The full text on this page is automatically extracted from the file linked above and may contain errors and inconsistencies.
Federal Reserve Bank of Chicago
Credit Misallocation and Macro Dynamics
with Oligopolistic Financial Intermediaries
Alessandro T. Villa
September 8, 2022
WP 2022-41
https://doi.org/10.21033/wp-2022-41
Working papers are not edited, and all opinions and errors are the
responsibility of the author(s). The views expressed do not necessarily
reflect the views of the Federal Reserve Bank of Chicago or the Federal
Reserve System.
*
Credit Misallocation and Macro Dynamics with
Oligopolistic Financial Intermediaries
Alessandro T. Villa*
September 8, 2022
Abstract
Bank market power shapes firm investment and financing dynamics and hence affects the transmission of macroeconomic shocks. Motivated by a secular increase in
the concentration of the US banking industry, I study bank market power through the
lens of a dynamic general equilibrium model with oligopolistic banks and heterogeneous firms. The lack of competition allows banks to price discriminate and charge
firm-specific markups in excess of default premia. In turn, the cross-sectional dispersion of markups amplifies the impact of macroeconomic shocks. During a crisis, banks
exploit their market power to extract higher markups, inducing a larger decline in real
activity. When a “big” (i.e., non-atomistic) bank fails, the remaining banks use their
increased market power to control the supply of credit, worsening and prolonging the
recession. The results suggest that bank market power could be an important concern
when formulating appropriate bail-out polices.
JEL Codes: D43, E44, G12, G21, L11.
Keywords: Dynamic Financial Oligopoly, Endogenous Financial Markups, Heterogeneous Firms, Firm Dynamics, Micro-Founded Financial Frictions, Price Discrimination.
* Federal
Reserve Bank of Chicago. Mailing address: 230 S. La Salle St., Chicago, IL 60604, USA. Email:
alessandro.villa@chi.frb.org. Acknowledgments: I am indebted to my advisors Andrea Lanteri, Lukas
Schmid, Giuseppe Lopomo, Francesco Bianchi, and Pietro Peretto for their support and advice. I would
also like to thank Joel David, François Gourio, Leonardo Melosi, Pablo D’Erasmo, Nicola Persico, Stijn G.
Van Nieuwerburgh, Dean Corbae, Vincenzo Quadrini, Robert Barsky, Roberto Steri, Fabio Schiantarelli,
Matthias Kehrig and Cosmin L. Ilut for their helpful comments and encouragements. I benefited from
fruitful conversations with numerous seminar participants at Computing in Economics and Finance 2019,
BFI Macro Finance Research Program 2020, Econometric Society/Bocconi World Congress 2020, University
of Surrey, Duke University, University of Essex, Boston College, Chicago Fed, University of Washington,
Federal Reserve Board and Penn State University. Disclaimer: The views expressed in this paper do not
represent the views of the Chicago Fed or the Federal Reserve System. I declare that I have no relevant or
material financial interests that relate to the research described in the paper. All errors are my own.
1
1
Introduction
The banking industry has become increasingly concentrated over the past two decades, with
the asset market share of the five largest US banks rising from 26% in 1996 to 50% in 2018.
Moreover, the Lerner index increased from 0.2 in 1996 to 0.33 in 2014, pointing to a sizable
increase in markups.1 A large and influential literature has studied the interactions between
financial markets, firm, and aggregate dynamics but has typically assumed perfectly competitive financial intermediaries; thus, it does not speak to this trend of increasing banking
sector consolidation and markups.2
In this paper, I study the role of imperfect competition in the financial intermediation
sector for firm investment and financing dynamics, as well as for the transmission of macroeconomic shocks. I develop a novel dynamic general equilibrium model that incorporates
an oligopolistic financial sector with heterogeneous firms. Imperfect competition enables
financial intermediaries to charge firm-specific markups that depend on the idiosyncratic
characteristics of the firms to which they lend. In particular, banks exert a higher degree
of market power on firms that are more financially constrained and have a high marginal
productivity of capital. These firms have worse outside options (e.g., a high cost of non-bank
finance) and one additional unit of investment in physical capital can contribute significantly
to their future production; hence, they exhibit a higher and less elastic demand for credit.
The resulting dispersion of markups (i) induces credit – and thus capital – misallocation,
reducing aggregate productivity and (ii) plays a significant role in shaping the transmission
of macroeconomic shocks.
During a crisis, banks exploit their market power to extract higher markups, inducing
a larger decline in real activity. Notably, since my model features non-atomistic banks, I
can study market structure changes in the intermediation sector (e.g., the failure of a large
intermediary). When a single “big” bank fails, surviving banks utilize their increased level
of market power to control the supply of credit contributing to amplifying and prolonging
the recession. The results suggest that banks’ market power should be an important source
of concern for policymakers deciding whether to bail out a large intermediary.
Succinctly, the model works as follows. Firms make optimal capital structure decisions
1
See Appendix A.4. See Corbae and D’Erasmo (2021) for related and more detailed evidence.
See, for instance, Bernanke and Gertler (1989), Covas and den Haan (2011), Jermann and Quadrini
(2012), Khan and Thomas (2013), and, more recently, Elenev, Landvoigt, and Van Nieuwerburgh (2021).
2
2
by balancing equity and debt financing, generating an endogenous dynamic demand for
loans. Banks are large (i.e., non-zero mass) players and firms are a continuum of followers
in a Stackelberg fashion (i.e., each financial intermediary takes firms’ dynamic demand for
loans as given and competes to supply funding to each individual firm). Intermediaries
make strategic decisions by internalizing the effect of their actions on present and future
banks, firms’ decisions, and on the aggregate economy. In such an environment, the bank’s
optimal equilibrium choice of loan supply is determined by solving so-called generalized Euler
equations.3 Each generalized Euler equation is an otherwise standard Euler equation, except
that it contains a firm-specific elasticity that measures the sensitivity of the future interest
rate with respect to the current supply of loans. The model generates firm-specific credit
spreads that accrue to banks, which include default premia and markups.
My analysis proceeds in two main steps. First, I develop a stylized two-period model
that I use to derive analytical insights on the role of oligopolistic intermediaries for firm and
aggregate dynamics. Second, I build a quantitative infinite-horizon version of the model that
I use to gauge the quantitative importance of these connections. I calibrate the model to
match several financial and macroeconomic variables using Federal Deposit Insurance Corporation (FDIC) data on Commercial and Industrial (C&I) Loans. The calibrated model
yields an annualized aggregate financial markup of approximately 24bps, in excess of default
premia. This translates into an average TFP loss of approximately 0.2%, suggesting that
imperfect competition in the financial sector can have a sizable effect on long-run macroeconomic outcomes. I then use the calibrated model to investigate the role that banks’ market
power plays in the transmission of three aggregate shocks: (i) a credit quality deterioration
(in the model, a sudden increase in the aggregate firms’ default probability), (ii) a “big” bank
failure (in the model, a sudden exit of a large bank), and (iii) a cut to the bank funding rate.
I conduct the experiments by comparing the dynamic response of the oligopolistic economy
against one with perfectly competitive financial intermediaries. I find that in each of these
cases, the endogenous cross-sectional dispersion of markups plays a key role in shaping —
and in particular, amplifying – the economy’s response to the exogenous shock.
An increase in the aggregate firms’ default probability induces a higher proportion of
3
Note that the optimal fiscal policy literature uses generalized Euler equations and Markov perfect equilibria in macroeconomics (Klein and Rı́os-Rull, 2003; Krusell, Martin, and Rı́os-Rull, 2004; Klein, Krusell,
and Rı́os-Rull, 2008; and Clymo and Lanteri, 2020).
3
more financially constrained firms with a high marginal productivity of capital. A more
concentrated banking sector can control the supply of credit more tightly by extracting
higher markups from these firms, leading to higher credit spreads. This mechanism allows
banks to compensate for the larger losses due to defaults, but it leads to a larger decline
in real activity, amplifying the recession. Quantitatively, when the aggregate firms’ default
rate matches that observed in the Great Recession, bank market power induces a larger peak
output decline of about 30% relative to the case of perfect competition.
When the economy also experiences the failure of a large intermediary, there are two key
effects: (i) in the short run, the change in market structure lowers the supply of credit to
firms, slowing down the economy; and (ii) in the long run, the resulting increase in banks’
market power further amplifies and prolongs the recession. When a “big” bank fails, the
surviving banks extend more credit to firms in order to capture the market share of the
defaulted bank. However, the speed of this adjustment is dampened by the decreased level
of competition. As a result of both credit constraints and market power, the aggregate
volume of credit drops sharply in the short run. In the long run the economy stabilizes
at a lower level of total credit, which results in less investment, output, and productivity.
The result suggests that banks’ market power may be an important source of concern for
policymakers deciding whether to bail out a “big” bank.
Finally, following a cut to the bank funding rate, the oligopolistic intermediation sector
acts as a potent transmission channel. Financial intermediaries exploit markups to dampen
the aggregate loans’ interest rate response. Quantitatively, these effects dampen the peak of
the expansion by about one-third relative to the competitive benchmark.
Related Literature This paper is mainly related to two strands of literature: (i) firm
dynamics in the face of financing frictions and (ii) macroeconomics with financial intermediaries. Indeed, one of the paper’s main contributions is to link the first literature, which
typically assumes a perfectly competitive credit market, with the second, which typically abstracts from firm dynamics and the heterogeneous effects of imperfectly competitive financial
intermediaries across firms with varying characteristics.
Credit Markets and Firm Dynamics. An important literature has studied the impact
of credit market frictions (e.g., borrowing constraints) on firm and aggregate dynamics, but
4
typically assumes that firms face a perfectly competitive credit market. My paper contributes to this line of work by jointly studying firms’ financing and investment decisions in
a credit market characterized by imperfectly competitive, non-atomistic banks that compete
strategically and focusing on how banks’ market power shapes the cross-sectional behavior
of firms. Classical papers in this literature are Kocherlakota (2000); Gomes (2001); Cooley
and Quadrini (2001); Cordoba and Ripoll (2004); Hennessy and Whited (2005); Hennessy
and Whited (2007); Covas and den Haan (2011); and Jermann and Quadrini (2012). Khan
and Thomas (2013) and Khan, Senga, and Thomas (2016) study models of heterogeneous
firms in a dynamic stochastic general equilibrium environment in which firms can source
their financing from a perfectly competitive intermediation sector.
Relatedly, the dynamic financial oligopoly I develop generates endogenous firm-level financial frictions that lead to time-varying second moments, such as the dispersion of loan
rates (directly linked to the dispersion of marginal products of capital) and aggregate productivity. In agreement with other work and empirical findings (e.g., Lanteri 2018 and David,
Schmid, and Zeke 2022), the model generates an increasing dispersion of loan rates during
recessions and hence, an increasing dispersion of marginal products of capital that shapes
the dynamic behavior of aggregate productivity. Thus, the model uncovers a new channel
for credit (hence, capital) misallocation linked to banks’ market power.4
Burga and Céspedes (2022) empirically estimate the effect of changes in bank market
power by exploiting a merger episode using a sample of small Peruvian firms and find that,
in agreement with the predictions of the model, the change in bank market structure results
in (i) a reduction of capital concentrated among small firms with a high marginal return
and (ii) an increase in capital misallocation.5 To conclude, another related literature in firm
dynamics studies financing constraints and irreversibility (e.g., Caggese 2007).
Macroeconomics with Financial Intermediaries. Several papers analyze the role of
financial intermediaries in macroeconomics, either with a focus on banks’ imperfect competition (e.g., Corbae and D’Erasmo 2021) or focusing on the interaction between credit con4
Another literature studies constrained optimal dynamic contracts in partial equilibrium (e.g., Albuquerque and Hopenhayn 2004; Clementi and Hopenhayn 2006; Brusco, Lopomo, Ropero, and Villa 2021).
5
The empirical literature about relationship lending – Rajan and Zingales (1998); Black and Strahan
(2002); Cetorelli and Gambera (2001); Cetorelli (2004); Cetorelli and Strahan (2006); and Cetorelli and
Peretto (2012) (theoretically)– is discussed in detail in Subsection 4.3.
5
straints and the financial intermediation sector (e.g., Elenev, Landvoigt, and Van Nieuwerburgh 2021).
There has been increasing interest in macroeconomics in analyzing the role of market
power.6 Corbae and D’Erasmo (2021) are among the first to investigate the effects of imperfect competition in loan markets by building a rich quantitative model of banking industry
dynamics to study the effects of financial regulations. My paper complements their seminal work by embedding an imperfectly competitive banking sector in a heterogeneous firm
environment, in which each firm makes optimal capital structure decisions and each bank
extracts endogenous firm-specific markups. Hence, the focus of my work is on the impact of
bank market power on macroeconomic outcomes with endogenously evolving heterogeneity
in borrower types.
Elenev, Landvoigt, and Van Nieuwerburgh (2021) investigate financial intermediaries’
capital requirements in a model with both financially constrained firms and intermediaries.
Similarly to my paper, their model focuses on the sudden and persistent fall in macroeconomic outcomes and credit supply during the financial crisis. My work complements their
analysis by investigating the role of intermediary market power which, through time-varying
endogenous firm-specific markups, leads to the amplification of macroeconomic shocks. As
a direction for future research, it would be interesting to combine intermediaries’ market
power with collateral constraints in order to investigate the economic mechanism through
which they interact and quantitatively disentangle the two joint effects.
Other papers studying the interaction between credit constraints and financial intermediation include Kiyotaki and Moore (1997) and Gertler and Karadi (2011).7 He and
Krishnamurthy (2013) introduce a stochastic model that explains how intermediary capital
affects risk premia variation. Rampini and Viswanathan (2018) propose a dynamic model
whereby financial intermediaries provide a superior collateralization service to households.
In constrast to these papers, my focus is on the transmission of macroeconomic shocks when
intermediaries have market power.
Last, recent papers have highlighted a key role of intermediary market power in shaping
6
For example, see Gutiérrez and Philippon (2016); Farhi and Gourio (2018); De Loecker, Eeckhout, and
Unger (2020); Corhay, Kung, and Schmid (2020) and Berger, Herkenhoff, and Mongey (2022).
7
Other relevant papers are He and Krishnamurthy (2013); Brunnermeier and Sannikov (2014); and
Rampini and Viswanathan (2018).
6
the transmission of monetary policy shocks.8 For example, Wang, Whited, Wu, and Xiao
(2022) analyze the impact of banks’ deposit market power on the loan maturity structure
and assess the relevance (for the transmission of monetary policy) of banks’ market power
in both the deposit and loan markets. The salient distinctive feature of my approach is
that it allows for endogenous firm-specific markups, creating an equilibrium cross-sectional
dispersion of markups, that I show is a key transmission channel of macroeconomic shocks.
Outline. The rest of the paper is organized as follows. Section 2 presents a stylized version
of the model with analytical insights. Section 3 describes the quantitative model and discusses various aspects of its solution in detail. Section 4 explains the calibration and results
of the oligopolistic stationary equilibrium. Section 5 illustrates the results of the three aforementioned macroeconomic shocks, such as the failure of one “big” bank and its interaction
with the production sector’s default rate. Section 6 concludes.
2
Stylized Model
In this section, I analyze a two-period model designed to provide preliminary intuition for
the quantitative model presented in Section 3. An oligopolistic banking sector interacts with
a continuum of heterogeneous firms in the presence of idiosyncratic total factor productivity
(TFP) and default shocks. I provide analytical results on the effects of an increase of the
number of banks B on several financial and macroeconomics variables of interest (including
aggregate loans, interest rates, expected returns on equity, physical investment, aggregate
leverage, dispersion of capital, dispersion of loan interest rates, dispersion of expected returns,
and aggregate TFP). In this stylized version of the model, there are two dates denoted
t = 0, 1.
