ISSN (Print) 0473-453X

Discussion Paper No. 1062

ISSN (Online) 2435-0982

AN ANALYSIS OF MONETARY POLICYIN A MONETARY SEARCH MODELWITH NON-UNITARY DISCOUNTING

Daiki Maeda

Revised July 2020
September 2019

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

An Analysis of Monetary Policy in a Monetary Search
Model with Non-unitary Discounting∗
Daiki Maeda†
July 10, 2020

Abstract
Based on ﬁndings in the behavioral economics literature, I incorporate nonunitary discounting into a monetary search model to study optimal monetary policy.
I apply non-unitary discounting, that is, discount rates that are diﬀerent acrossgoods. With this extension to the model, I ﬁnd that there are cases where optimalmonetary policy deviates from the Friedman rule.

Keywords: Non-unitary discounting; Search; Friedman rule
JEL classiﬁcation: E52, E70

1Introduction

Recent research on behavioral economics ﬁnd that subjective discount rates diﬀer between
goods (hereafter, following Hori and Futagami (2018), the phenomenon is denoted by∗

I would like to thank Koichi Futagami, Makoto Hanazono, Hirokazu Ishise, Noritaka Kudoh, Tarishi
Matsuoka, Toshiki Miyashita, Kohei Okada, Yoshiyasu Ono, Akihisa Shibata, Takashi Shimizu, Katsuya
Takii, Wataru Tamura, Makoto Watanabe, the Search Theory Workshop participants at Nagoya University, the conference participants at Osaka University, and the 2019 Japanese Economic Association Spring
Meeting participants in Musashi University for their helpful comments and suggestions on a prior versionof the paper. I would especially like to thank Tatsuro Iwaisako and Takeshi Ojima for their constructive
remarks and advice on an earlier version of this paper. All remaining errors are mine. This research
is ﬁnancially supported by the Japan Society for the Promotion of Science through Grants-in-Aid for
Scientiﬁc Research (S) Grant No. 15H05728 and the International Joint Research Promotion Program(Osaka University).
†
Institute of Social and Economic Research, Osaka University, 6-1 Mihogaoka, Ibaraki, Osaka, 5670047, Japan. E-mail address: u035866j@alumni.osaka-u.ac.jp

1

“non-unitary discounting”). For example, based on an experiment conducted in Uganda,
Ubfal (2016) reports the existence of non-unitary discounting.
The problem of non-unitary discounting is that it causes time-inconsistent preference.
For instance, suppose that an individual lives inﬁnitely and has the following preferencein period t:

Ut =

∞
∑

βci u(ct+i ) + βli v(lt+i ),i=0where c is consumption, l is leisure, and u(·) and v(·) are the utility functions of consumption and leisure. The diﬀerence between this and the standard models is that thediscount factors of consumption and leisure are diﬀerent. This means that at t = 0, theindividual has the following preference:

U0 = u(c0 ) + v(l0 ) + βc u(c1 ) + βl v(l1 ) + βc2 u(c2 ) + βl2 v(l2 ) + · · · ,(1)

and at t = 1, the individual has a preference of

U1 = u(c1 ) + v(l1 ) + βc u(c2 ) + βl v(l2 ) + βc2 u(c3 ) + βl2 v(l3 ) + · · · .

(2)I focus on the marginal rate of substitution between consumption and leisure in period 1. From (1), the individual’s marginal rate of substitution in period 0 is given by[βc u′ (c1 )]/[βl v ′ (l1 )]. On the other hand, from (2), in period 1, it is given by u′ (c1 )/v ′ (l1 ).
Because the marginal rate of substitutions between consumption and leisure in t = 0 and
t = 1 are diﬀerent, the individual’s preference changes from t = 0 to t = 1. Therefore,
the preference of non-unitary discounting has time-inconsistency.
Because non-unitary discounting has time-inconsistent preference, an individual’s intertemporal decision making may diﬀer from the standard model without non-unitary
discounting. This is shown by Hori and Futagami (2019) and Ohdoi et al. (2015),
which study the general equilibrium model without money. Moreover, these studies showthat the resource allocation is both diﬀerent from the standard model and ineﬃcient. I

