ISSN (Print) 0473-453X

Discussion Paper No. 1085

ISSN (Online) 2435-0982

FINANCIAL CRISISAND SLOW RECOVERY
WITH BAYESIAN LEARNING AGENTS

Ryo Horii
Yoshiyasu Ono

March 2020

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

Financial Crisis and Slow Recovery
with Bayesian Learning Agents∗
Ryo Horii† and Yoshiyasu Ono‡
March 31, 2020

Abstract
In a simple continuous-time model where the learning process affects the willingness to hold liquidity, we provide an intuitive explanation of business cycle asymmetry and post-crisis slow recovery. When observing a liquidity shock, individuals
rationally increase their subjective probability of re-encountering it. It leads to an
upward jump in liquidity preference and a discrete fall in consumption. Conversely,
as a period without shocks continues, they gradually decrease the subjective probability, reduce liquidity preference, and increase consumption. The recovery process
is particularly slow after many shocks are observed within a short period because
people do not easily change their pessimistic view.
Keywords and phrases: Bayesian Updating, Liquidity Preference, Markov Switching, Asymmetric Cycles, Persistence
JEL Classifications: E32, E41, D83.

∗

The authors are grateful to V.V. Chari, Kazuo Mino, Akihisa Shibata, and seminar participantsat the University of Tokyo, the Tokyo Institute of Technology, the Delhi School of Economics, and theSummer Workshop on Macroeconomics held at the University of Auckland for their helpful comments andsuggestions. This research is supported by JSPS KAKENHI Grant 15H01948, 15H05728, 15H03329 and
16K13353, and partly funded by the Behavioral Economics Research Center at ISER, Osaka University.
An earlier version of this paper was circulated as “Learning, Liquidity Preference, and Business Cycle.”†

Correspondence: ISER, Osaka University, 6-1 Mihogaoka, Ibaraki-shi, Osaka, 567-0047, JAPAN; E-

mail: horii@econ.jpn.org‡

ISER, Osaka University; E-mail ono@iser.osaka-u.ac.jp1

1

Introduction

This paper considers a situation where economic agents hold money to prepare for a possible aggregate liquidity shock, such as the global financial crisis of 2008 and the recent
Coronavirus shock of 2020. In normal times, the benefit of holding liquidity is limited because agents can settle transactions by relying on short-term credits, such as credit cards.
However, once a financial crisis occurs, and obtaining credits becomes difficult, holding
money stock yields an explicit benefit. Bank runs are another example of such liquidity
shocks. Usually, people need not hold much cash since they can easily withdraw deposits.
When a bank run occurs, cash has advantages over deposits. However, since people cannot
foresee when a crisis occurs, they have some incentive to hold money all the time.
The expected benefit of holding money depends on the subjective probability of encountering liquidity shock. The probability is not necessarily constant over time.1 For
example, during the global financial crisis of 2007-2008, many banks failed in succession,and the shortage of liquidity occurred across the globe, one after another. By observing
the situation, firms and individuals must have learned that the economy is more likely tobe hit by another shock in the near future, and therefore have incentives to hold more liquidity in preparation. Similar patterns happened during the collapse of bubbles in Japanin the 1990s, European debt crisis of 2010-2012, and also in the recent crisis caused byCoronavirus (COVID-19) pandemic of 2020.
To formalize this learning process and examine the resulting dynamics of the economy,
we introduce Markov-switching states in a money-in-the-utility model. In the model,
agents receive utility from holding money only when the liquidity shock occurs. There are
two unobservable states, stable and unstable, between which the economy goes back andforth according to a Markov process. The liquidity shock follows a Poisson process in eachstate with different arrival rates. By observing whether or not the shock occurs at each
1

Atolia et al. (2018) suggest that liquidity shocks, or tight credit conditions, are more likely to beobserved during economic downturns (see Figure 1 in their paper). Their paper also quantitatively showsthat the financial frictions (i.e., liquidity shocks) are only active occasionally and that they are a significant
source of macroeconomic volatility over the business cycle.

2point in time, agents rationally update the subjective probability using Bayes’ law andrevise the time paths of money holding. In this setting, when agents observe the shock,they strengthen the belief that they are in the unstable state, increase their subjective
probability for re-encountering the shock, and raise liquidity preference. Conversely, if theshock does not occur for a while, they gradually increase the belief of being in a safer state,
reduce the shock probability, and lower liquidity preference.
To map the evolution of liquidity preference to macroeconomic dynamics, we combine
the learning process with a minimal macroeconomic setup with incomplete nominal priceadjustments. The higher the subjective probability of meeting the shock, the more willingthe agents become to hold money, sacrificing their consumption. The resulting movements
in aggregate consumption further intensify fluctuations in liquidity preference. When people lower consumption due to high liquidity preference, the shortage in aggregate demand
will push down the inflation rate. Lower inflation reduces the cost of holding money,
further intensifying the liquidity preference. Furthermore, since people anticipate that
employment will decline after a liquidity shock, they want to increase savings by holding
more money if they believe the shock is likely. Both of these mechanisms strengthen theconnection among the belief of agents, their liquidity preference, and consumption.
This paper contributes to the literature of business cycles in two aspects. First, it provides a natural explanation for the observed asymmetry of the business cycles. Empirical
works found that Macroeconomic variables tend to move quicker in contraction than in
expansion.2 However, in typical linearized business cycle models, the economy fluctuates
symmetrically around the steady state. In a continuous-time model, our model shows thatthe movements of the output and employment are asymmetric since, while sudden discrete
shocks reveal bad news, the good news is known only from the length of the period without
2

The observation that the business cycles are asymmetric dates back to Keynes (1936). Neftci (1984),

Sichel (1993), and Acemoglu and Scott (1997) are among the empirical works that identified the asymmetric movement of output. Later studies focused on the relative degree of asymmetry among the key
macroeconomic variables. See, for example, Sichel (1993), Hansen and Prescott (2005), McKay and Reis
(2008), and Ferraro (2018).

3

shocks. The realized pattern of output is asymmetric even when the underlying Markovswitching model is symmetric. The asymmetry occurs because of the way the informationis brought to the agents.
The model also provides an intuitive mechanism for the high persistence of output loss
from the sizeable financial crisis.3 We show that the persistence is higher after the economy
is hit by multiple shocks in a short time, while the recovery is relatively quick if a shock is
an isolated event. This difference occurs because each realized shock only partially reveals
information about the true state. As long as the liquidity shock occurs sparsely in time,
people view the shock as a mere accident. Although consumption decreases temporarily, it
quickly recovers as the subsequent observation of no shock confirms this optimistic view.
However, when the shock occurs many times for a short while, people are convinced thatthey are in the less stable state. In this case, the recovery process takes a long time, evenafter the shock ceases. The recovery speed of consumption is initially slow because theirpessimistic belief is so strong that it is not easily overturned by the gradually revealed
information from the absence of shocks. The recovery process gradually accelerates as thebelief gets weaker and more sensitive to new information, and eventually slows down again
when they are quite sure that they are in the safer state.4

1.1

Related Studies

There have been various attempts to explain fluctuations of macroeconomic variables bycombining unobservable regime changes and Bayesian updating agents. When the signal is3

Cerra and Saxena (2008) presented robust evidence that the substantial output loss from financial

is highly persistent. Reinhart and Reinhart (2010) found that economic growth is notably slower in thedecade following a macroeconomic disruption and provided evidence of several “double dips.” Ikeda andKurozumi (2019) documented the slow recovery after the financial crisis of 1997.
4

Those results contrast with the previous studies of information cycles by Caplin and Leahy (1993) andZeira (1994), where a shock, once it occurs, immediately reveals the real state. Even when several shocks
come in a bunch, only the first one conveys relevant information. Thus, the number of shocks that agentsobserve does not affect their behavior. Boldrin and Levine (2001) also consider a related model, but theyrule out the possibility that the shock continually occurs.

