Chukyo University Institute of Economics
Discussion Paper Series
February 2024

No. 2201
Time inconsistency of a Central Bank in a Monetary
Search Model with Agent's Time inconsistency

Daiki Maeda

Time Inconsistency of a Central Bank in a Monetary
Search Model with Agent’s Time Inconsistency*
Daiki Maeda
February 8, 2024

Abstract
I incorporate the time-inconsistent preference for hyperbolic discounting into amonetary search model following Lagos and Wright (2005) and use it to analyze two
economies. One economy consists of sophisticated agents who understand their time
inconsistency, whereas the other consists of naı̈ve agents who do not understand theirtime inconsistency. I extend previous analyses of this topic by considering two monetary policy rules: inﬂation targeting under which the target variable is the inﬂation
rate and nominal growth rate targeting under which the target variable is the growthrate of gross domestic product. Through this analysis, I show that inﬂation targeting isa time-inconsistent monetary policy rule in the economy consisting of the naı̈ve agentseven if there is no uncertainty.

Keywords: Hyperbolic discounting; Monetary search model; Monetary policy; Inﬂation targeting
JEL classiﬁcation: E52; E70* I would like to thank Ryoji Hiraguchi and the participants of the 2022 Fall Meeting of the Japanese

Economic Association at Keio University and Nagoya Macroeconomics Workshop for their helpful comments
and suggestions on the previous version of this paper. I am ﬁnancially supported by a grant-in-aid from
Zengin Foundation for Studies on Economics and Finance.
 School of Global Studies, Chukyo University, 101-2 Yagoto-Honmachi, showa-ku, Nagoya, Aichi, 4668666, Japan. E-mail: d-maeda@mecl.chukyo-u.ac.jp1

1

Introduction

Central banks suﬀer time inconsistency because of the trade-oﬀ between inﬂation and output.
If the central bank has the discretion to decide on monetary policy, it has an incentive toraise inﬂation from its pre-planned rate to increase output. Since high inﬂation hurts agents’
welfare, many studies suggest policy rules to avoid it (e.g., inﬂation targeting under which
the target variable is the inﬂation rate). However, previous studies analyze only the central
bank’s time inconsistency and not that of the agent. If agents have time inconsistency, they
may change their planned behavior, meaning that the central bank may also need to change
its predetermined monetary policy. Therefore, in this study, I investigate monetary policy
rules in the economy with agents’ time inconsistency.
In the presented theoretical analysis, I incorporate hyperbolic discounting, a well-known
discounting method that causes time inconsistency, into the monetary search model provided by Lagos and Wright (2005), called the LW model hereafter. I consider two types of
economies. One consists of sophisticated agents who understand their time inconsistency(called the sophisticated agent economy hereafter) and the other consists of naı̈ve agents
who do not understand their time inconsistency (called the naı̈ve agent economy hereafter).
These economies do not face macroeconomic shocks.
This study’s main contribution is to extend previous analyses of this topic by considering
two monetary policy rules: inﬂation targeting under which the target variable is the inﬂation
rate and nominal growth rate targeting under which the target variable is the growth rateof gross domestic product (GDP). I show that inﬂation targeting is a time-inconsistentmonetary policy rule in the naı̈ve agent economy because the target of the inﬂation rate
is not achieved by the central bank that maximizes agents’ welfare. Because of this time
inconsistency of the central bank, this result shows that its monetary policy may not be
trusted by agents and that inﬂation targeting may be unable to fulﬁll its role, which stabilizes
agents’ expectations. This result is not obtained in the sophisticated agent economy andwhen the central bank adopts nominal growth rate targeting.
2

I review related literature to bring a perspective. The standard models discuss monetary
policy rules by deﬁning or deriving a loss function. This increases if the inﬂation rate or
the output gap increases. Since there are several pieces of literature using the loss function,
I study a recent paper. Billi (2017) discusses the optimality of GDP targeting when thenominal interest rate hits the zero lower bound by using the New Keynesian model. Anotherliterature that discusses the central bank’s targeting rules is reviewed by Walsh (2010).
Some studies examine policy rules or the stabilization of monetary policy by using themonetary search model. Berentsen and Waller (2011) show that price-level targeting is the
optimal policy rule for the short term. If the current price deviates from the long-term
price path due to shocks, the central bank controls the amount of money to be injected
to return the price to the target level. Berentsen and Waller (2015) discuss that when
the probability of a successful transaction is determined by the number of agents entering
the market, the optimal stabilization monetary policy can be derived. Boel and Waller(2019) discuss the optimal stabilization policy at the zero lower bound when agents faceidiosyncratic preference shocks. However, while these studies focus on the buyer’s liquidity
constraint, which is relaxed when the central bank injects money, my study focuses on theagent’s forecast error due to their time-inconsistent preference.
Some studies assess monetary policy rules in the economy in which agents or ﬁrms cannot
correctly predict the future. Ball, Mankiw, and Reis (2005) develop a model in which none of
the ﬁrms can immediately obtain the required information to set their prices, as in Mankiw
and Reis (2002). They indicate that price-level targeting is the more optimal policy thaninﬂation targeting. Agliari, Massaro, Pecora, and Spelta (2017) study a New Keynesianmodel in which there are costs to obtain information on economic states, as in Sims (2003).
They also prove that the achievement of the inﬂation rate target depends on the costs.
Angeletos and La’O (2020) study a model in which the signals of the economic state diﬀer
by ﬁrm type, as in Morris and Shin (1998). They show that nominal GDP targeting is moredesirable than price-level targeting. In addition, Grauwe (2011), Ho, Lin, and Yeh (2021),3and Molnar (2020) incorporate heuristics into the New Keynesian model. However, the causeof the agent’s forecast error in these models diﬀers from those in my model.
The following studies examine monetary policy using a behavioral economics model.
Graham and Snower (2008, 2013) and Maeda (2018, 2023) study the case of hyperbolic
discounting. Hori, Futagami, and Morimoto (2021) and Futagami and Maeda (2023) study
non-unitary discounting in which discount rates diﬀer between goods. Hiraguchi (2018)studies a model in which agents have the temptation to spend all their money. However,these studies do not discuss the monetary policy rules presented in my study.
The rest of the paper is structured as follows. Section 2 explains the setting of themodel. Section 3 solves the agent’s optimization problem. Section 4 derives the equilibriumand discusses the optimality of inﬂation targeting. Section 5 concludes.

