Discussion Papers In Economics
And Business

Optimal disaster-preventive expenditure in a dynamic andstochastic model

Takumi Motoyama
Discussion Paper 15-03-Rev.

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANOptimal disaster-preventive expenditure in a dynamic andstochastic model

Takumi Motoyama
Discussion Paper 15-03-Rev.
October 2016

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANOptimal disaster-preventive expenditure in a dynamic andstochastic model ∗Takumi Motoyama †Graduate School of Economics, Osaka University
October 2016

Abstract
The purpose of this study is to present an analytical framework for publicly optimal disasterpreventive expenditure. We examine the optimal policy combination of tax rate, disaster-preventive
expenditure, and productive government expenditure in a neoclassical growth model, in which natural disasters occur stochastically and partially destroy existing capital. Based on this model, we candecompose the welfare eﬀect of raising preventive expenditure into three eﬀects: the damage reduction, crowding out, and precautionary eﬀects. By identifying these marginal benefits and costs, we
obtain the policy conditions that maximize household welfare. Furthermore, we show that optimal
prevention is increasing in disaster probability, and by using a numerical example, we show that there
is an inverse U-shaped relationship between the expected growth rate and disaster probability.
Keywords: Natural disasters, Disaster-preventive expenditure, Optimal policy
JEL Classification Codes: E13; H4; Q54; Q58∗

I am grateful to Koichi Futagami and Tetsuo Ono as well as the participants at the 2014 Japanese Economic Association
Spring Meeting held at Doshisha University, and an antonymous referee for their helpful comments on an earlier version ofthis paper. I especially thank Tatsuro Iwaisako for his invaluable suggestions concerning this research. Needless to say, allremaining errors are my responsibility.
†
Graduate School of Economics, Osaka University, 1-7 Machikaneyama, Toyonaka, Osaka 560-0043, Japan. E-mail:pge028mt@student.econ.osaka-u.ac.jp.

1

Introduction

In recent years, increasing global economic damage has been caused by a growing number of natural
disasters such as earthquakes, floods, and hurricanes, as Cavallo and Noy (2011) point out. Figure1(a) shows the worldwide trend of natural disaster occurrences and their economic damage in 1975–2015 and Figure 1(b) shows those of the United States. In these figures, the dotted lines indicate
the regression lines of the number of natural disasters and the solid lines indicate those of economic
damage. 1 At the total level, Figure 1(a) shows that both the number of natural disasters and economic
damage have increased over time. In Figure 1(b), we see that in the United States, natural disasters,
especially hurricanes, frequently occur, with the number of disasters and economic damage also rising
as in Figure 1(a). These figures imply that the influence of natural disasters on economic behavior has
become increasingly important, and it is necessary to consider natural disasters as an economicallycrucial stochastic event. Moreover, because the requisite manner of governmental intervention is also
an economically meaningful question, we must discuss the intervention of governments to cope withnatural disasters.
Then, how can the government intervene in natural disasters? There are two measures to copewith natural disasters: mitigation (or abatement) and adaptation. Mitigation is a method of controlling root generating disasters. For example, mitigation may include the reduction of CO2 emissions
and purchase of eco-friendly goods so as not to accumulate pollutants. This results in the decreased
occurrence of climate change and disasters such as abnormal meteorology and hurricanes. On thecontrary, adaptation is a method of decreasing the damage caused by disasters when they occur. Adaptation includes the refinement of buildings, reinforcement of disaster-resistant construction codes, andpreparation of survival food. Naturally, each method is equally important for tackling natural disasters. In fact, the report of the Intergovernmental Panel on Climate Change (IPCC, 2007); “Climate

1

The data for each figure are obtained from the Emergency Events Database (EM-DAT) at http://www.emdat.be/. According to the EM-DAT website, the natural disasters included in these figures satisfy at least one of the following criteria:
1. 10 or more people dead;
2. 100 or more people damaged;
3. Declaration for a state of emergency; or
4. Call for international assistance.

1

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(b) Data on the United States

Figure 1: Time trend of disaster risk
Change 2007: Synthesis Report” states as follows.
“There is high confidence that neither adaptation nor mitigation alone can avoid all
climate change impacts; however, they can complement each other and together cansignificantly reduce the risks of climate change.”
Both adaptation and mitigation are required to cope with disasters; in fact, there are many representative examples of research on mitigation and empirical studies of the damage caused by disasters.
However, there is little theoretical literature and few empirical studies of adaptation, especially fromthe viewpoint of macroeconomic theory. As Figures 1(a) and 1(b) show, the occurrence of disasters2and resulting damage are increasing; therefore, it is crucial for us to focus on adaptation as well as mitigation. Furthermore, as IPCC says,2 there are many adaptation options, but there is little knowledge
about the eﬀects of adaptation on macroeconomic performance, such as consumption, savings, andeconomic growth rates as well as other non-climate policies such as tax rates and productive government expenditure. Thus, it is necessary to include other non-climate policies in addition to adaptation
policy and to establish the eﬀect of adaptation expenditure on macroeconomic performance and otherpolicies. To treat these problems, we require a dynamic general equilibrium model with a government
and a stochastic disaster shock.
Therefore, we construct such a macroeconomic model in which disasters destroy the existing capital stock stochastically and in which the government implements adaptive expenditure and productive
expenditure. We consider productive government expenditure, following Barro (1990), as well as
adaptive expenditure in order to establish the eﬀect of adaptation on the other non-climate policies.
Hereafter, for apposition we use the word “prevention” instead of “adaptation.” In this model, a Poisson stochastic disaster shock is adopted, that is, disasters occur at a certain constant probability ineach period. In the model, we assume that households cannot diversify disaster risks through disasterinsurance, 3 and by adopting a constant relative risk aversion (CRRA) instantaneous utility function,
we can take account of household behavior toward risk. The benevolent government implements its
policy by levying taxes on households. There are two policy alternatives in this model: disasterpreventive expenditure and productive government expenditure. Since disaster prevention has theproperties of public goods, the government finances disaster -preventive expenditure, which allows it
to reduce the damage to physical capital caused by disasters. On the contrary, productive government2

In addition, the IPCC makes the following two statements.
“A wide array of adaptation options is available, but more extensive adaptation than is currently occurring
is required to reduce vulnerability to climate change. There are barriers, limits and costs, which are not fully
understood.”“In several sectors, climate response options can be implemented to realise synergies and avoid conflicts
with other dimensions of sustainable development. Decisions about macroeconomic and other non-climate
policies can significantly aﬀect emissions, adaptive capacity and vulnerability.”3

According to data provided by the Munich Reinsurance Company, only about 25 % of the economic losses of the 10
costliest natural disasters in 1980 –2015 (as an aside, eight of these disasters occurred after 2004) was insured. Indeed,even when we choose the top 10 natural disasters ordered by insured loss, only 38 % of economic losses was insured.
These figures imply that disaster insurance market is still developing and that the assumption of no disaster insurance
market is justified. These data are available at https://www.munichre.com/touch/naturalhazards/en/natcatservice/annualstatistics/index.html.

3expenditure increases the productivity of final goods.
Based on this model, we can decompose the welfare eﬀect of raising preventive expenditure into
the following three eﬀects: the damage reduction, crowding out, and precautionary eﬀects. First,
the damage reduction eﬀect is the positive welfare eﬀect in which higher preventive expenditure
increases a household’s expected disposable total income by reducing the damage caused by disasters,
thereby raising the household’s consumption and savings. Second, the crowding out eﬀect is caused
because higher preventive expenditure requires productive government expenditure to decrease andthe tax rate to increase to finance it. This decreases the household’s consumption and savings andhas a negative impact on its welfare. Finally, the precautionary eﬀect is related to the household’srisk averse behavior. If the household is relatively risk averse and there is no disaster insurance
market, it saves more in the case of natural disasters. Such savings are called precautionary savings.
Aside from the decrease in savings because of the reduction in income from the damage caused bydisasters, the saving level under disaster risk is higher than that under no disaster risk. On the contrary,higher preventive expenditure can reduce the household’s precautionary savings through a decrease indisaster risk. Since welfare under no disaster risk is higher than that under disaster risk, a decrease in
precautionary savings through higher preventive expenditure improves the household’s welfare. Thiseﬀect is generated since we adopt CRRA utility rather than logarithmic utility and there is no disasterinsurance, and is one of the interesting implications obtained from this model. The government mustchoose the optimal policy by equalizing these eﬀects, and we can obtain the optimal policy conditionsin that they maximize the household’s intertemporal utility. Furthermore, we can show the existenceof the optimal policy under some assumptions.
From the comparative statics of optimal prevention with respect to disaster probability, we can obtain two interesting results. One is that in addition to the direct eﬀect, the precautionary eﬀect causes
the optimal preventive expenditure to rise as disaster probability increases. This is because higher
disaster probability shifts household consumption to savings in case disaster occurs, and it is optimalfor the government to invest in disaster-preventive expenditure in order to encourage households toreallocate their savings to consumption. This eﬀect results from the precautionary savings motive,
which cannot be captured by a static model or a dynamic model with an instantaneous log-utility

4

function. The other interesting result is that although the crowding out eﬀect has a negative impact
on the increase in preventive expenditure in response to disaster risk since higher disaster probability
reduces household expected intertemporal income and makes the crowding out eﬀect of preventive
expenditure severe, this negative eﬀect never dominates the other two positive eﬀects.
Further, we can decompose the eﬀect of an increase in disaster probability on the expected growthrate into five channels: two positive eﬀects and three negative eﬀects. As disaster probability rises, (i)the expected disaster loss decreases since prevention increases and (ii) a household’s saving increases
owing to the precautionary savings motive; hence, the economic growth rate increases. By contrast,(i) the expected disaster loss increases since disaster probability rises, (ii) the crowding out eﬀect
increases to finance preventive expenditure, and (iii) precautionary savings decrease since prevention
increases, which results in a decrease in the economic growth rate. Thus, the overall eﬀect is ambiguous. In a numerical example, we obtain an inverse U-shaped relationship between the expected
growth rate and disaster probability.
The remainder of this paper is organized as follows. In Section 2, the related literature is introduced. In Section 3, we describe the behavior of each component of this economy: a householdthat consists of a worker and investor, a firm, a government, and a disaster shock. In Section 4, we
solve the investor’s and the government’s intertemporal utility maximization problems and describe
the model’s equilibrium. In Section 5, we show the existence of the optimal policy and interpret its
conditions. In addition, we conduct comparative statics of the optimal preventive expenditure withrespect to disaster probability. Furthermore, we obtain the expected economic growth rate and calculate its value using a numerical example, by changing disaster probability. Section 6 summarizes thestudy.

