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Climate Defaults and Financial Adaptation

WP 23-06

Toàn Phan
Federal Reserve Bank of Richmond
Felipe Schwartzman
Federal Reserve Bank of Richmond

Climate Defaults and Financial Adaptation
Toàn Phan & Felipe Schwartzman∗
March 27, 2023

We analyze the relationship between climate-related disasters and
sovereign debt crises using a model with capital accumulation, sovereign
default, and disaster risk. We find that disaster risk and default risk
together lead to slow post-disaster recovery and heightened borrowing
costs. Calibrating the model to Mexico, we find that the increase in
cyclone risk due to climate change leads to a welfare loss equivalent
to a permanent 1% consumption drop. However, financial adaptation
via catastrophe bonds and disaster insurance can reduce these losses by
about 25%. Our study highlights the importance of financial frictions
in analyzing climate change impacts.
Keywords: climate change; disasters; sovereign default; emerging
markets; growth.
JEL classification codes: Q54, F41, F44, H63, H87.

The Federal Reserve Bank of Richmond; and We thank Tamon Asonuma and Enrico Mallucci for their thoughtful
discussions. For valuable comments, suggestions, and support, we also thank Laura
Bakkensen, Lint Barrage, Toni Braun, Grey Gordon, Igor Livshits, Leo Martinez, Bruno
Sultanum, Nico Trachter, Russell Wong, and the conference/seminar participants at the
IMF Climate-Related Natural Disasters: Macroeconomic Effects and Policy Responses
conference, the IMF Sovereign Debt workshop, the UCLA Climate Adaptation Symposium,
Cambridge University Mini-conference on Climate Change, FRB Richmond Climate Change
Economics workshop, FRB Richmond Sovereign Debt Week workshop, FRB System brown
bag, FRB Richmond brown bag, FRB CREST brown bag, and the Osaka University ISER
seminar. All errors are ours. The views expressed here are those of the authors and not
necessarily of the Federal Reserve Bank of Richmond or the Federal Reserve System.




Climate change is projected to alter the frequency and severity of weatherrelated disasters, such as hurricanes, in many economies. How quickly a country can recover from such disasters depends on its ability to attract foreign
capital. This is likely to prove challenging for many emerging economies, given
their relatively high risk of debt crises and costly access to external finance
in times of need. These financial challenges may amplify the costs of rising
climate-related risks, but they may also suggest a potential role for financial
market developments in helping countries adapt to the changing climate.
To systematically analyze these challenges and opportunities, we develop
a tractable and quantifiable growth model of a small open economy with an
exogenous climate-related disaster risk and an endogenous sovereign default
risk. The framework allows us to quantify the welfare implications of changes
in disaster risks and the potential benefits of financial adaptation strategies.
Crucially, the model generates implications that are consistent with several
critical empirical observations for emerging economies. First, adverse weather
shocks such as hurricanes can cause long-lasting adverse macroeconomic effects, with declines in GDP and national income that are persistent for many
years (Mendelsohn et al. 2012; Bakkensen and Mendelsohn 2016; Hsiang and
Jina 2014), and the damage tends to be more severe and long-lasting in countries with less financial development and insurance coverage (Bakkensen and
Barrage 2022; Von Peter et al. 2012). Second, these shocks have adverse financial effects. Ex-post, natural disasters often lead to increased borrowing
costs and a higher likelihood of a subsequent sovereign debt crisis (Klomp
2015, 2017; Asonuma et al. 2018). Ex-ante, countries with greater exposure
to climate-related disaster risks generally face higher borrowing costs, all else
being equal (Kling et al. 2018; Barnett et al. 2020; Beirne et al. 2020).
Our model can explain these stylized facts through an endogenous propagation mechanism that hinges on the movements of a sensitive default risk, which
depends not only on the level of debt issuance, as in standard sovereign debt
models, but also on the level of physical capital investment. By destroying a


country’s capital, a bad disaster shock increases the risk of default and the
interest rate spread as functions of debt issuance. This shift forces the country
to reduce borrowing, further depressing future output and investment. The
lowered borrowing capacity keeps investment low and borrowing costs high,
therefore creating a feedback loop that can lead to a persistent reduction in
capital stock and output in the aftermath of the disaster. Together, the model
generates persistent economic damage from adverse weather shocks. It also
predicts that, all else equal, countries with higher exposure to bad weather
shocks face higher borrowing costs and that the realization of a disaster raises
borrowing costs and the probability of a sovereign debt crisis.
To quantify the economic impacts of weather shocks and the welfare consequences of climate change, we calibrate the weather shock process to cyclones,
a critical and well-studied climate-related peril (Nordhaus 2010; Mendelsohn
et al. 2012). We calibrate the model economy to Mexico, an important emerging economy with significant cyclone exposure (Juarez-Torres and Puigvert
2021). Our quantitative analysis reveals that after a large cyclone strike, the
endogenous default risk can significantly delay the recovery by at least two
decades. This finding aligns with the best estimates from the empirical literature (Hsiang and Jina 2014, 2015) and highlights the long-lasting impact of
climate-related disasters on vulnerable economies. One key driver of this delay
is the looming threat of future default-triggering cyclone shocks, exacerbating
borrowing costs and further complicating the post-disaster recovery process.
In addition, we utilize our model to examine how financial frictions affect
the welfare implications of climate change. Drawing on the well-known climatology predictions developed by Emanuel et al. (2008), we assume that cyclone
activity in the Atlantic basin will increase by 10% by the end of the century
under the “business as usual” climate scenario. Our structural model allows
us to calculate the welfare loss due to such changes, which we estimate to be
equivalent to a permanent drop in consumption of approximately 1%. This
value is significant in comparison with, for instance, well-known measures of
the benefits of completely eliminating business cycle fluctuations.1

See Lucas (1987) for an early version of this calculation, which estimates the cost to be


