The effects of capital requirements on good and bad risk-taking.
Date: 2020-06-05
Institutions: European Systemic Risk Board
Subjects: EU banking union, economic model, financial regulation, financial requirements, financial risk, type of business

Working Paper Series 

No 104 / November 2019 

The effects of capital requirements 
on good and bad risk-taking 

by 
N. Aaron Pancost 
Roberto Robatto 

 
 
 
Abstract

We study optimal capital requirement regulation in a dynamic quantitative model

in which nonfinancial firms, as well as households, hold deposits. Firms hold deposits

for precautionary reasons and to facilitate the acquisition of production inputs. Our

theoretical analysis identifies a novel general equilibrium channel that operates through

firms’ deposits and mitigates the cost of increasing capital requirements. We calibrate

our model and find that the optimal capital requirement is 18.7% but only 13.6% in a

comparable model in which only households hold deposits. Our novel channel accounts

for most of the difference.

Keywords: deposit insurance, capital requirements, idiosyncratic risk, safe assets
JEL Classifications: E21, G21, G32

1

1 Introduction

After the 2007–2008 financial crisis, the revision of the regulatory framework of financial
intermediaries has become a central topic of discussion by regulators and academics. The
Basel III accord has tightened bank regulation with the aim of reducing the likelihood and
depth of financial crises. One of the key sets of rules at the center of this debate is capital
requirements, namely, limits on the fraction of debt that banks can use to finance their
investment. Appealing to the Modigliani and Miller (1958) theorem, Admati and Hellwig
(2013) argue that capital requirements should be raised even further to eliminate banks’
bad risk-taking, namely, the moral hazard induced by government guarantees and implicit
too-big-to-fail subsidies.1 One argument against tighter capital requirements is that banks
are special because their liabilities are valued for their safety and liquidity (DeAngelo and
Stulz, 2015). Under this view, raising capital requirements reduces excessive risky lending
by banks but at the cost of reducing the supply of valuable bank deposits.

A recent literature analyzes these trade-offs in a quantitative, general equilibrium frame-
work. While most analyses use models in which only households hold deposits (Van den
Heuvel, 2008; Davydiuk, 2017; Begenau, 2018), deposits in practice are held not only by
households but also by firms.

In this paper, we ask whether and to what extent accounting for the deposits held by
nonfinancial firms affects the determination of the optimal capital requirement. We derive
a general equilibrium model in which—similar to the literature—government guarantees
induce banks to take excessive risks. Capital requirements limit such a risk, but they also
reduce the supply of deposits. Crucially, though, the cost of reducing the supply of deposits
depends not only on how households value deposits but also on how the lower supply affects
firms’ behavior.

We find that the optimal capital requirement is substantially higher than in comparable
models with only households’ deposits. This result arises from a novel general equilibrium
effect related to the role played by firms’ deposits. Firms are subject to idiosyncratic risk
and hold deposits for precautionary reasons. In particular, deposits promote risky but so-
cially valuable production by facilitating the acquisition of production inputs. We refer to
the channel by which deposits stimulate firms’ activities as good risk-taking.
If financial
regulation is tightened and the availability of deposits decreases, firms will demand fewer
inputs. However, in general equilibrium, input prices will decrease, partially offsetting firms’
desire to hire fewer inputs. This additional effect mitigates the direct costs of increasing

1Some regulators have made a case for similar rules as well; see, for example, the Minneapolis Plan

discussed by Kashkari (2016).

2

capital requirements, thereby raising the optimal requirement relative to other quantitative
models that omit firms’ deposits.

The impact of our channel is quantitatively relevant. After presenting a baseline model
to describe the results, we extend it to include several quantitative features. The optimal
capital requirement in the quantitative model is 18.7% and thus substantially higher than
in many models in the literature. Importantly, the channel we highlight is large even in the
context of our model, in which some features are different from those in many related papers.
To make this argument, we solve for the capital requirement that would be chosen under two
alternative formulations: one in which all the deposits are held by households, and another
one in which both households and firms hold deposits but we shut down our novel general
equilibrium channel. When only households hold deposits, the optimal capital requirement
is 13.6%, that is, 5.1 percentage points lower than in the full model. Our novel channel
accounts for a large part of the difference: the capital requirement that maximizes welfare
when both households and firms hold deposits but we shut down our channel is 15.3%.

Aside from our novel approach at modeling the liquidity value of deposits for firms, other
aspects of our work build heavily on the existing literature that quantifies optimal capital
requirements. We follow the pioneering approach of Van den Heuvel (2008) in embedding the
analysis of capital requirements regulation in a dynamic, quantitative, general equilibrium
model, and some elements of both our theoretical and quantitative analyses are based on
Van den Heuvel (2008), Davydiuk (2017), and Begenau (2018).2

The firms’ side of our model builds on a framework introduced by Quadrini (2017), in
which deposits are held by firms subject to uninsurable idiosyncratic shocks to the produc-
tivity of their employees. The wage bill must be paid independently of the realization of the
shocks, and thus the productivity risk is borne by the firm. The effect of these shocks is to
reduce the firm’s labor demand relative to an economy without idiosyncratic shocks. Because
deposits are safe, they reduce the volatility of firms’ cash flows, so that a high availability
of and return on deposits increases firms’ willingness to hire workers. We call this channel
“good risk-taking” because increasing labor demand is in general socially valuable but not
fully exploited because of the idiosyncratic risk. The mechanism by which more cash avail-
able to a firm increases its labor demand is in line with the causal evidence in Benmelech,
Bergman and Seru (2015), who show that firms with more long-term debt maturing in any
given year—and thus likely to have less cash available—reduce their labor force by more

2In addition, we abstract from some features that have been studied in other papers, such as the in-
teractions with the shadow banking system (Begenau and Landvoigt, 2017; Dempsey, 2017), the possible
optimality of time-varying capital requirements (Davydiuk, 2017; Malherbe, 2017), the effect of legacy assets
(Bahaj et al., 2016; Bahaj and Malherbe, 2018), and the interaction with economic growth (Nguyễn, 2014).

3

than their peers.3

Tighter capital requirements reduce firms’ labor demand in the model; however, the
ultimate impact on welfare depends on what happens to the equilibrium level of employment,
which is affected not only by labor demand but also by labor supply. To clarify this point,
we analyze an extreme case of our model in which labor is in fixed supply. In this case, a
drop in labor demand reduces wages but, because employment is fixed, does not affect the
component of welfare that depends on the labor market.4 Under a proper calibration of
the key labor market parameters, tightening capital requirements reduces both wages and
employment. Nonetheless, our main message is unaltered because the drop in wages lowers
the wage bill of firms, thereby mitigating the welfare cost of the higher capital requirements.
That is, a bad idiosyncratic shock does not preclude a firm from financing future projects
even if it has fewer deposits.

We argue that government guarantees and capital requirements are important for firms’
deposits.
Indeed, even if some firms might have high deposit balances, the government
has been covering deposits and other bank liabilities well beyond the deposit insurance
limit. For instance, Veronesi and Zingales (2010) calculate that Paulson’s equity infusion
plan implemented during the Columbus Day weekend of October 2008—at the height of
the financial crisis—entailed a transfer of $21-$44 billion from taxpayers to the holders of
liabilities of some of the largest US banks. This is a large number when compared to the $90
billion borne by the Deposit Insurance Fund between 2007 and 2013 (Davison and Carreon,
2010) and is likely a lower bound on the overall taxpayer-subsidized transfers because of the
numerous interventions at that time.

Capital requirements in our model limit the negative impact of subsidized deposit in-
surance. More generally, though, deposit insurance can also be interpreted as any explicit
or implicit government guarantee on bank debt (e.g., the Temporary Liquidity Guarantee
Program set up by the Federal Deposit Insurance Corporation in 2008 and the too-big-to-fail
guarantee that resulted in the Paulson plan studied by Veronesi and Zingales, 2010). We
follow a common approach in the literature that imposes deposit insurance and motivates it
with its role in preventing runs, as in Diamond and Dybvig (1983), but does not explicitly
include runs nor analyze the optimality of such a policy.5

3Benmelech, Bergman and Seru (2015) build on the approach of Almeida et al. (2012), that is, on the
assumption that cross-sectional variation in long-term debt maturing at any point in time is exogenous to
corporate outcomes at that time.
Indeed, variations in maturing long-term debt among firms are likely
exogenous to market conditions and investment opportunities because the portion of maturing debt was
issued prior to the year of maturity.

4Total welfare is affected, though, because changing capital requirements alters the bad risk-taking of

banks even when labor is in fixed supply.

5We note that fully eliminating deposit insurance might not be optimal in our framework because it would

4

We motivate firms’ aversion to idiosyncratic risk with an agency problem, following a
literature that has highlighted the importance of these frictions (Panousi and Papanikolaou,
2012; Glover and Levine, 2015, 2017). Firms are owned by households that can fully diversify
away their exposure to idiosyncratic risk by holding a well-diversified portfolio of equity.
However, each firm is run by a manager who holds only an equity stake in the firm she
manages and thus is exposed to the idiosyncratic risk of the firm. As a separate contribution
of the paper, we introduce some modeling assumptions that make the model tractable and
allow us to ignore managers’ consumption when performing welfare analysis, without altering
In addition, even if idiosyncratic shocks create
the implications of the agency friction.
heterogeneity across firms, the equilibrium in our model depends only on aggregate firms’
wealth, and other moments of the firm size distribution are irrelevant. As a result, we can
easily aggregate firms’ behavior, maintaining the main focus on financial regulation and
solving the model in general equilibrium. This last feature is particularly important for
our novel channel; richer structural corporate finance models with agency frictions, such as
Nikolov and Whited (2014), are instead typically solved in partial equilibrium.

Another implication of our model is related to the measurement of the welfare cost of
In a model with deposits in the utility function, Van den Heuvel
capital requirements.
(2008) shows that the welfare cost can be computed using a simple formula that depends
only on variables that are easily observable. In particular, the liquidity premium—that is,
the spread between deposit rates and rates on similarly low-risk but less liquid investments—
plays a prominent role in that formula. We show that an additional key input is required to
compute the welfare cost in our model, namely, how firms’ input prices respond to changes
in regulation. This element is more difficult to quantify in the data without appealing to
the structure and simulation of a model.
In our theoretical analysis, we emphasize two
extreme parameterizations: one in which the deposit premium plays the most prominent
role in assessing the welfare cost of capital requirements, as in Van den Heuvel (2008), and
one in which it provides no information at all.6

2 Literature review

A recent and growing literature has embedded the analysis of capital requirements into quan-
titative general equilibrium models. This literature includes Van den Heuvel (2008), Chris-

make deposits unsafe, thereby increasing rather than reducing the volatility of firms’ cash flows. We leave
the joint determination of optimal deposit insurance and capital requirement regulation to future research.
6In the macro-monetary literature, some papers show that liquidity premia might not be sufficient statis-
tics for policy analysis. For instance, Li, Rocheteau and Weill (2012) show that assets with the same liquidity
value can have different liquidity premia.

5

tiano and Ikeda (2013), Corbae and D’Erasmo (2014), Elenev, Landvoigt and Van Nieuwer-
burgh (2018), Nguyễn (2014), Gertler, Kiyotaki and Prestipino (2016), Begenau and Land-
voigt (2017), Davydiuk (2017), Dempsey (2017), and Begenau (2018). Egan, Hortaçsu and
Matvos (2017) develop a structural model of the US banking sector and argue that capital
requirements should be at least 18% to avoid bank runs. Most of the literature, however,
Indeed,
tends to be qualitative or to abstract from general equilibrium (Thakor, 2014).
several papers, such as Berger and Bouwman (2013), Mehran and Thakor (2011), and Dia-
mond and Rajan (2001) provide results about the role of bank capital and changes in capital
requirements that are identified in the cross section, abstracting from general equilibrium
effects.

A related paper by Allen et al. (2018) studies the effect of government guarantees on
banks’ choices of liquidity and investments in risky projects. Both our paper and that of
Allen et al. (2018) point out that higher risk-taking can be a good consequence of financial
regulation. In their paper, more risk-taking by banks is associated with greater liquidity
In our paper, more risk-taking by firms has a positive impact, in general, on
provision.
employment and output.

Our approach for modeling firms’ risk builds on Quadrini (2017), who also emphasizes
the role of bank liabilities for insurance purposes.7 There are, however, two main differences.
First, he focuses mainly on banks’ risk-taking and crises, whereas our focus is on the impact
of financial regulation on nonfinancial firms’ risk-taking.8 Second, we extend his model so
that we can reinterpret the agents subject to idiosyncratic risk as firms run by managers who
cannot diversify away idiosyncratic risk because of an agency friction. This extension facili-
tates the welfare analysis and the comparison of the model with the data, and it represents
a separate contribution of our paper. Indeed, the firms’ building block of our model can be
used to study other questions at the intersection of corporate finance and macroeconomics.
This paper is also related to a literature that studies financial intermediaries as suppliers
of safe assets, such as Stein (2012), Magill, Quinzii and Rochet (2016), and Diamond (2016).
This literature builds on the ideas of Gorton and Pennacchi (1990) and Dang et al. (2017),
in which the debt of banks is riskless to enhance its liquidity value or to overcome an in-
formational friction. Bank debt is valuable in our model for a related but slightly different
reason: there is a demand for securities that are uncorrelated with the idiosyncratic risk of
firms. Some of these papers also perform policy analyses in models with only households’
deposits. While their theoretical results are not affected by our novel channel, the corre-

7The approach used by Quadrini (2017) at modeling firms’ risk builds on Arellano, Bai and Kehoe (2011).
8Dindo, Modena and Pelizzon (2018) derive a demand for banks’ liabilities for insurance purposes similar
to Quadrini (2017) and ours (although in a continuous-time model), but the regulation of intermediaries
that they study always reduces welfare.

6

sponding quantitative policy stance might need to be tilted by imposing stricter limits on
financial intermediaries.

Our paper relates to a large literature on firms’ exposure to idiosyncratic risks. Smith and
Stulz (1985) and Froot, Scharfstein and Stein (1993) derive theories in which firms do not
completely hedge away idiosyncratic risks because external financing is costly. Minton and
Schrand (1999) provide empirical evidence that firms that are more exposed to idiosyncratic
risk invest less and have higher costs of external financing than other firms, consistent with an
inability or unwillingness to completely insure against idiosyncratic shocks. In this setting,
idiosyncratic shocks are costly to the firm. In addition, we build on the results of Panousi
and Papanikolaou (2012) and Glover and Levine (2015, 2017), and we assume that firms are
run by managers that are subject to the idiosyncratic risk of the firm they run. Thus, as
pointed out by Panousi and Papanikolaou (2012), risk aversion of managers plays a role in
shaping firm decisions. In our model, this implies that deposits provide insurance against
the idiosyncratic risk, consistent with the finding of Bates, Kahle and Stulz (2009) of a
rising trend in firms’ cash holdings since the 1980s, which they attribute in large part to an
increasing precautionary motive.

More generally, our work is part of a broader literature that relates financial interme-
diaries, firms’ idiosyncratic risk, and the labor market. Bacchetta, Benhima and Poilly
(forthcoming) show a negative relation between corporate cash holdings and employment in
the data, which they rationalize with external liquidity shocks to firms. Bigio (2010) and
Jermann and Quadrini (2012) study how firms’ financial frictions affect the macroeconomy,
including labor demand. Donaldson, Piacentino and Thakor (forthcoming) analyze a model
in which households borrow from banks and the possibility of default interacts with the
firms’ output and employment.

3 Model

3.1 Environment

Time is discrete and infinite. There are four main players in the economy: firms run by
managers, banks, households, and the government. There is also a bank-financed production
sector, representing the assets in which banks invest, but it will play a minor role. Before
presenting the details of the model, we briefly summarize the main structure.

We follow Quadrini (2017) in assuming that firms are subject to idiosyncratic risk and
hold deposits for precautionary reasons, but we depart somewhat from his model. In partic-
ular, each firm is run by a risk-averse manager who acts in her own interest because of an

7

agency friction. This friction motivates the manager’s holding of an undifferentiated stake in
the firm she runs, and thus an exposure to firm-specific idiosyncratic risk. In contrast, firms’
owners (i.e., households) fully diversify away the idiosyncratic risk by holding equity stakes
in all firms. These assumptions are motivated by the results of Panousi and Papanikolaou
(2012), who document that managerial compensation affects firms’ investments in response
to idiosyncratic risk, and Glover and Levine (2015, 2017), who calibrate structural models of
firm investment and show that managers respond more to firm-specific shocks in comparison
to what shareholders would choose in the absence of agency frictions.

