in this paper are the interdependence between banks’and …rms’capital structure, due
to the intermediation role played by the …rst ones, the fact that deposits are also useful for
transaction purposes and earn so a liquidity premium, and the presence of costs of default.
Diamond and Dybvig (1983) and Diamond (1984) show that bank deposits are the optimal form of funding for banks that provide liquidity insurance to depositors or delegated
monitoring for investors. Gale (2004) extended the Diamond-Dybvig model to include bank
capital that provides additional risk sharing between risk neutral investors (equity holders)
and risk averse depositors.
Our model assumes there are direct costs of default that reduce the value of the bankruptcy estate. The empirical literature shows that these costs can be substantial for both
banks and non-…nancial …rms (see James, 1991; Andrade and Kaplan, 1998; Korteweg,
2010). More recent work suggests that these estimates may understate the true costs of
default (Almeida and Phillipon, 2007; Acharya, Bharath and Srinivasan, 2007). We assume
that these costs are true deadweight costs as distinct from the …re sale “losses” that are
actually transfers of value (cf. Gale and Gottardi, 2015).
For other models of banks’capital structure, see DeAngelo and Stulz (2013) and Gornall
and Strebulaev (2015).
The empirical literature on the relationship between a bank’s capital structure and its
market value is not large. Flannery and Rangan (2008) examined changes in banks capital
structure in the previous decade. Mehran and Thakor (2011) found a positive relationship
between bank value and bank capital in a cross section of banks. Gropp and Heider (2010)
found that the determinants of bank capital structure were similar to those of non-…nancial
…rms, although the levels of equity are di¤erent.
Our competitive equilibrium model is then related to the literature on the theory of the
…rm in incomplete markets, developed by Diamond (1968), Ekern and Wilson (1974), Radner
(1974), Drèze (1974), and Grossman and Hart (1979). In the earlier literature, …rms are only
funded by equity. The value of a …rm is then determined using the marginal valuations of
its owners. For example, a …rm that chooses a production plan y would have market value
v=
rui (xi ) y
krui (xi )k
where xi is consumer i’s consumption bundle, rui (xi ) is the vector of i’s marginal utilities,
and consumer i owns a positive share of the …rm. Our assumption of complete markets for
debt and equity ensures the existence of equilibrium prices for all possible securities, even
those that are not traded in equilibrium. A similar approach is in Makowski (1983) and
Allen and Gale (1988, 1991). An alternative to the complete markets approach is to assume
that only traded securities are priced, but that …rms have rational conjectures about the
value a security would have if a small amount of it were introduced. This approach was used
by Hart (1979), for example, and appears to give the same results as the complete markets
approach under su¢ ciently strong regularity conditions.
The existence of intermediaries and the costs of default in our model make the pricing of
assets more complicated than in a stock market economy. Because a …rm’s debt is held by
5
banks and default can occur at the …rm level, the bank level, or both, the value of a …rm’s
debt will depend on banks’willingness to pay for it, which in turn depends on the banks’
capital structure and the consumers willingness to pay for the debt and equity of banks. See
also Bisin, Gottardi and Ruta (2014) on the pricing of securities when intermediaries are
present.
There is a large theoretical literature on the role of bank capital in preventing risk shifting
or asset substitution (Stiglitz and Weiss, 1981; Martinez-Miera and Repullo, 2010).
2
An equilibrium model of banks’capital structure
2.1
Endowments and technologies
There are two dates, indexed t = 0; 1, and a …nite number of states of nature, s = 1; :::; S.
The true state is unknown at date 0 and revealed at date 1. The probability of state s at
date 0 is denoted by s > 0, for s = 1; :::; S.
There are two goods, a non-produced capital good and a produced consumption good.
Consumption is produced subject to constant returns to scale using capital goods as the
only input. There is a …nite number of technologies, indexed j = 1; :::; n, for producing the
consumption good. Using technology j, one unit of capital at date 0 produces Ajs > 0 units
of consumption at date 1 in state s.
There is a continuum of identical consumers with unit mass. Each consumer has an
initial endowment of k0 = 1 units of capital at date 0. There is no initial endowment of
consumption.
2.2
Firms
We assume that each active …rm can invest in only one of the n technologies available.
Because production is subject to constant returns to scale, there is no loss of generality
in assuming that each …rm uses one unit of capital. Then the amount invested in each
technology can be adjusted by varying the number of …rms using it.
Each …rm chooses its technology and its capital structure. The …rm’s capital structure is
determined by the face value of the debt it issues. The face value of the debt is denoted by `
and is assumed to belong to a …nite7 interval L = [0; `max ]. Since productivity shocks are the
only source of uncertainty, the technology choices made by …rms determine the level of risk
in the economy, while their capital structure choices determine how this risk is distributed
between equity and debt returns.
In the environment considered, …rms are ex ante identical but may choose di¤erent technologies and capital structures. The choice of each …rm is referred to as the …rm’s type. The
set of …rm types is denoted by F L N , with generic element (`; j), where N = f1; :::; ng
indicate the set of available technologies.
7
For simplicity of the analysis we discretize the interval [0; `max ] so that the set L has only …nitely many
points.
6
The securities issued by a …rm of type f = (`; j) 2 F are debt and equity, with payo¤
vectors denoted by adf ; aef 2 RS+ RS+ and de…ned by
adf s =
`
if Ajs `
A
f js if Ajs < `
(1)
and
aef s =
Ajs
0
`j if Ajs `
if Ajs < `
(2)
for any state s and type f = (`; j). The parameter 0
f < 1 describes the recovery ratio
in the event of default (hence the default costs are 1
f , per unit of ouput). Although
we allow for generality the recovery ratio to depend on the …rm’s type f; this plays no role
in the analysis and in most applications f is set independent of f . Note that the payo¤
vectors adf and aef are uniquely de…ned by the type f = (`; j) via the equations (1) and (2).
Firms choose their technology and capital structure so as to maximize their pro…ts, or
equivalently the value of the securities, debt and equity, that they issue to fund the purchase
of one unit of capital. Since …rms are subject to constant returns to scale, pro…ts must be
zero in equilibrium. In other words, the market value of the securities issued is just enough
to …nance the purchase of capital goods. Types of …rms that cannot earn a zero pro…t will
not operate in equilibrium.
Securities issued by …rms are sold on competitive markets. In line with our completeness
assumption, there is a price associated with the securities issued by each possible type of …rm,
given by the vector qF = qdF ; qeF 2 RF+ RF+ , where qdF is the subvector of debt prices and
qeF is the vector of equity prices. The market value of a …rm of type f is qfd +qfe . We normalize
the price of capital goods to be equal to one. Then, in equilibrium we have qfd + qfe 1 for
any f 2 F — otherwise the demand for capital goods would be unbounded— and only …rm
types that achieve zero pro…ts, qfd + qfe = 1, will operate in equilibrium.
2.3
Banks
Banks provide funds to …rms by purchasing the debt issued by …rms (we assume they are
not allowed to invest in …rm equity). This purchase is …nanced by issuing debt (or deposits)
and equity. Thus each bank chooses its capital structure and its debt portfolio. A bank’s
capital structure is determined by the level of deposits it chooses to issue. We denote the
face value of deposits by d 2 D, where D is a …nite interval [0; dmax ]. The bank’s portfolio
is described by a vector x 2 RF+ , where xf
0 denotes the number of type-f …rms whose
debt is held by the bank. Since the banks’technology is subject to constant returns to scale,
we
P can assume without loss of generality that each bank’s portfolio is normalized so that
f 2F xf = 1. In other words, the bank invests in a portfolio of debt issued by …rms that
corresponds to claims issued against one unit of capital goods. Let X RF+ denote the set
of admissible debt portfolios.
All banks have the same access to the …rms’debt and the same funding opportunities.
Their portfolios and capital structures may di¤er, however. We refer to a bank’s portfolio
7
x and capital structure d as its type. The set of bank types is B = X D. A bank of type
b = (xb ;d) 2 B has a portfolio xb and issues two securities, debt and equity, with payo¤
vectors adb ; aeb 2 RS+ RS+ de…ned by
adbs =
and
aebs =
if xb adF s d
if xb adF s < d
(3)
d if xb adF s d
if xb adF s < d;
(4)
d
b
x
x adF s
0
adF s
for every state s, where the vector adF s is de…ned by
adF s = adf s
f 2F
for every state s. The recovery rate 0
1 is a constant and may or not depend on the
b
bank’s type b 2 B.
The problem of each bank is to select its portfolio and its capital structure to maximize
its pro…ts, given by the di¤erence between its market value, that is the value of the liabilities
it issued, and the value of the portfolio it acquired. The bank takes as given the price of
all debt claims issued by …rms, qdF 2 RF+ ; as well as the prices of all types of securities the
d
RB
banks can issue, qB = qdB ; qeB 2 RB
+ , where qB is the subvector of deposit prices and
+
e
qB is the subvector of equity prices. More formally, each bank will choose its type b 2 B
to maximize market value minus the cost of the assets it acquired, qbd + qbe qdF xb . In
equilibrium, the maximum pro…t will be zero, that is, qbd + qbe qdF xb , and only banks that
achieve zero pro…ts, qbd + qbe = qdF xb , will be active.
