ISSN (Print) 0473-453X

Discussion Paper No. 1239

ISSN (Online) 2435-0982

INSURANCE AGAINSTAGGREGATE SHOCKS

Takuma Kunieda
Akihisa Shibata

April 2024

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

Insurance against Aggregate Shocks
Takuma Kunieda*

Akihisa Shibata

Kwansei Gakuin University

Kyoto University

April 5, 2024

Abstract
Although many studies in macroeconomics have examined the role of insurance inthe presence of income risk, whether aggregate shocks are insurable has not been suﬃciently investigated. We present a simple two-period general equilibrium model to show
the conditions under which insurance against aggregate shocks works in an economy
with constant-elasticity-substitution (CES) production technology and the GreenwoodHercowitz-Huﬀman (GHH) utility function (Greenwood et al.,1988). Our theoreticalinvestigation clariﬁes that only when agents are heterogeneous in their ability or initial
wealth can aggregate shocks be insurable. From our quantitative investigation, we ﬁndthat (i) agents with lower ability enjoy greater welfare improvement from insurance,
and as agents’ ability increases, the welfare improvement diminishes, (ii) agents enjoy
greater welfare improvement when the damage from disasters is more severe and when
the frequency of disasters is greater, and (iii) although the welfare improvement increases as agents’ initial wealth increases, the impact of a diﬀerence in agents’ initial
wealth on the diﬀerence in the contribution of insurance is very moderate.

Keywords: aggregate shocks, heterogeneous agents, state-contingent claims, incomplete
market.
JEL Classiﬁcation Numbers: D52, G12* Corresponding author. Professor at Kwansei Gakuin University. Address: School of Economics, Kwansei

Gakuin University, 1-155 Uegahara Ichiban-cho, Nishinomiya 662-8501, Hyogo, Japan; Phone: +81 798 54
6482, Fax: +81 798 51 0944, E-mail: tkunieda@kwansei.ac.jp
 Professor at Kyoto University. Address: Institute of Economic Research, Kyoto University, Yoshidahonmachi, Sakyo-ku, Kyoto 606-8501, Japan; E-mail: shibata@kier.kyoto-u.ac.jp1

1

Introduction

Can aggregate shocks be insured against? This question is crucial for humankind since the
COVID-19 crisis. The COVID-19 pandemic caused worldwide lockdowns and disrupted daily
life in societies. As a consequence, it damaged the world economy almost simultaneously. 1
We can say that the COVID-19 pandemic was an aggregate shock to the world. Althoughmany studies have examined the role of insurance in the presence of income risk, whether
aggregate shocks are insurable has not been suﬃciently investigated. In particular, howmuch insurance against aggregate shocks can contribute to welfare improvement has notbeen studied thus far in the literature. This paper is the ﬁrst attempt to ﬁll this gap.
One can classify stochastic shocks into idiosyncratic shocks and aggregate shocks. Idiosyncratic shocks aﬀect each agent randomly and independently. Therefore, an insurance
contract for idiosyncratic shocks works well and increases agents’ expected utility by smoothing consumption between stochastic states so that agents who received ex-post high income
can ﬁnance agents who received ex-post low income. In contrast, aggregate shocks are unlikely to be insurable because when such shocks aﬀect all agents in the same direction, no
one seems to oﬀer or accept an insurance contract. As such, most studies that analyzed therole of insurance in the context of macroeconomics have focused mainly on insurance against
idiosyncratic shocks. Cochrane (1991) and Mace (1991) argued that if there is a complete
insurance market for idiosyncratic shocks, each agent can achieve full risk sharing for theidiosyncratic shocks facing them. Then, individual consumption will no longer respond to
idiosyncratic shocks. However, individual consumption will still move in the same direction
in response to aggregate shocks since there is no way to diversify risks among agents.
The above arguments hold if agents in an economy are homogeneous. If agents areheterogeneous, aggregate shocks may have idiosyncratic eﬀects on agent behavior, such as theendogenous labor supply, in response to the shocks. In this case, insurance may be eﬀective
against aggregate shocks. The objectives of this paper are to show the conditions under which
insurance against aggregate shocks is eﬀective in general equilibrium in an economy with aconstant-elasticity-substitution (CES) production function and the Greenwood-HercowitzHuﬀman (GHH) utility function (Greenwood et al., 1988) and to quantitatively investigate
how much insurance against aggregate shocks improves economic welfare (the insurance
contribution). Although we consider a simple two-period general equilibrium model, we
1Barrett et al. (2021) reported the signiﬁcant impacts of the COVID-19 pandemic on per capita GDP
and total factor productivity (TFP) worldwide. Bloom et al. (2023) reported a decrease in TFP of 5% in
2020-21 in the United Kingdom.

2

obtain rich results both theoretically and quantitatively.
From our theoretical investigation, we clarify that only when agents are heterogeneousin their ability and initial wealth can aggregate shocks be insurable. Intuitively, the variation in their ability and initial wealth generates the variation in their total income, whichcauses idiosyncratic eﬀects of aggregate shocks. More concretely, although all agents face
the same wage rate, wage income is determined by the product of agent ability and the wage
rate. While aggregate shocks aﬀect the wage rate, a diﬀerence in agents’ ability magniﬁes adiﬀerence in their wage income. Furthermore, whereas aggregate shocks aﬀect the interest
rate that all agents face, capital income is the product of the interest rate and initial wealth.
Then, a diﬀerence in agents’ initial wealth magniﬁes a diﬀerence in their capital income.
These idiosyncratic eﬀects of aggregate shocks make the marginal rate of substitution of
consumption in the disaster state for consumption in the normal state uneven across agents.
Then, the Pareto optimal equilibrium is unachievable in an economy without insurance. In
other words, incomplete markets endogenously occur.
Our quantitative analyses also indicate signiﬁcant results. First, by comparing indirect
utility with insurance to that without insurance, we ﬁnd that other things being equal,
agents with lower ability enjoy greater welfare improvement from insurance, and as agents’ability increases, the welfare improvement is diminished. Second, agents enjoy greater welfare
improvement when the damage from disasters (aggregate shocks) is more severe and when
the frequency of disasters is greater. Third, although the welfare improvement increases asagents’ initial wealth increases, the impact of a diﬀerence in agents’ initial wealth on thediﬀerence in the contribution of insurance is very moderate.
Several researchers have examined the relation between insurance and aggregate shocks
in the literature on general equilibrium theory. The most relevant study is by Krueger andLustig (2010). They clariﬁed the conditions under which, no matter whether an insurance
market for idiosyncratic shocks exists, a no-trade equilibrium in the insurance market for aggregate shocks occurs, and the macroeconomic implications remain unchanged in an economy
where agents face both idiosyncratic and aggregate shocks. In other words, they identiﬁed
conditions that allow for the correct analysis of macro dynamics at the aggregate level under the assumption of a representative household, even in the presence of incompleteness ininsurance markets for individual shocks.2 These conditions can be summarized as follows:(1) the instantaneous utility function is homothetic, (2) idiosyncratic and aggregate shocks
are independent, and (3) the capital income share is constant. Mailer and Mailar (2001) and
2

Their result can be regarded as an extended version of the aggregation theorems that allow for the useof representative agent models, in line with the works of Gorman (1953, 1961) and Rubinstein (1974).