4noisy, agents slowly change their beliefs, making the effect of a regime change a more gradual one than in the case where the state is perfectly observable. Andolfatto and Gomme(2003) and Sill and Wrase (1999), for example, demonstrate that monetary policy has
long-lasting effects when it periodically switches between low and high monetary growth
in an unobservable way. Other papers examine how agents react to unobservable changes
in investment opportunities. Chalkley and Lee (1998) show that recovery from a recession
is protracted when risk aversion of agents prevents them from acting promptly on receiving good news. Potter (2000), Veldkamp (2005), and Nieuwerburgh and Veldkamp (2006)
explain slow recovery generated by an endogenous flow of information. If agents have a
pessimistic belief, their activities are low, generating less public information, and therefore
good news is only slowly revealed. This paper proposes an alternative mechanism, where
recovery is slow not because the information is scarce in recession, but because people’s
strong beliefs dwarf the significance of new, favorable information.
Lorenzoni (2009) and Boz et al. (2011) examine related problems, where agents receive
the signal on the TFP, but cannot distinguish the transitory and permanent components.
Alpanda (2019) extends those studies and incorporates the Markov switching between highand low TFP growth regimes. In these studies, persistent adverse productivity shocks
are the ultimate cause of the period of low growth. We complement this literature byconsidering a mechanism where the economy stagnates for an extended period, even when
the production capacity is unaffected.
Specifically, we consider a situation where the decline in output is caused by the shortage of aggregate demand, which is a result of people’s strong preference for money, particularly in turbulent times. In this respect, our setting is related to Caballero and Farhi(2018), who develop a framework where excessive demand for safe assets causes stagnation.
However, to deliver the macroeconomic implications from the learning process as clearly as
possible, we keep our model minimal by assuming a reduced-form relationship between the
output gap and the inflation rate, based on a simplified version of Ono (2001), Ono and
Ishida (2014), and Illing et al. (2018). These models employ a non-stochastic money-inutility function, where the marginal benefit of holding money has a strictly positive lower
5bound. They have shown that, depending on the parameters for the liquidity preference,the economy can be in persistent stagnation, where output and consumption never recoverby themselves. In contrast, this paper examines the possibility that the economy recovers,
albeit slowly, if the absence of the shocks gradually lowers liquidity preference through thelearning process.
The remainder of the paper is organized as follows. Section 2 presents a model ofBayesian learning in which a representative household faces an aggregate liquidity shock.
In Section 3, we examine the effect of fluctuations in belief on the optimal behavior of thehousehold and derive the existence, uniqueness, and other properties of the stationary dynamics. Section 4 shows the pattern of movements of aggregate variables and examines the
speed of the recovery process. The role of the learning process in generating an asymmetric pattern is discussed in Section 5. Section 6 summarizes and concludes. Mathematical
proof and numerical procedures are provided in the appendices.

2

Learning ProcessThis section sets up a continuous-time model of Bayesian learning in which a representative
household faces an aggregate liquidity shock that occurs at random discrete points in time.5
We consider a situation in which the arrival rate of the shock changes over time, although
it is unobservable to the household. In that setting, this section establishes the following
results. (i) When a shock occurs, the household discretely strengthens its belief that thecurrent state of the economy is unstable. (ii) During the period without the shock, thehousehold gradually (or continuously) weakens that belief. (iii) Once the household holds
a strong belief that they are in the unstable state, however, the belief can change only
slowly, at least for some time. Those results are utilized in later sections to investigate the5

We use a continuous-time model in order to highlight the difference between the change in belief when

bad news arrives and when there is no such news. This strategy is similar to Driffill and Miller (1993)
and Zeira (1999). In their models, however, the unobservable state is time-invariant, and the uncertainty
eventually vanishes.

6

pattern of fluctuations in the household’s liquidity preference and consumption.
Let us describe the precise settings of the model. There are two underlying states withdifferent probabilities of the shock, called states H and L. In state i ∈ {H, L}, the shock
occurs with probability θi per unit of 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 to state L with Poisson probability pH per unit of time whereas
state L changes to state H with probability pL . We assume that the shock occurs much
more frequently in state H than in state L and that the state change is a rare event whencompared to the shock in state H. Formally,
Assumption 1 θH − θL > pH + pL .
By observing whether the shock occurs or not the household continuously revises its
subjective shock probability in a Bayesian manner. Let θt ∈ {θH , θL } denote the true shockprobability at time t, which is unknown to the household. Using information available up
L
to time t, it forms a belief that current θt is θH with probability λH
t and θ with probability

λLt . Obviously,
λLt + λHt = 1 for all t.

(1)

In order to find how the household updates λit from t to t + ∆t,6 we first obtain thesubjective probability that the shock does not occur between t and t + ∆t for given λit .
It is denoted by Probt S(t,t+∆t] = ∅ , where Probt [·] 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 ∅ the empty set. Since the underlying state is either H or L at time
t + ∆t, this probability is divided into two components, Probt S(t,t+∆t] = ∅ ∩ θt+∆t = θHand Probt S(t,t+∆t] = ∅ ∩ θt+∆t = θL .
Each of the two components is further divided into two probabilities. The former isthe 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 state is L and6

Time interval ∆t is taken to be so short that the probability that the liquidity shock and a statechange coexist in the interval is negligible.

7the state changes to H during the interval.’ It is7



L
L
Probt S(t,t+∆t] = ∅ ∩ θt+∆t = θH = 1 − (θH + pH )∆t λH
t + (p ∆t)λt .

(2)

Similarly, the latter isProbt S(t,t+∆t] = ∅ ∩ θt+∆t = θL = 1 − (θL + pL )∆t λLt + (pH ∆t)λH
t .

(3)

Summing up (2) and (3) yields
Probt S(t,t+∆t] = ∅ = 1 − θte ∆t,(4)where θte represents the expected (or subjective) probability of the shock per unit of timeat time t,
L L
θte ≡ θH λH
t + θ λt .

(5)

Let us consider how the representative household updates its belief if it eventuallyfinds that the shock did not occur during (t, t + ∆t]. In this case the information that
S(t,t+∆t] = ∅ is added to its knowledge. Thus, using Bayes’ law we find updated subjective
probability λit+∆t to be
λit+∆t ≡ Probt+∆t θt+∆t = θi = Probt θt+∆t = θi |S(t,t+∆t] = ∅Probt S(t,t+∆t] = ∅ ∩ θt+∆t = θi=.
Probt S(t,t+∆t] = ∅8Since the numerator is given by (2) or (3) and the denominator by (4), λHt+∆t equals
 H
H
H
1
−
(θ
+
p
)∆t
λt + (pL ∆t)λLt
λH
=
.
t+∆t
1 − θte ∆t

From this equation we derive the time derivative of λH
t as
λ̇H
t =

H
λH
t+∆t − λt
L Llim= (θte − θH − pH )λH
t + p λt .
∆t→0
∆t

(6)

Next, consider the case where a shock occurs during (t, t + ∆t]. Since
Probt S(t,t+∆t] 6= ∅ ∩ θt+∆t = θi = θi λit ∆t for i ∈ {L, H},7

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

8 L
λt+∆t is analogously obtained. From (1) it equals 1 − λHt+∆t .

8(7)

the probability that the shock occurs is



e
L L
Probt S(t,t+∆t] 6= ∅ = θH λH
t + θ λt ∆t = θt ∆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 is
θi λit−θi λit0≡,e
t →t− θ e0θt−t

λit = 0lim

(9)where subscript t− represents the state just before t.9 Finally, we obtain the dynamics ofsubjective probability θte . From (1) and (5),λHt =θte − θL,
θH − θLλLt =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 − θH − 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 ∈ θL , θH , wherepθL + θH + pL + pH − (θH + pH − θL − pL )2 + 4pL pH∗∈ (θL , θH ).
θ ≡
2

(12)

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

θte = θL + θH −θL θHe≡ h(θt−)eθt−

for t ∈ S(0,∞) .

