Discussion Paper No. 601

LEARNING, LIQUIDITY PREFERENCE,
AND BUSINESS CYCLE

Ryo Horii and Yoshiyasu Ono

March 2004

The Institute of Social and Economic Research
Osaka University
6-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan

Learning, Liquidity Preference, and Business Cycle∗
Ryo Horii†and Yoshiyasu Ono‡
February 2004

Abstract
This paper examines a mechanism of liquidity-preference fluctuations causedby changes in people’s belief about a random liquidity shock. When observing
the shock, they rationally update their belief so that the shock probability ishigher; consequently they raise liquidity preference and reduce consumption.
As the period without the shock lasts, they become more optimistic so thatthey gradually lower liquidity preference and increase consumption. The recovery pattern depends on the realized frequency of the shock: when the shock
occurs many times in succession, the consumption recovery is first slow, gradually accelerates and eventually slows down, tracing an ‘S’-shaped curve.

JEL Classifications: D83, E41, E32
Keywords: Bayesian Learning, Liquidity Preference, Precautionary Motive,
Business Cycle, Markov Switching∗

The authors are grateful to Kazuo Mino, Akihisa Shibata, and seminar participants at the

University of Tokyo for their helpful comments and suggestions.
†

Graduate School of Economics, Osaka University, 1-7 Machikaneyama, Toyonaka, Osaka 560-

0043, 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

Why do people hold money although money does not generate interest earnings? In
the literature a wide variety of reasons for liquidity holding have been discussed, suchas the transaction motive, the speculative motive, status preference, etc. This paper
focuses on the precautionary motive. People hold money since they prepare for a
possible aggregate liquidity shock, such as bank runs or sudden malfunction of thefinancial system. By observing whether or not such a shock actually occurs at eachpoint in time they rationally update the subjective probability of the shock, causing
their liquidity preference to fluctuate. The fluctuation in turn aﬀects the optimal
time path of consumption and generates a demand-driven business cycle. This paper
examines such a business-cycle mechanism.
If transactions are properly settled by checks or credit cards, people need not holdmoney stock. However, if a financial crisis occurs and checks or credit cards do notproperly work, holding money stock yields an explicit benefit. Bank runs can also
be regarded as another example of such liquidity shocks. As long as banks normallyoperate, people need not hold money since they can easily withdraw money wheneverthey buy something. Thus, at normal times holding money does not directly benefit
people. If a bank run occurs, however, people cannot withdraw money so that they
have to use the money that they hold. On this occasion holding money generates
utility.
Since such crises cannot exactly be anticipated, people hold money even when
it is unnecessary. Such liquidity preference depends on how exactly people know
about the probability of the shock at each point in time. If they know the exact
probability and it does not change, the expected gain of holding money is constantover time. Thus, they behave as if they had a deterministic utility function of moneyas originally assumed in Sidrauski (1967). If they do not know the exact probability,
whether the shock occurs or not today conveys some information for people to guess2

the probability of the shock tomorrow. More precisely, when people observe theshock, they update their subjective probability for meeting with it again and raiseliquidity preference. If the shock does not occur for a while, they gradually reduce
the probability and lower liquidity preference.
In order to highlight such a fluctuation in liquidity preference we impose a rather
extreme assumption: people receive utility from holding money only when the liquidity shock occurs. We also assume that there are two states with diﬀerent probabilitiesof the shock and that people do not know which state they are in. The shock follows
a Poisson process in each state. By observing whether or not the shock occurs at eachpoint in time people rationally update the subjective probability using Bayes’ law
and revise the time paths of consumption and money holding. If nominal wage adjustment is sluggish in this setting, employment also fluctuates over time as liquidity
preference varies. In this way we obtain a demand-driven business cycle.
In the literature of business cycles, there are a few attempts to explain cyclicalmovements of macroeconomic variables by combining unobservable regime changes
and Bayesian updating agents.1 Chalkley and Lee (1998) and Potter (2000) examine
how agents react to unobservable changes in an investment opportunity. Andolfatto
and Gomme (2003) and Sill and Wrase (1999) focus on the situation where monetary
policy, which is not directly observable, periodically switches between low and highmonetary growth. In these models time is discrete and in each period agents receive
a noisy signal with respect to the current state. Since the signal is noisy, agentsslowly change their belief, making the eﬀect of a regime change more persistent thanin the case where the state is perfectly observable.
Besides the diﬀerence in motivation, our specification diﬀers from them in thatagents receive a signal (namely, observe the liquidity shock) only at sporadic points in
1

Hamilton (1989) presents a diﬀerent but somewhat related approach when viewing businesscycles as regime changes. He assumes that it is not agents but econometricians that cannot directly
observe regime changes, and develops a procedure to estimate the timing of regime changes.

3continuous time and form their belief depending upon when and how frequently theshock occurs.2 As long as the liquidity shock occurs sparsely in time, people view theshock as a mere accident. Consumption decreases only by a small amount and thenquickly recovers. However, when the shock occurs many times for a short while,
people are convinced that they are in the more dangerous state and significantly
reduce consumption. In this case, the recovery process takes a long time because aperiod without the shock reveals information about the state little by little and thus
people gradually lower liquidity preference. The recovery of consumption is thenfound to be initially slow, gradually accelerates and again slows down, tracing an‘S’-shaped curve.3
After modelling the belief-updating behavior of households in the next section,
we examine liquidity-preference fluctuations and the optimal consumption behavior
in section 3. Section 4 derives the existence, uniqueness and other properties of thestationary equilibrium path and presents the cyclical movements of the belief andconsumption. Section 5 summarizes and concludes.

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
2

There are some continuous-time models in which agents update their belief based on discrete

signals although the unobservable state is time invariant. Zeira (1999) investigates bubbles in thestock market in which agents know that bubbles will crash some day but do not know exactly when.
Driﬃll and Miller (1993) examine the inertia of the inflation rate under sluggish nominal price adjustment when agents are learning about the probability of sporadic exchange-rate realignments. In
these models, however, uncertainty vanishes in the long run and recurrent cycles are not generated.
3This mechanism provides a possible explanation of why it took so long a time for consumer

confidence in Japan to recover after experiencing a succession of bank failures in the 1990s.