3

Mailer et al. (2005) demonstrated that one could analyze the dynamics of income and wealthdistribution in an economy inhabited by a representative household by adopting settings thatsatisfy all the above conditions. The model presented by Clemens et al. (2020), an extensionof the Mailer et al. (2005) model, explicitly introduced insurance against aggregate shocks,
but Clemens et al. (2023), which is the published version of Clemens et al. (2020), discarded
such insurance because the presence or absence of insurance against aggregate shocks does
not aﬀect the equilibrium allocation in their model because it meets all three conditions.
What is commonly shared in these studies is that limited attention has been given to thecompleteness or incompleteness of insurance markets against aggregate shocks. However,is insurance against aggregate shocks truly insigniﬁcant in macroeconomics? Again, thisquestion is crucial for us because we all know the consequences of the COVID-19 crisis thatnegatively impacted almost the whole world. Since it is already known that insurance against
aggregate shocks is not signiﬁcant when the three conditions derived by Krueger and Lustig(2010) hold, we need to relax some of these conditions to investigate the role of insurance
against aggregate shocks. To minimize deviations from previous research, we relax only
condition (3). As previously described, only when agents are heterogeneous in their ability
and initial wealth are aggregate shocks insurable. Although this theoretical ﬁnding is one of
the contributions of our study, no-trade equilibrium in the insurance market for aggregate
shocks occurs if the Cobb-Douglas production technology is employed in our model. Thisis because the Cobb-Douglas production technology satisﬁes Krueger and Lustig’s third
condition. We instead adopt a CES production technology. Our investigation is the ﬁrst in
the literature in that we address the role of insurance markets for aggregate shocks.
The remainder of this paper is organized as follows. In the next section, we present a twoperiod general equilibrium model in which state-contingent claims for aggregate shocks are
introduced. Section 3 derives equilibrium, and section 4 explores whether insurance against
aggregate shocks achieves the Pareto optimum and clariﬁes the mechanism that induces anincomplete market for aggregate shocks although they aﬀect all agents in the same direction.
In section 5, we perform quantitative analyses. Section 6 concludes this paper.

2

Model

Consider a two-period economy inhabited by a continuum of agents whose population isnormalized to 1. Each agent is born with a certain endowment (initial wealth) in the ﬁrst
period, which varies between agents. The heterogeneity of agents originates not only fromindividual-speciﬁc endowments but also from their labor productivity. In the ﬁrst period,4

they invest in real assets and/or purchase or issue state-contingent claims. In the second
period, they acquire a return from real assets, clear state-contingent claims, and consume
all the income.
2.1

Production

A representative ﬁrm produces ﬁnal goods from capital and eﬀective labor with a constantelasticity-substitution (CES) production function:1

θi f (k, h) := θi [bk σ + (1 − b)hσ ] σ ,(1)where k is aggregate capital, h is aggregate eﬀective labor, and b ∈ (0, 1) and σ ∈ (−∞, 1)
are parameters, where b becomes a capital share of the total output when the production
function reduces to the Cobb-Dauglas production function as σ → 0. 1/(1 − σ) is theelasticity of substitution between capital and eﬀective labor. θi (i = 1, 2) is the total factor
productivity (TFP), which is a random variable such that 0 < θ1 < θ2 , being realized withprobability πi ∈ (0, 1) where π1 + π2 = 1. One may imagine that a disaster occurs when alow-productivity shock, θ1 , is realized.
Capital and labor are paid their marginal products because the markets of production
factors are competitive. Then, we have1−σw(θi ) := θi (1 − b) [bk σ + (1 − b)hσ ] σ hσ−1

(2)

and
1−σ

r(θi ) := θi b [bk σ + (1 − b)hσ ] σ k σ−1 ,(3)where w(θi ) and r(θi ) are the wage and interest rates, respectively, when θi is realized.
2.2

AgentsConsider an agent, say, agent j ∈ Ω where Ω is the whole set of agents, who is endowed with
the GHH type of utility function. Agent j maximizes the expected utility as follows.
max

2
∑
i=1

πi

xj (θi )1−γ − 1
1−γ

5

(4)

subject to
aj +

2
∑

pi mj (θi ) ≤ Ij ,

(5)

i=1

and
cj (θi ) ≤ r(θi )aj + ej lj (θi )w(θi ) + mj (θi )(6)for i ∈ {1, 2}. xj (θi ) in Eq. (4) is composed of consumption felicity and labor infelicity suchthat xj (θi ) := cj (θi ) − Ψlj (θi )1+χ /(1 + χ) with Ψ > 0, where cj (θi ) and lj (θi ) are consumption
and labor, respectively, and χ ∈ (0, ∞) is the inverse of Frisch elasticity of labor supply. If
γ = 1, (xj (θi )1−γ − 1)/(1 − γ) is reduced to ln xj (θi ). Inequality (5) is the budget constraint
in the ﬁrst period where aj is a real asset, mj (θi ) is a state-contingent claim with pi being
its price, and Ij is agent j’s initial endowment. The aggregation of aj becomes aggregate
capital in the economy. Short sales of aj (i.e., aj < 0) are allowed. Agents who purchase(sell) one unit of state-contingent claims at a price of pi in the ﬁrst period are paid (pay) one
unit of ﬁnal goods in the second period. Inequality (6) is the budget constraint in period 2
when θi is realized, where ej is agent j’s ability and ej lj (θi ) is individual eﬀective labor.
2.3

First-order conditions

The ﬁrst-order conditions are given by(lj (θi )1+χcj (θi ) − Ψ1+χ)−γ= λij ,

(7)

(1 − pi r(θi )) λji − pi r(θi′ )λij = 0,

(8)

πi

′

and

(
−πi

lj (θi )1+χcj (θi ) − Ψ1+χ)−γΨlj (θi )χ + λij ej w(θi ) = 0,(9)where λij is the Lagrange multiplier with (i, i′ ) = (1, 2) or (2, 1). Eqs. (7), (8), and (9) are
the ﬁrst-order conditions with respect to consumption, state-contingent claims, and labor,
respectively.
Eqs. (7) and (9) yield[ej w(θi )lj (θi ) =Ψwhich is the individual labor supply.