As shown in Figure 2, h(θ) satisfiesh(θH ) = θH , and θe < h(θe ) < θH for all θe ∈ (θL , θH ).

9

(13)

µ_et
µH

µ¤
0

µ

L

µet

g(µet )

Figure 1: Movement of belief during the period without a shock.

µet
µH
e
h(µt¡0
)

h(µ¤)

0

µL µ¤

µH

µet¡0

Figure 2: Movement of belief when the shock is observed.

10Equations (11) and (13) respectively describe the dynamics of θte without and with theshock. They jointly show that θte fluctuates within interval θ∗ , θH . A liquidity shock
is a rare event, and therefore causes a discrete change in people’s expectations about thepresent state once it occurs. As function h(θe ) is located above the 45-degree line in Figure
2, the more often people observe the shock, the more strongly they believe that they arein state H, and hence θte becomes closer to θH .
Conversely, in the absence of the shock people gradually become more and more optimistic and confident that the economy is in state L. Thus, their subjective probability ofthe shock gradually declines, converging to θ∗ .10 However, the U-shape of function g(θte )implies that the speed of adjusting belief is slow when θte is near θH . Note that θte ≈ θH isequivalent to λHt ≈ 1 from (10), which means that the precision of the prior belief is quite
high (i.e., people are quite sure that the current state is H). In that case, any additional
information has only a small impact on the posterior belief.

3

Liquidity Preference and Consumption Behavior

In this section, we investigate how evolutions in the representative household’s beliefs affecttheir liquidity preference and consumption behavior. We introduce a simple model in which
liquidity holding generates utility only when the shock occurs. Since the household cannot
anticipate the timing of the shock precisely, they always hold liquidity to prepare for theshock. In this setting, we derive the stationary dynamics in which the household optimallychooses money holding and consumption as a function of their belief on the probability ofshock, θte .
9Mathematically, θt− is the limit of θτ as τ approaches t from the left. θt− is different from θt when

belief of the household changes discretely at time t. Similar notations are used, for example, in a textbook
by Dockner et al. (2000, Chapter 8).
10 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.

11

In their optimization, the household weighs the cost of holding real money against its
benefit. The cost is the sum of the inflation rate and the real interest rate, and the benefit
is the expected marginal utility from real money holding. Focusing on each element inthis trade-off, this section proceeds in several steps. In the first subsection, we present aminimal setup in which the consumption behavior of the household affects the inflation
rate. In the second subsection, we describe the preference of the household as a variantof the money-in-the-utility model and explain how the belief of the household affects the
expected benefit from real money holding. The third subsection examines the household’sintertemporal optimization under uncertainty. We derive a modified version of the Keynes–
Ramsey rule, which gives the relation between the real interest rate and the growth rateof consumption. The fourth subsection describes the optimality condition for the choice
between consumption and money holdings. Combining the results from the previous four
subsections, the final subsection derives a difference-differential equation that characterizes
the stationary dynamics and establishes its essential properties.

3.1

Production Side

Since our focus is on the behavior of the household, we employ a minimal setting for the
production side of the economy, where aggregate consumption affects the inflation rate.
The economy is inhabited by the representative households with measure one. Eachhousehold lives infinitely and supplies labor to the representative firm. The firm produces
homogeneous final goods from labor on a one-to-one basis. The goods market is fully competitive, and the nominal price Pt coincides with the nominal marginal cost of production,i.e., the nominal wage level Wt . There is no investment in our model, and therefore all the
produced goods are used for consumption, ct . This means that total labor demand `t is
the same as consumption ct .
At each point in time, the household supplies labor elastically. However, if the level of
employment is higher than the natural level, which is normalized to one, there is upward
pressure on the nominal wage level. Conversely, if the employment is less than one, there
is a downward force. For simplicity, instead of explicitly introducing the adjustment cost12

of nominal wages, we assume a reduced-form relationship between employment `t and the
speed of nominal wage increase Ẇt /Wt :Ẇt /Wt = f (`t − 1),

f 0 (·) ≥ 0.

(14)

Intuitively, the nominal wage level rises slower (or falls faster) when there is less demandfor employment.11 From Pt = Wt and ct = `t , (14) implies that the inflation rate isdetermined as a function of aggregate consumption ct :πt ≡ Ṗt /Pt = f (ct − 1).

(15)

Equation (15) can be interpreted as a Phillips curve: a reduced-form relationship between
the output gap ct − 1 and the inflation rate.12 In this interpretation, potential output of
this economy is normalized to one.

3.2

Preference of the Household

We consider a stochastic version of the money-in-the-utility model that incorporates therandom liquidity shock. The representative household gains utility from consumption, ct .
In addition, when the shock occurs, they gain utility in proportion to their real money
balances mt . Their expected utility EUt is therefore given by


Z ∞
X
EUt = Et u(cτ )e−ρ(τ −t) dτ +βmτ e−ρ(τ −t)  ,t(16)

τ ∈S(t,∞)where constant ρ is their subjective discount rate. Function u(·) represents the instantaneous felicity from consumption, which satisfies the Inada conditions. Constant β specifies
11

Our specification is related to Schmitt-Grohé and Uribe (2017), where the maximal rate of nominal

wage decline is an increasing function of the unemployment rate. For simplicity, this paper assumes thatthe relationship holds when the nominal wage moves either direction. In a separate study, we considered
the case where there is no upward friction in the nominal wage adjustments, and confirmed that the twomain results in this paper (business cycle asymmetry and slow recovery) are retained.
12

Equation (15) is essentially the same as the Phillips curve in Eggertsson et al. (2019, equation 26)

when the output gap is negative.

13

the marginal benefit from holding real money balances, and therefore βmτ e−ρ(τ −t) gives
the discounted utility from money if a liquidity shock is encountered at date τ . It issummed over τ ∈ S(t,∞) , where S(t,∞) represents the discrete set of dates on which theshock occurs.13
Since there is no capital in this economy, money is the only asset that is in positive net
supply. The household also has access to the complete set of contingent claims for future
goods. However, since all households are identical, money is the only asset that they holdafter all arbitrage opportunities are exploited. There is no tax or transfers, and the only
source of income for the household is labor. Since the real wage is always unity, their labor
income is simply given by employment `t . With these settings, the flow budget equation
can be written asṁt = `t − πt mt − ct .

(17)

Given the evolution of the subjective shock probability derived in Section 2, the representative household chooses the time paths of consumption and money holding to maximize
the expected utility (16) subject to (17).
Objective function (16) depends only on ct , mt , and the expected pattern of the shock.
From the perspective of economic agents in the model, the last one is fully described by θte ,since θte is a sufficient statistic for the current and future values of unobservable θt . Note
that, although the budget constraint (17) depends on `t and πt , they are both determinedby ct on the realized path: `t = ct and πt = f (ct − 1) from (15). Therefore, the household’s
problem can be defined by ct , mt , and θte . Then, given the recursive structure of the model,
the movement of ct on the path of stationary dynamics must completely be expressed asa function of mt and θte .14 Furthermore, since objective function (16) and constraint (17)13This setting is obviously a simplifying approximation. In reality, a financial crisis does not instantly0

terminate but continues for a few days so that a household receives, say, utility flow β̃mτ 0 e−ρ(τ −t) throughout τ 0 ∈ [τ, τ + T ]. Since each period of financial crisis is typically short, we approximate the total utility
received from money holding during one occurrence of the shock by βmτ e−ρ(τ −t) , where β ≡ β̃T .
14This strategy for finding stationary dynamics is analogous to that of Lucas (1978), who analyzes thedetermination of equilibrium price behavior under an exogenous production shock that follows a Markov

14

are both linear in mt , the optimal choice of ct should be independent of the level of mt .15
These consideration reveals that, on the stationary path, ct should be a function only ofθte :ct = C(θte ) for all t.

