Low Interest Rate Policy and Financial
Stability
David Andolfatto
Aleksander Berentsen
Fernando Martin
The views expressed here are our own and should not be attributed to the Federal Reserve
Bank of St. Louis or the Federal Reserve System.
Motivation
Does low-interest-rate monetary policy (LIMP) promote …nancial stability
or instability? There are opposing views.
One view is that LIMP induces “reach-for-yield” into higher risk assets.
Promotes instability, esp. if …nanced with short-term debt (Stein, 2013).
Another view is that LIMP encourages banks to hold more cash reserves.
Promotes stability, since banks more likely to become narrow banks
(Cochrane, 2014).
What we do
The question we address requires a model of money and banking.
Money is modeled as an exchange medium (Lagos-Wright, 2005).
Dynamic general equilibrium framework (easy to make DSGE).
Steady-state analysis so R policy equivalent to
policy (Fisher e¤ect).
Bank is modeled as risk-sharing arrangement (Diamond-Dybvig, 1983).
Demandable debt is a solution to private information problem, but opens
the door to bank-runs (multiple equilibria).
Model
Discrete time, in…nite horizon, subjective discount factor :
Each period divided into 3 subperiods: morning, afternoon, evening.
Two groups of agents: investors and workers.
Investors face i.i.d. shock that determines type within period.
impatient/patient want consumption in afternoon/evening.
denotes fraction of impatient, shock realized in afternoon.
Model
Investors produce output in morning (consumer and capital goods) and
consume in afternoon/evening (depending on type).
Capital goods k yield afternoon output k or evening output Rk (assume
0
< 1 < R):
Workers want to consume in morning, able to work in afternoon.
Afternoon output can be costlessly stored to the evening.
Since < 1; afternoon output supplied more e¢ ciently by workers relative
to scrapping capital produced by investors.
Model
Investor and worker preferences, resp.
E0
1
X
t=0
t
h
i
y0;t + u(c1;t) + (1
E0
1
X
t=0
h
t c
0;t
y1;t
)u(c2;t)
i
Resource constraints (assuming no scrapping),
c0;t + kt = y0;t
c1;t
y1;t
(1
)c2;t
Rkt
E¢ cient allocation
Investors produce consumer and capital goods in morning.
Consumer goods used to reward workers in the morning.
Capital goods used to …nance investor consumption in the evening.
Workers produce output in the afternoon to service impatient investors.
E¢ ciency implies risk-sharing among investors (workers are risk-neutral).
Steps in our analysis
Step 1: Competitive monetary eqm with ex post securities market.
Investors accumulate money and capital in the morning, then trade
money for capital in the afternoon after realizing their types.
Markets are incomplete along two dimensions.
No ex ante insurance market & ex post securities market closed with
probability
(market freeze). Industry standard is
= 1 (illiquid
capital).
Liquidating capital when sec. mkt. closed means scrapping (…resale).
Steps in our analysis
Step 2: Construct a Diamond-Dybvig bank assuming no bank runs.
Demonstrate two dimensions along which welfare improved.
(i) improved risk-sharing and (ii) bypass sec. mkt. (avoid …resales).
Demonstrate that (i) above entails fractional reserve banking.
Bank funds assets (cash + capital) with liabilities redeemable on demand
for cash.
Optimal contract anticipates that only fraction
but fractional reserves implies bank run eqm.
will withdraw early,
Steps in our analysis
Step 3: Construct Diamond-Dybvig bank assuming runs occur w.p. :
We think of societies as potentially di¤ering in
(social fragility).
Bank o¤ers investors deposit contracts that maximizes ex ante welfare for
a given parameter con…guration ( ; ; ):
We identify regions in the parameter space where narrow banking is preferred.
We investigate how these regions depend on interest (in‡ation) rate policy.
Steps in our analysis
Step 4: In an extension, we plan to specify more general “belief functions.”
i.e., let
be a function of economic variables.
Jeremy Stein speci…cation
tween money and capital.
John Cochrane speci…cation
ratio.
is decreasing function of yield di¤erential be-
is decreasing fuction of currency-deposit
Workers
Workers in our model behave the same in all versions of the model.
Workers have linear preferences, their decisions essentially pin down eqm
pricing relationships.
a denote price of output in morning and afternoon, resp.
Let pm
;
p
t
t
The following no-arbitrage-condition must hold: 1=pat = =pm
t+1 ; or
m
m
n
(pat=pm
t ) = pt+1 =pt = = ( = ) = R
Rn is the nominal interest rate and Rn = = is the Fisher equation.
Monetary equilibrium with securities market
New money injected as lump-sum transfers to workers each morning.
Let pkt denote price of capital in security market (if it is open) and let
pkt =pat:
Let (~
c1; c~2) denote allocation when the securities market is open (^
c1; c^2)
when closed.
Let kI denote sales of capital by impatient depositor, let kP denote purchases of capital by patient depositor (assuming market is open).
Monetary equilibrium with securities market
Investor maximizes chooses consumption and x; k; kI ; kP to maximize
x
k + (1
) [ u(~
c1) + (1
)u(~
c2)] + [ u (^
c1) + (1
subject to
( = )x + kI +
k
( = )x
kI
k
kI
0
kP + R(k + kP )
( = )x
c1
kP
0
c2
)u(^
c2)]
Monetary equilibrium with securities market
Lemma: In a stationary monetary equilibrium,
R for any
2 [0; 1).
