Imperfect Information Transmission from Banks to Investors:
Real Implications
PRELIMINARY
Nicolás Figueroa, Oksana Leukhina, Carlos Ramírez
November 3, 2013
The 2003-2007 economic expansion, which preceeded the 2007 …nancial crisis, marked the
time period of rapid growth of markets for collateralized debt obligations. Several empirical papers document that securitization contributed to laxer screening standards, therefore
proposing a factor that contributed to the crisis. We develop a general equilibrium model
which allows us to study information transmission in secondary loan markets and screening
e¤ort at origination. Originating banks have the ability to collect soft information about the
borrowers, but cannot credibly pass that information on to investors. They may, however,
choose to employ an imperfect rating technology. The price di¤erential on assets with a good
rating and assets with a bad rating emerges as the main determinant of screening e¤ort at
origination. The screening e¤ort is suboptimal because the rating technology is imperfect.
Both the rise in collateral values and the increase in asset complexity unambiguously weaken
screening intensity. The mechanism provides new insight into the following pre-crises observations: (1) more intense use of ratings, (2) historically low spreads between high yield and
investment grade securities, (3) the rise in default probability conditional on investment grade
rating, (4) the rise in the fraction of assets receiving an investment grade rating. The two
suggested policies, mandatory rating and mandatory rating disclosure, are counterproductive,
both leading to further resource misallocation.
JEL Codes:
Keywords: Information Production, Creidt Misallocation, Loan sales, Securitization, Rating
Shopping
Acknowledgements:
We thank seminar participants at the University of Washinton, Carnegie Mellon University, Ponti…cia
Universidad Católica de Chile, Midwest Macro Conference 2013.
Nicolas Figueroa: Instituto de Economía, Ponti…cia Universidad Católica de Chile; Oksana Leukhina: University of
Washington; Carlos Ramírez : Carnegie Mellon University
1.
Introduction
The 2003-2007 economic expansion, which preceded the 2007 …nancial crisis, marked the time period of
rapid growth of markets for collateralized debt obligations (CDOs). The annual volume of new collateralized loan obligations (CLOs), for example, grew from less than 20 billion before 2003 to more than 180
billion in 2007 (Bord and Santos, 2011), while more than 1 trillion of CDOs were outstanding in 2005 with
50% of their collateral in loans (Lucas et al. 2006). The research on real implications of these markets is
just beginning to take o¤, and a lot remains ununderstood, as explained by Gorton and Metrick (2011).
Several empirical papers document that the rise in securitization contributed to laxer screening standards, therefore proposing a factor that contributed to the …nancial crisis. Purnandam (2010) uncovers
that banks with greater involvement in the originate-to-distribute (OTD) market prior to 2007 originated
excessively poor-quality mortgages, which led to higher default rates among the borrowers. Keys et al
(2010, 2011) explore the variation of loans to borrowers with credit score around 620. As a rule of thumb,
loans associated with a FICO score of 620+ are easier to distribute. The two papers …nd that loans with
the FICO score of 620+, which should actually be of slightly better credit quality than those at 620-,
tend to default within two years of origination at a rate 10-25% higher. Finally, Bord and Santos (2011)
compare loans sold to CLOs with loans originated by the same banks but not sold to CLOs. Everything
else equal, loans sold to CLOs at the time of their origination are more likely to default.
These empirical papers support the economists’public opinion regarding the consequences of securitization on the originator’s incentives to screen their borrowers (Stiglitz, 2007, Blinder, 2007). Nonetheless,
the theoretical mechanism behind these empirical …ndings is not well understood. Why does the presence
of markets for loans lead to less than e¢ cient screening intensity? In other words, why does the price fail
to adjust to the point of inducing an e¢ cient allocation of resources? The potential culprit we focus on is
adverse selection in secondary loan markets. Originating banks have the ability to collect soft information
about the borrowers, but cannot credibly pass that information on to investors.
We develop a general equilibrium model that allows us to study the interaction of information production in secondary markets and screening intensity at origination. There are borrowers, investors, and
banks. All banks sell loans to investors in secondary markets in order to raise funds. In other words, we
take the presence of secondary markets for loans as given, and do not attempt to explain their emergence
as in, for example, Parlour and Plantin (2008). Banks have access to an imperfect rating technology (at
a cost), which reveals the true asset type with probability greater than one half but less than one. Banks
may choose to rate their assets and to reveal/hide the assigned rating.
Our main …ndings are as follows. First of all, it is the price di¤erential on loans with a good rating
and loans with a bad rating that disciplines screening acitivity at origination. Therefore, understanding
what determines this price di¤erential is crucial for understanding resource allocation in the economy.
If the rating technology were perfect, i.e. the true type would be revealed with certainty, the …rst best
outcome would be achieved in our economy. However, because the rating technology is imperfect, the
price di¤erential is not large enough to induce the e¢ cient level of screening. Interestingly, we …nd that
the more intense use of rating technology by holders of bad loans reduces clarity in the market, therefore
reducing the price di¤erential and screening activity in the economy.
We …nd that a rise in collateral values, an increase in the fraction of repaying borrowers, and a decrease
2
in the rating technology precision, all unambiguously reduce screening e¤ort at origination. The case of
a lower rating precision deserves further comment. This exercise is aimed to re‡ect the increase in the
asset complexity. The direct implication is that, since mistakes are more likely to happen, banks with
loan baskets of poor quality shop for rating more intensely. This strategic behavior reduces clarity in the
market, reduces the price di¤erential on loans with a good and a bad rating, and relaxes incentives for
loan screening at origination.
Our model provides new insight into the following pre-crises observations: (1) laxer lending standards,
(2) more intense use of ratings (shopping for rating), (3) the rise in default probability conditional on
investment grade rating (in‡ation rating), (4) historically low spreads between high yield and investment
grade securities, (5) the rise in the fraction of assets receiving an investment grade rating. The …rst fact
is common to all expansions (see Asea and Blomberg, 1998; Berger and Udell, 2004; Lown and Morgan
2006, Rajan 1994), and was exacerbated by the rise of secondary markets in the pre-crisis expansion (see
evidence given above).
The second fact refers to the idea that issuers expend resources to ensure their asset/tranch receives
a high rating. They may solicit ratings from several agencies and disclose only the best rating, they
expend resources to acquire information on how to best structure their security, etc. Gri¢ n, Nickerson
and Tang (2013) provide empirical evidence for rating shopping in the pre-crises period. The remaining
observations provide corroborating evidence for the intense use of rating shopping. Several theoretical
models can rationalize in‡ation rating via rating shopping (Skreta and Veldkamp 2009; Faure-Grimaud,
Peyrache, and Quesada 2009; Farhi, Lerner, and Tirole 2011). What is new here is the feedback e¤ect on
screening e¤ort at loan origination stage. Surprisingly, due to this feedback e¤ect, the role of mandatory
rating disclosure policy is reversed.