Preferences. There are B identical banks, each owned by a continuum of identical savers;
hence there are B representative savers. Each saver’s preferences are represented by the
8
Malamudy and Schrimpf (2017); Drechsler, Savov, and Schnabl (2017); Li, Loutskina, and Strahan
(2019); and Wang, Whited, Wu, and Xiao (2022).
7
following linear utility function:
Cb,0 + β · Cb,1 ,
where Cb,t is the saver’s consumption at time t and β ∈ (0, 1) is the discount factor.9
There is a continuum of firms, each owned by a continuum of identical entrepreneurs;
hence there is one representative entrepreneur. The entrepreneur is risk-neutral and has
preferences represented by the utility function:
CE,0 + β · CE,1 ,
where CE,t is the entrepreneur’s aggregate consumption at time t.
Ownership Structure. Each representative saver owns a bank. In equilibrium, the saver
is indifferent between financing the bank’s loans with debt or equity. The representative
entrepreneur owns the entire mass of firms j ∈ [0, 1]. Each firm j is characterized by its
state vector
x(j) ≡ {{lb (j)}B
b , rl (j), k(j), z(j), I(j)},
where lb (j) denotes the firm’s loan by bank b, rl (j) is the interest rate (charged by all banks),
k(j) is the firm’s capital stock, z(j) is the firm’s productivity, and I(j) is an indicator function
that takes value 1 if the firm has not defaulted. Let φ(x) denote the density function of firms
in the economy.
Each saver cannot own any firm’s equity and thus needs to save through banks.
Markets. There are five markets in the economy: banks’ debt, banks’ equity, firms’ loans,
firms’ equity, and the market for the representative good.
Banks’ equity and debt markets. Each saver b invests in the production sector by supplying
9
The risk-neutrality assumption in the stylized model is made to simplify the analysis and isolate effects
that do not depend on the savers’ risk-aversion. In the quantitative model of Section 3 the savers are
risk-averse.
8
equity or debt to banks and faces the budget constraints:
Cb,0 + pb · Sb,1 + Db,1 = (pb + πb,0 ) · Sb,0
Cb,1 = πb,1 · Sb,1 + RD · Db,1 ,
where pb , Sb,0 , Sb,1 , Db,1 , πb,0 , and πb,1 are, respectively, the bank’s share price, the share
holdings at t = 0, 1, the debt holdings at t = 1, and the bank’s profit at t = 0, 1. Bank b
demands equity and debt from saver b, in order to finance loans to firms.
Firms’ equity market. The entrepreneur invests in the production sector by supplying equity
to the firms and faces budget constraints:
Z
I · (p0 + d˜0 ) · S0 dΦ
[I · p0 · S1 + (1 − I) · p0 · S1 ] dΦ =
CE,0 +
Z
CE,1 =
Z
I · d˜1 · S1 dΦ,
where p0 , S0 , S1 , d˜0 , and d˜1 are, respectively, the share price, share holdings at t = 0, 1, and
the dividend of each firm (net of equity issuance cost) at t = 0, 1. Firms demand equity
from, or distribute dividend to, the entrepreneur. If a firm decides to issue equity, it incurs
a quadratic equity issuance cost at t = 0 (with λ0 being a positive constant):
λ(d0 ) =
λ0 d20
2
if d0 ≤ 0
0
if d0 > 0
,
where d0 is a firm dividend at t = 0, defined below. The convexity of λ(.) captures the idea
of increasing marginal underwriting cost, or the increasing threat posed by a moral-hazard
problem when a greater amount of equity is demanded.
Firms’ loan market. A finite (and exogenous) number B of (identical) banks supply loans
to a continuum of firms. Each bank b = 1 . . . B can issue non state-contingent loans lb,1 to
each firm. Loans are due for repayment in the next period, unless the firm defaults. A firm
j takes the interest rate r1 (j) as given and chooses how much to invest and how much to
borrow from each bank. Banks take each firm’s demand schedule as given and compete à la
Cournot, i.e, simultaneously and independently choose their loan portfolios.
9
Goods market. The representative entrepreneur and the B representative savers demand
goods supplied by all firms.
Technology. In each period t = 0, 1, the output yt (j) produced by each firm j ∈ [0, 1] is
given by the production function yt (j) = zt (j) · kt (j)α , where 0 < α < 1.
Shocks. At time 0, firms are heterogeneous with respect to their capital stock k0 and
their idiosyncratic total factor productivity (TFP) z0 . At time 1, there are two types of
idiosyncratic shocks: the firm can default, with exogenous probability 1−ρ and, if it survives,
z1 realizes according to z1 = ρz z0 + ξ1 where ξ1 ∼ N (0, σz2 ) and ρz > 0.
Timing. All decisions are taken at t = 0. Given the initial distribution of firms with pdf
φ(x0 ) (and cdf Φ(x0 )), the timing is as follows: (1) each firm produces output y0 = z0 k0α ;
R
(2) each bank finances its supply of loans, lb (x0 ) dΦ(x0 ), by issuing equity and/or debt;
(3) each firm takes the interest rate R1 (x0 ) as given and chooses how much to invest and
the amount of loan to demand from each bank; (4) banks take each firm’s demand schedule
as given and compete with each others to supply the loans. The outcome is a contract
establishing: loan amount lb (x0 ), interest rate R1 (x0 ), and the new level of capital k1 (x0 );
P
and (5) firms distribute dividends d0 = z0 k0α + (1 − δ)k0 − k1 + B
b lb to the entrepreneur.
At t = 1, the 1 − ρ mass of defaulting firms exits the market. For the surviving firms, z1
is realized and: (1) firms produce output y1 = z1 k1α ; (2) firms repay their outstanding debt
R
P
ρR1 (x0 )l1,b (x0 ) dΦ(x0 )
plus interest R1 (x0 ) · B
b lb (x0 ); (3) each bank distributes its profit
P
to the saver; and (4) firms distribute dividend d1 = z1 k1α + (1 − δ)k1 − R1 B
b l1,b to the
entrepreneur.
2.1
Agents’ Optimization Problems
The representative saver b maximizes its intertemporal utility subject to their budget constraint, yielding an Euler equation that pins down the price of banks’ equity: ∀b :
βπb,1 =
pb,0 . The representative entrepreneur maximizes its intertemporal utility subject to their
10
budget constraint, yielding an Euler equation that pins down the price of each firm’s equity:
ρβE0
d1
= 1 − λd (d0 ),
p0
(1)
where p0 is the price of the share of a firm at time 0. Firms maximize the net present value
of dividends d0 + β · E0 [I · d1 ], where dividends in each period are given by:
d0 =
z0 k0α
+ (1 − δ)k0 − k1 +
B
X
l1,b ,
d1 =
z1 k1α
+ (1 − δ)k1 − R1
b
B
X
l1,b .
b
The firm’s first-order condition with respect to capital requires that the future interest rate
equals the expected marginal productivity of capital net of depreciation:
R1 = E0 1 + αz1 k1α−1 − δ .
(2)
The firm’s optimality condition with respect to the loan requires that the discounted
future expected interest rate be one net of the equity issuance cost:
ρβR1 = 1 − λd (d0 ).
(3)
Banks’ strategies map firm characteristics (x0 ) onto the current quantity and future interest
rate of loans. Given the probability density function φ(x0 ) (and cumulative distribution
function Φ(x0 )), each bank b chooses l1,b (x0 ) to best respond to other banks’ strategies
l1,−b (x0 ), such that
Z
max
l1,b (x0 )
π=−
Z
l1,b (x0 ) dΦ(x0 ) + β
ρR1 (x0 )l1,b (x0 ) dΦ(x0 ),
subject to equations (1), (2), and (3) for all firms in the distribution.
Each bank’s best response is characterized by the following generalized Euler equation (GEE)
∀x0 :
where
∂R1 (x0 )
∂l1,b (x0 )
∂π
∂R1 (x0 )
= −1 + ρβ
l1,b (x0 ) + ρβR1 (x0 ) = 0,
∂l1,b (x0 )
∂l1,b (x0 )
(4)
can be determined by the implicit function theorem on equations (2) and (3).
Equation (4) is a generalized Euler equation because it contains the derivative of the firm’s
11
policy functions. Each bank best responds by internalizing the effect of loans on the firms’
capital choice
∂k1
∂l1,b
as well.
Inverse Elasticity. For ease of notation, I drop the dependency of all optimal choices from
x0 . The inverse elasticity implicitly contained in the GEE
of the term
∂R1
.
∂l1,b
∂R1 l1,b
∂l1,b R1
requires the determination
From equation (2)
∂R1
α−2 ∂k1
= E0 α(α − 1)z1 k1
∂l1,b
∂l1,b
it is clear that the inverse elasticity depends on the expectation of the second derivative
of the production function. Note that this also captures the effects of banks’ decisions on
the investment choice of each firm (through the term
∂k1
).
∂l1,b
The previous equation can be
rewritten as
∂R1 l1,b
∂k1 l1,b
= E0 (α − 1)k1−1 · αz1 k1α−1 ·
.
| {z } ∂l1,b R1
∂l1,b R1
(5)
MPK
Note that the term α − 1 is always negative. This yields an inverse elasticity
is always negative. Banks exert higher market power when
∂R1
∂l1,b
·
l1,b
R1
∂R1
∂l1,b
·
l1,b
R1
that
is smaller. The formula
suggests that banks incorporate two components in their decision making when they extend
loans to firms.
First, the higher the marginal productivity of capital (MPK) of a firm, the higher the
markup that banks can extract (since for that firm the marginal value of one unit of investment is higher than for an established firm with high capital and low MPK).
Second, banks think strategically by internalizing the effects their actions have on firms’
investment decisions. This second effect is captured by the cross-elasticity
∂k1
∂l1,b
·
l1,b
.
R1
Expressions for the two cross-derivatives can be found jointly by taking the total derivatives of equations (2) and (3):
∂R1
1 − ρβR1
=
,
∂l1,b
ρβl1,b
∂k1
1 − ρβR1
1
=
·
.
∂l1,b
ρβl1,b
α(α − 1)E0 [z1 ]k1α−2
In equilibrium, for the mass of financially constrained firms (d0 < 0), the degree of
imperfect competition (number of banks B) matters. For each firm, the equilibrium is a
12
∗
vector (k1∗ , R1∗ , l1,b
, p∗0 ) such that equations (1), (2), (3), and (4) hold simultaneously. For
the mass of firms that, in equilibrium, is not financially constrained, the degree of imperfect
competition does not matter. For these firms, the solution is given by (k1∗ , R1∗ , p∗0 ) such that
equations (1), (2), and (3) hold simultaneously. Note that for these firms, the Modigliani∗
Miller theorem holds; hence, l1,b
is undetermined.
2.2
Characterization of the Equilibrium
I now describe intuitively the main mechanism that drives the analytical results presented in
this section. First, Equation (5) suggests that the higher the marginal productivity of capital
(MPK) of a firm, the higher the marginal value of one unit of loan for that firm that translates
in a lower inverse elasticity
∂R1
∂l1,b
·
l1,b
.
R1
Second, as explained above, the degree of imperfect
competition (number of banks B) matters only for the mass of financially constrained firms.
Hence, banks endogenously exert a higher degree of market power on firms that are financially
constrained and with a high marginal productivity of capital. Intuitively, these firms have
worse outside options (e.g., a high cost of non-bank finance) and one additional unit of
investment in physical capital can contribute significantly to their future production; hence,
they exhibit a less elastic demand for credit. An imperfectly competitive financial sector
internalizes that the same financial resources are more valuable for this type of firms and for
their future growth paths; therefore, can charge higher markups.
This mechanism leads financially constrained firms to grow slower, the higher the degree
of imperfect competition. As a result, the dispersion of marginal productivity of capital is
higher when there are fewer banks B in the economy. At the same time, a lack of competition
in the financial intermediation reduces aggregate productivity, since firms grow toward their
efficient level of capital on slower trajectories. This intuitive mechanism is at the base of
Proposition I. See Appendix C.2 for the proof.
13
Proposition I
Assume that the distribution φ(x0 ) is such that there is a non-zero measure of financially constrained firms:a
Z
∗
1[d0 (x0 , k1∗ , l1,b
) ≥ 0] dΦ(x0 ) < 1.
P=
A higher number of banks (i.e., a higher B) has the following effects:
1. aggregate loans per bank
R
lb∗ dΦ decreases;
2. average loan interest rate
R
Rl∗ dΦ decreases;
3. aggregate physical investment
R
k1∗ − (1 − δ)k0 dΦ increases;
4. aggregate share of expected returns
5. aggregate loans
R PB
b
6. aggregate leverage
7. aggregate TFP
R
R PB
b
R
IE [d∗1 ] /p∗ dΦ decreases;
lb∗ dΦ increases;
lb∗ /k1∗ dΦ increases;
k1∗α dΦ/
8. variance of capital
R
R
α
k1∗ dΦ increases;
R
k1∗2 dΦ − ( k1∗ dΦ)2 decreases;
9. variance of loan interest rates
10. variance of expected returns
R
R
R
Rl∗2 dΦ − ( Rl∗ dΦ)2 decreases;
(IE [d∗1 ] /p∗ )2 dΦ −
R
2
IE [d∗1 ] /p∗ dΦ decreases.
a
For subpoints 7, 8, 9 and 10, I assume that the mass of financially constrained firms 1 − P are
all ex-ante identical.
3
Quantitative Model
In the two-period model, banks’ choices are static. In the infinite-horizon model, each bank
faces a dynamic problem that: (i) depends on the same bank’s future strategies and other
banks’ current and future strategies, and (ii) is subject to all firms’ dynamic demand for loans;
also both the current and future distributions of firms matter. The equilibrium concept used
14
in this section is a Markov perfect equilibrium (e.g., Maskin and Tirole, 2001). Specifically,
I characterize the equilibrium using generalized Euler equations in a similar fashion to the
optimal fiscal policy literature (see, for instance, Klein and Rı́os-Rull, 2003; Krusell, Martin,
and Rı́os-Rull, 2004; Klein, Krusell, and Rı́os-Rull, 2008; and Clymo and Lanteri, 2020).
In this section, I build a dynamic framework to study firms’ financing-investment decisions
when banks are big, strategically interact with each other, and face idiosyncratic firms’
default risk.10 Households derive utility from a non-durable consumption good, own the
shares of the banks, and supply deposits. Banks issue debt and use both their internal
resources and debt to purchase firms’ loans. Firms make investment decisions, taking into
account the fact that debt provides a tax shield and issuing new equity is increasingly costly.
The key feature of the framework is the simultaneous presence of strategic interactions among
financial institutions, general equilibrium, macroeconomic shocks, and heterogeneous firms.
Note that each firm stipulates an idiosyncratic contract with the banks: in equilibrium, banks
have different degrees of market power on each single firm in function of its idiosyncratic
characteristics.
I will now describe the model and proceed to define the stationary oligopolistic equilibrium, in which all aggregates quantities and prices are constant over time. I overcome
the computational challenge by proposing algorithms to solve for the oligopolistic stationary
equilibrium and the related transitional dynamics in the presence of strategic interactions,
general equilibrium, and heterogeneous firms. The algorithms are discussed in detail in
Appendix B.
3.1
Environment
Time is discrete t = 0, 1, . . . and the horizon is infinite.
Preferences. There is an exogenous number B of identical banks. Each bank is owned
by a continuum of identical and infinitely lived savers, equivalent to B representative savers.
Each saver b ranks stream of consumption Cb,t according to the following lifetime utility
10
Specifically, I interpret the financial intermediation sector as a succession of decision makers – one at
each date t – without commitment to future realized quantity of loans supplied.
