778

T. KUNIEDA AND K. NISHIMURAof Benhabib and Nishimura’s model, in which consumption goods are produced in
the ﬁrst sector and intermediate goods are produced in the second sector. In ourmodel, agents receive an idiosyncratic productivity shock in each period when producing capital goods. Those who have higher productivity purchase intermediate
goods and initiate an investment project to produce capital goods. They face ﬁnancial constraints and can borrow only up to a certain proportion of their savings
when initiating an investment project. Meanwhile, those who have lower productivity lend their all savings in the ﬁnancial market. Therefore, borrowers and lenders
endogenously appear in equilibrium.
In our model, the extent of ﬁnancial constraints measures the extent of ﬁnancial
development, following the literature (e.g., Aghion and Banerjee, 2005). In equilibrium, whereas capital accumulation is promoted and the gross product in the steadystate monotonically increases as the ﬁnancial sector becomes better developed, indeterminacy of equilibria is more likely to occur if the consumption sector is morelabor intensive from the social perspective but less labor intensive from the private
perspective relative to the intermediate sector. Given these results, we introduce
a sunspot variable in the model and examine whether ﬁnancial development ampliﬁes or contracts sunspot ﬂuctuations. Our ﬁndings are as follows. Under highlabor intensity in the consumption good sector from the social perspective, ﬁnancial
development is more likely to magnify sunspot ﬂuctuations, whereas under high labor intensity in the intermediate good sector from the social perspective, ﬁnancial
development is more likely to contract sunspot ﬂuctuations.
The remainder of this paper proceeds as follows. In section 2, we present themodel in which there are two production sectors in the economy, with each sector subject to sector-speciﬁc production externalities, and in which inﬁnitely lived
agents face ﬁnancial constraints. In section 3, equilibrium in the model is derived.
We also characterize a steady state of the dynamical system in this section. In
section 4, the dynamic property in the neighborhood of the steady state is investigated, and the condition for the economy to exhibit indeterminacy of equilibria is
obtained. In section 5, we introduce a sunspot variable in the model and explore
whether ﬁnancial development ampliﬁes sunspot ﬂuctuations. Section 6 concludes
the paper.
2. ModelConsider a closed economy that consists of an inﬁnitely lived representative ﬁrm
and a continuum of inﬁnitely lived agents with the total population normalized toone. The economy goes in discrete time indexed by t from t = 0 to t = +∞.
The representative ﬁrm produces both consumption and intermediate goods with
diﬀerent technologies in each period. The intermediate goods are assumed to be
numeraire throughout the analysis. The inﬁnitely lived agents are potential capital
producers but receive uninsured idiosyncratic productivity shocks in each periodthat aﬀect productivity in capital production.
2.1. Agents.
2.1.1. Timing of events. At the beginning of period t, an agent earns incomes: awage income, returns to her savings, and a lump-sum proﬁt from the representative

FINANCIAL DEVELOPMENT AND SUNSPOT FLUCTUATIONS

779

ﬁrm. The market for consumption goods in period t opens at the beginning of theperiod and is closed before an idiosyncratic productivity shock in period t is realized.
Accordingly, the agent must make a decision about consumption and savings at the
beginning of period t without knowing her productivity in capital production. At
the end of period t, the agent receives an idiosyncratic productivity shock. The
agent can adopt two saving methods: lending her savings in the ﬁnancial market
or initiating an investment project. The agent chooses one of the saving methods
with knowing her productivity. Lending one unit of savings in period t yields aclaim to rt+1 units of intermediate goods in period t + 1, where rt+1 is the grossreal interest rate, whereas purchasing one unit of intermediate goods in period t for
the investment project creates Φt units of capital used in period t + 1. One unitof capital is sold at price qt+1 to the production sector in period t + 1. Capital is
perishable in one period. Although the agent can borrow in the ﬁnancial market,
she faces a ﬁnancial constraint, which implies that she can only borrow up to acertain proportion of her savings.
Φt is the productivity shock on the probability space (Ω, F, P ), where Ω is asample space, F is a σ−algebra on Ω, and P is the probability measure.1 Φt is afunction of ωt ∈ Ω, the support for which is given by [0, η] where η ∈ (0, ∞). The
cumulative distribution function of Φt is given by G(Φ) := P ({ωt ∈ Ω | Φt (ωt ) ≤Φ}), where {ωt ∈ Ω | Φt (ωt ) ≤ Φ} ∈ F. G(Φ) is continuously diﬀerentiable on the
support and time-invariant. Φ0 , Φ1 , · · · , are independent and identically distributed
across both agents and time (the i.i.d. assumption). Because there is no insurance
market for productivity shocks, no one can insure against low productivity. Deﬁnethe history of ωt as ω t−1 = {ω0 , ω1 , ..., ωt−1 }. Then, we obtain the probability space(Ωt , F t , P t ), which is a Cartesian product of t copies of (Ω, F, P ) in which ω t−1 isan element of Ωt . An individual who experiences this history can be identiﬁed byω t−1 .
2.1.2. Utility maximization. An agent’s expected lifetime utility in period t is given
by
"∞
#
X
E
δ τ −t cτ (ω τ −1 ) ω t−1 ,τ =twhere δ ∈ (0, 1) is the subjective discount factor and cτ (ω τ −1 ) is consumption.
E[.|ω t−1 ] is an expectation operator given the history ω t−1 . The agent in period t
maximizes her lifetime utility subject to
(2.1)
(2.2)
(2.3)

pτ cτ (ω τ −1 ) + xτ (ω τ ) + bτ (ω τ )= qτ Φτ −1 (ωτ −1 )xτ −1 (ω τ −1 ) + rτ bτ −1 (ω τ −1 ) + wτ + πτbτ (ω τ ) ≥ −λaτ (ω τ −1 )xτ (ω τ ) ≥ 0

for τ ≥ t. In (2.1), wτ is a wage income, πτ is a proﬁt obtained from the production
sector, and pτ is the price of consumption goods. Additionally, xt (ω t ) is an intermediate good used for an investment project, and bτ (ω t ) is lending if bτ (ω t ) > 0
1One can assume Ω = [0, 1].