2think that non-unitary discounting aﬀects not only the real economy but also the monetary economy. If the individual’s preference has time-inconsistency, the decision making
around consumption is aﬀected. Because money is held for the purpose of consumption,
non-unitary discounting may also aﬀect demand for money. Therefore, in this paper, I
investigate the eﬀect of non-unitary discounting on the monetary economy. As mentioned
before, the resource allocation of an economy with non-unitary discounting is ineﬃcient.
This means that monetary policy may improve welfare. Therefore, the aim of this paper
is to discuss optimal monetary policy.
To achieve the above aims, I introduce non-unitary discounting to a monetary searchmodel provided by Lagos and Wright (2005). In the Lagos-Wright model, each period
is divided into two subperiods. In the ﬁrst subperiod, a decentralized market is open
and individuals must search for a partner with which to trade. Moreover, money is theintermediary of trade in the market. In the second subperiod, a centralized market is open
and perfectly competitive. In each market, individuals obtain utility from consumption
and disutility from supplying goods (labor). In my model, I assume that the discountrates of consumption and supplying the goods in each subperiod are diﬀerent.
Through this extension, I show that the Friedman (1969) rule may be not optimal.
This tends to occur in the following two cases. The ﬁrst case is when the discount rate ofthe disutility from producing goods in the decentralized market is low. In this case, thesum of the discounted disutility from producing is emphasized more. Therefore, welfare
is improved by decreasing the production of goods. Consumption in the decentralized
market can be decreased by raising the inﬂation rate because an increase in the inﬂation
rate raises the cost of holding money to pay for expenditures in the decentralized market.
Therefore, welfare is improved by increasing inﬂation and thus decreasing consumption.
The second case is when the discount rate for the utility of the consumption of goods
traded in the decentralized market is suﬃciently low. The demand for money is higher
than in the unitary discounting case because the individuals want to consume more thenext time they experience the ﬁrst subperiod because of the low discount rate of goods.
In this case, the consumption of goods in the decentralized market is higher than the3

optimal one. As in the ﬁrst case, the high inﬂation rate decreases the money demand andconsumption in a decentralized market, and it improves welfare. The Friedman rule is thatthe inﬂation rate should be minimized to reduce or eliminate loss from holding money. In
both of the above cases, higher inﬂation improves welfare, leading to the conclusion that
the Friedman rule is not optimal.
There are several studies that use experiments and interviews to show that the discount rate diﬀers between goods. Odum and Rainaud (2003) study the diﬀerence in the
discount rates of hypothetical money, food, and alcohol. Additionally, Odum et al. (2006)
investigate this diﬀerence in hypothetical money and food. Attema et al. (2018) study
the diﬀerence in the discount rates of health and money. There are several theoreticalstudies analyzing a monetary economy by employing an element of behavioral economics.
Hori and Futagami (2018) study the monetary economy with non-unitary discounting.
However, they assume that households face a cash-in-advance constraint. My model has
a ﬁner micro-foundation for money demand than their model. Hiraguchi (2018) incorporates temptation preference into the Lagos-Wright model whereby the agents have a desire
to spend all their money, and they experience disutility from suppressing that desire. The
author shows that the Friedman rule may be not optimal. Although my model is similar
to his model, the behavioral economic assumption is diﬀerent. Moreover, the following
studies analyze an economy with hyperbolic discounting: Gong and Zhu (2009), Boulware
et al. (2013), Graham and Snower (2008, 2013), and Maeda (2018a, 2018b). There arealso many theoretical studies that discuss the optimal monetary policy by using a searchmodel, including Aruoba and Chugh (2010), Bhattacharya et al. (2005), Berentsen, Camera and Waller (2007), Berentsen, Racheteau and Shi (2007), Faig and Jerez (2007), Heet al. (2008), Hiraguchi and Kobayashi (2014), Hu et al. (2009), Jeong (2015), Lagos
and Rocheteau (2005), Li (1995), Shi (1997, 2001), and Williamson (2006). These studies
basically focus on market structures and bargaining but do not focus on the discounting
rate.
The remainder of this paper is organized as follows: Section 2 provides the model.
Section 3 addresses the individual’s optimization. Section 4 obtains the equilibrium.

4Section 5 presents an analysis of government policy. Section 6 concludes the paper.

