Download Benchmarking and Comparing Entrepreneurs with Incomplete

Finnish Economic Papers – Volume 20 – Number 2 – Autumn 2007
Benchmarking and Comparing Entrepreneurs with
Incomplete Information*
Juha-Pekka Niinimäki
Department of Economics, Helsinki School of Economics,
PB 1210, FIN-00101 Helsinki, Finland; e-mail: [email protected]
and
Tuomas Takalo
Monetary Policy and Research Department, Bank of Finland,
PO Box 160, FI-00101 Helsinki, Finland; e-mail: [email protected]
This paper studies how the creation of (ex ante) benchmarks and rankings can be
used to provide information in financial markets. Although an investor cannot
precisely estimate the future returns of an entrepreneur’s projects, the investor can
mitigate the incomplete information problem by comparing different entrepreneurs
and financing only the very best ones. Incomplete information can be eliminated
with certainty if the number of compared projects is sufficiently large. Because the
possibility to make benchmarks and comparisons favours centralised information
gathering, it creates a novel rationale for the establishment of a financial intermediary. (JEL: G21, G24)
1. Introduction
It is widely felt that incomplete information can
explain the existence of financial intermediaries
(Diamond, 1984 and Boyd and Prescott, 1986).
There are several means by which specialised
* We thank Andreas Wagener (the co-editor) and the two
referees for extensive comments. A previous version had a
third coauthor, Klaus Kultti, who dropped out due to time
constraints. We thank him for useful discussions. We also
thank Bengt Holmström, Ari Hyytinen, Marco Pagano and
audience in numerous seminars for helpful comments. The
paper was partly written while Takalo was visiting IDEI,
Toulouse. Financial support from Jenny and Antti Wihuri
Foundation, Säästöpankkien tutkimussäätiö and Yrjö Jahnsson Foundation is gratefully acknowledged.
outside investors can ease information frictions:
ex ante monitoring (Broecker, 1990), interim
and ex post monitoring (Holmström & Tirole,
1997; Diamond, 1984), collateral requirements
(Bester, 1985) and long-term lending relationships (von Thadden, 1995).1 The analysis is extended in this paper, which explores how making benchmarks and comparisons of potential
investment projects can help to mitigate – even
eliminate – the incomplete information problem
1
Freixas & Rochet (1997) and Gorton & Winton (2003)
provide thorough surveys of this literature.
91
Finnish Economic Papers 2/2007 – Juha-Pekka Niinimäki and Tuomas Takalo
in financial markets, and how this benefit of
benchmarking is conducive to centralised financial intermediation.
To conduct ex ante monitoring properly, outside investors ought to gather comprehensive
information on each entrepreneur or firm applying for funding and on their projects. For example, the skills and experience of the entrepreneur or firm’s management, the quality of the
business plan, the tangible and intangible assets,
the market potential of the products, and the
cost efficiency should all be examined in detail.
Based on this information, investors can evaluate the expected returns of the proposed project
and provide finance at appropriate (risk-based)
pricing. It is, however, not easy to assess expected returns precisely or to price risks correctly. In particular, in new markets where investors have little, or no, prior experience and
entrepreneurs’ assets are intangible, it is virtually impossible to make precise evaluations. In
more familiar sectors, finance for new entrepreneurs with no track record is difficult to price.
Even if entrepreneurs or firms seeking outside
finance are well known to the investors, changes in the economic environment may hamper
the rating of applicants and hence their funding.
In this paper we argue that investors can
overcome the difficulties in ex ante monitoring
by making benchmarks and comparisons. Although incomplete information problems in
funding entrepreneurs and their projects relate
to individual characteristics, learning from other entrepreneurs’ projects can help to uncover
the true distribution of project characteristics in
the economy. As a result, investors can compare
and rank the projects and choose the very best.
These projects will succeed with above-average
probability. Hence by making benchmarks and
comparisons, investors gather valuable information and thereby boost their investment yields.
Moreover, if an investor compares sufficiently
many entrepreneurs, the incomplete information
problem will be eliminated with certainty.
As an example, consider 10 de novo entrepreneurs from the same narrow high-tech sector.
Such entrepreneurs typically have fresh prototype products, their primary assets consist of
human capital and intellectual property rights,
92
and they lack funds for the investments required
to commercialise the prototypes. Any outside
investor will certainly find it a demanding task
to assess the value of a prototype, its expected
sales revenues, marketing strategy and ability of
entrepreneur, and so on. However, if the investor would contact each of the 10 firms and investigate their prototypes, intellectual property
portfolios, and entrepreneurial talent and then
compare these, she may discover crucial differences between entrepreneurs’ business plans.
The investor can rank the entrepreneurs and finance only the best ones. As a result, the investor can be quite confident that the best entrepreneurs’ products and plans are of sufficiently
high quality that the entrepreneurs will be able
to repay the funding.
To benefit from the creation of benchmarks
and rankings, however, an outside investor
needs to evaluate numerous entrepreneurs, even
if she can finance only one. With multiple investors operating in isolation, each entrepreneur
is evaluated and compared several times. This
duplicates the costs of information gathering.
Wasteful duplication could be avoided if the investors joined together to establish a financial
intermediary, which would evaluate each loan
applicant, compare and rank them, and finance
only the best ones. Each applicant is evaluated
only once, and inefficient duplication is eliminated. Consequently, the possibility to make
benchmarks and comparisons creates a novel
rationale for centralised financial intermediation and delegated information gathering. In this
respect our paper is related to the literature on
the role of information provision in explaining
financial intermediaries. Beyond the seminal
contribution by Leland and Pyle (1977), our argument is closely related those of Diamond
(1984), Ramakrishnan & Thakor (1984), and
Boyd & Prescott (1986). As in these papers, a
major advantage of forming a financial intermediary in our model is to reduce monitoring
costs. Finally, our intermediary engages in asset
transformation as in Diamond (1984) and Boyd
& Prescott (1986). Our work thus adds to the
long string of literature that extends the basic
insights on financial intermediaries as delegated
monitors and information gatherers into various
dimensions (see e.g. Krasa & Villamil, 1992,
Finnish Economic Papers 2/2007 – Juha-Pekka Niinimäki and Tuomas Takalo
Winton, 1995, Cerasi & Daltung, 2000, Hellwig, 2000, and Niinimäki, 2001).
Although the intermediary emerging from
our analysis is bank-like and we treat entrepreneurs seeking outside finance as loan applicants,
most of the analysis deals with a single investor
and needs not specify the form of the financial
contract. Moreover, the benefits of benchmarking are most evident in the finance of new ideas
in new markets, where debt contracts are less
predominant. Our paper therefore touches the
literature on entities that finance innovation
such as private equity and venture capitalists
(for an authoritative survey of venture capital
finance, see Gompers & Lerner, 2004). In particular, our study sheds light on the question of
why and how venture capitalists benchmark
their projects as in Bergemann & Hege (2002).
In contrast to Bergemann & Hege (2002), which
emphasises the value of ex post benchmarking
in alleviating moral hazard, project comparison
in our model provides ex ante benchmarks that
mitigate the incomplete information problem.2
Another connection with the works of Bergeman & Hege (1998, 2002) is that making benchmarks and comparisons can be seen as a special
form of costly learning or experimentation in
financial markets. Although benchmarks and
comparisons provide information, they are costly to make, since an investor must monitor numerous entrepreneurs to gather information.
Thus the investor has to weigh the opportunity
cost of monitoring yet one entrepreneur against
the future informational benefits, as in the theory of experimentation. The key difference versus the learning and experimentation literature
is that information derived from making benchmarks and comparisons exploits the differences
between entrepreneurs and does not require observations on the same entrepreneur over time.
2
Our model and some of the motivating examples have
also many similarities with Tirole (2006, Section 3.2.6).
Like Bergemann & Hege (2002) Tirole shows how ex post
benchmarking can be used to reduce moral hazard in entrepreneurial finance. We thank a referee for pointing out this
link to Tirole (2006). Our focus on ex ante benchmarking
also marks a difference to the related literature on relative
performance measurement stemming from Holmström
(1979). On performance benchmarking in practice, see e.g.
Siems & Barr (1998), Balk (2003) and Courty & Marschke
(2004).
Indeed, in our model no additional information
is gained by observing the same entrepreneur
more than once. In contrast, the experimentation and relationship lending literature (see e.g.
Sharpe, 1990, Boot, 2000, and von Thadden,
2004) emphasises that a bank can gather information trough multiple contacts with the same
borrower over time. As a result, the bank
achieves comparative information advantage in
lending against competitive investors. It has
also been shown in the literature (e.g. Pagano &
Jappelli, 1993) that banks can sometimes optimally share borrower specific information
among themselves.3
Since the problem of asymmetric information
and alternative suggestions to eliminate it play
important roles in the theory of finance, we have
mostly considered our analysis from this point
of view: a loan applicant knows the type of his
project, but the type is unobservable to investors. Alternatively, since there is no adverse selection in our model, it is possible to assume that
neither the loan applicant nor the investors know
the quality of the project. Each entrepreneur will
always search finance for his project since the
project incurs no costs to him. Whether or not a
loan applicant knows the quality of his project,
he will always search for finance and investors
aim to evaluate the project quality.4
3
In practice, learning by lending is combined with
benchmarking in credit scoring programs (see Thomas,
2000; Berger & Frame & Miller, 2005, and Blochlinger &
Leippold, 2006). Using a statistical program, a bank compares the information of a loan applicant to the credit performance of former borrowers with similar profiles. A
credit scoring program awards points for each factor that
helps predict who is likely to pay back a loan. A total
number of points – a credit score – helps evaluate how creditworthy a loan applicant is. Hence, credit scoring programs
offer an important instrument to handle information within
banks. Since the quality of the credit scoring system is increasing in the number of former borrowers, the programs
may generate information-releted economics of scale in
banking. Additionally, established banks with credit scoring
programs may have an information advantage against de
novo banks. Benchmarks and comparisons complete information that can be achieved using credit scoring and longterm lending relationships.
4
We thank an anonymous referee for pointing out the
possibility of recasting the model in terms of incomplete but
symmetric information.
93
Finnish Economic Papers 2/2007 – Juha-Pekka Niinimäki and Tuomas Takalo
The paper is organised as follows. In sections
2–3 we develop the main ideas and present the
costs and benefits of comparing using a simple
model with one investor and at most two entrepreneurs. In section 4 we consider a more general environment where the number of potential
entrepreneurs can increase without a bound. In
section 5 we allow for multiple investors and
show how comparing can explain the existence
of financial intermediaries. Section 6 concludes.
2. The economy
2.1 Financial market participants
In the basic model there are, N ∈Z +, risk-neutral
entrepreneurs (= loan applicants) and one riskneutral investor. In section 5 we allow for multiple investors. The entrepreneurs lack funds,
but each has a project that requires a fixed startup investment of unit size. The funding can be
obtained from the investor (she), who has a unit
of capital but no project of her own. The project
of an entrepreneur (he) is good with probability
g and bad with probability 1 – g. Project quality
is unobservable to outsiders and, without risk of
confusion, we refer to good and bad entrepreneurs. A good project yields a transferable income Y with certainty, and a bad project only
generates a non-transferable private benefit B to
the entrepreneur.
We assume that contacting an entrepreneur is
costly. This cost, denoted by c, includes, e.g.,
the costs of waiting or searching for an entrepreneur, evaluating and monitoring his project,
making the funding decision, and writing the
funding contract or informing of the rejection
of funding application. The cost occurs when
the investor contacts and monitors an entrepreneur, and it cannot be avoided. A contact provides an informative signal about the entrepreneur’s type, which will be specified in the next
subsection.5
5
Although we believe that unavoidable contacting and
monitoring costs are empirically relevant, the assumption
is essentially a short-cut. We could have regarded c as a
signal extraction expense and assumed that it can be avoided but only at the cost of not receiving the signal. This
94
Since there is only one investor we assume
that the investor has full bargaining power. Besides simplifying the analysis, the assumption
is convenient when we look more closely at the
benefits of centralised financial intermediation
with multiple investors in section 5, since the
assumption implies that investors have no incentive to form a financial intermediary to gain
market power. The assumption also means that
the form of financial contract is indeterminate:
the investor can seize the entire output of a good
project, Y, by driving the entrepreneur to the
zero profit level. It is assumed that Y – r – c > 0
where r ≥ 1 denotes the economy’s risk-free interest rate (investor’s opportunity cost). Hence
a good project has a positive net present value.
In contrast, a bad entrepreneur is assumed to
have a project with negative net present value,
i.e., B – r < 0.6
2.2 Signals
Upon contacting an entrepreneur, the investor
receives a signal on the entrepreneur’s type.
Each signal is informative in itself but, more
importantly, they can be used as a benchmark
with which subsequent signals will be compared. The signal can originate from any information reflecting profitability of the entrepreneur’s project. For brevity, we assume the signal
can take only three values: s1, s2, and s3 where
s1 > s2 > s3. Besides entrepreneur’s type, the
value of a signal depends on the state of the
world:
• With probability h, the state of the world is
high, in which case a good entrepreneur’s signal is invariably s1 and a bad entrepreneur’s
signal is invariably s2.
• With probability 1– h, the state of the world
is low, in which case a good entrepreneur’s
signal is invariably s2 and a bad entrepreneur’s signal s3.
Then, on average, the value of a signal is larger
in a high state of the world than in a low state
would have complicated the analysis by adding one layer
to the decision problem of the investor.
6
Strictly speaking we do not need the assumption that
B < r, but it is more appropriate to label entrepreneurs bad
if their projects have negative net present value.
Finnish Economic Papers 2/2007 – Juha-Pekka Niinimäki and Tuomas Takalo
of the world. For simplicity, we do not allow the
state of the world to affect the project return but
only the value of the signal. Because the investor cannot receive the signal s3 from a good entrepreneur and the signal s1 from a bad entrepreneur, the signals s1 and s3 are perfectly informative in that they fully reveal both the state of the
world and type of loan applicant. Signal s1 tells
the investor that the state of the world is high
and the entrepreneur is good. Similarly, signal
s3 says that the state of the world is low and the
entrepreneur is bad. After observing either s1 or
s3, the investor operates under perfect information.
Signal s2 is more interesting. It indicates a
bad entrepreneur in a high state of the world or
a good one in a low state. Although the investor
can use the signal s2 to update her prior beliefs
about the state of the world and entrepreneur’s
type, incomplete information remains after observing such a signal. After observing s2 from
the first loan applicant, the investor must accept
or reject the application without knowing the
entrepreneur’s type, or she can use the first signal as a benchmark and gather more information by contacting another entrepreneur and
comparing him with the benchmark. If the second signal is s1, the comparison of s1 with the
benchmark s2 reveals that the state of the world
is necessarily high and that the first entrepreneur is bad and the second is good. If the second signal is s3, the investor learns that the state
of the world is low, the first entrepreneur is
good and the second is bad.
If the investor receives s2 from the second
applicant, we still have incomplete information.
However, even in this case, seeking a second
loan applicant and comparing with the benchmark yields useful information, since the investor can update her beliefs about the state of the
world and entrepreneur’s type. Thus, although
the investor still must make a lending decision
under incomplete information, she is better informed.
Note that it is optimal to contact the loan applicants sequentially since this ensures that each
contact provides some useful information and
no costs of contacting are wasted. Since contacting a new loan applicant is costly, the investor encounters an optimal stopping problem
each time she receives the signal s2. We nonetheless assume for brevity that the contacts take
place fast enough so that the state of the world
remains unchanged and that previous contacts
can be recalled. As a result, if the first signal is
s2 but the second is s3, the investor optimally
returns to the first entrepreneur and grants him
a loan.
In sum, the signalling technology is simple:
there are only three signals and they are informative. Two of the signals are actually very informative: good and bad signals perfectly reveal
the borrower type. Despite this simple signalling technology, it is more informative to compare several entrepreneurs than just to observe
the same entrepreneur over time, since the signals can vary according to the state of the world.
Subsection 3.4 cites examples where signals
depend on the state of the world.
2.3 The timing of events
The investor’s decision problem has two stages.
In the first stage, the investor who may have
contacted some entrepreneurs previously faces
three options. First, she can exit the credit market without lending to anyone. Second, she can
grant a loan to any of entrepreneurs she has
contacted previously. Finally, she can pool the
previous signals with her prior to form a benchmark and invest c to acquire more information
by comparing a new loan applicant with the
benchmark. In the second stage, the investor
will collect her payoffs from exit or lending decisions. In case the investor decides to contact
a new entrepreneur, she will receive a signal on
the entrepreneur’s type and updates her belief
about the entrepreneur’s type and the state of
the world. In this case the investor again faces
the aforementioned three options.
3. Benchmarking and comparing with
two loan applicants
In this section, we first consider a loan market
with perfect information. We then further develop concepts in the context of a simple example where there is only one loan applicant in the
economy. Finally, we introduce a second loan
95
In this section, we first consider a loan market with perfect information. We then further develop
concepts in the context of a simple example where there is only one loan applicant in the economy.
