Put Your Money where Your Mouth Is: An Equilibrium Model
Relating Bank Scale to Bank Quality
Tianxi Wang
April, 2012
Abstract
In a framework where bank …nancing plays a certi…cation role, the paper shows
that if each and every bank enlarges credit supply, then each and every’s asset
quality falls. It identi…es scale produces two forces shaping quality: the increasing
return to fund scale in obtaining screening expertise and the equilibrium e¤ect. In
equilibrium, …nancial intermediation arises naturally, and direct …nance and intermediated …nance coexist in a uniquely determined allocation. The paper sheds new
lights on the leverage and industry organization of the banking sector.
Key Words: Certi…cation; Asset Scale; Asset Quality; Increasing Return to
Scale; Leverage; Industrial Organization
JEL: D53, D82, G21, G30, L10
Email: wangt@essex.ac.uk. Correspondence: Economics, University of Essex, Colchester, CO4 3SQ,
UK. Fax: +44 (0) 1206 872724. I am indebted to Dmitri Vinogradov for his enormous help with the
exposition. I thank Sanjay Banerji, Hans Gersbach, Esa Jokivuolle, Weimin Liu, Xuewen Liu, John
Moore, Jean Rochet, Zhen Song, Tuomas Takalo, Huainan Zhao, and seminar participants at Zhejiang
University, Fudan University, Essex University, Bank of Finland, ETH Zurich, Nottingham Business
School, for their helpful coments.
1
1
Introduction
The recent crisis arouses grave concerns about banks’ asset quality in general and risk
taking in particular. The focus, so far, is on capital adequacy ratio, unsurprisingly,
given its determining role in preventing banks from picking knowingly ine¢ cient, highrisk projects (see e.g. Jensen and Meckling 1976). The paper, however, takes a di¤erent
angle, the scales of banks’assets. It shows that independent of banks’capitalization, their
sizes play a decisive role for their asset quality and risk taking, by determining the level
at which they know the qualities of the projects. With this exercise, the paper sheds
new lights on many empirically well documented phenomena regarding the leverage and
industrial organization of the banking sector.
To study the choice of knowledge level by banks, the paper considers two types of
investors: households and bankers. Households are always uninformed of the types of entrepreneurs’projects. Bankers, by contrast, can obtain the expertise of screening projects
to some accuracy by paying the according costs. Bankers’ fund, which intends to capture equity capital of banks, is scarce. Entrepreneurs depend mainly on households’fund
for …nancing. But they will not get any if they do not convince the households of the
quality of their projects. They do it by obtaining enough fund from the bankers who
have attained the expertise and screened the projects with it. That is, for certi…cation
purpose, entrepreneurs need to demand a su¢ cient amount of the informed capital (i.e.
expertised bankers’fund),1 which drives it to earn a higher return rate than is earned by
the uninformed capital (e.g. households’fund).
The gap in return rate gives bankers the incentive to obtain screening expertise and
become informed. This incentive, the paper …nds, is shaped by two forces that the scales
of funds supplied by bankers bring forth.
1
Consistent with the certi…cation role of bank …nance, empirical studies well document that obtaining
bank …nance induces signi…cant movements of the borrower …rm’s stock price; see James (1989), Mikkelson
and Partch (1988), Lummer and McConnell (1989), Best and Zhang (1993), and Billett et. al. (1995),
Maskara and Mullineaux (forthcoming), and Ross (forthcoming). Moreover, the certi…cation role is not
con…ned to commercial banks. For example, Goldman Sachs’s purchase of $375 million of Facebook
shares sends the value of the company to $50 billion, more than 100 times of its annual earning $472
million; see "Facebook move lucrative for Goldman" and "Facebook", on Financial Times, the 4th and
9th of Jan., 2011, respectively.
2
The …rst is the increasing return to fund scale in obtaining screening expertise. If a
banker, by paying the cost, obtains the expertise of some level, then she knows the types of
projects to the corresponding accuracy, whereby all the fund she invests earns the return
rate of the informed capital at the accuracy. The larger the scale of the fund she invests,
the bigger the pro…t she earns by becoming informed;2 thus the bigger the incentive of
her to spend in screening expertise and the higher the level of screening expertise she
obtains. This type of increasing return to application scale in acquiring a technology or
expertise exists much widely. For instance, learning a widely spoken language delivers a
bigger bene…t than learning a thinly spoken one; and it is more pro…table to become a
top comedian where the audience is bigger.
The force alone commands that if all banks expand its credit supply, then all their asset
quality would be enhanced. But there is the second, counter force. That is the equilibrium
e¤ect: the ampler the aggregate supply of the informed capital by bankers, the lower the
return rate it earns, which dampens all bankers’incentives to become informed.
If each and every banker increases her supply of fund, the scale of each banker and
the aggregate supply of bank …nance will both increase and then the two forces are in
con‡ict. The second force dominates, the paper …nds, and in net the expertise level of
each and every banker falls, with which the asset quality of each and every declines. This
is consistent with the two trends which the years preceding the recent crisis witnessed:
the scale of banks’lending colossally expanded, while the loan quality monotonically fell.3
Especially, Mian and Su… (2009) document that securitization, the main way of enlarging
credit supply that time, led both a "sharp drop" in the interest rate and a "signi…cant
more" increase in the default rate.4
The paper derives a number of results regarding the leverage and industrial organiza2
Williamson (1986b) and Hauswald and Marquez (2006) consider other cases of increasing return to
scale in relation to information acquisition.
3
See Demyanyk and Van Hemert (forthcoming), Ja¤e et al (2009), Keys, et al (2010), Loutskina (forthcoming), Mayer et. al (2009), Mian and Su… (2009), Pozsar et. al (2010), Purnanandam(forthcoming);
especially, see Demyanyk and Van Hemert (forthcoming), Mayer et. al (2009), and Mian and Su… (2009)
for the monotonic quality fall.
4
For a closer examination on why securitization weakens the incentives of banks to monitor borrowers
or to acquire soft information, see Parlour and Plantin (2008) and Rajan et al (2008).
3
tion of the banking sector.
First, the increasing return to scale gives big banks an advantage over small banks.
Thus, the bigger the bank, the higher the level of screening expertise, evidence of which is
o¤ered by Ross (forthcoming) who shows that …nance by the dominate US banks certi…es
the borrowers are of higher quality and lower risk, and by Hao (2003) who documents that
bigger banks issue loans of smaller yields, suggesting they pick higher quality borrowers.
Second, the paper …nds a complementarity between leverage and screening expertise,
whereby leverage enhances the scale advantage of big banks. On the one hand, the higher
is a banker’s level of screening expertise, the more she can borrow. On the other hand, the
more she borrows, the larger the scale of the fund she deploys, which, due to the increasing
return to scale, drives her to choose a higher level of screening expertise. One immediate
implication is that bigger banks are leveraged at higher ratios, which is empirically well
documented.5
Third, common shocks allowing bigger leverage for all banks edge small banks out
and stimulate consolidation of the banking sector. Therefore, the upward trend of bank
leverage is a driving force for the trend of consolidation in the banking sector. Both trends
are empirically well documented,6 but the paper is the …rst that theoretically establishes
a causal link between them, as far as we know.
The paper looks into the asset quality of banks from a new angle by showing that their
sizes play a decisive role for their asset quality. It is thus related to the literature that
explains why loose lending standards are associated with lending booms (and vice versa);
see Rajan (1994), Ruckers (2004), and Dell’Ariccia and Marquez (2006). A di¤erence is
that in the present paper, scale expansion drives quality fall, whereas in this literature,
both are driven by other elements, such as career concerns of bank managers in Rajan
(1994), the distribution of borrowers’ quality in Ruckers (2004), or the distribution of
5
See Liang and Rhoades (1991), McAllistera and McManus (1993), Akhavein et al. (1997), Demsetz
and Strahan (1997), and Berger (1998) among others.
6
For the former, see Berger et al (1995) (…gure 1, for US over 1840-1990), Saunders and Wilson (1995)
(…gures 4-6, for UK, Canda and US over 1893-1991) Hortlund (2005) (…gure 2; for Sweden 1870-2001),
Miles et al (2011) (for UK over 1880-2004). For the latter, see Berger, Kashyap and Scalise (1995) (for
US over 1979-1994), Saunders and Wilson (1995) (…gures 2 and 3, for Canada and UK over 1893-1991),
Berger, Demsetz and Strahan (1999) (for US over 1988-1997), Jones and Critch…eld (2006) (for US over
1984-2003).
4
information in Dell’Ariccia and Marquez (2006).
The paper contributes to the literature that examines in general equilibrium banks’
provision of services that help …rms with information frictions; see Allen et. al. (forthcoming), Besanko and Kanatas (1993) (BK hereafter), Cantillo (2005), Holmstrom and Tirole
(1997) (HT hereafter), Morrison and White (2005), and Williamson (1986a). These papers
have other focuses than causal links between asset scale and asset quality. Furthermore,
the service examined in the present paper is certi…cation that helps with unobservable
types of projects,7 whereas it is ex ante monitoring that helps with moral hazards, in Allen
et. al. (forthcoming), BK, HT, and Morrison and White (2005), and is ex post monitoring or restructuring in Cantillo (2005) and Williamson (1986a), helping with costly
state veri…cation à la Townsend (1979). Lastly, the paper features that direct …nance and
intermediated …nance coexist: in equilibrium, households’ fund ‡ows to entrepreneurs
both directly and via bankers and the allocation between them is uniquely determined.
In contrast, while in BK and HT entrepreneurs (or …rms) also receive fund from both
households and banks, as in the present paper, BK consider no intermediated …nance (i.e.
no banks issuing liability to households), and HT admit that in their paper the allocation
between direct …nance and intermediated …nance is indeterminate.
