GCOE Discussion Paper Series
Global COE Program

Human Behavior and Socioeconomic Dynamics
Discussion Paper No.67

Zero Nominal Interest Rates, Unemployment,
Excess Reserves and Deflation in a Liquidity Trap

Ryu-ichiro Murota and Yoshiyasu Ono

July 2009GCOE Secretariat
Graduate School of Economics

OSAKA UNIVERSITY

1-7 Machikaneyama, Toyonaka, Osaka, 560-0043, Japan

Zero Nominal Interest Rates, Unemployment,
Excess Reserves and Deflation in a Liquidity
Trap
Ryu-ichiro Murota∗ and Yoshiyasu Ono†

Abstract: We present a dynamic and monetary model that consistently
explains such various phenomena as unemployment, deﬂation, zero nominalinterest rates and excess reserves held by commercial banks. These phenomena are commonly observed during the Great Depression in the UnitedStates, the recent long-run stagnation in Japan, and the worldwide ﬁnancial
crisis triggered by the US subprime loan problem of 2008. We show that an
excessive liquidity preference leads to a liquidity trap and thereby generates
the phenomena.

JEL Classification Codes: E12; E31; E41; E51
Keywords: Liquidity Trap; Zero Nominal Interest Rate; Persistent Unemployment; Excess Reserve; Deﬂation∗

Address: Faculty of Economics, Kinki University, 3-4-1 Kowakae, Higashi-Osaka, Osaka 577-8502, Japan. E-mail: murota@eco.kindai.ac.jp.
†
Address: Institute of Social and Economic Research, Osaka University, 6-1 Mihogaoka,
Ibaraki, Osaka 567-0047, Japan. E-mail: ono@iser.osaka-u.ac.jp. Ono’s research is ﬁnancially supported by the Grants-in-Aid for Scientiﬁc Research, JSPS.

1

1

Introduction

There are lots of common phenomena during the Great Depression in the
United States and the recent long-run stagnation in Japan, both of which
Krugman (1998) mentions as a liquidity trap. During the Great Depression,for example, the US unemployment rate rose from 3.2 percent in 1929 to apeak of 25.2 percent in 1933 and stayed higher than 10 percent for several
years. Deﬂation occurred, nominal interest rates declined to nearly zeroand excess reserves held by commercial banks expanded.1 Under the recent
stagnation in Japan the unemployment rate rose from 2.1 percent in 1990
to a peak of 5.4 percent in 2002 and is still around 4.0 percent (see ﬁgure1). Deﬂation has continued (ﬁgure 2), and nominal interest rates have beennearly zero (ﬁgure 3).2 And commercial banks held extensive excess reserves,which increased up to about six times as much as required by the Bank ofJapan (ﬁgure 4).
These phenomena have again been observed in the United States since2008. The seasonally adjusted unemployment rate has sharply increased from
4.9 percent in January 2008 to 7.2 percent in December 2008, and a monthly
change in the consumer price index for all urban consumers has been negative
or zero from August to December in 2008 (see the Bureau of Labor Statisticsof the US Department of Labor). The federal funds rate has been decreased
to nearly zero and excess reserves have massively expanded from 1.991 billion
dollars in August 2008 to 767.412 billion dollars in December 2008 (see the1See, e.g., Mankiw (2002, chapter 11, table 11-2) and Orphanides (2004, ﬁgures 1 and 2)
for the transitions of price levels, unemployment rates and nominal interest rates. For thefall in nominal interest rates, see also Temin (1976, chapter 4, ﬁgures 4 and 5). Moreover,see Orphanides (2004, ﬁgure 3 (b)) and Wilcox (1984, table 1) for the increase in excess
reserves.
2
See, e.g., Fujiki, Okina and Shiratsuka (2001, ﬁgure 1) and Orphanides (2004, ﬁgure 4)
for the movements of the policy interest rate and nominal rates of interest on government
bonds.