We also investigate mandatory rating and mandatory rating disclosure (as called for by Dodd-Frank)
as two policies that could be implemented in these markets. Both policies are counterproductive. The intuition is due to the interaction of information production in loan markets and screening e¤ort, something
which has not been previously modeled in the literature on rating shopping.
In the case of mandatory rating, banks with low quality assets increase their rating activity. This
lowers the probability that a good rating actually implies high quality asset. Clarity is reduced in the
loan market, which discourages screening e¤ort at origination and leads to a greater degree of resource
misallocation. With mandatory rating disclosure, banks selling a low quality asset are encouraged to
rate their assets, because the lack of rating can no longer be interpreted as the bad luck with the rating
agency. Therefore, rating intensity intensi…es, once again reducing the level of clarity in the market for
loans, reducing the expected return to screening and rating an asset, and therefore further aggrevating
the level of resource allocation in the economy.
2.
Related Literature
We also contribute to the literature on information in asset markets, through the analysis of how information transmission is endogenously altered through the strategic actions of loan originators. In our
model, there is a natural rate of information garbling, given by the imperfect nature of rating agencies.
3
However, this natural rate is augmented by banks holding bad loans who rate their claims, strategically
hiding bad ratings, and therefore decreasing the informativeness of these ratings. Skreta and Veldkamp
(2007) also model the idea of shoppings for ratings, and also study the interaction between ratings and
asset prices. However, they assume that investors are naive, and when they observe signals, they do not
consider the possibility that the asset issuer is hiding some of them.
As in Holmstrom and Tirole (1997), we assume that banks are not the same as investors. Banks have
access to a special technology, which allows them to screen the type of project they are …nancing.1 Banks
then are not simply a channel through which savings are allocated to borrowers, but …rms that use the
technologies they are endowed with to maximize pro…ts. As with any …rm, the use of the screening and
rating technologies is dictated by prices, which determine the banks’screening activity and hence credit
allocation and aggregate productivity.
In our model, as in Allen and Gale (2000), those endowed with funds must go through someone else
that is endowed with technology necessary to invest the funds. In Allen and Gale (2000), excessive risk
taking was generated because loan originators shifted the risk (due to limited liability) to those endowed
with the funds. In our model, it leads to misallocation of resources towards less productive producers.
The two are not the same, as it is not clear how risk correlates with average productivity of projects.
...
3.
Model
We consider an economy with fully rational investors, borrowers and banks. We model borrowers in
a very general way, and therefore, our setup is applicable to any …nancial market where the repaying
ability of borrowers cannot be costlessly observed. The main features modelled here is the informational
opaqueness of the borrower. Our setup thus applies to most …nancial markets, with an exception of
…nancial markets for large …rms with well-established reputation. Borrowers’informational opaqueness
is particularly acute in markets for startups and …rst home buyers.
Potential borrowers desire to borrow. Investors desire to save, and banks alone have the technology to
screen and identify repaying borrowers. Banks raise funds by selling loans to investors. We assume loan
markets are competitive.
3.1.
Potential Borrowers
There is a continuum of measure 1 of potential borrowers, each of whom seeks …nancing in the amount of
1 unit of funds. Potential borrowers are of unobservable type
and 1
0
respectively. Type
2 fG; Bg; represented in proportions
pays the return W to the bank with certainty, which represents either
the full loan repayment or bank collections in the case of default.2
Assumption 1 We assume WG > 1 > WB :
1
2
0
In Holmstrom and Tirole (1997) they can monitor the project they …nance.
In turn, bank collections are comprised of collateral sales or asset collections.
4
3.2.
Banks
There is a continuum of measure 1 of pro…t-maximizing banks, heterogeneous in their screening cost
k
F [0; 1] ; which is unobserved to investors. Each bank faces its own pool of potential borrowers of
types
2 fG; Bg; in proportions
0
and 1
0.
Banks have the option of using the screening technology at
the cost k; which ensures …nancing of a good project ( = G). Alternatively, they can …nance a borrower
chosen at random.
Banks must sell their loans to raise funds. We take the presence of secondary markets for loans as
given, and do not attempt to explain their emergence (see Parlour and Plantin, 2008). Instead, we aim
to examine the interaction of secondary markets for loans with screening incentives at origination, and
therefore, impose that all funds must be raised through loan sales to investors, or in the case of investment
banks, through attracting investor funds with a particular rate of return.
Once the borrower is …nanced, the loan type
is revealed to the bank. Banks are unable to costlessly
reveal this information to investors, but they have access to a simple objective rating technology at
the cost c; common to everyone, which reveals the true loan type with probability r > 12 :3 This rating
technology provides a credible way to transmit some information regarding loan quality to investors.
Because banks with better loan assets have a greater probability of obtaining a good rating, these ratings
are valuable signals in loan markets, and loans rated as good will sell at a premium. We assume that
only good ratings are revealed, which can be easily justi…ed by the availability of free bad ratings. Prices
on loans with the good rating (GR) and no rating (N R) are denoted by PGR and PN R : It is these prices,
taken as given by banks, that discipline the banks’behavior with respect to employing the screening and
rating technologies. In section XXX, we show that if r = 1; prices are su¢ ciently di¤erent to induce
an e¢ cient outcome. However, as long as r < 1, the incentives to both screen and rate the claims are
weakened, and insu¢ cient level of screening emerges, which leads to resource misallocation.
With the rating technology, we aim to capture the costly process the bank may engage in, which
results, with some positive probability, in the enhancement of the perceived value of their asset. The
process of getting all of the three rating agencies to assign a AAA rating to a large share of your loan
basket may be a costly process of …guring out the tranching system, the requirements set by each rating
agency, and hiring consultant …rms to assist with this process. The assumption of r > 1 simply captures
the idea that banks with better assets will have a greater chance at succeeding in the production of a
signal that will induce their loan to sell at a premium.
3.3.
Investors
Investors save by purchasing assets in competitive markets for loans. Since markets are competitive,
prices on loans with a given rating i 2 fGR; N Rg ; which we denote by PGR and PN R respectively, are
given by their expected payo¤s. Note that the screening productivity k of loan originators is unobservable
to investors, and therefore prices are only conditioned on the signal. The expected payo¤s depend on
investors’beliefs regarding the true fraction of good loans introduced into the economy. In turn, the true
3
In this context, r could be related with asset complexity or rating shopping, see for example Skreta and Veldkamp
(2009).
5
proportion of good loans in the market is determined by the screening decisions of banks, of all types k,
at the stage of loan origination.
3.4.
Timing, Banks’Strategies, and Equilibrium
In this subsection, we explain the banks’decision making in greater detail and introduce our notion of
equilibrium. The model period can be subdivided into four stages as follows:
1. Banks choose screening intensity and originate loans;
2. Loan quality is revealed to originators;
3. Banks choose rating intensity;
4. Banks and investors trade in competitive loan markets.
In the last stage, banks reveal only good signals, and therefore competitive prices are conditioned on
either the presence of a good rating or the lack of it. The rating decision of the bank depends on these
prices.