2 Related literature
As mentioned in the Introduction, theoretical studies on adaptation are scarce, especially optimal
disaster prevention, but some do exist. An early study of reaction to natural disasters is Lewis andNickerson (1989), which considers private protection under deterministic disaster damage. With regard to studies of adaptation in a stochastic environment in which the damage caused by disasters is

5

uncertain, Kane and Shogren (2000), Anbarci et al. (2005), and Cohen and Werker (2008) constructstatic models. The first study analyzes adaptation and mitigation simultaneously given a disaster occurrence probability distribution and the last two studies focus on optimal disaster prevention given acertain disaster probability. In Anbarci et al. (2005), when an earthquake occurs, the agent may die
and disaster -preventive expenditure can decrease mortality. Cohen and Werker (2008) study the difference between prevention and reconstruction and show that moral hazard exists from reconstruction.
With regard to a dynamic model with adaptation policy, to the best of my knowledge, Bretschger andValente (2011), De Zeeuw and Zemel (2012), Brechet et al. (2013), and Tsur and Withagen (2013)
construct such a model. De Zeeuw and Zemel (2012) and Tsur and Withagen (2013) study the dynamics of adaptation (the former authors also study mitigation) under a stochastic once-and-for-all
disaster (or catastrophe). In such a setting, unlike our setting in which natural disasters may occur
in each period, a maximum principle can be used and the calculation becomes easier. Brechet et al.
(2013) study the dynamics of adaptation and mitigation with a stock of pollutants rather than stochastic climate change and so do Bretschger and Valente (2011). In contrast, Rietz (1988), Barro (2009),
Gourio (2012), and Ikefuji and Horii (2012) construct a dynamic model that includes natural disasters as a stochastic event. The first three studies concentrate on asset pricing, especially on solving
a risk-premium puzzle, and the last study obtains an optimal carbon tax, such that it maximizes the
economic growth rate under a full-coverable disaster shock. However, their motives are not to obtain
optimal prevention, and hence, these terms do not appear in their model. If we regard preventive expenditure as a maintenance cost, which is a similar idea in that both can decrease the depreciation rateof the capital stock, then some studies exist on optimal levels of maintenance, such as Rioja (2003)
and Kalaitzidakis and Kalyvitis (2004). Two important diﬀerences between our study and theirs are
as follows. First, we take account of the Poisson process of disaster occurrence, and hence, our model
can consider household behavior toward uncertainty. Second, we consider a trade-oﬀ between preventive and productive government expenditure, unlike these studies. Neither of these studies treatsuch problems. A recent study that is close to ours is Barro (2015). In his model, which is similar
to Barro (2009), the damage caused by disasters occurs stochastically, and he considers an optimal
consumption/savings problem with utility maximizing environmental protection, which decreases the6probability of a natural disaster occurring. The diﬀerence between Barro’s study and ours is that his
model adopts a Lucas-tree asset pricing model with an Epstein-Zin-Weil utility function, and hence,the (intrinsic) economic growth rate is exogenous. In our model, on the contrary, the economic growthrate is also endogenous.

3

The environment

3.1

A household

This model is characterized by discrete time and a closed economy. In this economy, there are two
types of infinitely-lived households, which each comprise a worker or an investor. The number ofworkers is normalized to one and that of investors is also one. They have the same utility functions
and discount factors. However, their behavior and income are diﬀerent. A worker supplies one unit oflabor inelastically and earns wage w from a firm. We assume that the worker consumes all of his/herincome after paying tax (i.e., the worker does not save). 4 On the contrary, an investor earns income
by selling his/her capital remaining from the previous period. The price of capital is denoted by r,which is equivalent to the net interest rate. Here, we assume that the investor’s capital is held in theform of machinery, and hence, there is a risk that some of his/her capital is damaged by a natural
disaster. 5 In addition, we assume that there is no disaster insurance market to compensate for theinvestor’s damaged machinery, that is, disaster risk cannot be pooled and diversified. 6 Therefore, theinvestor must respond to disaster risk by self-insurance or an adjustment of savings. In this setting,the consumption of worker cw is derived as
cwt = (1 − τt )wt ,(1)

where τt is the tax rate for both the worker and the investor in period t. 74This setting enables us to obtain closed-form consumption and savings functions under a general equilibrium modelwith a CRRA utility function. Such a setting is assumed in some studies on optimal tax. The best-known study that adopts
this setting is Judd (1985).
5
In reality, an investor’s savings are held in monetary assets, which are not damaged by disasters, and a firm purchasesmachinery by borrowing savings from the investor before disaster risk is realized. Hence, the agent who faces uncertainty
is not the investor but the firm. However, with regard to our calculation, this setting is essentially equivalent to the setting
we use in this model. Therefore, we adopt the model described in the text.
6
This assumption results from the statement presented in footnote 3. However, we consider how the existence of acompetitive disaster insurance market aﬀects an investor’s behavior in Subsection 5.4.
7It may be assumed that the tax rate for the worker and investor should be separated. However, since one of the tax
rates is represented by the linear combination of the other tax rate at the optimal policy, (i.e., the tax rate for the worker is a7

The investor’s problem is somewhat cumbersome. His/her purpose is to maximize his/her expected utility under a budget constraint. The expected utility for the investor in period 0 is givenbyU(cIt ) = E0

∞
∑
t=0

1−γ

βt

cIt
,
1−γ

(2)

where cIt is the investor’s consumption in period t, β ∈ (0, 1) indicates a discount factor, and E0
represents an expectation operator in period 0. The reason why an expectation operator exists in (2)is that we take account of natural disasters that occur stochastically. An instantaneous utility function
is characterized by a CRRA utility function, where γ indicates the parameter of relative risk aversion.
Hereafter, we assume γ > 1, that is, the investor is relatively risk averse, which is plausible basedon both intuition and the findings of empirical studies. The budget constraint faced by the investor isgiven byat+1 = [1 + (1 − τt )rt − δ]st − cIt .

(3)

In (3), at+1 is the capital carried over to the next period, which has not yet been aﬀected by disaster
risk, and δ ∈ (0, 1) is the depreciation rate of capital. In addition, st is the resultant capital defined as(6) later.

3.2

A firm

A representative firm exists in this economy. It produces final goods, whose price is set as a numeraire,by using machinery from investors and labor from workers. In addition, productive government
expenditure positively aﬀects the firm’s production. Its production technology is given by
Yt = AKtα (Lt Gt )1−α , A > 0, α ∈ (0, 1),(4)where Yt is final good production in period t, Kt represents the capital stock in period t, Lt is the labor
force in period t, and Gt represents productive government spending in period t. The purpose of thisfirm is to maximize profit. Since the firm can employ Kt and Lt given Gt and use them before disasterrisk occurs, it faces no uncertainties.

positive linear function of the tax rate for the investor), there is little eﬀect on the result by assuming the same tax rate forthe worker and investor.

8

3.3

A government

Next, we turn to the government’s behavior. The government is benevolent and its purpose is tomaximize total household welfare. The government can control household utility by levying taxes
and using the tax revenue to implement its policy. The government faces two policy alternatives
in period t: disaster-preventive expenditure Ht and productive government expenditure Gt . In thismodel, following Barro (1990), we assume that Gt is a certain ratio of output, that is, Gt = gt Yt
for gt ∈ (0, 1). The government knows this rule and controls gt when choosing the optimal policy,
while the household and firm do not know this rule and take Gt as given. Preventive expenditure
can reduce the damage to physical capital caused by disasters. Here, we assume that damage has adecreasing relationship with respect not to Ht but to the preventive expenditure–production ratio ht ,where Ht = ht Yt . This is plausible for the following reasons. In this setting, as the absolute valueof preventive expenditure Ht increases, damage decreases, while when output level Yt increases, allother things being equal, damage increases. The former is natural and the latter would be true since if
Yt is large, the economy is filled with machinery, buildings, and so on. Then, these are destroyed in achain reaction when a disaster occurs, which could magnify the damage ratio. Let us assume that the
government’s budget balances in each period. Then, the budget constraint becomesτt (rt st + wt ) = (gt + ht )Yt .

(5)

Hereafter, we denote the policy {τt , ht , Gt } as πt .

3.4Disaster shock

We define a disaster shock as follows. 8• A disaster may occur after the firm finishes producing, households finish consuming and saving,
and the government finishes levying taxes and implementing its policy in each period.
8

In addition to the following settings, we can introduce the stochastic severity caused by disasters. In this case, we
capture the degree of severity by introducing a random variable d, which is distributed with probability density function
ϕ(d) ⊂ [d, d] and cumulative distribution function Φ(d). However, introducing this term has little eﬀect on the following
calculation, and the comparative statics with respect to severity resembles that with respect to disaster probability. Therefore, we omit this term. To introduce such severity, this version of the conditions for optimal policy can be obtained byreplacing D(h) with ED(h, d) in the equation below, where E remains in the latter since the damage caused by disaster D(h)contains a random variable d after the disaster occurs.