2

Board of Governors of the Federal Reserve System).3
Under such serious stagnation people tend to hold more cash and deposits
than required for transactions –i.e., people have various motives of holding
cash or deposits besides the transaction motive. Otani and Suzuki (2008) ﬁnd
that the volume of banknotes in circulation held by Japanese households for
non-transaction motives increased from around 1 or 5 trillion yen in 1995
to around 30 trillion yen in 2007, that 70 percent of the total increase inbanknotes in circulation from 1995 to 2007 was due to the increase in nontransaction demand, and that people held 120 trillion yen out of 310 trillion
yen in demand deposits for such motives in 2007.
We analyze the phenomena that occurred during the stagnation periodsin the US and Japan, using a model where both cash and interest-bearing
deposits yield utility of liquidity à la Romer (1985) and Jones et al. (2004).4
We ﬁnd that there are a certain value of the marginal utility of cash anda certain amount of full-employment output that divide the case of persistent unemployment from that of full employment. If the marginal utility of
cash is less than the critical value, or equivalently if full-employment output
is smaller than the critical amount, full employment obtains and nominalinterest rates are positive in a steady state. If otherwise, nominal interest
rates are zero and the marginal beneﬁt of cash exceeds that of consumption,
which makes people consume less than required to reach full employment andcauses persistent unemployment and deﬂation to occur. Since it is indiﬀerent
3

The Federal Open Market Committee decided to control the federal funds rate in arange from zero to 0.25 percent in December 16, 2008.
4
There are other approaches that consider liquidity of both cash and deposits. For
example, Walsh (1984) extends a cash-in-advance model to a model where individuals
must have cash and deposits in advance of buying goods. Simonsen and Cysne (2001)
present a model in which both cash and deposits reduce transaction costs. However,these models are not suitable for treating money demand motivated by factors other thantransactions.

3

for commercial banks to hold reserves and interest-bearing assets under thezero nominal interest rates, they hold excess reserves, which lowers monetary
aggregates and the money multiplier.
In the literature there are two well-known hypotheses about the cause of
the Great Depression: the spending hypothesis (viz. the IS eﬀect) and themoney hypothesis (viz. the LM eﬀect).5 Our paper emphasizes the monetaryside of an economy and shows that excessive demand for cash yields persistentstagnation, as mentioned by the money hypothesis. However, the stagnation
occurs – not because excessive demand for cash reduces monetary aggregates,
raises interest rates and decreases investment – but because it absorbs purchasing power that would otherwise be directed toward consumption. Thus,
our paper is more in conformity with the spending hypothesis than with themoney hypothesis.
The stagnation mechanism of our model also diﬀers from that of theliquidity trap model of Krugman (1998). By assuming rigid prices in theﬁrst period and perfect price adjustment that realizes full employment in
the second period, he insists that deﬂation and zero nominal interest rates
raise real interest rates and therefore reduce current consumption as a resultof people’s intertemporal decision making.6 Thus, stagnation is a short-run
phenomenon in his setting. In our paper, in contrast, excessive demand for
cash yields a shortage of consumption as a result of people’s intratemporal
preference between consumption and liquidity of cash and stagnation is a
steady-state phenomenon.
The remainder of our paper is organized as follows. Sections 2 and 3
5

See, e.g., Mankiw (2002, chapter 11) for a brief description on the two hypotheses.
It seems that this mechanism is not consistent with Japan’s recent slump. In fact,
Baba et al. (2005, ﬁgure 4) show that Japan’s real interest rates did not remarkably risefor the years 1991-2003 and slowly declined for the mid 1990s-2003.
64

describe the structure and the dynamics of the model respectively. Whereas
section 4 discusses the case where full employment is reached in a steadystate, section 5 analyzes the case of persistent unemployment. In this casedeﬂation occurs, nominal interest rates decline to zero, commercial banks
hold excess reserves and the money multiplier approaches unity. Section 6
concludes. Stability analyses are set out in appendices.

