Discussion Papers In Economics
And Business

Learning, Inflation Cycles, and Depression

Ryo Horii and Yoshiyasu Ono

Discussion Paper 06-14

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANLearning, Inflation Cycles, and Depression

Ryo Horii and Yoshiyasu Ono

Discussion Paper 06-14
May 2006
この研究は「大学院経済学研究科・経済学部記念事業」
基金より援助を受けた、記して感謝する。
Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANLearning, Inflation Cycles, and Depression∗
Ryo Horii† and Yoshiyasu Ono‡
Version: May 12, 2006

Abstract
This paper constructs a model that describes inflation cycles and prolonged
depression as generated by the learning behavior of households who face a random liquidity shock in which money is needed. Households update the subjective probability of the shock based on the observation and change their liquidity
preference accordingly. In this setting, we first derive a stationary cycles under
perfect price adjustment, which is characterized by periods of gradual inflationand sudden sporadic falls of the price level. When the nominal stickiness isintroduced, the liquidity shock is followed by a period of depression in whichunemployment exists and deflation occurs gradually. Depression is deep andprolonged when the economy has experienced a long period of boom before
encountering a liquidity shock.
JEL Classifications: D83, E41, E32
Keywords: Bayesian Learning in Continuous Time, Hamilton-Jacobi-Bellman
Equations, Markov Modulated Poisson Processes, Partial Delay Diﬀerential
Equations, Liquidity Preference.

∗

The authors are grateful to Jun-ichi Itaya, Yasutomo Murasawa, and seminar participants at

Osaka Prefecture University and JEA Annual Meeting Fall 2004 held at Okayama University for
their helpful comments and suggestions. This research is partially supported by JSPS Grant-inAid for Young Scientists (B) 16730097, and is a part of the Osaka University 21st Century COE
program, the Ministry of Education, Science, Sports and Culture.
†

Graduate School of Economics, Osaka University, 1-7 Machikaneyama, Toyonaka, Osaka 5600043, JAPAN; E-mail: horii@econ.osaka-u.ac.jp‡

Institute of Social and Economic Research, Osaka University, 6-1, Mihogaoka, Ibaraki-shi,
Osaka, 567-0047, JAPAN; E-mail: ono@iser.osaka-u.ac.jp1

1

Introduction

This paper constructs a model that describes the cyclical movement of price level
inflation and the possibility of prolonged depression as generated by the learning
behavior of households. The representative household experiences a utility loss whena liquidity shock occurs, and the loss is smaller if she has a larger amount of money.
The liquidity shock is randomly generated by a Poisson process, and the Poissonparameter, or the probability of the shock, changes unobservably following a Markov
process (i.e., the shock occurs according to a certain Markov modulated Poissonprocess). The household updates the subjective probability of the shock based on
the observation of the actual occurrences of the shock.1
In this setting, we first derive a stationary cycles under perfect price adjustment,
which is characterized by the periods of gradual inflation and sudden sporadic deflations. The price level must jumps down when the liquidity shock occurs because theinformation delivered by the news of the liquidity shock is large enough to change
the household’s belief discretely. Such downward jumps are, however, unlikely tooccur in the actual economies where various nominal frictions prevent the price level
and/or the nominal wage to fall too rapidly; e.g., labor unions, moral issues, etc.
Thus, we extend the benchmark model to incorporate the nominal stickiness that1

An example of such events is a bank run; even though people usually do business using checks,

once a bank run occurs their transactions cannot be settled without money. This example suggeststhat the desire to hold money depends on the expectation about the possibility of such events in the
future. When people observe a bank run, they guess that the underlying circumstance for banks
is bad and therefore another bank may fail in near future. Thus, in this event they increase thedemand for money to cope with another similar event. Given that the supply of nominal money
cannot be changed instantly, the strong liquidity preference pushes up the value of money in termsof goods and therefore deflation occurs. Conversely, if banks operates successfully for a long time,
people guess that the circumstance is stable and that it is unlikely for a bank run to occur in near
future. Then, the money demand decreases, causing inflation.

2

puts an upper bound on the rate of deflation.
In the modified setting where price and thus the real money supply cannot jump,
the representative household cannot increase the real money holding instantly when
the subjective shock probability jumps up. The expected marginal utility of money
holding stays high and the household decrease the amount of consumption to equalize
the marginal rate of substitution between money and goods to the cost of holding
money. This means that the liquidity shock causes a depression rather than a discrete
fall in the price level. The drop in output is followed by a gradual recovery, in which
price falls gradually. Once the price level falls to a certain threshold, the equilibriumin the goods and money markets is regained and then gradual inflation occurs.
The important finding is that the magnitude of depression is larger when theeconomy experiences a long period without the liquidity shock before encountering
one because in that case price must fall significantly to regain the equilibrium in themoney market. On the other hand, a economy with recurrent shocks harms little
from an addition shock (e.g., a bank run) because the price level is already near
the lowest level. The inflation rate immediately before the crush is not necessarily
higher in the former case than the latter; specifically, the inflation rate converges tothe rate of money growth (which should be the long-run average) when the economy
luckily proceeds without a crush for a very long time. Thus, our result implies thatan economy performing well for a long time is vulnerable to get into the long period
of depression even when its contemporary inflation rate is low.
There exist a number of earlier studies that analyzed the macroeconomic behavior
when an underlying state is only partially observable and information are revealed
gradually (e.g., Andolfatto and Gomme 2003; Chalkley and Lee 1998; Horii and
Ono 2005; Potter 2000; Sill and Wrase 1999 ). Those studies typically constructmodels in a way that the current belief regarding the underlying state determines
macroeconomic variables at each date. This is also the case in our model when
price adjustments are complete: i.e., the current belief on the risk of financial crises
3uniquely determines the current price level so that the demand and the supply of thereal money balance equalize.
However, when nominal stickiness is introduced, the price level cannot completely
follow the path that corresponds to the realized evolution of the belief. As a result,the current price level is determined not only by the current belief on the underlying
state of the economy, but also by the history of beliefs that people held in thepast. Accordingly, whether an economy experiences inflation and full employment,
or deflation and unemployment, is not solely determined by how frequently financial
shocks have occurred recently, but also by how the economy (i.e., the price level) has
adapted to such occurrences.
The organization of the paper is as follows. In section 2, we describe the process of
the liquidity shock and the evolution of the belief that is updated based on Bayes’ law.
Section 3 presents a benchmark model in which the price level is perfectly adjusted
and shows the pattern of inflation cycles. The nominal stickiness is introduced in
Section 4 to investigate the pattern of the crush and recovery. Section 5 concludes
the paper. The proofs of all Lemmata are collated in Appendix.

