Download Financial Crises and Systemic Bank Runs Roberto Robatto PRELIMINARY AND INCOMPLETE

Financial Crises and Systemic Bank Runs
in a Dynamic Model of Banking
Roberto Robatto∗
University of Chicago
PRELIMINARY AND INCOMPLETE
May 13, 2013
Abstract
I present a model of financial crises and systemic bank runs in an infinite-horizon, monetary model of banking, where deposits contracts are specified in nominal terms and there is
asymmetric information about idiosyncratic shocks that hit the balance sheets of banks. The
model has a multiplicity of equilibria: in the good equilibrium, all banks are always solvent
despite the shocks, and “money is a veil”; in the bad equilibrium, an endogenous deflationary
spiral depresses nominal prices and drives some banks to insolvency. The bad equilibrium
displays a flight to liquid assets, a drop in the velocity of money and a run on some banks,
similar to some crucial events of the 2008 US financial crisis and of the Great Depression.
Monetary injections may rule out the possibility of a crisis, but depending on the assumptions
about monetary policy and the type of interventions they might even worsen the outcome of
a bad equilibrium. Capital requirements can instead always rule out the bad equilibrium.
∗
Email: [email protected]. I am extremely greatful to Fernando Alvarez, Veronica Guerrieri, Robert Lucas
and Harald Uhlig for their suggestions and guidance, and seminar participants at EIEF, University of Chicago and
Macro-Financial Modeling Group Meeting for their comments. I am also greatful to the Becker-Friedman Institute
and the Macroeconomic Modeling and Systemic Risk Research Initiative for financial support.
1
1
Introduction
The failure of Lehman Brothers in September 2008 has been followed by a series of events that
many observers1 have defined as a “run” on the Repo market and on other institutions without
deposit insurance, the so-called “shadow banking system”. The run was a systemic event, in the
sense that it affected many financial institutions simultaneously. The Federal Reserve reacted
vigorously to this fact, implementing some “unconventional” monetary policy operations. The
monetary injections did not imply any increase in the price level, resulting instead in the drop of
the money multiplier and of the velocity of money. The Great Depression displayed a similar drop
of money velocity and of the money multiplier, but the lack of monetary injections by the Fed
resulted in a drop of the price level.
The literature about panic-based runs, based on the seminal work of Diamond and Dybvig (1983),
has often focused on real models with no role for fiat money2 . However, money injections by
central banks cannot be analyzed in such real models: the objective of this paper is to develop a
monetary models where systemic bank runs can take place and where the role of monetary policy
and money injections can be analyzed, consistently with the drop of the money multiplier and the
velocity of money. To replicate these monetary facts, I use an infinite-horizon formulation typical
of “macro” models. Moreover, I view this model as a first step in the direction of building “macro”
infinite-horizon models of bank runs that can be compared with the data, even though this model
is still too simple and I leave a more precise quantitative analysis to future research.
The key frictions in the model are represented by asymmetric information (households observe
with some delays the balance sheet of banks), a role for money (cash-in-advance constraint, nominal banking contracts) together with preference shocks about the timing of consumption, and a
sequential service constraint. In the even of an (idiosyncratic) shock that hit banks, the model has
two equilibria: a “good” equilibrium where prices and aggregate quantities are not affected by the
shock, and a “bad” equilibrium that displays runs on some banks, a drop of the velocity of money
and of the money multiplier, and a drop in nominal prices (deflationary spiral), consistent with
1
2
See e.g. Gorton and Metrick (2012).
There are some exception, such as the work of Allen, Carletti, and Gale (2012), that I discuss later.
2
some crucial events experienced by the US in the fourth quarter of 2008 and during the Great
Depression. Depending on the assumptions about the central bank and the type of intervention,
a monetary authority that injects money in the economy may rule out the crisis, but it might
surprisingly also worsen it. A “large enough” capital requirements for banks can instead always
rule out the bad equilibrium, and by a Modigliani-Miller argument this policy is costless in the
model that I consider.
The model combines the approach of Diamond and Dybvig (1983), where households face a liquidity risk, with a cash-in-advance constraint, in an infinite-horizon economy as in standard macromodels. All agents are ex-ante identical, but some of them turn out to have high marginal utility
of consumption (impatient) while some others have low marginal utility of consumption (patient).
Consumption must be financed with money because of the cash-in-advance constraint, but there is
an opportunity cost of holding money, represented by the return from holding a productive asset
(tree). Households deposit some of their wealth in banks, which in turn invest in trees and hold
enough money to satisfy the withdrawals by impatient depositors. By pooling the liquidity risk of
households, banks can provide a higher return to patient agents and allow to achieve the first-best
outcome. Banks are subject to idiosyncratic shocks; such shocks are observed with some delays
by households, so it is not possible to immediately distinguish which banks have been hit by a
bad shock. This is however not a problem, because banks have net worth that acts as a “buffer”
against bad shocks, therefore there exist an equilibrium in which all banks are always solvent.
Unfortunately, this is not the unique equilibrium in the model; though in general equilibrium
everything happens simultaneously, let me try to tell a story about the “bad” equilibrium.
In the bad equilibrium, households believe that all banks experience an additional drop in their net
worth, therefore the banks hit by the bad idiosyncratic shock are believed to be insolvent; however,
households cannot identify which are the insolvent banks, because of the asymmetric information.
As soon as more precise information about insolvent banks will be available, there will be a run on
such insolvent banks. At that point, agents that have high marginal utility of consumption might
be “last in line” during the run, therefore withdrawing no money and being unable to consume
(because of the cash-in-advance constraint). Anticipating this possibility, households are willing
3
to hold some precautionary amount of money, that is however unspent for agents that turn out to
be “patient”. If money supply is fixed, the unspent cash depresses nominal prices, including the
price of assets. Since banking contracts are expressed in nominal terms, the low nominal asset
price decreases the net worth of banks, confirming the original expectation of households. In such
bad equilibrium, some banks experience both a drop in the price of assets and a negative shock,
therefore they become insolvent. The insolvent banks would have otherwise been solvent if the
nominal price of assets had not decreased, because their net worth would have acted as a buffer
against the negative shock. Their net worth is instead not enough to act as a buffer against both
the negative shock and the drop in the price of assets.
Comparison with recent literature
The model combines some features that are already incorporated in some very recent literature,
but with a novel approach. Allen, Carletti, and Gale (2012) present a model in the tradition of
Diamond and Dybvig (1983), but with nominal banking contracts. They model a crisis as an
increase in the fraction of impatient households: the central bank can thus provide more money
and the price level goes up to clear the market, with no change in the velocity of money. This
model can be a description of banking crisis that are related to an increase of the price level, while
it fails to capture the drop of the money multiplier and of money velocity that we observed during
both the Great Depression and the 2008 US financial crisis.
Gertler and Kiyotaki (2012) present an infinite-horizon macro model with bank runs, but with
two crucial differences: they have real demand-deposit contracts (so they cannot capture any
fact related to the flight to liquidity and the drop in velocity) and an inefficient liquidation of
trees3 by banks in the event of a run (an assumption which is similar to Diamond and Dybvig
(1983)). In the event of a run, banks sell their trees on the market and households have to pay
a “management cost” to hold such trees; the management cost depresses the demand of trees by
households, resulting in a low equilibrium price and in the possibility of multiple equilibria. The
model of Martin et al. (2011) is similar in spirit to the one of Gertler and Kiyotaki (2012): they
3
In their model, they denote the productive assets as “capital”.
4
have a real model and there is heterogeneity in the ability of managing assets. A key difference
between my model and this literature is that I do not need to assume heterogeneity in the ability
of managing assets.
Structure of the paper
In Section 2, I present the basic features of the model. Next, in Section 3, I present the social
planner problem: I will later use the first-best outcome as a benchmark. In Section 4, I present
a model where there are no banks, and a friction prevents the decentralized economy to achieve
the first best. In the economy with banks of Section 5, there exist a good equilibrium where the
allocation is optimal, as shown in Section 7. However, for some parameter values, there exist also
a “bad” equilibrium where some banks become insolvent endogenously and are subject to runs: an
example of this equilibrium is provided in Section 8.
2
The model: basics
Time is discrete and each period is divided into two parts, a “day” and a “night”. I use the
notation “t, d” to denote a variable that is known during the day of period t, and the notation
“t, n” to denote a variable which is known during the night of time t. I will also refer to the
“morning” as the beginning of the day, denoted by “t, m”; no decisions will be taken during the
morning, but I introduce the morning to clarify the timing and to describe more easily the relevant
state variables for each player in the economy.
2.1
Preferences
The economy is populated by a unit mass of identical households indexed by h ∈ [0, 1]. Each
household consists of a continuum, of unit measure, of individual members, indexed by i ∈ [0, 1];
therefore, each individual in the economy is identified by h, i.
5
Consumption takes place at night. The utility that agent h, i derives from a stream of consumption
n
o∞
h,i
Ct,n
is given by:
t=1
∞
X
h,i
β t εh,i
t,n log Ct,n
t=1
where εh,i
t,n is a preference shock that is realized at the beginning of t, n. The preference shock can
take two values:
εh,i
t,n =



ε̄ (impatient) with probability κ
(1)


ε (patient) with probability 1 − κ
where:
0 < ε < 1 ≤ ε̄
The preference shock is iid over time and across members of the household, and I impose the
normalization:
E
εh,i
t,n
=1
(2)
Within each household, the law of large numbers applies, therefore a fraction κ of members is
impatient and a fraction 1 − κ is patient. The utility function of the household is given by:
∞
X
β
t
Z h
i
h,i
εh,i
log
C
t,n
t,n di
t=1
2.2
Assets in fixed supply
There are two assets in fixed supply in the economy:
• there is a supply M t of money, which matters because of a cash-in-advance constraint; unless
I specify otherwise, I assume that money supply is fixed, M t = M , for most of the paper;
• there is a supply ā of trees; each tree produces a constant dividend δ per period, which is the
only consumption good in the economy; fruits produced by trees are harvested and traded
at night.