780

T. KUNIEDA AND K. NISHIMURA

and borrowing if bτ (ω t ) < 0. For capital production, a linear technology is assumed such as Φτ −1 (ωτ −1 )xτ −1 (ω τ −1 ), which is capital produced in period τ by anagent who initiates a project in period τ −1 drawing productivity Φτ −1 . In any case,at (ω t−1 ) := xt (ω t ) + bτ (ω t ) is the agent’s savings in period t. When the agent makes
a decision about consumption, ct (ω t−1 ), and saving, at (ω t−1 ), in period t, she does
not know her productivity Φt (ωt ) in capital production, as previously explained.
However, as noted from the expression at (ω t−1 ) = xt (ω t ) + bt (ω t ), the agent knowsΦt (ωt ) when she makes a portfolio decision about investing in a project, lending,
and/or borrowing in period t. Eq. (2.1) is eﬀective for τ ≥ 1. It is assumed thatthe ﬂow budget constraint is given by a0 = q0 X0 + w0 + π0 − p0 c0 in period 0, where
X0 is the initial capital endowment that is commonly distributed across agents.
The agent faces a ﬁnancial constraint given by inequality (2.2).2 Inequality (2.2)
implies that an agent can borrow in the ﬁnancial market only up to a certain
proportion of her savings, which is also regarded as her net worth. λ ∈ (0, ∞)measures the extent of ﬁnancial constraints; i.e., the ﬁnancial constraint is morerelaxed as λ becomes greater. Inequality (2.2) is converted into bτ (ω τ ) ≥ −µxτ (ω τ ),where µ = λ/(1 + λ) ∈ (0, 1). This constraint is more convenient than inequality(2.2), and we use it henceforth. The ﬁnancial market approaches perfection as µ
goes to 1. As in the literature, µ is assumed to measure the degree of ﬁnancial
development.3 Finally, we impose a nonnegativity constraint given by inequality(2.3) on the purchase of intermediate goods.
2.1.3. Optimal decision within a period. We deﬁne a new variable as ϕt := rt+1 /qt+1 .
Agents know the productivity shocks for capital production realized in period t when
they make an optimal portfolio decision about investment, lending, and borrowing
in the same period. Therefore, an agent who receives Φt > ϕt borrows up to
the limit of the ﬁnancial constraint and purchases intermediate goods to start aninvestment project, whereas an agent who receives Φt ≤ ϕt lends all her savings in
the ﬁnancial market and obtains interest of rt+1 .4 One notes that ϕt is the cutoﬀ
for the productivity shocks that divide agents into lenders and borrowers in periodt, and the agent’s optimal decision within a period is given by(0
if Φt (ωt ) ≤ ϕt
(2.4)
xt (ω t ) =at (ω t−1 )if Φt (ωt ) > ϕt ,
1−µ
and
(2.5)


t

bt (ω ) =at (ω t−1 )if Φt (ωt ) ≤ ϕtµat (ω t−1 ) if Φt (ωt ) > ϕt .
− 1−µ2Many researchers assume this type of ﬁnancial constraint in the literature. See, for instance,

Aghion et al. (1999), Aghion and Banerjee (2005), Aghion et al. (2005), and Kunieda and Shibata(2016).
3See, for instance, Aghion and Banerjee (2005).
4Agents who receive Φ = ϕ are indiﬀerent between initiating a project and lending in thettﬁnancial market. We assume they lend their savings in the ﬁnancial market.

FINANCIAL DEVELOPMENT AND SUNSPOT FLUCTUATIONS

781

From the portfolio decision given by (2.4) and (2.5), the ﬂow budget constraint (2.1)
is rewritten as(2.6)

aτ (ω τ −1 ) + pτ cτ (ω τ −1 ) = Rτ (ωτ −1 )aτ −1 (ω τ −2 ) + wτ + πτ ,where Rτ (ωτ −1 ) := max{rτ , (qτ Φτ −1 (ωτ −1 ) − rτ µ)/(1 − µ)}. The maximization of
the agent’s lifetime utility subject to (2.6) yields the Euler equation as follows:


(2.7)
pt+1 = δE Rt+1 (ωt )|ω t−1 pt .
The necessary and suﬃcient optimality conditions for the lifetime utility maximization problem are given by the Euler equation (2.7) as well as the transversality
condition limτ →∞ δ τ E[at+τ (ω t+τ −1 )/pt+τ |ω t−1 ] = 0.
2.2. Production. The representative ﬁrm uses the following two Cobb-Douglas
production technologies:F̄ 1 (lt1 , zt1 , ¯lt1 , z̄t1 ) = A(lt1 )a1 (zt1 )b1 (¯lt1 )α1 (z̄t1 )β1 , 0 < a1 , b1 , α1 , β1 < 1
for intermediate goods, andF̄ 2 (lt2 , zt2 , ¯lt2 , z̄t2 ) = B(lt2 )a2 (zt2 )b2 (¯lt2 )α2 (z̄t2 )β2 , 0 < a2 , b2 , α2 , β2 < 1
for consumption goods, where ai +αi +bi +βi = 1 and ai +bi =: e ∈ (0, 1) for i = 1, 2.
In the production technologies, lti and zti are labor and capital inputs, respectively.
The components of production externalities with respect to labor and capital are
given by ¯lti and k̄ti , respectively. Note that both labor and capital exhibit positive
externalities to the production technologies because both αi and βi are positive.
Although ¯lti and k̄ti are exogenous when the ﬁrm solves the proﬁt maximization
problem, it holds that lti = ¯lti and kti = k̄ti in equilibrium.
The proﬁt maximization problem that the ﬁrm faces is given as follows:(2.8)
max Πt := F̄ 1 (l1 , z 1 , ¯l1 , z̄ 1 ) + pt F̄ 2 (l2 , z 2 , ¯l2 , z̄ 2 ) − qt zt − wt lt ,lt1 ,lt2 ,zt1 ,zt2

t

t

t

t

t

t

t

t

where zt = zt1 + zt2 is the aggregate capital and lt = lt1 + lt2 is the population of
agents. It is assumed that lt is constant and normalized to lt = 1. The ﬁrst-orderconditions for the proﬁt maximization are given by 1 1−θ1 2 1−θ2zz(2.9)
Aa1 1t= pt Ba2 2t= wt .
lt
lt
and(2.10)

 2 −θ2 1 −θ1zzt= pt Bb2 2t= qt ,
Ab1 1lt
lt

where ai + αi =: θi ∈ (0, 1) for i = 1, 2. From Eqs. (2.9) and (2.10), we have
(2.11)

zt1 =b1 w t 1l
a 1 qt t

zt2 =b2 w t 2l .
a 2 qt tand(2.12)

782

T. KUNIEDA AND K. NISHIMURA

Again from Eqs. (2.9) and (2.10), we also have
(2.13)1−θ1
θ −θ1

wt = Ψpt 2=: w(pt )and(2.14)−θ1
θ −θ1

qt = Λpt 2=: q(pt ),1 θ2 −1
2 1−θ1 1/(θ2 −θ1 )where Ψ = [(Aa1θ1 b1−θ)(Baθ22 b1−θ)]and Λ = [(Aaθ11 b11−θ1 )θ2
1
2
2 −θ1 1/(θ2 −θ1 )(Baθ22 b1−θ) ].
2
One may say that if a1 /b1 < a2 /b2 ⇐⇒ a1 /(e − a1 ) < a2 (e − a2 ) ⇐⇒ a1 < a2 ,the consumption good sector is labor intensive relative to the intermediate good
sector from the private perspective and if θ1 /(1 − θ1 ) < θ2 /(1 − θ2 ) ⇐⇒ θ1 < θ2 , theconsumption good sector is labor intensive relative to the intermediate good sector
from the social perspective.

