Financial frictions

Martin Ellison

MPhil Macroeconomics, University of Oxford

1 Introduction

In the previous lecture we saw that the standard representative agent macroeconomic model

runs into problems when trying to explain the simultaneous existence of a high equity pre-

mium and a low risk free rate. There have been many attempts to amend the representative

agent model to rectify these problems. In this lecture we focus on the role of financial in-

termediaries and financial frictions. Compared with the standard representative agent model

where financial intermediation happens costlessly and perfectly, we will be interested in mod-

els where there are imperfections in the efficiency with which financial intermediaries channel

funds from savers to investors. We will look at four models in particular, two in which there

are moral hazard problems, one where there is adverse selection and one where there is asym-

metric information and monitoring costs. Whilst the models are different in terms of the

mechanisms they highlight, they all stress the importance of the evolution and holdings of

net worth in the economy. This is important when we want to have interactions between the

macroeconomy and financial markets. In the representative agent models we typically only

have causality from the macroeconomy to financial variables, but in models with financial

frictions we have important feedback from financial variables to the macroeconomy. By the

end of this lecture we will have explored four mechanisms whereby a fall in asset values leads

to a drop in financial intermediation and a jump in interest rate spreads.

2 Key readings

The main reference for this lecture is a preliminary paper “Government Policy, Credit Markets

and Economic Activity” by Christiano and Ikeda (2012, mimeo), which presents the four

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models we will discuss. All the models are presented in 2-period versions for pedagogical

reasons, but can easily be extended into the infinite horizon if required. The first model is a

simplified version of the models of moral hazard in “Financial Intermediation and Credit Policy

in Business Cycle Analysis” by Gertler and Kiyotaki (2011, Handbook of Monetary Economics)

and “A Model of Unconventional Monetary Policy” by Gertler and Karadi (2009, mimeo).

These are proving very popular amongst policymakers and researchers trying to understand

the recent crisis. The third model with adverse selection goes back to “The Allocation of

Credit and Financial Collapse” by Mankiw (1985, Quarterly Journal of Economics). The

fourth model is the classic Bernanke, Gertler and Gilchrist “The Financial Accelerator in a

Quantitative Business Cycle Framework” (1999, Handbook of Macroeconomics) model of the

financial accelerator, which was the most common workhorse model prior to the start of the

financial crisis. The Christiano and Ikeda (2012) paper not only presents the models but also

discusses the appropriateness of different policy responses in each model.

3 Key concepts

Financial imperfections; Moral hazard; Adverse selection; Costly state verification; Financial

accelerator

4 Moral hazard and absconding

The first model we examine focuses on moral hazard problems in financial intermediaries,

following Gertler and Karadi (2009) and Gertler and Kiyotaki (2011). The key assumption is

that financial intermediaries can do a runner, absconding with the money that depositors have

placed in their care. This provides an incentive for financial intermediaries to default on their

obligations to depositors. To make sure this does not happen all the time, it is assumed that

financial intermediaries have some net worth of their own which they commit to investment

projects alongside the funds of depositors. Then, if the financial intermediary does default it

can only expropriate a proportion of its own net worth. This provides an incentive for financial

intermediaries not to default on their obligations. To give a simple example, a bank could

default after taking in deposits but in doing so would not be able to abscond with the worth

of the real estate (branches, head office and so on) owned by the bank.

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4.1 Households

Households in the model consist of some ‘bankers’ and some ‘workers’, with perfect insurance

within the household such that both bankers and workers have the same level of consumption

in each of two periods. The behaviour of the household is then standard, with a first period

budget constraint c1 + d ≤ y that restricts period one consumption c1 plus deposits d to be

less than an endowment per household member of y goods. The endowment has to be either

consumed in the first period or placed on deposit - it cannot be used by bankers directly. In

the second period the budget constraint is c2 ≤ Rd+π, where R is the return paid on deposits

and π is the profit brought home by the bankers. We assume that the household treats R as

given and π as a lump sum transfer. In other words, neither the worker nor the banker takes

into account that their individual actions will affect the rate of return or bank profitability. We

assume throughout that bankers are randomly matched to households so there is infinitessimal

probability that a household will be matched with exactly a banker from their own household.

Combining the budget constraints defines the household intertemporal budget constraint

c1 +c2R≤ y +

π

R

The utility function of the household is u(c1) + βu(c2), and we adopt the standard CRRA

form for the utility function u(c) = c1−γ

1−γ. The interesting case for us requires 0 < γ <

1, since this ensures in equilibrium that the substitution effect dominates the income effect

such that where the return R increases there is an increase in deposits d. The solution of

the household optimisation problem is characterised by the Euler equation for consumption

u′(c1) = βRu′(c2). In full, the household problems implies

c1 =y + π

R

1 + (βR)1γ

R

d = y − c1

c2 = Rd+ π

for given y, β, R and π. The first two of these variables are exogenous to the model, but R

and π are to be determined in equilibrium in the financial market.

4.2 Firms

The role of firms in the model is to produce good for consumption in period 2. To do this,

they sell securities s to bankers and use the proceeds to buy goods in the first period which

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they turn into capital and produce sRk goods in period 2, with Rk the return on privately

issued securities that is fixed exogenously. Firms are perfectly competitive and make no profit

so pass sRk revenue back to the bankers.