The certi…cation service naturally drives up …nancial intermediation: due to the service, bankers earn the rate of the informed capital on the asset side, while on the liability
side, they repay the rate of the uninformed capital; the gap in between is the pro…t margin
of …nancial intermediation. As such the paper contributes to the literature that endogenizes …nancial intermediation from the …rst principles; see the seminar works of Diamond
and Dybvig (1983), Diamond (1984), and Diamond and Rajan (2001), and see Gorton
and Winton (2003) for a survey.
The paper’s …nding that bigger banks could enlarge scales further via leverage is related
to increasing dominance which is extensively examined by the industrial organization
literature; see Flaherty (1980), Gilbert and Newbery (1982), Vickers (1986), Budd, Harris
7
The idea that banks are informed and o¤er the borrowers a service of certi…cation can be traced
back, at least, to Booth and Smith (1986), but this literature does not relate it to bank …nancing. Note
that certifying a …rm’s quality by obtaining capital from a third, informed party (i.e. banks) is related
to but di¤erent from signalling by the …rm’s retaining a share of its own assets, on which the literature
traces back to Leland and Pyle (1977), with a nice survey provided by Daniel and Titman (1995).
5
and Vickers (1993), Cabral and Riordan (1994), Klepper (1996), Athey and Schmutzler
(2001), Halbheer et al (2009), Cabral (2002, 2011), Besanko et al (2011) among others.
The paper progresses as follows. Section 2 sets up the basic model, with which Section
3 examines the certi…cation role of bank …nancing. To the basic model, Section 4 adds
the accuracy choice by bankers and thereby shows that scale up, quality down. Section
5 further encompasses bankers’leverage to show the coexistence of intermediated …nance
and direct …nance. Section 6 considers the case of heterogenous bankers and thereby spells
out the implications on the industrial organization of the banking sector. Some further
discussions are put in Section 7. And Section 8 concludes. All proofs are relegated in
Appendix.
2
The Basic Model
There are two dates with no discount, today for contracting and …nance, tomorrow for
repayment and consumption.
The economy consists of three sectors: entrepreneurs, bankers, and households. All
agents are risk neutral and protected by limited liability. Entrepreneurs have projects
but no funds. Bankers and households have funds but no projects. Entrepreneurs are
of measure [0; 1]
[0; 1], and bankers of measure [0; 1].8 The idea is that each banker
serves a large number of entrepreneurs so that she can rely on the Law of Large Numbers
to smooth out uncertainty to the demand for her fund (hereafter we refer to bankers
with female pronouns, entrepreneurs with male ones). We interpret bankers as those who
run banks in the interest of equity,9 and households as the investors who have no time or
expertise to evaluate the pros and cons of the risky investment in entrepreneurs. However,
the business of …nancial intermediation is not considered until Section 5, whence bankers
are closer to what the name means in real life.
Fund is either invested in entrepreneurs’projects or in a risk free asset of which the
gross return rate is 1. The project of an entrepreneur requires an investment of $B; and
8
For readers who feel uneasy with a continuum of continuum, here a continuum of measure 1 could
mean to represent, for example, one thousand; then there are one million entrepreneurs and one thousand
bankers.
9
We assume away the agency problems between the management and equity of banks
6
returns $Z if it succeeds and nothing if it fails. High projects succeed with probability q
and low ones with probability q:
1 > q > q > 0:
Thus V
qZ
B is the net present value (NPV) of a high type, V
qZ
B that of a
low type. The proportion of high types is denoted by n, that of low types by n = 1
n.
We assume that high type projects have a positive NPV and that on average projects
have a negative NPV, that is,
V >0
nV + n V :
(1)
The latter assumption makes bankers’certi…cation service indispensable for generating a
positive social value, as will be seen. Moreover, assume that if an entrepreneur expects
to get zero pro…t, he will not …nance and run the project.10 The assumption serves
only to simplify the exposition, without which low type entrepreneurs could want to
run the projects anyway, from which they have nothing to lose, and consequently some
uninteresting equilibria would arise. The assumption would not be necessary (i.e. these
equilibria would not arise) if we let entrepreneurs have a small amount of capital of their
own or incur a little disutility from running the projects, either of which would impose a
cost on low types to mix with high types.
All bankers are identical in Sections 2 through 5 and the case of heterogenous bankers
is examined in Section 6. Each banker has K > 0 units of fund. A unit is de…ned as the
sum total of a continuum of measure 1 of dollars; that is, out of 1 unit of fund, $1 can be
invested in each and every of a measure 1 continuum of projects.11 We assume bankers’
fund is scarce:
K <Vn
q
q
q
:
(2)
This inequality, we will show, commands that the fund of bankers earns a higher return
rate than the fund of households.
Households, in aggregation, have abundant fund. As a result, not all of their fund
is absorbed by entrepreneurs and part of it ‡ows to the risk free asset. It follows that
households are satis…ed with expected gross return rate 1.
10
A possible explanation is that he incurs disutility in running the projects and has lexicographic
preference over the pro…t and the disutility.
11
Following footnote 9, if a continuum of measure 1 represents 1,000, then 1 unit means 1,000 dollars.
7
Households only know the prior distribution of the types, but bankers have the expertise to evaluate them. In the basic model, we assume bankers perfectly observe the
types. What matters of this assumption is not that the observations are noiseless – if
banker observe noisy signals of the types, then just rede…ne the signals as the types and
all the analysis carries on –but that the accuracy of the observations is given, the choice
of which is be examined in Section 4.
Given the knowledge di¤erence between bankers and households, the fund provided
by bankers is called the informed capital, that provided by households the uninformed
capital. To focus on the sector of bankers, we assume that an entrepreneur does not know
his type before being evaluated by a banker, but after that he knows the evaluation.
Contracting between Entrepreneurs and Bankers
If bankers do not provide certi…cation service, that is, their information on entrepreneurs’types is not communicated to households, then by assumption (1), households will
not invest in entrepreneurs. As bankers’fund is scarce, most high type projects will be
forsaken, which is a huge social loss. In this economy, bankers can provide certi…cation
service only by investing enough of their own fund. They could not do it with word of
mouth, by simply announcing the evaluations of projects to households, because the economy is essentially static and reputation mechanisms are assumed away.12 On the other
hand, if a banker invests enough of her own fund in a project, she must truly think it
good.
For the purpose of certi…cation, therefore, the entrepreneur-banker contract must involve the investment of the banker’s fund. The contract is characterized by two variables
(I; Q), where I is the amount of her fund to be invested and Q the amount of revenue
to be repaid when the project succeeds (on its failure, by limited liability, no one gets
anything). We call the ratio
Q
I
F as the face rate of return, because it is the return
rate that reads from the contract. A contract (I; Q) could be represented as (I; F ); where
F =
12
Q
:
I
In fact, even under repeated interactions, reputational mechanisms will not work if q is small enough,
that is, if the observed signal, success or failure, is very noisy about the true quality; see the literature
on reputation under imperfect monitoring, e.g. Abreu et al (1986) and Abreu et al (1988).
8
Given face rate F , the expected return rate of the banker is qF if the borrower is
a high type and is qF if he is a low type. To get the same expected return rate R,
the banker charges face rate R=q to a low type and a lower one, R=q; to a high type.
If the banker cannot collude with the entrepreneur, given R; the contract to a low type
is (I; R=q) and that to a high type is (I; R=q) for any investment amount I, however
small it is; and therefore, obtaining even one cent of bank …nance at a low interest rate
certi…es the borrower is of high quality. To prevent such a unrealistic scenario, we assume
that besides the investment contract, an entrepreneur could have other contracts with the
banker, for example, for consulting services, and it is too costly for households to check
all the contracts between the two. Under the assumption, for small I, contract (I; R=q)
does not certify the entrepreneur is a high type, because he might (over)pay I(R=q
R=q)
to the banker in another contract. But, given R, contract (I; R=q) with a a su¢ ciently
large I certi…es high types, we will show.
After an entrepreneur secures an investment contract from a banker, he goes to the
market for households’fund. He shows before them the contract, trying to convince them
that his project is of high type. If they are convinced, they accept face rate 1=q (namely,
for $1 given up today they get back $1=q tomorrow if the project succeeds). Certainly,
the projects of which no bankers are willing to take a stake are of quality below the prior
average and are hence not invested by households.
Timing of Events and De…nition of Equilibrium
The timing of today’s events is as below.
1: Each banker posts R, the expected return rate she is to charge for her fund.
2: Each entrepreneur goes to one banker and is evaluated by her, with the evaluation
observed by both sides.
3: The entrepreneur submits his demand for her fund. If she accepts it, she signs with
him a contract which speci…es the amount of her fund to be invested (I) and the face rate
of return (F ).
4: Entrepreneurs, showing the contracts with the bankers if available, go to the market
for households’fund.
5. A project is started if the entrepreneur has got $B altogether.
9
In equilibrium, we will show, all the bankers charge the same return rate, R. We can
thus de…ne a general equilibrium as below, where for a variable x; we use x
b to denote its
value in equilibrium.
b I;
b L)
b is an equilibrium if
De…nition 1 Pro…le (R;
b a high type entrepreneur demands $Ib
(i): given the price of the informed capital is R,
b
of the informed capital and a low type $L;
b clears the
(ii): given the demands of the two types of entrepreneurs are as above, R
b = K:
market for the informed capital: nIb + nL
3
Certi…cation by Bank Financing
In this section, we show that there is a unique equilibrium in which high types are sorted
b whereby all the socially
out by the amount of the informed capital invested, that is, Ib 6= L;
e¢ cient projects are …nanced.
Let us start with how an individual banker provides certi…cation service.