Finally, we characterize the potential effects of financial adaptation in mitigating the impacts of climate change. We examine two forms of adaptation:
the utilization of disaster insurance and the issuance of catastrophe (CAT)
bonds. We find that, by combining the different advantages of the instruments,
their adoption can recover approximately 25% (a quarter) of the welfare loss
from climate change.
Intuitively, adopting insurance allows the country to smooth consumption
and net worth across disaster and nondisaster states. The country can then
quickly rebuild its capital stock, leading to greater overall wealth and reducing
the impact of climate change. However, the gains from insurance are balanced
by the fact that the country has to use its already constrained debt capacity to
pay the insurance premium in good times. On net, these two forces translate
into a slight increase in country wealth and capital over the long run, but not
nearly enough to compensate for the losses due to climate change. However,
insurance has no direct effect on the country’s incentive to default in disaster states as long as the payments from insurance contracts, which are the
sovereign country’s assets, cannot be seized by foreign creditors in the case of
default (Bulow and Rogoff 1989).
In contrast, CAT bonds decrease the debt burden in disaster-stricken states,
reducing the default risk for any level of debt. The model produces a “CATissuance Laffer curve,” whereby expected debt repayment and bond price are
nonmonotonic functions of the fraction of bond issuance that is CAT. While
a moderate share of CAT bonds can be beneficial for both lenders and the
borrowing country, an excessive amount can result in unnecessarily higher
borrowing costs.2
The two instruments are, therefore, complements rather than substitutes.
On the one hand, insurance provides countries with resources to speed up
just 0.05% in consumption equivalent terms for the United States. Jordà et al. (2020) revise
the cost to be about 15% in the post-WW2 era, primarily due to disasters and mini-disasters
in consumption growth dynamics.
This result is reminiscent of Krugman (1989) discussion of the 1980s debt crisis in
Latin America, that “just as governments may sometimes actually increase tax revenue by
reducing tax rates, creditors may sometimes increase expected payment by forgiving part
of a country’s debt.”


disaster recovery but does little to help avoid default costs. On the other
hand, CAT bonds help the country avoid costly defaults in disaster states but
provide little insurance.
Related literature. Our paper adds to the rapidly growing theoretical climate economics literature, pioneered by Nordhaus (1994) and Nordhaus and
Boyer (2000), and more recently Golosov et al. (2014) and Hassler et al.
(2021), among others. While the importance of studying the economic effects
of climate-related disasters has been emphasized in several studies, including
Weitzman (2009), Nordhaus (2010), Cai and Lontzek (2019), Bansal et al.
(2019), Cantelmo et al. (2020), and Hong et al. (2020),3 our contribution is
the first model to study how climate change affects welfare via the endogenous
and dynamic interplay between investment and sovereign default risk. Our
paper also integrates best estimates of cyclone damages from the empirical
climate economics literature, including Nordhaus (2010), Mendelsohn et al.
(2012), Hsiang and Jina (2014), Hsiang and Jina (2015), and Bakkensen and
Barrage (2022).4
Our paper connects the climate economics literature to the growing literature on sovereign default, including Eaton and Gersovitz (1981), Bulow and
Rogoff (1989), Aguiar and Gopinath (2006), Arellano (2008), Yue (2010), Mendoza and Yue (2012), Bai and Zhang (2012), Chatterjee and Eyigungor (2015),
Hatchondo et al. (2016), Phan (2016, 2017b), Park (2017), Dovis (2019),
Bianchi et al. (2019), Asonuma and Joo (2020), and da Rocha et al. (2022).5
A related paper is Mallucci (2022), which introduces hurricane risk and CAT
bonds into a quantitative endowment economy framework with long-term debt
carefully calibrated to a sample of Caribbean countries and provides a refined
discussion of the impact of disaster risk on spreads. In studying disastercontingent bonds, our paper is also related to those that analyze the potential

Also see Ikefuji and Horii (2012) and Müller-Fürstenberger and Schumacher (2015).
For a review of the empirical climate-economics literature in general, see Dell et al.
(2014). Also see Greenstone et al. (2013), Auffhammer (2018), Colacito et al. (2019) and
references therein.
For a comprehensive literature review, see Aguiar et al. (2016).


effects of making sovereign debt more state-contingent, such as Grossman and
Van Huyck (1988), Alfaro and Kanczuk (2005), Adam and Grill (2017), and
Borensztein et al. (2017).6 Our paper’s relative contribution to this literature
is a tractable model that allows for (analytical and quantitative) characterizations of how disaster risks affect debt, default risk, and investment dynamics.
Our framework offers a practical approach to simplifying the complex dynamics between endogenous investment and sovereign default, which have posed
significant challenges in the existing literature (Gordon and Guerron-Quintana
Our paper is organized as follows. Section 2 provides the model environment and several analytical characterizations. Section 3 provides the calibration exercise and several numerical results. Section 4 analyzes financial
adaptation strategies. Section 5 concludes.


Model and analytical characterizations

Consider a small open economy with a representative sovereign government.
The economy produces a single consumption good from capital Kt and labor—
which is supplied inelastically and normalized to be one—using a Cobb-Douglas
production function:
Yt = (e−xt dt Kt )α (At )1−α , 0 < α < 1,
where At is total factor productivity (TFP). We assume for simplicity that At
is identically and
follows a random walk, i.e., the growth shock gt := log AAt+1
independently distributed (i.i.d.) according to a distribution Φg .7
See also Grossman and Van Huyck (1993), Braun et al. (1999), Phan (2017a), Asonuma
et al. (2018), and Hatchondo et al. (2022).
It is straightforward to extend the model to allow the growth shocks to follow a Markov
process. We keep the simpler process as it captures the most salient quantitative facts with
significant gains in terms of tractability.


Weather shocks. Exogenous variables xt and dt represent the extensive and
intensive margins of a stochastic process for weather shocks. The dummy xt is
one if there is a bad weather shock (e.g., a cyclone strike) and zero otherwise.
The continuous variable dt ≥ 0 denotes the intensity of the damage to the
capital stock. In the baseline analysis, we assume that the probability of a
bad weather shock in each period is a constant Pr(xt = 1) = p and that the
damage dt is i.i.d. according to a distribution Φd with support over [0, ∞).
(We will later consider how climate change alters the distribution of weather
Preferences. To more fully capture the welfare effects of the weather jump
process, we assume that the representative government maximizes Epstein and
Zin (1989) recursive preferences:

 1−ι  1
1−γ 1−γ 1−ι
Vt = Ct1−ι + βEt Vt+1
where ι is the inverse intertemporal elasticity of substitution and γ is the
relative risk aversion coefficient (if ι = γ, then the preferences collapse to
the constant relative risk aversion specification). Following the macrofinance
literature (e.g., Bansal and Yaron 2004; Cai and Lontzek 2019), we focus on
the relevant parameter range where ι < 1.
Sovereign borrowing. The country can borrow from risk-neutral international lenders by issuing one-period noncontingent bonds. Each unit of bond
is a promise to repay one unit of the final consumption good in the subsequent
period. However, the country cannot commit to this promise. We assume that
default is costly: it leads to a deadweight loss of a fraction `t of the country’s
output Yt . Following Aguiar et al. (2016), we assume a procyclical fractional
loss `t = `(gt ), where
¯ ψg0 ,
`(g 0 ) = `e

, ψ ≥ 0, `¯ > 0.