Proposition: A stationary monetary equilibrium implements the …rst best allocation if and only if = 0, = and u0(c1)c1 = u0(c2)c2.
Result above shows that monetary equilibrium implements the …rst best
allocation in only a razor-edge case.
Sec. mkt. always open, Friedman rule, and (e.g.) log preferences.
In general, welfare gains will be possible under FRB system.
Fractional reserve bank
Bank o¤ers depositors a security that overcomes absence of liquid secondary market for capital.
This new security–a demand deposit liability–grants depositors the option
to convert deposits for cash; permits depositors to avoid …resale events
that in‡icted investors in monetary economy.
But even if secondary market is always open (
improve welfare by o¤ering superior risk-sharing.
= 0) a bank can still
Banks help complete the missing ex ante insurance market.
Fractional reserve bank assuming no bank run
Banks o¤er depositors a contract that maximizes
max f x
k + u(c1) + (1
( = )x
Rk + [( = ) x
)u(c2)g
c1
c1]
(1
c2
c1
)c2
0
Proposition: In a FRB equilibrium absent bank runs, the Friedman rule =
implements the …rst-best allocation and the currency-to-deposit ratio
x=(x + k) increases/decreases with in‡ation as u00(c)c=u0(c) 7 1:
Financial fragility
Well-known that demandable debt structures can be “fragile” (su¤er from
multiplicity of equilibria).
Unique implementation possible with su¢ ciently rich contract space.
We assume that banks cannot credibly commit to suspend withdrawals
when cash reserves are exhausted.
Then a “bank run” equilibrium is possible, with bank run de…ned as:
patient investors misrepresenting themselves as impatient en masse.
Social fragility
We index societies by a parameter
Formally,
2 [0; 1]:
is probability of “sunspot” that triggers a bank-run.
We think of as indexing the propensity of a society to “act on rumors”
and coordinate on bad outcomes.
Low
societies are inherently stable. High
societies inherently unstable.
Distinction between inherent instability and manufactured instability.
Fractional reserve banking when bank runs are expected
Bank o¤ers depostors contract that maximizes
(
x k + (1
) [ u(c1) + (1
)u(c2)]
+ (1
) [ u(~
c1) + (1
)u(~
c2)] + [u( c^1) + (1
( = )x
Rk + [( = ) x
c1]
( = )x + kB + (k kB )
( = )x + kB + (k kB ) + (R
)kP
( = )x + k
( = )x + Rk
k kB
( = )x + kB + (k kB )
kP
c2
)u(^
c2)]
c1
(1
)c2
c~1
c~2
c^1
c^2
0
0
c1 0
)
Fractional reserve banking when bank runs are expected
Proposition: In a FRB system with 2 (0; 1], then banks are generally able
to meet all their obligations in the event of a run if the securities market
is open. Formally, c~1 = c1 and c~2 = c2.
When secondary market is available, orderly disposal of capital is possible,
so bank run does not necessarily result in …resales.
Proposition: A FRB system with 2 (0; 1] implements the …rst-best allocation if and only if = 0, = and u0(c1)c1 = u0(c2)c2.
Franchise value of bank W B ( ; ; ) is decreasing in ( ; ; ):
Fractional reserve banking when bank runs are expected
Bene…t of FRB is higher in high in‡ation (high interest rate) economy.
High
implies low real return on cash (FRB economizes on cash).
Bene…ts of fractional reserve banking are greatly diminished in socially
unstable societies (high economies).
Especially in economies with very illiquid capital markets (high
economies).
FRB may be dominated by a narrow banking structure in low ; high ;
economies.
Narrow bank
Narrow bank: all demandable debt is backed 100% with cash reserves.
Unlike FRB, narrow bank can meet short-term obligations in every state
of the world.
Consequently, runs are impossible (at cost of excess reserves, poorer
risk-sharing).
WN( )
max f x
k + u(c1) + (1
( = )x
Rk + [( = ) x
c1]
W N ( ) is independent of ;
c1
(1
)u(c2)g
)c2
and decreasing in in‡ation :
The choice of banking regime
Proposition. There exists ^( ; ) satisfying W B (^; ; ) = W N ( ) such
that W B (^; ; ) > W N ( ) for < ^ and W B (^; ; ) < W N ( ) for
> ^: Moreover, ^( ; ) is strictly increasing in :
Inherently fragile societies prefer more stable banking regimes.
Inherently stable societies prefer more unstable banking regimes.
Moreover, high in‡ation economies encourage unstable banking regimes.
Interests of investors not necessarily aligned with those of workers.
The belief function
In models with multiple equilibria, the “belief parameter” can be anything
and still be consistent with rational expectations.
Nothing in theory dictates
being purely exogenous.
We could in principle assume any belief function
= B (x); where x
includes model parameters and endogenous variables.
We have experimented with two belief functions, one which implies a
decreasing function of
(Stein) and one which implies a decreasing
function of (Cochrane).
When reach-for-yield promotes instability
Extensions
Model “liquidation costs” more realistically (OTC market structure, disruption in payment system,...).
Introduce a role for bank capital (equity), study Basel III regulations (capital adequacy ratio).
Consider multibank model with interbank market.
Multibank model across two countries, with and without a common currency.