15
function:
∞
X
βbt · u(Cb,t ),
(6)
t=0
where βb ∈ (0, 1) is the saver’s discount factor, and uc > 0, ucc < 0.
There is a continuum of firms, each owned by a continuum of identical infinitely lived
entrepreneurs, equivalent to one representative entrepreneur. The entrepreneur ranks stream
of consumption CE,t according to the following lifetime utility function:
∞
X
βEt · u(CE,t ),
(7)
t=0
where βE ∈ (0, 1) is the entrepreneur’s discount factor. Note that I treat the two discount
factors identically (i.e, βb = βE ≡ β) in the paper with the exception of Subsection 5.3.
Ownership Structure. Each representative saver owns a bank. In equilibrium, the saver
is indifferent between financing the bank’s loans with debt or equity. The representative
entrepreneur owns the entire mass of firms j ∈ [0, 1]. Each firm j is characterized by its
state vector
x(j) ≡ {{lb (j)}B
b , rl (j), k(j), I(j)},
where lb (j) denotes the firm’s loan by bank b, rl (j) is the interest rate (charged by all banks),
k(j) is the firm’s capital stock, and I(j) is an indicator function that takes value 1 if the
firm has not defaulted. Let φ(x) denote the density function of firms in the economy.
Each saver cannot own any firm’s equity and thus needs to save through banks.
Markets. There are six markets in the economy: banks’ debt, banks’ equity, firms’ loans,
firms’ equity, interbank market, and the market for the representative good.
Banks’ equity and debt markets. Each saver b invests in the production sector by supplying
equity or debt to banks and faces the budget constraint:
Cb,t + pb,t · Sb,t+1 + Db,t+1 = (pb,t + πb,t ) · Sb,t + RD,t · Db,t
16
where pb,t , Sb,t , Sb,t+1 , Db,t , Db,t+1 , RD,t and πb,t are, respectively, the bank’s share price, the
share holdings at t and t + 1, the bank’s debt holdings at t and t + 1, the interest rate on
the bank’s debt, and the bank’s profit at t. Bank b demands equity and debt from saver b,
in order to finance loans to firms.
Firms’ equity market. The entrepreneur invests in the production sector by supplying equity
to the firms, and faces budget constraints:
Z
CE,t +
Z
[I · pt · St+1 + (1 − I) · pt · St+1 ] dΦ =
I · (pt + d˜t ) · St dΦ
where pt , St , St+1 , and d˜t are, respectively, the share price, share holdings at t, and the dividend of each firm (net of equity issuance cost) at t. Firms demand equity from, or distribute
dividends to, the entrepreneur. If a firm decides to issue equity, it incurs a quadratic equity
issuance cost λ(dt ) (see Section 2), where dt is a firm dividend at time t, defined below.
Firms’ loan market. A finite (and exogenous) number B of (identical) banks supply loans
to a continuum of firms. Each bank b = 1 . . . B can issue non-state-contingent loans lb,t+1 to
each firm. Loans are due for repayment in the next period, unless the firm defaults. A firm
j takes the interest rate rl,t+1 (j) as given and chooses how much to invest and how much to
borrow from each bank. Banks take each firm’s demand schedule as given and compete à
la Cournot, i.e, simultaneously and independently choose their loan portfolios. The process
determines the total amount of loans banks supply to each firm which, together with the
firm’s demand schedule, pins down the firm-specific interest rate rl,t+1 (j). At time t, each
bank and firm commit to such an interest rate.
Interbank market. A bank b can lend Mb,t to other banks that will be repaid in the following
period at rate rM,t+1 . Since all banks are identical, in equilibrium ∀b : Mb,t = 0.
Goods market. The representative entrepreneur and the B representative savers demand
goods supplied by all firms.
Technology. In each period t, the output yt (j) produced by each firm j ∈ [0, 1] is given
by the production function yt (j) = zt (j) · kt (j)α , where 0 < α < 1.
Shocks. At time t the firm can default with exogenous probability 1 − ρ. A new mass
of firms re-enters the economy with exogenous characteristics x0 so that the total mass is
17
constant over time. I relax this assumption in Appendix A.1.
Government. The government imposes proportional taxes τ on all firms’ production.
Firms can deduct loan interest and depreciated capital from their taxes. Government runs
a balance budget constraint. That is, the government uses the aggregate revenue from taxes
R α PB
Tt = τ
zt kt − b=1 rl,t lb,t − δkt dΦ to finance an exogenous government expenditure
that exactly balances Tt at each point in time.
Timing. The aggregate state space of the economy at time t is
B
Xt ≡ {{Db,t }B
b , rD,t , {Mb,t }b=0 , rM,t , B, ρ, φ(xt )}.
Given Xt , the timing is as follows: (1) a mass 1 − ρ of firms defaults, (2) each surviving firm
R
produces output yt = zt ktα ; (3) each bank finances its supply of loans, lb,t+1 (xt ) dΦ(xt ),
by issuing equity and/or debt; (4) each firm takes the interest rate rl,t+1 (xt ) as given and
chooses how much to invest and the amount of loan to demand from each bank; (5) banks
take each firm’s demand schedule as given and compete with each others to supply the loans.
The outcome is a contract establishing: loan amount lb,t (xt ), interest rate rl,t+1 (xt ), and new
P
level of capital kt+1 (xt ); (6) firms distribute dividends d = (1 − τ ) [zk α − b rl lb ] + τ δk − ĩ
to the entrepreneur; (7) bank b distributes profit to saver b. To simplify notation, in the
following subsections I avoid explicitly sub-scripting each variable with time t and t + 1, but
it is understood that (x, X) refers to (xt , Xt ), and (x0 , X 0 ) refers to (xt+1 , Xt+1 ).
3.2
Household: Saver
I now describe the saver’s problem in recursive form. Let VS (X) be the value function of the
saver with debt holdings D = [D1 . . . DB ] and equity holdings S = [S1 . . . SB ] in each bank
b. This function satisfies the following functional equation:
VS (X) = max
0
0
S ,D
u(Cb ) + β · VS (X 0 )
18
(8)
subject to the budget constraint:
Cb + pb · Sb0 + Db0 = (pb + πb ) · Sb + RD · Db .
(9)
The left-hand side of budget equation (9) reports the saver’s expenditures: household
aggregate consumption, banks b’s equity and debt purchases. The right-hand side reports
the saver’s resources: bank b’s equity holdings and debt.
0
The saver takes the future banks’ debt market rate rD
as given, together with future
banks’ profits, and purchases banks’ debt and equity according to:
p0b + πb0
pb
∀b :
1 = MS0 ·
∀b :
0
1 = MS0 · RD
,
(10)
(11)
u (C 0 )
where MS0 ≡ β ucc (Cbb ) , and πb is the profit of bank b distributed as a dividend to the saver.
3.3
Firms
I now characterize firm j’s problem in recursive form. For convenience, I omit the index
notation j. Let VF (x, X) be the value function of the firm j with loan holdings l = [l1 . . . lB ]
from each bank b and capital k. This function satisfies the following functional equation:
VF (x, X) = max
0
{lb0 }B
b=1 ,k
d − λ(d) + IE [I 0 · ME0 · VF (x0 , X 0 ) | (x, X)] ,
subject to
k 0 = k(1 − δ) + i
X
i = ĩ +
(lb0 − lb )
b
"
#
d = (1 − τ ) zk α −
X
rl lb + τ δk − ĩ,
b
where ME0 is the discount factor of the entrepreneur, as described in the following subsection.
Each firm takes the future loans’ market rate rl0 as given and finances itself through internal
financing (production and equity issuance) and external financing (loans from banks). The
19
first-order condition with respect to k 0 is
i
h
α−1
− δ · (1 − λd (d0 )) | (x, X) .
1 − λd (d) = IE I 0 · ME0 · 1 + (1 − τ ) z 0 αk 0
(12)
The first-order condition with respect to lb0 is
1 − λd (d) = IE [I 0 · ME0 · (1 + (1 − τ )rl0 ) · (1 − λd (d0 )) | (x, X)] .
3.4
(13)
Household: Entrepreneur
I now describe the entrepreneur’s problem in recursive form. Let VE (X) be the value function
of the representative entrepreneur with shares holding S(S). This function satisfies the
following functional equation:
VE (X) = max0
S(·)
u(CE ) + β · VE (X 0 )
(14)
subject to the budget constraint:
Z
CE +
0
0
Z
I · p · S + (1 − I) · p(x0 ) · S (x0 ) dΦ =
˜ · S dΦ,
I · (p + d)
(15)
where d˜ is the firm-specific dividend d net of the equity issuance cost λ(d). Hence, each
firm’s share value is priced according to:
"
#
0
˜0
p
+
d
1 = IE I 0 · ME0 ·
| (x, X) ,
p
(16)
u(C 0 )
where ME0 ≡ β u(CEE ) .
3.5
Banks
A bank b chooses the new level of debt to demand from the saver and the new level of loans
to offer to each firm. Formally, the strategy space is defined as:
Sb0 (x, X) ≡ {Db0 (X), lb0 (x, X)}.
20
The new amount of debt issued (∆Db0 = Db0 − Db ) and internal financing F is chosen to
provide enough coverage for the change in interbank lending and aggregate loans:
F+
∆Db0
=
∆Mb0
Z
+ρ
∆lb0 (x, X) dΦ.
(17)
I now describe the bank’s problem in recursive form. Let Vb (X) be the value function of a
bank b. This function satisfies the following functional equation:
Vb (X) =
max
0 }, M 0 , {l0 (x,X),r 0 (x,X)}
{Db0 ,rD
b
b
l
πb + MS0 (X, X 0 ) · Vb (X 0 )
(18)
subject to: (i) equation (17), (ii) the household’s interest rate-quantity schedule jointly
defined by equation (10) and (11), (iii) each firm’s interest rate-quantity schedule jointly
defined by equations (12), (13). Bank b’s profit πb is given by:
Z
πb = ρ
rl · lb dΦ + rM Mb − rD Db − F.
(19)
0
(X) and rl0 (x, X) adjust consistently with the interest rate-quantity
Future market rates rD
schedules. Each bank b issues bank debt according to a generalized Euler equation:
0
0
1 = MS0 (X, X 0 ) · RD
(X, X 0 ) · (1 + ηD
(X, X 0 )) ,
0
is the inverse elasticity
where ηD
0
∂RD
∂Db0
·
Db0
0
RD
(20)
between debt and its rate. In principle, equation
(20) is a best response function that captures the trade-off that a bank faces issuing new debt.
Every new unit of debt increases today financing capacity but needs to be repaid tomorrow
0
at the contracted interest rate. Moreover, since ηD
is non-negative, when a bank issues new
debt it is also increasing the market rate of deposits, incurring an additional future marginal
0
cost. In equilibrium, ηD
is zero, as implied by equations (20) and (11). Without aggregate
risk, savers and banks are completely indifferent to the financing structure of the banks.
A similar generalized Euler equation arises from the loans’ first-order condition:
1 = IE [I 0 · MS0 (X, X 0 ) · Rl0 (x, X, x0 , X 0 ) · (1 + ηl0 (x, X, x0 , X 0 )) | (x, X)] ,
where ηl0 (x, X, x0 , X 0 ) ≡
∂Rl0
∂lb0
·
lb0 (x,X)
Rl0 (x,X,x0 ,X 0 )
(21)
< 0 is the firm-specific inverse elasticity between
21
loans and their rates. Note that equation (21) is a functional equation that depends on the
idiosyncratic characteristics of each firm. See Appendix C.1 for details on how to calculate
ηl0 (x, X, x0 , X 0 ). Equation (21) is a best response function that captures the trade-off that a
bank faces issuing a new unit of loan to a specific firm. Every new unit of loan decreases
the current bank’s dividend but produces a marginal income tomorrow at the contracted
interest rate. Moreover, since ηl0 is non-positive, when banks issue new loans they are also
decreasing the future market rate of loans, incurring a marginal loss in the future. Note that
banks best respond internalizing the effects that their actions have on aggregate quantities
and all firms’ choices (e.g., if a bank changes the quantity of loan offered to a firm, that
firm might decide to re-optimize and adopt a different capital structure as a function of the
credit market conditions. Banks internalize all these effects in their decisions). Equation
(21), together with an Euler equation that regulates the banks’ behavior on the interbank
market
0
1 = MS0 (X, X 0 ) · RM
(X, X 0 ),
(22)
captures the decision making behavior of each bank. The outcome of the game played
by the banks at time t is a contract that pins down the firm-specific intermediation mar0
(X, X 0 ). In principle, this margin can be decomposed into: (i)
gin Rl0 (x, X, x0 , X 0 ) − RD
0
(X, X 0 )) and (ii) debt’s infirm-specific loan’s intermediation margin (Rl0 (x, X, x0 , X 0 ) − RM
0
0
(X, X 0 )). Note that the debt intermediation margin
(X, X 0 ) − RD
termediation margin (RM
0
0
0
(X, X 0 ). The following paragraph explains the
(X, X 0 ) = RM
= 0, hence RD
is zero, since ηD
decomposition of the loan’s intermediation margin in greater detail.
Loan’s Intermediation Margin The spread between the firm-specific loan’s rate and
the interbank market rate can be obtained by combining equations (21) and (22):
0
(X, X 0 ) = IE [I 0 · Rl0 (x, X, x0 , X 0 ) · (1 + ηl0 (x, X, x0 , X 0 )) | (x, X)] .
RM
22
(23)
The generalized Euler equation (23) can be manipulated
Markup
z
Rl0 (x, X, x0 , X 0 )
−
0
RM
(X, X 0 )
MC
}|
=−
1
|
+
{ z
ηl0 (x, X, x0 , X 0 )
+ ηl0 (x, X, x0 , X 0 )
{z
(1) Rents
}|
{
1
· 0
MS (X, X 0 )
}
1
1
1−ρ
· 0
·
,
0
0
0
ρ
1 + ηl (x, X, x , X ) MS (X, X 0 )
|
{z
}
(2) Risk Premia
to reveal a non-linearly separable decomposition of the loan’s intermediation margin in (1)
markup over marginal cost and (2) risk premia.11
Since ηl0 is non-positive, and the reciprocal of MS0 is the interbank market rate, this
formula is an inter-temporal markup rule (markup over marginal cost, the marginal cost
0
0
(X)). The more banks are introduced in the economy, the more the
(X) or RD
being RM
financial sector becomes competitive and the inverse elasticity ηl0 tends to decrease in module;
hence, the expected loan’s rate tends to the bank funding rate.
3.6
Oligopolistic Equilibrium
The government aggregate income from taxes is:
Z
T =τ
zk α −
B
X
!
rl (x, X) · lb (x, X) − δk dΦ.
b=1
The aggregate resource constraint of the economy is:
B
X
Z
Cb + CE +
Z
i(x, X) + λ(x, X) dΦ + T =
zk α dΦ.
(24)
b=1
The total production of the economy on the right-hand side of equation (24) can: (i) be
consumed by the savers, (ii) be consumed by the entrepreneur, (iii) be used for aggregate
investment in physical capital (in case some dividends are negative, some resources are spent
0
to pay the equity issuance cost λ), (iv) or be paid in taxes. The inverse elasticity ηD
is zero
It is non-linearly separable in the sense that risk premia contains ηl0 (x, X, x0 , X 0 ) which, in turn, contains
the default probability.
11
23
and ηl0 (x, X, x0 , X 0 ) of equation (21), is calculated as described in Appendix C.1.
A formal definition of the notion of Recursive Stationary Oligopolistic Equilibrium is
presented in Definition 3.1, and its extension to the dynamic case is discussed in Section 5.