4.3 Bankers

Bankers are endowed with N goods in the first period. We will refer to this as the net worth

of bankers. In the simplified 2-period model we discuss here it is treated as an exogenous

variable, but in a multiperiod model it will change over the time as the banker makes profits

or losses each period. In a fully specified DSGE model with many periods N becomes an

important state variable that means financial markets play a role in propagating shocks in the

economy. We do not get that feature in our 2-period setup, but we can exogenously change

N and see what effect this will have. The bankers accept deposits d and purchase securities s

from firms. The banker takes return to deposits R and the return to securities Rk as given,

and chooses the amount of deposits to accept from depositors to maximise

π = maxd

(sRk −Rd

)

The banker will always purchase the maximum quantity of securities possible, given its own

net worth and the deposits it takes in, because securities pay a positive return with certainty.

In other words, N + d = s.

4.4 Equilibrium

If there are no financial frictions in the economy then equilibrium is characterised by (i) the

household solving its utility maximisation problem, (ii) the banker solving its profit maximi-

sation, (iii) market clearing and (iv) non-negativity constraints c1, c2 > 0. The benchmark

equilibrium with no financial frictions is easy to characterise since it requires R = Rk. Oth-

erwise, if R > Rk the banker would set d = 0 or if R < Rk the banker would take in an

infinite amount of deposits. The condition R = Rk is sufficient to characterise the equilibrium

allocation c1, c2, R, π. Note that this is the first best allocation with the optimal amount of

deposits d. It is a simple exercise to show that it would be chose by a social planner.

We now make the equilibrium more interesting by introducing a moral hazard problem. In

particular, we follow Gertler and Karadi (2009) and Gertler and Kiyotaki (2011) in assuming

that the banker can default after receiving the payment sRk from firms in period 2. If the

banker chooses not to default then the behaviour of the equilibrium is as in the no financial

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frictions equilibrium as before. In they do default, then a banker can abscond with a fraction

θ of their total assets. The fraction 1 − θ is returned to depositors. The allocation of assets

to the banker and depositor on default is therefore

θRk(N + d) to bankers

(1− θ)Rk(N + d) to depositors

The banker will not default if the profits when not defaulting exceed the profits when default-

ing, i.e. if

(N + d)Rk −Rd︸ ︷︷ ︸Profit when no default

≥ θRk(N + d)︸ ︷︷ ︸Profit when default

The above equation identifies the nature of the trade off faced by bankers. If they default

then they gain because they do not have to pay Rd to the depositors, but they lose because

they only obtain a fraction θ of the return to securities (N+d)Rk. The fundamental tension is

between getting the full benefit of financing firms (when not defaulting) and avoiding paying

depositors (when defaulting). Note the key role here of the net worth of bankers N . An

increase in N increases the left hand side of the inequality more than the right hand side of

the inequality, so increasing N means that a banker is less likely to default. This is because

the banker loses some of their net worth when defaulting - the more of their own assets they

commit to the project the less likely they are to want to default.

We are interested in symmetric equilibria where default does not happen even though

bankers face a moral hazard problem.1 As before in the case of no financial imperfections,

the banker chooses the level of deposits d to take in and takes the returns Rk and R as given.

Since the banker is in a symmetric equilibrium with no default, it cannot accept a level of

deposits that would give it an incentive to default. If it did so, then depositors would instantly

withdraw their deposits and take them to another bank. In other words, the banker is subject

to a no default condition and faces the following problem

π = maxd

[(N + d)Rk −Rd

]

s.t.

(N + d)Rk −Rd ≥ θRk(N + d)

The equilibrium is characterised as before by (i) household solving its utility maximisation

problem, (ii) banker solving its profit maximisation problem, (iii) market clearing and (iv) non-

negativity constraints. The problem of the banker can be solved by setting up a Lagrangean

1We assume restrictions on Rk and θ that guarantee (N + d)Rk −Rd ≥ θRk(N + d) in equilibrium.

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L =(N + d)Rk −Rd+ λ[(N + d)Rk −Rd− θRk(N + d)

]

which has a first order condition

Rk −R =λ

1 + λθRk > 0

If the no default constraint does not bind in equilibrium then λ = 0 and Rk = R as in

the equilibrium with no financial imperfections. However, when the moral hazard problem

is sufficiently large the constraint starts to bind, λ > 0 and Rk − R > 0 so a spread opens

up between the exogenous return Rk to securities and the endogenous return R to deposits.

The return R is less than socially optimal so deposits d are less than socially optimal and

the household does not save as much as they should.2 In this way the presence of frictions in

financial markets imposes real costs on the economy and a reduction in social welfare.

An increase in the net worth of bankers N helps to ameliorate the problems caused by the

moral hazard friction. As we said before, increasing net worth makes it less likely that the

banker will default. In the Lagrangean, this means that the no default constraint binds less

tightly so through the first order condition an increase in N causes a fall in λ and R rises closer

to Rk. Two things are worthy of particular note here. Firstly, if we did have a model with

more than 2 periods then we would have a model with a dynamic financial accelerator with

negative shocks causing a fall in net worth in one period being propagated to the following

period by depressed net worth of bankers. Secondly, the focus on net worth of bankers partly

explains why central bankers pay so much attention to the health of bank balance sheets and

spent so many resources helping bankers to repair their balance sheets after the recent financial

crisis.

