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Interventions in markets with adverse
selection: Implications for discount
window stigma
WP 17-01
Huberto M. Ennis
Federal Reserve Bank of Richmond
Interventions in markets with adverse selection: Implications for
discount window stigma∗
Huberto M. Ennis
Research Department
Federal Reserve Bank of Richmond
January 9, 2017
Federal Reserve Bank of Richmond Working Paper No. 17-01
Abstract
I study the implications for central bank discount window stigma of the model by Philippon and
Skreta (2012). I take an equilibrium perspective for a given discount window program instead
of following the program-design approach of the original paper. This allows me to narrow the
focus on the model’s positive predictions. In the model, firms (banks) need to borrow to finance
a productive project. There is limited liability and firms have private information about their
ability to repay their debts. This creates an adverse selection problem. The central bank can
ameliorate the impact of adverse selection by lending to firms. Discount window borrowing is
observable and it may be taken as a signal of firms’ credit worthiness. Under some conditions,
firms borrowing from the discount window may pay higher interest rates to borrow in the market,
a phenomenon often associated with the presence of stigma. I discuss these conditions in detail
and what they suggest about the relevance of stigma as an empirical phenomenon.
JEL classification: E51, E58, G21, G28
Keywords: Banking, Federal Reserve, Central Bank, Policy, Lender of last resort
∗
I would like to thank Doug Diamond and the participants at a LAEF workshop in April 2016, the 2016 SED
meetings, and seminars at Penn State University and the Richmond Fed for comments and discussions. I also would
like to thank David Min for his help with the computation of the example and Thomas Noe for answering my
questions about his paper. All errors are my own. The views expressed in this article are those of the author and
do not necessarily represent the views of the Federal Reserve Bank of Richmond or the Federal Reserve System.
Author’s email: huberto.ennis@rich.frb.org
1
1
Introduction
In the leading article of the February 2012 issue of The American Economic Review, Thomas
Philippon and Vasiliki Skreta study optimal interventions in markets with adverse selection. At the
outset, the authors emphasize that, within the context of their model, “taking part in a government
program carries a stigma” (see the abstract of the paper). However, there is no explicit discussion of
the issue of stigma in the paper. In this article, I study in detail the implications of the PhilipponSkreta model for the incidence of stigma in one of the most prominent government programs directed
at financial markets: the central bank’s discount window.
Discount window stigma is a relevant topic. It refers to the reluctance of banks to borrow from
the central bank for fear of being regarded as in weak financial conditions as a result. Both policymakers and academic economists express concern about this issue on a regular basis. Former
Federal Reserve Chairman Ben Bernanke, for example, often cites stigma as an important consideration when designing policies (see also Fischer (2016)). When in May 2015 two U.S. senators
introduced a bill that was aimed at limiting the emergency lending powers of the central bank,
Bernanke characterized the bill as a “mistake.” The main reason for his argument was that the
bill would make the stigma associated with borrowing from the central bank more prevalent. He
warns us that “the stigma problem is very real, with many historical illustrations” and suggests
that, for example, Northern Rock was a victim of the kind of developments that give rise to stigma
in financial markets.
Gorton (2015), in his review of U.S. Treasury Secretary Geithner’s account of events during
the 2007-08 financial crisis (Geithner (2014)), highlights the critical role played by stigma in the
design of the policy responses to address perceived stresses in liquidity markets. Both Geithner and
Gorton, like Bernanke, believe that stigma was a real concern that could significantly compromise
the effectiveness of interventions.
The incidence of discount window stigma in the U.S. financial markets has also received some
attention in the academic literature. On the empirical side, a well-known example is Furfine (2003).
He studies data from before and after the Federal Reserve’s move, in 2003, to change policy and
transform the discount window into a standing facility (i.e., lending at a penalty rate with no
questions asked). Furfine finds that there was a lot less discount window borrowing happening
after the change in policy than what one would have predicted by looking at the distribution of fed
funds trades before the change. In the spirit of his earlier work (Furfine (2001)), he also confirms
that, under the new policy, the amount of borrowing in the market at rates higher than the discount
window rate was still very significant. He concludes from these findings that there is unambiguous
evidence of stigma at the Fed’s discount window.1
1
Klee (2011) discusses selection effects that can complicate the measurement of discount window stigma using
market interest rates, with an application to the 2007-08 financial crisis in the U.S.
2
More recently, Armantier et al. (2015) study discount window stigma during the 2007-08 financial
crisis. Their work is especially valuable because, contrary to Furfine (2003), Armantier et al. (2015)
do not rely on data on (supposed) interbank loans extracted, using Furfine’s methodology, from the
record of all daily Fedwire funds transfers. This is important since Furfine’s strategy for identifying
interbank loans has been recently shown to be not very reliable (Armantier and Copeland (2015)).
Armantier et al. (2015), instead, use data from bids submitted by banks to the Term Auction
Facility (TAF), a lending facility put in place by the Fed between December 2007 and March 2010.
Using this data, they find strong evidence of discount window stigma during the financial crisis.
Effectively, they find that many banks were willing to pay significantly higher interest rates to
borrow from the TAF than the rate they would have had to pay to borrow from the discount
window. As a result, banks were willing to accept (and indeed experience) significant extra cost
in terms of interest payments in order to avoid the stigma associated with borrowing from the
discount window.2
The amount of theoretical work addressing discount window stigma is sparser. Three recent
articles on the subject are Philippon and Skreta (2012), Ennis and Weinberg (2013), and La’O
(2014).3 Philippon and Skreta (2012) tackle the general question of how to optimally design
government programs aimed at intervening in financial markets. While some form of stigma can
certainly be present in their setup, they do not provide a thorough discussion of the nature of stigma
in that environment. The work by Ennis and Weinberg (2013) is more narrowly focused on the
issue of stigma at the discount window and is aimed at identifying specific features of an economic
environment where stigma, as is often described, can actually occur in equilibrium. The model in
Ennis and Weinberg (2013) is very different from the model in Philippon and Skreta (2012) and
the mechanisms that give rise to stigma in the two models are quite different. Finally, La’O (2014)
studies a model of predatory trading (a la Brunnermeier and Pedersen (2005)) where banks are
reluctant to borrow funds because such an action may become a signal of financial weakness: an
illiquid bank seeking to take a loan fears that other traders, realizing that the bank is weak, would
exploit that information to trade against it. Interestingly, stigma in La’O’s model is associated not
just with borrowing from the discount window, but also with borrowing from the interbank market.
2
Using data also from the crisis and the 2012 court-mandated disclosure of discount window activity by some
banks, Kleymenova (2016) finds no evidence of discount window stigma in those banks’ cost of funding in capital
markets (instead of the interbank market). However, she finds that banks’ behavior changes in ways consistent with
stigma as a result of increases in discount window disclosure requirements. Anbil (2015) and Vossmeyer (2016) are
two other recent empirical studies of the incidence and impact of stigma in financial markets. Both these studies take
a more historical perspective and use data from the Great Depression.
3
Very recently, Gauthier et al. (2015), Li, Milne, and Qiu (2016), and Gorton and Ordo˜ez (2016) also discuss
n
models where discount window stigma plays a role. The models in Gauthier et al. (2015) and Li, Milne, and Qiu
(2016) are very related and, in both models, borrowing from the discount window may represent a signal of the
inability of the bank in question to repay its debts. In Gorton and Ordo˜ez (2016), discount window activity, if
n
discovered, signals the quality of the asset-in-place held by a bank, which makes the bank more vulnerable to run-like
phenomena in the future.
3
See Lowery (2014) for an interesting discussion of La’O’s model.
The research discussed by Philippon and Skreta (2012) is deep and difficult. The paper has
also taken a prominent place in the literature and is often cited, along with Ennis and Weinberg
(2013), as representing the available formal explanation for the phenomenon of discount window
stigma (see, for example, Armantier et al. (2015, p. 318)). Since the Philippon and Skreta (2012)
model can produce insights that are relevant for how to think about discount window stigma, it
seems worth pursuing a better understanding of the mechanisms in the model that give rise to such
stigma and that are not fully discussed in the original paper. This is the objective of this paper.