2

The model

There is a continuum of agents with a unit measure. The agents live inﬁnitely. Time isdiscrete and goes from t = 0 to ∞. Each period has two subperiods: “day” and “night.”
Each subperiod has a goods market and the goods traded in them cannot be carried overfrom one subperiod to the next.
Before the agents enter the day subperiod, they are divided into “sellers” and “buyers.”
The agents become sellers with the probability n ∈ (0, 1) and buyers with the probability
1 − n. During the day, the sellers produce and sell goods and the buyers buy and consume
them. To focus on the agent’s time inconsistency, I simply assume that the goods producedby each seller are the same, following Berentsen, Camera, and Waller (2007). Then, themarket is perfectly competitive1 . This assumption seems to be the same as a centralized
market. However, I call this market a decentralized market (DM) following the original
1

Random matching and Nash bargaining can be incorporated into the decentralized market to determine
the quantity and price of goods as in the original LW model. However, the diﬀerence between the original
setting and setting of the perfectly competitive, decentralized market in my model with time inconsistency isqualitatively the same as the diﬀerence in the model without time inconsistency because the Nash bargaining
process is not relevant to intertemporal decision-making.

4

setting of the LW model. I assume that there are no methods of trade record-keeping.
Therefore, the buyers have to use money to buy the goods.
At night, all the agents simultaneously decide the consumption, labor supply, and money
holdings to be carried over to the next period. These markets are the same as in the standard
model (i.e., a centralized, perfectly competitive market). I call the goods market at nighta centralized market (CM). In this market, a unit of goods is produced by a unit of labor.
Since I assume that the labor market is perfectly competitive and the good traded in theCM (i.e., the CM good) is a numeraire good, the real wage is equal to one.
Money is divisible and storable, but intrinsically useless. Mt is the amount of money
issued before period t. The central bank issues and transfers money to the agents equallyat night via lump-sum transfers. I deﬁne Tt as the transfer of money to agents. Then,
Tt = Mt+1 − Mt .
Next, I explain the agent’s preference. The agent obtains utility from consumption anddisutility from production and labor supply in each subperiod. I assume that the utility
function is additively separable and quasilinear:

U (qtb , qts , xt , ht ) = u(qtb ) − c(qts ) + U (xt ) − ht ,(1)

where u(q) is the utility from consuming q units of goods and c(q) is the cost of producing q
units of goods during the day. U (x) is the utility from consuming x units of goods and h is thedisutility of supplying h units of labor during the night. The functions u, c, and U are twice
continuously diﬀerentiable and satisfy u(0) = c(0) = 0, u′ > 0, c′ > 0, U ′ > 0, u′′ < 0 c′′ ≥ 0,
U ′′ < 0, u′ (0) = U ′ (0) = ∞, and u′ (∞) = U ′ (∞) = 0. h is positive and has the upper boundh̄. As mentioned in the Introduction, future utility is hyperbolically discounted. Therefore,
lifetime utility in period t is denoted by

Zt = u(qtb ) − c(qts ) + U (xt ) − ht + β

∞
∑
i=1

5

[ b
]
s
δ i u(qt+i) − c(qt+i) + U (xt+i ) − ht+i ,(2)

where δ ∈ (0, 1) is the long-term discount factor. Since I assume that β ∈ (0, 1], the agents
evaluate current utility rather than future utility. In other words, β is a parameter denoting
present bias.
Behavioral economics studies consider two types of agents. First, sophisticated agents
understand that their preference changes over time. In this model, they know that their future selves will also have the present bias parameter β and resolve the optimization problem.
In other words, they know that their future selves will maximize (2). Second, naı̈ve agents
do not understand that their preference changes over time. In this model, they understandthat their future selves will not have present bias and their current decision is not changedby their future selves. In other words, they understand that their future selves will maximize∑∞ ibsi=1 δ U (qt+i , qt+i , xt+i , ht+i ). I summarize the agents’ types as follows.
Deﬁnition 1. In this model, sophisticated and naı̈ve agents predict their future behavior as
follows
1. Sophisticated agents predict that their future selves will maximize (2).
2. Naı̈ve agents predict that their future selves will maximize

∑∞

j
b
s
i=1 δ U (qt+i , qt+i , xt+i , ht+i ).

In this study, I consider two cases: all the agents are sophisticated and all the agents arenaı̈ve.

3

The agent’s optimization problem

3.1

The current problem

Before I discuss the agent’s optimization problem, note that I omit the variables’ subscript,
t, and add +1 to the variables in the next period.
I start by solving the seller’s problem by using the Bellman equation. An individual
experiences disutility c(q s ) by producing the goods during the day (i.e., DM goods) and6

experiences net utility U (x) − h during the night. I deﬁne the seller’s problem as follows:
V0s (m) = s max [−c(q s ) + U (x) − h + βδV (m+1 )],q ,x,h,m+1(3)where m denotes money holdings. V (m) denotes the expected value obtained from future
consumption and production (i.e., the sum of the expected discounted future utility). The
deﬁnition of V (m) is diﬀerent for sophisticated and naı̈ve agents, as explained after thissubsection. The agent obtains nominal income pq s , where p is the nominal price of the DM
goods from production during the day. When ϕ denotes the real price of money (the inverseof the price of the goods in the CM), real income is given by ϕpq s . As mentioned before, thereal wage at night is one. Thus, the seller’s budget constraint is denoted by

h = x + ϕm+1 − ϕpq s − ϕ(m + T ).

