Bank Size, Leverage, and Financial Downturns

Chacko George

Abstract

I construct a macroeconomic model with a financial sector in which the severity of

financial downturns depends on size differences in the banking sector and average bank

leverage. In my model, a more concentrated banking sector deepens and prolongs fi-

nancial downturns for two reasons: returns on assets decrease more, and banks sharply

decrease the liquidity supply in the interbank market. Calibrated to match the observed

long-term concentration in the banking sector, the model shows that a typical adverse

financial shock leads to a 1% sharper downturn compared to an economy with a homo-

geneous banking sector. I further find that the increasing concentration in US banking

has increased the volatility of output in the real economy in response to similarly sized

financial shocks.

1 Introduction

A bank’s size matters for the way it conducts its day-to-day operations. Even if management

were identical across banks, a bank’s size would still affect the level of risk faced by its

partners. Big means a bank has more assets to keep it afloat in case it gets into trouble,

all else equal, and therefore a loan to a big bank is more likely to be repaid than a loan to

a small bank.

These differences matter for downturns. Banks rely on other banks for funds to efficiently

meet day-to-day investment demands. Because of the lower risks lenders face, big banks

rely on these funds more than small banks. In a financial downturn, lending between banks

decreases, so big banks suffer both from the initial hit from the downturn and from a second

round effect from the drop in lending between banks. The drop in demand decreases the

returns on lending, which decreases lending further. The magnitude of the long-term effects

of downturns is thus a function of the makeup of the entire banking sector.

In this paper, I build a model to capture this story. Since banks of different sizes are

connected through lending to each other (interbank lending), the composition of banks by

size will matter for the magnitude of the total drop in lending. My model, economy contains

1

a banking sector and a market for interbank lending. The banking sector in the model is

composed of many banks which differ in one dimension, which I call intermediation ability.

Due to an agency problem, banks are constrained in their borrowing from the interbank

market. More able banks are larger, are more leveraged (take on more assets for a given level

of net worth1) and borrow more from the interbank lending market. As a result, larger banks

will be less borrowing constrained than smaller banks, and will rely on interbank lending

more. A fall in the net worth of all banks will cause large banks to sharply decrease their

demand for interbank loans, decreasing the return on interbank lending, and tightening

borrowing constraints in the future.

I compare the linearized impulse response of the model with the response of a similar

model with a homogeneous banking sector. When all banks in both economies experience

the same initial drop in net worth, the economy in this model experiences a sharper drop

in output, and takes longer to return to pre-crisis levels.

Next, I use the model to study the impact of long-term changes in concentration in

the US banking industry on the magnitude of potential downturns. Over the last two

decades, concentration has made bigger banks bigger relative to small banks (higher size

variance), but also decreased the likelihood of being a big bank relative being a small bank

(left skewness). I calibrate the model to match features of the bank size distribution in

1986 and 2006. When faced with a similarly sized financial shock, the 1986 model economy

experiences a smaller drop in output.

This paper follows a growing literature in macroeconomics considering the impact of

a financial sector on financial crises. The model in this paper is based largely on the

model of [5], who incorporate an interbank lending market in a macroeconomic model with

a banking sector composed of identical banks. The agency problem in that model keeps

aggregate investment below its efficient level in normal times, and amplifies the adverse

effects of financial shocks in downturns. [2] expands the model to include a heterogeneous

banking sector, and are able to create an economy in which banking crises endogenously

arise. Banks expand their balance sheets during good times, leading to large downturns

when they do arise.

My paper differs from this related work in two ways. First, my focus is different: I am

more interested in the connection between banking industry characteristics and downturns

(and aim to adapt the model to the industry more closely in future work). Second, my

model is different: leverage, or the assets banks hold per dollar net worth, will differ with

1The difference between assets and liabilities of the bank. For this paper, think of this as the value one

would place on the bank itself.

2

size. The addition of this extra margin will deepen and lengthen recessions relative to either

of the other models. It will also turn out that this margin will cause borrowing constraints

to tighten with the variance of the size distribution - the larger the highest bank is relative

to the smallest, the tighter the borrowing constraints, the deeper the recession.

Concentration in the banking sector has been steadily increasing over the last two

decades, and a great deal of research has been generated to consider its effects on the

likelihood and severity of banking sector crises. Researchers are still divided in their opin-

ion on the matter, however. One camp argues that concentration makes the economy less

prone to crises because large banks are also more diversified, while the other says that large

banks are less disciplined by competition than smaller banks, make poorer lending choices,

and lose more in a downturn.. [1] performs a reduced-form, cross-country study, finding

that concentration is associated with fewer crises. On the other hand, the model of [3]

predicts that banks in less competitive environments charge higher interest rates to firms,

which induces firms to take on greater risk. [7] performs a cross-country study and finds

that higher concentration is associated with the higher fragility of the largest five banks.

Though my model predicts that the increased concentration in the US banking industry

leads to worse potential downturns, it does not predict that this will always be the case - if

the concentration increase happens without much growth in dispersion, potential downturns

can actually be better. This may partly explain the mixed results of the concentration

literature.

The paper proceeds as follows. I begin by presenting the model framework, describing

its solution, and discussing some general properties. Then, I lay out the two quantitative

exercises described above and briefly analyze them. A third section concludes.

2 Model

My model builds on the DSGE model of [5]. Banks will differ in a single quantity, which

I call intermediation ability. Ability differences will generate differences in investment de-

mand, which will be satisfied through the reallocation of funds through an interbank lending

market. This reallocation will then generate size differences between banks.

In order to abstract away any distributional consequences other than what I am inter-

ested in, I specify the model in a way that the size distribution does not endogenously evolve

over time.2 To do this, I will have to assume that an individual bank’s size this period will

2This is an assumption I plan to relax in future work.

3

only affect its size next period through aggregate quantities, and not through its individual

choices.

In particular, I consider an infinite-horizon economy based comprised of a continuum of

islands a ∈ [0, 1]. On each island, there is a representative household which supplies labor,

a representative firm which produces consumption goods from capital goods and labor, and

a representative bank which takes funds from the households and lend them to the firms.

Households and firms will be identical across islands, but banks will not.

I will denote a single island by a, and an ability type by κ. Because banks with the same

ability type will choose the same quantities regardless of the specific island they live on, I

will abuse notation and refer to two different things with the same symbol: the quantity

x on island a is called x(a), but the quantity x on all islands with the same ability type

(a sum across islands) is called x(κ). In case of confusion, x(κ) = x(κ(a)) ∀a ∈ Aκ, where

Aκ = a : κ(a) = κ. The mass of any set of islands A will be given by the measure µ(A).

2.1 Banks

There are many infinitely lived, risk-neutral, competitive banks on each island. On each

island, banks raise deposits from households on their island, raise interbank loans from

banks on all other islands, and invest in the firms on their island. All banks on the same

island are identical, but banks on different islands are not.

Bank differences come purely through differences in intermediation ability. After they

receive deposits but before they borrow interbank loans, all the banks on an island a receive a

random ability draw κ(a). Draws are assumed to be distributed with cumulative distribution

function κ(a) ∼ F (·). I make several assumptions on the distribution of draws:

Assumption 1. The random ability draws for each island a, κ(a) ∼ F (·), have the following

properties:

• E(κ(a)) = 1

• Any draw is real, κ(a) ∈ R.

• Draws have bounded support, so that κ(a) ∈ [κ, κ].

• The distribution of draws F admits a density function, denoted p(·).

The mean 1 assumption is not required for the existence of equilibrium, but is required

for the existence of a steady state equilibrium. Notice that the distribution of ability draws

does not depend on t, the period in which they were drawn. This is an important implicit

4

assumption - the distribution of ability draws will be identical and independently distributed

over time.

Banks with higher ability get better returns per dollar invested in firms3. I introduce

ability into the model as a productivity term in a simple bank production function, so that

if banks are producing assets S valued at price Q from liabilities and net worth L:

QS = κL (1)

Given our assumptions on the distribution of ability draws, we will be able to solve the

model in terms of two types of representative banks: those that represent the individual

banks on a single island a, and those that represent all of the banks with the same ability

in a period κ. I will refer to the first type as ”island representative”, and the second type

as ”type representative”.

Banks raise deposits in an economy-wide deposit market. At the time they visit the

deposit market, banks are identical, so that all island representative banks offer the same

deposit rate Rt to raise deposits Dt from all households. Every bank will thus demand the

same quantity of deposits, which means the type representative will hold dt(κ) = p(κ)Dt.