Instead of the program-design approach taken by Philippon and Skreta (2012), I study perfect
Bayesian equilibrium for a given discount window program in place. This different perspective
allows me to focus the attention more narrowly on the issue of stigma. I also highlight the implications of multiplicity of equilibria for the predictions of the model. As is often the case with
perfect Bayesian equilibrium, off-equilibrium beliefs can play an important role in determining outcomes and, in many situations, stigma is an off-equilibrium phenomenon in the model. However,
for some configurations of the discount window program, banks borrowing from the central bank
are regarded as less likely to repay their debts in the market and hence charged a higher interest
rate on private loans. I discuss in detail the nature of this outcome and its robustness within the
context of the model.
The paper is organized as follows. In Section 2, I introduce the economic environment and the
equilibrium concept. In Section 3, I first describe by way of introduction the equilibrium of the
model when there is no discount window lending. After that, I analyze equilibria when the central
bank makes discount window loans at a given (fixed) rate and when the central bank restricts the
size of the loans that is willing to provide to firms. In each case, I discuss the implications for
discount window stigma of each situation. I provide some concluding remarks in Section 4.
2
The model
I work with the same economic environment that Philippon and Skreta (2012) use in their paper.4
The main difference in the analysis is that Philippon and Skreta consider the optimal design of the
intervention while I restrict attention to the perfect Bayesian equilibrium of the model, taking the
structure of the discount window (i.e., the government program) as given.
4
In a related environment, Tirole (2012) also studies interventions in markets with adverse selection. However,
Tirole’s model is less amenable to a discussion of stigma because agents interact either with the government or with
the market, but never with both (however, see the recent extension of Tirole’s model to several rounds of play in
Che, Choe, and Rhee (2015)). Other examples of recent papers that discuss interventions in markets with adverse
selection are Fuchs and Skrzypacz (2015), Moreno and Wooders (2016), Camargo, Kim, and Lester (2016), and Chiu
and Koeppl (forthcoming).
4
2.1
Environment
There are three time periods t = 0, 1, 2, a set of risk-neutral investors who do not discount the
future, a continuum of firms, and a central bank. In this context, firms should be thought of as
¯
banks. Each firm has cash c0 at time 0 and an asset that pays a random return a ∈ [0, A] at time
2. The initial asset owned by firms is of heterogeneous quality. Let θ be the type of the asset,
¯
and the distribution of asset quality across firms be given by H(θ) for θ ∈ Θ ⊂ [θ, θ], with density
¯
h(θ). An asset of type θ has a random return with distribution FA (a | θ) and density fA (a | θ).
Firms privately know the type of their initial (legacy) asset. For simplicity, I refer to a firm that
has initial assets of quality θ as a firm of type θ.
At time 1, each firm has an opportunity to make a new investment. The cost of the new
¯
investment is x, and it delivers a random return v ∈ [0, V ] at time 2, independent of a and θ.
Assume that E[v] > x > c0 , so that investing produces a positive expected net present value,
and firms need external funding to be able to invest. At time 1, a market for funds opens where
firms can borrow from investors. The market functions as follows: knowing their type θ, each firm
proposes a debt contract (l, r), and any investor can accept to fulfill that contract by making a loan
of size l to the firm at a (gross) interest rate r. Investors compete for contracts and have unlimited
resources (“deep pockets”).
At time 2, creditors of a firm only observe its total income y. More specifically, creditors cannot
observe whether the firm invested or not at time 1 and cannot discriminate between the income
coming from new investment and other income of the firm.
One way to capture that the legacy asset of a type θ firm is “more productive” than the legacy
asset of another firm of type θ is by assuming that the distribution of cash flows for a firm with asset
type θ first order stochastically dominates the distribution of cash flows for a firm with asset type
θ . In an environment closely related to this one, Nachman and Noe (1994) use an even stronger (in
the sense of implying stochastic dominance) order of cash flows: conditional stochastic dominance,
which allows them to establish the optimality of debt contracts.5 While I directly impose the debt
structure on contracts, I keep the stronger assumption in place to be consistent with the previous
literature.
Like Philippon and Skreta (2012), I adopt the approach of Nachman and Noe (1994) and assume
¯ ¯
conditional stochastic dominance directly over the cash flow y = a + v ∈ [0, A + V ], where the
distribution function of y is given by the convolution of the distributions of a and v.6 In the
current setting, conditional stochastic dominance amounts to the same as hazard rate dominance.
5
Nachman and Noe (1994) do not assume that firms have private information about the quality of legacy assets.
In assuming private information, Philippon and Skreta (2012) –and this paper– follow Myers and Majluf (1984).
6
Ideally, one would want to make assumptions over the distribution of a, the return on legacy assets, and then
derive implications for the distribution of cash flows y. For simplicity, however, the literature has imposed assumptions
directly over y. This is also the approach followed here.
5
The hazard rate of the distribution of y is λY (y | θ) = fY (y | θ)/[1 − FY (y | θ)]. Then I assume
¯ ¯
that for all (y, θ) ∈ [0, A + V ] × Θ, we have that fY (y | θ) > 0 and λY (y | θ) is decreasing in θ.
Philippon and Skreta (2012) call this condition the strict monotone hazard rate property. When
this property is satisfied, assets with higher θ dominate assets with lower θ in the conditional
stochastic dominance sense.
To simplify notation, it is useful to define the function:
min(y, rl)fY (y | θ)dy,
ρ(θ, rl) =
(1)
Y
Since the min function inside the integrand is nondecreasing, and strictly increasing for some values
of y that occur with positive probability, we have that the assumed stochastic dominance order
implies that ρ is an increasing function of θ. This property will be important for characterizing
¯ ¯
equilibrium. Note also that ρ(θ, rl) < rl for all rl > 0 since fY (y | θ) > 0 for all (y, θ) ∈ [0, A+V ]×Θ.
Now, let l0 = x − c0 and define r0 as the solution to ρ(θ, r0 l0 ) = l0 and r0 as the solution to:
¯
¯
¯ ¯
l0 =
ρ(θ, r0 l0 )dH(θ)
¯
(2)
Θ
Clearly, we have that r0 < r0 . Assume, also, that:
¯
¯
¯ ¯
l0 − x + E[v] − ρ(θ, r0 l0 ) < 0.
(3)
As will become clear later, in the absence of a discount window, this last condition guarantees that
there is not an equilibrium where all types invest. Since investment has positive net present value
for all types, when not all types invest in equilibrium there is an economic inefficiency that the
central bank may try to reduce by lending via a discount window. This possibility is the focus of
attention in the paper by Philippon and Skreta (2012), and it is also the focus of attention in this
paper.
2.2
The discount window policy
The central bank may decide to put in place a lending facility (discount window) that allows firms
(“banks”) to obtain loans from the central bank at time 0. A discount window loan is a pair (m, R),
where m is the size of the loan and R is the (gross) interest rate to be paid back to the central
bank at time 2.
In principle, the central bank could try to organize its lending so as to provide different loan
contracts to firms of different types. Philippon and Skreta (2012) consider this possibility and show
that there are no gains in this environment from offering menus of debt contracts if the objective of
the central bank is to increase the level of investment at minimum cost. In fact, menus may induce
unwelcomed multiplicity. Here, for simplicity, we restrict attention to discount window policies
that specify a unique interest rate for all loans granted by the central bank to indistinguishable
6
borrowers. This is mainly consistent with common central-bank practices where discrimination is
implemented in a very coarse manner, if at all.
Again following Philippon and Skreta (2012), I assume that investors at time 1 can observe
whether a given firm has borrowed from the discount window at time 0. In reality, discount
window activity in the U.S. is not made public by the central bank. Instead, every two weeks,
each Reserve Bank reports only the total amount borrowed in that period. However, it is often
maintained that in many cases market participants are able to combine information from different
sources to effectively identify discount window borrowers (see, for example, Duke (2010)).7
I assume that the objective of the central bank is (exclusively) to fund firms that are looking to
invest. Hence, any relevant discount window policy satisfies:
m ≤ l0 ≡ x − c0 ,
(4)
and we restrict attention to these policies in the analysis below.
2.3
Payoffs
Firms need to decide whether to borrow from the discount window at time 0 and whether to borrow
from the market and invest at time 1. Following Philippon and Skreta (2012), I assume that the
discount window claim is junior to the claim originated from firms’ borrowing in the market.8
The payoff of a firm that decides to borrow m from the discount window and l from the market
is given by:
(c0 + m + l − x · i + a + v · i − min{c0 + m + l − x · i + a + v · i, Rm + rl})fV (v)fA (a | θ)dvda, (5)
A
V
where i takes values in the set {0, 1}, with i = 1 indicating that the firm decided to invest and
i = 0 indicating that the firm is not investing. Note here that the assumption is that firms cannot
hide cash, and if they have cash at t = 2, they have to use it to repay their debt. For this reason, if
the firm does not spend the cash borrowed at t = 0 and t = 1, then those funds, m and l, become
part of the observable cash flow at t = 2 as indicated inside the bracket associated with the min
sign in equation (5).