Finnish Economic Papers 2/2007 – Juha-Pekka Niinimäki and Tuomas Takalo
Finally,
we to
introduce
a secondthe
loan
applicant
to demonstrate
the value ofinformation
benchmarking.
applicant
demonstrate
value
of bench3.2 Incomplete
with one loan
marking.
applicant
If the economy consists of just one entrepreneur, the investor has to decide first whether to
incur
andfirms
contact
and then
Under
perfect
information
the
investor
can
sepUnder perfect information the investor can separate good
fromc bad
and the
grantentrepreneur
a loan to a good
arate good from bad firms and grant a loan to a whether to grant a loan to the entrepreneur givher updated
belief
about
goodWe
one.
We assume
that
under
perfect infor­
one.
assume
that under
perfect
information,
lendingen
is profitable
to the
investor,
i.e.,the entrepreneur’s
mation, lending is profitable to the investor, type. With probability hg, the signal is s1. With
this signal, the investor knows that the entreprei.e.,
11
neur is good and she grants him a loan. With
(1) g (Y − r ) − c > 0 .
probability (1–h)(1–g) the signal is(1)s3 and the
entrepreneur's
type,that
but the
incomplete
information
remains. In suc
investor knows entrepreneur is bad
11
As explained, g in (1) is the probability of con- and she does not grant a loan. With probability
11
grant a loan in the presence of uncertainty about
h (1–g) + (1–h)g
the signal is s2 and the investor the type o
tacting a good entrepreneur. In such case the not
As
explained,
g in the
(1) is
the probability
of contacting
goodupdate
entrepreneur.
In
such
case
the information
investor type,
but11
incomplete
remains. In suc
10 aentrepreneur's
can
hertype,
belief
about
entrepreneur’s
investor
can reap
entire
output of the
project,
loan to the investor
with
one entrepreneur
in the economy
is
entrepreneur's
type,
but
incomplete
information
remains.
Ingive
suc
Y ; r denotes the opportunity cost of invested but incomplete information remains. In such a
can reap the entire output of the project, Y ; r denotes the
opportunity
cost
of
invested
capital
and
c
not
grant
a
loan
in
the
presence
of
uncertainty
about
the
type
o
case,
the
mayremains.
or may
not
grant
a loan
capital
the cost
of a contact.
first
entrepreneur's
type,
buthas
incomplete
information
In
such
aby
case,
the investor
the
cost and
of a ccontact.
Suppose
first thatSuppose
the economy
only
oneainvestor
entrepreneur,
who
is contacted
not
grant
loan in the presence
of uncertainty
about
the typema
o
that the economy has only one entrepreneur, in the presence of uncertainty about the type of
loan to the investor with one entrepreneur in the economy is give
entrepreneur.
Thus
the
whoinvestor.
is contacted
the investor.
probabilnot
a loan
inlearns
the presence
uncertainty
entrepreneur.
{hgwith
(Yabout
) +the
} v
= max
−one
rvalue
[ha(type
1of
− gaof
) +loan
(1in− the
hto
) geconomy
]the
v R ( s 2Thus
) − cis, 0the
the
With by
probability
g, grant
theWith
investor
that
the
entrepreneur
represents
good
type.
loan
toVof
the
entrepreneur
give
N =1 investor
ity g, the investor learns that the entrepreneur investor with one entrepreneur in the economy
loaninvestor
to the
investor
with
entrepreneur
in the
economy
is given
by
is
by a good
represents
good type.
The
finances
The
investorafinances
the project
and
makes
profit
Y −one
r −given
c , since
project
always
succeeds.
the project and makes profit Y – r – c, since a
V N =1 = max{hg (Y − r ) + [h(1 − g ) + (1 − h) g ]v R ( s 2 ) − c, 0}
With
1-g, succeeds.
the entrepreneur
proves to be bad.
investor
does
lend
good probability
project always
With probability
where
N{=1
(ofY −Vnot
Vsubscript
hg
ris) +the
[h(number
1 −toghim.
) + of
(1 In
−entrepreneurs
h) g ]v R ( s 2 ) −in
c, the
0} e
(2) The
N =1 = max
1–g, the entrepreneur proves to be bad. The in(Yrisk-free
} bears
V N =1 = max
− r ) + [hinterest
(1 − g ) +rate
(1 − of
h) gthe
]v Reconomy
( s 2 ) − c, 0and
contrast,
the investor
hisInendowment
at{hg
the
vestor does
not lendinvests
to him.
contrast,
the
investor invests his endowment at the risk-free where subscript N=1 of V is the number of entrepreneurs in the e
the
cost of
contacting.
Thus, theand
investor’s
returns
− c . Inv subscript
sum, with N = 1
probability
g− the
investor of en( gVof
s 2is)Vthe
Yis
rthe
, 0}number
where
interest
rate
of the economy
bears the
costare where
subscript
N=1{pof
number
of entrepreneurs in the e
R (s 2 ) = max
of contacting. Thus, the investor’s returns are trepreneurs in the economy and
probability
1-g
− cnumber
. The expected
returns ofinthe
makes profits Y − r − c and with
where
N=1she
of loses
V is the
of entrepreneurs
theinvestor
economy and
– c. In sum, with probability
gsubscript
the investor
makes profits
with isprobability
1–g to (1).
(3) v R (s 2 ) = max{p ( g s 2 ) Y − r , 0}
r ) − c >and
positive owing
amount
to g (Y −Y – r – c
0 , which
{punder
v R (s 2 )of= amax
( g s 2 )uncertainty,
Y − r , 0} given the signal s 2 . In (3
loan
she loses – c. The expected returns of the inves- is the value
is
the
value
of
a
loan
under
uncertainty,
tor amount to
g (Y – r) – c > 0,
which
is
positive
Suppose now that the
entrepreneurs until
a good given
{p( gcontact
v R (investor
s 2 ) = maxcan
s 2 ) Y −numerous
r , 0}
the signal s2. In (3)
owing to (1).
Suppose now
thatThe
the expected
investor value
can contact
is the
valueisof ga (loan
entrepreneur
appears.
of the each
contact
) − c >uncertainty,
0 to her. Ifgiven
Y − runder
the the signal s 2 . In (3
(1 − huncertainty,
)g
the value of a loan under
given the signal s 2 . In (3
numerous entrepreneurs until a good entrepre- is
(
)
=
p
g
s
(4)
2
h(1 − g )to+ (1[g−(Yh)−g r ) − c ] +
neur appears.
expected value
the each expected
profits given
economy
has nTheentrepreneurs,
theofinvestor’s
amount
is the value
of a loan under
uncertainty,
the signal s 2 . In (3)
contact is g (Y – r) – c > 0 to her. If the economy
(1 − h) g that signal s2 indihas
the conditional
probability
(1 − ng )[entrepreneurs,
g (Y − r ) − c ] + (the
1 − ginvestor’s
) 2 [g (Y − r ) expected
− c ] + .... . , is
which
pis( gequal
s2 ) =to
(g1 )−+h()1g−As
profits amount to [g (Y – r) – c] + (1–g)[g (Y – r) cates pa(good
entrepreneur.
h
(
1
−
h) g(3) shows,vR(s2)
g s2 ) =
indicates a good ent
conditional
hprobability
(1 − investor
g ) + (1 −that
hfinances
) gsignal sa2 project
– c] + (1–g)2[g (Y – r) – c] +… . , which is equal to(1 −is
ish)the
positive
since
the
g
p ( g s2 ) =
only if its NPV is positive. Note that the posih(1 − g ) positive
+ (1 − h) gsince the investor finances a project only if its NPV is p
tive value of vR(s2) has nothing to do with lim(
−
)
−
g
Y
r
c
[1 − (1 − g ) n ]
.
s 2 context
indicates a good ent
is
theliability,
conditional
probability
signal
ited
which
plays nothat
role
in this
g
of
v R (conditional
sthe
nothing
to do
withher
limited
s 2 liability,
indicates
a goodplay
ent
is
the
probability
that
signal
2 ) has
since
investor
uses
only
own
funds. which
positive
since
thevalue
investor
finances
a project
only
if
its NPV is p
Similarly,
the
of
a
loan
given
by
(2)
is
This is always positive when
(1)
is
satisfied.
s 2 investor
indicates
afunds.
good Similarly,
shows,
is the conditional probability
that signal
investor
uses
her
ownfinances
the As
value
of
a is
loan
positive
the
aentrepreneur.
project
only
if
its(3)NPV
p
positivesince
sinceonly
the
investor
contacts
the entrepre(s 2 ) has
of
v R only
nothing
to doreturns
with limited
liability,
which play
neur
if
the
expected
from
the
con- that
This is always positive when positive
(1) is satisfied.
since the investorof
finances
project
only
if its
NPV
islimited
positive.
Note
the play
posif
investor
contacts
the
entrepreneur
only
the
expected
returns
(s 2 positive.
) ahas
nothing
to
doand
with
liability,
tactv Rare
From
(1)
(2)
it isif clear
that which
investor usesinformation
only her own funds. the
Similarly, of
the value of a loan
of v (s ) has nothing toincomplete
do with limited liability,reduces
which playsvalue
no role the
in this context
3.1 Perfect
Perfectinformation
information
3.1
(1) and (2)
it is
clear
incomplete
information
va
investor
uses
only
herthat
own
funds. Similarly,
the reduces
value ofthe
a loan
3.2
with one loan applicant investor contacts the entrepreneur only if the expected returns f
96 Incomplete information investor
uses only her own
funds.information
Similarly,
value of never
a only
loanhas
by (2)
is returns
positive
perfect
investor
to risk
of losing
her fc
investor
contacts
thethe
entrepreneur
ifgiven
the
expected
(1) and has
(2) ittoisdecide
clear that
reduces the va
If the economy consists of just one entrepreneur, the investor
first incomplete
whether to information
incur c
investor contacts the entrepreneur
only
the expected
from
thesignal
contact
posit
information
takes
suchincomplete
riskreturns
when information
the
first
is sare
and
vR
(1)
and (2)
itshe
isifclear
that
reduces
val
2 the
perfect
information
the
investor
never
has
to
risk
of
losing
her
c
and contact the entrepreneur and then whether to grant a loan to the entrepreneur given her updated
(1) and (2) it is clear that perfect
incomplete
information
reduces the
value
opportu
information
the investor
never
hasoftothe
risklending
of losing
her c
R
2
v(s3 ) = g (Y − r ) − c > 0 .
v(s3 ) = g (Y − r ) − c > 0 .
Note that the value of a loan stems entirely from 13
the possibility of c
Finnish Economic Papers 2/2007 – Juha-Pekka Niinimäki and Tuomas Takalo
Note that the value of a loan stems entirely
theispossibility
of case
comparing
the entrepre
The firstfrom
signal
s . In this
the investor's
prob
2
lending opportunity: with perfect information Note
max{g(Y – r) – c,
by (1),
v(s3 ) that
= g (the
Y −value
r ) − c0}.
. g (Y – r) – c > 0
of> a0As
loan
stems entirely from
the possibility
of c
13
signal
is s 2 . In this case the investor's problem is more complica
the investor never has the risk of losingThe
herfirstwe
obtain
second signal is also s 2 , the investor cannot be sure about the e
capital (r ) whereas with incomplete information
The first signal is s 2 . In this case the investor's prob
v(sinvestor
− r ) − cbe
> 0sure
. about the entrepreneurs' type and
(5)
she takes such a risk when thesecond
first signal
s 2 , the
signal is
is salso
2
3 ) = g (Y cannot
updatethat
herthe
beliefs.
The
value
of a loan
underfrom
uncertainty
when both
Note
value
of
a
loan
stems
entirely
possibility
of ce
and vR(s2) is strictly positive.
cannot bethesure
about the
second signal is also s 2 , the investor
value
ofuncertainty
a loan stems
entirely
from are s 2 is given
update her beliefs. TheNote
valuethat
of athe
loan
under
when
both signals
Theoffirst
signal
is the
In under
this case
the investor's
prob
the possibility
comparing
2 . entrepreneurs.
update
her beliefs.
The
value of
as loan
uncertainty
when both
3.3 Incomplete information with two loan
Note
that
the value
of
a loan
stems
entirely
from the possibilit
The
first
signal
is
s
.
In
this
case
the
inves2
applicants
(s 2 , s 2 )isis= also
)Y − r ,If0cannot
}the second
v Rsignal
max{p (,g the
s 2 , sinvestor
2
be sure about the e
second
tor’s problem
morescomplicated.
2
signal cannot
is s 2 . Inbethis
case the investor's
the first
investor
sure
With two entrepreneurs, the investorv R must
(s 2 , s 2 first
) = maxsignal
{p(g s 2 ,iss 2 )also
Y − r ,s02}, The
update
her
beliefs.
The
value
of
a
loan
under
uncertainty
when both
decide whether to contact either of the entrepre- aboutvthe
(s 2entrepreneurs’
, s 2 ) = max{p (g stype
, s 2 )Yand
− r can
, 0} only upR
2
, the
investor
cannot
second
signalThe
is also
beliefs.
values 2of
a loan
under
un- be sure about
neurs. If she contacts one of them, she has to date her
where, analogously to (4),
decide whether to grant a loan to him or use him certainty when both signals are s2 is given by
as a benchmark. By creating the
benchmark
the to (4), update her beliefs. The value of a loan under uncertainty when
where,
analogously
(s 2 , s 2 ) = max
p (g s 2 , s 2 )Y − r , 0}
(6) vanalogously
investor can contact a second entrepreneur and where,
R
to {(4),
compare him with the benchmark. Then the in(1 − h) g 2
g s2 , s2 ) =
to (4),2
vestor must decide whether to grant a loan to where,p(analogously
h(1 − g ) + (1 − h) g 2
(1 − h) g 2v R (s 2 , s 2 ) = max{p (g s 2 , s 2 )Y − r , 0}
either or neither of the two entrepreneurs.
Since
p (g s , s ) =
the investor has only one unit of capital, 2he 2can h(1where,
− g ) 2 analogously
+ (1 − h) g 2 to (4),(1 − h) g 2
(7) p(g s2 , s2 ) =
not finance both projects even when both of
h(1 − g ) 2 + (1 − h) g 2
them are good.7 If the first signal is either s1 or is the conditional probability that signal s indicates a good entre
analogously
to (4), that signal2s2 indis3, the investor’s decision problem is straight- is thewhere,
conditional
probability
(
1 − h)after
g a2 good
s 2 indicates
after two ob
is the
conditional probability
forward so we first characterise
them.
cates a that
goodsignal
entrepreneur
twoentrepreneur
observaAgain,pv(gR (ss22,,ss22)) =is positive
since
the investor
finances a project onl
2
(12−) gis) 2positive
(1 − hsignal
) gsince
+that
The first signal is s1. If the signal from the is
tions.
Again, vR(sprobability
in- a good entre
the conditional
s 2 the
indicates
2h, s
) is positive
Again, knows
v R (s 2 , sthat
the investor
finances
a project
first entrepreneur is s1, the investor
vestorsince
finances
a project
only
if 2its only
NPVif its
is NPV is positiv
2
If the signal received
the second entrepreneur is
(1 −the
) ginvestor
hfrom
he is good. Since the investor has only one unit Again,
positive.
v R (s 2p, (sg2 )s2is, spositive
since
finances
a project onl
)
=
2
2
(1 − entrepreneur
1 − h) gis2 entrehfrom
g )the
+ (second
of capital, there is no need to contact the second
the signal
received
s1 , the investor know
If the signalIfreceived
from
the second
is dealing
with a good
entrepreneur
and grants
a loan. aIfgood
the secon
s 2 second
indicates
entre
the conditional
probability
that
signal
entrepreneur, and the investor grants a loan to is
preneur
is sIf1, the
the
investor
knows
she
is entrepreneur
signal
received
fromthat
the
is
the first entrepreneur, which isyields
Y – r
with
dealing
with
a
good
entrepreneur
and
grants
a
dealing with a good entrepreneur and grants a loan. If the second contact yields s 3 , th
knows
is low, the
entrepreneur
isfinances
bad andathe
first onl
en
(sthe
Again,
vthe
, s 2state
is positive
since
thesand
investor
certainty.
loan.
Ifthat
contact
yields
investor
R with
2second
3, the
is
dealing
a) good
entrepreneur
grants
a loan. Ifproject
the secon
s
indicates
a
good
is
the
conditional
probability
that
signal
2 is
The first signal is s3. If theknows
signal
received
knows
that
the state isis low,
the the
entrepreneur
that
the state isthus
low,grants
the
entrepreneur
bad and
first entrepreneur
is good.
a If
loan
tosignal
the first
entrepreneur.
When
the
first
signalTh
the
received
from
the
second
entrepreneur
isi
upon contacting the first entrepreneur is s3, the knows
bad and
the
first
entrepreneur
is
good.