2

The Model

The private sector consists of a representative ﬁrm, a representative commercial bank and a representative household. There are two assets, money
and government bonds. Money is held by the household and the commercialbank whereas all government bonds are held by the commercial bank.7

2.1

The Representative Firm and the Government

We assume that the representative ﬁrm uses only labor to produce a commodity and that the labor productivity is constant, which we call y. Given
real wage wt the ﬁrm chooses labor demand ltd as follows:


Wt
d
,
lt = 0 if y < wt ≡Pt
0 < ltd < ∞ if y = wt ,(1)ltd = ∞ if y > wt ,where Wt and Pt are the nominal wage and the nominal commodity pricerespectively.
A monetary authority keeps nominal monetary base Mt constant:
Mt = M.
7

In Japan most of government bonds are indeed held by Japanese ﬁnancial institutions.

5A ﬁscal authority ﬁnances payments of interest on government bonds, Rt Bt ,where Bt and Rt denote nominal government bonds and the nominal rate ofinterest on government bonds respectively, by issuing new government bonds
Ḃt and collecting lump-sum tax τt :Ḃt + Pt τt = Rt Bt ,where the initial stock of nominal government bonds, B0 , is historically given.
Let πt (≡ Ṗt /Pt ), bt (≡ Bt /Pt ) and rt (≡ Rt − πt ) denote the inﬂation rate,
real government bonds and the real rate of interest on government bonds
respectively. Then the government’s real budget equation isḃt + τt = rt bt .
The ﬁscal authority follows a Ricardian rule:τt = rt bt + θ(bt − b), θ > 0, b > 0,
where b is a target level of real government bonds.8 This rule yields thefollowing simple dynamic equation of bt :ḃt = −θ(bt − b),(2)

implying that bt converges to b.

2.2

The Representative Commercial Bank

The representative commercial bank collects nominal deposits Dt and holds
government bonds Bt so as to gain interest revenues Rt Bt . As assumed in
Walsh (1984) and Romer (1985), the monetary authority requires the bank
8

This rule is imposed basically for the validity of the no-Ponzi game condition of thegovernment. See, e.g., Woodford (2001) and Schmitt-Grohé and Uribe (2007) for similarassumptions.

6to keep the reserve-deposit ratio higher than a ﬁxed level, which we shall call:
MtB ≥ Dt , 0 <  < 1,(3)

where MtB denotes nominal reserve holdings. The bank maximizes its proﬁt:
Rt Bt − RtD Dt ,where RtD is the nominal rate of interest on deposits, subject to reserve
requirement (3) and the balance sheet:
Bt + MtB = Dt .

(4)

The Lagrange function for the proﬁt-maximization problem is given by
Lt = Rt Bt − RtD Dt + κt (MtB − Dt ) + δt (Dt − Bt − MtB ),where κt and δt are the Lagrange multipliers associated with (3) and (4)respectively. The ﬁrst-order conditions are
Rt = δt ,κt = δt ,RtD = δt − κt ,(5)

κt ≥ 0, MtB − Dt ≥ 0, κt (MtB − Dt ) = 0.
When Rt > 0, the bank reduces reserves to the minimum level and consequently (3) is binding (κt > 0). From (3)-(5), we therefore obtain
RtD = (1 − )Rt > 0, Dt =Bt
, MtB =
Bt .
1−
1−

(6)

When Rt = 0, it is indiﬀerent for the bank to hold reserves and interestbearing assets. Thus, (3) is not binding (κt = 0) and no equation of (6) isvalid. From (5) where κt = 0, we ﬁnd
Rt = RtD = 0.
We formally state this property:
7Lemma 1. Rt = RtD = 0 if and only if (3) is not binding.