3. Equilibrium
A competitive equilibrium is given by sequences of prices {wt , qt , pt , rt+1 } for allt ≥ 0 and allocation {ct (ω t−1 ), at (ω t−1 ), kt (ω t ), bt (ω t )} for all t ≥ 0 and ω t , and{zt , lt } for all t ≥ 0, so that (i) for each ω t , the consumer maximizes her lifetime
utility from time t onward, (ii) the representative ﬁrm maximizes its proﬁts in eachperiod, and (iii) the consumption and intermediate good markets, the ﬁnancial
market, the capital market, and the labor market clear.5
3.1. Market clearing conditions. In each period, aggregate consumption is equal
to the production of consumption goods, and thus, the consumption good market
clearing condition is given by
Z
(3.1)
Ct :=
ct (ω t−1 )dP t (ω t−1 ) = F 2 (zt2 , lt2 ),

Ωt
2
2
2
2
2
2
where F (zt , lt ) = F̄ (lt , zt , lt2 , zt2 ).

Eq. (2.4) implies that the intermediate goods
are purchased by agents who draw higher productivity such that Φt (ωt ) > ϕt .
Accordingly, the intermediate good market clearing condition is given byZ(3.2)
xt (ω t )dP t+1 (ω t ) = F 1 (zt1 , lt1 ),Ωt ×(Ω\Ξt )where Ξt = {ωt ∈ Ω|Φt (ωt ) ≤ ϕt } and F 1 (zt1 , lt1 ) = F̄ 1 (lt1 , zt1 , lt1 , zt1 ). In the ﬁnancial
market, the aggregation of all debts and lending becomes zero, and thus, it holdsthatZ
bt (ω t )dP t+1 (ω t ) = 0.
(3.3)
Ωt+1

Capital is produced by agents who draw higher productivity such that Φt (ωt ) > ϕt
as seen in (2.4) and used by the representative ﬁrm. Then, we have the capital
market clearing condition as follows:
Z
1
2
(3.4)
zt+1 + zt+1 =Φt (ωt )xt (ω t )dP t+1 (ω t ).
Ωt ×(Ω\Ξt )5To be precise, ω −1 is empty because c is not subject to any history of the stochastic events.
0FINANCIAL DEVELOPMENT AND SUNSPOT FLUCTUATIONS

783

Finally, the labor market clearing condition is given by
lt1 + lt2 = 1.

(3.5)

3.2. Gross product in equilibrium. From (2.11), (2.12), (3.5), and zt1 + zt2 = zt ,the production functions are expressed by
F 1 (lt1 , zt1 ) =a2 q(pt )zt − b2 w(pt )e(a2 − a1 )pt F 2 (lt2 , zt2 ) =a1 q(pt )zt − b1 w(pt ).
e(a1 − a2 )

(3.6)
and
(3.7)

Eqs. (3.6) and (3.7) rewrite the gross product, Yt = F 1 (lt1 , zt1 ) + pt F 2 (lt2 , zt2 ), as
follows:
q(pt )zt + w(pt )
(3.8)
Yt =
.
e
3.3. Cutoﬀ. The ﬁnancial market clearing condition (3.3) determines ϕt in equilibrium as shown in Proposition 3.1.
Proposition 3.1. The cutoﬀ, ϕt , is given by(3.9)

G(ϕt ) = µ.

Proof. See the Appendix.
Since the cumulative distribution function is strictly increasing over the support,
ϕt is uniquely determined as ϕt = G−1 (µ) =: ϕ∗ and increases as µ increases. This
relationship between ϕ∗ and µ implies that as the ﬁnancial constraint is relaxed,
the number of lenders increases and the number of capital producers (borrowers)
decreases. As the number of capital producers decreases, ineﬃciency regarding theallocation of intermediate goods is corrected because the intermediate goods areintensively used by higher-productivity agents when ϕ∗ increases.
3.4. Dynamical system. We obtain two lemmata that are useful for aggregating
the ﬂow budget constraint (2.6) across all agents.
Lemma 3.2.

Z
Rt (ωt−1 )at−1 (ω t−2 )dP t (ω t−1 ) = q(pt )zt .

(3.10)Ωt

Proof. See the Appendix.
One notes from Lemma 3.2 that the total income from all agents’ savings is
eventually equal to the value of total capital in the economy. By using Lemma
3.2, we aggregate the ﬂow budget constraint (2.6) across all agents and obtain therelationship between the total saving and the intermediate goods produced in period
t in Lemma 3.3 below.
Lemma 3.3.

Z

(3.11)Ωtat (ω t−1 )dP t (ω t−1 ) = F 1 (lt1 , zt1 )784

T. KUNIEDA AND K. NISHIMURA

Proof. See the Appendix.
The intermediate goods are used to produce capital by the higher-productivity
agents, i.e., those who draw Φt (ωt ) greater than ϕt , and thus, the i.i.d. assumption,
Lemma 3.3, and Eq. (2.4) yield capital zt+1 as in Proposition 3.4.
Proposition 3.4.
(3.12)

zt+1 =

where H(ϕ∗ ) =Rη

H(ϕ∗ ) 1 1 1
F (lt , zt ),1−µϕ∗ Φt (ωt )dG(Φ).

Proof. See the Appendix.
Substituting Eq. (3.6) into Eq. (3.12) yields
H(ϕ∗ ) a2 q(pt )zt − b2 w(pt )(3.13)
zt+1 =,1−µe(a2 − a1 )From Eqs. (3.8) and (3.13), we have
(3.14)

Yt+1 = −H(ϕ∗ )q(pt+1 )w(pt+1 )(a2 Yt − w(pt )) +.
e(1 − µ)(a1 − a2 )e

The expected return, E[Rt+1 (ωt )|ω t−1 ], can be computed by using G(ϕ∗ ) = µand ϕ∗ = rt+1 /q(pt+1 ) in Proposition 3.5 below.
Proposition 3.5.
(3.15)

E[Rt+1 (ωt )|ω t−1 ] = q(pt+1 )H(ϕ∗ ).
1−µ

Proof. See the Appendix.
Eq. (2.7) is rewritten by Eqs. (2.14) and (3.15) as follows: θ2 −θ1θ2 −θ1θ2ΛδH(ϕ∗ )(3.16)
pt+1 =(pt ) θ2 =: s(pt ).
1−µ3.5. Steady state. From Eq. (3.16), we obtain the price of consumption goods inthe steady state, p̄, as θ2ΛδH(ϕ∗ ) θ1 −1(3.17)p̄ =.
1−µIt follows from Eqs. (2.13), (2.14), (3.13), and (3.17) that the capital stock, z̄, inthe steady state can be computed as1
1−θ1 
δH(ϕ∗ ) θ1b2 ΨΛ θ11−µ(3.18)z̄ =.
a2 + δe(a1 − a2 )Eqs. (2.13), (2.14), (3.14), and (3.17) yield the gross product, Ȳ , in the steady state
as follows: 1−θ1δΛH(ϕ∗ ) θ1
1 + δ(a1 − a2 ).
(3.19)
Ȳ = Ψ
a2 + δe(a1 − a2 )1−µ