2

Liquidity Shock and Bayesian Learning

We use a continuous-time model in which a representative household faces an aggregate liquidity shock that follows an exogenous Poisson process. Liquidity holding
generates utility when the shock actually occurs, but does not while the shock does
not occur. Since when the shock occurs cannot exactly be anticipated, even during
the period without it the household holds liquidity so as to prepare for it.
There are two underlying states with diﬀerent probabilities of the shock, calledstates H and L. In state i ∈ {H, L} the shock occurs with probability θi per unit
time, where θH > θL > 0. The household cannot directly observe the current state
but knows that the state evolves according to a Markov process: state H changes to4

state L with Poisson probability pH per unit time whereas state L changes to state
H with probability pL .
By observing whether the shock occurs or not she continuously revises her subjective shock probability in a Bayesian manner. Let θt denote the true shock probability
at time t, which is unknown to her. Using information available up to time t, sheforms a belief that θt = θH with probability ft (θH ) and θt = θ L with probabilityft (θL ). Obviously,
ft (θL ) + ft (θH ) = 1 for any t.

(1)

In order to find how she updates ft (θ i ) from t to t + ∆t,2 we first obtain the subjective probability that the shock does not occur between t and t+∆t for given ft (θi ).
¤£It is denoted by Ft S(t,t+∆t] = φ , where Ft [·] is a probability operator based on information available at t, S(a,b] is the set of dates on which the shock actually occurs

during (a, b], and φ the empty set. Since the underlying state is either H or L at time£¤t+∆t, this probability is divided into two components, Ft S(t,t+∆t] = φ ∩ θt+∆t = θH£¤
and Ft S(t,t+∆t] = φ ∩ θt+∆t = θL .
Each of the two components is further divided into two probabilities. The former

is the sum of the probability that ‘the state is H at time t and neither the state change
nor the shock occurs during the interval’ and the probability that ‘the present stateis L and the state changes to H during the interval.’ It is3¢£¤ ¡Ft S(t,t+∆t] = φ ∩ θt+∆t = θ H = 1 − (θH + pH )∆t ft (θH ) + pL ∆tft (θL ).

(2)

Similarly, the latter is¢£¤ ¡Ft S(t,t+∆t] = φ ∩ θt+∆t = θL = 1 − (θL + pL )∆t ft (θL ) + pH ∆tft (θ H ).
2

Time interval ∆t is taken to be so short that the probability that the liquidity shock and astate change coexist in the interval is negligible.
3(3)

Throughout the paper we ignore the second-order term of ∆t and higher because ∆t → 0.

5

Summing up (2) and (3) yields£¤
Ft S(t,t+∆t] = φ = 1 − θte ∆t,(4)where θte represents the expected (or subjective) probability of the shock per unit
time at time t:θte ≡ θH ft (θH ) + θL ft (θL ).

(5)

Let us consider how the representative household updates her belief if she eventually finds that the shock did not occur during (t, t + ∆t]. In this case the informationthat S(t,t+∆t] = φ is added to her knowledge. Thus, using Bayes’ law we find updated
subjective probability ft+∆t (θ i ) to be£¤£¤ft+∆t (θi ) ≡ Ft+∆t θt+∆t = θi = Ft θt+∆t = θi |S(t,t+∆t] = φ£¤
Ft S(t,t+∆t] = φ ∩ θt+∆t = θi£¤=.
Ft S(t,t+∆t] = φ

Since the numerator is given by (2) or (3) and the denominator by (4), ft+∆t (θH )equals4¡¢
1 − (θH + pH )∆t ft (θH ) + pL ∆tft (θL )ft+∆t (θ ) =.
1 − θte ∆t
H

From this equation we derive the time derivative of ft (θH ):ft+∆t (θH ) − ft (θ H )dft (θH )
= lim
∆t→0
dt
∆t

(6)

= (θte − θ H − pH )ft (θ H ) + pL ft (θL ).
We next consider the case where the shock occurs during (t, t + ∆t]. Since£¤
Ft S(t,t+∆t] 6= φ ∩ θt+∆t = θi = θ i ft (θ i )∆t for i ∈ {L, H},(7)

the probability that the shock occurs is¤ ¡¢£Ft S(t,t+∆t] 6= φ = θ H ft (θ H ) + θ L ft (θ L ) ∆t = θte ∆t,
4ft+∆t (θ L ) is analogously obtained. From (1) it equals 1 − ft+∆t (θ H ).