We model firms using a simpler approach than in typical structural corporate finance
models with agency frictions, such as Nikolov and Whited (2014), for two reasons. First,
our approach allows us to keep the firm side of the model tractable and focus on financial
regulation. Second, we can easily solve the model in general equilibrium, which is crucial for
our results. Many papers in the corporate finance literature model firms’ dynamics in rich
partial equilibrium settings, and extending those frameworks to general equilibrium would
cloud our message.

Banks collect deposits from firms, invest in a bank-financed sector, and are subject to
idiosyncratic shocks that make a fraction of them insolvent. Deposits at insolvent banks are
made whole by the government through deposit insurance.

Households consume, supply labor, and own firms and banks. We will impose some
parameter restrictions so that the objective of financial regulation will be solely that of
maximizing the welfare of households, without any consideration for the welfare of managers.
We first convey our results in a simple model that includes only the main elements. In
particular, the simple model has no aggregate risk, linear utility from consumption, exoge-
nous risk of banks’ investments, an ad hoc rule to determine firms’ dividends, and no deposits
held by households. We relax these assumptions in the quantitative model of Section 5.

3.1.1 Firms and managers

There is a continuum of firms that are owned by households. Each firm is run by a manager
who behaves in her own interest because of an agency friction and owns an equity stake in
the firm.9

The role of the manager is to choose the amount of labor li

the firm. The total output produced by the firm is zi
The productivity zi

t that is hired every period by
t+1 is the firm’s productivity.
t+1 is subject to an idiosyncratic shock realized at the beginning of t + 1.

t, where zi

t+1li

9The contract between the firm and the manager is imposed exogenously, but we could derive it endoge-
nously as the best contract that makes the manager unwilling to divert resources away from the firm for
personal use. However, this would complicate the exposition and would not affect the results.

8

t+1 is realized after the manager has chosen the labor input li

Crucially, zi
t and has committed
to pay the wage bill wtlt, where wt is the wage. For future reference, we will denote (cid:22)z to be
}
the average realization of the shock: Et

zi
t+1
In addition, the firm holds deposits di
t at time t in the banking system, which earn a
t at t + 1. The government provides full deposit insurance, so deposits are safe.
t depends on the

return Rd
At time t, the manager takes the deposits di
history of managers’ choices and shocks realized before time t.

t as predetermined. That is, di

= (cid:22)z.

{

At the beginning of t + 1, the total wealth available to the firm is given by

(

)

xi
t+1 =

zi
t+1

(cid:0) wt

t + Rd
li

t di
t;

(1)

where the first term on the right-hand side denotes the profits obtained by hiring workers
(which can be negative if the productivity shock zi
t+1 is low) and the second term denotes the
gross return on deposits. Thus, if the realization of the productivity shock zi
t+1 is low—in
particular, if it is lower than the wage wt—the firm must tap its deposit balance, Rd
t, to
pay its workers.

t di

A fraction (cid:11) of the wealth xi

t+1 is paid out as dividends, and the remaining fraction 1 (cid:0) (cid:11)
is retained by the firm, where (cid:11) is an exogenous parameter (we will endogenize (cid:11) in the full
model of Section 5). Thus, deposits evolve according to di

t+1 or, using (1),

[(

)

t+1 = (1 (cid:0) (cid:11)) xi
]

t+1 = (1 (cid:0) (cid:11))
di

zi
t+1

(cid:0) wt

t + Rd
li

t di
t

:

(2)

In this simple model in which (cid:11) is exogenous, all results are unchanged if we make such a
parameter time varying and firm specific, but we instead keep it constant to simplify the
exposition.10 Note that (cid:11) is not chosen by the manager and, in this sense, can be understood
as a part of the contract between the manager and the shareholders; we return to this point
in Section 5 when we endogenize the choice of (cid:11).

We emphasize that the manager’s choice of labor li

t is actually a joint decision of how
many workers to hire and how many deposits to hold at t+1, subject to equation (2). Indeed,
given prices, equation (2) shows that the choice of li

t determines deposits at di

t+1.

We parameterize the equity stake of the manager by (cid:20), so that the manager’s dividends
t+1. The

t+1 and dividends paid to shareholders (i.e., to households) are (cid:11)(1 (cid:0) (cid:20))xi

are (cid:11)(cid:20)xi

10More precisely, we could replace (cid:11) with (cid:11)i

∫

t, subject to the restriction that the average value is the same

in the cross section, that is,

(cid:11)i

tdi = (cid:11) for all t.

9

manager consumes all her dividends each period,11 so her consumption ci

t+1 is

[(

)

]

ci
t+1 = (cid:11)(cid:20)

zi
t+1

(cid:0) wt

t + Rd
li

t di
t

;

(3)

which produces a utility value log ci
given by

t+1. Dividends paid to shareholders (i.e., households) are
]

[(

)

t+1 = (cid:11)(1 (cid:0) (cid:20))
(cid:25)i

zi
t+1

(cid:0) wt

t + Rd
li

t di
t

:

We can thus formalize the problem of the manager. The manager solves

(

)

V m
t

di
t

= max
t+1; di
t; ci
li

t+1

{

(

)}

(cid:12)mEt

log ci

t+1 + V m
t+1

di
t+1

(4)

(5)

subject to (2) and (3), and where (cid:12)m 2 (0; 1) is the discount factor of the manager and Et
denotes the expectation with respect to the idiosyncratic productivity shock zi

t+1.

Since the manager hires workers li

t before the realization of the productivity shock zi

and thus chooses the wage bill wtli
tuates over time. For any given choice of li
t+1, dividends (1 (cid:0) (cid:11)) xi
xi
versa.

t before knowing zi

t, a high value of zi

t+1, and manager’s consumption ci

t+1,
t+1, the manager’s consumption fluc-
t+1 implies that firms’ wealth
t+1 will be high as well, and vice

Crucially, the manager has the ability to control the volatility of her consumption ci

t+1
through her choice of li
In particular, her consumption volatility is increasing in labor.
t.
In principle, the manager can choose li
t = 0, which would imply deterministic consumption
t+1 = (cid:11)(cid:20)Rd
ci
t > 0, and
thus be exposed to idiosyncratic risk. The return on deposits and the supply of deposits from
banks, both of which are related to capital requirements regulation, will affect the manager’s
desired exposure to such idiosyncratic risk.

t; however, the manager will find it optimal to hire some workers, li

t di

The following proposition characterizes the optimal choice of managers. It shows that
firm i’s labor demand, li
t. Thus, although individual firms
will grow and shrink as they receive different sequences of idiosyncratic shocks, we can solve
for the equilibrium by easily aggregating the firms’ building block of the model because the
firm-size distribution does not affect aggregate variables. All proofs are in Appendix A.

t, is proportional to its deposits di

Proposition 1. (Managers and firms’ labor demand) Manager i’s optimal choice of labor is

11This assumption is without loss of generality because of some parameter restrictions that we will impose

in Section 3.3.

li
t = ϕtdi
t;

(6)

10

where ϕt is independent of di

t and satisfies the following first-order condition:

{

0 = Et

(

zi
t+1

(cid:0) wt
)

zi
t+1
(cid:0) wt

ϕt + Rd
t

}

;

(7)

The first-order condition (7) governs firms’ decisions to hire workers. The firm’s labor
demand is lower in comparison to what it would be in an economy in which the manager
is able to diversify away her exposure to the firms’ idiosyncratic risk. That is why we refer
to the willingness of firms to hire workers as the “good” risk-taking decision, because any
force that creates an incentive for firms to hire more workers brings the economy closer to
the first best.

The result of Proposition 1 implies that our model is consistent with some evidence
documented by the corporate finance literature. First, as in Benmelech, Bergman and Seru
(2015), our model generates a labor demand that is increasing in the safe financial assets
held by the firm. Indeed, Benmelech, Bergman and Seru (2015) document a causal, positive
effect of cash held by the firm (i.e., safe financial assets) on the firm’s labor force. Second,
as in Opler et al. (1999), riskier cash flow (i.e., higher variance of zi
t+1) implies that the firm
is willing to hold more cash. In addition, firms that do well (i.e., firms hit by a sequence of
good values of zi
t+1) accumulate cash internally, and firms that experience losses (i.e., firms
with bad realizations of zi

t+1) experience decreases in cash.

We emphasize that the result of Proposition 1 is independent of the equity stake (cid:20) of
the manager, provided that (cid:20) > 0. When solving for the equilibrium, we will exploit this
feature to simplify the welfare analysis. In particular, we will impose parametric restrictions
which imply that managers’ consumption is arbitrarily small and all dividends will be paid
to shareholders (i.e., we will assume that (cid:20) is arbitrarily small) while at the same time
preserving the implications of the agency friction on firms’ behavior (see Section 3.3).

We will derive our theoretical results under a general specification of the stochastic
t+1. However, in the simulations, we will use the following functional form:
t+1 = 0 with probability 1 (cid:0) pz. This implies

process for zi
t+1 = 1=pz with probability pz 2 (0; 1), and zi
zi
that (cid:22)z (cid:17) Et

will be normalized to one.

{

}

zi
t+1

3.1.2 Banks and bank-financed sector

Banks live for a single period: each bank is set up at time t and is liquidated at the beginning
of time t + 1.12 Each newly created bank receives an amount nt of net worth from its

12This assumption is made without loss of generality because our model does not include adjustment costs

on banks’ size nor costs to raise equity.

11

shareholders (i.e., households). Then, the bank collects deposits dt and uses the resources
nt + dt to purchase physical capital kt.
In the economy as a whole, capital accumulates
endogenously, similar to a standard real business cycle model.

At the beginning of t + 1, each bank is hit by an idiosyncratic quality shock ", with
Ef"g = 1. This shock captures the fact that some banks experience higher or lower losses
on their loans and investments in comparison to the average bank in the economy. Formally,
after the shock, the stock of capital held by a particular bank is "kt; the total stock of capital
in the banking sector as a whole is unchanged because the shock " is idiosyncratic. Banks
lend the physical capital "kt to a bank-financed sector, which then returns the undepreciated
fraction 1(cid:0)(cid:14) of the physical capital plus a return rt+1 to the banks. Thus, bank assets can be
interpreted as loans to some producers that rely on banks for financing—different from those
analyzed in Section 3.1.1—or as loans to other borrowers unrelated to production (e.g., home
mortgages). Banks’ profits are given by the cash flow "kt (1 (cid:0) (cid:14) + rt+1) net of the repayment
Rd
t is the gross return on deposits. Profits are bounded below by a
limited liability constraint: banks with negative profits pay zero to shareholders and default
on their depositors.

t dt to depositors, where Rd

Banks face a capital requirement (cid:16) that limits their ability to raise deposits. That is,

their equity ratio nt=kt must be weakly larger than the regulatory requirement (cid:16).

In this simple model, the risk of the idiosyncratic shock " is exogenous. However, there
is still a sense in which banks can increase their risk-taking to exploit the deposit insurance
subsidy: even though the risk per unit of capital is fixed, the total risk-taking of a bank
depends on the size kt of the bank’s balance sheet. In the equilibrium of our baseline model,
banks take advantage of the deposit insurance subsidy by expanding beyond what would
be socially optimal. Thus, deposit insurance increases the total amount of banks’ bad risk-
taking even though the risk per unit of capital is exogenous. In the quantitative model of
Section 5, banks will also be able to increase their level of risk per unit of capital, but the
qualitative results will be the same.

We can now formalize the problem of banks. Given nt, the bank’s problem is

∫ {

"kt (1 (cid:0) (cid:14) + rt+1) (cid:0) Rd

t dt

max
kt;dt

s.t.

}

+

dF (")

kt = dt + nt
nt (cid:21) (cid:16)kt

(8)

(9)

(10)

where (9) is the budget constraint, (10) reflects the capital requirement, f(cid:1)g+ = max f(cid:1); 0g is

12

(

the positive part of banks profits, and F ((cid:1)) is the CDF of banks’ idiosyncratic productivity
shocks. Our theory results hold for a generic F ((cid:1)), but for our simulations, we will assume
that " is log-normally distributed with mean E (") = 1 and variance (cid:27)2; that is, log " (cid:24)
N
. The deposits at banks that receive a low value of the productivity shock
", such that "kt (1 (cid:0) (cid:14) + rt+1) < Rd
t dt, are fully repaid to depositors thanks to the deposit
insurance intervention.

2 (cid:27)2; (cid:27)2

(cid:0) 1

)

When shareholders invest in banks’ equity, they invest in a mutual fund that diversifies
t+1 is given by

its holdings of equity over all banks in the economy. The return on equity Rn

Rn

t+1 =

1
nt

∫ {

"kt (1 (cid:0) (cid:14) + rt+1) (cid:0) Rd

t dt

}

+

dF (") :

(11)

Equation (11) implies that "t+1, the highest value of " at which banks default on their
depositors, is implicitly defined as

Rd

t dt = "t+1kt (1 (cid:0) (cid:14) + rt+1) :

(12)

Subsidized deposit insurance creates an incentive for banks to lower their lending rate
rt+1, thereby increasing the amount of physical capital in the economy. The logic of this
result is the usual combination of limited liability and a lack of responsiveness of the deposit
rate Rd
t because of deposit insurance. The overaccumulation of capital arising from the
distortion of subsidized deposit insurance entails a welfare loss.

From a mechanical perspective, the problem of banks can be solved as follows. Because in
equilibrium the return on deposits will be less than the return on banks’ capital, the capital
requirement (10) is always binding. As a result, banks’ technology is constant returns to
scale. Given nt, (10) can be used to compute the size kt of the banks’ assets, and then the
budget constraint (9) can be used to solve for deposits dt. Finally, the equilibrium value
of nt is determined such that (12) holds, given the return on equity demanded by banks’
shareholders in equilibrium.

To better understand the behavior of banks, we can combine (11), (9), and (10) to obtain

∫ 1

"t+1

" (1 (cid:0) (cid:14) + rt+1) dF (") = (cid:16)Rn

t+1 + (1 (cid:0) (cid:16)) Pr

{

" (cid:21) "t+1

}

Rd
t :

(13)

Equation (13) equalizes the benefits and costs of purchasing an extra dollar of physical capital
for a bank. The marginal benefit is the gross return " (1 (cid:0) (cid:14) + rt+1), only in the states in
which the bank is not in default, that is, for " (cid:21) "t+1. The marginal cost of financing the
purchase corresponds to the cost of raising more deposits and more equity. A fraction (cid:16) of

13

the purchase requires equity, for which households require a return Rn
t+1. The remaining
fraction 1 (cid:0) (cid:16) can be financed with deposits, which require a return Rd
t ; the bank internalizes
the cost of deposits only in the states of the world in which it remains solvent, which happens
with probability Pr

{

}

.

" (cid:21) "t+1

We close this section by describing the problem of the bank-financed sector. These
players rent capital kt from banks and use it for production, according to the production
function Ak(cid:13)
t , with (cid:13) 2 (0; 1) and A > 0. The output can be interpreted as production by
bank-financed firms or as housing services, if kt is interpreted as mortgages. The profits (cid:25)bf
t+1
(where bf stands for “bank financed”) are obtained by maximizing

(cid:25)bf
t+1 = max

kt

Ak(cid:13)
t

(cid:0) rt+1kt;

which implies the first-order condition

A(cid:13)k(cid:13)(cid:0)1

t = rt+1:

(14)

(15)

The profits (cid:25)bf
technology, and they are distributed lump sum to households.

t+1 are positive because bank-financed players use a decreasing returns to scale

3.1.3 Households

Households are infinitely lived agents with linear utility of consumption ct and convex disu-
tility of labor supply lt. This quasi-linear specification allows us to easily characterize their
choices, but we will use a more general utility function in the quantitative model of Section
5. Households supply the labor used by firms and the equity to banks, and they earn the
profits generated by the firms run by managers, the banks, and the bank-financed firms.

Households’ utility is given by

1∑

t=0

(cid:12)tu (ct; lt) =

2

(cid:12)t

4ct (cid:0) (cid:12)(cid:31)

3

5 ;

1+ 1
(cid:17)
l
t
1 + 1
(cid:17)

1∑

t=0

(16)

where (cid:31) > 0, (cid:17) > 0, and (cid:12) 2 (0; 1). Households choose labor at time t but do not provide
that labor until the beginning of period t + 1 when production actually occurs; thus we
discount the disutility of labor chosen at t with an additional (cid:12) in equation (16).