FINANCIAL DEVELOPMENT AND SUNSPOT FLUCTUATIONS

785

The extent of ﬁnancial market imperfections measured by µ aﬀects the price ofconsumption goods, capital accumulation, and the gross product in the steady state
as demonstrated in Proposition 3.6 below.6
Proposition 3.6. As the ﬁnancial constraint is relaxed, the following hold:• If the production of the intermediate goods is more labor (capital) intensive
than that of the consumption goods from the social perspective, i.e., θ1 > θ2
(θ1 < θ2 ) from the social perspective, the price of consumption goods in thesteady state decreases (increases), i.e., ∂ p̄/∂µ < 0 (∂ p̄/∂µ > 0).
• The capital stock in the steady state increases, i.e., ∂ z̄/∂µ > 0.
• The gross product in the steady state increases, i.e., ∂ Ȳ /∂µ > 0.
Proof. See the Appendix.
We can intuitively understand the results of Proposition 3.6. As the ﬁnancial
constraint is relaxed, allocative eﬃciency with regard to the use of intermediate
goods is corrected. As the ﬁnancial constraint is relaxed, less productive agents are
excluded from capital production activities, and capital producers who draw higher
productivity use more intermediate goods. Then, aggregate productivity regarding
capital production in the economy becomes higher. Accordingly, total productionof capital in the steady state, z̄, increases. Capital and labor are complementary inproduction in both production sectors. Therefore, it is straightforward to show that
if z̄ increases, the capital price decreases because the marginal product of capitaldecreases, whereas the wage rate increases because the marginal product of labor
increases. If the capital price decreases and the wage rate increases, the relative
price of ﬁnal goods in the labor-intensive sector increases, and the relative priceof ﬁnal goods in the capital-intensive sector decreases. This means that the priceof consumption goods, p̄, decreases (increases) as µ increases if θ1 > θ2 (θ1 < θ2 ).
This reverse Stolper-Samuelson property helps us understand the eﬀect that therelaxation of the ﬁnancial constraint has on the price of consumption goods, p̄, inthe steady state.7
4. Local dynamics
By inserting Eq. (3.16) into (3.14), we obtain a dynamical system in this economy
with respect to Yt and pt as follows: Yt+1 = J(Yt , pt ) θ2 −θ1θ2 −θ1(4.1)∗
 pt+1 = ΛδH(ϕ ) θ2 (pt ) θ2 ,1−µ6Throughout the analysis, we exclusively focus on the case of imperfect specialization in which

the economy consistently produces both intermediate and consumption goods. One can show thatthe steady state that we have derived exists in the area of imperfect specialization. By continuity,
both intermediate and consumption goods are produced in the neighborhood of the steady state.
7Note that the causality from the prices of input goods to the prices of ﬁnal goods is opposite
to that of the standard Stolper-Samuelson theorem, in which the prices of ﬁnal goods aﬀect theprices of input goods. In a closed economy without any aggregate shocks, the prices of input goods
are determined by the extant capital and labor together with technologies, and thus, the prices of
input goods aﬀect the prices of ﬁnal goods.

786

T. KUNIEDA AND K. NISHIMURA

where

H(ϕ∗ )q(s(pt ))w(s(pt ))(a2 Yt − w(pt )) +.
e(1 − µ)(a1 − a2 )e
The linearization of the dynamical system (4.1) around the steady state yields!− δe(aa12−a2 ) Jp (Ȳ , p̄)
Yt+1 − Ȳ
Yt − Ȳ(4.2)=,
pt+1 − p̄
pt − p̄01 − θθ12
J(Yt , pt ) = −where Jp (Y, p) := ∂J(Y, p)/∂p. From Eq. (4.2), we obtain the eigenvalues, κ1 andκ2 , of this dynamical system as follows:a2(4.3)
κ1 = −
δe(a1 − a2 )
and
θ1
(4.4)
κ2 = 1 − .
θ2Depending upon the parameter values, the eigenvalues are characterized in Lemma
4.1 below.
Lemma 4.1. The eigenvalues, κ1 and κ2 , are characterized as follows:• The value of κ1 is determined by the subjective discount factor and the parameter values related to the factor intensity from the private perspective as
follows:– a1 − a2 < 0 if and only if κ1 > 1.
– 0 < eδ < a2 /(a1 − a2 ) if and only if κ1 < −1.
– 0 < a2 /(a1 − a2 ) < eδ if and only if −1 < κ1 < 0.
• The value of κ2 is determined by the parameter values related to the factor
intensity from the social perspective as follows:– 0 < θ1 < 2θ2 if and only if −1 < κ2 < 1.
– 0 < 2θ2 < θ1 if and only if κ2 < −1.
Proof. See the Appendix.
We obtain various results regarding the local stability of the steady state. In
particular, we focus on the case in which the steady state is totally stable and indeterminacy of equilibrium occurs so that we can consider self-fulﬁlling businessﬂuctuations. It is said that equilibrium is indeterminate if there exists a continuumof competitive equilibrium. In particular, in this model, indeterminacy of equilibrium occurs if there exists a continuum of initial prices of the consumption goods
for any given initial capital, z0 , such that each one of them is consistent with acompetitive equilibrium.
Proposition 4.2. It holds that the steady state, (Ȳ , p̄), is totally stable, and thus,
equilibrium is indeterminate around the steady state if and only if 0 < a2 /(a1 −a2 ) <eδ and 0 < θ1 < 2θ2 .
Proof. From Lemma 4.1, it follows that −1 < κ1 < 0 and −1 < κ2 < 1 if andonly if 0 < a2 /(a1 − a2 ) < eδ and 0 < θ1 < 2θ2 . Therefore, the steady state,(Ȳ , p̄), is totally stable if and only if 0 < a2 /(a1 − a2 ) < eδ and 0 < θ1 < 2θ2 .
Since z0 is predetermined and p0 can jump, if the steady state is totally stable,