6
Trees can only be traded during the day (this is an important assumption). I normalize the price
of money to one, and I denote Qt to be the price of a tree and Pt to be the price of consumption
good.
3
Social planner
Which allocation would be chosen by a social planner that can observe the realization of the
preference shock4 ? The solution to the social planner problem represents the first-best benchmark
that I will use for comparison in the rest of the paper.
The social planner puts a weight λht on the utility of household h, but he weights equally all the
members of an households. The problem is given by:
Z
λht
max
h,i
{Ct,n
}(h,i)∈[0,1]×[0,1]
h∈[0,1]
Z
h
i
h,i
εh,i
log
C
d (h, i)
t,n
t,n
i∈[0,1]
subject to:
Z
h∈[0,1]
Z
h,i
Ct,n
d (h, i) = δā
i∈[0,1]
where δā is the total quantity of fruits produced in the economy. The optimal allocation of
consumption of members i and j of household h satisfies:
h,i
Ct,n
h,j
Ct,n
=
εh,i
t,n
εh,j
t,n
h
h
or, denoting Ct,n
(ε̄) to be the consumption of impatient members of household h, and Ct,n
(ε) to
be the consumption of patient:
h
Ct,n
(ε)
= ε/ε̄.
h
Ct,n (ε̄)
4
The social planner choice is not subject to the cash-in-advance constraint, so money does not matter in this
Section.
7
4
Economy with no banks
In this Section, I analyze a decentralized economy where there are no banks.
4.1
Nominal return on trees
For future reference, it is useful to describe the nominal return on trees, which is given by:
1 + rt =
Qt+1 + δPt
Qt
To understand this expression, consider the following: with 1$, you can buy 1/Qt trees; each
tree produces δ fruits at night that can be sold at price Pt (thus generating proceeds
1
δPt )
Qt
and
each tree can be sold tomorrow at price Qt+1 ; therefore every dollar invested gives a gross return
Qt+1 +δPt
.
Qt
If both Pt and Qt are constant (which will be the case in this Section), I can drop the subscript t
and then there is no inflation:
1+r =
Q + δP
P
=1+δ
Q
Q
⇒
r=δ
P
Q
Let 1 + R be the real interest rate; if there is no inflation, the real interest rate is equal to the
nominal interest rate, R = r.
4.2
Household problem
The consumption choice of each individual h, i (that happens at night) is subject to a cash-inadvance constraint, and the allocation of money within the household can be made only during the
day. In what follows, the preference shock is partially unobservable. I assume that the preference
shock is observable by the members of the same household, but it is not observable by other
households or banks. The combination of:
• the cash-in-advance constraint for each h, i
8
• the external unobservability of the preference shock for each h, i
• the fact that money cannot be reallocated within the household at night
creates a crucial friction. Agents that turn out to be patient have a liquidity need (namely, they
need money), but no intra-household transfer of money is possible at night and no Arrow-Debreu
security based on the realization of the preference shock can be traded with members of other
households because of the external unobservability of the shock.
Here is the exact description of the timing for household h ∈ [0, 1]:
• household h starts t, m with wealth Aht,m composed by money and trees;
h
h
• household h can buy trees aht,d and money Mt,d
subject to the budget constraint Mt,d
+aht Qt ≤
Aht,m ;
h,i
• the cash is distributed to the members of the household: each member h, i gets Mt,d
units
R h,i
h
of money, subject to Mt,d di = Mt,d
;
• the night begins; the preference shock for each member of the household is realized;
• each household member h, i can buy consumption good subject to the cash-in-advance conh,i
h,i
straint Pt Ct,n
≤ Mt,d
• household members gather fruits and sell them on the market (they cannot consume their
own fruits)
• the wealth of t + 1, d is given by:
Aht+1,m
=
aht,d Qt+1
+
aht,d δPt
+
Z h,i
− Pt Cth,i di
Mt,d
where aht,d Qt+1 is the value of trees in t + 1, d; the second term aht,d δPt represents the proceeds
of selling fruits on the market at night; and the last term is unspent cash.
9
h
The household chooses the consumption level Ct,n
(ε̄) for impatients, and the consumption level
h
Ct,n
(ε) for patients. Since all the members of an household are identical during the day (they all
have the same probability of being patient or impatient at night), they get the same amount of
cash:
h,i
h
Mt,d
= Mt,d
for all i ∈ [0, 1]
Thus, the household problem is given by:
vt Aht,m =
max
h ,ah ,C h (ε̄),C h (ε),Ah
Mt,d
t,n
t+1
t,d t,n
h
h
κε̄ log Ct,n
(ε̄) + (1 − κ) ε log Ct,n
(ε) + βvt+1 Aht+1,m
subject to:
h
Mt,d
≥ 0 aht ≥ 0 (non negativity constraints)
h
Mt,d
+ aht,d Qt ≤ Aht (budget constraint)
h
h
Pt Ct,n
(ε̄) ≤ Mt,d
(cash in advance constraint for impatients)
h
h
(cash in advance constraint for patients)
Pt Ct,n
(ε) ≤ Mt,d
h
h
Aht+1,m = aht,d Qt+1 (1 + rt ) + (1 − κ) Mt,d
− Pt Ct,n
(ε) (law of motion of wealth)
(3)
Because of the law of large numbers within the household, a fraction κ of the members will be
h
h
impatient and will consume Ct,n
(ε̄), and a fraction 1 − κ will be patient and will consume Ct,n
(ε).
In the law of motion of wealth, equation (3), I have already imposed that the cash-in-advance
constraint is binding for impatient (otherwise the household would be better off by holding less
h
h
money and more trees): the last term of (3) is unspent cash, which is equal to Mt,d
− Pt Ct,n
(ε)
per impatient member, and there is a mass 1 − κ of such agents.
10
The solution to the household problem is given by5 :
h
Ct,n
h
Ct,n
Aht,m
(ε) =
[ε (1 + rt ) (1 − β)]
Pt
Aht,m ε̄κ (1 + rt ) (1 − β)
(ε̄) =
Pt
r+κ
and the optimal consumption ratio for household h is given by:
h
Ct,n
(ε)
=
h
Ct,n (ε̄)
ε
ε̄κ
rt +κ
The first best in this model can be achieved by a path of money supply that delivers zero nominal
interest rates, rt = 0 for all t. But if money supply is exogenously fixed at M t = M , the nominal
interest rate is not zero, as I show below. Therefore, the first best cannot be achieved, and there
is a distortion in the intra-household consumption allocation. This consideration motivates the
introduction of banks, that allows to achieve the first best even in an economy with fixed money
supply.
4.3
Large households, cash-in-advance constraint and infinite-horizon
economy: a comment
As an important remark, notice that, since rt ≥ 0:
κ
≤1
rt + κ
With a constant money supply, rt > 0 so impatient agents consume too much and impatient agents
consume too little in comparison to the first best. This implication is the opposite of traditional
banking model such as Diamond and Dybvig (1983), where impatient agents consume too little
5
I guess and verify that the value function is given by:
vt Aht =
1
log Aht + Ξt
1−β
where Ξt is a term that does not depend on the wealth Aht but it can be time-dependent.
11
and patient agents consume too much compared with the first-best.
The difference is related to two types of risk that I consider in the model: the consumption risk
and the liquidity risk.
• The consumption risk is the fact that an impatient individual has high marginal utility of
consumption at t, n. If individuals were “on their own” and were not part of a large household,
then impatient agents would have to rely on their savings to consume more today, reducing
their ability to responds to future consumption shocks. This type of risk would produce the
effects that we observe in Bewley-type economies (extra-savings to self-insure against the
shocks). But the large household is exactly insuring individuals against this type of risk, so
in equilibrium there are no savings for self-insurance reason.
• Liquidity risk: at night, the consumption decision of individuals is subject to a cash-inadvance constraint; the cash-in-advance constraint applies to each individual separately, but
the household allocates money to its members during the day, before the realization of the
preference shock. Therefore, patient agents have “too much” money at night, and impatient
agents have “too little” money at night (in comparison to their desired/unconstrained decisions). Since a dollar today is worth more than a dollar tomorrow (because of discounting)
and patient agents have a lot of cash at night that gives zero return if unspent, patient agents
consume more with respect to the first-best.
The liquidity risk would disappear if the household could allocate the money across its
members at night, so that impatient agents would get more money then patient agents.
Moreover, the infinite-horizon model naturally delivers the result that some money is unspent in
equilibrium in this economy, while it would have been harder to get this result in a 3-periods model.
Therefore the model is suited to to be used as a guideline to think about empirical facts about
money and monetary policy, such as the money injections by the FED or the Friedman-Schwartz
hypothesis about the Great Depression.
12
4.4
Aggregation
The following Proposition summarizes the equilibrium in the economy with no banks.