A household that starts with wealth at solves the problem

V h
t (at) = max
ct;lt;nt

ct (cid:0) (cid:12)(cid:31)

1+ 1
(cid:17)
l
t
1 + 1
(cid:17)

+ (cid:12)V h

t+1 (at+1)

14

subject to the budget constraint

ct + nt (cid:20) at

(17)

and to an upper bound on the amount of hours worked, lt (cid:20) (cid:22)l. Wealth at t + 1 is given by
∫

at+1 = wtlt + ntRn

t+1 +

t+1di + (cid:25)bf
(cid:25)i

t+1

(cid:0) Tt+1:

(18)

At time t, the household chooses how to allocate its wealth at between consumption ct and
investment in bank equity nt, and it chooses the labor supply lt taking as given the wage wt.
At t + 1, the wealth is the sum of its labor income wtlt, the gross return on banks’ equity
ntRn
t+1 distributed by firms and by the bank-financed
sector, net of lump-sum taxes Tt+1.13

t+1, and the profits

t+1di and (cid:25)bf
(cid:25)i

∫

The linear utility from consumption implies that households’ value function is linear as

well, that is,

V h

(

)′

(a) = 1. Thus, their labor supply curve is given by

wt = (cid:31) (lt)

1
(cid:17) ;

(19)

and the supply of banks’ net worth nt is fully elastic (i.e., they are willing to supply any
amount) provided that the return on equity Rn

t+1 satisfies

Rn

t+1 =

1
(cid:12)

:

(20)

Note that households’ choice of nt affects the accumulation of physical capital kt, given firms’
deposits

di
t di and banks’ budget constraint (9).

∫

3.1.4 Government

The government taxes households in order to ensure that depositors at failed banks at the
beginning of t+1 are made whole. The government seizes output at failed banks (who return
zero to their equity holders) to partially defray the expenses of paying back depositors.

The total amount of tax to be collected is

Tt+1 =

∫

"t+1

(cid:0)1

2

3

6
4

Rd
t dt
|{z}
owed to depositors

(cid:0) "kt (1 (cid:0) (cid:14) + rt+1)
|
}
{z
collected from banks

7
5 dF (") ;

(21)

13Note that we allow households to choose only investments in banks’ equity claims because that is a key
element of our model. In principle, we could enrich households’ portfolio choice by allowing them to trade
claims on firms’ dividends; however, that would not change any of the results and would simply add more
notation because of our representative household assumption.

15

t dt is the amount owed to depositors, and "kt (1 (cid:0) (cid:14) + rt+1) for " < "t+1 denotes the

where Rd
assets of failed banks.

3.2 Equilibrium definition and aggregation of firms’ decisions
}

{

Given initial conditions di
and
f"g, an equilibrium is a collection of firm policies, bank policies, household policies, and
government taxes such that

0 for all i and a0, and exogenous stochastic processes for

zi
t

1. Firms’ deposits di

t, managers’ consumption ci

t, and managers’ choice for labor demand

li
t solve (5);

2. Banks’ choices for capital kt and deposits dt solve their problem (8);

3. Households’ choices for supply of labor lt and net worth nt maximize their utility (16);

4. The government taxes households lump-sum and uses the proceeds to pay depositors

at failed banks according to equation (21);

5. The wage wt and the return on deposits Rd

t clear the labor and deposit markets,

respectively.

The aggregate resource constraint will hold by Walras’ law. Nonetheless, we state it to

clarify the mechanics of the model:

ct + kt (cid:0) (1 (cid:0) (cid:14))kt(cid:0)1 (cid:20) (cid:22)zlt(cid:0)1 + Ak(cid:13)

t(cid:0)1;

∫

li
t(cid:0)1 di is total labor. The resources produced each period are the output
where lt(cid:0)1 =
(cid:22)zlt(cid:0)1 of manager-run firms and the output Ak(cid:13)
t(cid:0)1 of the bank-financed sector; recall that the
output of manager-run firms depends on labor choices made at t (cid:0) 1. These resources are
used for consumption ct or to invest in the stock of capital kt, net of the undepreciated stock
(1 (cid:0) (cid:14))kt(cid:0)1 from the previous period, as in the standard growth model.

When computing the equilibrium, we must aggregate the firms’ building block of the
model. However, as noted in Section 3.1.1, firms’ labor demand does not depend on the
distribution of deposits across firms, a result that simplifies the analysis considerably. Thus,
total labor demand lt is

∫

lt =

li
t di

= ϕtdt ;

16

(22)

where the second line uses the result li
t from Proposition 1 and defines dt to be total
di
deposits across firms, dt =
t di. Similarly, we can easily aggregate over firms’ law of
motion for deposits. To do so, we first impose the restriction

t = ϕtdi

∫

(cid:11) = 1 (cid:0) (cid:12):

(23)

The restriction in (23) can be justified in either of two ways. First, (23) arises endogenously
in the nonstochastic steady state of the full model of Section 5, in which we endogenize firms’
dividend policy. Second, as an alternative justification, we note that (23) is necessary and
sufficient to obtain a benchmark result in which Modigliani-Miller holds for banks if we shut
down deposit insurance (see Appendix B). Then, using (2) and (23), the law of motion of
aggregate deposits dt is

dt+1 = (cid:12)

[
((cid:22)z (cid:0) wt) lt + Rd

t dt

]

;

(24)

where (cid:22)z denotes the mean of the idiosyncratic firm productivity zi

t+1.

3.3 Welfare

When computing optimal capital requirements, we should, in principle, account for the wel-
fare of both households and managers. However, to do so we must take a stand on the weight
to be assigned to households and managers in the welfare function—a task that would signif-
icantly complicate the analysis. To address this issue, we note that, in practice, the fraction
of managers in the population is small relative to households, and their compensation is
small relative to firms’ profits. Thus, in this section we formalize the notion that total con-
sumption and total welfare are well approximated by households’ consumption and welfare,
respectively, so that we can compute the optimal capital requirement by looking solely at
the welfare of households.

We define the welfare function W to be the sum of households’ welfare and managers’
∫

welfare:

W = V h

0 (a0) + (cid:18)

0 (di
V m

0) di;

(25)

where (cid:18) > 0 is a Pareto weight that defines managers’ contribution to total welfare in
comparison to households. The welfare analysis depends, in principle, on the Pareto weight
(cid:18) and the fraction (cid:20) of dividends paid to managers. However, we discipline these two
parameters in such a way that (i) welfare depends only on households’ utility, W = V h
0 (a0);
(ii) the implication of the agency friction and the easy aggregation of the firms’ building block
of the model, derived in Proposition 1, continue to hold; and (iii) aggregate consumption
depends only on households’ consumption.

17

To discipline (cid:18), we note that the fraction of managers in the population is, in practice,
small. As a result, we consider (cid:18) to be arbitrarily small so that the economy-wide welfare
depends only on households’ utility: W = V h
In Appendix C, we provide a simple
extension of the model in which we formalize the measure of managers relative to households,
take the limit of such measure to zero, and show that W = V h
0 (a0) even if we set the Pareto
weight (cid:18) to one (i.e., even if each manager has the same Pareto weight as each household).
To discipline (cid:20), we note that managers’ compensations are small relative to the overall
dividends paid by firms. As a result, we choose (cid:20) to be arbitrarily small, which implies that
all dividends are paid to households.14

0 (a0).

We summarize the implications for choosing (cid:18) and (cid:20) to be arbitrarily small in the fol-

lowing proposition.

Proposition 2. (Parameter restrictions and welfare) If (cid:20) ! 0 and (cid:18) ! 0 (with (cid:18)=(cid:20) constant
along the limiting sequence), manager i’s optimal choice in Proposition 1 is not affected,
! 0, profits distributed to households, defined in
manager i’s consumption converges to ci
(4), become

t+1

[(

)

]

(cid:25)i
t+1

! (cid:11)

zi
t+1

(cid:0) wt

ϕt + Rd
t

di
t;

and the economy-wide welfare function W becomes equal to the value function of households,
W ! V h

0 (a0).

The key result of Proposition 2 is that the importance of managers vanishes for welfare
purposes (i.e., their consumption converges to zero) without affecting the key first-order
condition that governs their labor demand, (7). The implication of Proposition 2 is that
we can just focus on the welfare of households when evaluating financial regulation, even
though the model exhibits a behavior driven by an agency friction.

4 Theoretical results

This section derives the main theoretical results of the paper. We begin by showing that our
model generates a liquidity premium on deposits (Section 4.1). Then, we study the effects
of capital requirements on firms’ good risk-taking (Section 4.2), and we examine the welfare
implications of ignoring our novel channel when setting the capital requirement (Section 4.3).
Because what we call the “bad risk-taking” channel is a standard mechanism in the literature

14Setting an arbitrarily small (cid:20) drives managers’ consumption to zero. However, we show in Appendix C
that when we formalize the measures of managers and take the limit as this measure goes to zero, managers’
consumption remains positive and well defined even if (cid:20) ! 0.

18

(Van den Heuvel, 2008; Begenau, 2018), we defer discussion of how banks’ bad risk-taking
interacts with firms’ deposits to Appendix D.

Throughout this section, we illustrate the theory with numerical examples to highlight
how good risk-taking and bad risk-taking are traded off in the determination of the optimal
capital requirements. In Section 5, we calibrate a quantitative model to better estimate the
magnitudes of the effects we describe in this section.

4.1 Liquidity premium on deposits

We begin by showing that our model generates a premium on the return on deposits that is
equal to their marginal private value (i.e., the marginal value from the firms’ point of view).
We focus on the steady state, as we do for most of our theoretical analysis, and denote
steady-state values by dropping the time subscript.

Proposition 3. (Deposit premium) Assume that V ar
premium is

)

(

zi
t+1

> 0. In steady state, the deposit

0 <

1
(cid:12)

(cid:0) Rd =

∫

2

(

4

{

}

Et

zi
t+1

(cid:0) w

)

(cid:12)
(cid:12)
(cid:12)
(cid:12)
(cid:12)

@li
t
@di
t

3

5 di

w fixed

= ((cid:22)z (cid:0) w) ϕ;

(26)

where the right-hand side of (26) denotes the marginal private value of deposits.

The deposit premium is positive because deposits provide insurance to firms’ managers
against the exposure to idiosyncratic risk. Such a premium is then equal to the marginal
private value of deposits (i.e., the right-hand side of equation (26)). To understand this
expression, consider the following. If each firm began time t with an additional $1 of deposits,
managers would take on more risk (i.e., hire more workers) and firm i would earn a profit
(cid:0) wt per additional worker hired. Such profits would then be valued by households
zi
t+1
according to their constant unitary marginal utility of consumption.

Crucially, the marginal private value of deposits is computed by taking the wage w as
given. In the next sections, we show that the optimal capital requirement should instead
be set by taking into account the response of the wage. This consideration gives rise to a
wedge between the private value of deposits (i.e., the value of deposits from the firms’ point
of view) and the social value (i.e., the value of deposits for the regulator), which then reduces
the welfare cost of increasing capital requirements.

Proposition 3 also provides a direct link between labor market outcomes and the deposit
premium. Indeed, equation (26) shows that the deposit premium is related to the wage w,

19

Frisch elasticity: (cid:17) ! 1

Frisch elasticity: (cid:17) ! 0

wt

wt

Labor supply

Labor supply

Labor demand

lt

Labor demand

lt

Figure 1. Equilibrium in the labor market. Left panel: the labor supply curve
is infinitely elastic and thus pins down the wage. Right panel: the labor supply
curve is inelastic and thus pins down the equilibrium value of employment.

which in turn is affected by the utility parameter (cid:31). In Section 5, we will use this result to
calibrate (cid:31) to match the deposit premium in the data.

4.2 Capital requirements and good risk-taking

This section presents the main result about the interaction between capital requirements
and firms’ good risk-taking. The effects of modifying capital requirements differ dramatically
depending on how firms’ input prices—in our model, wages—adjust in response to the policy
change.

In this simple model, we show next that the equilibrium response of wages to policy
changes depends on the Frisch elasticity of labor supply. This elasticity is defined as the
percentage change in households’ labor supply in response to a 1% change in the wage,
keeping consumption constant. Using the first-order condition of households (19), we obtain
that

Frisch elasticity of labor supply = (cid:17)

(recall that (cid:17) is one of the parameters that affects the disutility of labor).

To further clarify the role of the Frisch elasticity of labor supply, Figure 1 plots the
demand and supply in the labor market in two extreme cases. In the left panel, the Frisch
elasticity of labor supply is (cid:17) ! 1 (i.e., labor supply is fully elastic and households have lin-
ear disutility from labor). In this case, the wage is essentially fixed because the households’
first-order condition (19) implies wt = (cid:31). Thus, if firms demand more inputs in response to
policy changes, the wage does not change, and the effect of financial regulation on employ-
ment is maximal. The right panel plots the other extreme case, in which the Frisch elasticity

20

is (cid:17) ! 0 (i.e., labor supply is fixed). In this case, financial regulation produces an impact
only on wages wt and has no effect on employment lt.

In the rest of this section, we formalize the above results and then provide an illustrative
example. We maintain our focus on the two extreme cases with fully elastic labor supply
(i.e., (cid:17) ! 1) and fixed labor supply (i.e., (cid:17) ! 0).

Tightening capital requirements reduces deposits, which in turn makes it harder for
firms to self-insure against idiosyncratic risk.15 In response, firms reduce labor demand. The
extent to which this change in labor demand affects equilibrium labor lt depends on the
Frisch elasticity of labor supply, (cid:17). If (cid:17) ! 1 (i.e., labor supply is fully elastic), any change
in labor demand is transmitted one-for-one into changes in the equilibrium value of labor,
If (cid:17) ! 0 (i.e., labor supply is fixed), changes in labor demand produce only changes
lt.
in the wage, wt, but no changes in employment. We summarize this result in the next
proposition, focusing on a comparison across steady states and denoting the steady-state
value of endogenous variables by dropping the time subscript.

Proposition 4. (Capital requirements and good risk-taking) Assume that there exists a
steady-state equilibrium with l > 0. Then:

• If the Frisch elasticity of labor supply is (cid:17) 2 (0; 1), then @w=@(cid:16) < 0 and @l=@(cid:16) < 0;

• If the Frisch elasticity of labor supply is (cid:17) ! 1, then @w=@(cid:16) = 0 and @l=@(cid:16) < 0;

• If the Frisch elasticity of labor supply is (cid:17) ! 0, then @w=@(cid:16) < 0 and @l=@(cid:16) = 0.

The first implication of Proposition 4 is that the marginal social value of deposits differs
from the marginal private value for all (cid:17) < 1.16 The marginal private value of deposits
from the point of view of each firm is implicitly computed by taking the wage as given, but
the social value accounts for the fact that the wage may adjust in general equilibrium as
the supply of deposits changes. To clarify this point, consider the limiting case in which
the labor supply is fixed (i.e., (cid:17) ! 0).
In this case, an increase in the availability of
deposits—arising from, for instance, lower capital requirements—triggers an increase in the
wage that exactly offsets the private benefits of the additional deposits. As a result, changing
capital requirements does not affect firms’ good risk-taking or the equilibrium value of output

15Lemma 6 in Appendix A shows that the steady-state level of deposits, denoted by d, satisfies @d=@(cid:16) < 0

for all (cid:17) 2 (0; 1) as well as in the limit as (cid:17) goes to either zero or infinity.

16This reasoning requires deposits to be above a certain threshold, which depends on parameters, so that
firms hire workers in equilibrium; hence, we require l > 0 as an assumption of the proposition. If the quantity
of deposits is too small, firms cannot insure against idiosyncratic risk, and thus they might decide not to
hire any workers.

21

Figure 2. The figure plots total welfare W ((cid:16)) for values of the capital re-
quirement (cid:16) ranging from 3% to 20%, starting from an initial steady-state where
(cid:16) = 8%. Parameter values are (cid:12) = 0:95, pz = 0:952, (cid:14) = 0:1, A = 0:52, (cid:27) = 0:063,
(cid:31) = 0:96, and (cid:13) = 0:3. We choose A, (cid:27), and (cid:31) to induce a deposit premium of
2%, a default probability of 10%, and an equilibrium consumption value of 1. The
dotted line assumes a fully elastic labor supply (i.e., (cid:17) ! 1); the solid line as-
sumes a fixed labor supply (i.e., (cid:17) ! 0), where labor supply is fixed at (cid:22)l = 0:555.
Details on these calculations are in Appendix E.

produced by firms. This is not to say that capital requirements do not produce any effect at
all. As we discuss in Appendix D, even with fixed labor supply, capital requirements affect
banks’ bad risk-taking and the size of the banking sector.