Lemma 1 Among the entrepreneurs who come to a banker charging R > 1; the high types
demand
I(R) =
V
R qq
(3)
1
dollars of her fund, whereas the low types demand L(R) = 0. Thus the high types are
certi…ed out. The ex ante expected pro…t of the entrepreneurs is
(R) = nV
R( qq
1)
R qq
1
:
(4)
The key to the lemma is that high types can sort themselves out by demanding I
I(R) dollars of the banker’s fund –then as her fund is more expensive than households’
(R > 1), they only demand I(R); the minimum amount needed for sorting, with the short
fall …nanced by cheap households’fund. To see an intuition for the sorting, consider a low
type who mimics by demanding $I of the banker’s fund also. He cannot hide his type from
the banker, who observes it. Therefore, he must give her a surplus of I(R
1). But by
mimicking, he cheats the households and convinces them to …nance the shortfall, $(B I);
at face rate 1=q: As his project succeeds actually with probability q; the households lose
10
1
q=q dollar for each dollar invested and overall (B
entrepreneur. He, therefore, gets by mimicking V + (B
is non-positive if I
Note that
0
q=q) dollars to the
I) (1
I) (1
q=q)
I(R
1); which
I(R):
(R) < 0; that is, the higher the rate a banker charges, the smaller the
pro…t to her entrepreneur customers. Therefore, in equilibrium all the bankers charge the
b Then, Condition (ii) of De…nition 1, that of market clearing, commands:
same rate, R.
With the help of Lemma 1,
b + nL(R)
b = K:
nI(R)
q
b = ( nV + 1):
R
q K
(5)
b > 1. Put back to Lemma 1, in equilibrium high types
As K is small by assumption (2), R
demand Ib =
K
;
n
b = 0; and the ex ante pro…t of entrepreneurs is
low types L
b=
q
q
q
(nV + K):
(6)
The equilibrium expounded above is unique: in any equilibrium, by Lemma 1, high
b I;
b R)
b as
types are certi…ed out and all the bankers charge the same price, which gives (L;
above. To summarize:
Proposition 1 For any K, there is a unique equilibrium where Ib =
K
;
n
b = 0 and R
b is
L
given by (5), and all the high types are certi…ed out by bank …nance and thus invested.
The analysis so far shows that given bankers’level of screening expertise in terms of the
accuracy in which they observe the types, the asset size of bankers (measured by K) does
not matter for the asset quality, because bankers only invest in the projects they evaluate
as high types (that, however, is not technically given but driven by competition). Yet, we
show below, size matters for the choice of expertise level and hence for asset quality.
4
Size, Screening Accuracy, and Asset Quality
From this section onwards, bankers’level of screening expertise is not …xed, but chosen by
them, and the level is measured by the accuracy of their evaluations, denoted by p 2 [0; 1],
modelled below. Thus to the timing of the basic model, we add:
11
Stage 0. Each banker chooses expertise level p, which is publicly observed hereafter.
(From it onwards, stages 1 through 5 are the same as in Section 2, namely, bankers post
R - then entrepreneurs come to be screened by bankers and sign investment contracts with
them if available - then entrepreneurs seek fund from households - …nally the entrepreneurs
who have gathered $B start their projects.)
The assumption of the level being publicly observed delivers the bene…t that the
asymmetric information between bankers and households is concerned only with the evaluations, not with the expertise level. The alternative assumption that p is private information of the banker is to be discussed in Subsection 7.2.
The expertise level is modelled as follows. On each project a banker evaluates, she
observes a noisy signal se = g or b, which represents a good or bad evaluation, according
to the following distribution:
Pr(e
s = bje
q = q) = (p); Pr(e
s = gje
q = q) = (p):
Namely, low types receive bad evaluations with probability (p) and high types receive
good evaluations with probability (p): Across the projects she screens, the signals arrive
independently.
Here p 2 [0; 1] captures the accuracy of the banker’s evaluations and thereby represents
the level of her screening expertise. We do not need a speci…c form of (p) and (p); the
paper’s analysis is valid so long as they satisfy properties enlisted below with the following
notations.
Given accuracy p; let qg (p) denote the posterior probability of success of a project
conditional on being evaluated good; qb (p) the posterior probability of success of an evaluated bad project; and ng (p) the probability of obtaining a good evaluation.13 Thus,
Vg (p)
qg (p)Z
B denotes the NPV of a evaluated good project; S(p)
ng (p)Vg (p)
denotes the ex ante social surplus of a project if it is …nanced only when being evaluated
good; and d(p)
qg (p)=qb (p) denotes the quality dispersion between evaluated good and
bad projects.
The paper’s analysis only requires the following …ve properties, which are thus assumed:
13
That is, qg (p) =
n (p)q+n(1
n (p)+n(1
(p))q
(p)) ;
qb (p) =
n(1
(p))q+n (p)q
n(1
(p))+n (p)
12
and ng (p) = n
(p) + n (1
(p)):
P0: qg (0) = qb (0) = n q + n q; the prior mean, and qg (1) = q and qb (1) = q ; that is,
the evaluations are not informative at all at zero accuracy and noiseless at full accuracy.
P1. qg0 (p) > 0; that is, the more accurate the evaluations, the higher the probability
in which evaluated good projects succeed.
P2. d0 (p) > 0; that is, the more accurate the evaluations, the bigger the di¤erence in
quality between evaluated good and bad projects.
P3. d(p) > 1 for p > 0; that is, this di¤erence is always positive so long as the banker
is a little informed.
P4. S 0 (p) > 0 and S 00 (p)
0; that is, the marginal social bene…t of more accurate
evaluations is positive but decreasing.
All these are reasonable properties for a proper modelling of accuracy and are satis…ed
for those often used in the relevant literature.14
The banker needs to spend C(p) in order that her evaluations are of accuracy p.
Function C( ) is convex over interval [0; 1] and satis…es C 00 ( ) > 0; C(0) = C 0 (0) = 0; and
C 0 (1) = 1: And C 0 (p) = o( (1 1p)2 ) at p
1; 15 which ensures the cost does not grow too
fast with p. Note that the C(p) is the cost of obtaining screening expertise of accuracy p,
not that of observing the type of one project to the accuracy; that is, with C(p) invested,
on each project to which the expertise is applied the evaluation is of accuracy p.16 We
could add a marginal cost of evaluating a project and let it a¤ect the accuracy of the
signal on the project also, which, however, would not qualitatively change the paper’s
results.
Moreover, to simplify the exposition, we assume C(p) is laid not out of the banker’
fund, but out of the revenue of investing in entrepreneurs; that is, having chosen accuracy
p, she still has K units of fund in hand.17 Should C(p) be extracted out of K; none of
the papers’results would qualitatively change.
14
The two often used examples are (p) = p; (p) = 1 and (p) = (p) =
p+1
2 :
They both satisfy all
the properties P0-P5.
15
That is, C 0 (p)(1 p)2 ! 0 if p ! 1:
16
This investment in expertise (rather than in acquiring information project by project) is also present
in Williamson (1986b) and Hauswald and Marquez (2006).
17
If the expenditure to improve screening accuracy mainly consist in recruiting …nancial experts, then
the assumption means the experts are paid not out of the initial fund, but out of the investment revenue,
which is indeed in the interest of the banker.
13
Let p is de…ned by
Vg (p) = 0:
That is, the evaluated good projects have a positive NPV if and only if the accuracy is
beyond p. Therefore, any screening expertise of an accuracy below p is useless. A banker,
if not choosing an accuracy beyond p; will choose p = 0.
In this section we focus on the accuracy choice by bankers at stage 0. Instead of
tediously writing down the full de…nition of subgame perfect Nash equilibrium, we take
a shortcut. The choice of accuracy is determined by the pro…t associated with each level
of accuracy p, which is in turn determined by the return rate charged, R; given the
accuracy p, which in turn by bankers’competition for entrepreneurs. Of this competition
the market conditions to bankers are summarized by a single variable, the ex ante pro…t
to entrepreneurs in equilibrium, b ; because a banker can attract entrepreneurs to her
only if her deal (p; R) o¤ers them a pro…t no less than b and in equilibrium she will not
o¤er more. Therefore, bankers’choices of accuracy at stage 0 is ruled by the equilibrium
de…ned below, where bankers are indexed by j 2 [0; 1]:
De…nition 2 A pro…le of (fb
pj gj2[0;1] ; b ) forms an equilibrium, if:
(i) given entrepreneurs expect to get b ; banker j chooses accuracy pbj :
(ii) given banker j has chosen pbj for j 2 [0; 1]; entrepreneurs expect to get b :
There is always a unique equilibrium. Depending on the values of parameters, it is
either a symmetric equilibrium where pbj all equal some pb > p; or a mixed equilibrium
where pbj equals either pb or 0 (i.e. bankers are indi¤erent between being informed and
being uninformed). To see that, let us examine how a banker, given the market conditions
summarized by b , decides on her screening accuracy. For this exercise, the following
remark is useful.
To the basic model of Section 2, here we add the ex ante stage for accuracy choice,
the subgame after which has been analyzed in Section 3. In order to use the results there
for …nding the equilibrium here with backward induction, we only need to rede…ne the
signals bankers observe here, g or b, as the types there, high or low.
14
4.1
The Decision of an Individual Banker on Screening Accuracy
To …nd out what rate R the banker charges after choosing accuracy p, we need to …nd
out what pro…t her deal (p; R) will give to the entrepreneurs who come to her.
Lemma 2 Facing deal (p; R); the expected pro…t of the entrepreneurs is
(R; p) = S(p)
(d(p) 1)R
;
d(p)R 1
(7)
and the evaluated good entrepreneurs demand
V (p)
:
Rd(p) 1
I(R; p) =
(8)
Note that this pro…t decreases with the rate charged, but it increases with the accuracy
of the banker, p, because a higher p not only generates a higher social surplus (S 0 (p) > 0);
but also widen the quality dispersion (d0 (p) > 0); both bene…ting the entrepreneurs.18
Therefore, if obtaining a higher accuracy, the banker can charge a higher return rate,
while still attracting entrepreneurs to come. This gives her the incentive to increase
accuracy.