This specification implies that the country has more incentive to default in lowgrowth states. Such a cyclical cost of default is a standard assumption in the
sovereign debt literature as it helps the model match the cyclical movements
of spreads observed in data (Aguiar et al. 2016).
As in Adam and Grill (2017), we assume that the defaulting country can
immediately regain access to the debt market. An important advantage of this
assumption is that it greatly enhances the tractability of the model, allowing
us to characterize the equilibrium bond price in closed form and numerically
solve the optimization problem with both capital accumulation and strategic
default without the traditional curse of dimensionality (Gordon and GuerronQuintana 2018). A potential disadvantage is that we may miss the quantitative
implications of the cost of credit exclusion; for example, we may underestimate
the level of debt that can be sustained in equilibrium. However, it has been
well known in the literature that credit exclusion alone plays a limited role in
sustaining debt in equilibrium (Bulow and Rogoff 1989; Aguiar et al. 2016).8
Optimization problem. In each period t, after the realization of the growth
shock and the weather shock, the country chooses (i) to repay or to default
on its outstanding debt obligation, (ii) the value of new bonds issues, and (iii)
new capital investment. We detrend all variables by the productivity level
At and denote the detrended variables with lowercase letters (e.g., vt := AVtt ,
kt := K
). The recursive optimization problem can be written concisely with
just one state variable—the country’s net worth:


= max c
kn ≥0,bn


+ βE


v(max{m0R , m0D })1−γ e(1−γ)g

i 1−γ



subject to a budget constraint:
c = m − kn + q(bn , kn )bn ,

Alternatively, we can interpret the cost as a reduced form for various sources of default
costs, including the effects of temporary exclusion from international credit markets. This
interpretation is arguably reasonable in our context, since in the calibration we make each
period five years, which is close to the average duration of exclusion.


where bn and kn denote the new bond issuance and next period capital, and q
denotes the bond price schedule (determined below). The detrended debt and
capital positions in the subsequent period (after the next period’s shocks have
been realized) are given by:

b0 = e−g bn
k0 = e

−x0 d0 −g 0

kn .


The country’s net worth, conditional on either debt repayment or default, is
given by:
m0R = (k 0 )α + (1 − δ)k 0 − b0


m0D = (1 − `(g 0 ))(k 0 )α + (1 − δ)k 0 − 0.


The country’s net worth next period is then simply m0 = max{m0R , m0D }. It is
straightforward from (5-6) that the country chooses to default if and only if
its debt over GDP exceeds the output lost fraction `(g 0 ):



> `(g 0 ).
(k )

Given specification (1), the country defaults if and only if the weather-adjusted
x0 d0 , which captures the damage from the weather
growth term g̃ 0 := g 0 − 1−α+ψ
shock, is below an endogenous default threshold ḡ(bn , kn ) := 1−α+ψ
ln `k
¯ α:

default ⇐⇒ g 0 −
x0 d 0 <
ln ¯ α .
1 − α + ψ `kn
} |
g̃ 0


ḡ(bn ,kn )

The difference between the ḡ and g̃ 0 can be interpreted as the distance to
default. Note from (7) that the default threshold ḡ is increasing in bn and decreasing in kn . This implies that the default risk increases with more debt and
decreases with the next period’s capital stock. The expression also highlights
the role of cyclical default costs. As ψ increases (so that default costs increase


more rapidly with the growth shock g 0 ), the default threshold ḡ becomes less
sensitive to changes in debt and capital stock.
Figure 1 illustrates how the disaster risk affects the tail of the distribution
of growth shocks. It plots the histograms associated with the probability
distributions of g 0 and g̃ 0 (when the TFP growth shock g 0 follows a normal
distribution and the damage intensity d0 follows a Weibull distribution). It
is immediate to see that for a given threshold ḡ (illustrated by the dashed
vertical line), the tail event g 0 < ḡ has a smaller probability than the tail event
g̃ 0 < ḡ, implying that the presence of the bad weather shocks increases the
likelihood of default.


Figure 1: Effects of weather shocks on growth. Histograms of growth shock
x0 d0 , which
g 0 (blue) and of weather-adjusted growth shocks g̃ 0 := g 0 − 1−α+ψ
captures the damage of the weather shock (red).

Bond price schedule. The equilibrium bond price schedule is a function
that specifies the price per unit of bonds issued by the country given its choices.
In the competitive credit market with risk-neutral lenders, where lenders rationally expect the possibility of default, the schedule is given by:
q(bn , kn ) =

1 − s(bn , kn )
, ∀bn , kn ,


where r is the world risk-free interest rate (assumed to be a constant for
simplicity), and s is the sovereign default spread, defined as the probability of
s(bn , kn ) = Pr[m0R < m0D ],
with repayment and default net worth values m0R , m0D as specified in (3-6).
The function s, which is key to our analysis, is the spread between the
price of a risk-free bond (1/(1 + r)) and the price of a bond issued by the
country. A nice feature of the tractability of our model is that we can derive
a closed-form expression for s by using the default decision specified in (7):
s(bn , kn ) = Pr [g̃ 0 < ḡ(bn , kn )]
= (1 − p)Φg (ḡ) + pEd0 Φg ḡ +




Equation (9) shows how the disaster risk affects the spread schedule. Intuitively, default is a tail event, which happens when the economic conditions
are sufficiently bad, making the burden of repaying larger than the output loss
of default. The presence of the disaster risk effectively changes the growth
shock from g 0 to g̃ 0 . Since the distribution of g̃ 0 has a fatter left tail than
the distribution of g 0 , it follows that the disaster risk raises the probability of
default (for any given choice of debt bn and next period capital kn ).
The shape of the spread schedule s determines the sensitivity of the economy’s borrowing cost to disaster risk. The following proposition analytically
characterizes the elasticity of s:
Proposition 1. The equilibrium spread schedule s is positively elastic in debt
issuance bn :
bn ∂s
h(ḡ) > 0,
s ∂bn
and negatively elastic in investment kn :
kn ∂s
h(ḡ) < 0,
s ∂kn


where the ratio h̃(ḡ) is defined as: h̃(ḡ) =

(1−p)φg (ḡ)+pEd0 φg (ḡ+ 1−α+ψ
d0 )
(1−p)Φg (ḡ)+pEd0 Φg (ḡ+ 1−α+ψ
d0 )