Rating Decision
We …rst analyze the rating decision. Since prices on loans are conditioned only on the rating signal,
all banks that hold a given type of loan will make the same decision with respect to employing the rating
technology. We allow for a mixed strategy, and denote the probability with which banks with loans of
type
employ the rating technology by f .
A bank that holds a loan of type B chooses to employ the rating technology if the expected gain in the
loan price strictly exceeds the cost of using the technology. The loan sold with no rating if the expected
gain falls short of the cost, and the mixed strategy is possible otherwise. Formally, we have
fB = 1
[(1
fB = 0
r) PGR + rPN R ]
if
fB 2 (0; 1)
PN R > c;
:::
< c;
:::
= c:
(1)
We restrict attention to the range of parameter values that ensure that, in equilibrium, banks selling
loans of high quality optimally choose to employ the rating (fG = 1):
rPGR + (1
r)PN R
c > PN R :
(2)
Screening Decision
We now turn to the screening decision. Denote by R the expected payo¤ to the bank from …nancing a
basket of borrowers of type
2 fB; Gg. If the rating technology is employed by the bank that loaned to
type G, with probability r, this bank receives type G rating on their loan, reveals it, and sells it for PGR .
With probability 1
r, this bank obtains the type B rating, hides it and sells the loan at price PN R . If
the rating technology is not used, the bank sells its loan for PN R with certainty. The situation for banks
6
…nancing type B borrowers is analogous, except these banks receive a type G rating with probability
1
r. Hence, the expected payo¤s to …nancing type
RG = rfG PGR + (1
RB = (1
borrower are given by
rfG )PN R
r)fB PGR + (1
(1
fG c
(3)
r)fB )PN R
The expected payo¤ to a bank that chooses not to screen is
payo¤ to a bank that chooses to screen is RG
1;
0 RG
fB c
1:
+ (1
0 )RB ;
(4)
while the expected
k. Therefore, there exists a cuto¤ k = (1
0 )(RG
RB )
that identi…es the bank that is indi¤erent between the two choices. Banks endowed with a screening
technology k < k introduce only good projects into the economy, while the rest of the banks …nance
good projects with probability
0.
Hence, the measure of type G projects …nanced, which we denote by
(PGR; PN R ) to emphasize the dependence on asset prices, is given by
(PGR; PN R ) = F (k) + (1
3.5.
F (k))
0:
(5)
Investors
In the previous section, we analyzed the decision of banks that took asset prices PGR and PN R as given,
and optimally chose how to select their borrowers and whether or not to use the rating technology. Given
these prices, we proceeded to derive the measure of total funds allocated to productive projects (equation
5 above). In this section, we describe how these prices are determined. In short, these prices must re‡ect
investors’ beliefs about expected payo¤ to the loan that sells with a good rating or no rating attached
to it. These beliefs, in turn, depend on the measure of good projects …nanced, i.e., asset prices depend
on the allocation of funds to productive projects, so we write PGR ( ) and PN R ( ). In what follows, we
proceed to explicitly derive this dependence of asset prices on credit allocation.
Since investors do not have the ability to observe the borrowers’type, they invest in assets created by
banks using the asset rating, or lack of it, to gain some information about returns. Investors are willing
to pay a premium for claims with the good rating. The size of that premium, of course, depends on the
beliefs such a rating induces about the quality of a project4 .
Recall that banks know the type
2 fG; Bg of the project they …nanced and must decide whether or
not to employ the rating agency. For a given proportion of good projects in the economy
and strategies
(fG ; fB ), the consistent beliefs by investors must satisfy (whenever possible):
Pr( = GjGR) =
Pr( = GjN R) =
rfG
;
rfG + (1
)fB (1 r)
[(1 fG ) + (1
[(1 fG ) + (1 r)fG ] + (1
(6)
r)fG ]
)[(1
fB ) + fB r]
:
(7)
The belief that the asset with the good rating is of high quality is given by the fraction of good projects
4
This premium could be interpreted as Du¢ e’s (2009) lemon premium, where an o¤er by a bank to sell a loan is associated
with a drop in price, since it is assumed that the bank has private information. There is also the moral hazard premium,
where a sale is associated with a drop in price since the bank has less incentives to control the credit risk of the loan, which
we do not consider in this paper.
7
among those that received a good signal, i.e., among the ranked good loans with the correct signal and
the ranked bad loans with the incorrect signal. Similarly, the belief that the asset with no rating is a
high quality asset is given by the fraction of good loans among those with no rating.
Competition among investors then implies that prices are determined as the expected payo¤ of an
asset:
PGR ( ) = WG Pr( = GjGR) + WB [1
Pr( = GjGR)] =
W Pr( = GjGR) + WB ;
(8)
PN R ( ) = WG Pr( = GjN R) + WB [1
Pr( = GjN R)] =
W Pr( = GjN R) + WB ;
(9)
where
W = WG
WB . Note that we emphasize the dependence of prices on the fraction of good
projects …nanced.
3.6.
Equilibrium
Up to now we discussed how prices paid on assets with good rating and no rating attached to them determine banks’screening and rating choices, and therefore the proportion of credit allocated to productive
entrepreneurs. We also discussed how, in turn, the fraction of credit allocated to productive entrepreneurs
a¤ects the prices paid on loans with good or no rating. Finding the model equilibrium involve …nding the
…xed point solution to (5), after explicitly incorporating into it the dependence of asset prices on credit
allocation to productive entrepreneurs. We de…ne the equilibrium below.
De…nition 1 An equilibrium is de…ned by a proportion of good projects in the economy
egy fB (
{RG (
), beliefs {Pr( = GjGR)(
); RB (
); Pr( = GjN R)(
)}, prices {PGR (
); PN R (
, a strat-
)} and revenues
)} such that (1)-(9) are satis…ed and
= F (k(
)) + (1
F (k(
)))
0
(10)
We now proceed to characterize the strategies (fG ; fB ) for a given proportion of good projects in
the economy . This involves solving for a …xed point. For given beliefs Pr( = GjGR); Pr( = GjN R),
equations (2) and (1) characterize the optimal behavior of banks. In fact, this conditions can be rewritten
as
c < r W [Pr( = GjGR)
c < (1
Pr( = GjN R)],
r) W [Pr( = GjGR)
Pr( = GjN R)]:
If the relevant inequality os satis…ed, the bank rates the loan with probability f = 1 because the gain
associated with obtaining a good signal outweighs the cost. If the inequality is reversed, f = 0 is optimal.
If it holds with equality, a mixed strategy is optimal, so f 2 (0; 1). But these strategies (fG ; fB ) in turn
determine the rational beliefs Pr( = GjGR); Pr( = GjN R) of investors.
It is easy to see that if r >
1
2,
the strategies {fG 2 (0; 1) ; fB = 1}, {fG 2 (0; 1) ; fB 2 (0; 1)} and
{fG = 0; fB = 1} are never optimal. Moreover, we will not consider the equilibrium {fG = 0; fB = 0},
8
that can always be rationalized as an equilibrium for some arbitrary out of equilibrium beliefs, since it is
not interesting.