The return rate charged as a function of accuracy is to be found by equalizing the
pro…t the entrepreneurs get from the banker,
(R; p); to the pro…t they would get from
any other bankers, b ; from which it follows:
R(p; b ) =
d(p) b
b
(d(p)
1)S(p)
:
(9)
Note, however, that the banker can always choose not to compete for entrepreneurs
and invest her fund in the risk free asset with gross return rate 1. Hence, the gross return
rate of her fund is maxfR(p; b ); 1g; while the opportunity cost rate is 1. Therefore, by
obtaining accuracy p, her economic pro…t is
18
K (maxfR(p; b ); 1g
1)
C(p):
Intuitively,the former bene…ts because entrepreneurs get a fraction of the social surplus, the latter
because the bigger the quality dispersion, the more easily can high types certify themselves out.
15
With (9), the decision problem of the banker on the choice of accuracy, p; is:
max K (maxf
0 p 1
d(p) b
b
(d(p)
1)S(p)
; 1g
1)
C(p):
(10)
The solution to this problem is either some p > p (so S(p) > 0 and then R > 1); or
p = 0, or both. However,
Lemma 3 It is impossible that in equilibrium all bankers choose p = 0.
Intuitively, if all the other bankers choose to be uninformed, thus unable to provide
certi…cation service, then the particular banker is left with all the entrepreneurs as her
customers and will get a huge pro…t by serving them all with screening expertise of some
accuracy p > p.
The equilibrium is thus either symmetric, where bankers’ fund earns a nonnegative
net return and bankers all choose the same p > p, or mixed, where bankers’fund earns 0
net return and bankers are indi¤erent between choosing some p > p and being uninformed
(p = 0). We focus on the former case, because it is more likely to capture the situation of
real life banks in the decade preceding the recent crisis: bankers’fund means to represents
bank equity, on which the average return rate in the decade was approximately 20%.19
However, for theoretical completeness, we discuss the mixed equilibrium in Subsection
7.1.
4.2
The Symmetric Equilibrium: Scale up, Quality down
In the symmetric equilibrium, all the bankers choose to spend in screening expertise,
driven by R > 1: Then, problem (10) becomes:
( b ; K)
max K (
0 p 1
b
d(p) b
(d(p)
1)S(p)
1)
C(p):
(11)
Let its solution denoted by (K; b ); which then satis…es the following …rst order condition:
19
Kb
1)S 0 (b
p) + d0 (b
p)(S(b
p) b )
= C 0 (b
p):
2
[d(b
p) b (d(b
p) 1)S(b
p)]
(d(b
p)
See "Promises that proved ultimately empty", The Financial Times, Jan. 9, 2012.
16
(12)
This equation elaborates condition (i) of De…nition 2. It is essentially the incentive compatibility constraint for bankers to spend in screening expertise, whereas
( b ; K)
0 is
the individual rationality constraint (IR), which implies S(p) > 0, that is, pb > p:
Now elaborate condition (ii) of De…nition 2.
Lemma 4 If all the bankers have chosen accuracy pb > p; then the equilibrium pro…t of
entrepreneurs, b ; is:
p) 1
b = d(b
(S(b
p) + K):
d(b
p)
(13)
The symmetric equilibrium, (b
p; b ), is the solution of the simultaneous equations of
(12) and (13).
Proposition 2 (i): The simultaneous equations of (12) and (13) has a unique solution,
(b
ps ; b s ): Therefore, the symmetric equilibrium uniquely exists if and only if
namely, the IR is satis…ed.
(ii):
If
db
ps
dK
( b s ; K)
0,
< 0, that is, the equilibrium accuracy decreases with K.
( b s ; K) < 0; we establish the existence of a unique mixed equilibrium and char-
acterize it in Section 7. Thus, the equilibrium always uniquely exists.
To understand result (ii), note that K denotes both the individual stock and aggregate
stock of bankers’fund. Hence, an increase in K by one unit means that each and every
banker increases her fund stock by one unit. This change will generate two e¤ects upon
bankers’ incentive to improve screening accuracy, both formalized below, which drive
Proposition 2(ii).
Lemma 5
@ (K; b )
@K
@ (K; b )
@K
> 0 and
@ (K; b )
@b
< 0; where (K; b ) is the solution to problem (11).
> 0 captures the increasing return to fund scale in improving screening exper-
tise: The more fund a banker has, the more extra pro…t she earns from an increment in
the interest rate, and hence the bigger the marginal pro…t of improving accuracy; indeed,
the left hand side (LHS) of (12), which represents the marginal pro…t, is in proportion to
K.
17
@ (K; b )
@b
< 0 captures the equilibrium e¤ect, which works in the opposite direction: The
bigger the aggregate stock of the informed capital (K), the higher the equilibrium pro…t
to entrepreneurs ( b ) (by 13), which reduces the marginal pro…t of a higher accuracy
to all bankers as follows. Given p, a banker screens a mass
entrepreneurs, from each and every of whom, she gets S(p)
K
I(p;R(p; b ))
N (p; b ) of
b ; the di¤erence of the
social value of his project minus the pro…t she gives up to him. The marginal pro…t then
consists of two parts: Np0 (p; b )(S(p)
b ); from the expansion of scale, and N (p; b )S 0 (p);
from the widening of pro…t margin. Both parts diminish with b : The former is because
the pro…t margin, S(p)
b ; diminishes with b : The latter is because so does the scale:
the higher the pro…t to the entrepreneurs ( b ); the lower the rate charged (R); then the
more of the banker’s fund demanded by each evaluated good entrepreneur for certi…cation
purpose (IR0 < 0 by 8) and hence the smaller the scale.
The two e¤ects are thus in con‡ict. In net, the proposition asserts, the latter one
dominates. Hence if all bankers increase their fund by one unit, the screening accuracy of
all goes down. As a result, the default risk of all their assets goes up: although bankers still
invest only in evaluated good projects, with less accurate screening, a smaller proportion
of them is actually good.20 Here comes a main result of the paper: if each and every bank
expands its credit supply, then the quality of each and every bank’s assets goes down.
An obvious way for bankers to expand credit supply is to borrow from households,
which is examined in the next section, where we show that with all bankers expanding
credit supply via leverage, their screening accuracy all goes down and hence so does their
asset quality, as suggested by Proposition 2 (ii).
5
Leverage of Bankers
So far bankers do not borrow from households and what they invest is their own fund,
that is, they are running wholly-equitied banks. In this section, we add the stage when
bankers borrow from households, which we assume occurs before they choose their screen20
Mathematically, the default probability of the projects they invest is 1
ps
qg0 db
dK
> 0 because
qg0 (
) > 0 (by P1).
18
qg (b
ps ) and
d(1 qg (b
ps ))
dK
=
ing level.21
In this model economy, bankers are driven to borrow. On becoming informed, bankers
earn the rate of the informed capital, which is above 1, on the asset side, while they repay
households with the rate of the uninformed capital, which is 1, on the liability side. The
rate gap gives the pro…t margin of borrowing.22 In fact, individual bankers take this pro…t
margin as given, and so long as it is positive, they want to borrow as much as possible.
But if they borrow too much, they will be susceptible to risk shifting problems, as in
Jensen and Meckling (1976).
5.1
The Risk Shifting Problem and Leverage Ratio
The analysis hitherto does not depend on how the risks of individual projects correlate.
To bring forth the risk shifting problem, in this section we assume that the risks of projects
are correlated and cannot be diversi…ed away.23 To simplify the exposition, actually, we
assume the risks of projects are perfectly correlated. Speci…cally, ex ante there are three
possible states of the world, f ; 1; 2g, with probability 1
q; q
q, and q: In state ; no
projects succeed; in state 1, only high type projects succeed and low types fail; and in
state 2, both types succeed. So high type projects succeed in both states 1 and 2, thus
with probability q; and low types succeed in state 2 only, thus with probability q:
Suppose a banker borrows D units fund from households at face rate f , namely, by
repaying them overall Df whenever possible. When D is too large, the banker could
invest in evaluated bad projects at a lower expected return rate but a higher face rate
than would be obtained by investing in evaluated good projects.
Let F be the face rate of investing in evaluated good projects, that is F
R
;
qg
and F 0
be that of investing in evaluated bad projects, which succeed with probability qb : Then,
for some
21
22
2 (0; 1);
F 0 = ((1
)
qg
+ )F;
qb
(14)
If it instead occurred afterwards, the results of this section would not be qualitatively a¤ected.
In the paper, bankers are assumed not to issue outside equities to laymen, possibly due to some
friction of costly state veri…cation in the manner of Townsend (1979), Diamond (1984), and Gale and
Hellwig (1985).
23
Otherwise, as each banker …nances a continuum of projects and issues debt contracts to households,
the risks on her asset side will be completely diversi…ed away and no risk shifting problems will arise.
19
because F 0 lies somewhere between F and
qg F
.
qb
On the one hand, the banker rejects any
face rate below F ; on the other hand, evaluated bad entrepreneurs reject any face rate
above
qg F
;
qb
which gives the banker the same expected rate R, and thus by the argument
for Lemma 1, makes them get 0 pro…t by mimicking good entrepreneurs.