6(8)

which is consistent with (4). From Bayes’ law dividing (7) by (8) gives the updated
subjective probability that θt+∆t = θi under the condition that the shock occurs
during (t, t + ∆t]. It isθ i ft−0 (θi )θ i ft0 (θi )≡,et →t−0θte0θt−0

ft (θi ) = 0 lim(9)where subscript t − 0 represents the state just before t.
Finally, we obtain the dynamics of subjective probability θte . From (1) and (5),θte − θ L,
ft (θ ) = H
θ − θLθH − θteft (θ ) = H.
θ − θLH

L(10)

Substituting (6) and (10) into the time derivative of (5) yields the time derivative of
θte in the case where the shock does not occur at time t:θ̇te = (θte − θL − pL )(θte − θ H − pH ) − pL pH ≡ g(θte )for t ∈/ S(0,∞) ,(11)

which satisfies£¤g(θ) Q 0 ⇐⇒ θ R θ∗ for any θ ∈ θ L , θ H , wherepθL + θH + pL + pH − (θH + pH − θL − pL )2 + 4pL pH∗θ ≡∈ (θ L , θ H ).
2

(12)

Similarly, by substituting (9) and (10) into (5) we obtain the value of θte as a functionein the case where the shock does occur at time t.
of θt−0θte = θL + θH −θL θHe≡ h(θt−0)eθt−0

for t ∈ S(0,∞) ,(13)

which satisfiesh(θH ) = θH , and θe < h(θe ) < θ H for all θ e ∈ (θL , θH ).
Equations (11) and (13) describe the dynamics of θte with and without the shock
respectively. It continuously declines as long as the shock does not occur, but discretely jumps upward when it occurs.5 Intuitively, in the absence of the shock people
5¡¤
From (11) and (13), we find that θte is trapped within interval θ ∗ , θH in the long run. Since

we are interested in the long-term behavior of the economy, it is assumed throughout this paper¤¡that θte is always within θ∗ , θH . Under this assumption, θte always declines while the shock does

not occur.

7

gradually become more and more optimistic and confident that the economy is instate L. Thus, their subjective probability of the shock gradually declines and converges to θ ∗ .6 Due to the U-shape of function g(θte ), the speed of adjusting theirbelief is slower when θte is near either θ ∗ or θ H than when it is in the middle.
Conversely, when the shock is observed, people discretely change their expectation
about the present state. Since h(θe ) > θ e , the more often people observe the shock,the more strongly people believe that they are in state H, and hence θte becomescloser to θH . In this way θte fluctuates between θ∗ and θH .

3

Inflation Cycles under Perfect Price Adjustment

The economy is inhabited by a continuum of infinitely lived homogeneous householdswith measure one. Each household is endowed a unit labor at each point time andsupplies labor inelastically to the labor market. A representative firm produces goods
from labor. The output is y`t , where y > 0 is a constant and `t is labor input. Since
total labor supply is one, the aggregate output would be y each time as long as full
employment obtains. The monetary authority issues a constant amount of nominal
money stock, the size of which is normalized to one.7 Goods are perishable and thus
cannot be stored. The households will not borrow or lend among themselves because
the are identical. The firm has no value because of its linear production technology
and perfect competition. Therefore, money is the only asset in this economy.
At each date, the representative household gains utility u(ct ) from consumption,

6 e
θt never becomes lower than θ∗ (> θL ) since people take into account the possibility that state

L might have changed to state H even though the shock does not occur.
7

This assumption is made only for the simplicity of the description of the model and notations.

The results to follow are essentially the same even when the nominal money growth rate is positiveand constant. In that case the price level would not be stationary, and therefore we need tonormalize the price level by by dividing by the nominal money supply.

8

where instantaneous felicity function u(·) is twice diﬀerentiable, u0 (·) > 0, and satisfies the Inada conditions. In addition, she experiences utility loss v(mt ) < 0 according
to her real money holding mt at the dates at which the liquidity shock occurs. We
assume v 0 (m) > 0, which means that the size of utility loss is small when her realmoney holding is large. Function v(·) also satisfies v 00 (m) < 0, limm→0 mv 0 (m) > 0,and limm→∞ v 0 (m) = 0. Her expected utility EUt is therefore given by


Z ∞
X
EUt = Et u(cτ )e−ρ(τ −t) dτ +v(mτ )e−ρ(τ −t)  ,t(14)

τ ∈S(t,∞)where ρ is her subjective discount rate and S(t,∞) is the set of future dates at which
the shock occurs.
Let pt denote the price of consumption good, and Wt the nominal wage at t. The
household maximize (14) under the budget constraint
Ṁt = Wt − pt ct .

(15)

Since there is no friction in the market, full employment always obtains and thus
Wt = pt y for all t. Thus, in the real terms, (15) becomesṁt = y − πt mt − ct
mt = mt−0 /Πtwhen pt changes continuously,

when pt jumps,

(16)
(17)

where πt ≡ ṗt /pt and Πt ≡ pt /pt−0 .
There is no steady state in equilibrium at which the price level stays constantfor all t because the expectation about the probability of the liquidity shock changes
forever according to (11) and (13) and the decisions of household depend on θte . We
instead search for a stationary equilibrium dynamics in which pt evolves as a functionof θte . Specifically, we search for a function p(·) that satisfies8
pt = p(θte ) for all t.
8This approach is similar to Lucas (1978).

9

(18)

Then, the inflation rate can also be written as a function of θte ,p0 (θte ) eπt =g(θt ) ≡ π(θte ),ep(θt )p(h(θte ))Πt =≡ Π(θte ).
ep(θt )(19)

We are interested in a monetary equilibrium path in which money has a positive
value. Suppose that pt0 = ∞ for some date t0 , which means that money has novalue at t0 . Then, it follows that pt = ∞ for all t ≥ t0 since otherwise an arbitrage
opportunity arises: consumers can obtain an arbitrary amount of money at datet0 at no cost and then sell money (i.e., purchase goods) at a date in which pt isfinite to increase their expected utility. Since θte evolves within (θ ∗ , θH ) recurrently,(18) implies that if p(θ∞ ) = ∞ for some θ ∞ ∈ (θ ∗ , θH ) then p(θe ) = ∞ for allθe ∈ (θ∗ , θH ). That is, if there is such θ ∞ , then pt = ∞ for all t and therefore moneyis never demanded. Since the nonmonetary equilibrium dynamics is obvious and isof no interest here, we assume that9Assumption 1 p(θe ) ∈ (0, ∞) for all θ e ∈ (θ∗ , θ H ).
Let U (θ e , m) denote the value function of a household with belief θe and realmoney holding m. Then, the bellman equation for the problem of (11), (13), (14),(16) and (17), combined with (18) and (19), ish
U (θ e , m) = max u(c)∆t + (θe ∆t)v(m00 )
c

+

ªi
©
1
(1 − θe ∆t)U (θe0 , m0 ) + (θ e ∆t)U (h(θ), m00 ) ,1 + ρ∆t(20)

where θe 0 = θe + g(θe )∆t, m0 = m + (y − π(θe )m − c)∆t, and m00 = m/Π(θ e ). Taking
the limit ∆t → 0 in (20) yields the Hamilton-Jacobi-Bellman (HJB) equation for the
problem:h¡¢ρU (θe , m) = max u(c) + θ e v(m/Π(θ e )) + U (h(θe ), m/Π(θe )) − U (θe , m)ci¡¢
+ g(θe )Uθ (θ e , m) + y − π(θ e )m − c Um (θ e , m) .