9• If a disaster occurs, a certain ratio of the existing capital stock denoted by Dt ∈ [0, 1] is destroyed. In this study, we call this ratio the damage ratio.
• The probability of a disaster occurring is given by p, which is a constant and exogenous value.
We call this the disaster probability. The investor and government know this probability precisely. Based on these assumptions, we can write the next level of capital st+1 as(1 − Dt )at+1 with probability p,
st+1 = at+1
with probability 1 − p.

(6)

• When the government pays preventive expenditure in period t, the damage ratio lowers to somedegree. That is, when the government incurs preventive expenditure in period t, the damage tothe capital stock reduces. Here, we assume that preventive expenditure in period t aﬀects only
the damage ratio in period t and there are no eﬀects thereafter.
From the fourth assumption, we can write the damage ratio in period t as a function of ht , such as
Dt = D(ht ). As for the relationship between the damage ratio and preventive expenditure–production
ratio, we assume the following. 9
Dt = D(ht ) ∈ (0, 1), D′ (ht ) < 0, D′′ (ht ) ≥ 0.

(7)

The second and third properties indicate that the damage ratio function is decreasing, twice diﬀerentiable, and weakly convex with respect to the preventive expenditure–production ratio. The convexityof the damage ratio function implies a decrease in the marginal eﬀectiveness of preventive expenditure. Hereafter, we abbreviate the subscripts t and t + 1 for notational simplicity. These equations
enable us to solve optimal policy π∗ .

9For example, these properties are satisfied under the following fractional function:D(ht ) =d̂
1 − d̂−d̂ht − d̂,()
where d̂ = − 1 − d̄ /d̄ for d̄ ∈ (0, 1). In this specification, d̄ is the upper limit of the damage ratio D(0) = d̄, while the lower
limit of damage ratio is zero D(1) = 0. In the numerical example presented in Subsection 5.3, we use this specification.

104Model solution

4.1

Equilibrium

In this subsection, we define the equilibrium of this economy. First, we obtain the market equilibrium.
That is, given the announced policy π = {τ, h, G} and initial savings s0 ,1. A worker consumes cw as (1) and an investor maximizes his/her intertemporal utility. Then, theworker’s and investor’s consumption functions and the investor’s savings function are obtained,
2. A firm chooses K and L to maximize its profit,
3. Demand for and supply of inputs are equal, that is,
s = K, L = 1,

(8)

4. The budget constraint for the government (5) is satisfied.
Second, given the household’s consumption and savings functions, initial conditions s0 , and thegovernment’s budget constraint (5), the government chooses optimal policy π∗ ≡ {τ∗ , h∗ , g∗ } such thatit maximizes the sum of the investor’s and worker’s intertemporal utility 10 given by
[
]
∞
∑
1 1−γ
1 1−γ
t
E0
β
c +
c
.
1−γ w
1−γ I
t=0
Finally, under the optimal policy, we can obtain the household’s optimal level of consumption andsavings {c∗w , c∗I , ã∗ }.

4.2 Investor’s problem
To obtain the optimal policy, we must first solve the investor’s utility maximization problem by maximizing (2) subject to (3), given π and initial savings s0 . Denoting V(s) as the investor’s value function,
we can describe the investor’s problem in period t as

 1−γ





 cI
+
βEV(
s̃)
,
V(s) = max 1 − γ{cI ,ã} (9)subject to ã = [1 + (1 − τ)r − δ]s − cI ≡ Rs − cI , given π, s0 .
10This procedure is used in, for example, Glomm and Ravikumar (1994) and Rioja (2003). Under this rule, a problem
exists in which a government has an incentive to change its policy after the household and firm finish their behavior.
However, by assuming that the government commits its announced policy, we ignore this problem. In addition, see also
Ljungqvist and Sargent (2004, Chapter 15) for this problem.

11

In (9), the term E indicates the expectation operator in period t and the superscript ˜ refers to thevalue of the variables in the next period. The term R ≡ [1 + (1 − τ)r − δ] summarizes the return on
savings. By solving this dynamic programming problem, we can obtain the investor’s savings andconsumption functions analytically, as shown in Proposition 1.
Proposition 1 By solving (9), we can obtain the investor’s savings and consumption functions asfollows:[]1/γã = R1/γ β(p(1 − D(h))1−γ + 1 − p)s ≡ (Rρ(h))1/γ s ≡ σ(τ, h)s,(10)cI = [R − σ(τ, h)]s,(11)where ρ(h) ≡ β(p(1 − D(h))1−γ + 1 − p) is a risk-adjusted discount factor and σ(τ, h) ≡ (Rρ(h))1/γ is
the investor’s savings rate for current savings.
Proof.
See Appendix A.1 for deriving (10). Then, substituting (10) into (3) yields (11).

Note that the savings function is aﬀected by disaster probability p and preventive expenditure h.
The intuition of this is discussed below. Moreover, to make consumption positive, we have R − σ > 0.
As shown in Proposition 3 described in Subsection 5.1., this inequality must hold under the optimal
policy and hence, we confine the case to that in which the investor’s consumption is positive.
Before moving onto the government problem, we check the signs of the derivatives of the savings
rate σ(τ, h) with respect to disaster probability p and each policy variable. We can check the sign ofthe derivative of the savings rate on the tax rate τ as∂σ(τ, h)< 0,∂τwhich is natural since a higher tax rate decreases the investor’s disposable income and savings. In
contrast, raising the tax rate has an ambiguous eﬀect on the investor’s consumption since savings
decrease, while disposable income decreases, and we do not know which eﬀect is dominant. However,when an increase in the tax rate decreases consumption, this condition can be written as σ < γR.
Hence, if consumption is positive, that is, R > σ, this is satisfied. Positive consumption is guaranteed
under the optimal policy, as shown in Proposition 4, and a higher tax rate also decreases the investor’s

12

consumption. The derivative of σ(τ, h) on p is as follows:∂σ(τ, h)> 0.
∂pSince the investor is relatively risk averse (γ > 1) and cannot diversify risk through disaster insurance,s/he must mitigate disaster risk by self-insurance. Then, s/he has an incentive to save in the caseof disasters because s/he prefers consumption smoothing between a disaster-stricken state and nodisaster state, which is known as precautionary savings. 11 12 On the contrary, the eﬀect of preventive
government expenditure on the savings rate is∂σ(τ, h)< 0.
∂hThis derivative has the opposite sign to dσ(τ, h)/d p because an increase in h reduces the expected lossof disaster pD(h), and the investor need not prepare more than before and thus saves less. We call thiseﬀect the precautionary eﬀect since higher preventive expenditure dilutes the investor’s precautionarysavings motives if s/he is relatively risk averse.

4.3 Firm’s problem and equilibrium condition
The representative firm maximizes its profit by choosing K and L given its factor price and policy.
The familiar conditions for profit maximization are given byr = αAK α−1 (LG)1−α ,w = (1 − α)AK α L−αG1−α .
By substituting (8) and the profit maximization conditions into (1), (10), and (11), we can obtain the
consumption levels of both agents and the investor’s savings function under the market equilibrium

11

See, for example, Sandmo (1970) for precautionary savings.
Moreover, if γ = 1, which is the logarithmic preference, disaster probability does not aﬀect the household’s savings
because a higher p decreases the expected return on savings, while disposable income also decreases. Under the logarithmic
preference, a change in disaster probability does not aﬀect the household’s savings owing to an oﬀset of income and thesubstitution eﬀect. The amount of precautionary savings is equivalent to that of a decrease in savings from a decrease in
expected intertemporal income.
1213

as follows:
cwE = (1 − τ)(1 − α)AK αG1−α ,cEI = [R − σ(τ, h)]K, where R = 1 + (1 − τ)αAK α−1G1−α − δ,(12)ãE = σ(τ, h)K, where R is defined above.
By using (12), the government budget constraint (5) is reduced tog = τ − h.

(13)

4.4 Government’s problem
Given the equilibrium condition (12) and under the budget constraint for the government (13), thegovernment solves the following dynamic programming problem by expressing V g (K) as its valuefunction. Note that the government knows G = gY, and the production function for the government
becomes Y = (Ag1−α )1/α K ≡ B(g)K. Moreover, for the government, R = (1 − τ)αB(g) + 1 − δ andw = (1 − α)B(g)K. Considering these, the government’s dynamic programming problem is given by
]
}
{[
1 1−γ
1 1−γ
g
g
c +
c
+ βEV (K̃) ,V (K) = max
π
1−γ w
1−γ I
(14)
subject to (12) and (13), given K0 .
We define the consumption propensity for the government c̄ as follows:
[

]
]
1 1−γ
1 1−γ
1 [
1
cw +
cI
=
[(1 − τ)(1 − α)B(g)]1−γ + (R − σ)1−γ K 1−γ ≡c̄K 1−γ . (15)
1−γ
1−γ
1−γ
1−γ

Then, the dynamic programming problem (14) is reduced to
}
{
1
1−γ
g
g
c̄K
+ βEV (K̃) ,V (K) = max
π
1−γ

(16)

subject to ã = σK, g = τ − h, given K0 .
By substituting g = τ − h into B(g), this problem contains only two endogenous variables, τ and h.
From the calculation presented in Appendix A.2, we can obtain the first-order conditions with respect14

to {τ, h} as 13
1
∂c̄
∂σ
∂σ
K −γ + βpV ′g ((1 − D)ã)(1 − D)+ β(1 − p)V ′g (ã)
= 0,
1−γ
∂τ
∂τ
∂τ
[
]
1
∂σ
−γ ∂c̄′gK+ βpV ((1 − D)ã) (1 − D)
1−γ
∂h
∂h
∂σ
− βpV ((1 − D)ã)D (h)σ + β(1 − p)V (ã)
= 0,
∂h
′g