6

] χ1
,

(10)

Lemma 1. The following two equations hold:p1 r(θ1 ) + p2 r(θ2 ) = 1

(11)

λ2j
p2
= .
1
λj
p1

(12)

and

Proof. Eqs. (11) and (12) directly follow from Eq. (8).
There are two equations in Eq. (8) for the combination of (i, i′ ) = (1, 2) or (2, 1). For λ1j
and λ2j to exist, however, Eq. (11) must hold. In this case, one of the two equations in Eq.
(8) is redundant, and they reduce to one equation, Eq. (12). This situation occurs because
agents have three investment opportunities for the two stochastic states: one real asset andtwo state-contingent claims. Due to this situation, agents’ rational decisions on how muchthey purchase and sell state-contingent claims become indeterminate.
From Eqs. (7) and (12), it follows that(p2 π1 )− γ1(lj (θ1 )1+χcj (θ1 ) − Ψ1+χ

)= (p1 π2 )

− γ1

(

lj (θ2 )1+χcj (θ2 ) − Ψ
1+χ

)
.

(13)

Furthermore, Eqs. (6) and (10) yield
1+χ
χΨ− χ
lj (θi )1+χ= r(θi )aj +[ej w(θi )] χ + mj (θi ).
cj (θi ) − Ψ
1+χ
1+χ
1

3

(14)

Equilibrium

A competitive equilibrium is given by prices, {pi , r(θi ), w(θi )} for i = 1, 2, and allocation,{(aj , mj (θi ), cj (θi ), lj (θi ))} for i = 1, 2 and j ∈ Ω, such that given {pi , r(θi ), w(θi )} each agent
solves their utility maximization problem, given {r(θi ), w(θi )} the representative ﬁrm solves
its proﬁt maximization problem, and capital, state-contingent claim, and labor markets all
clear.
3.1

Capital and insurance markets

The insurance market clearing condition is given by∫
mj (θi )dj = 0
j∈Ω

7

(15)

for i = 1 and 2. The aggregate capital is obtained as follows:∫k=aj dj.

(16)j∈Ω

From Eqs. (5), (15), and (16), it follows thatk = I,
where I :=

∫

I dj
j∈Ω j

(17)

is the aggregate initial endowment. Eq. (17) is a supply of capitaland Eq. (3) yields a demand for capital. Then, given eﬀective labor, h, the capital market
clearing condition is given by1−σr(θi ) = θi b [bI σ + (1 − b)hσ ] σ I σ−1 .
3.2

(18)

Labor market

From Eq. (10), the supply of aggregate eﬀective labor is obtained as follows:∫1+χ

ej lj (θi )dj = Ψ− χ e χ w(θi ) χ ,1h(θi ) =

1

(19)

j∈Ω

[
where e :=

∫

1+χ
χ

e
j∈Ω j

χ
] 1+χ

dj

. Note that aggregate eﬀective labor, h, is subject to the realiza-tion of productivity shock, θi . Since given aggregate capital, k, Eq. (2) yields a demand for
aggregate eﬀective labor, inserting Eq. (19) in Eq. (2) produces the labor market clearing
condition as follows:[w(θi ) = θi (1 − b) bk + (1 − b)(Ψ
3.3

σ

1
−χ

e

1+χ
χ

1
χ

w(θi ) )

σ

] 1−σ
σ

1+χ

(Ψ− χ e χ w(θi ) χ )σ−1 .
1

1

(20)

Existence and uniqueness of equilibrium wage and interest rates

By inserting Eq. (19) in Eq. (18) and Eq. (17) in Eq. (20), we have
[

1
−χ

1+χ
χ

1
χ

] 1−σ
σ

r(θi ) = θi b bI + (1 − b)(Ψ e w(θi ) )
I σ−1
[
] 1−σ
1 1+χ
1
σ
−χ
σ
χ
χ
= θi b b + (1 − b)(Ψ e w(θi ) /I)
σ

8

σ

(21)

and
[
] 1−σ
1 1+χ
1
1 1+χ
1
σ
w(θi ) = θi (1 − b) bI σ + (1 − b)(Ψ− χ e χ w(θi ) χ )σ(Ψ− χ e χ w(θi ) χ )σ−1
[
] 1−σ
1 1+χ
1
σ
= θi (1 − b) b(Ψ− χ e χ w(θi ) χ /I)−σ + (1 − b),(22)respectively. Given θi , Eqs. (21) and (22) determine the equilibrium wage and interest rates.
Since the right-hand side of Eq. (22) is a decreasing function, it is straightforward to showthat the equilibrium wage rate is uniquely determined. Furthermore, once the equilibrium
wage rate is determined, the equilibrium interest rate is also uniquely determined by Eq.
(21). Henceforth, we denote the wage and interest rates in equilibrium by w∗ (θi ) and r∗ (θi ).
It holds that w∗ (θ1 ) < w∗ (θ2 ) because if θi increases, the demand for labor increases, as
shown in Eq. (2). Additionally, r∗ (θ1 ) < r∗ (θ2 ) because r∗ (θi ) increases with w∗ (θi ) and θi ,as shown in Eq. (21).
Although it is diﬃcult to obtain the equilibrium wage and interest rates explicitly from
Eqs. (21) and (22), some comparative statistics are useful for understanding the situationsof the capital and labor markets. First, Eqs. (2) and (19) prove that ∂w∗ (θi )/∂Ψ > 0 and∂h∗ (θi )/∂Ψ < 0. From Eq. (19), it follows that the increase in a labor reluctant parameter,
Ψ, shifts the eﬀective labor supply curve to the left, and thus, the wage rate increases andthe eﬀective labor decreases in equilibrium. Second, from Eq. (21) and ∂h∗ (θi )/∂Ψ < 0, we
can prove that ∂r(θi )/∂Ψ < 0 because if the supply of eﬀective labor decreases, the marginal
product of capital decreases as the complementarity between capital and eﬀective labor ispresent in the CES production function. Third, it is straightforward that ∂r(θi )/∂I < 0and ∂w(θi )/∂I > 0 by considering the marginal products of capital and eﬀective labor,
respectively. All the above consequences hold regardless of the realized values of θi .
Here, we present a lemma useful for the analysis in what follows.
Lemma 2. It holds thatr∗ (θi )w∗ (θi )

1+χ
χ

(
=

b
1−b

)(

1+χ

Ψ− χ e χI

Proof. Eq. (23) follows from Eqs. (21) and (22).