9

(21)

We also rule out the possibility that p(θe ) = 0 for some θe ∈ (θ ∗ , θH ) because the value ofconsumption good never becomes zero from u0 (·) > 0.

10

Diﬀerentiating the right hand side of (21) with respect to c gives the first orderconditionc) = Um (θe , m),
u0 (e

(22)

where ec denotes the optimal amount of consumption. Since θte and mt evolves ac-

cording to (11) and (16), equation (22) shows that the movement of consumption ischaracterized by¡¢
d 0ct ) = g(θte )Umθ + y − π(θ e )m − e
c Umm
u (e
dt

for t ∈/ S(0,∞) ,(23)

abbreviating the arguments for U (·, ·) functions when they are (θ e , m). From theenvelope theorem, (21) can be diﬀerentiated with respect to m at c = e
c to give¡¢c Umm(ρ + π(θe ) + θ e )Um =g(θ e )Uθm + y − π(θe )m − e¢¡+ θ e Π(θ e )−1 v 0 (m/Π(θe ) + Um (h(θe ), m/Π(θ e )) .

(24)

By substituting (22) and (23) for (24), we can eliminate the value function from it
to obtain the Euler equation,d 0
v 0 (mt /Π(θ e )) + u0 (ec00t )ct ) = (ρ + π(θ e ) + θe )u0 (ect ) − θ eu (edtΠ(θe )

for t ∈/ S(0,∞) ,(25)

where ec00t represents the optimal amount of consumption when the state changes to(h(θte ), m/Π(θte )).

Since all households are symmetric, the goods and money markets clear whenect = y, mt = p(θte )−1

for all t.

(26)

Function p(·) is determined so that the household’s demand for goods and money
satisfies (26) given the path of price, (18). Substituting (26) into (25) yields a
condition that must be satisfied for all possible values of θe ,¡¢
ρ + π(θ e ) = θe Π(θe )−1 v 0 (p(h(θ e ))−1 ) + θe Π(θe )−1 − 1 .

(27)

The left hand side represents the cost of holding money: the utility loss from postponing consumption plus the capital loss caused by inflation. In the other side are11the expected benefits of holding money: the first term is the expected utility fromholding money, whereas the second term represents the expected capital gain by the
downward jump in the price level (the upward jump in the value of money) when
the liquidity shock occurs. Thus, (27) shows that function p(·) is determined so thatthe cost and the benefit of holding money are equalized with each other.
From (19) and (27), we obtain a (delay) diﬀerential equation for p(·).
p(θ e )γp (θe ), whereg(θe )
e
0
e −1
0
e
e
e p(θ ) v (p(h(θ )) ) + u (y)γp (θ ) ≡ −(ρ + θ ) + θ.
p(h(θe ))u0 (y)p0 (θe ) =(28)

Note that γp (θ e ) gives the growth rate of pt (i.e., the inflation rate) that must holdin absence of the liquidity shock as a function of θte . Since functions u, v, g, h are
already known, (28) is an autonomous diﬀerential equation with respect to functionp(·). The following lemma gives a boundary condition with which function p(·) ispinned down.
Lemma 1 Under Assumption 1 and transversality condition10
lim Et e−ρ(T −t) u0 (cT )mT = 0

T →∞for all t,(29)function γp (·) must satisfy limθe →θ∗ γp (θe ) = 0.
proof: in appendix
Note that pt converges to zero as θte → θ∗ if γp (θ e ) is negative at the limit. Thenthe real money holding of a household, mt = p−1
t , increases unboundedly in a waythat the transversality condition is violated. Conversely, the positive limiting valueof γp (θ e ) means that pt becomes arbitrarily large. In this case, the value of money
vanishes in a finite time period, which violates the assumption of the monetary
equilibrium.

10

Operator Et represents the expectation based on the information available to agents at date t.

12(a) The shape of function P (θe )(b) Evolution of inflation rate

Figure 1: Inflation cycles without nominal frictions

The stationary dynamics of a monetary equilibrium can be calculated from (28)and the boundary condition given by Lemma 1. Panel (a) of Figure 1 shows therepresentative shape of function p(·) against θ e , which is downward sloping.11 A
large value of θte means that people anticipates that the liquidity shock occurs with ahigh probability. In that situation, their liquidity preference is high. Thus, to clear
the market for money, the value of money must be suﬃciently high in relative to thevalue of good, which means a low price level.
During the period without the liquidity shock, θte gradually declines and pt increases. Panel (b) of Figure 1 shows the evolution of inflation rate against timeas θte moves from θ H to θ ∗ . Inflation accelerates temporarily when the households
adjusts their belief responding to observing no shock for a certain time length, but
it gradually falls to the rate of nominal money growth, which is zero in this case,as the economy converges to the most optimistic state. When the liquidity shock
occurs, θte jumps up. Then pt jumps down so that the (θte , pt ) pair is always on the
curve depicted in panel (a). Thus, the dynamics of the economy is characterized by

11

In all examples presented in this paper, we specify u(c) = ln c, v(m) = −m−1 , y = 1, ρ = .05,

θH = .5, θ L = .05, pH = .025 and pL = .0025. We have confirmed that our results are robust tochanges in parameter values.