2.3

The Representative Household

Following Romer (1985) and Jones et al. (2004), we assume that the representative household’s utility depends on consumption ct , real cash holdings
H
H
H
mHt (≡ Mt /Pt ) where Mt is the nominal value of mt , and real deposit

holdings dt (≡ Dt /Pt ). The household maximizes the following lifetime utility:
 ∞
0




u(ct ) + V (mHt , dt ) exp(−ρt)dt,

where ρ (> 0) is the subjective discount rate, u(ct ) is the utility of consumption and V (mHt , dt ) is the utility of liquidity. They satisfy normal properties:u (·) > 0, u (·) < 0, u (0) = ∞, u (∞) = 0;

 2


∂ V
∂V
≡ Vm (·, ·) > 0,
2 ≡ Vmm (·, ·) < 0,
∂mH
∂mH
t
t


Vm (0, dt ) = ∞, Vm (∞, dt) = 0;

 2

∂ V
∂V
≡ Vd (·, ·) > 0,
≡ Vdd (·, ·) < 0,
∂dt
∂dt 2
H
Vd (mHt , 0) = ∞, Vd (mt , ∞) = 0.

(7)

V (mH
t , dt ) is further assumed to be linear homogeneous and hence marginalHH
9
utility of cash Vm (mHt , dt ) and that of deposits Vd (mt , dt ) depend on mt /dt :H
Vm (mHt , dt ) ≡ vm (xt ), Vd (mt , dt ) ≡ vd (xt ),where xt is the cash-deposit ratio deﬁned such that
xt ≡

mH
t
.
dt

9

(8)

Similarly, Jones et al. (2004) assume a constant elasticity of substitution function of
cash and deposits.

8The properties given in (7) imply vm (xt ) and vd (xt ) to satisfy
(xt ) < 0, vm (0) = ∞, vm (∞) = 0;vmvd (xt ) > 0, vd (0) = 0, vd (∞) = ∞.

(9)

The household’s assets at consist of cash mHt and deposits dt :at = mHt + dt .

(10)Note that there is no equity since the ﬁrm value is zero under the linear
technology. The ﬂow budget equation isȧt = rtD dt − πt mH
t + wt lt − ct − τt ,(11)

where rtD is the real rate of interest on deposits:rtD ≡ RtD − πt(12)and lt is the realized amount of labor supply. The household’s labor supply is inelastic and normalized to unity, but it may not be fully employed.
Therefore, realized labor supply lt is determined by the short side of labordemand and supply:
lt = min ltd , 1 .

(13)

The ﬁrst-order conditions for the utility-maximization problem subject
to (10) and (11) areu (ct ) = λt ,−πt λt + vm (xt ) = γt ,rtD λt + vd (xt ) = γt ,λ̇t − ρλt = −γt ,

9

(14)

where λt is the co-state variable associated with (11) and γt is the Lagrange
multiplier associated with (10). The transversality condition islim λt at exp(−ρt) = 0.

t→∞

(15)

From (12) and (14) we obtainρ + η(ct )ċtvm (xt )vd (xt )+ πt = = RtD + ,ct
u (ct )u (ct )

(16)

where η(ct ) ≡ −u (ct )ct /u (ct ). The left-hand side of (16) is the intertemporal
marginal rate of substitution of consumption measured in nominal terms, themiddle is the marginal beneﬁt of cash given by the intratemporal marginal
rate of substitution between cash and consumption, and the right-hand sideis the marginal beneﬁt of deposits given by the sum of the nominal rateof interest on deposits and the intratemporal marginal rate of substitution
between deposits and consumption.

3

The Dynamics

We obtain the dynamics of the present economy from all agents’ behavior
presented in the previous section. The money market equilibrium conditionis
MtH + MtB = M.

(17)

Adjustment of nominal wage Wt is assumed to be sluggish:Ẇt= α(ltd − 1),
Wt

(18)

where α (> 0) is exogenous and constant and W0 is historically given.10 In
contrast, commodity price Pt perfectly adjusts. Thus, the second equation
10

Many economists have long explored how nominal or real wages adjust. Although thisissue is still interesting and important, we assume a simple adjustment process because10

of (1) is valid at any point in time:Wt= y for ∀t,
Pt

(19)

and the commodity market is always in equilibrium
ct = ylt ,(20)

lt = ltd ≤ 1.