4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 tostate L with Poisson probability pH per unit time whereas state L changes to state
H with probability pL . We assume that the shock occurs much more frequently instate H than in state L and that the state change is a rare event when compared tothe shock in state H. Formally,
Assumption 1 θH − θ L > pH + 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,4 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 .
4

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.

5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 is5¢£¤ ¡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 ).

(3)

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 )equals6

¡
¢
H
H
1
−
(θ
+
p
)∆t
ft (θH ) + pL ∆tft (θL )ft+∆t (θH ) =.
1 − θte ∆t

5Throughout the paper we ignore the second-order term of ∆t and higher because ∆t → 0.

6ft+∆t (θ L ) is analogously obtained. From (1) it equals 1 − ft+∆t (θ H ).

6

From this equation we derive the time derivative of ft (θH ):dft (θH )ft+∆t (θH ) − ft (θ H )
= 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,(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),ft (θ H ) =θte − θ L,
θH − θLft (θL ) =θH − θte.
θH − θL(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)

Under Assumption 1, this function has an ‘U’-shape as illustrated in figure 1. The
figure shows that£¤g(θ) Q 0 ⇐⇒ θ R θ∗ for any θ ∈ θ L , θ H , where
p
L
H
L
H
θ
+
θ
+
p
+
p
−
(θH + pH − θL − pL )2 + 4pL pHθ∗ ≡∈ (θ L , θ H ).
2
7

(12)

Similarly, by substituting (9) and (10) into (5) we obtain the value of θte as aein the case where the shock does occur at time t.
function of θt−0

θte = θL + θH −θL θHe≡ h(θt−0)eθt−0

for t ∈ S(0,∞) .

(13)

As shown in Figure 2, h(θ) 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 to steady state value θ∗ as long as the shock does
not occur, but discretely jumps upward when it occurs.7 Intuitively, in the absenceof the shock people gradually become more and more optimistic and confident thatthe economy is in state L. Thus, their subjective probability of the shock gradually
declines and converges to θ ∗ .8 Due to the U-shape of function g(θte ), the speed ofadjusting their belief 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 ) is located above the 45-degree line in Figure 2,
the more often people observe the shock, the more strongly people believe that they
are in state H, and hence θte becomes closer to θH . In this way θte fluctuates betweenθ∗ and θ H .

7¤¡
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.

8 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.

8

3

Liquidity Preference and Consumption Behavior

In this section, we investigate how the aforementioned fluctuations in the representative household’s belief aﬀects her liquidity preference and consumption behavior.
Before doing that, we briefly describe the basic structure and production side of theeconomy.
The economy is inhabited by the representative households with measure one.
Each household is infinitely lived and supplies labor to a representative firm. The
firm inputs only labor to produce the commodity according to yt = yxt , where yt
and xt are respectively the level of output and that of labor input.9 Input-output
coeﬃcient y is assumed to be constant. Since there is no investment in our setting
and thus consumption ct equals total commodity demand, total labor demand xt is
xt = ct /y.

(14)

In this economy, money aﬀects real variables through the sluggishness of nominal
wage adjustment. Instead of explicitly introducing the adjustment cost of nominal
wages, we simply assume that there is a reduced-form relationship between labordemand xt and the rate of nominal wage adjustment Ẇt /Wt ,ω 0 (·) ≥ 0, ω(0) = 0,Ẇt /Wt = ω(xt − 1),(15)i.e., the rate of nominal wage adjustment is an increasing function of labor demandin excess of the ‘natural’ level, the latter being normalized to unity. Given Wt , theperfect adjustment of commodity price Pt always yields
Wt /Pt ≡ wt = y,9(16)Labor supplied by each household is potentially diﬀerentiated. In that case ‘yt = yxt ’ should

be considered as the outcome when the representative firm employs the same amount of labor fromeach household.

9

which shows that real wage wt is constant. Thus, from (15), the inflation rate isdetermined as a function of commodity demand ct :πt ≡ Ṗt /Pt = ω(ct /y − 1).

(17)

Next, we describe the demand side. We extend the money-in-the-utility-function
framework to include the random liquidity shock. The representative household gains
utility only from consumption ct when the shock does not occur. However, when it
occurs, she gains utility not only from consumption but also from money holding mt .
Her expected utility EUt is therefore given by


Z ∞
X
EUt = Et u(cτ )e−ρ(τ −t) dτ +βmτ e−ρ(τ −t)  ,t(18)

τ ∈S(t,∞)where ρ is her subjective discount rate, β is a constant specifying the marginal utilityof money holding, and S(t,∞) is the set of future dates on which the shock occurs.
Instantaneous felicity from consumption u(·) is twice diﬀerentiable and satisfies the
Inada conditions.
The household chooses assets among money and contingent claims for futurecommodities. However, since all households are identical, money is the only asset
that they hold after all arbitrage opportunities are exploited.10 We assume that
nominal money supply is constant and that there is no tax-cum-subsidy.11 Thus, theflow budget equation isṁt = wt xt − πt mt − ct .

(19)

Having the belief mentioned in the previous section, the representative householdchooses the time paths of consumption and money holding so as to maximize theexpected utility (18) subject to (19).
10

Since the profits of firms are always zero in the present setting, the ownership of firms has novalue.
11

The assumption of constant money supply is not essential. In fact, as shown below, the levelof money stock does not aﬀect the household behavior in the present setting.

10

Objective function (18) depends on only ct , mt and the expected pattern of theshock, the last of which is fully described by θte since θte is a suﬃcient statistic forθt that governs the current and future probabilities of the shock. Constraint (19)depends on wt , xt and πt , all of which are determined by only ct on the equilibrium path, as seen from (14), (16) and (17). Therefore, from the perspective of thehousehold that determines ct , the current status is fully summarized by mt and θte .
Thus, given the recursive structure of the model, the movement of ct on the pathof stationary dynamics must completely be expressed as a function of mt and θte .12
Furthermore, since objective function (18) and constraint (19) are both linear in mt ,the optimal choice of ct is independent of the level of mt .13 Thus, ct should be afunction of only θte :ct = C(θte ) for all t.

