DP2021-11

A Search and Bargaining Model ofNon-degenerate Distributions of
Money Holdings
Kazuya KAMIYASo KUBOTA

Revised August 18, 2021

A Search and Bargaining Model of Non-degenerate
Distributions of Money Holdings∗
Kazuya Kamiya†and So Kubota‡
March 31, 2021

Abstract
We study a standard search and bargaining model of money, where goods are traded
only in decentralized markets and distributions of money holdings are non-degenerate inequilibria. We assume fixed costs in each seller’s production, which allows an analyticalcharacterization of a tractable equilibrium. Each Nash bargaining solution satisfies
pay-all property, where the buyer pays the whole amount of cash as a corner solution,and the seller produces goods as the interior solution. In the stationary equilibrium,
the aggregate variables, such as total production and the number of matchings, areexpressed by given parameters, i.e., determinate. On the other hand, individual-levelvariables are indeterminate. Distributional monetary policies are effective in both theshort-run and the long-run.

1

Introduction

We propose a search and bargaining model of money and study a tractable equilibrium with
non-degenerate distributions of divisible money holdings. The model is a straightforward
extension of the basic search models with indivisible money such as Kiyotaki and Wright∗

We thank Nobu Kiyotaki, Tai-Wei Hu, Ian King, and seminar participants at Kobe and Hitotsubashi for
their comments and suggestions. This work is financially supported by the Zengin Foundation for Studies
on Economics and Finance in 2019.
†
Research Institute for Economics & Business Administration (RIEB), Kobe University (E-mail:kkamiya@rieb.kobe-u.ac.jp)‡
School of Political Science and Economics, Waseda University (E-mail: kubotaso@waseda.jp)1

(1993), Trejos and Wright (1995), and Shi (1995). That is, divisible money and goods aretraded on an environment of only random-matching and Nash bargaining. The crucial factor
is fixed costs in production, which generate a tractable equilibrium with a simple structure ofmoney holding distributions. We analytically characterize the equilibria, prove the existence,and evaluate distributional monetary policies both in the short-run and the long-run.
Originated from Kiyotaki and Wright (1989), search theory provides a solid microfoundation of money and implications for real-world economic phenomena. However, the literature
has struggled to handle the problem of heterogeneous money holdings. It is a fundamental consequence of the random matching assumption, where different amounts of money
are exchanged in each trade. But the characterization of dynamic equilibrium requires tokeep track of the distribution of money as a state variable. To overcome the problem, theworkhorse (so-called the third generation) models impose assumptions to make money holding distribution degenerate. The major approaches are large household model (Shi (1997))
and alternation between search and centralized markets (Lagos and Wright (2005)).
We consider a standard search model where agents randomly match and exchange goods
and money through Nash bargaining. We construct a particular type of tractable equilibriumthat satisfies Pay-All property. In each Nash bargaining problem, the production is decided
as the interior solution. On the other hand, the cash payment is solved as a corner solution,that is, the buyer pays all the money holding. On top of that, a simple non-degeneratemoney holding distribution arises. There is a mass of agents at 0 money holding, and theothers save different positive amounts. The transitional dynamics are also straightforward.
A positive money holder pays all cash in each matching and will carry 0 in the next period.
A no-cash agent takes a role as a seller and will save a positive amount. Indeed this model
has a full non-degenerate distribution; however, the individual-level dynamics is as simple
as the first and second-generation money search models.
The pay-all equilibrium is constructed by fixed costs in the production. The well-known
difficulty in dealing with the non-degenerate money holding distribution comes from eachagent’s dual roles. If each agent possibly becomes both a buyer and a seller, the mappingfrom the current to next distribution becomes almost untraceable. In pay-all equilibrium,
there is a cut-off level of money holding where each agent always becomes only a seller(buyer) if the money holding is lower (higher) than this level. This cut-off emerges by the
fixed costs. A seller has no incentive to trade with a partner who has a sufficiently small
2amount of money because it does not cover the fixed costs. Then, agents are sorted into
buyers and sellers according to their money holdings.
We find some interesting properties of the stationary equilibrium. That is, the aggregate variables are determinate, i.e., the macroeconomic measures such as total production,
the number of matching, and the Benthamian welfare are uniquely determined. However,individual-level variables are indeterminate: there is a continuum of steady-state distributions of money holdings, productions, consumptions, and individual lifetime utilities. In thissense, the equilibrium is similar to the first and second-generation models in the aggregate
level but parallel to Green and Zhou (1998) regarding the individual level. About the quantity theory of money, the equilibrium satisfies both neutrality and superneutrality of money
if cash is injected proportionally to each agent’s money holding.
We examine two types of distributional monetary policy. One is conducted on the steadystate while keeping the total money supply and the money holding distribution unchanged.
In each match, the government levies a per-unit tax (subsidy) depending on the amount ofgoods, and distributes (collects) money depending on after-trade money holding. The tax
rate is chosen so that the government budget is balanced each period. We show that thispolicy improves welfare, although the direction depends on the parameters. Intuitively, it
modifies the intra-temporal condition of Nash bargaining depending on the bargaining power
and moves the equilibrium toward Hosios condition.
Another policy is a temporal expansion of the money supply. The same amount of money
is injected to all agents in one period. Then the same amount is subtracted in the next period
from agents holding sufficient amount of money. It also causes a short-run improvement inrelative buyer/seller allocation. Notably, there are multiple transition paths, that is, thepolicy’s welfare effect is unpredictable. Since the individual-level allocation is indeterminate,
the transition paths caused by the redistribution policy is also uncertain.
There are some models in the literature which succeed in characterizing the non-degenerate
distribution of money holdings. The initial contribution is made by Green and Zhou (1998).
They consider a random-matching model with divisible money where each exchange is closed
by a take-it-or-leave-it offer, which is a special case of Nash bargaining. The key result isthe real indeterminacy of stationary equilibrium1 : there exists a continuum of steady-state
1
The general framework is constructed by Kamiya and Shimizu (2006). Note that the indeterminacy
emerges even if goods are divisible. See Kamiya and Shimizu (2007), Kamiya and Shimizu (2013), andKubota (2019) for the conditions of the indeterminacy.

3where each one has a different real allocation. Although the result itself is theoreticallyappealing, this property obstructs applied researches. Another important model is Menzio
et al. (2013), which eliminates one-to-one Nash bargaining and instead assumes a competitive
search environment. They construct a block-recursive structure of equilibrium, which makes
a simple transitional process of agents on the non-degenerate distribution. However, theconstruction of equilibrium still relies on the centralized labor market. Recently, Rocheteau
et al. (2018) analyze a discrete distribution caused by delayed money holding adjustment on
Lagos-Wright model. Another notable approach is numerical methods conducted by Molico(2006), Chiu and Molico (2010), and Chiu and Molico (2020). Finally, Camera and Corbae(1999) consider the countable amount of money. In contrast, the novelty of our approach is(i) tractable equilibria constructed only on a random-matching market with Nash bargaining, (ii) analytical characterization and proved existence of equilibria, and (iii) analyses of
distributional monetary policy.
The next section introduces the economic environment. Then, Section 3 illustrates theequilibrium and its characterization. The existence of the equilibrium is proved in Section 4.
Section 5 and 6 consider distributional monetary policy in the long-run and the short-run,respectively. Section 7 concludes this paper.

2

Economic Environment

Time is discrete and time horizon is infinite, as denoted by t = 1, 2, · · · . Goods are divisible.
There is a continuum of agents with measure one. Each agent can produce her production
goods with a cost (disutility) functionus (x) =
−d − cx

if x > 00

if x = 0,

where x is an amount of goods, d > 0 is a fixed cost, and c > 0 is a unit cost. Each agent
cannot consume her production good for eliminating double-coincidence of wants. However,4

she can consume some others’ production good with a temporal utility function

ub (x) =k + x

if x > 00

if x = 0,

where k > 0 is a given constant. This fixed utility is interpreted as an appetizer.2 Theoretically, k is introduced so that v(0) > 0 in the equilibrium. These assumptions about utility
and cost functions will be discussed in Section 3.2.
Each agent discounts her future payoff with a discount factor β ∈ (0, 1). Money isperfectly divisible, and its total supply in the economy is fixed at M > 0. Each agent’s
money holding m is a real number with an upper-bound. We assume this upper-bound as
1 for normalization, that is, m ∈ [0, 1]. This assumption allows us to ignore the behavior ofvery rich agent who exists only off-path, and then, simplifies the proof of the existence ofequilibrium (Zhou (1999)).
The timeline in each period is as follows. In the beginning of each period, agents observe
the current economy-wide money holdings distribution. Then, pairwise random matchingoccurs with a probability 2α ∈ (0, 1). In each matching, each agent observes the partner’s
money holding. One agent becomes a seller with probability 12 , and the other becomes abuyer. After that, bargaining over the price and the amount of goods proceeds throughNash bargaining with the buyer’s bargaining power θ ∈ (0, 1). At the end of each period, thematching resolves and proceeds to the next period. Let H be a money holdings distribution(a Borel measure) on R+ . In a meeting of a buyer (a seller) with money holding mb (ms ),the Nash bargaining decides the amount of goods x(mb , ms , H) and the amount of moneyp(mb , ms , H) in the trade. No trade, x(mb , ms , H) = p(mb , ms , H) = 0, is also a solution
when the joint surplus cannot be positive. The stationary monetary equilibrium is defined
as follows.
Definition 1. Let v be a function on R+ . A pair (H, v) is called a stationary monetary
equilibrium if
2

It might be considered that a certain fixed amount of goods, of which utility is k, is produced by the
fixed cost d.

5

Figure 1: Money holding distribution in Pay-All Equilibrium

Indeterminate under two restrictions# 0"! "# $ = 1 − #(0)!
"determinate! $"# $ = +
!