Banks on any island are able to borrow or lend to banks on any other island.4 Instead

of characterizing loans between individual banks, I will be interested in the net loans made

between islands, which we can characterize as loans made between the representative banks

on each island. Ability and amount borrowed are observable, so we can characterize all

interbank loans with a contract specifying the interest rate and the borrowing quantity,

(Rbt, bt(a)), where a is the island where the borrower banks live.5

In equilibrium, all banks drawing the same ability will have the same demands for

interbank loans, banks on any island with the same ability type will hold the same quantity

3Ability differences, and their correlation with size, can be motivated by several stories: bigger banks

provide more services (e.g. consulting services, business contacts) that firms value, but don’t manifest as

differences in loan interest rates. These services reduce costs for firms, and the firms realize higher returns

on projects as a result. Bigger banks may also have more productive investment hunters, so that per hour

of bank employee labor, bigger banks find a higher number of investment opportunities.4Including their own, but because banks on the same island have the same ability, they will have the

same demand for investment, and therefore no need to reallocate deposit funds among themselves5More precisely, an interbank loan contract is bilateral, that is, we should specify (Rbt(a, a

′), bt(a, a′)),

where a, a′ are the island where the borrower and lender banks live, respectively. However, since there

are a continuum of potential lenders, lenders compete with each other for borrowers across all islands. All

potential lenders then offer the same interest rate, since otherwise a borrower would go to another lender.

The borrowed amount will be borrower-dependent because of the financial friction I describe below. Then

the debt contract will have the property that (Rbt(a, a′), bt(a, a

′)) = (Rbt, bt(a)).

5

of loans, so bt(a) = bt(a′) if κ(a) = κ(a′). Calling this common quantity bt(κ), the total

quantity of loans on islands of a particular type will be∫a:κ(a)=κ bt(a)dµ(a) = p(κ)bt(κ).

Ability increases bank demands for investment. Because deposits are made before ability

is realized, deposits will not be allocated according to ability, and therefore a reallocation

of funds after ability is realized can improve the profits of all banks. This reallocation is

the primary function of the interbank lending market.

Banks lend a mixture of capital and consumption goods to the firms on the same island.

Firms can use these loans to buy new capital goods in period t, and then use the old and

any new capital to produce consumption goods in period t + 1. The firm then gives the

bank its output (less wages) along with its undepreciated capital goods, also at time t+ 1.

The bank is potentially infinitely-lived, but faces a borrowing constraint that I will

present below. In order to prevent the bank from saving its way out of this constraint,

a constant proportion σ of banks on each island exit every period. Upon exit, a bank

transfers its earnings to the household on its island. Second, new banks enter to replace the

old banks; in order to ensure that these banks have something to invest with, these banks

receive a ”start-up transfer” equal to a constant fraction ξ of the total assets of firms.

Banks carry wealth from period to period in the form of net worth, defined as the payoff

from assets less deposits and interbank loans. The net worth of the island a representative

bank at time t is given by:

nt(a) = [Zt + (1− δ)Qt(a)]ψt(σ + ξ)st−1 −Rt−1dt−1 −Rbt−1bt−1 (2)

The first term gives the island representative bank’s returns on investments in the firm:

Zt is the economy-wide representative firm’s gross profits from investments (per unit in-

vested), Qt(a) is the price of capital on island a, and st−1 is the units of capital held by

the firm in period t − 1. The second term gives the bank’s repayments for deposits and

interbank loans: bt−1 is the funds borrowed on the interbank market last period, and dt−1

is the deposits made by households last period.

Banks carry wealth from period to period in the form of net worth, defined as the payoff

from assets less deposits and interbank loans. For any island receiving ability draw κ, there

will be a positive measure of islands with the same ability draw.6 Since the distribution is

6To truly ensure that this is the case, I should have used the following setup: specify the set of islands over

the space [0, 1]×[0, 1], where a specific island is referred to by two coordinates, a = (a1, a2). Now assume that

in even periods, all islands with same first coordinate receive the same draw, so that κ(a1, a2) = κ(a1, a2),

and in odd periods, all islands with the same second coordinate receive the same draw. Then for any island

with a particular κ, there will be a positive measure of islands with the same κ. Since [0, 1]× [0, 1] has the

6

assumed to be iid across periods, this set of islands will be a microcosm of the economy in the

previous period; the net interbank loan repayment will be 0, and the total capital installed

last period will just be a fraction of aggregate capital: bt−1(κ) = 0, st−1(κ) = p(κ)Kt. Then

the net worth for all islands with the same ability will be

nt(κ) = [Zt + (1− δ)Qt(κ)]ψt(σ + ξ)p(κ)Kt − p(κ)σRt−1Dt−1 (3)

The first term gives the type-representative bank’s returns on investments in the firm: Zt

is the economy-wide representative firm’s gross profits from investments (per unit invested),

Qt(κ) is the price of capital on all islands with ability κ, and Kt is the capital installed

by all firms in the economy in period t− 1. The second term gives the bank’s repayments

for deposits and interbank loans: bt−1 is the funds borrowed on the interbank market last

period, and dt−1 is the deposits made by households last period.

ψt is a shock to the quality of capital, and the key source of uncertainty in this model;

it typically takes value 1. This kind of shock is crucial to the model, because in order to

precipitate a ”financial crisis” in the model, we need a way to exogenously affect the value of

net worth. In addition, because the price of capital is endogenous in this model, this shock

serves as an exogenous trigger for asset price dynamics that were an important feature of

recent economic downturns.

Without intermediation ability banks should balance their the dollars invested in assets

with the dollars taken as liabilities and equity, i.e. deposits, interbank loans, and net

worth. With intermediation ability, banks balance assets against the intermediated value of

liabilities and equity. We can summarize the balance sheet of the type-representative bank

with a flow of funds constraint:

Qt(κ)st(κ) = κ [nt(κ) + dt(κ) + bt(κ) ] (4)

We can now state the bank’s objective function. The type representative bank maxi-

mizes the expected value of its net worth upon exit calculated at the end of each period.

Since households receive this net worth upon the bank’s exit, we discount it by Λt,t+i, the

stochastic discount factor of the economy-wide representative household7:

Vt(κ) = Et,κ

∞∑i=1

(1− σ)σi−1Λt,t+int+i(κ) (5)

same cardinality as [0, 1], I leave this setup out of the main text.7The household on each island should value dividends at Λt,t+i(a), the stochastic discount factor of the

island representative household. However, because of an assumption I make later, expected returns on each

island will be equal at the beginning of each period. This will imply that any maximum of the problem in

the text will also be a maximum of the more precise problem.

7

We can obtain a Bellman equation from this infinite sum

Vt(κ) = Et,κΛt,t+1

[(1− σ)nt+1(κ) + σmax

dt+1

(max

st+1(κ),bt+1(κ)V (st+1(κ), bt+1(κ), dt+1)

)](6)

2.1.1 Financial Friction

In every period, banks are supposed to repay depositors and banks they borrowed from in

the previous period. Instead of repayment, however, banks can choose to default on their

loans, in which case they take a fraction θ ∈ [0, 1] of the funds. Depositors and potential

lender banks know this, and though default will not occur in equilibrium, it will impose a

limit on the amount of interbank loans any bank can borrow in a period.8

In the model, this friction will impose an incentive constraint on bank assets. If

V (st(κ), bt(κ), dt(κ)) is the maximized value of Vt, the incentive constraint then reads

V (st(κ), bt(κ), dt(κ)) ≥ θQt(κ)st(κ) (7)

On the other hand, there will be no friction between banks and firms on the same island:

banks will be able to enforce full repayment of loans made to firms.

2.1.2 Additional Assumptions

I make two additional assumptions to ensure a tractable solution to the model. First, in this

version of the model, I would like to make the ability distribution in any period independent

of the ability distribution in previous periods. Even if ability draws are independent across

periods, the interbank loans from the previous period will make the expected returns from

assets unequal. In this case, the history of previous draws will matter for bank returns

today. To prevent this, I follow [5] and allow individual banks to arbitrage these return

differences away, before their new ability type is realized:

Assumption 2. At the beginning of a period, individual banks are allowed to physically

move to another island, that is, they will be allowed to transfer net worth to another island

in exchange for a reduction in interbank debt.

8I interpret this ”running away” as capturing a bankruptcy cost: if a borrower bank decides to declare

bankruptcy, creditors can capture some, but probably not all, of the repayments they are owed. Second,

despite the perhaps unrealistic form of the constraint, it generates the more realistic property that the

maximum a bank can borrow depends on its size.