2.4
Consumers
All consumers have VNM preferences over consumption at t = 1 described by
X
s u (c1s + c2s ) ;
(5)
s
where c1s denotes the consumption in state s that can occur immediately, as it is paid for
with deposits, while c2s denotes the consumption which is paid for with the yields of equity,
and occurs with some delay. The constant 0 < < 1 captures the cost of this delay. The
speci…cation of the preferences re‡ects the assumption that deposits serve as money, whereas
equity does not. The delay (or equivalently, transaction) costs involved in converting equity
into “cash”are measured by the parameter 2 (0; 1).8 The function u : R+ ! R, describing
8
The speci…cation is a reduced-form representation of the greater convenience of using deposits for consumption compared to equity. A shareholder who wants to convert shares into consumption must pay a
commission to sell the shares. Dividends are paid infrequently and must be converted into deposits before
they can be spent. This time delay reduces the value of the consumption because of discounting.
8
the utility of total consumption in any state s; c1s + c2s , is assumed to be increasing, concave
and continuously di¤erentiable.
Each consumer can use the revenue obtained by selling his endowment of capital at
date 0 to purchase debt and equity issued by banks and equity issued by …rms. Thus
consumers cannot purchase …rm debt, but can do it indirectly by holding the debt and
equity of banks which purchase …rm debt. A consumer’s portfolio is described by a vector
d e
RB
z (zF ; zB ) 2 RF+ RF+ RB
+ , where zb ; zb denotes the consumer’s demand for debt
+
and equity issued by banks of type b and similarly for zfd ; zfe . The set of feasible portfolios
is denoted by Z and de…ned to be the set of portfolios z such that zfd = 0 for all f 2 F .
Letting q (qF ; qB ), the consumer chooses a consumption bundle c = (c1 ; c2 ) 2 RS+ RS+
and a portfolio z 2 Z to maximize
S
X
U (c)
s u (c1s
+ c2s )
s=1
subject to the budget constraints,
c2 =
q z 1;
X
c1 =
zbd adb ;
X
b2B
zbe aeh
X
+
f 2F
b2B
2.5
zfe aeh :
Equilibrium
An allocation is described by a consumption bundle, c, and a portfolio, z, of the representative consumer, and a distribution of banks over the set of possible bank types = ( b )b2B ,
and a distribution of …rm types = ( f )f 2F . Formally, the allocation is an array (c; z; ; ).
An allocation is attainable if
X
(6)
f = 1;
f 2F
X
b xb
(7)
= ;
b2B
zbd = zbe =
zfe =
and
c=z a=
X
b2B
b;
(8)
8f 2 F;
f;
zbd adb ;
8b 2 B;
X
b2B
zbe aeb +
(9)
X
f 2F
zfe aef
!
:
(10)
The …rst attainability condition (6) says that the …rms collectively use the entire one unit
of the capital good in the consumers’endowments. The second (7) says that banks hold in
9
their portfolio all the debt issued by …rms. The third and fourth conditions, (8) and (9) say
that consumers hold all the deposits and equity issued by banks and all the equity issued by
…rms. Finally, the last condition, (10), restates the relationship between consumption and
the payo¤ of the portfolio held by consumers.
An equilibrium consists of an attainable allocation (c; z; ; ) and a price system q such
that
(i) f > 0 only if f solves the …rm’s problem, given the prices q;
(ii) b > 0 only if b solves the bank’s problem, given the prices q;
(iii) (c; z) solves the consumer’s problem, given the prices q.
Note that equilibrium condition (i) is equivalent to
f
> 0 =) qfd + qfe = max qfd + qfe = 1;
f 2F
for any f 2 F . Similarly, equilibrium condition (ii) is equivalent to
b
> 0 =) qbd + qbe
qdF xb = max qbd + qbe
b2B
qdF xb = 0;
for all b 2 B.
3
Constrained e¢ ciency
In this section we show that, in the environment considered, analogues of the Fundamental
Theorems of Welfare Economics hold. Since we restricted the available securities to be debt
and equity and de…ned so attainable allocations as those that can be implemented using
debt and equity, the appropriate welfare concept is constrained Pareto e¢ ciency, rather
than Pareto e¢ ciency.
We say that an attainable allocation (c ; z ; ; ) is constrained Pareto e¢ cient, or
constrained e¢ cient, for short, if there does not exist an attainable allocation (c; z; ; )
such that U (c) > U (c ). Formally, this is the case if and only if (c ; ; ) solves the
problem
S
X
max
s u (c1s + c2s )
s=1
subject to the constraints9
X
b
(11a)
= 1;
b2B
c=
X
b
adb ; aeb +
b2B
X
f
0; aef :
(12a)
f 2F
9
In (11a), P
(12a) we used conditions (8) and (9) to substitute for the consumers’ portfolio z and (7)
together with f 2F xf = 1 to substitute for f in the other attainability constraints.
10
Proposition 1 Let (c ; z ;
Pareto e¢ cient.
;
; q ) be an equilibrium. Then (c ; z ;
;
) is constrained
The argument of the proof is standard, and exploits the fact that markets for all the possible types of securities that can be issued by …rms and banks are competitive and complete.
Also, note that the set of attainable consumption vectors satisfying (11a), (12a) is convex
and this allows us to establish the following:
Proposition 2 Suppose that (c ; z ; ;
exists a price vector q such that (c ; z ;
4
) is a constrained e¢ cient allocation. Then there
; ; q ) is an equilibrium.
Banks’equilibrium capital structure
In the economy described the issue of equity entails a cost because it reduces the portion
of the cash‡ow generated by …rms’ projects that is available to be paid as yield of liquid
assets (deposits). This occurs directly in the case of banks and indirectly for …rms, since a
higher level of equity in …rms implies that the level of assets available for banks to purchase
will be lower, that is, the level of intermediation will be lower. Equity however also entails
a bene…t because it provides a bu¤er against the risk of default and default is costly. Since
the productivities of the available technologies and the returns on the assets in the banks’
portfolios are random, the higher the level of equity in …rms or banks the lower the probability
that they default. Moreover, the intermediation role of banks implies that the riskiness of
the assets in their portfolio depends on the amount of equity issued by …rms: the higher this
amount, the lower the riskiness of banks’assets. This shows that …rm equity does “double
duty,” in the sense that, by providing a bu¤er against default by the …rm, it also helps to
prevent default by the bank that lends to the …rm.
We should point out that two other choices by …rms and banks also play a role here.
The choice of the technology by each …rm contributes to determine the level of risk faced
by individual …rms as well as the level of aggregate risk in the economy. Moreover, a bank’s
choice of its portfolio of …rms’debt a¤ects the bank’s probability of default. As we will see in
what follows, the possibility of diversifying the portfolio holding debt of di¤erent instruments
constitute another instrument available to banks to reduce their probability of default.
While it is clear that the cost of equity described above is higher the higher is the liquidity
premium 1= ; and the bene…ts of equity are higher the higher are the costs of default 1
b,
1
f ; the more interesting and less obvious issue we intend to analyze is the relationship
between the capital structure of …rms and banks, that is whether equity should be higher in
…rms or banks. The answer to this question depends in an important way on the e¤ects of
the linkages between …rms and banks induced by the intermediation activity of the latter,
that is the double duty played by …rms’ equity and the choice of the portfolio of debt of
di¤erent types of …rms. As we will see in this section, in these e¤ects the nature of the
uncertainty, and in particular the relationship between the shocks to the productivities of
the di¤erent technologies, plays a crucial role.
11
4.1
An environment with no bank equity
As we said in the Introduction, ACM derive the remarkable result that in equilibrium banks
fund their loans with deposits only in a model that is similar to ours and in the extreme case
where …rms’returns are perfectly correlated and identically distributed. The case of perfectly
correlated and identically distributed returns corresponds in our model to the special case of
a single technology, n = 1. We can demonstrate an analogous result under weaker conditions.
Instead of perfect correlation and identical distributions10 , we assume that technologies are
co-monotonic in the following sense:
De…nition 3 Technologies are said to be co-monotonic if Ajs
and j = 1; :::; n.
1
< Ajs , for every s = 2; :::; S
This assumption ensures that the productivities of all available technologies are increasing
in s, hence productivities are strongly, positively correlated. As a consequence, an increase
in s reduces defaults for all types of …rms and banks and we get a very sharp result, that
banks default if and only if some …rms default. Under the co-monotonicity condition, the
double duty role of …rm equity is thus particularly e¤ective and it turns out that the pro…t
maximizing - and also e¢ cient - choice of …rms and banks is to concentrate equity in the
…rms:
Proposition 4 Assume that technologies are co-monotonic. Then if (c ; z ;
equilibrium, the value of bank equity is zero for all b 2 B such that b > 0.
;
; q ) is an
It is interesting to point out that this result holds irrespectively of what is the relative
size of the costs of default for …rms (1
f ) and banks (1
b ), only the above property
of the structure of the uncertainty faced by the economy matters. A formal proof of the
proposition is found in the appendix. Here we provide an outline o the main steps. The
proof proceeds in a number of steps. Let `b denote the face value of …rms’debt xb held by
type b banks and let `b (s) denote the actual amount paid by …rms to debtholders in state s.