7
Armantier et al. (2015, p. 318) discuss in detail the various aspects that influence observability in the U.S.
system. Ennis and Weinberg (2013) consider a model where discount window activity is observed only with some
probability.
8
In the U.S., discount window lending is collateralized and, in general, not the most junior claim in banks’
portfolios. In footnote 15 of their paper, Philippon and Skreta (2012) argue that assuming that the government is a
junior creditor is without loss of generality for their purposes. I will discuss below how this issue matters for stigma.
7
2.4
Equilibrium concept
I study the Perfect Bayesian Equilibrium of the model. Define the functions i(θ), which maps the
set Θ to {0, 1}. When i(θ) = 0 the firm of type θ does not invest and when i(θ) = 1 the firm of
type θ invests. Similarly, define the functions m(θ) and l(θ) mapping Θ to R+ that tells us how
much a firm of type θ decides to borrow from the discount window and the market, respectively.
We denote with B(θ | l, m) the beliefs of the market (i.e., investors) about the value of θ when
the firm borrows m from the central bank and l from the market.
Given a discount window policy (m, R), an equilibrium is a set of functions {i∗ (θ), l∗ (θ), m∗ (θ)},
market interest rates r∗ (l; m), and beliefs B ∗ (θ | l, m) such that the following conditions hold:
(1) Individual Rationality: The functions i∗ (θ), l∗ (θ), m∗ (θ) maximize the objective of the firm
given the interest rates r∗ (l; m) and R;
(2) Break-even: Given market beliefs, the interest rates r∗ (l; m) satisfies the condition:
ρ(θ, r∗ (l; m)l)dB ∗ (θ | l, m) = l;
(6)
Θ
(3) Belief consistency: Beliefs are consistent with Bayes’ rule whenever the values of l and m
are observed in equilibrium.
Condition (6) tells us that the expected repayment associated with a loan of size l in the market
is equal to the value of the loan. This condition is the result of competition among risk-neutral
investors who do not discount the future. The condition also reflects the fact that all investors
share the same level of information and, hence, have the same (on equilibrium) beliefs.
The Perfect Bayesian Equilibrium concept places no constraints on off-equilibrium beliefs; that
is, beliefs over θ when the values of l and m are not chosen in equilibrium. As it is well known,
the freedom to set off-equilibrium beliefs in an unrestricted way can produce multiple equilibria.
One approach often used in the signaling literature is to consider refinements, such as the ChoKreps intuitive criterion (Cho and Kreps (1987)). Nachman and Noe (1994) use the stronger D1
refinement and make it part of their definition of equilibrium. Philippon and Skreta (2012) do not
discuss refinements in their paper.
2.5
Definition of stigma
It is important to be clear about what is meant by the word “stigma.” For example, very recently,
Gorton (2015) discusses discount window stigma and defines it as “a bank’s reluctance to go to
the discount window because of fears that depositors, creditors, and investors will view this as
a sign of weakness, causing its borrowing costs to rise or maybe generating a bank run.” This is
broadly consistent with the interpretation of the term “stigma” used by Bernanke (2008) and, more
8
recently, Armantier et al. (2015).9
In terms of “observables,” it is often taken as evidence of stigma the fact that some banks are
willing to borrow from the market at rates (much) higher than the rates that they could obtain at
the discount window (Furfine (2003)).
In the model studied in this paper, the manifestation of stigma depends on the equilibrium
configuration. For example, in some situations firms that borrow from the discount window are
perceived as representing a higher repayment risk than firms that borrow from the market. However,
when there are no firms borrowing simultaneously from the discount window and the market (as
is the case in Proposition 2), there is no explicit stigma cost associated with borrowing from the
discount window. In fact, in these situations, firms that borrow from the market and firms that
borrow from the discount window all incur the same interest rate cost.
In other equilibria, firms do pay higher rates in the market when also borrowing from the
discount window. In those situations, some firms will borrow only from the market, even though
they could access the discount window at a lower rate. But, because the size of discount window
loans is exogenously restricted in that case, it is again the case that in equilibrium the average
interest cost for a firm borrowing from the discount window (and the market) is the same as that
for a firm borrowing only from the market.
In general, definitions of stigma come in the form of a mixture of a set of observations that
would be associated with the phenomenon and an often partial explanation of the origin of those
observations. Here, the model will allow us to map certain observables, such as interest rate
differentials, to the mechanisms in the model that generate those observables. Whether one decides
to call the phenomenon “stigma,” or something else, becomes less important.
3
Equilibrium
In this section, I study equilibrium with and without a discount window. In both situations, when
there is some borrowing happening in the market, the equilibrium (net) interest rate in the market
has to be positive. We demonstrate this in the following lemma.
Lemma 1. In any equilibrium with an active market for private loans, we have that r∗ (l, m) > 1
for all l > 0 and all m.
Proof. The result follows from applying the break-even condition and the fact that ρ(θ, rl) < rl
9
Bernanke (2008) says: “the efficacy of the discount window has been limited by the reluctance of depository
institutions to use the window as a source of funding. The “stigma” associated with the discount window, which
if anything intensifies during periods of crisis, arises primarily from banks’ concerns that market participants will
draw adverse inferences about their financial condition if their borrowing from the Federal Reserve were to become
known.”
9
whenever rl > 0 since we then have that:
r∗ (l, m)ldB ∗ (θ | l, m) = r∗ (l, m)l,
ρ(θ, r∗ (l, m)l)dB ∗ (θ | l, m) <
l=
Θ
Θ
which implies that r∗ (l, m) > 1.
3.1
Equilibrium without a discount window
When the central bank’s discount window is not active, there is an equilibrium where all firms of
types below a given threshold take a loan in the market and invest, and all firms of types above
that threshold do not borrow and do not invest. Define the threshold value θ∗ ∈ Θ as the solution
to the following equation:
l0 − x + E[v] − ρ(θ∗ , r∗ l0 ) = 0,
(7)
where the interest rate r∗ is the one that satisfies:
θ∗
θ
¯
ρ(θ, r∗ l0 )
dH(θ)
= l0 .
H(θ∗ )
(8)
Figure 1 plots an example of the locus of values of θ∗ (horizontal axis) and r∗ (vertical axis) that
satisfy conditions (7) and (8), separately. The intersection of the two curves identify the values of
interest for θ∗ and r∗ in our equilibrium analysis.10
Using the conditions on parameters assumed in Section 2.1, the following lemma shows that θ∗
lies in the interior of the set Θ. Furthermore, we have that equations (7) and (8) imply r∗ > 1.
¯
Lemma 2. θ < θ∗ < θ and r∗ > 1.
¯
Proof. First, we show that θ∗ lies in the interior of the set Θ. Suppose this is not the case and
instead θ∗ = θ, then by equation (8) we have that r∗ = r0 and hence ρ(θ, r∗ l0 ) = l0 . But, then,
¯
¯
¯
¯
since E[v] > x, this contradicts equation (7). Now suppose that θ∗ = θ, then equation (8) implies
r∗ = r0 and condition (3) (in Section 2.1) immediately implies a contradiction of equation (7).
¯
¯ ¯
Clearly, for the pair (θ, r0 ) we have that l0 − x + E[v] − ρ(θ, r0 l0 ) > 0 and for the pair (θ, r0 )
¯ ¯
¯ ¯
¯ ¯
we have that l0 − x + E[v] − ρ(θ, r0 l0 ) < 0. Since both expressions (7) and (8) are continuous in
¯
(θ∗ , r∗ ), we have that there is a solution to the system of equations (7) and (8) with θ∗ ∈ (θ, θ) and
¯
r∗ ∈ (¯0 , r0 ). That r∗ > 1 follows directly from Lemma 1 and the fact that equation (7) implies
r
¯
∗ l > 0.
that r 0
With the values of θ∗ and r∗ that solve equations (7) and (8), we are now ready to establish the
following result:
10
The example considers that H(θ) is a uniform distribution for values in the interval [−0.8, 0.8] and y has a Beta
distribution with parameters 2 + θ and 2 − θ. The values for the other parameters are listed at the top of the figure:
l0 = 0.25, x = 0.27, and E[v] = 0.285.