The
inthat the state is low, the entrepreneur is bad and
the first en
(
)
Again,
v
s
,
s
is
positive
since
the
investor
finances
a
proje
entrepreneur is bad with certainty.
the second
vestor
grants
loan to
entrepreR
2
2a When
thus If
grants
a loan to the
firstthus
entrepreneur.
thethe
firstfirst
signal
is s 2 , the value of a lo
written as
is
dealing
with
a good
entrepreneur
and value
grants
a loan.
If thesignal
seconi
contact also yields s3, the investor does not lend. thus
neur.
When
first
signal
is s2, the
of
athe first
grants
a the
loan
to the
first entrepreneur.
When
If the
If the second signal is s2, the investor
as signal received from the second entrepren
written ascan elim- loan can be written
inate the incomplete information problem by written
knows that
and the first en
as the state is low, the entrepreneur is bad
14 a loan. If the
with
a(sgood
entrepreneur
and
grants
comparing signals. Because the first signal, s3, (8) isv(sdealing
)
{
)
(
)
(
)
(
=
max
v
,
t
Y
−
r
+
1
−
t
v
s
,
s
2
R
2
2
2
R
2
2 ) − c, 0} .
reveals that the state of the world is low, the thus grants a loan to the first entrepreneur. When the first signal i
)(h+ s(12the
)v[1R−(sisp2 ,(low,
}g .) is the probability
v(sa2 )worthy
= max{v Rwhere
t 2 t(Y =− rpthat
second entrepreneur with s2 must be
is bad and that
the fi
)−gt +2state
(c1, −0entrepreneur
hs 2s)2−)]the
(s 2 ), knows
2
s 2 ) = max{v R (s 2 ), t 2 (Y − r ) + (1 − t 2 )v R (s 2 , s 2 ) − c, 0} .
borrower. Since the probability that the second writtenv(as
grants
loan
to the signal.
first
entrepreneur.
When the first si
wherethus
t2 = p(h
| sa2)g + [1– p(h
| s 2)](1– g)
entrepreneur emits s2 is g, the value of a loan eliminated
by using
the second
Here is the
when s3 is the first signal is given by v(s3) = probability that incomplete information can be
writtenby
as using the second signal. Here
eliminated
v(s 2 ) = max{v R (s 2 ), t 2 (Y − r ) + (1 − t 2 )v R (s 2 , s 2 ) − c, 0} .
7
By assuming that an investor has only a unit of capital
in this section, we will emphasise that the investor’s sole
reason to contact numerous entrepreneurs is to gather more
information. As a result, it is possible to explore optimal
stopping in information gathering. The analysis is extended
in section 5, where the economy has numerous investors and
entrepreneurs.
(9)
h(1 − g )
= 1 − p (g s 2 )
h(1 − g ) + (1 − h) g
v(s 2 ) = max{v R (s 2 ), t 2 (Y − r ) + (1 − t 2 )v R (s 2 , s 2 ) − c, 0}
p(h s 2 ) =
is the conditional probability that the state is
high when the investor observes signal s2. As it
stands,
(8) showsprobability
the three choices
the when the inve
is
the conditional
that thefaced
state by
is high
stands, (8) shows the three choices faced by the 97
investor after obse
First, if p (g s 2 )Y − r < 0 implying v R ( s 2 ) = 0, and if t 2 (Y − r
investor does not lend at all and v(s )=0. That is, the investor can a
is the conditional probability that the state is high when the investor observes signal s 2 . As it
stands, (8)
shows the
three 2/2007
choices –
faced
by the investor
after and
observing
s 2 Takalo
from the first contact.
Finnish
Economic
Papers
Juha-Pekka
Niinimäki
Tuomas
investor after observing s2 from the first contact. comparing entrepreneurs and updating her beFirst, if p (g s 2 )Y − r < 0 implying v R ( s 2 ) = 0, and if t 2 (Y − r ) + (1 − t 2 )v R (s 2 , s 2 ) − c < 0 , the
First, if p(g | s2) Y – r < 0
implying vR(s2) = 0, and liefs. The cost of benchmarking is on the RHS
if t2(Y – r) + (1– t2)vR(s2, s2) – c < 0, the investor of (10). Besides the cost of contacting, the opinvestor
at allv(s
and
v(s2)=0. That is, the portunity
investor can
achieve
zero returns by
omitting
cost
of benchmarking
stems
from the
does
not does
lendnot
at lend
all and
2) = 0. That is, the
investor 8can achieve zero returns by omitting possibility to grant a loan under uncertainty
lending. Otherwise, the maximisation problem on the right-hand side of (8) reflects the investor's
lending.8 Otherwise, the maximisation problem about the type of the first entrepreneur. For the
g > 1/2,
it isdecides
easy totosee
why
part under
i) of Propon
the right-hand
of (8)the
reflects
inves- case
choice
of whether side
to contact
secondthe
entrepreneur.
If the
investor
grant
a loan
tor’s choice of whether to contact the second osition 1 must be true, since then Y – r > vR(s2,
) ≥ vR(s2),payoff
i.e., the
payoff
entrepreneur.
If the
investor
decides
to grant a hers2expected
(s2 ) .benchmarking
From (8)
uncertainty after
contacting
only
one entrepreneur,
is given
by vfrom
R
loan under uncertainty after contacting only one clearly exceeds the value of the loan under uncertainty.
entrepreneur,
her
payoff the
is given
we observe that
theexpected
investor contacts
secondby
entrepreneur
if But it turns out that the same concluvR(s2). From (8) we observe that the investor sion holds even for g < 1/2, so the benefits exceed the cost of benchmarking when the cost of
contacts the second entrepreneur if
contacting is sufficiently low.
As part ii) suggests, the investor may
(10) prefer
(10) t 2 (Y − r ) + (1 − t 2 )v R (s 2 , s 2 ) ≥ c + v R (s 2 ) .
to grant a loan under uncertainty about the type
In Appendix we prove that the investor’s opti- of the first entrepreneur over using him as a
mal lending strategy after observing s2 satisfies benchmark if the cost of contacting is suffi9
In Appendix
we prove that
the investor's optimal lending
afterstrategy
observingmay
s 2 satisfies
the
followingAconditions:
cientlystrategy
high. The
also be the
optimal
with moderate or even with low contacting cost
9
following conditions:
Proposition
1 Suppose
that the first loan ap- if the first entrepreneur, given signal s2, is good
plicant’s type is unknown (signal s2). Then,
with sufficiently high probability, that is, when
g is close to one or h is close to zero. In such
i) if the contacting cost (c) is sufficiently low,
circumstances, incomplete information causes a
the investor
contactsthat
a second
appliProposition
1. Suppose
the firstloan
loan applicant's
type isnuisance
unknown (signal
). Then, benefits of
minor
and thes2 expected
cant and compares him with the first one,
16
benchmarking do not cover the cost of contactii) if c is sufficiently high and if either the
ing.
16
prior probability that a loan applicant has a
8
Partcontact
iii) are
ofPart
Proposition
iii) of Proposition
1sunk
canbe
alsoseen
be seen from (10
Thegood
investor
can
achieve
zero
returns,
since
the
sunk
costs
of
the
first
not
included
in v (1s 2 )can
. Thealso
project (g) is sufficiently high or if
from (10). The investor prefers not to lend at all
prior
probability of a Part
highiii)
state
(h) is
coststhe
appear
in (11).
of Proposition
1 can
also
be
(10).
The
investor
to lend
after
signal
s2 seen
ifs2 the
expected
return
on
aa loan
iffrom
the
expected
return
onprefers
loan not
under
uncera
all
after
signal
9
sufficiently
low to
renderand
theTakalo,
net present
A working
paper version
(Niinimäki
2007) includes
extensive
numerical
examples
showing
that
all
result
under uncertainty is not positive and the cost of
value of a loan under
uncertainty
thepositive
expected satisfying
return
onequation
a loan
under uncertainty
is
not positive
and the cost o
after signal
2 if simultaneously
exist for some plausibleall
parameter
values swhile
(1).
contacting
is
hightotorender
render
a search
contacting
is sufficiently
sufficiently
high
a search
for a second en
(vR(s2) > 0), the investor grants a loan to the
for a second entrepreneur unprofitable.
first applicant.contacting is sufficiently high to render a search for a second entrepreneur unprofitable.
a loan.
Weabove
have above
The value The
of avalue
loan.ofWe
have
deter-determined t
iii) if c is sufficiently high and if either g is
mined the conditional value of a loan for the
sufficiently low or h sufficiently
to We
The value high
of a loan.
have
above determined
thefirst
conditional
value
of a loan for th
three
possible
values
the
signal.
The
initial
three
possible
valuesof of
the first
signal.
The value of a
render vR(s2) ≤ 0, the investor does not lend
initialThe
value
of avalue
loanof– athe
investor’s
expected
three possible values of the first signal.
loan
– theapplicant
investor's
return
at all.
beforebefore
sheinitial
hasshe
contacted
either
loan
isexpected
then given
by
return
has contacted
either loan–apThe LHS of (10) shows
the has
benefits
of bench– is– then
given
before she
contacted
either loanplicant
applicant
is then
givenby
by
marking: with positive probability (t2) the second contact will probably eliminate the incom- (11) V = hg (Y − r ) + [h(1 − g ) + (1 − h )g ]v( s ) + (1 − h )(1 − g )
2
N =2
plete information problem and, even if it does
(
)
(
)
(
)
(
)(
)
V
=
hg
Y
−
r
+
[
h
1
−
g
+
1
−
h
g
]
v
(
s
)
+
1
−
h
1
−
g
v
(
s
)
−
c
.
(11)
not, the investor can gatherN =more
information by 2
2
3
(5),and they giv
The investor can achieve zero returns, since the sunk InIn(11)
(11) v(s
v( s22) )and
andv(s
v(3s)3 )are
aregiven
givenby
by (8)
(8) and
and (5),
and
they
give
the
values
of
a
loan
to
the
investor
costs of the first contact are not included in v (s2). The sunk
costs appear in (11). In (11) v ( s 2 ) and v( s 3 ) are given when
by (8)the
andfirst
(5), signals
and theyare
give
a loanofto the investo
s the
andvalues
s . Theofvalue
9
the first signals are s22 and s33 . The value of a loan after
A working paper version (Niinimäki and Takalo, 2007) awhen
loan after the investor receives
signal s1 from
includes extensive numerical examples showing that all re first entrepreneur is simply Y–r. These
s 2 and sthe
whenparameter
the first values
signalswhile
are simul3 . The value of a loan after the investor receives signal s1 from
sult exist for some plausible
the first entrepreneur is simply Y-r. These values of loan
values­ of loans are weighted by the appropriate
taneously satisfying equation (1).
8
98
the first entrepreneur is simply Y-r.
These values
loans are weighted
the appropriat
probabilities:
withof probability
hg the bysignal
of the
probabilities: with probability hg the signal of the first entrepreneur is s1 , with
probability [h(1 − g ) + (1 − h )g ] the signal is s2, and with probabi
probability [h(1 − g ) + (1 − h )g ] the signal is s2, and with probability (1-h)(1-g) it is s3.
That the investor's optimal strategy under in
Finnish Economic Papers 2/2007 – Juha-Pekka Niinimäki and Tuomas Takalo
probabilities: with probability hg the signal of
the first entrepreneur is s1, with probability
[h(1– g) + (1– h)g] the signal is s2, and with
probability (1–h)(1–g) it is s3.
That the investor’s optimal strategy under incomplete information can involve benchmarking can be seen from (11). After contacting the
first entrepreneur, the investor can gather more
information by using the first signal as a benchmark, seeking a second entrepreneur and comparing him with the benchmark. If the signal
from the first entrepreneur is ’better’ than the
signal of the second, the first proves to be worthy of finance and vice versa. If the received
signals are identical, the investor can update her
prior beliefs on the state of the world and types
of entrepreneurs, but the incomplete information problem remains. The investor will create
a benchmark and search for a second entrepreneur if the benefits outweigh the opportunity
costs of doing so. If the cost of contacting is
sufficiently large, the investor may want to grant
a loan to the first entrepreneur without benchmarking and comparing, although there is higher risk of credit loss than for a loan under uncertainty after contacting the second entrepreneur. Alternatively, the investor will not grant a
loan at all. If VN = 2 ≥ 0 the loan market opens up.
This occurs if c is not too high. If VN = 2 ≥ 0, (1)
also holds, but not necessarily vice versa.
We summarise the benefits of benchmarking
based on the above analysis as follows:
Proposition 2 By using the first entrepreneur as
a benchmark and comparing the second entrepreneur with him, the investor can gather more
information and raise the expected return on her
loan. If the signals received from the entrepreneurs differ, incomplete information is eliminated, and the investor can grant the loan to the
good entrepreneur. Even if both signals are the
same (s2) and incomplete information obtains,
the investor can update prior beliefs on types of
entrepreneurs.
In spite of all the benefits of using the first entrepreneur as a benchmark, the investor’s lending decision is still less efficient than under
perfect information. From (1) and (11), we see
that four inefficiencies remain: i) The first en-
trepreneur is good, but his type is unobservable
to the investor because of signal s2. The investor
may waste resources by searching for a second
entrepreneur or she may inefficiently exit the
credit markets; ii) The investor encounters two
good entrepreneurs, but the entrepreneurs’ types
remain unobservable. If p(g | s2, s2)Y < r, the investor makes a mistake and denies a loan; iii)
The investor contacts a bad entrepreneur who
signals s2 and who inefficiently receives a loan,
if c is sufficiently high and p2(g | s2)Y > r (part ii)
of Proposition 1); iv) The investor contacts two
bad entrepreneurs with s2, and she grants a loan
if p2(g | s2, s2)Y > r.
3.4 Discussion
The model with one investor, two entrepreneurs,
and three signals is admittedly limited. In the
subsequent sections we will allow for multiple
entrepreneurs and financiers. While the extension of the model to arbitrary number of signals
is beyond the scope of this paper, it is clear that
the basic insight about the value of making
benchmarks and comparisons would not change
in so far the signal space is finite.10
To gain intuition for the upcoming results and
illustrate the value of benchmarking, we illustrate few examples.
• Variation of signals across sectors: Financial
ratios reveal valuable information to investors, but a part of this information is sectorspecific. What is regarded as a good ratio of
asset turnover, administration costs, gross
profit or solvency will vary across industries.
Thus, even when the financial ratios of a sector are fixed in time, the critical values of financial ratios differ across sectors. Hence, if
an investor who has no previous experience
in the sector evaluates a firm, the firm’s financial ratios do not reveal its true financial condition to the investor. To evaluate the firm
10
Increasing the number of intermediate signals that do
not perfectly reveal the borrower type would make the information asymmetry friction costlier to resolve but would
not otherwise affect the results. If there were uncountable
measure of signals, having access to many borrowers would
not eliminate the information asymmetry. However, even in
that case benchmarking and comparing would be valuable.
99
Finnish Economic Papers 2/2007 – Juha-Pekka Niinimäki and Tuomas Takalo
properly, the investor must compare the financial ratios of the firm with the financial ratios
of the other firms in the very same sector.
Only after studying sufficiently many firms,
will the investor understand the correct meaning of financial ratios in the sector.
• Variation of signals over time: The investor
may have previous experience in the sector,
but the signal value fluctuates over time, making the information value of a single signal
modest. For instance, a good firm might earn
handsome profits if the industry is booming
or moderate profits if the industry is in recession, whereas a bad firm might earn moderate
profits during a boom or fail in a recession.
Hence, whether moderate profit indicates a
good or a bad firm would depend on the stage
of the industry cycle. Without comparing, the
investor might make a wrong decision by
granting a loan to a bad firm with moderate
profits.
• Variation of signals across regions: Production costs and market potential differ from
one region to another. The same wage cost
per employee, for example, may indicate a
good, efficient firm in Area A, or a bad, inefficient firm in Area B. To be able to evaluate
the competitiveness of a firm in its local region, the investor needs to compare the firm’s
characteristics with its local competitors.
• Variation of signals in research and development: The progress of research and development is stochastic almost by definition. At
first glance, the prototype of an entrepreneur
may seem to be promising. However, careful
comparison with the other entrepreneurs’ prototypes or products in the sector may reveal
that the prototype is lagging. Financing the
first entrepreneur is likely to be a mistake
since its product will hardly be commercially
viable.
• Variation of signals according to the industry
dynamics. Multiple equilibria are pervasive in
network industries. A network firm’s profit
may be negative, but contrasting the firm with
other firms in the industry may reveal that the
firms are engaged in fierce competition for
dominance where all firms are making losses,
and that the firm is leading the competition
for a dominant position in the market. Once
100
in the dominant position, the firm will be able
to raise its price and make substantial profits.
Denying the firm finance would probably be
a mistake.