Proof. Appendix A.1.
The first claim of Proposition 1 confirms a standard result in the sovereign
debt literature: the equilibrium spread increases in debt issuance. The newer
part is the second claim, which states that the spread decreases in the next
period capital. This is a direct corollary of the fact that the default threshold
b̄ is decreasing in kn (leading to a smaller default region according to (7)).
Figure 2 illustrates the dependence of the spread on capital (for a simple
case where the TFP growth shock g 0 follows a normal distribution, and the
damage intensity d0 is deterministic). Figure 2a plots s as a function of bn and
kn , with two cross-sectional cuts at two different levels of next-period capital.
Figure 2b plots the two curves associated with these two cross-section cuts,
showing s as a function of debt issuance conditional on two different levels of
next-period capital. The solid line is associated with a higher capital level and
the dashed line with lower capital. It is clear how the spread curve shifts to
the right as kn moves from the low level to the high level, meaning that for
each amount of debt issuance bn , the spread is higher when there is less capital
in the next period. Furthermore, note how the spread function is steeper (the
default probability is more sensitive to an increase in debt issuance bn ) when
investment kn is lower.
The interplay among capital, debt, and spreads gives rise to the vicious
feedback loop, illustrated in Figure 2c. As the capital stock falls, spreads
increase, making debt more costly, and limiting capital accumulation. This
vicious cycle will play an important role in the propagation of weather shocks
in our quantitative analysis in Section 3.
Finally, we can use the framework to derive the qualitative impact of climate disasters on spreads. This provides a qualitative sense of the role of
disasters and the effects of their increased frequency. This is summarized in
the following proposition:
Proposition 2. The spread schedule increases in the weather shocks’ frequency


(a) Spread surface s(bn , kn ) as given by (9).


high k

low k


(b) Spread as a function of debt issuance bn at a low and a high
level of investment kn .


Capital ↓

Default risk & spread ↑

(c) Feedback loop between declined investment and heightened default risk, which slows down the post-disaster recovery.

Figure 2: The dependence of spread on capital.

= −Φg (ḡ) + Ed0 Φg ḡ +
d > 0,


and increases (in the first-order stochastic dominance sense) in the distribution
of the shock’s damage:
f osd

Φ̂d ≥ Φd ⇒ s(·, ·|Φ̂d ) ≥ s(·, ·|Φd ).
Proof. Appendix A.2.
Intuitively, an increase in the frequency or intensity of the weather shock
raises the thickness of the left tail of the weather-adjusted growth shock g̃ 0 and
thus leads to an increase in the default risk.


Quantitative analysis

We now examine the implications of the model in a quantitative setting. The
disaster type we have in mind is a cyclone, which not only is an important
source of climate-related risk, but also has been extensively studied in both
the climate science literature (Emanuel et al. 2008) and the economics literature (Hsiang and Jina 2014; Bakkensen and Barrage 2022). The country that
we calibrate the model to is Mexico, an emerging economy that is subject
to substantial cyclone risk and whose business cycles are well studied in the
macroeconomics literature.



Basic parameters. We take each period to equal five years, which is appropriate for our focus on the recovery dynamics from large disasters. The use
of five-year periods affords us a degree of tractability. At such horizons, capital adjustment costs are less likely to be an important part of the dynamics.
Furthermore, the implied five-year maturity of the model’s one-period debt is
a better approximation of the average maturity of EMs’ sovereign bonds than
that in a quarterly or annual model. The five-year period also allows us to
abstract from autocorrelation in TFP growth rates, further reducing the state

Given that disasters destroy both physical and human capital (Bakkensen
and Barrage 2022), we follow Mankiw et al. (1992) and set α = 2/3 as a simple
way to capture both types of capital jointly. More recently, Dietz and Stern
(2015) also adopted this parametrization to reflect the capital damage due to
climate shocks.
We set β = 0.965 ; namely, the discount factor is 0.96 per year, which
is a standard value in business cycle models (but higher than typical values
in endowment-economy sovereign debt models without disaster risks such as
Arellano 2008). We set depreciation rate δ to a standard value of 10% per year.
We set the foreign risk-free interest rate to 1% per year, in line with recent
global trends. As usual, the fact that β(1 + r) < 1 implies that the sovereign
in the model is impatient relative to the market and will seek to borrow to tilt
consumption and investment forward in time. For the Epstein-Zin preference
parameters, we set ι = 0.5 and γ = 4 as in Gourio (2012).
We choose the parameters µg and σg governing the mean and standard
deviation of TFP growth shocks based on the quarterly values employed by
Aguiar and Gopinath (2006, 2007). For the fractional output loss from default,
based on Aguiar et al. (2016), we set the level parameter `¯ at 0.07% and the
curvature parameter ψ at 7. These parameter values imply that in the ergodic
steady state of the model, the average debt-to-annual-GDP ratio is about 35%,
and the average annual spread is about 1.6%, which are reasonable estimates
for Mexico during the 2010s.
Weather shock parameters. In order to calibrate the risk and size of the
weather shock, we rely on Hsiang and Jina (2014, 2015)’s well-known analysis
of an extensive and carefully constructed global data set of exposure to tropical
cyclones/hurricanes/typhoons during 1950-2008.9
On the extensive margin, we interpret the model’s weather shock dummy
x as corresponding to whether a country experiences a cyclone landfall in any
period. We set the strike probability p (which determines the Poisson rate of

Tropical cyclones are also known as “tropical storms” or “hurricanes” in the Atlantic
Ocean, “typhoon” in the Pacific Ocean, and “cyclones” in the Indian Ocean. As in the
empirical climate literature, we will refer to all of them simply as cyclones.


arrival for cyclone landfall) to match the average annual probability of 58%
that at least one cyclone (of any size) makes landfall.10 Since a model period
is five years, the implied value for p is
p = 1 − (1 − 0.58)5 = 0.9869.
This number implies that the country is almost guaranteed to be hit by at
least one cyclone (big or small) over a five-year period.
On the intensive margin, we interpret the damage parameter d as corresponding to the country’s cumulative damage from cyclone activity over a
five-year period. Specifically, we assume that d = ω × µ, where ω is the cumulative intensity (measured in the unit of wind speed, m/s) of the cyclones that
make landfalls in a period, and µ is the marginal damage to the capital stock
of each additional m/s of wind speed, normalized by the area of the country.
Specifically, we set
µ = 0.0895%/α = 0.1343%,
to match Hsiang and Jina (2014)’s estimate that each additional m/s of wind
speed per unit area causes a cumulative output loss of 0.0895% after five
We assume that the maximum wind speed of a cyclone follows a Weibull
distribution, which is known to be a good empirical approximation (Bakkensen
and Barrage 2022). From Hsiang and Jina (2014)’s cyclone statistics, we
set the scale parameter to 6.579, in order to match Mexico’s current annual
average cyclone activity of 6.6 m/s per unit area. We set the shape parameter
to 0.993, in order to match the ratio of 19.5/39.2 between the strength of a
90th percentile cyclone and that of a 99th percentile cyclone. We then get the
distribution of ω by the convolution of the Weibull distribution for the intensity
of each cyclone that makes landfall in a period. Thus, the distribution of ω