!

$

0

%

!
#

1. Bellman equation is consistently constructed asZv(m) = α
Z
+α

 b


u x(m, ms , H) + βv m − p(m, ms , H) dH(ms ) su x(mb , m, H) + βv m + p(mb , m, H) dH(mb ) + (1 − 2α)βv(m),2. H is a stationary distribution of the process under the transition p(mb , ms , H),
3. x(mb , ms , H) ≥ 0 and p(mb , ms , H) ≥ 0 solve each Nash bargaining problem.
4. v(m) > 0 for all m ≥ 0 and v(m) is strictly increasing.
The last condition represents the individual rationality and positive equilibrium value of
fiat money.

3

Pay-All Equilibrium

We consider an equilibrium with the following money holdings distribution. The support ofthe distribution H is {0} ∪ [z, Z], where 0 < z < Z < 1. Let H(0) ∈ [0, 1] be the measure of6

agents without money and H be a function satisfying
Z Z
dH = 1 − H(0)(1)

mdH = M

(2)

z

Z Z
z

A Nash bargaining problem between a buyer with mb and a seller with ms is as follows:• if there exists (x, p) such that x > 0, 0 ≤ p ≤ mb , and both the buyer and seller’s
surpluses are non-negative, i.e., k + x + β [v(mb − p) − v(mb )] ≥ 0 and −d − cx +β [v(ms + p) − v(ms )] ≥ 0, then the trade (x∗ , p∗ ) is determined by(x∗ , p∗ ) = arg max {k + x + β [v(mb − p) − v(mb )]}θ {−d − cx + β [v(ms + p) − v(ms )]}1−θ
x,ps.t. x > 0, 0 ≤ p ≤ mb ,(3)

• otherwise x∗ = p∗ = 0.
This formulation assumes that the fixed utility and cost emerge only if positive amount ofmoney and good are traded.
We construct an equilibrium where (i) on the equilibrium-path, Nash bargaining is agreed
only if a buyer holds m ∈ [z, Z] and a seller has m = 0, otherwise any trade does not cover
the fixed costs for a seller. Moreover, we focus on the equilibrium where (ii) the monetary
payment is binding at p∗ = mb , while the good production x∗ is determined as the interior
solution. For simplicity, let m be the buyer’s money holding. Then, the first-order condition
with respect to x is
(1 − θ)c{k + x∗ + β[v(0) − v(m)]} = θ{−d − cx∗ + β[v(m) − v(0)]}.

(4)Note that, under the condition (i), we can write x∗ = x(m).
We call a stationary monetary equilibrium with (i) and (ii) a Pay-All equilibrium, because
each buyer spends all amount of money holding. In parallel with this, (i) and (ii) arecollectively called the Pay-All property. The Bellman equation in the pay-all equilibrium is7

written as follows:
v(m) = αH(0)[k + x(m) + βv(0)] + [1 − αH(0)]βv(m)

(5)

Z Z
[−d − cx(m) + βv(m)]dH + {1 − α[1 − H(0)]}βv(0)v(0) = α

(6)

z

3.1

Characterization

If a pay-all equilibrium exists, it satisfies the properties summarized in the following lemmas.
x0 (m)
Define the consumption elasticity of discounted value as ξ(m) = 0. It must be constantv (m)in the equilibrium.
Lemma 1.
ξ(m) =βθ(1 − c)
+β
c

(7)

Proof. By (4),cx(m) = θ[−d − βv(0)] − (1 − θ)c[k + βv(0)] + θβv(m) + (1 − θ)cβv(m),⇒ cx0 (m) = θβv 0 (m) + (1 − θ)cβv 0 (m) = β[θ(1 − c) + c]v 0 (m),which leads to (7).

Next lemma derives another condition about ξ(m).
Lemma 2.
ξ(m) =

1−β
+β
αH(0)

(8)

Proof. The derivative of (5) isv 0 (m) = αH(0)x0 (m) + [1 − αH(0)]βx0 (m).
It is rearranged to (8).

8These two equations about ξ(m) determine H(0). That is, the measure of non-money holders
H(0) is explicitly pinned down by parameters. On the other hand, we will later show that
H(m) for m ∈ [z, Z] is indeterminate.
Lemma 3.
H(0) =(1 − β)cαβθ(1 − c)(9)

Proof. By Equation (7) and Equation (8).

The measure of agents with positive amount of money, 1−H(0), is also uniquely determined.
Since they are potential buyers, the following corollary holds.
Corollary 1. The number of matching αH(0)(1 − H(0)) is constant.

In the equilibrium, the aggregated discounted value of the potential buyers and sellers are
RZv(m)dH and H(0)v(0), respectively. The next lemma shows that the ratio is constant.
z
Lemma 4.
RZzv(m)dH

H(0)v(0)=θ(1 − θ)c(10)

Proof. Appendix.

On top of that, the discounted value of no-money holder is uniquely derived.
Lemma 5.
v(0) =

ck − dβ(1 − c)(11)

Proof. Appendix.

By Lemma 4, the total discounted value of buyers immediately follows.
9Corollary 2.
Z Z

v(m)dH =z

ck − dβ(1 − c)θH(0)(1 − θ)c(12)Note that, as will be shown, the shape of v(m) is indeterminate in the equilibrium. That is,although the aggregated discounted value is unique, the distribution of welfare is uncertain.
RZ
Finally, define the total welfare as W = H(0)v(0) + z v(m)dH. By Lemma 5 andCorollary 2, it is also constant.
Lemma 6.
W =

ξH(0)v(0)β(1 − θ)Proof. Appendix.

3.2

Discussion about the model’s assumptions

Both the fixed and linear terms of the utility and production functions drastically make theequilibrium tractable. The fixed costs d provides an incentive of sellers to decline tradeswith buyers who hold small amount of money. If the revenue is sufficiently small, it does notcover the fixed costs d. Then, the equilibrium money holding distribution H(m) is divided
into 0 and sufficiently large positive amount m ≥ z. It makes agents to play only one
role, either buyer or seller, depending on money holding m. As a consequence, each agent
alternates between buyer and seller, which makes a tractable transition of money holdings.
The fixed utility k is assumed to overcome the fixed costs d and assure the agent’s incentive
to participate the market, v(0) > 0.
The linearity of variable costs and utility are also crucial for tractability. They make theconsumption elasticity of discounted value, ξ(m), constant in Lemma 1. Then, this property
helps to pin down H(0) in Lemma 3. This uniqueness of H(0) makes Pay-All equilibrium
overcome the real indeterminacy of divisible money in random-matching search models in
the literature3 . Green and Zhou (1998) derive both distributional and aggregate-level inde3

The indeterminacy arises due to some identity hidden in monetary exchange. See, for the case of finite

10terminacy of real allocations. This indeterminacy is one-dimensional which is characterized
by the H(0). The Pay-All equilibrium erases this type of indeterminacy by anchoring H(0).
Although our model still has indeterminacy of money holding distribution among buyers,
the determinate H(0) makes the aggregated allocation determinate. The transition of agentsin Pay-All equilibrium is consistent with, so called, the second generation models such as
Trejos and Wright (1995) and Shi (1995), where the total money supply is 1−H(0). As in ourmodel, each seller becomes a buyer with probability α[1−H(0)], and each buyer turns to be aseller with probability αH(0). Moreover, the Pay-All equilibrium decides only the quantityof goods as an interior solution of Nash bargaining as in the second generation models.
RZ
1v(m)dH(m).
Define the average discounted sum of utilities of buyers as v buyer = 1−H(0)zThen, Pay-All equilibrium can be interpreted as an indivisible money model with v(0) for
each seller and v buyer for each buyer. In short, given that buyers are aggregated as a kindof one representative buyer with value v buyer , Pay-All equilibrium is in line with the second
generation models, where H(0) is exogenously given. As Trejos and Wright (1995) andShi (1995) do not show Green and Zhou (1998)’s indeterminacy, the allocation in Pay-All
equilibrium becomes also determinate in the aggregate level.
Note that, although the aggregated allocation is indeterminate, the individual-level indeterminacy among buyers is not nominal but real. As will be shown in the proof of theexistence of Pay-All equilibrium, it requires only inequality conditions about the shape of thevalue function. Combinations of both indeterminate v(m) and H(m) lead to distributional
indeterminacy of each buyer’s real allocation and individual welfare.

4

ExistenceIn this section, we show the existence of a continuum of pay-all equilibrium by construction.
That is, we first present a candidate for a pay-all equilibrium (v, H) and show that it isindeed an equilibrium. The following linear function is a candidate for a value function:v(m) =


v(0) + am

for m ∈ [0, m∗ ),v(0) + F + bm,

for m ∈ [m∗ , ∞],(13)

support of money holdings distribution, Kamiya and Shimizu (2006) and Kamiya and Shimizu (2007), andfor the case of infinite support, Kamiya (2019).

11

Figure 2: Value function

only sell

only buy

𝑣(𝑚)
𝑏

𝐹
𝑣(0)
0

𝑎
𝑚

∗

𝑧

𝑚
𝑍

𝑚
ഥ

where m∗ > 0, b > a > 0, and F > 0 will be determined later. A candidate for an equilibriummoney holding distribution H satisfies that :the support of the distribution H is {0} ∪ [z, Z],(14)

where z, and Z are parameters satisfying (1), (2) andm∗ ≤ z < Z ≤ 1 − m∗ .

(15)

The above (v, H) is shown to be a pay-all equilibrium if the parameters are in some range.
Note again that H(0) and v(0) are unique.