The second takeaway from Proposition 4 is that the existence of a deposit premium in
the data is not sufficient to conclude that deposits have a positive marginal social value. In
our model, deposits display a positive premium (i.e., 1=(cid:12) (cid:0) Rd > 0, as shown in Proposition
3) even if they have a zero marginal social value. This is because the return on deposits Rd
is determined by the firms’ first-order conditions, which account only for private marginal
values.

The last and key implication of Proposition 4 is that the existence of a wedge between the
marginal and social values of deposits has crucial implications for determining the optimal
capital requirement. If deposits’ marginal social value is indeed less than private values, the
availability of an additional dollar of deposits is not very important. As a result, capital
requirements can be set at a somewhat high level to limit the negative effects of subsidized
deposit insurance on banks’ bad risk-taking.

To illustrate this final point, we compare two numerical examples, where the Frisch
elasticity is (cid:17) ! 1 or (cid:17) ! 0. We consider an economy that is initialized at a steady state

22

2468101214161820capital requirement (%)0.9920.9930.9940.9950.9960.9970.9980.9991welfarewith a capital requirement (cid:16) = 8% and where, at t = 0, the regulator permanently changes
the capital requirement to a different (cid:16). We then compute welfare W ((cid:16)) as the economy
transitions and reaches the new steady state.17 Figure 2 plots W ((cid:16)) for capital requirements
ranging from 3% to 20%. We stress that we have constructed the examples so that both
economies with (cid:17) ! 1 and (cid:17) ! 0 have the same deposit premium of 2% in the initial steady
state with an 8% capital requirement.

The optimal capital requirement differs substantially between the two extreme cases for
the Frisch elasticity (cid:17). With (cid:17) ! 1 (i.e., fully elastic labor supply), the private and social
values of deposits are equalized, and the optimal capital requirement is 4.8%. With (cid:17) ! 0
(i.e., fixed labor supply), the marginal social value of deposits is zero, and the optimal capital
requirement is much higher, at 8.7%. In addition, the sensitivity of welfare to changes in
In Appendix D, we explain why
capital requirements is much lower in the second case.
the optimal capital requirement under a fixed labor supply is not too high despite the zero
marginal social value of deposits.18

4.3 Capital requirements and welfare

The previous analysis has compared economies with different Frisch elasticities of labor sup-
ply. We now discuss how to compute the effect of our novel channel in a given economy.
Indeed, in the quantitative analysis of Section 5, we want to quantify the importance of in-
cluding firms’ deposits and the good risk-taking channel on the determination of the optimal
capital requirement.

We consider two approaches to compute the importance of firms’ deposits and the good
risk-taking channel. The first approach is to compare the optimal capital requirement with
that of another model in which deposits are held by households only, rather than firms. Since
this comparison is conceptually simple, we will undertake it directly in the quantitative model
of Section 5. In this section, we explore a second approach that illustrates the magnitude
of our channel without resorting to a different model. Specifically, we compute the optimal
capital requirement obtained by shutting down our novel channel that operates through
wages and which is described in Proposition 4. To do so, we replace the path for wages that
arises after changing the capital requirement with the one that would arise if the capital
requirement had not changed, compute counterfactual paths for consumption and labor,
and calculate an alternative measure of welfare, which we denote as cW ((cid:16)). Crucially, we

17Welfare W ((cid:16)) is equal to the value function of households (see Proposition 2); we emphasize its depen-

dence on the capital requirement (cid:16) chosen at t = 0 by the regulator.

18The logic of this result is similar to that in Begenau (2018), who shows that tighter capital requirements
have two opposite effects on bank lending. These two forces generate a trade-off even if changes in capital
requirements have no effect on welfare through the supply of deposits. See Appendix D for more details.

23

compute cW ((cid:16)) by shutting down only our novel channel in the context of the labor market,
without altering the way capital requirements affect banks’ bad risk-taking.19

We consider an economy that is initially in a steady state with a capital requirement
(cid:16)(cid:0)1, and we solve for the equilibrium as the regulator changes the capital requirement to
some different (cid:16) at t = 0. We then compute the welfare that arises under the new capital
requirement (cid:16), denoted by W ((cid:16)), and the welfare cW ((cid:16)) that does not account for our novel
channel. We next describe how we compute cW ((cid:16)).

To compute the welfare cW ((cid:16)) that does not include our new channel, we construct a
fictitious path for consumption and labor, f^ct((cid:16)); ^lt((cid:16))g1
t=0, that would arise had wages not
adjusted in response to the new capital requirement. More precisely, we limit the effect of
this experiment on employment, without affecting the way capital requirements affect banks’
bad risk-taking. Thus, our fictitious consumption and labor f^ct((cid:16)); ^lt((cid:16))g1
t=0 are constructed
as follows:

• We compute firms’ labor demand ^lt given (i) the correct equilibrium paths of firms’
deposits fdt((cid:16))g1
t=0 that arise under the new
capital requirement, but (ii) assuming that firms face the wage ^wt((cid:16)) = w(cid:0)1 for all t,
that is, the wage that would prevail if the capital requirement had remained constant;

t=0 and the return on deposits fRd

t ((cid:16))g1

• We compute consumption ^ct that would arise using (i) the correct equilibrium value of
the output of the bank-financed sector and (ii) the output produced by firms if they
had employed the fictitious amount of workers ^lt, rather than the true equilibrium
amount lt.

We then evaluate the utility of households at the path of consumption and labor f^ct((cid:16)); ^lt((cid:16))g1
which gives us the welfare measure cW ((cid:16)) (see Section 3.3 for the link between welfare and
households’ utility). With this approach, we are able to isolate the effect of our channel on
welfare through the labor market, without contaminating our measure with the way capital
requirements affect banks.

t=0,

Figure 3 illustrates the result of this section for an economy with a Frisch elasticity of
labor supply (cid:17) = 1.20 The solid line in Figure 3 plots the correct welfare W ((cid:16)). The optimal
capital requirement in this economy is (cid:16) (cid:3) = 6:8%; we normalize W ((cid:16)) to one at such a
value.21 The dashed line in Figure 3 plots the welfare cW ((cid:16)) that does not include our novel

19In the quantitative model of Section 5, we show that the two approaches described here—using a model
with only households’ deposits versus shutting down our channel that acts through wages—deliver approxi-
mately the same optimal capital requirement, well below the true optimum.

20This is in line with standard values employed in macro-labor models and, for our purposes, is the most

conservative value that is in line with the empirical evidence of Chetty et al. (2011).

21In this simple model, welfare is measured in units of consumption because of the linearity of the utility

function in ct.

24

Figure 3. The solid line plots the full welfare W ((cid:16)) for values of the capital
requirement (cid:16) ranging from 3% to 20%, starting from an initial steady state
where (cid:16)(cid:0)1 = 8%. The dashed line plots the welfare cW ((cid:16)) that does not include
the effects of our novel channel operating through the labor market and wages.
Parameter values are (cid:12) = 0:95, pz = 0:95, (cid:14) = 0:1, A = 0:52, (cid:27) = 0:063, (cid:31) = 0:96,
(cid:13) = 0:3, and (cid:17) = 1.

channel. The welfare measure cW ((cid:16)) peaks at ^(cid:16) = 4:1%, which is significantly lower than
the correct capital requirement (cid:16) (cid:3) = 6:8%. Because our novel channel reduces the cost of
increasing capital requirements, the optimal capital requirement is much higher than the
one that maximizes the component of welfare that does not include our effect. In addition,
a welfare analysis based on cW ((cid:16)) overestimates the losses of setting capital requirements
too high. All the qualitative results derived here will also hold in the quantitative model of
Section 5.

5 Quantitative model

In this section, we extend the theoretical model with the objective of performing a quan-
titative analysis.
In particular, we undertake two main exercises. First, we solve for the
optimal capital requirement in a quantitative version of the model. Second, we quantify
the contribution of our novel channel by computing the optimal capital requirement that
abstracts away from changes in wages or deposits held by firms. In particular, we follow two
approaches to quantify our contribution: we compute (i) the capital requirement that would
be chosen if we shut down the effects of our novel channel that operates in the labor market,
similar to Section 4.3; and (ii) the capital requirement that would be chosen in a comparable
model in which only households hold deposits.

25

2468101214161820capital requirement (%)0.9750.980.9850.990.9951welfareWe find that the optimal capital requirement in the full quantitative model is 18.7%, the
capital requirement that would be chosen in the full model absent our novel channel is 15.3%,
and the capital requirement that maximizes welfare if all deposits are held by households is
13.6%. Thus, the main reason why the optimal regulation is tighter in our model is related
to our novel channel acting through wages in general equilibrium, rather than who holds the
deposits.

We extend the baseline model along five dimensions: (i) we endogenize firms’ dividend
policy, rather than setting it exogenously; (ii) we introduce aggregate risk; (iii) we allow
banks to risk-shift by giving them access to a technology that increases their idiosyncratic
risk, subject to a cost; (iv) we give households a constant relative risk aversion (CRRA)
utility from consumption, rather than linear utility; and (v) we give households utility from
holding deposits.22 We briefly describe these features and then present the calibration and
results.

We solve the model using nonlinear global projection methods that allow for rich dynam-
ics and occasionally binding constraints. We use this approach to match some key features of
the time series of bank default rates, which are low in normal times but spike during crises.
Our quantitative model allows banks to endogenously engage in more “bad risk-taking” by
making riskier and less productive loans, but they do so less than half of the time in the
calibrated model. Thus, a solution approach based on standard perturbation methods would
not adequately capture this key feature of the data.

One realistic feature we do not include in the quantitative model are firms’ holdings of
other safe assets, such as government bonds. Such a feature would substantially complicate
the analysis but would not change the main results, so long as the supply of such assets
is not systematically correlated with changes in capital requirements; this assumption is
used, for instance, in Begenau (2018). Our claim is based on two arguments. First, firms’
holdings of other safe assets are small in comparison to deposits (see Appendix F). This
statement is based on data from the Flow of Funds and, thus, accounts for privately held
firms; using statistics that account for private firms is the right approach for our purposes
because we use a general equilibrium model of the whole economy, and privately held firms
in the United States account for about two-thirds of aggregate employment (Dinlersoz et al.,
2018). Second, and more importantly, the fact that firms hold additional safe assets does
not alter our main results as long as the model is calibrated to match the deposit premium
in the data. As shown in Proposition 3, the deposit premium is equal to firms’ marginal

22We have explored simpler quantitative versions of the model in which households do not hold deposits,
and as expected, the optimal capital requirement in those models is even higher than what we document
here.

26

value of banks’ deposits, and thus it captures firms’ willingness to hire out of an extra dollar
of deposits. Thus, the labor market effects of a change in capital requirements that modifies
the supply of deposits depend on (i) the liquidity premium and (ii) how wages adjust in
equilibrium. Inframarginal holdings of other safe assets will not affect the response of firms’
hiring decisions to a change in capital requirements, so long as the liquidity premium is
calibrated to match the data.

5.1 Extended model

Firms’ dividend policy. We endogenize firms’ dividend policy by allowing firms’ share-
holders (i.e., households) to choose optimally the amount of dividends to be paid every
period, thereby relaxing (23). The choice of letting households rather than managers choose
dividends is motivated by the results of La Porta et al. (2000)—who find that firms in
countries with good legal protections for shareholders choose dividend policies that are con-
sistent with shareholders’ rather than managers’ preferences—and the fact that we calibrate
the model to the US economy.
In practice, shareholders can easily monitor firms’ cash,
and in common-law countries such as the United States (where all shareholders—including
minority ones—enjoy a high level of legal protection), they can influence dividend payments.
For instance, for public firms, activist investors are likely to take actions if cash and div-
idends are managed in a way that hurts shareholders. To prevent such actions, managers
might follow policies that are most beneficial for shareholders in the first place. In contrast,
monitoring the optimality of firms’ core management decisions—including those related to
the firms’ labor force—is more difficult. Thus, core management decisions are more likely
to be subject to agency frictions, as in our model. From a modeling perspective, letting
households choose the dividend policy implies that the agency friction affects only firms’
hiring policies, thereby making the effects of such a friction more transparent.

The fraction of firms’ wealth paid out as dividends is now possibly time varying and firm
t, rather than (cid:11). The optimal (cid:11)i
specific and thus denoted by (cid:11)i
t is chosen so that the stream of
dividends maximizes the value of the firm from the households’ point of view (i.e., using the
household discount factor).23 The next proposition shows that the dividend policy implies
the same (cid:11)i
t = (cid:11)t for all i. This follows from the simple structure
of our model, which allows an easy aggregation across firms. The proposition also describes
the equilibrium condition that pins down (cid:11)t. The inability of the manager to differentiate
away firms’ idiosyncratic risk is unchanged, and so is the first-order condition (7).

t at all firms, and thus (cid:11)i

23The value of (cid:11)i

t can still be understood as part of the contract between shareholders and the manager.

27

Proposition 5. The optimal dividend policy from the point of view of households implies
that (cid:11)i

t = (cid:11)t for all i, and (cid:11)t is chosen so that
) [

{

(

}

]

1 = Et

(cid:12)

((cid:22)z (cid:0) wt) ϕt + Rd
t

;

(27)

(cid:3)t+1
(cid:3)t

where (cid:3)t is households’ marginal utility of consumption.

The choice of (cid:11)t can depend on aggregate shocks or changes in capital requirements
realized at time t. This implies that deposits dt can vary in response to aggregate shocks
or changes in capital requirements. This is in contrast to the baseline model, in which (23)
implied that firms’ deposits were essentially predetermined at t (cid:0) 1. In addition, we note
that the optimal dividend policy in Proposition 5 collapses to (cid:11) = 1 (cid:0) (cid:12) in the nonstochastic
steady state, which is the same condition as (23).

Because (cid:11)t is in general time varying, the law of motion for deposits (24) is replaced by

df
t+1 = (1 (cid:0) (cid:11)t+1)

[
((cid:22)z (cid:0) wt) ϕt + Rd
t

]

df
t ;

t denotes the total amount of deposits held by firms. We will continue to denote dt

where df
to be the deposits of banks, and we will denote dh

t to be the deposits held by households.

Aggregate risk. We introduce aggregate risk by assuming that the productivity of the
bank-financed sector, now denoted by At+1, follows an AR(1) process in logs. The law of
motion of At+1 is

log At+1 = (1 (cid:0) (cid:26)) log (cid:22)A + (cid:26) log At + (cid:27)A"A

t+1;

where (cid:22)A; (cid:27)A > 0 and "A

t+1

(cid:24) N (0; 1). We replace A in equations (14) and (15) with At+1.

Banks’ risk choice. We allow banks to increase their idiosyncratic volatility by paying
a convex cost, similar to Van den Heuvel (2008) and Begenau (2018). Limited liability and
the deposit insurance intervention together imply that bankers will avail themselves of this
technology, although it is socially suboptimal. Specifically, we assume that banks can choose
a probability pt (cid:21) 0 such that their idiosyncratic shock is " = 0 with probability pt and
"=(1 (cid:0) pt) with probability 1 (cid:0) pt, where " is drawn from the same distribution F (") as in the
baseline model. Without a convex cost of choosing pt, bankers would let pt get arbitrarily
close to 1; we assume that bankers pay a cost (cid:21) (pt) dt, where (cid:21) ((cid:1)) is an increasing and convex

28

function, specified below. The banker’s problem becomes

max
kt;dt;pt

(1 (cid:0) pt) Et

∫ {

"
1 (cid:0) pt

kt (1 (cid:0) (cid:14) + rt+1) (cid:0) Rd

t dt (cid:0) (cid:21) (pt) dt

dF (")

(28)

}

+

subject to pt (cid:21) 0 and to the same budget and capital requirement constraints as before,
(9) and (10). For failed banks, the cost (cid:21) (pt) dt reduces the value of assets seized by the
government to partially defray the expenses of paying back depositors and, thus, increases
the taxes that must be collected to fund deposit insurance. Formally, we include (cid:21) (pt) dt on
the right-hand side of equation (21).24

The endogenous risk chosen by banks affects welfare and the optimal regulation through
two channels. First, the cost (cid:21) (pt) dt reduces total output and ultimately households’ con-
sumption. Second, the ability to choose pt > 0 increases the private value of physical capital
kt to banks and therefore exacerbates the overinvestment induced by deposit insurance.
Both channels increase the social value of increasing capital requirements, and higher capital
requirements will lead banks to choose a lower value of pt.