9

1

)1−σ
w∗ (θi )− χ .
σ

(23)

3.4

Aggregation

By using Eqs. (15)-(17), we aggregate Eq. (14) across agents to obtain the following equation:

(

∫
j∈Ω

lj (θi )1+χcj (θi ) − Ψ
1+χ

)

1+χ
χΨ− χ
dj = r (θi )I +[ew∗ (θi )] χ
1+χ
1

∗

(24)

for i ∈ {1, 2}, where we have used the equilibrium wage and interest rates. Aggregating bothsides of Eq. (13) and substituting Eq. (24) into the resulting equation, we have
(
p2 π1

1+χ
χΨ− χ
∗
r (θ2 )I +[ew∗ (θ2 )] χ
1+χ
1

)γ

(
= p1 π2

1+χ
χΨ− χ
∗
r (θ1 )I +[ew∗ (θ1 )] χ
1+χ
1

)γ
.

(25)

Proposition 1 below presents the price of the state-contingent claim.
Proposition 1. The price of the state-contingent claim in equilibrium, p∗i , is given by
(
πi
p∗i =

∑

−1

∗

χ
r (θi′ )I + χΨ[ew∗ (θi′ )]
1+χ

(

(2,1)

1+χ
χ

)γ

1
−χ

πi r∗ (θi ) r∗ (θi′ )I + χΨ[ew∗ (θi′ )]
1+χ

1+χ
χ

)γ

(26)

(i,i′ )=(1,2)

where (i, i′ ) = (1, 2) or (2, 1).
Proof. Eq. (26) follows from Eqs. (11) and (25).
Two remarks concerning Proposition 1 are in order. First, given the same probabilities
for states 1 and 2, i.e., π1∗ = π2∗ , p∗1 > p∗2 holds because r∗ (θ2 ) > r∗ (θ1 ) and w∗ (θ2 ) > w∗ (θ1 ).
We obtain this outcome because agents acquire a higher income in state 2 than in state 1,
and thus, agents who beneﬁt from state 2 demand the state-contingent claim of state 1 morethan that of state 2 to smooth consumption between the two stochastic states. Second, if
the probability with which state 1 occurs is very small, the outcome of p∗1 > p∗2 does notnecessarily hold because if the probability is so small, agents care much less about it. In thiscase, p∗1 < p∗2 is more likely to occur even though the income in state 2 is higher than thatin state 1.
For the exposition below, we deﬁne two variables such thatχΨ− χ 1+χχΨ− χ 1+χBj =ej χ and B =
e χ .
1+χ
1+χ
1

Note that

∫
j∈Ω

1

Bj dj = B, which is the average of Bj . Since one can regard Bj as another10

measure of agents’ ability, we also call Bj and B agents’ ability and its average, respectively,
unless there is confusion. According to Eqs. (5), (11), (13), and (14), it follows thatr∗ (θ2 )mj (θ1 ) − r∗ (θ1 )mj (θ2 )
(
)
(
)
1+χ
1+χ
− γ1 ∗− γ1 ∗− γ1 ∗− γ1 ∗∗∗∗∗
χχ
[p1 π2 ] r (θ2 ) − [p2 π1 ] r (θ1 ) Ij + [p1 π2 ] w (θ2 )− [p2 π1 ] w (θ1 )
Bj
.
=
1
1
[p∗1 π2 ]− γ p∗1 + [p∗2 π1 ]− γ p∗2(27)
By aggregating Eq. (27), we obtain
(

)
)
(
1+χ
1+χ
1
1
1
1
[p∗1 π2 ]− γ r∗ (θ2 ) − [p2∗ π1 ]− γ r∗ (θ1 ) I + [p∗1 π2 ]− γ w∗ (θ2 ) χ − [p∗2 π1 ]− γ w∗ (θ1 ) χ B = 0.
(28)

Proposition 2. The state-contingent claims purchased or sold by agent j in equilibrium,m∗j (θ1 ) and m∗j (θ2 ), satisfy the following equation:)11[p∗2 π1 ]− γ r∗ (θ1 ) − [p∗1 π2 ]− γ r∗ (θ2 ) (Bj I − BIj )()
r∗ (θ2 )m∗j (θ1 ) − r∗ (θ1 )m∗j (θ2 ) =.
− γ1 ∗− γ1 ∗∗∗[p1 π2 ] p1 + [p2 π1 ] p2 B(

(29)

Proof. Eq. (29) follows from Eqs. (27) and (28).
If agents are homogeneous in Eq. (29), i.e., if Bj = B and Ij = I for all j, it holds thatm∗j (θ1 )/r∗ (θ1 ) = m∗j (θ2 )/r∗ (θ2 ). This equation and Eq. (11) imply that purchasing statecontingent claims and holding real assets are completely indiﬀerent. Then, in any case,agents acquire a return, r∗ (θi ), when θi is realized. In this case, one naturally considers thata no-trade equilibrium (m∗j (θ1 ) = m∗j (θ2 ) = 0 for all j ∈ Ω) occurs for the state-contingent
claims. Therefore, for insurance against aggregate shocks to be eﬀective, some agents shouldbe heterogeneous such that Bj ̸= B or Ij ̸= I.
Corollary 1. Eq. (29) is equivalent tor∗ (θ2 )m∗j (θ1 ) − r∗ (θ1 )m∗j (θ2 )()( b ) ( 1+χ )1−σ ∗
1+χ
σ
σ
−χ
−χ
∗
∗
∗
χ
B
w
(θ
)
−
w
(θ
)
(Bj I − BIj )
(w
(θ
)w
(θ
))
1
2
1
2
1−b
χ
(
)
=
.
1+χ
1+χ
I 1−σ I + p∗1 Bw∗ (θ1 ) χ + p∗2 Bw∗ (θ2 ) χ(30)

Proof. See Appendix A.
Note from Corollary 1 that if the production technology is of the Cobb-Douglas type, i.e.,11σ → 0, or if the labor supply is inelastic, i.e., χ → ∞, it holds that m∗j (θ1 )/r∗ (θ1 ) =m∗j (θ2 )/r∗ (θ2 ), and again, a no-trade equilibrium is a natural outcome as in the case ofhomogeneous agents.