Proposition 4.1. The nominal prices of consumption goods and of trees in the economy with no
banks (superscript “N B”) are given by:
P NB =
M [(1 − β) (1 − κ) ε + βκ]
βāδ ε̄κ
QN B =
M [(1 − β) (1 − κ) ε + βκ]
ā (1 − β) ε̄κ
The real price of a tree is given by:
QN B
β
=δ
N
B
P
1−β
(4)
and the nominal interest rate is given by:
1+r =
1
β
(5)
Since the price level is constant, the real interest rate 1 + R is equal to the nominal interest rate:
1+R=
1
β
(6)
To understand these results, it is useful to take the limit as ε → 0, so the prices simplify to6 :
κM
δā
(7)
β κM
1−β a
(8)
P NB =
QN B =
When ε → 0, the price level is given by (7): in the aggregate, only a fraction κ of money is
spent, therefore the price level is given by the total amount of money spent κM , divided by total
consumption āδ. Equation (7) can be rearranged to look like the equation of exchange “money
6
Using assumption (2), the limit ε → 0 implies ε̄κ → 1.
13
times velocity equal prices times output” (where “output” is the total amount of fruits produced
in the economy, δā):
M κ = P N B (δā)
therefore κ is also the velocity of money.
The price of trees, equation (8), has a simple interpretation: a tree allows to gather fruits at night
and sell them on the market; the total proceeds from selling fruits on the market is κM , and one
tree generates revenue
κM
a
per period (the total proceeds divided by the “number” of trees in the
economy); the revenues from the tree are then discounted at rate β.
Taking the limit as κ → 1, the prices converge to the value that would arise in a frictionless
economy (with no preference shocks). Notice that the real price of a tree (4), the nominal interest
rate (5) and the real interest rate (6) are independent of ε and κ, therefore they are not affected
by the frictions.
5
Economy with banks
There is a unit mass of banks indexed by b ∈ [0, 1]. During the day, household h can deposits in all
h
and trees aht,d ). At night, each member
banks in the economy (in addition to holding money Mt,d
h, i is assigned by the head of the household to a particular bank7 b, therefore each member of
h,i
the household h, i has access to only one bank in the economy. Agent h, i can thus withdraw Wt,n
money from a particular bank b (withdrawals are governed by a sequential service constraint).
Therefore, individual h, i can buy consumption good subject to:
h,i
h,i
h
Pt Ct,n
≤ Mt,d
+ Wt,n
During the day, banks collect deposits and keep only a fraction of such deposits as money, investing
the difference in trees (this is a fractional banking system). In “normal times”, banks hold all the
7
I also impose the constraint that household h can assign at most one member h, i to a bank b. Under this
restriction, the agents assigned to a particular bank b are members of different households, and thus they do not
coordinate on the actions to be taken, in particular they do not coordinate in the event of run.
14
h
money during the day (so Mt,d
= 0 for all h); at night, agents withdraw the amount of money
that they need to finance their consumption expenditure, therefore all the cash is spent and the
equilibrium allocation in the economy with banks achieves the first best. On the contrary, if
h
> 0 and
households expect that there will be runs at night, then household h wants to hold Mt,d
distributes it equally to all members during the day: if individual h, i is impatient but “last in
h,i
line” during a run (Wt,n
= 0), she can nonetheless consume
h
Mt,d
Pt
> 0. However, patient agents
don’t value much consumption and they want to carry some money to t + 1, d if ε is sufficiently
small, thus some cash is unspent in the economy. Since some cash is unspent, nominal prices drop:
both Pt and Qt are smaller then in an economy with no runs. If the demand-deposit contract is
specified in nominal term, as I will assume, the real value of deposits of banks increases, driving
some bank to insolvency.
I consider the limit as ε goes to zero, therefore patient agents don’t want to consume: this simplifies
considerably the analysis and the exposition:
Assumption 5.1. The value of the preference shock for impatients satisfy:
ε → 0.
5.1
Shocks to trees
Households and banks that hold trees are subject to a shock (I will consider a one-time-unanticipated
b
h
shock) denoted by ξt,m
for banks and ξt,m
for households. When the shock hits the economy, I
assume that:
b
h
, ξt,m
∈ ξ, ξ¯
ξt,m
and, conditional on the shock hitting the economy:
b
P r ξt,m
= ξ¯ = p
b
P r ξt,m
=ξ =1−p
15
Z
b
ξt,m
db
Z
=0
h
ξt,m
dh = 0
(the last assumption is justified by a result in Al-Najjar (2004)). By a law of large number, p and
1 − p are also the fraction of banks that are hit by the bad shock and the good shock.
When the shocks hit the economy, households know that banks and other households have been
h
.
hit by the shock; but each household observes only its own shocks ξt,m
Next, I describe banks choices and behavior, and then I formally analyze the problem of households,
which is just an extension of Section 4.
5.2
Banks
Banks collect deposits from households and hold money and trees8 . At every point in time, a
snapshot of the balance sheet of banks looks as follow:
Assets
Liabilities
Value of Trees
Deposits
Money
Net worth
In the good equilibrium, banks have enough net worth to make sure that deposits are risk-free:
the shocks that hit trees are thus “absorbed” by net worth and the value of deposits is not affected.
5.2.1
Restrictions on the demand-deposit contract offered by banks
I impose two restrictions on the demand-deposit contract offered by banks:
1. the contract must be in nominal terms;
2. banks cannot distribute fruits to depositors (fruits must be sold on the market).
Both restrictions are motivated by the nature of demand-deposit contracts in the real world.
Demand-deposit contracts are usually expressed in nominal terms, and in practice banks don’t
8
Banks also gather the fruits and sell them on the market.
16
distribute perishable consumption good but they rather allow withdrawals of money that can be
used to buy consumption goods9 .
Consistently with the above restriction, I define deposits as follow:
Definition 5.2. A deposit at bank b is a claim to obtain one unit of money from bank b, or the
equivalent value in trees if the deposit is redeemed during the day.
Next, I describe banks choices and behavior, and then I formally analyze the problem of households.
Here I describe a model where the idiosyncratic shock can hit banks only once, and where banks
“die” exogenously and they cannot pay dividends or issue new equity10 . But the results can be
extended to the case in which banks can pay dividends and issue new equity, and where shocks
can hit the balance sheet of banks every period11 .
5.2.2
Banks: timing and notation
The timing of banks is as follow.
• The state variables for bank b at t, m are:
•
b
;
money: Mt−1,n
•
b
b
is the idiosyncratic shock);
(where ξt,m
trees: abt−1,d + ξt,m
•
b
deposits Dt−1,n
b
The net worth in the morning Nt,m
(“m” stands for “morning”) is given by:
b
b
b
b
Nt,m
= abt−1 + ξt,m
Qt + Mt−1,n
− Dt−1,n
9
The second restriction is also in line with the theoretical assumption which is usually imposed in economy
with a cash-in-advance constraint, where households cannot consume their own endowment but they must buy the
consumption good on the market. If I had not imposed such condition, banks could directly distribute fruits at
night and money would not be essential.
10
The fact that banks “die” is just a modeling trick to pay dividends to households.
11
When shocks can hit banks every period, I must allow for recapitalization to allow banks that are hit by bad
shocks to rebuild a buffer against future shocks. This actually allows me to talk about banks recapitalization or
the lack of thereof during a financial crisis, because of asymmetric information, and the role of the government
in equity infusion in banks, which is a relevant event of the 2008 crisis (this part of the paper is still a work in
progress).
17
• With probability 1 − λ, bank b survives and operate in t, d; with probability λ, bank b is
liquidated (the liquidation process is described in Section 5.2.3).
• Each surviving bank is splitted into
1
1−λ
> 1 banks, therefore the measure of banks is constant
at 1; the net worth of a bank b0 that is originated from splitting a bank b is given by:
0
b
Nt,d
=
b
Nt,m
1
1−λ
b
= Nt,m
(1 − λ)
I will also impose a restriction on λ such that, in the good equilibrium, the net worth of
banks is constant over time.
• During t, d, bank b ∈ [0, 1]:
•
b
takes as given net worth minus dividends Nt,d
;
•
promises to pay a return rtb on deposits that are not withdrawn at t, n;
•
b
b
issues deposits Dt,d
, and holds trees abt,d and money Mt,d
subject to the budget con-
straint:
b
b
b
abt,d Qt + Mt,d
≤ Dt,d
+ Nt,d
and to the non-negativity constraints:
b
abt,d ≥ 0 Mt,d
≥0
(9)
b
the objective of the bank is to maximize Et,d max 0, Nt+1,d
as described in Section
5.2.4.
b
• At t, n, bank b faces withdrawals Wt,n
subject to the feasibility constraint:
b
b
0 ≤ Wt,n
≤ Mt,d
b
If desired withdrawals by depositors of bank b exceeds Mt,d
, then I say that bank b is subject
to a run: in this case, withdrawals are governed by a sequential service constraint, and the
18
position of depositors in the line is random. If a bank is subject to a run, it will be liquidated
in t + 1, d as described in Section 5.2.3.
• After withdrawals have taken place, bank b sells fruits on the market getting proceeds abt,d δPt ;
the end-of-night stock of money of bank b is thus given by:
b
b
b
Mt,n
= Mt,d
− Wt,n
+ abt,d δPt
(10)
= (money at day) − (withdrawals) + (proceeds of selling fruits)
The end-of-night deposits of bank b are given by:
b
b
b
Dt,n
= Dt,d
− Wt,n
1 + rtb
= (deposits at day − withdrawals at night) (1 + return)
b
b
b
where the return Dt,d
− Wt,n
rt is paid by bank b by giving new claims to depositors.
• The relevant states for bank b in t + 1, d are given by:
b
b
b
Mt,n
, Dt,n
, abt,d + ξt+1,m
b
is the shock.
where ξt+1,m
I impose the restriction that all banks are initially identical.