It is also convenient to use the notation b and f to indicate the types of banks and …rms
that are active in equilibrium, that is, such that b > 0 and f > 0.
Step 1: For all active banks’types b = (xb ;d ); either `b > d and the value of bank equity
is positive, or `b = d and the value of bank equity is 0.
It will never be optimal for a bank to choose d > `b , because this implies the bank will always
be in default and will incur unnecessary default costs. If d = `b , then it is clear that there
is nothing left over for the bank’s equity holders, even if the bank and all of the …rms that
borrowed from it are solvent. On the other hand, if d < `b , then there will be some state, for
example, s = S, in which all …rms are solvent and pay the face value of their debt and the
10
ACM also assume that consumers are risk neutral and are exogenously divided into depositors and
shareholders. These assumptions are unnecessary in our framework, where we allow for risk aversion and
consumers can hold both debt and equity.
12
return to equity is `b d > 0. Since limited liability ensures the payment to equity holders
is non-negative in every state, this is enough to prove that the value of equity is positive.
Step 2: For each active …rm’s type f = (` ; j ), there exists a state sf such that …rm f
is solvent if and only if s sf . Similarly, for each bank’s type b there exists a state sb
such that the bank is solvent if and only if s sb .
Firm f is solvent if and only if Aj s ` . Then the …rst claim follows from the fact that
Ajs is increasing in s and that the …rm is solvent in at least one state. The latter property
holds since, by a similar argument as for banks in step 1, it will never be optimal for a …rm
to choose a face value of debt ` > AjS (otherwise the …rm will always be in default and incur
unnecessary costs of default). Next note that the revenue of …rms (net of bankruptcy costs)
is increasing in s and from this it follows that the amount repaid to bank b is non-decreasing
in s. This is su¢ cient to establish the second claim.
Step 3: For each active bank’s type b the face value of deposits satis…es d = `b (sb ).
If d < `b (sb ), the bank has the option of increasing the face value of deposits without
increasing the probability of default. Since the bank is already in default in states s < sb ,
increasing the face value of deposits will not change the amount of consumption received by
deposit holders or equity holders in states s < sb . In states s
sb , on the other hand,
an increase in the face value of deposits will transfer consumption to the bank’s depositors
from the bank’s shareholders. There is a representative consumer, so shareholders and
depositors are the same individuals, and since one unit of consumption from equity’s returns
is worth
units of consumption from deposit’s returns, this transfer will increase their
welfare, contradicting the constrained e¢ ciency of equilibrium. Hence in equilibrium we
must have d = `b (sb ).
Step 4: For all types of banks b that are active in equilibrium equity has no value: d = `b .
If d < `b , contrary to what we want to prove, there must be at least one …rm that is
bankrupt in state sb . Otherwise, all …rms would repay the full amount of their debt and we
would have `b (sb ) = `b > d , contradicting Step 3. But if one of the …rms is bankrupt in
state sb , this …rm has the option of reducing the amount of debt it issues by an amount such
that the …rm no longer defaults in state sb . This will leave the yield of deposits unchanged
and will have two e¤ects. First, there will be a transfer to the …rm’s shareholders from the
bank’s shareholders, but this will have no e¤ect on total consumption or welfare. Second,
by eliminating the bankruptcy costs in state sb the change will increase the returns to the
…rm’s shareholders and is an unambigous gain in welfare. This contradiction shows that we
cannot have d < `b in equilibrium. Thus, d = `b and we have shown that the value of
bank equity must be zero in equilibrium.
Step 1 of the proof shows that default by one or more of the borrowing …rms is always a
necessary condition for the lending bank to default, because a bank will never set the face
value of deposits d higher than the face value of the borrowers’debt. This clearly illustrates
the fact that …rm equity always does “double duty,” serving as a bu¤er against both bank
default and …rm default.
13
The co-monotonicity assumption is stated as a property of the productivity of all the
technologies available in the economy. It is easy from the proof of Proposition 4 that what
su¢ ces for the validity of this result is that an analogous property holds for the yields of
the debt claims in the portfolio chosen by a bank. For any bank portfolio x, let the set of
technologies represented in the portfolio be denoted by J (x) and de…ned by
J (x) = j = 1; :::; n : x(`;j) > 0 for some ` :
Then we say that the portfolio x is co-monotonic if the set of technologies J (x) is comonotonic in the usual sense. The following corollary is then immediate.
Corollary 5 In any equilibrium (c ; z ; ; ; q ), the value of equity is zero for any bank
b (with b > 0) whose portfolio xb is co-monotonic.
The corollary gives us no information about the conditions under which a co-monotonic
portfolio will be chosen in equilibrium, beyond the conditions of Proposition 4. It merely
emphasises that bank capital is not needed as long as the bank does not diversify its portfolio
outside a set of co-monotonic technologies. In particular, if a bank does not diversify its
portfolio and only holds debt of one type of …rms, the bank will have zero equity (its portfolio
is trivially monotonic).
Proposition 4 states that banks use debt …nancing exclusively. It does not say anything
about the capital structure of …rms, however. For example, it does not claim that …rms will
issue equity to reduce their default risk. The …rms’choice of capital structure will depend
on model parameters, such as the recovery rates of banks and …rms, b and f . The higher
the default costs, other things being equal, the higher one expects the …rms’equity to be.
The only certainty is that banks will use no equity in equilibrium.
Proposition 4 is also silent on the variety of capital structures and technologies used
by …rms in equilibrium, as well as on the portfolio choice by banks. Because of the nonconvexities that are an essential part of the model, we allow the full use of the convexifying
e¤ect of large numbers in order to ensure the existence of an equilibrium. Many types of …rms,
distinguished by their capital structures and (in the case of …rms) technology choices, as well
as many types of banks, distinguished by their portfolio choice, are potentially active in
equilibrium. To make the analysis tractable and say more about the properties of equilibria,
it will be useful to consider some special cases where the number of types is limited. We
consider a few of these cases in the next Section 4.2.
To sum up the …ndings of this section, an important implication of the co-monotonicity
assumption is that under this condition there is little scope for diversi…cation in the choice
of banks’ portfolio. As argued above, defaults in …rms of di¤erent types are positively
correlated, and so they are with defaults in banks. Thus banks’ ability to reduce their
default probability by diversifying their portfolio of loans among …rms of di¤erent types is
quite limited, the use of equity is the main bu¤er to limit default. The above result shows
that the e¢ cient allocation of equity is at the top of the chain linking …rms and banks. The
result in Proposition 4 also allows to identify the non co-monotonicity of …rms’technologies
14
- and hence of banks’portfolios - as a necessary condition for bank equity to have positive
value in equilibrium.
Hence we can say that, in order for banks to issue equity in equilibrium, the bene…ts of
diversi…cation, that is, the possibility of reducing banks’default probability by diversifying
their portfolio among debt issued by …rms with di¤erent technologies, must be present and
exploited by banks. We turn then in the next section to identify environments where this
indeed happens.
4.2
Optimal bank equity
In this section we assume that consumers are risk neutral, which simpli…es the determination
of the equilibrium market prices of all the assets and the choice problem of …rms and banks.
Under this assumption, for an attainable allocation (c ; z ; ; ) to be constrained e¢ cient
we need that c and any point b in the support of
satisfy
S
X
s
(c1s + c2s ) =
s=1
S
X
s
adb + (aeb + xb aeF ) ;
s=1
In other words, the yield of the assets issued by each active bank together with the yield
of the corresponding …rms’ equity (xb aeF ) must allow to achieve the maximum expected
utility in equilibrium. We shall use this property repeatedly in what follows.
Aggregate and Idiosyncratic uncertainty To allow for the possibility that some of the
risk in …rms’returns can be reduced by diversifying banks’portfolio, we consider the case
where the set of available technologies features both an idiosyncratic and an aggregate risk
components. More speci…cally, there are in…nitely many technologies, and the return (per
unit of capital) of technology j = 1; ::: is given by Aj = j a, where j and a are random
variables with c.d.f.s F and G, respectively.11 Both F and G are C 2 and satisfy
F ( 0 ) = 0; F ( 1 ) = 1; and F 0
j
> 0 for
0
<
j
<
1,
F 0 ( 0) = 0 = F 0 ( 1)
and
G (a0 ) = 0; G (a1 ) = 1; and G0 (a) > 0 for a0 < a
a1 :
j
The random variable is i.i.d. across all available technologies j, and E j = 1. The law
of large numbers holds so that the cross sectional distribution of the idiosyncratic shock is
given by F with probability one. Thus the random variable j describes a purely idiosyncratic
shock, while a describes an aggregate shock that hits all technologies in the same way. It
is clear that the available technologies are now non co-monotonic (though productivities are
still positively correlated).
11
It is convenient here to allow the number of available technologies to be in…nite, and the support of the
productivity variables to be also in…nite.
15
In what follows we analyse the capital structure decisions by banks. To this end, as will
become clearer in what follows, we need to specify some properties of the capital structure
choices by …rms. Here we will suppose that all …rms choose the same capital structure,
whatever their technology choice.12 Since all available technologies are ex ante identical, it
is then natural that in equilibrium each …rm chooses a di¤erent technology and each bank
chooses a fully diversi…ed portfolio, with an equal amount held of the debt issued by …rms
choosing the di¤erent technologies13 .