5 Moral hazard and effort

The second model we look at introduces a moral hazard problem between bankers and firms,

rather than between bankers and depositors as in the first model. In particular, it is assumed

that a banker can choose how much effort to make when buying securities from firms. If

the banker makes a lot of effort then they can identify good quality securities and be pretty

confident that the securities will pay a high return. In contrast, if the banker makes only

2This result relies on the assumption that 0 < γ < 1, although even if this was not true the equilibrium in

an economy with financial frictions would not have the socially optimal level of deposits.

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little effort then they are likely to end up holding bad securities that only pay a low return.

The banker’s resulting incentive to make an effort to identify good securities is tempered by

effort being costly, which can be thought of as the time cost of the effort of identifying good

securities.3

Households in the model have the same endowment process as in the first model, such

that optimal consumption decisions in periods 1 and 2 must satisfy the Euler equation for

consumption u′(c1) = βRu′(c2). We continue to assume 0 < γ < 1 when working with a

utility function of the CRRA form, to restrict attention to the interesting case whereby an

increase in the return on deposits leads to an increase in funds deposited. To get the moral

hazard problem running, it is necessary that depositors and bankers cannot sign contracts

that condition on the level of effort made by bankers. We satisfy this condition by assuming

that effort is unobservable, an assumption that does not sit easily in the framework of the first

model where there is one-to-one matching between a depositor and a banker. In such a world

it is difficult to argue that the depositor cannot observe the effort made by the banker. To

circumvent this problem, we assume that depositors place their deposits with mutual funds,

which then pass on the deposits to all bankers in the economy. With mutual funds diversified

across all bankers, it is easier to argue that it is difficult for depositors to monitor bankers.

Mutual funds are completely diversified across bankers so are not in a position to monitor the

effort of each and every banker.

5.1 Bankers

Bankers have an endowment N of goods in the first period. They receive deposits d from the

mutual funds and combine them with their endowment to purchase securities s = N + d from

firms. The quality of the securities can either be good or bad, with the probability that the

securities purchased by the banker being good depending on the amount of effort e made by

the banker. For simplicity it is assumed that the probability of purchased securities being

good p(e) is linear in effort, i.e.

p(e) = a+ be

where b > 0 so p′(e) = b > 0 and p′′(e) = 0. The parameters of the model are restricted so that

0 < p(e) < 1 in equilibrium. Good securities pay a certain return Rg and bad securities pay

a certain return Rb so mean return on bank assets is p(e)Rg + (1− p(e))Rb and the variance

3A similar trade-off is at the heart of “Too Many Dragons in the Dragons’ Den” (Ellison and Giannitsarou,

2010), available for download from my webpage.

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of the return on bank assets is p(e)(1 − p(e))(Rg − Rb)2. We assume p(e) > 0.5 so that the

variance of the return falls when the banker makers more effort.

5.2 Observable effort benchmark

The natural benchmark against which to assess how the second moral hazard problem affects

equilibrium is a model in which the effort of banks is observable. In that case, the deposit

contract between mutual funds and bankers can be conditioned on the effort of the banker

and there will be no financial market imperfections. The loan contract between the banker

and mutual fund stipulates (d, e, Rdg, Rdb ), where Rdg and Rdb are conditional returns paid out

if the securities purchased by the bank turn out to be good or bad respectively. The mutual

funds themselves take deposits d from households and pay a return R which it treats as given.

They are competitive and so any contract between mutual funds and bankers must satisfy a

zero profit condition

p(e)Rdgd+ (1− p(e))Rdbd = Rd

otherwise if profits were positive bankers would set d→∞ which would exhaust the deposits

of households, or if they were negative they would set d = 0 and achieve zero profit. The

banker needs to ensure they have enough resources to pay the mutual funds irrespective of

whether the securities they purchase are good or bad. In other words, there are two cash

constraints

Rg(N + d)−Rdgd ≥ 0

Rb(N + d)−Rdbd ≥ 0

which have to be satisfied by the optimal contract. It is possible to show that these cash

constraints in practice either never bind or if they do bind they bind in the bad state of nature

when securities turn out to be bad. In defining the problem of the banker it is then sufficient

to only consider the second cash constraint. If the banker has enough resources to cover their

commitments when the securities it has purchased are bad, then it will automatically have

sufficient resources to cover commitments if it ends up with good securities.

The maximisation problem of the banker/mutual fund involves choosing (d, e, Rdg, Rdb ) to

maximise its return (less effort cost) subject to the zero profit condition for mutual funds and

the cash constraint to have sufficient resources to cover its commitments in the bad state of

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the world.

maxd,e,Rdg,R

db

λ{p(e)(Rg(N + d)−Rdgd) + (1− p(e))R

b(N + d)−Rdbd)}−1

2e2

s.t.

p(e)Rdgd+ (1− p(e))Rdbd = Rd

Rb(N + d)−Rdbd ≥ 0

The constant λ is the marginal utility of consumption in the household of the banker, taken

as given by the banker. The cost of effort is modeled as a quadratic increasing function. The

Lagrangean of the problem is4

L = λ{p(e)(Rg(N + d)−Rdgd) + (1− p(e))R

b(N + d)−Rdbd)}−1

2e2

+µ[p(e)Rdgd+ (1− p(e))R

dbd−Rd

]+ ν

[Rb(N + d)−Rdbd

]