Assumption 5.3. All banks are initially identical at t = 0:
0
b
b
M0,n
= M0,n
0
b
b
D0,n
= Dt,0
0
ab0,d = ab0,d
for all b, b0 ∈ [0, 1]
for all b, b0 ∈ [0, 1]
for all b, b0 ∈ [0, 1]
19
5.2.3
Liquidation and bankruptcy
If a bank is subject to a run at t, n or if the bank dies at t + 1, m, then the bank is liquidated in
t+1, m. During the bankruptcy process, the assets (money and trees) are distributed to depositors,
pro-rata.
More precisely, the liquidation works as follow:
b
b
Qt+1 ;
+ abt,d + ξt+1,m
• the value of assets for bank b at the beginning of t+1, m is given by Mt,n
b
• if the value of assets is ≥ Dt,n
, depositors get 100% of their value;
• if the value of assets is less than the value of deposits, then the bank is bankrupt and
depositors get a return r̂tb on their deposits12 , where r̂tb is defined by:
b
b
Dt,d
− Wt,n
b
b
1 + r̂tb = Mt,n
+ abt,d + ξt+1,m
Qt+1
The relevant case of bankruptcy is the situation in which r̂tb < 0: in this case, the return on
deposits is negative, while the return from holding one unit of money at night is zero. At t, n,
if depositors knows that r̂tb < 0 for bank b, then all depositors of bank b prefer to withdraw as
much as possible at night, causing a run on the bank and triggering the liquidation of bank b in
t + 1, m.13 .
12
I use r̂tb to denote the return paid at t + 1, m on deposits of banks under liquidation; such return is formally
measurable with respect to t + 1, d only, because it depends on Qt+1 ; however, I use the subscript t on the return
r̂tb for consistency with the notation of the promised
return on deposits rtb which is known at t, d.
13
b
In case the bank is insolvent Nt+1,m < 0 but 0 ≤ r̂tb < rtb , depositors have no incentive to run because
holding money gives just a zero return, while holding deposits give a positive return (even though smaller than
the promised return). I will show later that the case 0 ≤ r̂tb < rtb will not be an equilibrium outcome under the
assumption that the shock can take two values. However, to be precise about technology, I need to specify what
happens to insolvent banks that, if liquidated, would give a return 0 ≤ r̂tb < rtb . To simplify the analysis, I just
introduce the following law: if a bank is insolvent (negative net worth) it is not legally allowed to operate, and it is
liquidated, following the liquidation process described before. A bank b such that 0 ≤ r̂tb < rtb is insolvent, therefore
it should be liquidated according to this law. The law is however implemented by a government that has the same
information as households, therefore if households cannot identify “bad banks” then the government cannot identify
the bad banks either.
20
5.2.4
Optimal choice of banks
Before presenting the problem of banks, it is useful to state the following two assumptions. The
first one is a technological assumption, while the second one is an equilibrium selection rule. I
should have introduced the equilibrium selection rule after the complete description of the economy,
but it is helpful to state it here because it simplifies considerably the problem of banks.
Assumption 5.4. (Law large numbers) Each bank has infinitely many depositors; the law of large
numbers holds at the level of each bank: for bank b, a fraction κ of depositors is impatient (ε̄), and
a fraction 1 − κ of depositors is patient (ε).
Assumption 5.5. (Minimum-banks-usage) If households are indifferent between several choices
of deposits, I consider equilibria in which they always choose the smallest quantity of deposits that
maximize their utility (and invest the rest in trees directly).
Under Assumption 5.5, impatient agents always wants to withdraw all the deposits available to
them, in order to finance their consumption expenditure. Therefore, Assumptions 5.4 and 5.5 imply
b
that desired withdrawals from bank b are at least κDt,d
. In “normal times”, when only impatient
b
depositors withdraw, desired withdrawals at bank b will be exactly equal to κDt,d
(because only
impatient agents withdraw), while in the event of a run on bank b, when all agents would like to
b
withdraw their deposits, desired withdrawals from bank b are equal to Dt,d
.
b
Given Nt,d
, the problem of bank b is given by:
max
b ,M b
rtb ,abt ,Dt,d
t,d
b
Et,d max 0, Nt+1,m
subject to:
b
b
b
+ Nt,d
abt,d Qt + Mt,d
≤ Dt,d
(budget constraint)
b
b
b
b
b
b
− Wt,n
+ abt δPt − Dt,d
− Wt,n
1 + rtb
Nt+1,m
= abt,d + ξt+1,m
Qt+1 + Mt,d
|
{z
} |
{z
} |
{z
}
value of trees in t+1
money at the end of the night
(11)
deposits at the end of the night
and the constraint arising from competition among banks14 :
14
Households have to satisfy the constraint of choosing a function that maps each member h, i into a bank in b
21
Dtb



≥ 0
0
if rtb ≥ maxb0 ∈[0,1] rtb |b0 6= b


0
otherwise
(12)
Equation (11) is the law of motion of wealth of bank b. Equation (12) constrains the choices of
deposits and return: if bank b offer a return rtb which is below the return offered by other banks
rtD (b), it will not get any deposit.
The next proposition characterize the solution to the bank problem15 ..
Proposition 5.6. The optimal money holding during the day is given by:
b
b
Mt,d
= κDt,d
and:
0
• rtb = min maxb0 ∈[0,1] rtb |b0 6= b , rt
b
• if rtb = rt , any Dt,d
is a solution; if rtb < rt , Dtb = +∞
0
The relevant case (that happens in equilibrium) is maxb0 ∈[0,1] rtb |b0 6= b = rt . In this case, to
show that the optimal choice is given by rtb = rt :
b
b
b
b
b
Et,d Nt+1,m
= abt,d Qt+1 + Mt,d
− Wt,n
+ abt,d δPt − Dt,d
− Wt,n
1 + rtb
b
b
b
Using Mt,d
= κDt,d
and the fact that depositors are willing to withdraw at least κDt,d
(thus
such that, for all i, j ∈ [0, 1], the bank of h, i is different from the bank of h, j. Since there exists bijective functions
that maps the closed interval [0, 1] into the open interval (0, 1] as I describe later when I analyze the problem of
households, a bank that deviates from offering the equilibrium contract rtb = rt to offering a worse contract gets no
deposits. Therefore, despite the constraint that there cannot be more than 1 member of household h in island b at
night, banks act in a perfectly competitive setting.
15
A “good” bank (with positive net worth) will always choose to have enough money at night to satisfy withdrawals
from patient depositors, otherwise it will be liquidated in t + 1, m for sure. There is however a problem with a “bad”
bank that has negative net worth; in this case, the bank knows that, at night, households will find out that the
banks is insolvent and therefore there will be a run for sure. So during the day the bank is indifferent between any
choice, because the payoff is anyway zero. However, if I introduce a small (→ 0) probability of another shock that
makes the bank solvent again, then the bank will choose to have the right amount of money to satisfy withdrawals
from impatient because it would consider just the state of the world in which it survives.
22
b
b
b
Wt,n
= Mt,d
= κDt,d
):
Et,d
b
Nt+1,m
=
Qt abt,d
Qt+1
Qt
+
abt,d Qt
Pt
b
δ
− Dt,d
(1 − κ) 1 + rtb
Qt
or:
Et,d
b
Nt+1,m
=
Qt abt,d
Qt+1 + δPt
b
−Dt,d
(1 − κ) 1 + rtb
Qt
|
{z
}
=1+rt
b
b
b
b
b
:
= κDt,d
and using again Mt,d
− Mt,d
+ Dt,d
Using the budget constraint of the bank Qt abt,d = Nt,d
b
b
b
b
Et,d Nt+1,md
= Nt,d
+ Dt,d
(1 − κ) (1 + rt ) − Dt,d
(1 − κ) 1 + rtb
or:
b
b
b
Et,d Nt+1,m
= Nt,d
(1 + rt ) + Dt,d
(1 − κ) rt − rtb
Net worth in t + 1, m is decreasing in rtb , and banks are not willing to offer a return on deposits
which is above rt ; moreover, a return rtb < rt would generate positive profits for bank b if the bank
is able to collect deposits at this return: since there are no frictions in the banking sector, banks
must make zero profits and in equilibrium rtb = rt for all banks.
If initial net worth is positive, the (expected) net worth in t +1 will be bigger because of the return
on trees:
b
b
d
b
b
Nt,d
> 0 ⇒ Et,d Nt+1,m
= Nt,d
(1 + rt ) > Nt,d
r =r
t
t
I will impose a restriction on λ so that when a surviving bank is split into
1
1−λ
> 1 new banks the
net worth in t + 1, d remains constant.
If initial net worth is instead negative, the insolvency of bank b widens:
d
b
b
b
b
= Nt,d
(1 + rt ) < Nt,d
Nt,d
< 0 ⇒ Et,d Nt+1,m
r =r
t
t
b
That’s because the bank promises to repay 1+rt on every dollar not withdrawn, but if Nt,d
< 0 the
bank does not have enough trees to honor the promise. Therefore, the fruits sold on the market
23
are not enough to repay the return on all the deposits, widening the insolvency of the bank.
5.2.5
Fraction of depositors served during a run
b
of depositors is
If all depositors of bank b try to withdraw their funds at night, only a fraction ft,d
served (“f ” stands for “first in line”). The fraction of depositors served is given by:
b
ft,d
=
b
Mt,d
b
Dt,d
b
is also the probability of being “first in line”
From the point of view of a depositor of bank b, ft,d
in the event of a run.
5.3
5.3.1
Household problem
Information about banks
b
and the balance sheet of
When the idiosyncratic shocks hit the economy, households observe ξt.m
banks only at night t, n. Thus, at night, all banks that offer a return r̂tb < 0, if any, are subject to
a run, therefore they are liquidated in t + 1, d and deposits not withdrawn experience a loss (this
will be the case in the “bad” equilibrium).