• The profitability of a firm varies with the industry equilibria. Efficient firms may, for example, earn supernormal profits if the firms
in an industry collude and moderate profits if
they compete. Less efficient firms may earn
moderate profits if the industry is in collusion
or zero profits if there is competition. Moderate profits may signal an efficient firm if the
firms compete or an inefficient firm in case of
collusion.
• Technological cycles change the meanings of
signals. The propensity to patent changes over
time depending on the legal and technological
environment. For example, it is known that
innovations can come in waves: after a breakthrough invention it is easier to make follow
up innovations. A successful innovative firm
may possess dozens of patents after a breakthrough, having had only a handful before. A
less innovative firm, having struggled to obtain one patent before the breakthrough, may
easily obtain a handful of patents after the
breakthrough. For an outsider, it is hard to
know whether a handful of patents indicates
an innovative or an unsuccessful firm.
4. Benchmarking and comparing with
N loan applicants
So far we have assumed that the investor can
contact at most two firms. Although a complete
characterization of a more general case with an
arbitrary number of entrepreneurs is beyond the
scope of this study, in this section we briefly
confirm that the insights gained from the two
entrepreneur case apply for a larger pool of loan
applicants.
We first note that incomplete information can
be eliminated with certainty when the number
of contacted entrepreneurs approaches infinity.
Recall that incomplete information obtains only
if the investor receives signal s2. Consequently,
when the number of compared entrepreneurs
grows, incomplete information still obtains after n contacts only with probability (1– h)gn +
N
N
2
N
invest in information gathering, taking into21
account the costs, we write then initial value of a loan
toand
(1 − h) g + h(1 − g ) n . With probability 1-h theanalogously
stateVisN low
with
has
((11)
)g ]vinvestor
= hg
Y−
ras) +probability
[h(1 − g ) + (1g− hthe
N ( s 2 ) + (1 − h )(1 − g ) v N
21
analogously to (11) as
(1 − h )(n1 −Takalo
V N = 2/2007
hg (Y − –
r ) + [h(1 − g ) + (1Niinimäki
− h )g ]v Nn ( sand
g ) v N ( s3 ) − c ,
n Finnish Economic
n
2 ) + Tuomas
Papers
(1 − h) gonly
+ h(good
1 − g )entrepreneurs.
. With probability
the Juha-Pekka
state
is low
g high
the
21 contacted
where
subscript
Nand
denotes
the
number
of investor
entrepreneurs.
With1-h
probabilities
h and
(1with
− g)probability
thetotal
state
is
and thehas The f
n
n
n
n + h(1 − g ) . With probability 1-h the state is low and with probability g the investor has
(
1
−
h
)
g
h(1– g) . With probability 1-h the state is low21 (13) V N = hg (Y − r ) + [h(1 − g ) + (1 − h )g ]v N ( s 2 ) + (1 − h )(1 − g ) v N
captures
(13)
possibility
that
the first
is s1 , prom
where
subscriptthe
denotes
number
of signal
entrepreneurs.
The f
contacted
only
goodinvestor
entrepreneurs.
probabilities
hNprobabilities
and
(1 −the
g) ntotal
the
andstate
with
probability
gn the
has
con-With
investor
compared
only
bad
entrepreneurs.
Because
g n state
and is(1high
− g) n and
arethe
bability 1-h the
is low
andhas
with
grn) the
investor
has
(
(
)
(
)
(
)(
)
Vprobability
=
hg
Y
−
+
[
h
1
−
g
+
1
−
h
g
]
v
(
s
)
+
1
−
h
1
−
g
v
(13)
n
N
N
2
N ( s3 ) − c ,
21
contacted
entrepreneurs.
With
probabilities
h and the
(1 −total
g) the
stateofisentrepreneurs.
high and theThe first term in the ri
tacted
onlyonly
goodgood
entrepreneurs.
With
probabiliwhere
subscript N denotes
number
n
nthird
(13)
captures
the
possibility
that
the arise
firsthas
signal
s , where
prom
immediately.
The
second
and
term
from
the
n iscases
(1 − hh)and
gdecreasing
+investor
h(1 −ngthe
) nhas
.n,
With
probability
1-hbad
the state
is
low
and
with
probability
g the
the
investor
nis
ties
(1– g)
state
high
and
the
inveswhere
subscript
N
denotes
total
compared
only
entrepreneurs.
Because
probabilities
g n number
and
(1n−of
in
the
probability
that
incomplete
information
obtains
approaches
zero,
when
→g)∞ . are1
neurs. With probabilities h and (1 − g) the state is high and the
n
(13)
possibility
that
the
signal
is
s1 are
, prompting
the investo
compared
only1-hbad
entrepreneurs.
The
first
term
in
where
subscript
Ninvestor
denotes
the
total
number
of entrepreneurs.
The
investor
has
compared
only
bad is
entrepreneurs.
Because
probabilities
gfirst
and
(1 −the
g) nright-hand
h) g n + h(1 −tor
g ) nhas
. With
probability
theentrepreneurs.
state
low captures
andBewith the
probability
g n The
the
has
immediately.
second
and
third
term
arise
from
the
cases
where
n
n
n
.
As
discussed
in
the
previous
se
The
first
signal
is
s
cause
probabilities
g
and
(1– g)
are
decreasing
side
of
(13)
captures
the
possibility
that
the
first
3
contacted
only
good
entrepreneurs.
With
probabilities
h
and
(
1
−
g)
the
state
is
high
and
the
ninvestor
n
Even
if
the
has
not
received
two
different
signals
over
the
first
n
contacts,
decreasing
in
n,
the
probability
that
incomplete
information
obtains
approaches
zero,
when
n
→
∞
.
subscriptgN denotes
number of entrepreneurs. The first term in the right-hand side of
bad entrepreneurs. Because where
probabilities
and (1the
− g)totalare
(13)
captures
the
possibility
that
the
first
signal
is
s
,
prom
in
n, the probability
that incomplete
information
is
s
,
prompting
the
investor
grant
a
loan
immediately.
Thesignal
second
and
third
term
arise
from
the
cases
where
the
first
signal
is
decreasing
in n, the probability
that incomplete
information
obtains
approaches
zero,
when
n
→
∞
.
1
n
1
acted only good entrepreneurs. With probabilities h and (1 − g)
the state
is that
high
theis s . Asis discussed
in
the previous
se
first
signal
investor
knows
theand
entrepreneur
bad
with
certainty.
This ca
n and
nthis,
3To see
obtains comparing
approaches
zero,
when
n → ∞.
immediately.
The
third
term
arise
can
inEven
practice
render
the deficiency
of information
insignificant.
consider
if the
the
investor
received
twoThe
different
signals
over
the
first
n
contacts,
investor
has compared
only
bad
entrepreneurs.
Because
probabilities
g
and
(
1
−
g)
are
(13)
captures
possibility
thatnot
first
signal
is ssecond
,
prompting
the
investor
grant
a
loan
that incomplete
information
obtains
approaches
zero,
when nhas
→
∞ .the
1
if thehas
investor
has nottwo
received
two
different
signals
overthe
the
first
nthe
contacts,
Even if theEven
investor
not received
dif- The
from
the
cases
where
first
signal
isarise
s2 orfrom
s3. the cases
immediately.
The
second
and
third
term
in
previous
section,
if thewhere
first
first
signal
n is s 3 . As discussed
n
stor has compared
only
badthe
entrepreneurs.
Because
probabilities
g knows
andn-1
(1is
−the
g)
are
if
theprevious
second
entrepreneur
releasessTo
, the
the
investor
canThis
elimi
investor
that
entrepreneur
is.sin
bad
with
certainty.
ca
ferent
signals
over
the
first
n
contacts,
comparThe
first
signal
s
.
As
discussed
pre2 see
contacting
nth
entrepreneur
that
yields,
like
all
contacts,
signal
The
probability
comparing
can
in
practice
render
the
deficiency
of
information
insignificant.
this,
consider
3
or has not received
twoindifferent
signals over
firstand
n contacts,
decreasing
n, the
probability
thatthe
incomplete
information
approaches
zero,
n→∞
immediately.
The
second
third
term ariseobtains
from the
cases where
the when
first 2signal
is . s 2 or s 3 .
comparing
can
in
practice
render
the
deficiency
of
information
insignificant.
To
see
this,
consider
ing can in practice render the deficiency
in- that
vious
if the
signal
investor
3, the
discussed
theused
previous
The
first
signal
is certainty.
sis3 .sAs
investorofknows
thesection,
entrepreneur
isfirst
bad
with
This
caninbe
as a bes
iflike
theallsecond
entrepreneur
ss2with
the cerinvestor
can elimi
easing
in n, the
probability
that incomplete
information
obtains
approaches
zero,
→second
∞ .releases
s 3 , the investor
do
comparing
signals.
If ncontacts,
the
is, also
n when
formation
insignificant.
Toentrepreneur
see
this,
consider
knows
that
the
entrepreneur
issignal
bad
contacting
the
nth
that
yields,
previous
n-1
signal
er the deficiency
of information
insignificant.
To
see
this,
consider
2 . The probability
(
1
)
−
h
g
Even
if
the
investor
has
not
received
two
different
signals
over
the
first
n
contacts,
. As discussed
in the previous
section,
if the first signal is s 3 , the
first
signal
isall
s3=previous
contacting
entrepreneur
that
yields,
like
n-1
contacts,
signal
s
.
The
probability
(
)
p
g
S
or
thatthe
thenth
entrepreneur
isThe
good
is
2
contacting
the
nth
entrepreneur
that
yields,
like
tainty.
This
can
be
used
as
a
benchmark
so
that
2
,
n
if the second entrepreneur
releases
s 2 ,entrepreneur
the investoriscan
incomplete
investor
bad eliminate
with certainty.
This ca
(1comparing
(1 first
− h)over
g n +knows
hsignals.
− nthat
g ) nnIfthe
Even
investor
has
not received
different
signals
the
contacts,
s 3 , there
the investor
do
the second
signal
is also
contacting.
Thus
the
value
ofs2a,consider
loan
when
are N pot
all ifprevious
n–1
contacts,
signal
ss2two
. The
probaifcontinue
the second
entrepreneur
releases
the
investhat yields,
like
allthe
previous
n-1
contacts,
signal the
The
probability
comparing
can
ininvestor
practice
render
of
information
insignificant.
To
see
this,
(
1
)
−
h
g
2 . deficiency
n with certainty. This can be used as a benchmark so that
knows
thep(g
entrepreneur
is) gbad
(
)
p
g
S
=
or
thatentrepreneur
the entrepreneur
isthat
good
is
(
1
−
h
bility
that
the
is
good
is
|
S
)
=
tor
can
eliminate
incomplete
information
by
2
,
n
n
n is alsoreleases
2, n signals.
if(1Ifthe
s 2 , the does
investor
can elimi
s 3 , the investor
not lend
and
comparing
the
signal
g Sinformation
or
the entrepreneur
good is p(of
−signal
h)second
gsecond
+see
hs (1entrepreneur
− g )Thus
2, n ) =
n first
ncontacting.
paring can inthat
practice
render
the isdeficiency
insignificant.
this,
is
written
as
the
value
of
a loan
comparing
signals.
Ifconsider
the
second
signal
iswhen
also there are N pot
(1 − hall
) g previous
+continue
h(1 −
gn-1
)To
3 can be
contacting
that
yields,
like
contacts,
signal
s
.
The
probability
(1 − hthe
) g n nthifentrepreneur
2
the
s 2 , the investor can eliminate incomplete information by
p (g S 2 , n ) =
or second entrepreneur releasess3comparing
, the
investor
does
lend
may
may
s , theentrepreneu
investor d
the
signal
is or
also
contacting.
Thus
thesignals.
value
ofIfanot
loansecond
whenand
there
are
N
potential
− h) g n + h(1 −that
g ) nyields, like all previous
1 continue
acting the nth(1entrepreneur
n-1
contacts,
The
probability
first
signal sis2 . scontacting.
as the value of a3(12)
not signal
continue
Thus
3 can be written
.
p (g S 2, n ) =
n
(1 − h)is
g also s3 , the investor does not lend and may or may not
comparing signals.
second signal
h If1the
or
that the entrepreneur is good
there
N potential
1 +is p(g S(first
1n ) n is s3 nloan
signal
can
bewhen
written
as areThus
2 , n )−=
continue
contacting.
the value entrepreneurs
of a loan when there are N pot
n
1
+ hwhen
(1 −
)v(s3 , s3 ),as0}, (12)
v Ng()sthe
max{signal
g (Y − is
r) −
ccan
+ (1 −beg written
1 1 − (h1 − gh) g (1 − h. ) g and
s
3 ) =first
(
)
=
p
g
S
3
2
,
n
(12) is good
. 1 the value
(12)
p (g Sis
(continue
g S 2, n ) =contacting.
orof a loan when there are N potential entrepreneurs
the entrepreneur
2 , n )p=
and when the
nn
h (111−+h) ghnn Thus
+ (h(1 − 1g))
first signal is s 3 can be written as
+
−
1
(
1
)
.
(12) (14) v N (s3 ) = max{g (Y − r ) − c + (1 − g )v(s3 , s3 ), 0},
1− h g 1− h g
first signal is s 3 can be written as
− 1) n
{g (Y − r ) − c + (1 − g )v(s3 , s3 ), 0},
v N (s3 ) = max
S 2,n, s=2, 2s ,…, s
,..., sis2,n the
is the
vector of signals
from
n entrepreneurs.
(12) it
where vs(2s 3received
, s 3 ) is the
value
of a loan afterFrom
two observations
of s
2 ,1 , s 2 ,1
22, n where pSwhere
s2, 1
(2, n
. vector of
(12)
g S = 
2, n ) =
h
1
signals s2 received
From where vv N(s(3s, s
is the
ofc a+ (loan
)3=) max
{g (value
Y − r) −
1 − g )after
v(s3 , stwo
1
+.= s n, entrepreneurs.
− 1 )sn is the
1from
(s 1 ,...,
3 ), 0},
where
vector
of
signals
s3,2sthe
received
from
nof
entrepreneurs.
From
(12)aitloanofat s
(12)
p (g S 2, n ) =
) ninvestor
where
v(sas
is
the
value
aFrom
loan
after
two
observations
]
whereit Sis
= [s 2,1 , Ssthat
s 22gh,,1ng < is
vector
of signals
sobservations
received
from
entrepreneurs.
(12)
itnot
negative,
can
always
decide
to
grant
2that
,2n,...,
2g
, 2the
2 ,probability
nprobability
3indicates
3of
(12)
is
evident
if
,
the
<
the
that
signal
s
a
good
entrepreneur
is
decreasing
evident
if
s
.
Note
that
v (
s
, s
)
cannot
be
−
1
2
,
n
2
,
2
3
3
3
2
2
h
1
n
v N (s3 ) = max{g (Y − r ) − c + (1 − g )v(s3 , s3 ), 0},
(14)
+
−
1
(
1
)
signal
s2 indicates
entrepreneur
investor
decide
whereFrom
v(s 3(12)
,iss 3 ) itnegative,
is the valueasofthe
a loan
after can
two always
observations
of not
s3 . Note that v(s
the vector ofthat
signals
from an good
entrepreneurs.
g
1 −sh2 received
1 if ifgg > negative,
as
the
investor
can
always
decide
to n.
grant
loan
1 1 , the probabils3 at.B
the ainvestor
continues
comparing
if thenot
signal
decreasing
inSimilarly,
n.if Similarly,
<
probability
signal
sloan
indicates
aare
good
entrepreneur
isfirst
decreasing
is
evident
to(1),
grant
at
all.s2 Since
also
g <that
, the probability
that signal sthat
indicates
a2 good
entrepreneur
is decreasing
is evident
that
with
good
isg (Y – r) – c > 0
increasing
in
In a is
in n.
2if g > 2 , 2the probability that2 entrepreneurs
ity that entrepreneurs with s2 are negative,
good is as
in-the by
(1),
the
investor
continues
comparing
if
the
(s 3 ,always
where vcan
s 3 ) is the
value
loan after
observations
decide
notoftoa grant
a loantwo
at all.
Since alsoofg (s
obability that signal s 2 indicates a good entrepreneur is decreasinginvestor
where
S 2,ninin
= n.
s 2,In
s the
is
the
of signals
sstationary,
from
ncontinues
entrepreneurs.
From
(1),
the
investor
comparing
if(12)
the itfirst
signal
is s .B
1 vector
1 s 2 , n when
creasing
n, the
approaches
first
signal
is
sare
the
investor’s
problem
we
can
easily
solve
(14)
recursively.
Thisin
yields
1Similarly,
2g
, 2 ,...,
2 received
3. Because
> limit
,
the
probability
that
entrepreneurs
with
s
good
is
increasing
in
n.