(4)

Substituting (4) into (3), I obtain

V0s (m) = max[ϕpq s − c(q s )] + max[U (x) − x] + ϕ(m + T ) + max[−ϕm+1 + βδV (m+1 )].
s
q

x

m+1

(5)
Since the DM is perfectly competitive, the seller behaves with price ϕp as given. Then, theﬁrst-order condition of q s is
ϕp = c′ (q s ).

(6)

From the second term of (5), the ﬁrst-order condition of x is given by
U ′ (x) = 1.

(7)

Let x∗ denote the value of satisfying (7). Moreover, I assume that U (x∗ ) > x∗ is satisﬁed for7

x∗ to have a positive value. From the last term of (5), I ﬁnd that m+1 does not depend on mor on whether the individual is a seller or a buyer during the day. The ﬁrst-order conditionof m+1 is presented in the next subsection.
Next, I solve the buyer’s problem. Since the buyers are anonymous, they have to holdmoney before they consume it. Therefore, the buyer faces the following constraint:

pq b ≤ m.

(8)

The buyer’s budget constraint is given by

h = x + ϕm+1 + ϕpq b − ϕ(m + T ).

(9)

As for the seller’s problem, the value function of the buyer is calculated by

V0b (m) = max [u(q b ) − ϕpq b ] + max[U (x) − x] + ϕ(m + T ) + max[−ϕm+1 + βδV (m+1 )].
pq b ≤m

x

m+1

(10)
From (10), the buyer’s optimal conditions of x and m+1 are the same as those of the seller.
If (8) binds, q b = m/p. If it does not bind, q b = q ∗ , which satisﬁes u′ (q ∗ ) = ϕp. Hereafter, I
consider the case in which (8) binds2 .

3.2

The sophisticated agent’s expectation and money holdings

I need to know the future value V (m) to determine money holdings. V (m) is the sum of the
expected discounted future utility as follows:

V (m) = E

∞
∑

]
[ b
s
) + U (xt+i ) − ht+i ,) − c(qt+iδ i u(qt+i

(11)

i=0

2

I do not focus on the case in which (8) does not bind because the agent’s welfare in this case is no higher
than the case in which (8) binds.

8

where E is an expectation operator. Since the probabilities of the individuals becoming
sellers and buyers are n and 1 − n, respectively, I can rewrite (11) as follows:bs
V (m) = (1 − n)u(qso) − nc(qso) + [(1 − n)U (xbso ) + nU (xsso )] − [(1 − n)hbso + nhsso ] + δV (mso,+1 ),(12)
where the superscripts b and s, respectively, denote the values when the individual will
become a buyer and a seller in the future, and the subscript so denotes a sophisticated agent’s
expected value. From deﬁnition 1, individuals predict that their future selves maximize (2);that is, the predicted future objective function is the same as the current one. This meansthat their future selves’ behavior is the same as their behavior in the present. Since q s , x
and m+1 do not depend on m (see Section 3.1), I can rewrite the last terms of (5) and (10)as follows:max[−ϕm+1 + βδ{(1 − n)v(m+1 ) + ϕ+1 m+1 }],
m+1

(13)

where v(m) ≡ u(q b ) − ϕpq b , and I omit the variables that do not depend on m+1 . The
optimal condition of money holdings is denoted by[]bu′ (q+1)ϕ= βδ (1 − n)+n .
ϕ+1ϕ+1 p+1(14)

Using the real price of the DM goods obtained by (6), I can rewrite (14) as follows:[]b)u′ (q+1ϕ= βδ (1 − n) ′ s + n .
ϕ+1
c (q+1 )(15)

The left-hand side of (15) is the gross inﬂation rate. Therefore, (15) expresses the relationship between the inﬂation rate and output. As explained in Section 4, the market-clearing
condition of the DM is given by q b = 1−nq s . Moreover, I assume u′′ < 0 and c′′ ≥ 0.
n
Therefore, output decreases if the inﬂation rate increases.
93.3

The naı̈ve agent’s expectation and money holdings

From deﬁnition 1, the naı̈ve agents predict they would maximize

∑∞

j
b
s
i=0 δ U (qt+j+i , qt+j+i , xt+j , ht+j+i )in the future. This is the same as maximizing V (m). Therefore, I can rewrite (11) as follows:

V (m) =

max

b ,q s ,x
qna
na na ,hna

[E

∞
∑

]
[ b
s
) + U (xna,t+i ) − hna,t+i ,δ i u(qna,t+i) − c(qna,t+i

(16)

i=0

where the variables added to the subscript na denote the naı̈ve agent’s expected value. As in
Subsection 3.1, I ﬁrst solve the future seller’s problem. The future seller’s value is denotedby

V s (mna ) = smax

qna ,xna ,hna ,mna,+1s[−c(qna) + U (xna ) − hna + δV (mna,+1 )].

(17)

Substituting the seller’s budget constraint, (4), into (17), I obtainss
V s (mna ) = max[ϕna pna qna− c(qna)] + max[U (xna ) − xna ] + ϕna (mna + Tna )
s
qna

xna

+ max [−ϕna mna,,+1 + δV (mna,+1 )].
mna,+1(18)

From this equation, I obtain the following optimal conditions:s
ϕna pna = c′ (qna
),

(19)

U ′ (xna ) = 1.