Lemma 6 A banker who borrows D; obtains accuracy p and charges R(p) does not invest
in evaluated bad projects if and only if
D
K
(1
(q
q)f
1
)R(p)
d(p)
1
)R(p)
(1 d(p)
L(p):
(15)
If the inequality holds, the banker repays her debt with probability q:
By the lemma, the expected return rate to the households who lend to bankers is qf:
Households are satis…ed with expected return rate 1. Hence
1
f= :
q
5.2
(16)
The Characterization of Equilibrium with Leverage
b the leverage ratio, L;
b
Proposition 3 (i): In equilibrium, the amount of borrowing, D;
and the screening accuracy, pbL ; solve for (D; L; p) of the following simultaneous equations:
(17)
D=K L
1
S(p)
d0 (p) S(p)
(
+ 1)[S 0 (p) +
(
d(p) K + D
d(p) d(p) 1
L=
K
1)( S(p)
+ 1)
K
(d(p)
q q
d(p)2
q
(d(p)
1)
:
D)] = C 0 (p)
(18)
(19)
(ii): The expansion of credit supply by all bankers through leverage lowers all their
screening accuracy and their asset quality: pbL < pbs :
(iii): Households’ fund ‡ows to entrepreneurs both directly and via bankers, that is,
direct …nance and intermediated …nance coexist.
Equation (17) holds because given the positive rate gap, each banker wants to borrow
as much as possible, thus reaching the upper bound commanded by the risk shifting
problem.
20
The formula of (18) shows that debt capital, D; plays the same role as equity capital, K, in determining equilibrium choice of accuracy. The accuracy choice, intuitively,
results from the balance between two forces: the increasing return to fund scale and the
equilibrium e¤ect, in both of which D and K play the same role. The latter is driven by
the increasing of b with the aggregate supply of the informed capital, to which how it is
…nanced by debt and equity is irrelevant. For the former, by Lemma 6, the probability
of debt being repaid, …xed at q; is independent of the accuracy choice, p. Hence so is the
cost of debt (D), as is the (opportunity) cost of equity (K). On the other hand, debt
and equity contribute in equal terms to a banker’s revenue ((K + D)R) and thus to her
marginal revenue ((K + D)R0 (p)): Therefore, D and K contribute in equal terms to the
increasing return to fund scale, which implies that, given b ; a banker’s choice of accuracy
is
p = (K + D; b ):
(20)
As borrowed capital plays the same role as equity capital in determining equilibrium
choice of accuracy, the expansion of credit supply through leverage produces the same
e¤ect as an increase in K does, which, by Proposition 2, is the lowering of accuracy. So
arises result (ii).
Result (iii) arises because entrepreneurs want bank …nance only to the extent of certifying they are evaluated good by bankers, which is in turn because bank …nance, as the
informed capital, is more expensive than the uninformed capital, otherwise bankers have
no incentives to spend in screening expertise and become informed.
6
Heterogeneous Bankers
In this section, we remove the assumption of identical bankers and deliver two points. (i)
The increasing return to fund scale advantages big banks: the bigger the bank, the higher
the level of screening expertise and the asset quality. (ii) There is a complementarity
between leverage and screening expertise: the higher the level of screening expertise,
the more the bank can borrow; then the larger the scale of its fund, which, due to the
increasing return to fund scale, feedbacks to drive the bank to choose a higher expertise
level. This complementarity strengthens the scale advantage explained by point (i). These
21
two points bear on the industrial organization of the banking sector.
So far all bankers have the same amount of fund. Now in this section we consider
the case where banker j has fund Kj for j 2 [0; 1]: For banker j, her problem of deciding
on accuracy is sill described by (11), with K replaced by Kj . Therefore, her choice of
accuracy is
pj = (Kj ; b ):
(21)
We put the remainder of the characterization of the equilibrium in the proof of the
following lemma, which extends Proposition 2(ii) to the case of heterogenous bankers.
Lemma 7 If fKj gj2[0;1] are increased in such a way that leaves bankers’ market shares
unchanged, then the screening accuracies of all bankers go down.
But the focus of the section is the comparisons across banks within an economy. By
(21) and Lemma 5, if Kj > Ki ; then pj > p: This is our point (i). Intuitively, within
an economy, all the banks face the same market conditions (characterized by b ); the
comparison between them is thus not subject to the general equilibrium e¤ect, but ruled
by the increasing return to fund scale only; and therefore, the larger the scale, the higher
the level of screening expertise.
Let us further incorporate the borrowing by bankers. Then by (20), banker j’s choice
of accuracy is pj = (Kj + Dj ; b ); where her borrowing Dj = Kj L(pj ) by (17). Therefore,
Substitute (16) for f into (15),
pj = (Kj (1 + L(pj )); b ):
L(p) =
(1
(1
q
)
q
1
)R(p)
d(p)
:
1
(1 d(p)
)R(p)
(22)
(23)
And L0 (p) > 0 because both d0 (p) > 0 and R0 (p) > 0: Intuitively, the higher the screening
accuracy (p), the starker the di¤erence in quality between evaluated good and bad projects
(d) and the higher the rate charged (R); both leading to a bigger value destruction by
risk shifting (R
qb F 0 = (1
1
)R
d
by 14) and therefore a smaller incentive to do that,
which allows for a larger leverage ratio.
22
Equation (22) then points to a complementarity between screening accuracy and leverage: A higher accuracy (p) induces a higher leverage (L) due to L0 (p) > 0; while a higher
leverage induces a higher accuracy due to the increasing return to fund scale. This is our
point (ii). The complementarity implies:
Proposition 4 If Kj > Ki ; then the leverage ratio Lj > Li :
The proposition delivers:
Prediction (a): The bigger the bank, the higher the leverage ratio.
It is empirically supported by Liang and Rhoades (1991), McAllistera and McManus
(1993), Akhavein et al. (1997), Demsetz and Strahan (1997), and Berger (1998). This
literature, however, explains it via the bene…t of diversi…cation á la Diamond (1984).
These two lines of argument diverge at the implication on the link between the size
of a bank and the quality of individual loans it issues. The argument of diversi…cation
implies no links between the two, whereas in the present paper loans of bigger banks are
of higher quality, because the borrowers are evaluated good to a bigger accuracy, due to
the increasing return to fund scale. That is,
Prediction (b): The bigger the bank, the higher the quality of its individual assets
(measured by the default probability) and the higher the level of certi…cation it provides in
the sense that obtaining its fund means a higher quality.
It is empirically supported by Ross (forthcoming) who documents that the loans from
three US dominant banks (J.P. Morgan Chase, Bank of American, and Citigroup) induce
the borrowers’stock prices to jump higher and are issued at lower interest rates and "less
likely to be protected by a borrowing base", and the stock price jump is larger when
the borrower is opaque, altogether suggesting that these banks "provide a higher level of
certi…cation". Also Hao (2003) documents an inverse correlation between bank size and
loan yield spread, suggesting bigger banks pick higher quality borrowers.
Finally, the increasing return to scale commands small banks are edged out of the
banking business. Banker i’s pro…t,
( b ; Ki ); de…ned in (11), is increasing with the scale
23
of her fund, Ki : If the scale is too small such that
( b ; Ki )
0; then the banker is edged
out of business and choose to be uninformed. Leverage, by Proposition 4, ampli…es the
increasing return to scale and exacerbates the disadvantage of small banks. Moreover,
they are squeezed further if due to some technological or institutional shocks, all banks
could be leveraged more. This occurs, in the paper, with a decrease in
is the common factor for all bankers’leverage. The paper …nds that j @L
j=
@
increases with p and thus with the scale of the banker. That is, if
1
q=q; which
(1
(
(1
1
)R(p)
d(p)
1
)R(p))2
d(p)
decreases, the bigger
the bank, the larger is the increment in leverage ratio and the further is its scale advantage
augmented, consequently the more small banks are edged out and the more concentrated
the banking sector becomes. That is:
Prediction (c): A common shock allowing bigger leverage for all banks edges small
banks out. Therefore, the upward trend of bank leverage is a driving force for the trend of
consolidation in the banking sector.
Both trends are well empirically documented.24 Especially, consistent with the paper’s
explanation, one important reason Berger, Demsetz and Strahan (1999) attribute to the
consolidation in the US over 1988-1997, when the number of banks falls by 30%, is the
improvement in the …nancial conditions, like low interest rates, which makes it easier to
raise debt for M&As; and Berger, Kashyap and Scalise (1995) …nd deregulation of deposit
ceiling rate contributes to the consolidation over 1979-1994 because it allows banks to
absorb more deposit.
7
Further Discussions
In this section we characterize for the case of identical bankers the equilibrium when it is
mixed and discuss the assumption that the screening accuracy of a banker is her private
information.
24
The former is documented by Berger et al (1995) (…gure 1, for US over 1840-1990), Saunders and
Wilson (1995) (…gures 4-6, for UK, Canda and US over 1893-1991), Hortlund (2005) (…gure 2; for Sweden
1870-2001), and Miles et al (2011) (for UK over 1880-2004), the latter by Berger, Kashyap and Scalise
(1995) (for US over 1979-1994), Saunders and Wilson (1995) (…gures 2 and 3, for Canada and UK over
1893-1991), Berger, Demsetz and Strahan (1999) (for US over 1988-1997), and Jones and Critch…eld
(2004) (for US over 1984-2003).
24
7.1
The Case Where the Equilibrium Is Mixed
By Proposition 2, if
( b s ; K) < 0; the symmetric equilibrium does not exist, so in
equilibrium not all the bankers invest to be informed, whereas by Lemma 3, it is not
possible that no bankers invest. Therefore, the equilibrium, if it exists, must be mixed,
with
2 (0; 1) fraction of bankers choosing to be informed while the remainder to be
uninformed. The equilibrium is characterized as follows.
First, ex ante, bankers are indi¤erent between the two choices, that is,
( b m ; K) = 0;
(24)
where b m is the ex ante pro…t to entrepreneurs in the mixed equilibrium.
Second, those bankers who choose to be informed solve the same problem, (11), and
their choice of accuracy is, as denoted, (K; b ): With b = b m in the equilibrium, the
equilibrium accuracy is then
pbm = (K; b m ):
(25)
Third, in the mixed equilibrium, the aggregate informed capital is K instead of K:
Therefore, by (13), the equilibrium pro…t to entrepreneurs, b m ; satis…es:
pm ) 1
b m = d(b
(S(b
pm ) + K)
d(b
pm )
(26)
The mixed equilibrium, ( b m ; pbm ; ); is characterized by the simultaneous equations of
(24), (25) and (26).