2

The model

In this section, I explain the environment. Time is discrete and runs from t = 0 to ∞.
There exists a continuum of inﬁnitely lived agents with a unit measure. Each period is
divided into two subperiods. The ﬁrst subperiod is called “day” and the second is called“night.”In the day subperiod, the individuals become either a “seller,” who produces goods
with probability n ∈ (0, 1), or a “buyer,” who consumes it with probability 1 − n, before
they enter a market1 . The market is perfectly competitive2 . Following previous studies,
I call the market a decentralized market or DM. The buyers must use money to trade
because it is assumed that buyers and sellers are anonymous.
In the night subperiod, the individuals determine consumption, the supply of labor,
and the amount of money carried over to the next period at the same time. In other words,
the market is a standard centralized perfectly competitive market. I call the market acentralized market or CM. One unit of a good in this market is produced by inputting
one unit of labor.
Money is divisible and storable but intrinsically useless. Mt denotes the amountof money issued before period t, and the growth rate of money is γ − 1. Therefore,
Mt+1 /Mt = γ. Money is issued in the night subperiod, and all money is transferred toindividuals equally. When we deﬁne T as the transfer of money to individuals, we obtain
Tt = (γ − 1)Mt .
In the rest of this section, I explain the assumption about the utility of the individual.
The individual obtains utility from consumption and disutility from production and labor1

In the original setting of the Lagos-Wright model, the individuals enter the market and search fortheir partner. Then, they ﬁnd whether they become a seller or buyer. This point diﬀers from my model.
However, my model’s setting can obtain the same result as the Lagos-Wright model when the pricebargaining power of the buyer is equal to one. Because my model’s setting is simpler than the LagosWright model, I use the simpler one. I show that my model’s setting can obtain the same result as theLagos-Wright model in Appendix C.
2
There are previous studies in which the seller or buyer is determined exogenously, and the market isperfectly competitive. For example, Berentsen, Camera, and Waller (2007).

5

supply in each subperiod. As in the Lagos-Wright model, we assume that the utility
function is additively separable and quasilinear; that isU(qtb , qts , xt , ht ) = u(qtb ) − c(qts ) + U (xt ) − ht ,(3)

where u(q) is the utility from consuming q units of the goods and c(q) is the cost of
producing q units of the goods in the day subperiod. U (x) is the utility from consuming
x units of the goods, and h is the disutility of suppling h units of labor in the nightsubperiod. The functions u, c, and U are twice continuously diﬀerentiable and satisfyu(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 upper bound h̄. The key assumption ofthis paper is that the discount rates for the (dis)utility of consumption and production
are diﬀerent. Therefore, I consider the following lifetime utility:
∞
∑
[ t
]
t
t
t
Z=
βdb u(qtb ) − βdsc(qts ) + βcb
U (xt ) − βcs
ht ,

(4)

t=0

where βbd denotes the discount factor of u(q b ), βds denotes the discount factor of c(q s ), βcb
denotes the discount factor of U (x), and βcs denotes the discount factor of h. As mentioned
in the introduction, non-unitary discounting causes time inconsistency. Therefore, it isimportant to determine whether the individuals are sophisticated or naive. In this paper,
we assume that all individuals are sophisticated.

3

Individual’s optimization

As mentioned in the previous section, there are two states associated with an agent: aseller and a buyer. In this section, we seek the optimal behavior in each state. After
that, we seek the amount of money brought to the next period. In terms of notations, I
omit the subscript t, which represents added variables, except for the case in which it isneeded. Moreover, I add the variable in the next period to +1, for example, z+1 .

6

3.1

Seller’s problem

First, we seek the seller’s optimal behavior. The individual who becomes a seller in periodt can obtain money and experience disutility, c(q s ), in the day subperiod. The seller also
experiences net utility, U (x) − h, in the night subperiod. Therefore, the seller’s valuefunction, which she maximized, is given by

V0s (m) = s max [−c(q s ) + U (x) − h + V (m+1 )],q ,x,h,m+1

(5)

where
V (m+1 ) ≡ βdb Vdb (m+1 ) − βds Vds (m+1 ) + βcb Vcb (m+1 ) − βcs Vcs (m+1 ).

(6)

V (m) is the value that is obtained from future consumption and production. Each element, Vdb (m), Vds (m), Vcb (m), and Vcs (m), denotes the sum of the expected discounted
present value of consuming and producing goods in the future day subperiod, and consuming and supplying labor in the future night subperiod, respectively. These are givenby

Vdb (mt ) = E

∞
∑

j
b
βdb
u(qt+j
),

(7)

j
s
βds
c(qt+j
),

(8)

j
βcb
U (xt+j ),

(9)

j=0

Vds (mt ) = E
Vcb (mt ) = E

∞
∑
j=0
∞
∑
i=0

Vcs (mt ) = E

∞
∑

j
βcs
ht+j ,(10)i=0where E is the expectation operator. The seller’s budget constraint in the night subperiod
is given by

h = x + ϕm+1 − ϕpq s − ϕ(m + T )

7

(11)

where ϕ is the real price of money and p is the nominal price of the goods in the day
subperiod. Substituting (11) into (5), we obtain

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

x

m+1

(12)
Because we assume that the DM is perfectly competitive, the seller treats ϕp as given.
Therefore, the ﬁrst order condition of q s is given by
ϕp = c′ (q s ).