(18)

Since θte fluctuates within interval (θ∗ , θH ], as shown in Section 2, we only need to characterize the shape of function C(θte ) in this interval. In the following it is assumed thatθte ∈ (θ∗ , θH ] unless otherwise noted.

3.3Intertemporal Optimization

Under complete financial markets, the representative household sells and buys contingent
claims to future goods to maximize its lifetime utility. This subsection shows that, eventhough those contingent claims are in zero net supply, their equilibrium prices are uniquelydetermined once function C(·) is given. We later use this information to find the opportunity cost of holding real money.
Let 1 − µ(θte )∆t denote the price of the claim to a unit of good at t + ∆t measured
in terms of goods at t under the condition that the shock does not occur between t andt + ∆t.16 Note that it is a function of θte because the value of the claim depends on theprobability with which the contingent event occurs.17 If the shock does not occur during
the interval, consumption increases from C(θte ) to C(θte + g(θte )∆t) since θte changes byprocess.
15

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 household in thesimplest setting. When the marginal utility of holding real money balances is variable, we actually findthat the dynamics of ct depends on both θte and mt . We do the analysis for such a general case in a
separate paper.
16

We utilize a method analogous to Ikefuji and Horii (2012, Appendix 8) in solving the continuous-time

optimization with the possibility of jumps. They considered the household problem under the risk of being
hit by natural disasters.
17

In addition, θte affects consumption and therefore the marginal utility of consumption.

15

the amount of g(θte )∆t, as shown by (11). Since the probability that the shock does notoccur during this interval is 1 − θte ∆t, as given 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 derive
C 0 (θte )eeµ(θt ) = ρ + θt +γ(C(θte ))g(θte )eC(θt )

for all θte ,(19)

where γ(c) ≡ −u00 (c)c/u0 (c) represents the degree of risk aversion.
Analogously, let ν(θte )∆t denote the price of the contingent claim to a unit of good at
t + ∆t under the condition that the shock does occur between t and t + ∆t. When theshock occurs, consumption jumps from C(θte ) to C(h(θte )), as seen from (13). Since theshock probability is θte ∆t, as shown by (8), the first-order condition 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 .

(20)

Next, let us derive the rate of return on a risk-free asset, such as a riskless bond. By
denoting by rt the real interest rate of a risk-free asset, the price of a risk-free claim to
the future good at t + ∆t can be written as e−rt ∆t . Since the claim is equivalent to theasset of the synthesis of the claim contingent on the absence of the shock whose price is
1 − µ(θte )∆t and that conditional on its presence whose price is ν(θte )∆t, the no-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.
Substituting (19) and (20) into (21) yields: 0C(θte )u (C(h(θte )))0 e
ert − ρ + θt
−1 .
C (θt ) =γ(C(θte ))g(θte )u0 (C(θte ))
16

(21)

(22)

Applying (11) and (18) to (22) leads to the dynamics of ct (≡ C(θte )) while the shock does
not occur

 0

1
u (C(h(θte )))ċte=rt − ρ + θt−1ct
γ(ct )
u0 (ct )for t ∈/ S(0,∞) ,(23)

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 thegrowth 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, it tries to reallocate consumption from the present to the future, raising the
growth rate of consumption during the period without the shock.

3.4

Choice between Consumption and Money

Having examined the household’s intertemporal optimization of consumption, we now turnto the optimal choice between consumption and money holding. By holding one unit ofreal money balances 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 increase inreal money holding raises the household’s utility (16) by β if the shock occurs. Since thesubjective probability that the shock occurs between t and t + ∆t is θte ∆t, the increasein the expected utility is βθte ∆t. The marginal benefit should equal the marginal coston the optimal path, which yields the first-order condition between money holding andconsumption:r t + πt =βθteu0 (ct )for all t.

(24)

Equation (24) implies the well-known property that the marginal rate of substitution between consumption and real money holding equals the nominal rate of interest.
Substituting (15) into (24) and rearranging terms yields
rt =

βθte
− f (ct − 1) ≡ R(θte , ct ).
u0 (ct )17(25)

Intuitively, function R(·) represents the household’s preference for liquidity, defined by the
expected utility gain minus capital loss from holding money for a unit of time. Given theamount of consumption, the preference for liquidity is stronger when the subjective shockprobability is higher; i.e.,Rθ (θe , c) ≡ ∂R(θe , c)/∂θe =β> 0 for all θte > 0 and ct > 0.
u0 (ct )(26)

We restrict our attention to a reasonable case where function R(·) in (25), or equivalently utility function u(·) and price adjustment function f (·), satisfies two additional
properties:
Assumption 2 limc→0 R(θH , c) < ρ and limc→∞ R(θ∗ , c) > ρ.
Assumption 3 Rc (θe , c) is continuous and positive for all θe ∈ [θ∗ , θH ] and c > 0, whereRc (θe , c) ≡ ∂R(θe , c)/∂c.
If the household’s preference for liquidity, given by R(·), is stronger than its preference
for present consumption ρ, it postpones consumption and holds more money. Therefore,in Assumption 2 the first condition implies that this household prefers consumption toliquidity holding when its consumption is quite low even if it expects the highest shockprobability θH . The second condition implies that the household prefers liquidity to consumption when its consumption is sufficiently high even if it expects the lowest shockprobability θ∗ . Assumption 3 extends this relationship between ct and R(·) to a smoothand monotonic 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.

3.5

Stationary Dynamics

Now we are ready to determine the pattern of stationary dynamics, represented by function
C(·). Equation (25) requires that the rate of return from a risk-free asset (riskless bonds),
rt , should be equalized to R(·) for all t. Also, rt must be consistent with Euler equation18

(22). From these two equations, the optimizing household behavior is summarized as 0
u (C(h(θte )))C(θte )eee
0 e
R(θt , C(θt )) − ρ + θt
−1
C (θt ) =γ(C(θte ))g(θte )u0 (C(θte ))(27)
for all θte ∈ (θ∗ , θH ].
Function C(·) is determined so that it satisfies differential equation (27). 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, if ċt /ct remains negativeas θte → θ∗ , ct converges to zero, violating the transversality condition. We rule out suchpaths by imposing a boundary condition,18
 0
u (C(h(θe )))− 1 = 0,lim R(θ , C(θ )) − ρ + θ
θe →θ∗
u0 (C(θe ))
e

e

e

(28)

under which ċt /ct given by (23) approaches zero as θte → θ∗ .
Equations (27) and (28) determine the whole shape of C(θte ) within interval θ∗ , θH .
Note that (27) is not a usual difference equation, but a difference-differential equation:19
the gradient of function C(·) at θe depends on the value of function C(·) not only at θe ,but also at h(θe ) 6= θe . This is a natural consequence since an occurrence of the shock
discretely changes the belief and consumption of the household, and their optimization
always takes such a possibility into account. Although analysis of this type of system is
not straightforward, in Appendix A we obtain its important properties: existence of thesolution, its uniqueness, boundedness, and the sign of gradient of function C(θe ).
Proposition 1 Under the assumptions 2 and 3, there exists a unique function C(θe ) thatsatisfies (27) and (28). It is strictly downward sloping for any θe ∈ (θ∗ , θH ] and has positiveand finite upper and lower bounds c and c that are given byR(θ∗ , c) = ρ, R(θH , c) = ρ.
Proof. See Appendix A.
18

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

19

It is also called a delay difference equation.