Since the shock is unanticipated, households behave as if trees were riskless; and when the shocks
h
ξt,m
hit households, the effect of shocks on households is just a redistribution of wealth.
The assumption that deposits are claims that are redeemable on demand (together with the lack
of information during t, d about the shocks in t − 1, n) allows me to obtain a financial crisis. At
t, d it is impossible to immediately check if a bank is actually solvent or insolvent unless a large
enough group of depositors demand to redeem their claims and no new wealth is deposited at a
bank. The result would be totally different if all banks where required to give back to households
trees and money at the beginning of t, d. In this case, all insolvent banks would be spotted and
only “good” banks would be active in t, d with no effects on prices and no effects on banks that
are not hit by the shock.
24
Let RU Nt,d be the expectation about the fraction of banks that will be subject to a run at t, n:
RU Nt,d = Et,d fraction of banks subject to run, with r̂tb < 0
The variable RU Nt,d is crucial in the mechanism of the multiplicity of equilibria. When the shocks
hit the economy, if all households believe RU Nt,d = 0 then the economy is in the good equilibrium.
Otherwise, if:
Z
RU Nt,d =
where I (·) is the indicator function (thus
b
I ξt−1,n
= ξ db = 1 − p
R
b
I ξt−1,n
db is the fraction of banks hit by the bad
shock ξ) an additional equilibrium might exist, depending on parameters and initial conditions.
From the point of view of a single member of an household, the object RU Nt,d represents also the
probability that a bank will be liquidated and bankrupt in t + 1, d.
5.3.2
Household problem
I can now state the household problem in the economy with banks. Household h takes as given
the initial wealth Aht,m :
h
h
h
h
h
Aht,m = Mt−1,n
+ Dt−1,n
+ aht−1,d + ξt,m
Qt + qt Nt−1,d
+ πt,m Nt−1,d
(13)
h
h
where Mt−1,n
is the quantity of money at the end of t − 1, n; Dt−1,n
is the quantity of deposits
h
h
at the end of t − 1, n; aht−1,d + ξt,m
is the quantity of trees owned by the household; Nt−1,d
are
the claims on the proceeds of liquidation of banks and qt is the price of a claim; and πt,m are the
dividends paid to households from the liquidation of banks that exits, per unit of claim.
Let:
h
ωt,n
n
o
b(h,i)
h,i
=
εh,i
,
r̂
,
W̃
t,n
t,n
t,n
i∈[0,1]
be a realization of idiosyncratic shocks εh,i
t,n for the members of household h, a realization of returns
25
b(h,i)
r̂t,n
for each bank b of member h, i, and a limit on withdrawal:
h,i
W̃t,n
=



0
if the bank of h, i is subject to a run, and h, i is ”last in line”


h
Dt,d
otherwise
During the day, the household faces a portfolio choice of splitting the wealth into money, deposits,
trees and claims on dividends. At night, the household choose, for each memeber i ∈ [0, 1], a value
h,i
h,i
h
h
for withdrawals Wt,n
ωt,n
and a value of consumption Ct,n
ωt,n
contingent on the realization of
h
the the state ωt,n
:
vt Aht,m =


max
h ,D h ,ah ,N h
Mt,d
t,d t,d
t,d
|
{z
Eωt,n
h
}
day
Z
max
h,i
h,i
h
h
{Wt,n
(ωt,n
),Ct,n
(ωt,n
)}i∈[0,1] ,Aht+1,m
|
h,i
h
εh,i
ωt,n
di + βvt+1 Aht+1,m
t,n log Ct

}
{z
night
subject to:
h
Mt,d
≥ 0 (non-negativity constraint on money)
h
h
h
Mt,d
+ Dt,d
+ aht,d Qt + qt Nt,d
≤ Aht,m
h,i
h,i
h
Wt,n
ωt,n
≤ W̃t,n
h,i
h,i
h
h
h
Pt Ct,n
ωt,n
≤ Mt,d
+ Wt,n
ωt,n
(budget constraint)
(limit on withdrawals)
for all i ∈ [0, 1]
(cash in advance constraint)
h
because the realization
for all i ∈ [0, 1]. The wealth in t + 1, given by Aht+1,m , is not indexed by ωt,n
b(h,i)
of εh,i
t,n and of the return r̂t,n
follows a law of large number within the households. The law of
motion of wealth (which is just the same as equation (13), one period ahead):
h
h
h
h
h
+ πt+1,m Nt,d
+ aht−1,d + ξt+1,m
Qt+1 + qt+1 Nt,d
Aht+1,m = Mt,n
+ Dt,n
(law of motion of wealth)
where:
h
Mt,n
=
aht,d δ
h
Dt,n
=
Pt +
Z h
h
Mt,d
h
Dt,d
−
+
Z h,i
Wt,n
h,i
Wt,n
h
ωt,n
26
h
ωt,n
−
Pt Cth,i


h
ωt,n
i
b(h,i)
di 1 + rt
di
di
(14)
(15)
b(h,i)
and rt
is the actual return paid by the bank serving member h, i of the household. The money
h
at the end of the night Mt,n
are given by the sum of the proceeds of selling fruits aht,d δ Pt , plus
R h,i h h,i
h
h
di. The value of deposits at the end of the
ωt,n
+
Wt,n ωt,n − Pt Ct,n
unspent cash Mt,d
h
is equal to deposits not withdrawn, plus the return.
night Dt,n
During the day, household h also chooses how to allocate its members across the continuum of
banks. In the relevant case in which all banks offer the return rtb = rt , households are indifferent
about the allocation of their members on their islands, so I impose that member h, i is matched
with bank b = i. In the out-of-equilibrium case in which bank b deviates from the equilibrium
choice rtb = rt and offers rtb < rt (say, without loss of generality, that b = 0 so rt0 < rt ) the
household can allocate its members as follow:
island (i) =




1/2




for i = 0
1/(n + 1)






i
for i = 1/n , n = 2, 3, ...
for i ∈ S
where:
1 1
S := [0, 1] \ 0, , , ...
2 3
This function is bijective and satisfy the restriction that household h can assign at most 1 member
to bank b, and no depositor is assigned to bank b = 0. This allocation of members, in the outof-equilibrium case in which bank b = 0 deviates, supports perfect competition in the banking
sector.
5.3.3
Households choices
Let me impose the condition that all banks promise to pay a return on deposit rtb = rt . Moreover,
I restrict the analysis to the relevant case in which all insolvent banks are identical, thus:
r̂tb = r̂t
for all b ∈ [0, 1]
27
b
ft,d
= ft,d
for all b ∈ [0, 1]
At night, a member h, i of household h will face the following:
1. with probability 1 − RU Nt,d , the bank of h, i will pay the promised return rt ; therefore,
h
and if agent h, i is impatient
depositor h, i can withdraw any amount of money Wth,i ≤ Dt,d
h
because of Assumption 5.5;
(εh,i = ε̄) she will withdraw all the available deposits Dt,d
2. with probability RU Nt,d , the bank of h, i will be bankrupt in t + 1, m (paying a return r̂t ,
and restricting the analysis to the case r̂t < 0), therefore the optimal choice is to run and
h,i
try to withdraw Wth,i = Dt,d
no matter whether h, i is patient or impatient:
(a) with probability ft,d , individual h, i will be “first in line” so she will be able to withh
draw any amount of money Wth,i ≤ Dt,d
; if individual h, i is impatient she will buy
h
h
consumption for an amount Dt,d
+ Mt,d
, otherwise she will carry the money to t + 1, d;
(b) with probability 1 − ft,d , individual h, i will be “last in line”, so she will not be able to
h
Mt,d
.
Pt
withdraw, Wth,i = 0; in this case, agent h, i can consume at most
Since a law of large number holds at the level of each household, a fraction 1 − RU Nt,d of members
will face solvent banks, a fraction RU Nt,d ft,d will be first in a run and a fraction RU Nt,d (1 − ft,d )
will be last in line in a run. Thus, the problem of households becomes:
(
vt Aht,m =
max
h , ah , D h , C h,RU N , Ah
Mt,d
t,n
t+1
t,d
t,d
(1 − RU Nt,d ) (κε̄) log
+ RU Nt,d ft,d (κε̄) log
h
h
Mt,d
+ Dt,d
Pt
!
h
h
Mt,d
+ Dt,d
Pt
+
+
+ RU Nt,d (1 − ft,d ) (κε̄) log
28
!
h
Mt,d
Pt
!
)
+ βvt+1 Aht+1,m
h
subject to aht,d ≥ 0, Mt,d
≥ 0 and:
h
h
h
Aht+1,m = aht,d + ξt+1,m
Qt+1 (1 + rt ) + qt+1 Nt,d
+ πt+1,d Nt,d
+
h
h
+ (1 − RU Nt,d ) Dt,d
(1 − κ) (1 + rt ) + (1 − κ) Mt,d
+
h
h
+RU Nt,d ft (1 − κ) Mt,d
+ Dt,d
+
h
h
(1 + r̂t ) + (1 − κ) Mt,d
+RU Nt,d (1 − ft ) Dt,d
where I have implicitly assumed that, in each household, a fraction ft of members are first in line
in the event of a run, and a fraction 1 − ft is last in line during a run.