The first order conditions determine the nature of the optimal contract. We look at the first

order conditions with respect to Rdg, Rdb and e

−λp(e) + µp(e) = 0

−λ(1− p(e)) + µ(1− p(e))− ν = 0

λp′(e)[(Rg −Rb)(N + d)− (Rdg −R

db)d]− e+ µp′(e)(Rdg −R

db)d = 0

The first two of these conditions imply µ = λ and ν = 0 so the cash constraint is not

binding in the equilibrium where effort is observable. The conditional payments Rdg and Rdb are

indeterminate in equilibrium. There is an equilibrium where payments are state contingent

Rdg = Rg and Rdb = Rb, but there may also be an equilibrium where deposit rates are not

state-contingent so that Rdg = Rdb = R if N is sufficiently large. The third first order condition

determines the optimal level of effort.

e = λb(Rg −Rb)(N + d)

5.3 Unobservable effort

If the effort of the bank is unobservable then it is no longer possible to condition the contract

between mutual funds and bankers on effort. Instead, we imagine a situation where the mutual

4Note that the second constraint has been transposed compared with Christiano and Ikeda (2010) to ensure

that both Lagrange multipliers are positive.

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funds draw up a contract on (d,Rdg, Rdb) and the banker chooses effort e. In choosing effort,

the banker has the same objective as before except they do not worry about the constraints

that profits for the mutual funds have to be zero and that there is a cash constraint that has

to be satisfied in the bad state of the world. We assume that the mutual funds worry about

these things and only offer contracts that satisfy those constraints. The banker takes d,Rdg, Rdb

as given and chooses e to solve the maximisation problem

maxeλ[p(e)(Rg(N + d)−Rdgd) + (1− p(e))R

b(N + d)−Rdbd)]−1

2e2

The first order condition is

λp′(e)[(Rg −Rb)(N + d)− (Rdg −R

db)d]− e = 0

which is the same as that in the observable effort case above, except µ = 0 because the banker

does not worry about the zero profit condition for mutual funds.

We now turn to the problem of the mutual funds, who choose (d,Rdg, Rdb) to maximise

the same objective as the bankers but taking into account that the effort of bankers will be

determined by their first order condition for optimal effort. The Lagrangean of the problem

of mutual funds is then

L = maxd,Rdg,R

db

λ[p(e)(Rg(N + d)−Rdgd) + (1− p(e))R

b(N + d)−Rdbd)]− 1

2e2

+µ[p(e)Rdgd+ (1− p(e))R

dbd−Rd

]

+η[λp′(e)

[(Rg −Rd)(N + d)− (Rdg −R

db)d]− e]

+ν[Rb(N + d)−Rdbd

]

which is identical to that in the observable effort case apart from the additional constraint

which is indexed by the Lagrange multiplier η. The first order conditions for Rdg, Rdb and e are

−λp(e) + µp(e)− ηλp′(e) = 0

−λ(1− p(e)) + µ(1− p(e)) + ηλp′(e)− ν = 0

λp′(e)[(Rg −Rb)(N + d)− (Rdg −R

db)d]− e

+µp′(e)(Rdg −Rdb)d+ η

[λp′′(e)

[(Rg −Rd)(N + d)− (Rdg −R

db)d]− 1] = 0

The first two of these conditions combine to give µ = λ+ ν and νp(e) = ηλb as p(e) is linear

in e. The effort constraint can be used to substitute out for e in the first order condition for

effort to give

(λ+ ν)b(Rdg −Rdb)d− η = 0

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We distinguish between two different cases. The first characterises “normal times” in that

the bankers are assumed to have sufficient net worth N that the cash constraint Rb(N + d)−

Rdbd ≥ 0 does not bind in equilibrium. In this case ν = 0 because the cash constraint does

not bind and νp(e) = ηλb implies η = 0 as well. It is clear that if ν = η = 0 then from the

above equation it must be the case that Rdg −Rdb = 0 and the equilibrium is characterised by

non-contingent payments. Imposing further that the mutual funds makes zero profit we have

that R = Rdg = Rdb . To summarise, equilibrium in normal times is characterised by bankers

making non-contingent payments to the mutual funds. The degree of effort made by bankers

in normal times is defined by imposing Rdg = Rdb on the first order condition for effort to obtain

e = λb(Rg −Rb)(N + d)

This level of effort is the same as that which resulted in the model with effort observable

so will be socially optimal. Intuitively, in normal times bankers have sufficient incentives to

make an effort when purchasing securities because they are investing enough of their own net

worth N in the securities. The moral hazard problem is not present as the bankers choose the

optimal amount of effort anyway. In normal times they have sufficient funds that it is in their

own self-interest.

The second case we distinguish is one of “abnormal times” where the cash constraint is no

longer binding. Since (λ + ν)b(Rdg −Rdb)d− η = 0 must still hold in equilibrium and we have

that neither Lagrange multiplier is zero it follows that

Rdg −Rdb =

η

λ + νbd

and the returns Rdg and Rdb to the mutual funds become conditional. We therefore see that

financial markets where bankers have low net worth are characterised by a spread between

returns. To see the effect of this on the effort made by bankers, return to the bankers’ first

order condition for effort and write

e = λb[(Rg −Rd)(N + d)− (Rdg −R

db)d]< λb

[(Rg −Rd)(N + d)

]

to see that the level of effort when effort is not observed is lower than that in the model where

effort is observed. This is the key cost imposed by the moral hazard problem in this model.