8

To see how this works, consider two islands, aL and aH . aL has low expected returns

when it enters the period, that is, the representative bank has high interbank debt obli-

gations, and aH has high expected returns. When given the opportunity, an individual

bank on aL decides to move to aH . It currently holds assets (investments) in firms on aL,

interbank debt to banks on aH , and deposits (economy-wide). Before it moves, it trades

its assets with another bank on aL for more interbank debt to aH . It then takes its net

worth and deposits and moves; this reduces the interbank debt of aL to aH but maintains

the total asset held in aL firms. In equilibrium, this process will continue until returns are

equalized.

Another assumption has to do with the capital goods that carry over from previous

periods. Intermediation ability also applies to the undepreciated capital that banks carry

between periods - banks re-allocate, or re-intermediate, the undepreciated portion of the

capital stock among firms on the same island. In equilibrium, banks on low ability islands

will only invest enough to maintain the undepreciated capital stock. Re-intermediation of

capital will leave some islands with higher production capacity than others. In order to

ensure that we can maintain a simple law of motion for capital, I assume that these changes

to production capacity are realized as adjustments to the production schedule of the firm

in the same period:

Assumption 3. When existing capital is intermediated, any gains (losses) in the production

possibilities of that capital are realized as an increase (decrease) in output of the firm in the

same period.

We will see that in equilibrium, this assumption will allow higher ability banks that

don’t purchase new capital to lend more than lower ability banks.

2.2 Households

Households9 on each island a are infinitely-lived, supplies up to one unit of labor per period,

saves in the form of deposits, and consumes consumption goods. The household makes

its deposit and labor supply decision before the banks on its island realize their ability.

Preferences over consumption, labor, and deposits (ct(a), lt(a), dst (a)) in each period are

represented by

9More precisely, the household is composed of a continuum of members (of measure 1). This continuum

is divided into f workers and 1 − f bankers. Workers supply labor to firms, and return their wages to the

household. Similarly, all bank profits are returned the household each period. A household with this form

allows the representative household form above.

9

Et

∞∑i=0

βi[ln(ct+i(a)− γct+i−1(a))− χ

1 + ϕ(lt+i(a))1+ϕ

](8)

where β ∈ (0, 1) is the discount factor and γ ∈ [0, 1) is a habit formation parameter10.

ϕ is the inverse elasticity of labor supply.

Households on each island save by making riskless one-period deposits dst in the economy-

wide deposit market11 at the interest rate Rt. Deposits made last period are repaid at the

beginning of this period.

The household also owns the bank, and receives dividends from exiting banks every

period Πt(a). The household will earn wage wt(a) for each unit of labor it supplies. Then

the household’s budget constraint is

ct(a) = wt(a)lt(a) + Πt(a) +Rt−1dst−1(a) + dst (a) (9)

In order to simplify the household’s problem further, I assume that workers are freely

mobile:

Assumption 4. In each period, workers can supply up to 1 unit of labor to a firm on any

island.

With this assumption, wages will be constant across islands, so that wt(a) = Wt on any

island a.

Since the prices faced by households Rt, Wt are identical across islands, and preferences

are aggregable, we can represent the island households with an economy-wide representative

household, with preferences

Et

∞∑i=0

βi[ln(Ct+i − γCt+i−1)−

χ

1 + ϕ(Lt+i)

1+ϕ

](10)

with budget constraint

Ct = WtLt + Πt +Rt−1Dt−1 +Dt (11)

10These preferences exhibit habit formation when γ ∈ (0, 1), a feature that is included for comparison to

other models in the literature. The model is not fundamentally different when we turn off habit formation,

i.e. set γ = 0.11We could alternatively restrict the household to only making deposits in the banks on their island; with

the timing assumptions below, nothing about the model changes. I maintain this formulation for ease of

explanation.

10

Taking first order conditions, we first work out a condition for aggregate deposits:

Rt(EtΛt,t+1) = 1 (12)

and a condition for aggregate labor:

WtEt(uCt) = χ(Lt)ϕ (13)

where uCt ≡ (Ct− γCt−1)−1− βγ(Ct+1− γCt)−1 is the marginal utility of consumption

and Λt,t+1 ≡ β uCt+1

uCtis defined as the household’s stochastic discount factor.

The economy-wide representative household is useful because the decisions of banks and

firms only depend on household decisions only through the aggregate deposits and labor

chosen. Since banks draw deposits from a national deposit market, only the economy-wide

supply of deposits Dt matters for determining the level of individual bank deposits. Since

firms essentially draw labor from an economy-wide labor market, only the total supply of

labor Lt matters for firm labor demands.

2.3 Firms

There are many identical, competitve firms on each island. Each firm borrows funds from a

bank on its island and uses them to operate a constant returns to scale technology, producing

basic goods from capital goods and labor. The representative firm on island a chooses kdt

and ldt to maximize the production function:

yt(a) = Atkdt (a)αldt (a)1−α (14)

where At is the (economy-wide) total factor productivity of the firm, and kdt and ldt are

capital and labor demanded by the firm, respectively.

Undepreciated capital may differ across islands coming into a period. I denote the

capital held by banks coming into the period by kt(a).

If capital demand exceeds the level of undepreciated capital, that is, kdt (a) > kt(a),

firms go to the new capital market to augment their capital stock. Call the demand for new

capital by firms on island a it(a).

Since all firms use the same, constant returns to scale technology, the optimal capi-

tal/labor ratio is constant across firms. We can then represent the island firms with an

economy-wide representative firm, with production function

Yt = At(Kt)α(Lt)

1−α (15)

11

We’ll denote aggregate output, labor, and capital by Yt, Lt, and Kt, respectively.

Firms choose labor to equate the economy-wide wage with the marginal product of

labor:

Wt = (1− α)yt(a)

ldt (a)= (1− α)

YtLt

(16)

The bank expects per capital profit zt(a) for each unit of capital purchased with its loan.

Then firms choose capital to equate this with the marginal product of capital:

zt(a) = αAt

(ldt (a)

kdt (a)

)1−α

= αAt

(LtKt

)1−α= Zt (17)

The above equation implies that, in any equilibrium, zt(a) cannot differ across islands.

I call this common per unit profit Zt, so that zt(a) = Zt for all islands.

The economy-wide representative firm is useful because the decisions of the type rep-

resentative bank depend on firm decisions only through the capital that firms install on

the island in the previous period, but this capital only depends on the aggregate capital in

the previous period. Household labor decisions depend on firm decisions, but because the

labor market is economy-wide, the economy-wide representative firm’s problem is enough

to characterize household decisions as well.

2.4 Capital Goods Producers

When firms decide to expand their existing capital stock, they travel to a central (economy-

wide) market for new capital. The market is perfectly competitive, but as in [5], all pro-

ducers face adjustment costs12. Capital goods producers sell new capital to firms for the

price Qit. These producers then choose It to maximize

Et

∞∑τ=t

[QiτIτ − [1 + f(IτIτ−1

)Iτ ] (18)

From the solution to the profit maximization problem, the new capital price should

satisfy

Qit = 1 + f

(ItIt−1

)+

(ItIt−1

)f ′(

ItIt−1

)− EtΛt,t+1

(It+1

It

)2

f ′(It+1

It

)(19)

Last, I will make an assumption that ensures that islands that don’t visit the new capital

market will have a capital price that can freely adjust:

12This assumption is maintained for comparison with [5]

12

Assumption 5. Once installed, capital cannot be transferred back to the common capital

market or another island and sold.

When we combine the household budget constraint with the equation for firm output

and new capital, we can write an economy-wide resource constraint:

Yt = Ct +

(1− f

(ItIt−1

))It (20)

Second, with the assumptions above, we will be able to write a simple law of motion for

capital:

Kt+1 = ψt+1 (It + (1− δ)Kt) (21)

2.5 Timing

Basically, a period is divided into two parts: before and after intermediation ability is real-

ized. Deposits are made in the first part, while firm loans, interbank loans, and investment

purchases are made in the second.

From the beginning to the end of period t:

• ψt and At are realized

• Some banks move, equalizing expected returns

• Some banks exit (with probability σ), pay dividends to households

• New banks enter in their place

• Deposits Rt−1dt−1 and interbank loans Rbt−1bt−1 are repaid

• Households make deposits

• Households make labor supply decisions

• Ability types are realized

• Simultaneously...