(21)

where from (13)Note that y can be taken as full-employment output since the representative
household’s labor endowment is unity.
From (18)-(21) we ﬁnd πt to be
πt = α




ct
−1 .
y

(22)

Since Mt is constant at M, the dynamic equation of real money balances
mt (≡ M/Pt ) isṁt
= −πt = −α
mt




ct
−1 .
y

(23)

From the second and third equations of (6), (8), the second equality of(16), and (17), cash-deposit ratio xt ismt−  ≡ x(mt , bt ) if RtD > 0,bt
xt = x̃ if RtD = 0,xt = (1 − )(24)

our purpose is not to analyze why wages are rigid but to show why unemployment persists
even though wages continue to adjust, as was the case under the Great Depression of
the 1930s and Japan’s recent slump. If wages instantaneously adjust so that demandalways matches supply in the labor market, the possibility of unemployment is intrinsically
avoided. However, it can be shown that there is no equilibrium path under such perfect
wage adjustment if a liquidity preference for cash is excessive and/or if full-employment
output is large (see section 5). Moreover, it is known that the original Phillips curve holds
well in Japan (see, e.g., Smith, 2008).

11

where x̃ satisﬁes
vm (x̃) = vd (x̃).

(25)

Note that x̃ is uniquely determined under the properties in (9). From (24) and(25), we ﬁnd that in contrast to the standard money-in-the-utility-function
model the marginal utility of cash is positive even when nominal interest
rates RtD and Rt are zero. It is because from (16) both cash and deposits
yield utility of liquidity and RtD merely ﬁlls the gap between the liquidity
premiums of the two.
The ﬁrst equality of (16), (22) and (24) give the dynamic equation of ct :ctvm (x(mt , bt ))ċt−1 +− ρ if RtD > 0,
= σ(ct ) −αctyu (ct )



(26)
ct
vm (x̃)ċtD−1 + 
− ρ if Rt = 0,
= σ(ct ) −αctyu (ct )
where σ(ct ) ≡ 1/η(ct ). (2), (23) and (26) characterize the dynamic behaviorof the present economy.

4

The Full Employment Steady State

This section shows that full employment obtains and nominal interest rates
R and RD are positive in a steady state ifρ>vm (x̃)
.
u(y)

(27)

In the full employment steady state, from (20), (23) and the ﬁrst equationof (26),c = y, l = 1, π = 0, ρ =vm (x∗ )
,
u(y)

(28)

where x∗ denotes the cash-deposit ratio in this state and is uniquely determined owing to the properties of vm (·) in (9).
12Since vm(x) < 0 and vd (x) > 0 as shown by (9), equations (25) and (27)and the last equation of (28) yieldx∗ < x̃, ρ =vm (x∗ )vm (x̃)vd (x̃)vd (x∗ )>=>.
u (y)u (y)u (y)u(y)Therefore, from (6) and (16), the nominal interest rates are positive:
RD = (1 − )R =vm (x∗ ) vd (x∗ )− > 0.
u (y)u (y)(29)

Since the cash-deposit ratio is given by the last equation of (28) and lemma
1 implies that reserve requirement (3) is binding in this state, (2), (6), (8)and (17) yieldb = b, d =(x∗ + )bb
x∗ b
b
, mH =, mB =
, m=
.
1−
1−
1−
1−

(30)

Using d, mH and m given in (30) we ﬁnd the money multiplier to be larger
than unity:
MH + DM
mH + dx∗ + 1
=
= ∗
> 1.
m
x +

Moreover, as shown in appendix A, there is a unique path converging to
the full employment steady state. Thus we obtain the following proposition:
Proposition 1. The economy converges to the full employment steady state
represented by (28)-(30) if (27) holds.