We parameterize the cost of increasing pt above zero using the function

(cid:21) (p) = (cid:21)p

1
(1 (cid:0) p)1+(cid:23) ;

(29)

where (cid:21) and (cid:23) are parameters. This cost function has two advantages over a more traditional
one, such as a quadratic function. First, because banks will default in equilibrium, what
matters to them is their expected cost, which is proportional to (1 (cid:0) pt) (cid:21) (pt), using (28).
(cid:0)1 term, the expected marginal cost of increasing pt would
If (cid:21) (pt) were lacking a (1 (cid:0) pt)
be nonmonotone, and the total expected cost would vanish as pt ! 1.
In contrast, the
formulation in (29) guarantees that the expected marginal cost of increasing pt is monotone
and goes to infinity as pt ! 1. Thus, (29) ensures that we avoid multiple equilibria or
implausibly high choices of pt. Second, as we describe below, equation (29) will imply that
banks optimally set pt = 0 in many states of the world, allowing for rich, nonlinear dynamics.
As a result, bank default will be highly nonlinear, replicating the large variations observed
in the data between normal times and crises. We will return to the mechanics of banks’
choices of pt when we describe the calibration of the parameters (cid:21) and (cid:23).

Households’ utility and households’ deposits. We give households a CRRA utility
from consumption and utility from holding deposits dh
t . We follow the literature by inter-

24The cost (cid:21) (pt) dt can be interpreted as an obligation undertaken by the bank at time t, such as the cost
to manage loans to riskier borrowers, which must be paid to recover the value of the loans "kt (1 (cid:0) (cid:14) + rt+1).

29

preting the utility from deposits as arising from their use in transactions, owing to their high
liquidity.25

Thus, the value function of households becomes

V h
t (at) = max
ct;lt;nt

(

(cid:0) 1

c1(cid:0)(cid:13)c
t
1 (cid:0) (cid:13)c

+  

)

(1(cid:0)(cid:13)d)

dh
t =ct
1 (cid:0) (cid:13)d

(cid:0) (cid:12)(cid:31)

1+ 1
(cid:17)
l
t
1 + 1
(cid:17)

{

}

+ (cid:12)Et

V h
t+1 (at+1)

:

(30)

The parameter (cid:13)c is households’ coefficient of relative risk aversion, (cid:13)d governs the elasticity
of deposits demand, and   governs the level of deposits demand. The budget constraint of
households is now given by ct + nt + dh
t

(cid:20) at, and the law of motion of wealth is

at+1 = wtlt + ntRn

t+1 + dh

t Rd

t +

∫

t+1di + (cid:25)bf
(cid:25)i

t+1

(cid:0) Tt+1:

(31)

The specification in equation (30) for how deposits enter households’ utility is similar to that
in Begenau (2018).

5.2 Calibration and simulation

We calibrate the model under an 8% capital requirement and then study the welfare effect
of changing the requirement. We divide the parameters into two groups. The first set of
parameters, reported in Panel A of Table 1, includes parameters that are normalized (i.e.,
(cid:22)z and (cid:22)A are normalized to one) and others that are set to values in line with the related
literature. The second set, reported in Panel B, is calibrated toward the indicated data
moments.

Panel A of Table 1 reports the fixed parameters for which we do not have a calibration
target. The model is calibrated at an annual frequency, and thus we set the discount factor
(cid:12) to 0:95. Standard values are also used for the depreciation rate of capital ((cid:14) = 0:1), the
Cobb-Douglas coefficient of capital ((cid:13) = 0:3), households’ risk aversion ((cid:13)c = 1, that is, log
utility), and the autocorrelation of productivity shocks ((cid:26) = 0:95). A key parameter is the
Frisch elasticity of labor supply, (cid:17), which we set to 1, in line with the standard approach in
the macro-labor literature; this is also the most conservative value according to the estimates
of Chetty et al. (2011).

Panel B of Table 1 reports parameters that we calibrate toward the indicated data mo-
ments. Because the model is nonlinear and we solve it globally, we do not exactly match all

25In principle, it is possible to think about the effect of deposits in providing insurance against idiosyncratic
risk to households, as we do for firms. However, we abstract from this channel here and leave it for future
research.

30

Panel A: Set Parameters

Parameter Value

Parameter Value

(cid:17)
(cid:12)
(cid:14)

1
0.95
0.1

(cid:13)
(cid:13)c
(cid:26)

0.3
1
0.95

Parameter
(cid:22)A
(cid:22)z
zi

Value

1
1
f1=pz; 0g

Parameter Value Target

Value (data) Value (model)

Panel B: Calibrated Parameters

(cid:31)
pz
(cid:27)
(cid:23)
(cid:21)

(cid:13)d
(cid:27)A

1.439 Deposit premium Rf (cid:0) Rd
0.944
0.03125 Avg. bank default probability

Continuers employment growth

Std. dev. bank default probability
p90 bank default probability

0.14
1.03409
0.0229 Deposits held by firms
1.03
Deposit volatility
0.0062 Volatility log GDP

Std(d=c)
Std(log GDP )

1.92%
2.50%
0.76%
1.05%
2.26%
33.3%
1.27
1.77%

1.92%
2.51%
0.76%
1.04%
2.27%
33.2%
1.27
1.78%

Table 1. Calibrated Parameter Values

moments. In addition, each parameter affects all moments, but Panel B of Table 1 indicates
target moments that are particularly affected by each parameter.

We choose the disutility of labor, (cid:31), to match the average deposit premium in the data,
exploiting the results of Proposition 3. There, we show that the deposit premium is de-
creasing in the wage and increasing in the managers’ risk-taking decision ϕt introduced in
Proposition 1. Increasing (cid:31) both raises the wage and reduces ϕt, and thus (cid:31) has a direct
effect on the average deposit premium. We choose a deposit premium target of 1.92%, which
is the midpoint of the 3.16% deposit premium measured by Van den Heuvel (2008) and the
0.68% deposit premium of Davydiuk (2017).

We set the parameter that governs firms’ idiosyncratic shocks, pz, to match the average
employment growth rate of firms that expand their labor force, corresponding to firms that
receive the high value of zi
t+1 in the model. This average employment growth rate is defined
as

gi
t+1

(cid:17)

1
2

bli
(cid:0) li
(
t+1
t
bli
t+1 + li
t

);

31

 
where li

t = ϕtdi

t (as in Proposition 1) and

bli
t+1 = ϕt+1di

t+1

(cid:12)
(cid:12)
(cid:12)

zi
t+1>0

[(

= ϕt+1 (1 (cid:0) (cid:11)t+1)

)

]

(cid:0) wt

ϕt + Rd
t

di
t:

1
pz

Haltiwanger, Jarmin and Miranda (2013) show that for all but the youngest firms, this
growth rate is around 2.5%.

We calibrate (cid:27), (cid:23), and (cid:21) to match three moments of the time-series variation in the default
probability of banks. We measure bank default probabilities using the FDIC’s Historical
Statistics on Banking from 1975 to 2016. We define the default probability of banks as the
number of bank failures in a given year reported by the FDIC, divided by the total number
of banks covered by the FDIC. This yields an average default probability of 0.76%, as in
Davydiuk (2017), which is mostly affected by (cid:27). We also target the standard deviation and
the 90th percentile of the time-series distribution of the bank default probability; these two
moments are mostly affected by (cid:23) and (cid:21), respectively.

To clarify the calibration of (cid:27), (cid:23), and (cid:21), consider the first-order condition of banks with
respect to the choice of pt. Denoting (cid:24)t to be the Lagrange multiplier of the nonnegativity
constraint pt (cid:21) 0, we have

Rd

t + (cid:24)t = (cid:21)

1 + ((cid:23) (cid:0) 1) pt
(1 (cid:0) pt)1+(cid:23)

:

(32)

At pt = 0 and (cid:24)t = 0, the marginal cost of increasing pt is (cid:21). Thus, (cid:21) represents the lowest
level of Rd
t at which banks choose to increase pt above zero. We can thus calibrate (cid:21) so that,
most of the time, Rd
t < (cid:21) and thus banks choose pt = 0, allowing the model to replicate
low default probabilities in normal times. In particular, our choice of (cid:21) implies that 72.5%
of the time, banks choose pt = 0.26 Once pt > 0, its elasticity with respect to Rd
is then
t
determined by the parameter (cid:23).27 By varying (cid:27), (cid:21), and (cid:23) jointly, we are able to match the
average, standard deviation, and 90th percentile of bank default probability that we observe
in the data.

We calibrate the parameters that govern how deposits enter households’ utility,   and
(cid:13)d, to match the average fraction of deposits held by firms and the volatility of the deposit-
consumption ratio, respectively. As described in Appendix F, we find that firms hold about
one-third of the total deposits held jointly by households and firms (i.e., we exclude deposits
held by governments, foreigners, nonprofits, and the financial sector). To derive this statistic,

26The fact that pt is often 0 is what allows the parameter (cid:27) to primarily affect the average default

probability, though of course (cid:27), (cid:21), and (cid:23) all jointly affect this moment.

27In particular, the elasticity is negatively related to (cid:23), so that a lower value of (cid:23) raises pt, given Rd
t .

32

we use data from the Flow of Funds (FF) and the Survey of Consumer Finances (SCF).28
In addition, we compare the total deposits from the FF and SCF with data on deposits
from the FDIC and the National Credit Union Administration, and the two measures are
consistent with each other. We then set (cid:13)d to match the volatility of d=c, as in Begenau
(2018). To estimate this volatility in the data, we use the HP-filtered ratio of total deposits
at FDIC-insured banks to consumption expenditures for nondurable goods and services, for
the period 1981-2017. We choose data starting from 1981 to maximize the length of the data
sample while, at the same time, focusing on a period of time in which banks were allowed
to pay interest at least on some deposit instruments.29

Finally, we choose the standard deviation of the productivity process, (cid:27)A, to match the
volatility of the HP-filtered log of GDP. We focus on the 1981-2017 period for consistency
with the other macro data used to calibrate the household utility parameter (cid:13)d.

5.3 Results

To compute the welfare effects of changing capital requirements, we begin by simulating an
economy with an 8% capital requirement, a value in between the 7.26% used by Davydiuk
(2017) and the 9.25% used by Begenau (2018). We then change the capital requirement to
a new level, (cid:16), and compute the welfare as the economy transitions to the new stochastic
steady state.

To put our results in context, we then compute two additional results: the welfare that
arises in a comparable model in which only households hold deposits, and the welfare that
arises in our full model if we shut down our novel channel that acts through wages. This
second approach generalizes Section 4.3 by computing counterfactual paths for consumption
and labor, holding the wage fixed at the path that would arise without any change in reg-
ulation. The main takeaway is that our novel channel accounts for a large fraction of the
difference between the results of our full model and those of a model with only households’
deposits.

The solid line in Figure 4 reports welfare in consumption-equivalent percentage units
for our benchmark quantitative model for capital requirements ranging from 7% to 30%,
assuming that the economy begins with an 8% capital requirement. The optimal capital
requirement in this economy is 18.7%. This is much higher than the average capital require-
ment of 6% found by Davydiuk (2017) and the 12.4% optimal capital requirement computed

28As noted before, the FF accounts for privately held firms and thus produces results that might differ

from those based on Compustat.

29Our data target is slightly different from Begenau (2018) because we calibrate the model to an annual

frequency, whereas Begenau (2018) uses a quarterly frequency.

33

Figure 4. The solid line plots welfare W ((cid:16)), the dashed line plots the wel-
fare cW ((cid:16)) obtained by shutting down our novel general equilibrium effect that
acts through wages, and the dotted line plots the welfare in an economy in
which deposits are held only by households. Parameter values for the full model
used to compute W ((cid:16)) and cW ((cid:16)) are in Table 1, and parameter values for the
model with household deposits only are in Appendix E.4. Welfare is defined in
consumption-equivalent units. We compute welfare by averaging across 1,000
simulated economies for 500 periods each, where each of the 1,000 simulations is
initialized by drawing a point in the state space of the ergodic distribution of the
(cid:16) = 8% economy.

by Begenau (2018), and slightly higher than the 17% of Begenau and Landvoigt (2017) and
the 18% required to eliminate runs in Egan, Hortaçsu and Matvos (2017).30

The dotted line in Figure 4 plots the welfare in a comparable model in which only
households hold deposits. We calibrate this model to the same moments in Panel B of Table
1, which requires adjusting some parameters, and so that deposits at banks in the initial
steady state are the same as those of the full model (see Appendix E.4 for more details).
The optimal capital requirement in this model is 13.6%. This is substantially lower than
the optimal requirements in our full model when firms hold deposits (i.e., 5.1 percentage
points lower), but in line with the 12.4% derived by Begenau (2018) in her model with only
households’ deposits and endogenous banks’ risk-taking, and the Basel III “fully-phased
in” level of 14%–15% (Basel Committee on Banking Supervision, 2017). We also note two

30Although Egan, Hortaçsu and Matvos (2017) find that the optimal capital requirement is 39%, they

prefer to focus on the 18% requirement that eliminates major welfare losses due to runs.

34

51015202530capital requirement (%)-0.5-0.4-0.3-0.2-0.100.10.20.30.4welfare (consumption equiv, %)additional results that arise if one analyzes welfare through the lens of a model with only
households’ deposits, which are both captured by the difference between the dotted and solid
lines in Figure 4. First, using a model with only households’ deposits overstates the welfare
losses of setting capital requirements too high. Second, this approach understates the welfare
benefit of setting capital requirements optimally; that is, the welfare gain to transitioning
to the optimal capital requirement in the full model in which firms hold deposits is about
twice as high as in the model with only household deposits.

To investigate the source of the difference between welfare in our full model and that
in the model with only households deposits, we perform an exercise along the lines of Sec-
tion 4.3. That is, we compute the welfare cW ((cid:16)) that would arise in our full model with both
households’ and firms’ deposits if we shut down our novel general equilibrium effect that acts
through wages. This is constructed similar to Section 4.3 and is represented by the dashed
line in Figure 4 (see Appendix E.4 for more details).

The welfare cW ((cid:16)) that does not include our novel channel closely tracks the welfare of
the model with only households’ deposits. This implies that, absent our novel channel, ac-
counting for firms’ deposits alone would not alter the welfare analysis of capital requirement
regulation very much. Because of the channel we identify, however, accounting for firms’
deposits has important implications for financial regulation. In response to a higher capital
requirement, input prices (in our model, the wage) decrease, partially offsetting the costs of
the tighter requirement and driving the wedge between the dashed and solid lines in Fig-
ure 4. The welfare cW ((cid:16)) peaks at the capital requirement (cid:16) = 15:3% and, thus, our channel
accounts for about two-thirds of the difference between the 18.7% optimal requirement in
the full model and the 13.6% in the model with only households’ deposits. In addition,
our channel accounts almost entirely for the difference in the welfare gains from regulating
capital requirements optimally (i.e., the difference between the maximum values of the solid
and dotted lines). In other words, ignoring firms’ deposit holdings and our novel channel un-
derstates both the optimal capital requirement and the welfare gains of the optimal financial
regulation.

6 Conclusion

We have presented a model to study capital requirement regulation in which both households
and firms hold deposits. Our analysis differs from most of the existing literature, which
focuses on households’ deposits. Because of a novel general equilibrium effect related to
firms’ deposits, the optimal capital requirement is substantially higher than in comparable
models that only allow households to hold deposits.

35

This paper opens up several directions for future research. First, depending on how firms’
idiosyncratic productivities evolve over time, the value of deposits to firms might be time
varying. This effect might give rise to important implications for the analysis of time-varying
regulation above and beyond those analyzed by Davydiuk (2017) and Malherbe (2017), in
which the value of deposits is constant over time and time variation in optimal capital
requirements comes from time variation in banks’ overinvestment. Second, we have followed
the literature in assuming complete deposit insurance, but Egan, Hortaçsu and Matvos (2017)
show that only about half of all deposits in the United States are in fact FDIC insured. Our
In our model,
model can be employed to study the optimal degree of deposit insurance.
eliminating deposit insurance altogether increases firms’ cash-flow volatility, and thus the
optimal outcome is likely to be partial deposit insurance.