This probability is derived from Hsiang and Jina (2014)’s estimate that the annual
probability of a top 10th percentile cyclone is 5.8%.
We use this global average estimate since Hsiang and Jina (2014) do not report countryspecific estimates. They argue that the global estimate is very robust and applies to various
subsamples of countries in their data.


represents the distribution of the cumulative cyclone activity over a five-year
Table 1 summarizes our parameter choices.

period length
capital share
discount factor
world interest rate
inverse elasticity of substitution
risk aversion
mean TFP growth
std of TFP growth
default cost constant
default cost curvature
cyclone strike probability
marginal output damage
shape of Weibull distribution
scale of Weibull distribution



5 years
 Mankiw et al. (1992)
1 − 0.9
Standard RBC values
1.015 − 1 
Gourio (2012)

1.00620√− 1
Aguiar and Gopinath (2007)
0.0213 20 
Aguiar et al. (2016)
Hsiang and Jina (2014)
0.993 

Table 1: Calibrated parameters.


Numerical results

We now show the quantitative implications of the calibrated model: longlasting economic effects of cyclone shocks and significant welfare consequences
of increased cyclone activity due to climate change.

Propagation of cyclone shock

We examine the impulse responses of the economy to a one-time one-standarddeviation shock to cyclone activity. We generate the responses from the average of one million simulation paths where, in each path, we simulate the
economy until it reaches the ergodic steady state. Then in a period labeled
0, there is a one-time unanticipated shock that raises the period’s cumulative

area-weighted cyclone activity from its ergodic steady-state average value of
24m/s by one standard deviation to 39m/s. We interpret this shock as representing the increased cyclone activity in period t = 0 due to the strike of a
large cyclone.12
Figure 3 plots the impulse in the first panel and the responses of aggregate
detrended variables in the remaining panels. To facilitate interpretation, given
that each period of the model is five years, we divide the spread and default
frequency by five and multiply the debt-to-output by five to report them in
annual terms.

Figure 3: Impulse responses of detrended variables to an unanticipated onestandard-deviation shock to cyclone activity. Responses are generated from
an average of one million simulation paths. In each path, the economy first
reaches the ergodic steady state. Then in a period labeled 0, there is a one-time
unanticipated shock that raises the period’s cumulative area-weighted cyclone
activity by one standard deviation. Variables on the bottom row have been
annualized for easy interpretation.

For comparison, the maximum sustained wind speed of a category 5 hurricane is 70m/s
or higher.


Panels two and three plot the impulse responses of the detrended capital
stock kt , output flow yt , net worth mt , and consumption ct . All the aggregate economic variables dip at t = 0, due to the damaging impact of the
cyclone activity shock on the capital stock. Importantly, the model’s internal
propagation mechanism predicts that the economy will slowly recover in the
aftermath (t ≥ 1), with the half-life of the recovery path toward the steady
state of about 20 years. This slow recovery endogenously arises due to the
vicious feedback loop between the increased demand for borrowing to rebuild
the damaged capital stock and the heightened default risk discussed in Section
To see this further, panel four plots the impulse response of the debt over
output ratio, and panel six plots the response of the spread. The ratio increases
mechanically upon impact at t = 0, as the output denominator decreases.
However, the ratio remains elevated for many years afterward, reflecting the
need to issue more bonds to rebuild the capital stock. The increase in bond
issuance leads to a lower bond price, as reflected by the widening spread. This
is a vicious cycle at work: the increase in borrowing to raise funds to rebuild
increases the default risk and reduces the revenue per bond issued, which then
necessitates more bond issuance, further exacerbating the cycle.
Panel six also plots the response of the frequency of realized sovereign defaults (shown by the dashed line), which jumps upon impact at t = 0 due
to the unexpected nature of the shock, but remains persistently high afterwards.13 The predictions of the persistent increases in the spread and the
default frequencies are consistent with the aforementioned empirical evidence
(Klomp 2015, 2017).
Note that in a counterfactual environment without financial frictions, foreign credit would flow more into the country in the immediate aftermath of the
cyclone shock (due to the increase in the marginal product of capital), and the
economy would converge back to the steady state trend after just one period.
Instead, our model predicts the opposite: foreign capital flows would decrease,

The path of the spread is smoother than the path of the default frequency because the
former is the expectation of the latter.


as shown by the decline in the capital inflows over output ratio depicted in
panel six.14 Also note that this procyclical capital inflows over output ratio
(or equivalently, countercyclical trade balance over output ratio) is consistent
with well-known empirical patterns of business cycles in emerging economies
(Uribe and Schmitt-Grohé 2017). The decline in the capital inflows also helps
explain why consumption drops more than net worth as shown in panel three.
Overall, the impulse responses underscore the importance of financial frictions in explaining the delayed recovery from cyclone strikes as well as the
increased spreads and frequencies of debt crises as documented in the empirical literature.

Welfare cost of increased cyclone risk

We examine the potential welfare implications of a shift in the distribution
of cyclone activity due to climate change. According to projections based on
the Intergovernmental Panel on Climate Change (IPCC)’s A1B scenario, the
intensity of cyclone activity in the Atlantic basin that contains Mexico is expected to increase by an average of 10.3% by 2090 (Emanuel et al. 2008). We
model this shift by increasing the scale parameter of the Weibull distribution
for cyclone intensity by 10%. Figure 4a shows the resulting first-order stochastic shift in the distribution of cyclone activity in our model. The blue dashed
lines show the probability density function and the cumulative density function for the cyclone activity in the baseline scenario (Table 1), while the red
solid lines show those in the increased cyclone risk scenario (Emanuel et al.
We measure the implied welfare change for the representative agent in the
country by
E+ [v+ (m)]
∆w := 1 −
Here, v(m) denotes the lifetime utility for a given net worth level m (as defined
in (2)), and E[v(m)] denotes the expected lifetime utility given the distribuWe define the capital inflows as qt bnt − bt , where recall that bnt is the debt issuance at t.
Hence, the capital inflows are simply the negative of the trade balance or net export.