In an ideal world, it is wise to allow the banker to be the residual claimant on the project. In

other words, it is good to allow the agent choosing how much effort to make to reap the full

benefit of making their effort. In normal times when the equilibrium is characterised by non-

contingent returns, the banker has to pay R = Rdg = Rdb to the mutual funds irrespective of

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whether they purchase good or bad securities. The incentive for the banker to make the effort

to find good securities is fully preserved as the banker receives the full benefit of purchasing

good securities. However, in abnormal times it is not possible to support an equilibrium with

noncontingent returns. Loosely speaking, one can think of the banker as not having sufficient

funds to pay out a non-contingent return to the mutual funds in a bad state of the world. In

such a situation the optimal contract has to be adjusted so that the banker pays less in the

bad state of the world and more in the good state of the world, i.e. returns are contingent.

Whilst this is good for ensuring the banker is always able to satisfy their cash constraint, it

does cause problems because the banker is no longer the “residual claimant”. Put simply,

there is less of an incentive for the banker to make the effort to find good securities if they

know that conditional on purchasing good securities they will have to pay a higher return to

mutual funds. Some of the return to effort is lost to the banker as the additional contingent

return they have to pay mutual funds. The differential Rdg − Rdb weakens the incentive for

effort and the financial friction has real welfare costs in the economy.

6 Adverse selection

The third model we discuss highlights adverse selection problems in investment decisions and

is by Mankiw (1986). The key margin of interest is the number of bankers who decide to make

investments in risky projects. As we shall see, the optimal number of projects to invest in

depends as usual on the production technology (the marginal rate of transformation). However,

the number of projects invested in when there is an adverse selection problem depends not

only on the production technology but also the net worth of bankers. This leads to too few

projects being invested in and returns to deposits being too low when there is an adverse

selection.

The household in this model consists of workers and bankers, with the measure of bankers

being e. Household members perfectly insure each other such that the consumption of worker

and banker members of the household is equal. The household receives an endowment y of

goods in the first period as in the previous two models, so faces a first period budget constraint

c1 + d = y where d are deposits at mutual funds. Bankers earn profits per capita of π, so the

second period budget constraint of the household is c2 ≤ Rd+ eπ. The household has CRRA

preferences and their Euler equation for consumption is c−γ1 = βRc−γ2 with γ > 0.

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6.1 Bankers

Bankers have an endowment N < 1, which they can either invest in a mutual fund or invest

in a risky project. If the banker invests their endowment in a mutual fund they receive a

certain return RN in the next period. The risky project available is indivisible, needing an

investment of one unit of goods in period 1. Since the endowment N < 1 it follows that an

banker wanting to make the risky investment needs to borrow 1 − N from the mutual fund.

The rate of interest on loans is r, to be determined in equilibrium. Bankers take the deposit

rate R and the loan rate r as given.

The risky investment project on offer to a banker in the first period pays a random return

θ > 0 with random probability p in the second period. With probability 1 − p the risky

project pays nothing in the second period. The random variables θ and p are drawn from a

distribution F (θ, p) and are private information to the banker and the household to which the

bank belongs. The mutual funds know the distribution function F (θ, p) but do not know the

values of θ and p for a particular project. In other words, the banker knows the return and

risk associated with their own project, whereas the mutual funds only know the return and

risk associated with projects in general.

To derive analytical results, Mankiw (1985) imposes a strong restriction on the distribution

function F (θ, p). In particular, he imposes the condition θp = θ̄ which means that all projects

have the same expected return pθ+ (1− p)0 = θ̄. There are some projects that are very risky

yet pay a high return and some projects that are low risk but pay a low return. Reducing the

number of random variables in this way means we can consider only the probability of success

of the project p to be random and let the payoff satisfy θ = θ̄/p. We assume for simplicity

that p is distributed according to a uniform distribution on [0, 1].

The choice of the banker is between investing in their project or not. The expected return

of investing is p(θ− r(1−N))+ (1− p)0 = θ̄− pr(1−N), which the banker compares against

the certain return of RN which they receive if they do not invest in their project. The banker

will choose to invest if

θ̄ − pr(1−N) ≥ RN

so the projects that are invested in will be those with a low probability of success. We define

a critical probability p̄(r) so all projects that are invested in satisfy 0 < p < p̄(r) where

p̄(r) =θ̄ −RN

1−N

p̄(r) is by definition the fraction of bankers activating their projects. Since p is a uniform

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distribution p̄(r) =∫ p̄(r)0

dp is also the total quantity of banker investment in the economy.

The average value Π(r) = 12p̄(r) is the average value of p amongst bankers investing in their

projects.

6.2 Mutual funds

Mutual funds operate in a competitive market so make zero profits in equilibrium. They invest

in all bankers wanting to invest in their projects, so are fully diversified and have costs and

revenues that are non-stochastic. The cost of a unit of funds to mutual funds is the return R

they pay to depositors. The income from a unit of funds is the loan rate r times the average

value Π(r) = 12p̄(r) of p amongst bankers investing in their project. The zero profit condition

is

Π(r)r = R

so the spread between loan and deposit rates is

r

R=

2

p̄(r)> 2

The definition of the critical probability p̄(r) at which the bankers invests in projects implies

R =θ̄

2−N

so the deposit rate is determined by the investment technology θ̄ and the net worth of bankers

N . The loan rate is also determined by the same variables as Π(r)r = R. The adverse selection

problem is at the heart of both these results because the revenue

Π(r)r =1

2

θ̄ −RN

1−N

of the mutual fund is completely independent of the rate of interest it charges on its loans.