10
l0 =0.25, x =0.27, E[v] =0.285
1.7
Equation (7)
Equation (8)
1.6
1.5
r$
1.4
1.3
1.2
1.1
1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
3$
Figure 1: Equilibrium
Proposition 1 (Equilibrium without discount window). When the discount window is not active,
¯
there is an equilibrium where: (1) l∗ (θ) = l0 for all θ ≤ θ∗ < θ and zero otherwise; (2) i∗ (θ) = 1
¯
for all θ ≤ θ∗ < θ and zero otherwise; (3) the market interest rate is equal to r∗ ; and (4) the
market beliefs B(θ | l0 ) = H(θ)/H(θ∗ ) for all θ ≤ θ∗ and zero otherwise.
Proof. The crucial step in the proof is to verify that the proposed functions l∗ (θ) and i∗ (θ) satisfy
individual rationality given the interest rate r∗ . Belief consistency is immediate given the strategies
followed by firms. The break-even requirement follows directly from the definition of r∗ in equation
¯
(8). That θ∗ < θ, and hence not all firms invest in equilibrium, follows from Lemma 2.
To see that l∗ (θ) and i∗ (θ) are individually rational, we start by showing that for all types θ
the strategy of borrowing l0 at rate r∗ and not investing is dominated by not borrowing and not
investing. The payoff from borrowing and not investing is given by:
[c0 + l0 + a − min(c0 + l0 + a, r∗ l0 )]fA (a | θ)da,
(9)
A
which can be simplified to obtain the following inequality:
¯
A
[c0 + a − (r∗ − 1)l0 ]fA (a | θ)da <
a
¯
(c0 + a)fA (a | θ)da,
A
where a = r∗ l0 − c0 − l0 and the right-hand side of the inequality is the payoff from not borrowing
¯
and not investing. So, a firm that decides not to invest also does not borrow.
Now, note that to be able to invest, a firm needs to borrow from the market at least l0 . If the firm
11
borrows exactly l0 when it decides to invest, then it would decide to invest whenever the following
inequality holds:
[c0 + l0 − x + a + v − min(a + v, r∗ l0 )]fV (v)fA (a | θ)dvda ≥
A
(c0 + a)fA (a | θ)da,
A
V
which can be simplified to:
l0 − x + E[v] − ρ(θ, r∗ l0 ) ≥ 0.
(10)
Recall that ρ is a strictly increasing function of θ. Then, by the definition of θ∗ in equation (7) we
have that equation (10) holds for all θ ≤ θ∗ and does not hold for any θ > θ∗ . This confirms that
conditional on a firm borrowing l0 , the decision function i∗ (θ) in the statement of the proposition
satisfies individual rationality.
In principle, there are several specifications of off-equilibrium beliefs that can sustain l∗ (θ) as an
equilibrium. A simple case is when beliefs are such that for all l > l0 we have that B(θ | l) = 1
if θ = θ and zero otherwise. That is, if a firm were to ask for a loan greater than l0 , investors
¯
would believe that the firm is of type θ.11 Given these beliefs, the break-even condition implies
¯
that investors will charge an interest rate rl that satisfies:
¯
Y
min(y + l − l0 , rl l)fY (y | θ)dy = l.
¯
¯
Using that ρ(θ, r0 l0 ) = l0 , it is easy to show that r0 l0 = rl l0 + (rl − 1)(l − l0 ) and the payoff to
¯ ¯
¯
¯
¯
a firm of taking a loan l > l0 at rate rl is the same as the payoff to a firm of taking a loan l0 at
¯
rate r0 . Now, from expressions (7) and (8) we have that ρ(θ, r∗ l0 ) < l0 , which implies that r∗ < r0 .
¯
¯
¯
Hence, taking a loan l0 at rate r∗ gives a higher payoff to the firm than taking a loan l > l0 at rate
rl . We conclude that the decision function l∗ (θ) of the statement of the proposition is individually
¯
rational under the proposed beliefs system.
In principle, there could be more than one solution {θ∗ , r∗ } to equations (7) and (8). Using any
of those solutions, we can construct an equilibrium like the one described in Proposition 1. This
multiplicity is due to the adverse selection effects present in the model and is readily recognized in
Philippon and Skreta (2012). They, however, concentrate their attention on the equilibrium with
the highest value of θ∗ , denoted θD . The corresponding value of r that together with θD solve the
system of equations (7) and (8) is denoted by rD .
3.2
Equilibrium with a simple discount window policy
Seeking to attain a higher level of investment than in the situation without intervention, suppose
that the central bank sets the interest rate charged at the discount window R < rD and stands
11
Note that no firm would ask for a loan lower than l0 because then investors would know that the firm is not
investing and would demand a high interest rate, making borrowing not optimal for the firm.
12
ready to make loans of size l0 to any firm that wishes to borrow. It is easy to see that if R ≤ 1,
then m∗ (θ) = l0 for all θ and i∗ (θ) = 1 for all θ. In other words, when the central bank provides
discount window loans at a negative net interest rate, all firms take the maximum loan at the
discount window and all firms invest in equilibrium.
While a discount window policy that sets its (net) interest rate to negative values (R ≤ 1)
attains the maximum level of investment, it also involves significant subsidies to borrowers. For
this reason, the central bank may want to consider rates that increase investment without going all
the way to the maximum amount. These policies involve interest rates in the range between unity
and rD .12
When R ≥ 1 equilibrium is more complicated as not all firms may borrow and invest. Next,
we study different possible equilibria in this case. One key observation when considering these
equilibria is that, once R > 1 holds, no firm would borrow at the discount window with the
intention of not investing.
Lemma 3. When R > 1, there is no equilibrium with m∗ (θ) > 0 and i∗ (θ) = 0 for some θ.
∗
Proof. From Lemma 1 we know that r∗ (l; m) ≡ rlm > 1 for all l > 0 and all m. Now suppose that
m∗ (θ) = m > 0 and i∗ (θ) = 0 for some θ. The payoff of the firm is:
¯
A
∗
(c0 + l + m + a − rlm l − Rm)fA (a | θ)da,
a
¯
∗
where a = rlm l + Rm − c0 + l + m. But, then, we have:
¯
¯
A
[c0 + a −
∗
(rlm
¯
A
− 1)l − (R − 1)m]fA (a | θ)da <
(c0 + a)fA (a | θ)da ≤
a
¯
a
¯
(c0 + a)fA (a | θ)da,
A
where the last term is the payoff of the firm that does not borrow and does not invest.
As Philippon and Skreta (2012) readily recognized, given a simple discount window policy (l0 , R),
there are multiple equilibria where different subsets of firms borrow from the government. As it
turns out, these different equilibria can have different implications for the extent to which stigma
plays a role in the equilibrium. We analyze first the equilibrium discussed by Philippon and Skreta
(2012) in their implementation section and, after that, we study other possible equilibria.
3.2.1
The Philippon-Skreta equilibrium
Suppose the central bank offers discount window loans of size l0 at interest rate RT ∈ (1, rD ).
Philippon and Skreta (2012) propose one equilibrium where firms with relatively low values of θ
12
Policies that do not achieve the maximum possible investment can be easily motivated by assuming that there is
a cost involved in government revenue collection (such as, for example, distortionary taxes).
13
borrow from the government. Define θT as the solution to:
l0 − x + E[v] − ρ(θT , RT l0 ) = 0,
and θP as the solution to:
θT
ρ(θ, RT l0 )
θP
dH(θ)
= l0 .
H(θT ) − H(θP )
(11)
(12)
Note that such a θP ∈ [θ, θT ] exists because:
¯
θT
ρ(θ, RT l0 )
lim
θP →θT
θP
dH(θ)
= ρ(θT , RT l0 ) > l0 ,
− H(θP )
H(θT )
where the second inequality holds by equation (11) since we have that ρ(θT , RT l0 ) = l0 −x+E[v] > l0
and,
θT
ρ(θ, RT l0 )
θ
¯
dH(θ)
< l0 ,
H(θT )
because condition (3) holds and, with RT ≤ rD , we have that θT > θD . Since the left-hand side of
equation (12) is continuous in θP , the intermediate value theorem implies that such a θP ∈ [θ, θT ]
¯
exists.