When the face value of …rms’debt is given by `, the yield of a fully diversi…ed portfolio
of …rms’debt in aggregate state a is denoted by ` (a) and given by:
8
R 1
>
if 1 `=a
< f 0 adF = f a
R `=a
` (a) =
adF + ` (1 F (`=a)) if 0 < `=a < 1 ; :
f
0
>
:
`
if `=a
0
Note this yield only depends on the aggregate state a, since the idiosyncratic risk is fully
diversi…ed away in such a portfolio. When a is low (a 1 `) …rms default in all idiosyncratic
states and the yield is increasing in a, `0 (a) = f , while …rms never default when a is
0
su¢ ciently high (`
0 a) and the yield is invariant with respect to a, ` (a) = 0. For
intermediate values of the aggregate shock, 0 < a` < 1 , …rms only default in the low
aggregate states and the yield is strictly increasing in a.14 Also under the stated assumptions
on F , both ` (a) and `0 (a) are continuous.
We proceed next to characterize the pro…t maximizing choice of banks’debt, described
by the face value of their deposits d, in this environment. To this end, it is convenient to
notice that any pair of levels of banks’debt d and of …rms’debt ` induces two partitions of
the set of aggregate states. Let ab denote the smallest value of a a0 such that ` (a) d,
that is, a bank issuing debt d is solvent. Note that ab is well de…ned, since there is always
at least one state where a bank is solvent15 . Since ` (a) is monotonically non-decreasing,
as we showed above, the bank will also be solvent for all a > ab . Similarly, let af denote
the smallest value of a
a0 such that 0 a
`; that is, no …rm with debt ` defaults, for
all realizations of the idiosyncratic shock , whenever such value of exists. Again the same
property is true for all a > af : If there is no value of a in the interval [a0 ; a1 ] satisfying this
property, we set af = 1 by convention.
Notice that the value of bank equity is positive if and only if ab < af AND ab < a1 . It
is clear that the condition is su¢ cient for equity to have positive value, since for all a ab
a bank is solvent but, as we saw above, for a < af not all …rms are solvent and the yield of
the bank’s assets is strictly increasing in a: `0 (a) > 0. In that case we have ` > d and the
12
We provide later some conditions under which this property holds.
Such a diversi…cation does not a¤ect the expected yield of …rms’debt in the banks’portfolio, but clearly
reduces its variability since the idiosyncratic component of the return can be fully diversi…ed.
14
For a formal proof of this and the next claim see the proof of Proposition 6 in the Appendix.
15
Since default is costly, it is never optimal for a bank to choose a level of debt d such that the bank
defaults in each aggregate state a.
13
16
expected return on bank equity is
Z a1
(` (a) d) dG =
ab
Z
af
(` (a)
d) dG + (`
d) (1
G (af )) :
ab
To see that the condition is also necessary, note that ab = af implies that ` = d so that
banks are solvent whenever …rms are solvent and the yield of debt is so constant and equal
to d; while ab > af implies d > `, so that banks are always insolvent.: Also, if af
a1 we
have `=a1
and
a
=
a
and
the
level
of
bank’
s
debt
is
any
d
`
(a
)
:
0
b
1
1
We can then show:
Proposition 6 Assume in…nitely many, ex ante identical technologies are available with
aggregate and idiosyncratic productivity shocks, as described above. Then, if in equilibrium
all …rms have the same capital structure (whatever their technology) the value of bank equity
is strictly positive.
The proof shows that in a situation where bank equity is zero a marginal decrease in
banks’debt always increases the market value of banks since, by reducing the default costs
it allows to increase the payo¤ of bank debt. The result shows that bank equity may have
a comparative advantage over …rm equity when banks, by diversifying their portfolio can
reduce, but not completely eliminate the risk in their portfolio.
We consider next a simpler environment, where the bene…ts of diversi…cation are similar
to the ones of the situation examined above, but we can characterize the properties of the
capital structure chosen by banks and …rms in equilibrium. There are n technologies and a
…nite number S = n + 2 of states of nature. The probability of state s is denoted by s and
given by
8 1 "
for 1 s n
< n
for s = n + 1 :
=
(13)
s
:
"
for s = n + 2
The productivity of technology j in state s is then
8
aL if s = j n
>
>
<
aM if s 6= j n
Ajs =
;
aM if s = n + 1
>
>
:
aH if s = n + 2
(14)
where 0 < aL < aM < aH .
If a state s 2 f1; ::; ng occurs, one and only one technology (s) is hit by a negative shock,
the productivity of all other technologies is at the ’normal’ level aM : On the other hand,
if state n + 1 occurs, no technology is hit by a negative shock, while if state n + 2 realizes
all technologies are hit by a positive shock. Since all the states 1; ::; n are equiprobable, all
technologies are ex ante identical. Also, note that again there is both aggregate and purely
idiosyncratic uncertainty: we can view a change from state i 2 f1; ::; ng to another state
17
j 2 f1; ::; ng as a purely idiosyncratic shock, while a change to state n + 1 constitutes a small
aggregate shock and a change to n + 2 a big aggregate shock that is also a macroeconomic
or systemic shock.
We are able to show, for the speci…cation described in (13), (14), an important and
useful property: it is optimal for banks to choose a simple portfolio, that is, one in which
the …rms with the same technology have the same capital structure. Recall such a property
was instead assumed in Proposition 6. This will allow to characterize and illustrate the
properties of optimal capital structures in the sequel.
Proposition 7 When …rms’technologies are as in (13), (14), it is always optimal for a bank
to choose a simple portfolio, that is, one containing only the debt of …rms with no default
risk (`j = aL ), or containing only the debt of …rms with small default risk (`j = aM ), or
containing only the debt of …rms with high default risk (`j = aH ).
Given the structure of the productivity shocks a …rm may choose to set its debt level at
a su¢ ciently low level that the …rm never defaults, alternatively at an intermediate level so
that the …rm only defaults when hit by a negative shock or again at a high level so that the
…rm defaults unless it is hit by a positive shock. In these three cases we can then say that
…rms’debt is safe, risky or very risky. Unfortunately, there is little intuition to explain the
result stated above. The proof proceeds by considering the various possible cases regarding
the portfolio composition and the debt level of banks, showing that in each of them pro…ts
are higher with simple portfolios.
The speci…cation in (13), (14) also incorporates a number of interesting cases. At one
extreme, as + " ! 1, it converges to the case of co-monotonic technologies (there is only
aggregate risk and all the n technologies are in fact identical in such case): by Proposition 4,
the value of bank equity will be zero in any equilibrium with +" = 1. At the other extreme,
as ! 0 and " ! 0, we have the case of pure idiosyncratic risk: in each of the n states that
occur with positive probability, exactly one technology yields aL and the remainder yield
aM . In this case too, bank equity has zero value, but for rather di¤erent reasons than in
the co-monotonic case. The following proposition characterizes the competitive equilibrium
(and, by Propositions 1 and 2, also the constrained e¢ cient) allocations.
Proposition 8 Assume …rms’ technologies are as in (13), (14) and = " = 0. Then in
equilibrium we have one of the two folowing cases:
n
(i) if n 1f aL > aM ; each bank lends exclusively to …rms with safe debt (`j = aL ) and16
sets its level of deposits at d = aL , or
n
(ii) if n 1f aL < aM ; each bank lends exclusively to …rms with risky debt (`j = aM ) and
chooses a fully diversi…ed portfolio (holding 1=n units of the debt of …rms choosing technology
j, for each j 2 f1; ::; ng) and sets
d=
n
1
n
aM +
16
1
n
f aL :
Since the yield of the debt of …rms with safe debt is always the same, whichever their technology, the
composition of the bank’s portfolio (with regard to the debt of these …rms) is clearly irrelevant).
18
In either case bank equity has zero value.
When = " = 0, there are then two kinds of possible equilibria, that di¤er with regard to
the debt level of …rms, and of whether …rms may default or not. In each of them the return
on the portfolio of banks is however risk free. A risk free portfolio should always be 100%
funded with deposits because equity is costly ( < 1) and there is no need to have an equity
bu¤er when the portfolio is risk free.
Between the two extremes = " = 0 and + " = 1 discussed above, that is when there
is both aggregate and idiosyncratic risk, we …nd a role for both …rm equity and bank equity,
as we show in the following result:
Proposition 9 Assume …rms’technologies are as in (13), (14). Then if
n
n
f
1
aL < a M :
there exist numbers
> 0 and " > 0 such that, for any 0 < <
and 0 < " < " ,
in equilibrium each bank chooses a fully diversi…ed portfolio of debt of risky …rms (xb =
n
0; n1 ; 0 j=1 ) and the face value of deposits to satisfy
d=
n
1
n
aM +
1
n
f aL ;
so that bank equity has positive value.