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Appendix

A Proofs

Proof of Proposition 1. We conjecture and later verify that the value function has the form

(

)

V m
t

di
t

=

(cid:12)m
1 (cid:0) (cid:12)m log

(

)

di
t

+ (cid:4)t;

(33)

where (cid:4)t is independent of di
condition (7). Since li
verified, obtaining
{

t. Then, the problem of the manager implies the first-order
t, the conjecture about the value function can be

t is independent of di

(cid:4)t =

(cid:12)m
1 (cid:0) (cid:12)m

log ((cid:20)(cid:11)) +

(cid:12)m
1 (cid:0) (cid:12)m log (1 (cid:0) (cid:11)) +

1

1 (cid:0) (cid:12)m Et log

[(

)

]}

zi
t+1

(cid:0) wt

ϕt + Rd
t

:

(34)

Proof of Proposition 2. When taking the limit as (cid:20) ! 0 and (cid:18) ! 0, we have: (i) Proposition
1 is not affected because (7) is independent of (cid:20) and (cid:18); (ii) the results about ci
t+1
follow from (3) and (4); (iii) to show that W ! V h
0 (a0), we use the results in the proof
! 0. In particular, the last result uses the fact that
of Proposition 1 to obtain (cid:18)V m
0
(cid:18)(cid:4)t ! 0 as both (cid:18) and (cid:20) tend to zero (keeping (cid:18)=(cid:20) constant), given the definition of (cid:4)t in
(34).

t and (cid:25)i

di
0

)

(

{

}

}

}

{

{

zi
t+1

Proof of Proposition 3. To show Rd < 1=(cid:12), we first establish that w < Et
. First, note
that w (cid:20) Et
, otherwise firms would make negative profits on average and thus will
not hire any workers, and the labor market will not clear. Next, assume by contradiction
that w = Et
t+1) > 0 imply that ϕ = 0
and thus firms will not hire any workers, and the labor market will not clear. Thus, the only
possible case is w < Et
. This result and equation (24) evaluated in steady state imply
Rd < 1=(cid:12) and 1=(cid:12) (cid:0) Rd = ((cid:22)z (cid:0) w)ϕ. Finally, the right-hand side of (26) can be computed
using li

. Equation (7) and the assumption that V ar(zi

g = (cid:22)z, establishing the result.

zi
t+1

zi
t+1

zi
t+1

}

{

t = ϕdi

t and Etfzi

t+1

To prove Propositions 4 and 8, we first state and prove the following intermediate lemma.

41

Lemma 6. If there exists a steady-state equilibrium with l > 0, we have

@d
@(cid:16)

< 0;

@Rd
@(cid:16)

< 0;

@w
@(cid:16)

< 0;

@ϕ
@(cid:16)

> 0;

@l
@(cid:16)

< 0;

for all (cid:17) 2 (0; 1). Moreover, the sign of @d=@(cid:16) is also preserved in the limit as (cid:17) ! 0 and
(cid:17) ! 1, whereas @Rd=@(cid:16) < 0 as (cid:17) ! 0 but @Rd=@(cid:16) ! 0 as (cid:17) ! 1.

Proof of Lemma 6. We totally differentiate (7), the law of motion of deposits (2) evaluated
in steady state and integrated over i, the labor demand equation l = ϕd, and the first-order
condition of banks (13). Thus, we obtain a system of four equations in four unknowns, where
the unknowns are @w=@(cid:16), @ϕ=@(cid:16), @Rd=@(cid:16), and @d=@(cid:16). To derive the results, it is useful to
define the following variables:

8
<

:
8
<

:
8
<

:
{

A (cid:17) E

B (cid:17) E

C (cid:17) E

D (cid:17) E

9
=
; > 0
9
=
; > 0
9
=
; > 0

]
2

Rd
[
ϕ (z′ (cid:0) w) + Rd

]

2

(z′ (cid:0) w)2
[
ϕ (z′ (cid:0) w) + Rd

]

2

(cid:0)

[

(z′ (cid:0) w)
ϕ (z′ (cid:0) w) + Rd

1
ϕ(z′ (cid:0) w) + Rd

}

> 0:

The signs of A, B, and D hold because the argument of the expectation is positive for all
states. The sign of C follows from three remarks. First, C is a cross-partial derivative of the
manager’s objective function with respect to Rd and ϕ, that is,

@
@Rd

@ (manager’s objective function)
@ϕ

= C;

where

{

(

)}

(manager’s objective function) = (cid:12)mEt

(cid:18) log ci

t+1 + V m
t+1

di
t+1

:

Second, since the objective function of the manager is concave in ϕ (i.e., the first-order
condition with respect to ϕ pins down a maximum), then its derivative with respect to ϕ is
locally decreasing in ϕ. Third, (7) implies a constant ratio ϕ=Rd, and thus a marginal increase
in Rd implies an increase in ϕ. Thus, C must be positive when evaluated at equilibrium
values.

42

We can then solve the system of four equations in four unknowns described before. For

deposits, we obtain
(

@d
@(cid:16)

= (cid:0)

(cid:0) Rd
[

1
(cid:12)
∫ 1
A(1(cid:0)(cid:13))(cid:13)
" "dF (")
(d2(cid:0)(cid:13))(1(cid:0)(cid:16))(cid:13)(cid:0)1

∫ 1
" dF (") +

∫ 1
A(1(cid:0)(cid:13))(cid:13)
" "dF (")
(d1(cid:0)(cid:13))(1(cid:0)(cid:16))(cid:13)(cid:0)1

]

dD +

(cid:17)(B+C((cid:22)z(cid:0)w))
w1(cid:0)(cid:17)(cid:31)(cid:17)

+ (1 (cid:0) (cid:16))

) (

dD +

∫ 1
" dF (") ϕ

(cid:17)(B+C((cid:22)z(cid:0)w))
w1(cid:0)(cid:17)(cid:31)(cid:17)
[
A ((cid:22)z (cid:0) w) + Bϕ

] < 0;

)

where the inequality follows from the fact that both the numerator and the denominator
on the right-hand side are positive, using Rd < 1=(cid:12) and w < (cid:22)z (which must both hold,
otherwise firms would make negative profits) and the signs of A, B, C, and D established
before. The sign of the derivative is preserved in the limit as (cid:17) ! 0 and (cid:17) ! 1, using the
labor supply equation (19).

Similarly, for the return on deposits, we obtain

@Rd
@(cid:16)

= (cid:0)

(

∫ 1
A(1(cid:0)(cid:13))(cid:13)
" "dF (")
(d2(cid:0)(cid:13))(1(cid:0)(cid:16))(cid:13)(cid:0)1

(

)
A ((cid:22)z (cid:0) w) + Bϕ

ϕ

(

1
(cid:12)

(cid:0) Rd
)

)

∫ 1
" dF (") +
∫ 1
" dF (") ϕ

∫ 1
A(1(cid:0)(cid:13))(cid:13)
" "dF (")
(d1(cid:0)(cid:13))(1(cid:0)(cid:16))(cid:13)(cid:0)1
[

+ (1 (cid:0) (cid:16))

A ((cid:22)z (cid:0) w) + Bϕ

] < 0;

dD +

(cid:17)(B+C((cid:22)z(cid:0)w))
w1(cid:0)(cid:17)(cid:31)(cid:17)

and taking the limit as (cid:17) goes to zero or 1, we can establish the respective result.

For the wage, we obtain

(

)
B + C ((cid:22)z (cid:0) w)

ϕ

(

@w
@(cid:16)

= (cid:0)

(

∫ 1
A(1(cid:0)(cid:13))(cid:13)
" "dF (")
(d2(cid:0)(cid:13))(1(cid:0)(cid:16))(cid:13)(cid:0)1

dD +

(cid:17)(B+C((cid:22)z(cid:0)w))
w1(cid:0)(cid:17)(cid:31)(cid:17)

1
(cid:12)

(cid:0) Rd
)

)

∫ 1
" "dF (")

A(1(cid:0)(cid:13))(cid:13)

∫ 1
" dF (") +
∫ 1
" dF (") ϕ

+ (1 (cid:0) (cid:16))

(d1(cid:0)(cid:13))(1(cid:0)(cid:16))(cid:13)

[

A ((cid:22)z (cid:0) w) + Bϕ

] < 0;

and for the manager’s risk-taking choice ϕ we obtain

@ϕ
@(cid:16)

=

∫ 1
A(1(cid:0)(cid:13))(cid:13)
" "dF (")
(d2(cid:0)(cid:13))(1(cid:0)(cid:16))(cid:13)(cid:0)1

(

ϕD
(

(cid:0) Rd

1
(cid:12)

∫ 1
" dF (") +
)

)

A(1(cid:0)(cid:13))(cid:13)

∫ 1
" "dF (")

(d1(cid:0)(cid:13))(1(cid:0)(cid:16))(cid:13)

dD +

(cid:17)(B+C((cid:22)z(cid:0)w))
w1(cid:0)(cid:17)(cid:31)(cid:17)

+ (1 (cid:0) (cid:16))

∫ 1
" dF (") ϕ

[

A ((cid:22)z (cid:0) w) + Bϕ

] > 0:

Finally, using l = ϕd, totally differentiating with respect to (cid:16), and using the previous

results, we obtain

(

1
(cid:12)

(cid:0) Rd
(

∫ 1
" dF (") +

A(1(cid:0)(cid:13))(cid:13)

∫ 1
" "dF (")

@l
@(cid:16)

= (cid:0)

∫ 1
A(1(cid:0)(cid:13))(cid:13)
" "dF (")
(d2(cid:0)(cid:13))(1(cid:0)(cid:16))(cid:13)(cid:0)1

dD +

(cid:17)(B+C((cid:22)z(cid:0)w))
w1(cid:0)(cid:17)(cid:31)(cid:17)

) [

ϕ

]

(cid:17)(B+C((cid:22)z(cid:0)w))
w1(cid:0)(cid:17)(cid:31)(cid:17)
[

∫ 1
" dF (") ϕ

A ((cid:22)z (cid:0) w) + Bϕ

] < 0:

(d1(cid:0)(cid:13))(1(cid:0)(cid:16))(cid:13)
)

+ (1 (cid:0) (cid:16))

43

Proof of Proposition 4. For (cid:17) 2 (0; 1), the results are shown in Lemma 6. For the case
(cid:17) ! 1 and (cid:17) ! 0, the results follow as corollaries of the proof of Lemma 6, using the labor
supply (19) and taking the appropriate limits with respect to (cid:17).

Proof of Proposition 5. Define the value of the firm before dividends as V f
, from the
t
point of view of shareholders. This value corresponds to the present discounted stream
of dividends, discounted using the stochastic discount factor of households, and where the
choice of dividends is made optimally to maximize households’ utility. Let (cid:11)i
t be the fraction
of firm i wealth paid out as dividends. Then,

xi
t

(

)

(

)

V f
t

xi
t

= max
(cid:11)i
t

t xi
(cid:11)i

t + (cid:12)Et

{(

(cid:3)t+1
(cid:3)t

)

(

}

)

V f
t+1

xi
t+1

;

where (cid:3)t is the households’ marginal utility of consumption and

xi
t+1 =

(

zi
t+1
[(

)

(cid:0) wt

lt + Rd
)

t dt

]

=

=

[(

zi
t+1

(cid:0) wt

ϕt + Rd
t

)

dt
] (

)

zi
t+1

(cid:0) wt

ϕt + Rd
t

1 (cid:0) (cid:11)i
t

xi
t;

and the last line uses dt =
the manager.

)

(

1 (cid:0) (cid:11)i
t

xi
t. Note that ϕt is taken as given because it is chosen by

The first-order condition with respect to (cid:11)i

t implies

{(

(cid:3)t+1
(cid:3)t

1 = (cid:12)Et

) [(

)

] (

zi
t+1

(cid:0) wt

ϕt + Rd
t

V f
t+1

}

)

)′ (

xi
t+1

(35)

where the marginal value of the firm can be computed recursively using the envelope condi-
tion

)′ (

(

V f
t

)

xi
t

= (cid:11)i

t + (cid:12)Et

{(

(cid:3)t+1
(cid:3)t

) (

) [(

)

] (

1 (cid:0) (cid:11)i
t

zi
t+1

(cid:0) wt

ϕt + Rd
t

V f
t+1

)′ (

}

)

xi
t+1

:

44

The previous two equations imply

)′ (

(
V f
t

)

xi
t

= 1 for all t, and thus (35) simplifies to

{(

(cid:3)t+1
(cid:3)t

) [(

)

]}

zi
t+1

(cid:0) wt

ϕt + Rd
t

:

1 = (cid:12)Et

We then note that zi
t+1 is i.i.d. and independent of all other endogenous variables at t + 1,
and thus it can be replaced with its expectation (cid:22)z, obtaining (27). In addition, the condition
that pins down (cid:11)i

t is the same for all firms, and thus (cid:11)i

t = (cid:11)t for all i.

B Benchmark with no shocks: Modigliani-Miller

This appendix characterizes the equilibrium in a version of the model in which we shut
down all shocks—the idiosyncratic shocks on firms and those on banks. The objective is to
establish a benchmark that can be used as a comparison for the analyses that follow.

An implication of shutting down the shocks to banks is that there is de facto no deposit
insurance. This is because banks’ profits are fully deterministic and thus no bank fails in
equilibrium, implying that deposit insurance disbursements are zero.

In this benchmark scenario, Modigliani-Miller holds in the sense that the equilibrium
is independent of how banks’ assets are financed. We view this result as a “check” that
our framework allows for the Modigliani-Miller theorem to hold once we shut down deposit
insurance and the uninsurable idiosyncratic risk of firms. The restriction on (cid:11) imposed in
(27) is essential to obtain such a result.

To overcome the indeterminacy of Modigliani-Miller, we assume that capital require-
ments are imposed anyway by the regulator, even though there is no deposit insurance
disbursement, and that they are satisfied with equality: nt=kt = (cid:16).

(cid:17) 1 and " (cid:17) 1.
Proposition 7. (Benchmark equilibrium without any shock) Suppose zi
Given (cid:16), the equilibrium is characterized by prices Rd = Rn = 1=(cid:12), w = 1; banking variables

t+1

[(

)

](cid:0) 1

1(cid:0)(cid:13)

k =

(cid:0) (1 (cid:0) (cid:14))

1
(cid:12)

1
A(cid:13)

, d = k (1 (cid:0) (cid:16)), and n = (cid:16)k; labor l =

(

)

(

)
1=(cid:31)

(cid:17); taxes T = 0; and

consumption of households c =

1=(cid:31)

(cid:17) + Ak(cid:13) (cid:0) (cid:14)k.

Proof. Since there are no idiosyncratic shocks to firms, the first-order condition (7) implies
w = 1, which in turn implies the equilibrium value of labor stated in the proposition, using
(19). The law of motion of deposits, (2), evaluated in steady state (i.e., at di
t+1) and
using the restriction on (cid:11) in (23), implies Rd = 1=(cid:12). Given this result, and since no bank fails
in equilibrium because shocks are shut down, equation (13) implies 1 (cid:0) (cid:14) + r = 1=(cid:12) which,

t = di

45

together with the first-order condition of bank-financed players, (15), implies the value of
capital k stated in the proposition. The value of deposits and equity follows from the bank’s
budget and capital requirement constraints, (9) and (10). Taxes T = 0 follow from the fact
that no bank fails in equilibrium, and consumption follows from Walras’ law.

C Formalizing the measure of managers in the econ-

omy

In this appendix, we provide a simple extension to the baseline model in which we formalize
the measure of managers in the economy, and we show that, when taking such measure to
be small, the social welfare function includes only households’ welfare, as in Proposition 2.
Crucially, this is the case even if the Pareto weight on managers and households is the same.
Our objective is to formalize the idea that (i) each firm has a different manager, (ii)
managers’ compensations are small relative to firms’ total dividends, and (iii) managers are
a small fraction of the population. At the same time, we also want to (iv) avoid the problem
of the baseline model that managers’ consumption is driven to zero as (cid:20) ! 0. Formalizing
these concepts in a tractable manner presents some challenges, which we discuss briefly. In
the baseline model of Section 3, the measure of firms is (implicitly) normalized to one. There,
the measure of managers is one as well, so that each firm has a different manager, but then
we run into the problem that the size of households (also normalized to one) is the same
as that of managers. If we instead try to reduce the measure of managers to make it small
relative to households, we run into the problem of not having “enough” managers to run all
the firms in the economy. Formally, these problems arise because of the need to work with
a continuum of agents for tractability reasons.