4

Optimality

The fact that aggregate shocks are insurable contradicts our intuition from the ﬁrst welfare
theorem. Does the insurance plan for aggregate shocks that satisﬁes Eq. (29) or (30) achieve
Pareto optimal equilibrium? If so, what is the mechanism through which the insurance planworks for aggregate shocks? To elaborate on these points, we begin by considering a social
planner problem.
4.1

Social planner problem

Suppose that a social planner solves the following social welfare maximization problem:
∫
φj

max

( 2
∑

xj (θi )1−γ − 1
πi
1−γ
i=1

j∈Ω

subject to

[

∫

)
dj

)σ ] σ1(∫

cj (θi )dj ≤ θi bI + (1 − b)
σ

j∈Ω

(31)

ej lj (θi )dj

(32)

j∈Ω

for i ∈ {1, 2}, where φj ∈ (0, ∞) is the weight parameter on each agent’s expected utility. We
verify that there are prices that support the social planner problem’s ﬁrst-order conditions
being consistent with competitive equilibrium.
The ﬁrst-order conditions of the social planner problem with respect to cj (θi ) and lj (θi )are given by

(
j

φ πi

lj (θi )1+χcj (θi ) − Ψ
1+χ

)−γ
= λi

(33)

and
(

)−γ
lj (θi )1+χ
− φ πi cj (θi ) − ΨΨlj (θi )χ
1+χ
[(∫)σ ] 1−σ(∫σiσ+ λ ej θi (1 − b) bI + (1 − b)ej lj (θi )dj
j

j∈Ω

)σ−1
ej lj (θi )dj

= 0,

(34)

j∈Ω

respectively, where λi is the Lagrange multiplier. By setting λi = pi and from Eq. (33), we

12

obtain the following equation:(p2 π1lj (θ1 )1+χcj (θ1 ) − Ψ
1+χ

)−γ

(
= p1 π2∫which is identical to Eq. (13). Since h(θi ) =[lj (θ2 )1+χcj (θ2 ) − Ψ
1+χ

)−γ

)σ ] 1−σ(∫σw(θi ) = θi (1 − b) bI + (1 − b)(35)e l (θ )dj, we can letj∈Ω j j i

(∫
σ

,

)σ−1

ej lj (θi )dj

ej lj (θi )dj

j∈Ω

(36)

j∈Ω

in Eq. (34). Then, Eqs. (33) and (34) yield[ej w(θi )lj (θi ) =
Ψ

] χ1
,

(37)

which is also identical to Eq. (10). These consequences mean that competitive equilibriumwith state-contingent claims achieves the Pareto optimal outcome.
4.2

Source of incomplete markets

We have found that state-contingent claims are crucial in achieving the Pareto optimumin our model when aggregate shocks aﬀect the economy. In this section, we focus on themechanism that induces an incomplete market for aggregate shocks, although they aﬀect allagents in the same direction.
Remark 1. It follows from Eq. (33) that for agent j, the marginal rate of substitution of
consumption in state 1 for consumption in state 2 is given by [π1 xj (θ1 )−γ ]/[π2 xj (θ2 )−γ ] =λ1 /λ2 in the Pareto optimum, which means that the marginal rate of substitution being equal
across agents is the necessary condition for the Pareto optimum.
Based on Remark 1, we examine whether each agent’s marginal rate of substitution
without state-contingent claims is uneven across them. Consider a situation in which there
are no state-contingent claims. Since Eq. (10) holds regardless of the presence of statecontingent claims, Eq. (14) with aj = Ij becomes1+χ
lj (θi )1+χχΨ− χ
xj (θi ) = cj (θi ) − Ψ= r(θi )Ij +[ej w(θi )] χ .
1+χ
1+χ
1

13

(38)

From Eqs. (2), (3), and (19), it follows that(w(θi ) =1−bbχ
) 1+χ−σ(1+χΨ− χ e χ1) (σ−1)χ
1+χ−σ(1−σ)χ

χ

I 1+χ−σ r(θi ) 1+χ−σ .

(39)

Substituting Eq. (39) into Eq. (38), we can rewrite Eq. (38) as
xj (θi ) = cj (θi ) − Ψ
whereχΨ− χ 1+χ
δj :=
ej χ
1+χ
1

(

1+χ
lj (θi )1+χ= r(θi )Ij + r(θi ) 1+χ−σ δj ,
1+χ

1−b
b

1+χ
) 1+χ−σ(1+χΨ− χ e χ1) (σ−1)(1+χ)
1+χ−σ

I

(1−σ)(1+χ)
1+χ−σ

(40)

.

(41)

The marginal rate of substitution without state-contingent claims is deﬁned as
(
)−γ
1+χ
π1 r(θ1 )Ij + r(θ1 ) 1+χ−σ δj
R(Ij , δj ) := (
)−γ .
1+χ
1+χ−σ
δj
π2 r(θ2 )Ij + r(θ2 )(42)

According to Remark 1 and Eq. (42), if R(Ij , δj ) is independent of j, state-contingent claims
are unnecessary for the economy to achieve the Pareto optimum.
When the production technology is of the Cobb-Douglas type (σ → 0) or when thelabor supply is inelastic (χ → ∞), we can compute the marginal rate of substitution as
R(Ij , δj ) = [π1 r(θ1 )−γ ] / [π2 r(θ2 )−γ ], which is independent of j. In this case, the marginal
rate of substitution is even across agents, and thus, the economy achieves Pareto optimalequilibrium without state-contingent claims. This outcome is consistent with the case inwhich we take σ → 0 or χ → ∞ in Eq. (30) of Corollary 1, and state-contingent claims
do not play any role in insuring against aggregate shocks in competitive equilibrium. Furthermore, when agents are homogeneous in their ability and initial wealth such that ej = e
and Ij = I for all j ∈ Ω, it follows that R(Ij , δj ) = R(I, δ), where δ is deﬁned such that
one replaces ei with e in Eq. (41). In this case, R(Ij , δj ) is also independent of j, and thus,
the economy achieves Pareto optimal equilibrium without state-contingent claims. This outcome is consistent with the case in which Bj and Ij in Eq. (30) are replaced by B and I,
respectively. Again, in this case, state-contingent claims do not play any role in insuring
against aggregate shocks in competitive equilibrium.
The above assessments imply that if agents are heterogeneous in their ability and initial
wealth in an economy with generic CES production technology and an endogenous labor
supply, the marginal rate of substitution of consumption in state 1 for consumption in state14