To solve the problem, I guess that the value function is given by:
vt Aht,m =
1
log Aht,m + Ξt
1−β
where Ξt is a time-varying terms that does not depend on Aht,m . The problem can be solved in
closed form, and the solution is given by:
h
Dt,h
= ηt Aht,m
h
Mt,d
= µt Aht,m
h
Qaht,d + qt Nt,d
= Aht,m (1 − ηt − µt )
where the terms ηt and µt are defined by:
ηt = (β − 1) (1 + rt )
µt =
(1 − ft ) RU Nt,d
+
(1 − RU Nt,d ) [(ft,d − 1) (κ + r̂t ) − κrt + rt ] + (1 − ft ) (κ + r̂t )
(1 − RU Nt,d )
(16)
+
(−RU Nt,d ) [(ft,d − 1) r̂t,d + rt,d ] − κ (1 − RU Nt,d ) (1 + rt )
(1 − β) (1 − ft ) RU Nt,d (1 + rt )
(1 − RU Nt,d ) [(ft − 1) (κ + r̂t ) + rt (1 − κ)] + (1 − ft ) (κ + r̂t )
(17)
and under the assumption that the following arbitrage condition holds (households must be indif-
29
ferent between holding trees or claims on proceeds from liquidated banks):
qt+1 + Et,d (πt+1,m )
Qt+1 + δPt
=
qt
Qt
The choices of the households are proportional to wealth, therefore they can be easily aggregated
R
and there exists a representative household. Let At,m = Aht,m dh be the aggregate initial wealth
of households; then the economy-wide choices of deposits, money, consumption for “first in line”,
and trees holding by households are given by (the superscript “H” stands for “households”):
H
Dt,d
H
M t,d
Z
≡
Z
≡
h
dh = ηt At,m
Dt,d
h
Mt−1,n
dh = µt At,m
H
QāH
t + qt N t,d = At,m (1 − ηt − µt )
I can also define the end-of-period holdings of money and deposits for household h, using (14) and
(15):
h,RU N
h
h
h
− κPt Ct,n
Mt,n
= aht,d δ Pt +Mt,d
[(1 − RU Nt,d ) (1 − κ) + RU Nt,d (1 − ft,d ) (1 − κ)]+RU Nt,d ft Dt,d
h
h
h
Dt,n
= (1 − RU Nt,d ) (1 − κ) Dt,d
(1 + rt ) + RU Nt,d (1 − ft ) Dt,d
(1 + r̂t )
and integrating by h:
H
M t,n
Z
h
Mt,n
dh
1 − ηt − µt
= At,m
δPt + mt ((1 − RU Nt,d ) (1 − κ) + RU Nt,d (1 − ft,d ) (1 − κ)) + RU Nt,d ft,d (ηt − κct )
Qt
≡
H
Dt,n
Z
=
h
Dt,n
dh = ηt At,m [(1 − RU Nt,d ) (1 − κ) (1 + rt ) + RU Nt,d (1 − ft ) (1 + r̂t )]
h
If the economy is in a stationary equilibrium, the value of a claim Nt,d
is equal the present
30
discounted stream of the proceeds from the liquidation of banks:
Z
∞
1 X τ
b
qt =
β λ Nt+τ,m
db
N τ =1
where N is the total supply of claims.
5.4
Market clearing conditions
The following market clearing conditions must hold in equilibrium:
• goods market clearing:
Z Z
h,i
Ct,i
d (h, i)
āδ =
and multiplying both sides by Pt and rearranging:
Z
āδPt = κ
h
h
h
h
h
(1 − RU Nt,d ) Mt,d
+ Dt,n
+ RU Nt,d ft,d Mt,d
+ Dt,n
+ RU Nt,d (1 − ft,d ) Mt,d
dh
or:
āδPt = κAt,m [(µt + ηt ) (1 − RU Nt,d + RU Nt,d ft,d ) + RU Nt,d (1 − ft,d ) µt ]
(18)
• I consider a symmetric allocation where banks get the same amount of deposits, Dt,d ≡
0
b
b
Dt,d
= Dt,d
for all b, b0 ∈ [0, 1]; the market clearing condition for deposits at t, d is given by:
Dt,d ≡
H
Dt,d
Dt,n ≡
H
Dt,n
Z
=
b
Dt,d
db
and similarly for the night:
Z
=
b
Dt,n
db
• money market clearing:
H
M t,d
Z
+
31
b
Mt,d
db = M
b
b
and using the optimality condition for banks Mt,d
= κDt,d
:
H
M t,d + κDt,d = M
H
and the definitions M t,d = µt At,m and Dt,d = ηt At,m :
At,m (µt + κηt ) = M
(19)
• market clearing in the claims on proceeds from liquidated banks (let N is the total supply
of claims):
Z
h
Nt,d
dh = N
• dividends paid to households:
Z
πt,m N = λ
b
max 0, Nt,m
db
(recall that πt,m is the amount of dividends per unit of claim).
6
6.1
Equilibrium
Aggregate state
The state of the economy is given by the initial states of banks and by a sunspot:
ψt =
n
abt−1,d
+
b
ξt−1,n
,
b
Mt−1,n
,
b
Dt−1,n
b∈[0,1]
, sunspott
o
The quantities that determine the initial wealth of households can be computed as follow:
aH
t−1,d
Z
=a−
b
abt−1,d + ξt−1,n
db
32
H
M t−1,n
Z
=M−
H
Dt−1,n
and
R
Z
=
b
Mt−1,n
db
b
Dt−1,n
db
h
dh = N from the relative market clearing condition.
Nt−1,d
Definition 6.1. Given the initial state of the economy ψ1 that satisfies Assumption 5.3 and an
b h exogenous process for the shock ξt,m
and
ξt,m b∈[0,1] that satisfies the requirement of Section
b∈[0,1]
5.1, an equilibrium is a collection of:
h
by households (day)
• money, deposit holding, trees holding and holding of claims Nt,d
• withdrawals and consumption by each household member (night)
• money, deposits, trees holding and promised return on deposits by banks (day)
• liquidation returns r̂tb for all b ∈ [0, 1]
such that:
• household maximizes utility and have rational expectations about RU Nt,d
• banks maximize their objective function
• markets clear
I impose a restriction on λ, so that in good equilibrium the net worth of households is constant.
Assumption 6.2. The parameter λ satisfies:
λ=1−β
Moreover, I impose that the initial net worth of banks is “large enough” so that, in the good
b
equilibrium, all banks are always solvent, even if they are hit by the bad shock ξt,n
=ξ
33
Assumption 6.3. The initial conditions of banks and the parameters of the model satisfy:
ab0,d
6.2
+ξ
β M
1 − β ā
b
b
+ M0,n
− D0,n
≥ 0.
(20)
Understanding the multiplicity of equilibria
In the following Sections, I will provide examples of the multiplicity of equilibria. Before doing
that, I want to highlight what is the source of such multiplicity, even though the model is too
complicate to show formally that there exist a “bad” equilibrium under certain parameters.
In this Section, I perform a partial equilibrium exercise about the money market. The market
clearing condition in the money market is given by:
Z
h
Mt,d
dh
Z
+
b
Mt,d
db = M
or, as I have shown previously:
At,m (µt + κηt ) = M
Because of log utility, household members consume a constant fraction of their wealth, and the total
wealth of households in the economy is given by At,m . Therefore, money demand is proportional
to At,m . Moreover, because the wealth of each households depends on trees aht−1,d , then At,m is
increasing in the price of trees Qt and therefore money demand is increasing in Qt as well, as
shown by the black upward sloping straight line in the figure below (money supply is constant at
M ). Setting r̂t < 0 (I take the limit as r̂t ↑ 0), I can show that16 :
∂ (money demand)t,d >0
∂RU Nt,d
RU Nt,d =0, Qt =Q∗
When there is a marginal change in RU Nt,d from 0 to some value “close to 0” (but positive), money
demand evaluated at Qt = Q∗ moves up, as emphasized in the picture below. If some banks are
16
The result is shown in the Appendix, under the assumptions (βκ (2 + β) − 1) > 0 and ā (1 − β) + βκξ > 0,
which are satisfied for a large subset of the parameter space.
34
insolvent in the economy, there are two effects:
h
• households want to hold some money Mt,d
> 0 because some of the members of the household
will end up last in line during a run, with no possibility to withdraw;
• households want to hold less deposits at banks (because some banks are insolvent and therefore the realized return will be negative in some banks), so banks hold less money.
At Qt = Q∗ , the household still wants its members to consume a constant fraction of wealth,
therefore money demand must go up overall. I can also show that, at Qt = 0, money demand
moves down when lt,d marginally increases from zero to a positive value17 . Therefore, by continuity,
money demand must cross money supply at some point where Qt < Q∗ . Even though this is a
partial equilibrium analysis, the idea in general equilibrium is the following: because of the drop
in Qt , the wealth of households goes down, so households want to consume less and the money
market clear.
money
demand, RU Nt,d = 0
M
supply
demand, RU Nt,d > 0
Qt
Qt
0
17
Q
∗
However, I do not have a clear intuition of why this is the case, since Qt = 0 is not an equilibrium.
35
7
Good equilibrium
A “good equilibrium” where banks are always solvent and the economy reaches the first-best always
exist.
Proposition 7.1. There exist an equilibrium such that:
b
≥ 0 for all b and for all t, and households have rational expecta• all banks are solvent, Nt,m
tions about that:
RU Nt,d = 0 for all t, d
• consumption is equalized across the members of the same type of household h, for all h and
for all t:
h,i
h,j
Ct,n
= Ct,n
h,j
⇐⇒ εh,i
t,n = εt,n
• the aggregate prices are constant, Pt = P ∗ , Qt = Q∗ and qt = q ∗ , where:
M
P =
āδ
∗
β M
Q =
1 − β ā
∗
R
∗
q =β
b
N1,m
db
N
b
of banks is given by:
where the initial value of net worth N1,m
b
b
b
N1,m
= ab0 Q∗ + M0,n
− D0,n
• the nominal interest rate r∗ is given by:
r∗ =
1
−1
β
The proof is provided in the Appendix. In the good equilibrium, households are indifferent among
h
any Dt,d
≥ Aht 1−β
, where the right-hand side is the minimum amount which is required to finance
κ
h
the consumption of impatient, but I impose the solution Dt,d
= Aht,m 1−β
based on Assumption 5.5.