To overcome these issues, we proceed as follows. We now assume that each manager is
part of a family with (cid:22) members (e.g., if (cid:22) = 2, you can think of each manager as having
a spouse that is not part of the labor force). Since the compensation paid by firm i to the
manager is a fraction (cid:20) of the dividends (cid:25)i
t+1, each member of the family receives an equal
t+1=(cid:22) of it. We denote ci
share (cid:20)(cid:25)i
t+1 to be the consumption of each member of the family.
As a result, total consumption (cid:22)ci
t+1 of the family is financed each period with the total
compensation (cid:20)(cid:25)i

t+1, so that equation (3) is replaced by31

(cid:22)ci

t+1 = (cid:20)(cid:25)i

t+1;

(36)

and the total utility of the family is (cid:22) log ci

t+1.

31Recall that we do not allow managers to save.

46

With this structure, we can now redefine the problem of the manager, (5). We now
to be the utility of the whole manager’s family, which is given by

denote V m

(

)

di
t

t

(

)

V m
t

di
t

= max
t+1; di
t; ci
li

t+1

{

(

)}

(cid:12)mEt

(cid:22) log ci

t+1 + V m
t+1

di
t+1

subject to (2) and (36). We can then show that the first-order condition (7) is unchanged
using the same approach employed in the proof of Proposition 1, that is, conjecturing and
then verifying that the value function has the form

(

)

V m
t

di
t

= (cid:22)

(cid:12)m
1 (cid:0) (cid:12)m log

(

)

di
t

+ (cid:22)(cid:4)t;

(37)

where (cid:4)t is defined in equation (34).

As a final step, we show that if the fraction of dividends paid to managers is small (i.e.,
(cid:20) ! 0) and managers are small in comparison to households (i.e., if the size of each manager’s
family (cid:22) ! 0), we recover the result that total welfare depends only on households’ welfare:
W ! V h
0 (a0). The welfare function is given by (25) and, different from the analysis in
Section 3.3, we now do not take a stand on the Pareto weight (cid:18). This means, for instance,
that (cid:18) could be set to one to have an equal weight on households and managers.

We consider the limit as (cid:20) ! 0 and (cid:22) ! 0, with (cid:20)=(cid:22) constant along the limiting sequence.
Then, equation (36) implies that consumption ci
t+1 of each manager’s family member is not
affected by the limit—indeed, (36) implies ci
(cid:25)i
t+1, and (cid:20)=(cid:22) is constant along the
t+1 =
limiting sequence. We also note, using (37), that the utility value per member of the family,
V m
of each family goes to
t
zero as (cid:20) ! 0 and (cid:22) ! 0. This last result allows us to establish our main result, namely,
that the welfare function (25) depends only on households’ utility in the limit. Formally,

=(cid:22), is not affected by the limits, and the total value V m

(cid:20)=(cid:22)

di
t

di
t

(

)

)

(

)

(

t

W ! V h

0 (a0) for any Pareto weight (cid:18) > 0

as (cid:20) ! 0 and (cid:22) ! 0.

D Capital requirements and bad risk-taking

In this appendix, we clarify the effects of capital requirements on banks’ choices. First,
we highlight that higher capital requirements can either increase or decrease bank lending.
This result is essentially the same as in Begenau (2018), but it interacts with firms’ good
risk-taking in our model. Second, we explain why a capital requirement that is too high

47

Figure 5. Both lines in the figure plot the steady-state level of physical capital
k for values of the capital requirement (cid:16) ranging from 3% to 20%. Parameter
values are (cid:12) = 0:95, pz = 0:952, (cid:14) = 0:1, A = 0:52, (cid:27) = 0:063, (cid:31) = 0:96, and
(cid:13) = 0:3. We choose A, (cid:27), and (cid:31) to induce a deposit premium of 2%, a default
probability of 10%, and an equilibrium consumption value of 1. The dotted line
assumes a fully elastic labor supply (i.e., (cid:17) ! 1); the solid line assumes a fixed
labor supply (i.e., (cid:17) ! 0), where labor supply is fixed at (cid:22)l = 0:555. Details on
these calculations are in Appendix E.

is not optimal even if the marginal social value of deposits is zero. Finally, we link all
these theoretical results with some empirical evidence that supports the existence of a wedge
between the private and social value of deposits.

Tightening capital requirements produces two effects on the amount of bank lending k,
which we label the leverage effect and the funding effect. The leverage effect forces banks
to be financed proportionally more with equity (i.e., to reduce their leverage). Since equity
is more expensive than deposits, banks react by reducing total lending k. The funding
effect is related to how the deposit rate changes in response to a modification of capital
requirements, as noted by Begenau (2018). As tighter capital requirements make deposits
more scarce, depositors might be willing to accept a lower return Rd. Such a lower return
reduces the cost for banks to fund an additional dollar of loans, increasing k.

The leverage and funding effects are also related to the Frisch elasticity of labor supply,
(cid:17). Recall that firms hold deposits to insure against the need to pay wages in the event of a
bad productivity shock. Since wages are affected by the Frisch elasticity (cid:17), firms’ demand
for deposits is tied to this elasticity as well. When the Frisch elasticity is (cid:17) < 1, both the
leverage and funding effects are at work. If instead the Frisch elasticity is (cid:17) ! 1 (i.e., if
labor demand is fully elastic), only the leverage effect operates; in this case, wages do not

48

2468101214161820capital requirement (%)1.11.151.21.251.31.351.41.451.51.551.6steady-state krespond to changes in capital requirements, and neither do the private benefits of holding
deposits.

We summarize these points in the following proposition.

Proposition 8. (Capital requirements and bad risk-taking) If (cid:17) < 1, increasing the capital
requirement (cid:16) produces two effects:

• Leverage effect. Fixing Rd, banks’ debt-to-equity ratio d=n decreases and banks’ lending
rate r increases; this effect reduces the equilibrium value of physical capital k and thus
reduces the size of the banking sector;

• Funding effect. The deposit rate Rd decreases; this puts downward pressure on the
banks’ lending rate r, generating an increase in the equilibrium value of physical capital
k and thus increasing the size of the banking sector.

If instead (cid:17) = 1, only the leverage effect operates.

Proof. Combining (15), (20), and (13) evaluated at the nonstochastic steady state, we obtain

∫ 1

(

"

"

1 (cid:0) (cid:14) + A(cid:13)k(cid:13)(cid:0)1

)

dF (") = (cid:16)

1
(cid:12)

∫ 1

+ (1 (cid:0) (cid:16)) Rd

dF (")

"

Totally differentiating with respect to (cid:16), we have

∫ 1

(

"

"

A(cid:13)((cid:13) (cid:0) 1)k(cid:13)(cid:0)2 @k
@(cid:16)

)

dF (") =

[

1
(cid:12)

∫ 1

]

(cid:0) Rd

dF (")

+ (1 (cid:0) (cid:16))

"

@Rd
@(cid:16)

∫ 1

"

dF (") :

Fixing Rd, and since the term in square brackets on the right-hand side is positive (because
Rd < 1=(cid:12) in equilibrium), then capital k drops when (cid:16) marginally increases (leverage effect).
The funding effect follows from the fact that Rd weakly decreases, as established in

Lemma 6 (see Appendix A).

An implication of Proposition 8 is that the effect of increasing capital requirements on
total bank lending k is ambiguous if (cid:17) < 1. If instead (cid:17) ! 1, bank lending k unambiguously
decreases. Figure 5 illustrates these results with a numerical example. The figure plots the
steady-state value of bank assets k for various values of capital requirement (cid:16) and for the
two extreme cases of (cid:17) ! 1 and (cid:17) ! 0.

When (cid:17) ! 1 (i.e., dotted line in Figure 5), higher capital requirements always reduce k
because only the leverage effect is at work. When (cid:17) ! 0 (i.e., solid line), both the leverage
and funding effect are at work.
In this case, lending by banks (i.e., capital k) decreases
for levels of the capital requirement up to 11.4%. However, as the capital requirement

49

is increased above 11.4%, banks’ lending k goes up. This is because the leverage effect
dominates for values of (cid:16) below 11.4%, whereas the funding effect dominates as (cid:16) is pushed
above 11.4%.

The existence of the leverage effect explains why the optimal capital requirement is not
too high even when the marginal social value of deposits is zero. In such a case, the objective
of the capital requirement is solely that of offsetting the negative effects of subsidized deposit
In particular, in this simple
insurance, with no consideration for the effects on deposits.
model, deposit insurance gives rise to an incentive for banks to lend too much.
In the
numerical example in Figure 5, the best way to offset deposit insurance is to target the
minimum level of k that can be achieved, corresponding to a 11.4% capital requirement.
However, the optimal capital requirement (computed in Figure 2) is 8.7% because it accounts
for the transition from the old to the new steady state.

When mapping the theory to the data, Begenau (2018) notes that the return on deposits
Rd responds to changes in the quantity of deposits. In both her model and ours, this feature
gives rise to the funding effect, which runs counter to the conventional intuition that tighter
capital requirements reduce bank lending. Crucially, in our model, the existence of a funding
effect is associated with a Frisch elasticity of labor supply (cid:17) < 1, which in turn produces a
wedge between the marginal and private value of deposits, as shown in Proposition 4. This
wedge lowers the cost of increasing capital requirements, as we describe in Section 4.3.

E Solution method

In this section we describe the numerical method for solving the numerical examples in
Section 4 and Appendix D, and the quantitative model in Section 5. We take all parameters
as given and constant.

E.1 Steady state of the baseline model of Section 3

We describe how we solve for the steady state of the baseline model of Section 3. We use
this method to generate initial conditions to perform the policy experiments represented in
Figures 2 and 3, and to produce Figure 5. We denote steady-state variables by dropping all
time subscripts.

First, because the deposit premium is positive, as shown by Proposition 3, the capital

50

constraint always binds. Thus, equations (9) and (10) together imply that

n = (cid:16)k
d = (1 (cid:0) (cid:16)) k:

Equation (15) can be used to solve for k,

and equation (24) implies that

] 1
1(cid:0)(cid:13)

;

[

(cid:13)A
r

k =

Rd =

1
(cid:12)

(cid:0) ((cid:22)z (cid:0) w) ϕ:

(38)

(39)

(40)

(41)

To solve for the steady state, we guess a value for w. Given this value for w, we solve
t+1 can only take the two values of 0 and 1=pz, we have

equation (7) for ϕ, noting that since zi
ϕt in closed form:

ϕt = Rd
t

1 (cid:0) wt
(
1
pz

(cid:0) wt

):

wt

(42)

We then have Rd from equation (41). We then plug (12), (20), and (39) into (13) and,
using the value of Rd just computed, we solve for r. In particular, since " is log-normal with
mean 1, we have

∫ 1

"

"dF (") = 1 (cid:0) (cid:8)

Pr f" (cid:21) "g = 1 (cid:0) (cid:8)

{

{

1
(cid:27)
1
(cid:27)

log " (cid:0) 1
2
1
2

log " +

(cid:27)

(cid:27)

}

}

;

;

where (cid:8) f(cid:1)g denotes the standard normal CDF. Given r, we can find n, d, and k from
equations (38)-(40). Finally, we search over values of w 2 (0; (cid:22)z) to satisfiy equation (19),
given l = ϕd (from Proposition 1) computed using the implied values of d and ϕ.

E.2 Changing capital requirements at t = 0: numerical examples

of Section 4

We now describe how we solve the model to compute the welfare plotted in Figures 2 and 3.
We perform the following policy experiment. Given the economy in steady state with an 8%

51

capital requirement, we vary the capital requirement to a new level, ranging from (cid:16) = 3% to
(cid:16) = 20%. For each new level of (cid:16) in this range, we solve for the equilibrium transition path
to the new steady state. We then compute welfare over this path, W ((cid:16)), which is equal to
the value function of households V h

0 (a0), as discussed in Section 3.3.32

Assume the economy is in steady state at t = 0. At the start of t = 1, a new capital
requirement (cid:16) is announced.33 Deposits dt are predetermined at time t(cid:0)1 from equation (24).
Thus, d1 = d0.

We solve the model at each t recursively. Suppose we know dt. Then (9) and (10)
imply kt = dt
, and rt+1 comes directly from equation (15). We then plug equation (12)
1(cid:0)(cid:16)t
into equation (13) and solve numerically for Rd
t , we plug equation (19) into
equation (7) and solve numerically for ϕt (and then recover wt). Then, given ϕt, wt, and Rd
t ,
we use equation (24) to compute dt+1, and we move to the next t.

t . Given Rd

Consumption at t + 1 can be computed using the resource constraint of the economy.
The amount of resources available at the beginning of t + 1 is given by the output produced
by firms, (cid:22)zlt, plus the output produced by the bank-financed sector, Ak(cid:13)
t , and is used for
consumption ct+1 and investments. Since investments can be expressed as kt+1 (cid:0) (1 (cid:0) (cid:14)) kt,
we have

[

]

ct+1 = (cid:22)zlt + Ak(cid:13)
t

(cid:0)

= (cid:22)zϕtdt + Ak(cid:13)
t

kt+1 (cid:0) (1 (cid:0) (cid:14)) kt
(cid:0) (cid:14)kt (cid:0) (kt+1 (cid:0) kt) ;

(43)

where the second line uses lt = ϕtdt from Proposition 1 and rearranges.

t ((cid:16)); dt((cid:16)); kt((cid:16))g1

To compute the welfare cW ((cid:16)) that does not include our novel channel, we proceed as
follows. Let fRd
t=0 be the equilibrium value of the return on deposits, de-
posits, and capital obtained by changing the capital requirement to (cid:16) at t = 0. As a first
step, we compute the level of firms’ risk taking ^ϕt((cid:16)) that would be chosen by a firm that
faces the correct return on deposits Rd
t ((cid:16)) and the “wrong” wage w(cid:0)1 (i.e., the wage that
would arise if the capital requirement had not changed). To do so, we solve for ^ϕt((cid:16)) using
t ((cid:16)). Second, we compute the labor demand ^lt((cid:16))
(7) evaluated at wt = w(cid:0)1 and Rd
using ^ϕt((cid:16)) and the correct equilibrium value of deposits: ^lt((cid:16)) = ^ϕt((cid:16))dt((cid:16)). Third, we solve
for the consumption ^ct((cid:16)) that would arise if employment is ^lt((cid:16)), using (43) evaluated at

t = Rd

32Although the economy only approaches the new steady state asymptotically, we find that 1,000 time

periods is sufficient for welfare to converge.

33Although we assume that the capital requirement is changed once and for all at t = 0, this solution
method in this simple model with quasi-linear utility works if (cid:16) is time varying and follows any deterministic
path.

52

lt = ^lt((cid:16)) and at the correct equilibrium value of capital kt((cid:16)):

^ct+1((cid:16)) = (cid:22)z^lt((cid:16)) + Akt((cid:16))(cid:13) (cid:0)

]
[
kt+1((cid:16)) (cid:0) (1 (cid:0) (cid:14)) kt((cid:16))

:

We then compute welfare cW ((cid:16)) by evaluating households’ utility at f^ct((cid:16)); ^lt((cid:16))g1

t=0.

E.3 Quantitative model

Households’ and banks’ problems. The first-order conditions for lt and nt for prob-
lem (30), after plugging in the budget constraint, are

wtEt f(cid:3)t+1g = (cid:31) (lt)1=(cid:17)
(

{

1 = Et

(cid:12)

)

}

Rn

t+1

(cid:3)t+1
(cid:3)t

(44)

(45)

where (cid:3)t is the marginal utility of consumption, given by

2

(cid:3)t = c

(cid:0)(cid:13)c
t

41 (cid:0)  

)

1(cid:0)(cid:13)d

(

dh
t
ct

3

5 :

1
c1(cid:0)(cid:13)c

t

Equation (44) would be a standard labor supply curve if it were not for the expectation on
the left-hand side. This expectation appears in (44) because wages are earned at t + 1 but
labor is chosen at t. The expected return condition for bank equity (45) is standard.

The first-order condition for households’ deposits is

(

dh
t
ct

c1(cid:0)(cid:13)c

t

(cid:0)  

)(cid:0)(cid:13)d

{

)

1(cid:0)(cid:13)d

= Et

(cid:12)

(

dh
t
ct

} (

)

Rf
t

(cid:0) Rd
t

(cid:3)t+1
(cid:3)t

(46)

[

{

}](cid:0)1

where Rf
t

(cid:17)

Et

(cid:12) (cid:3)t+1
(cid:3)t

. The left-hand side is the marginal benefit of an additional $1

of deposits for households—additional utility from deposits—while the right-hand side is the
marginal cost—deposits earn less than the risk-free rate. In this sense,   is the “intercept”
of households’ deposit demand, and (cid:13)d controls the slope of their demand curve.