– Bank net worth is calculated

– Loan demands are made by firms to banks

– Banks make loans to firms st(a)

– Investment capital demands it(a) are made by firms to capital goods producers

13

– Capital goods producers sell total new capital It at price Qit

– Interbank loans bt(a) are made

– Firms adjust production schedules to reflect any gains from re-intermediation

– Firms produce yt(a)

• Firms pay Zt per unit capital to banks, Wt to workers

2.6 Equilibrium Conditions

To characterize equilibrium, we need to understand the behavior of:

• Agents

– Representative bank for islands of each type, denoted κ

– Economy-wide representative household

– Economy-wide representative firm

– Capital goods producers

• Markets

– Deposits (common to all islands)

– Labor (common to all islands)

– Interbank Loans (common to all islands)

– Assets on each island

– New capital (common to all islands)

Definition 1. A recursive competitive equilibrium in ability types consists of a sequence

of economy-wide prices Pt ≡ (Rt+i, Rbt+i,Wt+i, Zt+i, Qit+i)

∞i=0 , a sequence of type-specific

prices Pκt ≡ (Qt+i(κ))∞i=0,

a sequence of economy-wide quantities Qt ≡ (Kt+i, Ct+i, It+i, Yt+i, Lt+i, Lt+i, Dt+i)∞i=0,

a sequence of type-specific quantities Qκt ≡ (bt+i(κ), st+i(κ), dt+i(κ), it+i(κ))∞i=013 such that:

(Individual Optimization) for each t,

• (bt(κ), st(κ), dt(κ)) maximizes the representative bank’s expected value (5) subject to

their flow of funds constraint (4) for each ability type κ

13Note that type-specific quantities are functions xt(κ) : [0, 1]→ R

14

• (Ct, Lt, Dt) maximizes the economy-wide representative household’s expected utility

(10)

• (Kt, Lt) maximizes the economy-wide representative firm’s profits (15) on each island

a

• It maximizes capital goods producer profits (18)

(Market Clearing) and for each t,

• Rt clears the market for deposits, so that Dt =∫a dt(κ(a))p(κ(a))da

• Wt clears the labor market, so that Lst = Ldt

• Rbt clears the market for interbank loans, so that∫a bt(κ(a))p(κ(a))da = 0

• Qit clears the market for new capital, so that It =∫a it(κ(a))p(κ(a))d

• Qt(κ) clears the markets on each island (and each ability type) for assets, so that

st(κ(a)) = (1− δ)kt (κ(a)) + it(κ(a))

2.7 Solution

I solve the model using a method presented in [5]. First, solve the island representative

bank’s problem for the difference between the returns from investment and the deposit

interest rate; there will be at least one such spread which clears the interbank lending

market. I will focus on situations where only one such spread exists. Once this spread is

known, solve the household and firm problems; only one combination of spread and either

return or deposit interest rate will solve the problems of these agents as well. We’ll then

solve for the remaining quantities by aggregating up to the ability type level.

The island representative bank maximizes net worth, a linear function of assets, deposits,

and loans. Because of this property, the bank’s value function will also turn out to be linear.

The problem then boils down to solving for the coefficients of this value function.

First, guess that in every period a bank guided by a linear function of shares, deposits,

and interbank loans will maximize its expected net worth; we’ll verify this guess later.

Specifically:

V (st(a), bt(a), dt) = νstst(a)− νbtbt(a)− νtdt (22)

15

In what follows, I restrict attention to equilibria with interior solutions to the above

guess, so that banks can only hold nonzero quantities of all choice variables - only these

equilibria will have positive interbank lending.

Next, use the guess to set up a Lagrangian for the bank’s problem. First, substitute the

flow of funds constraint into the guess to reduce the number of choice variables to two:

V (dt, bt(a)) = νstst(a)− νtdt(a)− νbtbt(a)

=κ(a)νstQt(a)

(nt(a) + dt(a) + bt(a))− νtdt(a)− νbtbt(a)

=κ(a)νstQt(a)

nt(a) + (κ(a)νstQt(a)

− νt)dt(a) + (κ(a)νstQt(a)

− νbt)bt(a) (23)

Note nt(a), the bank’s net worth, is essentially a state variable. Next, substitute the

guess into the incentive constraint, and use it to construct a Lagrangian:

L(dt, bt(a)) =

((1 + λt(a))

κ(a)νstQt(a)

− λt(a)κ(a)θ

)nt(a)

+

((1 + λt(a))(

κ(a)νstQt(a)

− νt)− λt(a)κ(a)θ)

)dt

+

((1 + λt(a))(

κ(a)νstQt(a)

− νbt)− λt(a)κ(a)θ

)bt(a)

where λt(a) is the multiplier on the incentive constraint in period t for the representative

bank from island a.

We obtain the following first order conditions for dt and bt(a):

νt = νbt (24)

κ(a)θλt(a)

1 + λt(a)= κ(a)

νstQt(a)

− νbt (25)

Using the above, we can also rearrange the borrowing constraint into a form that will

prove useful later. If bt(a) is the maximum a bank will borrow, then

bt(a) ≤ bt(a) =

κ(a)νstQt(a)

− κ(a)θ

νbt − κ(a)νstQt(a)

+ κ(a)θnt(a)− dt(a) ≡ φt(a)nt(a)− dt(a) (26)

φt(a) is an important quantity. It is related to the leverage ratio (denoted L(a)), or the

ratio of shares held to net worth; L(a) = κ(a)(1 + φt(a)). This means L(a) is a convex,

increasing functions of κ. This property arises because intermediation ability affects banks

in two ways: first, intermediation ability decreases the cost of shares directly through its

16

effect on the flow of funds constraint; second, ability increases the continuation value relative

to the ”running away” value, loosening the borrowing constraint.

If the guess is correct, the coefficients νst and νbt will be equal to the marginal value of

shares and interbank loans. Shares produce profits from firms tomorrow, which increases

the value of tomorrow’s net worth. Deposits and interbank loans today reduce tomorrow’s

net worth through repayment. Because net worth is a linear function of each of these

quantities, this marginal value will be constant. Calculating the value of these coefficients

amounts to calculating the marginal effect of increasing these quantities on net worth.

Note that the above Lagrangian is also equal to the bank’s maximized value written in

terms of net worth. Substituting the FOCs into the Lagrangian, we get

V (dt, bt(a)) = (1 + λt(a))νbtnt(a) (27)

If we iterate this one period forward and then plug this into the Bellman equation, we

obtain:

νstst − νtdt − νbtbt = Et,a′Λt,t+1(1− σ)nt+1(a′) + σ(1 + λt+1(a

′))νbt+1nt+1(a′) (28)

We can obtain the value of each coefficient by taking the partial derivatives of both sides

of the above equation with respect to each of the variables bt(a), dt(a), st(a):

νbt = RbtEt,a′Λt,t+1Ωa′t+1 (29)

νt = RtEt,a′Λt,t+1Ωa′t+1 (30)

νst = Et,a′Λt,t+1ψt+1(Ωa′t+1(Zt+1 + (1− δ)Qa′t+1) (31)

where

Ωa′t+1 = 1− σ + σ(1 + λt+1(a

′))νbt+1 (32)

As long as ability draws next period are independent of the draw this period, these

coefficients do not depend on ability type or level of either of the choice variables. They

also maximize the value of island representative version of the original objective function

(5):

Vt(a) = Et,a′∞∑i=1

(1− σ)σi−1Λt,t+int+i(a′) (33)

To see this, consider the effect of dt on the original bank objective function. dt only

affects the bank objective when the bank exits. A fraction (1− σ) of banks exit from each

island every period. An increase in dt affects net worth in period t+1 directly, by increasing

17

the repayment in that period. Thus, banks that exit in period t + 1 will have lower net

worth. The decrease in t+ 1 net worth decreases st+1 through the flow of funds constraint,

which turn reduces the net worth in period t + 2; if banks exit then, they will also have

lower net worth. The full effect of the change in deposits can then be written as

dVtddt

= (1− σ)∂nt+1

∂dt+ σ(1− σ)

∂nt+1

∂dt

∂st+1

∂nt+1

∂nt+2

∂st+1

+ σ2(1− σ)∂nt+1

∂dt

∂st+1

∂nt+1

∂nt+2

∂st+1

∂st+2

∂nt+2

∂nt+3

∂st+2+ ... (34)

The partial effects ∂st+i∂nt+i

are captured by the quantity (1 + λt+i(a′))νbt+i. This means

that the coefficient νt completely summarizes the direct effect of dt on the island objective

function. The same argument holds for bt(a). Because of this, we can conclude:

Proposition 1. A set of choices (d∗t+i, b∗t+i(a))∞i=0 that maximizes the linear value function

guess (22) for each i will also maximize the bank objective (33).