2 is uneven without state-contingent claims. In this case, a situation of incomplete markets
endogenously occurs, and the Pareto optimal outcome is unachievable.
4.3

Insurance design

Since agents have three investment opportunities for the two stochastic states, how much theypurchase or sell state-contingent claims is indeterminate. Whereas their optimal programis given by Eq. (29) or Eq. (30) such that the market clearing condition (15) should besatisﬁed, an extra equation is necessary to pin down the state-contingent claims that agentspurchase or sell. We consider typical insurance designs in this section.
The sign of σ plays a crucial role in the insurance plan because it determines whetherw∗ (θ1 )− χ − w∗ (θ2 )− χ is positive or negative in Eq. (30). Many empirical studies haveσσsuggested various values of the elasticity of substitution between capital and labor (e.g.,
Klump et al., 2007, 2012; Chirinko, 2008; León-Ledesma et al., 2010). Among others,
Gechert et al. (2022) performed a meta-analysis with 3186 observations of the elasticity of
substitution obtained from 121 prior studies. They reported that the mean of the elasticity
of substitution between capital and labor is 0.3.3 According to their suggestion, it follows
that 1/(1 − σ) = 0.3, which means σ = −7/3 < 0. Then, we assume that σ is negative.
Assumption 1. σ < 0.
Under Assumption 1, since w∗ (θ1 ) < w∗ (θ2 ), it holds that w∗ (θ1 )− χ − w∗ (θ2 )− χ < 0.
σσ

Example 1
One of the possible ways to discern the amount of state-contingent claims is to have the net
purchase of the state-contingent claims zero in the ﬁrst-period budget constraint (5) so thatthe following equation holds:
p1 mj (θ1 ) + p2 mj (θ2 ) = 0.

3

(43)

Gechert et al. (2022) conducted a meta-analysis, correcting for publication bias, using variation acrossindustries, and including information on the ﬁrst-order condition for capital.

15

Under this insurance design, each agent’s net purchase of state-contingent claims in the ﬁrst
period is 0. From Eqs. (30) and (43) with Eq. (11), it follows thatm∗j (θ1 ) =

p∗2

(
)
( b ) ( 1+χ )1−σ ∗
1+χ
σ
σ
−χ
−χ
∗
∗
∗
χ
w
(θ
)
B
(w
(θ
)w
(θ
))
−
w
(θ
)
(Bj I − BIj )
1
1
2
2
1−b
χ
(
)
1+χ
1+χ
I 1−σ I + p∗1 Bw∗ (θ1 ) χ + p∗2 Bw∗ (θ2 ) χ
(44)

and
m∗j (θ2 ) = −

p∗1

(
)
( b ) ( 1+χ )1−σ ∗
1+χ
σ
σ
−χ
−χ
∗
∗
∗
χ
w
(θ
)
−
w
(θ
)
(Bj I − BIj )
B
(w
(θ
)w
(θ
))
1
2
1
2
1−b
χ
(
)
.
1+χ
1+χ
I 1−σ I + p∗1 Bw∗ (θ1 ) χ + p∗2 Bw∗ (θ2 ) χ(45)Under the insurance design embodied by Eqs. (44) and (45), the insurance market clearing∫
condition (15) is satisﬁed because j∈Ω (Bj I − BIj ) dj = 0. Whether m∗j (θi ) is positive or
negative depends upon the sign of Bj I − BIj . Suppose that Assumption 1 holds. Then, if
Bj − B > (B/I)(Ij − I), it holds that m∗j (θ1 ) < 0 and m∗j (θ2 ) > 0. An agent with Bj − B >(B/I)(Ij − I) sells the state-contingent claim for state 1 (disaster state) and purchases
the state-contingent claim for state 2 (normal state) in the ﬁrst period. As agents’ ability
increases and initial wealth decreases, they are more likely to desire to receive payments in thenormal state while compensating for damages in the disaster state. Conversely, if Bj − B <(B/I)(Ij −I), it holds that m∗j (θ1 ) > 0 and m∗j (θ2 ) < 0. An agent with Bj −B < (B/I)(Ij −I)
sells the state-contingent claim for the normal state and purchases the state-contingent claim
for the disaster state. As agents’ ability decreases and initial wealth increases, they are more
likely to desire compensation in the disaster state while paying in the normal state.
Example 2
There are other insurance designs that are satisﬁed with Eq. (29) or (30). The following
design indicates that agents are prepared for the disaster state without holding the satecontingent claim for state 2.
)(( b ) ( 1+χ )1−σ ∗
1+χ
σ
σ
−χ
−χ
∗
∗
∗
χ
−
w
(θ
)
(Bj I − BIj )
B
(w
(θ
)w
(θ
))
w
(θ
)
2
1
2
1
1−b
χ
[
]
m∗j (θ1 ) =1+χ
1+χ
I 1−σ r∗ (θ2 ) I + p∗1 Bw∗ (θ1 ) χ + p∗2 Bw∗ (θ2 ) χ

(46)

and
m∗j (θ2 ) = 0.
16

(47)

Table 1: Parameter Values
Parameter
Elasticity of substitution between capital and eﬀective labor
Capital share when σ → 0
Relative risk aversion with respect to xj (θi )Inverse of Frisch elasticity
Parameter of labor reluctanceParameter of agent ability distribution
Parameter of agent ability distribution
TFP in disaster state
TFP in normal stateValue

σ = −7/3
b = 0.33
γ = 1.1
χ = 0.4545
Ψ = 0.043
n = 3.2034
α = 0.5557
θ1 = 0.01511
θ2 = 0.3719

Source/Target
Gechert et al. (2022)
RBC literaturenormalization
Clemens et al. (2023)
See Appendix B
n = 1.001(1 + χ)/χSee Appendix B
w(θ1 ) = 0.1w(θ2 )See Appendix B

Again, suppose that Assumption 1 holds. Then, if Bj − B > (B/I)(Ij − I), it holds thatm∗j (θ1 ) < 0. An agent with Bj − B > (B/I)(Ij − I) sells the state-contingent claim for state
1 (disaster state). As agents’ ability increases and the initial endowment decreases, they
are more likely to compensate for damages in the disaster state. Conversely, if Bj − B <(B/I)(Ij − I), it holds that m∗j (θ1 ) > 0. An agent with Bj − B < (B/I)(Ij − I) purchases
the state-contingent claim for state 1. As agents’ ability decreases and the initial endowment
increases, they are more likely to desire compensation in the disaster state.
Notably, if the insurance is designed such that Eq. (29) or (30) holds together with themarket clearing conditions, any insurance design can achieve the Pareto optimum, although
the optimal insurance design is indeterminate. The quantitative analysis in the next section
visualizes the role of the state-contingent claims in achieving the Pareto optimum.