12

Proposition 1. If the following conditions are satisfied,
1−c
1−β
<
,
αβθ
c

(16)

ck > d,

(17)

θH(0)
> 1,
(1 − θ)[1 − H(0)]c(1 − θ)ck + θd(φ − 1)v(0) >,[(1 − θ)c + θ]βhi
1−H(0)(1−θ)ck+θd
(φ
−
1)
v(0)
−
M
[(1−θ)c+θ]βi < 2d,h(1−θ)(2−c)χ
1 + 1−H(0)M
2[(1−θ)c+θ]
φ≡

(18)
(19)

(20)

(1−β)cαβ(1−θ)[1−H(0)]ck−dwhere H(0) = αβθ(1−c), v(0) = β(1−c), and χ ≡ αβ(1−θ)[1−H(0)]+1−β, then there exist a contin-

uum of stationary pay-all equilibrium. Moreover, there exist parameters c, d, k, α, β, θ, andM satisfying the above conditions.
Proof. Appendix.
The first condition means that the unit cost c is sufficiently small and it assures H(0) < 1.
The second conditions means that, given c and d, the fixed utility k is sufficiently large andit assures v(0) > 0. The third and the forth conditions assure that the slope of the valuefunction is positive. The last condition guarantee that if both a buyer and a seller have small(large) amount of money, then they do not trade.
In the proof, we first derive conditions on (m∗ , a, b, F ) that Nash bargaining reaches
a deal if ms < m∗ and mb ≥ m∗ , and breaks down otherwise and the pay-all property
holds. Moreover, we find H and M which can be money holdings distribution and a money
supply. Then we confirm that all the conditions on (m∗ , a, b, F, H, M ) are satisfied under thepremises. Finally, we verify the consistency of (13) with (5) and (6). We also show that acontinuum of (m∗ , a, b, F, H, M ) can be equilibria.

5

The bargaining power and welfare

The Hosios condition, a condition on the bargaining power θ for efficiency, is often discussed
in the literature of search models. In this section, we discuss the impact of θ on the total
welfare W , i.e., ∂W.
∂θ
13

First, note that, for given the other parameters, a change in θ may violates the conditions
for the existence of a pay-all equilibrium. Namely, the pay-all property and/or H(0) ∈ (0, 1)
could be violated. In what follows in this section, we assume for simplicity that a pay-all
equilibrium exists for all θ given the other parameters.
RZ
Recall that W = H(0)v(0) + z v(m)dH. From Lemmas 3 and 5, the first term in the(1−β)c ck−d
RHS is αβθ(1−c). Thus ∂H(0)v(0)< 0. From Corollary 2 and Lemma 3, the second termβ(1−c)∂θin the RHS is

ck − dβ(1 − c)(1−β)cθ αβθ(1−c)!

(1 − θ)c

.

Thus
∂
∂θ

!

Z Z
v(m)dH

> 0.

(21)

z

Below, we investigate which dominates the other.
From Lemmas 1, 3, and 6, W is written as

W =

ξv(0)
β(1 − θ)The derivative ofH(0) =

θ(1−c)+1c

v(0)(1 − β)c
αβ 2 (1 − c)1θ(1−θ)θ(1 − c)
+1
c



1
θ(1 − θ)


.

(22)

with respect to θ is2θ − 1 (θ(1 − c) + c) 1 − c
1
1
+
= 2
((1 − c)θ2 + 2cθ − c).
2cc θ(1 − θ)cθ (1 − θ)

θ2 (1 − θ)2Since(1 − c)θ2 + 2cθ − c = 02θ
has a negative solution and a positive solution θ = c−c√c and θ = c+c√c . Note that c = (1−θ)
2holds at the positive solution. Thus, the following proposition holds.
Proposition 2. ∂W
is positive, if θ > c+c√c , and it is negative, if θ < c+c√c .
∂θNote that W is minimum at θ̄ = c+c√c , if a pay-all equilibrium exists at θ̄. Moreover,from (22), W → ∞ as θ → 1, if a pay-all equilibrium exists around θ = 1.
14

6

Distributional monetary policy in the long-run: achange in an effective bargaining power

Here, we consider a distributional monetary policy which keeps the total money supply andthe money holding distribution overtime. The government imposes tax and provide subsidy
so as to balance the budget. In each trade, they are transferred in the different way. Tax iscollected as linear per-unit tax tx, where t is tax rate and x is the amount of good sold. Then,the collected money is injected to each seller depending on after-tax money holding. This
amount is g(m − tx), where g is the gross rate. Note that if t < 0, then it can be considered
as a subsidy rate. It will be shown that the policy changes the power of the bargaining
defined below as an effective bargaining power. Therefore, a Hosios-type condition on thebargaining power can be derived as in the previous section, and the policy can improve thewelfare.
We assume the budget balance of the government. That is,Z0=Z(g − 1)mdH − gtx(m)dH.

Thus
Z(g − 1)M = gt

x(m)dH.

(23)

As in the original case, the system is characterized by three equations: the first-order condition of Nash bargaining problem, buyer’s value function, and seller’s value function. Under
pay-all equilibrium, the first one is obtained by the following problem:



 1−θ
max{k + x + β[v(0) − v(m)]}θ −d − cx + β v g(m − tx) − v(0).
xHere, the seller’s money holding after the trade depends on g and t. Then, the first-ordercondition is






(1−θ) c + βgtv 0 g(m − tx) {k+x+β[v(0)−v(m)]} = θ −d − cx + β v g(m − tx) − v(0) .
(24)

15

The Bellman equation for buyer is unchanged from (5). Whatever the policy is, each buyer
pays the entire cash in pay-all equilibrium.
The value function of the seller is
Z Zv(0) = α−d − cx(m) + βv g(m − tx) dH + {1 − α[1 − H(0)]}βv(0)

(25)

z

We assume a linear value function: v(m) = v(0) + F + bm hereafter. As shown in the proofof existence, we can take such a linear value function. This assumption makes the systemdrastically tractable because (25) becomes identical to the non-policy case.
Lemma 7. If the value function is linear, (25) coincides with (6)Proof. Appendix.
From the linearity of the value function, v 0 g(m − tx) is a constant b. In what follows
we use v 0 instead of b. Therefore, (24) becomes(1−θ) [c + βgtv 0 ] {k +x+β[v(0)−v(m)]} = θ −d − cx + β v g(m − tx) − v(0) . (26)Therefore, compared to the case without policy, the difference is only the first order conditionof the Nash bargaining (26).
Recall that the first order condition of the Nash bargaining without policy derived in
Equation (4) is(1 − θ)c{k + x + β[v(0) − v(m)]} = θ {−d − cx + β [v(m) − v(0)]} .
Thus, the effective bargaining power of a buyer in the case of policy t, denoted by θe (t), is
defined as follows:(1 − θ)[c + βgtv 0 ][1 − θe (t)]c=.
θe (t)
θLet G = gt. Note that G0 = g + t dg
, and thus, at g = 1 and t = 0, G0 = g + t dg
= 1.
dt
dt
Then, the following proposition holds for any real number t.

16

Proposition 3. Suppose G0 > 0. Then ∂W> 0 holds, if and only if∂tc + βGv 0
c + βGv 0
√
√
<
θ
<
.
c − c + βGv 0c + c + βGv 0> 0 holds, if and only if
In particular, in the case of t = 0, ∂W
∂t
θ<

c
√ .
c+ c

Proof. Appendix.
The key implication is that, if t = 0, the condition becomes identical with Proposition 2.
A small government intervention changes the bargaining power between buyer and seller. In
this case, θe (0) = θ, then an increase in t leads to a decrease in θe (t). But, globally, it also
depends on the shape of value function. Note that if the absolute value of t is not large, then
0

0

√
√
G0 > 0 holds, since G0 = 1 at t = 0. Note also that, if c−c+βGv< 0, then c−c+βGv<θ
c+βGv 0c+βGv 0can be replaced by 0 ≤ θ. In particular, this applies in the case of t = 0.

7

Distributional monetary policy in the short-run

In this section, we consider one-time helicopter drop of money. At the beginning of periodt, the central bank unexpectedly announces the following policy.
• The central bank injects the same τ unit of money to everybody at the beginning of
period t.
• The central bank will deprive τ unit of money at period t + 1. If an agent’s money
holding is m < τ , it will deprive m.
The collection of money at period t + 1 is for keeping Pay-All equilibrium. Otherwise, therange of money holding distribution diverges to infinity4 . For simplicity, assume that thedensity function h(m) associated with H(m) exists for m ∈ [z, Z].
4Suppose the maximum money holding is mmax at period t − 1. At the beginning of period t, sellers also
hold τ . After the trade, the maximum amount increases to mmax + τ . If there is no reduction in money
holding, the maximum amount changes to mmax + 2τ at the end of period t + 1. It will eventually diverges
and exceed the limit of money holding 1.