Proposition 2 (Philippon-Skreta equilibrium with a discount window). When the discount window offers loans of size l0 at interest rate RT ∈ (1, rD ), there is an equilibrium where: (1)
m∗ (θ) = l0 for all θ < θP and zero otherwise and l∗ (θ) = l0 for all θ ∈ [θP , θT ] and zero otherwise; (2) i∗ (θ) = 1 for all θ ≤ θT and zero otherwise; (3) the market interest rate equals RT ;
and (4) the market beliefs B(θ | l0 , 0) = H(θ)/[H(θT ) − H(θP )] for all θ ∈ [θP , θT ] and zero
otherwise.
Proof. If the firm decides to invest, then it must pick l and m such that l + m ≥ l0 . Consider the
case when the firm investing chooses l + m = l0 . Given the equilibrium interest rate in the market,
the payoff of a type θ firm is:
[y − min(y, RT m + RT l)]fY (y | θ)dy,
and, hence, the payoffs from choosing m = l0 or l = l0 (with l + m = l0 ) are the same.
Assume, as we did in the proof of Proposition 1, that off-equilibrium beliefs for l > l0 and m = 0
are given by B(θ | l, 0) = 1. Then, just as in the proof of Proposition 1, break-even conditions
¯
imply that rl l − (l − l0 ) = r0 l0 and, since RT < r0 , firms have no incentives to deviate and borrow
¯
¯
¯
more than l0 when borrowing from the private market. When a discount window is available, we
need to also consider the situation when m = l0 and l = l0 . Again, assume that B(θ | l, l0 ) = 1 in
¯
this case. Since the break-even condition implies that rlm > 1, no firm will choose to deviate to
m = l0 and l = l0 .
Given Lemma 3, we have that a firm of type θ would choose i∗ (θ) = 1 if and only if:
[y − min(y, RT l0 )]fY (y | θ)dy ≥
14
(c0 + a)fA (a | θ)dy,
which is equivalent to:
E[v] − ρ(θ, RT l0 ) ≥ c0 = x − l0 .
Hence, from the definition of θT and the fact that ρ(θ, RT l) is increasing in θ, we have that i∗ (θ) = 1
for all θ ≤ θT and zero otherwise.
Given that all firms with θ ≤ θT choose to invest and that firms that invest are indifferent between
any feasible choice of l and m such that l+m = l0 , we have that m∗ (θ) = l0 for θ ≤ θP and l∗ (θ) = l0
for θ ∈ [θP , θT ] satisfy individual rationality. Since only firms with θ ∈ [θP , θT ] borrow from the
market, belief consistency implies that B(θ | l0 , 0) = H(θ)/[H(θT ) − H(θP )] for all θ ∈ [θP , θT ], as
required. Finally, by the definition of θP in equation (12), the break-even condition is immediately
satisfied.
There are other equilibria with similar characteristics to the one studied in Proposition 2 but
where firms with higher values of θ than θP borrow from the central bank. Following Philippon
and Skreta (2012), consider a function p : Θ → [0, 1] and assume that for each value of θ a firm
borrows from the discount window with probability p(θ). The case studied in Proposition 2 is that
for which p(θ) = 1 if θ < θP and zero otherwise. However, there are many other possible functions
p(θ) for which the break-even condition in the private market would be consistent with the interest
rate RT . Each of those different functions induce an equilibrium with a market interest rate RT
and some firms borrowing from the discount window. For the issue of stigma, as we will discuss
later, all these equilibria have similar implications since the average quality of the pool of firms
borrowing from the discount window is in each case the same.
Note that in the equilibrium of Proposition 2 it is important that the discount window offers
loans only of size l0 . If firms could choose government loans of different sizes, then in principle
there could be profitable deviations from the equilibrium strategies. Firms may be able to lower
their total funding costs by borrowing less from the market. This is the case because the discount
window rate is not adjusting to changes in the underlying probability of repayment. Then, a firm
taking a small loan from the market may induce a high probability of repayment for that loan.
This, in turn, lowers the corresponding interest rate and may result in a reduction on the total
interest cost from borrowing l0 . We study this case in more detail in section 3.3.1.
3.2.2
Other equilibria
Suppose again that the interest rate at the discount window is RT ∈ (1, rD ) and the central bank
offers loans of size l0 at the discount window. This discount window policy is the same as the one
in place in the equilibrium described in Proposition 2. Interestingly, there is another equilibrium
consistent with that policy, which we describe next.
15
Proposition 3 (Equilibrium with an inactive private market). When the discount window offers
loans of size l0 at interest rate RT ∈ (1, rD ), there is an equilibrium where: (1) m∗ (θ) = l0 for
all θ ≤ θT and zero otherwise and l∗ (θ) = 0 for all θ ∈ Θ; (2) i∗ (θ) = 1 for all θ ≤ θT and zero
otherwise; (3) the private market for loans is inactive.
Proof. Suppose firms borrowing in equilibrium only borrow from the discount window. We verify
this is the case later in the proof. Since RT > 1, firms only borrow from the discount window if
they plan to invest. A firm planning to invest, then, borrows l0 from the discount window at rate
RT . A firm would invest if and only if:
E[v] − ρ(θ, RT l0 ) ≥ c0 = x − l0 .
Hence, given the definition of θT in equation (11) and the fact that ρ(θ, RT l) is increasing in θ, we
have that i∗ (θ) = 1 for all θ ≤ θT and zero otherwise.
It remains to verify that no firm would want to borrow from the private market. Assume that
off-equilibrium beliefs are such that B(θ | l, m) = 1 for all l > 0 and all m. There are two cases
¯
to consider. First, if a firm borrows l0 from the discount window and some extra funds l from the
market, then the firm will be able to pay back the private loan with probability one and the breakeven interest rate is equal to one. The firm, then, is indifferent between playing the equilibrium
strategy or deviating to this alternative. The second case is when the firm does not borrow from
the discount window and instead takes a loan of size l ≥ l0 from the market. Following similar steps
as in the proof of Proposition 1 we can show that the firm would be indifferent between taking a
loan of size l > l0 at rate rl and a loan of size l0 at rate r0 . Since r0 > RT , we have that the firm
¯
¯
¯
would prefer to take a loan of size l0 at rate RT from the discount window.
Off-equilibrium beliefs are rather extreme in the proof of this proposition. In particular, investors
believe that any firm asking for a loan in the market has legacy assets of the lowest type. This was
just used for simplicity. The arguments in the proof of Proposition 3 still go through for many other
systems of off-equilibrium beliefs. For example, even if investors believe that any firm borrowing
from the market, and not from the discount window, is a random draw from the relevant set of
firms, the equilibrium configuration in Proposition 3 is still an equilibrium. Here, the “relevant set
of firms” is those firms that would find the strategy of borrowing from the market and investing
more attractive than not borrowing and not investing.
To understand this claim note that the break-even condition implies that the net interest cost
for the firm of borrowing and investing is the same regardless of whether the firm borrows from
the market l > l0 or exactly l0 . From equations (7) and (8), the relevant firms are those for which
θ ≤ θD and the borrowing cost is r∗ l0 = rD l0 . Given that rD > RT , firms will prefer to borrow from
the discount window rather than from the market, which confirms the equilibrium of Proposition
3 where the private market is inactive.13
13
Note that this equilibrium cannot be refined away using the intuitive criterion: if a firm deviates and borrows
16
3.2.3
Implications for stigma
In the equilibrium of Proposition 2, firms borrowing from the market are considered to be less risky
(in the sense that they are more likely to repay their debts) than those borrowing from the discount
window. One might want to interpret this situation as representing a form of stigma. However,
it is important to note that both firms borrowing from the market and from the discount window
pay the same interest rate for their borrowed funds.
Also, importantly, the way to generate more investment in the model is to get more risky
firms to borrow from the discount window. This selection effect allows the composition of firms
borrowing from the market to change in the direction of lower (repayment) risk — relative to a
situation where all firms are borrowing from the market. In other words, if there is any stigma
in Proposition 2, it is a reflection of the strategy used by the central bank to increase investment
and enhance efficiency. It could hardly be called an unintended consequence or an impediment to
obtaining better policy results, which is often the argument made when discussing discount window
stigma in policy circles.14
The forces at work in the equilibrium of Proposition 2 surface more clearly in the case when the
central bank offers a limited amount of funds to each firm asking for a loan. Specifically, suppose
ˆ
ˆ
the central bank offers loans of size m < l0 at an interest rate R, with R ∈ (1, rD ). Then, if a firm
ˆ
wants to borrow from the discount window and invest, it would have to complement that borrowing
with a loan from the private market of size at least l0 − m.