5

Quantitative analysis

To see the eﬀect that the state-contingent claims for aggregate shocks have on indirect utility,
we perform numerical exercises in this section.
5.1

Parameters

The parameter values that we use in the numerical analysis are listed in Table 1. As discussed
in the previous section, Gechert et al. (2022) suggested that the elasticity of substitution
between capital and labor is 1/(1 − σ) = 0.30. Following their report, we set σ = −7/3.
We set b = 0.33, which becomes a capital share of the output assumed in the standard
real business cycles literature when the production function reduces to the Cobb-Douglas
production function as σ → 0. γ is related to the coeﬃcient of relative risk aversion, which
17Figure 1: Agent ability versus expected utility

is given by γcj (θi )/xj (θi ) in our model. There is mixed evidence for the coeﬃcient of relativerisk aversion and it is subject to agent ability in our setting. Thus, it is diﬃcult to pin down
γ. Greenwood et al. (1988) used two alternative values, γ = 1.001 and γ = 2.0. In our
exercise, we employ a value relatively closer to the former and set γ = 1.1, which leads tothe case of near-log utility.4
We assume that individual-speciﬁc labor productivity is ej = j and follows a Pareto
distribution, whose cumulative distribution function, G(j), is given by

G(j) =( )n 1− αif α < j 0otherwise,

j

(48)

where α > 0. We assume that n > (1 + χ)/χ so that e can be well deﬁned. From Eq. (48),the density function of j is G′ (j) = nαn j −n−1 ; thus, e can be computed as
[∫ ∞
e=

j

1+χ
χ

n −n−1· nα j

χ
] 1+χ

dj

α
4

[

nχ
=
nχ − (1 + χ)

χ
] 1+χ

α.

(49)

Although we examined some other alternative values of γ, the main results were essentially unchanged.

18

Figure 2: Agent ability versus insurance contribution to expected utility (%)

Accounting for the average working hours in the United States, Clemens et al. (2023) set
χ = 0.4545. Thus, we use the same value. The values of n and α determine the averagelabor productivity, but n should be greater than (1+ χ)/χ. Then, we set n = 1.001(1+ χ)/χ.
We still must determine the values of Ψ, α, θ1 , and θ2 . To determine Ψ, α, and θ2 , we use
the data for per worker GDP, per worker capital, and the annual average working hours,
which are obtained from the Penn World Table, version 10.01. We elaborate on the details
in Appendix C.
5.2

Results

We examine what would happen if the wage rate in a disaster state became 10% of that ina normal state, i.e., w∗ (θ1 ) = 0.1w∗ (θ2 ) as a benchmark case, which might be extreme but
could occur. Figure 2 presents the relationship between agent ability and indirect utility
with agents’ initial wealth remaining as the average per capita capital from 2011 to 2019.
The two panels in the left column show the benchmark case. The panels in the middle
and right columns present the cases in which w∗ (θ1 ) = 0.01w∗ (θ2 ) and w∗ (θ1 ) = 0.01w∗ (θ2 )for robustness checks, respectively. The upper and lower rows show the cases in which the19Figure 3: Initial wealth versus expected utility

probabilities of disaster occurrence are π1 = 0.02 and π1 = 0.01, respectively. Although
we assume a two-period model, one may regard the two periods as a year. Then, one canconsider that a disaster occurs once every ﬁfty years if π1 = 0.02 (since 1/0.02 = 50) andonce a hundred years if π1 = 0.01 (since 1/0.01 = 100). The solid (dotted) line in Figure 1
indicates the indirect utility when an aggregate shock is (not) insured against.
As shown in Figure 1, the indirect utility without insurance is less than that with insurance in all cases despite the agent’s ability. Whereas our theoretical investigation naturally
expects this outcome, the numerical results show that labor allocation may not be optimal
when an aggregate shock occurs in an economy with heterogeneous agents. It is natural thatas an agent’s ability increases, the indirect utility with and without insurance increases. It
is also natural that as the wage rate in a disaster state decreases, the indirect utility decreases. Furthermore, as the probability of disaster occurrence decreases, the indirect utility
increases. Notably, the contribution of insurance to indirect utility is greater for agents withlower ability than for agents with higher ability. In particular, the contribution of insurance
almost vanishes for agents with suﬃciently high ability. We elaborate on this point in Figure
2 by observing the percentage change in indirect utility from the case without insurance to20

Figure 4: Initial wealth versus insurance contribution to expected utility (%)

that with insurance.

∑2

ι 1−γ
−1]/(1−γ) (ι = a, b) is an indirect utility wherei=1 πi [(xj (θi ) )a (b) stands for “insured” (“uninsured”) and xj (θi )ι is the value determined in equilibrium.

Suppose that EU ι :=Figure 2 presents the percentage contribution of insurance to indirect utility, measured by
100 × (EU a − EU b )/EU b . As shown in Figure 2, when the agent’s ability is low, there are
signiﬁcant insurance contributions. In particular, at the lowest agent ability (j = 0.5557),
the insurance contribution approximately ranges from 2% to 10% in the once-in-ﬁfty-years
case (π1 = 0.02) and from 1% to 4.5% in the once-in-a-hundred-years case. As expected fromthe analysis in Figure 1, however, the insurance contribution shrinks as the agent’s ability
becomes high. Observing the three cases in each panel, we ﬁnd that greater damage to thewage rate induces a greater insurance contribution. Comparing the left and right panels, wenote that a greater frequency of disasters causes a greater insurance contribution.
We next examine the relationship between agents’ initial wealth and indirect utility. As
seen in Figure 3, we observe that the indirect utility increases with initial wealth in all
panels. Whereas the indirect utility uninsured is less than that insured as expected, we
note from the comparison between the upper and the lower rows that a higher frequency

21

of disasters produces greater welfare improvement. Furthermore, severe disaster damage
induces a greater contribution of insurance, which we ﬁnd by examining the three columns
from left to right. We can conﬁrm the last by observing the three cases in each panel in
Figure 4, which presents the percentage contribution of insurance. Figure 4 also indicatesthat the percentage contribution moderately increases with agents’ initial wealth.