Solving for the …xed point allows us to determine the optimal (fG ; fB ) as functions of
, and we
summarize the results in the lemma below.
Lemma 1 For a given measure of good projects
in the economy and de…ning c~
c
W,
the banks’rating
decisions can be summarized as follows:
fG = 1 and fB = 1 , 0
c~ <
[r
(1
[r
(2r
(1 r)(1
1 r
fG = 1 and fB 2 (0; 1) ,
fG = 1 and fB = 0 ,
fG 2 (0; 1) and fB = 0 , r(1
)
(1
)(1 r)(2r 1)
;
(2r 1)][ (2r 1) + (1 r)]
(1 r)(1
)(1 r)(2r 1)
c~ <
1)][ (2r 1) + (1 r)]
1 r
)
r(1
)
c~ <
;
1 r
r(1
)
:
c~ <
1 r
(11)
)
;
(12)
(13)
Proof. See the appendix.
We only consider equilibria in which banks …nancing productive entrepreneurs choose to employ the
rating technology with certainty. We thus do not consider the last case. Since in the other three cases
fG = 1, from now on we consider fG ( ) = 1. The restriction to this type of equilibria implies that an
r W
W rc .
equilibrium exists only if and
3.7.
Equilibrium and E¢ cient Allocation
This economy is composed of four types of agents: entrepreneurs, banks, investors and rating agencies.
For simplicity, we will assume that the rating cost c is simply a transfer from the banks to the rating
agency, and that implementing an idea by entrepreneurs has zero costs apart from 1 unit of capital that
must be used for production.
If we consider that each good project produces YG and each bad project only YB (recall that WG and
WB is what banks receive), the total output of the economy, given that only banks with k
k screen is
given by
Y = F (k)YG + (1
F (k))(
0 YG + (1
0 )YB )
Z
k
kdF (k)
1
0
A constrained e¢ cient allocation is given by a screening level k that maximizes total output. It
corresponds to the choice of a benevolent social planner who owns the banking technology and chooses
optimally the level of screening. We make the assumption that screening is cheap from a social point of
view:
Assumption 2 (1
0 )(YG
YB )
1
The social gain of screening by a bank is (1
0 )(YG
YB ), since with probability
0,
a non-screening
bank would get a good project anyway. The assumption states that even the worst bank (which has a
screening cost of 1) should be screening in an e¢ cient allocation.
9
In an equilibrium given by
, total output is given by
Y
=
YG + (1
=
[YG
)YB
YB ] + YB
Z
Z
k(
)
kdF (k)
0
k(
1
)
kdF (k)
1
0
We can immediately conclude that equilibrium is ine¢ cient
Proposition 1 In an e¢ cient allocation, k ef = 1 (or equivalently
equivalently
ef
= 1). In equilibrium k < 1 (or
< 1), as long as r < 1:
Proof: The fact that k ef = 1 follows directly from the previous assumption. Suppose we have r < 1:
If in equilibrium
= 1; then PGR (
(k = 0), and therefore
=
0,
) = PN R (
) = WG , leading to no bank engaging in screening
which is a contradiction.
Note that if there are no informational frictions (r = 1), an equilibrium with
= 1 is possible. In
such a case, Pr(GjN R) cannot be de…ned through Bayes rule (there are no bad loans and all good loans
get a good rating). Therefore, if investors hold o¤ bad beliefs indicating that a no-rating implies that
the loan is bad with probability 1, prices would be PGR (
banks to screen if (1
0 )(WG
WB )
) = WG andPN R (
) = WB , inducing all
1. This implies that even if screening is costly, e¢ ciency can
be achieved. What induces distortions is the fact that banks must resell their claims to investors, and in
doing so information is transmitted imperfectly (r < 1). Since this allows to shift part of the burden of
their decisions, an ine¢ cient use of the screening technology appears, inducing to a misallocation of the
productive resources.
4.
Existence and uniqueness of the equilibrium
We have already characterized the equilibria. Since the critical condition is given by (10) we de…ne:
H( ) = F (k( )) + (1
F (k( )))
(14)
0
With this, an equilibrium is determined by a …xed point of the function H( ).
In order to prove existence and uniqueness of the equilibrium, it is enough to prove that H is a
contraction, which we do in the next proposition
Lemma 2 Conditions under which H( ) is a contraction
sup F 0 (k) and A =
De…ning f
k2[0;1]
a) If c~
jH 0 (
(1
)j
r)(2r
1)then if
c~r(1 r)
(2r 1)(1 r c~(2r 1)) ,
r W (1
2
c)f
0 ) (1 r r~
(1 r)2
1 and there exists a unique equilibrium.
10
we have:
1 and
f (1
0)
r(1
2
W (2r 1)2 2
r)(A+4( 14
q
1
4
r(1 r)A
r(1 r)A))
1,
b) If (1
r)(2r
1) < c~
(1
r) then if
r W (1
2
c) f
0 ) (1 r r~
(1 r)2
unique equilibrium.
1, jH 0 ( )j
1 and there exists a
Proof: See appendix.
There are simpler conditions under which an equilibrium exists, but the conditionjH 0 ( )j
1 greatly
simpli…es the analysis of the comparative statics. Just to have existence, it is enough to note that H( )
is continuous and H(0) =
r(WG WB )
(WG WB rc)
H
0 > 0.
r(WG WB )
r(WG WB rc) .
Then, the only condition we must impose is that, in addition, if
Now, we do the comparative statics of the equilibrium
5.
with respect to the set of parameters.
Comparative statics of the equilibrium
Since in equilibrium not all banks screen (and therefore not all projects are good), and since the rating
technology is imperfect r < 1, the amount of clarity in the market may change. In fact, a good rating
does not necessarily imply that the loan was made to a good project. A measure of clarity is given by
Pr( = GjGR)
Pr( = GjN R):
It is not di¢ cult to see, from equations (6) and (7), that clarity is decreasing in fB . Intuitively, the
more banks who …nanced a bad claim try to ”cheat” the system, the less valuable is a good rating as
a signal. Also, it is easy to see that clarity is increasing in r. A more precise rating agency (or less
complicated assets) makes for a more valuable signal.
Clarity is critical to determine the screening activity of banks. Ceteris paribus, an increase in clarity
leads to an increase in the screening activity of banks. In fact, the cuto¤ level for screening
k = (1
0 )[(r
(1
r)fB )
W (Pr( = GjGR)
Pr( = GjN R))
c(1
fB )]
increases with clarity.
It is crucial to note that in an equilibrium with fB (
) 2 (0; 1), the condition that equates, for a …rm
holding a bad loan, the payo¤ or rating and not rating the loan, is equivalent to
Pr( = GjGR)
Pr( = GjN R) =
c
W (1
r)
:
Moreover, we can see immediately that
As
W goes up, clarity goes down
As the quality of the rating agency r goes up, or the cost of screening c goes up, clarity goes up.
How does this happen? The rating activity of banks owning bad loans (fB ) adjusts in order to keep
the amount of confusion constant if there are changes. For example, as
decreasing clarity and keeping the equality Pr( = GjGR)
11
W goes up, fB also changes,
Pr( = GjN R) =
W (1
r)=c.