In
in n. Similarly,
if,where
g
>
probability
that
entrepreneurs
with
s
are
good
is
increasing
n.
n.
if
the limit
when
n
approaches
infinity,
incomplete
informational
is
removed
with
certainty:
If In be 3
2
2
2v (s , s ) is2 the value of a loan after two observations of s . Note that v (s , s ) cannot
3
3
3
3
3
infinity,
incomplete
informational
is
removed
is
stationary,
we
can
easily
solve
(14)
recurnegative,
ascomparing
the investor
can first
decide
to grant a loan at
the investor
continues
the
signal
is snot
re
S 2,n = sthat
signals
s(1),
n entrepreneurs.
Fromif (12)
italways
3 .Because the inves
2 ,1 , sentrepreneurs
2 , 2 ,..., s 2 , n is the
2 received
robability
withvector
s 21 areofgood
is increasing
in from
n. In stationary,
can
easily
solve
(14)
recursively.
This yields
sively.
This=we
yields
1
lim
with
certainty:
Ifngg < , Sthe
p(g
|and
S2, n) = 0,
and
<when
probability
that
signal
s 2pindicates
a1is.good
entrepreneur
is
decreasing
is
evident
iflim
the
limit
when
approaches
infinity,
incomplete
informational
removed
with
certainty:
If
the
n
approaches
infinity,
incomplete
informational
is
removed
with
certainty:
If
2
g that
< 12 ,limit
p
g
=
0
,
if
g
>
,
lim
g
S
We
will
conclude
this
as
follows.
negative,
as
decide not
g , n the investor can always
g , n to grant a loan at all. Since also g (Y − r ) − c >0 by
n → ∞
2
n − >∞
n −>∞
(1),
the
investor
continues
comparing
if
the
first
signal
is s3 .B
1
stationary,
we
can
easily
solve
(14)
recursively.
This
yields
N −1
limprobability
p(g | S2, nis
) = 1.
We will
conclude
< 1 2 , the
that
signal
s indicates
a good
vident
that incomplete
ifif gg > infinity,
informational
removed
with
certainty:
If entrepreneur is decreasing
ª1 follows.
− (1 − g ) in ºn. In
1 0 , and if g > 21 , lim p (g 1S
n → ∞ p (1g S
)
)
=
=
1
.
We
will
conclude
this
as
g
>
,
the
probability
that
entrepreneurs
with
s
are
good
is
increasing
ingn.< Similarly,
if
g
<
,
lim
p
g
S
=
0
,
and
if
g
>
,
lim
p
g
S
=
1
.
We
will
conclude
this
as
follows.
g
,
n
g
,
n
2 , lim
2
(
)
[
]
=
−
−
(
)
.
v
s
g
Y
r
c
(15)
2
g ,n
g ,3n signal is s3 .Because
2 investor
the investor's
problem is
continues
comparing
if theN first
2 (1), the
2
«
»
n − >∞
n −>∞
n − >∞
n −>∞
this as follows.
stationary,
we can easily solve
This yields
¬ (14)grecursively.
N −1 ¼
)
if
g > 12 , lim
p(g> S1 g,,nthe
= 1probability
. We will conclude
this as follows.
g
that
entrepreneurs
with
s
are
good
is
increasing
in
n.
In
Similarly,
if
ª
º
(
)
−
−
1
1
g
2
2
n −>∞
[isg (first
− r ) − c ]«with certainty:
v N (sThis
Yremoved
the limit when nstationary,
approaches
infinity,
incomplete
informational
» . If bound,
3 ) = yields
we can
easilyofsolve
(14) recursively.
N −signal
1
RemarkRemark
1 When 1.
the
number
of
contacted
and
In
words,
if
s3g, thewithout
investor
When
the
number
contacted
and
evaluated
ª1 the
º ¬ isgrows
− (loan
1 − g )applicants
¼
]«with certainty:
− r ) − ccontacting
v N (s3 ) = [gis
(Yremoved
loan applicants
grows without
a bound,
continues
and
the entre» . comparing
limit when evaluated
n approaches
infinity, incomplete
informational
If
1
1
g grows
g <problem
lim
p g S gthe
0 , and
if number
g > 2 , islim
p gevaluated
S gpreneurs
1loan
. We
will conclude
as follows.
Remark
1.
When
of
contacted
and
bound,
¬applicants
¼ thiswithout
, n1.=number
,n =
2 , the
N −1 2. The bound,
the
information
until
she
encounters
When
the
and
evaluated
loan
applicants
of incomplete
information
is eliminated
with
certainty.
n −Remark
nof
>problem
∞ of incomplete
−elim> ∞contacted
ª1 − (1signal
−grows
g ) ºswithout
1of contacted
1
(
[
− c ]«
v N this
sthe
gfollows.
(Y −isr )increasing
inated
with
certainty.
value
of
loan
in
the
» . total
and
evaluated
loan
applicants
grows
without
bound,
3)=
,
lim
p
g
S
=
0
,
and
if
g
>
,
lim
p
g
S
=
1
.
We
will
conclude
as
g ,n
N −1
2
2
g
n − >∞
ninformation
−>∞
ª1 − (1 −with
º of entrepreneurs in¬ the economy,
theg ,nproblem of incomplete
is eliminated
certainty.
)
gnumber
¼ since
the problem vofN incomplete
is
eliminated
with
certainty.
(s3 ) = [g (Y −information
]
(15)
r) − c «
».
g
The above
discussion dismisses the costs
increasingly likely that the investor
mation is eliminated
with certainty.
¬ of it becomes
¼
The1.above
dismisses
costs
contacting.
the
investor
contacts
n entrepreneurs,
contacting.
IfWhen
the discussion
investor
contacts
n the
entreprewill findloan
aIfgood
entrepreneur
and
canbound,
grant a the
Remark
the number
of contacted
andofevaluated
applicants
grows
without
n
[
]
[
[
]
]
[
]
(
)
(
(
)
(
)
(
)
(
)
(
)
(
neurs, the total cost of contacting amounts to
)
)
loan.
The above
discussion
dismisses
theevaluated
costs of contacting.
If the investor
contacts bound,
n entrepreneurs, the
mark 1. When
ofofcontacted
and
grows
total
cost
contacting
amounts
n loan
c .costs
Toapplicants
investigate
thesignal
investor’s
how
The
above
discussion
dismisses
the
of
contacting.
Ifwithout
the investor
n ofentrepreneurs,
n c.the
To number
investigate
the
investor’s
optimal
choice
The
first
is soptimal
inchoice
section
3,
if much
the tothe
2. Ascontacts
the
problem
of
incomplete
information
istoeliminated
with
certainty.
the costs of of
contacting.
If the
investorincontacts
n entrepreneurs,
the
how
much
to invest
information
gathering,
signal is optimal
s2, and choice
the second
signal
total
cost
of contacting
amounts
to n c . To
investigate first
the investor’s
of how
muchistodifferproblem of incomplete
information
iscontacting
eliminated
with
total
cost ofthe
to ninic . Toent
investigate
optimal information
choice of howismuch to
taking into
account
costs, weamounts
writecertainty.
the
(either sthe
s3), incomplete
1 orinvestor’s
to n c . To investigate
optimal choice
of how
tial value the
of ainvestor’s
loan analogously
to (11)
as much toeliminated and the investor can make an optimal
lending
However,
if the second signal
The above discussion dismisses the costs of contacting.
If the decision.
investor contacts
n entrepreneurs,
the
is also s2, the investor can update her priors.
above discussion dismisses the costs of contacting. If the investor contacts n entrepreneurs, the
total cost of contacting amounts to n c . To investigate the investor’s optimal choice of how much to
101
cost of contacting amounts to n c . To investigate the investor’s optimal choice of how much to
comparing
canis ineliminated
practice
render
deficiency
of
information
(either
s1 orinformation
s 3 ), incomplete
information
andmake
the
can can
make
an her insignifican
lending
decision.
However,
if theand
second
signal
iscan
also
s 2the
, investor
the
update
ferent (either s1different
or s 3optimal
), incomplete
is eliminated
the investor
an investor
decreasing in n, the probability that incomplete information obtn
t (either s1 or s 3 ), incomplete information is eliminated
and thetheinvestor
can entrepreneurs.
makethat
an yields,
only
good
With
probabilities
h and (1 − g)
contacting
nth
like
all previous
optimal
lending
decision.
However,
thecontacted
second
signal
is
alsoof
, entrepreneurs
the investor
can
update
her n-1
priors.
Subsequently,
the investor
grant
to entrepreneur
either
releasing
signal
s 2 ,1contacts, sig
2 update
imal lending decision.
However,
if the second
signalif can
is
also
s 2a, loan
the
investor
cansthe
her
(
)
=
.
p
g
S
if the
investor has
2 , n not received two differ
Finnish Economic Papers 2/2007 – Juha-Pekka Niinimäki andEven
Tuomas
Takalo
h s 2,n1 is the
S 2,n = s 2,1 ,Because
s 2n, 2 ,...,
n v
lending decision. However, if the second signal is also investor
s 2 , the investor
can
update
her1bad where
(
)
=
.
p
g
S
has
compared
only
entrepreneurs.
probabiliti
2 , n the
(1 −a hthird
priors.
Subsequently,
theexit
can grant
a loan
to either
of
signal
s1)2 g+, loan ( − 1 )
can decide
to
thetocredit
market
without
lending,
or entrepreneurs
she can
to compare
hpproceed
ors. Subsequently,
the she
investor
can
ainvestor
loan
ofa the
releasing
signal
, )render
(g(Ss122,nreleasing
theentrepreneurs
entrepreneur
is good
is
gor
1 − hof informatio
Subsequently,
thegrant
investor
can either
grantthat
loan
to comparing
To understand
Proposition
recall
can
practice
the deficiency
n from (12)
− 1=) n3,
1 +in
(
1
)
−
h
g
+
h(1 − g1 ) n
h
g
−
1
Subsequently, the
investor
can grant
a loan
to the
either
thesignal
entrepreneurs
,that
either
ofdecide
the
entrepreneurs
releasing
that
p (gprobability
| Ssignal
) issnot
ina nthird
when
gg < <the
,. the probabi
is
evident
that
2, be
2, nproceed
2decreasing
decreasing
inreleasing
n,
incomplete
information
2 obtains appro
applicant.
she
takes
lastofoption,
she swill
in
athe
similar
but
identical
position
asif
after
she can
toIfexit
the
credit
market
or
she
can
to compare
loan
can decide to exit
market
without
lending,
or without
shewithout
can lending,
proceed
to compare
a third
loan
she the
cancredit
decide
to exit
the credit
market
Thus,
if granting
a loan
to thethat
firstyields,
entrepreneur
contacting
the nth
entrepreneur
like all previous n-1
[
]
decide to exit the
credit
market
without
lending,
or
sheshe
canwill
to
compare
loan
lending,
or she
can
proceed
compare
aproceed
third
sthird
isor
unprofitable
(p (her
greceived
| s2beliefs.
)Y < r),
itg will
second
contact:
either
theto
incomplete
information
isEven
eliminated
she canhas
update
2not
ifathe
investor
not
two
different
signals
> 12 , the probab
n.
Similarly,
ifthe
applicant.
she
takes
option,
be in
asignalling
similar
but
identical
position
as
after
plicant. If she takes
lastIfoption,
she the
willlast
in a similar
but not
as
afterwhere
thein
S 2,n = s 2,1 , sentrepre2 , 2 ,..., s 2 , n is the nvecto
loanthe
applicant.
If she
takesbethe
last option,
sheidentical
remainposition
unprofitable
for
all subsequent
(
1
)
−
h
g
Sneurs
=inswith
,as
s 2she
vector
signals
ss2 received
from
nt. If she takes the
last
she
will
benot
in identical
a after
similar
but
not where
identical
(ghave
) =information
pout
Softhat
that
entrepreneur
is is
good
is
,=
n the
2 ,1or
, 2after
2 ,update
nalso
will
be option,
in
a similar
position
as
s1,...,
.can
Itsthe
turns
an
exit,
Inbut
general,
the
investor
encountered
who
emitted
2 , nof
(has
)2position
second
contact:
either
the
incomplete
information
her
beliefs.
comparing
practice
render
the
deficiency
. the
giscan
S 2eliminated
2entrepreneurs
2 , n insignifi
, ncan
the
limit
when
n
approaches
infin
ond contact: either
the incomplete
information
is eliminated
or pshe
update
herhn-1
beliefs.
(
1
)
(
1
−
h
g
+
h
−g
1
1
n isevident
after the second contact: either the incomplete yielding
not optimal
In probability
< 2 ., the
that if ifgg < − 1 )is
1 + zero (payoff,
contact: either the
incomplete
information
is
eliminated
or
she
can
update
her
beliefs.
1
h< 2g,the
−gg < information
eliminated
orbethe
she
can as
update
her
thethat
case
probability
thatallasignal
new
s 2 , congeneral,
after
investor
has
encountered
n-1entrepreneur
entrepreneurs
have
emitted
thethat
probability
s 2 indicates
a go
is
evident
if1of
the value
of
a loanhas
can
written
the
nth
yields,
previous
n-1 contacts,
1 like
swho
In general,
after
theisIninvestor
encountered
n-1contacting
entrepreneurs
who
have
emitted
2 , g < 2 , lim p g S g , n = 0 , and if g >
1
n − >∞
beliefs.
tact eliminates incomplete
information
exceeds
In general, after the investor has encountered n-1 entrepreneurs who have emitted1 s 2 , in n. Similarly, if g > 2 ,n the probabilit
after
thebe
investor
g and, thus
return from
theInvalue
of a loan
written has
as encountered
> 2expected
, the probability
that entrepreneurs
in n. Similarly,
if gthe
(1each
− h) gnew with s 2
value of a loan can
begeneral,
written
as can
or
the entrepreneur
is
is p(g S 21, n ) = (recall
n-1 entrepreneurs who have emitted s2, that
the valcontact
g (Y – r) – c > 0
(1)).
The n infinity
.
p (gexceeds
S 2good
,n ) =
theh limit
when
(1 n− h)nsg2 napproaches
)
+(16)
h(1 − gfrom
e of a loan can be written as v(S
is the vector
of1 signals
received
n entr
(S 2,n−1 ), t nwhere
(Y − r ) S+2(,1n −= t nsinvestor
)2v,1(,Ss22,,n2 ),...,
max{v Ras
− thus
cs}2,n compares
−1 ) =
ue of a loan can2,nbe
written
entrepreneurs
until
in1+
( − 1 )incomplete informational i
the limit when n approaches
infinity,
heliminated.
g
complete information1 −isRemark
1.
When
the
number
of co
g < 12 , lim p g(16)
S g ,n = 0 , and if g > 12
1
(16) (1 − t n ) vthat
(
}
−− rcis)}+evident
S 2,nif) −Ifcgg > n
−
>
∞
,
the
investor’s
behaviour
is
different.
<
probability
that
signal
s
indicates
a
good
ent
{v R (Sv2(,Sn−21,)n,−t1n)(=Y max
)
v(S 2,n −1 ) = max
− r ){+v R(1(S−2t,nn−)1v),(tSn2(,Y
(16)
n
g < 12 , lim p g2 S g ,n 21
= 0 , and if g > 12 , lim p2 g S g ,n = 1 . We will
n −such
n − >of
>∞
∞number
In
circumstance
the
optimal
of
the
problem
incomplete
informatio
(S 2,n−1 ) = max{vR (S 2,n−1where
), t n (Y −vr )(S+ (1 −)t=n )max
v(S 2,n ) − c }
(16)
1 under uncertainty after n-1
{p(g S 2.n−1in)Yn n.− rSimilarly,
,0} p
is(gthe
value
aisloan
always
finite,
verified by with
then s are goo
= > 12of, the
. that as
Scomparisons
R
2 , n −1
probability
entrepreneurs
2if
,n ) g
n
h
(
1
−
h
)
g
+
h
(
1
−
g
)
.
With
probability
1-h
the
state
low
probability
g the
where
S
=
s
,
s 1is,...,
s andiswith
the vector
of signals
s22 investo
receive
2 , n proposition.
where vR (S2, n–1) = max{p (g | S2, n–1)Y–r, 0} is the following
1 + 2,1 (2, 2 − 1 )2n,n
h
g
−
1
(
)
{
(
)
}
v
S
=
max
p
g
S
Y
−
r
,
0
is
the
value
of
a
loan
under
uncertainty
after
n-1
where
n–1limit
Remark
1.that
When
the number of
RSloan
2 , n −under
1 and uncertainty
2+sig­
[1 when
)](1 − g ) isafter
where t2.n −=1 after
p (h Sof
gsloan
−under
p (h Sn2,uncertainty
the
probability
incomplete
max{of
ps(2gasignals
n-1
ere v R (S 2,n −1 ) =value
2 , n −1a)
n −1approaches
infinity,
incomplete
informational
is conta
remo
2. n −1 )Y − r ,0} is then value the
1 h and (1 − g) n the state is high and
contacted
only good
With
probabilities
The
above discussion
dismisses
theapp
c
nals
and
where
t
(
h
| S2, n–1)g + [1–p (
h
|entrepreneurs.