13gradual inflation with sporadic and discrete falls in the price level.
At each event of the liquidity shock, price level must jump down in order to clear
the increased liquidity demand induced by the change in people’s belief. However,
we rarely observe such a discrete fall in the price level in the aggregate economy;
although we do sometimes observe a discrete fall in the prices of certain goods, theaggregated general price level tends to fall only slowly. One explanation for this is
the existence of a (downward) nominal stickiness in the price level caused by labor
unions, menu costs, moral issues, and the all other factors discussed in the literature.
If the price cannot jump downward, our model predicts that the demand for money
exceeds for the supply, and, by Warlas’ law, a demand shortage occurs in the goods
and labor market. The next section investigates this possibility.

4

Depressions with Nominal Stickiness

Consider an economy similar to the one introduced in the previous section, with aonly diﬀerence in that the price level cannot fall faster than a certain rate,12ṗt /pt ≥ −δ,δ ∈ (0, ∞).

(30)

The discrete fall in the price level derived in the previous section implies that the
instantaneous rate of inflation is minus infinity. Condition (30) rules out this possibility by assuming that δ is finite. The immediate consequence of this restriction is asfollows. Even when the liquidity preference jumps up following the liquidity shock,cannot increase instantly. The representative household
the real money supply p−1tthen feels that money at their hands is more valuable than before, and consequently
reduces the demand for goods. It creates a shortage in the goods market, which
12

Condition (30) is equivalent to assuming that Ẇt /Wt ≥ −δ because the competition of firms

ensures pt = yWt for all t. We could also assume a symmetric restriction such as ṗt /pt ∈ [−δ, δ].
This would make the analysis a little complicated without changing the final results.

14

cannot be cleared instantly because of the nominal stickiness in pt .
This consideration implies that there are several diﬀerences in the pattern of
economic dynamics from the case analyzed in the previous section. Now there isno one-for-one relationship between the price level and the belief because pt cannot
jump when θte changes discretely. Thus, the state of the economy is described not
only by θte but by the pair of (θte , pt ). Consumption does not necessarily coincidewith the full employment output y. We can imagine two possibilities at each point
in time. First, constraint (30) is not binding and full employment obtains (wt = y).
The second possibility is that (30) is binding, i.e., ṗt /pt = −δ, and unemployment
exists (wt < y).
Which one of these possibilities occurs depends on the state of the economy,
summarized by (θte , pt ). It is natural to guess that, for a given level of θte , there isa level of pt at which the money market clears and full employment obtains. Let
us denote this critical level by p(θte ). Price level pt cannot be below the thresholdp(θ e ) since there is no upward stickiness in the price level and thus can be adjusted
instantly if pt < p(θe ). Similarly to the previous section, if there is some θ∞ ∈ (θ∗ , θH )such that p(θ∞ ) = ∞ then pt = ∞ for all t. We rule out this trivial path by assuming
that
Assumption 2 p(θe ) ∈ (0, ∞) for all θ e ∈ (θ∗ , θ H ).
Unemployment occurs when constraint (30) is binding, i.e., when p > p(θe ). In
this case, the economy experience deflation at the rate of δ. If (30) is not binding,full employment obtains and the price level evolves so that equilibrium conditionp = p(θe ) is maintained. Let us denote by C(θe , p) aggregate demand for goods atstate (θ e , p). Then,e

C(θ , p)



 =y

 <y

15

if p = p(θe ),(31)
e

if p > p(θ ).

The inflation rate for a given state can be summarized as 0 ep (θ ) e
g(θ ) if p = p(θe ),e)p(θeπ(θ , p) =
 − δ if p > p(θ e ).

(32)

The representative household maximize (14) under budget constraint (15). Since

the final good production is competitive and it uses only labor as input, the nominal
income of the representative household coincides with the nominal aggregate demand
for goods, Wt = pt C(θte , pt ) for all t. The budget constraint can thus be written asṁt = C(θte , pt ) − π(θte , pt )mt + ct

(33)

as long as pt evolves continuously, and (17) if pt jumps. From (30), price level pt
never jumps down, and it jumps up only when θte jumps up to h(θte ) and pt is smaller
than the new market clearing price level, p(h(θte )). Let us denote the value functionof the household by U (θe , p, m), which now depends on the current value of p because
it aﬀects the aggregate demand and thus her income. The Bellman equation for thisproblem ishU (θ , p, m) = max u(c)∆t + (θ e ∆t)v(m00 )
e

c

ªi
©
1
(1 − θe ∆t)U (θe0 , p0 , m0 ) + (θ e ∆t)U (h(θe ), p00 , m00 ) ,+1 + ρ∆t(34)

where θe 0 = θe + g(θ e )∆t, p0 = p + π(θe , p)p∆t, m0 = m + (C(θe , p) − π(θe , p)m + c)∆t,p00 = max{p(h(θ)), p}, and m00 = (p/p00 )m. Taking the limit of ∆t → 0 in (34) yields
the HJB equation,h¡¢ρU = max u(c) + θe v(m00 ) + U (h(θ e ), p00 , m00 ) − U + g(θ e )Uθ
c
¡
¢ i
e
e
e
+ π(θ , p)pUp + C(θ , p) − π(θ , p)m − c Um ,(35)

where the arguments of function U (·, ·, ·) and its partial derivatives are abbreviatedc) = Um (θe , p, m),when they are (θ e , p, m). The first order condition for (35) is u0 (ewhere ec is the optimal amount of consumption. Then, the envelope condition is¡¢(ρ + π(θe , p) + θ e )Um =θe (p/p00 ) v 0 (m00 ) + Um (h(θe ), p00 , m00 ) + g(θe )Uθm
(36)
¡
¢
e
e
+ π(θ , p)pUpm + C(θ, p) − π(θ , p)m − ec Umm .
16