(20)

Since θte fluctuates within interval (θ∗ , θH ], as shown in the previous section, weonly need to characterize the shape of function C(θte ) in this interval.14 To this end
we examine the first-order conditions for the household’s optimizing behavior.
Let 1 − µ(θte )∆t denote the price of the claim to a unit of the commodity at t + ∆tmeasured in terms of the commodity at t under the condition that the shock does
not occur between t and t + ∆t. Note that it is a function of θte because the value

12This strategy for finding stationary dynamics is analogous to Lucas (1978) who analyses thedetermination of equilibrium price behavior under an exogenous production shock that follows a
Markov process.
13

The linearity of the utility function with respect to mt is assumed primarily for showing how the

fluctuation in liquidity preference aﬀects the consumption path of the utility-maximizing householdin the simplest setting. When the marginal utility of holding money is variable, we actually findthat the equilibrium dynamics of ct depends on both θte and mt . It substantially complicates theanalysis but does not aﬀect our main results, such as the pattern of recovery.
14

In the following, it is assumed that θte ∈ (θ ∗ , θ H ] unless otherwise noticed.

11

of the claim depends on the probability with which the contingent event occurs.15 If
the shock does not occur during the interval, consumption increases from C(θte ) toC(θte + g(θte )∆t) since θte changes by the amount of g(θte )∆t, as shown by (11). Since
the probability that the shock does not occur during this interval is 1 − θte ∆t, asgiven by (4), the first-order condition between the present and future consumption
under the condition that the shock does not occur is(1 − µ(θte )∆t)u0 (C(θte )) = (1 − θte ∆t)u0 (C(θte + g(θte )∆t))e−ρ∆t .
From this equation in which ∆t → 0 we deriveµ(θte ) = ρ + θte +C 0 (θte )γ(C(θte ))g(θte ) for all θte ,C(θte )

(21)

where γ(c) ≡ −u00 (c)c/u0 (c).
Analogously, let ν(θte )∆t denote the price of the contingent claim to a unit of
the commodity at t + ∆t under the condition that the shock does occur between t
and t + ∆t. When the shock occurs, consumption jumps from C(θte ) to C(h(θte )), as
seen from (13). Since the shock probability is θte ∆t, as shown by (8), the first-ordercondition between the present and the future consumption in this case isν(θte )∆t · u0 (C(θte )) = θte ∆t · u0 (C(h(θte )))e−ρ∆t .
By making ∆t → 0 in this equation we findν(θte ) =θte u0 (C(h(θte )))u0 (C(θte ))for all θte .

(22)

Next, let us consider the arbitrage between these contingent claims and a risk-free
asset. Let rt be the real interest rate of a risk-free asset, then the price of a risk-freeclaim to the future commodity at t + ∆t is e−rt ∆t . Since the claim is equivalent tothe asset of the synthesis of the claim contingent on the absence of the shock whose

15

Note also that consumption (which aﬀects the marginal utility of consumption) depends on θte .

12

price is 1 − µ(θte )∆t and that conditional on its presence whose price is ν(θte )∆t, theno-arbitrage condition requires
e−rt ∆t = 1 − µ(θte )∆t + ν(θte )∆t.
As ∆t → 0, it reduces to
rt = µ(θte ) − ν(θte ) for all t.

(23)

Substituting (21) and (22) into (23) yields
0

C (θte ) =

µ 0
·
¶¸
C(θte )u (C(h(θte )))ert − ρ + θt
−1 .
γ(C(θte ))g(θte )u0 (C(θte ))(24)

Applying (11) and (20) to (24) leads to the dynamics of ct (≡ C(θte )) while the shock
does not occur:
µ 0
·
¶¸
1
u (C(h(θte )))ċte=rt − ρ + θt−1ct
γ(ct )
u0 (ct )for t ∈/ S(0,∞) ,(25)

which is the Keynes-Ramsey rule in the present setting. Note that it is the same as
the standard one except for the third term in brackets of the right-hand side. This
term represents a jump in the marginal utility caused by the shock. If ct declines
after the shock, causing the marginal utility of consumption to increase, this term is
positive and thus the growth rate of ct is higher than would obtain in the standard
Ramsey model. That is, since the representative household anticipates a possible
increase in the marginal utility of consumption, she tries to reallocate consumption
from the present to the future, raising the growth rate of consumption during theperiod without the shock.
Having examined the household’s intertemporal optimization of consumption, we
now turn to the optimal choice between consumption and money holding. By holding
a one-unit money between time t and t + ∆t, the household loses (rt + πt )∆t units ofconsumption, or equivalently (rt +πt )u0 (ct )∆t units in terms of utility, when comparedto holding a unit of risk-free asset during this period. At this cost a one-unit increase13

in real money holding raises the household’s utility (18) by β if the shock occurs.
Since the subjective probability that the shock occurs between t and t + ∆t is θte ∆t,
the increase in the expected utility is βθte ∆t. The marginal benefit should equal the
marginal cost on the optimal path, which yields the first-order condition between
money holding and consumption:
rt + π t =βθteu0 (ct )for all t.

(26)

Equation (26) implies the well-known property–i.e., the marginal rate of substitution between consumption and real money holding equals the nominal rate of interest.
Substituting (17) into (26) and rearranging terms, the real interest rate is written as
a function of θte and ct :¶ct
− 1 ≡ R(θte , ct ), satisfyingyβRθ (θ e , c) ≡ ∂R(θ e , c)/∂θe = 0> 0 for all θte > 0 and ct > 0.
u (ct )βθtert = 0−ωu (ct )µFrom (24) and (27), the optimizing household behavior is summarized asµ 0·¶¸C(θte )u (C(h(θte )))
0 e
e
e
e
C (θt ) =R(θt , C(θt )) − ρ + θt
−1
γ(C(θte ))g(θte )u0 (C(θte ))

(27)

(28)

for all θte ∈ (θ∗ , θ H ].
Function C(·) is determined so that it satisfies diﬀerential equation (28). To pin
down C(·), however, we also need a boundary condition. If ċt /ct remains positive as
θte approaches steady-state value θ ∗ , then ct unboundedly explodes. Conversely, ifċt /ct remains negative as θte → θ ∗ , ct converges to zero, violating the transversality
condition. We rule out such paths by imposing a boundary condition:16¶µ 0
u (C(h(θe )))lim R(θ , C(θ )) − ρ + θ− 1 = 0,
θ e →θ ∗u0 (C(θ e ))eeeunder which ċt /ct given by (25) approaches zero as θte → θ∗ .
16

Throughout this paper, we use operator ‘lim’ to denote the right-hand limit.