The above result provides su¢ cient conditions for bank equity to have positive value in
equilibrium. The …rst condition is that the size of the idiosyncratic, negative shock aM aL
is su¢ ciently large, so that, when = " = 0, in equilibrium …rms choose risky debt and
banks a fully diversi…ed portfolio (as in case (ii) of Proposition 8). The second condition is
that and " are both positive but not too large, that is, we are su¢ ciently close to the case
of purely idiosyncratic uncertanty so that …rms’and banks’debt level and banks’portfolio
are the same as when = " = 0: Note that, with debt levels as speci…ed above, banks’
shareholders receive a positive return in states s = n + 1; n + 2, whereas …rms’shareholders
receive a positive return in s = n + 2. Hence condition " > 0 ensures that both bank equity
and …rm equity have positive value, while if 0 < <
and " = 0 bank equity has positive
value and …rm equity has zero value.
To gain some understanding of why banks choose in equilibrium a positive level of equity
in the situation described in the above proposition, note that …rms choose a risky debt level
(`j = aM ). Hence a …rm choosing technology j defaults when state s = j occurs, that is,
when the …rm is hit by a negative shock. If such a …rm were to have an equity bu¤er to
be protected against default in all states, it would have to reduce its debt level `j from aM
to aL . Given the relatively small likelihood of state s = j; this would be rather expensive
relative to its bene…ts. On the other hand, the introduction of such an equity bu¤er in
19
a diversi…ed bank, which lent the same amount to …rms using each of the technologies
j = 1; :::; n would be both more bene…cial and less costly. This is because the bank feels the
e¤ect of the idiosyncratic shocks hitting the …rms with a probability that is n times higher.
In other words, a shock that is unlikely to a¤ect a single …rm may be quite likely to a¤ect
a diversi…ed bank. Also, the e¤ect of the shock on the yield of the banks’assets would be
smaller. Hence the reduction in the bank’s debt level needed to ensure that the bank never
defaults when some …rm is hit by an idiosyncratic shock is also much smaller. We begin so
to see why banks may choose in equilibrium to hold a capital bu¤er against this shock even
though …rms do not …nd it worth holding a capital bu¤er against it.
In the table below we summarize the properties of the equilibrium regarding the value of
bank and …rm equity in the di¤erent cases discussed above17 :
Parameters
0
< ;0<"<"
0< < ;"=0
"+ =1
"+ =0
Value of Bank equity Value of Firm equity
POSITIVE
POSITIVE
POSITIVE
ZERO
ZERO
UNCERTAIN
ZERO
POSITIVE
Note that it is not clear whether …rm equity has positive value in the monotonic case "+ = 1.
There are two possible types of equilibria. In one, `j = aM and the value of …rm equity is
positive. In the other, `j = aH and the value of …rm equity is zero.
The speci…cation in (13), (14) also allows us to better see the connection between portfolio diversi…cation and the presence of positive bank equity. After Corollary 5, at the end
of Section ??, we noted that bank equity cannot have a positive value without portfolio
diversi…cation by banks. In other words, bank equity has no role as a bu¤er unless the bank
faces heterogeneous risks, associated to …rms with di¤erent technologies. At the same time,
one can have diversi…cation without a positive value of bank equity, as we saw happens in
the limiting case = " = 0. More precisely, Proposition 8 identi…es two situations where in
equilibrium bank equity has a zero value. In that case no equity bu¤er is needed since the
yield of the portfolio of …rms’debt held by a bank is riskless. But there are also situations
where diversi…cation could be bene…cial but still, in equilibrium banks choose not to diversify
and have both a positive probability of default and a zero value of equity:
Proposition 10 Assume …rms’ technologies are as in (13), (14) and " = 0. Then there
exists
> 0 such that, for all
< < 1, in equilibrium each bank lends to …rms using
a single technology j and having a debt level `j = aM . The face value of banks’ debt is also
d = aM and both banks and …rms default when the return to technology j is aL .
The above result shows that, even if diversi…cation is possible, it is not always optimal. The
bank’s objective is non-convex, because of the presence of default costs, and there are also
negative consequences, and not only bene…ts attached to diversi…cation. The bottom line is
17
We maintain here the condition
n
f
n 1
aL < aM :
20
that diversi…cation of banks’investments - and the non co-monotonicity of banks’portfolios
- is a necessary but not su¢ cient condition for bank equity to have positive value.18
The single-technology case We conclude by examining the special case where a single
technology is available, n = 1: Let As > 0 denote the output per unit of capital in state
s = 1; :::; S of the only technology available. In this case we can always re-order the states so
that As < As+1 for all s and the co-monotonicity assumption is trivially sati…ed. Hence, from
Proposition 4 we know that the bank’s capital structure will have zero equity in equilibrium.
In what follows we characterize the properties of equilibria, and in particular of the capital
structure and the portfolio chosen by banks when n = 1, hence also in a situation where
co-monotonicity holds.
Since equity is expensive, …rms will not hold more equity than necessary to avoid default.
This means that the debt level of each …rm will be equal to As , for some state s. Thus, the
set of …rm types is simply given by the set of possibleP
debt levels: F = fA1 ; :::; AS g. A
portfolio for a bank is a vector x = (x1 ; :::; xS ) such that Ss=1 xs = 1. From the argument of
the proof of Proposition 4, we know that a bank of type b that is active in equilibrium sets
the face value of its deposits at d = `b , that is at a level equal to the face value of the debt
in the bank’s portfolio. Since the bank is paid the face value of all the loans in its portfolio,
the types of …rms with a positive weight in the bank’s portfolio must all be solvent in the
states s sb where bank b is solvent. This implies that no
in the bank’s portfolio has
P…rm
sb
borrowed ` > Asb , so xs = 0 for s > sb . We have so d = s=1 xs As :
By the risk neutrality of consumers we know that the portfolio xb = (x1 ; :::; xsb ; 0; :::; 0)
of each active bank together with the induced value of the yield of the assets issued by the
bank must allow to achieve the consumers’ maximum expected utility. That is, xb must
solve the problem
PS
d
+ (aeb +P
x aeF ) =P P
max
x
P
P s=1 s ab P
P
Ai )
s
i s s xi (As
s<sb s b
i<s xi As +
s sb s
i x i Ai +
i s x i Ai + f
(15)
PS
subject to the constraint s=1 xs = 1.
In the case of a single technology, a portfolio x is simple if xm = 1 for some state s = m.
In the case of a simple portfolio, we also have sb = m and d = Am . The value of the
consumers’expected utility in this case is
b f
m
X1
s=1
s As
+
S
X
s Am
+
s=m
S
X
s
(As
Am )
Um :
s=m
18
On a related line, ACM provide an example with two available technologies with i.i.d. returns where
they show that in equilibrium banks specialize in lending to …rms using the same technology. Thus, even
though technologies are non co-monotonic and there is so more scope to diversify across technologies, banks
still …nd it optimal to avoid such diversi…cation. Since each bank specializes in a single technology and
chooses so a co-monotonic portfolio, it is again optimal for banks to use 100% debt …nancing and so bank
equity has zero value.
21
Note that the expression of the expected utility in (15) for a general portfolio x = (x1 ; :::; xsb ; 0; :::; 0)
can be rewritten as follows:
(
!
)
sb
X
X
X
X
X
xi
+
Ai )
b
s f As +
s Ai
s Ai +
s (As
i=1
sb >s>i
which is strictly less than
(
sb
X
X
xi
b
i=1
sb >s>i
s i
s f As
!
+
X
s sb
s Ai
s i
+
X
s i
s i
s
(As
)
Ai )
=
sb
X
x i Ui
i=1
if the portfolio is not simple, that is, if xs > 0 for some s < sb . But this means that, for some
s, the expected utility Us of a simple portfolio xs = 1 is greater than the expected utility of
a general portfolio x, contradicting the assumed constrained e¢ ciency of x. Thus, we have
proved the following:
Proposition 11 If there is a single technology, n = 1, then in any competitive equilibrium
banks choose simple portfolios.
This is turn implies that in equilibrium banks choose a portfolio consisting only of debt
of …rms with a face value of debt ` = As , for some s, and set then also the face value of
deposits at d = As :
5
Conclusion
We have presented a classical model of competitive equilibrium with banks acting as intermediaries between productive …rms and conssumers. Both banks and …rms can raise funds
by issuing debt and equity. In the environment considered the Modigliani Miller Theorem
does not hold, that is, the capital structure of …rms and banks is determinate because banks’
debt (deposits) - and hence also, indirectly, …rms’ debt - enjoy a liquidity premium but
also generates the risk of bankruptcy and this is costly. In this context, we have established analogues of the fundamental theorems of welfare economics, showing that equilibria
are constrained e¢ cient and that constrained-e¢ cient allocations are decentralizable. Thus,
equilibrium capital structures are privately and socially optimal— they maximize the market
value of the …rm or bank and they are consistent with constrained e¢ ciency. The importance
of the general equilibrium theory is that it shows how risk is allocated optimally between
the corporate and …nancial sectors and how the capital structure of banks relate to that of
…rms.
This may be a …rst step towards a theory of bank capital regulation, but it is not by any
means the last word. We alluded brie‡y to a number of missing factors in the introduction.
Asymmetric information is one of them. Banks are opaque and depositors and equity holders
alike may be uncertain about the risk of the bank’s portfolio. In the economics literature,
22
risk shifting and asset substitution are one of the justi…cations often cited for capital requirements. Moral hazard, associated with the possibility of bank bailouts in the event of default,
is another one as it may also distort capital structure decisions. But capital requirements
may not be su¢ cient to control excessive risk taking, because the bank’s management may
not operate in the interests of shareholders. Even if the interests of the top management are
aligned with shareholders’interests, it is not clear that the top management is aware of and
able to control risk taking by highly incentivized managers at lower levels.