Note that the RHS of (36) depends on whether p jumps in the event of the liquidity
shock. As long as function p(·) is weakly downward sloping, p ≥ p(θe ) ≥ p(h(θe ))and therefore p00 = p and m00 = m. The following analysis focuses on this case andwe leave for Appendix the analysis of the case of p < p(h(θe )).
Substituting the first order condition, its time derivative, and the conditions forthe representative household, e
c = C(θe , p) and m = p−1 , into (36) yields the Euler

equation,Ãd 0u (C(θe , p))ρ − dt 0
u (C(θe , p))!

v 0 (p−1 )+ π(θ , p) = θ 0+ θeu (C(θe , p))
e

e

µ 0
¶
u (C(h(θ e ), p))− 1 (37)
u0 (C(θe , p))

for all t ∈/ S(0,∞) . Equation (37) has an interpretation similar to (27). The cost ofholding money, given by the LHS, is the sum of time preference and inflation. The
benefit is the sum of the direct utility gain and the expected capital gain measured
in terms of utility when a shock occurs and consumption jumps down.
Functions p(·) and C(·, ·) are determined so that equation (37) holds for all possible pairs of (θe , p). Let us first consider the case of full employment, where p = p(θ e ),C(θ e , p) = y and π(θe , p) = p0 (θe )g(θe )/p(θ e ) from (31) and (32). Substituting thesefor (37) gives a diﬀerential equation that determines the form of function p(·):
p0 (θ e ) =p(θe )γp (θe ),
g(θ)

where

v 0 (p(θe )−1 ) + u0 (C(h(θe ), p(θ e )))γp (θ ) = −(ρ + θ ) + θe.
u0 (y)e(38)

eFunction γp (θe ) in (38) represents the growth rate of pt ≡ p(θte ) as a function ofθte assuming that p ≥ p(θ e ). The correct expression for γp (θe ) when p < p(θe ) is
given by (43) in Appendix A. The diﬀerence between (28) and (38) lies in the factthat consumption is adjusted in the occurrence of the liquidity shock when nominal
stickiness exists, while adjustment is done fully by the price level when price iscompletely flexible. A boundary condition for function p(·) is given by the following
lemma.

17

Lemma 2 Under Assumption 2 and transversality condition (29), function γp (·)
must satisfy limθe →θ∗ γp (θe ) = 0
proof: in appendix
Suppose that limθe →θ∗ γp (θ e ) < 0, which implies that p(θe ) approaches 0 as θe converges to θ∗ . This means that the level of p must also converges to 0 since it evolves
according either to ṗ/p = p(θ e ) < 0 or to ṗ/p = −δ < 0 as θ e → 0. Lemma 1 showsthat the transversality condition is violated in this case. In addition, the limiting
value of γp (θ e ) as θe → θ∗ should not be positive since otherwise Assumption 2 isviolated. Thus, function p(·) must satisfy diﬀerential equation (38) and boundary
condition limθe →θ∗ γp (θ e ) = 0.
Next, consider the case of unemployment, where π(θe , p) = −δ. Substituting thisfor (37) gives a partial diﬀerential equation of function C(·, ·),u0 (C(θe , p))
γu0 (θe , p), whereu00 (C(θ e , p))v 0 (p−1 ) + u0 (C(h(θe ), p))
γu0 (θe , p) = ρ − δ + θe − θe.
u0 (C(θe , p))g(θe )Cθ (θe , p) − pδCp (θe , p) =(39)

In (39), γu0 (θe , p) is the rate of change in marginal utility. Combined with the boundary condition C(θ e , p(θe )) = y for all θ e , this partial diﬀerential equation determines
the shape of function C(·, ·) for all (θe , p) ∈ {(θ e , p)|p > p(θe ))}.
Since p(·) and C(·, ·) are interrelated as described above, they are determined
simultaneously so that they satisfy the system of partial diﬀerential equations, (38)and (39), along with two boundary conditions specified above. Figure 2 shows a
representative shape of function C(·, ·) in (θe , p) space, where the solid curve on theedge represents function p(·). Observe that function p(·) is downward sloping inθe . The reason behind it is the same as the reason for the property of p(·) in theprevious section: when θe is large, people’s liquidity preference is high and thus alow price level (a high relative price of money to goods) is required to equalize themoney demand to the money supply. The height of curved surface indicates thevalue of function C(θ e , p) at each state in region p > p(θ e ). C(θ e , p) is equal to y18

Figure 2: Representative shapes of function C(θe , p) and function p(θe )on the curve of p = p(θe ) and gets smaller as the pair (θ e , p) moves to the directionof north-east. That is, a pair of high θte and high pt implies a combination of highliquidity preference, a low relative price of money to goods, and a small supply ofreal money stock. In that case, the excess demand for money is huge and therefore
the aggregate demand for goods (and thus employment) is small.
The pattern of the dynamics of the economy is described as follows. If the current
price level pt is higher than the market clearing price level p(θte ), then price gradually
falls and consumption grows according to"#¡ −1 ¢
0
0 00
p
+
u
v
(c
)
u0 (ct )ṗttt= −δ, ċt = 00
ρ − δ + θe − θte,0ptu (ct )u (ct )

(40)

where c00t ≡ C(h(θe ), pt ). As long as no shock occurs, the pair (θe , pt ) follows (40)until it reaches the market clearing line pt = p(θte ) in a finite time. From that timeon, consumption stays constant and the price level rises so that the pair traces the19(a) phase diagram in (θe , p) space(b) phase diagram in (c, p) space

Figure 3: Inflation cycles with nominal stickinessmarket clearing line,0 00
v 0 (p−1
ṗtt ) + u (ct )= −(ρ + θte ) + θte,pt
u0 (y)

ct = y.