FINANCIAL DEVELOPMENT AND SUNSPOT FLUCTUATIONS

787

it follows that for any given initial capital, z0 , there exists a continuum of initial
prices of the consumption goods, each one of which together with z0 determines Y0
by Eq. (3.8) and produces a sequence {Yt , pt }∞t=0 in competitive equilibrium, andthus, equilibrium is indeterminate.
□
Proposition 4.2 implies that indeterminacy of equilibrium is more likely to occur
when the factor intensity is reversed between the private and social perspectives.
The intuition behind this appearance of the continuum of competitive equilibria
when the factor intensity is reversed is as follows. Suppose that θ2 > θ1 anda2 < a1 , which satisfy the condition for indeterminacy. Under this condition, theconsumption sector is labor intensive from the social perspective and the intermediate sector is labor intensive from the private perspective. Consider a competitive
equilibrium. To examine whether the neighborhood of this competitive equilibrium
can also be another competitive equilibrium, suppose that the price of consumption goods pτ becomes inﬁnitesimally greater than that of the original competitive
equilibrium in a certain period τ . Since the consumption sector is labor intensivefrom the social perspective, the greater price of consumption goods increases thewage rate and decreases the capital price (the Stolper-Samuelson property). Accordingly, as noted from Eqs. (2.11) and (2.12), the capital-labor ratios in both
sectors must increase. Again, note from Eqs. (2.11) and (2.12) that since a2 < a1
and e − a2 = b2 > b1 = e − a1 , it must hold that zt2 /lt2 (the capital-labor ratio in
the consumption good sector) is greater than zt1 /lt1 (the capital-labor ratio in theintermediate good sector) for all t ≥ 0. Therefore, the only way for both zt2 /lt2 andzt1 /lt1 to increase is that both zt2 and lt2 decrease and both zt1 and lt1 increase. Then,in period τ + 1, output in the consumption good sector decreases, whereas that in
the intermediate good sector increases. As a result, the price of consumption goods
increases in period τ + 1. Because we have assumed the inﬁnitesimal increase in theprice of consumption goods in period τ , this outcome is consistent with the dynamic
equation of pt in Eq. (4.2), and thus, the neighborhood of a competitive equilibrium
can be another competitive equilibrium. In contrast, when a2 = θ2 > θ1 = a1 , itmust hold that zt2 /lt2 is smaller than zt1 /lt1 for all t ≥ 0, and then, the only way
for both zt2 /lt2 and zt1 /lt1 to increase is that both zt2 and lt2 increase and both zt1
and lt1 decrease. Accordingly, the output in the consumption good sector increases,
whereas that in the intermediate good sector decreases in period τ + 1. As a result,the price of consumption goods decreases in period τ + 1. This outcome contradicts the dynamic equation of pt in Eq. (4.2), and thus, the neighborhood of acompetitive equilibrium cannot be another competitive equilibrium.
5. Self-fulfilling business fluctuations
One notes from Lemma 4.1 that the stability of the steady state is independentof the extent of ﬁnancial constraints, as it is aﬀected only by the private and social factor intensities and the subjective discount factor. Although the extent ofﬁnancial constraints does not aﬀect the stability of the steady state, it would amplify endogenous business ﬂuctuations caused by extrinsic uncertainty. It is wellknown that indeterminacy equilibrium in dynamic general equilibrium models is apotential cause of sunspot ﬂuctuations. Therefore, we focus on the case in which

788

T. KUNIEDA AND K. NISHIMURA

indeterminacy in equilibrium arises, i.e., 0 < a2 /(a1 − a2 ) < eδ and 0 < θ1 < 2θ2 asshown in Proposition 4.2.
Assumption 5.1. (i) 0 < θ1 < 2θ2 and (ii) 0 < a2 /(a1 − a2 ) < eδ.
5.1. Sunspot variable. A sunspot variable, ϵt , is introduced in our model, whichis assumed to follow a white noise process with mean 0 and variance σ 2 and to be
independent and identically distributed across time. The support for ϵt is [−ϵ̄, ϵ̄],where ϵ̄ is assumed not to be too large so that equilibrium with sunspots can exist.
Each agent faces a common sunspot variable in each period, and ϵt is independentof Φt′ (ωt′ ) for all t ≥ 0 and t′ ≥ 0. The history of sunspot events until period t is
written as ϵt = {ϵ0 , ϵ1 , ..., ϵt }.
The sunspot variable, ϵt , is realized at the beginning of period t, and thus, eachagent makes a decision about consumption and saving given information about thehistory of sunspot events, ϵt . Then, an agent’s expected lifetime utility in period t
is given by
"∞
#
X
τ −t
τ −1 tt−1 tE
δ cτ (ω,ϵ ) ω ,ϵ ,
τ =t
t−1
t
where E[.| ω , ϵ ] is the expectation operator given ω t−1 and ϵt . Although the cut-oﬀ, ϕt , is no longer equal to rt+1 /qt+1 , the concrete expression of ϕt is unnecessary.
This is because from the ﬁnancial market clearing condition, Proposition 3.1 stillholds, and eventually, we obtain ϕt = ϕ∗ .8 Then, the individual return to saving,
Rt+1 in equilibrium, is given by(
rt+1
if Φt (ωt ) ≤ ϕ∗
(5.1)
Rt+1 (ωt ) =q(pt+1 )Φt (ωt )−rt+1 µif Φt (ωt ) > ϕ∗ ,1−µThe Euler equation is obtained as
(5.2)



Rt+1 (ωt ) t−1 t1= δEω ,ϵ .
pt
pt+1

The Euler equation (5.2) can be computed as follows:
1
δH(ϕ∗ )
q(pt+1 ) t
=
E
ϵ .
(5.3)
pt
1−µ
pt+1
5.2. Sunspot ﬂuctuations and ﬁnancial development. From Eq. (5.3), it isassumed that the sunspot variable ϵt+1 is introduced as in the following.
1
δH(ϕ∗ ) q(pt+1 )(5.4)=+ ϵt+1 .
pt
1−µ
pt+1
One can verify that Eq. (5.3) can be obtained by taking the expectation for bothsides of Eq. (5.4) with ϵt given. Eq.(5.4) can be rewritten as θ2 −θ1θ2 −θ1θ2ΛδH(ϕ∗ )(5.5)
pt+1 =(pt ) θ2 =: s(pt ; ϵt+1 ).
(1 − µ)(1 − pt ϵt+1 )tt8For reference, we can obtain ϕ = rtt+1 E[1/pt+1 |ϵ ]/E[q(pt+1 )/pt+1 |ϵ ] when the agents are

subject to the sunspot variable.

FINANCIAL DEVELOPMENT AND SUNSPOT FLUCTUATIONS

789

From (3.14) and (5.5), we obtain(5.6) Yt+1 = −H(ϕ∗ )q(s(pt ; ϵt+1 ))w(s(pt ; ϵt+1 ))(a2 Yt − w(pt )) +=: J(Yt , pt ; ϵt+1 ).
e(1 − µ)(a1 − a2 )e

The linearization of (5.5) and (5.6) around the steady state with ϵt+1 = 0 leads to
Yt+1 − Ȳ(5.7)
pt+1 − p̄ 1+θ2 −2θ1 !θ1H(ϕ∗ )a2− δe(a1 −a2 ) Jp (Ȳ , p̄)
Yt − Ȳ −Θ 1−µ=+ θ2 −θ1  ϵt+1 ,pt − p̄∗01 − θθ1
θ1
θ2 −θ1 H(ϕ )
2

θ1

where
1+θ2 −2θ1

Ψ(Λδ) θ1
Θ=
eθ2



1−µ

(a1 − a2 )(1 − (1 − θ1 )eδ) + α1
a2 + eδ(a1 − a2 )

.