(20)

Since (20) is the same as (7), xna = x∗ .

10

The future buyer’s problem is denoted byb
Vna(mna ) =maxb ≤m
pna qnanabb[u(qna) − ϕna pna qna] + max[U (xna ) − xna ] + ϕ(mna + Tna )xna+ max [−ϕmna,+1 + δV (mna,+1 )],mna,+1(21)

s.t. (8) and (9).

bTherefore, since I assume that (8) binds, qna= mna /pna . The optimal condition of xna isdenoted by (20).
Next, I ﬁnd the expected money holdings, mna,+1 . I can rewrite (16) as follows:bs) − nc(qna) + [(1 − n)U (xbna ) + nU (xsna )] − [(1 − n)hbna + nhsna ] + δV (mna,+1 ).
V (mna ) = (1 − n)u(qna(22)
As in the current problem, I can rewrite the last terms of (18) and (21) as follows:max [−ϕna mna,+1 + δ{(1 − n)v(mna,+1 ) + ϕna,+1 mna,+1 }].

mna,+1(23)b= mna /pna , the optimal condition of the expected money holdings is denoted byIf qna[]b)u′ (qna,+1ϕna= δ (1 − n)+n .
ϕna,+1ϕna,+1 pna,+1(24)

I obtain the expected values of the naı̈ve agents from conditions (19), (20), and (24) and use
them to ﬁnd current money holdings. The current problem is the same for the sophisticatedand naı̈ve agents. Therefore, I solve the problem corresponding to (13), which is denoted bymax[−ϕm+1 + βδ{(1 − n)vna (m+1 ) + ϕna,+1 m+1 }],
m+1

(25)

b
b
b
where vna (m+1 ) = u(qna) − ϕna,+1 pna,+1 qna,+1
and qna= m+1 /pna,+1 . I do not have to replace11

mna,+1 with m+1 because the current agents can decide current nominal money holdings.
Solving (25), I obtain]bu′ (qna,+1)= βδ (1 − n)+n .
ϕna,+1ϕna,+1 pna,+1
ϕ

[

(26)

To determine the agents’ expectations, I need their price expectations. The prices are determined in each market. Therefore, I also need an agent’s expectation of the other agents’behavior. I make the following assumption by referring to Gabrieli and Ghosal (2013), Ojima(2017), and Futagami and Maeda (2023).
Assumption 1. A naı̈ve agent predicts that the other agents also behave as naı̈ve agents.
From the above assumption and the agents’ utility functions being the same, the DM goods’
price is given by (19). Substituting this into (24) and (26), I obtain
[

]
′ b
)
u
(q
ϕna
na,+1
= δ (1 − n) ′ s+n ,ϕna,+1c (qna,+1 )
[
]
′ b
)
u
(q
ϕ
na,+1
= βδ (1 − n) ′ s+n .
ϕna,+1c (qna,+1 )

(27)
(28)

These equations express the negative relationship between the expected inﬂation rate andexpected output, as in Subsection 3.2.

4

Equilibrium and policy

4.1

Market-clearing conditions

Since I assume that the probabilities of becoming a seller and a buyer are n and 1 − n,respectively, and there is a continuum of agents with a unit measure, n sellers and 1 − n12buyers exist. Therefore, the market-clearing condition of the DM is denoted by(1 − n)q b = nq s .

(29)Hereafter, I deﬁne q as the buyer’s consumption. Therefore, q s = [(1 − n)q]/n. Since theproductivity of labor is one and the consumption of the CM goods is denoted by (6), themarket-clearing condition of the CM is denoted by(1 − n)xb + nxs = (1 − n)hb + nhs ,(30)

where xj j ∈ b, s is the consumption of the CM good by the buyer and sellers. Since theagents’ money holdings must be equal to the money supply, I obtainm = M.

4.2(31)

Policy rules

In this study, I consider two monetary policy rules: inﬂation targeting and nominal growthrate targeting. First, following previous studies on the LW model, I deﬁne ϕ/ϕ+1 − 1 as theinﬂation rate in this model because ϕ is the reciprocal number of the price of the numeraire
good. Therefore, setting the inﬂation rate as in the following equation is termed inﬂation
targeting herein:
ϕ
= Π,
ϕ+1

(32)

where Π is constant and larger than zero.
Second, since the agents in this economy consume the DM and CM goods and there are13

no investment goods, nominal GDP is given by

GDP ≡ pq +

1 ∗
x.
ϕ

(33)

CM goods consumption is given by x∗ because all the agents consume x∗ units of CM goods
from (7). As nominal growth rate targeting aims to keep the growth rate of (33) constant,the policy rule is given by1
p+1 q+1 + ϕ+1x∗pq + ϕ1 x∗

= g,

(34)

where g is constant and larger than zero.

4.3

The equilibrium path in the sophisticated agent economy

To ﬁnd the equilibrium path in the sophisticated agent economy, I deﬁne the equilibria as
follows.
Deﬁnition 2. Given a monetary policy rule, the equilibria in the sophisticated agent economy consist of the quantities {q, x, m+1 , h} and prices {ϕ, p}, which satisfy the following
conditions.
1. The optimal conditions of the agent’s behavior are given by (6), (7), and (15).
2. The agents’ budget constraints are given by (4) and (9).
3. The market-clearing conditions are given by (29)–(31).
4. Constraint (8) binds.