Definition 3.1. A Recursive Stationary Oligopolistic Equilibrium is a Markov perfect
equilibrium where i) the banks’ debt holdings {Db }B
b=1 and the relative market rate RD ; ii)
B
the banks’ share holdings {Sb }B
b=1 and the relative market prices {pb }b=1 ; iii) the interbank
debt holdings {Mb }B
b=1 and the relative market rate RM ; iv) the saver’s consumption Cb and
the entrepreneur’s consumption CE ; v) the distribution φ(x); vi) the policy functions: k 0 (x),
l0 (x) and Rl0 (x); are such that i) the saver’s problem is solved–i.e, equations (10) and (11);
ii) the entrepreneur’s problem is solved–i.e, equation (16) holds; iii) each firm’s problem is
solved–i.e, equations (12) and (13); iv) each bank is best responding to all other banks–i.e,
equations (20), (21) and (22) hold; v) and all markets clear: 1) the good market clears –i.e.,
equation (24) holds; 2) each bank’s equity market clears –i.e., ∀b : Sb = 1; 3) each firm’s
equity market clears –i.e., S(x) = 1; 4) the interbank market clears –i.e., ∀b : Mb = 0.
4
Calibration
In this section, I calibrate the Stationary Oligopolistic Equilibrium of the model to match
the credit spreads of the Commercial & Industrial (C&I) Loans.12 C&I credit spreads are
calculated as the difference between the weighted-average C&I effective loan rate and the
3-Month T-bill rates.13 I use the C&I Loans charge-off rates as a proxy for default risk,
which can be retrieved from the loan performance archive of the FDIC Quarterly Banking
Profile.14 I choose this asset class for two reasons: (i) it is an important component of US
banks’ balance sheets (as of February 2022, the amount outstanding is USD 2.48 trillion)
and (ii) it is composed of short-term loans, making maturity premium a minor concern.
4.1
Data
As shown in Table IV in Appendix A.4, data reveals a positive correlation (significant at
the 1% level) between C&I credit spreads and banks’ asset market concentration, even after
12
C&I Loans: https://fred.stlouisfed.org/series/BUSLOANS.
Weighted-average C&I effective loan rate: https://fred.stlouisfed.org/series/EEANQ.
14
FDIC QBP: https://www.fdic.gov/analysis/quarterly-banking-profile/index.html.
13
24
controlling for default risk (identified with the charge-off rate on C&I Loans), quantity of
loans outstanding, and average maturity. Moreover, the correlation between C&I credit
spreads and the quantity of C&I Loans outstanding is also significant at the 1% level and
negative, consistent with the sign of the model’s inverse elasticity ηL between loan and loan’s
rate contained in equation (23). In Table IV, banks’ asset market concentration is calculated
as the market share of the five largest US banks, which can be obtained by aggregating the
assets of each single bank from the FDIC aggregate balance sheets.
Figure 1 reports the quarterly net charge-off to loans, used as a proxy for default rate.15
Figure 1 also reports the aggregate time series for C&I credit spreads.16 As a proxy for
the interbank market rate (rM ), I use the 3-Month T-bill rates. Note that there is not a
quantitatively significant difference when the Fed funds rates are used instead of the 3-Month
T-bill rates. Hence, credit spreads reported in Figure 1 are calculated as interest rates on
C&I Loans (rL ) minus 3-Month T-bill rates (rM ). Consistently with the model, interest
expenses and holdings are converted in real terms.
The counterparts in the model of the aggregate credit spreads reported in Figure 1 are
equilibrium firm-specific (i) risk premia and (ii) markup, according to the generalized Euler
equation (23). As previously shown, these two components are not linearly separable. In
order to separate the aggregate markup from risk premia, I solve the Recursive Stationary
Oligopolistic Equilibrium with an increasing number of banks.
15
Charge-off rates are broadly consistent with default rates reported by Moody’s and Standard & Poor’s,
but have the advantages to refer specifically to C&I Loans.
16
Alternatively to using the weighted-average C&I effective loan rate, it is possible to compute credit
spreads from the interest rates obtained by combining the FDIC aggregate balance sheets and income statements. Hence, obtain interest rates on loans as the ratio between interest income and loans. This procedure
yields a closely similar time series.
25
Figure 1. C&I Loans Data
Notes: The figure shows the evolution over time of the (annualized) credit spreads (top panel), the
(annualized) net charge-off to loans (central panel) and the average maturity (bottom panel) of the C&I
Loans. The average annualized credit spread between 1997 and 2017 is ∼2.2%. The average annualized
net charge-off to loan between 1997 and 2017 is ∼0.84%. The average maturity between 1997 and 2017
is ∼1.4 years.
I choose the number of banks in the economy to match an annualized credit spread of
2.16%, obtained as the average of the time-series illustrated in Figure 1. Hence, the aggregate markup can be identified as the difference between (i) credit spreads in the calibrated
economy and (ii) credit spreads under perfect competition. This procedure is illustrated in
detail in the following subsection and yields an annualized aggregate markup of 0.24%.
4.2
Calibration of the Oligopolistic Equilibrium
I now describe the choices of parameters values for preferences for the production sector and
the number of banks. Table II in Appendix A.2 summarizes all parameter values.
Household Preferences. A period in the model coincides with a quarter, consistent with
the frequency of interest expenses in the data. I set β = 0.9936 to match a quarterly
26
stationary bank’s debt rate (or interbank debt rate) of 0.64%, which is consistent with the
average of the 3-month T-bill rates calculated in the time window considered in Figure 1,
and reflects the prolonged period of low interest rates. I use a constant relative risk aversion
utility function
u(C) =
C 1−γ
1−γ
γ ≥ 0 ∧ γ 6= 0
,
log(C) γ = 1
where the parameter γ is the degree of relative risk aversion. I use a unitary γ in the baseline
calibration. I benchmark the baseline calibration against the same equilibrium calculated
with γ = 5 in Table III in Appendix A.2.
Firms. I set a depreciation rate δ equal to 2.5%, consistent with an annualized depreciation
rate of 10%. The effective capital share is set to 0.41, obtained as the average of the time
series from 1990 to 2018 reported by the Bureau of Labor Statistics, which reflects the secular
decline in the labor share. Firms’ income tax is set to 24%, obtained as the average from
1990 to 2018 of the ratio between taxes on corporate income and corporate profit (both
time series obtained from FRED). I use the C&I Loans charge-off rates to identify the risk
of default, set to 0.21% (quarterly) consistent with the average of the time series reported
in Figure 1. The equity issuance cost λ0 is set to 1.5 to match an annualized frequency
of equity issuance of 0.04. The frequency of equity issuance is computed from a sample of
non-financial, unregulated firms from Compustat. Note that, more generally, λ0 can be seen
as a non-bank financing cost. For robustness (and to capture the idea that firms could have
other outside options and not just equity issuance), I benchmark the baseline calibration
against the same equilibrium calculated with λ0 = 0.75 in Table III. I also fix a maximum
age N̄ in the life cycle of the firm. In particular, I consider an average lifespan of a company
of 19.5 years, as calculated using the Standard and Poor’s 500 Index.17 Note that 19.5 years
corresponds to 78 quarters; hence, N̄ is such that:
N̄
X
age · ρ
age
= 78 ·
age=0
N̄
X
ρage .
age=0
17
This is consistent with an average of 19.6 years between 1997 and 2017 computed from the Standard and
Poor’s 500 Index available here: https://www.statista.com/statistics/1259275/average-company-lifespan/.
27
This yields a maximum age of N̄ = 167 quarters. In other words, the oldest firms in the
model have an age of about 42 years. In each period, I also assume that a new mass of firms
replaces the mass of firms that defaulted 1 − ρ, starting from age zero with zero capital and
zero debt. I relax this assumption in Appendix A.1.
Banks. The number of banks is calibrated to match the average aggregate credit spread
(calculated as the average of the data reported in Figure 1). For the sake of transparency,
Figure 12 in Appendix A.3 illustrates the process as a function of the number of banks. A
model with two banks predicts a credit spread of 2.2%, close to the value in the data of 2.16%
(calculated as the average of the time-series reported in Figure 1). The figure also shows
that, as the number of banks increases, the model converges to the perfectly competitive
case with a credit spread of 1.96%. Hence, the model predicts an annualized aggregate
markup of 0.24%. In Table I, I report the corresponding aggregate stationary equilibrium
moments. As explained, the aggregate credit spread and the frequency of equity issuance
are targeted by controlling the number of banks and the equity flotation cost, respectively.
All remaining moments are untargeted and reported for validation. The model performs
fairly well on untargeted moments such as capital-to-GDP ratio and investment-to-capital
ratio.18 Notably, the financial frictions that arise from the strategic interactions among
intermediaries render the debt adjustment cost to capital ratio significantly smaller (0.2%)
than in the perfectly competitive case (0.9%) and toward the value observed in the data
(0.1%). The more banks are introduced in the economy, the cheaper the interest rate on
debt. For this reason, a more competitive oligopoly induces firms to substitute internal
financing with external resources. As a result, leverage increases with the number of banks
in the economy and tends to stabilize in a perfectly competitive financial market. Note
that, even under perfect competition, firms still make well-defined optimal capital structure
decisions because of the presence of the tax shield. Notably, the calibrated economy yields
a leverage of 30%, close to the average in the data, which is 34% between 1997 and 2018.
Leverage in the data is computed as the non-financial corporate business debt as a percentage
of the market value of corporate equities, retrievable from FRED.
18
Capital and investment in the data are computed from nonresidential current-cost net stock of fixed
assets, retrievable from the U.S. Bureau of Economic Analysis.
28
Table I. Stationary Equilibrium and Quarterly Moments
Description
Moment
Model
Monopoly Duopoly
R
Credit Spread
r
(x,
X)dΦ
−
r
0.70%
0.55%
M
R l
Freq. of Equity Issuance
(d(x, X) < 0)dΦ
3%
1%
Capital to GDP Ratio
K/Y
10.9
11.1
Investment to K Ratio
I/K
3.24%
3.23%
R
0.1%
0.2%
Debt Adjust. to K Ratio B R ∆lb0 (x, X) dΦ/K
Market Leverage
B lb (x, X)/VF (x, X) dΦ
17%
30%
Data
P.C.
0.49%
0%
11.3
3.21%
0.9%
132%
0.54%
1%
11.8
3.5%
0.1%
34%
Notes: This table reports the targeted and untargeted aggregated quarterly moments in function of
the number of banks. The aggregate credit spread and the frequency of equity issuance are targeted
by controlling the number of banks and the equity flotation cost, respectively. Ohter moments are
untargeted. Column monopoly refers to an economy with 1 bank. Column duopoly refers to an economy
with 2 banks. Column P.C. refers to an economy with 10 banks, as a proxy for perfect competition.
For illustration purposes and robustness, in Table III, I report the moments in the calibrated stationary equilibrium benchmarked against two cases: (i) an equity flotation cost
reduced from λ0 = 1.5 (calibrated) to λ0 = 0.75 (left column) and (ii) a risk-aversion increased from γ = 1 (calibrated) to γ = 5 (right column). Note that the risk-aversion γ plays
a role in the stationary equilibrium since it enters all firms elasticities ηl0 (x, X, x0 , X 0 ).
First, the calibrated credit spreads are stable. Moreover, when the credit spreads are
subtracted from the credit spreads calculated under perfect competition in the three corresponding cases, I get markups of about 24bps, 24bps, and 25bps, respectively. A higher γ
increases the consumption smoothing desire of the entrepreneur, with an effect on markups
similar to the equity issuance cost λ0 . The difference being that γ acts at the level of the
aggregate dividend and the equity issuance cost acts at a firm-level. A higher γ increases
banks’ markups by 1bp. Hence, markups are stable as well.
Second, a lower λ0 intuitively increases the aggregate frequency of equity issuance as
the firms’ outside option of issuing equity becomes more appealing. Although in aggregate
terms the stationary credit spread remains stable, as an additional check, I also run the
macroeconomic shocks presented in Section 5 with λ0 = 0.75. The results are qualitatively
and quantitatively similar to those presented in Section 5 with the calibrated model.
29
4.3
Credit Misallocation in the Stationary Equilibrium
I now describe the key properties of the stationary equilibrium of the calibrated model, with
a greater focus on the role of strategic interactions. Figure 2 reports the life cycle of a
firm in the stationary equilibrium. Firms can reach their capital objectives by (i) investing
internal resources, (ii) issuing equity or (iii) demanding external financing resources on the
loan market. A more concentrated banking sector reduces the credit availability in the
economy. Firms with a high marginal productivity of capital and fewer outside options,
likely smaller or highly leveraged, exhibit higher credit demand. Therefore, this type of firm
is more exposed to the negative effects of the lack of competition in the banking sector. This
is the same intuition captured by equation (5) in the stylized model of Section 2.
Along the life cycle of firms, since markups are endogenous in the cross-section of firms,
banks endogenously exert a higher degree of market power on firms with a high marginal
productivity of capital and lower internal resources; hence, fewer outside options. These firms
need banks’ credits and would otherwise incur an equity issuance cost to finance their growth,
in case their current production alone would not be sufficient to sustain the desired level of
physical investment. An imperfectly competitive financial sector internalizes that the same
financial resources are more valuable for firms with a higher marginal productivity of capital
and fewer outside options (e.g., a higher equity issuance cost) and, therefore, can charge
higher markups. This creates a mechanism of endogenous financial friction, as captured
by the central panel of Figure 2. Through this mechanism, the lack of competition in the
financial sector not only induces credit misallocation that forces firms to grow slower, but
also induces lower aggregate productivity. This outcome is illustrated in Figure 3. Notably,
Burga and Céspedes (2022) use a dataset of small firms from Peru to estimate the effect of
banks’ market power. In particular, they exploit a merger episode and find that the change
in banks’ market structure results in “a contraction of capital among small firms with high
marginal returns, which increases capital misallocation,” in agreement with the predictions
of (i) the central panel of Figure 2 and (ii) the dispersion of marginal products of capital,
σ(rL ), reported on the left-axis of Figure 3.19
19
Relatedly, also Cavalcanti, Kaboski, Martins, and Santos (2021) study the effects of intermediation costs
and market power on the dispersion of credit spreads using Brazilian data.
30
Figure 2. Stationary Equilibrium and Firms’ Life Cycle
90
5.5%
70
80
5%
60
70
4.5%
50
60
4%
50
40
3.5%
40
30
3%
30
2.5%
20
20
2%
10
10
0
1.5%
0
0
20
40
1%
0
20
40
0
20
40
Notes: This figure reports the equilibrium policies for loan quantity (left panel) and loan interest rate
(right panel), along the life cycle of a firm in the stationary equilibrium. Younger firms need financing
resources to reach their capital objectives, and since markups are endogenous in the cross-section, banks
naturally extract higher markups from this type of firms, characterized by a higher marginal productivity
of capital and low internal resources. This creates a mechanism of endogenous financial friction, as
captured by the central panel. X-axes report the firms’ age.
Figure 3. Stationary Equilibrium and Credit Misallocation
1.5
1.002
1
1.4
0.998
1.3
0.996
0.994
1.2
0.992
1.1
0.99
1
1
2
3
4
5
6
7
8
9
0.988
10
Notes: This figure reports the standard deviation of interest rates and TFP, both calculated as a ratio to
those obtained in an economy with 10 banks. The figure illustrates the mechanism: lack of competition
in the financial sector induces misallocation of credits, which is linked to misallocation of capital and a
reduced productivity.
Figure 13 in Appendix A.3 reports the inverse elasticities ηl0 (x, X, x0 , X 0 ) contained in the
generalized Euler equation (21), along the life cycle of a firm in the stationary equilibrium.