Therefore, in any equilibrium where the banks that …nanced a bad project play a mixed strategy, a
change in economic conditions has an important indirect e¤ect on
. This indirect e¤ect plays through
the changes induced in the screening activities of banks due to confusion. We now show how this plays
a role in the comparative statics of equilibrium.
We begin analyzing the changes introduced by a change in
W . We interpret an increase in
W
as either an increase in the productivity of good projects or an increase in the relationship between the
performance of the project and performance of the …nancial asset (for example investment through equity
instead of only bonds). Intuitively, an increase in
therefore increasing
W should increase the measure of banks that screen,
. In the lemma below, however, we see that this e¤ect is exactly canceled by an
increase in the rating activity of banks with bad loans (fB ), which in turn increases confusion.
Lemma 3 Comparative statics of
with respect to
W
In equilibrium
a) If fB (
) 2 (0; 1) then
( ) does not change with
b) If fB (
) 2 f0; 1g then
( ) is increasing in
W
W
Proof: See appendix
The e¤ect of r is unequivocal. As r increases, there are more incentives to screen and, moreover,
confusion is decreased. Therefore, both e¤ects point in the same direction.
Lemma 4 Comparative statics of
In equilibrium
with respect to r
is strictly increasing in r
Proof: See appendix
Figures 1 and 2 illustrate the numerical experiment of lowering r, i.e. lowering rating precision. As it
falls from 0.9 to 0.88, the strategic behavior on the part of lemon holders intensi…es. For a precise enough
rating, banks holding bad loans never use this technology, and for a low enough level of precision, everyone
shops for rating. Clarity decreases in the market through the direct e¤ect of signals becoming less precise.
But in the region where shopping for rating is induced, clarity reduces faster. This is also the region
where the measure of good loans in these loan markets declines most rapidly. In the same region, the
fraction of assets with a good rating actually increases despite the fact that the measure of the good loans
declines. Therefore, this comparative statics in particular helps rationalize several phenomena observed
prior to the …nancial crisis: (i) Laxer screening standards, as identi…ed by the aforementioned empirical
papers; (ii) More intense use of rating (shopping for rating); (iii) Historically low spreads between high
yield and investment grade securities; (iv) The rise in default probability conditional on investment grade
rating; (v) The rise in the fraction of assets receiving an investment grade rating.
If the screening cost c increases, there is a direct e¤ects on
, given by the fact that is now less
attractive to screen, since after obtaining a good project, in equilibrium the rating cost is incurred with
probability 1 and this is now more expensive. Of course, this e¤ect disappears if fB = 1, since then the
12
quality of the project is irrelevant for the rating decision. Moreover, there is an indirect e¤ect through
confusion, since a change in c may induce an e¤ect in fB . We have the following result:
Lemma 5 Comparative statics of
with respect to c
In equilibrium,
a) If fB (
) = 0 then
( ) decreases with c.
b) If fB (
) 2 (0; 1) then
c) If fB (
) = 1 then
( ) increases with c.
( ) does not change with c.
Proof: See appendix
Finally, we present what is maybe the most surprising result. An increase in
e¤ect on
0
has a positive direct
. Ceteris paribus, it can only increase the proportion of good projects in the economy.
However, there is also a direct e¤ect on the incentives to screen. The better the ex-ante distribution of
projects, the less incentives a bank has to spend resources on screening.
The next condition comes from the interplay of both e¤ects. The positive direct e¤ect amounts to
1
F (k), since an increase in
0
lifts the quality of the projects selected by all the banks that do not
0 )Fk (k)
screen. The indirect e¤ect is negative and equal to (1
R, and correspond to the marginal
amount of banks that stop screening (Fk (k) R) times the quality loss due to this (1
Lemma 6 Comparative statics of
De…ning
R
RG
RB , we have that
1
with respect to
0
is increasing in
0
F (k( ))
Fk (k( ))
(1
0 ).
if and only if
0)
R:
(15)
Proof: See appendix
As we can see from the previous results, the characteristics of the equilibrium, and how it reacts to
changes in the di¤erent parameters, depend crucially on fB (
equilibrium rating activity fB (
). It is the natural top ask when the
) belongs to (0,1). The next result gives a general characterization of
fB as a function of . We show how, for low levels of , fB is interior and increasing on , then fB = 1,
then it is again interior, but decreases in , and …nally it is 0.
Why is this so? Remember that the rating activity of banks with bad loans (fB ) is by the di¤erence
P r(GjGR)
P r(GjN R), which indicates the informativeness of the signal. It is not di¢ cult to see that,
as a function of , this is a single-peaked function, increasing for small values of
values of . A good rating does not increase much the beliefs on a good loan if
and decreasing for high
is small, since there
are very few good loans out there anyway. Then, for these low values, an increase in
has a positive
e¤ect, making investors value the signal of a good rating more. The opposite happens for high values of
, where an increase in
deteriorates the value of a good rating, since most projects are good anyway.
Lemma 7 Characterization of fB ( )
De…ning c~ = cW and f^B ( ) as the smaller solution to the quadratic equation
13
fB2 [(1
)2 (1
a) If c~
(1
De…ning
r)~
c]
fB (1
r)(2r
2
r)[ (1
r) + c~(1
2 r)] + r [(1
)(1
r) + c~ r(1
r )] = 0 (16)
1)
and
1
)(1
2
(2r
as the real solutions to the equation:
1)(1
r
c~(2r
1)) + (2r
if
2 [0;
1]
1)(~
c(2r
1)
(1
r)) + c~r(1
r)
0
we have that
8
>
f^B ( )
>
>
>
< 1
– fB ( ) =
>
f^B ( )
>
>
>
: 0
if
2[
if
2
if
2
1;
2]
1 r c~
[ 2 ; 1 r r~c ]
c~
r c~
[ 11 rr r~
c ; r(1 c~) ]
– Moreover, fB is increasing in [0;
b) If (1
r)(2r
1) < c~
(1
1]
and decreasing in [
1 r c~
2 ; 1 r r~
c ].
r), and considering
max
=
1
2 (1
(1
r)2
r)2
c(2r
c2
1)(1
r)
we have that
– fB ( ) =
(
f^B ( )
if
0
if
r c~
r r~
c]
1 r c~
r c~
[ 1 r r~c ; r(1
c~) ]
2 [0; 11
2
– Moreover, fB is increasing in [0;
max ]
and decreasing in [
1 r c~
max ; 1 r r~
c ].
Proof: See appendix
The following …gures illustrate the previous result .
6.
6.1.
Policy Experiments
Mandatory Rating
Consider a policy where all loans made by banks must be rated. In this case, the possibility of not showing
the signal is irrelevant: if everybody gets a signal, not showing one is equivalent to having received a bad
one.
In our model, this is therefore equivalent to setting fB = fG = 1. Let’s denote the proportion of good
projects under this mandatory rating regime as
M R.