14

(29)

Equations (28) and (29) determine the whole shape of C(θte ) within interval¡ ∗ H¤
θ , θ . Once it is determined, applying the dynamics of θte given by (11) and(13) to it provides the dynamic path of ct .

4Business Cycle

This section investigates the shape of the consumption path determined by (28) and(29). Note that the right-hand side of diﬀerential equation (28) includes C(h(θ))
along with θ and C(θ), implying that we cannot simply illustrate a phase diagram.17
Therefore, we theoretically examine basic properties of function C(θ) and then numerically obtain a typical shape of it.
For the tractability of the analysis we assume the following two properties regarding u(·) and ω(·), which are stated in terms of R(·) defined by (27).
Assumption 2 limc→0 R(θ H , c) < ρ and limc→∞ R(θ∗ , c) > ρ.
Assumption 3 Rc (θe , c) is continuous and positive for all θe ∈ [θ∗ , θ H ] and c > 0,
where Rc (θe , c) ≡ ∂R(θe , c)/∂c.
Intuitively, function R(·) represents the household’s preference for liquidity–i.e.,
the expected utility gain minus capital loss from holding money for a unit time,
as shown by (27). If it is stronger than her preference for present consumption ρ,
she postpones consumption and holds more money. Therefore, in Assumption 2
the former property implies that she prefers consumption to liquidity holding when
her consumption is quite low even if she expects the highest shock probability θ H .
The latter property implies that she prefers liquidity holding to consumption when
her consumption is suﬃciently high even if she expects the lowest shock probability

17

In mathematical term, this type of equation is called a diﬀerence-diﬀerential equation or a delay

diﬀerence equation.

15θ∗ . Assumption 3 extends this relationship between ct and R(·) to a smooth andmonotonic one under a given θte –i.e., as consumption increases, liquidity preference
rises as long as the state of expectation about the shock is unchanged.
Under the two assumptions we find the existence, and a few properties, of function
C(θ e ) that satisfies (28) and (29):
Proposition 1 Under Assumptions 2 and 3, there exists a unique function C(θ e )that satisfies (28) and (29). It is strictly downward sloping for any θe ∈ (θ∗ , θH ] andhas positive and finite upper and lower bounds c and c that are given byR(θ∗ , c) = ρ, R(θH , c) = ρ.

(30)

Proof. See Appendix A.
Given initial belief θ0e and the history of the liquidity shock S(0,t] , the path of θteis uniquely determined by (11) and (13). Therefore, the uniqueness of function C(·)
implies that of the consumption path: ct = C(θte ). The negative relationship between
θte and ct provides an intuitive figure of the dynamics. Figure 1 shows θ̇te < 0 for
any (θ∗ , θ H ], implying that subjective shock probability θte gradually declines while
the shock does not occur. Thus, preference for liquidity gradually decreases andconsumption grows. When the shock occurs, people discretely increase θte to h(θte ),as illustrated in Figure 2, causing a negative jump in consumption to occur. If theshock does not occur for a while, people again gradually become optimistic and raise
consumption. In this way, consumption persistently fluctuates within finite interval[c, c].
To obtain a typical shape of function C(θe ) and the dynamics of consumption
more clearly, we numerically analyze the dynamics by assumingu(ct ) = log ct and ω(xt − 1) = α · (xt − 1) where α is constant.

16

(31)

In this setting, Assumptions 2 and 3 reduce to the following:18βyθ ∗ > α,

ρ > α.

In the numerical calculation we choose parameter values so that these conditions aswell as Assumption 1 are satisfied.19
Figure 3 illustrates the shape of C(θte ) obtained from the numerical analysis. It
is in fact downward sloping from c∗ ≡ C(θ ∗ ) to cH ≡ C(θH ). If the shock does notoccur while θte moves from θH to θ∗ according to equation (11), C(θte ) moves along
the solid curve in Figure 4. Note that consumption first grows slowly, gradually
accelerates, and eventually slows down again as it approaches c∗ –i.e., it traces an
S-shaped trajectory.
The intuition behind is clear from the U-shape of function g(θte ) as depicted in
Figure 1. If people strongly believe that they are in state H and thus θte is very closeto θ H , they do not significantly alter their pessimistic view for a while. In fact, thespeed of change in θte , given by |g(θte )|, is then small and thus consumption increases
very slowly. As the period without the shock lasts, θte decreases and |g(θte )| increases,
as Figure 1 shows. People become more and more optimistic and thus θte declines
faster, which accelerates the recovery speed of consumption. As θte approaches tosteady-state value θ∗ , people become quite confident that they are in state L, andhence an additional period without the shock provides little information. |g(θte )|approaches zero and the growth rate of consumption converges to zero.
18

Using a money-in-utility model without uncertainty Ono (1994, pp.86-88; 2001) shows that in

the case where βy > α and ρ > α there is a unique saddle-stable path. Furthermore, the pathaccommodates a persistent demand shortage when ρ < βy, whereas it reaches a full-employment
steady state when ρ > βy. The present condition is the same as his condition except that theformer includes shock probability θ ∗ .
19Specifically, θH = .4, θL = .05, pH = .025, pL = .1, y = 1, ρ = .05, α = .025, and β = .4.

Under these parameter values, we obtain θ ∗ ≈ .069, c∗ ≈ 1.3 and cH ≈ .60. The details of thenumerical procedure are described in Appendix B.