(a) Change in the distribution of cyclone activity. The dashed lines
plot the PDF and CDF of cyclone activity in the baseline calibration
(Table 1). The solid red lines plot those when the scale parameter
of the Weibull distribution for cyclone activity increases by 10%.

(b) Changes in the welfare functions v(m) and the ergodic distributions of
net worth m. The thick blue dashed line and the thick red solid line plot
the sovereign country’s welfare functions v(m) and v+ (m) in the baseline
calibration and in the increased cyclone risk scenario, respectively. The
blue shaded area (with a thin dashed border) labeled P DF (m) and the red
shaded area (with a thin solid border) labeled P DF+ (m) plot the kernel density estimation of the ergodic distributions of m in the baseline calibration
and in the increased cyclone risk scenario, respectively.

Figure 4: Welfare implications of the increase in cyclone activity due to climate


tion of net worth over the ergodic steady state, under the baseline scenario.
Similarly, v+ (m) and E+ [v+ (m)] denote those under the increased cyclone risk
scenario. Note that by taking into account the ergodic distributions of the
state variable m, ∆w measures the long-run welfare changes, hence appropriately taking into account the long-term nature of the shift in cyclone risk due
to the gradual process of climate change. Furthermore, given the Epstein-Zin
preference specification, ∆w is also equal to the measure of welfare change in
consumption equivalent units that is standard in the macroeconomics literature since Lucas (1987). That is, ∆w is equal to the fraction of consumption
that the representative agent in the country is willing to permanently give up
(i.e., in all time periods and in all states of the world) in order to avoid the
increase in cyclone activity.
There are two components of the welfare effect of the increase in cyclone
risk. First, there is a downward shift of the value function from v to v+ at
any given level of net worth m. And second, there is a leftward shift in the
distribution of m in the ergodic steady state.
To visualize the first component, the thick lines in Figure 4b plot v and
v+ over the relevant range of m (three standard deviations around the ergodic
averages). The figure clearly shows that the welfare function shifts from the
thick blue dashed line representing v to the thick red solid line representing
v+ .
Figure 5 provides a closer look at this shift in v. There, the red solid line
+ (m)
as a function of the net worth state variable m
plots the loss term 1 − vv(m)
(shown over the same range of m as in Figure 4b). The plot shows that, for
any given net worth m, an increase in cyclone activity leads to a shift in the
welfare function that is equivalent to about a 0.6% to 0.7% permanent loss in
However, the downward shift in the welfare function is only part of the
story. This is because the wealth of the country (measured by net worth m)
will endogenously adjust over time to the change in cyclone activity, which
also will lead to a change in the ergodic distribution of m. Since more intense
cyclones make it harder for the country to accumulate assets, the ergodic

distribution of m experiences a first-order stochastic dominance shift for the
worse. Visualizing this distributional shift, the blue and red shaded areas in
Figure 4b plot the ergodic distribution of m in the baseline scenario and in
the increased cyclone risk scenario, respectively.
Hence, over the long run, the welfare effect consists of not only the shift
in v, but also the shift in the distribution of m, as appropriately captured in
the definition of ∆w. The first row in Table 2 reports the long-run changes.
The first column shows that the leftward shift in the ergodic distribution of
m due to increased cyclone activity leads to a reduction of the average net
worth by nearly one percentage point (0.93%). The second column shows
that, by taking into account the distribution shift in m, the change in longrun welfare ∆w is about 1%. In other words, our calibrated model suggests
that the representative agent in the country is willing to permanently give up
more than 1% of consumption in order to avoid the increase in cyclone activity.
To get a sense of the economic magnitude, a welfare loss of 1% in consumption equivalent terms is significantly greater than the conventional benchmark
estimate of merely 0.05% for the cost of business cycle fluctuations in the
United States (Lucas 1987, 2003). This is because the increased cyclone risk
not only amplifies the short-run fluctuations of consumption, but also results in
larger cumulative output losses from cyclone shocks, due to the model’s propagation mechanism with financial friction as shown in Section 3.2.1. Therefore,
the welfare effects of increased cyclone activity are qualitatively comparable
to the “mini” economic disasters that Jordà et al. (2020) account for in their
revised measurement of the cost of business cycles. In particular, when taking
such disasters into account, they revise the impact of fluctuations to be about
15% in consumption equivalent terms in the post-World War II era.
Finally, it is worth emphasizing that our welfare analysis solely considers
the cost of heightened cyclone risk and does not incorporate other facets of
climate change, such as rising average temperatures, sea level rise, or increased
risks associated with heat waves, droughts, flooding, and so on. Conversely,
our current calculation does not account for potential physical or financial
adaptations to climate change, the latter of which we will now explore further.

Figure 5: Shifts in welfare functions due to increased cyclone risk and due
to financial adaptation, measured in consumption equivalent terms. The red
solid line plots the loss term due to increased cyclone risk 1 − v+ (m)/v(m)
over the relevant range of net worth m. The other lines plot the gain term due
to financial adaptation 1 − vi (m)/v+ (m), where i ∈ {insurance, catastrophe
bond, or both} represents different forms of financial adaptation.
Change in the ergodic average of
Net worth m Lifetime utility v
Due to increased cyclone risk
Due to financial adaptation
CAT bond





Table 2: Long-run changes in ergodic steady-state averages of net worth m
and lifetime utility v. The changes from increased cyclone risk are calculated
as percent changes relative to the corresponding averages in the baseline calibration (Section 3.1). The changes from financial adaptation are calculated as
percent changes relative to the corresponding averages in the calibration with
increased cyclone risk (Section 3.2.2).



Financial adaptation

As climate change increases the strength of cyclones, we should expect adaptation to occur. One form of adaptation, which has been the focus of much
of the literature, is physical adaptation, such as building or retrofitting structures to be more resilient to flooding (see, e.g., Barrage 2015; Fried 2022 and
references therein). Another form of adaptation, which we focus on here, is
financial adaptation, meaning the use of more sophisticated financial instruments in order to better cope with the change in climate-related risks.15 We
consider two relevant forms of financial adaptation.