κ
36
For a more general economy with ε ≥ 0, the following result holds: [PROPOSITION TO BE
CHECKED]
Proposition 7.2. The good equilibrium of the economy with banks reaches the first-best outcome
that would be chosen by a social planner that put Pareto weights λht =
Ah
t,d
At,d
on the utility of
household h, for all 0 ≤ ε < 1
8
Financial crisis and runs: an example
In this Section, I provide a numerical example of an equilibrium that displays a financial crisis. I
choose the following numerical values for the parameters:
and using the restriction
R
Parameter
Value
Parameter
Value
β
0.985
M
1
κ
0.5
a
1
δ
0.34
N
1
ξ
0.2abt−1
ξ¯
0.022abt−1
b
db = 0, the probability p of a good shock is given by 0.9; therefore
ξt,m
10% of banks are hit by the bad shock, and 90% of the banks are hit by the good shock. Moreover,
I set initial conditions so that the restriction in Assumption 6.3 is satisfied with equality.
The shock hits the economy in t, m; to construct the bad equilibrium in t, I conjecture that:
RU Nt,d = 1 − p
and the equilibrium is given by:
• net worth of good banks = 0.01; net worth of bad banks = −0.2
• return r̂tb on insolvent banks = −0.14
• price of trees Qt = 65 < Q∗ = 80
37
• price level Pt = 2.2 < P ∗ = 2.9
• money holding of households, as a fraction of wealth: µt = 0.6% > 0
• return on trees = 0.22 > r∗ = 0.0125
Since the return on insolvent banks is r̂tb = −0.14 < 0, then the conjecture is correct and this is
indeed an equilibrium.
b
Nt,d
Pt
0.2
2.9
b
= ξ¯
ξt,n
0.01
2.2
0
1
2
-0.2
0
1
2
3
b
ξt,n
time
3
=ξ
time
2
M1
Qt
1.6
80
deposits
1.2
65
M
1
0
1
2
3
money held by
time
0.4
households
0
8.1
1
2
3
time
Fraction of banks hit by the bad shock and depth of a crisis
How is the depth of the crisis related to the parameters of the model? In this Section, I relate the
depth of the crisis to the idiosyncratic shock. The larger is the fraction of banks hit by the bad
shock ξ (which is the same as the probability 1 − p), the worse is the crisis (in terms of drop in
prices and distortions in the allocation of consumption):
38
Fraction of
banks hit by
Q1
R
h
Mt,d
dh
Deposits
M1
r̂
b
Nt,m
good
ξ
banks
Good
80
0
2
2
n/a
0.22
1.4%
74
0.14
1.7
1.85
0
0.12
10%
65
0.4
1.2
1.6
-0.14
0.01
equilibrium
>10%
no equilibrium
If less then 1.4% of banks are hit by the bad shock, the probability of a run is very small from the
point of view of the households; therefore households don’t have much incentive to hold money for
precautionary reasons, and prices are not affected much, so the return r̂t is actually positive (and
there is no bad equilibrium). If more then 10% of banks are hit by the bad shock, the drop in Qt
is so large that all banks become insolvent, including banks hit by the good shock. In this case,
either all banks are liquidated and new banks enter the market so there is no crisis, or if e.g. it
takes one period to set up new banks, the outcome during the crisis is similar to the outcome in
the economy with no banks.
I have also analyzed the role of the value of ξ on the crisis. It turns out that the value of ξ does
not matter much, because the higher is ξ, the higher is net worth to counteract the shock (by
Assumption 6.3), therefore the equilibrium values of prices and quantities are not affected much
by the value of ξ.
9
Monetary policy
In this Section, I consider the effects of a monetary authority that injects money in the economy.
When the shock hit the economy at time t, the central bank increases money supply from M to
39
M (1 + ρt ). The monetary injection can be used to buy trees directly, or the monetary authority
can provide liquidity to banks. In t + 1 the crisis is over, and the money supply goes back to the
pre-crisis level M . The central bank gets a return from the monetary expansion, getting either a
direct return on trees or charging an interest to banks for the provision of liquidity; such returns
are rebated to households with lump-sum transfers.
9.1
Liquidity facility: discount window lending
First, I consider the case in which the central bank set up a “liquidity facility”, and only banks
have access to it. Banks can borrow money from the central bank during the day, and they will
have to pay back 1 + rt dollars for every dollar withdrawn.
In case some banks are insolvent in the economy, I have to take a stand on the ability of the
central bank to recover loans to failed banks. I consider two extreme cases: in the first scenario,
the central bank has the same ability as the private sector to recover its loans, therefore it faces a
return r̂t on loans to insolvent banks; in the second scenario, the central bank is able to recover
the full value of loans (and of the returns), and depositors split the value of assets after the central
bank has been repaid.
9.1.1
Central bank with the same ability as the private sector to recover loans
In this case, the monetary authority faces losses on loans to banks that turns out to be insolvent:
b
Dt,d
−
b
Wt,n
+ (loans from CB)t,d
b
b
1 + r̂tb = Mt,n
+ abt,d + ξt+1,m
Qt+1
(where “CB” stands for central bank). Figure 1 plot the price level Pt , the price of trees Qt and the
return on insolvent banks r̂t as a function of the monetary policy intervention ρt . The outcome is
monotone in ρt : the higher is the monetary intervention, the higher are the prices and the return.
In the example, if ρt > 0.33, the solution plotted is actually not an equilibrium, because the return
r̂t is positive and therefore there will be no runs. Thus, under the assumption that the central
40
bank has the same ability as the private sector to recover loans to banks, the central bank can rule
out the bad equilibrium if it responds aggressively (with a “large enough” monetary injection) to
the crisis.
Moreover, notice that in the economy without intervention the real price of a tree goes up:
Qt
Q∗
= 29.3 > ∗ = 27.2
Pt
P
which is a different way to see that if the monetary authority promises to react to the crisis by
increasing money supply and committing to take losses on insolvent banks, the financial crisis is
just an out-of-equilibrium outcome. However, the model can be extended in a way that could
produce a drop in the real price of trees. Notice that banks holding of trees drop:
abt,d of good banks = 0.009
abt,d of bad bank = 0.006
while, in the periods before the shocks hit the economy, trees holding of banks are 0.0125 per bank.
That’s because households hold money directly, therefore they reduce their holding of deposits; and
with less deposits, banks buy less trees. If I extend the model and I add some costs of holding trees
for households above a certain threshold (e.g. a management cost such as Gertler and Kiyotaki
(2012) or a lower ability to deal with productive assets such as in Brunnermeier and Sannikov
(2012)), then the real price of trees should drop in the bad equilibrium because households are the
marginal agents that price such assets. With this formulation, banks would be able to get a higher
return from holding trees than households, but households would still have incentives to hold trees
directly because they are a riskless investment, while holding trees through banks is risky because
some of the banks are insolvent. With such a drop in the real price of trees, monetary policy can
still have some positive effects if the central bank reacts to the crisis by injecting money in the
economy; but if many banks have been hit by the bad shock ξ (so the crisis without intervention
is very severe) monetary policy alone might not be able to avoid the multiplicity of equilibria.
41
Figure 1: Central bank lending to banks (CB faces losses on insolvent banks)
Price Level
P_t
2.9
2.8
2.7
2.6
2.5
2.4
2.3
0.2
0.4
0.6
0.8
0.6
0.8
r
Price of Trees
Q_t
80
78
76
74
72
70
68
0.2
0.4
r
Return on insolvent banks
rhat
0.2
0.4
0.6
- 0.05
- 0.10
The green straight lines represents the value of P ∗ and Q∗ .
42
0.8
r
The next two sections describes other type of monetary intervention that, even in the event of an
increase of the real price of trees, might not be able to rule out a crisis.
9.1.2
Central bank is able to recover the full value of loans
Under this assumptions, the results are surprisingly different. When the central bank can recover
the full value of loans, the return r̂t that depositors face on insolvent banks is given by:
b
b
− Wt,n
Dt,d
b
b
Qt+1 − (loans from CB)t,d (1 + rt )
+ abt,d + ξt+1,m
1 + r̂tb = Mt,n
Figure 2 plots the result in this case18 . Looking at the blue line, you can see that the effect of
the monetary injection increases monotonically in the interval 0 ≤ ρt ≤ 0.84. However, for values
ρt ≥ 0.5, there exists an additional “bad” equilibrium (the red line in Figure 2), which is worse than
the other one in the sense that it displays a larger drop in prices for a given monetary intervention.
Moreover, in this additional equilibrium, the return on insolvent banks is always r̂t < 0, therefore
the central bank might not be able to rule out the crisis if it is able to recover the full value of
loans.
What’s the idea behind the “very bad” equilibrium that arises with this monetary intervention?
During the day, households reduce their deposits and hold some money directly; households know
that they will face some losses on some deposits because some bank is insolvent. Such losses
depends on the degree of insolvency of “bad” banks, and on how many deposits are withdrawn
during the day (in an insolvent bank, losses are shared across deposits that are not withdrawn).