17

Figure 3: Money holding distribution

At the beginning of Period 𝑡
𝐻𝑡−1 (0)𝐻𝑡 (𝜏)ℎ𝑡−1 (𝑚)ℎ𝑡 (𝑚)

𝑚
0

𝑧 𝑧+𝜏

𝜏

𝑚
#

𝑍 𝑍+𝜏

Period 𝑡 trades
𝐻𝑡 (𝜏)
%&$'(

𝐻$
%&$'(

𝐻$

ℎ𝑡 (𝑚)

(𝜏)

%&$'(

ℎ$

𝑚

(0)
𝑚
0

𝑧 + 𝜏 𝑧 + 2𝜏𝜏

𝑍 + 𝜏 𝑍 + 2𝜏 𝑚#

At the beginning of Period 𝑡 + 1
𝐻𝑡)* (0)
%&$'(

𝐻$
%&$'(

𝐻$

ℎ𝑡)* (𝑚)

(𝜏)

%&$'(

ℎ$

𝑚

(0)
𝑚
0

𝜏

𝑧

𝑧+𝜏

𝑍 + 𝜏 𝑍 + 2𝜏 𝑚#

18

Given Pay-All equilibrium, the transition of the distribution is described as follows. Figure 3 draws an example of a triangle shape distribution.
At period t − 1 and before, the stationary money holding distribution satisfies(1−β)c> 0,• Ht−1 (0) = αβθ(1−c)• ht−1 (m) ≥ 0 for all m ∈ [z, Z],
and no agents hold m 6∈ {0} ∪ [z, Z] amount of money.
At the beginning of period t, τ is injected. Then, the distribution shifts to the right byτ:(1−β)c> 0,• Ht (τ ) = Ht−1 (0) = αβθ(1−c)• ht (m) = ht−1 (m − τ ) ≥ 0 for all m ∈ [z + τ, Z + τ ].
In period t, suppose that the Pay-All property still holds in the bargaining. Then, eachbuyer m ∈ [z + τ, Z + τ ] pays all the amount of money. Each seller finds a buyer withprobability α[1 − Ht (τ )], and each buyer makes a matching with αHt (τ ). Let Htaf ter be themoney holding distribution after the trade in period t. Then,• Htaf ter (0) = αHt (τ )[1 − Ht (τ )] = αHt−1 (0)[1 − Ht−1 (0)]• Htaf ter (τ ) = Ht (τ ){1 − α[1 − Ht (τ )]} = Ht−1 (0){1 − α[1 − Ht−1 (0)]}ter• For m ∈ [z + τ, z + 2τ ), haf(m) = [1 − αHt (τ )]ht (m) = [1 − αHt−1 (0)]ht−1 (m − τ )tter(m) = αHt (τ )ht (m − τ ) + [1 − αHt (τ )]ht (m) =• For m ∈ [z + 2τ, Z + τ ], haft

αHt−1 (0)ht−1 (m − 2τ ) + [1 − αHt−1 (0)]ht−1 (m − τ ).
ter
• For m ∈ (Z + τ, Z + 2τ ], haf(m) = αHt (τ )ht (m − τ ) = [1 − αHt−1 (0)]ht−1 (m − 2τ )tFinally, at the beginning of period t + 1, money holdings of all agents except m = 0 are
declined by τ . The distribution is sustained stationary hereafter.
• Ht+1 (0) = Htaf ter (0) + Htaf ter (τ ) = αHt−1 (0)[1 − Ht−1 (0)] + Ht−1 (0){1 − α[1 − Ht (τ )]} =Ht−1 (0).
ter• For m ∈ [z, z + τ ), ht+1 (m) = haf(m + τ ) = [1 − αHt−1 (0)]ht−1 (m)
t

19

ter
• For m ∈ [z+τ, Z], ht+1 (m) = haf(m+τ ) = αHt−1 (0)ht−1 (m−τ )+[1−αHt−1 (0)]ht−1 (m).
tter
• For m ∈ (Z, Z + τ ], ht+1 (m) = haf(m + τ ) = αHt−1 (0)ht−1 (m − τ )t

The measure of non-money holder is unchanged between old and new stationary distribution: H(0) = Ht+1 (0) = Ht−1 (0). The distribution of positive money holders spreads out.
Because of the indeterminacy of the money holding distribution h(m) for m ≥ z, the new
distribution holds a stationary Pay-All Equilibrium with the same social welfare. Then, we
can consider Period t allocation is on a transition path between the two steady-states. Letv(0) ≡ vt−1 (0) = vt+1 (0). Note that vt−1 (m) and vt+1 (m) for m ≥ z are possibly different by
indeterminacy. Consider the bargaining problem of a seller holding τ and a buyer holding
mt = mt−1 + τ at period t:
max {k + xt + β[vt+1 (0) − vt+1 (mt−1 )]}θ {−d − cxt + β[vt+1 (mt−1 + τ ) − vt+1 (0)]}1−θxt

In the first term, the buyer holds 0 at Period t + 1 given Pay-All property. If the bargainig
fails, the buyer looses τ unit of money and hold mt+1 = mt−1 + τ − τ = mt−1 unit of money,In the second term, the seller obtains mt = mt−1 + τ by the trade at Period t and looses
τ in the next, then mt+1 = τ + mt−1 + τ − τ = mt−1 + τ in total at Period t + 1. In caseof no trade, the τ unit of money holding needs to be returned to the central bank, thenmt+1 = mt−1 = 0. The first-order condition isθ {−d − cxt + β[vt+1 (mt−1 + τ ) − vt+1 (0)]} = (1 − θ)c {k + xt + β[vt+1 (0) − vt+1 (mt−1 )]}⇔ cxt (mt−1 ) = β(1−θ)c[vt+1 (mt−1 )−vt+1 (0)]+βθ[vt+1 (mt−1 +τ )−vt+1 (0)]−[(1−θ)ck+θd]= βθvt+1 (mt−1 + τ ) + β(1 − θ)cvt+1 (mt−1 ) − β[(1 − θ)c + θ]v(0) − [(1 − θ)ck + θd]
Let the total production at period t as Xt .
Z Z
Xt = αH(0)xt (mt−1 )ht−1 (mt−1 )dmt−1z=αβH(0)c

 Z Z
θ
vt+1 (mt−1 + τ )ht−1 (mt−1 )dmt−1
z

20


+
−

αβH(0)cZ Z

(1 − θ)c

vt+1 (mt−1 )ht−1 (mt−1 )dmt−1zαβH(0)[1 − H(0)]{β[(1 − θ)c + θ]v(0) + [(1 − θ)ck + θd]}c

Now, we want to show that Xt is larger than the steady-state amount of production, X ≡Xt−1 = Xt+1 . By the linear utility and cost functions, it implies the welfare improvement by
the short-run monetary policy.
αβH(0)
X = Xt+1 =
c

−

(

)

Z Z+τ

[θ + (1 − θ)c]

vt+1 (mt+1 )ht+1 (mt+1 )dmt.

zαβH(0)[1 − H(0)]{β[(1 − θ)c + θ]v(0) + [(1 − θ)ck + θd]}cThen, the condition Xt > X is equivalent to
Z Z

Z Z
vt+1 (mt−1 + τ )ht−1 (mt−1 )dmt−1 + (1 − θ)cθzvt+1 (mt−1 )ht−1 (mt−1 )dmt−1z

Z Z+τ> [θ + (1 − θ)c]

vt+1 (mt+1 )ht+1 (mt+1 )dmt+1z

By the transition of the money holding distribution,
Z Z+τ
vt+1 (mt+1 )ht+1 (mt+1 )dmt+1z

Z Z= [1 − αH(0)]Z Z+τvt+1 (mt−1 )ht−1 (m − τ )vt+1 (mt−1 )ht−1 (mt−1 )dmt−1 + αH(0)
z

z+τ

Z Z

Z Z

= [1 − αH(0)]vt+1 (mt−1 )ht−1 (mt−1 )dmt−1 + αH(0)zvt+1 (mt−1 + τ )ht−1 (m)zTherefore,
Xt > X ⇔Z Z{θ − [θ + (1 − θ)c]αH(0)}vt+1 (mt−1 + τ )ht−1 (m)
z

21

Z Z
> {[θ + (1 − θ)c][1 − αH(0)] − (1 − θ)c}

vt+1 (mt−1 )ht−1 (m)z"Z⇔ {θ[1−αH(0)]−(1−θ)cαH(0)}Z

Z Z
vt+1 (mt−1 + τ )ht−1 (m) −z#
vt+1 (mt−1 )ht−1 (m) > 0z(27)
Since the value function is strictly increasing in Pay-All equilibrium, vt−1 (mt−1 + τ ) >vt−1 (mt−1 ) for all mt−1 . Hence, the second term is positive. The sign of the productionchange depends on the first-term.
Xt > X⇔ θ[1 − αH(0)] > (1 − θ)cαH(0)(1 − β)c(1 − β)c2⇔θ 1−> (1 − θ)βθ(1 − c)βθ(1 − c)⇔ θ2 β(1 − c) − θ(1 − β)c > (1 − θ)(1 − β)c2
⇔ θ2 β(1 − c) > (1 − β)c2 + θ(1 − β)c(1 − c)⇔ θ2 β(1 − c) > (1 − β)c[c + θ(1 − c)]This condition holds for sufficiently large β.
Intuitively, the policy raises the production by changing each seller’s future welfare.
The money holding distribution will be diverged at t + 1. Sellers at t − 1 will hold m ∈[z + τ, Z + τ ] at t + 1. They are larger than the average in the steady state after t + 1. It
makes more incentive for production in Nash bargaining since the seller’s threat point value
β[v(0) − v(m + τ )] decreases.
Interestingly, this positive welfare improvement is indeterminate. The welfare improvement depends on the second term of Equation (27). Hence, the quantitative difference hinges
on the shape of the value function vt+1 (m). Therefore, the individual-level indeterminacy
leads to the aggregate level indeterminacy in the welfare improvement by the distributionalmonetary policy.