19

(29)

Proposition 1 states that there is a negative relationship between belief θte and consumption ct = C(θte ). When the subjective probability of the shock θte rises, the expected benefitof holding money increases, and therefore the household reduces consumption. Also, there
is an endogenous dynamic mechanism that reinforces this relationship. Since a higher
subjective probability of the shock θte reduces consumption C(θte ), labor demand declines,
and hence the inflation rate falls. It reduces the cost of holding money, and therefore further strengthens the liquidity preference.20 Furthermore, the household understands that
their income (= aggregate consumption) C(θte ) will fall discretely if a shock occurs in the
future. Therefore, a higher subjective possibility of a liquidity shock θte leads to a stronger
incentive to save current income in the form of money.21 In this way, the fact that function
C(·) is downward sloping further strengthens the effect of the belief of the household on
liquidity preference and consumption.

4

Time Evolution of the Economy

Utilizing the results obtained in the previous sections, this section considers the time
evolution of aggregate real variables in the economy. We first show that the pattern ofevolution of the economy is characterized by sporadic falls in consumption and output,
followed by gradual recovery toward a steady-state level. Following this, we examine thespeed of the recovery process.
Recall that, in Section 2, we have shown that the time path of belief θte is uniquelydetermined by initial belief θ0e and the history of the liquidity shock S(0,t] . Then, in Section
3, we have derived a mapping from belief θte to consumption ct . Combining those results
naturally gives the time path of ct . For concreteness, let us consider log utility function
20

Observe that equation (25) is increasing in ct . Strictly speaking, there is also an opposing effect since afall in consumption increases the marginal utility from consumption, and therefore reduces the preference
for money relative to consumption. In Assumption 3, we assumed that the latter effect is weaker than thepositive effect explained in the text.
21

This is confirmed from the fact that the additional term in the modified Keynes–Ramsey equation(23) is positive.

20

c
c¤
C(µe )
cH

µH

µ¤

µe

Figure 3: A typical shape of function C(θte ). Parameters: θH = .4, θL = .05, pH = .025, pL = .1,
ρ = .05, α = .025, and β = .4. Under these parameter values, we obtain θ∗ ≈ .069, c∗ ≈ 1.3 and cH ≈ .60.

u(ct ) = log ct and a linear Phillips curve πt = f (ct − 1) = α · (ct − 1), where α > 0 is aconstant. In this setting, Assumptions 2 and 3 simplify to βθ∗ > α and ρ > α, respectively.22 Then, choosing parameter values so that these conditions as well as Assumption 1
are satisfied, we can numerically calculate the shape of function C(·). Figure 3 illustrates
a typical shape of the function (Details of the numerical procedure are described in Appendix B.) It confirms the statement of Proposition 1 that the curve is downward sloping
and ranges between finite values, c∗ ≡ C(θ∗ ) and cH ≡ C(θH ).
As the belief of the household θte changes, the amount of consumption, and therefore
aggregate output, move along this curve. As discussed in Section 2, the law of motion
for belief θte is different depending on whether a shock is observed or not at a particularpoint in time t. For most of the time, the household does not observe a shock, andbelief θte follows a deterministic path given by (11). As illustrated in Figure 1, the time
derivative of θte is negative for any (θ∗ , θH ], implying that subjective shock probability
22

Using a money-in-utility model without uncertainty, Ono (1994, pp.86–88; 2001) shows that in thecase where β > α and ρ > α there is a unique saddle-stable path. Furthermore, the path accommodates apersistent demand shortage when ρ < β, whereas it reaches a full-employment steady state when ρ > β.
The present condition is the same as Ono’s condition except that the former includes the shock probability
θ∗ .

21

c
c¤
C(µe )

C(h(µe ))
cH
0

10

20

time

Figure 4: Time path of consumption recovery. This figure is obtained in two steps. First, thetime path of θte is calculated by integrating (11) with respect to time. Second, the calculated time pathsof θte and h(θte ) are applied to function C(·), which is numerically calculated as in Figure 3. Parameters
are the same as in Figure 3.

θte gradually falls toward θ∗ . Thus, the preference for liquidity gradually declines andconsumption grows toward c∗ ≡ C(θ∗ ) since function C(·) is downward sloping. Conversely,at discrete points in time, the household observes a shock and discretely increases θteaccording to (11). Since function C(·) is downward sloping, this causes a downward jumpin consumption. As illustrated in Figure 2, belief θte approaches θH when shocks occur
in succession. Accordingly, consumption can fall to cH ≡ C(θH ). Since the shock occurs(only) at sporadic points in time, the economy experiences sporadic falls and gradual
recoveries in consumption and output. In this way, ct persistently fluctuates within finite
interval [cH , c∗ ].
The speed at which consumption recovers toward the steady-state level c∗ depends on
how quickly the household changes its beliefs. The solid curve in Figure 4 depicts thetime evolution of ct , assuming that ct starts from the lowest level cH and that no shock
occurs after that. Observe that consumption first grows slowly, gradually accelerates, andeventually slows down again as it approaches c∗ —i.e., it traces an S-shaped trajectory.
To see why ct follows the S-shaped trajectory, recall that the time derivative of θteis given by U-shape function θ̇te = g(θte ), as depicted in Figure 1. We can interpret the22dynamics in three phases.
1. When people strongly believe that they are in state H and thus θte is very close to θH ,then Figure 1 shows that |g(θte )| is small. This implies that they do not significantly
alter their pessimistic view for a while, and therefore consumption increases only
slowly.
2. As the period without the shock lasts, θte decreases and |g(θte )| increases. As theirbelief gets weaker, they become more willing to revise it based upon the fact that they
do not observe shocks for a while, and therefore θte declines more quickly. Accordingly,
the recovery speed of consumption accelerates.
3. As θte approaches to steady-state value θ∗ , people become quite confident that they
are in state L, and hence an additional period without the shock provides little
information. Therefore, |g(θte )| falls to zero and the growth of consumption slows
down.
Figure 4 shows that ct recovers toward c∗ in the absense of liquidity shocks. However,a liquidity shock will certainly occur before ct reaches c∗ . At the moment the shock isobserved, consumption falls from C(θte ) to C(h(θte )), as depicted by vertical arrows in
Figure 4. Then, the recovery process ‘restarts’ from the point that corresponds to thedecreased level of consumption (as indicated by horizontal arrows) and consumption again
traces the solid curve. Note that a single shock can only send ct from c∗ to an intermediate
value, from where the economy can recover relatively quickly. However, if the shocks
occur repeatedly, the subjective probability successively increases, and C(θte ) approaches
the lowest level cH . After the shocks cease, consumption recovers along the S-shaped
trajectory, as explained above.
Figure 5 presents an example of the realized time paths of θt , θte and ct , which are
obtained by simulating the Markov process of the underlying state and the Poisson process
for the shock. Consumption in fact traces an S-shaped path, especially after a bunch of
liquidity shocks make it close to the lowest value cH .

23

µH

µt
µet

µL
0

c¤
y¹

25

50

75

100

25

50

75

100

ct

cH
0

Figure 5: An example of the realized paths of actual state θt , belief θte , and consumption
ct . Parameters are the same as in Figure 3.