We have also ignored the fact that policymakers have focused on the role of bank capital
as a bu¤er, rather than as an incentive mechanism. Bank capital provides a cushion that
shields depositors and bond holders from losses but, more importantly from the point of
view of politicians, it also makes politically unpopular bailouts unnecessary.
So, there is much to be done in order to develop a satisfactory microfoundation for
capital regulation. But the recognition that e¢ cient capital structures in the banking and
corporate sectors are interrelated and determined by general equilibrium forces is a …rst-order
requirement for any sensible theory.
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26
Appendix
Proof of Proposition 1
The proof is by contradiction. Suppose, contrary to what we want to prove, that (c ; z ; ; )
is constrained ine¢ cient. Then there exists a feasible allocation (c; z; ; ) such that U (c) >
U (c ). Condition (iii) of the de…nition of equilibrium implies that c lies outside the representative consumer’s budget set. That is, q z > 1, where
!
X
X
X
c=z a=
zbd adb ;
zbe aeb +
zfe aef :
b2B
b2B
f 2F
Now the equilibrium optimality condition for banks implies that
qbd + qbe
for any b 2 B. Then
X
b2B
qbd zbd +
X
qdF
qbe zbd =
b2B
xb ;
X
b
qbd + qbe
b2B
X
d
b qF
xb
b2B
=
X
d
f qf
f 2F
P
because attainability requires that zbd = zbe = b , for any b 2 B, and b2B
any f 2 F .
Similarly, the equilibrium optimality condition for …rms implies that
qfd + qfe
for any f 2 F . But this implies that
X
q z =
qbd
b+
b2B
X
qfd
X
+
f 2F
X
=
f,
for
1;
qbe
b+
b2B
f
b xbf
X
X
qfe
f
f 2F
qfe
f
f 2F
f
= 1:
f 2F
This contradicts our initial hypothesis and proves that the equilibrium allocation must be
e¢ cient.
Proof of Proposition 2
27
Let C denote the set of attainable consumption vectors. Then the supporting hyperplane
theorem tells us that there exists a non-negative price vector p such that
c = sup fp
p
c : c 2 Cg :
Without loss of generality, we normalize prices so that p c = 1. Now de…ne the securities’
prices q as follows:
qbd
qbe
qfe
= p
= p
= p
adb ; 0 ;
(0; aeb ) ;
0;aef ; and
qfd
= 1
qfe :
For any (c; ; ) satisfying (11a) and (12a),
1 = p
p
c
c
X
= p
=
X
adb ; aeh +
b
b2B
b
=
0; aef
f
f 2F
qbd + qbe
+
b2B
X
X
X
!
d
f qf
f 2F
b
qbd + qbe
+
X
qdF :
b xb
b2B
b2B
It follows from this inequality that, for any bank b, with portfolio xb ,
qbd + qbe + xb qdF
=) qbd + qbe + xb
=)
qbd
+
qbe
1;
qdF
1
qeF
xb
1;
0:
In other words, no bank can earn positive pro…ts. But active …rms must earn zero pro…ts in
equilibrium, because
1 = p
c
X
= p
=
X
b
adb ; aeh +
+
qbe
b2B
b
qbd
=
+
X
f 2F
b
f
0; aef
f 2F
b2B
X
X
qbd + qbe
+1
e
f qf
X
f 2F
b2B
28
d
f qf
!
so
X
X
qbd + qbe
b
= 0:
f 2F
b2B
Then b > 0 implies qbd + qbe
By de…nition, we have
d
f qf
xb qdF = 0.
qfd + qfe = 1;
so all …rms are value maximizing.
Finally, the optimality of the representative consumer’s choice follows from the fact that
U (c) > U (c ) implies that p c > p c . Any portfolio z that generates a consumption
bundle c must therefore have a value greater than 1 because we have de…ned security prices
so that
q z = p c:
But this implies that z does not belong to the budget set fz : q z
1g.
Proof of Proposition 4
In state s, the holders of equity in bank b receive
8
< X
max
f (s) xbf min f` ; Af s g
:
f =(` ;j )
9
=
d; 0
;
units of consumption and the holders of deposits receive
8
9
<
=
X
(s)
min
d;
(s)
x
min
f`
;
A
g
bf
f s
b
f
:
;
f =(` ;j )
units of consumption, where
f
and
b
Let
(s) =
1
f
(s) =
if `
Af
if ` > Af
1
f
P
if d
Pf
if d > f
`b =
f
f
X
s
s
(s) xbf min f` ; Af s g
:
(s) xbf min f` ; Af s g
xbf `f
f =(` ;j )
denote the face value of the debt owed to bank b and let
X
`b (s) =
f (s) xbf min f` ; Af s g
f
29
denote the actual amount repaid to bank b in state s.
Step 1: The value of bank equity is positive if and only if `b > d .
The dividends paid by the bank are non-negative in each state because of limited liability.
Then it is clear that the value of equity is strictly positive if and only if `b (s) > d in at least
one state s. We can assume, without essential loss of generality, that `
Af S and d
`b .
We claim that the value of equity is positive if and only if `b > d . To see this, note …rst
that, if `b
d , the payment to equity holders (as de…ned above) is zero in each state and,
second, if `b > d , the payment must be positive in state S at least, because in that state
each …rm f repays the face value of its debt, the bank receives `b and the equity holders
receive `b (S) d = `b
d > 0.
Step 2: For each f , there exists a state sf such that …rm f is solvent if and only if
s sf . Similarly, for each bank b there exists a state sb such that the bank is solvent if
and only if s sb .
A …rm f is insolvent in state s if and only if `f > Af s . Let sf be the smallest state such
that f is solvent. There must be such a state because the …rm is solvent at least in state
S. The fact that Af s is increasing in s implies that f is solvent if and only if s sf .
Similarly, we can show that there is a state sb such that bank b is solvent if and only if
s sb . Let sb denote the smallest state in which the bank is solvent. There must be such a
state because the bank is solvent at least in state S. It is clear that `b (s) is non-decreasing
in s because Af s is increasing in s for every i and f (s) is non-decreasing in s, for every
f . Let
db (s) = b (s) min fd ; `b (s)g
denote the payment to deposit holders in state s. Then it is clear that db (s) is non-decreasing
in s because `b (s) and b (s) are non-decreasing. From this observation it follows that the
bank is solvent if and only if s sb .
Step 3: The face value of deposits satis…es d = `b (sb ).
To prove this claim, we have to consider two cases. First, suppose that sb = 1 and d <
`b (1).Then increasing d to d + " < `b (1), say, will increase the payout to depositors by "
in every state and reduce the payout to equity holders by the same amount. This increases
e¤ective consumption by (1
) " in each state, contradicting the constrained e¢ ciency of
th equilibrium.
The second possibility is that sb > 1 and d < `b (sb ). Suppose the bank increases
deposits by " > 0 to d + " < `b (sb ). This will not have any e¤ect in states s < sb because
the bank is in default, depositors are receiving `b (s) < d and equity holders are receiving
nothing. In states s sb , on the other hand, the net e¤ect will be an increase in e¤ective
consumption of (1
) ". This again contradicts constrained e¢ ciency.
Step 4: Bank b’s equity has no value: d = `b .
Suppose to the contrary that d = `b (sb ) < `b . Then there must exist at least one …rm that
is bankrupt in state sb . Otherwise, the …rms’repayment would be `b (sb ) = `b . Suppose
that …rm f is bankrupt in state sb and consider the e¤ect of reducing the borrowing of f
30
by an amount " > 0 such that `f " = Af sb . This change has no impact on the viability of
the bank in state sb because …rm f is now paying Af sb in state sb , instead of f Af sb .
So the bank is solvent in states s sb as before. Also, the change does have an e¤ect on
the solvency of the …rm, as it is now solvent in state sb and hence in all states s sb .
Note that none of the payo¤s to the debt and equity of the bank or the …rm change
in states s < sb . Moreover, there are no changes to the payo¤ to bank’s debt (deposits)
changes in states s sb because the bank is solvent in all these states and hence pays d
to depositors. The changes in payo¤ a¤ect only the returns to equity in the states s sb .
Consider …rst the equity of the bank. In states s
sb , the payo¤ of bank’s equity will
increase because of the increase in the …rm’s repayment, that is the change in the payo¤ of
bank’s equity is equal to
b ;s
Af
=
sb
f
Af
sb
Af
`
if sb
if sf
s
s < sf
s
;
because the …rm will pay Af sb in each state and was previously paying
sb
s < sf and ` in states sf
s.
The return to the equity of …rm f is increased by Af s Af sb for all s
in the …rm’s equity is so
f ;s
=
Af
`
Af s b
Af s b
s
if sb
if sf
s < sf
s
f
Af
s
in states
sb ; the change
:
The transfer between bank equity holders and …rm equity holders has no e¤ect on total
consumption. The net increase in consumption is the sum of b ;s and f ;s , which is
b ;s
+
f ;s
=
=
Af
sb
Af
Af
f
sb
s
f
Af s + Af s Af
` +`
Af s b
Af
s
0
if sb
if sf
if sb
if sf
sb
s < sf
s
s < sf
s
:
Thus, the net gain for equity holders is the saving in default costs (1
f ) Af s in the
states sb
s < sf . The possibility of such a gain contradicts the constrained e¢ ciency of
equilibrium.