If few deposits are withdrawn during the day, the losses of an insolvent banks are spread across
“many” deposits, therefore the loss per dollar of deposit in t + 1 is small. If many deposits are
withdrawn during the day, the losses of an insolvent bank are spread across “few” deposits, therefore
the loss per deposit in t + 1 is large. The uniqueness or multiplicity of the bad equilibrium is then
18
The outcome is qualitatively similar if the CB can recover only the value of loans but it cannot recover the
return, so r̂t is defined by:
b
b
b
b
Dt,d
− Wt,n
1 + r̂tb = Mt,n
+ abt,d + ξt+1,m
Qt+1 − (loans from CB)t,d
43
a numerical issue: the equilibrium has to satisfy several non-negativity constraints (on money
holding, trees holding, etc), and under the policy assumptions of this Section there are values of
ρt such that the constraints are satisfied.
9.1.3
Central bank buys trees directly
If the central bank buys trees directly on the market and sell fruits at night, the outcome is
similar to the case when the monetary authority has the ability to recover the full value of loans.
Increasing money supply counteract the effects of the crisis on the nominal variables, and a “large
enough” ρt might rule out the crisis. However, for some value of ρt , there exists an additional
“bad” equilibrium where the return r̂t < 0 and the crisis cannot be avoided, see Figure 3.
10
Capital requirements19
In this Section, I analyze the role of net worth in ruling out the bad equilibrium. In the example
without any monetary intervention, a capital requirement of 16.7% of the value of assets is required
to offset the bad shock ξ in the good equilibrium. A slightly larger capital cushion can also rule
out the bad equilibrium, because a larger net worth is a buffer against both the bad shock ξ and
the drop in Qt :
Capital requirements
Fraction of
to rule out
banks hit by
bad equilibrium
ξ
16.8% of value of
1.4%
trees
10%
19
23% of value of trees
In this section I use the term “capital” to denote net worth.
44
Figure 2: Central bank lending to banks (CB does not face losses on insolvent banks)
Price Level
P_t
2.9
2.8
2.7
2.6
2.5
2.4
2.3
0.2
0.4
0.6
0.8
0.6
0.8
r
Price of Trees
Q_t
80
78
76
74
72
70
68
66
0.2
0.4
r
Return on insolvent banks
rhat
0.2
0.4
0.6
- 0.2
- 0.4
- 0.6
- 0.8
- 1.0
The green straight lines represents the value of P ∗ and Q∗ .
45
0.8
r
Figure 3: Central bank buys trees directly
Price Level
P_t
3.0
2.9
2.8
2.7
2.6
2.5
2.4
2.3
0.2
0.4
0.6
0.8
0.6
0.8
r
Price of Trees
Q_t
80
78
76
74
72
70
68
0.2
0.4
r
Return on insolvent banks
rhat
0.2
0.4
0.6
- 0.2
- 0.4
- 0.6
- 0.8
The green straight lines represents the value of P ∗ and Q∗ .
46
0.8
r
In this model, a policy of forcing banks to hold more capital is actually costless, because of a
Modigliani-Miller argument.
11
Future work
I plan to extend the analysis as follow. On the theoretical side, I want to analyze the model
with banks for the case ε > 0, without taking the limit as such value converges to zero. I plan
also to include some simple modeling features that are able to produce a drop in the real price of
trees. If, with these small modifications, the real price of a tree drops in the bad equilibrium, then
monetary policy might not be enough to prevent and avoid a crisis, even in the scenario where it
faces losses on insolvent banks. A natural question is thus to allow for recapitalization of banks,
and to ask whether or not households are willing to inject new equity into banks. I conjecture
that the answer to this question is “no”, because of asymmetric information: households would
be willing to inject equity only in good banks, because any dollar invested in a bad bank is a
transfer to depositors. But asymmetric information prevents households from distinguishing good
and bad banks. If this is the case, then I could analyze the role of the government in injecting
equity into banks, therefore the model could be used to study the recapitalization of banks that
the US government undertook in 2008-09. I could then compare the reaction of asset prices in the
model with those described by Veronesi and Zingales (2010) after the government intervention in
the financial sector. I conjecture that the price of a claim on banks net worth and the price of a
claim on a more senior debt (such as bonds) would react similarly20 : if this were the case, then
asymmetric information would result to be a crucial frictions that is consistent with many relevant
facts of the recent US financial crisis.
Finally, the reaction of banks to 19th century banking panics could be analyzed in this framework
20
In the model without intervention, the price qt of claims on dividends of banks drop in the event of a crisis;
moreover, if I allow banks to issue bonds (bonds would be claims with the same seniority as deposits, they would
be issued in t, d and pay a fixed amount in t + 1, d but they would not allow any withdrawal at night) I conjecture
that all such bonds would trade at a discount in the event of a crisis, because some banks are insolvent, therefore
the bonds issued by such banks would be worth less then 100% of their face value; because of the asymmetric
information, even the bonds of “good” banks would be traded at a discount. Therefore, a recapitalization of the
whole financial sector would push both qt and the price of bonds up.
47
as well. Banks formed “coalitions” (clearinghouses) that were insuring the deposits of all the
banks; moreover, clearinghouses were dealing with information about the balance sheet of banks
in a different way during panics and in “normal times”, adding another piece of evidence to the
importance of such friction. In the model, a large enough coalition of banks could create an
insurance against the idiosyncratic shock, therefore ruling out the bad equilibrium.
References
Al-Najjar, Nabil I. 2004. “Aggregation and the law of large numbers in large economies.” Games
and Economic Behavior 47 (1):1–35.
Allen, F., E. Carletti, and D. Gale. 2012. “Money, Financial Stability and Efficiency.” .
Brunnermeier, Markus and Yuliy Sannikov. 2012. “A macroeconomic model with a financial sector.”
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48
Appendix
A
Money demand, partial equilibrium analysis
I impose the following Assumption:
Assumption A.1. The parameters satisfy:
βκ (2 + β) − 1 > 0
and:
ā (1 − β) + βκξ > 0
I can now state and prove the result.
Proposition A.2. Fix
• Qt+1 = Q∗
• Pt = P ∗
• r̂t = 0
b
= Q∗ ξ for all b ∈ [0, 1]
• Nt,m
Then money demand:
DEM AN D
M t,d
= At,m (κηt + mt )
has the properties:
DEM AN D ∂M t,d
∂RU Nt,d
>0
RU Nt,d =0,Qt
=Q∗
RU Nt,d =0,Qt
=Q∗
DEM AN D ∂M t,d
∂RU Nt,d
<0
49
Proof. Using the expression for µt and mt :
DEM AN D ∂M t,d
∂RU Nt,d
=
RU Nt,d =0,Qt =Q∗
M (1 − κ) (βκ (2 + β) − 1)
(1 + β) κ
which is positive under Assumption A.1. Also:
DEM AN D ∂M t,d
∂RU Nt,d
RU Nt,d =0,Qt =0
M (1 − κ) ā (1 − β) + ξβκ
=−
āκ2
which is negative under Assumption A.1.
B
Good equilibrium
Proof of Proposition 7.1.
Proof. Using the equilibrium prices and:
1 + r∗ =
Q∗ + δP ∗
Q∗
you get:
r∗ =
1
−1
β
so you can verify that, under Assumption 6.2:
r∗
λ=
1 + r∗
As long as the economy is not hit by the shocks, the net worth of surviving banks is constant:
0
b
b
Nt,d
= Nt,m
(1 − λ) =
b
b
Nt−1,d
(1 + r∗ )
Nt,m
b
=
= Nt−1,d
1 + r∗
1 + r∗
50
Therefore, as long as the shock has not hit the economy:
b
b
Nt,m
= N1,m
>0
for all banks; the inequality follows from Assumption 6.3 because:
b
b
b
N1,m
= ab0,d Q∗ + M0,n
− D0,n
β M
β M
b
b
b
b
b
b
≥0
− D0,n
= a0,d
+ M0,n − D0,n > a0,d + ξ
+ M0,n
1 − β ā
1 − β ā
ξ̄
b
¯ then Nt,m
= ξ,
> 0. For banks such
When the shock hits the economy, for banks such that ξt−1,n
ξ
b
that ξt−1,n
= ξ, the net worth is non-negative Nt,m ≥ 0 because of Assumption 6.3:
b
b
b
Nt,m
= abt−1,d + ξ Q∗ + Mt−1,n
− Dt−1,n
b
b
= abt−1,d Q∗ + Mt−1,n
− Dt−1,n
+ξQ∗
{z
}
|
b
=N1,m
as shown few lines before
b
:
and using the expression for Q∗ and for N1,m
b
Nt,m
=
ab0
+ξ
β M
1 − β ā
b
b
+ M0,n
− D0,n
≥0
where the last inequality follows from Assumption 6.3. Because the shock is idiosyncratic and
no banks is insolvent, the overall amount of net worth of banks in the economy is unchanged,
therefore:
Z
b
db
Nt,m
Z
=
b
N1,m
db
and qt is constant. Using stationarity, the price of a claim is given by:
R b
R b
db
N1,m db
β λ N1,m
qt =
=β
1−β
N
N
where the last equality follows from the Assumption λ = 1 − β.
51
Finally, from the problem of households:
µt = 0
and only impatient agents withdraw at night. Therefore all cash in the economy is spent and the
price P ∗ is given from the market clearing condition in the fruits market:
āδP ∗ = M .
To show the value of Q∗ , note that the overall value of wealth in the economy is:
At,m = Qt ā + M
and a fraction β of wealth is saved in trees:
βAt,m = āQt
Combining the two equations, you get that the price of trees is constant at Qt = Q∗ and:
Q∗ =
β M
.
1 − β ā
52