Intuitively, the mutual fund gains extra revenue from increasing r as bankers have to pay

back more but the high quality projects (in terms of their probability of paying back their

loan) are no longer invested in so the average probability p̄(r) of them paying also falls. In

the adverse selection equilibrium these two effects exactly offset each other. This implies that

deposit rates and loan rates are completely determined by the features of the deposit and loan

market, not for example by preferences as is usual in first best allocations.

14

6.3 Equilibrium

To fully characterise equilibrium with adverse selection, start with loan market clearing

ep̄(r) = d+ eN

where ep̄(r) is investment, d is deposits and eN is net worth of bankers. The income of bankers

is

π =

∫ p̄(r)

0

[θ̄ − pr(1−N)

]dp+

∫ 1

p̄(r)

NRdp = p̄(r)[θ̄ − Π(r)r(1−N)

]+ (1− p̄(r))NR

so total household income in the second period is

Rd+ ep̄(r)[θ̄ − Π(r)r(1−N)

]+ e(1− p̄(r))NR = ep̄(r)θ̄

which is a particularly neat expression because Π(r)r = R and d = ep̄(r) − eN . The rest of

the equilibrium is

c1 =y+eN

(βR)1γ +θ̄θ̄ R = θ̄

(2−N)

c2 =y+eN

(βR)1γ +θ̄

(βR)1

γ θ̄ r = 2eR (βR)−1γ θ̄+1

y+eN

6.4 Social optimum

The benchmark against which the equilibrium under adverse selection should be compared is

the first best allocation that would be selected by a social planner. The social planner chooses

the mass ep∗ of bankers who will invest in their projects, needing d+ eN resources to do so.

The planner is indifferent between which bankers activate their projects since the expected

return on all projects is identical and the social planner can diversify away the risk in any one

project. The optimisation problem of the social planner is

maxc∗1,c∗2,p∗,d∗

u(c∗1) + βu(c∗

2)

s.t.

c∗1 + d∗ ≤ y

ep∗ ≤ d∗ + eN

c∗2 ≤ ep∗θ̄

which has solution

15

Social optimum Adverse selection equilibrium

c∗1 =y+eN

(βθ̄)1γ +θ̄θ̄ c1 =

y+eN

(βR)1γ +θ̄θ̄

c∗2 = c1(βθ̄)1

γ c2 = c1(βR)1

γ

ep∗ = y+eN

(βθ̄)1γ +θ̄

(βθ̄)1

γ ep̄(r) = y+eN

(βR)1γ +θ̄

(βR)1

γ

The social optimum is different to the allocation in the adverse selection equilibrium be-

cause R = θ̄2−N

�= θ̄. In particular, the return to deposits R in the adverse selection equilibrium

is too low, which tempts households to consume too much in the first period and c1 > c∗

1. The

resources being passed to the second period are too low, with ep̄(r) < ep∗ as not enough

bankers invest in their projects p̄(r) < p∗. Second period consumption is then too low c2 < c∗

2.

With first period consumption too high and second period consumption too low, the adverse

selection problem distorts the intertemporal trade-off and social welfare is lower than it should

be. Note that the net worth of bankers N plays a crucial role in creating the distortion. If

the endowment of bankers was N = 1 so they could invest in their their own project then

R = θ̄2−N

= θ̄ and the adverse selection equilibrium is also efficient. The market fails under

adverse selection because the price mechanism is unable to give sufficient incentives for bankers

to activate their projects and drive up the return to deposits. Remember that the revenues of

the mutual fund are independent of the loan rate r, so there is no incentive for them to lower

the loan rate to induce more bankers to invest in their projects.

7 Asymmetric information and monitoring costs

The final model to be considered is the famous financial accelerator of Bernanke, Gertler

and Gilchrist (1999). This model has not surprisingly been very influential in the policies

of Chairman Bernanke over the last couple of years. The asymmetric information problem

is caused by bankers having the possibility of faking bankruptcy in the second period and

reneging on their loan payment obligations to mutual funds. Mutual funds are able to offset the

incentives for bankers to fake bankruptcy by subjecting them to a monitoring process, which

reveals whether a banker really is bankrupt or not. Monitoring is costly to the mutual funds

and these frameworks are often know as “costly state verification” (CSV) models. Financial

markets do not work perfectly because there is asymmetric information (between the banker

and the mutual fund) that is costly to reveal.

16

7.1 Bankers

Bankers have net worth N in the first period. They then augment this net worth with a loan B

from mutual funds to produce output in period 2. The size of the loan B is a choice variable of

bankers. It is assumed that bankers sign a standard debt contract with mutual funds, repaying

ZB and keeping the rest if possible and losing everything in the event of bankruptcy. For now

we will assume that the banker only declares bankruptcy if absolutely necessary - later we will

see how mutual funds put mechanisms in place to make sure that the banker does not fake

bankruptcy. In period 2 the banker produces ω(N + B)rk with rk an exogenous aggregate

return determined by technology. Production is subject to an idiosyncratic productivity shock