ˆ
Let π ∈ (0, 1) and define as rS the break-even interest rate when investors are providing loans
of size (1 − π)l0 to all firms of type θ ≤ θP , where θP is the solution to equation (12). In other
words, rS solves:
θP
ρ(θ, rS (1 − π)l0 )
θ
¯
dH(θ)
= (1 − π)l0 ,
H(θP )
(13)
where the seniority of private debt is implicitly recognized. Note that, in general, rS depends on π.
As before, the idea is to consider a situation where the government intends to increase investment by
providing discount window loans of size πl0 anticipating that the resulting configuration of interest
rates and credit will generate a given, targeted amount of investment. In particular, assume that the
government’s target is that all firms with θ ≤ θT decide to invest. For the purpose of comparison,
assume the value of θT is given by the solution to equation (11). The following proposition spells
out the details of this case.
Proposition 4 (Equilibrium with limited discount window lending). When the discount window
from the private market claiming to be a high-θ type and investors believe it, hence lowering the interest rate, then
all other firms with lower values of θ would have similar incentives to deviate. This logic undermines the power of
the intuitive criterion more generally in this model.
14
Interestingly, Gorton and Ordo˜ez (2016) take a similar perspective on the “problem” of stigma and show that,
n
in their model, stigma is desirable as it allows the government to implement the optimal policy during crisis.
17
ˆ
offers loans of size m = πl0 with π < 1 at an interest rate R = [RT − rS (1 − π)]/π, there is
ˆ
an equilibrium where: (1) m∗ (θ) = m for all θ ≤ θP and zero otherwise; and l∗ (θ) = l0 − m
ˆ
ˆ
for all θ ≤ θP , l∗ (θ) = l0 for all θ ∈ (θP , θT ], and l∗ (θ) = 0 for all θ > θT ; (2) i∗ (θ) = 1 for
all θ ≤ θT and zero otherwise; (3) there are two market interest rates, r∗ (l0 − m, m) = rS and
ˆ ˆ
r∗ (l0 , 0) = RT ; (4) the market beliefs are: B(θ | (1 − π)l0 , πl0 ) = H(θ)/H(θP ) for all θ ≤ θP and
B(θ | l0 , 0) = H(θ)/[H(θT ) − H(θP )] for all θ ∈ (θP , θT ].
Proof. As in the proof of Proposition 2, assume that B(θ | l, m) = 1 for all l ∈ {(1 − π)l0 , l0 } and
/
¯
m ∈ {0, πl0 }. Then, an argument similar to the one used there shows that those firms that decide
to invest will choose to borrow either (l, m) = (l0 , 0) or (l, m) = (πl0 , (1 − π)l0 ). Furthermore, since
ˆ
Rπl0 + rS (1 − π)l0 = RT l0 , the cost of borrowing from the discount window and the market to
invest is the same as the cost of borrowing only from the market. It follows from equation (11)
that all firms with θ ≤ θT will decide to borrow and invest; that is, i∗ (θ) = 1 for all θ ≤ θT . Since
firms that invest are indifferent between borrowing from the discount window or not, we have that
the decision rules:
πl0
for θ ≤ θP
0
otherwise,
(1 − π)l0
for θ ≤ θP
m∗ (θ) =
and
l∗ (θ) =
l0
for θ ∈ (θP , θT ]
0
θ > θT ,
are individually rational. Belief consistency follows immediately from the decision rules m∗ (θ)
and l∗ (θ). Finally, the break-even conditions hold since θP satisfies equation (12) and rS satisfies
equation (13).
Perhaps a natural question to ask is why would the central bank choose to limit the size of the
loans provided to firms. A common consideration in policy circles when evaluating credit market
interventions is the extent to which the proposed policy crowds out too much of private activity,
creating what has been called a “footprint” concern (see, for example, Potter (2015)). In the simple
model of this paper, unfortunately, justifications for the “footprint” concern cannot be explicitly
evaluated.
The equilibrium in Proposition 4 produces some interesting implications for thinking about
discount window stigma. There are two situations to consider, depending on whether rS is higher
ˆ
or lower than RT . Figure 2 plots rS and R as a function of π, where higher values of π correspond
to larger discount window loans. As can be seen in the figure, for low values of π we have that rS is
greater than RT and for high values of π the opposite is true. This gives rise to the two situations
under consideration.
18
ˆ
The dependence of R on π is more complicated because both direct and indirect effects (through
ˆ
rS ) play a role in this case. The function R(π) can be interpreted as the locus of central-bank policies
ˆ
(R, π) that are consistent with implementing a level of investment that has all firms with θ ≤ θT
investing. In other words, if the central bank fixes a particular rate at the discount window, then
ˆ
the inverse of the function R(π) plotted in Figure 2 gives the size of the discount window loan that
the central bank should offer to firms in order to implement the desired level of investment (that
corresponds to θT ). Interestingly, note that for high values of the discount window rate (values
above RT ) there are two possible sizes of the discount window loan that implement the same level
of investment in the economy.
l0 =0.25, x =0.27, E[v] =0.285
1.3
1.25
Interest Rate
1.2
1.15
1.1
1.05
1
rS
^
R
RT
0.95
0.9
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
:
Figure 2: Interest rates
Going back to the implications for stigma, we have that when rS > RT , firms borrowing at
the discount window pay an interest rate in the market that is higher than the one paid by firms
borrowing only in the market (RT ). Since rS converges to a value higher than RT when π converges
to zero (compare expressions (12) and (13) with π = 0), we know that this case is possible when
the central bank provides discount window loans that are relatively small.15 Of course, the interest
ˆ
rate on the discount window loan, R, would have to be lower than RT for firms to remain indifferent
between borrowing from the discount window or from (only) the market.
This situation, then, has firms that borrow from the discount window paying higher interest
15
Philippon and Skreta (2012, p. 18) show that, for a given RT , there is a minimum value of π consistent with
ˆ ≥ 1, which is required if the central bank wants to restrict borrowing only to firms that are planning to invest.
R
This is also evident in Figure 2.
19
rates in the market than the rates paid by the firms not borrowing from the discount window.
Also, firms that borrow only from the market borrow at a rate (RT ) that is higher than the rate
ˆ
they would obtain at the discount window (R). These two outcomes are often associated with the
perception that the discount window is subject to stigma.16
For higher values of π, Proposition 4 is consistent with a situation where rS < RT . In such
cases, firms borrowing from the discount window actually pay in the market an interest rate that
is lower than the rate paid by firms not borrowing from the discount window. Concurrently, the
discount window rate is higher than the rate paid by firms that only borrow from the market. So,
firms borrow from the discount window even though the interest rate for borrowing l0 in the market
would be lower.
The key to understanding this result is to note that, by borrowing from the discount window,
a firm lowers the amount of the loan that it needs to obtain in the market. Since discount window
loans are junior claims, by reducing the size of the loan obtained in the market, the firm is able
to reduce repayment risk and, in consequence, reduce the interest rate paid on that portion of the
total amount borrowed (i.e., the part that is covered with a loan in the private market). Repayment
risk is transferred from the market loan to the discount window loan, but since the interest rate
at the discount window is an administered rate — and does not adjust with changes in the level
of repayment risk — the shift in risk reduces the total cost of borrowing for the firm. Hence, even
though borrowing from the discount window appears to be more expensive than borrowing only
from the market, some firms borrow from the window in equilibrium.
3.3
Other discount window policies
The discount window policies considered so far have many features that do not align well with the
way the discount window is structured in the U.S. For example, the Fed does not specify a size of
the loans that can be obtained at the discount window. While a minimum bid size was imposed for
the Term Auction Facility during the time it was in operation (2007-2010), this type of loan-size
restrictions are not imposed at the regular discount window (Armantier et al. (2015)).
A second feature that is at odds with how the discount window is run in the U.S. is that loans
from the Federal Reserve are fully collateralized. This is important because the assumed seniority
of private loans (relative to discount window loans) in the model is probably not the best way to
capture the institutional arrangements prevailing in the U.S. economy.17 Seniority matters crucially
16
Note that the Philippon-Skreta model does not consider the possibility of firms borrowing from the discount
window at time 0 to then lend to other firms in the market at time 1. Whether firms would want to engage in
this kind of intermediation is not obvious: lending firms would need to compete with deep-pocketed investors in the
market at time 1. Since the interest rate at the discount window is positive, the face value of the funding cost of
those firms is higher than the investors’ cost. However, in principle, firms intermediating funds may not be able to
pay back their discount window loans in some situations, lowering the expected value of their funding cost.