The sunspot variable that aﬀects the price of consumption goods, ϵt+1 , has an
impact on the gross product. Formally, we have Proposition 5.2 below.
Proposition 5.2. Given Yt and pt , the sunspot variable, ϵt+1 , negatively aﬀects thegross product, Yt+1 , in the neighborhood of the steady state, i.e., ∂Yt+1 /∂ϵt+1 < 0.
□

Proof. The claim follows from (5.7) because Θ > 0.

The sunspot variable, ϵt+1 , has a negative impact on Yt+1 in both cases in which
θ2 > θ1 and θ2 < θ1 . In the case in which θ2 > θ1 , the consumption good sector ismore labor intensive than the intermediate good sector from the social perspective.
In this case, if a positive extrinsic shock (ϵt+1 > 0) increases pt+1 in (5.5), the wage
rate increases and the capital price decreases (the Stolper-Samuelson property). The
increase in the wage rate has a positive eﬀect on the gross product in period t + 1
as seen in the second term of the right-hand side of Eq. (5.6), whereas the decreasein the capital price has a negative eﬀect on the gross product, as seen in the ﬁrst
term of the right-hand side of Eq. (5.6) because a2 Yt − w(pt ) is negative. The latter
negative eﬀect is always stronger than the former positive eﬀect. On the other hand,in the case in which θ2 < θ1 , the intermediate good sector is more labor-intensive
than the consumption good sector from the social perspective. Therefore, if an
extrinsic shock (ϵt+1 > 0) increases 1/pt+1 in (5.5), which is a relative price of theintermediate goods, the wage rate increases and the capital price decreases (again,
the Stolper-Samuelson property). As in the case in which θ2 > θ1 , the increase in
the wage rate has a positive eﬀect on the gross product, whereas the decrease in thecapital price has a negative eﬀect on the gross product. Again, the latter negativeeﬀect is always stronger than the former positive eﬀect. In either case, a positive
extrinsic shock decreases the gross product.
With Yt and pt given, we obtain the variance of Yt+1 from Eq. (5.7) as follows:(5.8)

V (Yt+1 ) := Θ2H(ϕ∗ )1−µ 2(1+θ2 −2θ1 )
θ1

σ2.

790

T. KUNIEDA AND K. NISHIMURA

H(ϕ∗ )/(1 − µ) is an increasing function with respect to µ as proven in the proofof Proposition 3.6. Therefore, whether ﬁnancial development ampliﬁes or contracts
sunspot ﬂuctuations depends on the factor intensity from the social perspective.
Proposition 5.3. Suppose that Assumption 1 holds. Then, the following hold:• ∂V (Yt+1 )/∂µ > 0 if θ2 > 2θ1 − 1.
• ∂V (Yt+1 )/∂µ < 0 if θ2 < 2θ1 − 1.
Proof. H(ϕ∗ )/(1 − µ) is an increasing function with respect to µ. Therefore, theclaims follow from Eq. (5.8).
□
Fig. 1 shows the regions of θ1 and θ2 that represent whether the eﬀect that thesunspot variable has on the gross product is ampliﬁed or contracted. As seen inthis ﬁgure, when the labor intensity in the consumption good sector from the social
perspective is very large, it is more likely that the eﬀect of the sunspot variable ismagniﬁed, whereas when the labor intensity in the intermediate good sector fromthe social perspective is very large, it is more likely that the eﬀect of the sunspot
variable is contracted.

Figure 1. The eﬀect of ﬁnancial development on the gross productvariance. Notes: In the variance-expanded region, the variance of thegross product is expanded as µ increases, whereas in the variancecontracted region, the variance of the gross product is contracted asµ increases.
The sunspot variable, ϵt+1 , that appears in (5.5) perturbs both Eqs. (5.5) and(5.6), which are the dynamic equations with respect to pt and Yt , respectively. Ourconcern was whether ﬁnancial development ampliﬁes sunspot ﬂuctuations with theperturbations being magniﬁed. Proposition 5.3 implies that the answer to this question depends on the factor intensities from the social perspective. As far as ϵt+1
is not so large, the dynamic course of (Yt , pt ) does not diverge for any initial consumption price, p0 , and any gross product, Y0 under Assumption 1. More precisely,(Yt , pt ) converges to the stationary state in the neighborhood of the steady state,(Ȳ , p̄), when the sunspot variable is realized in each period. Given this situation, one

FINANCIAL DEVELOPMENT AND SUNSPOT FLUCTUATIONS

791

should check whether the dynamic behavior of (Yt , pt ) around the steady state satisﬁes the transversality condition, limτ →∞ δ τ E[at+τ (ω t+τ −1 , ϵt+τ )/pt+τ |ω t−1 , ϵt ] = 0,
because the sunspot variable is extrapolated in the system. In the Appendix, we
conﬁrm that the transversality condition is actually satisﬁed.
6. Conclusion
Does ﬁnancial development amplify sunspot ﬂuctuations? To answer this question, we have investigated a two-sector dynamic general equilibrium model withsector-speciﬁc production externalities and ﬁnancial frictions. As the ﬁnancial sector is well developed, capital accumulation is promoted and the gross product inthe steady state monotonically increases. On the other hand, if the consumption
sector is less labor intensive from the private perspective but more labor intensivefrom the social perspective relative to the intermediate sector, it is more likely forindeterminacy of equilibria to occur. By introducing a sunspot variable in this situation, we demonstrate that ﬁnancial development is more likely to amplify sunspot
ﬂuctuations when the labor intensity in the consumption good sector from the social
perspective is very large, whereas ﬁnancial development is more likely to contract
sunspot ﬂuctuations when the labor intensity in the intermediate good sector fromthe social perspective is very large.
One possible direction in which to extend from the current model is to investigate
open economies with the basic setup remaining unchanged. Whether the results
remain the same if a small open economy or a large open economy is assumed is not
obvious at all because not only international trade but also ﬁnancial trade acrosseconomies with ﬁnancial frictions must be taken into account. This question is left
for future research.
Appendix
Proof of Proposition 3.1. Inserting (2.5) into the ﬁnancial market clearing condition(3.3) yields
ZZµat (ω t−1 )dP t+1 (ω t ) −at (ω t−1 )dP t+1 (ω t ) = 0,
1
−
µ
t
t
Ω ×Ξt
Ω ×(Ω\Ξt )or equivalently,
(A.1)
Z Z
at (ω
Ξt

t−1

t

)dP (ω

Ωtt−1µ)dP (ωt ) −
1−µ

Z

Z
at (ω t−1 )dP t (ω t−1 )dP (ωt ) = 0.
Ω\Ξt

Ωt

where Ξt = {ωt ∈ Ω|Φt (ωt ) ≤ ϕt }. The i.i.d. assumption rewrites Eq. (A.1) as
Z
Z h
Z
Z ϕt
µ
at (ω t−1 )dP t (ω t−1 )dG(Φ) = 0,
at (ω t−1 )dP t (ω t−1 )dG(Φ) −
1 − µ Ωtϕt
Ωt0from which we obtain

Z
t−1
t t−1
at (ω )dP (ω ) G(ϕt ) −ΩtThe last equation yields G(ϕt ) = µ.