S2, n–1
)]
Proposition
4 If
g > ,
investor
stops
comRemark
1.
When
the
number
of
contacted
and
evaluated
loan
g
<
the
probability
that
signal
s
indicat
is
evident
that
if
n = p 2
2
v R (S 2,n −1 ) = max{p (g S 2.n −1 )Y − r ,0} is the value of a loan under uncertainty after n-1
(1–g)
is theand
probability
that
incomplete
paring
new
loan
applicants
even
ifincomplete
incomplete
the
of
information
(
)
[
(
)
]
(
)
s
signals
where
t
=
p
h
S
g
+
1
−
p
h
S
1
−
g
is
the
probability
that
incomplete
1infor1problem
information
will
be
eliminated
by
the
nth
contact.
Here
2
n
2
,
n
−
1
2
,
n
−
1
lim probability
p g S g ,n =that
0 , and
if g > 2 , lim p g S g ,n = 1 .n We will conclunis
signals and where t n = p (h S 2,n −1 )g + [1 − p (h S 2,n −1 )](1 −gg<) 2is, the
incomplete
cost
of contacting
amounts
nc
n −problem
> ∞bad
> ∞is eliminated
mation will be eliminated
by the
nth contact.
remains.
the
of incomplete
certainty.
investor has
compared
only
Because
probabilities
and
(1 −tog)w
> 1 ,total
then −probability
thatgwith
entrepreneurs
ininformation
n. entrepreneurs.
Similarly,
if ginformation
als and where t n = p (h S 2,n −1 )g + [1 − p (h S 2,n −1 )](1n −−1 g ) is where
the probability
incomplete
S 2,n = s 2,1that
, s 2, 2 ,...,
s 2,n is the2 vector of signals s 2 received from n e
information
will h(1be
by
the
nth Here contact.
Here
− g ) by eliminated
ormation
will
the
Here p(hbeS 2,n −1 ) =eliminatedn −1
)limit
=
1 − p (g nth
S 2On
is contact.
the
conditional
that the in
state
is
the
one
hand,
p (g |probability
S2, nobtains
) is increasing
nzero,
by
, n −1incomplete
n−
1
decreasing
in
n,hthe
probability
that
information
approaches
when
n→
the
when
n
approaches
infinity,
incomplete
informat
h
(
1
−
g
)
+
(
1
−
)
g
1
tion
will
be
eliminated
the is evident
nth that
contact.
Here
n −1by
The
above
discussion
dismisses
the
costs
(12),
if
g > .
The
probability
approaches
unity
g
<
,
the
probability
that
signal
s
indicates
a good
if
2
n −1
h
(
1
−
g
)
2
hp(1(h−Sg2,)n −1 ) =
) probability
= 1if−conditional
p (g The
S 2,n −1.
iswhen
the
conditional
probability
that
the
stateover
isreturn
discussion
the
of evaluated
contacting.
If first
the
investor
1above
n isnumber
largedismisses
enough,
andcosts
the
expected
When
the
oftwo
contacted
and
loan
applicants
n −1p (g S
nis
−1 theRemark
h S 2,n −1 ) =
= 1−
that
is
Even
the
different
signals
the
n cont
1
(1 −2,nh−)1 g)signals
g has
< 12 ,not
limreceived
p the
g S gstate
previous
are
s 2investor
.
, n = 0 , and if g > 2 , lim p g S g , n = 1 . W
h(h1(−1 −g )gn)−1n −+1 high
(1 − hwhen
) ghn(1−1−allg )n-1 +
total
cost
of
contacting
to
nc . T
n − > ∞ 1under uncertainty, p (g | S2, n)Y – r,
n − > ∞amounts
from
a
loan
apg
>
,
the
probability
that
entrepreneurs
with
s
in
n.
Similarly,
if
= 1 − p (g S 2,n −1 ) is the conditional probability that the2 state is
2 are
−1 ) =
n −1
n −1
total
cost
of
contacting
amounts
to
n
c
.
To
investigate
the
investor’s
h(1 − g ) is+ (the
1 − hconditional
)g
Y – r.
On
the other
hand,
the
expected
probabilitycan
that
staterender
is ofproaches
comparing
in the
practice
the deficiency
of information
insignificant.
To see this, cono
problem
incomplete
information
is eliminated
with
certainty.
high when
all n-1
signals are sthe
2 .
h when all n-1 previous
signals
areprevious
sprevious
.
return
from
contacting
a
new
loan
applicant
is
high when
all n–1
signals
are
s
.
2
2 limit when n approaches infinity, incomplete informational
the
is re
always
lower
than
Y – r – c.
Thus,
when
n
is
The
value
of
the
loan
(16)
could
be
solved
hen all n-1 previous signals are s 2 . contacting the nth entrepreneur that yields, like all previous n-1 contacts, signal s 2 . The probab
large enough,
thethe
investor
granting
backwards, beginning with the termination payRemark
1. When
numberprefers
of contacted
and aevaluated lo
g < 12 , lim p g S g ,n = 0 , and if g > 12 , lim p g S g ,n = 1 . We will con
n − loan
n −>∞
>∞
n of
The above
dismisses
the
costs
contacting.
If
the
investor contac
under
uncertainty
to
contacting
and
comoff vR (S2, N) = max{p (g | S2, N)Y–r, 0}. Because
thediscussion
(1 − h) g
(g problem
S 2, n )a=new
ortheisnumber
that the entrepreneur
good is pthe
paring
loan napplicant,
and
of with certai
problem is non-stationary,
however, issolving
of incomplete
information
eliminated
n
(1 − h) gto +n ch(.1To
)
− ginvestigate
of contacting
amounts
compared
loan applicants
is finite. the investor’s optimal
(16) completely is a messy exercisetotal
andcost
does
[
[
]
]
(
[
)
]
(
(
)
[
(
(
]
(
(
)
)
)
]
)
[
(
)
)
(
(
)
)
An implication of Proposition 4 is that it can
not yield substantial insights. Nonetheless, from
beWhen
optimal
grant ofa contacted
loan already
the firstloan applica
(16) we obtain a few general results that
are 1.
Remark
the to
number
andtoevaluated
above discussion
the about
costs ofhis
contacting.
If the in
entrepreneur
even ifdismisses
uncertainty
type
intuitive but tedious to prove. In the remainder The
1
remains,
as
in
the
two-entrepreneur
economy
of
of the section we presentp (the
results
(Proposi.
(1
g S 2, n ) =
the problem of incomplete
information is eliminated with certainty.
h for1 total
of contacting
to nsuch
c . To
investigate the inves
It is easyamounts
to see that
a lending
tions 3 and 4) and give heuristic arguments
1+
( −section
1 ) n cost 3.3.
1 − h g strategy is optimal if the first loan applicant is
them (for the formal proofs of the propositions,
good with a sufficiently high probability or if
see Niinimäki and Takalo, 2007).
The above discussion
dismisses
the costs of
contacting. If the investor con
the contacting
is sufficiently
costly.
Proposition 3 If the net present value of a loan
[s(s2,12,) ≤ 0)
] is ofthecontacting
S 2,n (v
= R s 2, 2 ,...,
s 2,ncost
vector of signals
from n entrepreneurs.
From
(1
total
amountss 2toreceived
n c . To investigate
the investor’s
optim
under uncertainty iswhere
negative
and
the
prior probability that a loan applicant has a 5. Multiple investors and financial
good project is less
than one
theprobability
< 12 ),
, the
that signal s 2 indicates a good entrepreneur is decrea
is evident
thathalf
if g(g < intermediation
investor continues to contact and compare new
Sectionsthat
3 and
4 show howwith
making
benchmarks
loan applicants until
problem of
g > 12 , the probability
entrepreneurs
s 2 are
good is increasing in
in the
n. Similarly,
if incomplete
and comparisons is beneficial even if there is
information is eliminated.
only oneincomplete
investor ininformational
the credit market.
If there
the limit when n approaches infinity,
is removed
with certaint
are multiple investors, however, an arrangement
102
g < 12 , lim p(g S g ,n ) = 0 , and if g > 12 , lim p(g S g ,n ) = 1 . We will conclude this as follows.
n − >∞
n −>∞
Remark 1. When the number of contacted and evaluated loan applicants grows without bo
Finnish Economic Papers 2/2007 – Juha-Pekka Niinimäki and Tuomas Takalo
wherein each investor separately collects information has many shortcomings: If an investor
contacts one loan applicant and receives s2, the
investor operates under incomplete information.
The investor can gather more information by
using the first signal as a benchmark and comparing signals from subsequent loan applicants
with it. But this is costly. Each investor needs to
contact several loan applicants even if each investor has only a unit of capital and she can
grant only one loan. Each loan applicant is then
contacted several times although each time the
monitoring provides the very same information.
Moreover, an investor cannot be sure ex ante
that comparing several entrepreneurs will eliminate incomplete information. In this section we
show how investors can overcome these shortcomings by establishing a coalition, financial
intermediary as in Boyd & Prescott (1986) and
Diamond (1984).
For brevity, we focus on the case where there
are (very) large numbers of entrepreneurs and
investors. Suppose that an investor establishes
a financial intermediary and invests her endowment in it as equity capital. This investor is
called the banker. The other investors, labelled
as depositors, invest their endowments in the
intermediary. The banker offers a standard deposit contract to the depositors, promising to
pay fixed interest rD on deposits that is senior to
payments on equity capital.
Let us proceed under the assumption that the
banker contacts loan applicants and monitors
them on the behalf of the other investors and
check later that the contract is incentive compatible. Since the intermediary has a lot of funds
and there are many entrepreneurs, it is likely
that the banker contacts two different types of
loan applicants, which eliminates incomplete
information. Even if all the intermediary’s loan
applicants are similar, the banker can be quite
confident about the true state of the world and
types of the loan applicants. In the limit, when
the financial intermediary is sufficiently large,
the incomplete information will be eliminated
with certainty (Remark 1). Furthermore, the
banker does not evaluate a loan applicant twice,
and the duplication of information gathering is
eliminated. To put it differently, a sufficiently
large financial intermediary achieves the very
same optimal solution that is achieved under
perfect information. A conclusion follows:
Proposition 5 If the investors join together and
establish a financial intermediary, the duplication of information provision cost is avoided.
When the intermediary is sufficiently large, incomplete information will be removed with certainty.
This result resembles that of Diamond (1984).
In both models, the financial intermediary is
bank-like and eliminates the duplication of information provision, thereby cutting the costs
of lending. Nonetheless, the models differ in
some important aspects: In Diamond (1984),
centralised information provision is profitable,
since a single investor can finance only a small
fraction of a project. Thus many investors are
needed to finance the whole project. If each investor monitored the project, the cost of monitoring would be duplicated. The duplication can
be eliminated by establishing a financial intermediary, which monitors the project only once.
On the contrary, in our model, centralised information provision is profitable even if a single
investor can finance the whole project! If a single investor financed only one project, she
should contact numerous loan applicants to
gather information. With multiple investors, the
same loan applicants would be contacted by
several investors and the cost of contacting
would be duplicated. Useless duplication can be
avoided by establishing a centralised intermediary. In our case the banker monitors each loan
applicant at most once.
The exact number of contacted loan applicants depends on the ratio of good loan applicants to investors. Let F ∈Z + denote the number
of investors. When F and N (the number of entrepreneurs) are very large, the realized share of
contacted good loan applicants is fixed, g, owing to the law of large numbers. If gN > F, there
are more good projects than investment funds in
the economy. Since each contact is costly, the
banker stops contacting upon encountering F
good loan applicants. Hence the optimal number
of contacted loan applicants is given by N* =
F/g where N* < N. On the contrary, when gN ≤ F,
good projects are scarcer, and the banker opti103
29
Finnish Economic Papers 2/2007 – Juha-Pekka
Niinimäki
Tuomas
Takalo
applicants
who and
proved
to be
good. The rest of the funds are
mally contacts all loan applicants because the applicants who proved to be good. The rest of
. Finally,
( F −in1)rthe
option (the
F −α
gN *)rare
D denotes
expected return from a contact is positive (recall
funds
invested
outsidepayments
option on deposit
28
(1)), i.e., N* = N.
(F– α gN*)r. Finally­, (F–1)rD denotes payments
using
rD given
bymanipulation
the Remark 2, using
(17) can
28 the
Pushing the
ondefinition
deposits.ofAfter
some
thebe rewritten a
28 argument for centralised finan1
cial intermediation
we canand
also
definition
of rD given by the Remark 2, (17) can
intermediary
has F investorsto(the
F −limit,
1 depositors
thedebanker),
F Π is the profit share of a single
termine
the
deposit
rate.
Let
Π
denote
the
total
be
rewritten
as
1
has the
F investors
and the
nvestors28
( F − 1intermediary
depositors and
banker), (F1FΠ− 1is depositors
the profit share
of banker),
a single F Π is the profit share of a single
profit
of
the
intermediary.
Because
the
intermeinvestor.
(18) π B = Max { rD + N * [c − (1 − α )g (Y − r )],0 } .
diary has F investors (F–1 depositors and the
investor. 1
theprofit
profitshare
shareofofa asingle
single in- This shows that the banker’s income is increasositors and the banker),
banker), F ΠΠ isis the
ing in α and is thus maximised when α =1. That
vestor.
is,tothe
contacts
andreturn,
monitors
optimal
Remark 2. A sufficiently large intermediary can
anbanker
investor
a fixed
of the in
Thispay
shows
that
the banker's
income
is increasing
Į and is thus maxim
amount
of
entrepreneurs.
In
this
case
the
bankRemark
2 A
sufficiently
large
intermediary
can
Remark
2.
A
sufficiently
large
intermediary
can
pay
to
an
investor
a
fixed
return,
of
fficiently large intermediary can pay to an investor a fixed return, of
ers
income
is
r
+
cN*
which
exactly
equals
to
pay1 to an investor
a
fixed
return,
of
banker contacts and monitors
the optimal amount of entrepreneurs.
In t
D
N*
[
rD ≡ lim F Π = r +
g (Y − r ) − c )] .
the
opportunity
cost
of
the
equity
capital
inF →∞
F
N
*
N*
1
[g (Y − r ) − c )] .
r+
is rD+cN*
whichinexactly
equals to intermediary
the opportunity and
cost of
[g (Y − rcan
) − cr)D]pay
.≡ Flim
vested
the financial
thethe equity cap
F Π
ermediary
an= investor
→to
∞
F a fixed return, of
F
(non-monetary) costs of contacting. Just as in
intermediary
and (1984)
the (non-monetary)
of contacting.
Diamond
the incentivecosts
problem
betweenJust as in Di
The result follows from the law of large numthethe
bank
its depositors
is At
eliminated
thanks
The result
from thethe
lawidea
of large
numbers (1984).
and it utilises
ideaand
of Diamond
(1984).
the
bersfollows
and it utilises
of Diamond
problemtobetween
the bank
and its
depositors is eliminated thanks to the
thethe
law
theoflarge
numbers.
At the
beginning
ofutilises
period,
each
investor
deposThe
result
follows
from
the law
of
large
numbers
and it utilises
theofidea
Diamond
(1984). At the
rom the law of large
numbers
and it
the
idea
of Diamond
(1984).
At
beginning
of unit
period,
eachintermediary.
investor deposits
unit conin the intermediary.
Thethat
banker
contacts
theno incentive
Finally, note
investors
have
its her
in the
The her
banker
Finally, note that investors have no incentive to form an
beginning
of
period,
each
investor
deposits
her
unit
in
the
intermediary.
The
banker
contacts
thepower,
to
form
an
intermediary
to
gain
market
tacts
the
optimal
number
of
loan
applicants,
N*,
, each investor deposits her unit in the intermediary. The banker contacts the
optimal
numbergood
of loan
applicants,
N *something
, finances is
good
projects
and,
if something they
is left,
puts
the all the sursince
by
assumption
can
extract
finances
projects
and,
if
left,
numbers and it utilises the idea of Diamond (1984). At the power, since by assumption they can extract all the surplus from e
number
of
applicants,
Nsomething
* , finances
projects
if somethingThe
is left,
puts the
oan applicants,optimal
N * ,the
finances
projects
is good
left,plus
puts
the and,
from
entrepreneurs.
tendency
towards
puts
rest ofgood
theloan
funds
inand,
the ifoutside
option.
rest of the funds in the outside option. At the end of the period, the loans yield a net return of
centralised
financial
intermediation
arises
soleAt
the
end
of
the
period,
the
loans
yield
a
net
towards
centralised
financial
intermediation
arises
solely
from economi
sits her unit in the intermediary. The banker contacts the
rest
ofAtthe
in the period,
outside
option.
the
end
the
period,
the loansofyield
return of
the outside option.
thefunds
end
of
the
loansAtyield
a netofly
return
ofeconomics
from
scalea innetcomparing.
return
of
N* (
gY– c)
and
the
outside
option
N * ( gY − c) and the outside option yields ( F − gN *) . The intermediary can then pay a return of rD
(F– gN*)r.