First, I obtain the equilibria when the monetary policy rules are inﬂation targeting. Since

14

the policy rule of inﬂation targeting is given by (32), I can rewrite (15) as follows:[]u′ (q+1 )Π = βδ (1 − n) ′ ( 1−n ) + n ,c n q+1(35)

IF
where q s = [(1 − n)q]/n from (29). I deﬁne q̄so
as the value of DM goods consumption thatIF
satisﬁes (35). I ﬁnd that q̄so
is unique because u′′ < 0, c′′ ≥ 0, u′ (0) = ∞, and u′ (∞) = 0.

If the central bank promised that the gross inﬂation rate from one period ago to the currentIF
period is equal to Π, current DM goods consumption is equal to q̄so.

Because constraint (8) binds, I can rewrite the left-hand side of (15) in the equilibrium
as follows:()
c′ 1−n
q
q M+1
m+1
ϕpq
= ′ ( 1−nn ).
ϕ+1 p+1 q+1 m
c n q+1 q+1 M(36)

Substituting this into (15) and calculating it, I obtain the equilibrium path of the DM goods
as follows:
(
c

′

[
]
)
)
(
q+1 βδ
1−n1−nu′ (q+1 )
′
q q=c
q+1
(1 − n) ′ ( 1−n ) + n ,nnγc n q+1(37)

IF. Therefore, I obtain
where γ ≡ M+1 /M . Since (35) is satisﬁed, for all periods, q = q̄so

γ = Π.

(38)

From the above discussion, I obtain the following proposition.
Proposition 1. If the central bank adopts inﬂation targeting, the equilibrium is characterized
by the following values:IF
and DM goods supply is given by q s =1. DM goods consumption is given by q b = q̄so
1−n IFq̄ ,n so

2. CM goods consumption is given by x = x∗ ,15

3. The labor supplies of the seller and buyer in the CM are given by()
1 − n IF 1 − n IF
h =x −c
q̄
q̄ ,
n so
n so
(
)
1 − n IFb∗′
IFh =x +cq̄so q̄so
,
n
s

∗

′

(39)
(40)

4. The gross growth rate of the money supply is given by γ = Π,
5. The gross growth rate of the nominal price of the DM goods, p, is given by p+1 /p = Π.

Proof. I only show points 3 and 5 because the other values are discussed before this proposition.
IFq̄ IF , x = x∗ ,First, I show that point 3 is correct. Substituting (6), q b = q̄so
, q s = 1−nn so(31), and M+1 = γM into (4) and (9), I obtain (39) and (40).
Next, I show point 5 is correct. From (6), I obtain p = c′ (q s )/ϕ. Since q s = 1−nq̄ IF forn so
all t (38) is satisﬁed, I obtain the gross growth rate of p:()
IFc′ 1−n
q̄
p+1
ϕ
so
n
)
= ′ ( 1−n= Π.
IFpc n q̄so ϕ+1(41)

Next, I obtain the equilibria when the monetary policy rule is nominal growth rate
targeting. Substituting (6) and q s = 1−nq into (34) and solving the equation for ϕ/ϕ+1 , Inobtain the following equation:()
c′ 1−nq q + x∗ϕn
= g ′ ( 1−n ).
ϕ+1c n q+1 q+1 + x∗

(42)

Substituting (42) into (15), I obtain
(
c′

]
) [
)
( (
)
′
u
(q
)
1−n
1 ′ 1−n+1q q=
cq+1 q+1 + x∗ βδ (1 − n) ′ ( 1−n ) + n − x∗ .
ngn
c n q+1
16(43)

From this equation, I obtain the following proposition.
Proposition 2. If the central bank adopts nominal growth rate targeting, the positive DM
GDPgoods consumption in the steady state, q̄so
, is unique.

Proof. If the economy is in a steady state and DM goods consumption is positive, I canrewrite (43) as follows:u′ (q̄ GDP )g/(βδ) − n)=( 1−nso,GDP1−nc′ n q̄so

(44)

GDP
GDP
where q̄so
is DM goods consumption. q̄so
is unique because u′′ < 0, c′′ ≥ 0, u(0) = ∞,and u′ (∞) = 0.
I also obtain the following proposition about the stability of the steady state.
Proposition 3. The steady state is unstable when DM goods consumption in the steady stateGDP.
is equal to q̄so

Proof. Totally diﬀerentiating (43), I obtain
)
)]
[ (
(
1−n
dq+1
g
1−n
1−n
′
′′
=
q
q+c
q
c
dq
βδ
n
n
n
}
[{ (
)
(
)} {
′
1
−
n
1
−
n
1
−
n
u
(q
)
+1
×
c′′
q+1
q+1 + c′q+1
(1 − n) ′ ( 1−n ) + nnnnc n q+1
(
)
(
) ]−1
{ (
)
}
1−n
′′
′
′
′′ 1−n
u
(q
)c
q
q
−
u
(q
)c
1
−
n
+1
+1
+1
+1
n
n
. (45)+ c′
q+1 + x∗ (1 − n)
{ ( 1−n )}2′nc n q+1
From (43), one of the steady states is q = q+1 = 0. When the above equation is evaluatednear this steady state, I obtain
dq+1|q=q+1 =0 = 0
dq

(46)

because u′ (0) = ∞. In the other steady state, ϕ/ϕ+1 = g from (42). Therefore, from (15), I
17

obtain[]GDP)
ϕu′ (q̄sog== βδ (1 − n) ′ ( 1−n GDP ) + n .
ϕ+1
c n q̄so(47)

GDP
Using this relationship, (45) when q = q+1 = q̄so
can be rewritten by

dq+1
GDP
|
dq q=q+1 =q̄so

(
)
(
) −1
)
}
{ (
′′ GDP ′ 1−n GDP′ GDP ′′ 1−n GDP)c n q̄sou (q̄− u (q̄so )cq̄
1 − n GDP
GDP
n so
[ (
q̄so
+ x∗ (1 − n) ( so=  1 + c′q̄so)()]  .
)
21−n1−n GDP1−n GDP1−n GDP
n
′
′′
′
c
+c
c
q̄
q̄
q̄
q̄
n

so

n

n

so

n

so

(48)
GDP ′
Because u′ > 0, c′ > 0, u′′ < 0 and c′′ ≥ 0, u′′ (q̄so)c)( 1−n GDP )(GDP ′′ 1−n GDPq̄− u′ (q̄so
)c
q̄
<
n so
n so