The bank’s problem (28), after plugging in equation (29) and the constraints and rear-

ranging, becomes

∫ {

max
pt

Et

" (1 (cid:0) (cid:14) + rt+1) (cid:0) (1 (cid:0) pt) Rd

t (1 (cid:0) (cid:16)) (cid:0) (cid:21)pt

}
(1 (cid:0) pt)(cid:23) (1 (cid:0) (cid:16))

+

dF (")

53

 
for which the first-order condition is equation (32), which we solve numerically for pt. Notice
also that the default threshold "t+1 becomes

"t+1 = (1 (cid:0) pt) (1 (cid:0) (cid:16))

Rd
t + (cid:21) (pt)
1 (cid:0) (cid:14) + rt+1

:

The left-hand-side of equation (32) is the marginal benefit of increasing pt, while the right-
hand side is the marginal cost. The marginal cost is (cid:21) at pt = (cid:24)t = 0 and rises to infinity as
pt ! 1, so if Rd
t > (cid:21), (cid:24)t = 0 and there is a single solution to equation (32) for pt; otherwise
(cid:24)t = (cid:21) (cid:0) Rd

t and the optimal pt = 0.

Numerical solution method.
In this economy there is a single exogenous state variable,
At, and two endogenous state variables, which we denote Yt and Xt. The variable Yt is total
output available for consumption and future investment. The variable Xt is total wealth
held by firms before they make dividend payments. That is,

t(cid:0)1 + Atk(cid:13)

Yt (cid:17) ϕt(cid:0)1df
[
((cid:22)z (cid:0) wt(cid:0)1) ϕt(cid:0)1 + Rd

Xt (cid:17)

t(cid:0)1 + (1 (cid:0) (cid:14)) kt(cid:0)1 (cid:0) (cid:21)
]

df
t(cid:0)1:

t(cid:0)1

pt(cid:0)1
(1 (cid:0) pt(cid:0)1)1+(cid:23)

(

)

t(cid:0)1 + df
dh

t(cid:0)1

Notice that Xt is predetermined at t (cid:0) 1, much like dt was predetermined at t (cid:0) 1 in the
model of Section 3. However, in this model (cid:11)t is a choice variable of firms, so that df
t satisfies

df
t = (1 (cid:0) (cid:11)t) Xt;

where the dependence of df
capital constraint (10) becomes

t on the state at t comes from its dependence on (cid:11)t. The bank’s

nt (cid:21) (cid:16)kt
(

= (cid:16)

nt + dh

t + df

t

)

:

We solve the model globally on a grid of values for the state variables. Instead of using

Yt and Xt, however, we define

!t (cid:17) log Xt (cid:0)  0 (cid:0)  1 log Yt;

where  0 and  1 are constants, and we solve the model over log Yt and !t. Solving the model
on a grid for !t rather than log Xt is more accurate because log Yt and log Xt are correlated

54

in equilibrium, so that solving on a grid of log Y and log X would include node points that
occur with extremely low probabilities in the ergodic distribution. In addition, instead of
using equation (46) to solve for households’ deposits directly, it is more convenient to instead
solve for

(cid:7)t (cid:17)  

)

1(cid:0)(cid:13)d

(

dh
t
ct

1
c1(cid:0)(cid:13)c

t

:

Given ct, there are simple one-to-one mappings between (cid:7)t, (cid:3)t, dh
t =ct. However,
in practice (cid:7)t is easier to interpolate between node points, which we must do to compute
expectations of (cid:3)t+1=(cid:3)t, as described below. In a similar vein, for fixed values of the right-
hand side of equation (46), the left-hand side is monotonically decreasing in dh
t =ct, so we
solve it in dh
t and (cid:7)t at each node point. For the
model to be sensible, we require (cid:7)t 2 (0; 1). To enforce this, rather than interpolate (cid:7) as a
Chebyshev polynomial, we instead interpolate ^(cid:7), where

t =ct and back out the implied values of dh

t , and dh

^(cid:7) (cid:17) log

(

)

:

(cid:7)
1 (cid:0) (cid:7)

This ensures ^(cid:7) can range between (cid:0)1 and 1 even as (cid:7) remains in (0; 1), not only on the
node points but between them as well.

We choose six points for log At using the method of Rouwenhorst (1995), and for each
t , wt, ϕt, ^(cid:7)t, and log ct as fifth-order Chebyshev polyno-
log At point we approximate (cid:11)t, Rd
mials on a grid of 36 points for log Yt and !t. In particular, at each of the 6 (cid:2) 36 = 216
node points, we solve equations (7) and (44) for wt and ϕt, given df
t , and equa-
tions (27) and (45) for Rd
t and (cid:11)t, given pt, ϕt, and wt. Consumption ct comes from the
household’s budget constraint

t and Rd

ct = Yt (cid:0) kt

= Yt (cid:0) 1 (cid:0) (cid:11)t
1 (cid:0) (cid:16)t

Xt;

which we then use to solve equation (46) for ^(cid:7)t. We then solve equation (32) for the optimal
pt, setting pt = 0 if Rd

t < (cid:21).

)

(

This procedure requires a guess for
t ; dh
t

to solve for (ϕt; wt) and a guess for (ϕt; wt; pt)
, in addition to expectations of (cid:3)t+1, at each node point. Given Rd
t ,
to solve for
we solve for (wt; ϕt) by searching over values of wt to satisfy equation (19) after plugging in
equation (42).

(cid:11)t; Rd

(cid:11)t; Rd
t

(

)

55

(

)

We solve equations (27) and (45) for

(cid:11)t; Rd
, given (wt; ϕt) and using a guess for the
t
interpolated log c and ^(cid:7) functions of the state (i.e., evaluated at any state) that allows us to
compute expectations of (cid:3)t+1 at time t for each node point. We then solve equation (46) for
and using the guess for pt. Interpolating log c and ^(cid:7) on
dh
t =ct given our solution for
the grid rather than c and dh allows us to ensure that consumption and the marginal utility
of consumption are always positive, not just at the node points but at every point between
them as well.

(cid:11)t; Rd
t

(

)

)

After solving for Rd
t at each node point, we update the value of pt by inverting equa-
(
, which we linearly interpolate on a grid of 1,000 points for Rd
Rd
tion (32) to get pt = p
t
t
from (cid:21) to 1.06. We have pt = 0 whenever Rd
t < (cid:21). This “kink” at Rd
t = (cid:21) is why we do
not interpolate pt using Chebyshev polynomials at the node points, as we do for the other
Rd
dynamically during simulations,
endogenous state variables; instead we compute pt = p
t
since this function depends only on parameters and Rd
t .

)

(

We start with a guess for the value of each variable at each node point from the quasi-
linear utility model and iterate until the maximum percentage change across the six endoge-
from iteration to iteration is zero to four decimal places
nous variables
in absolute value. We use level changes (not percentages) for p since p is zero at some node
points.

Rd; ϕ; w; (cid:11); p; dh

{

}

After solving the model equations, we simulate 2,500 economies for 700 periods each
in order to verify that the model remains within the assumed bounds for the endogenous
state variables. Each simulation starts at the steady state of the quasi-linear utility model.
To estimate the implied moments of the model, we compute averages over time and over
the 2,500 simulated economies, throwing away the first 200 observations of each simulation
to reduce dependence on the initial state. We simulate the model using the full Gaussian
distribution for log A, linearly interpolating Chebyshev polynomials between the node points
in the Markov chain, rather than simulating the Markov chain approximation of log A itself.
To compute welfare for a change to (cid:16), we simulate 1,000 economies, using the (cid:16) = 8%
model, for 400 periods. We then take the 2,500 points in the final period as the initial point
for each value of (cid:16) we consider; that is, we discard the first 399 points and assume that at
t = 401, the new capital requirement is (cid:16). This implies no change to A, log Y , or !, but it
does change the endogenous policy functions (cid:11), ϕ, w, and Rd as a function of the state. We
then simulate 1,000 economies of 500 periods each for 100 values of (cid:16) between 0.07 and 0.3
to compute welfare.

For the model parameters reported in Table 1, we use  0 = 0:35 and  1 = (cid:0)0:22. For
these values of  0 and  1, we find that for values of (cid:16) ranging from 7% to 30% we can solve
the model on a grid of (1:3; 1:7) for log Y and ((cid:0)0:15; 0:15) for !. For the higher values of

56

(cid:16) in the full model, we need to decrease the lower bound for ! a bit, to -0.169.

E.4 Additional Welfare Calculations

This section describes the computation of welfare in the model with only households’ deposits
(i.e., the dotted line in Figure 4) and the computation of welfare in the full model when
shutting down the novel channel that acts through wages (i.e., the dashed line in Figure 4).
To compute the dotted line in Figure 4, we solve a version of the model without firm
deposits and calibrate it to the same moments as in Table 1. To do so, we assume that
pz = 1, so that firms face no idiosyncratic risk. As a result, equation (7) simplifies to wt = 1,
and equation (44) pins down the level of labor lt. To calibrate the model, we use the same
fixed parameters in Panel A of Table 1, and we target the same moments of Panel B of
Table 1, albeit with two differences. First, because we set pz = 1, we ignore the continuers
employment growth target. Second, because firms do not hold deposits in this model, we
now set   to match the deposit premium and (cid:31) to match the average quantity of labor as in
the full model. Notice that, by construction, this implies that the two models will have the
same average quantity of deposits as well. This calibration implies (cid:31) = 0:324, (cid:27) = 0:0311,
(cid:23) = 0:1435, (cid:21) = 1:03404,   = 0:0815, (cid:13)d = 2:43, and (cid:27)A = 0:00635.

The model with only households’ deposits has only a single endogenous state, Yt. How-
ever, for consistency, we solve it on the same grid for log Y on which we solve the full model
(i.e., 6 Chebyshev zeroes in [1:3; 1:7]). Apart from the lower dimension, we solve this model
exactly as we solve the full model, that is, as described in Appendix E.3.

To compute the welfare cW ((cid:16)), which is represented by the dashed line in Figure 4,
we follow an approach similar to that of Section 4.3. The complication is that the full
quantitative model includes aggregate risk and, thus, the wage wt in the baseline scenario—
in which the capital requirement remains unchanged at the 8% level—is not constant over
time. More precisely, the path of fwtg depends on the path of the exogenous productivity
fAtg. Nonetheless, we can follow the approach of Section 4.3 as long as we “integrate over” all
the possible paths of At. First, note that when we change the capital requirement from 8% to
(cid:16) 2 [7%; 30%], we draw 1,000 exogenous paths of At, denoted by
for j = 1; : : : ; 1; 000,
and then we compute the overall welfare by averaging out across the welfare in each path.34
To compute cW ((cid:16)), we follow the procedure described in Section 4.3 for each path j, and
}
A(j)
then we average out the results. That is, for any given j, we use the exogenous path
t

A(j)
t

{

}

{

{

}

to compute the path of the wages
unchanged at 8%. Then, for each t, we compute ^ϕ(j)

that would arise if we keep the capital requirement
t using equation (42) evaluated at w(j)

t

w(j)
t

34More precisely, we use the same 1,000 paths of At for all values of (cid:16).

57

and at the correct return on deposits. We then use ^ϕ(j)
t
labor and consumption, ^l(j)
path j. Finally, we average out across j’s, obtaining cW ((cid:16)).

and ^c(j)
t

to derive the counterfactual paths of
, which we use to compute welfare along the exogenous

t

F Deposits held by firms and households

To gather data about households’ holdings of deposits, we use the Survey of Consumer
Finances (SCF).35 We measure deposits as the sum of transaction accounts and certificates of
deposit. Transaction accounts include savings, checking, and money market deposit accounts,
money market funds, and call or cash accounts at brokerages. Prepaid debit cards are
included starting in 2016. The SCF is administered every three years. We use the data from
the last survey, 2016, but we also analyze the previous waves starting in 1989.36

For firms, we use data from the Flow of Funds (FF). For comparison with the SCF, we
use data for the third quarter of the years in which the SCF is administered because the
majority of the SCF data is collected between May and December. We consider nonfinancial
noncorporate and corporate holdings of checkable deposits and currency, total time and
savings deposits, and money market fund shares. In our baseline calculations, we include
money market fund shares to make the results comparable with the households’ transaction
accounts in the SCF. However, as described below, we compute a robustness check in which
we exclude money market fund shares for both households and firms, and the results are
virtually unchanged.

In 2016, firms’ deposits as a fraction of deposits held by both firms and households is
about 1/3 (33.88%), and households’ deposits is 2/3. The result is virtually identical in
2013 (33.95%). As shown in Figure 6, the share of firms’ deposits has increased over time,
in line with the results of the literature that documents the increase in firms’ cash holdings
In levels, firms held $2.86 trillion and households held
(Bates, Kahle and Stulz (2009)).
$5.58 trillion. If we remove money market funds, the 1/3 versus 2/3 number is essentially
unchanged. In 2016, firms’ holdings of money market funds are $0.58 trillion (FF), and retail
money market funds are $0.89 trillion (Cipriani and La Spada (2017)). Assuming that retail
funds are held by households, the shares of firms’ and households’ deposits become 32.72%
and 67.28%, respectively.37

35We do not use the Flow of Funds data to measure households’ deposits because estimates for this sector

are largely residuals and are derived from data for other sectors.

36The SCF is also available before 1989, but the sample design and the core of the SCF questionnaire have

only changed in minor ways since 1989.

37As noted by Cipriani and La Spada (2017), this number is computed using data on a subset of money

market funds that covers about 90% of the total net assets. Thus, our estimate is slightly conservative.

58

Figure 6. The solid line plots firms’ deposits as a fraction of deposits held
by firms and households. The dashed line plots households’ deposits as a frac-
tion of deposits held by firms and households. The dotted line plots firms’ and
households’ deposits as a fraction of deposits at FDIC-insured banks.

For completeness, we also note that deposits are held not only by households and firms.
In the FF data, other holders include state and federal governments, the nonprofit sector,
non-US holdings, and the financial sector itself. Because these agents either do not hold
deposits in our model or are not even modeled, we do not consider their holdings of deposits.
We also compare the data we have used from the SCF and the FF with total deposits
at FDIC-insured banks, as reported by the FDIC. The share of deposits held jointly by
households and firms as a fraction of total deposits at FDIC-insured banks is 73.66% in
2016; as shown by the dotted line in Figure 6, this share was lower in 1989 and reached
its peak in 2001. While other agents hold deposits in practice, as noted above, firms’ and
households’ deposits account for a very large fraction of deposits at FDIC-insured banks. In
addition, even if deposits at FDIC-insured banks do not represent the universe of deposits,
they do account for most of them.
In 2016, deposits at FDIC-insured banks are $11.46
trillion, whereas deposits at federally insured credit unions are $1.1 trillion (National Credit
Union Administration data) and assets managed by money market mutual funds are slightly
less than $3 trillion (Cipriani and La Spada, 2017).

Finally, we note that firms’ holdings of cash-like securities are mostly in deposits at
banks. For instance, in 2016, investments in commercial paper are $0.15 trillion and those
in Treasury securities $0.11 trillion.

59

Imprint and acknowledgements 

We thank Juliane Begenau (discussant), Itay Goldstein (discussant), Gregor Matvos, Oliver Levine, Skander 
Van den Heuvel, Diego Vega San Martin (discussant), and Kairong Xiao (discussant) for helpful conversations 
and suggestions, as well as seminar participants at the University of Wisconsin-Madison, the 2018 ASSA 
annual meeting, the 2018 MFA annual meeting, the University of Texas at Austin McCombs School of Business, 
the 2018 EEA-ESEM meeting, the 2018 FDIC Annual Bank Research Conference, the 2018 FMA annual 
meeting, the 2018 Southern Finance Association Annual Meeting, UCLA, FIRS, Australian National University, 
SED, and the second Columbia Workshop in New Empirical Finance. Ikuo Takey and Vivian Wong provided 
excellent research assistance. Any remaining errors are our own. Financial support from the Wisconsin Alumni 
Research Foundation (WARF) is gratefully acknowledged. 

Author name 
Universtiy of Texas at Austin, Austin, United States; email: aaron.pancost@mccombs.utexas.edu 

Author name 
University of Wisconsin-Madison, Madison, United States; email: robatto@wisc.edu 

© European Systemic Risk Board, 2019 

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Note: 
The views expressed in ESRB Working Papers are those of the authors and do not necessarily reflect 
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affiliated. 

ISSN 
ISBN 
DOI 
EU catalogue No 

2467-0677 (pdf) 
978-92-9472-118-1 (pdf) 
10.2849/469416 (pdf) 
DT-AD-19-018-EN-N (pdf)