22

8

Conclusion

This paper shows that a class of standard search and bargaining models of money has
analytical characterizations of equilibrium aggregate variables, i.e., the welfare, the measureof non-money holders, and the other aggregate variables are uniquely expressed by given
parameters. Due to the analytical characterization, the standard comparative statics canbe applied and the effects of policies can be well investigated. The model has two notablefeatures. First, it does not have centralized markets, as in Lagos and Wright (2005), sothat only the effect of policies on the decentralize market can be extracted. Second, theassumption of large households, as in Shi (1997), is not made so that the effect of parametersand policies on money holdings distributions can be investigated.
A long-run and a short-run distributional monetary policies are investigated. In the longrun policy, the government levies a per-unit tax (subsidy) depending on the amount of goods,
and distributes (collects) money depending on after-trade money holdings in each period,
where they are chosen so that the government budget is balanced. Intuitively, it modifies
the intra-temporal condition of Nash bargaining changing the effective bargaining power andmoves the equilibrium toward the Hosios condition on the efficiency. In the short-run policy,
the government injects the same amount of money to all agents only in one period. Then thesame amount is subtracted in the next period. Intuitively, the policy raises the production
by changing the future welfare of each seller, and it leads to a short-run improvement in the
total welfare.
In the literature, long-term changes in the amount of money, such as the Friedman rule,
are often investigated. This type of policy is not covered in this paper, because our existence
proof cannot be applied to such policies. In the existence proof, we need an upper bound anda lower bound of the support of money holdings distributions. However, in such policies,
they converge to infinity or to zero. Of course, an analysis of this type of policies is an
important future research topic.

23

References
Camera, Gabriele and Dean Corbae (1999), “Money and price dispersion.” International
Economic Review, 40, 985–1008.
Chiu, Jonathan and Miguel Molico (2010), “Liquidity, redistribution, and the welfare costof inflation.” Journal of Monetary Economics, 57, 428–438.
Chiu, Jonathan and Miguel Molico (2020), “Short-run dynamics in a search-theoretic modelof monetary exchange.” Review of Economic Dynamics.
Green, Edward J and Ruilin Zhou (1998), “A rudimentary random-matching model withdivisible money and prices.” Journal of Economic theory, 81, 252–271.
Kamiya, Kazuya (2019), “On the real determinacy and indeterminacy of stationary equilibria
in monetary models.” Discussion paper series.
Kamiya, Kazuya and Takashi Shimizu (2006), “Real indeterminacy of stationary equilibria inmatching models with divisible money.” Journal of Mathematical Economics, 42, 594–617.
Kamiya, Kazuya and Takashi Shimizu (2007), “On the role of tax subsidy scheme in money
search models.” International Economic Review, 48, 575–606.
Kamiya, Kazuya and Takashi Shimizu (2013), “Dynamic auction markets with fiat money.”
Journal of Money, Credit and Banking, 45, 349–378.
Kiyotaki, Nobuhiro and Randall Wright (1989), “On money as a medium of exchange.”
Journal of political Economy, 97, 927–954.
Kiyotaki, Nobuhiro and Randall Wright (1993), “A search-theoretic approach to monetary
economics.” The American Economic Review, 63–77.
Kubota, So (2019), “Market structure and indeterminacy of stationary equilibria in a decentralized monetary economy.” Discussion paper series.
Lagos, Ricardo and Randall Wright (2005), “A unified framework for monetary theory andpolicy analysis.” Journal of Political Economy, 113, 463–484.

24

Menzio, Guido, Shouyong Shi, and Hongfei Sun (2013), “A monetary theory with nondegenerate distributions.” Journal of Economic Theory, 148, 2266–2312.
Molico, Miguel (2006), “The distribution of money and prices in search equilibrium.” International Economic Review, 47, 701–722.
Rocheteau, Guillaume, Pierre-Olivier Weill, and Tsz-Nga Wong (2018), “An heterogeneousagent new-monetarist model with an application to unemployment.” Technical report,
National Bureau of Economic Research.
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Economic Theory, 67, 467–496.
Shi, Shouyong (1997), “A divisible search model of fiat money.” Econometrica: Journal of
the Econometric Society, 75–102.
Trejos, Alberto and Randall Wright (1995), “Search, bargaining, money, and prices.” Journalof political Economy, 103, 118–141.
Zhou, Ruilin (1999), “Individual and aggregate real balances in a random-matching model.”International Economic Review, 40, 1009–1038.

25

Appendix

The proof of Lemma 4
By (5),

(1 − β)(1 − θc)v(m) = (1 − θ)c{k + x(m) + β[v(0) − v(m)]}
αH(0)


⇒

(1 − β)(1 − θc)
αH(0)

Z Z

Z Z
(1 − θ)c{k + x(m) + β[v(0) − v(m)]}dH (A.1)

v(m)dH =zz

By (6),Z Z

Z Z[−d − cx(m) + βv(m)]dH + βv(0) − αv(0) = αzβv(0)dHz

Z Z{−d − cx(m) + β[v(m) − v(0])}dH + βv(0)

=α
z


⇔


Z Z
(1 − β)θ
v(0) = θ{−d − cx(m) + β[v(m) − v(0])}dH
α
z

(A.2)

Integrating the FOC of Nash bargaining solution, (4), derives
Z Z(1 − θ)c[k + x(m) + β[v(0) − v(m))]dHz

Z Zθ{−d − cx(m) + β[v(m) − v(0)]}dH=z

Substitute Equation (A.1) and (A.2) into (A.3), then

(1 − β)(1 − θ)c
αH(0)

Z Z
z




(1 − β)θ
v(m)dH =v(0),αwhich leads to the Lemma .

26

(A.3)

The proof of Lemma 5
By (5),


1−β
H(0)


⇔


v(m) = α{k + x(m) + β[v(0) − v(m))]}(1 − β)cH(0)v(m) = α{ck + cx(m) + βcv(0) − βcv(m)}⇔ α[ck + βcv(0) − βcv(m)] −(1 − β)cH(0)v(m) = α[−cx(m) + βcv(m)]= α[d − d − cx(m) + βv(m) − β(1 − c)v(m)]= α[−d − cx(m) + βv(m)] + α[d − β(1 − c)v(m)]Then,α[−d − cx(m) + βv(m)](1 − β)c= αck + αβcv(0) −v(m) − αd + αβ(1 − c)v(m)]H(0)
(1 − β)c= αβ(1 − c) −v(m) + α[ck − d + βcv(0)].
H(0)
By integrating both sides of (A.4), we get
Z Z[−d − cx(m) + βv(m)]dHαz(1 − β)c= αβ(1 − c) −
H(0)

Z Z
v(m)dH + α[1 − H(0)][ck − d + βcv(0)].
zBy (6),{1 − β + α[1 − H(0)]β}v(0)(1 − β)c= αβ(1 − c) −
H(0)

Z Z
v(m)dH + α[1 − H(0)][ck − d + βcv(0)].
z⇔ {1 − β + α[1 − H(0)](1 − c)β}v(0)

27

(A.4)



(1 − β)c= αβ(1 − c) −
H(0)

Z Z
v(m)dH + α[1 − H(0)](ck − d).
z

By Lemma 4,{1 − β + α[1 − H(0)](1 − c)β}v(0)(1 − β)c= αβ(1 − c) −
H(0)




θH(0)v(0)+ α[1 − H(0)](ck − d).
(1 − θ)cThen,{1 − β + α[1 − H(0)](1 − c)β}(1 − θ)cv(0)= [αβ(1 − c)H(0) − (1 − β)c] θv(0) + α[1 − H(0)](1 − θ)c(ck − d).
Thusv(0) =αc(1 − H(0))(1 − θ)(ck − d).
c(1 − β) + αβ(1 − θ)(1 − c) − (αβ(1 − c)(c + θ − cθ))H(0)(1−β)cSubstituting H(0) = αβθ(1−c)
into the denominator of the above, we obtainc(1 − θ)(cβ + θαβ − c − cθαβ)c.
c(1 − β) + αβ(1 − θ)(1 − c) − (1 − β)(c + θ − cθ) =
θ
θ
Since
1 − H(0) =cβ + θαβ − c − cθαβ,αβθ(1 − c)we obtainv(0) =

ck − d.
β(1 − c)The proof of Proposition 1
The steps of the proof
1. Assuming the pay-all property, we derive some properties of the Nash bargaining so28

lution.
2. We present a candidate for (v, H).
3. We derive the conditions which guarantee the pay-all property. It can be seen that acontinuum of endogenously determined variables, including (v, H), satisfy the conditions.
4. We show that the conditions in Step 3 are satisfied under the assumptions in Proposition 2,
5. Finally, we show the existence of equilibria. That is, (i) (v, H) in Step 2 is an equilibrium, i,e., H is stationary and v satisfies Bellman equation, and (ii) there exist
parameters satisfying the assumptions in Proposition 2,

Step 1: Some properties of the Nash bargaining solution under the pay-all property
This step derives some properties the Nash bargaining solution in the case that a meetingof a buyer holding mb and a seller holding ms , where the buyer pays all mb and the amountof goods traded in the bargaining is positive. Note that in this step we do not assume thelinearity of v. The results will be used in the follwoing Steps. In these steps, the off-path
meetings will be also investigated, where agents hold non-equilibrium amount of money,
Lemma A.1. Suppose the pay-all property is satisfied, i.e., x(ms , mb , H) > 0 and q(ms , mb , H) =mb , thencx(ms , mb , H) = (1−θ)cβ[v(mb )−v(mb −q)]+θβ[v(ms +q)−v(ms )]−[(1−θ)ck+θd]. (A.5)
Moreover, the seller’s surplus is(1 − θ) β[v(ms + q) − v(ms )] − cβ[v(mb ) − v(mb − q)] + (ck − d)] .