ω ∼ F where F is the lognormal cumulative distribution function with support (0,∞) and

standard deviation σ. The standard debt contract requiring repayment of ZB if possible

means there will be a cutoff level of productivity ω̄ at which the banker has insufficient funds

to meet their loan obligations and so is forced to declare bankruptcy

ω̄(N +B)rk = ZB

The expected return to the banker is calculated by integrating the return for all values of

the idiosyncratic productivity shock above the cutoff.∫∞

ω̄

(ω(N +B)rk − ZB

)dF (ω) =

∫∞

ω̄

(ω − ω̄) (N +B)rkdF (ω) = NLrk∫∞

ω̄

(ω − ω̄) dF (ω)

where L = N+BN

is leverage. Bankers choose B to maximise this objective. To aid notation

later we write∫∞

ω̄

(ω − ω̄) dF (ω) =

∫∞

ω̄

ωdF (ω)− ω̄

∫∞

ω̄

dF (ω) = 1−G(ω̄)− ω̄(1− F (ω̄)) = 1− Γ(ω̄)

where G(ω̄) =∫∞

ω̄ωdF (ω) and Γ(ω̄) = G(ω̄) + ω̄(1− F (ω̄)).

7.2 Mutual funds

Mutual funds receive deposits from households and make loans to bankers so they can produce

goods in the second period. They are perfectly competitive. To prevent bankers from faking

bankruptcy they commit to monitoring any bankers who declare themselves bankrupt. Whilst

this is sufficient to ensure truth-telling on the part of bankers, it is costly in that the mutual

fund is only able to recover a fraction 1− µ of the assets of a bankrupt banker. The mutual

fund receives revenues from successful bankers, who have sufficiently high productivity to be

17

able to repay their loan, and some revenue from unsuccessful bankers who declare bankruptcy.

The mutual fund’s costs arise from its need to pay a return to depositors. Zero profits requires

(1− F (ω̄))︸ ︷︷ ︸proportion of

loans that are

successful

ZB + (1− µ)︸ ︷︷ ︸share of assets

recovered from

unsuccessful loans

ω̄∫

0

ω(N +B)rkdF (ω)

︸ ︷︷ ︸average assets in

unsuccessful loans

= RB︸︷︷︸paid to

depositors

In terms of our G(ω̄) and Γ(ω̄) notation, the zero profit condition is

(Γ(ω̄)− µG(ω̄))rk(N +B)

B= R

7.3 Bankers again

We are now in a position to define and solve the optimisation problem of the banker, whose

contract with mutual funds maximises its objective subject to a zero profit condition for

mutual funds. The banker chooses the quantity of loans B and the cutoff ω̄ probability at

which they declare bankruptcy. Since leverage L is defined by L = N+BN

and net worth N is

given, we can also think of the banker choosing the leverage ratio L instead of the quantity of

loans. The maximisation problem is

maxL,ω̄L (1− Γ(ω̄))

s.t.

L =1

1− rk

R(Γ(ω̄)− µG(ω̄))

where the constraint is the zero profit condition for mutual funds, written in terms of leverage.

The constraint can be substituted directly in the objective to obtain 1−Γ(ω̄)

1−rk

R(Γ(ω̄)−µG(ω̄))

and there

is only one first order condition, for ω̄

1− F (ω̄)

1− Γ(ω̄)=

rk

R[1− F (ω̄)− µω̄F ′(ω̄)]

1− rk

R[Γ(ω̄)− µG(ω̄)]

where we have used that G′(ω̄) = ω̄F ′(ω̄) and Γ′(ω̄) = G′(ω̄)+1−F (ω̄)− ω̄F ′(ω̄) = 1−F (ω̄).

This equation determines the cutoff productivity level ω̄ as a function of rk

R, which are taken

as given by the banker. Once ω̄ has been calculated, leverage L and the interest rate Z are

L = 1

1− rk

R(Γ(ω̄)−µG(ω̄))

Z = ω̄rk LL−1

18

7.4 Households

Households follow our standard pattern by placing deposits B out of their first period endow-

ment y at mutual funds so that first period consumption is c1 +B ≤ y. In the second period

the household receives the return to its deposits RB = rk(N +B) (Γ(ω̄)− µG(ω̄)), the profits

brought back by bankers π = NLrk (1− Γ(ω̄)) and zero profits from mutual funds. It follows

that the second period budget constraint is

c2 ≤ rk(N +B) [1− µG(ω̄)]

The household has preferences u(c1) + βu(c2) so the optimal allocation satisfies the Euler

equation for consumption u′(c1) = βRu′(c2). If the utility function is of the usual CRRA form

then consumption as a function of the return R on deposits satisfies

c1 =y + π

R

1 + (βR)1α

R

c2 = c1 (βR)1

α

Note that the coefficient of relative risk aversion is now α instead of γ. This change in notation

is made to maintain consistency with Christiano and Ikeda (2012).

7.5 Equilibrium conditions and calibration

The equilibrium of the model is characterised by profit maximisation by mutual funds and

bankers, and utility maximisation by the households. We repeat them here for completeness.

c2 = c1 (βR)1

α Euler equation for consumption

c2 = rk(N + B) [1− µG(ω̄)] Aggregate resource constraint in period 2

c+B = y Aggregate resource constraint in period 1

1−F (ω̄)1−Γ(ω̄)

=rk

R[1−F (ω̄)−µω̄F ′(ω̄)]

1− rk

R[Γ(ω̄)−µG(ω̄)]

First order condition for optimal contract

N+BN

= 1

1− rk

R(Γ(ω̄)−µG(ω̄))

Mutual fund zero profit condition

The five equations simultaneously determine the five endogenous variables

c1, c2, R, ω̄, B

It is not possible to proceed from here analytically so we calibrate the parameters of the model

and present numerical results. The calibration is taken from Christiano and Ikeda (2012).