MR
14
We directly obtain that
@
MR
@r
@
>0
@c
=0
@ MR
@ W
>0
MR
@
MR
@
0
> 0 if and only if inequality (15) holds.
The …rst result is direct from the observation that PGS
PBS is decreasing in fB , and therefore the
RHS of equation
= F (k( )) + (1
F (k( )))
0
is smaller under mandatory rating. The other results follow directly from the comparative statics in
the benchmark model and the fact that fB = 1.
6.2.
Mandatory Disclosure
We now consider a policy where banks choose whether or not to rate their loans, but in case they do,
they must disclose the signal they got. In this case. In this case, there are three distinct prices because
this policy allows investors to di¤erentiate between assets with good signals, bad signals and no signals
PGS , PBS and PN S .
A bank with a high quality asset chooses to employ the rating technology (fG = 1) if and only if
rPGS + (1
r)PBS
c > PN S :
(17)
Similarly, a bank that …nanced a type B project chooses to employ the rating technology (fB = 1) if and
only if
(1
r) PGS + rPBS
c > PN S :
(18)
The expected payo¤ to the bank from …nancing a project of type
2 fB; Gg is then
RG = fG [rPGS + (1
RB = fB [(1
r)PBS
c] + (1
fG )PN S
1;
(19)
r)PGS + rPBS
c] + (1
fB )PN S
1:
(20)
For a given proportion of good projects in the economy
and strategies (fG ; fB ), the consistent beliefs
by investors must satisfy (whenever possible):
Pr( = GjGS) =
Pr( = GjBS) =
Pr( = GjN S) =
rfG
rfG + (1
)fB (1 r)
(1 r)fG
(1 r)fG + (1
)fB r
(1 fG )
(1 fG ) + (1
)(1 fB )
(21)
(22)
(23)
where Pr( = GjGS) is the belief that the asset with good rating is of high quality, Pr( = GjBS) is the
belief that the asset with bad rating is of high quality and Pr( = GjN S) is the belief that the asset with
15
no rating is of high quality.
Competition among investors implies that prices are determined as the expected payo¤. Thus,
where
PGS = WG Pr( = GjGS) + WB [1
Pr( = GjGS)] =
W Pr( = GjGS) + WB ;
(24)
PBS = WG Pr( = GjBS) + WB [1
Pr( = GjBS)] =
W Pr( = GjB) + WB ;
(25)
PN S = WG Pr( = GjN S) + WB [1
Pr( = GjN S)] =
W Pr( = GjN S) + WB ;
(26)
W = WG
WB .
The next lemma shows that under mandatory disclosure, it is never the case fB = 0 (unless that
fG = 0 too, a case we do not consider). The intuition is simle. If fB = 0 and fG > 0, then any signal
indicates a good loan. The prices generated by those beliefs would always give incentives for banks with
a bad loan to rate their loan.
Lemma 8 Let
be the measure of good projects in the economy. Banks’rating decisions can be summa-
rized as follows:
r(1 r)
;
[ r + (1
)(1 r)][ (1 r) + (1
)r]
r(1 r)
c~ < 1
[ r + (1
)(1 r)][ (1 r) + (1
)r]
fG = 1 and fB = 1 , c~ <
fG = 1 and fB 2 (0; 1) ,
Proof. It is obvious that fG
(27)
(28)
fB from (17) and (17). We …rst prove that if fB = 0, then fG = 0 too.
In fact, if fB = 0 and fG > 0, we have that PGS = PBS = 1. From here, it is direct to compute that
(1
r) PGS + rPBS
c > PN S as long as
W > c (which is true by assumption).
The remaining cases are: (a) fG = fB = 1, (b) fG = 1 and fB 2 (0; 1) and (c) fG = fB = 0, but we
do not consider the last one (for that, we assume that out of equilibrium beliefs for a good or bad signal
are that the loan in good).
Consider case (a), which implies PN = WB . It then follows rPG + (1
fG = fB = 1 then c needs to be smaller than rPB + (1
c < (1
, c~ <
r)PG
r)PB > rPB + (1
PN . The above argument implies
r)PG + rPB
PN = (1 r)(PG PN ) + r(PB
r(1 r)
)(1 r)][ (1 r) + (1
)r]
[ r + (1
r)PG . If
PN )
now consider (b), we need to solve the following equation for fB
(1
W r
r + (1
)fB (1
r)
r)
+r
(1
W (1 r)
r) + (1
)fB r
=c
After solving the last equation and imposing fB > 0 we get c~ < 1. If we impose that the positive root
needs to be smaller than one we get
[ r+(1
r(1 r)
)(1 r)][ (1 r)+(1
)r]
< c~
What can we say about the comparison between the proportion of good projects under mandatory or
voluntary disclosure? Let’s denote by
Proposition 2
MD
MD
the proportion of good projects under mandatory disclosure.
.
16
Proof. In both cases, under mandatory and voluntary disclosure, the equilibrium proportion of good
projects is given by a …xed point of the operator H( ). Since we assume that this operator is a contraction,
we only need to assume that H is bigger (pointwise) under voluntary disclosure. Recalling that H( ) =
F (k( ))(1
0)
+
0,
it is enough to prove that k is bigger (pointwise) under voluntary disclosure.
We know that k is decreasing in fB , so it is enough to prove that fB is smaller (pointwise) under
voluntary disclosure. In both cases (mandatory and voluntary disclosure) fB is simply a truncation
between the solution to the indi¤erence condition of type B and the natural limits 0 and 1. For mandatory
disclosure, it is the solution to:
(1
r)
r + (1
r
)fB (1
r)
+
(1
r
r) + (1
)fB r
= c~
While for voluntary disclosure it is the solution to:
(1
r)
r + (1
r
)fB (1
(1
r)
(1
r) + (1
r)
)(1
fB + fB r)
= c~
It is easy to see (the second term is positive in the …rst equation and negative in the second) that fB
is bigger under mandatory disclosure
7.
Conclusion
We develop a general equilibrium model that allows us to study the interaction of information production
in secondary markets and screening intensity at origination. The model provides insight into why screening
e¤ort at origination is less than optimal, and shows that screening e¤ort unambiguously decreases as a
result of a rise in collateral values, an increase in the fraction of repaying borrowers, and a decrease in the
rating technology precision. The model provides new insight into several pre-crises empirical observations.
The roles of mandatory rating disclosure policy and mandatory rating policy are found to be di¤erent
from the common wisdom.
More empirical work is needed on this subject. In terms of policy, we plan to analyze a restriction on
the amount of loans that can be sold in secondary markets.
17
A.