(A.6)

and the buyer’s surplus is(θ/c) β[v(ms + q) − v(ms )] − cβ[v(mb ) − v(mb − q)] + (ck − d)
29

(A.7)

Proof. The first-order condition with respect to x is−(1 − θ)c[k + x + β(v(mb − q) − v(mb ))] + θ[−d − cx + β(v(ms + q) − v(ms ))] = 0.
Thuscx = (1 − θ)c[β(v(mb ) − v(mb − q))] + θβ[v(ms + q) − v(ms )] − [(1 − θ)ck + θd].
Using the above, the seller’s surplus is−d − cx + β(v(ms + q) − v(ms ))= −d−(1−θ)cβ[v(mb )−v(mb −q)]−θβ[v(ms +q)−v(ms )]+[(1−θ)ck+θd]+β[v(ms +q)−v(ms )]= (1 − θ) β[v(ms + q) − v(ms )] − cβ[v(mb ) − v(mb − q)] + (ck − d)] .
Similarly, the buyer’s surplus is
k + x + β(v(mb − q) − v(mb ))= k+(1−θ)β[v(mb )−v(mb −q)]+(θ/c)β[v(ms +q)−v(ms )]−[(1−θ)k+(θ/c)d]+β(v(mb −q)−v(mb ))= (θ/c)β[v(ms + q) − v(ms )] − θβ[v(mb ) − v(mb − q)] + θk − (θ/c)d= (θ/c) β[v(ms + q) − v(ms )] − cβ[v(mb ) − v(mb − q)] + ck − dStep 2: The candidates for an equilibrium value function and an equilibrium money holding
distribution(1−β)cck−dFirst, note that v(0) = β(1−c)
and H(0) = αβθ(1−c)
are already given. The candidate for30

an equilibrium value function is as follows:

v(m) =


v(0) + am

if m < m∗ ,v(0) + F + bm

if m ≥ m∗ ,(A.8)

where a, b, F, and m∗ are parameters.
The candidate for a equilibrium money holding distribution satisfies Equation (14):the support of the distribution H is {0} ∪ [z, Z].
In the following steps, it will be shown that H, z, and Z can be freely chosen, i.e., indeterminate, if they satisfy (1), (2), and Equation (15)m∗ ≤ z < Z ≤ 1 − m∗ .

Next, we derive the relationship between the slopes a and b, The slope a is the marginallife-time utility of money for sellers, and b is that for buyers. We show that a < b. A seller
needs to wait at least one period for being a buyer and purchasing goods; hence, the gainfrom holding money is discounted by the time preference.
Lemma A.2. The coefficients a and b satisfy
a = χb,(A.9)

αβ(1−θ)[1−H(0)]where χ ≡ {αβ(1−θ)[1−H(0)]+1−β}< 1.

Proof. Consider a meeting in which a seller holds ms < m∗ and a buyer holds mb ∈ [z, Z].
Note that from (15), the money holding of the seller after the trade does not exceeds the
maximum money holding 1. By (A.6),
Z[−d − cx(ms , mb , H) + β(v(ms + mb ) − v(ms ))]h(mb )dmb + βv(ms )v(ms ) = α
mb ∈[z,Z]

31

Thus

R
ams = α(1 − θ) mb ∈[z,Z] β[v(ms + mb ) − cv(mb )]h(mb )dmb + α(1 − θ)[h(Z) − h(z)][−βv(ms ) + cβv(0) −R= α(1 − θ) mb ∈[z,Z] βv(0) + F + b(ms + mb ) − c[v(0) + F + bmb ]}h(mb )dmb−α(1 − θ)[1 − H(0)][aβms + (βc − 1)v(0) + ck − d] + βams + βv(0)= α(1 − θ)[1 − H(0)](1 − c)β[v(0) + F ] + α(1 − θ)[1 − H(0)]βbms + α(1 − θ)(1 − c)βb(M/m)−α(1 − θ)[1 − H(0)](aβms + (βc − 1)v(0) + ck − d) + βams + βv(0)Comparing the coefficients of ms in the LHS and the RHS yields
a = α(1 − θ)[1 − H(0)]β(b − a) + βa.

Step 3: The conditions on the parameters which guarantee the pay-all property We firstpresent a sufficient condition that there is no trade if ms + q(ms , mb , H) < m∗ or ms ≥ m∗ .
Lemma A.3. Supposeb(1 − m∗ ) < d.

(A.10)

Then the bargaining can reach the agreement if only if ms < m∗ < ms + q(ms , mb , H).
Proof. Suppose ms < m∗ < ms + q(ms , mb , H) does not hold and the bargaining reaches
the agreement with x(mb , ms ) > 0 and q(ms , mb , H). The seller’s surplus does not contain
F . Then the maximum increase in the discounted utility by earning money is am∗ , b(Z) orb(1 − m∗ ). By a < b and (15), b(1 − m∗ ) is the largest among them. Then, the maximum
amount of the surplus does not exceed−d + b(1 − m∗ ),which is negative under (A.10). Note that −d is the cost of production in the case of
x = 0.

32

We next present a condition that the all-pay property holds in a bargaining between a seller
without money and a buyer with mb ≥ m∗ .
Lemma A.4. If[(1 − θ)c + θ]βF > (1 − θ)ck + θd,(A.11)

then x(ms , mb , H) > 0 and q(ms , mb , H) = mb hold in the bargaining between a seller with
ms = 0 and a buyer with mb ≥ m∗ .
Proof. If x(ms , mb , H) = 0 in the Nash bargaining, then q(ms , mb , H) = 0, i.e., no trade,and thus the surpluses of the agents are zero. As shown below, they can be positive and thebuyer and the seller trade.
If x(ms , mb , H) > 0, then by Lemma (A.1), q(ms , mb , H) = mb . Indeed, x(ms , mb , H) >0 will be shown below, and the surpluses are positive.
From Lemma (A.1),
cx = (1 − θ)cβ(F + bmb ) + θβ(F + bmb ) − [(1 − θ)ck + θd]= [(1 − θ)c + θ]β(F + bmb ) − [(1 − θ)ck + θd].
From (A.11), this is positive and thus x(ms , mb , H) > 0.
[(1 − θ)c + θ]β(F + bm∗ ) > [(1 − θ)ck + θd]
The seller’s surplus is positive, because−d − cx + β(v(mb ) − v(0))= (1 − θ) {(1 − c)β[F + b(mb − m∗ )] + (ck − d)} > 0
holds by mb ≥ m∗ . Since the seller’s surplus is positive, so is the buyer’s one. Thusx(ms , mb , H) > 0 and q(ms , mb , H) = mb hold.

33

Next, we show that the pay-all property holds in the bargaining between a seller with ms <m∗ and a buyer with mb ∈ [z, Z].
Lemma A.5. If (A.11) and[(1 − θ)c + θ]βF < (1 − θ)ck + θd + (1 − θ)(2 − c)aβm∗ − [(1 − θ)c + θ]bβ(Z − m∗ ) (A.12)
Hold, then, in the bargaining between a seller with ms < m∗ and a buyer with mb ∈ [z, Z],x(ms , mb , H) > 0 and q(ms , mb , H) = mb hold.
Proof. If x(ms , mb , H) = 0 in the Nash bargaining, then q(ms , mb , H) = 0, i.e., no trade,and thus the surpluses of the agents are zero. As shown below, they can be positive and thebuyer and the seller trade.
If q satisfies ms +q < m∗ , then from Lemma (A.3), it can not be q(ms , mb , H). Therefore,
below we consider the case ms +q ≥ m∗ . As for a buyer, we consider the cases (i) mb −q < m∗and (ii) mb − q ≥ m∗ .
Case (i): If x(ms , mb , H) > 0, then from Lemma (A.1), q(ms , mb , H) = mb holds. Indeed,x(ms , mb , H) > 0 will be shown below, and the surpluses are positive. From Lemma (A.1),
cx = (1 − θ)cβ(F + bmb ) + θβ [F + b(mb + ms ) − ams ] − [(1 − θ)ck + θd]= [(1 − θ)c + θ]β(F + bmb ) + θ(b − a)βms − [(1 − θ)ck + θd]hold. From (A.11), this is positive, and thus x(ms , mb , H) > 0.
The seller’s surplus is−d−cx+βv(mb +ms )−βv(ms )) = (1−θ) [(1 − c)β(F + bmb ) + (b − a)βms + (ck − d)] > 0.
(A.13)
Similarly, the buyer’s surplus is positive.
Case (ii): In this case, q ≤ mb − m∗ and ms + q > m∗ . Note that the buyer’s surplus
does not contain −F . The seller’s surplus is−d − cx + βv(ms + q) − βv(ms ) = −d − cx + aβms + βF + βb(ms + q)34

< −d + βF + (b − a)βms + bβ(mb − m∗ )(A.14)

Below. we show that the surplus in (A.14) is smaller than that in (A.13), i.e., Case (ii) does
not occur in the bargaining. The inequality is expressed as:−d+βF +(b−a)βms +bβ(mb −m∗ ) < (1−θ) [(1 − c)β(F + bmb ) + (b − a)βms + (ck − d)] .
Since the RHS is equal to(1 − θ)(1 − c)βF + (1 − θ)(1 − c)bβmb + (1 − θ)(b − a)βms + (1 − θ)(ck − d),
The above inequality is equivalent to[(1 − θ)c + θ]βF < (1 − θ)ck + θd + bm∗ − θ(b − a)βms − [(1 − θ)c + θ]bβmb . (A.15)
Since ms < m∗ ,[(1 − θ)c + θ]βF < (1 − θ)ck + θd + bβm∗ − θ(b − a)βm∗ − [(1 − θ)c + θ]bβmb (A.16)
is a sufficient condition for (A.15). Since mb < Z,[(1 − θ)c + θ]βF < (1 − θ)ck + θd + [a + (1 − θ)b]βm∗ − [(1 − θ)c + θ]bβZ
is a sufficient condition for (A.16), and it is equivalent to[(1−θ)c+θ]βF < (1−θ)ck +θd+aβm∗ +[(1−θ)(1−c)−θ]bβm∗ −[(1−θ)c+θ]bβ(Z −m∗ ).
(A.17)
Since b > a,[(1−θ)c+θ]βF < (1−θ)ck +θd+aβm∗ +[(1−θ)(1−c)−θ]aβm∗ −[(1−θ)c+θ]bβ(Z −m∗ )is a sufficient condition for (A.17), and it is equivalent to (A.12).