(13)

From the second term of (12), the ﬁrst order condition of x is given by
U ′ (x) = 1.

(14)

Let x∗ denote the value of satisfying (14). Moreover, we assume that U (x∗ ) > x∗ issatisﬁed for x∗ to have a positive value. From the last term of (12), we ﬁnd that m+1
does not depend on m or on whether the individual becomes the seller or buyer in the day
subperiod. This is a property of the Lagos-Wright model. Because of this property, we
do not have to consider each individual’s state. Although we should seek the ﬁrst ordercondition of m+1 , we will obtain it in subsection 3.3 because V (m+1 ) depends not only
on the seller’s behavior but on the buyer’s, which is discussed in the next subsection.

3.2

Buyer’s problem

In this subsection, we seek the buyer’s optimal behavior. Because buyers and sellers areanonymous, buyers need to hold money before they consume. Therefore, buyers face thefollowing constraint:

pq b ≤ m.

8(15)

The buyer’s budget constraint in the night subperiod is given by

h = x + ϕm+1 + ϕpq b − ϕ(m + T ).

(16)

Calculating as in the seller’s problem, we obtain the value function, which the buyer
maximizes, as follows:

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

x

m+1

(17)
From this equation, we ﬁnd that the optimal conditions of x and m+1 are the same as
those of the seller. The value of q b is determined by whether (15) is binding. If (15) is
not binding, pq b < m, then the individual determines q b to satisfy u′ (q b ) = ϕp. If (15) isbinding, q b = m/p. Furthermore, from the ﬁrst term of (17), the buyer has no incentive
to increase the consumption in the DM to satisfy u′ (q b ) < ϕp in the binding case because
we assume that u′′ (q b ) < 0. Therefore, the following condition is always satisﬁed:u′ (q b ) ≥ ϕp.

3.3

(18)

Money holdings

To determine the amount of money brought to the next period, we must obtain the future
value, V (m+1 ). From the deﬁnitions of V (m+1 ), (6), and (7) to (10), we can express the
future value as follows:sb) + βds Vds (m+1 )]) + βdb Vdb (m+2 )] − βds [nc(q+1
V (m+1 ) =βdb [(1 − n)u(q+1+ βcb [(1 − n)U (xb+1 ) + nU (xs+1 ) + βcb Vcb (m+2 )]− βcs [(1 − n)hb+1 + nhs+1 + βcs Vcs (m+2 )],(19)

where m+2 denote money holdings after two periods, xb(s) is the consumption, and hb(s) islabor supply in the CM when the individual becomes the buyer (seller). The variables of9

(19) are the future quantity. Therefore, it is important to understand how the individual
predicts her future behavior. In this paper, we assume that the individual is sophisticated.
That individual knows that she cannot control her future behavior. In other words, her
future self maximizes (12) and (17) just as her current self does3 . From subsections 3.1
and 3.2 which describe current behavior, q s , x and m+1 do not depend on m. Since thiss
implies that q+1, x+1 , and m+2 do not also depend on m+1 , (19) is rewritten as follows:

V (m+1 ) = (1 − n)v(m+1 ) + βcs ϕ+1 m+1 + constant term,

(20)

b
b
where v(m+1 ) ≡ βdb u(q+1) − βcs ϕ+1 p+1 q+1
is the beneﬁt for the current individual when

the individual becomes the buyer in the next period, constant term is a term that does
not depend on m+1 , and (20) has been substituted as the budget constraint of the nightsubperiod, (11) and (16). Substituting this equation into the last term of (12) and (17),and omitting the terms that do not depend on m+1 , we obtainmax[−ϕm+1 + (1 − n)v(m+1 ) + βcs ϕ+1 m+1 ].
m+1

(21)

If (15) is binding, we obtain the ﬁrst order condition as follows:bu′ (q+1)ϕ= (1 − n)βdb+ nβcs .
ϕ+1
ϕ+1 p+1(22)

We can also consider the cases where (15) is not binding. However, we can show thatthese cases are not equilibrium, as in Appendix A. Therefore, in subsequent sections, weonly consider the binding case where (22) is satisﬁed.