Lemma 8 If
( b s ; K) < 0; then the mixed equilibrium exists uniquely.
Proposition 2(i) and Lemma 8 imply that the equilibrium always exists and is unique.
b 1) = q nV
Note that in Section 2, banker’s economic pro…t is K(R
q
q q
K
q
and decreasing
with K, which is driven by the equilibrium e¤ect that the more amply is the informed
capital supplied, the cheaper it becomes and the smaller the pro…t to bankers. Suggested
by this feature, the symmetric equilibrium is likely to be the case when K is small, the
mixed equilibrium the case when K is big. Intuitively, when the aggregate bankers’fund is
scarce, the competition between bankers is weak, the interest rate of the informed capital
is high, and all this fund is transformed into the informed capital. With the bankers’
25
fund becoming more and more abundant, not all of it become informed capital, bringing
us into the domain of the mixed equilibrium
In the mixed equilibrium, the change of screening accuracy with respect to K is not
monotonic any more, as was described in Proposition 2(ii).
Lemma 9
db
pm
dK
< 0 if and only if
for which a su¢ cient condition is:
Sb
bm
bm
(log R)0p
<
;
(log S)0p
(27)
Sb
);
Sb b m
b > min(2;
R
(28)
b is its value in the equilibrium
where R as a function of p and b is de…ned by (9), and R
and Sb is the equilibrium social surplus of a project.
An intuition for (28) is that as we saw after Proposition 2, the decrease in screening
accuracy with K is driven by the general equilibrium e¤ect; and in the mixed equilibrium,
the strength of this e¤ect could be measured by the expected return rate of the informed
capital, R. To see this point, bear in mind that the e¤ect is due to the increasing of the
ex ante pro…t to entrepreneurs, b ; with K. Suppose K increases by K: Were b …xed,
the pro…t of bankers who invest in screening expertise would be increased by (R
1) K
by the envelop theorem. However, in the mixed equilibrium, their pro…t is settled at 0.
Therefore, this supposed increment all goes to entrepreneurs (as households always break
even). That is, the increases in b is roughly (R
1) K; and is thus proportional to R
1:
Therefore, if R is big enough, as (28) is satis…ed, the general equilibrium e¤ect will be
so strong as to overwhelm the force of the increasing return to scale, and thus push the
screening accuracy down.
Actually, this intuition suggests that
db
pm
dK
> 0 if K is large enough; therefore, overall,
the equilibrium screening accuracy as a function of aggregate bankers’fund is in a U shape.
To see that inequality (27) will be reversed, thus
a large K implies a small
db
pm
dK
> 0; when K is large, note that
; the proportion of bankers who invest to be informed, and
26
thus a big 1 ; the mass of the entrepreneurs served by each banker. Therefore, from each
project screened, a banker only gets a small surplus (in order to compensate the cost of
investing in screening). That is, Sb
be reversed.
7.2
bm
0 and b m
b whereby inequality (27) will
S;
Unobservability of Screening Accuracy
Above we assume every banker’s screening accuracy is publicly observed. This assumption
is made mainly to gain tractability, but also because it reasonably captures real life.
There are reasons to believe that a bank’s screening accuracy is public information
in real life. First, this accuracy could be recovered from the data of its balance sheet,
particularly the default rate of the loans it made to the real sectors. Second, it is re‡ected
in the reputation of the bank, particularly their reputation of hunting good deals. Third,
if the spending in screening expertise consists mainly in hiring reputably talented bankers,
the relocation of them is often exposed by the media, for example, Financial Times.
If the accuracy is assumed private information of the banker, the major di¢ culty is that
the equilibrium will be very sensitive to o¤ equilibrium assumptions. Suppose a banker
thinks all the other bankers obtain accuracy pb: For it to be the equilibrium accuracy ,
the banker must get more from choosing the same accuracy , pb; than from choosing an
o¤-equilibrium accuracy . The o¤-equilibrium pro…t, however, is very sensitive to what is
assumed on how the banker invests her residual fund if she could not …nd enough evaluated
good projects from the entrepreneurs coming to her in equilibrium.25 Two possibilities
have been explored. One is that the banker could lower the face rate a little, by which she
attracts more entrepreneurs than is supposed in equilibrium (i.e. a continuum of measure
25
If p is private information, in and o¤ the equilibrium path, the banker screens a measure 1 continuum
of entrepreneurs, of whom ng (p) are evaluated good, each demanding $I(b
pe ); such that ng (b
p)I(b
p) = K:
Note the demand only depends on pbe ; laymen’s rational expectation of pb (all this is to convince them)
and hence does not change with the actual screening quality of the banker. Therefore, if ng (p) < ng (b
p);
the overall demand of her capital by all the evaluated good entrepreneurs, ng (p)I(b
p); is smaller than
K and she needs to consider how to invest the residual K
ng (p)I(b
p) units of her fund. By contrast,
if p is publicly observed, as we studied, in the subgame where ng (p) < ng (b
p); there will be measure
continuum of entrepreneurs coming to the banker such that ng (p)I(p) = K, that is, the overall demand
of her capital equals K on and o¤ equilibrium path.
27
1); the other is that she has to invest the residual fund in the risk free asset. The two
assumptions lead to opposite results.26
Another di¢ culty in connection with the assumption of the accuracy being private
information is that however a banker invests her residual fund if she has chosen an o¤equilibrium accuracy, her asset will be di¤erent from that of all the other bankers (who
obtain the equilibrium accuracy). Then what stops households (and entrepreneurs) inferring from this di¤erence her actual accuracy? If they do, the accuracy will, in e¤ect,
become observable to households and entrepreneurs, as was assumed.
8
Conclusion
In a general equilibrium framework where bank …nancing certi…es the good type of the
borrower, the paper examines the choice of banks on the level of screening expertise and
identi…es two causal links between scale and quality of bank assets. One is the increasing
return to fund scale in attaining screening expertise, which commands that the bigger
the bank, the higher the level of its screening expertise and thus the higher the quality
of its assets. The other is the general equilibrium e¤ect, which commands that the larger
the aggregate supply of bank …nance, the smaller the rate it earns, which dampens the
incentives of all banks to spend in screening expertise and consequently lowers all their
asset quality. If each and every bank enlarges credit supply simultaneously, the two forces
are in con‡ict; the paper shows that the latter force dominates and thus in net the asset
quality of each and every bank goes down.
Furthermore, the paper shows that banks are driven to enlarge credit supply through
leverage, by the gap in rate between the informed capital and the uninformed capital.
In the paper this intermediated …nance stands side by side with the direct …nance of
entrepreneurs by households.
Lastly, the paper …nds a positive feedback between a bank’s leverage and its level of
screening expertise: a higher level of the expertise enables it to be leveraged more and
to enlarge its asset scale further, which, due to the increasing return to scale, feeds back
to drive it to choose a higher level of screening expertise. This positive feedback has
26
The detailed analysis could be obtained upon request.
28
important implications on the industrial organization of the banking sector. Particularly,
it implies that leverage uprising is a force driving consolidation in the sector.
Appendix: The Proofs
For Lemma 1:
Proof. Two considerations pin down I(R): First, high types can sort themselves out
by demanding I
I(R) of the banker’s fund. If a low type mimics by demanding $I of
her fund, then the banker, knowing the low quality of his project, will charge him face
rate
R
q
and ask him to repay $ RI
when he succeeds. By mimicking, the low type at best
q
convinces households that his project is of high type, so that they are willing to …nance
the shortfall, $(B
I); at face rate 1q ; thus for face value $ B q I . Therefore, by mimicking,
the low type obtains, in the case of success, Z
RI
q
B I
:
q
Mimicking delivers zero pro…t
to him if
Z
RI
q
B
I
q
0,I
I(R):
(29)
Second, as R > 1; the banker’s fund is more expensive than households’. Therefore, the
high types only demand the minimum of the amounts by which they can sort their type
out (and …nance the short fall with the cheap households’fund). By (29), they demand
exactly I = I(R):
Low types, as their types is sorted out, will get zero pro…t if they enter …nancing.
Then, they do not even enter …nancing as we assumed, that is, they demand nothing of
the banker’s fund.
By demanding I(R) of the banker’s fund, a high type sorts himself out and is not
mixed with low types, whereby he surrenders 0 surplus to household investors, but a
surplus of I(R)(R
1) to the banker. (Note that If the high type were mixed with low
types, households would demand a surplus from him to compensate their loss in the low
types mixed with him.) His expected pro…t is thus V
I(R)(R
1); which times n; the
probability of being among high types, gives the ex ante pro…t of the entrepreneurs who
come to the banker.
For Lemma 2:
29
Proof. Note that for entrepreneurs coming to a particular banker who o¤ers (p; R),
the circumstance in this section is isomorphic to that studied in Section 2, by rede…ning
the signal observed, g or b, as the type. Therefore, we have:
(MapB): The results concerned with individual bankers, such as (3), (4), and (29),
can be applied here by substituting high type there with good evaluation here, low type
with bad evaluation, n with ng ; q with qg , q with qb ; and V with Vg :
Applying (MapB) to (4), we …nd the ex ante pro…t of the entrepreneurs coming to
her,
(R; p); is
ng Vg
R( qqgb
1)
R qqgb
1
Then with notations S(p) = ng (p)Vg (p) and d(p) =
:
qg (p)
;
qb (p)
(7) arises. Similarly, applying
(MapB) to (3), we …nd the demand of the evaluated good entrepreneurs is as given by
(8).