(41)

As θe approaches θ∗ , the price level converges to p∗ ≡ p(θ∗ ), and inflation rate
converges to zero.
Figure 3 shows the pattern of movement of (θe , p) pair. Once a liquidity shock
occurs, the pair (θe , pt ) jumps toward east. Since pt cannot jump immediately, theeconomy follows (40) until it reaches the market clearing line and then again follows(41). The liquidity shock may occur even before the economy goes back to the full
employment phase. If the shock occurs many times in a short while, θte changes
considerably while giving little time for the price level to adjust through deflation.
Then, consumption must decrease by large and, moreover, it takes long time for the
economy to regain full employment.
This problem becomes more serious if there had long been no shocks observed: inthat case, the price level have gone up to near the highest level p∗ ≡ p(θ∗ ) before theshock occurs and thus the process of adjustment becomes a long way. Conversely,
when an economy is experiencing the liquidity shock regularly, then pt should always
be near the lowest level pH ≡ p(θ H ) and thus it will take a short time for the priceto adjust even after an avalanche of shocks. This mechanism explains why once a20

(a) actual price level (pt ) and market clearing price (pt )(b) consumption

Figure 4: Simulated time paths of pt , pt ≡ p(θte ), and ct
long-time booming economy experiences large negative shocks it has to go througha long and deflationary period of depression, while recovery seems not so diﬃcult
when the boom period preceding that was a short one.

5

Conclusion

This paper presents a theory of inflation cycles and prolonged depression based onhouseholds’ learning behavior. The model is constructed in a number of steps. First,
we analyzed the process of Bayesian learning in continuous time by the representative
household who observes a shock that follows a certain Markov modulated Poissonprocess. Second, using Hamilton-Jacobi-Bellman equations, we investigated the evolutions of consumption and money holding of the household who behaves rationallybased on his belief about the state of the economy. Third, we derived a stationary
cycle under perfect price adjustments in terms of a delay diﬀerential equation (or,21sometimes called a diﬀerence-diﬀerential equation), and demonstrated that the pricelevel would experience sporadic downward jumps in such a setting. Forth, we extended the model to incorporate nominal stickiness and derived stationary dynamics.
In this case, since the belief and the slow-moving real variables cannot correspond
one-to-one, we obtained a system of partial delay diﬀerential equations, by which
we have shown that the economy experience a prolonged depression if and only if it
have enjoyed a long tranquil periods before it experiences successive occurrence ofshocks.

AppendixA

Analysis of the case of p < p(h(θe ))If pt < p(h(θte )), p00 = p(h(θTe )) and m00 = mp/p(h(θte )) in (34), (35) and (36). Substituting the first order condition, its time derivative, and the equilibrium conditions,ec = C(θe , p) and m = p−1 , into (36) yields the Euler equation,
!
Ã
d 0
e
u
(C(θ
,
p))
+ π(θe , p)
ρ − dt 0eu (C(θ , p))¶µpv 0 (p(h(θ e ))−1 )u0 (y)
p
e
e
=θ
+θ
−1
p(h(θe )) u0 (C(θe , p))p(h(θe )) u0 (C(θe , p))(42)

for all t ∈/ S(0,∞) . Substituting p = p(θe ), C(θ e , p) = y and π(θ e , p) = p0 (θ e )g(θe )/p(θe ),from (31) and (32), for (42) gives the growth rate of pt during the period of full employment:
γp (θe ) = −(ρ + θe ) + θ ep(θ e ) v 0 (p(h(θe ))−1 ) + u0 (y).
p(h(θ e ))u0 (y)(43)

When unemployment exists (i.e., p ≥ p(θ e )), the rate of change in marginal utility is
obtained by substituting π(θ e , p) = −δ for (42),γu0 (θe , p) = ρ − δ + θe + θe22pv 0 (p−1 ) + u0 (y).
p(h(θe )) u0 (C(θ e , p))

(44)

B

Proof of Lemmata

ns
ns
ns
Let θt,T, cns
t,T , pt,T and mt,T denote respectively the values of θT , cT , pT and mT

conditional on that no shock occurs between t and T . Then, the probability that no´³ R
Tshock occurs between t and T is given by exp − t θtns dτ .
The transversality condition (TVC) can be written as limT →∞ Et Vt,T = 0, where

Vt,T ≡ e−ρ(T −t) u0 (cT )mT and Et denotes the expectation taken upon the information
available at t. Since u0 (cT )mT ≥ 0 for all T ,
µ Z T
¶
ns
ns
θt dτ e−ρ(T −t) u0 (cns
Et Vt,T ≥ exp −t,T )mt,T ≡ Vt,T .

(45)

t

ns
is a deterministic variable given theNote that while Vt,T is a random variable, Vt,T

information available at t. From (45), a necessary condition for the TVC isns≤ 0.
lim Vt,T

T →∞

(46)

ns
Diﬀerentiating (45) with respect to T and using equilibrium condition mns
t,T = 1/pt,T

yield
ns
du0 (cns
dpns
dVt,T
/dT
t,T )/dTt,T /dT
ns
=
−θ
−
ρ
+
.
−
t,T
ns
Vt,T
u0 (cns
pns
t,T )
t,T

(47)

Lemma 1
0 ns
Without nominal stickiness, cns
t,T = y for all T and thus du (ct,T )/dT = 0. From (28),
ns
ns
(dpns
t,T /dT )/pt,T = γp (θT ). Substituting these into (47) yields
ns
ns
ns −1
dVt,T
p(θt,T) v 0 (p(h(θt,T)) ) + u0 (y) nsns= −θt,T
Vt,T .
nsdT
p(h(θt,T))u0 (y)

(48)

ns
ns
ns
) = pns
Using p(θt,Tt,T = 1/mt,T and the definition of Vt,T in (45), equation (48) reduces
RT
ns
ns
ns
ns
/dT = − exp(− t ρ+θt,v
dv)θt,TZ(h(θt,T)), where Z(θ) ≡ (v 0 (p(θ)−1 ) + u0 (y)) /p(θ).
to dVt,T

Integrating this diﬀerential equation with respect to T from t to ∞ and using thefact that Vt,tns = u0 (y)mt givens
= u0 (y)mt −lim Vt,T
T →∞