17

Once the liquidity shock occurs, however, their consumption jumps downward
since the subjective shock probability jumps upward, raising their liquidity preference. Consumption falls from C(θte ) to C(h(θte )), the latter being depicted by thedashed curve in Figure 4 (where vertical arrows express the magnitude of each fall).
Thereafter the recovery process ‘restarts’ from the point that corresponds to the decreased level of consumption (as indicated by horizontal arrows) and consumption
again traces the solid curve. If the shock continually occurs for a short period, thesubjective probability successively increases and C(θte ) approaches the lowest level
cH . Thereafter, consumption recovers along the S-shaped trajectory, as mentioned
above.
Finally, by simulating the Markov process of the underlying state and the Poissonprocess for the shock, we numerically obtain an example of the realized time pathsof θt , θte and ct . Figure 5 illustrates them. Consumption in fact traces an S-shaped
path especially after it vastly declines. Since the inflation rate is given by (17) andy is located between c∗ and cH under the present parameter values, in the recovery
process serious deflation initially occurs, then its rate reduces, and eventually a boom
comes and inflation arises after c exceeds y. If the shock does not frequently occur
and hence consumption does not seriously decline, the recovery process starts before
deflation occurs.
It is also worth noting that realized booms and depressions do not exactly matchthe underlying state of the economy but follow the subjective probability that people have in mind. Even when the economy switches to state H and thus the trueprobability of the shock jumps up, people do not increase money holding until they
actually observe it. Analogously, even if the true shock probability jumps down, they
still keep strong liquidity preference and thus the recovery speed is very slow for awhile once they become very pessimistic.

18

5

Conclusions

Liquidity preference depends on people’s belief about how frequently they encountercrises in which liquidity is needed. This paper has examined the way they update
the belief based on Bayesian inference and its eﬀect on their preference for liquidity
holding over consumption, in a circumstance where the economy shifts between two
unobservable states with diﬀerent probabilities of the liquidity shock. Each timethey observe the shock, they raise their subjective probability of being in the moredangerous state and increase preference for holding money over consumption. The
longer the period without the shock lasts, the larger probability people attach to thesafer state and increase preference for consumption over money holding. With incomplete nominal wage adjustment, such movements in liquidity preference generatedemand-driven business cycles.
The magnitude and persistence of fluctuations in consumption depend on therealized frequency of the shock, which does not necessarily match the underlying
state of the economy. As long as the shock occurs sparsely in time, it has only aminor eﬀect on the belief and economic recovery after it is fast. However, if people
observe the shock many times for a short while, they hold a strong belief of being in
the more dangerous state and reduce consumption a lot. Once it occurs, it takes along time for them to reverse their belief and increase consumption. In this process,the recovery speed is first slow, then gradually accelerates, and eventually declines,
tracing an S-shaped curve.

19

Appendix A: Proof of Proposition 1
Proof of C(θ) to be downward sloping20
Before starting the proof we define D(θ):µ 0¶
u (C(h(θ)))D(θ) ≡ R(θ, C(θ)) − ρ + θ−1 .
u0 (C(θ))(32)

Since D(θ) is the expression in brackets of (28) and (11) shows g(θ) to be negative
for all θ ∈ (θ ∗ , θH ],C 0 (θ) S 0 ⇐⇒ D(θ) T 0.

(33)

Using function D(θ) we first prove
Lemma 1 Suppose that C(θ) satisfies (28) and that there exists θ0 ∈ [θ ∗ , θH ) satisfying limθ→θ0 D(θ) = 0. Then, under Assumption 3, C(θ) is strictly downward
sloping for all θ ∈ (θ 0 , θH ].
Proof: If Lemma 1 does not hold and hence C(·) is weakly upward sloping somewherein (θ0 , θH ], either of the following must be the case.
(i) There exists some θA ∈ [θ0 , θH ) such that limθ→θA D(θ) = 0 and C 0 (θ) ≥ 0 for
all θ ∈ (θA , θH ].
(ii) There exist some θA ∈ [θ 0 , θH ) and θB ∈ (θ A , θH ) such that limθ→θA D(θ) =D(θB ) = 0, C 0 (θ) ≥ 0 for all θ ∈ (θA , θB ], and C 0 (θ) ≤ 0 for all θ ∈ (θB , θH ].
Intuitively, if the lemma is false, we can choose interval (θ A , θB ] in which function
C(θ) is weakly increasing, θA is either a local minimum or θ 0 , and θ B is either a local
maximum or θH . If there are multiple intervals of such, we choose the rightmost one.
We shall find neither (i) nor (ii) to be valid.

20

In Appendices A and B we use θ instead of θte to minimize notation.

20

We first show that case (i) leads to a contradiction. Since Rc > 0 from Assumption
3 and Rθ > 0 from (27), in case (i)lim R(C(θ), θ) < R(C(θH ), θH ).

θ→θ A

(34)

Since h(θA ) ∈ (θA , θH ), limθ→θA C(θ) ≤ C(h(θA )) whereas C(θH ) = C(h(θH )) sinceh(θH ) = θ H from (13). Since applying these properties and (34) to (32) implies 0 =limθ→θA D(θ) < D(θH ) in case (i), from (33) we find C 0 (θH ) < 0, which contradicts(i).
In case (ii) limθ→θA C(θ) ≤ C(θ B ). Since Rc > 0 from Assumption 3 and Rθ > 0
from (27), this inequality implies
lim R(C(θ), θ) < R(C(θ B ), θB ).

θ→θ A

(35)

Further, h(θA ) is located in either (θ A , θ B ] or (θB , θH ). If h(θA ) ∈ (θA , θ B ],then C(h(θA )) ≥ limθ→θA C(θ) since we suppose C 0 (θ) ≥ 0 for all θ ∈ (θA , θ B ].
Contrastingly, C(h(θ B )) ≤ C(θ B ) since h(θB ) ∈ (θB , θH ) and C 0 (θ) ≤ 0 for allθ ∈ (θB , θH ]. Using these inequalities, (32) and (35) we find limθ→θA D(θ) < D(θB ),which contradicts case (ii).
If h(θA ) ∈ (θ B , θ H ), then because h0 (θ) > 0 from (13), we find θA < θB < h(θA ) <h(θB ) < θH . In case (ii), this means
lim C(θ) ≤ C(θB ) ≥ C(h(θ A )) ≥ C(h(θB )).