Proofs
Proof. Lemma 8
a) If fB = fG = 1 we have
fB = 1 , (1
r) W
fG = 1 , r W
Pr > c
Pr > c
But we know that
r W
Pr > (1
r) W
Pr > c
Then a su¢ cient condition is given by
(1
r) W
Pr > c , (1
r) W
(1 r)
r) + r(1
r
)(1 r)
(1
)
(1
)(2r 1)
r) W
>c
( r + (1 r)(1
))( (1 r) + r(1
))
(1
)(2r 1)
>c
r) W
(r
(2r 1))( (2r 1) + (1 r))
, (1
, (1
r + (1
>c
b) If fG = 1 and fB 2 (0; 1) we have
fB 2 (0; 1) , (1
r) W
fG = 1 , r W
Pr = c
Pr > c
Rewriting the …rst equation we have,
(1
r) W
Pr = c , (1
r) W
, (1
r) W
r + (1
[ r + (1
r
)(1
(1 r)
=c
r)fB
(1 r) + (1
)(1 (1 r)fB )
(1
)(r (1 r)fB )
=c
)(1 r)fB ][ (1 r) + (1
)(1 (1 r)fB )]
Solving the last equation for fB we impose that fB > 0. Doing this we have,
fB > 0 ,
>
p
W2
c(2 r
, c<
W
2 r2
1)
(1
2 W2
2r
6 Wc r +
W (1 r)
r)(1
)
(1 r )
18
W2
2
+ 2 Wc +
W2
2
+ 4 W cr2
Imposing that fB < 1 we have
fB < 1 ,
<
p
2c(1
, c>
, c>
W2
2 r2
2 W2
2r
W2
6 Wc r +
2
+ 2 Wc +
W2
2
+ 4 W cr2
r)(1
) + 2c r +
Wr
W +c
2
(1
)( + 2 r
3 r)
W
4 r(1
)(1 r) r(1 r)
(1
)
(1
)(2r 1)(1 r)
W
(r
(2r 1))( (2r 1) + (1 r))
c) If fG = 1 and fB = 0, we have
fB = 0 , (1
r) W
, c>
Pr < c
1
r)
1 r
W (1
We also know
fG = 1 , r W
Pr < c
1
, c<r W
1 r
d) If fG 2 (0; 1) and fB = 0 we have
fG 2 (0; 1) , r W
Pr = c
, 0<c=r W
1
r) + 1
(fG (1
fG ) + (1
<1
)
Then,
fG > 0 , c > r(1
) W
r(1
)
W
1
r
fG < 1 , c <
Proof. Lemma 2
Using the de…nition of H( ) we have,
H( ) = F (k) + (1
a) If fB = 0 we have
R=
As a consequence H =
b) If fB 2 (0; 1) we have
As a consequence H =
2 [0;
1)
when c~
(1
F (k))
0
r W (1 )
(1 r ) .
Fk (1
) H = Fk (1
@ R
=
@
r(1
r)
W
2
0 ) [ (1 r )2 ]
Then
0)
@k
= Fk (1
@
r)(2r
RB )
@
r(1 r) W
.
(1 r )2
0
1. Thus, 8 2 [ 11
c(r (1 r)fB )
. Then @@ R =
1 r
2 @fB
Fk (1
0 ) c @ . Using lemma
R=
2 @(RG
0)
r c~
r c~
r r~
c ; r(1 c~) ],
1
B
c @f
@ .
1 we know that fB is weakly increasing if
1). In addition we know that fB is also increasing if
19
H
2 [0;
max )
when (1
r)(2r
1) < c~
(1
r). Therefore, H
0
1 in those intervals.
On the other hand, we know that if fB 2 (0; 1) the following equation is satis…ed:
P =
c
1 r.
Di¤erentiating this equation we obtain:
@fB
@
=
)H
(1
r)
2
Fk (1
0) c
(1
)(1 r)
=
=
where A~
1
)(1
2
0)
W (1
r + (1
(
1 r
~
B
1 r
+ B~ 2 )
r
~
A
( A~r2
(
(r
(1
r)fB )
(1
r
r
r)2
(1
~
r)fB and B
)(1
(r
(1
r)fB )
r
~
A
( A~r2
1 r
~
B
+ 1B~ 2r )
r~
c)f
(1
)
)
[
c(1
(1
2
0)
][
1 r c~
1
1 r r~
c)
r
r
][f ]
1
r) + (1
)(1
(1
r)fB ).
Thus, if the previous equation is satis…ed we have that H( ) is a contraction in the intervals where fB
is weakly decreasing in , i.e. when c~
c~
(1
r), 8 2 [
r + (1
R=
r) and ~b
)(1
As a consequence H =
happens when c~
( r+(1
(1
On the contrary, i.e.
Fk (1
r)(2r
1
2
r) + r(1
2 (2r
we have,
2
0)
=
r(1
r)( (2r
f (1
=
r(1
0
W (2r 1)2 r(1
2
2
a(~ ) b( ~ )
)2
r)( (2r
1
W (2r
q
1
4
+ 4( 14
q
2
1) 2( 14
W (2r
@ R
@
=
1) <
(2r 1)2 W (1 2 )r(1 r)
a
~2~b2
where
c~
1)(1 r c~(2r 1))
1
2
Therefore, if
r)(1
1)2 2(
c~
1)(1 r c~(2r 1))
2
0)
Then
and when (1 r)(2r
).
1
Fk (1
)) .
1)2 W (1 2 )r(1 r)
.
a
~2~b2
1
2 [ 2 ; 2 ).
1), 8
Fk (1
H
(2r 1)2 W (1 )
)(1 r))( (1 r)+r(1
(1
0)
1 r c~
2 ; 1 r r~
c)
1), 8 2 [
1 r c~
max ; 1 r r~
c ).
c) If fB = 1 we have
a
~
(1 r)(2r
+ 4( 14
2
the H
0
1. This
1)
c~r(1
(2r 1)(1 r
c~r(1
(2r 1)(1 r
r)
c~(2r 1)) )
r)
2
c~(2r 1)) ))
c~r(1 r)
(2r 1)(1 r c~(2r 1)) )
c~r(1 r)
2
(2r 1)(1 r c~(2r 1)) ))
1
Thus, if the previous equation is satis…ed we have that H( ) is a contraction in the intervals where
fB = 1 and
1
2,
i.e. when (1
r)(2r
1) < c~
(1
r), 8 2 [
Proof. Lemma 3
The equilibrium proportion of good projects
satis…es
( W ) = H(
Therefore
( W ); W )
@
H W
=
@ W
1 H
20
1
1 ; 2 ).
Since H( ) is a contraction we only have to analyze the sign of H
H( ; W ) = F (k( ; W )) + (1
Thus, H( ; ) is increasing in
Moreover, we know that:
F (k( ; W )))
W if and only if k( ; ) is increasing in
k( ; W )) = (1
0 )(RG (
= (1
0 )((r
; W)
(1
0
W Moreover, we know that
RB ( ; W ))
r)fB ( ; W )) W (P ( = GjGS)( ; W )
P ( = GjN S)( ; W ))
Therefore
W.
(1
fB ( ; W )c))
W enters through two channels
Directly, in an obviously positive way. The bigger the stakes, the more incentives to lend well.