4Equilibrium

In this section, we discuss the equilibrium in this model. It is deﬁned as follows.
Deﬁnition 1. Equilibria consist of the quantities, q s , q b , x, h, and m+1 , and the prices, ϕ3There is another solution in which the individual can control her future behavior. In such a case,she is considered naive. The solution in this case may be diﬀerent. However, this paper focuses on a
sophisticated agent.

10and p, which satisfy
1. given the policy variable, γ, the individual behaves,
2. the current behavior satisﬁes (11), (13), and (22),3. (15) is binding,
4. the current individual predicts future behavior in the same way as current behavior,
5. and the following market clearing conditions are satisﬁed:(a) The DM goods(1 − n)q b = nq s ,(23)x∗ = (1 − n)hb + nhs ,(24)

m = M.

(25)(b) The goods in the CM(c) Money

Following this deﬁnition, the inﬂation rate in equilibria is given byϕu′ (q+1 )= (1 − n)βdb ′ ( 1−n ) + nβcs .
ϕ+1
c n q+1(26)

where q is the amount of the consumption in the DM in equilibria. Because (15) is binding
and m = M , pq = M . Then, we can rewrite (26) as follows:ϕpq
M+1u′ (q+1 )= (1 − n)βdb ′ ( 1−n ) + nβcs .
ϕ+1 p+1 q+1 M
c n q+1(27)

Because M+1 /M = γ and the seller’s optimal condition, (11), is satisﬁed in the equilibria,

11

from (27), we obtain the dynamics of q as follows:
(
c′

[
]
(
)
)
′
c′ 1−n
q
q
1−n
u
(q
)
+1
+1
n +1
q q=
(1 − n)βdb ′ ( 1−n ) + nβcs ,nγ
c n q+1(28)

From this equation, we can draw the phase diagram shown in Figure 1 4 . From the ﬁgure,

Figure 1: Phase diagram of q
we ﬁnd that there are two steady states. One is q = 0. The other is q = q̄ > 0. We
also ﬁnd that there are two types of equilibria in this model. One is that the economy
approaches q = 0. The other is that the economy initially jumps to q = q̄. The existenceof two equilibria is also seen in the Lagos-Wright model. This is a property of the model
introducing ﬁat money. This paper focuses on the latter equilibrium because it is realistic.
Assumption 1. The equilibrium in which the economy initially jumps to q = q̄ is chosen.
If we accept this assumption and combine (26) and (28), we obtainu′ (q̄)ϕ= γ = (1 − n)βdb ′ ( 1−n ) + nβcs .
ϕ+1
c n q̄

4

See Appendix B for an explanation of how the ﬁgure is drawn.

12

(29)

However, q̄ and γ have restrictions. Substituting (11) into (18), we obtainu′ (q̄)( 1−n ) > 1.
c′ n q̄

(30)
′

(q)
= 1 is not
Notice that (30) does not include the equality because the case where c′ u1−n
( n q)
an equilibrium5 . From (21), we obtain the restriction of the inﬂation rate. Because theindividual does not need to have more money than her desired expenditure, the marginal
beneﬁt when she becomes a buyer in the next period, v ′ (m), is not negative. Therefore,
if −ϕ + βcs ϕ+1 > 0, from (21), the marginal beneﬁt of holding money is positive. This
causes inﬁnite money demand. To hold the money market clearing condition, (25), thefollowing relationship must hold:ϕ≥ βcs .
ϕ+1

(31)

From (30) and (31), we obtain the restriction of γ:

γ≥γ

where

γ = max[βcs , (1 − n)βdb + nβcs ].

(32)

In some cases, γ = γ cannot hold. Therefore, (32) is a necessary condition for theequilibrium to exist6 .

5

Policy analysis

In this subsection, we discuss the optimal monetary policy. First, I must deﬁne a welfare
function. It is diﬃcult to deﬁne this function because preferences change as time goesby when there is time-inconsistency. In this paper, the welfare function is given by thesum of the expected values at the beginning of a period or before the individuals ﬁnd out

5This is shown in Appendix A.
We can also obtain the dynamics of the nominal price of the goods in the DM. Using (13), (29), andq = q̄ for all periods, it is given by p+1 /p = γ.
613whether they will become the seller or the buyer; that is,

W (m) ≡ nV0s (m) + (1 − n)V0b (m).