θ→θ A

Thus,

µ
¶
u0 (C(h(θ)))
u0 (C(h(θB )))− 1 ≥ max 0, lim−1 .
u0 (C(θB ))θ→θA u0 (C(θ))

Applying this property and (35) to (32) yields limθ→θA D(θ) < D(θB ), which contradicts case (ii). Thus, anyway case (ii) results in a contradiction.

¥

From (32), boundary condition (29) is equivalent to limθ→θ∗ D(θ) = 0. By regarding θ∗ as θ 0 in Lemma 1, we find C(θ) that satisfies (28) and (29) to be strictly
downward sloping for all θ ∈ (θ ∗ , θH ].
21

The existence of upper and lower bounds for C(θ)
We first show c and c to be unique and well defined. Since θ∗ < θ H and Rθ > 0 from(27), under Assumption 2
lim R(θ∗ , c) ≤ lim R(θH , c) < ρ < lim R(θ ∗ , c) ≤ lim R(θH , c).

c→0

c→0

c→∞

c→∞

Applying this property and Assumption 3 to the intermediate value theorem implies that there are unique and positive c and c satisfying (30). Furthermore, sinceR(θH , c) > R(θ∗ , c) = ρ = R(θH , c), Assumption 3 implies c > c.
Next, we prove the following lemma.
Lemma 2 Suppose that C(θ) satisfies (28) and that there exists θ0 ∈ [θ ∗ , θH ) satisfying limθ→θ0 D(θ) = 0. Then, under Assumptions 2 and 3, C(θ) ∈ [c, c] for allθ ∈ (θ 0 , θH ].
Proof: As shown by Lemma 1, the last term in (32) is positive when θ → θ0 andhence
lim R(θ, C(θ)) < ρ.

θ→θ 0

(36)

From the first equation of (30), the positivity of Rθ given by (27), and Assumption
3, R(θ, c) > ρ for all θ ∈ [θ ∗ , θH ] and c > c. Thus, (36) implies limθ→θ0 C(θ) ≤ c.
When θ = θH , the last term in (32) equals zero since h(θ H ) = θ H from (13). Since
C 0 (θ) = D(θ)/g(θ) from (28) and (32), C 0 (θ H ) ≤ 0 from Lemma 1, and g(θH ) < 0
from (11), we find D(θH ) ≤ 0. Applying these properties to (32) yields
R(θH , C(θH )) ≥ ρ.
Comparing this property with the second equation of (30) and using Assumption
3 yield C(θH ) ≥ c. Furthermore, the monotonicity of C(θ) from Lemma 1 implies
C(θ) ∈ [C(θH ), limθ→θ0 C(θ)] ⊆ [c, c] for all θ ∈ (θ0 , θ H ].

¥Under condition (29), θ∗ satisfies the requirement for θ0 in Lemma 2. Thus,
C(θ) ∈ [c, c] for all θ ∈ (θ∗ , θ H ].
22

The uniqueness of C(θ)
e cH ) be the solution to diﬀerential equation (28) that satisfies the following
Let C(θ,boundary condition:e H , cH ) = cH ,
C(θ

(37)

where cH (> 0) is an arbitrary constant. We can solve diﬀerential equation (28)backward from θ = θH with boundary condition (37), because h(θ) is larger than θand thus C(θ) and C(h(θ)) are already known when we calculate the gradient of C(·)
e cH ) is uniquely determined within interval (θ∗ , θ H ].
at θ.21 Therefore, function C(θ,e
Using function C(·),
boundary condition (29) can be rewritten ase cH ) = 0, wherelim D(θ,θ→θ ∗e c )) − ρ + θe c ) ≡ R(θ, C(θ,
D(θ,
H

H

Ã

!
e
u0 (C(h(θ),
cH ))
−1 .
e cH ))
u0 (C(θ,

(38)
(39)

e cH ).
e cH ) is the expression in brackets of (28) with C(θ) being replaced by C(θ,D(θ,e and D(·)
e have the following properties:22
Functions C(·)

ec (θ, cH ) > 0 and (b) D
e c (θ, cH ) > 0 for all θ ∈Lemma 3 Under Assumption 3, (a) C

e c (θ, cH ) >(θ∗ , θH ] and cH > 0. In addition, (c) there is a constant, Dc , such that D
e cH ) ∈ [c, c].
Dc > 0 whenever C(θ,Proof: By rearranging terms in (28),HHeee cH ) = g(θ) γ(C(θ, c ))Cθ (θ, c ) = −g(θ)Mθ (θ, cH ),D(θ,e cH )
C(θ,
´
³
H
0 e
H
where M (θ, c ) ≡ ln u C(θ, c ) .

(40)
(41)

Diﬀerentiating (40) with respect to cH yields

21

e c (θ, cH )/g(θ).
Mθc (θ, cH ) = −D

The numerical analysis follows this way. See Appendix B for it.

22 e
e cH )/∂cH . D
e c (θ, cH ) and other partial derivatives are defined likewise.
Cc (θ, cH ) ≡ ∂ C(θ,

23

(42)

Diﬀerentiating (39) with respect to cH givese c (θ, cH ) = Φ(θ, cH ) + Ψ(θ, cH ), whereD

e cH ))
u0 (C(θ,HeΦ(θ, c ) ≡ Rc (θ, C(θ, c ))Mc (θ, cH ),00
Heu (C(θ, c ))e¢u0 (C(h(θ),
cH )) ¡Mc (h(θ), cH ) − Mc (θ, cH ) .
Ψ(θ, cH ) ≡ θe cH ))
u0 (C(θ,H

ec (θH , cH ) = 1, diﬀerentiating (41) with respect to cH yields
Since (37) implies C
Mc (θ H , cH ) = u00 (cH )/u0 (cH ) < 0.