Indirectly, through changes in fB
We have to consider two cases. First, suppose that fB (
GjN S)) =
c
1 r,
therefore
k( ; W )) = (1
Noting that this is independent of
0 )c
2r
1
W (P ( = GjGS)
P( =
1
r
W the result follows. Now, suppose that fB (
fB is constant in a neighborhood of the equilibrium
is increasing in
) 2 (0; 1). Then
) 2 f0; 1g. Then
, and only the direct e¤ect remains , therefore k
W . The result follows.
Proof. Lemma 4
Exactly as we did in the previous lemma, we have that
@
Hr
=
@r
1 H
Then, if H( ) is a contraction in
we only have to analyze the sign of Hr . In addition, we know that:
H( ; r) = F (k( ; r)) + (1
F (k( ; r)))
0
Thus, H( ; ) is increasing in r if and only if k( ; ) is increasing in r. Moreover,
k( ; r) = (1
= (1
0 )(RG (
0 )[(r
; r)
(1
RB ( ; r))
r)fB ( ; r)) W (P ( = GjGS)( ; r)
P ( = GjN S)( ; r))
(1
fB )c]
Therefore r enters through two channels
A direct e¤ect in the …rst term, obviously positive. A bigger precision increases the incentives to
loan to a good project.
An indirect e¤ect through fB .
21
We again consider two cases. If fB 2 (0; 1), then
have k =
2r 1
1 r c 0,
W (P ( = GjGS)
P ( = GjN S)) =
c
1 r
and we
which is increasing in r, and the result follows. Now, if fB 2 f0; 1g, we know that fB
is constant in a neighborhood of fB (
), so we have:
@(RG RB )
= (1 + fB ) W P + (r
@r
(1
r)fB ) W
@ P
>0
@r
which concludes the proof.
Proof. Lemma 5
Exactly as we did in the previous lemma, we have that
@
Hc
=
@c
1 H
Then, if H( ) is a contraction in
we only have to analyze the sign of Hc . In addition, we know that:
H( ; c) = F (k( ; c)) + (1
F (k( ; c)))
0
Thus, H( ; ) is increasing in c if and only if k( ; ) is increasing in c. Moreover,
k( ; c) = (1
= (1
0 )(RG (
0 )[(r
; r)
(1
RB ( ; c))
r)fB ( ; c)) W (P ( = GjGS)( ; c)
We consider three cases. If fB 2 (0; 1), then
k=
2r 1
1 r c 0,
W (P ( = GjGS)
P ( = GjN S)( ; c))
P ( = GjN S)) =
c
1 r
(1
fB )c]
and we have
which is increasing in r, and the result follows. If fB = 1, then fB = 1 in a neighborhood
of the equilibrium, and we have
k = (1
0 )(2r
1) W (P ( = GjGS)
P ( = GjN S))
which does not depend on c (remember that fB is constant in a neighborhood). Finally, if fB = 0,
then fB = 0 in a neighborhood of the equilibrium, and we have
k = (1
0 )[(2r
1) W (P ( = GjGS)
P ( = GjN S))
which is strictly decreasing in c, so the result follows.
Proof. Lemma 6
We have:
( 0 ) = H( (
H 0
@
)
=
@ 0
1 H
22
0 );
0)
c]
Then, if H( ) is a contraction we only have to analyze the sign of H 0 . In addition, we know that:
H( ) = F (k) + (1
Thus, if k is increasing in
0
F (k))
0
then H( ) is also increasing. Then,
@H
@ 0
= 1 + f(1
0 )Fk (k)
= (1
F (k)) + (1
= (1
F (k))
@k
@ 0
F (k)g
0 )Fk (k)
@k
@ 0
Using the de…nition of k we have:
@H
@ 0
(1
0 )Fk (k)
R
As a consequence
If
then
is weakly increasing in
0
1
0
1
1
F (k)
Fk (k)
1
R
1
F (k)
Fk (k)
1
R
0
Otherwise,
then
is weakly decreasing in
0
Proof. Lemma 7
a) Using (16), it is easy to see that the smaller solution (the other is always greater than 1), is smaller
or equal than 1 if and only if:
2
(2r
1)(1
De…ning A = (2r
r
1)(1
c~(2r
r
1)) + (2r
c~(2r
1)(~
c(2r
1)) and C = c~r(1
quadratic form that is 0 at
1;2
Since c~
(1
(2r
r)(2r
1),
1)(1
r
r
1;2
c~(2r
are real,
1))
1)
>
1
1
=
2
r
1
4
(1
r)) + c~r(1
r)
r), and noting that A
0
0, we have a
C
A
(30)
2 (0; 1) and, in addition,
(2r
r
1)(1
23
r
c~r)
(29)
>
1
1
r
c~r
>
1 r c~
(1 r c~r)2
therefore
1 r c~
1 r c~ .
2
Therefore, fB ( ) < 1 if and only if
2 [0;
Using proposition 1(c), it is easy to see that
S
1 r c~
2 ; 1 r r~
c ).
1 r c~
fB = 0 if
1 r r~
c
1)
(
and that fB = 1 if
2[
1;
2 ].
To see where fB is decreasing or decreasing, we note that the solution to (16) is given by:
fB ( ) =
1 r + 2~
c r
2
c~ +
p
2 r2
2
c~(1
2r
6~
c r+
r)(1
)
which has a unique stationary point, that is a maximum since
since fB (
if
2
2(
1 ) = 1 and fB ( 2 ) = 1 we get that max 2 [ 1 ; 2 ].
1 r c~
r c~
[ 1 r r~c ; r(1
c~) ], it is easy to see that fB ( ) increases in
1 r c~
2 ; 1 r r~
c ].
b) If (2r
1)(1
2
r) < c~
(2r
1)(1
(1
r
max )
+ 2~
c + c~2 + 4~
cr2
@fB
@ (0)
=
r
c
r
1 r
(31)
> 0. Moreover,
Using that and the fact that fB = 0
if
2 [0;
1)
and decreases in
(1
r)) + c~r(1
if
r). Then, we have that
c~(2r
1)) + (2r
and therefore fB ( ) < 1 8 2 [0; 11
2 [0;
2
and decreasing in
if
r c~
r r~
c ).
max
1)(~
c(2r
1)
r) > 0
Then, it is easy to see that fB ( ) is increasing in
2(
1 r c~
0 ; 1 r r~
c ].
In addition, fB = 0 if
if
1 r c~
1 r r~
c.
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25
Figure 1: A Lower Rating Precision
1.5
f (mu*)
0.8
f zero(mu*)
0.7
B
1
B
0.6
0.5
0
0.91
0.9
0.89
0.88
0.5
clarity
0.4
P
0.3
0.91
GR
0.9
r
-P
NR
0.89
0.88
0.89
0.88
r
0.85
0.8
0.84
0.78
frac
GR
mu*
0.83
0.76
0.82
0.74
0.81
0.72
0.8
0.91
0.9
0.89
0.88
r
0.7
0.91
0.9
r
26
Figure 2: A Lower Rating Precision
1.1
1.08
1.06
1.04
1.02
.
1
0.98
0.96
0.94
0.92
Prob |GS
G
0.9
0.91
0.905
0.9
0.895
r
27
0.89
0.885
0.88