17
Seniority of claims is less clear when taking a consolidated government perspective in the presence of deposit
20
for the results coming out of the model. This is clear in Proposition 4 and will become clear also
when we discuss other discount window policies in this section.
Another important aspect of U.S. discount window policy is that the administered interest rate
is required to be a penalty rate: that is, a rate that is equal to a benchmark market interest rate
(or policy rate) plus a positive premium. At the time of writing, discount window lending in the
U.S. (primary credit) is offered at an interest rate that is equal to (the top of the range for) the
target policy rate plus 50 basis points. In the model, no such requirement is imposed and, in fact,
for the result in Proposition 2 it is important that the interest rate at the discount window can
be equal to the market rate. As we saw in Proposition 4, the model is consistent with equilibrium
situations where the interest rate at the discount window is higher than the rates in the private
market, but seniority of private market loans is crucial for such results.
Of course, there are other features of the Federal Reserve’s discount window policy that are
not addressed by the model. For example, access to the regular discount window is contingent on
the regulatory assessments of the financial conditions of the bank at the time of borrowing. This
supervisory power may be the source of an informational advantage on the part of the Federal
Reserve. By contrast, in the model the central bank has the same information as the market about
the quality of the legacy assets (and the loan repayment ability) of firms.18
In what follows, we discuss implications of the model when modified to capture more closely U.S.
institutional features. The results tend to underscore the significant challenges that the PhilipponSkreta model encounters as a framework for rationalizing the discount window outcomes observed
in U.S. financial markets.
3.3.1
No loan-size restrictions
When the discount window offers loans of any size m ≤ l0 at a given interest rate R, firms have to
decide how much to borrow from the market and how much from the discount window. To make
this decision, firms have to know the interest rate that investors would charge for loans of different
sizes. In other words, firms need to know a price function and, given that price function, they will
choose the size for their private loan.
As is clear from the break-even condition (6), market interest rates depend on investors’ beliefs.
The perfect Bayesian equilibrium concept restricts only on-equilibrium beliefs, while the full system
insurance. While discount window loans are fully collateralized, it is often the case that the losses experienced by the
insurance fund depend on the ability of the failing bank to borrow from the discount window before failing (Goodfriend
and Lacker (1999)). The extra liquidity available to the bank is often used to pay back uninsured depositors, making
them effectively senior claimants relative to the consolidated government.
18
See, for example, Rochet and Vives (2004) for a model of the discount window where the central bank has an
information advantage over market lenders due to its supervisory powers. In Ennis and Weinberg (2013) the central
bank actually has less information than private banks, and the same information than outside investors.
21
of beliefs (on and off equilibrium) determines the patterns of lending observed in equilibrium,
through the price function. This delicate interaction between beliefs and equilibrium creates the
possibility of multiple equilibrium configurations.
For concreteness, and to get a better sense of the forces at work in the model, we will study
one particular equilibrium. In this equilibrium, investors believe that any firm borrowing from the
market, regardless of how much it asks to borrow, is a random draw from the set of firms investing
in equilibrium.
Suppose that all firms with legacy assets of type θ lower than a threshold value θ∗∗ are expected
to invest. Using the break-even condition, we obtain the interest rate function r∗∗ (m) that satisfies
the equation:
θ∗∗
ρ[θ, r∗∗ (m)(l0 − m)]
θ
¯
dH(θ)
= l0 − m,
H(θ∗∗ )
(14)
for all values of m ≤ l0 . Note that this equation is equivalent to equation (13) and, as illustrated
in Figure 2, the function r∗∗ (m) is decreasing.
Suppose now that firms take as given the pricing function r∗∗ (m) when they decide how much
to borrow from the discount window and the market. As we will confirm later, firms that decide
to invest will borrow exactly the amount l0 ; that is, m + l = l0 where m is the amount borrowed
at the discount window and l the amount borrow from the market. Then, we have that firms will
choose m to solve:
{y − min[y, Rm + r∗∗ (m)(l0 − m)]}fY (y | θ)dy,
max
m
Y
which is equivalent to minimizing total funding costs (private plus discount window loans); that is:
min Rm + r∗∗ (m)(l0 − m).
m
(15)
Denote by m∗∗ the solution to this problem. Note that m∗∗ is independent of θ, so all investing
firms will choose to borrow the same amount from the discount window (and from the market).
Finally, the threshold value θ∗∗ is given by the equation:
l0 − x + E[v] − ρ[θ∗∗ , Rm∗∗ + r∗∗ (m∗∗ )(l0 − m∗∗ )] = 0,
(16)
which is the counterpart of equation (7) in this case.
Consider a function r∗∗ (m) and values of θ∗∗ and m∗∗ that jointly solve equations (14), (15),
and (16). Then, we have the following result:
Proposition 5 (Equilibrium with flexible discount window lending). When the discount window
offers loans of any size m ≤ l0 at an interest rate R, there is an equilibrium where: (1) m∗ (θ) = m∗∗
and l∗ (θ) = l0 − m∗∗ for all θ ≤ θ∗∗ and zero otherwise; (2) i∗ (θ) = 1 for all θ ≤ θ∗∗ and zero
otherwise; (3) the market interest-rate function is r∗ (l0 − m, m) = r∗∗ (m); (4) the market beliefs
are: B(θ | l0 − m, m) = H(θ)/H(θ∗∗ ) for all θ ≤ θ∗∗ and all m ≤ l0 .
22
Proof. Consider the following system of beliefs: for all m ≥ 0 and l = l0 − m, let B(θ | l, m) =
H(θ)/H(θ∗∗ ) for all θ ≤ θ∗∗ and zero otherwise; for all m ≥ 0 and l > l0 − m, let B(θ | l, m) = 1.
¯
Given these beliefs, for all m ≥ 0 and l = l0 − m, the break-even condition for investors (14) implies
that r∗ (l0 − m, m) = r∗∗ (m). If l ≥ l0 − m, then the break-even condition for investors under the
proposed beliefs is:
Y
min{y + l + m − l0 , rlm l}fY (y | θ)dy = l,
¯
¯
which determines the value of rlm . Following similar steps as in the proof of Proposition 1, we can
¯
show that r∗∗ (m)(l0 − m) < rlm l − (l + m − l0 ), which implies that the funding cost associated
with a loan of size l > l0 − m is higher than the funding cost of a loan of size l = l0 − m.
Hence, firms take loans in the private market of size l0 − m. From this we conclude that, given
the pricing function r∗∗ (m) and the fact that m∗∗ solves equation (15), firms investing will choose
m∗ (θ) = m∗∗ and l∗ (θ) = l0 − m∗∗ . Firms not investing can always repay as much as they
borrowed, and given that r∗∗ (m) > 1, borrowing to not invest is not optimal. Then, according
to equation (16), all firms with θ ≤ θ∗∗ , and only those firms, will choose to invest (that is,
i∗ (θ) = 1 for all θ ≤ θ∗∗ and zero otherwise). Finally, given firms decisions, we have that the beliefs
B(θ | l0 − m∗∗ , m∗∗ ) = H(θ)/H(θ∗∗ ) for all θ ≤ θ∗∗ satisfy Bayes rule.
l0 =0.25, x =0.27, E[v] =0.285
0.3
3 $$ =0.11 R=1.2 m**=.092
$$
3 =0.36 R=1.15 m**=.0973
Total funding cost
0.295
0.29
0.285
0.28
0.275
0.27
0
0.05
0.1
0.15
0.2
0.25
m
Figure 3: Optimal discount window loan
Figure (3) plots an example of the objective function from problem (15): the total funding costs
as a function of the size of the discount window loan m. When the optimal choice of m is interior
(as it is the case in the figure), we have that R > r∗∗ (m∗∗ ) and there are two forces at play in
23
the determination of the optimal value of m. On one side, by borrowing more from the discount
window and less from the market, firms can shift repayment risk away from market transactions
and, in that way, lower the interest rate and the borrowing costs associated with private loans.
On the other side, since discount window borrowing is more expensive, borrowing more from the
discount window and less from the market tends to increase the total cost of borrowing.