35

Step 4: The conditions in the Lemmas
Lemma A.6. Under the assumptions in Proposition 1, there exists a continuum of (z, Z, H, m∗ , M, a, b, F )satisfying the following conditions in the previous Lemmas:• b>0• (15): m∗ ≤ z < Z ≤ 1 − m∗
• H is a Borel measure on R satisfying that the support is {0} ∪ [z, Z],R
and mdH = M

R

dH = 1 − H(0),• (A.10): b(1 − m∗ ) < d• (A.11): [(1 − θ)c + θ]βF > (1 − θ)ck + θd
• (A.12): [(1 − θ)c + θ]βF < (1 − θ)ck + θd + (1 − θ)(2 − c)aβm∗ − [(1 − θ)c + θ]bβ(Z − m∗ )Proof. For simplicity, we denote x(m) = x(0, m, H). Assuming x(m) > 0, we first calculate
F . From the first order condition,−d − cx(m) + β(v(m) − v(0)) =(1 − θ)cθ[k + x + β(v(0) − v(m))].

On the other hand,RZv(0) = α z [−d − cx(m) + βv(m)]dH + (1 − α(1 − H(0)))βv(0)RZ= α z [−d − cx(m) + β(v(m) − v(0))]dH + βv(0).
Thus, from (5),1v(0) =
α
1−β

=

=

Z Z
[−d − cx(m) + β(v(m) − v(0))dHz(1 − θ)c
θH(0)

Z Z

(1 − θ)cθH(0)

[v(0) + F + bm]dHz{M b + [1 − H(0)][v(0) + F ]} .

36

Thus

⇔ [1 − H(0)]F =θH(0) − (1 − θ)c[1 − H(0)](1 − θ)c
v(0) − M bθH(0)holds. Define φ ≡ (1−θ)[1−H(0)]c. Then, from the above,F = (φ − 1) v(0) −M
1 − H(0)b,and thusb=

1 − H(0)[(φ − 1) v(0) − F ] .
M(A.18)For some ε > 0, (A.11) can be rewritten as[(1 − θ)c + θ]βF = (1 − θ)ck + θd + ε.

(A.19)

Substituting Equation (A.19) into (A.18) derives
1 − H(0)(1 − θ)ck + θd + εb=(φ − 1) v(0) −.
M[(1 − θ)c + θ]β

(A.20)For a sufficiently small δ > 0, let m∗ = 12 − δ, Z̄ = 21 − 3δ , z = 12 − 2δ3 . Then Z̄ − m∗ = 23 δ.
We can choose M satisfying[1 − H(0)]z < M < [1 − H(0)]Z̄.

(A.21)

Then there exists a measure H on [z, Z̄] satisfying

RZzdH = M and

Next, we show that there exists b > 0 satisfying (A.12). Letχ≡αβ(1 − θ)[1 − H(0)]αβ(1 − θ)[1 − H(0)] + 1 − β

37

RZ
z

dH = 1 − H(0).

Then the sum of the last two terms in the right hand side of (A.12) is denoted by LTT:
LT T = (1 − θ)(2 − c)χβm∗ b − [(1 − θ)c + θ]β(Z − m∗ )b
= (1 − θ)(2 − c)χβ 21 − δ b − [(1 − θ)c + θ]β 23 δ b= (1 − θ)(2 − c)χβ 2b − (1 − θ)(2 − c)χβ + [(1 − θ)c + θ]β

2
3



δb.

Note that the conditions(A.11) and (A.12) are.
[(1 − θ)c + θ]βF = (1 − θ)ck + θd + ε,[(1 − θ)c + θ]βF < (1 − θ)ck + θd + LT T
If, for some ε > 0 and γ > 1,
(1 − θ)(2 − c)χβ

b
= γε,
2

(A.22)

then both conditions hold for sufficiently small δ > 0. Solving (A.22) for ε and substituting
it to (A.20) yields


1 − H(0) b=(φ − 1) v(0) −M(1 − θ)ck + θd + (1 − θ)(2 − c)χβ[(1 − θ)c + θ]β

Thus



1+

1 − H(0)M(1 − θ)(2 − c)χ
[(1 − θ)c + θ]



1
2γ





1 − H(0)(1 − θ)ck + θd=(φ − 1) v(0) −M[(1 − θ)c + θ]βFor b > 0, φ must be larger than one and(φ − 1)v(0) >(1 − θ)ck + θd.
[(1 − θ)c + θ]β

38

b

 
b
2γ

.

Finally, b(1 − m∗ ) < d holds for a γ close to one and a sufficiently small δ > 0 ifhi(1−θ)ck+θd(φ − 1) v(0) − [(1−θ)c+θ]βhi < 2d.
(1−θ)(2−c)χ
1 + 1−H(0)M
2[(1−θ)c+θ]

1−H(0)M

Step 5: The existence of equilibria
Below, we show that the linear value function considered in Proposition 1 satisfies the
Bellman equation (5) and (6), (Lemmas A.8 and A.9) and the money holdings distribution
presented in the previous step is stationary. (Lemma A.7) That is, there exist equilibria. Notethat there exist a continuum of pay-all equilibria, since (z, Z, H, m∗ , M, a, b, F ) satisfying theconditions in Lemma A.6 is also a continuum.
In the following three Lemmas, we assume the assumptions in Proposition 1, and thus
all the Lemmas in the previous Steps can be used.
Lemma A.7. Under the assumptions in Proposition 1, (m∗ , z, Z, H) given in the previous
step is stationary.
Proof. In the distribution H, some agents do not have money, and the other agents havemoney in [z, Z]. From Lemma A.3, a buyer without money does not trade. Similarly, aseller with money in [z, Z] does not trade. Thus, on an equilibrium path, a trade occurs
only in the case that a seller does not have money and a buyer has money mb ∈ [z, Z]. From
Lemma A.4, the pay-all property holds and, after the trade, the seller has money mb ∈ [z, Z]
and the buyer does not have money. Therefore, the money holdings distribution remains thesame.
Lemma A.8. Under the assumptions in Proposition 1, the linear value function defined as
Equation (13) is consistent with Equation (5).
Proof. We substitute Equation (13) to the right-hand side of Equation (5) and then check
it is actually v(m). For convinience, Equation (5) isv(m) = αH(0)[k + x(m) + βv(0)] + [1 − αH(0)]βv(m)= αH(0){k + x(m) + β[v(0) − v(m)]} + βv(m)39Consider the buyer’s surplus from the bargaining, k + x(m) + β[v(0) − v(m)]. Under pay-all
equilibrium, the buyer pays m. By Equation (A.7), this buyer’s surplus is
k + x(m) + β[v(0) − v(m)] =θβ[v(m) − v(0)] − cβ[v(m) − v(0)] + ck − d
c

=


θ
β(F + bm) + cβ(F + bm) + ck − dc=θ
[(1 − c)β(F + bm) + (ck − d)] .
cThen, the right-hand-side of Equation (5) is
αH(0){k + x(m) + β[v(0) − v(m)]} + βv(m) θ= αH(0)[(1 − c)β(F + bm) + (ck − d)] + βv(m),cby Lemma 3,=α(1 − β)cαβθ(1 − c)θ[(1 − c)β(F + bm) + (ck − d)] + βv(m)cck − d= (1 − β) F + bm ++ βv(m),β(1 − c)by Lemma 5,= (1 − β) [F + bm + v(0)] + βv(m),= (1 − β)v(m) + βv(m),= v(m),
which is Equation (5).
Lemma A.9. Under the assumptions in Proposition 1, the linear value function defined as
Equation (13) is consistent with Equation (6).
Proof. Before the derivation of Equation (13), we first consider the seller’s surplus. By

40

Equation (A.6),= −d − cx(m) + β[v(m) − v(0)]= (1 − θ) β[v(m) − v(0)] − cβ[v(m) − v(0)] + (ck − d)] ,given Equation (13),= (1 − θ)[(1 − c)β(F + bm) + ck − d],= (1 − θ)[(1 − c)β(v(0) + F + bm) − (1 − c)βv(0) + ck − d],by Lemma 5,= (1 − θ)[(1 − c)βv(m) − (ck − d) + ck − d],= (1 − θ)(1 − c)βv(m).

(A.23)

Now we show the consistency. Note that Equation (6) is
Z Z[−d − cx(m) + βv(m)]dH + {1 − α[1 − H(0)]}βv(0).

v(0) = αzHere, we substitute Equation (A.23) to the right-hand side of Equation (6) and derive v(0).
The right-hand side is equivalent to
Z Z

Z Z[−d − cx(m) + βv(m)]dH + βv(0) −αzαβv(0)dHz

Z Z{−d − cx(m) + β[v(m) − v(0)]}dH + βv(0).

=αzBy Equation (A.23), it is equivalent to
Z Z(1 − θ)(1 − c)βv(m)dH + βv(0).