′

(17)

(18)

′g

where the superscript ′ implies a derivative. The marginal condition of tax rate (17) is interpreted asfollows. The second and third terms of (17) are the marginal disutility from the reduction in savings
because of a reduction in income from raising the tax rate. The first term includes the marginal
disutility from the reduction in consumption because of a reduction in income and the marginal benefit
from raising tax, which means that the levied tax is used as productive government expenditure. Theinterpretation of (18) is similar to that of (17). The second and fourth terms are the marginal disutility
from raising h on savings. A decrease in savings comes about from two eﬀects. The first is a reductionin the return on savings because of the crowding out eﬀect. Higher preventive expenditure means
less productive expenditure given the tax rate to finance h, and this leads to a decrease in the returnon savings R. The second eﬀect is that higher preventive expenditure lessens an investor’s saving
incentives because of the precautionary eﬀect, as mentioned in Subsection 4.2. Higher preventive
expenditure increases expected savings and this shifts the investor’s savings to consumption. The
third term of (18) is the direct eﬀect of preventive expenditure which captures the following benefit.
Higher preventive expenditure decreases the damage caused by disasters by D′ (h), and we obtainthe marginal utility by βV ′g ((1 − D)ã)σ × D′ (h) because of an increase in expected intertemporal
income. With probability p, we can enjoy this utility, and hence, this term is the marginal utility
from the direct eﬀect of raising preventive expenditure. The first term summarizes the change in thehousehold’s consumption, which comes from two eﬀects. The first is a decrease in the worker’s andinvestor’s consumption because of the crowding out eﬀect, while the second is an increase in theinvestor’s consumption from the precautionary eﬀect.
For simplicity, hereafter, we assume the full depreciation of physical capital, that is, δ = 1. Then,

13

For notational simplicity, we omit the variable of the function such that B(g) → B if it causes no confusion.

15

from the calculation in Appendix A.2, we can obtain the optimal policy conditions as follows:[(1 − α))]γR − σc̄ασ[(1 − τ)B] + (R − σ) α+[(1 − τ)B′ − B] = 0,γR
(R − σ)γR
] 1−α
[(
σ )
− γ+
[(1 − α)(1 − τ)B]1−γ + (R − σ)−γ γR
R−σα(τ − h)[] F(h)+ γ(R − σ)−γ + [(1 − α)(1 − τ)B]1−γ (R − σ)−1 σ
= 0,
ρ
−γ

1−γ

−γ

(

(19)

(20)

where, for convenience, we define function F(h) as follows:
F(h) ≡ βp−D′.
(1 − D)γ(21)

Function F(h) indicates the marginal direct benefit from raising preventive expenditure, and this is adecreasing function since, from (7),()
D′D′′ (1 − D) + γD′ 2∂=−βpF (h) = −βp< 0.
∂h (1 − D)γ(1 − D)1+γ′

This means that the marginal direct benefit from raising preventive expenditure is decreasing. Hereafter, we assume the following to ensure that an inner solution of optimal preventive expenditure
exists.
Assumption 1 The damage ratio function (7) satisfies the following.
γ−1
R
F
+
> lim .
h→0 ρ
1 + limh→0 (R − σ) h→0 σ

α lim

Assumption 1 states that a marginal direct benefit under zero preventive expenditure must be suﬃciently large to satisfy the above expression. Otherwise, there may be a case in which it is optimal
not to invest in preventive expenditure since there is little marginal benefit even under zero preventive
expenditure. Note that imposing this assumption is, in fact, less restrictive than imposing the Inada
condition of the damage ratio function, limh→0 D′ (h) → ∞.
By simultaneously solving (19) and (20), we can obtain the optimal policy pair {τ∗ , h∗ }. Note thatthe value of the pair, if any, is state- and time- independent since the return on savings R is constantbecause of the AK production structure and hence, (19) and (20) do not contain any state variables.
From the budget constraint for the government (13), the optimal productive expenditure–production

16

ratio g∗ becomes τ∗ − h∗ . Given optimal policy π∗ , the resulting consumption and savings functionsarec∗w = (1 − τ∗ )(1 − α)B(g∗ )K,c∗I = [R∗ − σ(τ∗ , h∗ )]K with R∗ = (1 − τ∗ )αB(g∗ ), σ(τ∗ , h∗ ) = (R∗ ρ(h∗ ))1/γ(22)ã∗ = σ(τ∗ , h∗ )K with σ(τ∗ , h∗ ) defined above.

5

Optimal disaster-preventive expenditure

5.1

Existence and conditions of the optimal policy

At first glance, optimal conditions (19) and (20) are too complicated to deal with. However, these
conditions can be simplified as shown in the following proposition.
Proposition 2 Assume R−σ ≥ 0. Then, from (19), the optimal policy satisfies the following equation:(1 − τ)B′ = B ⇔ τ = αh + 1 − α → τ = τ(h).

(23)

By using (23), (20) becomes[] F
γ(R − σ)1−γ + [(1 − α)(1 − τ)B]1−γ zρ[]= (γ + z)[(1 − α)(1 − τ)B]1−γ + (R − σ)−γ γR

1,α(1 − h)(24)

where z = σ/(R − σ) is the savings–consumption ratio for the investor and all tax rates τ in (24) areevaluated at τ = τ(h). The optimal preventive expenditure h∗ satisfies (24).
Proof.
First, we prove that the first term of (19) must be positive under non-negative consumption R−σ ≥0. When we use the definition of c̄ and rewrite the first term of (19), it becomes
[

]
ασ
1+
[(1 − α)(1 − τ)B]1−γ + α(R − σ)−γ .
(R − σ)γR

Hence, if R − σ ≥ 0, this term is positive. Therefore, by dividing both sides of (19) by the first termof (19), we obtain (23).
Next, we show that under (23), (20) becomes (24). Considering B = (Ag1−α )1/α , the first equationof (23) implies the second one and this implies g = (1 − α)(1 − h) and (1 − τ) = α(1 − h). By defining17

z ≡ σ/(R − σ) as the investor’s savings–consumption ratio and substituting τ − h = (1 − α)(1 − h) andz into (20), we obtain (24). The first condition, (1 − τ)B′ = B, implies that the government chooses τ such that a marginalincrease in the household’s total income from more productive government expenditure is equal to amarginal decrease in that from raising the tax rate. According to (23), if h = 0, τ = g = 1 − α, whichis equivalent to the result of Barro (1990); in addition, when the government finances one unit ofpreventive expenditure, α% is collected by raising the tax rate and (1 − α)% is collected by decreasing
productive expenditure. The second equation of Proposition 3, (24), which is the most important
equation in this model, summarizes the marginal benefit and disutility from preventive expenditure.
The left-hand side (LHS) of (24) is the marginal utility and the right-hand side (RHS) is the marginal
disutility from preventive expenditure. The LHS of (24) summarizes the marginal utility from two
eﬀects, namely the precautionary eﬀect and a net direct eﬀect. The first term implies a marginalincrease in utility from an increase in the investor’s consumption since higher preventive expenditurecauses an investor to shift savings to consumption owing to the precautionary eﬀect. The second
term, the net direct eﬀect, consists of two terms. One is the increase in utility from the direct eﬀect,
while the other is the marginal disutility from a decrease in savings because of the precautionaryeﬀect. Interestingly, the sum of these two eﬀects becomes positive, and hence, it is categorized on
the LHS and we refer to it as net since it includes the marginal disutility from the precautionaryeﬀect. 14 The RHS of (24) is marginal disutility, which comes from the crowding out eﬀect. The
first term is the decrease in the worker’s consumption, the second is the decrease in the investor’s
consumption, and the third term is the decrease in the investor’s savings because of the crowding out
eﬀect. 15 By equalizing the marginal benefit and cost of raising preventive expenditure, we obtain

14
A marginal decrease in utility from a decrease in savings because of the precautionary eﬀect is captured by c̄σ(1 −γ)/γ(R − σ) ∗ F/ρ, while a marginal increase in utility from the direct eﬀect is given by c̄σ/(R − σ) ∗ F/ρ. Hence, the sumof these two terms becomes c̄σ/γ(R − σ) ∗ F/ρ > 0. See Appendix A.2 for why these terms represent the precautionaryeﬀect and direct eﬀect.
15
If logarithmic utility is adopted, which is the case of γ = 1, (24) can be rewritten as the following simple form.

F(h) =1.
α(1 − h)We obtain such a simple form under γ = 1 as the precautionary eﬀect does not exist. In such a case, at the same time, thedirect eﬀect and crowding out eﬀect are not as complex as in this model. In fact, in the above equation, the LHS implies a
direct eﬀect, while the RHS implies a crowding out eﬀect.

18

the optimal preventive expenditure ratio. The presence of the precautionary eﬀect implies that, if wewere to consider the optimal prevention under the static model or instantaneous logarithmic utility,
underinvestment in prevention in contrast to under CRRA utility could occur.
The important point is whether there exists h∗ that satisfies (24). To identify the existence of theroot of (24), we define R(h) ≡ (1−τ(h))αB(g(h)), where g(h) ≡ (1−α)(1−h) and impose the following
assumption.
Assumption 2 When h = 0, the investor’s consumption is positive, that is,
R(0)

γ−1

{ [
] 1 }γ−12
1−α α> ρ(0), ⇔ α A(1 − α)> β(p(1 − D(0))1−γ + 1 − p).