Since banks on islands with the same ability draw will make the same choices for dt and

bt(a), this result extends to all banks with the same ability κ.

2.8 Capital Prices

From the the FOC for bt(a) in the previous section, we can see that if the representative

banks on island a is not borrowing constrained (which implies λt(a) = 0), it must be that

the price of capital on its island satisfies

Qt(a)

κ(a)=νstνbt≡ Qnt (35)

Further, assumption 5 also implies that the price of capital on each island is free to

adjust to a different value on each island. Make another guess which we can verify later:

Qt(κ) = κνstνbt

= κQnt ∀κ ∈ L (36)

where L is the set of ability types for which banks are not borrowing constrained.

If the price of capital has the above form, an unconstrained bank will be indifferent

between re-lending its existing capital stock to the firms on its island and lending the con-

sumption good equivalent of the existing capital stock on the interbank lending market.

18

In order to ensure that the law of motion for capital has a simple form, I add the follow-

ing assumption to the model, though without explicitly adding a constraint to the bank’s

problem above14.

Assumption 6. st(κ) ≥ kt(κ) ∀κ

The FOC for bt(a) also tells us that constrained banks view assets as more valuable

than interbank lending, implying that these banks also desire higher investment. If it were

free to adjust, the price of capital would be pushed up. However, because there is just one

price in the market for new capital, the price gets pinned down:

Qt(κ) = Qit ∀a ∈ B (37)

where B is the set of islands on which banks are not borrowing constrained.

2.9 Ability Cutoff

Generally, high ability banks will borrow, and be borrowing constrained, while low ability

banks will lend. To see this, first notice that the flow of funds constraint reduces the number

of bank choice variables to two, and since deposits are chosen before ability type is realized,

the only choice the bank makes after ability is realized is the level of interbank loans.

Proposition 2. Consider an equilibrium with strictly positive lending. In each period t,

there exists a κ∗t such that:

• for all banks with κ < κ∗t , bt(κ) = −(nt(κ) + dt −Qnt kt(κ)), that is, the bank will lend

its net worth and deposits less the cost of refinancing the entire existing capital stock

on its island.

• for all banks with κ > κ∗t , bt(κ) = bt(κ), that is, the bank will borrow up to its borrowing

constraint and use the funds to purchase assets.

Proof. Consider the term for borrowing in equation (23), and let us first assume

(κνstQt(κ)

− νbt) > 0

In this case, the bank gets positive value for every dollar it borrows, and it will choose

to borrow as much as it can, i.e. until bt(κ) = bt(κ). Because all constrained banks face

capital prices Qt(κ) = Qit, this implies that κ > νbtνstQit.

14Since banks are indifferent between re-lending and not re-lending, they face no implicit costs by choosing

to fully re-lend.

19

If ( κνstQt(κ)

− νbt) ≤ 0, the bank gets positive value for every dollar it lends (negative

borrowing). By assumption 6, the bank will only lend funds left over after re-lending its

existing capital stock, i.e. bt(κ) = −(nt(κ) + dt(κ)−Qnt kt(κ)). We know that capital prices

on these islands are smaller than the new capital market price, so κ ≤ νbtνstQt(κ) ≤ νbt

νstQit.

Call κ∗ ≡ νbtνst

. Then, for any bank with ability κ > κ∗, borrowing is strictly more

profitable than lending on the interbank market. For any bank with ability κ ≤ κ∗, lending

is weakly more profitable than borrowing.

The ability differences between banks in this model generate differences in borrowing

and investment behavior. The above proposition tells us that these differences are easily

summarized with one equilibrium object, the ability cutoff.

In equilibrium, lower ability interbank lenders will lend less than higher ability interbank

lenders.

Proposition 3. Consider two banks on different islands with types κ and κ′ and bt(κ), bt(κ′) <

0. If κ < κ′, then bt(κ) < bt(κ′).

Proof. The cost of refinancing the entire existing capital stock kt(a), net of the benefit

derived in the form of net worth, is given by

Qt(κ)

κkt(κ)−Qt(κ)kt(κ)

Because of assumption 3, any differences in refinancing costs are reflected in output

in the same period. This allows banks to lend the consumption good equivalent of the

differences.

For unconstrained (lending) islands, this simplifies to

Qnt kt(κ)(1− κ)

This is a decreasing function of κ. Thus, for interbank lenders the cost of refinancing

the capital stock is decreasing in κ. By the flow of funds constraint, the amount left over

for lending to other islands is then increasing in κ.

2.10 Parameter Restrictions

Some sets of parameters will not admit equilibria with positive lending. I make three

restrictions on parameters to prevent these cases.

First, we need to restrict κ, the lowest possible intermediation ability, to ensure that

banks on all islands can fund reinvestment of their entire capital stock. Renegotiation of

20

the remainder of the capital stock means that each share of this portion costs Qt(κ)κ . For

high productivity islands, this is a boon; the cost of these shares is smaller than that of

the fixed capital. For low productivity islands, this is a burden; their poor managerial skill

causes them to lose some of their net worth through this process. Thus, if we require that

the entire existing capital stock will be reinvested, we must ensure that the ability of the

worst bank is not so low that it cannot cover the cost. That is, we should ensure that κ

satisfies

(1− δ)p(κ)Kt ≤κ

Qt(κ)nt(κ) + p(κ)Dt (38)

which can be simplified to

(1− κ)Qnt (1− δ)Kt ≤ ZtKt −Rt−1Dt−1 +Dt (39)

In steady state, this equation becomes

(1− κ)Qn(1− δ) ≤ Z + (1− 1

β)D

K(40)

Second, the borrowing constraint equation only acts as an upper bound on borrowing if

the denominator of φt(a) is positive, that is, νbt >κνstQit− κθ. Though equilibria exist if the

inequality is reversed, all banks in these equilibria are unconstrained, which would make

the incentive constraint useless. I avoid this case by choosing κ and θ so that

θ >νstQit− νbt

κ(41)

Third, because HHs also consider the borrowing constraint when making deposits in

any bank, we can have a case where some banks are bound by their borrowing constraint

before their ability is realized. Though an equilibrium exists in this case, ability will have

no effect on the investment decisions of most banks. I therefore want to avoid this case by

ensuring that no bank takes on so many deposits that it cannot borrow any more once its

ability is realized.

φt(κ)nt(κ) ≥ dt ∀κ (42)

2.11 Interbank and Deposit Market Clearing

Given the solution above, we will see that clearing the interbank market automatically

clears the market for deposits. First, consider the interbank lending market. Call B the

set of ability types with representative banks that borrow, and L the set of all types that

21

lend. The sum of all interbank lending by types in L should equal the sum of all interbank

borrowing by types in B:∫κ∈B

b(κ)p(κ)dκ =

∫κ∈B

b(κ)p(κ)dκ =

∫κ∈B

(φt(κ)nt(κ)− dt(κ)) dκ

= −∫κ∈L

b(κ)p(κ)dκ =

∫κ∈L

(nt(κ) + dt −Qt(κ)kt(κ)) dκ (43)

With this in hand, we can rewrite the interbank lending market condition in terms of

aggregates:

Dt =

∫ κ

κ∗p(κ)φt(κ)((Zt + (1− δ)Qit)Kt(σ + ξ)− σRt−1Dt−1)dκ

−∫ κ∗

κp(κ)((Zt + (1− δ)Qt(a))Kt(σ + ξ)− σRt−1Dt−1)dκ

+

∫ κ∗

κp(κ)

Qt(a)

κ(a)(1− δ)Ktdκ (44)

Where Dt is aggregate deposits and Kt is the aggregate capital stock.

In order to pin down investment, we need to consider the market for new capital alone.

We know that the aggregate investment and existing capital held by firms on borrower

islands is equal to It + (1 − δ)Kt∑κ

κ∗ p(κ), and that capital is demanded in the form of

interbank borrowing and deposits. Noting that borrowing islands will borrow the maximum

possible, we can replace the borrowing constraint into the flow of funds equation:

st(κ) = it(κ) + kt(κ) =κ

Qt(κ)(nt(κ) + dt(κ) + bt(κ))

=κ

Qit(1 + φt(κ))nt(κ) ∀κ ∈ [κ∗, κ] (45)

Finally, integrate both sides over the set B to get the condition in terms of aggregates:

Qit

(It + (1− δ)Kt

∫ κ

κ∗p(κ)

)=

κ∑κ∗

κ(1 + φt(κ))nt(κ)dκ (46)

Equations that characterize this equilibrium are given in appendix A.