(33)

I think that this welfare function is reasonable. Since the individuals in all periods face
the problem, (12) and (17), and they can be both a seller and a buyer, deﬁning (28) asa welfare function that the government maximizes is not wrong. Calculating each value,(7) to (10), in the equilibrium and substituting them into (33), the value of (33) in theequilibrium is given by
(
)
c 1−n
q̄
u(q̄)
U (x∗ )x∗n
W = (1 − n)−n+−.
1 − βdb
1 − βds
1 − βcb 1 − βcs(34)

From this equation, we obtain the following lemma about welfare.
Lemma 1. q ∗∗ maximizes the welfare function, (34), where q ∗∗ satisﬁes
1 − βdbu′ (q ∗∗ )( 1−n ) =.
1 − βds
c′ n q ∗∗(35)

Proof. The ﬁrst order condition of (34) is given by (35). Since u′′ (q b ) < 0 and c′′ (q s ) ≥ 0,
the suﬃcient condition is satisﬁed. Therefore, q ∗∗ maximizes the welfare function.
This lemma implies that if the discount factor of consumption in the DM, βdb , is high(low)–or that of the production in the DM is low (high), βds –the welfare maximizing levelof consumption in the DM, q ∗∗ , is high (low). This is intuitive. A high (low) βdb denotes
a high (low) evaluation of future consumption, and a low (high) βds denotes a low (high)evaluation of future disutility from production. This means that the welfare function,(34), is increased by an increase (decrease) in consumption and production. Therefore, if
βdb is high (low) and βds is low (high), q ∗∗ is also high (low).
Next, we discuss the optimal policy. Since we have obtained the welfare maximizing
level of q and the relationship between q and γ, (29), we easily obtain the optimal valueof γ as follows.
Proposition 1. The optimal policy is given by
14

[
]
1−βdb
1. γ = max (1 − n)βdb 1−βds + nβcs , βcs if βdb < βds ,∗2. γ ∗ → γ if βdb ≥ βds .
Proof. In the ﬁrst case, (35) is larger than one. From (30), q ∗∗ is feasible. Substituting(35) into (29), we obtain

γ = (1 − n)βdb

1 − βdb+ nβcs .
1 − βds(36)

If this equation is larger than βcs , it is the optimal policy because (32) is satisﬁed. Otherwise, βcs , which is the lower limit of the inﬂation rate, is optimal.
In the second case, because (35) is smaller than one, q ∗∗ is not feasible. Moreover,)(q ∗ = 1, because u′′ (q b ) < 0 and c′′ (q s ) ≥ 0. If
q ∗∗ > q ∗ , where q ∗ satisﬁes u′ (q ∗ )/c′ 1−nn()
so, the optimal policy is that q gets closer to q ∗ . Substituting u′ (q ∗ )/c′ 1−nq ∗ = 1 inton(29), we obtain:

γ = (1 − n)βdb + nβcs .

(37)

The larger of (37) and βcs is the optimal policy. This is the second case.
We compare the optimal policy with the Friedman rule to evaluate the optimal policy.
The Friedman rule is that the inﬂation rate should be minimized to minimize loss whenholding money. In this model, since the lower limit of the inﬂation rate is given by (32),
we can deﬁne the Friedman rule as follows.
Deﬁnition 2. The Friedman rule is a policy of minimizing the inﬂation rate; that is
γ = (→)γ.
From this deﬁnition, we obtain the following proposition.
1−βdb
> 1.
Proposition 2. The Friedman rule is not optimal if βdb < βds and ββdb
cs 1−βds

Proof. To show Proposition 2, we have to show that γ ∗ > γ. In the case of this proposition,
1−βdb
1−βdb
the optimal policy is γ ∗ = (1 − n)βdb 1−β+ nβcs > βcs because ββdb
> 1. Hence, we
cs 1−βds
ds

15

show that γ ∗ is higher than (1 − n)βdb + nβcs to show that γ ∗ > γ. This is true because(∗

γ − (1 − n)βdb + nβcs = βdb)
1 − βdb− 1 > 0,1 − βds(38)

where βdb < βds . By the above, Proposition 2 is shown.
I also show that the Friedman rule is optimal in other cases. It is obvious that the
1−βdbFriedman rule is optimal if βdb ≥ βds from Proposition 1. In the case where ββdb
≤ 1,cs 1−βdsγ ∗ = βcs from Proposition 1. This means that γ ∗ ≤ γ. Therefore, the Friedman rule is
optimal.
This proposition implies that the Friedman rule tends not to be optimal in the following
two cases. The ﬁrst case is when the discount factor of production in the DM, βds , ishigh and that of consumption in the DM, βdb , is low. In this case, since the individualemphasizes the disutility from production more than the utility from consumption in theDM, the consumption maximizing welfare, q ∗∗ , is small. Consumption in the DM can be
decreased by raising the inﬂation rate because an increase in the inﬂation rate raises thecost of holding money to pay the expenditure in the DM. Therefore, welfare is improved
through an increase in inﬂation and thus a decrease in consumption. The second case iswhen the discount factor of consumption in the DM, βdb , is high and that of labor in theCM, βcs is low7 . In this case, the money demand is strong because the current individuals
consume more in the next period. In some case, the individuals hold more money than
the quantity needed to achieve optimal consumption. As mentioned before, an increase
in inﬂation raises the cost of holding money. Therefore, welfare is improved by raising
the inﬂation rate until the individual’s money holding is optimal. In both of the above
cases, a higher inﬂation improves welfare; thus, we can conclude that the Friedman rule
is not optimal.