This completes the proof that bank equity has no value.
Proof of Proposition 6
We show …rst that, for intermediate values of …rms’ debt 0 < a` < 1 , ` (a) is strictly
increasing. We have in fact:
Z `=a
`
`
`
`
0
0
` (a) = f
dF + f `F 0
`F
a a2
a a2
0
Z `=a
2
`
`
0
= f
dF + (1
)
F
> 0:
f
a
a
0
31
Moreover,
0
lim ` (a) =
a!`=
f
1
lim `0 (a) = (1
a!`=
Z
1
dF + (1
f) F
0
( 1 ) ( 1 )2 =
f;
0
f) F
0
( 0 ) ( 0 )2 = 0
0
R
where the …nal equalities follow from our assumption that 01 dF = 1 and that F 0 ( 0 ) =
F 0 ( 1 ) = 0. This shows that `0 (a) is continuous and well de…ned at `=a = 0 and `=a = 1 .
Suppose that ab = af and af a1 : Note that, by the de…nition of af we have 0 af = ` =
`(af ), while from the continuity of ` (a) we get ` (ab ) = d. We show in what follows that a
small decrease in d to d0 (holding ` constant), corresponding to a decrease in ab to a0 , always
increases bank pro…ts. At a0 we have
Z `=a0
`
0
0
d = `(a ) = f
a0 dF + ` 1 F
;
a0
0
and, by the property established above,
`0 (a0 ) = 0:
lim
a0 !ab =`=
0
Consider then the market value of a bank with debt level d0 = ` (a0 ):
Z a0
Z af
0
0
(` (a) ` (a0 )) dG + (` ` (a0 )) (1
` (a) dG + ` (a ) (1 G (a )) +
b
G (af )) :
a0
a0
Di¤erentiating with respect to a0 and evaluating then the derivative at a0 = af yields:
Z af
0
0
0
0
0
0
0
0
0
0
0
0
0
G (a )) ` (a ) G (a )
(` (a ) ` (a )) G (a )
`0 (a0 ) dG
b ` (a ) G (a ) + ` (a ) (1
= b ` (af ) G0 (af ) + `0 (af ) (1 G (af )) ` (af ) G0 (af )
`0 (af ) (1 G(af )
= ( b ` (af )
` (1
) ` (af )) G0 (af ) + (1
) `0 (af ) (1 G (af ))
= ( b 1) `G0 (af ) < 0;
a0
(`
` (af )) G0 (af )
because `0 (af ) = 0 and d = ` = ` (af ). Thus, a small decrease in d and ab holding ` constant
must increase the bank’s market value, contradicting the equilibrium conditions.
It remains then to consider the case af a1 so that `=a1
0 and ab = a1 . In that case,
the bank’s debt level is any d `(a1 ): The bank’s market value is
Z a1
` (a) dG
b
a0
If we consider again a marginal decrease of ab to a0 (equivalently, of d below `(a1 )); its market
value changes as follows:
Z a0
Z a1
0
0
` (a) dG + ` (a ) (1 G (a )) +
(` (a) ` (a0 )) dG:
b
a0
a0
32
`0 (a0 )
Di¤erentiating again this expression with respect to a0 and evaluating then the derivative at
a0 = a1 yields:
Z a1
0
0
0
0
0
0
0
0
0
0
0
0
0
`0 (a0 ) dG
G (a )) ` (a ) G (a )
(` (a ) ` (a )) G (a )
b ` (a ) G (a ) + ` (a ) (1
0
0
= b ` (a1 ) G (a1 ) + ` (a1 ) (1 G (a1 ))
= ( b ` (a1 ) ` (a1 )) G0 (a1 ) + `0 (a1 ) (1
= ( b 1) ` (a1 ) G0 (a1 ) < 0;
a0
0
` (a1 ) G (a1 )
G (a1 ))
Hence in this case too banks’market value can be increased by lowering d below `(a1 ), that
is, by having equity.
Proof of Proposition 7.
Banks holding risk free debt In the case of a bank holding only P
the debt of …rms
with `j = aL , the portfolio x is indeterminate subject to the constraint j xLj = 1. The
optimal capital structure for the bank is to issue the maximum amount of deposits, d = aL .
The expected utility generated by the bank and the …rms whose debt it holds will be
d + " (aH
d) +
(aM
d) + (1
= aL + " (aH
aL ) +
(aM
= aL + " (aH
aL ) +
+ (1
n
")
aL ) + (1
")
")
n
1
n
1
aL d
n
n
n 1
n 1
aM
aL
n
n
(aM
1
aM +
aL ) :
Banks holding safe and risky debt Now suppose that the bank lends to a mixture
of safe and risky …rms. We can focus here on the loands made by the bank to …rms with
technology j but possibly di¤erent capital structure. Suppose that units of capital are
invested in safe …rms with `j = aL and 1
units of capital are invested in risky or very
risky …rms, that is, …rms that have a capital structure `j 2 faH ; aM g. There is no need
to distinguish safe …rms according to the technology they use: from the point of view of
banks and shareholders, who hold their debt and equity, they are identical. Let xH
j and
xLj denote the fraction of 1 P invested in …rms with technology j and `j equal to aH and
M
aM , respectively. Note that nj=1 aH
j + aj = 1 and that the amounts invested in …rms with
technology j and `j equal to aH and aM are aH
) and aM
), respectively.
j (1
j (1
Suppose the bank chooses a level of deposits d. Since the safe banks pay aL for sure, the
bank will fail if and only if payment from the risky banks is less than d
aL . Now suppose
that we split the bank into two banks, one of which funds safe …rms and the other funds
risky …rms. The safe bank invests one unit in safe …rms and issues deposits dS = aL and
M n
the risky bank invests one unit in a portfolio xH
of risky …rms and issues deposits
j ; xj
j=1
R
d . The expected utility from the safe bank is denoted by U S and the expected utility from
the risky bank is denoted by U R . What is the di¤erence between these two banks and the
33
combined bank we started with? Note that the risky bank will default if and only if the
uni…ed bank defaults with a positive probability. If there is no probability of default, there is
no di¤erence in the expected utility generated by the two structures. On the other hand, if
there is a positive probability of default, the separated banks will generate a higher expected
utility, because the safe bank does not default whereas the combined bank does default in
some states. In fact, the gain in expected utility by separating the banks is precisely the
S
probability of default muliplied by the default cost (1
b ) db .
Thus, either there is no gain from mixing safe and risky debt in the banks portfolio or, if
the mixture of safe and risky debt causes the bank to default with positive probabilty, there
is a loss.
Safe banks holding risky debt Now suppose that a bank chooses a portfolio xb =
n
M
M
xH
where xH
j ; xj ; 0
j is the number of …rms of type j with `j = aH , xj is the number
j=1
of …rms of type j with `j = aM , and we assume that no …rms with `j = aL are included. The
portfolio x has no impact in states s = n + 1; n + 2 because all technologies have identical
payo¤s in these states. Now consider the states s = 1; :::; n and let j denote the repayment
of all …rms when type j has productivity aL . Then
X
M
M
xH
aM + x H
f + xi
f aL :
j =
i
j + xj
i6=j
Without essential loss of generality, we can order the types of …rms so that j
j+1
for j = 1; :::; n 1. The bank wants to maximize the face value of deposits subject to
the no-default constraint d
1 . To do that, it must choose a portfolio xb such that
1
H
M
xj ; xj = 0; n for all j = 1; :::; n, that is, a simple portfolio. Having done so, the value
of deposits it can safely issue is
d=
n
1
n
aM +
1
n
f aL :
The expected utility generated by the bank and the …rms whose debt it holds will be
d + (1
= d+
(aM
")
n
1
1
n
d) :
aM +
n
d) + " (aH
f aL
d +
(aM
d) + " (aH
d)
Risky banks holding risky debt We split the analysis in two parts, considering …rst
that case where d aM and, second, the case where d > aM .
i) Suppose that there is a positive probability of the bank defaulting, but that d aM .
This means that default only occurs in states s = 1; :::; n. As before, the portfolio xb is
irrelevant in states s = n + 1; n + 2 so we restrict attention to the states s = 1; :::; n. With
our usual convention that j
j+1 , there exists a technology k such that, d >
j for
34
j = 1; :::; k and d
j for j = k + 1; :::; n. (The bank will never choose to default with
probability one). The expected utility of the bank’s depositors and shareholders will be
1
n
b
k
X
j
+
n
j=1
1
d+
n
n
n
X
k
1
n
d
j
j=k+1
1
n
=
because
b
k
X
j
+
j
+
n
j=1
b
k
X
j=1
n
1 X
d+
n
n j=k+1
k
n
1 X
n j=k+1
j;
< 1. Now
1
n
b
k
X
j=1
j
1
=
n
1
=
n
+
b
k
X
X
j=1
f
M
+ xM
aM + x H
i
j + xj
xH
j
f
+ xM
aM +
j
1
+
n
b
k
X
xH
j
f
+ xM
aM
j
j=2
b
k
X
j=2
k
X
M
xH
j + xj
f aL
j=1
j=k+1
1
n
f aL
i6=j
n
X
b
xH
i
!