2.12 Steady State Properties

In this section, I discuss two relevant basic properties of the model. For illustration, consider

the case of the model where ability is distributed according to a discrete uniform distribution

around 1, and the economy is in steady state.

22

Figure 1: Marginal value from interbank borrowing (dash) and marginal value from holding

shares (solid) by banks of different abilities

Figure 1 shows the marginal value from lending on the interbank market (dash line) and

the marginal value from investing in firms. This is essentially a visualization of proposition

2. The value of lending is the same for all banks, since all banks receive the same interest

rate for loans of any size. The value from investing/lending to firms, on the other hand,

increases with bank intermediation ability. In any equilibrium with positive lending, the

two lines have to cross - if the interbank lending line was always under the investment line,

no bank would be willing to lend, and if the interbank lending line was always above the

investment line, no bank would be willing to borrow. The ability level at which these two

lines cross is the ability cutoff - for any bank with ability above the cutoff, the marginal

value from investing is higher than the value from lending, so it becomes profitable for the

bank to borrow and invest.

Ability increases the intensity with which banks borrow or lend in the interbank market.

Figure 2 shows interbank borrowing versus ability in a typical case of the model. All banks

with ability smaller than the ability cutoff lend on the interbank market, and because

higher ability banks are more efficiently able to refinance their existing capital stock, they

have more dollars left over for lending, and therefore lend more on the interbank lending

market than lower ability banks. (Since the vertical axis represents borrowing, lending is

23

Figure 2: Interbank borrowing (left) and investment (right) by banks of different abilities

represented with a negative y-coordinate.)

Banks with ability higher than the ability cutoff decide to borrow, buy new capital, and

expand the production capacity of the firms on their island. The difference in borrowing

between banks just above and just below the cutoff can be divided into two components.

The first component represents a reduction in lending - banks just below the cutoff make

these loans. The second component represents borrowing by banks up to the point that

their borrowing constraint binds.

Moreover, the cap on borrowing increases in κ at an increasing rate. Figure 3 shows

leverage as a function of ability. The function is convex, implying that the average leverage

in the economy is higher than the leverage of the average bank. As we will see later, this

property will generate a connection between the variance of the ability distribution and

the marginal value of interbank lending. The differences in leverage are what ultimately

generate the size differences we’re nterested in.

Interbank lenders lend different quantities based on their intermediation ability. This is

a result of both assumptions 5 and 6. The cost of refinancing the capital stock is smaller for

more able banks, and since all lending banks refinance their capital stock, more able banks

have more funds left over to lend out to other banks.

The right panel of figure 2 shows the value of investments (measured in units of con-

24

Figure 3: Leverage and Intermediation Ability

sumption goods) by banks of different abilities. All banks reinvest their existing capital

stock - capital prices on lending islands adjust to ensure that this is optimal. Once ability

increases above the cutoff, we see a jump in the quantity of investments as banks suddenly

switch from a ”lend and refinance” strategy to a ”borrow and buy new capital” strategy. As

ability increases, borrowing limits increase, and because the price of new capital is pinned

down by the new capital market, this translates into higher investment.

2.12.1 Tightening the Financial Friction

If there were no financial friction in the model, no bank would be constrained in its borrow-

ing. Depositors would make deposits in all banks as before, but once ability is realized, the

interbank lending market would funnel all deposits to the most able bank. This bank would

transform these loans into assets, which firms would use to purchase the highest amount of

capital possible.

Introducing the friction limits the maximum the most able bank can borrow through

the interbank lending market. By limiting the maximum the bank can borrow, funds that

would go to the most able bank go to less able banks. Economy-wide investment and average

returns banks obtain on investments decrease.

To visualize this, compare the steady state from the previous case with the steady state

25

Figure 4: Interbank borrowing for low friction (dash) and high friction (solid) by banks of

different abilities

that obtains after an increase in the level of financial friction, θ. In figure 4, the steady

state borrowing and investment curves from the model above are plotted along with steady

state borrowing and investment after increasing θ from 0.4, in the previous section, to 0.5.

Among interbank borrowers, a change in θ has both an ”intensive” and an ”extensive”

effect on steady state behavior. The intensive effect is a change in leverage for all banks:

the value of the leverage ratio φt(a) decreases as θ increases, so any borrower that continues

to be a borrower will face a tighter constraint on the level of their borrowing, and thus will

not be able to invest as much. Moreover, the leverage curve gets less steep, so an increase

in ability results in less additional borrowing for higher θ.

On the extensive side, as the friction increases and demand for interbank loans drops,

the price of borrowing should decrease, at least relative to the value from lending to firms.

This decrease in price causes some banks that were interbank lenders to become borrowers

and invest. The increase in demand for borrowing that results from this extensive movement

will offset some of the initial fall.

Both of these effects result in a decrease in the volume of interbank loans that are made

in the economy. This is what we should expect - demand falls and supply falls with it.

The magnitude of a given change in the level of the friction decreases as the initial

26

friction increases. For parameter values for which steady state equilibria exist, an increase

in the level of the friction always results in a decrease in total interbank borrowing. But

the size of this decrease itself decreases as the initial level of the friction increases. Since

the leverage ratio increases at an increasing rate in the friction, a higher initial friction

level essentially means that we are starting higher up on the leverage ratio curve, where the

increases are larger.

Among interbank lenders, we see a decrease in the slope of the line that determines how

much lenders lend; more able lenders still lend more than less able ones, but the size of the

difference between the two decreases as the level of the friction increases. This is because

the steady state level of capital decreases as the friction increases, and because the cost of

refinancing is a linear function of the steady state level of capital, the slope decreases.

2.12.2 Dispersion

The difference in ability between the largest and the smallest ability types, or dispersion,

affects the model even if the mean and other parameters don’t change. As dispersion

increases, the intensive and extensive effects of financial frictions tend to be amplified.

To illustrate this point, consider two uniform ability distributions with the same mean

but different variances. Figure 5 compares steady state asset holdings for two discrete uni-

form distributions, one with κ(a) ∼ Udnif [0.8, 1.2] and κ(a) ∼ U

dnif [0.85, 1.15]. Widening

the distribution mirrors a change in the financial friction on both the intensive and extensive

side. The leverage curve for the wider distribution is flatter, implying borrowing constraints

are tighter. The ratio of the ability cutoff to the highest ability type is lower for the wider

distribution, implying there are more low ability banks acting as borrowers/investors than

for the narrower distribution.

Both effects can be tied to an increase in the spread between the return on interbank

lending and the return on assets. An increase in this spread increases the shadow value of

defaulting on interbank loans and deposits, tightening the borrowing constraint. In turn,

the value of default pushes the cost of borrowing up, decreasing average returns on assets,

decreasing demand for interbank loans, and pushing down the value of interbank lending

as a proportion of the highest ability type. More banks enter the borrower side of the

interbank market as a result.

The spread increases in the first place because the average marginal value of holding net

worth tomorrow increases as the distribution gets wider. To see this, recall the equation for

the marginal value of interbank lending:

27

Figure 5: Effect of Dispersion on Investment

Ωt+1(κ′) = 1− σ + σ(1 + λt+1(κ

′))νbt+1 (47)

The term λt+1(κ′) represents the shadow value of net worth next period, and it increases

(with the leverage curve) at an increasing rate in κ′.

Banks facing a wider ability distribution every period face the possibility of drawing very

high ability next period. But very high ability means the bank will be very constrained - it

will value any thing that loosens its borrowing constraint, i.e. net worth, a lot. And because

the value of the multiplier increases at an increasing rate in κ, the bank’s expectation of

how constrained it will be also increases. Thus, the average marginal value of net worth

increases.

To see how an increase in the marginal value of net worth affects the spread, consider

the difference νbt − κνst in the solution above:

νbt − κνst = Et,κ′Λt,t+1Ωt+1(κ′)(1− κψt+1(Zt+1 + (1− δ)Qt+1(κ

′))) (48)

For a bank that borrows this period, the term κ(Zt+1 + (1 − δ)Qt+1(κ′)) − 1) > 0 as

long as the restriction (39) holds in every period. This means that as Ωt+1(κ′) increases,

the spread above increases. To put this another way, if the marginal value of net worth

28

increases, the relative costs of repaying interbank loans next period increase. As a result,

default becomes a relatively more attractive option.

The connection between dispersion and efficiency comes from the bank’s assessment of

its own future opportunities based on the performance of other banks today. In the model, a

high ability bank can become a low ability bank next period - there is no connection between

types across time. This is certainly unrealistic. However, if we interpret bank ability as

a measure of sector-specific returns on bank-firm relationships, a bank’s expectations of

future returns should depend, at least partly, on the average performance of other sectors

of the economy today.