31
These elasticities, which are endogenous and depend on current and future firm-level characteristics, are directly linked to the equilibrium trajectories of markups. A lower inverse
elasticity translates into a higher financial markup. Furthermore, Figure 13 in Appendix
A.3 shows that the higher the concentration of the banking sector, the lower the inverse
elasticities and the longer it takes for a firm to reach its efficient level of capital. Under
perfect competition, the inverse elasticity would be constant in the cross-section of firms
(long-lived firms still experience a non-zero elasticity because of the tax shield).20 A more
concentrated banking sector can extract higher rents out of financially constrained firms with
a high marginal productivity of capital. This mechanism endogenously creates slower growth
trajectories as a function of the banks’ market structure, as shown in the central panel of
Figure 2. The intensity of this mechanism is time-varying, acting as an amplification channel
in the transition paths of the macroeconomic shocks reported in Section 5.
To conclude this subsection, it is important to highlight that an extensive empirical
literature has investigated the impact of bank competition on the cross-section of firms.
Early seminal contributions include Rajan and Zingales (1998); Black and Strahan (2002);
Cetorelli and Gambera (2001); Cetorelli (2004); and Cetorelli and Strahan (2006). Although
evidence is mixed, the general consensus is that, consistent with my model, banks’ market
power reduces the total amount of credit available in the economy, but importantly this effect
is not constant across firms. This same literature has suggested that two countervailing forces
might be in play and would explain the mixed results.
On the one hand, younger firms exhibit higher credit demand and, therefore, are more
exposed to the negative effects of lack of competition in the banking sector than established
firms. This is the classic industrial organization perspective (see, Freixas and Rochet 1997)
and is fully captured by my model.
On the other hand, a more concentrated banking sector has an incentive to sustain one of
its established clients and refrain from extending credit to young firms. The less competitive
the conditions in the credit market, the lower the incentive for lenders to finance newcomers
as documented by Petersen and Rajan (1995) and Cetorelli and Strahan (2006). Although
my model does not focus on capturing complex features of relationship banking, the desire
20
For the sake of comparison, other papers (e.g., Jamilov and Monacelli 2021 and Wang, Whited, Wu, and
Xiao 2022) capture banks’ market power via constant elasticity of substitution. Differently, my approach
allows for endogenous firm-specific markups, which is key to the transmission of macroeconomic shocks.
32
for inter-temporal smoothing of banks’ profits and households’ consumption – embedded in
the dynamic contract of equations (12), (13), and (21) – captures the fact that creditors,
when contracting markups, take into account the expected stream of future dividends of the
firms, as well as their own future profits. This important economic force is consistent with
Petersen and Rajan (1995) and Cetorelli and Strahan (2006). In my calibrated model, as is
clear from Figure 2, the first force dominates.
5
Macroeconomic Dynamics
This section analyzes the role that banks’ market power plays in the transmission of macroeconomic shocks. The section includes three macroeconomic shocks: (i) a credit quality
deterioration (in the model, a sudden increase to the aggregate firms’ default probability),
(ii) a sudden change to the market structure of the banking sector (in the model, a sudden
decrease of the number of banks in the economy), and (iii) a cut to the bank funding rate.
I compute the transitional dynamics of the model initialized at the stationary equilibrium
as defined in Section 4. Then, I hit the economy with unexpected aggregate shocks and,
depending on the type of shock, the economy converges to the old or a new stationary equilibrium in the long run. Several papers assume agents did not foresee the aggregate shocks
of the Great Recession (e.g., Guerrieri and Lorenzoni, 2017). Along the transitional dynamics, after the shock, I assume all agents can perfectly foresee the paths of all aggregate
variables. In order to compute the equilibrium dynamics, I find sequences of: (i) aggregate
savers’ consumption {Cb,t }Tt=0 , (ii) aggregate entrepreneurs’ consumption {CE,t }Tt=0 , and (iii)
firms’ distributions {φt (xt )}Tt=0 ; such that both households maximize utilities, all markets
clear in each period and the firms’ distributions evolve according to: (i) the firms’ policy
functions, (ii) the banks’ generalized Euler equations (21) and (iii) the idiosyncratic default
shocks. See Appendix B for additional computational details. The main computational challenge is to solve for an equilibrium path simultaneously characterized by general equilibrium,
banks’ strategic interactions, and heterogeneous firms: (i) firms’ decisions are affected by
other firms’ decisions through the aggregate variables, (ii) banks’ decisions are affected by
each single firm’s decisions (banks issue idiosyncratic firm-level optimal contract), aggregate
variables (banks internalize the effects that their actions have on aggregate dividends), and
other banks’ decisions (each bank is best responding to other banks).
33
5.1
Credit Quality Deterioration
This section investigates the effects of banks’ market power when the aggregate firms’ default
probability is hit by an aggregate shock that pushes it to suddenly increase as shown in Figure
14 in Appendix A.3.21
Figures 4 and 5 report the dynamic responses calculated (i) with the calibrated oligopolistic banking sector of Section 4 (solid line) and (ii) with the corresponding perfectly competitive banking sector (dashed line). As discussed in Section 4.3, a more concentrated banking
sector can extract higher rents out of the financially constrained firms with a high marginal
productivity of capital. In the stationary equilibrium, this mechanism endogenously creates
slower growth trajectories as a function of the banks’ market structure. In the dynamics,
this mechanism of endogenous financial frictions interacts with the higher default probability, generating a higher credit demand that drives up markups as shown in Figure 15
in Appendix A.3. Therefore, during the shock, credit spreads rise more under oligopolistic
competition than under perfect competition, as shown in the right panel of Figure 4.
Moreover, as shown by the left panel of Figure 4, total loans decline less under oligopolistic
competition. Banks exploit their market power not only to extract higher markups, but
also to boost their profits. Through this mechanism, banks effectively exploit their market
power to face the increased losses due to the higher default rate, generating countercyclical
financial markups and bank profits. In particular, during the Great Recession, the annualized
aggregate C&I credit spread (see data in Figure 1) spiked up by a magnitude comparable to
the one reported in the right panel of Figure 4. In summary, when credit quality deteriorates,
a concentrated banking sector exploits its market power to extract higher markups as shown
in Figure 15. This mechanism induces a larger and more persistent decline in real activity
in terms of aggregate investment and output, as shown in Figure 5.
21
The shock is calibrated on a magnitude similar to that of the Great Recession.
34
Figure 4. Credit Quality Shock and Financing
0%
4%
-0.5%
3.5%
-1%
3%
-1.5%
2.5%
-2%
2%
-2.5%
-3%
1.5%
0
10
20
30
40
50
0
10
20
30
40
50
R
Notes: This figure reports the transitional dynamics of aggregate loans B lb,t (xt , Xt ) dΦt (in % change
from the initial stationary equilibrium aggregate value) and the annualized aggregate credit spread
R
rl,t (xt , Xt ) dΦt − RM,t (Xt ) (in % level) following an unexpected credit quality shock (as reported in
Figure 14 in Appendix A.3). X-axes report time t.
Figure 5. Credit Quality Shock and Real Activity
2%
0%
0%
-0.2%
-2%
-0.4%
-4%
-6%
-0.6%
-8%
-0.8%
-10%
-12%
-1%
-14%
-1.2%
-16%
-18%
-1.4%
0
20
40
0
20
40
Notes: This figure reports the transitional dynamics (in % change from the initial stationary equilibrium
R
R
aggregate values) of aggregate physical investment it (xt , Xt ) dΦt and aggregate output yt (xt , Xt ) dΦt
following an unexpected credit quality shock (as reported in Figure 14 in Appendix A.3). X-axes report
time t.
Finally, note that the decline in real activity captured by Figure 5 further constrains
35
firms by restraining households’ capacity to support firms with equity issuance. The lack of
outside options creates a vicious cycle that further reduces the interest rate-quantity loan
elasticities ηl0 (x, X, x0 , X 0 ) and boosts banks’ markups. Throughout the paper, I assume that
the defaulting mass of firms is replaced, at each date t, by an equal mass of new entrant firms.
In Appendix A.1, I relax this assumption. As shown by Figure 11 in Appendix A.3, when
the firms’ default rate increases but not all firms immediately re-enter the market (in the
spirit of the Great Recession), then imperfect competition in the financial intermediation
sector leads to a lower amplification effect at the peak but is much more persistent. See
Appendix A.1, for more details.
5.2
Changes in Bank Market Structure
The salient ingredient of my framework is the presence of non-atomistic financial intermediaries. Thanks to this feature, I can study market structure changes, such as banking industry
consolidation and a bank failure. I view the number of banks in the economy as exogenous,
in the sense that this industry has historically been heavily regulated and my primary purpose is to study its effect on the real economy. Among other examples, the reader can think
of the acquisition of Wachovia by Wells Fargo in 2008 (following a government-forced sale
to avoid Wachovia’s failure).
Through the lens of the model, a “big” bank is a financial firm with non-zero mass.
Figures 6 and 7 report the dynamic response to a shock that combines: the credit quality
deterioration shock of Section 5.1 with a market structure shock (i.e., the market structure
changes from 2 banks to 1 bank). At the beginning of date t, when the failure occurs, firms
that do not default repay their entire amount of loans outstanding to each bank. All loans are
repaid before the market structure change, since agents trade one-period securities. Hence,
there is a market structure change.22
22
Initially, I assume there is one continuum of households that trades shares of both banks. Note that,
in the model, the bank failure is not equivalent to a merger, because banks are not consolidated and the
bank that stops its activity does not pay dividends. Since Modigliani-Miller holds on the bank side, savers
are indifferent between equity and deposits. I assume a Debt-to-Equity Ratio of 1, roughly in agreement
with US data. When the bank fails, savers recover deposits but not equity. In order to focus the paper, I
omitted the results but I also simulated a M&A between two “big” banks. In the long run, it yields identical
results to the bank failure. In the short run, it significantly softens the impact with a peak GDP drop of
approximately 3%, instead of almost 7%, as shown in Figure 7 for the bank failure.
36
After the change, all agents make decisions according to the new structure of the economy.
Succinctly, when the number of banks changes the model captures two ideas: (i) in the short
term, the effects of market power of the surviving bank contributes to lowering the supply
of credit to firms, further slowing down the economy and (ii) in the long run, the heightened
banks’ market power further contributes to amplifying and prolonging the recession and,
ultimately, leads to a lower level of available credit, less investment, output, and productivity.
My approach offers a different angle – related to market power – complimentary to typical
arguments (e.g., avoid a bank run) to explain why governments may want to bail out a “big”
bank.
As shown in the left panel of Figure 6, when a “big” bank fails, the surviving bank
starts to slowly extend more credit to firms in order to capture the market share of the
defaulted bank. However, the speed of this adjustment is dampened by the decreased level
of competition among surviving banks. The surviving bank’s market power interacts with
credit constraints, yielding a sharp drop in the aggregate volume of credit in the short run.
Figure 6. Market Structure and Financing
3%
4.5%
2%
4%
1%
3.5%
0%
3%
-1%
2.5%
-2%
2%
-3%
1.5%
0
20
40
60
0
20
40
60
R
Notes: This figure reports the transitional dynamics of aggregate loan per bank lb,t (xt , Xt ) dΦt (in
% change from the initial stationary equilibrium aggregate value) and the annualized aggregate credit
R
spread rl,t (xt , Xt ) dΦt −RM,t (Xt ) (in % level) following an unexpected credit quality shock (as reported
in Figure 14 in Appendix A.3) and, simultaneously, a market structure change. X-axes report time t.
37
Figure 7. Market Structure and Real Activity
0%
-1%
-2%
-3%
-4%
-5%
-6%
-7%
0
10
20
30
40
50
60
Notes: This figure reports the transitional dynamics (in % change from the initial stationary equilibrium
R
aggregate values) and aggregate output yt (xt , Xt ) dΦt , following an unexpected credit quality shock (as
reported in Figure 14 in Appendix A.3) and, simultaneously, a market structure change. X-axis reports
time t.
Moreover, because of the general equilibrium effects and the reduced level of competition,
in the long run, the economy stabilizes at a lower level of volume of credit, which results in
less investment, output, productivity, and more credit and capital misallocation (as shown
by the next subsection). In this sense, my analysis suggests that banks’ market power may
be an important source of concern for policymakers deciding whether to bail out a “big”
bank.
5.2.1
Dispersion of Loan Rates and Aggregate TFP
The dynamic financial oligopoly generates firm-level endogenous financial frictions that create time-varying second moments, such as the dispersion of loan rates (directly linked to the
dispersion of marginal products of capital) and TFP. The left panel of Figure 8 reports the
dynamic of the standard deviation of loan rates, expressed in percentage levels. In agreement with empirical evidence (e.g., Bloom, Floetotto, Jaimovich, Saporta-Eksten, and Terry
2018 and David, Schmid, and Zeke 2022), the model produces a dynamic with an increasing dispersion of loan rates during recessions; hence, an increasing dispersion of marginal
productivity of capital which, in turn, shapes the dynamic behavior of aggregate TFP. The
38
right panel of Figure 8 reports the dynamic of aggregate TFP (calculated as the residual of
an aggregate production Yt = TFPt · Ktα ). Figure 8 suggests that, after the failure of a large
player, banks’ market power contributes significantly to the misallocation of credits (hence,
dispersion of marginal products of capital) and induces a persistent decline in aggregate
TFP.
Figure 8. Market Structure Shock and Credit Misallocation
14%
0.1%
12%
0%
10%
-0.1%
8%
-0.2%
6%
-0.3%
4%
-0.4%
2%
-0.5%
0%
-0.6%
0
20
40
60
0
20
40
60
Notes: This figure reports the transitional dynamics of the dispersion of loans’ interest rates
R 2
R
(in % levels), calculated as the square root of rl,t
(xt , Xt ) dΦt − ( rl,t (xt , Xt ) dΦt )2 , and aggregate TFP (in % change from the initial stationary equilibrium aggregate value), calculated as
R α
R
kt (xt , Xt ) dΦt /( kt (xt , Xt ) dΦt )α , following an unexpected credit quality shock (as reported in Figure
14 in Appendix A.3) and, simultaneously, a market structure shock. X-axes report time t.
5.3
Bank Funding Rate Cut
This subsection analyzes the effects of banks’ market power when the banks’ funding rate
suddenly decreases, as shown in the left panel of Figure 9.
Initially, the economy is in stationary equilibrium with RM = MS0−1 (X, X 0 ) = βb−1 , where
βb denotes the discount factor of the saver in equation (6).23 Following a cut to the bank
funding rate, the oligopolistic financial intermediation sector, which has the power to affect
aggregate prices, acts as a potent transmission channel. Oligopolistic financial intermediaries
exploit markups to dampen the aggregate loans’ interest rate response.
23
Note that βE in equation (7) is held fixed during the funding rate shock.
39
Figure 9. Funding Rate Cut and Loan Rate
0.9965
0.005%
0.996
0%
0.9955
-0.005%
0.995
-0.01%
0.9945
-0.015%
0.994
-0.02%
-0.025%
0.9935
0
20
40
60
80
0
20
40
60
80
Notes: The left panel of this figure reports the shock: following a sudden unexpected 25 basis points
increase in the βb discount rate of the banker (time t = 2), the economy mean-reverts to its original level.
After the unexpected shock, all agents can perfectly forecast the mean-reversion path. The right panel reR
ports the transitional dynamics of the annualized aggregate interest rate on banks’ loans rl,t (xt , Xt ) dΦt
(in % points with respect to the initial stationary equilibrium aggregate value), correspondent to the shock
reported in the left panel. X-axes report time t.
Figure 10. Funding Rate Cut and Real Activity
1.4%
0.12%
1.2%
0.1%
1%
0.08%
0.8%
0.6%
0.06%
0.4%
0.04%
0.2%
0%
0.02%
-0.2%
0%
-0.4%
-0.6%
-0.02%
0
20
40
60
80
0
20
40
60
80
Notes: This figure reports the transitional dynamics (in % change from the initial stationary equilibrium
R
aggregate values) of aggregate physical investment it (xt , Xt ) dΦt (left panel) and aggregate output
R
yt (xt , Xt ) dΦt , correspondent to the shock reported in the left panel of Figure 8. X-axes report time t.