0. Because the steady state is unique from Proposition 8, I can draw Figure 1. Therefore,

Figure 1: Phase diagram of qGDP.
the steady state is unstable when q = q+1 = q̄soThis proposition shows that there are two equilibria at which the economy converges to q = 0
18and initially jumps to the steady state at which q is positive. Hereafter, I assume that thelatter equilibrium is selected and obtain the following proposition.
Proposition 4. If the central bank adopts nominal growth rate targeting, the equilibrium ischaracterized by the following values:GDP
1. DM goods consumption is given by q b = q̄soand DM goods supply is given by q s =1−n GDPq̄,n so

2. CM goods consumption is given by x = x∗ ,3. The labor supplies of the seller and buyer in the CM are given by()
1 − n GDP 1 − n GDP
h =x −c
q̄
q̄
,
n so
n so
)
(
1 − n GDP
b
∗
′
GDP
q̄
h =x +c
q̄so
,
n so
s

∗

′

(49)
(50)

4. The gross inﬂation rate is given by ϕ/ϕ+1 = g,
5. The gross growth rate of the nominal price of the DM goods, p, is given by p+1 /p = g,
6. The gross growth rate of money is given by γ = g.

Proof. I only show point 6 because the other values are discussed before this proposition or
found by using the same method as in Proposition 1.
Since I assume that pq = M for all t, (p+1 q+1 )/(pq) = M+1 /M = γ. Therefore, thefollowing equation is satisﬁed in the steady state:γ=GDP
p+1 q̄so
p+1
= g.
=
GDP
pq̄so
p

19

(51)

From Propositions 1 and 4, I ﬁnd that the central bank achieves the target inﬂation rate
or target nominal growth rate, and monetary policy is time-consistent although the agents
have a time-inconsistent preference. This is because the sophisticated agents correctly predict
their behavior. Then, the agents do not change their behavior from the pre-planned behavior,and the central bank does not need to change its monetary policy plan.

4.4

The equilibrium path in the naı̈ve agent economy

From Assumption 1, all the agents behave the same. Therefore, the expected market-clearing
conditions are denoted bysb= nqna,(1 − n)qna(52)

x∗ = (1 − n)hbna + nhsna ,(53)

mna = Mna .

(54)b= mna /pna are satisﬁed, I can deﬁne the equilibria as
Since the above conditions and qna

follows.
Deﬁnition 3. The equilibria in the naı̈ve agent economy consist of the quantities{q, x, qna , xna , m+1 , mna,+1 , h, hna } and prices {ϕ, ϕna , p, pna }, which satisfy the following conditions:
1. The optimal conditions of the agent’s behavior are given by (6), (7), and (28).
2. The optimal conditions of the expected agent’s behavior are given by (19), (20), and(27).
3. The market-clearing conditions are given by (29)–(31).
4. The expected market-clearing conditions are given by (52) and (53).
5. Constraint (8) binds.
20

First, I discuss the case of inﬂation targeting. If the central bank adopts inﬂation targeting, the following equation is satisﬁed from (27):]IFu′ (q̄na,long)) +n ,
Π = δ (1 − n) ′ ( 1−n IF
c n q̄na,long
[

(55)

IF
where q̄na,longis the “long-run” expected DM goods consumption which satisﬁes (55) 3 . BeIF
cause (27) is satisﬁed after two periods from the current period, I call q̄na,long
the long-run

expected DM goods consumption. Following this deﬁnition, we can rewrite (27) in the caseof inﬂation targeting as follows:
(
c′

[
]
)
(
) IF
′ IF
q
δ
u
(q
)
1
−
n
1 − n IFna,longna,long) + n , (56)
= c′
q IF
q
q IF
(1 − n) ′ ( 1−n IFIFn na,long na,longn na,long γna,longc n qna,longIF
where γna,longis the long-run expected gross growth rate of money. Substituting (55) intoIF
this equation and solving it for γna,long, I obtainIFγna,long
=Π

(57)

IF
. Because
Next, I discuss the “short-run” expected DM goods consumption, q̄na,shortIF
ϕ/ϕna,+1 = Π, I obtain q̄na,shortfrom (28), which satisﬁes the following equation:[]IFu′ (q̄na,short)) +n .
Π = βδ (1 − n) ′ ( 1−n IF
c n q̄na,short(58)

From (27), the following relationship must be satisﬁed:
(
c′

[
]
(
)
) IF
′ IF
δ
)
q
u
(q
1 − n IF1−nna,longna,long) +n ,
q
q IF
= c′
q IF
(1 − n) ′ ( 1−n IFIFn na,short na,shortn na,long γna,shortc n qna,long
(59)

3 IF
q̄na,long is unique because u′′ < 0, c′′ ≥ 0, u′ (0) = ∞, and u′ (∞) = 0.
IFq̄na,longis constant because Π does not change over time.

21

This uniqueness means thatwhere γna,short is the short-run expected gross growth rate of money. Substituting (55) into
this equation and solving it for γna,short , I obtain
)

(

IF
q IF
qna,longc′ 1−nn na,longIF).
γna,short= Π ′ ( 1−nIFIFqna,shortc n qna,short

(60)

IF
Next, I discuss current DM goods consumption, q̄na
. I can rewrite (28) as (56):(c′

1 − n IF
q̄
n na

(

)
IF
= c′
q̄na

1 − n IFqn na,short

)

[
]
IF
IF
qna,shortβδu′ (qna,short)) +n .
(1 − n) ′ ( 1−n IFγ
c n qna,short

(61)

IF
q̄na
is determined to satisfy (61). Then, I obtain the following proposition.