3 Quantitative Exercises

In this section, I carry out two quantitative exercises using the model of the previous section.

First, I compare the response of this model with [5], a model with a homogeneous banking

sector, and show that recessions in this model are longer and deeper. Second, I consider

the impact of the increase in concentration that took place from 1986 to 2006 in the US

banking industry, and show that potential downturns are slightly deeper and longer. All

impulse responses are generated using the full set of equilibrium equations in Appendix A

with Dynare.

3.1 Calibration

The model demands the choice of 9 parameters, the adjustment costs function, and the

distribution of ability types. Five of the parameters control the standard preference and

technology shocks from from the literature: the discount rate β, the habit parameter γ, the

utility weight of labor χ, the share of capital in production α, and the depreciation rate δ.

These parameters are drawn from [5] and [4] and are given in Table 1 below.

Three of the parameters come from [5]. The parameter ϕ, the inverse elasticity of labor

supply, is chosen so that the Frisch elasticity is ten. I do this to ensure a good comparison

with [5]. The parameter ξ, the start-up transfer to new bankers, governs the average spread

between the interbank lending rate and the average return on assets. The parameter σ, the

exogenous probability of exit by a bank, is chosen so that the average bank survives for

approximately 15 years.

Adjustment costs take the quadratic form:

f

(ItIt−1

)=cI2

(ItIt−1

− 1)2 (49)

29

Parameter Value Target

Inverse Elasticity of Labor Supply ϕ 0.33 Gertler/Kiyotaki (2011)

Adjustment Cost Parameter cI 1.5 Gertler/Kiyotaki (2011)

Start-up Transfer ξ 0.002 Gertler/Kiyotaki (2011)

Probability of Bank Exit σ 0.982 Average Bank Age

Discount Factor β 0.99 Gertler/Kiyotaki (2011)

Habit Parameter γ 0.5 Gertler/Kiyotaki (2011)

Depreciation Rate δ 0.025 Christiano, Eichenbaum, Evans (2005)

Effective Capital Share α 0.36 Christiano, Eichenbaum, Evans (2005)

Utility Weight of Labor χ 5.584 Gertler/Kiyotaki (2011)

Table 1: Parameter Values

The parameter cI is chosen to match the inverse elasticity of net investment to the price

of capital in [5].

The parameter θ and the distribution of ability types are the parameters of interest in

what follows. Together, they govern the average leverage held by all banks and the bank

size distribution that is generated in the model. I will adjust these in the three experiments

that follow.

3.2 Amplification

As outlined above, dispersion amplifies the effects of financial frictions. An implication of

this is that the downturns generated in this model tend to be deeper and longer than those

generated in [5], where banks have just one size.

In figure 6, I initiate a downturn with a 1% drop in capital quality ψ and compare the re-

sponse of my model with that of [5]. The size of the drop is chosen as a reference. The ability

distribution in my model is distributed uniformly around 1, so that κ(a) ∼dUnif [0.9, 1.1]. I

assume ψ follows an AR(1) process centered around 1 with an autoregressive factor of 0.66.

I choose the parameter θ = 0.5 in both models. This is chosen because the parameter

restriction (41) is especially tight for the uniform distribution; choosing θ high ensures that

dynamics in the neighborhood of the steady state are linear. Unfortunately, the high θ

decreases the absolute impact of the capital quality shock in both models, but not in a way

to make the comparison invalid. The parameter choices generate an average leverage ratio

of 2 in my model, and an average leverage ratio of 1.5 in [5].

In [5], all banks are identical in ability; in my model, this would cause the level of inter-

30

Figure 6: Impulse responses to financial shock in the model (blue) and Gertler-Kiyotaki

(red)

bank lending to be indeterminate. Demand for interbank loans is created by exogenously

determining that a constant fraction πi of banks are the only ones allowed to purchase new

capital every period. In this comparison, then, I choose πi in the simulation of the model

of [5] so that in steady state, the fraction of banks that purchase new capital in my model

(i.e. those with ability higher than the cutoff) matches the fraction of banks that purchase

new capital in theirs.

On impact, output, employment, and investment drop relatively farther in this model.

This happens because average leverage in the many types economy is higher; as we saw in

figure 3, because leverage is convex in ability, dispersion tends to increase average leverage.

The negative impact of the financial shock on employment is small in both models. As

pointed out by [5], this may be due to the absence of labor market frictions in the model.

The recovery of output and consumption is longer in this model than in the case of [5].

This is due to the extensive effect in the interbank lending market. When the capital quality

shock hits, the ability cutoff decreases, and low ability banks become borrowers. This has

two offsetting effects on the recovery: first, these banks dampen the initial impact of the

shock by increasing investment demand; second, these banks push up the future value of

net worth, tightening borrowing constraints for all banks, and keeping the value low after

the shock dissipates. Thus, output downturns in this model will generally take longer to

return to steady state levels, even though they will not always be deeper.

31

Year GINI CV Kurtosis

1986 0.83 9.9 1589.33

2006 0.91 14.9 1796.16

Table 2: Size Distribution Facts

3.3 US Banking Concentration

Banks in the US have become increasingly concentrated in size over time. More precisely,

we observe two facts relevant to this discussion: the share of assets held by the largest banks

has increased, and the size of the largest banks has increased relative to smaller banks. In

what follows, I change the ability distribution to generate changes in the size distribution

in both of these facts, and then examine the impulse responses as a measure of the severity

of potential crises.

In this example, I match the bank size distribution in the first quarter of 1986 and

2006.15 I use quarterly commercial bank data to generate these facts.16 Table 2 shows the

three features I mention above. Inequality in asset size, measured by the GINI coefficient,

increases over the period. (This increase corresponds to an increase in the share of total

assets held by the largest four banks from 10% to 30%). The variance of asset sizes increases

dramatically, by more than a factor of 10. CV reported in the table below is the sample

coefficient of variation, or the ratio of the sample standard deviation to the mean. This is

a normalized measure of dispersion, and it also indicates an increase in dispersion over the

period. Last, the kurtosis, or heavy-tailed-ness, of the size distribution increases, indicating

that more of the variance is due to the size of the largest banks.17

[6] finds that the tail of the bank size distribution is best modeled with a Pareto distribu-

tion, while the remainder is best modeled with a lognormal distribution. For computational

convenience, I use truncated Pareto ability distributions to generate the necessary size dis-

tributions. In particular, I vary the width, Pareto shape parameter, and the parameter θ to

match the GINI coefficient, ratio of largest to average size, and the average leverage ratio

in the data. Table 3 gives the chosen parameters and fit for the model. Figure 7 shows the

15Because the ability distribution is exogenous in this model, we can only use it to analyze long-term

trends in the data.16Data is taken from the Report on Condition and Income, FFIEC 031 and 041, from the Chicago Fed

website [8].17Since the distributions in the model are assumed to be mean 1, the average bank is used as a reference.

32

Figure 7: Cumulative Share of Assets, Data and Model

fit of the size distribution in a picture. I am unable to match both the GINI and leverage

ratio very precisely. I choose ability distributions to produce an increase in both quantities,

which we observe in the data.

In this case, I initiate a downturn with a 5% drop in capital quality ψ and compare

the response of the two economies. The size of the drop is chosen in this case so that the

resulting drop in output for the 2006 distribution is approximately what was seen during

the 2008 crisis, that is, output drops by 2%. Following the calibration of [5], I again assume

ψ follows an AR(1) process centered around 1 with an autoregressive factor of 0.66.

The impulse responses for the two distributions are given in figure 8. The response

Year GINI LR Shape θ

1986 0.83 7.3

2006 0.91 10

Model 1986 0.62 10 5 0.4

Model 2006 0.74 14 5.5 0.4

Table 3: Data Moments to Match and Chosen Model Parameters

33

Figure 8: Impulse responses to financial shock in for low concentration (blue) and high

concentration regimes (red)

34

corresponding to the 2006 case is given in red. Output falls farther on impact - the change

in the distribution of bank sizes increases the average leverage in the economy, so banks

decrease their investment more in response to a drop in net worth.

4 Conclusion

In this paper, I construct a macroeconomic model with a heterogeneous banking sector, and

show that heterogeneity has consequences for downturns. In particular, mean-independent

changes in the distribution of bank sizes can mirror the effects of financial frictions.