Z ∞
t

µ Z T
¶
ns
ns
ns
exp −
ρ + θt,v dv θt,TZ(h(θt,T)) dT.
t23(49)

Fix a small constant a > 0 and define a closed interval A ≡ [h(θ∗ ), h(θ∗ + a)] ∈(θ∗ , θH ). Note that Assumption 1 implies that Z(θ) ∈ (0, ∞) for all θ ∈ (θ∗ , θ H ).
In addition, it is continuous in this interval because (28) implies that p(·) is differentiable. Thus, there exist finite constants Zmin ≡ minθ∈A Z(θ) ∈ (0, 1) andns∈ (θ∗ , θ ∗ + a) for all
Zmax ≡ maxθ∈A Z(θ) ∈ (0, 1). Whenever θte ∈ (θ∗ , θ ∗ + a), θt,TT ≥ t. and therefore there is upper and lower bounds for the second term in the
RHS of (49), given byµθ∗ Zmin (θ∗ + a)Zmax,
ρ + θ∗ + a
ρ + θ∗¶≡ (Imin , Imax ) ⊂ (0, ∞).

(50)

Now suppose that limθe →θ∗ γp (θe ) < 0. Then as θte converges to θ ∗ , pt → 0 andtherefore mt → ∞. However, this violates the TVC since conditions (46), (49) and(50) imply that the TVC requires mt ≤ Imax /u0 (y) whenever θte ∈ (θ ∗ , θ∗ + a).
Suppose conversely that limθe →θ∗ γp (θe ) < 0. Then as θte converges to θ ∗ , pt → ∞ns< 0.
and therefore mt → 0. For suﬃciently small mt , (49) and (50) imply limT →∞ Vt,Tns
is continuous in T , there should be a value of T ≥ t
Since Vt,tns = u0 (ct )mt > 0 and Vt,Tnsns
= 0. From the definition of Vt,Tin (45) this implies that mns
such that Vt,Tt,T = 0 andns) = pns
therefore p(θt,Tt,T = ∞, violating Assumption 1.

Lemma 2
We first derive a contradiction under assumption limθe →θ∗ γp (θe ) < 0. Fix a > 0 anddefine A = [h(θ∗ ), h(θ∗ + a)]. Then, from p(θ) ∈ (0, ∞) and its continuity, there
exists pmin ≡ minθ∈A p(θ) ∈ (0, ∞). The assumption limθe →θ∗ γp (θ e ) < 0 implies thatp(θ) → 0 as θ → θ∗ . Recall, in addition, that pt falls at the rate of δ wheneverpt > p(θ∗ ). Thus, there is a positive probability that (θte , pt ) pair satisfies θte < θ∗ + a
and pt ≤ pmin when the liquidity shock does not occur for a suﬃciently long while.
Suppose that the current (θte , pt ) pair satisfies the above inequalities. Thenns
pnst,T < p(h(θt,T )) for all T ≥ t, which means that the analysis in Appendix A applies.

24Substituting the results obtained in Appendix A into (47) yields


ns
ns
ns
 − θt,Tns− ρ + γu0 (θt,T) + δ if pns
/dT
dVt,T
t,T > p(θt,T ),
=
ns

Vt,T
 − θns − ρ − γp (θns ) if pns = p(θns ).
t,T
t,T
t,T
t,T

(51)

ns
Substituting (43) and (44) into (51) and using pns
t,T = 1/mt,T and the definition of
RT
ns
ns
ns
ns
ns
in it, equation (51) reduces to dVt,T/dT = − exp(− t ρ + θt,vdv)θt,TZ(h(θt,T)),
Vt,T

where Z(θ) ≡ (v 0 (p(θ)−1 ) + u0 (y)) /p(θ). Integrating this diﬀerential equation withrespect to T from t to ∞ and using the fact that Vt,tns = u0 (cnst,T )mt givens
= u0 (cnslim Vt,Tt,T )mt −
T →∞

Z ∞
t

µ Z T
¶
ns
ns
ns
exp −
ρ + θt,v dv θt,TZ(h(θt,T)) dT.

(52)tns) ∈ A for all T ≥ t and that there exists a finite constantNote that h(θt,T0 ns
0Z max ≡ maxθ∈A Z(θ). From cns
t,T ≤ y, u (ct,T ) ≥ u (y) for all T . Thus (46) and (52)jointly imply that(θ∗ + a)Z max
mt ≤.
(ρ + θ ∗ )u0 (y)(53)

While assumption limθe →θ∗ γp (θe ) < 0 implies that an arbitrarily large mt = 1/pt
realizes with a positive probability, the RHS of (53) is constant. Thus (53) andhence the TVC will be violated with a positive probability.
Next, assume conversely that limθe →θ∗ γp (θe ) > 0, which means that p(θe ) becomensns∈ (θ ∗ , θ∗ + a) and pns
arbitrarily large as θe → θ∗ . Then, θt,Tt,T = p(θt,T ) > pmax ≡maxθ∈A p(θ) for suﬃciently large T . In this case, Analysis in Section 4 applies andnsfull employment obtains. From mns
t,T = 1/pt,T and (38),
ns
0
ns
ns
ns
v 0 (mns
dmnst,Tt,T )mt,T + u (C(h(θt,T ), 1/mt,T ))mt,T
ns
ns
)mns
−
θ
= (ρ + θt,T
t,T
t,T
dT
u0 (y)(54)ns) → ∞ and therefore mns
for suﬃciently large T . As T → ∞, p(θt,Tt,T → 0. In
ns 0−1limm→0 v 0 (m)m < 0, where thethis case, (54) implies limT →∞ dmns
t,T /dT < θT u (y)latter inequality follows from the definition of v(·). These properties jointly implyns
ns
that there is a finite T such that mns
t,T = 0 and therefore p(θt,T ) = pt,T = ∞, violating

Assumption 2.
25

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26