5

The Unemployment Steady StateIn the previous section we ﬁnd that under (27) full employment is eventuallyreached, in other words, unemployment is a short-run phenomenon. However,
unemployment often lasts long, as did under the Great Depression in the
1930s and Japan’s lost decade. This section shows that if (27) is invalid, i.e.,ρ<vm (x̃),u (y)
13(31)

persistent unemployment occurs, the nominal interest rates are zero, deﬂation lasts, excess reserves are held by the commercial bank and the money
multiplier equals unity in a steady state.11 (31) implies that the marginal
beneﬁt of cash (the right-hand side) exceeds that of consumption (the lefthand side) if consumption is large enough to realize full employment. It isvalid if either labor productivity y (which equals full-employment output) or
marginal utility of cash vm is large enough.
From (9) and (31), x̃ in (25) and x∗ in (28) satisfyx∗ > x̃, ρ =vm (x̃)vd (x̃)vd (x∗ )vm (x∗ )
<
=
<
,
u(y)
u (y)u(y)u (y)

which implies that if x is determined so that it equates the marginal beneﬁtof cash to that of consumption under full employment then the nominal rateof interest on deposits must be negative:RD =vm (x∗ ) vd (x∗ )− < 0.
u (y)u (y)If RD < 0, however, the commercial bank can unlimitedly increase its proﬁt
by collecting deposits and holding reserves, which increases the bank’s demand for deposits to inﬁnity. RD therefore rises up to zero, and consequently
RD = 0, R = 0.

(32)

In this state excess reserves are held by the commercial bank, as stated bylemma 1, and the cash-deposit ratio isx = x̃,11(33)

We ignore the inﬁnitesimal possibility that neither (27) nor (31) is valid, that is,ρ=vm (x̃).
u (y)In the steady state of this case, the zero nominal interest rates and full employment go
together.

14

as shown by (24).
From (31) and (33), the last equation of (28) is obviously invalid andhence the full employment steady state is unreachable. From the second
equation of (26) we obtain

ρ+α

c
−1
y


=

vm (x̃),u (c)(34)

instead of the last equation of (28). Under (31) the left-hand side of (34) issmaller than the right-hand side if c = y. Thus, in order for c that satisﬁes(34) to exist uniquely within (0, y) the left-hand side of (34) must be larger
than the right-hand side if c = 0, i.e.,ρ > α.

(35)

From (31) and (35), the slope of the right-hand side of (34) must be steeper
than that of the left-hand side at the value of c satisfying (34):vm (x̃)u (c)α.
<−y[u (c)]2(36)

As shown in appendix B, (36) assures the saddle-path stability in the neighborhood of the present steady state and hence there is a unique path to reachthe steady state.
Let us summarize the present mechanism of persistent unemployment.
Since RD cannot be negative, it is stuck at zero, which makes x equal x̃
and vm (x) equal vm (x̃). Since the marginal beneﬁt of cash exceeds that ofconsumption (viz. the time preference rate) under full employment, as shownby (31), the household reduces consumption to less than full-employment
output y in order to hold more cash. Hence persistent unemployment arises(from (20) and (21) l = ld < 1 under c < y).1212

Ono (1994, 2001) ﬁnds that an aggregate demand shortage and unemployment occur

15

From (23) where c < y, deﬂation arises and real money balances continue
to increase:ṁ= −π > 0.
m(37)

Using (2), (4), (8), (17), (33) and (37) we ﬁnd
b = b, d =x̃ m + b
m+b
m − x̃b= ∞, mH == ∞, mB == ∞.
1 + x̃
1 + x̃
1 + x̃
(38)