Though dispersion tends to worsen the effects of financial frictions, a change in kurtosis

tends to better it. This is because, for the Pareto distributions I consider here, an increase

in kurtosis corresponds to an increase in the mass of banks with low ability. And just as

dispersion increased the average value of net worth tomorrow, kurtosis tends to decrease

the average value. This results in a decrease in the value of net worth tomorrow, a decrease

in the value of lending today, and a loosening of borrowing constraints.

Since both dispersion and kurtosis can increase when banking sector concentration in-

creases, the model does not make a clear qualitative prediction about the effects of the

increase in concentration. I point this out as a partial explanation of the mixed results in

the concentration-stability literature. If banking sector concentration comes in the form of

growth for banks of all sizes, the adverse effects of financial shocks will be mitigated by the

additional liquidity provided by small banks.

There are two simplifying assumptions that would be worth weakening in future work.

First, dropping assumption 2 will cause bank size next period to depend on its size this

period. If ability tomorrow doesn’t depend on ability today, but size carries to next period,

large banks can use their assets as a buffer against poor ability draws, but small banks would

be more sensitive to such changes. As a result, the average value of interbank lending would

not adjust as much as in the case of this model. Second, if banks defaulted in equilibrium,

downturns would have the added effect of increasing risk in interbank lending. This may

cause these markets to function even less well than before, exacerbating the frictions in the

economy even further.

I see this project as a (very) small step in a larger agenda of analyzing the macroeconomic

implications of changes (not just size differences) in the banking industry. So far, research

in this field has not placed much emphasis on the characteristics of individual banks and the

dynamic consequences from changes in those characteristics. Research in this vein would

be especially informative for the conduct of unconventional monetary policy.

35

References

[1] T. Beck, A. Demirguc-Kunt, and R. Levine. Bank concentration and fragility. The Risks

of Financial Institutions, page 193, 2007.

[2] Collard F. Boissay, F. and F. Smets. Booms and systemic banking crises. ECB Working

Paper, Available at SSRN 2131075, 2012.

[3] John H Boyd and Gianni De Nicolo. The theory of bank risk taking and competition

revisited. The Journal of Finance, 60(3):1329–1343, 2005.

[4] Lawrence J Christiano, Martin Eichenbaum, and Charles L Evans. Nominal rigidities

and the dynamic effects of a shock to monetary policy. Journal of Political Economy,

113(1):1–45, 2005.

[5] M. Gertler and N. Kiyotaki. Financial intermediation and credit policy in business cycle

analysis. Handbook of Monetary Economics, 3(3):547–599, 2010.

[6] H. Janicki and E. Prescott. Changes in the size distribution of us banks: 1960-2005.

FRB Richmond Economic Quarterly, 92(4):291–316, 2006.

[7] Gianni De Nicolo, Philip Bartholomew, Jahanara Zaman, and Mary Zephirin. Bank

consolidation, internationalization, and conglomeration: trends and implications for

financial risk. Financial markets, institutions & instruments, 13(4):173–217, 2004.

[8] Federal Reserve Bank of Chicago. Commercial bank data, 2013.

36

A Equilibrium Conditions

A.1 Full System

HH/Firms

Yt = AtKαt L

1−αt (50)

Kt = ψt(It−1 + (1− δ)Kt−1) (51)

Yt = Ct + (1 + f(ItIt−1

))It (52)

1 = EtΛt,t+1Rt (53)

uCt = (Ct − γCt−1)−1 − βγ(Ct+1 − γCt)−1 (54)

Λt,t+1 = βuCt+1

uCt(55)

Zt = αAt(LtKt

)1−α (56)

Qit = 1 + f(ItIt−1

) + (ItIt−1

)f ′(ItIt−1

)− EtΛt,t+1(It+1

It)2f ′(

It+1

It) (57)

χLϕt = (1− α)YtLtEtuCt (58)

To drop habit formation from the model, set γ = 0. To drop adjustment costs from the

model, set f( ItIt−1

) = 0 everywhere.

Exogenous Shock Processes

At = ρAAt−1 + εAt (59)

ψt = ρψψt−1 + εψt (60)

Bank Optimization

Qκt =κ

κct∀κ ∈ L (61)

νt = νbt (62)

νbt =κctQitνst (63)

λt(κ) =(κ− κct)νst

κθQit − (κ− κct)νst∀κ ∈ B (64)

λt(κ) = 0 ∀κ ∈ L (65)

Ωt(κ) = 1− σ + σνbt(1 + λt(κ)) (66)

φt(κ) =κ(νst − θQit)

νbtQit − κ(νst − θQit)

(67)

37

νbt = EtRtΛt,t+1(1− σ + σνbt+1) + EtRtΛt,t+1σνbt+1G0 (68)

νst = EtΛt,t+1Ψt+1[(1− σ + σνbt+1 + σνbt+1G0)Zt+1

+ ((1− σ)(

∫ κ

κct+1

p(κ)dκ) + σνbt+1G0)(1− δ)Qit+1

+ (1− σ + σνbt+1)(

∫ κct+1

κp(κ)κdκ)] (69)

G0 =

∫ κ

κct+1

p(κ)λt+1(κ) dκ (70)

Securities Market

Qit(It + (1− δ)Kt

∫ κ

κct

p(κ)dκ) = ((Zt + (1− δ)Qit)(σ + ξ)Kt − σRt−1Dt−1)G1 (71)

G1 =

∫ κ

κct

p(κ)κ(1 + φt(κ)) dκ (72)

Deposit Market

Dt = (Zt + (1− δ)Qit)(σ + ξ)Kt − σRt−1Dt−1)G2

− (Zt − σRt−1Dt−1)(

∫ κct

κp(κ)dκ)

− (1− δ)Qnt (σ + ξ)Kt(

∫ κct

κp(κ)κdκ) +Qnt (1− δ)Kt(

∫ κct

κp(κ)dκ) (73)

G2 =

∫ κ

κct

p(κ)κφt(κ) dκ (74)

38

A.2 Steady State

HH

I = δK (75)

C = [A(L

K)1−α − δ]K (76)

χLε = (1− α)A(L

K)−α

1− βγ1− γ

1

C(77)

L = (Z

αA)

11−αK (78)

R =1

β(79)

Λ = β (80)

Qi = 1 (81)

Bank Optimization

λ(κ) =(κ− κc)νs

κθ − (κ− κc)νs∀κ ∈ B (82)

λ(a) = 0 ∀a ∈ L (83)

1

κc= νs(1−

σ

1− σG0) (84)

νb = κcνs (85)

φ(κ) =κ(νs − θ)

νb − κ(νs − θ)(86)

Qn =1

κc(87)

Z =νsβνb− 1− δ

νb

((1− σ)(

∫ κ

κcp(κ)dκ) + σνbG0

)− (1− δ)Qn

νb

((1− σ + σνb)(

∫ κc

κp(κ)κdκ)

)(88)

Securities Market

σ

β

D

K= (Z + 1− δ)(σ + ξ)−

δ + (1− δ)(∫ κκc p(κ)dκ)

G1

(89)

Deposit MarketN i

K= (Z + (1− δ))(σ + ξ)− σ

β

D

K(90)

39

D

K=N i

KG2−

((Z + (1− δ)Qn)(σ + ξ)− σ

β

D

K

)(∫ κc

κp(κ)dκ

)−(1−δ)Qn(σ+ξ)

(∫ κc

κp(κ)κdκ

)+Qn(1− δ)

(∫ κc

κp(κ)dκ

)(91)

To solve the model, I first guess a productivity cutoff κc. Then the value from interbank

lending νb = κcνs, and the equation for interbank lending above becomes an equation in

one variable, which we can solve for parameters. We can then use the equation for νs to get

Z, and use the investing islands securities market equation to get DK . With this in hand, we

can test the deposit market clearing condition, by calculating both the right and left hand

sides of equation (21).

G0 =

∫ κ

κct+1

p(κ)(κ− κct+1)νst+1

κθQit+1 − (κ− κct+1)νst+1dκ (92)

G1 =

∫ κ

κct

p(κ)κκctνstκctνst − κνst + κθQit

dκ (93)

G2 =

∫ κ

κct

p(κ)κ(νst − θQit)κctνst − κνst + κθQit

dκ (94)

G0 =

∫ κ

κc

p(κ)(κ− κc)νsκθ − (κ− κc)νs

dκ (95)

G1 =

∫ κ

κc

p(κ)κκcνsκcνs − κνs + κθ

dκ (96)

G2 =

∫ κ

κc

p(κ)κ(νs − θ)κcνs − κνs + κθ

dκ (97)

40