µ(1 − G(ϕt )) = 0.
1−µ
□

792

T. KUNIEDA AND K. NISHIMURA

Proof of Lemma 3.2. It follows from Rt (ωt−1 ) = max{rt , (q(pt )Φt−1 (ωt−1 )−rt µ)/(1−µ)} thatZZq(pt )Φt−1 (ωt−1 ) − rt µRt (ωt−1 )at−1 (ω t−2 )dP t ω t−1 =max rt ,1−µΩtΩt× at−1 (ω t−2 )dP t (ω t−1 ) =: It
We deﬁne Ξt−1 = {ωt−1 ∈ Ω|Φt−1 (ωt−1 ) ≤ ϕt−1 } as in the proof of Proposition 3.1.
The use of ϕt−1 = rt /q(pt ) computes It as follows:ZIt =rt at−1 (ω t−2 )dP t (ω t−1 )ZΩt−1 ×Ξt−1

q(pt )Φt−1 (ωt−1 ) − rt µ
at−1 (ω t−2 )dP t (ω t−1 )
1
−
µ
t−1
Ω
×(Ω\Ξt−1 )ZZrt at−1 (ω t−2 )dP t−1 (ω t−2 )dP (ωt−1 )=Ξt−1 Ωt−1
Z
Z
rt µ
−
at−1 (ω t−2 )dP t−1 (ω t−2 )dP (ωt−1 )
1
−
µ
t−1
Ω\Ξt−1 ΩZ
q(pt )Φt−1 (ωt−1 )+at−1 (ω t−2 )dP t (ω t−1 ).
1
−
µ
t−1
Ω
×(Ω\Ξt−1 )

+

(B.1)

Because of the i.i.d. assumption, (B.1) is further computed as
Z ϕt−1ZIt =rt dG(Φ)
at−1 (ω t−2 )dP t−1 (ω t−2 )
0

Ωt−1

−

rt µ
dG(Φ)
ϕt−1 1 − µ

Z

Z h

Z
at−1 (ω t−2 )dP t−1 (ω t−2 )Ωt−1q(pt )Φt−1 (ωt−1 )at−1 (ω t−2 )dP t (ω t−1 )
1
−
µ
t−1
Ω
×(Ω\Ξt−1 )Zq(pt )Φt−1 (ωt−1 )=at−1 (ω t−2 )dP t (ω t−1 ),
1
−
µ
t−1
Ω
×(Ω\Ξt−1 )

+
(B.2)

where the second equality of (B.2) is obtained because G(ϕt−1 ) = µ. Capital producers invest xt−1 (ω t−1 ) = at−1 (ω t−2 )/(1 − µ), and then, (B.2) becomesZq(pt )Φt−1 (ωt−1 )It =(1 − µ)xt−1 (ω t−1 )dP t ω t−1
1−µΩt−1 ×(Ω\Ξt−1 )Z(B.3)= q(pt )Φt−1 (ωt−1 )xt−1 (ω t−1 )dP t (ω t−1 ).
Since zt =
(6.1)

R

Ωt−1 ×(Ω\Ξt−1 )Ωt−1 ×(Ω\Ξt−1 ) Φt−1 (ωt−1 )xt−1 (ωt−1 )dP t (ω t−1 ), (B.3) becomes

It = q(pt )zt .
□FINANCIAL DEVELOPMENT AND SUNSPOT FLUCTUATIONS

793

R
Proof of Lemma 3.3. Since Ωt πt dP t (ω t−1 ) = Πt , the use of Lemma 3.2 aggregates
Eq. (2.6) across all agents as follows:
Z
Z
t−1
t t−1
at (ω )dP (ω ) =[Rt (ωt−1 )at−1 (ω t−2 ) + wt + πt − pt ct (ω t−1 )]dP t (ω t−1 )ΩtΩt= q(pt )zt + wt + Πt − pt Ct .

(C.1)

From Eq. (2.8), we have F 1 (lt1 , zt1 ) + pt F 2 (lt2 , zt2 ) = q(pt )zt + wt + Πt . Additionally, from the market clearing condition for consumption goods, it follows thatpt F 2 (lt2 , zt2 ) = pt Ct . Then, Eq. (C.1) is transformed intoZ
(C.2)
at (ω t−1 )dP t (ω t−1 ) = F 1 (lt1 , zt1 ).
Ωt□Proof of Proposition 3.4. Capital is produced by agents who draw an individualspeciﬁc productivity, Φt (ωt ), greater than ϕt . Therefore, the i.i.d. assumption andLemma 3.3 compute zt+1 as follows:Z
zt+1 =Φt (ωt )xt (ω t )dP t+1 (ω t )ZΩt−1 ×(Ω\Ξt )Zat (ω t−1 ) t t−1
dP (ω )dP (ωt )1−µΩ\Ξt Ωt−1
Z hZ
Φt (ωt )=dG(Φ)
at (ω t−1 )dP t (ω t−1 )
1
−
µ
ϕt
Ωt−1
=

Φt (ωt )=H(ϕ∗ ) 1 1 1
F (lt , zt ),1−µwhere Ξt = {ωt ∈ Ω|Φt (ωt ) ≤ ϕt }, and H(ϕ∗ ) =in equilibrium.

Rhϕ∗ Φt (ωt )dG(Φ) because ϕt = ϕ∗□

Proof of Proposition 3.5. It follows from ϕt = rt+1 /q(pt+1 ) and ϕt = ϕ∗ that
q(pt+1 )Φt (ωt ) − rt+1 µE[Rt+1 (ωt )|ω t−1 ] = E max rt+1 ,
ω t−1
1−µ




Φt (ωt ) − ϕt µt−1
= q(pt+1 )E max ϕt ,
ω
1−µ
"Z ∗
#
Z h
ϕ
Φt (ωt ) − ϕt µ∗
= q(pt+1 )ϕ dG(Φ) +
dG(Φ)
1−µ
0
ϕ∗


ϕ∗ µ
H(ϕ∗ )∗∗∗= q(pt+1 ) ϕ G(ϕ ) −(1 − G(ϕ )) +1−µ1−µH(ϕ∗ )(D.1).
= q(pt+1 )1−µProposition 3.1 is used to derive the last equality.

□

794

T. KUNIEDA AND K. NISHIMURA

Proof of Proposition 3.6. From the inverse function theorem, it follows that
∂ H(ϕ∗ )−(1 − µ)ϕ∗ G′ (ϕ∗ )(∂ϕ∗ /∂µ) + H(ϕ∗ )=∂µ 1 − µ(1 − µ)2Rh∗∗ϕ∗ Φt (ωt )dG(Φ) − ϕ (1 − G(ϕ ))> 0.
=(1 − µ)2
The claims follow from this equation, Eqs. (3.17), (3.18), and (3.19).