Disaster insurance

We first consider disaster insurance. Bakkensen and Barrage (2022) report
significant differences across countries in the insurance coverage for disaster
damages, with developed countries having close to 55% of damages covered,
whereas low- and middle-income countries have coverage below 12%. Von Peter et al. (2012) show that countries with higher insurance coverage suffer
smaller output losses and experience faster recovery from disasters.
We consider an ideal scenario of a complete disaster insurance market,
where the country can purchase disaster insurance contracts within each period at actuarially fair prices from competitive foreign insurers. We allow
the country to purchase insurance after it observes the realized value of the
period’s TFP growth, but before the weather shock realizes and before the
default decision (Figure 6 illustrates the timing of events within each period
t). One interpretation of this timing assumption is that insurance contracts
are relatively short-term compared to debt contracts.

Such a form of adaptation may be a natural development to consider since, as hypothesized by Frame and White (2004), one might expect financial innovation to follow greater
demand for a particular financial product. For example, the Government Accounting Office (2003) reports an interview with a mutual fund manager where they state that “given
the small size of the catastrophe bond market, it did not make sense to hire experts in
hurricanes or earthquakes to monitor the market” but that “if the market for catastrophe
bonds expanded, the company would reconsider employing experts to better understand
these securities.”

































Figure 6: Timing of events within each period.
Importantly, we also assume that the country receives insurance payments
regardless of its default decision. This assumption reflects the fact that an
insurance contract is a financial asset, and it is legally complicated for foreign
investors to seize the defaulting countries’ assets (Bulow and Rogoff 1989).
Since the representative agent in the country is risk averse (the value function v is concave in m), and insurance is actuarially fair, the country will
optimally choose full insurance. That is, the country will choose to purchase
insurance so that its net worth is the same in disaster and nondisaster states
(m0 is the same across different realizations of x0 and d0 ). This then leads to a
simplified recursive optimization:


= max c
kn ,bn


i 1−ι
0 1−γ (1−γ)g 0 1−γ
+ βE v (E[max {mR , mD } |g ])

subject again to equations (3) through (6). Note that the continuation value
term v(E[m0 |g 0 ]) (where m0 = max{m0R , m0D }) reflects the facts that the country can insure against the weather shocks but not against the TFP shocks (i.e.,
while the net worth is the same across different realizations of the weather
shock, it still differs across different realizations of the TFP shock g 0 ).
As in Section 3.2.2, there are two components of the welfare effects of the
adoption of disaster insurance. First, there is a shift in the welfare function.
Figure 5 again visualizes this shift. There, the blue dash-dotted line plots
1 − vinsurance (m)/v+ (m), which is the difference between the welfare function






v+ (·) under the increased cyclone risk scenario (as studied in Section 3.2.2) and
the new welfare function vinsurance (·) under the scenario of increased cyclone
risk with a complete disaster insurance market that we are studying. For any
given level of m within the relevant range, the welfare gain is a little over
0.1% of permanent consumption. In other words, under the increased cyclone
risk scenario, the representative agent is indifferent between having access to a
complete disaster insurance market and having their consumption permanently
increase by a tenth of a percentage point.
Second, the adoption of disaster insurance also implies a change in the
distribution of net worth in the ergodic steady state. The second row in Table
2 shows the changes in the ergodic steady states in the increased cyclone
risk scenario with and without disaster insurance. In particular, on average,
net worth m is 0.14% larger in the long run thanks to disaster insurance,
leading to a gain in the long-run welfare measure of approximately 0.2% (i.e.,
1 − Einsurance [vinsurance (m)]/E+ [v+ (m)] = 0.2%). Compared to the first row in
the same table, the welfare gain from disaster insurance can undo about 20%
of the welfare loss due to increased cyclone risk.
Why isn’t the welfare gain from having access to an actuarially fair disaster
insurance market more substantial? The main reason is that while it allows
for a faster recovery after a bad cyclone shock, insurance is not free—the
country pays a premium to foreign insurers in each period. The insurance
premium payment reduces the country’s net worth and hence its ability to
borrow and invest in states without a bad cyclone shock. Nevertheless, on
average, disaster insurance still yields output gains due to the production
function’s concavity, resulting in a modest increase in wealth accumulation
(and thus a slight increase in average net worth) over the long term.


Catastrophe bonds

Next, we examine the utilization of catastrophe (or CAT) bonds. These bonds
operate like traditional bonds, except that their face values are automatically
reduced upon the occurrence of a trigger event, typically the onset of a natural


disaster exceeding a predetermined intensity threshold. The market for CAT
bonds has experienced substantial growth in the last decades (Artemis 2020).
However, with few exceptions such as Grenada, which issued bonds with explicit hurricane clauses following the extensive destruction of its capital stock
by Hurricane Ivan, CAT bonds have yet to be widely adopted by vulnerable
countries (Asonuma et al. 2018).
For simplicity, we model CAT bonds as bonds that have a face value of
zero if a cyclone hits and the realized damage is above a certain (exogenous)
¯ We maintain the assumption that the country
threshold (x = 1 and d > d).
loses a fraction of its output as long as any of its debt is defaulted upon.
Hence, should the country want to default, it will always default on all the
bonds and not just CAT or non-CAT bonds.
In each period, the country can choose how many regular bonds and how
many CAT bonds to issue. Let θ ∈ [0, 1] denote the fraction of CAT bonds in
total debt issuance. Then, the problem of the country becomes:
i 1−ι
0 1−γ
v(m)1−ι = max c1−ι + βE v(max {m0R , m0D })1−γ e(1−γ)g
kn ,bn ,θ

subject to
c = m − kn + q(bn , kn , θ)bn

0 0


b0 = (1 − T 0 θ)e−g bn , k 0 = e−x d −g kn
m0R = k 0α + (1 − δ)k 0 − b0 , m0D = (1 − `(g 0 ))k 0α + (1 − δ)k 0 − 0,
where T 0 is the dummy for a “trigger event”:
T 0 = x0 1d0 >d¯ 1bn ≥0 .
In words, the trigger event will be activated (and hence the next-period debt
b0 will be reduced by a fraction θ) if a sufficiently strong disaster hits (x0 = 1
¯ and if debt issuance is nonnegative (bn ≥ 0).
and d0 > d)
In addition, observe that q now represents the price of the entire bond
portfolio. It will be dependent on θ, not only because lenders will require

an insurance premium for disaster relief, but also because the utilization of
CAT bonds influences the country’s incentive to default. In particular, q now
1 − s (ḡ(bn , kn ), θ)
q(bn , kn , θ) =
where s denotes the spread as a function of the threshold ḡ as defined in (7)
and of the CAT fraction θ. In the simple case of d¯ = 0 (i.e., the trigger event
is simply the onset of any disaster), s has a straightforward expression:


d0 +
ln(1 − θ)
<0, reduced default risk

+ pθ 1 − Ed0 Φg ḡ +
d +
ln(1 − θ)

s(ḡ, θ) =(1 − p)Φg (ḡ) + pEd0 Φg 
ḡ +

>0, "CAT/insurance premium"