From (34) and (38), deﬂation continues to expand real cash and deposit
holdings but does not stimulate consumption –i.e., the Pigou eﬀect does notwork.13
From (17) and the second and fourth properties of (38) we obtain
1 − x̃b/m
MB= lim= 1, M H + D = M H + M B = M,m→∞ 1 + b/m
D(39)

which shows that the money multiplier approaches unity and monetary aggregates M H + D decline to monetary base M.
Since (4), (10) and (17) yielda (= mH + d) = m + b,
from (37) and (38) we ﬁnd
ȧ
ṁ/m
ḃ/b
ṁ
=
+
=
= −π.
a
1 + b/m m/b + 1
m
as a steady-state phenomenon if there is a positive lower bound for the marginal utility ofmoney. See also Matsuzaki (2003), Hashimoto (2004), Johdo (2006, 2008a, 2008b), Ono(2006), Rodriguez-Arana (2007) and Johdo and Hashimoto (2009) for the same type of
unemployment. Whereas they assume the marginal utility of money to remain positive,
we do not assume it and nevertheless persistent unemployment occurs. In our paper
the nonnegativity of nominal interest rates causes the marginal utility of money to stay
positive.
13
Ono (1994, 2001) also ﬁnds that the Pigou eﬀect does not work under insatiableliquidity preference.

16

Time diﬀerentiation of the ﬁrst equation of (14), the ﬁrst equality of (16)and the above property lead toċtvm (x̃)λ̇t ȧt+ − ρ = lim −η(ct ) − π − ρ = − < 0,t→∞ λt
t→∞
at
ct
u (c)
lim

implying that transversality condition (15) is satisﬁed although m permanently expands.14
We summarize the above discussion in the following proposition:
Proposition 2. The economy reaches the unemployment steady state characterized by (32)-(34), (37) and (38) if (31) and (35) are valid.
As observed during the Great Depression in the US and the recent stagnation in Japan, unemployment persists, leading to deﬂation, and nominalinterest rates RD and R are zero, causing the commercial bank to hold excess reserves. Moreover, the latter is supported by some empirical papers.
For example, Wilcox (1984) attributes 80 percent of the increase in excess
reserves during the Great Depression to a signiﬁcant fall in interest rates.
Ogawa (2007) reports a similar result in Japan of the late 1990s.
The result shown by (39) is also consistent with both stagnation periods.
For instance, Wheelock (1992, ﬁgure 10) illustrates a signiﬁcant fall in themoney multiplier during the Great Depression, and ﬁgure 5 exhibits a similar
phenomenon that occurred during Japan’s recent stagnation. In the present
steady state the contraction in monetary aggregates is not the cause but
the result of slump. This causality is opposite to the money hypothesis à laFriedman and Schwartz (1963) but the same as Temin (1976) and Krugman(1998).
14

Even if b does not follow (2) but continues to increase, the transversality condition
holds and the money multiplier remains unity as long as the expansion rate of b is lessthan that of m.

17

Note that (32)-(34), (37) and (38) are valid regardless of the amount of
M. Thus, even if the monetary authority supplies a larger amount of M,
all of the real variables remain unaﬀected and only nominal cash and excess
reserve holdings increase. It is indeed the case during Japan’s recent long-run
stagnation. The Bank of Japan expanded the monetary base from 65.7961
trillion yen in March 2001 to 108.7749 trillion yen in March 2006 but mostof the increased money were absorbed as excess reserves (see ﬁgure 4) while
neither aggregate demand nor employment was signiﬁcantly stimulated.

6

Concluding Remarks

We present a simple model where a representative household derives utility
from liquidity of both cash and deposits, a representative commercial bankis required to hold money as reserves, and nominal wages sluggishly adjust.
In this setting we ﬁnd that there are a certain value of the marginal utilityof cash and a certain amount of full-employment output that divide the caseof persistent unemployment from that of full employment.
If the marginal utility of cash is less than the critical value, or equivalently, if full-employment output is smaller than the critical amount, full
employment obtains and nominal interest rates are positive in a steady state.
If otherwise, typical phenomena of persistent stagnation that are allegedly
due to a liquidity trap obtain in a steady state. Nominal interest rates arestuck at zero and the commercial bank holds excess reserves. The marginal
beneﬁt of cash would be larger than that of consumption if the household
consumed enough to realize full employment. Therefore, the household reduces consumption so as to increase cash holdings, which creates an aggregate demand shortage, unemployment and deﬂation. These phenomena werecommon during the Great Depression in the US and the recent long-run
18