7

The small βcs induces a high consumption demand like βdb . If the individual becomes the buyer, she
has to increase the labor supply to obtain money for the future consumption (see (16)). The small βcs
means the future disutility from labor is small when becoming the buyer. That is, the cost of consumption
is low when βcs is small. Therefore, a small βcs induces a high consumption demand.

16

6

Concluding remarks

I developed a monetary search model that incorporates non-unitary discounting and analyzed the optimal monetary policy. Through the analysis, I showed that the Friedman
rule may not be optimal. The ﬁrst reason for this is that optimal consumption in the DM
is lower than the standard model with unitary discounting. Because an increase in theinﬂation rate raises the cost of holding money, the consumption is decreased. Therefore,
welfare is improved by raising the inﬂation rate. This case occurs when the discount rate
(factor) of production is low (high) and that of consumption in the DM is high (low)because the individual emphasizes disutility from production more than utility from consumption in the DM. The second reason is that the demand for money is strong enoughto increase future consumption. Then, the realized consumption is larger than the optimal one. Because a high inﬂation rate raises the cost of holding money, it is optimal.
This case occurs when the discount rate (factor) of consumption in the DM is low (high)and that of labor in the CM is high (low). The Friedman rule is a policy of minimizing
the inﬂation rate. Therefore, in both of the above cases, the Friedman rule may not be
optimal. By the above discussion, I conclude that the diﬀerence in the discount rates
signiﬁcantly aﬀects the optimality of the Friedman rule.

AppendixA

Proof of binding (15) in the equilibrium

Suppose that βcs ϕ+1 > ϕ. Then, from (21), we ﬁnd that the optimal behavior is m+1 = ∞.
Therefore, this case is not the equilibrium and βcs ϕ+1 ≤ ϕ in the equilibrium. Whenβcs ϕ+1 < ϕ, (15) is binding because the agents suﬀer losses if they hold more money thannecessary. Therefore, we check if in the case where βcs ϕ+1 = ϕ, there is no equilibrium or(15) binds. In this case, v ′ (m+1 ) = 0 is satisﬁed. This occurs in the following two cases.
The ﬁrst case is when ϕ+1 p+1 q ∗ ≤ m+1 , where q ∗∗∗ satisﬁes u′ (q ∗∗∗ ) = ϕp. In this case,the future individual does not increase consumption of the DM goods if m+1 increases.

17

Therefore, v ′ (m+1 ) = 0. When we seek the left-hand limit of v ′ (m+1 ) at ϕ+1 p+1 q ∗∗∗ , we
get

lim

m+1 →ϕ+1 p+1 q ∗∗∗ −0v ′ (m+1 ) = −ϕ+1 p+1 < 0.

Hence, the case where ϕ+1 p+1 q ∗∗∗ = m+1 is not the equilibrium, and the case whereβcs ϕ+1 < ϕ is the only equilibrium.
The second case is when m+1 = ϕpqmax , where qmax denotes the maximizing valueof v(m+1 ). The individual does not have an incentive to increase her present holding ofmoney because she does not want to consume more than qmax in the future. Therefore,(15) is binding in this case. By the above discussion, we show that (15) is binding in theequilibrium.
()
I also mention (30). Combining (13) and (23), in the equilibrium, ϕp = c′ 1−n
q̄
.
n
(
)
Therefore, u′ (q ∗ ) = c′ 1−nq ∗ is not the equilibrium. This is why (30) does not includenthe equality.