Proposition 5. If the central bank adopts inﬂation targeting and determines γ, the equilibrium is characterized by the following values:
1. Realized, short-run expected, and long-run expected DM goods consumption are given byIFIFIF, q̄na,shortand q̄na,long, while the realized, short-run expected, and long-run expectedq̄na

DM goods supply are given by 1−nq̄ IF , 1−nq̄ IF, and 1−n
q̄ IF
.
n na
n nas hortn na,long
2. Realized and expected CM goods consumption is given by x = xna = x∗ ,3. The realized, short-run expected, and long-run expected labor supply of the seller andIFIFIFbuyer in the CM are given by substituting q̄na
, q̄na,shortand q̄na,long
into the following

equations:
)
1−n
1−n
q
q,
h =x −cnn()
1−nh b = x∗ + c′
q q,
n
(

s

∗

′

4. The long-run expected gross growth rate of the money supply is given by γna,long = Π
and the short-run expected gross growth rate of the money supply is given by γna,short =c′ ( 1−n q IF
)qIF
Π c′ 1−nn qIFna,long qna,long,IF( n na,short ) na,short
22

5. The long-run expected gross growth rate of the nominal price of the DM goods is given
IF
by pIF
long,+1 /plong = Π and the short-run expected gross growth rate of the nominal priceIFc′ ( 1−nqna,long)nIFof the DM goods is given by pIF.
long,+1 /pshort = Π c′ ( 1−n q IF
na,short )nProof. I only show point 5 because the other values are discussed before this proposition or
found by using the same method as in Proposition 1.
From (6), I obtain p = c′ (q s )/ϕ. Since the long-run expected DM goods consumption iss
qnalong= 1−nq̄ IF
for all t , I obtain the gross growth rate of p:n na,long()
IFc′ 1−nq̄
p+1,long
ϕna,longn) na = Π.
= ′ ( 1−nIFplongc n q̄na,long ϕna,+1(62)Because p = c′ (q s )/ϕ, the short-run expected, long-run expected DM goods consumption areIFIFgiven by q̄na,shortand q̄na,long, and ϕ/ϕna,+1 = Π, I obtain the following equation:

)
(
IF
q̄
c′ 1−np+1,long
n na,long) Π.
= ′ ( 1−nIFpshortc n q̄na,short(63)

From this proposition, I do not obtain the current growth rate of money γ. Therefore, I haveto determine the policy rule to determine γ. I consider two cases. The ﬁrst case is the centralIFfor all t and uses γ to induce the realized gross inﬂation rate to Π.
bank knows that q b = q̄na

From constraint (8), I obtain p+1 /p = γ. Combining this equation and the seller’s ﬁrst-ordercondition (6), I also obtain ϕ/ϕ+1 = γ. Therefore, I obtain the following proposition.
IFfor all t and uses γ to induce the
Proposition 6. If the central bank knows that q b = q̄na

realized gross inﬂation rate to Π, γ = Π.
Proof. Because the gross realized inﬂation rate is ϕ/ϕ+1 is equal to Π and γ, Π = γ.
IFHowever, the central bank may not know that q b = q̄na
for all t. In other words, it predicts

that the agent’s behavior is given by Proposition 5. Therefore, I consider the second case in23

IF
which the central bank does not know that q b = q̄na
for all t. It adjusts the money supply

to maximize the agents’ welfare. One of the most likely welfare functions is the sum of the
lifetime utility of the sellers and buyers. However, I do not have to consider the future agents’behavior because this is given by Proposition 5. Hence, the welfare function considered by
the central bank is the sum of the current instant utility of the sellers and buyers as follows:(W ≡ (1 − n)u(q) − nc

)
1−n
q ,
n

(64)

where I omit utility from the CM because goods consumption and the labor supply in the
CM do not depend on monetary policy. Then, I obtain the following lemma.
Lemma 1. If q = q ∗ , (64) is maximized.
Proof. The ﬁrst-order condition of (64) is given by(′

u (q) = c

′

)
1−n
q .
n

(65)

Since ϕp = c′ (q s ) from (6), (65) means that q = q ∗ . Therefore, I obtain this proposition.

If the purpose of the central bank is to induce q to q ∗ , I obtain the following proposition.
Proposition 7.

1. To induce q to q ∗ , the central bank determines the gross growth rate

as follows:

IF
γna
=

c′

) IF
( 1−n IFq̄q̄n na,short( 1−n ) na,short Π.
c′ n q ∗ q ∗

(66)

IF
< Π.
2. γnaIF
Proof. I show that point 1 is correct. Substituting qna= q ∗ and (58) into (61) and solving

for γ, I obtain (66).
24

IF
IF
Next, I show point 2 is correct. From (55) and (58), q̄na,short< q̄na,long
because β < 1.
IF
IF
The maximized q is q ∗ . Therefore, q̄na,short< q̄na,long≤ q ∗ . Because this inequality meansIFIFc′ ( 1−nq̄na,short)q̄na,shortn
is smaller than 1, γop < Π.
that
1−n ∗
′
c ( n q )q ∗IF
This proposition means that the target of the inﬂation rate is not achieved because γna

deviates from the condition of Proposition 6. It is easier to obtain information on the historyof the inﬂation rate than to obtain information on consumption. Therefore, the agents
may not believe the central bank’s announcement, and the central bank cannot adopt themonetary policy which maximizes (64). If the naı̈ve agents do not doubt the central bank’s
announcement, the central bank can adopt the monetary policy. This discussion means
whether the optimal policy can be realized depends on how people form their expectations.
Therefore, inﬂation targeting is an unstable monetary policy rule in the naı̈ve agent economy.
Second, I discuss nominal growth rate targeting. As in (42), the expected inﬂation rate
is given by()
qna qna + x∗
c′ 1−nϕnan)= g ′ ( 1−n.
ϕna,+1c n qna,+1 qna,+1 + x∗