6

Concluding remarks

The COVID-19 pandemic has damaged the world economy. It is presumed that this kind ofcrisis occurs once a hundred years. Thus, this experience is rare for ordinary people whose
longevity is at most 80 or 90 years. Although we have hardly investigated insurance against
aggregate shocks in the macroeconomics literature thus far, this crisis made us notice that
it is crucial to consider such insurance. Our model suggests that an ex-post redistribution
performed by insurance can induce an ex-ante welfare improvement without reallocating
resources before a disaster (Example 1). Under this insurance program, agents with lower
ability can enjoy greater welfare improvement than agents with higher ability even if theinsurance program reallocates resources from agents with lower ability to agents with greater
ability if a disaster does not occur.
A caveat limits our ability to see the results obtained in this paper. We investigated a
two-period model. Although rich results have been derived, we do not know what would
happen if we extended the time horizon to inﬁnity, as in usual dynamic macroeconomic
models. In particular, it is interesting to study the eﬀect of insurance against aggregate
shocks when there is capital accumulation. This topic is worthy of future research.

Appendices
Appendix A
We ﬁrst derive Eq. (29) from Eq. (30). By using Eq. (23) in Lemma 2, the right-hand sideof Eq. (30) can be rewritten as(r∗ (θ2 )m∗j (θ1 ) − r∗ (θ1 )m∗j (θ2 ) =∗∗r (θ1 )w (θ2 )

1+χ
χ

∗

∗

− r (θ2 )w (θ1 )I + p∗1 Bw∗ (θ1 )

1+χ
χ

1+χ
χ

)

(Bj I − BIj )1+χ

+ p∗2 Bw∗ (θ2 ) χ

.

(A.1)

Eliminating I from the denominator in the right-hand side of Eq. (A.1) by using Eq. (28)
and applying Eq. (11) in Lemma 1 yield Eq. (29). To derive Eq. (30) from Eq, (29), we
22

follow the opposite of the above procedure.
Appendix B
To pin down the values of Ψ and α, we use the actual data for per worker GDP, per worker
capital, and persons’ working hours. We assembled the data for real GDP (rgdpna), realcapital (rnna), the number of workers (emp), and persons’ annual average working hours(avh) from the Penn World Table, 10.01 from 1950 to 2019, and prepared per worker real
GDP and per worker real capital for the same period.
The stochastic states are irrelevant for determining Ψ and α, so we omit θi from thevariables unless stated otherwise but add a time subscript, t, for further exposition. We∫
assume that the annual average working hours in year t are H̃t := j∈Ω lt,j dj. From Eqs. (10)and (48), we obtain(∫H̃t :=

lt,j dj =j∈Ω

or equivalently,

Since ht =∫e l djj∈Ω j t,jnχnχ − 1)(

αwt ) χ1
,
Ψ

( )(
)χ
Ψ
nχ − 1
wt =
H̃t .
α
nχ
1

(B.1)

1+χ

is computed as ht = (wt /Ψ) χ e χ , it follows from Eqs. (49) and(B.1) thatht = h̃t α,(B.2)

where h̃t = H̃t (nχ − 1)/(nχ − 1 − χ). Therefore, Eq. (B.1) is rewritten as
wt =

Ψ
α1+χ

(

nχ − 1 − χ
αh̃t
nχ

)χ
.

(B.3)

From Eqs. (1) and (2), we havewt =(1 − b)yt hσ−1t,bktσ + (1 − b)hσt(B.4)

where yt is per worker GDP and kt is per worker capital. Eliminating wt from Eqs. (B.3)
and (B.4) with (B.2), we have
Zt (Ψα−σ ) + V Ψ = yt h̃−1−χ
,
t

23

(B.5)

(

where
Zt :=

b
1−b

)(

(

and
V :=

nχ − 1 − χnχnχ − 1 − χnχ)χktσ h̃−σ
t

(B.6)

)χ
.

(B.7)

From Eqs. (B.5)-(B.7), it follows that(
Ψα−σ =V ar(yt h̃t−1−χ )

Ψ=

(B.8)

V ar(Zt )



and

) 21

1  −1−χyt h̃t−V(

V ar(yt h̃t−1−χ )) 12

V ar(Zt )
Zt  ,(B.9)

where the variance is taken for the period 1950-2019 and the overline represents the meanof variables over the same period. Eqs. (B.8) and (B.9) yieldα=1  −1−χ−yt h̃tV(

V ar(yt h̃−1−χ
)
t

) 12

V ar(Zt )(
Zt V ar(yt h̃−1−χ)t
V ar(Zt ))− 12  σ1

 .

(B.10)We use Eqs. (B.9) and (B.10) as the calibrated values of Ψ and α, respectively.
Next, we determine θ2 . Once one has obtained α, it follows from Eqs. (1) and (B.2) thatyt = θ2 W t ,(B.11)()1/σ
where Wt := bktσ + (1 − b)ασ h̃σt
. To determine θ2 , we ﬁrst prepared time series of yt
and Wt from 1950-2019 and created θ2 year by year. We assumed that the United States wasin a normal state (i.e., state 2) from 2011 to 2019. Then, we assembled the θ2 values over theperiod 2011-2019 to average them over the same period. We used the average value of θ2 . For
I = k in Eqs. (21) and (22), we averaged per capita capital over the period 2011-2019. Now
that we have had parameter values other than θ1 , we obtain w(θ2 ) and r(θ2 ) endogenously
from Eqs. (21) and (22). We set θ1 by using Eq. (22) such that w(θ1 ) = 0.1w(θ2 ), as notedin Table 1, to investigate a counterfactual situation. We also examine two other alternative
cases in which 0.01w(θ2 ) and 0.001w(θ2 ) for robustness checks.

24

Acknowledgments
We are grateful for the ﬁnancial support from Kwansei Gakuin University, the joint research
programs of KIER (Kyoto University) and ISER (Osaka University), and the Japan Society for the Promotion of Science Grants-in-Aid for Scientiﬁc Research (Nos. 20K01647,
20H05631, 23H00831).

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