Interestingly, then, when the central bank offers loans at a relatively high rate, firms may choose
to borrow some from the market and some from the discount window as a way to manage their
repayment risk in the dealings with private investors (the risk-sensitive counterparties in the model).
An observer may wonder why firms are borrowing from the discount window at an interest rate
higher than the one they are able to obtain in the market. The key to understanding this outcome
is to note that the interest rate on a private-market loan is increasing in the amount of the loan.
The ability to borrow from the discount window, then, gives firms flexibility to adjust the amount
of their private borrowing so as to respond to those price effects.
When the solution m∗∗ is interior, lower discount window interest rates are associated with
higher discount window loans: an intensive margin effect. In the figure, when R = 1.2 we have
that m∗∗ = 0.092 and when R = 1.15 we have that m∗∗ = 0.0973. Eventually, as the interest rate
on discount window loans becomes very low, the solution to problem (15) stops being interior and
investing firms choose to cover all of their funding needs with central-bank loans.
There is in the model also an extensive margin effect because the equilibrium value of θ∗∗ also
depends on the level of the discount window interest rate R. As shown in the figure, when the
discount window rate decreases (from 1.2 to 1.15), it becomes less expensive to fund investment and
more firms decide to invest; that is, the value of θ∗∗ increases (from 0.11 to 0.36, in the figure). In
this sense, both the intensive and the extensive margin move in the same direction: lower discount
window rates imply more lending.
3.3.2
Seniority and “penalty” rates
As we saw in propositions 4 and 5, in principle, the equilibrium of the model can be consistent with
situations where the interest rate charged at the discount window is higher than the rate charged in
the market. However, it is clear from our discussion of those propositions that the results depend
(critically) on the fact that discount window loans are junior to the loans obtained in the market.
In the U.S., discount window loans are fully collateralized and offered at a penalty rate. Collateralization implies that central bank loans are effectively senior to private loans (see footnote 17,
however). I demonstrate now that the equilibrium configurations described in Propositions 4 and
5, which are consistent with “penalty-like” discount window interest rate, are not possible when
discount window loans are senior to private loans.
The main change in the equilibrium conditions, relative to when private loans are senior claims,
24
is in the break-even condition. In particular, when discount window loans are senior, the expected
repayment function conditional on a given value of θ is:
min[max(0, y − Rm), rl]fY (y | θ)dy
ξ(θ, Rm, rl) =
Y
¯ ¯
A+V
min(y − Rm, rl)fY (y | θ)dy,
=
(17)
Rm
and this repayment function replaces ρ(θ, rl) in the investors’ break-even condition (6). For both
the configuration in Proposition 4 and in Proposition 5 the other equilibrium conditions remain the
same. The following proposition demonstrates that those proposed configurations are not consistent
with equilibrium when discount window loans are senior claims over private loans.
Proposition 6 (Discount window senior claims and “penalty” rates). The configurations proposed
in propositions 4 and 5 cannot be an equilibrium when discount window loans are senior claims
relative to private loans.
Proof. The proof is by contradiction. Start with the configuration in Proposition 4. First note that
¯
for any θP and θT with θ < θP < θT < θ, the following inequality holds:
¯
θP
¯ ¯
A+V
min(y, rl0 )fY (y | θ)dy
θ
¯
Rm
dH(θ)
<
H(θP )
θT
ρ(θ, rl0 )
θP
dH(θ)
.
− H(θP )
H(θT )
(18)
Now consider the case when the discount window offers loans, senior to private claims, at rate
R > 1. If an equilibrium with the configuration described in Proposition 4 were to exist, then,
given a feasible target investment level indicated with θT , there would be values θP , RT , and m
such that the following two conditions hold:
θP
ξ(θ, Rm, RT l0 − Rm)
θ
¯
θT
ρ(θ, RT l0 )
θP
dH(θ)
H(θP )
dH(θ)
− H(θP )
H(θT )
= l0 − m
(19)
= l0 .
(20)
We now show that these two conditions holding simultaneously imply a contradiction. To see this,
note that the two conditions imply that:
θP
¯ ¯
A+V
min(y, RT l0 ))fY (y | θ)dy
θ
¯
Rm
dH(θ)
= l0 +(R−1)m > l0 =
H(θP )
θT
ρ(θ, RT l0 )
θP
dH(θ)
,
H(θT ) − H(θP )
which stands in contradiction with inequality (18).
To show that the configuration in Proposition 5 cannot occur in equilibrium when m∗∗ > 0 and
the discount window loans are senior claims, note that the function r∗∗ (m) solves:
θ∗∗
ξ[θ, Rm, r∗∗ (m)(l0 − m)]
θ
¯
25
dH(θ)
= l0 − m,
H(θ∗∗ )
for all m ∈ [0, 1]. When m∗∗ is interior, we have that Rm∗∗ + r∗∗ (m∗∗ )(l0 − m∗∗ ) < r∗∗ (0)l0 and
ξ[θ, Rm∗∗ , r∗∗ (m∗∗ )(l0 − m∗∗ )] < ρ(θ, r∗∗ (0)l0 ) − Rm∗∗ . Then, we have that:
∗∗
l0 −m
θ∗∗
=
θ
¯
dH(θ)
ξ[θ, Rm , r (m )(l0 −m )]
<
H(θ∗∗ )
∗∗
∗∗
∗∗
θ∗∗
∗∗
ρ(θ, r∗∗ (0)l0 )
θ
¯
dH(θ)
−Rm∗∗ = l0 −Rm∗∗ ,
H(θ∗∗ )
which leads to a contradiction when R > 1.
When private loans are senior claims, Propositions 4 and 5 describe possible equilibrium situations where some discount window lending takes place at rates higher than the rate offered in
the market, a situation often associated with discount window stigma. However, in Proposition 6
I have shown that such situations are not consistent with equilibrium if discount window loans are
senior claims – arguably a more consistent representation of the way discount window loans are
granted in the U.S.
By comparison, the model in Ennis and Weinberg (2013) generates equilibrium discount window
lending at a penalty rate using a different mechanism. In that model, since some lenders can observe
information about the repayment ability of borrowers, the threat from the associated transmission
of information when trading in the market makes the discount window effectively the cheapest
funding alternative for those borrowers (even though it is offered at a penalty rate). This is also
the reason why, in equilibrium, the discount window is considered a negative signal about borrowers
in the Ennis-Weinberg model: some borrowers end up at the discount window because some lenders
observed their (poor) “quality” and decided not to lend to them.
4
Conclusions
There appears to be a relative consensus among policymakers that the Fed’s discount window
suffers from the ailment of stigma: the over-reluctance of borrowers to access the funds offered at
the facility. From the way discount window stigma is being discussed in policy circles, one might
conclude that we have a relatively good understanding of the theoretical underpinnings of the idea.
However, I am aware of only very few papers in the literature that present models where stigma
can endogenously arise. One of those models is the one analyzed in the article by Philippon and
Skreta (2012) to study optimal program design aimed at interventions in credit markets.
In this paper, I have investigated in detail the implications for discount window stigma of the
Philippon-Skreta model. While the analysis produced some interesting insights about the nature
of stigma, overall I conclude that the explanation of stigma provided by this model does not align
well with the ideas debated in policy circles. While the model is such that in general the average
“quality” of borrowers at the discount window can be low – and in that sense the discount window
could be considered “a sign of weakness”– there is no clear sense in which stigma reduces “the
efficacy of the discount window” (see the quotes in section 2.5). In fact, stigma-like effects in the
26
Philippon-Skreta model are the means by which the government enhances efficiency by promoting
more overall lending and investment. In other words, in the context of the model, one is left
thinking that stigma should perhaps be considered a good thing.
The primary objective here was to understand stigma in the model as it was set up by Philippon
and Skreta (2012). In such a model, there are several assumptions that appeared at odds with the
way the discount window works in the U.S. economy. Discount window loans in the U.S. are
effectively fully collateralized and offered at a penalty rate, for example. As we discussed, the
Philippon-Skreta model has a hard time simultaneously accommodating these features. At the
same time, discount window activity in the U.S. is not directly observed by market participants
and banks borrowing at the discount window can, in principle, intermediate funds by later lending
in the private market. As we have discussed, these are critical aspects of the problem that are not
part of the original model. Studying modifications of the theory that would better align it with
the U.S. reality while at the same time producing outcomes that improve our understanding of the
incidence of stigma is potentially a productive avenue for further research.
27
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@FedFRASER