40
Quantitatively, this effect is shown in the right panel of Figure 9. As a consequence,
the associated boom in output is dampened by roughly 30% at the peak. These effects are
shown in the left and the right panel of Figure 10 for aggregate investment and aggregate
output, respectively.
6
Conclusion
Motivated by several pieces of evidence, in this paper I study banks’ market power through
the lens of a new dynamic general equilibrium model that combines oligopolistic banks and
heterogeneous firms. The distinctive feature of my model is that financial intermediaries
charge firm-specific markups, practicing price discrimination. The lack of competition in
the financial sector creates an endogenous cross-sectional dispersion of markups, yielding a
mechanism of endogenous financial friction which induces misallocation of credits and reduces productivity. I find that this mechanism plays a significant role in the transmission
of macroeconomic shocks. Notably, since my model features non-atomistic banks, I can
study banks’ market structure changes, such as a “big” bank failure. When a “big” (i.e.,
non-atomistic) bank fails, surviving banks’ market power lowers the total supply of credit,
contributing to amplifying and prolonging the recession. In this sense, my analysis offers a
different angle, complimentary to typical arguments (e.g., avoid a bank run), to explain why
policymakers may want to bail out a “big” bank. A natural direction for future research is to
combine intermediaries’ market power with collateral constraints, in order to investigate the
economic mechanism through which they interact and disentangle the two joint effects. Another interesting direction for future research would consider integrating endogenous entry of
firms in this framework. The empirical evidence (e.g, Cetorelli and Strahan 2006) highlights
that in markets characterized by higher bank concentration, potential entrant firms face
greater difficulty in gaining access to credit, since banks have an incentive to favor incumbent firms. Taking a further step to incorporate forces of relationship lending in the model
would allow one to study if and how these forces affect the propagation of macroeconomic
shocks.
41
References
Albuquerque, R., and H. A. Hopenhayn (2004): “Optimal Lending Contracts and
Firm Dynamics,” The Review of Economic Studies, 71(2), 285–315.
Berger, D., K. Herkenhoff, and S. Mongey (2022): “Labor Market Power,” American Economic Review, 112(4), 1147–93.
Bernanke, B., and M. Gertler (1989): “Agency Costs, Net Worth, and Business Fluctuations,” American Economic Review, 79(1), 14–31.
Black, S. E., and P. E. Strahan (2002): “Entrepreneurship and Bank Credit Availability,” The Journal of Finance, 57(6), 2807–2833.
Bloom, N., M. Floetotto, N. Jaimovich, I. Saporta-Eksten, and S. J. Terry
(2018): “Really Uncertain Business Cycles,” Econometrica, 86(3), 1031–1065.
Brunnermeier, M. K., and Y. Sannikov (2014): “A Macroeconomic Model with a
Financial Sector,” American Economic Review, 104(2), 379–421.
Brusco, S., G. Lopomo, E. Ropero, and A. T. Villa (2021): “Optimal financial
contracting and the effects of firm’s size,” The RAND Journal of Economics, 52(2), 446–
467.
Burga, C., and N. Céspedes (2022): “Bank Competition, Capital Misallocation, and
Industry Concentration: Evidence from Peru,” Working Paper.
Caggese, A. (2007): “Financing constraints, irreversibility, and investment dynamics,”
Journal of Monetary Economics, 54(7), 2102–2130.
Cavalcanti, T. V., J. P. Kaboski, B. S. Martins, and C. Santos (2021): “Dispersion in Financing Costs and Development,” (28635).
Cetorelli, N. (2004): “Real Effects of Bank Competition,” Journal of Money, Credit and
Banking, 36(3), 543–558.
Cetorelli, N., and M. Gambera (2001): “Banking Market Structure, Financial Dependence and Growth: International Evidence from Industry Data,” The Journal of Finance,
56(2), 617–648.
Cetorelli, N., and P. F. Peretto (2012): “Credit Quantity and Credit Quality: Bank
Competition and Capital Accumulation,” Journal of Economic Theory, 147(3), 967–998.
42
Cetorelli, N., and P. E. Strahan (2006): “Finance as a Barrier to Entry: Bank
Competition and Industry Structure in Local U.S. Markets,” The Journal of Finance,
61(1), 437–461.
Clementi, G. L., and H. A. Hopenhayn (2006): “A Theory of Financing Constraints
and Firm Dynamics,” The Quarterly Journal of Economics, 121(1), 229–265.
Clymo, A., and A. Lanteri (2020): “Fiscal Policy with Limited-Time Commitment,”
The Economic Journal, 130(627), 623–652.
Cooley, T. F., and V. Quadrini (2001): “Financial Markets and Firm Dynamics,”
American Economic Review, 91(5), 1286–1310.
Corbae, D., and P. D’Erasmo (2021): “Capital Buffers in a Quantitative Model of
Banking Industry Dynamics,” Econometrica, 89(6), 2975–3023.
Cordoba, J.-C., and M. Ripoll (2004): “Credit Cycles Redux,” International Economic
Review, 45(4), 1011–1046.
Corhay, A., H. Kung, and L. Schmid (2020): “Competition, Markups, and Predictable
Returns,” The Review of Financial Studies, 33(12), 5906–5939.
Covas, F., and W. J. den Haan (2011): “The Cyclical Behavior of Debt and Equity
Finance,” American Economic Review, 101(2), 877–99.
David, J. M., L. Schmid, and D. Zeke (2022): “Risk-adjusted capital allocation and
misallocation,” Journal of Financial Economics, 145(3), 684–705.
De Loecker, J., J. Eeckhout, and G. Unger (2020): “The Rise of Market Power and
the Macroeconomic Implications*,” The Quarterly Journal of Economics, 135(2), 561–644.
Drechsler, I., A. Savov, and P. Schnabl (2017): “The Deposits Channel of Monetary
Policy*,” The Quarterly Journal of Economics, 132(4), 1819–1876.
Elenev, V., T. Landvoigt, and S. Van Nieuwerburgh (2021): “A Macroeconomic
Model With Financially Constrained Producers and Intermediaries,” Econometrica, 89(3),
1361–1418.
Farhi, E., and F. Gourio (2018): “Accounting for Macro-Finance Trends: Market Power,
Intangibles, and Risk Premia,” Brookings Papers on Economic Activity, Fall, 147–250.
Freixas, X., and J. Rochet (1997): “Microeconomics of Banking, Volume 1,” The MIT
Press, 1 edition.
43
Gertler, M., and P. Karadi (2011): “A model of unconventional monetary policy,”
Journal of Monetary Economics, 58(1), 17–34, Carnegie-Rochester Conference Series on
Public Policy: The Future of Central Banking April 16-17, 2010.
Gomes, J. F. (2001): “Financing Investment,” American Economic Review, 91(5), 1263–
1285.
Guerrieri, V., and G. Lorenzoni (2017): “Credit Crises, Precautionary Savings, and
the Liquidity Trap*,” The Quarterly Journal of Economics, 132(3), 1427–1467.
Gutiérrez, G., and T. Philippon (2016): “Investment-less Growth: An Empirical Investigation,” (22897).
He, Z., and A. Krishnamurthy (2013): “Intermediary Asset Pricing,” American Economic Review, 103(2), 732–70.
Hennessy, C. A., and T. M. Whited (2005): “Debt Dynamics,” The Journal of Finance,
60(3), 1129–1165.
(2007): “How Costly Is External Financing? Evidence from a Structural Estimation,” The Journal of Finance, 62(4), 1705–1745.
Ifrach, B., and G. Y. Weintraub (2016): “A Framework for Dynamic Oligopoly in
Concentrated Industries,” The Review of Economic Studies, 84(3), 1106–1150.
Jamilov, R., and T. Monacelli (2021): “Bewley Banks,” Working Paper.
Jermann, U., and V. Quadrini (2012): “Macroeconomic Effects of Financial Shocks,”
American Economic Review, 102(1), 238–71.
Khan, A., T. Senga, and J. K. Thomas (2016): “Default Risk and Aggregate Fluctuations in an Economy with Production Heterogeneity,” Working paper.
Khan, A., and J. K. Thomas (2013): “Credit Shocks and Aggregate Fluctuations in an
Economy with Production Heterogeneity,” Journal of Political Economy, 121(6), 1055–
1107.
Kiyotaki, N., and J. Moore (1997): “Credit Cycles,” Journal of Political Economy,
105(2), 211–248.
Klein, P., P. Krusell, and J.-V. Rı́os-Rull (2008): “Time-Consistent Public Policy,”
The Review of Economic Studies, 75(3), 789–808.
44
Klein, P., and J.-V. Rı́os-Rull (2003): “Time-Consistent Optimal Fiscal Policy,” International Economic Review, 44(4), 1217–1245.
Kocherlakota, N. R. (2000): “Creating Business Cycles through Credit Constraints,”
Federal Reserve Bank of Minneapolis Quarterly Review, Volume 24, No. 3.
Krusell, P., F. Martin, and J.-V. Rı́os-Rull (2004): “On the Determination of Government Debt,” https://www.sas.upenn.edu/ vr0j/slides/humboldt.pdf.
Lanteri, A. (2018): “The Market for Used Capital: Endogenous Irreversibility and Reallocation over the Business Cycle,” American Economic Review, 108(9), 2383–2419.
Li, L., E. Loutskina, and P. E. Strahan (2019): “Deposit Market Power, Funding
Stability and Long-Term Credit,” (26163).
Malamudy, S., and A. Schrimpf (2017): “Intermediation Markups and Monetary Policy
Pass-through,” Swiss Finance Institute Research Paper No. 16-75.
Maskin, E., and J. Tirole (2001): “Markov Perfect Equilibrium: I. Observable Actions,”
Journal of Economic Theory, 100(2), 191–219.
Petersen, M. A., and R. G. Rajan (1995): “The Effect of Credit Market Competition
on Lending Relationships,” The Quarterly Journal of Economics, 110(2), 407–443.
Rajan, R. G., and L. Zingales (1998): “Financial Dependence and Growth,” American
Economic Review, 88(3), 559–586.
Rampini, A. A., and S. Viswanathan (2018): “Financial Intermediary Capital,” The
Review of Economic Studies, 86(1), 413–455.
Wang, Y., T. M. Whited, Y. Wu, and K. Xiao (2022): “Bank Market Power and
Monetary Policy Transmission: Evidence from a Structural Estimation,” The Journal of
Finance, 77(4), 2093–2141.
45
ONLINE APPENDIX
This online appendix is organized as follows. First, Appendix A contains additional materials
such as additional robustness checks. Second, Appendix B is the computational appendix
that contains details about the algorithms to solve for (i) the stationary equilibrium and (ii)
the transitional dynamics. Third, Appendix C contains mathematical details.
A
Additional Material
This section is divided in four parts: (i) the first part contains the extension where I check
the effects of the entry rate to the credit quality deterioration shock described in Subsection
5.1, (ii) the second part contains additional tables, (iii) the third part contains additional
figures, and (iv) the fourth part contains additional empirics.
A.1
The Effects of the Entry Rate
Throughout the paper, I assume that the defaulting mass of firms is replaced, at each date
t, by an equal mass of new entrants firms. In this subsection, I relax this assumption. As
before, I let the mass of exiting firms evolves according to the evolution of the default rate
in the credit deterioration shock of Figure 14. Differently from before, I keep the entering
mass of firms constant to its initial stationary equilibrium value. As a consequence, along
the shock, the mass of firms in the production sector drops in the short-run and returns to
1 in the long-run. In the spirit of the parallelism with the Great Recession, this produces
(in the model) a decline in the mass of firms of about -3.2% at the lower peak, aligned with
the percentage decline in the number of firms between 2007 and 2009.24 In Figure 11, I refer
to this type of shock with the label Entry Mass ≤ Exit Mass. In contrast, the label Entry
Mass = Exit Mass refers to the shock already analyzed in the previous Section 5.1, whose
effects on real activity are reported in Figure 5.
24
See, for instance, https://www.federalreserve.gov/econres/feds/firm-entry-and-employment-dynamicsin-the-great-recession.htm.
46
Figure 11. Amplification of Credit Quality Shock and Firms’ Entry
0.5%
0.05%
0%
0%
-0.05%
-0.5%
-0.1%
-1%
-0.15%
-1.5%
-0.2%
-2%
-0.25%
-2.5%
-0.3%
-3%
-0.35%
0
50
100
150
200
0
50
100
150
200
Notes: This figure reports the amplification effects during the transitional dynamics (in % points) of
R
R
aggregate physical investment it (xt , Xt ) dΦt and aggregate output yt (xt , Xt ) dΦt , following an unexpected credit quality shock (as reported in Figure 14) with and without the assumption Entry Mass =
Exit Mass. The amplification effects are computed as follow. First, compute the transitional dynamics (in
% change from the initial stationary equilibrium aggregate values) with and without perfect competition,
as in Figure 5. Second, calculate the amplification effects as % points difference between the responses
under oligopolistic competition and under perfect competition. X-axes report time t.
Figure 11 suggests that, when the firms’ default rate increases but not all firms immediately re-enter the market, then imperfect competition in the financial intermediation sector
leads to a lower amplification effect at the peak but is much more persistent. The intuition
is linked to the economic mechanism discussed in the Section 5.1.
On the one side, a more concentrated banking sector extracts higher rents out of the
financially constrained firms with a high marginal productivity of capital, since these firms
have worse outside options (e.g., a higher equity issuance cost) and one additional unit of
investment in physical capital can contribute significantly to their production. When not
all defaulting firms immediately re-enter the market, there is a decline of smaller size firms
which tend to be more financially constrained and with a high marginal productivity of
capital. Hence, banks partly postpone the extraction of markups to the future, when there
will be a recovery.
On the other side, the additional decline in real activity due to the decline in the mass
47
of producing firms restrains the household’s capacity to support firms with equity issuance
as the aggregate dividend falls. This economic force tends to increase banks’ market power
in the short run.
A.2
Additional Tables
Table II. Parameter Values
Household
Time Discount
Risk Aversion
Firms
Depreciation Rate
Effective Capital Share
Corporate Tax Rate
Default Rate
Equity Flotation Cost
Banks
Number of Banks
Parameter
Value
Target/Source
β
γ
0.9936
1
Match Deposit Rate (Source: FDIC)
δ
α
τ
1−ρ
λ0
0.025
0.41
0.241
0.21%
1.5
Bureau of Economic Analysis
Bureau of Labor Statistics
Tax Corp. Income/ Corp. Profit (Source: FRED)
Quarterly Net Charge-off to Loan (Source: FDIC)
Internally calibrated (see Table I)
B
2
Internally calibrated (see Table I)
Notes: The table reports the parameter values.
Table III. Stationary Equilibrium and Moments Robustness
Description
Moment
Model
λ0 = 0.75 Calibrated
R
Credit Spread
r
(x,
X)dΦ
−
r
0.55%
0.55%
M
R l
Freq. of Equity Issuance
(d(x, X) < 0)dΦ
3.5%
1%
Capital to GDP Ratio
K/Y
11.13
11.13
Investment to K Ratio
I/K
3.23%
3.23%
R
Debt Adjust. to K Ratio B R ∆lb0 (x, X) dΦ/K
0.2%
0.2%
Market Leverage
B lb (x, X)/VF (x, X) dΦ 30.40%
30.41%
γ=5
0.56%
2%
11.12
3.23%
0.2%
30.23%
Notes: This table reports the same moments of Table I. The calibrated column is the same column of
Table I under Duopoly, with baseline parameters λ0 = 1.5 and γ = 1. The right column reports the
moments when γ = 5, everything else equal to the the calibrated column. The left column reports the
moments when λ0 = 0.75, everything else equal to the the calibrated column.