The intermediary
can the
then pay
, finances goodyields
projects
and, if something
is left, puts
* ( gY(−F c−) gN
and
the
outside
option yields
( F −pay
gN *) . The intermediary can then pay a return of rD
e outside optionaNyields
*)
. The
intermediary
canthe
then
return
of
r
per
deposit
unit
and
bankera return of rD
D
per deposit unit and the banker also obtains return rD on his investment. Clearly, equation (1)
obtains the
return
rD yield
on hisa investment.
6. Conclusion
6. Conclusion
At the end of also
the period,
loans
net return ofClearper obtains
deposit return
unit and
banker
also obtains
return
rD on (1)
his investment. Clearly, equation (1)
d the banker also
rD the
on his
investment.
Clearly,
equation
ly,
equation
(1)
implies
that
r
> r.
From
the
D
implies that rD>r. From the assumed market
structure, itasfollows that entrepreneurs earn zero profit.
In this paper we study ex ante benchmarking
ds ( F − gN *) . The
intermediary
can then
pay
a return
rstructure,
sumed
market
structure,
follows
thatofentrepreD
rD>r.
From
theitassumed
market
it follows
that entrepreneurs earn zero profit.
om the assumedimplies
marketthat
structure,
it follows
that entrepreneurs
earn zero
profit.
and
comparing
a source
of information in
neurs earn
zero profit.
In equilibrium
there is no incentive problem between
the bankerasand
the depositors.
In
this
paper
we
study
ex
ante
benchmarking
and comparing
financial
markets.
Making
and as a sourc
In
equilibrium
there
is
no
incentive
problem
btains
return
r
on
his
investment.
Clearly,
equation
(1)
D
equilibrium
therethe
is no
incentive
problem
between the banker and thebenchmarks
depositors.
ilibrium there is no incentiveInproblem
between
banker
and the
depositors.
comparing
it amount
Suppose
that thethe
banker
misbehaves
and contactsSuppose
only a fraction
α of the others
(optimalwith
number
of) loan to a simple
between
banker
and the depositors.
markets.
Makingαbenchmarks
andItnumber
comparing
others
with it amount
of the
learning.
has many
features
ket structure,
itSuppose
follows
that the
entrepreneurs
earn
profit.
that
banker
misbehaves
contacts
only
a ainstrument
that
banker
misbehaves
and
contacts
only
fraction
of
(optimal
of) loan
nker
misbehaves
and the
contacts
only
a fraction
αand
ofzero
the
(optimal
number
of)
loan
applicants.
In
this
way
she
can
not
only
save
some
costs
of
contacting
but
also
invest
an
extra
common
with
other
forms
of
learning
such
as
fraction α of the (optimal number of) loan aplearning.
It has
many
features
common
with
other
forms of learning
learning
by
doing,
experimentation
and
imitaincentive
problem
between
the
banker
and
the
depositors.
applicants.
In
this
way
she
can
not
only
save
some
costs
of
contacting
but
also
invest
an
extra
plicants.
In
this
way
she
can
not
only
save
some
way she can not only save some costs of contacting but also invest an extra
amountcosts
of the
funds
in the
outside
option. The
net revenue
– thehowever, retion.intermediary's
The other forms
of learning,
of intermediary's
contacting but
also
invest
an extra
experimentation
and
imitation.
The
other
forms– of
learning, however, re
funds
in
the
outside
option.
intermediary's
the
contacts only
aamount
fraction
αthe
of intermediary's
the
(optimal
of)in
loan
quire
many
observationsnetonrevenue
the same
entrepremediary's
funds
in the of
outside
option.
Thenumber
intermediary's
netoutrevenue
– The
the
amount
of
the
intermediary’s
funds
the
banker’s income - is at most (when the banker has contacted sufficiently
many
loan
applicants
to
be
neur
over timeover
whereas
comparing
exploits
the the differe
side option. The intermediary’s net revenue
the–contacted
same
entrepreneur
timeloan
whereas
comparing
banker’s
income
- is but
at most
(when
the
banker
many
applicants
toWe
be exploits
yatsave
of contacting
also
invest
an
extra
mostsome
(whencosts
the
banker
has contacted
sufficiently
many
loanhas
applicants
tosufficiently
be
differences
between
entrepreneurs.
show
the
banker’s
income
–
is
at
most
(when
the
able to interpret the signals correctly)
how
incomplete
be reduced
reducedbybymaking bench
banker has contacted sufficiently many loan We
ap- show
how
incompleteinformation
information can
can be
able toThe
interpret the signalsnet
correctly)
signals
correctly)
the outside
option.
revenue
– the cor- making benchmarks and comparisons. By complicants tointermediary's
be able to interpret
the signals
paring
sufficiently
entrepreneurs,
an incomparing
sufficiently
manymany
entrepreneurs,
an investor
can separate go
rectly)
anker has contacted sufficiently many loan applicants to be
vestor
can
separate
good
from
bad
loan
appli(17)
(17) π B = Max {YαgN * +( F − αgN *)r − ( F − 1)rD , 0 }
If every investor invests in information gathering, it wi
cants.
{
}
π
=
Max
Y
α
gN
*
+
(
F
−
α
gN
*)
r
−
(
F
−
1
)
r
,
0
(17)
Max {YαgN * + (F − αgN *)r − (BF − 1)rD , 0 }
(17)
IfD every investor invests in information gathSince the intermediary is very large, the realised ering, it will be inefficient, since loan applicants
are compared
several
timesowing
with to
zero informafractions
of different
are fixed
owing of different
Since the
intermediary
is veryasset
large,types
the realised
fractions
asset types
are fixed
tion
gain.
To
prevent
the
useless
duplication
of
to
the
law
of
large
numbers.
In
(17)
Y
α g
N*
repSince
the
intermediary
is
very
large,
the
realised
fractions
of
different
asset
types
are
fixed
owing
to
large,
realised
fractions of different asset types
owing to
} lending
αry
gNis*)very
r −the
( F law
− 1the
)rof
(17) are fixed
D , 0large
numbers.
In (17)
* represents
αgNcontacted
returns the
frominvestors
the contacted
loan join togethcomparing,
optimally
resents
returns
from Ythe
loanlending
large numbers.
(17) Yα
gN *the
represents
umbers. In (17)theYαlaw
gN *ofrepresents
lendingIn returns
from
contactedlending
loan returns from the contacted loan
104
realised fractions of different asset types are fixed owing to
gN * represents lending returns from the contacted loan
Finnish Economic Papers 2/2007 – Juha-Pekka Niinimäki and Tuomas Takalo
er and establish a financial intermediary, which
contacts each loan applicant only once. If the
intermediary can grow without bound, the problem of incomplete information can be eliminated with certainty without further investment
in information acquisition. This provides a novel rationale for centralised financial intermediation.
An implication of our model is that financial
institutions such as banks can become dominant
financiers of an industry by exploiting scale
economies in comparing. Once a bank has gathered information about firms in the industry by
comparing them, it is difficult for new financial
institutions to enter the market for finance of
the industry. The entrants should start the process of comparing from the beginning whereas
the incumbents know the firms and the signals.
Thus, even if the incumbents and entrants observe the same information, the incumbents may
have an information advantage since they are
able to interpret the information correctly. Only
in new industries will old and new financiers
compete on equal footing. In such circumstances the efficient use of benchmarks and comparisons can be the crucial determinant of financial institutions’ successes and failures.
Indeed, although we have emphasised the
role of comparing in credit markets and that the
intermediary arising from our analysis is banklike, comparing is perhaps even more relevant
for business angels, venture capitalists and other entities that focus on financing new high-tech
industries. As carefully documented by Gompers and Lerner (2004), such private equity financiers frequently encounter ideas for businesses in areas where there is little available information. The lack of track records for applicants
or a particular business area does not lead to the
collapse of markets for innovation finance, since
financiers seek many applications from the
same narrow area. In comparing applications, it
becomes evident that funding should be denied
to some business ideas. The most promising
projects receive the first stage financing. After
a new round of comparing, only the most successful firm can obtain second stage financing.
The next logical step is to endogenise the form
of financial contract so as to enable assessment
of the relative benefits of benchmarking and
comparing in private equity and debt finance.
Another interesting issue would be to introduce some tradeoffs to counter the tendency
towards bigger financial intermediaries predicted by our model. For example, investors should
be allowed to go beyond benchmarking and
comparing and use other strategies to mitigate
information asymmetry such as estimation of
cash flows. Finally, the model has been simplified by assuming that the state of the world has
no effect on the project output. This is a strong
assumption. Relaxing it and many other intriguing issues are left for future work.
QED
References
Balk, B. (2003). “The residual: on monitoring and benchmarking firms, industries, and economies with respect
to productivity.” Journal of Productivity Analysis 20(1),
5–47.
Bergemann, D., and U. Hege (1998). “Venture capital financing, moral hazard, and learning.” Journal of Banking & Finance 22, 703–735.
Bergemann, D., and U. Hege (2002). “The value of benchmarking.” In Venture Capital Contracting and the Valuation of High Tech Firms. Eds. J. McCahery and L. Renneboog. Oxford University Press, Oxford.
Berger, A., S. Frame, and N. Miller (2005). “Credit scoring and the availability, price, and risk of small business
credit.” Journal of Money, Credit and Banking 37(2),
191–222.
Bester, H. (1985). “Screening vs. rationing in credit markets with imperfect information.” American Economic
Review 75, 401–405.
Blochlinger, A., and M. Leippold (2006). “Economic benefit of powerful credit scoring.” Journal of Banking and
Finance 30(3), 851–873.
Boot, A. W. (2000). “Relationship banking: what do we
know?” Journal of Financial Intermediation 9, 7–25.
Boyd, J., and E. Prescott (1986). “Financial intermediarycoalitions.” Journal of Economic Theory 38(3), 211–
232.
Broecker, T. (1990). “Credit-worthiness tests and interbank
competition.” Econometrica 58(2), 429–452.
Cerasi, V., and S. Daltung (2000). “The optimal size of a
bank: costs and benefits of diversification.” European
Economic Review 44(9), 1701–1726.
Courty, P., and G. Marschke (2004). “Benchmarking performance.” Public Finance and Management 4(3), 288–
316.
Diamond, D. (1984). “Financial intermediation and delegated monitoring.” Review of Economic Studies 51,
393–414.
Freixas, X., and J-C. Rochet (1997). Microeconomics of
Banking. The MIT Press, Cambridge, Massachusetts.
Hellwig, M. (2000). “Financial intermediation with risk
aversion.” Review of Economic Studies 67, 719–742.
Holmström, B. (1979). “Moral hazard and observability.”
Bell Journal of Economics 10, 74–91.
105
the most successful firm can obtain second stage financing. The next logic
the most successful firm can obtain second stage
the most successful firm can obtain second stage financing. The next logical step is to endogenise
the form of financial contract so as to enable assessment of the relative b
the form of financial contract so as to enable ass
31
31
the form
of financial
contract
so –asJuha-Pekka
to enable assessment
theTuomas
relative Takalo
benefits of benchmarking
Finnish
Economic
Papers
2/2007
Niinimäkiofand
and comparing in private equity and debt finance.
and
comparing
in private equity and debt finance.
Holmström,
B., andin
J. private
Tirole (1997).
“Financial
intermeRamakrishnan, R., and A. Thakor (1984). “Information
and
comparing
equity
and
debt
finance.
the
most
successful
firm
can
obtain
second
stage
financing.
The
next
logical
step
is
to
diation,
funds firm
and the
sector.”
Quarterly
reliability
and anext
theory
of financial
intermediation.”
Rethe
mostloanable
successful
canreal
obtain
second
stage financing.
The
logical
step
is
tobeendogenise
endogenise
Another
interesting
issue
would
to introduce
some tr
Journal of Economics 112, 663–691.
view of Economic Studies 51,
415–432.interesting issue would
Another
Gombers, P., andAnother
J. Lerner interesting
(2004). The Venture
Capital
Sharpe,
S. (1990). “Asymmetric
information,
bank lending
issue
would
be
to
introduce
some
tradeoffs
to
counter
the
the
form
of financial
contract
as
enable
of
relative
benefits
of benchmarking
the
form
financial
contract so
so
as to
to
enable2ndassessment
assessment
of the
the
relativeintermediaries
benefits
benchmarking
Cycle.
TheofMIT
Press, Cambridge,
Massachusetts,
andbigger
implicit
contracts:
A stylizedofmodel
of customer
tendency
towards
financial
predicted
by our model.
tendency
financial intermediaries
Edition.
relationships.”
Journal towards
of Financebigger
45(4), 1069–1087.
tendency
towards
bigger
financial
intermediaries
predicted
by
our
model.
For
example,
investors
Gorton,
G., and A. Winton
(2003).
“Financial
IntermediaSiems, T., and R. Barr (1998). “Benchmarking the producand
in
equity
and
finance.
and comparing
comparing
in private
private
equity
and debt
debt
finance.
should
be allowed
go beyond
benchmarking
and comparing
and use oth
tion.” In Handbook of The Economics of Finance.
Eds.
tive to
efficiency
of U.Sbe
banks.”
Financial
Studies,
should
allowed
to goIndustry
beyond
benchmarking a
G. Constantinides,
M. to
Harris
and R. M.benchmarking
Stultz. Elsevier, and comparing
11–24.
should
be allowed
go
beyond
and
use
other
strategies
to
mitigate
interesting
would
to
some
tradeoffs
to
counter
the
Another
interesting issue
issue
would be
be
to introduce
introduce
some
tradeoffs
to and
counter
the
North Holland, Another
Amsterdam.
Thomas,
L. such
(2000).
“A
survey
of of
credit
behavioural
information
asymmetry
as
estimation
cash
flows.
Finally,
the mode
information
asymmetry
suchtoasconsumestimation of cash
Krasa, S., and A. Villamil (1992). “Monitoring the moniscoring: Forecasting
financial
risk of lending
information
asymmetry
such
as estimation
of cash flows.
Finally,
themodel.
model For
has
been simplified
by
tor: An incentive
structure for
a financial
intermediary.”
ers.” International
Journal
of Forecasting
16(2),
149–
tendency
towards
financial
intermediaries
by
our
example,
investors
tendency
towards bigger
bigger
financial
predicted
bythe
ourworld
model.
example,
investors
assuming thatpredicted
the
state of
hasFor
no effect
on the
project output. Thi
Journal of Economic
Theory 57,
197–221.intermediaries
172.
assuming
that
the
state
of
the
world
has no effect
Millon,
M., and
A. the
Thakor
(1985).
“Moral
hazard
in- on
Tirole,
J. (2006).
The Theory
ofisCorporate
Finance.
Princassuming
that
state
ofbeyond
theinformation
world
hasgathering
noand
effect
the
project
output.
This
a strong
assumption.
should
be
allowed
to
go
benchmarking
and
comparing
and
use
other
strategies
to
mitigate
formation
sharing:
A
model
of
eton
University
Press,
Princeton,
New
Jersey.
should be allowed to go beyond benchmarking
comparingtoand
use other
strategies
to mitigateThat and man
It would beand
illuminating
consider
the reserve
assumption.
agencies.” Journal of Finance 40, 1403–1422.
Von Thadden,ItE-L.
(1995).
“Long-term contracts,
short-the reserve a
would
be illuminating
to consider
Niinimäki,
J-P.
(2001).
“Intertemporal
diversification
in
term
investments
and
monitoring.”
Review
of Economic
It
would
be
illuminating
to
consider
the
reserve
assumption.
That
and
many
other
intriguing
issues
information
asymmetry
such
of
cash
flows.
Finally,
the
information
asymmetryJournal
such as
asofestimation
estimation
offor
cash
flows.
Finally,
the model
model has
has been
been simplified
simplified by
by
financial intermediation.”
Banking
and
FiStudies
62(4) , 557–575.
are left
future
work.
left(2004).
for future
work. information,
nance 25(5), 965–991.
Von Thadden,are
E-L.
“Asymmetric
are left for
future
work.
Niinimäki,
J.-P.,
Takalo
Benchmarking
and on
bankproject
lendingoutput.
and implicit
The winner’s
curse.”
assuming
that
the
of
the
has
This
is
assumption.
assuming
thatand
theT.state
state
of(2007).
the world
world
has no
no effect
effect
on the
the
project
output.
Thiscontracts:
is aa strong
strong
assumption.
comparing entrepreneurs with incomplete information.
Financial Research Letters, 11–23.
HECER Discussion Paper No. 173 (July, 2007).
It
be
to
the
reserve
That
and
intriguing issues
It would
would
be illuminating
illuminating
to consider
consider
the
reserve
assumption.