(43)

(44)

From (13), h(θH ) = θH and thus Ψ(θ H , cH ) = 0. Using this property, Assumption 3,(37), (43) and (44) we obtaine c (θ H , cH ) = Rc (θ H , cH ) > 0.
D

Now we extend this property to all θ ∈ (θ∗ , θH ]. To prove this, suppose otherwise.

e c (θ, cH ) > 0 for all θ ∈Then, there should be some θA ∈ (θ∗ , θ H ) that satisfies D

e c (θA , cH ) ≤ 0. This property, combined with (42), (44) and the(θA , θ H ] and D
negativity of g(θ) from (11), gives
A

H

H

H

Mc (θ , c ) = Mc (θ , c ) +

Z θH
θA

00

H

0

H

< u (c )/u (c ) < 0.

e c (θ, cH )/g(θ) dθ
DAssumption 3, (43) and (45) imply Φ(θ A , cH ) > 0. Similarly, from (42)Z h(θA )A
H
A H
e c (θ, cH )/g(θ) dθ > 0,
Mc (h(θ ), c ) − Mc (θ , c ) = −
D

(45)

(46)

θA

e c (θ A , cH ) >which means Ψ(θA , cH ) > 0. Substituting these results into (43) yields D
e c (θA , cH ) ≤ 0. Thus there is no such θA ,0, which contradicts the assumption that D
and therefore property (b) holds.
Property (b) and (42) implyMθc (θ, cH ) > 0 for all θ ∈ (θ∗ , θH ] and cH > 0.
24

(47)

From (44) and (47),Mc (θ, cH ) < u00 (cH )/u0 (cH ) < 0 for all θ ∈ (θ ∗ , θH ].

(48)

ec (θ, cH ) > 0 from (41), property (a) holds.
Since (48) is equivalent to C
Finally we prove property (c). Define

u0 (c)u00 (cH )Φ(θ, c) ≡ Rc (θ, c) 00.
u (c)u0 (cH )From Assumption 3, Φ(θ, c) is positive and continuous for all (θ, c) ∈ Θ ≡ [θ∗ , θ H ] ×[c, c]. Since Θ is a compact set, there exists Dc ≡ min(θ,c)∈Θ Φ(θ, c) > 0. Combinedwith (43), (47) and (48), this property implies
0 e
H
00 H
e cH )) u (C(θ, c ))u (c ) ≥ Dc
e c (θ, cH ) > Φ(θ, cH ) > Rc (θ, C(θ,D
e cH ))u0 (cH )u00 (C(θ,e cH ) ∈ [c, c].
whenever C(θ,¥We now prove the uniqueness of function C(θ) using Lemmata 2 and 3. Supposethat there are two distinct functions C1 (θ) and C2 (θ) both of which satisfy (28) and
H
H
H
H
e
(29). Let cH
1 ≡ C1 (θ ) and c2 ≡ C2 (θ ). Then C1 (θ) = C(θ, c1 ) and C2 (θ) =
∗ H
H
H
e cH
C(θ,
2 ) for all θ ∈ (θ , θ ]. Note that c1 6= c2 because we have assumed that C1 (θ)

and C2 (θ) are distinct functions. Since (29) is equivalent to (38), both functions
satisfye cH ) = lim D(θ,e cH ) = 0.
lim D(θ,
1
2
∗

θ→θ ∗

θ→θ

(49)

e cH ) ∈ [c, c] for all θ ∈ (θ ∗ , θH ]. Applying it toe cH ), C(θ,From Lemma 2, C(θ,12He cH ) ∈ [c, c] for all cH ∈ [cH
property (a) of Lemma 3 implies C(θ,1 , c2 ] and all θ ∈(θ∗ , θH ]. Thus, we can use property (c) of Lemma 3 to obtaine cH ) − D(θ,e cH )| > |cH − cH |Dc > 0,
lim |D(θ,
1
2
1
2

θ→θ∗

which contradicts (49).

25

(50)

The existence of C(θ)
e cH ).
We obtain another property with respect to D(θ,e c) < 0 and (b) D(θ,e c) > 0 for all
Lemma 4 Under Assumptions 2 and 3, (a) D(θ,θ ∈ (θ ∗ , θH ).

Proof: We first prove property (a). From (27) and (30),
R(θ H , c) = ρ and R(θ, c) < ρ for all θ ∈ (θ ∗ , θH ).

(51)

Since (13) implies h(θ H ) = θ H , substituting (37) and (51) into (39) yields
e H , c) = 0.
D(θ

(52)

eθ (θ H , c) = 0, and Rθ > 0 from (27), diﬀerentiating (39)
Since (40) and (52) imply Cwith respect to θ when θ = θH givese θ (θH , c) = Rθ (θH , c) > 0.
D(53)e c) < 0 for
Equations (52) and (53) show that there is a small ε (> 0) such that D(θ,all θ ∈ (θ H − ε, θ H ).

e c) to the whole interval of (θ∗ , θ H ). To see
We now extend the negativity of D(θ,e c) <
this, suppose otherwise. Then, there must be some θA ∈ (θ∗ , θ H ) such that D(θ,e A , c) ≥ 0. Since D
e < 0 ⇔ Ceθ > 0 from (40)0 for all θ ∈ (θA , θ H ) and that D(θAee H , c) = c.
e A , c) < C(h(θ), c) < C(θand the negativity of g(θ) in (11), we obtain C(θe A , c)) < R(θA , c) < ρ
With Assumption 3 and (51), these inequalities yield R(θ A , C(θe A , c) < 0, which is a contradiction.
and hence from (39) D(θNext we prove property (b). From Assumptions 2 and 3,R(θ, c) > ρ for all θ ∈ (θ ∗ , θH ].