Before closing this section, let us briefly consider the effect of the household’s belief
on other assets than money. On the equilibrium path, the expected rate of return fromany asset should be adjusted so that the household is always indifferent between holding
money and any other assets. In particular, as shown in equation (24), the nominal interest
rate on a risk-free asset should coincide with the expected benefit from money holding.
On the equilibrium path, the latter is a function of the household’s belief and can be
written as βθte /u0 (C(θte )). When the household observes a shock, its subjective shockprobability θe rises and consumption C(θte ) falls. As a result, both expected utility frommoney holding βθte and marginal utility from consumption u0 (C(θte )) rise, making the net
effect of a higher θte on the nominal interest rate ambiguous. Nonetheless, the real interest
rate tends to be upward sloping in θte , because the reduced aggregate consumption pushes
down the inflation rate. More precisely, we can show that rt is necessarily higher when θteis at its highest level θH than when it is at its lowest level θ∗ : From Rc > 0 and Proposition
1, R(θ∗ , C(θ∗ )) < R(θ∗ , c) = ρ = R(θH , c) < R(θH , C(θH )).

24

5Discussion: The Role of the Learning Process

So far, we have explained the movements of aggregate variables in terms of the information
brought by the realized shocks. Note that an occurrence of the shock is a piece of relevant
information, and therefore has a real effect on the aggregate economy because the household
does not know the actual state of the economy. If households precisely know the actual
state, occurrences of the shock do not alter their belief or behavior.
Then, why cannot the household learn the true state even in the long run? If the true
state of the economy, θt , is constant, cumulative observations of the shock will eventually
reveal the true state (i.e., θte → θt ), and further occurrences of the shock do not affect thehousehold’s beliefs and their behavior. In the economy considered, however, the true statechanges over time in an unobservable way, which prevents complete learning. In this sense,
the source of persistent fluctuations in economic variables is stochastic movements in theactual underlying state.
Nonetheless, the learning process plays an essential role in generating the previously
described pattern of aggregate movements. To see this, suppose that the household candirectly observe true state θt . This means that there is no learning process, and triviallyθte = θt . Recall that θte = θt takes either θH or θL . Since the discussion in Section 3.2
is still valid in this case, equation (18) implies that on the stationary path, ct takes only
two values. Let us denote them again by C(θH ) ≡ cH and C(θL ) ≡ cL , although theirvalues differs from those in Section 3. From the optimization of the household (analogous
to the analysis in Section 3.3 and Section 3.4), it can be shown that cH and cL are given
as a solution to certain simultaneous equations, and that cH is smaller than cL .23 This23

In particular, cH and cL must satisfypH

u0 (cH ) − u0 (cL )= R(θH , cH ) − ρ,
u0 (cH )and pLu0 (cL ) − u0 (cH )= R(θL , cL ) − ρ,
u0 (cL )where the LHS of each equation represents the expected growth rate of the marginal utility due to thestochastic movement of the underlying state, and the RHS represents the deviation of the risk free ratefrom the rate of time preference. From these equations, it can be easily confirmed that Assumption 3 will
be violated if cH ≥ cL .

25means that the behavior of households correspond one-to-one with the underlying state:once they observe that the economy switches to state H (L), they immediately increase(decrease) liquidity preference and decrease (increase) consumption.
In contrast, when households cannot directly observe the underlying state, realizedbooms and depressions do not precisely match the underlying state of the economy but
follow the subjective probability that people have in mind (see Figure 5). Even whenthe economy switches to state H, and thus the actual probability of the shock jumps up,
people do not increase liquidity preference until they observe the shock. Similarly, evenwhen the true shock probability jumps down, they still retain a strong liquidity preference
for a while.
More importantly, the learning process creates an asymmetry between the fall andrecovery of consumption. The fall of consumption occurs as soon as people observe a shock.
In contrast, the recovery is only gradual, especially once they become very pessimistic. Thispattern applies even when underlying θt follows a symmetric Markov process (i.e., when
pH = pL ). The aggregate movements have the asymmetric pattern because an occurrenceof the shock has a discrete amount of information, and this information is always a “badnews” in our model. In other words, the safer state (state L) is revealed only on the basisof “no news is good news.” This is why recovery occurs only gradually.
It is for the sake of simplicity that we considered only one type of shock, which delivers
information that favors less consumption. As well as bad news, we sometimes hear thegood news that delivers discrete amounts of positive information, e.g., a discovery of abreakthrough technology that might boost aggregate productivity. In a model that incorporates a positive signal, a rise of consumption (or a rise of investment, depending on themodel) would occur immediately after the signal is received. Consumption would gradually
fall when people do not receive such a signal for a while. An important task for further
work would be to study the relative importance of positive and negative discrete signals
in the real economy, or, more specifically, their frequency and the amounts of information
brought by their occurrences.

26

6Conclusion

Liquidity preference depends on people’s belief about how frequently they encounter crises.
This paper has examined the way they update the belief based on Bayesian inference and its
effect on their preference for liquidity holding relative to consumption. The analysis is donein a circumstance where the economy shifts between two unobservable states with different
probabilities of the liquidity shock. Each time they observe the shock, they raise theirsubjective probability of being in the less stable state and increase the liquidity preference.
The longer the period without the shock lasts, the larger the probability people attach
to the safer state and increase preference for consumption over money holdings. With
incomplete nominal wage adjustment, such movements in liquidity preference produce
fluctuations in aggregate consumption and output.
Even when the underlying state of the economy follows a symmetric Markov process, thepattern of fluctuations in aggregate variables is not symmetric. The fall occurs immediately
after observing the shock, whereas the recovery is gradual. Also, the speed of the recovery
process depends on the realized frequency of the shock. As long as the shock sparsely occurs
in time, it has only a minor effect on the belief, and hence the economic recovery is relatively
quick. However, if people observe the shock many times over a short period, they hold astrong belief of being in the less stable state and reduce consumption considerably. Once
that occurs, it takes a long time for them to reverse their belief and increase consumption.
In this process, the recovery speed is first slow, then gradually accelerates and eventually
settles down, tracing an S-shaped curve.
The asymmetry occurs because the occurrence of a shock is a discrete event, while therecovery is realized only by “no news is a good news,” which takes time. To highlight
this difference, we employed a continuous-time model of Bayesian learning, while mostpreceding models of business cycles are developed in discrete-time settings. Although ourstrategy was beneficial in making an intuitive explanation of the asymmetry, there aresome limitations. In particular, because we need to solve the continuous-time dynamics
with the possibility of jumps, the model will become quickly intractable if we incorporate

27

various factors that are common in linearized DSGE models.
To address the technical difficulty, we chose to make two simplifying assumptions inthis paper. First, we assumed a reduced-form relationship between the inflation rate andthe output gap, rather than explicitly specifying the process of price adjustments. Second,
the stochastic money-in-utility function in our model (equation 16) did not incorporate
the diminishing returns to money holdings. The latter simplification enabled us to derive
the consumption function that depends only on their belief, C(θte ). However, by thisassumption, the current analysis in effect abstracted from the role played the amount ofmoney supply. Still, we believe that our mechanism of learning in a continuous-time setting
gives an intuitive explanation of business cycle asymmetry and slow recovery, especiallywhen the cycle is driven by discrete events, such as a sudden collapse of key financial
institutions. We hope this paper serves as a starting point for future work that relaxes thelimitations.

Appendix A: Proof of Proposition 1
Proof of C(θ) to be downward sloping
In the appendices we use θ instead of θte to minimize notation. Before starting the proof we definefunction D(θ) by 0u (C(h(θ)))−1 .
D(θ) ≡ R(θ, C(θ)) − ρ + θ
u0 (C(θ))(30)

Since D(θ) is the expression in brackets of (27) and (11) shows g(θ) to be negative for all θ ∈(θ∗ , θH ],C 0 (θ) S 0 ⇐⇒ D(θ) T 0.