That
and many
many other
other
issues
Pagano,
M., and
T. Jappelli (1993).
“Information
sharing assumption.
Appendix
A: Proof of
Proposition
1 intriguing
in credit markets.” Journal of Finance 48, 1693–1718.
Appendix
A: Proof
of Proposition
Appendix
A: Proof
are
left
work.
are
left for
for future
future
work. of Proposition 1
1
Proof of part i): In the proof we use the following observation
Proof of part i): In the proof we use
Proof of part i): In the proof we use the following observation frequently
2
h(1 − g ) + (1 − h) g − g (1 − g ) = (1 − h) g 2 + h(1 − g ) .
(
)
h
1
−
g
+
(
1
−
h
)
g
−
g
(
1
−
g ) = (1 − h) g 2 + h
2
2
Appendix
A:
Proof
of
Proposition
1
(
)
(
)
h
1
−
g
+
(
1
−
h
)
g
−
g
(
1
−
g
)
=
(
1
−
h
)
g
+
h
1
−
g
.
(A.1)
Appendix: Proof of Proposition 1
Given (3), (6), and (10), it is profitable to contact the second applicant if
Given (3), (6), and (10), it is profitable to contact
Proof
ofproof
part
i):
Inuse
the the
proof
we use
use the
following
observation
frequently
Given
andthe
(10),
it isi):
profitable
to
contact
second
applicant
if
Proof
of (3),
part(6),
i)Proof
In
we
following
observation
frequently
of
part
In
the
proof
we
the
following
observation
frequently
t 2 (Y − r ) + (1 − t 2 )Max{p ( g s 2 , s 2 )Y − r , 0}≥ Max{p ( g s 2 )Y − r , 0} + c
(
) + (1 − t 2 )Max{p( g s 2 , s 2 )Y − r , 0}≥
t
Y
−
r
2
(A.2)
(11 −−{ggp)(22g.. s 2 )Y − r , 0}+ c
−−g
gr))+
++((1
1(1−
−− h
ht 2)) g
g)Max
−g
g{((1
1p (−
−gg
gs))2 ,=
=s((21
1)Y−
−h
h−))rg
g, 220}+
+≥h
hMax
(A.1)
(A.1) thh2(11(Y−
−
(A.1)
The inequality is satisfied if p ( g s 2 ) Y ≤ r for sufficiently low c. Hence
The inequality is satisfied if p ( g s 2 ) Y ≤ r for
Given
(3),
(6),
and
(10),
is
profitable
to
applicant
if
Given
to≤contact
contact
thesecond
second
applicant
Given(3),
(3),(6),
(6),and
and
(10), itit
it is
isifprofitable
profitable
to
contact
the
second
applicant
if if we focus on the case
p( g s 2 ) Y
r for the
sufficiently
low
c. Hence,
The
inequality
is (10),
satisfied
where p ( g s 2 ) Y > r . There are two possibilities, depending on wheth
are two possibilit
− r ) + (1 − t )Max{p
(g
ss 22 ))Y
−
pp (( gg ss 22 ))Y
−
,, 00} +
(A.2)
(A.2) tt 22 (Y
g ss 22 ,,two
Ypossibilities,
− rr ,, 0
0}≥
≥ Max
Max{depending
Ywhere
− rron
+pcc( g s 2 ) Yp (>grs. ,There
(A.2)
where pY( g−sr2 )+Y1>−rt 22. Max
Therep (are
whether
2 s 2 ) Y − r is
t 2 = p (h s 2 )g +
positive or negative. Hence,
Usingwe (9),
we
The
inequality
is is
satisfied
if p(g
| s2) Y ≤ r for sufficiently low c.positive
on therewrite
case
where
orfocus
negative.
Using
(9), we
p
(( g
ss 22 )) Y
≤
rr for
sufficiently
low
c.
Hence,
we
focus
on
the
case
The
inequality
satisfied
if
p
g
Y
≤
for
sufficiently
low
c.
Hence,
we
focus
on
the
case
The
inequality
is
satisfied
if
)g2) Y – r
t 2 =p(g
p (h | ss22, s
+ [1 − is
p (hpositive
s 2 )](1 − or
g ) negaas
positive
Using (9),depending
we rewrite
p(g
| s2) Y > r. or
Therenegative.
are two possibilities,
on whether
g (1 − g )
− g)
t 2 =possibilities, depending
. As
a g (1result,
if ) Yp (−grs 2 is
,s ) Y < r ,
where
p
g
rr .. There
are
on
where
p ((g(9),
g(1ss 22−))we
Y) >
>rewrite
There
are
two
possibilities,
on whether
whether pp (( gg ss 22 ,,.ssAs
Ya −result,
r ais 2 result, i
tive.
Using
t2 = p (h | two
s2)g + [1–p 22 ) As
gY
h(1 −(hg |) s+2)](1–g)
(1depending
− h) gas t 2 =
h(r1 ,− g )(A.2)
+ (1 − h)simplifies
g
t2 =
. As a result, if
p( g s 2 , s 2 ) Y <
to
h(1 − g ) + (1 − h) g
p (h s )g + [1 − p (h s )](1 − gg ) as
positive
1 − g ) rewrite
positive or
or negative.
negative. Using
Using (9),
(9), g (we
we
rewrite tt 22 =
(1 −ash )(A.1),
g
if p(g | s2, s2) Y < r, (A.2) simplifies to
using
(Y − r ) ≥= p hg(1s(122−−ghg)+g) 1 − p(YYh−−s 22rr )+≥1c−or,
using
to
Y −r
h(1 − g ) + (1 − h )g
(1 − h )hg(1 − g Y) +−(1r−+hc)gor, using (A.1),
g (1 − g )
(
)
(
)
(
)
(
)
h
1
−
g
+
1
−
h
g
h
1
−
g
+
1
−
h
g
(
)
Y
−
r
≥
to
g
(
1
−
g
)
g (1 − g )
(1 − g )aa+ (1 −result,
h )g
(A.1),
tth22(1=
.. hAs
if
=−tog ) + (1 − h )g
As
result,
if 2 pp (( gg ss 22 ,, ss 22 2)) Y
Y<
< rr ,, (A.2)
(A.2) simplifies
simplifies to
to
h
h((1
1−
−g
g )) +
+ ((1
1−
−h
h)) g
g
r (1 −32
h) g + h(1 − g ) + gY (h − g ) ≥2 [h(1 − g ) + 2(1 − h )g ]c .
r (1 − h) g + h(1 − g ) + gY (h − g ) ≥ [h(1 − g
(A.3) r (1 − h) g 2 + h(1 − g )2 + gY (h − g ) ≥ [h(1 − g ) + (1 − h )g ]c .
(A.3)
(11 −− hh )gg
g
g (1
1−
−g
g)
(
)
Y
−
r
≥
−
+
or,
using
(A.1),
to
Y
r
c
Y − rY(1– h)g
Y −ofr(A.3)
+ c or,
using (A.1), to
2 2 in the LHS
Adding
subtracting
Y≥(1h
(A.3)
yields
Adding
yields
(11 −− LHS
)and
h
+
−
h
h(1
1−
−g
gand
+ (1
1subtracting
−h
h )g
g
h−(1
1h−
−)gg
g )in+
+ the
h )g
g of
[
]
[[
]]
[
] [
22
22
2
(A.4) rrr (((111 −
(1(−hh h−−)ggg )2 ≥≥+[hhYgh
(11 −−(1gg−) ++g )(11≥−−[hhh(1)gg−]ccg ..) + (1 − h )g ]c .
−
+
h((11
−
+
gY
− hh
h))) gg
g2 +
+ hh
1−
− gg
g )) −
+ YgY
[
]
(A.4)
(A.3)
(A.3)
]
2
2
2
2
Since
p(gp |( sg2, s
(1– h)g
ofLHS
(A.4)ofis(A.4)
positive.
Hence,
(1 − the
Since
s 22,) Y < r
s 2 ) Y implies
< r implies
(1 − Yh < (1– h)g
) g 2 Y < (1 − + h(1– g)
h) g 2 + h r,
g ) LHS
r , the
is positive.
(A.4) holds if c is sufficiently low.
Hence, (A.4) holds if c is sufficiently low.
106
Now assume p ( g s2 , s2 ) Y > r . Equation (A.2) simplifies to
Y [t 2 − p ( g s 2 ) + (1 − t 2 ) p ( g s 2 , s 2 )] ≥ c .
(A.5)
[[
]]
2 2
2 22
2
Since
Sincep (pg( gs 2s,22s,2s)22 Y
) Y< <r rimplies
implies(1(−
1 −h)hg) gY2 Y< <(1(−
1 −h)hg) g +
+h(h1(−
1 −g )g ) r ,r the
, theLHS
LHSofof(A.4)
(A.4)isispositive.
positive.
2 2
(A.2) simplifies to
Y [t 2 − pNow
( g s assume
) + (1 − tp2 ()gp (sg2 , ss22 ,)sY2 )>] ≥r .cEquation
.
(A.5)
Hence,
Hence,(A.4)
(A.4)holds
holdsif2 ifc cisissufficiently
sufficientlylow.
low.
Finnish Economic Papers 2/2007 – Juha-Pekka Niinimäki and Tuomas Takalo
Y [t 2 −Now
p (1g−sassume
) g+2 (+1 h−p(1t(2g−)spg(),g2s s)2 Y, s 2>)]r .≥Equation
c.
(A.5)
h2assume
simplifies
toto to
Now
p ( g2s22 ,2sby
> r . (A.5)
Equation
(A.2)
simplifies
2) Y
Now
assume
p(g
| s2, s
(A.2)
simplifies
2
Since
(A.1),
can(A.2)
beto
further
simplified
1− t2 =
2) Y > r . Equation
h(1 − g ) + (1 − h) g
2
−)h))+g+(21(+−1 h−
(
1
t 2(1t)22p−)(pgg()gs 2s,by
≥ ≥c .c(A.5)
(A.5)
(A.5)
.
(A.5)
2s,2s)(A.1),
2]) ]
Since
can be further simplified to
22
2
2
Y [Y1t 2−[t−22t 2(−p1=(−pg(ggs)2sgh
c h) g
h(1 − g ) + ≥(1 −
,
(A.6)
h(1 − g ) + (1 − 2h)22g Y 2 22
h)hg) g + +h(h1(−1 −g )g )
(1(−1 −
Since
byby(A.1),
(A.1),
(A.5)
can
further
simplified
Since
t 2t 22=(1=− g ) gh
tototo
Since1 −1 −
(A.1),(A.5)
(A.5)can
canbebe
befurther
furthersimplified
simplified
(A.6)
)g+) +(1≥(−1 −hc )h,g) glow.
h(h1c (−1is−gsufficiently
which his(1true
if
− g ) + (1 − h) g Y
(1(−1 −gProof
)ggh
c c From (A.2) we see that a necessary condition for the optimality of a
) gh of part
≥ ≥ ,ii):
(A.6)
which is true if
c is sufficiently
, low.
(A.6)
(A.6)
h(h1(−1 −g )g+) +(1(−1 −h)hg) g Y Y
loan under Proof
uncertainty
the(A.2)
firstweentrepreneur's
type is
p ( g s for
− roptimality
> 0. When
2 ) Ythe
of part about
ii): From
see that a necessary
condition
of a
which
whichisistrue
trueifififcccisisissufficiently
sufficientlylow.
low.
which
true
sufficiently
loan
uncertainty about the first entrepreneur's type is p ( g s 2 ) Y − r > 0. When
p ( g sunder
2 ) Y − r > 0 , (A.2) implies that the loan under uncertainty is optimal if
ofofpart
From
condition
forforthe
Proof
partii):ii):
From
(A.2)
wesee
seethat
thatacondition
anecessary
necessary
condition
theoptimality
optimality
ofa a
Proof of part Proof
ii)
From
(A.2)
we
see(A.2)
that
awe
necessary
for
the optimality
of
a loan of
under
uncertainty about the first entrepreneur’s type is p(g | s2) Y – r > 0 When p(g | s2) Y – r > 0, (A.2) imp ( gunder
sunder
the{entrepreneur's
loan
) + (1the
)Max
}− c isisis optimal
p2 () gYs−2uncertainty
)rY
−>r 0≥,t(A.2)
−about
rimplies
−the
tthat
pentrepreneur's
( g sunder
, s )Yuncertainty
− r ,type
0type
(A.7)
2 (Y
2is
loan
about
first
p (pg( gs 2s)if22 Y) Y− −r r > >0.0. When
loan
uncertainty
first
When
plies
that
the loan
under
uncertainty
optimal
if2 2
Y{p−( gr −
optimality of the(A.7)
loan
Because( gthes 2 RHS
isr )smaller
)Max
(A.7)
)Y − rof≥ (A.7)
t 2 (Y −implies
+ (1 − t 2than
s 2c, s, 2auncertainty
)Ysufficient
− r , 0}− ccondition for
p (pg( gs 2s)p22 Y
) Y− −r r > >0 ,0(A.2)
, (A.2)
impliesthat
thatthe
theloan
loanunder
under uncertaintyisisoptimal
optimalif if
Because
the
RHS ofis(A.7)
isissmaller
a sufficient
condition
optimality
of the loan
r − c . As
a result, for
aforsufficient
condition
for
under
uncertainty
that
LHS ofthan
(A.7)Y – r – c,
exceed
− r − c ,Ya−sufficient
optimality
of the loan
Because
the RHS of
(A.7)the
smaller
than
{pY{(pexceed
p (pg( gs 2s)22Y)Y− −r is
≥r ≥tthat
−the
g( gs 2s,22s,2s)Y – r – c.
− −r ,r0,}0−As
(A.7)
) +(1(−1 −tof
)Max
}−c ccondition
under uncertainty
(A.7)
a result, a sufficient condition
for
−r )r+LHS
(A.7)
2 t(Y
2 t)22Max
22 (Y
22Y)Y
part
ii)
of
Proposition
1
to
hold
is
part
ii)
of
Proposition
1
to
is
under uncertainty is that the LHS of (A.7) exceed Y − r − c . As a result, a sufficient condition for
Because
r −c ,c a, asufficient
sufficientcondition
conditionforforoptimality
optimalityofofthe
theloan
loan
Becausethe
theRHS
RHSofof(A.7)
(A.7)isissmaller
smallerthan
thanY Y− −r −
(A.8)
(
)
c
≥
1
−
p
(
g
s
)
Y
.
(A.8)
2
part ii) of Proposition 1 to hold is
under
r −c .c As
. Asa aresult,
result,a asufficient
sufficientcondition
conditionforfor
underuncertainty
uncertaintyisisthat
thatthe
theLHS
LHSofof(A.7)
(A.7)exceed
exceedY Y− −r −
Condition (A.8) holds if c or p(g | s2) is sufficiently large. The latter occurs when g is close to one
(
)
c
≥
1
−
p
(
g
s
)
Y
.
(A.8)
Condition (A.8) holds
if c or p ( g s 2 ) is sufficiently large. The latter occurs when g is close to
one
2
or
h is
close
to zero (see
(9)).
part
ii)ii)
ofofProposition
1 1toto
hold
part
Proposition
holdisis
or
h of
is close
to zero
(see
(9)).
Condition
s 2 ) investor
is sufficiently
large.
The
lattertooccurs
when
g is close(A.8)
to
one
Proof
iii)
sif.2) Y – r < 0
does not
grant
a loan
the first
entrepreneur.
Thus,
c c≥part
s 2p(g
≥(1(−1(A.8)
−p (pgIf
( gholds
s)22)Y
) )| Y
.c or p ( gthe
(A.8)
if c is so high that (A.2) is violated, the investor does not grant a loan.
part
iii): If p ( g s 2 ) Y − r < 0, the investor does not grant a loan to the first
or h is close toProof
zero of
(see
(9)).
Condition
Condition(A.8)
(A.8)holds
holdsif ifc or
c orp (pg( gs 2s)22 )isissufficiently
sufficientlylarge.
large.The
Thelatter
latteroccurs
occurswhen
wheng gisisclose
closetotoone
one
entrepreneur. Proof
Thus, of
if cpart
is soiii):
high
the investor
investor does
does not
not grant a loan
loan.to the first
If that
p ( g(A.2)
s 2 ) Yis−violated,
r < 0, the
ororh hisisclose
closetotozero
zero(see
(see(9)).
(9)).
entrepreneur. Thus, if c is so high that (A.2) is violated, the investor does not grant a loan.
Proof
Proofofofpart
partiii):
iii):IfIf p (pg( gs 2s)22 Y) Y− −r r < <0,0,the
theinvestor
investordoes
doesnot
notgrant
granta aloan
loantotothe
thefirst
first
entrepreneur.
entrepreneur.Thus,
Thus,if ifc cisissosohigh
highthat
that(A.2)
(A.2)isisviolated,
violated,the
theinvestor
investordoes
doesnot
notgrant
granta aloan.
loan.
107