stagnation in Japan.
The present mechanism of stagnation is more in conformity with thespending hypothesis than with the money hypothesis although the monetaryside plays a crucial role in generating stagnation. The stagnation occurs– not because excessive demand for cash decreases monetary aggregates,
keeps an interest rate too high and prevents investment from increasing – but
because it absorbs purchasing power that would otherwise be directed toward
consumption and generates a shortage of aggregate demand. Furthermore,this mechanism is diﬀerent from such recent studies of liquidity traps as
Krugman (1998) who shows that a combination of deﬂation and zero nominalinterest rates increases real interest rates and decreases current consumption.

19

Appendix A The Stability of the Full Employment Steady StateIn the neighborhood of the steady state where b, m and c are given by(28) and (30), linearizing (2), (23) and the ﬁrst equation of (26) yields thecharacteristic equation:−θ − z

0

0

−z


σcvm
xb

u


σcvm
xm

u

0
αm
−
y


α vm 
σc − −  2 u − z
y
u

= 0,

where z is a characteristic root and xb and xm are partial derivatives of theﬁrst equation of (24):m1−
xb = −(1 − ) 2 < 0, xm => 0.
bb
One of the three characteristic roots isz = −θ < 0,and the other two satisfy

α vm 
αmσcvmxm
2= 0,+ 2 u z +z + σcyyuuwhere the constant term is negative. Thus, the characteristic equation has
one positive and two negative roots.
From (19), P0 is determined so that W0 /P0 = y, where W0 is historicallygiven. Then, m0 and b0 are given as m0 = yM/W0 and b0 = yB0 /W0 , where
y and M are exogenous and constant and B0 is historically given. Since c0
is jumpable and m0 and b0 are not, there is a unique path converging to thefull employment steady state mentioned in proposition 1.
20

Appendix B The Stability of the Unemployment Steady State
By replacing mt by qt (≡ 1/mt ) and linearizing (2), (23) and the second
equation of (26) in the neighborhood of the steady state where q = 0, b = b
and c satisﬁes (34), we obtain the following characteristic equation:−θ − z
0
0


α

0

c
−1 −z
y
0

0



0
= 0.


α vm
σc − −  2 u − zyu

The roots of this equation are




c
α vm 
− 1 < 0, z = σc − −  2 u > 0,z = −θ < 0, z = αyyuwhere the last inequality holds under (36). Note that (36) must hold under(31) and (35). Since c0 is jumpable but neither q0 (= 1/m0 ) nor b0 is,there is a unique path to reach the unemployment steady state described inproposition 2.

21

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25

Source: The Statistics Bureau, Ministry of Internal Aﬀairs and Communications, Japan.

Figure 1: The average rate of unemployment in Japan for the years 19902007.

26

Source: The Cabinet Oﬃce, Government of Japan.

Figure 2: The year-to-year percentage change in the GDP deﬂator in Japan
from January-March 1995 to July-September 2007.

27

Sources: The Bank of Japan.
Note: Ordinary deposits and installment savings correspond to demand andtime deposits respectively.

Figure 3: The nominal rates of interest on deposits from October 17, 1994
to September 24, 2007.

28

Sources: The Bank of Japan.

Figure 4: The monetary base, cash, and actual and required reserves in Japan
from January 1990 to October 2007.

29

Sources: The Bank of Japan.
Note: The left and right axes measure the ratios and the money multiplier
respectively.

Figure 5: The money multiplier, the cash-deposit ratio and the reservedeposit ratio in Japan from January 1990 to October 2007.

30