For Lemma 3:
Proof. If all bankers choose to be uninformed, then no projects will be …nanced
by assumption (1) and thus b = 0: We show that if b <
d(1) 1
S(1),
d(1)
a banker gets an
in…nitely large pro…t –impossible in equilibrium –in the sense that the value of problem
(10) is in…nite. Mathematically, that is obvious for b > 0; because the denominator in the
objective function, d(p) b (d(p) 1)S(p); equals 0, at some p0 < 1; and therefore the value
of the problem is in…nite. Let us provide an economic argument also. If b <
q q
nV
q
; then b =
d(p0 )
d(p0 )
1
S(p0 ) for some p0 2 [p; 1) because
d( ) 1
S(
d( )
d(1) 1
S(1)
d(1)
=
) is increasing as both
d0 ( ) > 0 and S 0 ( ) > 0: Then a banker can both charge R = 1; thus reaping an in…nitely
large pro…t, and attract all the entrepreneurs by giving them more than b . On her deal
(p; R); the entrepreneurs coming to her expect to obtain, by (7),
(R; p) =
which increases with p; decreases with R, and is thus always bigger than
limR!1 (R; p): Therefore, if she o¤ers p = p0 +
coming to her get more than b :
d( ) 1
S(
d( )
(1; p0 + ) =
) is increasing as noted above.
(d(p) 1)R
S(p);
d(p)R 1
d(p) 1
S(p) =
d(p)
< 1 and R = 1; the entrepreneurs
d(p) 1
S(p)jp=p0 +
d(p)
>
d(p0 ) 1
S(p0 )
d(p0 )
= b , as
For Lemma 4:
Proof. In the case of symmetric equilibrium, note that for the whole economy, the
circumstance from stage 1 onward is isomorphic to that studied in Section 2, if we map
30
the signal observed, g or b, to the type, high or low. Therefore, the results of Section 2
concerned with equilibrium, such as (5) and (6), can be applied here by substituting high
type with good evaluation, low type with bad evaluation, n with nbg ; q with qbg , q with qbb ;
and V with Vbg :27 Apply this mapping to (6), and we …nd the ex ante equilibrium pro…t
of entrepreneurs is
b = qbg
qbg
qbb
(b
ng Vbg + K);
which becomes, with notations S(p) = ng Vg and d(p) =
qg (p)
,
qb (p)
(13).
For Proposition 2:
Proof. for (i): The proof applies
@ ( b ;K)
@b
< 0; a property given in Lemma 5. It su¢ ces
to show that the following simultaneous equations of (b
p; b ) have a unique solution:
( b ; K)
p) 1
b = d(b
(S(b
p) + K)
d(b
p)
pb =
g(b
p; K)
Or equivalently, the following equation has a unique solution for p over [0; 1] :
p = (g(p; K); K):
This is equivalent to
f (p)
p
(g(p; K); K)
has a unique root over [0; 1], which we are going to prove by showing the following three
claims.
(a) f 0 > 0: Actually f 0 = 1
@g
@p
@ ( b ;K)
@b
@g
@p
> 1 because
> 0 since both d0 ( ) > 0 and S 0 ( ) > 0:
(b) f (1) > 0 : At p = 1; b = b 1
d(1) 1
(S(1) + K)
d(1)
>
@ ( b ;K)
@b
< 0 by Lemma 5, and
d(1) 1
S(1):
d(1)
For this value of b ,
the LHS of (12) < 1 with p ! 1; whereas the RHS goes to 1, and therefore its solution
( b 1 ; K) < 1: It follows that f (1) = 1
(c) f (p0 ) < 0; where p0 is the root of
( b 1 ; K) > 0:
g(1; 0) = g(p0 ; K):
First, p0 < 1: This, given
forward, as
27
@g
@K
@g
@p
> 0; is equivalent to g(1; 0) < g(1; K); which is straight-
> 0: Second, we show that since we assumed C 0 (p) = o( (1 1p)2 ) at p
For f (p) (a function of p), fb
f (b
p):
31
1;
(g(1; 0); K) = 1: For b = g(1; 0) =
1
,
(1 p)2
order of
d(1) 1
S(1);
d(1)
at p
1; the LHS of (12) is in the
whereas the RHS, by the assumption, is in the order of o( (1 1p)2 ) and thus
dominated by the LHS. It follows that the solution of problem (11) is p = 1: The two
points put together, f (p0 ) = p0
(g(p0 ; K); K) = p0
(g(1; 0); K) = p0
1 < 0.
By claims (a), the root of f (p) = 0 is unique. By claims (b) and (c), it exists in (p0 ; 1):
For (ii): In the symmetric equilibrium, the accuracy of all bankers, pbs ; is to be found
by substituting b from (13) into (12); that is, pbs is the root for p of the following equation:
d0 (p) S(p)
1 S(p)
0
(
+ 1)(S (p) +
(
d(p) K
d(p) d(p) 1
It implicitly de…nes pbs as a function of K:
K)) = C 0 (p):
(30)
Denote the left hand side term by X(p; K). Then, by implicit function theorem,
db
ps
dK
=
db
ps
dK
< 0; it su¢ ces to prove
@X
@K
[ ( @X
@p
@X
@K
C 00 )] 1 : Obviously
@X
@p
< 0 as S(p) > 0 and d0 ( ) > 0. To prove
d(p) 1
(S(p)
d(p)
C 00 < 0: Let g(p; K)
+ K), that is,
b = g(b
p; K): Then, X(p; K) = Y (p; g(p; K)); where Y (p; b ) denotes the left hand side
of (12). Therefore,
Lemma 5) and
@g
@p
@X
@p
=
@Y
@p
+
@Y @g
@ b @p
<
> 0: It follows that
@Y
@p
, because
@X
@p
C 00 <
@Y
@p
@Y
@b
< 0 (to be shown in the proof of
C 00 : And
@Y
@p
C 00 < 0 holds true,
as it is the second order condition of maximization problem (11).
For Lemma 5:
Proof. Denote the left hand side term of (12) by Y (p; b ) as we did in the last
@ ( b ;K)
@K
proof. Then, by implicit function theorem,
@Y
@b
[ ( @Y
@p
that
show
that
@Y
@p
@Y
@K
Y
K
=
C 00 < 0 at p = ( b ; K)
[ ( @Y
@p
C 00 )]
1
and
@ ( b ;K)
@b
=
@Y
< 0.
@b
(d(e
p) 1)S 0 (e
p) b
+
[d(e
p) b (d(e
p) 1)S(e
p)]2
> 0 and
d(1) 1
S(1)
d(1)
pe. Therefore, to prove the lemma, it su¢ ces to
The former is straightforward. To show the latter, note
d0 (e
p)
(S b ) b
;
[d b (d 1)S]2
1)S 0 (e
p) > 0 and
b ) b f[d b
both terms are to be shown decreasing
b
[d(e
p) b (d(e
p) 1)S(e
p)]2
decreases with b for
(shown in the proof of Lemma 3) and thus bigger than
For the second, d0 (e
p) > 0 and f [d b(S(d
(S
@Y
@K
C 00 )] 1 : The second order condition of the maximization problem (11) implies
with b : For the …rst, (d(e
p)
b >
=
(d
1)S]2 g0 , (S
b)b
1)S]2
g0b < 0 , f(S
2 b )[d b
(d
1)S] < 2d(S
only if S > b , namely if bankers get positive surplus S
screened and that d b
(d
b ) b g0 [d b
(d(e
p) 1)
S(e
p):
d(e
p)
(d
1)S]2 <
b ) b : Note that
>0
b from each and every project
1)S > 0 in equilibrium by Lemma 3. Therefore, the last
inequality of the chain holds true if S
2 b < 0: If S
32
2 b > 0; the left hand side of that
2 b )d b < d(S
inequality is smaller than (S
b ) b < 2d(S
b ) b ; the right hand side.
For Lemma 6:
Proof. Suppose out of K + D units of fund under her deployment (K of her own and
D borrowed) the banker invests M in evaluated bad projects and K + D
M in evaluated
good ones.
In state , no projects succeed and no one gets anything.
In state 1, high type projects succeed, but low types fail. By the law of large numbers,
out of all the evaluated good projects, the proportion of high types is Pr(e
q = qje
s = g),
s = b): Hence, in
while that proportion out of the evaluated bad projects is Pr(e
q = qje
state 1; proportion Pr(e
q = qje
s = g) of the investment in the evaluated good projects
succeeds, so does the proportion Pr(e
q = qje
s = b) of the investment in the evaluated bad
projects. Success delivers $F for each dollar put in the former investment and $F 0 for
each dollar put in the latter investment. Therefore, the revenue of the banker in state 1
is (K + D
M ) Pr(e
q = qje
s = g) F + M Pr(e
q = qje
s = b) F 0 . And her liability is Df:
Her pro…t is then:
maxf(K + D
s = g) F + M Pr(e
q = qje
s = b) F 0
M ) Pr(e
q = qje
Df; 0g
b (M ):
In state 2, all her projects succeed. A dollar invested in an evaluated good project
returns $F and a dollar invested in an evaluated bad one returns $F 0 . Hence the banker’s
pro…t is
M )F + M F 0
(K + D
Df
g (M ):
Altogether, her expected pro…t, by investing M 2 [0; K+D] units fund in the evaluated
bad projects, is
(M ) = (q
q)
b (M ) + q
g (M ).
This function has two local maximizer,
M = 0 and M = K + D, as follows.
If M is small enough such that
b (M )
0, then
(M ) = (K+D M )F f(q q) Pr(e
q = qje
s = g)+qg+M F 0 f(q q) Pr(e
q = qje
s = b)+qg D qf:
Note that (q
similarly (q
q) Pr(e
q = qje
s = g) + q = q Pr(e
q = qje
s = g) + q(e
q = qje
s = g) = qg and
q) Pr(e
q = qje
s = b) + q = qb : Therefore, in this case,
33
(M ) = (K + D)qg F
M (qg F
qb F 0 )
D qf and decreases with M because qg F > qb F 0 by (14) (namely risk
shifting brings about a lower expected return rate). Thus the maximum occurs at M = 0:
If M is not so small that
b (M )
= 0; then
(M ) = q((K + D)F + M (F 0
F)
Df )
and increases with M because F 0 > F by (14) (namely risk shifting brings about a higher
face rate). Thus the maximum occurs for this case at M = K + D:
To prevent the banker from investing in the evaluated bad projects, it commands
(0)
(K + D), which, by substituting (14) for F 0 and noting F =
R
,
qg
gives rise to
(15).