□

Proof of Lemma 4.1. (i) Regarding κ1 , because δ ∈ (0, 1) and e ∈ (0, 1), it follows
that eδa1 + (1 − eδ)a2 > 0 ⇐⇒ a2 > eδ(a2 − a1 ). Therefore, it holds that a2 − a1 >0 ⇐⇒ κ1 = a2 /[eδ(a2 −a1 )] > 1. Obviously, it holds that 0 < eδ < a2 /(a1 −a2 ) ⇐⇒
κ1 < −1 and 0 < a2 /(a1 − a2 ) < eδ ⇐⇒ −1 < κ1 < 0. (ii) Regarding κ2 ,0 < θ1 < 2θ2 ⇐⇒ −1 < κ2 < 1 and 0 < 2θ2 < θ1 ⇐⇒ κ2 < −1.
□
Conﬁrmation of the transversality condition. The local dynamical system,(5.7), can be rewritten as

 

 

κ1 0
B̃
Ỹt+τ
Ỹt+τ −1(E.1)=+
ϵt+1 ,
0 κ2
p̃t+τ
p̃t+τ −1D̃
where B̃ = −Θ[H(ϕ∗ )/(1−µ)]1+θ2 −2θ1
θ1+[H(ϕ∗ )/(1−µ)]θ2 −θ1θ1

θ2 −θ1
θ1[(Jp (Ȳ , p̄)(θ2 −θ1 )]/[θ1 (κ1 −κ2 )], D̃ = [H(ϕ∗ )/(1 − µ)][θ2 − θ1 ]/[θ1 (κ1 − κ2 )], Ỹt = Yt − Ȳ + [Jp (Ȳ , p̄)/(κ1 −κ2 )](pt − p̄), and p̃t = [1/(κ1 − κ2 )](pt − p̄). Using (E.1), we obtain
Jp (Ȳ , p̄)(κτ1 − κτ2 )(pt − p̄)
κ1 − κ2
τ
τ
X
X
τ −s
κτ1 −s ϵt+s
− Jp (Ȳ , p̄)D̃κ2 ϵt+s + B̃Yt+τ = Ȳ + κτ1 (Yt − Ȳ ) +
(E.2)

s=1

s=1

and
θ2 − θ1pt+τ = p̄ + κτ2 (pt − p̄) +
θ1

(E.3)



H(ϕ∗ )1−µ θ2 −θ1 Xτθ1κτ2 −s ϵt+s .

s=1

From (2.13), (3.6), (3.8), (E.2), and (E.3), it follows that1 )
F 1 (L1t+τ , Zt+τ
pt+τ
"
#
P
κτ −κτ
Ȳ + κτ1 (Yt − Ȳ ) + κ11 −κ22 (pt − p̄) + B̃ τs=1 κτ1 −s ϵt+s Jp (Ȳ , p̄)
a2
−
=−
P
a1 − a2κ1 − κ2p̄ + κτ2 (pt − p̄) + D̃(κ1 − κ2 ) τs=1 κτ2 −s ϵt+s

(E.4)
+

"

Ψ
p̄ + κτ2 (pt − p̄) + D̃(κ1 − κ2 )eτX

# 1−θ2κτ2 −s ϵt+s

θ2 −θ1.

s=1

In (E.4), note that |κ1 | < 1 and |κ2 | < 1, and the support for ϵt+1 is a closed interval.
1 , Z 1 )/p
Therefore, F 1 (Lt+τ
t+τ is bounded both below and above as τ → ∞, andt+τFINANCIAL DEVELOPMENT AND SUNSPOT FLUCTUATIONS

thus,
(E.5)


lim δ τ E

τ →∞

795


1 )
F 1 (L1t+τ , Zt+τϵt = 0pt+τ

By applying Lemma 3.3 to (E.5), it follows that
(E.6)
 1 1
R


t+τ −1 , ϵt+τ )dP t+τ (ω t+τ −1 )1 )F (Lt+τ , Zt+ττt
τtΩt+τ at+τ (ωδ E
ϵ =δ E
ϵ .
pt+τ
pt+τ
To compute the right-hand side of (E.6), we deﬁne ω t+τ −1,t−1 = {ωt , ..., ωt+τ −1 } ∈Ωt+τ \ Ωt . Then, the right-hand side of (E.6) can be computed asRt+τ −1 , ϵt+τ )dP t+τ (ω t+τ −1 )t+τ at+τ (ω
τ
δ E Ω
ϵt
pt+τ
"R R
#
t+τ −1 , ϵt+τ )dP τ (ω t+τ −1,t−1 )dP t (ω t−1 )t Ωt+τ \Ωt at+τ (ωΩ= δτ E
ϵt
pt+τ
"R
#
Z
t+τ −1 , ϵt+τ )dP τ (ω t+τ −1,t−1 )
a
(ω
t+τ
t
t+τ
Ω
\Ω
= δτ
E
ϵt dP t (ω t−1 )
p
t
t+τ
Ω


Z
at+τ (ω t+τ −1 , ϵt+τ ) t−1 t
τ
(E.7)
E
ω , ϵ dP t (ω t−1 ).
=δ
pt+τ
Ωt
From (E.5), (E.6) and (E.7), we obtainZ
at+τ (ω t+τ −1 , ϵt+τ ) t−1 t(E.8)
lim δ τE
ω , ϵ dP t (ω t−1 ) = 0.
τ →∞
pt+τ
Ωt
The ﬁnancial constraint implies that at+τ (ω t+τ −1 , ϵt+τ ) = bt+τ (ω t+τ , ϵt+τ )+kt+τ (ω t+τ , ϵt+τ ) > bt+τ (ω t+τ , ϵt+τ ) + µkt+τ (ω t+τ , ϵt+τ ) ≥ 0. Therefore, it follows
from (E.8) thatat+τ (ω t+τ −1 , ϵt+τ ) t−1 tτ
ω , ϵ = 0.
lim δ E
τ →∞pt+τThis is the transversality condition for each agent.
Acknowledgment
This work is ﬁnancially supported by the Japan Society for the Promotion of
Science, Grants-in-Aid for Scientiﬁc Research (Nos. 15H05729, 16H02026, and
16H03598). We are grateful to an anonymous referee for his/her invaluable comments to improve our manuscript. Of course, all remaining errors, if any, are ours.
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Manuscript received December 21 2019revised February 14 2020

T. Kunieda
School of Economics, Kwansei Gakuin University, 1-155 Uegahara Ichiban-Cho, Nishinomiya,
Hyogo, 662-8501, JAPANE-mail address: tkunieda@kwansei.ac.jp
K. Nishimura
Research Institute for Economics and Business Administration, Kobe University and RIETI, 2-1
Rokkodaicho, Nadaku, Kobe, 657-8501, JAPANE-mail address: nishimura@rieb.kobe-u.ac.jp