The equation implies that increasing the fraction of CAT bonds has two countervailing effects on the equilibrium spread function. On the one hand, it
reduces the risk of default in the state with a cyclone shock, generating a net
benefit for the lender since they will be able to retain the face value of regular bonds. On the other hand, lenders require an “insurance premium” in
exchange for the state contingency of the CAT bonds. In particular,
= p(1 − Φ(ḡ d ))

h̃ (ḡ)
−1 ,

where h̃(ḡ) was defined in Proposition 1. With the growth shocks g normally distributed, the function h(g) starts at zero and increases toward infinity. Therefore, there is a “CAT bond Laffer curve,” in the sense that the
spread decreases for small values of the CAT fraction θ and increases otherwise. Consequently, the country will not choose to set θ at a very low value,
since by increasing it, the country can get more insurance and issue debt at a
more favorable price.


Given the increased cyclone risk, what are the welfare implications of the
adoption of CAT bonds? As before, Figure 5 plots the shift in the welfare
function at any given value of net worth m in the relevant range, and the third
row in Table 2 shows the overall welfare change that takes into account not
only the shift in the welfare function, but also the shift in the distribution of
net worth.16 Both the figure and the table show that the welfare gain from
CAT bonds is smaller than that from disaster insurance. Specifically, Table
2 shows that the long-run gain in welfare from CAT bond issuance is only
about 0.05% in consumption equivalent terms (while the gain from disaster
insurance is 0.20%). One reason why CAT bonds fall short of insurance is
that the bonds only help hedge the risk of a large cyclone if the country does
not default. However, the incentive to default is heightened exactly when a
large cyclone strikes.17 A second reason is that the bonds have to be issued one
period in advance (while disaster insurance contracts can be purchased within
the same period). Since the country is impatient relative to foreign lenders,
the country will prefer to hedge the disaster risk using shorter-term disaster
insurance contracts than using longer-term CAT bonds. Finally, as pointed
out by Mallucci (2022), the introduction of CAT bonds reduces the borrowing
cost per unit of bond issuance, leading to more accumulation of debt, which
in turn leads to higher consumption in the short run but lower net worth and
hence lower consumption and utility in the long run.
Finally, we explore the welfare gain from the combination of having access
to a complete insurance market and being able to issue CAT bonds. The green
dashed line in Figure 5 plots the change in the welfare function at any given
value of m, and the last row of Table 2 shows the overall welfare change in the
long run. Table 2 shows that the combination of both forms of financial adaptation can increase the country’s welfare by 0.26% in consumption equivalent
In the quantitative exercise, we set the trigger threshold d¯ to be at the 90th percentile
of cyclone damage (or equivalently, the CAT clause is triggered if the cyclone activity in a
period is above the 90th percentile of its distribution).
To see this further, consider an extreme case of a country that always defaults in response to disasters above the CAT bond threshold. For that country, CAT bonds add little
insurance benefit.


units. That is, the combination can undo about a quarter of the total welfare
loss from the increased cyclone risk due to climate change. The fact that the
welfare gains from the financial adaptation instruments add up emphasizes the
ways in which the two instruments complement each other. On the one hand,
disaster insurance allows the country to smooth wealth across states and helps
the country rebuild capital after cyclone strikes. However, since the country
receives insurance payments regardless of its default decision, the presence of
insurance does not affect the default incentive and hence does not affect the
borrowing cost. On the other hand, CAT bonds are less useful for insurance
but help countries avoid costly defaults in disaster states and help reduce the
effect of the disaster risk on raising the country’s borrowing costs.



In summary, this paper provides a new analysis of the impacts of adverse
weather shocks on the debt and investment dynamics in emerging economies,
the welfare effects of climate change, and the effectiveness of financial adaptation strategies. We believe that the tractable model and its quantitative assessment can help policymakers better understand the channels through which
climate and financial risks interact. This insight, we hope, will enable them to
make well-informed decisions on approaches to reduce the impact of climaterelated natural disasters. Overall, our calibrated exercise underscores both the
nontrivial potential welfare gains and the limitations of financial adaptation.

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Omitted proofs


Proof of Proposition 1

Proof. First, let us prove (7). The country defaults if and only if m0R < m0D . From
net worth definitions (5) and (6), we know that this is the case if and only if the
debt over GDP is sufficiently high:

(k0 )α

> `(g 0 ). Given definitions (1), (3), and (4),

this is in turn equivalent to:

e−g bn


e−αx0 d0 −αg0 knα

¯ ψg ,
> `e

> e(1−α+ψ)g ,

which yields (7).
Given (7), the spread or probability of default is then given by (9):

s(bn , kn ) = (1 − p)Φg (ḡ(bn , kn )) + p


ḡ(bn , kn ) +
d dΦd (d0 ).

Taking partial derivatives:

= (1 − p)φg (ḡ) + p φg (ḡ +
d )dΦd (d )

= (1 − p)φg (ḡ) + p φg (ḡ +
d0 )dΦd (d0 )



1 − α + ψ bn
= −
1 − α + ψ kn

Combining these four equations yields the following expressions for the elasticity of
the spread function:
bn ∂ḡ
s ∂bn
kn ∂ḡ
s ∂kn
where h̃(ḡ) =


= −


(1−p)φg (ḡ)+p


φg (ḡ+ 1−α+ψ
d0 )dΦd (d0 )

(1−p)Φg (ḡ)+p


Φg (ḡ+ 1−α+ψ
d0 )dΦd (d0 )

, as stated in the proposition.

Proof of Proposition 2

Proof. Given (9), the partial derivative of s with respect to p is:
= −Φg (ḡ) +


d dΦd (d0 ).
ḡ +

Since the support of Φd is [0, ∞), it follows that


> −Φg (ḡ) + Φg (ḡ)dΦd (d0 ) = 0,

i.e., the spread is increasing in the probability of the bad weather shock, as desired.
Furthermore, suppose Φ̂d first-order stochastic dominates Φd . Let ŝ denote the
d0 )
spread function associated with the damage distribution Φ̂d . Since Φg (ḡ + 1−α+ψ

is increasing in d0 , it follows that E[Φg (ḡ +

1−α+ψ d )|Φ̂d ]

It then immediately follows that ŝ ≥ s, as desired.


≥ E[Φg (ḡ +

1−α+ψ d )|Φd ].