48
A.3
Additional Figures
Figure 12. Stationary Equilibrium and Credit Spreads
2.9%
2.8%
2.7%
2.6%
2.5%
2.4%
2.3%
2.2%
2.1%
2%
1.9%
1
2
3
5
10
20
Notes: This figure reports the average annualized credit spreads predicted by the model in various
stationary equilibria, calculated with an increasing number of banks. The figure shows that the model
with two banks predicts an annualized credit spread of about 2.2%, close to the value of 2.16% observed
in the data (calculated as average of the time series reported in Figure 1). Moreover, the more banks are
introduced in the oligopoly, the more the economy converges to the perfectly competitive case with credit
spread of about 1.96%. Hence, the model predicts an annualized aggregate markup of about 0.24%.
Figure 13. Stationary Equilibrium and Cross-Sectional Markups
-0.2
-0.3
B=1
B=2
B=3
B=10
-0.4
-0.5
-0.6
-0.7
-0.8
-0.9
0
5
10
15
20
25
30
35
40
0
Notes: This figure reports the inverse elasticities ηL
(x, X, x0 , X 0 ) contained in the generalized Euler
equations (21) along the life cycle of a firm in the stationary equilibrium. These are the elasticities of the
future loans’ interest rate with respect to loans’ quantity. The X-axes reports the firms’ age.
49
Figure 14. Annualized Charge-Off Rate 1 − ρ (2008Q1-2012Q2)
2.4%
2.2%
2%
1.8%
1.6%
1.4%
1.2%
1%
0.8%
0.6%
2
4
6
8
10
12
14
16
18
Notes: This figure reports the annualized net charge-off rates during the Great Recession. The X-axis
reports time t, expressed in quarters. The shock is represented by the dashed line: following a sudden
unexpected increase in the aggregate firms’ default probability (time t = 2) the economy mean-reverts to
its original level. After the unexpected shock, all agents can perfectly forecast the mean-reversion path.
Figure 15. Credit Quality Shock and Aggregate Markup
0.65%
0.6%
0.55%
0.5%
0.45%
0.4%
0.35%
0.3%
0.25%
0
10
20
30
40
50
Notes: This figure reports the transitional dynamics of the annualized aggregate markup (in % level)
following an unexpected credit quality shock (as reported in Figure 14). The evolution over time of
the aggregate markup is calculated as the difference between (i) the annualized aggregate credit spread
R
rl,t (xt , Xt ) dΦt − RM,t (Xt ) under the calibrated oligopoly and the (ii) the annualized aggregate credit
R PC
PC
spread rl,t
(xt , Xt ) dΦPC
t − RM,t (Xt ) under perfect competition. Note that the dynamics of both credit
spreads are reported in the right panel of Figure 4. The X-axis reports time t.
50
A.4
Additional Empirics
This section contains additional empirical evidences of imperfect competition in the banking
sector. Figure 16 shows the time evolution of the 5-Bank Asset Concentration for the United
States, which has been increasing both in terms of deposits and assets since 1995. This data
are obtained using the Federal Deposit Insurance Corporation (FDIC) summary of deposits
survey of branch from 1994 to 2018, from bank’s level data. Banks deposits and assets are
aggregated at the level of the holding bank and divided by the deposits and assets of the
entire industry. As an example, in 2018, the biggest five banks in terms of assets were: (1)
JP Morgan Chase & Co (14.45%), (2) Bank of America Corp. (11.73%), (3) Wells Fargo &
Company (11.17%), (4) Citigroup Inc. (9.32%) and (5) U.S. Bancorp (3.02%).
Figure 16. Market Share of the Top 5 US Banks
Notes: The figure shows the evolution over time of the deposit and asset market share of the top 5 US
banks. The source of the data is the FDIC summary of deposits survey of branch.
51
Figure 17. Lerner Index (US Banks)
Notes: This figure reports the Lerner index, which is the difference between output prices and marginal
costs divided by prices: 0 indicates perfect competition and 1 indicates monopoly. Output prices are
calculated as total bank revenue over assets. Marginal costs are calculated as estimated translog cost
function with respect to output. The source of the data is the World Bank Global Financial Development
Database.
Another evidence of imperfect competition in the banking sector is provided by the
Lerner Index. As shown by Figure 17, the Lerner index has been increasing. Moreover, it is
significantly different from zero, where zero is the perfect competition benchmark.
Another relevant statistics in this context is the Rosse-Panzar H index, also reported in
the World Bank Global Financial Development Database. The value has fluctuated around
0.45 between 2010 and 2015. It is a measure of the elasticity of bank revenues relative to
input prices: 1 indicates perfect competition, 0 (or less) indicates monopoly. Overall, these
evidences suggest that there is a significant degree of imperfect competition in the banking
sector.
To conclude the analysis, I now turn the attention to the data used to calibrate the
model. Table IV reports the effects that the market share of the top 5 US banks and the net
charge-off rates have on C&I credit spreads. As a reminder, credit spreads are calculated as
the difference between the weighted-average effective loan rate for all C&I Loans (RL ) and
3-Month T-bill rates (RM ). Two other measures are added to the analysis: (i) outstanding
quantity of C&I Loans ($tn) and (ii) the weighted-average maturity for all C&I Loans. Each
52
period t is a quarter between 1997Q2 and 2017Q2. The results of the following regression
RL,t − RM,t = β0 + β1 × C5,t + β2 × (1 − ρt ) + β3 × Lt + β4 × Mt ,
are reports in Table IV.
Table IV. Credit spread and Banks Market Concentration
Dependent variable:
Commercial & Industrial Loan Rates Spreads over intended federal funds rate
Market share of top 5 banks (%)
Net Charge-Off Rate (%)
(1)
(2)
(3)
0.040∗∗∗
0.053∗∗∗
(0.004)
(0.006)
0.056∗∗∗
(0.007)
0.337∗∗∗
(0.051)
0.295∗∗∗
(0.051)
0.272∗∗∗
(0.059)
C&I Loans ($tn)
−0.391∗∗∗
(0.139)
−0.345∗∗
(0.152)
−0.121
(0.157)
Maturity
Constant
Observations
R2
Adjusted R2
Residual Std. Error
F Statistic
0.434∗∗
(0.183)
0.384∗∗
(0.177)
81
0.644
0.635
0.291 (df = 78)
70.500∗∗∗ (df = 2; 78)
0.406∗∗
(0.179)
81
0.677
0.664
0.279 (df = 77)
53.802∗∗∗ (df = 3; 77)
81
0.680
0.663
0.280 (df = 76)
40.292∗∗∗ (df = 4; 76)
∗ p<0.1; ∗∗ p<0.05; ∗∗∗ p<0.01
Note:
Notes: There is a strong significant positive correlation between credit spread, banks market concentration, and net charge-off rate. The correlation with the quantity of C&I Loans outstanding is significant
and negative, consistently with the model (the elasticity between loan and loan rate is negative).
B
Computational Online Appendix
In this section, I describe the algorithm to solve both the stationary equilibrium and the
dynamics with the MIT shocks. I highlight the novel methodology used to solve for both General Equilibrium and strategic interactions. Since banks optimize over the optimal choices
of the firms, solving this problem using value function iteration would require to nesting two
value function iterations inside each other and iterating on the nested value function system
given guesses for the aggregate dynamics. Moreover, accounting for strategic interactions
with value function iteration would require to solving this system of two nested VFIs given
other banks’ strategies and finding the fixed point of the resulting policies. This brute force
53
approach is clearly not viable. To avoid this, I use projection methods jointly on the generalized Euler equations (21) and the loan firms optimality conditions (12) and (13). Hence, I
leverage the fact that the elasticities can be calculated applying the implicit function theorem as described in Appendix C.1. Note that the aggregate quantities are not only contained
in the discount factors, but also in the elasticities of the generalized Euler equations (see
Appendix C.1). In order to account for strategic interactions and solve the generalized Euler equations (which can also be interpreted as best response functions), I impose ex-post
symmetric strategies between banks after calculating the elasticity as described in Section
C.1; hence, I proceed to calculate the root of the resulting equation with time iteration and
projection methods, as any other Euler equation.25 Note also that the projection step with
time iteration can be efficiently computed parallelizing the calculation of the policy functions, fixing state variables. In particular, on a grid of K × L, such that K = [0, k1 , ..., kK̄ ]
and L = [0, lb,1 , ..., lb,L̄ ], this corresponds to have K̄ × L̄ parallel subproblems at each step
of the time iteration. The code is written in C/C++, each subproblem can be efficiently
parallelized using the OpenMP API specification for parallel programming. Moreover, each
subproblem is solved using the Levenberg-Marquardt algorithm contained in ALGLIB.
B.1
Oligopolistic Stationary Equilibrium
Here are the main steps to solve for the oligopolistic stationary equilibrium (see Definition
3.1). Create grids K = [0, k1 , ..., k̄] and L = [0, lb,1 , ..., lb,L̄ ]. Initialize the policy functions for
investment and loan to the solution of the corresponding steady-state model without firms
heterogeneity; i.e., ∀(k, lb ) ∈ K × L,
lb0 (k, lb ) = lb∗ and k 0 (k, lb ) = k ∗ . Create an iterator j
and set j = 0; hence, proceed as follow.
1. Guess an aggregate dividend D̃j =
R
d˜dΦ (e.g., use the steady-state dividend calcu-
lated without firm heterogeneity).26
2. Create an iterator w and set w = 0.
(a) Fork the program in k̄ × L̄ parallel threads. Each thread solves the subproblem
associated with a fixed pair (k, lb ) ∈ K × L.27
25
This is equivalent to finding the fixed-point of the banks’ strategies.
Note that the aggregate dividend D̃j is contained in the elasticity ηl of equation (21).
27
This can be achieved either by using POSIX Threads directly or, at a higher level, using OpenMP API.
26
54
0
0
0
(b) Each thread uses Levenberg-Marquardt to jointly solve for k w+1 , lbw+1 , Rlw+1 such
that the firms’ first-order conditions (12) and (13) hold, and such that the generalized Euler equation (21) hold, given future policy functions k
00
0
0
00
00 w
0
0
(k w+1 , lbw+1 ),
0
0
lb w (k w+1 , lbw+1 ), and Rl w (k w+1 , lbw+1 ).28 The elasticity ηl0 of equation (21) is
calculated according to equations (25), (26), (27), (28) and the condition of sym0
0
0
metry among bank’s strategies l1w+1 = ... = lbw+1 = ... = lBw+1 , which is imposed
ex-post.
(c) Wait for all parallel threads to complete their tasks and update the policy func0
0
k 0 (k, lb ) = k w+1 , lb0 (k, lb ) = lbw+1 , Rl0 (k, lb ) =
tions accordingly ∀(k, lb ) ∈ K × L,
0
Rlw+1 .29
0
0
0
0
(d) If the policy functions converged (i.e., max(sup |k w+1 − k w |, sup |lbw+1 − lbw |) < )
proceed to step 3. Otherwise, set w = w + 1 and restart from step 2.
3. The probability density function over age is given by
ρage
φ(age) = PN̄
age
age=0 ρ
.
Start from age = 0 and simulate the policy functions up to age N̄ . This yields a
mapping between φ(age) and φ(x).
4. Compute the implied aggregate dividend D̃j+1 =
R
d˜dΦ(x).
5. If the aggregate dividend converged (i.e., |D̃j+1 − D̃j | < ), the program terminates.
Otherwise, set j = j +1 and restart from step 2. Use a quasi-Newton method to correct
the guess of the aggregate dividend D̃j , given the implied aggregate dividend D̃j+1 .30
B.2
Transitional Dynamics
The economy is initially in its stationary equilibrium when all agents discover a sudden
change in a model parameter at t = 0. In order to compute the equilibrium dynamics, I
need to find sequences of: (i) aggregate saver’s consumption {Cb,t }Tt=0 , (ii) entrepreneur’s
28
I use k̄ = 10, L̄ = 10 and perform piece-wise bilinear interpolation.
The update requires dampening. For example, I update the policy function for capital according to
0
0
0
k w+1 = (1 − ω) · k w + ω · k w+1 , with ω = 0.01.
30
This update also requires dampening, similarly to the update of the policy functions.
29
55
aggregate consumption {CE,t }Tt=0 , and (iii) firms distributions {φt (x)}Tt=0 ; such that both
households maximize utilities, all markets clear in each period and the firms distributions
evolve according to: (i) the firms’ policy functions, (ii) the banks generalized Euler equations and (iii) the idiosyncratic default shocks. First, compute the two stationary equilibria
associated with the configuration of parameters before and after the shock, as described
previously.31 Second, create an iterator j and set j = 0; hence, proceed as follow.
j T
1. Guess sequences of: (i) aggregate saver’s consumption {Cb,t
}t=0 , (ii) entrepreneur’s
j
aggregate consumption {CE,t
}Tt=0 .32
2. Create an iterator t and set t = T − 1. Hence, use projection with backward time
iteration from t = T − 1 to t = 0. The policy functions at t = T , are the ones
associated with the ending stationary equilibrium, previously calculated. At each time
t proceed similarly to before.
(a) Fork the program in k̄ × L̄ parallel threads. Each thread solves the subproblem
associated with a fixed pair (k, lb ) ∈ K × L.
(b) Each thread uses Levenberg-Marquardt to jointly solve for k t+1 , lbt+1 , Rlt+1 such
that the firms’ first-order conditions (12) and (13) hold, and such that the generalized Euler equation (21) hold, given future policy functions k t+2 (k t+1 , lbt+1 ),
lbt+2 (k t+1 , lbt+1 ), and Rlt+2 (k t+1 , lbt+1 ), always performing piece-wise bilinear interpolation when needed. The elasticity ηl0 of equation (21) is calculated according
to equations (25), (26), (27), (28) and the condition of symmetry among bank’s
t+1
strategies l1t+1 = ... = lbt+1 = ... = lB
, which is imposed ex-post.
(c) Wait for all parallel threads to complete their tasks and update the policy functions accordingly ∀(k, lb ) ∈ K×L,
k t+1 (k, lb ) = k t+1 , lbt+1 (k, lb ) = lbt+1 , Rlt+1 (k, lb ) =
Rlt+1 .
3. Now, start from t = 0 and iterate forward up to t = T . The probability density
31
If there are not permanent change to the parameters, the two stationary equilibria coincides.
T should be long enough, so that after the shock the economy converges to its long-run stationary
equilibrium. In this paper, I use T=100 quarters or T=200 quarters, depending on the shock considered.
32
56
function over age is now time-varying
ρage
φt (age) = PN̄ t
age
age=0 ρt
.
At each time t, start from age = 0 and simulate the time t policy functions up to age
N̄ . This yields a mapping between φt (age) and φt (x).
j+1
4. For each time t, compute the implied aggregate dividend CE,t
=
R
d˜t dΦt (x), which is
j+1
=
consumed by the entrepreneur. Compute the implied aggregate bank’s profit Cb,t
R
πb dΦt (x), which is consumed by the saver.
5. If the sequences for aggregate consumptions converged; i.e.,
j+1
j
j+1
j
max(sup{|CE,t
− CE,t
|}Tt=0 , sup{|Cb,t
− Cb,t
|}Tt=0 ) < ,
the program terminates. Otherwise, set j = j +1 and restart from step 2, after heaving
updated the sequences with a heavy dampening parameter.
C
Mathematical Online Appendix
This section contains: (i) the calculation of the elasticity ηl in the generalized Euler equation
(21), and (ii) the proofs of the statements in the proposition of Section 2.