σ = 0.37 α = 1 y = 3.11 rk = 1.04 N = 1 β = 0.97

19

7.6 Numerical results

If µ = 0 then the mutual fund can recover all available funds if the banker declares bank-

ruptcy and state verification is not costly. This is the natural benchmark of the economy and

represents the first best solution. We therefore solve the model first with µ = 0 and then

increase its value to progressively introduce greater frictions in financial markets. The impact

on c1, c2, u(c1) + βu(c2) and R are shown below.

0 0.1 0.2 0.3 0.4 0.51.95

2

2.05

2.1

c1

µ0 0.1 0.2 0.3 0.4 0.5

2.1

2.12

2.14

2.16

2.18

c2

µ

0 0.1 0.2 0.3 0.4 0.51.435

1.44

1.445

1.45

1.455

1.46Welfare

µ0 0.1 0.2 0.3 0.4 0.5

1.02

1.04

1.06

1.08

1.1

1.12

1.14R

µ

The financial friction leads contrary to the previous model to an increase in the deposit rate

R, which provides too much incentive for the household to save so c1 falls. The extra saving

translates into higher consumption c2 in the second period. The deposit rate in the economy

with financial imperfection is too high, which contrasts with the deposit rate being too low

in the adverse selection equilibrium. The intertemporal optimisation decision is still though

distorted and welfare is decreasing in µ. The key to understanding this is that introducing

costly state verification as a market imperfection increases the probability that a banker will

declare bankruptcy in equilibrium. In other words, the cut-off probability ω̄ at which the

banker declares bankruptcy increases. Intuitively this makes sense as the contract between

the mutual fund and the banker becomes less favourable to the banker as the mutual fund has

20

to cover the costs of state verification.

0 0.1 0.2 0.3 0.4 0.50.5

0.52

0.54

0.56

0.58

0.6ω

µ0 0.1 0.2 0.3 0.4 0.5

1.02

1.04

1.06

1.08

1.1

1.12

1.14B

µ

0 0.1 0.2 0.3 0.4 0.52.02

2.04

2.06

2.08

2.1

2.12

2.14L

µ0 0.1 0.2 0.3 0.4 0.5

-0.02

0

0.02

0.04

0.06

0.08R-rk

µ

The increase in the probability of bankruptcy gives an additional incentive for the banker

to take on loans and increase leverage. If the investment becomes more risky then the banker

prefers to have more external finance than internal finance. Increasing leverage means that

the scale of production is higher under costly state verification, which implies that bankers

need more loans in period 1 and so the deposit rate must increase. To verify this intuition,

we can “freeze” the probability of bankruptcy at the value it takes when there is no market

imperfection and then introduce the market imperfection. Technically, this means replacing

the first order condition for the optimal contract with ω̄ = 0.5081. The behaviour of the

21

economy is then:

0 0.1 0.2 0.3 0.4 0.52.086

2.087

2.088

2.089

2.09

2.091

2.092

c1

µ0 0.1 0.2 0.3 0.4 0.5

2.08

2.085

2.09

2.095

2.1

2.105

c2

µ

0 0.1 0.2 0.3 0.4 0.51.448

1.45

1.452

1.454

1.456

1.458

1.46Welfare

µ0 0.1 0.2 0.3 0.4 0.5

1.02

1.025

1.03

1.035

1.04

1.045R

µ

Compared to the earlier result, we see now that the financial imperfection increases con-

sumption in the first period and decreases it in the second period. This is in accordance with

the first pass intuition that states the financial imperfection makes it harder to transfer re-

sources from period to period, so households prefer to consume in period 1 and do not save

for period 2. Notice that the deposit rate now falls as state verification becomes more costly.

0 0.1 0.2 0.3 0.4 0.5-0.5

0

0.5

1

1.5

2ω

µ0 0.1 0.2 0.3 0.4 0.5

1.018

1.02

1.022

1.024

1.026B

µ

0 0.1 0.2 0.3 0.4 0.5

2.019

2.02

2.021

2.022

2.023

2.024L

µ0 0.1 0.2 0.3 0.4 0.5

-15

-10

-5

0

5x 10

-3 R-rk

µ

22

This second figure confirms the earlier explanation. With ω̄ fixed, borrowing into period 2

falls and the spread between the deposit rate and return to capital opens up again, although

this time it is negative. Comparing these two figures with those where ω̄ was not fixed, we

see that the change in the probability of bankruptcy causes leverage to increase so much that

the equilibrium effects of costly state verification are reversed. This shows that leverage is

a strong and important propagation channel in the model. Whether it is strong enough to

dominate for all calibrations of the model remains an open question. In any case an increasing

deposit rate is problematic for social welfare since it implies the opening of a wedge R − rk

between interest rates. It should come as no great surprise that welfare losses accompany such

a wedge.

8 Conclusion

The four models in this lecture note stress different financial imperfections but have a common

cause in the lack of net worth of bankers. Christiano and Ikeda (2012) discuss the efficacy of

different policy options in each model. As a general rule, policies which improve the net worth

of bankers are usually beneficial. It is generally a good thing that bankers have sufficient

“skin in the game”. Another set of models argues that financial frictions are not real as in this

lecture, but instead are about liquidity and liquidity shortages. These models are discussed

in the next lecture.

23