This assumption should be satisfied since if h = 0, this model is similar to Barro’s (1990) model;
however, even in this case, the investor’s consumption should be positive. By combining Assumptions
1 and 2, we have the following proposition considering the existence of h.
Proposition 3 Given Assumptions 1 and 2, at least one root of (24), h∗ , exists, and this is less than ĥ,
where ĥ satisfies R(ĥ)γ−1 = ρ(ĥ).
Proof. See Appendix A.3.
Proposition 3 guarantees optimal preventive expenditure h∗ . Hence, under the optimal policy, theinvestor’s consumption must be positive since h∗ < ĥ. There are probably multiple candidates for h∗ ,but we do not treat this problem hereafter. Once h∗ is determined, the optimal tax rate is given byτ∗ = αh∗ + 1 − α and the optimal productive government ratio is g∗ = (1 − α)(1 − h∗ ) from (23).

5.2

Comparative statics with respect to disaster probability

Of the parameters in equation (24), we focus only on disaster probability p. Then, we can obtain thefollowing proposition.
Proposition 4 The optimal preventive expenditure ratio h∗ increases when disaster probability p
increases. That is, we have the following sign of the derivative:
∂h∗
> 0.
∂p

19

(25)

Proof. See Appendix A.4.
Although this is an intuitive result, the mechanism is somewhat complex. Recall that the optimal
condition for preventive expenditure can be divided into the direct, precautionary, and crowding out
eﬀects. An increase in disaster probability influences all three eﬀects. Hereafter, we consider theeﬀect of a rise in p. However, before considering these eﬀects, it would be useful to revisit the eﬀectof higher probability on the investor’s behavior. Higher disaster probability decreases the expectedintertemporal income of the investor, thereby causing his/her consumption and savings to decrease.
We call this eﬀect the income eﬀect of higher probability. On the contrary, in the event of a disaster,
the investor saves more than previously, meaning that his/her consumption decreases, whereas his/hersavings increase as disaster probability rises. We call this eﬀect the substitution eﬀect.
• The direct eﬀect
When disaster probability increases, the direct eﬀect also rises. In such a case, an income eﬀect
arises and the household’s savings and consumption decrease. Then, the favored policy is theone that can mitigate the expected damage of natural disasters since higher marginal utility
is derived owing to smaller consumption and savings and the concavity of the investor’s valuefunction. That is, since the investor experiences a more severe situation without any prevention,
it is optimal to invest in preventive expenditure through the direct eﬀect.
• The precautionary eﬀect
Higher disaster probability causes this eﬀect to rise, and thus the optimal preventive expenditure
increases. This eﬀect comes from the substitution eﬀect, that is, the rise in p shifts consumption
to savings owing to the precautionary eﬀect. Then, relatively scarce consumption makes themarginal utility of consumption increase because of the concavity of the utility function. Since
the government knows there is an increase in marginal utility from consumption, it invests inpreventive expenditure so that the investor reallocates savings to consumption.
• The crowding out eﬀect
Higher disaster probability worsens the crowding out eﬀect, which stems from the income effect. It decreases the investor’s expected income, and hence, consumption and savings decline.
20

Now that consumption and savings are lower than previously, the marginal cost of giving up
one unit of consumption and savings increases and the crowding out eﬀect is magnified.
As p increases, there is an increase in the marginal benefit, consisting of the direct and precautionary eﬀects, while there is also an increase in the marginal cost, which is the crowding out eﬀect.
16 Thus, the sign of ∂h∗ /∂p could be ambiguous. However, as shown in Appendix A.4, the first two

positive eﬀects must dominate the third negative eﬀect under the optimal policy. Moreover, since theoptimal preventive expenditure rises as disaster probability increases, the optimal tax rate increases∗∗dhby α ∗ dh
d p from (23), and productive government expenditure decreases by −(1 − α) ∗ d p .

5.3

Expected growth rate

Since this model eventually reduces to the AK model, endogenous growth arises. This model has astochastic disaster shock, and hence, we evaluate the growth rate with expectations. The expectedeconomic growth rate is defined as the growth rate of production, which is equivalent to that of thephysical capital stock according to the AK construction. The savings rate per unit of physical capitalis given as σ(τ∗ , h∗ ) under the optimal policy, and the expected growth rate is
Egt (h∗ ) = Et=

Kt+1 − Kt
Kt

p(1 − D(h∗ ))σ(τ∗ , h∗ )Kt + (1 − p)σ(τ∗ , h∗ )Kt − Kt
= (1 − pD(h∗ ))σ(τ∗ , h∗ ).
KtFurthermore, since R∗ is rewritten as α2 A1/α (1 − α)(1−α)/α (1 − h∗ )1/α ≡ Rc (1 − h∗ )1/α , by using thedefinitions of σ and ρ, we obtain the expected growth rate as
{
[
]}1/γ
Egt = Eg = (1 − pD(h∗ )) βRc (1 − h∗ )1/α p(1 − D(h∗ ))1−γ + 1 − p.

(26)Hereafter, as an important form of comparative statics, we examine how an increase in disasterprobability aﬀects the growth rate. Before considering the case of CRRA utility, for reference, we
consider the logarithmic utility case (γ = 1). In this case, the precautionary eﬀect vanishes and the
expected growth rate is given as follows:
Eg = (1 − pD(h∗ ))βRc (1 − h∗ )1/α .
16

In the first half of Appendix A.4, we show this result mathematically by using equation (24).

21Higher preventive expenditure aﬀects the expected growth rate through the following two channels.
The first eﬀect is that an increase in disaster probability aﬀects the first term (1 − pD(h∗ )). If p
increases, the expected loss increases given h∗ , while the damage caused by the disaster decreases
since preventive expenditure increases as p rises. Thus, the eﬀect of an increase in p on (1 − pD(h∗ ))is ambiguous: if the eﬀect of the decrease in damage dominates that of the increase in the expected
loss, (1 − pD(h∗ )) increases and vice versa. Whether (1 − pD(h∗ )) is increasing in p thus cruciallydepends on the shape of the damage ratio function. The second channel is the crowding out eﬀect,that is, greater prevention reduces the term (1 − h∗ )1/α . This is because greater prevention requires
more tax revenue and a decrease in productive government expenditure, each of which decreases
the return on capital R and reduces savings. This leads to a reduction in the expected growth rate.
Thus, if (1 − pD(h∗ )) is decreasing in p, the economic growth rate is also decreasing in p, while if(1 − pD(h∗ )) is increasing in p, there is an inverse U-shaped relationship between disaster probability
and the economic growth rate.
In the CRRA utility case, there is another eﬀect in addition to those shown in the logarithmic
utility case. The channel stems from the precautionary savings motive. As p increases, an investor
saves more in case of disaster, which enhances economic growth. On the contrary, s/he reduces
precautionary savings since higher p accompanies higher preventive expenditure and this decreases
the expected damage. The lower savings result in a decrease in the growth rate. Thus, generally,
whether precautionary savings increase is ambiguous: if precautionary savings increase (decrease) as
p rises, this eﬀect enhances (hampers) economic growth. In fact, in (26), the following five channelsof the eﬀect of an increase in p exist:


1/γ










∗ 
∗ 1/α ∗ 1−γ
Eg = (1 − p D(h )) 
βR
(1
−
h
)
p
[(1
−
D(h
))
−1]
+
1
.



c





| {z } |{z} |{z}|{z} |{z} 




(i)

(ii)

(iii)

(iv)

(27)

(v)

Thus, in the CRRA utility case, the positive eﬀects of an increase in disaster probability on the expected economic growth rate are twofold: (ii) a decrease in the expected damage because of anincrease in preventive expenditure and (iv) an increase in precautionary savings through a rise in disaster risk. On the contrary, the negative eﬀects are threefold: (i) an increase in the expected loss,(iii) the crowding out eﬀect caused by financing the higher preventive expenditure ratio, and (v) the22

decrease in precautionary savings through a decline in the expected economic loss because of thehigher preventive expenditure ratio. If the former two eﬀects dominate the latter three, Eg increases
as p rises, and vice versa.
Since the overall eﬀect of disaster probability on the economic growth rate is complicated, in what
follows, we resort to a numerical example. We focus on one state in the United States by using thedata presented in Figure 1(b), and we take one period as one quarter (three months). We assume thatthe damage function is D(h) = 1−d̂ d̂ − h−d̂ d̂ as in footnote 9. We set the benchmark parameter values as
β = (0.95)1/4 ≃ 0.987, γ = 2, α = 1/3, A = 4.076, p = 14/75 ≃ 18.7%, and d̂ = −0.25. We set
state-average quarterly disaster probability as about 18.7% since natural disasters occurred 28 times
in the United States in 2015 according to Figure 1(b), and thus average quarterly disaster probability
per state is 28/(50 × 3) ≃ 0.187. 17 The other parameter values except for A and the parameterson the damage ratio function are the usual parameter values. The value of d̂ is arbitrarily selectedas D(0) = 0.8, that is, when there is no prevention, 80% of physical capital is lost when disasters
occur. Total factor productivity A is set as the resulting economic growth rate, which becomes about(1.02)1/4 ≃ 1.005. Further, we consider the optimal preventive expenditure and expected growth rate
under the values of disaster probability from 5% to 55% in steps of 5 percentage points in addition
to 18.7%. Under this parameter, the probability that exceeds 60% entails no investor’s consumption,
R(0) − σ(0) < 0, which violates Assumption 2, and hence, we focus on a probability less than 55%.
In this setting, Assumptions 1 and 2 hold, and the optimal preventive expenditure under p = 0.187 isaround 10.79%.
Figures 2(a) and 2(b) are scatter diagrams between disaster probability and the optimal preventive
expenditure and between disaster probability and the expected growth rate, respectively. The values ofp and h∗ are measured by percentage and Eg is derived from (26). The optimal preventive expenditureis given by the value for which (24) holds, and the optimal preventive expenditure–production ratio
ranges from 3.5% at p = 5% to 22.15% at p = 55% as probability increases, as in Figure 2(a).
As disaster probability increases, the optimal preventive expenditure rises, which is consistent with
Proposition 4. The corresponding expected economic growth rate, which is derived from (26) with17

We assume that natural disasters independently and identically occur in each state and do not aﬀect more than one state
for simplicity.