B

The proof of drawing Figure 1

Totally diﬀerentiating (28), we obtain
[ (
)
)]
(
1−n
1−n
1−n
dq+1
′′
′
=γ c
q
q+c
q
dq
n
n
n
}
[{ (
)
(
)} {
′
u
(q
)
1
−
n
1
−
n
1
−
n
+1
×
c′′
q+1
q+1 + c′q+1
(1 − n)βdb ′ ( 1−n ) + nβcsnnnc n q+1
(
)
(
)
]
1−n
1−n
u′′ (q+1 )c′ n q+1 − u′ (q+1 )c′′ n q+1 −1)(+ q+1 (1 − n)βdb
.
(B.1)
q
c′ 1−nn +1
From (28), one of the steady states is q = q+1 = 0. When the above equation is evaluatednear this steady state, we obtain
dq+1|q=q+1 =0 = 0
dq

18

(B.2)

because u′ (0) = ∞. In the other steady state, since (29) is satisﬁed,
[
]−1
(
)
(
)
′ ¯ ′′ 1−n ¯u′′ (q̄)c′ 1−n
q̄
u
(
q̄
q̄
−
)c
dq+1
(n
)]
|q=q+1 =q̄ = 1 + q̄(1 − n)βdb ′ ( 1−n ) [ ′′ (n1−n ) 1−n.
′ 1−n q̄dq
c n q̄¯ c
q̄
q̄
+
c
γ
n
n
n

(B.3)

u′′ (q̄)c′ ( 1−n q̄ )−u′ (q̄¯)c′′ ( 1−n q̄¯)
Since u′ > 0, c′ > 0, u′′ < 0 and c′′ ≥ 0, c′ 1−n q̄¯ c′′ n1−n q̄ 1−n q̄+c′ n1−n q̄ γ < 0. Therefore,( n )[ ( n ) n( n )](B.3) is larger than 1. By the above discussion, we can draw Figure 1.

C

The case where the price and quantity of the goodsin the DM are determined by bargaining

In this appendix, we change the assumptions of the DM from this paper as follows. Eachindividual produces and consumes a kind of good in the DM. In the DM, individual isearches for the person who consumes the goods produced or produces the goods consumed
by individual i. λ ∈ (0, 1/2] denotes a probability of meeting a partner who can trade.
There is no double coincidence of wants. The buyer and seller determine the quantity, q,
and the price, p, of the DM good by Nash bargaining.
Because we do not change the other assumptions, the seller’s payoﬀ is given by (12)and the buyer’s one is given by (17). Therefore, the bargaining problem is given by:

max V0b (m)θ V0s (m)1−θ(C.1)

z ≤ m,(C.2)

where z ≡ pq and θ is the buyer’s bargaining weight. Since the buyer’s and seller’s problem
in the night subperiod does not change from subsection 3.1 and 3.2, x and m+1 do notdepend on whether the individual becomes the buyer, the seller or other. Therefore,
maximizing (C.1) is the same as maximizing the following equation:(u(q) − ϕz)θ (−c(q) + ϕz)1−θ ,

19

(C.3)

If (C.2) is binding, the solution is given by

ϕz = g(q) ≡θc(q)u′ (q) + (1 − θ)u(q)c′ (q).
θu′ (q) + (1 − θ)c′ (q)(C.4)

Notice that g ′ (q) > 0 because there are assumptions of instant utility. If (C.2) is notbinding, q = q ∗ because the optimal condition is u′ (q b ) = c′ (q s ).
If λ < 1/2, there is a probability of not meeting the partner. Then, we can rewrite(19) as follows:bs) + βds Vds (m+1 )]V (m+1 ) =βdb [λu(q+1) + βdb Vdb (m+2 )] − βds [λc(q+1+ βcb [U (x∗ ) + βcb Vcb (m+2 )]− βcs [λhb+1 + λhs+1 + (1 − 2λ)hn+1 + βcs Vcs (m+2 )],(C.5)

where hn is the labor supply when the individual cannot ﬁnd her partner, and I havesubstituted the optimal consumption in the CM, x∗ . Since

hn = x + ϕm+1 − ϕ(m + T ),we can rewrite (21) as follows:max [−ϕm+1 + λv(m+1 ) + βcs ϕ+1 m+1 ] .
m+1

(C.6)

Because the individual does not have any incentive to make her consumption larger thanq ∗ , as in the case of the competitive DM, z+1 = m+1 in the equilibrium. Then, from (C.4),
q ′ (m+1 ) = g′ (qϕ+1 ) . Using this equation, we obtain (22) in the case ofϕu′ (q+1 )= λβdb ′+ (1 − λ)βcs .
ϕ+1g (q+1 )(C.7)

If θ → 1, g(q) = c(q). Then, (C.7) is the same as (26). In other words, if the price andquantity are determined by bargaining, we can obtain the same result in this paper.

20

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