(31)

Using function D(θ) we first prove
Lemma 1 Suppose that C(θ) satisfies (27) 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 ∈ (θ0 , θH ].
Proof: If Lemma 1 does not hold and hence C(·) is weakly upward sloping somewhere in (θ0 , θH ],either of the following must be the case.

28

(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 exists some θA ∈ [θ0 , θH ) and θB ∈ (θA , 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 an interval (θA , θB ] in which function C(θ) is
weakly increasing, θA is either a local minimum or θ0 , and θB is either a local maximum or θH .
If there are multiple intervals of that, we choose the rightmost one. We shall find neither (i) nor(ii) to be valid.
We first show that case (i) leads to a contradiction. Since Rc > 0 from Assumption 3 andRθ > 0 from (26), in case (i),lim R(C(θ), θ) < R(C(θH ), θH ).

θ→θA

(32)

Since h(θA ) ∈ (θA , θH ), limθ→θA C(θ) ≤ C(h(θA )) whereas C(θH ) = C(h(θH )) since h(θH ) = θHfrom (13). Since applying these properties and (32) to (30) implies 0 = limθ→θA D(θ) < D(θH )in case (i), from (31) we find C 0 (θH ) < 0, which contradicts (i).
In case (ii) limθ→θA C(θ) ≤ C(θB ). Since Rc > 0 from Assumption 3 and Rθ > 0 from (26),this inequality implies
lim R(C(θ), θ) < R(C(θB ), θB ).

θ→θA

(33)

Further, h(θA ) is located in either (θA , θB ] or (θB , θH ). If h(θA ) ∈ (θA , θB ], then C(h(θA )) ≥limθ→θA C(θ) since we suppose C 0 (θ) ≥ 0 for all θ ∈ (θA , θB ]. In contrast, C(h(θB )) ≤ C(θB )since h(θB ) ∈ (θB , θH ) and C 0 (θ) ≤ 0 for all θ ∈ (θB , θH ]. Using these inequalities, (30) and (33)
we find limθ→θA D(θ) < D(θB ), which contradicts case (ii).
If h(θA ) ∈ (θB , θH ), 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(θA )) ≥ C(h(θB )).

θ→θA

Thus,


u0 (C(h(θB )))u0 (C(h(θ)))− 1 ≥ max 0, lim−1 .
u0 (C(θB ))θ→θA u0 (C(θ))
Applying this property and (33) to (30) yields limθ→θA D(θ) < D(θB ), which contradicts case(ii). Thus, anyway case (ii) results in a contradiction.

29

From (30), boundary condition (28) is equivalent to limθ→θ∗ D(θ) = 0. By regarding θ∗ as
θ0 in Lemma 1, we find C(θ) that satisfies (27) and (28) to be strictly downward sloping for allθ ∈ (θ∗ , θH ].

The existence of upper and lower bounds for C(θ)
We first show c and c to be unique and well defined. Since θ∗ < θH and Rθ > 0 from (26), under
Assumption 2,
lim R(θ∗ , c) ≤ lim R(θH , c) < ρ < lim R(θ∗ , c) ≤ lim R(θH , c).

c→0

c→∞

c→0

c→∞

Applying this property and Assumption 3 to the intermediate value theorem implies that there
are unique and positive c and c satisfying (29). Furthermore, since R(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 (27) and that there exists θ0 ∈ [θ∗ , θH ) satisfying limθ→θ0 D(θ) =0. Then, under Assumptions 2 and 3, C(θ) ∈ [c, c] for all θ ∈ (θ0 , θH ].
Proof: As shown by Lemma 1, the last term in (30) is positive when θ → θ0 and hence
lim R(θ, C(θ)) < ρ.

θ→θ0

(34)

From the first equation of (29), (26), and Assumption 3, R(θ, c) > ρ for all θ ∈ [θ∗ , θH ] and c > c.
Thus, (34) implies limθ→θ0 C(θ) ≤ c.
When θ = θH , the last term in (30) equals zero since h(θH ) = θH from (13). Since C 0 (θH ) ≤ 0
from Lemma 1, (31) implies D(θH ) ≥ 0. Applying these properties to (30) yields
R(θH , C(θH )) ≥ ρ.
Comparing this property with the second equation of (29) 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 , θH ].



Under condition (28), θ∗ satisfies the requirement for θ0 in Lemma 2. Thus, C(θ) ∈ [c, c] forall θ ∈ (θ∗ , θH ].

30

The uniqueness of C(θ)
e cH ) be the solution to differential equation (27) that satisfies the following boundary
Let C(θ,condition:e H , cH ) = cH ,
C(θ

(35)

where cH (> 0) is an arbitrary constant. We can solve differential equation (27) backward fromθ = θH with boundary condition (35), because h(θ) is larger than θ and thus C(θ) and C(h(θ))
e cH ) isare already known when we calculate the gradient of C(·) at θ.24 Therefore, function C(θ,uniquely determined within interval (θ∗ , θH ].
e
Using function C(·),
boundary condition (28) can be rewritten ase cH ) = 0, wherelim D(θ,

(36)

θ→θ∗

!

e cH ) ≡ R(θ, C(θ,e cH )) − ρ + θD(θ,eu0 (C(h(θ),
cH ))
−1 .
e cH ))
u0 (C(θ,(37)

e cH ) is the expression in brackets of (27) with C(θ) being replaced by C(θ,e cH ). FunctionsD(θ,e and D(·)
e have the following properties.25
C(·)
ec (θ, cH ) > 0 and (b) D
e c (θ, cH ) > 0 for all θ ∈ (θ∗ , θH ]Lemma 3 Under Assumption 3, (a) C
e c (θ, cH ) > Dc > 0 wheneverand cH > 0. In addition, (c) there is a constant, Dc , such that D
e cH ) ∈ [c, c].
C(θ,Proof: By rearranging terms in (27),H e
Hee cH ) = g(θ) γ(C(θ, c ))Cθ (θ, c ) = −g(θ)Mθ (θ, cH ),D(θ,e cH )C(θ,e cH ) .
where M (θ, cH ) ≡ ln u0 C(θ,

(38)
(39)

Differentiating (38) with respect to cH yields
e c (θ, cH )/g(θ).
Mθc (θ, cH ) = −D
24

The numerical analysis follows from this and is given in Appendix B.

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

31

(40)

Differentiating (37) with respect to cH givese c (θ, cH ) = Φ(θ, cH ) + Ψ(θ, cH ), whereD
0 eHe cH )) u (C(θ, c )) Mc (θ, cH ),Φ(θ, cH ) ≡ Rc (θ, C(θ,e cH ))
u00 (C(θ,eu0 (C(h(θ),
cH ))Ψ(θ, cH ) ≡ θMc (h(θ), cH ) − Mc (θ, cH ) .
e cH ))
u0 (C(θ,(41)

ec (θH , cH ) = 1, differentiating (39) with respect to cH yields
Since (35) implies C
Mc (θH , cH ) = u00 (cH )/u0 (cH ) < 0.