23

(a) Optimal preventive expenditure(b) Expected growth rate

Figure 2: Results of the numerical example
the optimal preventive expenditure–production ratio, is given as in Figure 2(b). The interesting pointis that the expected growth rate has an inverse U-shaped relationship with disaster probability in thisparameter setting. The two positive eﬀects dominate the three negative eﬀects at low probability,
while the inverse is true at high probability. Until disaster probability reaches the threshold, which isaround 40%, the economic growth rate rises continuously and vice versa.
This result is consistent with the empirical results that the eﬀect of disaster risk on economic
growth is empirically ambiguous. According to Albala-Bertrand (1993) and Skidmore and Toya
(2002), disaster risk increases with economic growth. On the contrary, Loayza et al. (2009) andNoy (2009) show that disaster risk is positively related to economic growth in some countries andnegatively related to economic growth in others. According to this model’s result, for low disaster
risk, the former positive relationship is supported since higher disaster risk entails a higher economic
growth rate. In addition, the latter negative relationship is realized in this model. In a country withlow (high) disaster risk, there is a positive (negative) relationship between risk and economic growth.

5.4 Discussion on disaster insurance
In this subsection, we discuss the case in which a perfectly competitive disaster insurance market
exists and assess the eﬀect on a household’s behavior. In particular, we consider the following disasterinsurance: if an investor pays qt × θt × at+1 to an insurance company, his/her asset is compensated24

by θt × at+1 from the company in case of disaster; in other words, qt denotes the insurance price andθt denotes the insurance coverage. An investor can choose coverage θt to maximize his/her lifetime
utility. Then, the resulting asset st+1 becomes as follows:[(1 − D(ht ))(1 − qt θt ) + θt ]at+1
st+1 = (1 − qt θt )at+1

with probability pwith probability 1 − p.

(28)

Since the insurance market is competitive, the expected profit of the insurance company becomes
zero, that is, qt = p. If the disaster insurance can perfectly insure an investor’s asset from disasterrisks, the resulting asset is equalized with and without disaster occurrence. That is, when we define
perfectly insuring coverage as θ̂, it must satisfy(1 − D(h))(1 − qθ̂) + θ̂ = 1 − qθ̂ ⇔ θ̂ =D(h).
1 + D(h)q(29)

Here, we assume that the insurance company can oﬀer coverage to the maximum of ϕθ̂, where ϕ ∈[0, 1] captures the incompleteness of disaster insurance reflecting measurement errors and so on.
ϕ = 1 means perfect disaster insurance, while ϕ = 0 means no disaster insurance. Moreover, as
described in footnote 3, for the top 10 natural disasters ordered by economic loss, ϕ ≃ 0.25. In thissituation, we can obtain the following proposition.
Proposition 5 If there is an insurance market in the economy, an investor chooses θt = ϕθ̂ andsavings are determined asã =(ρϕ R)1/γ
1 − pϕθ̂s ≡ σϕ s,(30)

where ρϕ is defined as () 1 + (1 − ϕ)D(h)[p(1 − D(h)) − 1] 1−γρϕ ≡ β  p+ 1 − p ,1 + (1 − ϕ)D(h)p(31)

and σϕ ≡ (ρϕ R)1/γ /(1 − pϕθ̂). In particular, if ϕ = 1 (perfect insurance), ρϕ = β and if ϕ = 0 (no[]disaster insurance), ρϕ = β p(1 − D(h))1−γ + 1 − p = ρ.
Proof. See the first half of Appendix A.5.
If ϕ = 0, which is the same case as in the previous section, Proposition 5 corresponds to Proposition
1. Otherwise, an investor demands the upper limit of coverage because s/he is risk averse and somedisaster risks can be diversified through the disaster insurance. In particular, when ϕ = 1, ρ = β and25

the investor’s resulting savings become s̃ = (βR)1/γ s irrespective of the occurrence of disasters sinceincome risk is perfectly diversified.
Then, how about the comparative statics of savings rate σϕ with respect to p and h? In thissetting, in addition to the precautionary eﬀect, there is another eﬀect of raising p and h on σϕ . Whenp increases, insurance payment pϕθ̂ increases through a rise in the insurance price. On the contrary,
if h increases, demand for insurance and insurance payment decreases since disaster risk decreases.
As a result, we have the following signs of the derivatives, although there is an extra eﬀect throughthe insurance market:
dσϕ
> 0,
dp

dσϕ
< 0.
dh

In summary, the signs are the same as those presented in the previous subsection, although thevalues of the derivatives are more complicated. 18 To consider these results, let us call dρϕ /di for
i = {p, h} the modified precautionary eﬀects and d(pθ̂)/di for i = {p, h} the insurance eﬀects. Then,
we have dρϕ /d p > 0 and the modified precautionary eﬀect increases savings as p increases, whiledρϕ /dh < 0 and the modified precautionary eﬀect decreases savings as h increases, which holds
from the same reasoning as before. In terms of the insurance eﬀects, we have ϕd(pθ̂)/d p > 0 and
ϕpdθ̂/dh < 0. When p increases, aside from the modified precautionary eﬀect, the insurance payment
increases, and in turn, net saving (saving after paying the insurance fee) decreases given the same ã;hence, to sustain the same level of net saving, ã must increase. The inverse relation holds in the caseof an increase in h. Thus, both the modified precautionary eﬀect and the insurance eﬀect increase(decrease) ã as p (h) increases. Since ϕ is small in reality and the values of the derivatives are toocomplicated, we impose ϕ = 0 to derive the optimal policy conditions.

6 Conclusion
In this study, we construct a macroeconomic model in which disasters destroy the existing capitalstock stochastically and we obtain an optimal policy, consisting of a tax rate, disaster-preventive expenditure, and productive government expenditure. In this model, the optimal policy conditions andthe existence of the optimal policy are shown in Propositions 2 and 3. According to the condition for18

See the latter half of Appendix A.5 for the values of the derivatives.

26

optimal preventive expenditure (24), the government must choose its optimal disaster-preventive expenditure such that the marginal utility from preventive expenditure, which is a net direct eﬀect and a
precautionary eﬀect, is equal to the marginal disutility from the crowding out eﬀect. The precautionary eﬀect is the eﬀect in which higher preventive expenditure decreases a household’s precautionarysavings through a decrease in disaster risk, and the government optimally sets the preventive expenditure to reallocate the precautionary savings into consumption. Since we assume the CRRA utility
function and that disaster risks cannot be diversified, the precautionary eﬀect arises. Furthermore, inthe comparative statics with respect to disaster probability, we obtain a positive relationship between
the optimal preventive expenditure and disaster probability. Higher disaster probability strengthens
the direct and precautionary eﬀects, and so does the crowding out eﬀect. Although the marginal benefit of preventive expenditure and marginal cost of preventive expenditure increase simultaneously, we
can show that the marginal benefit necessarily dominates the marginal cost. When we consider the
relationship between the expected growth rate and disaster probability, we decompose the eﬀect of
an increase of disaster probability into five channels: two positive eﬀects and three negative eﬀects.
Furthermore, we observe an inverse U-shaped relationship in the presented numerical example. Sucha non-monotonic relationship between disaster risk and economic growth is supported by the findingsof empirical studies such as Loayza et al. (2009) and Noy (2009).
Finally, we present some extensions of this model. One important extension would be to introduce mitigation in addition to prevention. Mitigation can reduce disaster probability and it would
be interesting to investigate whether each policy is substitutive or complementary. This model may
suggest that each policy is substitutive since lower disaster probability leads to less preventive expenditure as the comparative statics show. The other extension would be to make this model more usefulby using a numerical method. For example, defining government expenditure and preventive expenditure as stock variables and introducing government debt as a safe asset could alter or strengthen
our results. Moreover, it would be interesting to introduce technical change and human capital as
disaster-proof capital. These modifications would lead to many endogenous variables, and this would
be diﬃcult to solve by hand; hence, we would have to resort to a numerical calculation. Finally, as
briefly discussed in Subsection 5.4, the optimal policy in the disaster insurance market should also be27

considered. These research directions are left to future studies.

AppendicesA.1 Derivation of equation (10) from (9)Equation (9) is rewritten as{}(Rs − ã)1−γ
V(s) = max+ βpV((1 − D(h))ã) + β(1 − p)V(ã) .
ã
1−γ

(A.1)

1
Guess that V(s) = 1−γ(α1 s)1−γ , where α1 is an undermined variable. By substituting this into (A.1),

we obtain{}(α1 s)1−γ(Rs − ã)1−γ(α1 (1 − D(h))ã)1−γ(α1 ã)1−γ
= max
+ βp
+ β(1 − p)
.
ã
1−γ
1−γ
1−γ
1−γ

(A.2)

The first-order condition with respect to ã is(Rs − ã)−γ = βα1 (p(1 − D(h))1−γ + 1 − p)ã−γ ≡ α1 ρ(h)ã−γ ,
1−γ

1−γ

where ρ(h) ≡ β(p(1 − D(h))1−γ + 1 − p) is a risk-adjusted discount rate. Hence,ã = (β1 + 1)−1 Rs,−(1−γ)/γ

where β1 ≡ α1

(A.3)

ρ(h)−1/γ . Substituting (A.3) into (A.2) and multiplying (1 − γ) on both sides leadto{}
(α1 s)1−γ = (Rs − (β1 + 1)−1 Rs)1−γ + βp[α1 (1 − D(h))(β1 + 1)−1 Rs]1−γ + β(1 − p)[α1 (β1 + 1)−1 Rs]1−γ .
−γThe RHS of this equation can be simplified to R1−γ β1 (β1 + 1)γ s1−γ , and by equating the coeﬃcients,
1−γ

α1

−γ

= R1−γ β1 (β1 + 1)γ .