C.1
Firm-Specific Inverse Elasticity ηl
This subsection contains the calculation of the elasticity in the generalized Euler equation
(21). Combine the firms’ first-order conditions (12) and (13) to define functions
f (x, X, x0 , X 0 ) ≡ πk (k 0 ) − δ − Rl0 + 1 = 0,
g (x, X, x0 , X 0 ) ≡ ρ · ME0 · (1 − λ0d ) ((1 − τ )(Rl0 − 1) + 1) − 1 + λd = 0.
57
Compute the total derivatives of these two functions with respect to k 0 and lb0 . Hence, solve
the resulting linear system to get the following expression:33
fk0 · glb0 − flb0 · gk0
∂Rl0
=
.
0
∂lb
fRL0 · gk0 − fk0 · gRL0
Note that flb0 = 0, fRl0 = −1, and fk0 = πkk (k 0 ). Therefore, the elasticity ηl0 is given by
ηl0 =
fk0 · glb0
∂Rl0 lb0
lb0
=
−
,
∂lb0 Rl0
gk0 + fk0 · gRl0 Rl0
(25)
where the partial derivatives of the g function are
0
∂ME0
∂λd (d)
0
0
0 ∂λd (d )
0
+
1)
+
(1
−
λ
(d
))
((1
−
τ
)r
+
1)
−
ρM
((1
−
τ
)r
,
(26)
d
l
E
l
∂k 0
∂k 0
∂k 0
∂λd (d0 )
∂λd (d)
∂M 0
glb0 = ρ 0E (1 − λd (d0 )) ((1 − τ )rl 0 + 1) − ρME0
((1 − τ )rl 0 + 1) +
,
(27)
0
∂lb
∂lb
∂lb0
0
∂ME0
∂λd (d)
0
0
0 ∂λd (d )
gRl0 = ρ
(1
−
λ
(d
))
((1
−
τ
)r
+
1)
−
ρM
((1 − τ )rl 0 + 1) + ρME0 (1 − τ ) +
.
d
l
E
0
0
∂Rl
∂Rl
∂Rl0
gk0 = ρ
(28)
Note that equations (26), (27), and (28) contain all other banks strategies l−b = [l1 , ..., lB ]\{lb }
in the discount factor ME0 and in the marginal equity issuance costs λd (d), λd (d0 ).
C.2
Proofs
This section contains the proofs of the statements 1-10 in the main proposition of Section 2.
Proof. Statements 1,2 and 3. Note that the number of banks matters only for the financially
constrained firms, so that each integral can be rewritten as follow
∂
∂B
Z
∗
x dΦ =
Z
∂x∗
· 1[d∗1 < 0] dΦ +
∂B
Z
∂x∗
· 1[d∗1 ≥ 0] dΦ =
∂B
Z
∂x∗
· 1[d∗1 < 0] dΦ,
∂B
∗
where x∗ is a place holder for l1,b
, R1∗ , p∗0 and k1∗ . Hence, a sufficient condition to establish
R ∗
∗
∂
the sign of ∂B
x dΦ is to determine the sign of ∂x
· 1[d∗1 < 0]. Total differentiation of the
∂B
optimality conditions (1), (2) and (3) of the financially constrained firms yields the following
33
This is equivalent to apply the implicit function theorem.
58
linear system
κ1
κ2
∂R1∗
∂l1,b
−α(α − 1)E [z ]k ∗α−2 1
0 1 1
−1
ρβ
· κ2
0
B
∂k∗
1
Hence, the GEE implies
∂l1,b
∂B
∂R
1∗ = 0 .
∂B
∗
∂l1,b
∗
−l1,b
∂B
Note first that equation (2), for firms with d0 < 0, implies R1∗ =
∂R1∗
0
1−λ0 d0
ρβ
>
1
ρβ
for λ0 > 0.
< 0 for financially constrained firms. Hence, by concavity of
∂R∗
λ0 (α−2)
the production function and since 0 < α < 1, κ1 = ∂l1,b1 α(α−1)E
∗α−1 < 0. It also follows
0 [z1 ]k1
that κ2 = l∗1 α(α−1)Eλ0[z ]k∗α−2 − ρβ < 0. The determinant of the matrix is therefore:
0
1,b
D=
1
1
∂R1∗
∂R1∗
κ2 + κ1 B + κ2 α(α − 1)E0 [z1 ]k1∗α−2 B −
κ2 α(α − 1)E0 [z1 ]k1∗α−2 ρβ.
∂l1,b
∂l1,b
Direct inversion yields:
∂k∗
1
∂R1∗
κ l∗
∂l1,b 2 1,b
∂B
∂R
∂R∗
∗α−2 ∗
1∗ =
1
· D−1 .
κ
α(α
−
1)E
[z
]k
l
0 1 1
1,b
∂B ∂l1,b 2
∗
∂l1,b
∗α−2
∗
−l1,b (κ1 + κ2 α(α − 1)E0 [z1 ]k1 )
∂B
Note that if κ1 + κ2 α(α − 1)E0 [z1 ]k1∗α−2 > 0, then we can conclude that:
and
∗
∂l1,b
∂B
∂k1∗
∂B
> 0,
∂R1∗
∂B
< 0. This is equivalent to showing:
1 − ρβR1∗
λ0 (α − 2)
λ0
1
∗α−2
> 0.
∗α−1 + ∗ − ∗ ρβα(α − 1)E0 [z1 ]k1
∗
ρβl1,b α(α − 1)E0 [z1 ]k1
l1,b l1,b
Note that equation (1) of the optimalities of the constrained firm can be rewritten as
λ0
1 − ρβR1∗
1 − ρβR1∗
1
=
ρβ
+ λ0 .
∗
∗
ρβl1,b
ρβl1,b
α(α − 1)E0 [z1 ]k1∗α−1 k1∗−1
Using this equivalence the want to show can be rewritten as
1 − ρβR1∗
λ0 (α − 2)
λ0
1
∗α−2
∗α−1 + ∗ − ∗ ρβα(α − 1)E0 [z1 ]k1
∗
ρβl1,b α(α − 1)E0 [z1 ]k1
l1,b l1,b
1 − ρβR1∗
λ0
1
= ρβ
(α − 2)k1∗−1 + λ0 (α − 2)k1∗−1 + ∗ − ∗ ρβα(α − 1)E0 [z1 ]k1∗α−2 > 0.
∗
ρβl1,b
l1,b l1,b
59
<0
∗
Multiply everything by l1,b
> 0, to get:
(1 − ρβR1∗ )(α − 2)k1∗−1 + λ0 (α − 2)
l1∗
+ λ0 − ρβα(α − 1)E0 [z1 ]k1∗α−2 > 0.
∗
k1
Hence, use equation (3) of the optimalities of the constrained firms to back out an expression
∗
in function of R1∗ and k1∗ , and rewrite
for l1,b
(1 − ρβR1∗ )(α − 2)k1∗−1 + (α − 2)
1 − ρβR1∗ − λ0 (z0 k0α + (1 − δ)k0 − k1∗ )
+ λ0 − ρβα(α − 1)E0 [z1 ]k1∗α−2 > 0.
Bk1∗
The left-hand side can be rearranged as
(α − 2)
(1 − ρβR1∗ )(B + 1) − λ0 (z0 k0α + (1 − δ)k0 − k1∗ )
+ λ0 − ρβα(α − 1)E0 [z1 ]k1∗α−2
∗
Bk1
= (α − 2) [(1 − ρβR1∗ )(B + 1) − λ0 (z0 k0α + (1 − δ)k0 − k1∗ )] + λ0 Bk1∗ − ρβα(α − 1)E0 [z1 ]Bk1∗α−1 .
Divide by (α − 2) < 0 (changing sign because it is always negative), the previous want to
show is equivalent to show
(1 − ρβR1∗ )(B + 1) − λ0 (z0 k0α + (1 − δ)k0 ) +λ0 k1∗
{z
} |
{z
}
|
<0 when d0 <0
>0
α−2+B
α−1
E0 [z1 ]Bk1∗α−1 < 0.
− ρβα
α−2
α−2
| {z } |
{z
}
>0
<0 if B>1
Consider two cases. If B > 1 (oligopoly), this last inequality is always satisfied. For B = 1
(monopoly), the inequality collapses to
α−1
< 0.
(1 − ρβR1∗ )2 − λ0 (z0 k0α + (1 − δ)k0 ) + (λ0 k1∗ − ρβR1∗ )
|
{z
} |
{z
} |
{z
}α−2
|
{z
}
>0
<0
<0 when d0 <0
>0
Finally, rearrange the Euler ρβR1∗ = 1 − λ0 d0 to get
∗
) − 1,
λ0 k1∗ − ρβR1∗ = λ0 (z0 k0α + (1 − δ)k0 + l1,b
60
which yields the result
ρβR1∗ )2 +
(1 −
|
{z
}
<0 when d0 <0
α−1
α−1
∗
− 1 λ0 (z0 k0α + (1 − δ)k0 ) + λ0 l1,b
−1
< 0.
|
{z
} | {z } α − 2
α−2
| {z }
>0
<0
>0
Proof. Statements 4, 5 and 6. Following a similar logic as the one of equation (5) the proof
focuses in studying the signs of the optimal choices of the financially constrained firms.
∗
Hence, all equations that follow refer to those firms such that d0 (k0 , z0 , k1∗ , l1,b
) < 0.
For the expected return of the shares, equating the Eulers for loans and the price of
h ∗i
d
shares provides the following non-arbitrage condition E0 p∗1 = R1∗ . Hence, for financially
0
h ∗i
d
∂R∗
∂
constrained firms ∂B
E0 p∗1 = ∂B1 < 0, which is always negative by previous result.
0
Before studying the effect of the number of banks on leverage, first note that the effect
on total debt is ambiguous:
∗
∂l1,b
∂
∗
∗
B · l1,b
=B
+ l1,b
.
∂B
∂B
As shown previously, as the number of banks increases l1,b decreases. Plugging the formula
for
∗
∂l1,b
∂B
found previously can resolve this ambiguity:
∂
κ1 + κ2 α(α − 1)E0 [z1 ]k1∗α−2
∗
∗
∗
B · l1,b = l1,b 1 − B
= l1,b
1 −
∂B
D
since
∗
l1,b
> 0 and
∗
∂R1
κ
∂l1,b 2
(1−α(α−1)E0 [z1 ]k1∗α−2 ρβ )
Bκ1 +Bκ2 α(α−1)E0 [z1 ]k1∗α−2
> 0 =⇒
∂
B
∂B
1
1+
∗
∂R1
κ
∂l1,b 2
1−α(α−1)E0 [z1 ]k1∗α−2 ρβ
(
Bκ1 +Bκ2 α(α−1)E0 [z1 ]k1∗α−2
∗
· l1,b
> 0.
In order to prove that the leverage increases with the number of banks, it remains to
show that the following inequality is always satisfied for the financially constrained firms:
∗
∂ B · l1,b
=
∂B k1∗
∗
∂l1,b
1
B · l1,b ∂k1∗
∗
B
+ l1,b
−
·
∂B
k1∗
k1∗2
∂B
∗
l1,b
∂R1∗
∗α−2
∗
κ
Bκ
+
Bκ
α(α
−
1)E
[z
]k
−
B
∗
1
2
0 1 1
l1,b
k1 2 ∂l1,b
> 0.
= ∗ 1 −
k1
D
61
,
)
∂R∗
∗
Since l1,b
/k1∗ > 0, D > 0 and κ2 ∂l1,b1 > 0, this is equivalent to show:
−B
∗
∗
l1,b
l1,b
∂R1∗
∂R1∗
∂R1∗
∗α−2
ρβ
⇐⇒
−B
<
κ
−
κ
α(α
−
1)E
[z
]k
κ
< 1 − α(α − 1)E0 [z1 ]k1∗α−2 ρβ,
2
2
0 1 1
2
k1∗ ∂l1,b
∂l1,b
∂l1,b
k1∗
l∗
which is always true since −B k1,b∗ is always negative and 1 − α(α − 1)E0 [z1 ]k1∗α−2 ρβ is always
1
positive.
Proof. Statements 7, 8, 9 and 10. For statements 7, 8, 9 and 10, I assume that the mass of
financially constrained firms 1 − P are all ex-ante identical. For TFP, the want to show is
∂
∂B
E [k1∗α ]
(E [k1∗ ])α
=
h
i
∂k∗
αE k1∗α−1 ∂B1
(E [k1∗ ])α
−α
E[k1∗α ]E
h
∂k1∗
∂B
i
(E [k1∗ ])α+1
> 0.
This is equivalent to show that
∗
∗
∂k1
∗α−1 ∂k1
∗
∗α
E k1
E [k1 ] − E[k1 ]E
> 0.
∂B
∂B
Which is again equivalent to
k1∗α−1
∂k1∗
∂k ∗
(1 − P)(k1∗ (1 − P) + k̄P) − (k1∗α (1 − P) + k̄ α P) 1 (1 − P) > 0
∂B
∂B
⇐⇒ k1∗α−1 (k1∗ (1 − P) + k̄P) − (k1∗α (1 − P) + k̄ α P) > 0
⇐⇒ k1∗α (1 − P) + k1∗α−1 k̄P − k1∗α (1 − P − k̄ α P > 0
⇐⇒ k1∗α−1 k̄P − k̄ α P > 0
⇐⇒ k1∗α−1 > k̄ α−1 .
Since k1∗ < k̄, the last inequality is always verified.
For the dispersion of capital, the want to show is
∂ ∗
∂
∂ ∗
∗ 2
∗ 2
∗ 2
E (k1 − E [k1 ]) = E
(k − E [k1 ]) |d0 < 0 (1 − P) + E
(k̄1 − E [k1 ]) |d0 ≥ 0 P < 0,
∂B
∂B 1
∂B
where P is the mass of the firms not financially constrained and κ̄ is the optimal choice of
62
capital of the non financially constrained firms. Hence:
∂ ∗
∂ k̄
∂k ∗ ∂k ∗
E (k1 − E [k1∗ ])2 |d0 < 0 = 2(k1∗ − k1∗ (1 − P) − k̄P) 1 − 1 (1 − P) −
P
∂B
∂B
∂B
∂B
|{z}
=0
∂k ∗
= 2P(k1∗ − k̄) 1 P < 0.
∂B
Note that the last inequality follows from the fact that k1∗ < k̄, otherwise the mass of firms
1 − P would not be financially constrained.
∂k1∗
∂B
> 0 is positive from the previous proof. Note
that the second term is always negative
∂
∂
∗
∗ ∂ k̄1
< 0.
E
(k̄1 − E [k1∗ ])2 |d0 ≥ 0 = 2E
)
−
E
[k
(
k̄
−
E
[k
]
|d
≥
0
]
1
0
1
1
| {z } ∂B ∂B
∂B
|{z} | {z }
>0
=0
>0
Furthermore, note that R1∗ = E0 [1 + αz1 k1∗α−1 − δ] and
σ(R1∗ ) = σ 2 1 + αE0 [z1 ]k1∗α−1 − δ = α2 E20 [z1 ]σ k1∗α−1 .
Hence:
∂σ 2 k1∗α−1
∂σ 2 (R1∗ )
= α2 E20 [z1 ]
∂B
∂B
∗
α−1
∗α−1 ∂ k̄1α−1 ∂E k1∗α−1
∗α−2 ∂k1
∗α−1
α−1
|d0 ≥ 0 < 0.
(k̄
(α
−
1)
−
k
P
+
2E
−
E
k1
)
= α2 E20 [z1 ]
2P
(k
−
k̄
)
1
1
| 1
∂B
{z
} | {z }
|
{z
} | ∂B
{z } | ∂B
{z
}
<0
>0
<0
=0
<0
Equating the two Euler equations for the price of the shares of the firms and the price of
h ∗i
d
∂R∗
∂
the bonds yields: ∂B
E p∗1 = ∂B1 < 0.
0
63
@FedFRASER