X
xH
i
i6=j
35
f
1
n
b
k
X
M
xH
j + xj
f aL
j=2
M
+ xM
aM + x H
i
j + xj
!
f aL
!
j
d
1
n
+
n
X
b
xH
j
f
1
n
xH
j
b
f
+ xM
aM
j
j=2
< 1 and xH
j
1
=
n
f
b
+
k
X
j=2
M
xH
j + xj
f
k
f aL
f
+ xM
j
f aL
+ xM
aM +
j
xH
f
i
+
xM
i
j=k+1
n
X
xH
j
f
k
X
M
xH
j + xj
f aL
aM +
xH
j f
+
xM
j
aM
+ xM
aM +
j
k
X
M
xH
j + xj
f
;
!
!
f aL
j=1
xH
j
!
f aL ,
i6=j
1
n
M
xH
j + xj
j=1
X
n
X
b
f aL
!
j=2
j=k+1
1
+
n
1
=
n
xH
j
1
n
k
X
+ xM
aM + x H
i
j
f
+ xM
aM
j
n
X
M
xH
j + xj
j=1
j=k+1
k
X
k
1X X H
+
x
n j=2 i6=j i
because
+ xM
aM +
j
k
X
!
+ xM
aM :
j
j=1
Substituting this upper bound into the expression for expected utility, we obtain the inequality
1
n
b
k
X
j=1
1
n
+
b
j
n
1 X
+
n j=k+1
n
X
xH
j
f
j
aM +
+ xM
j
X
1
xH
n j=k+1 i6=j i
M
xH
j + xj
f aL
j=1
j=k+1
n
X
k
X
f
M
+ xM
aM + x H
j + xj
i
36
f aL
!
!
+
1X
n
k
n
j=1
xH
j
f
+ xM
aM
j
1
n
n
X
b
xH
j f
aM +
n
1 X
xH + xM
b)
j
n j=k+1 j
n
X
X
1
xH
+
n j=k+1 i6=j i
1
=
n
n
X
b
+
f
f aL
xM
i
M
xH
j + xj
1
n
b
k
X
j=1
1
n
+
< 1 and
b
b
f aL +
k
X
1
n
n
1X
f aL
+
xM
j
xH
j
Pn
j=k+1
n
1 X
+
n j=k+1
n
X
n
X
+
+ (1
f aL
<
f aL
!
+
Pn
j=k+1
+
1
n
n
xH
j
1X
n
k
!
f aL
+
k
X
M
xH
j + xj
f aL
j=1
k
+ xM
aM
j
!
+ xM
aM
j
f aL
f
f
xH
j
f
+ xM
aM
j
j=1
+ xM
aM . So
j
j
xM
j
aM
j=1
b f aL
f
M
xH
j + xj
M
xH
j + xj
xH
j f
n
1 X
xH
b)
n j=k+1 j
+ xM
aM
j
f
j=1
M
xH
j + xj
+ xM
aM + x H
i
j
f
!
n
k
xH
j
j=k+1
k
1
n
j
+
xM
j
j=1
X
1
xH
n j=k+1 i6=j i
because
+
aM +
j=k+1
n
X
xH
j
j=1
j=k+1
(1
+
+
xM
j
k
X
aM +
n
n
1 X X H
x
+
n j=k+1 i6=j i
k
n
Pn
aM =
1
n
b f aL
+
f
!
+
n
xM
i
1
n
aM +
xH
j f
+
xM
j
aM
!
aM ;
P
M
H
M
= 1. But the last expression in this series
because nj=1 xH
j f + xj
j=1 xj + xj
of inequalities is the representative consumer’s expected utility when the bank lends only to
…rms that use a single technology j and choose the capital structure ` = aM and the level of
deposits is d = aM . It is easy to check that for any other portfolio and face value of deposits
one of the inequalities is strict, so this is the unique policy that maximizes expected utility
in the states s = 1; :::; n when d aM and the probability of default is positive.
Now let us check that this policy is optimal in the states s = n + 1; n + 2. The expected
utility in states s = n + 1; n + 2 is
+"
For any other portfolio
"
(aM + (aH
+"
aM +
M
xH
j ; xj ; 0
n
j=1
aM )) :
, the bank will be in default in state s = n + 1 if
37
Pn
j=1
xH
j > 0 and the expected utility in states s = n + 1; n + 2 is
+"
b
n
X
xH
j
f
aM
+ xM
j
j=1
!
+
"
(aM + (aH
+"
aM )) <
+"
If
Pn
j=1
+"
aM +
"
(aM + (aH
+"
aM )) :
xH
j = 0, the bank is not in default in either state and the payo¤ is
n
X
j=1
xM
j aM
!
"
(aM + (aH
+"
+
aM )) =
+"
aM +
"
(aM + (aH
+"
Thus, the unique optimal policy is to set d = aM and xM
j = 1 for some j as long as d
aM ))
aM .
ii) Now consider the case in which d > aM . In that case, the bank always defaults
states s = 1; :::; n + 1. In states s = 1; :::; n, the expected utility will be
!
n
X
1X
M
M
xH
aM + x H
b
f + xi
f aL
i
j + xj
n j=1
i6=j
and in states s = n + 1; n + 2 it will be
+"
b
n
X
xH
j
f
+ xM
aM +
j
j=1
"
(d + (aH
+"
d)) :
The choice of P
d will be the maximum that allows the bank
solventP
in state s = n+2,
Pn to remain
n
H
M
H
H
M
that is, d = j=1 xj aH + xj aM . Letting x = j=1 xj and x = nj=1 xM
j , we can
rewrite the expected utility as
+"
b
xH
f
+ x M aM +
"
(d + (aH
+"
db ))
in states s = n + 1; n + 2 and
n
b
1
n
xH
f
+ x M aM +
1
n
f aM
in states s = 1; :::; n. This expression is linear in xH ; xM , so at least one of the extreme
points xH ; xM = (0; 1) or xH ; xM = (1; 0) must be an optimum. Since we assume that
d > aM , this case is only observed if xH = 1.
38
Proof of Proposition 8.
As shown in the text, if the representative bank lends to safe …rms (` = aL ) the expected
utility is
n 1
aL +
(aM aL ) :
n
If the representative bank issues safe debt (deposits) and lends to risky …rms (` = aM ), on
the other hand, the expected utility is
n
1
n 11
(aM
f aL +
f aL ) :
n
n
n n
It is strictly optimal to issue safe debt and lend to risky …rms if
aL +
Multiplying by
n
1
n
n
n 1
(aM
aM +
aL ) <
n
1
n
aM +
1
n
f aL
+
n
11
(aM
n n
f aL ) :
yields
n
n
1
1
aL + (aM
n
aL ) < a M +
n
1
f aL
+
1
(aM
n
f aL )
and collecting like terms gives us
n
n
1
1
n
1
f
aL
1
n
1
aL <
f
n
1
1
aM
n
This can be rewritten as
n
n
1
1
1
n
1
1
n
aL <
f
1
n
1
n
aM
which is equivalent to
n
(n 1)
n 1
or
n
n
aL <
f
f
n
(n 1)
n
aM
aL < a M :
1
Proof of Proposition 9
The argument in the proof of Proposition 7 left us with the following candidates for an
optimal bank policy.
1. The bank invests in …rms with ` = aL and d = aL . The …rms’ types are irrelevant
because …rm debt is risk free. The expected utility in equilibrium is
aL +
(1
")
n
1
n
aM +
39
1
aL + aM + "aH
n
aL :
1
2a The bank invests in …rms with ` = aM . The portfolio is de…ned by xM
j = n for all j and
n 1
1
the face value of deposits is d = n aM + n f aL . The expected utility in equilibrium
is
n
1
n
aM +
1
n
f aL
+ ( + ")
+"
"
aH
+"
aM +
n
1
n
aM
1
n
f aL
2b The bank invests in …rms with ` = aM . The portfolio is de…ned by xM
j = 1 for some j
and the face value of deposits is d = aM . The expected utility in equilibrium is
(1
")
n
b
1
n
1
n
aM +
f aL +
+ ( + ") aM + " (aH
aM ) :
3 The bank in invests in …rms with ` = aH . The portfolio is de…ned by xH = 1 (the
distribution over j is irrelevant) and the face value of deposits is d = aH . The expected
utility in equilibrium is
"aH +
Suppose that
converge to
b
f aM
+ (1
n
")
1
f aM
n
+
1
n
f aL
:
and " converge to zero. The expected utilities in the di¤erent cases
n
aL +
1
n
n
1
n
n
b
1
n
and
n
1
n
1
+
n
1
n
aL ;
aM +
f aL ;
aM
f aL +
n
b f
aM
1
n
aM +
1
aL ;
n
(Case 1)
(Case 2a)
;
(Case 2b)
(Case 3)
respectively. Finally, Proposition 9 guarantees that Case 2a dominates Case 1. Thus, Case
2a dominates all other cases for values of and " su¢ ciently close to zero.
40