By solving this equation with respect to α1 , we can obtain()−γ/(1−γ)α1 = R 1 − R(1−γ)/γ ρ(h)1/γ.
Under this value, β1 becomes
β1 = R−(1−γ)/γ ρ(h)−1/γ − 1.
Therefore, by substituting this value into (A.3), savings function (10) is derived, that is,ã = (Rρ(h))1/γ s.
28

A.2 The derivation of equations (19)–(20) from (16)Equation (16) is, when we consider an expectation,{}c̄V g (K) = max
K 1−γ + βpV g ((1 − D)ã) + β(1 − p)V g (ã) ,π
1−γ
subject to ã = σK, g = τ − h, given K0 .
By substituting the second constraint g = τ − h into B(g), this problem has only two endogenous variables τ and h. Then, the first-order conditions with respect to τ and h are obtained by diﬀerentiating(16), setting it to be equal to zero, and dividing both sides by K,
1
∂c̄
∂σ
∂σ
K −γ + βpV ′g ((1 − D)ã)(1 − D)+ β(1 − p)V ′g (ã)
= 0,
1−γ
∂τ
∂τ
∂τ
[
]
1
∂σ
−γ ∂c̄′gK+ βpV ((1 − D)ã) (1 − D)
1−γ
∂h
∂h
− βpV ′g ((1 − D)ã)D′ (h)σ + β(1 − p)V ′g (ã)

∂σ
= 0.
∂h

which are the same as (17) and (18). To obtain more tractable forms of these equations, we must
α2obtain a closed-form of the value function of V g (K). Here, we guess that V g (K) = 1−γ
K 1−γ + α3

for some undetermined variables α2 and α3 . Then, at the optimal policy, the derivatives of the valuefunction with respect to capital are given by∂V g (K)= α2 K −γ = c̄K −γ + β[p(1 − D)1−γ + 1 − p]α2 σ1−γ K −γ = c̄K −γ + βρα2 σ1−γ K −γ .
∂K
Therefore, by dividing both sides by K −γ and solving this equation with respect to α2 ,
α2 =

c̄
R
= c̄
> 0.
R−σ
1 − ρσ1−γ

(A.4)

α2
Hence, we obtain the value function as V g (K) = 1−γ
K 1−γ + α3 , where α2 is defined as (A.6). By

substituting this value function into the first-order conditions (17) and (18) and dividing both sides by
K −γ , we can obtain such equations as
1 ∂c̄
c̄ ∂σ
+
= 0,
1 − γ ∂τ R − σ ∂τ

(A.5)

1 ∂c̄
c̄ ∂σ
c̄σ F(h)++= 0,1 − γ ∂h R − σ ∂h R − σ ρ

(A.6)

29

where we use α2 ρσ−γ = c̄/(R − σ) and F(h) defined as (21). Next, we calculate a derivative of savingrate σ and c̄ with respect to τ and h. Each derivative is derived as follows:∂σ σ ∂R 1 ασ==[(1 − τ)B′ − B],∂τγ ∂τ R γR[][]∂σ σ ∂R 1 ∂ρ 1
σ
B′
F(h)
=
+
=
−(1 − τ)α + (1 − γ),∂h
γ ∂h R ∂h ρ
γ
R
ρ
[
]
∂c̄
∂
1−γ ∂
1−γ
1−γ
= (1 − α)[(1 − τ)B] + (R − σ)
∂τ
∂τ
∂τ
[
[
]]
∂σ
1−γ
−γ
′
−γ ∂R
= (1 − γ) (1 − α) [(1 − τ)B] [(1 − τ)B − B] + (R − σ)
−
∂τ ∂τ
)]
[
(
γR − σ1−γ−γ−γ[(1 − τ)B′ − B],= (1 − γ) (1 − α) [(1 − τ)B] + (R − σ) α
γR
[
]
∂c̄
∂
1−γ ∂ 1−γ
1−γ
= −[(1 − τ)(1 − α)]
B + (R − σ)
∂h
∂h
∂h
]]
[
[
∂σ
1−γ −γ ′−γ ∂R−= (1 − γ) −[(1 − τ)(1 − α)] B B + (R − σ)
∂h ∂h
[
[
(
)
]]
1 − γ F(h)1−γ −γ ′−γ′ σ − γR= (1 − γ) −[(1 − τ)(1 − α)] B B + (R − σ) (1 − τ)αB
−
σ
,
γR
γ
ρ
where note that ∂R/∂τ = α[(1−τ)B′ −B], ∂R/∂h = −(1−τ)αB′ , and ∂ρ/∂h = (1−γ)F(h). Furthermore,
if we assume δ = 1 as in the latter half of Subsection 4.4, R = (1 − τ)αB and B′ /B = (1 − α)/[α(τ − h)]from B ≡ (A(τ − h)1−α )1/α . Then, we obtain
[
[
]]
∂c̄
1−α
1 − γ F(h)1−γ 1 − α−γ= (1 − γ) −[(1 − τ)(1 − α)B]+ (R − σ)(σ − γR) −σ∂hα(τ − h)γα(τ − h)γρ
By substituting the derivatives with respect to the tax rate into (A.5), we obtain[()]
γR − σ
c̄ ασ
1−γ
−γ
−γ
(1 − α) [(1 − τ)B] + (R − σ) α[(1 − τ)B′ − B] +[(1 − τ)B′ − B]γR
R − σ γR
)
]
[
(
c̄ ασ
γR − σ1−γ−γ−γ+[(1 − τ)B′ − B] = 0.
= (1 − α) [(1 − τ)B] + (R − σ) αγRR − σ γR
This is the same as (19). Similarly, by substituting ∂σ/∂h and ∂c̄/∂h into (A.6),
[
]
1−α
1 1−α
F(h)
− [(1 − τ)(1 − α)B]1−γ+ (R − σ)−γ
(σ − γR) − (1 − γ)σα(τ − h)γ α(τ − h)
ρ
[
]
1−α
F(h)
c̄σ F(h)c̄ σ−+ (1 − γ)+= 0,+
R − σ γ α(τ − h)
ρ
R−σ ρ
30

ܴሺ݄ሻఊିଵ

ߩሺ݄ሻ

݄෠

ͳ

݄

Figure 3: The positional relationship between R(h)γ−1 and ρ(h)where note that −(1 − τ)αB′ /R = −(1 − α)/(α(τ − h)) in the first term in the bracket of the second
term. Rearranging these terms and multiplying by γ on both sides lead to[c̄σ ] 1 − α−γ[(1 − τ)(1 − α)B]1−γ + (R − σ)−γ (σ − γR) −R − σ α(τ − h)[c̄ ] F(h)− (1 − γ)(R − σ)−γ −
σ
= 0.
R−σ
ρ
By using the definition of c̄ ≡ [(1 − α)(1 − τ)B]1−γ + (R − σ)1−γ , each term is simplified to
[(
] 1−α
σ )
− γ+
[(1 − α)(1 − τ)B]1−γ + (R − σ)−γ γR
R−σα(τ − h)[] F(h)= 0.
+ γ(R − σ)−γ + [(1 − α)(1 − τ)B]1−γ (R − σ)−1 σρThis equation corresponds to (20).

A.3 Proof of Proposition 3
First, we show that under Assumption 2, R(h)γ−1 and ρ(h) intersect only once in h ∈ (0, 1). When weevaluate them at h = 1, we obtain R(1)γ−1 = 0 and ρ(1) = β(p(1 − D(1))1−γ + 1 − p) > 0. Then, thefirst and second derivative of ρ with respect to h are given byρ′ (h) = βp(1 − D(h))−γ (−D′ (h)) < 0,[]
γD′ (h)D′′ (h)′′′ρ (h) = ρ (h)+> 0.
1 − D(h) D′ (h)31Hence, ρ(h) is a decreasing and concave function. On the contrary, the first and second derivatives ofR(h)γ−1 ≡ G(h) are given by
1 − γ G(h)
< 0,
α 1−h
(
)
1 + α − γ G′ (h)G′′ (h) =R 0 ⇔ 1 + α Q γ.
α1−hG′ (h) =Thus, R(h)γ−1 is a decreasing function and whether it is convex or concave depends on the parameter.
Figure 3 shows that R(h)γ−1 and ρ(h) cross only once regardless of the concavity of G(h) and hence,
we can show that there must be one intersection. Next, we show that there is at least one root of (24)that is less than ĥ, which satisfies R(ĥ)γ−1 = ρ(ĥ). Equation (24) is written as
[
] [
]
F
1
F
γ(R − σ)−γ σ − αB = (γ + z)
− z [(1 − α)α(1 − h)B]1−γ ,ρα(1 − h)ρ

(A.7)

where note that 1 − τ = α(1 − h) and R/(α(1 − h)) = αB. In (A.7), we show that limh→0 LHS >limh→0 RHS from Assumption 1. Hereafter, to see the values of the LHS and RHS at h → ĥ, we
check the properties of ĥ. At h = ĥ, the followings is true:
1. R = σ, ⇔ z → ∞,
2.

∂ γ−1
∂h R

RF
> ∂ρ
∂h , ⇔ αB > ρ .