(42)

From (13), h(θH ) = θH and thus Ψ(θH , cH ) = 0. Using this property, Assumption 3, (35), (41)and (42) we obtaine c (θH , cH ) = Rc (θH , cH ) > 0.
D
Now we extend this property to all θ ∈ (θ∗ , θH ]. To prove this, suppose otherwise. Then, theree c (θ, cH ) > 0 for all θ ∈ (θA , θH ] and De c (θA , cH ) ≤ 0.
should be some θA ∈ (θ∗ , θH ) that satisfies DThis property, combined with (40), (42) and the negativity of g(θ) from (11), gives
Mc (θA , cH ) = Mc (θH , cH ) +Z θHe c (θ, cH )/g(θ) dθ
D

θA
00

H

0

(43)

H

< u (c )/u (c ) < 0.
Assumption 3, (41) and (43) imply Φ(θA , cH ) > 0. Similarly, from (40),A

HA

Z h(θA )H

Mc (h(θ ), c ) − Mc (θ , c ) = −θAe c (θ, cH )/g(θ) dθ > 0,D(44)e c (θA , cH ) > 0, which
which means Ψ(θA , cH ) > 0. Substituting these results into (41) yields D
e c (θA , cH ) ≤ 0. Thus there is no such θA , and therefore property
contradicts the assumption that D(b) holds.
Property (b) and (40) implyMθc (θ, cH ) > 0

for all θ ∈ (θ∗ , θH ] and cH > 0.

(45)

From (42) and (45),Mc (θ, cH ) < u00 (cH )/u0 (cH ) < 0

for all θ ∈ (θ∗ , θH ].

ec (θ, cH ) > 0 from (39), property (a) holds.
Since (46) is equivalent to C

32

(46)

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∗ , θH ] × [c, c]. SinceΘ is a compact set, there exists Dc ≡ min(θ,c)∈Θ Φ(θ, c) > 0. Combined with (41), (45) and (46),this property implies
0 e
H
00 H
e c (θ, cH ) > Φ(θ, cH ) > Rc (θ, C(θ,e cH )) u (C(θ, c ))u (c ) ≥ Dc ,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. Suppose that thereH
are two distinct functions C1 (θ) and C2 (θ), both of which satisfy (27) and (28). Let cH
1 ≡ C1 (θ )
H
H
H
∗ H
e
e
and cH
2 ≡ C2 (θ ). Then C1 (θ) = C(θ, c1 ) and C2 (θ) = C(θ, c2 ) for all θ ∈ (θ , θ ]. Note thatHcH
1 6= c2 because we have assumed that C1 (θ) and C2 (θ) are distinct functions. Since (28) is

equivalent to (36), both functions satisfyHe cH
elim D(θ,1 ) = lim∗ D(θ, c2 ) = 0.

θ→θ∗

θ→θ

(47)

e cH ), C(θ,e cH ) ∈ [c, c] for all θ ∈ (θ∗ , θH ]. Applying it to property (a) of
From Lemma 2, C(θ,12e cH ) ∈ [c, c] for all cH ∈ [cH , cH ] and all θ ∈ (θ∗ , θH ]. Thus, we can use
Lemma 3 implies C(θ,12
property (c) of Lemma 3 to obtain
H
H
H
e cH
e
lim |D(θ,1 ) − D(θ, c2 )| > |c1 − c2 |Dc > 0,

θ→θ∗

(48)

which contradicts (47).

The existence of C(θ)
e cH ).
We obtain another property with respect to D(θ,e c) > 0 for all θ ∈ (θ∗ , θH ).
e c) < 0 and (b) D(θ,Lemma 4 Under Assumptions 2 and 3, (a) D(θ,Proof: We first prove property (a). From (26) and (29),
R(θH , c) = ρ and R(θ, c) < ρ for all θ ∈ (θ∗ , θH ).

33

(49)

Since (13) implies h(θH ) = θH , substituting (35) and (49) into (37) yields
e H , c) = 0.
D(θ

(50)

eθ (θH , c) = 0, and Rθ > 0 from (26), differentiating (37) with respect
Since (38) and (50) imply C
to θ when θ = θH givese θ (θH , c) = Rθ (θH , c) > 0.
D(51)e c) < 0 for all θ ∈Equations (50) and (51) show that there is a small ε (> 0) such that D(θ,(θH − ε, θH ).
e c) to the whole interval of (θ∗ , θH ). To see this, suppose
We now extend the negativity of D(θ,e c) < 0 for all θ ∈ (θA , θH )otherwise. Then, there must be some θA ∈ (θ∗ , θH ) such that D(θ,e A , c) ≥ 0. Since D
e <0⇔Ceθ > 0 from (38) and the negativity of g(θ) in (11), weand that D(θA ), c) < C(θe A , c) < C(h(θee H , c) = c. With Assumption 3 and (49), these inequalities
obtain C(θe A , c)) < R(θA , c) < ρ and hence from (37) D(θe A , c) < 0, which is a contradiction.
yield R(θA , C(θNext we prove property (b). From Assumptions 2 and 3,R(θ, c) > ρ for all θ ∈ (θ∗ , θH ].

(52)e H , c) > 0 from (37) since h(θH ) = θH from (13). To prove that this inequalIt implies D(θity actually holds for whole (θ∗ , θH ], suppose otherwise. Then, there should be some θA ∈e c) > 0 for all θ ∈ (θA , θH ] and that D(θe A , c) ≤ 0. From (31), we find(θ∗ , θH ) such that D(θ,A ), c) > C(θe A , c) > C(h(θee H , c) = c. With Assumption 3 and (52), these inequalities implyC(θe A , c)) > R(θA , c) > ρ and therefore from (37) D(θe A , c) > 0, which is again a contradicR(θA , C(θ
tion.



Applying property (b) of Lemma 3 and Lemma 4 to the intermediate value theorem assures
e 0 , cH ) = 0.
that, for any given θ0 ∈ (θ∗ , θH ), there uniquely exists cH ∈ [c, c] that satisfies D(θThat is, there is a unique function, cH = ζ(θ0 ), satisfying
e 0 , ζ(θ0 )) = 0 for all θ0 ∈ (θ∗ , θH ).
D(θ

(53)

e ζ(θ0 )) is monotonic and bounded by [c, c] within interval
From Lemmata 1 and 2, function C(θ,e cH∗ ) is monotonicθ ∈ (θ0 , θH ]. By taking limit as θ0 → θ∗ , we conclude that C(θ) = C(θ,34and bounded for all θ ∈ (θ∗ , θH ], where cH∗ ≡ limθ0 →θ∗ ζ(θ0 ).26 This implies ct should neither
explode, implode nor oscillate. Thus ċt /ct → 0 as θte → θ∗ , which gives the validity of boundary
condition (28).

Appendix B: Numerical Procedure for 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.27 However, wecannot use them since the right-hand side of (27) contains C(h(θ)), which makes it impossible
to calculate the gradient of C(θ) before C(h(θ)) is determined. Since h(θ) is always larger thanθ, we cannot solve the differential equation forward from θ∗ , where the boundary condition is
given, toward θH . Instead, 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 thestarting point, cH ≡ C(θH ), is not predetermined. Thus, we have to find an appropriate initialvalue cH such that boundary condition (28) is eventually met when (27) is solved from it.28 Thismethod is in fact used when we prove proposition 1 in Appendix A, where we show that there
is a unique cH∗ ∈ [c, c] that satisfies this property. 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 (27) backward
starting from boundary value C(θH ) = cHi .
Step 3. If C(θ) exceeds c during the calculation, or if D(θ) defined by (30) remains positive0
when θ approaches θ∗ , let hi+1 = cH
i and li+1 = li . Conversely, if C (θ) becomes positive
26

Note that ζ(θ) is bounded by c and c from Lemma 2. In addition, by totally differentiating (53) andutilizing Lemma 1, we can prove that ζ(θ) is continuous and monotonic for all θ ∈ (θ∗ , θH ]. It means thatζ(θ0 ) does not oscillate as θ0 → θ∗ , hence the existence of cH∗ ≡ limθ0 →θ∗ ζ(θ0 ) is guaranteed.
27

See, for example, Judd (1998).

28

This method is usually called ‘monkey hunting’.

35during the calculation or if D(θ) remains negative when θ approaches θ∗ , let hi+1 = hi andH
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. Then admit cH =(hi + li )/2 as the solution.

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