=α
z

Z Z
= α(1 − θ)(1 − c)βv(m)dH + βv(0),z41

by Lemma 4,θH(0)v(0)+ βv(0)= α(1 − θ)(1 − c)β(1 − θ)c


 


1−c
= α
θH(0) + 1 βv(0),cby Lemma 5,
 


 
(1 − β)c1−cθ
+ 1 βv(0)= αcαβθ(1 − c)



1−β
=
+ 1 βv(0)β= v(0).
This is the left-hand side of Equation (6).
Finally, c = 0.3, d = 0.7, k = 40, α = 0.8, β = 0.9.θ = 0.9, and M = 0.5 satisfy theconditions in the proposition.
The proof of Lemma 7
By the government budget constraint (23), (25) is rewritten as
Z Zv(0) = α−d − cx(m) + β{v(0) + F + bg[m − tx(m)]} dH + {1 − α[1 − H(0)]}βv(0)

z

Z Z
=α



Z Z
−d−cx(m)+β[v(0)+F +bgm] dH −αzβbgtx(m)dH +{1−α[1−H(0)]}βv(0)
z

Z Z
=α


−d − cx(m) + β[v(0) + F + bgm] dH − αβb(g − 1)M + {1 − α[1 − H(0)]}βv(0)

z

Z Z
=α


−d−cx(m)+β[v(0)+F +bgm] dH −αβb(g−1)

zZ Z
mdH +{1−α[1−H(0)]}βv(0)
z

Z Z
=α



−d − cx(m) + β[v(0) + F + bm] dH + {1 − α[1 − H(0)]}βv(0)

z

42

Z Z
=α




−d − cx(m) + βv(m) dH + {1 − α[1 − H(0)]}βv(0)zThe proof of Proposition 3
First, We will derive key equations.
Lemma A.10.
H(0) =(1 − β)(c + (1 − θ)βgtv 0 )αβθ(1 − c)(A.24)

Proof. Shown below.
The seller’s population H(0) depends on g and t, that is, the policy alters the distribution of money holdings. By changing the seller’s marginal welfare, the policy affects therelative surplus in (24). Then, the associated distribution is also reshaped so as to keep thestationarity.
Lemma A.11.
RZzv(m)dH

H(0)v(0)=θ(1 − θ)(c + βgtv 0 )(A.25)

Proof. Shown below.
This Lemma shows that the shift in bargaining surplus also affects the ratio of discounted
RZ
values between the two groups. The numerator, z v(m)dH, and the denominator, H(0)v(0),represents the aggregate discounted sum of utilities of buyers and sellers, repsectively. The
long-run policy with g > 1 and t > 0 shifts welfare from buyers to sellers.
Lemma A.12.
v(0) =

ck − d.
β(1 − c)(A.26)

Proof. Shown below.
The proof of Lemma A.10

43

First, we derive the consumption elasticity of discounted value ξ(m) =x0 (m). By (24),v 0 (m)(c + (1 − θ)βgtv 0 )x0 = (θβ + (1 − θ)cβ + (1 − θ)β 2 gtv 0 )v 0 ,⇔ ξ(m) =βθ(1 − c)+βc + (1 − θ)βgtv 0

Since the buyer’s value function is unchanged, we can also derive ξ(m) as in (7). Therefore,βθ(1 − c)
1−β
+ β,
+
β
=
ξ(m)
=
c + (1 − θ)βgtv 0αH(0)which leads to the expression of H(0).
The proof of Lemma A.11
From (5)(1 − β)(1 − θ)c)v(m) = (1 − θ)c{k + x(m) + β[v(0) − v(m)]}
αH(0)


⇒

(1 − β)(1 − θc)
αH(0)

Z Z

Z Z
(1 − θ)c{k + x(m) + β[v(0) − v(m)]}v(m) =z(A.27)z

By (6),Z Z

Z Z[−d − cx(m) + βv(m)]dH + βv(0) − αv(0) = αzβv(0)dHz

Z Z{−d − cx(m) + β[v(m) − v(0])}dH + βv(0)

=α
z


⇔


Z Z
(1 − β)θ
v(0) = θ{−d − cx(m) + β[v(m) − v(0])}dHαz44(A.28)

Integrating the FOC of Nash bargaining solution, (24), derives
Z Z(1 − θ)(c + βgtv 0 )[k + x(m) + β[v(0) − v(m))]dHz

Z Zθ{−d − cx(m) + β[v(m) − v(0)]}dH=(A.29)z

Substitute Equation (A.27) and (A.28) into (A.29), then

(1 − β)(1 − θ)c βgtv 0 (1 − β)(1 − θ)c+αH(0)cαH(0)

Z Z
z




(1 − β)θ
v(m)dH =v(0),αwhich leads to the Lemma .
The proof of Lemma A.12
By (5),


1−β
H(0)


⇔


v(m) = α{k + x(m) + β[v(0) − v(m))]}(1 − β)cH(0)v(m) = α{ck + cx(m) + βcv(0) − βcv(m)}⇔ α[ck + βcv(0) − βcv(m)] −(1 − β)cH(0)v(m) = α[−cx(m)]= α[d − d − cx(m) + βv(m) − βv(m)]= α[−d − cx(m) + βv(m)] + α[d − βv(m)]Then,α[−d − cx(m) + βv(m)](1 − β)cv(m) − αd + αβ(1 − c)v(m)]= αck + αβcv(0) −H(0)(1 − β)c= αβ(1 − c) −v(m) + α[ck − d + βcv(0)].
H(0)

45

(A.30)

By integrating both sides of (A.30), we get
Z Z[−d − cx(m) + βv(m)]dHαz(1 − β)c= αβ(1 − c) −
H(0)

Z Z
v(m)dH + α[1 − H(0)][ck − d + βcv(0)].
zBy (6),{1 − β + α[1 − H(0)]β}v(0)(1 − β)c= αβ(1 − c) −
H(0)

Z Z
v(m)dH + α[1 − H(0)][ck − d + βcv(0)].
z⇔ {1 − β + α[1 − H(0)](1 − c)β}v(0)Z Z(1 − β)cv(m)dH + α[1 − H(0)](ck − d).
= αβ(1 − c) −
H(0)
z
Let

X=

βGv 0
1+
c


.

Then by Lemma A.11,{1 − β + α[1 − H(0)](1 − c)β}v(0)θH(0)v(0)+ α[1 − H(0)](ck − d)(1 − θ)cXθv(0)= [αβ(1 − c)H(0) − (1 − β)c]
+ α[1 − H(0)](ck − d).
(1 − θ)cX(1 − β)c= αβ(1 − c) −
H(0)



Then,
⇔ {1 − β + α[1 − H(0)](1 − c)β}(1 − θ)cXv(0)= [αβ(1 − c)H(0) − (1 − β)c] θv(0) + α[1 − H(0)](1 − θ)cX(ck − d).
Solving the above with respect to v(0), we obtain

46

d−ck
v(0) = Xcα (θ − 1) θ(c(β−1)−H(0)αβ(c−1))+Xc(θ−1)(−β+αβ(H(0)−1)(c−1)+1)(H(0) − 1) .
0)
Substituting H(0) = (1−β)(c+(1−θ)βGv, the RHS of the above is equal toαβθ(1−c)Xc(d−ck)(c−cβ+Gv 0 β−θαβ−Gv 0 β 2 −Gv 0 θβ+cθαβ+Gv 0 θβ 2 )β(c−1)(cθ+Xc2 −Xcθ−cθβ−Xc2 β+Gv 0 θβ+Xcθβ−Gv 0 θβ 2 +Xc2 θαβ+Gv 0 Xcβ−Xcθαβ−Gv 0 Xcβ 2 −Gv 0 Xcθβ+Gv 0 Xcθβ 2 )0Substituting X = 1 + βGv, intoc(c−cβ+Gv 0 β−θαβ−Gv 0 β 2 −Gv 0 θβ+cθαβ+Gv 0 θβ 2 )(cθ+Xc2 −Xcθ−cθβ−Xc2 β+Gv 0 θβ+Xcθβ−Gv 0 θβ 2 +Xc2 θαβ+Gv 0 Xcβ−Xcθαβ−Gv 0 Xcβ 2 −Gv 0 Xcθβ+Gv 0 Xcθβ 2 )we obtainc−cβ+Gv 0 β−θαβ−Gv 0 β 2 −Gv 0 θβ+cθαβ+Gv 0 θβ 2
1
= c+Gv
0β .
(c+Gv 0 β)(c−cβ+Gv 0 β−θαβ−Gv 0 β 2 −Gv 0 θβ+cθαβ+Gv 0 θβ 2 )Therefore,Xc(ck−d)v(0) = β(1−c)(c+Gv0 β) .

βGv 0Since X = 1 + c ,(ck−d)
v(0) = β(1−c).

The proof of Proposition 3
Define the total welfare as W = H(0)v(0) +RZzv(m)dH. Then, from Lemma A.11 andLemma A.12, it can be rewritten as
W = QH(0)v(0),(A.31)θwhere Q = 1 + (1−θ)(c+βGv
0) .
Then,
∂W
∂Q
∂H(0)
∂v(0)
=
H(0)v(0) +Qv(0) +
QH(0).
∂t∂t∂t∂t(A.32)

From the definition of Q,∂Q−θβG0 v 0=.
∂t
(1 − θ)(c + βGv 0 )2

(A.33)

From Lemma A.10,∂H(0)(1 − β)(1 − θ)βG0 v 0=.
∂tαβθ(1 − c)(A.34)
47

From (A.26),
∂v(0)
= 0.
∂t

(A.35)

Below, we obtain ∂W. From (A.33), (A.34), and (A.35),∂t∂Q
H(0)v(0) + ∂H(0)Qv(0) + ∂v(0)
QH(0)
∂t
∂t
∂t

=

βG0 v 0 (1−β)(Gv 0 β(1−θ)2 (2c+Gv 0 β)+c((c−1)θ2 −2cθ+c))αθ(1−c)(1−θ)(c+GV β)2(A.36)
This is positive, if and only if
Gv 0 β (1 − θ)2 (2c + Gv 0 β) + c((c − 1) θ2 − 2cθ + c) > 0.

(A.37)
0√andConsidering (A.37)= 0 as an equation with respect to θ, it has two solutions c−c+βGv
c+βGv 00
c+βGv√.
c+ c+βGv 0

0

0

√
√
Suppose G0 > 0. Then ∂W< θ < c+c+βGv, since> 0, if and only if c−c+βGv∂t
c+βGv 0c+βGv 0the coefficient of θ2 is negative.

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