See Figure 3 for the second statement, in which the slope of Rγ−1 is steeper than that of ρ at h = ĥ.
Considering these facts, we see what the LHS and RHS of (A.7) become at h → ĥ. At h = ĥ, thefollowing holds:−γ

lim LHS = γ lim (R − σ)

h→ĥ

h→ĥ

]
F
∗ lim σ − αB .
ρ
h→ĥ
[

The first term becomes infinity since (R − σ) → 0 from the first property of ĥ and γ > 0. The second
term becomes negative constant at h → ĥ since
[

]
F
F
RF
lim σ − αB = lim σ lim − α lim B = lim− α lim B < 0,
ρh→ĥh→ĥ h→ĥ ρh→ĥh→ĥ ρ
h→ĥwhere the second equality holds from the first property of ĥ and the first inequality holds from thesecond property of ĥ. Therefore, at h = ĥ, the LHS is negative infinity, that is, limh→ĥ LHS → −∞.

32

The limit of the RHS is derived as follows. First, we use the following expression:
(
)
(
)
(
)
1
F
1
F
z
R
RF
z
RF
z
−z =z
−
=
−
=
αB −
.
α(1 − h)ρα(1 − h) ρR α(1 − h)ρRρSecond, we rewrite the RHS of (A.7) as
[
(
)]
γ
z
RF
αB −
[(1 − α)α(1 − h)B]1−γ .
+α(1 − h) RρThen, as h converges ĥ, the RHS of (A.7) becomes
[ (
[
)]]
γ
z
RF
+ lim
lim
αB −
lim [(1 − α)α(1 − h)B]1−γ .
α(1
−
h)
R
ρ
h→ĥ
h→ĥ
h→ĥ
γ
Since limh→ĥ α(1−h)
and limh→ĥ [(1 − α)α(1 − h)B]1−γ converge to some positive constant, we have)][ (. Its limit is
only to check the value of limh→ĥ Rz αB − RF
ρ
(
)
z
RF
lim ∗ lim αB −→ ∞,ρh→ĥ R h→ĥ|{z} |{z}→ ∞positive constantwhere we use the fact that the second term converges to a positive constant with the second propertyof ĥ. These limits imply that the LHS is larger than the RHS at h = 0, while the RHS is larger thanthe LHS at h = ĥ. Since the LHS and RHS are continuous in h ∈ (0, ĥ), then there must be at least
one intersection with domain h ∈ (0, ĥ).



A.4 Sign of the derivative of the optimal preventive expenditure with respect to disasterprobability
We use equation (24) when we conduct the comparative statics. First, we can obtain the following
derivatives under fixed preventive expenditure.
∂ρ= β[(1 − D)1−γ − 1] > 0,
∂p
∂z
1 ∂ρ
= (1 + z)z
> 0,
∂p
γρ ∂p

∂σ
σ ∂ρ
=
> 0,
∂p γρ ∂p∂ F
βF
= 2 > 0,
∂p ρ
pρ

∂
z ∂ρ
(R − σ)−γ = (R − σ)−γ
> 0.
∂p
ρ ∂p

Then, the eﬀect of p rising on each term of equation (24) is obtained as follows:




γ (R − σ)1−γ +[(1 − α)(1 − τ)B]1−γ  z F ρ
 | {z }
|{z}
⊕
⊕




1


1−γ
−γ
,
= (γ + |{z}z )[(1 − α)(1 − τ)B] + (R − σ) γR| {z }  α(1 − h)

⊕

⊕

33

where ⊕ means that p rising increases the corresponding term. The above-mentioned five derivatives
imply that when p increases, the LHS of (24) must rise. To establish why the LHS shifts upward,
recall that the LHS represents the marginal utility of raising preventive expenditure and that it consistsof a precautionary eﬀect and a net direct eﬀect. Each eﬀect increases with p rising. On the contrary,
the eﬀect of a rise in p on the RHS is also positive. Since the RHS indicates the marginal cost ofraising preventive expenditure, that is, a crowding out eﬀect, a higher RHS indicates less preventive
expenditure. Note that p rising does not aﬀect the worker’s consumption at all since there is no
precautionary eﬀect or income eﬀect for the worker. The first (second) eﬀect on the RHS reflects a
crowding out eﬀect that decreases the investor’s consumption (savings). Thus, seemingly, there is an
ambiguous relationship between the optimal preventive expenditure h∗ and disaster probability p. In
what follows, by integrating these eﬀects, we check whether ∂h∗ /∂p is positive or negative. Aftersome calculations, the sign of optimal preventive expenditure with respect to p is obtained as follows.
19

∂h∗
R0⇔
∂p
(A.8)
[(
)
]
F αB
αB
βF
−γ
1−γ ∂ρ z(γ − 1)σ +(R − γσ) (R − σ) +[(1 − α)α(1 − h)B]+ c̄z 2 R 0.
ρ
σ
σ
∂p ρ
pρ
(
)
Recall that ∂ρ/∂p is positive. At first glance, the term (γ − 1)σ Fρ + αB
σ (R − γσ) could be negative
since the first term is positive, while the second term could be negative. However, under the optimal
policy, we can show that this term is also positive. To show this, since R > σ, the following holds:
[
]
F αB
F αB
F
(γ − 1)σ +(R − γσ) > (γ − 1)σ +(σ − γσ) = (γ − 1) σ − αB .
ρ
σ
ρ
σ
ρ
By multiplying both sides of (24) by R and solving this equation for σF/ρ, we obtainσF(γ + z)[(1 − α)(1 − τ)B]1−γ + (R − σ)−γ γR=
αB ≡ XαB.
ρ[γ(R − σ)1−γ + [(1 − α)(1 − τ)B]1−γ ](1 + z)

Since we can show that X ≥ 1, under the optimal policy h = h∗ ,
σ

F
> αB.
ρ

19
To obtain (A.8), we move the RHS of (24) to the LHS of (24) and diﬀerentiate the LHS with respect to disasterprobability p. Now that the resultant LHS indicates the diﬀerence between the marginal utility and marginal disutility frompreventive expenditure h, if this term increases as p increases, greater preventive expenditure must increase in response top. Inequality (A.8) is such a condition.

34

Therefore, the first term of (A.8) is positive and this means that the LHS of (A.8) must be positive.
Consequently, we can show that∂h∗
> 0. 
∂p

A.5. The proof of Proposition 5
To obtain the results of Proposition 5, we must solve the following dynamic optimization problem:{ I1−γ}[]c
V(s) = max+ βpV ((1 − D)(1 − qθ) + θ)ã + β(1 − p)V((1 − qθ)ã) ,θ∈[0,ϕθ̂],ã 1 − γ(A.9)

s.t. (1 − qθ)ã = Rs − cI , given s0 .
By following a similar procedure to that described in Appendix A.1, given θ, we can obtain an equation corresponding to (A.3) as(1 − qθ)ã = (1 + β1 )−1 Rs,−(1−γ)/γ

where β1 ≡ ρ−1/γ α1(A.10), ρ ≡ β[p(1−D+θ(1−qθ)−1 )1−γ +1− p], and α1 is an undetermined variable.

Again, by following a similar procedure to that described in Appendix A.1, the undetermined variable
becomes R(1 − R(1−γ)/γ ρ1/γ )−γ/(1−γ) and the optimal savings becomeã =(Rρ)1/γ≡ σϕ s.
1 − qθ

On the contrary, the derivative of (A.9) with respect to θ becomes[]q (Rs − (1 − qθ)ã)−γ ã − (1 − qθ)−γ Θ(α1 ã)1−γ ,[]where Θ ≡ β p[1 − D + θ(1 − qθ)−1 ]−γ (1 − D − q−1 ) + 1 − p . Hence,dRHS of (A.9)
R 0 ⇔ (1 − qθ)ã R (1 + β2 )−1 Rs,dθ−(1−γ)/γ

where β2 ≡ Θ−1/γ α1. Since (1) in equilibrium qt = p < 1 and (2) the investor’s savings ã

must not become zero, ρ > Θ from definitions and hence β2 > β1 . Then, the derivative of (A.9) withrespect to θ must be positive when (A.10) is satisfied. Thus, an investor demands the upper limit ofcoverage, that is, θ = ϕθ̂. Finally, by evaluating ρ at θ = ϕθ̂ and qt = p, we obtain equation (31) in
Proposition 5.

35

In what follows, we show the derivatives of ρϕ and pθ̂ with respect to p and h for the comparative
statics of savings rate σϕ . First, we obtain the following derivatives:()()2d 1 + (1 − ϕ)D[p(1 − D) − 1]1
= −(1 − ϕ)D
ϕD < 0,dp
1 + (1 − ϕ)pD
1 + (1 − ϕ)pD
(
)
(
)2 [
]
d 1 + (1 − ϕ)D[p(1 − D) − 1]1′
= −(1 − ϕ)D
1 + 2pD + (1 − ϕ)p2 D2 > 0.
dh
1 + (1 − ϕ)pD
1 + (1 − ϕ)pD
Then, the derivatives of ρϕ with respect to p and h can be calculated as follows:
]
[
1 dρϕ
1−γ
−γ dA
1−γ
−1 dA
= A − 1 + p(1 − γ)A=A
1 + p(1 − γ)A− 1 > 0,
β dp
dp
dp

where A ≡( 1+(1−ϕ)D[p(1−D)−1] )1+(1−ϕ)pDdρϕ
dA= βp(1 − γ)A−γ
< 0,
dh
dh
< 1. The derivatives of pθ̂ with respect to p and h are simply given by

(
)2
d pD
1
d pθ̂
=
=
D > 0,
dp
d p 1 + pD
1 + pD
(
)2
d
D
1
d pθ̂
=p
= pϕ
D′ < 0.
dh
dh 1 + pD
1 + pDThus, the comparative statics of σϕ in the latter half of Subsection 5.4 hold.

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