The argument above shows also that, so long as bankers do not invest in any evaluated
bad projects, their debt is fully repaid in both states 1 and 2; thus with probability q:
For Proposition 3:
Proof. (i): As was seen at the beginning of the section, each banker wants to borrow
as much as possible, given there is a positive rate gap. Therefore, each banker borrows
to the upper bound, that is,
D = K L(pe );
(31)
where pe is households’s expectation of the accuracy; it is a functions of pe because
borrowing is assumed to occur before the accuracy choice is made.28
A particular banker then has K + D units of fund in hand to invest in entrepreneurs.
Her economic pro…t is (K + D) qg F
D qf
K: The expected rate charged, R = qg F;
as was shown in Section 3, is given by (9), in order to attract entrepreneurs coming to
her. Her problem of choosing accuracy p is:
max(K + D)
p
d(p) b
b
(d(p)
1)S(p)
D qf
K
C(p):
The accuracy chosen, denoted by pbL ; satis…es the same FOC as (12) except that K there
is replaced by K + D here, that is,
It implies (20).
28
(K + D) b
1)S 0 (p) a0 (p)( b S(p))
= C 0 (p):
2
b
[a(p)
(a(p) 1)S(p)]
(a(p)
(32)
If borrowing occured after the investment (but before the charging of rate), then D would be a
function of p (namely D = KL(p)) in the banker’s problem of choosing the quality. The results would
be qualitatively the same, though the analysis would be more complex.
34
On the other hand, the aggregate fund under bankers’deployment is expanded through
leverage from K to K + D, by (13), the equilibrium pro…t of entrepreneurs is
pL ) 1
b = d(b
(S(b
pL ) + K + D):
L
d(b
p )
(33)
Note that both in (32) and in (33), which capture respectively the increasing return to
scale and the general equilibrium e¤ect, D plays the same role as K. Thus, the accuracy
choice depends only on the asset sizes of bankers, not on the composition of their liabilities.
Put (32) and (33) together. The same characterization of equilibrium accuracy as (30) is
derived, except that K there is replaced with K + D here, which gives (18).
The last piece needed to complete the characterization of the equilibrium is to …nd L:
For this purpose, we …nd R by substituting (33) into (9), which gives
b
b = 1( S
R
+ 1):
db K + D
This together with (15), (16), (31), and pe = pbL , gives (19).
b L;
b pbL ) is characterized by simultaneous equations of (31) (with pe
Altogether, (D;
replaced by pbL due to rational expectation), (18), and (19).
(ii): Equation (18), which determines pbL ; can be derived from (30), which determines
pbs ; by substituting K + D for K: And K + D > K because L > 0: Hence by Proposition
2(ii), pbL < pbs :
(iii): It su¢ ces to show that in equilibrium the demand of bankers’fund by evaluated
b (given by 8), is smaller than the investment need, B: This
good entrepreneurs, I(b
pL ; R)
b It follows from (a) R
b > 1 and (b) qbb Z < B: R
b > 1 because
is equivalent to qbb Z < RB:
otherwise no bankers will spend in screening expertise, that is, all bankers choose p = 0;
which, by Lemma 3, cannot arise in equilibrium. (b) is equivalent to that evaluated bad
projects have a negative NPV, which holds true, otherwise the project of prior mean
quality would have a positive NPV and assumption (1) would be violated.
For Lemma 7:
Proof. First we complete the characterization of the equilibrium for this case of
heterogenous bankers. The demand of each evaluated-good entrepreneur for banker j 0 s
fund, by subscripting variables in (8) with j, is
Ij (Rj ) =
Vj
:
Rj dj 1
35
And the banker will induce neither over-subscription to her fund nor under-subscription.
Therefore,
j nj Ij (Rj )
where
j
(34)
= Kj ;
is the mass of the entrepreneurs who come to banker j, out of whom, proportion
nj is evaluated good and each demands $Ij of her fund.
The ex ante pro…t to the entrepreneur coming to her,
ables in (7), is Sj
(dj 1)Rj
;
dj Rj 1
(Rj ; pj ); by subscribing vari-
which equals b in equilibrium, that is,
Sj
(dj 1)Rj
= b:
dj Rj 1
Given b ; the banker’s deal (pj ; Rj ;
j)
(35)
is characterized by the simultaneous equations
(21), (34), and (35), with b settled down by the following market clearing condition:
Z
j = 1:
Then, to prove the lemma, it su¢ ces to show that if
j
is given, then
dpj
dKj
< 0: pj is
determined by (21), or equivalently the FOC, (12), with K replaced by Kj ; that is,
Given
j,
Kj b
1)S 0 (p) + d0 (p)(S(p) b )
= C 0 (p):
2
b
[d(p)
(d(p) 1)S(p)]
(d(p)
b is related to Kj through (34) and (35):
b = d(p) 1 (S(p) + Kj )jp=p :
j
d(p)
j
(36)
(37)
Substitute it into (36), and then …nd pj is related to Kj through
j
d2 (p)
(
j S(p)
Kj
d0 (p) S(p)
+ 1)[S (p) +
(
d(p) d(p) 1
0
Kj
)] = C 0 (p):
j
The left hand side decreases with Kj : Therefore, in the same way we proved Proposition
2(ii), we can show
dpj
dKj
< 0:
For Lemma 8:
Proof. It su¢ ces to show (a) that the if
satis…es (24); and (b) the
For (a): Note
maxp
d(p) 1
S(p):
d(p)
( b s ; K) < 0; there is a unique b m that
found above is strictly between 0 and 1.
(K; b ) decreases with b : By Lemma 3,
(K; b ) = 1; if
Therefore, given (K; b s ) < 0; there exists some b m between maxp
36
<
d(p) 1
S(p)
d(p)
and b s such that
with b :
(K; b m ) = 0. Moreover, this b m is unique because
For (b): Let f (p; x)
d(p) 1
(S(p)
d(p)
is the root of f (b
pm ; x) = b m : Note
+ xK): Then
f (p; x) increases with x: Therefore, that 1 >
(K; b ) decreases
> 0 is equivalent to that f (b
pm ; 1) > b m >
f (b
pm ; 0): The latter inequality holds because b m > maxp
d(p) 1
S(p)
d(p)
= maxp f (p; 0) >
f (b
pm ; 0): The former, due to b m < b s and b s = f (b
ps ; 1); is implied by f (b
ps ; 1) < f (b
pm ; 1);
which is equivalent to pbs < pbm ; as fp0 > 0: And pbs < pbm , (K; b s ) < (K; b m ) ,Lemma 5
b s > b m ; which was shown in the …rst half of the proof.
For Lemma 9:
Proof. Let f (K; b ; p)
K ( d(p) b
b
1) C(p): Then (K; b ) = max0
(d(p) 1)S(p)
p 1
f (K; b ; p):
And pbm satis…es the FOC fp (K; b m (K); p) = 0; where b m (K) is implicitly de…ned by
(K; b m ) = 0: Therefore, by the implicit function theorem,
The second order condition implies that fpp < 0: Therefore,
fpK + fp b
By the implicit function theorem,
db
dK
=
= f b : Therefore, (38) is equivalent to fpK
b
fpK
<
fK
because fK > 0 and f b =
and
@Y
@b
db
pm
dK
=
db
pm
dK
< 0 is equivalent to
1
fpp
(fpK + fp b
db
< 0:
dK
K
b
(38)
: By the envelop theorem,
fp b
fb
(39)
< 0 (see the proof of Lemma 5). Note that fpK = fKp
(
Then,
fK
fb
=
B( b B)
:
K(d 1)S
(fK )0p
fK
0 , (B(1
Sp
S
b<
Rp
(S
R
b 0
))
S p
fK 0
) < 0:
fb p
d(p) b
1, where B
Therefore, (40) is equivalent to:
(
Furthermore, ( b
B) = (d
< 0 ,
b ) , (27):
1)(S
Sp
S
= fK and
;
= (log fK )0p and the RHS equals
(log f b )0p : Therefore, (39) is equivalent to (log fK )0p < (log f b )0p , (log
b
B
K
fp b ffKb < 0 ,
fp b = ( f b )0p : Then the LHS of (39) equals
Calculate and …nd fK =
db
):
dK
b <
fK 0
)
fb p
<0,
(d(p) 1)S(p), and f b = K
B( b B) 0
) < 0:
(d 1)S p
b ); which, since
37
(d 1)S
:
B2
(41)
b ). Therefore, (41) is equivalent to ( B(S
S
Bp
(S
B
(40)
b
B
b) 0
)p
<
= R; is equivalent to
To see the su¢ cient condition, (28), …rst, note that both d and S increases with p,
1)S: Therefore (41), and thus (27), is implied by (B( b
and so does (d
(b
2B)Bp < 0jBp <0 , b
Lemma 5, (R
B))0p < 0 ,
2B > 0jB= b , R > 2: Second, as we explained following
R
1)K = N (p; b )(S(p)
b ); where N is the measure of entrepreneurs
screened by a banker with accuracy p, and Np > 0: Therefore, Rp K > N Sp , which
Rp
b 1
R
b
S
:
b
S bm
implies
b>
R
>
Sp
b b
S
,
(log R)0p
(log S)0p
>
b 1
R
b
R
b
S
b b:
S
Then, (27) is implied by
b 1
R
b
R
b
S
b b
S
>
bm
b bm
S
,
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