(67)

Substituting this equation into (27), I obtain
(
c′

]
)
)
( (
) [
′
1−n
1 ′ 1−n
u
(q
)
na,+1
) + n − x∗ .
qna qna =qna,+1 qna,+1 + x∗ δ (1 − n) ′ ( 1−ncngnc n qna,+1(68)

From this equation, I obtain the following proposition.
Proposition 8. If the central bank adopts nominal growth rate targeting, the positive DM
GDPgood consumption in the steady state, q̄na,long, is unique,

Proof. If the economy is in a steady state and DM goods consumption is positive, I can25

rewrite (68) as follows:u′ (q̄ GDP )g/(δ) − n)
( 1−nna,long
=
,
GDP
1−n
c′ n q̄na,long

(69)

GDP
where q̄na,longis expected DM goods consumption under nominal growth rate targeting.
GDPq̄na,longis unique because u′′ < 0, c′′ ≥ 0, u(0) = ∞, and u′ (∞) = 0.

From this proposition and (27), I obtain[]
GDPu′ (q̄na,long)ϕna= δ (1 − n) ′ ( 1−n GDP ) + n = g.
ϕna,+1c n q̄na,long(70)

Substituting this equation into (28), I obtain
ϕϕna,+1

= βg.

(71)

From this equation, and (42), I obtain(c′

1 − n GDP
q̄
n na

)
GDP
q̄na
=β

( (
)
)
1 − n GDP′GDP∗
q̄na,long q̄na,long + x − x∗ .
c
n

(72)

GDP
GDP
is also constant in the steady state because q̄na,longisFrom this equation, I ﬁnd that q̄na

constant. The other variables are obtained as in the following proposition.
Proposition 9. If the central bank adopts nominal growth targeting and determines g, theequilibrium is characterized by the following values:
GDPGDP
and the curand q̄na,long
1. Realized and expected DM goods consumption are given by q̄na

rent, short-run expected, and long-run expected DM goods supply are given by 1−nq̄ GDPn na
and 1−nq̄ GDP ,n na,long
2. Realized and expected CM goods consumption is given by x = xna = x∗ ,26

3. The realized and expected labor supply of the seller and buyer in the CM are given by
GDP
substituting 1−nq̄ GDP and q̄na,long
into the following equations:
n na

(

)
1−n
1−n
h =x −c
q
q,
n
n
(
)
1−n
b
∗
′
h =x +c
q q,
n
s

∗

′

4. The realized and long-run expected gross inﬂation rates are given by ϕ/ϕ+1 = ϕna /ϕna,+1 =g and the short-run expected gross inﬂation rate is given by ϕ/ϕna,+1 = βg
5. The realized and long-run expected gross growth rates of the nominal price of the DM
goods are given by p+1 /p = pna,+1 /pna = g and the short-run expected gross growth rateGDPq̄na,longc′ ( 1−n)nof the nominal price of the DM goods is given by pna,+1 /p = c′ 1−nβg,
GDP( n q̄na )6. The realized and long-run expected gross growth rates of money are given by γ =γna,long = g and the short-run expected gross growth rate of money is given by γna,short =GDPGDPc′ ( 1−nq̄na,long)q̄na,longnβg.
1−n GDP′GDPc ( n qna )qna
Proof. This proposition is shown by the discussion before this proposition or found by using
the same method as in Proposition 1, Proposition 4, or Proposition 5. Therefore, I do notshow that this proposition is correct.
This proposition shows that the realized nominal growth rate is consistent with that expectedunder nominal growth rate targeting. Therefore, nominal growth rate targeting is a trusted
policy rule by the naı̈ve agents because it has no time inconsistency. This point diﬀers fromthe case of inﬂation targeting, which can only bind the expected variables (i.e., consumption
and the growth rate of money), even though nominal growth rate targeting can bind both the
expected and the current variables. Therefore, the central bank can determine and change
the current money supply from the past plan to adjust current consumption to the optimallevel under inﬂation targeting. This diﬀerence does not happen in the sophisticated agent
27

economy because the agent’s behavior is time-consistent even though the agent has a timeinconsistent preference. Therefore, the central bank in the sophisticated agent economy does
not need to change its pre-determined policy to adjust the agent’s time inconsistency.

5Conclusion

I have extended the monetary search model in which agents have present bias given byhyperbolic discounting. Through this model, in the naı̈ve agent economy, I have shown thatinﬂation targeting may be a time-inconsistent monetary policy rule because the target of theinﬂation rate is not achieved by the central bank which is a welfare maximizer, even thoughnominal growth rate targeting is a time-consistent rule. This result implies that inﬂation
targeting cannot play a role in stabilizing the agent’s expectation. Therefore, nominal growthrate targeting is a better rule than inﬂation targeting if we emphasize this role in the naı̈ve
agent economy. I have also shown that this diﬀerence is not found in the sophisticated agent
economy. This implies that inﬂation targeting can be adopted if the sophisticated agent
economy can be distinguished from the naı̈ve agent economy. However, as both types of
agents exist in the real-world economy, this is impossible. Hence, there is no positive reason
why the central bank adopts inﬂation targeting under agents’ time inconsistency.

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31