(54)

e H , c) > 0 from (39) since h(θ H ) = θH from (13). To prove that this
It implies D(θinequality actually holds for whole (θ∗ , θH ], suppose otherwise. Then, there should be26e c) > 0 for all θ ∈ (θA , θH ] and that D(θe A , c) ≤ 0.
some θA ∈ (θ ∗ , θH ) such that D(θ,Aee H , c) = c. With Assumption 3 ande A , c) > C(h(θ), c) > C(θFrom (33), we find C(θe A , c)) > R(θA , c) > ρ and therefore from (39)(54), these inequalities imply R(θ A , C(θe A , c) > 0, which is again a contradiction.
D(θ¥Applying property (b) of Lemma 3 and Lemma 4 to the intermediate value the-orem assures that, for any given θ 0 ∈ (θ∗ , θ H ), there uniquely exists cH ∈ [c, c] thate 0 , cH ) = 0. That is, there is a unique function, cH = ζ(θ0 ), satisfying
satisfies D(θe 0 , ζ(θ 0 )) = 0 for all θ 0 ∈ (θ∗ , θH ).
D(θ

(55)

e ζ(θ 0 )) is monotonic and bounded by [c, c]
From Lemmata 1 and 2, function C(θ,within interval θ ∈ (θ0 , θH ]. By taking limit as θ 0 → θ ∗ , we conclude that C(θ) =e cH∗ ) is monotonic and bounded for all θ ∈ (θ∗ , θ H ], where cH∗ ≡ limθ0 →θ∗ ζ(θ0 ).23
C(θ,This implies ct should neither explode, implode nor oscillate. Thus ċt /ct → 0 as
θte → θ∗ , which gives the validity of boundary condition (29).

Appendix B: Numerical procedure of finding C(·)
Our problem is generally called an IVP (initial value problem), which is usually
solved by finite diﬀerence methods, such as the Runge-Kutta method and the Euler
method.24 However, we cannot use them since the right-hand side of (28) contains
C(h(θ)), which makes impossible to calculate the gradient of C(θ) before C(h(θ))
is determined. Since h(θ) is always larger than θ, we cannot solve the diﬀerential
equation forward from θ∗ , where the boundary condition is given, toward θH . Instead,
23

Note that, ζ(θ) is bounded by c and c from Lemma 2. In addition, by totally diﬀerentiating(55) and utilizing Lemma 1, we can prove that ζ(θ) is continuous and monotonic for all θ ∈ (θ ∗ , θH ].
It means that ζ(θ0 ) does not oscillate as θ0 → θ ∗ , hence the existence of cH∗ ≡ limθ0 →θ∗ ζ(θ0 ) is
guaranteed.
24

See, for example, Judd (1998).

27

we can solve it backward from θH toward θ∗ , during which C(h(θ)) is already known
when we calculate the gradient of C(θ).
This strategy, however, involves another diﬃculty because the value of the function at the starting point, cH ≡ C(θH ), is not predetermined. Thus, we have to find
an appropriate initial value cH such that boundary condition (29) is eventually met
when (28) is solved from it.25 This method is in fact used when we prove proposition
1 in appendix A–i.e., we show that there is a unique cH∗ ∈ [c, c] that satisfies thisproperty. In the numerical analysis we calculate cH∗ in the following way:
Step 1. Let i = 0, h0 = c and l0 = c.
Step 2. Let cH
i = (hi +li )/2. Using the Euler method, solve diﬀerential equation (28)(to which (31) is applied) backward starting from boundary value C(θH ) = cHi .
Step 3. If C(θ) exceeds c during the calculation, or if D(θ) defined by (32) remains
0
positive when θ approaches θ ∗ , let hi+1 = cH
i and li+1 = li . Conversely, if C (θ)

becomes positive during the calculation or if D(θ) remains negative when θHapproaches θ∗ , let hi+1 = hi and li+1 = cH
i . Otherwise, ci is the solution.

Step 4. Let i = i + 1.
Step 5. Repeat steps 2-4 until hi and li get suﬃciently close to each other. Thenadmit cH = (hi + li )/2 as the solution.

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[2] Chalkley, Martin and In Ho Lee (1997), “Learning and Asymmetric Business
Cycles,” Review of Economic Dynamics, 1, 623-645.
25

This method is usually called ‘monkey hunting’.

28

[3] Driﬃll, John and Marcus Miller (1993), “Learning and Inflation Convergence inthe ERM,”Economic Journal, 103, 369-378.
[4] Hamilton, James D. (1989), “A New Approach to the Economic Analysis of
Nonstationary Time Series and the Business Cycle,” Econometrica, 57(2), 357384.
[5] Judd, Kenneth L. (1998), Numerical Methods in Economics, The MIT Press.
[6] Lucas, Robert E. Jr. (1978), “Asset Prices in an Exchange Economy,” Econometrica, 46(6), 1429-1445.
[7] Ono, Yoshiyasu (1994), Money, Interest, and Stagnation, Oxford University
Press.
[8] Ono, Yoshiyasu (2001), “ A Reinterpretation of Chapter 17 of Keynes’s General
Theory: Eﬀective Demand Shortage Under Dynamic Optimization,” International Economic Review, 42, 207-236.
[9] Potter, Simon M. (2000), “A Nonlinear Model of Business Cycle,” Studies in
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[10] Sidrauski, Miguel (1967), “Rational Choice and Pattern of Growth in a Monetary Economy,” American Economic Review, 57(2), 534-544.
[11] Sill, Keith and Jeﬀ Wrase (1999), “Exchange Rates, Monetary Policy Regimes,
and Beliefs,” Federal Reserve Bank of Philadelphia Working Paper, 99-6.
[12] Zeira, Joseph (1999), “Informational Overshooting, Booms, and Crashes,” Journal of Monetary Economics, 43, 237-257.

29

µ_et
µH

µ¤
0

µ

L

g(µet )

Figure 1: Movement of µet in absence of the shock

µet

µet
µH
e
h(µt¡0
)

h(µ¤)

0

L

µ µ

¤

µ

H

µet¡0

Figure 2: Movement of µet when the shock occurs

cc¤
C(µe )cHµ¤µH

Figure 3: The shape of function C(µet )

µe

c
c¤
C(µe )

C(h(µe ))cH
0

10

Figure 4: Dynamics of consumption

20

time

µH

µt
µet

µL
0

c¤
y¹

25

50

75

100

25

50

75

100

ct

cH
0

Figure 5: Realized paths of µt , µte and ct