ï»¿ WPS6216
Policy Research Working Paper 6216
Background Paper to the 2014 World Development Report
An Exploration of the Link between
Development, Economic Growth,
and Natural Risk
StÃ©phane Hallegatte
The World Bank
Sustainable Development Network
Office of the Chief Economist
October 2012
Policy Research Working Paper 6216
Abstract
This paper investigates the link between development, disasters grow with income, and they grow faster than
economic growth, and the economic losses from natural income at low levels of development and slower than
disasters in a general analytical framework, with an income at high levels of development. These findings
illustration on hurricane flood risks in New Orleans. are robust to a broad range of modeling choices and
It concludes that, where capital accumulates through parameter values, to the inclusion of risk aversion, and to
increased density of capital at risk in a given area, (i) variations in the decision-making framework (including
the probability of disaster occurrence decreases with the introduction of prospect theory's decision weights,
income; (ii) capital at riskâ€”and thus economic losses in biases in risk perception and myopic expectations). They
case of disasterâ€”increases faster than economic growth; show that risk-taking is both a driver and a consequence
(iii) increasing risk-taking reinforces economic growth. of economic development, that risk taking should not be
Economic growth and improved protection transfer indiscriminately suppressed, and that the world is very
risks from frequent low-intensity events to rarer high- likely to experience fewer but more costly disasters in the
impact events. In this context, average annual losses from future.
This paperâ€”prepared as a background paper to the World Bankâ€™s World Development Report 2014: Managing Risks for
Developmentâ€”is a product of the Office of the Chief Economist, Sustainable Development Network. It is part of a larger
effort by the World Bank to provide open access to its research and make a contribution to development policy discussions
around the world. Policy Research Working Papers are also posted on the Web at http://econ.worldbank.org. The author
may be contacted at shallegatte@worldbank.org.
The Policy Research Working Paper Series disseminates the findings of work in progress to encourage the exchange of ideas about development
issues. An objective of the series is to get the findings out quickly, even if the presentations are less than fully polished. The papers carry the
names of the authors and should be cited accordingly. The findings, interpretations, and conclusions expressed in this paper are entirely those
of the authors. They do not necessarily represent the views of the International Bank for Reconstruction and Development/World Bank and
its affiliated organizations, or those of the Executive Directors of the World Bank or the governments they represent.
Produced by the Research Support Team
1
An exploration of the link between development,
economic growth, and natural risk
ephane Hallegatte 1
StÂ´
Revised version - March 1, 2013
Keyword: Development; Economic growth; Risk; Natural disaster; Economic losses.
JEL: O10; O44; Q01; Q54.
1 Introduction
The damages caused by Hurricane Sandy in Haiti and in the US show that poor and rich
countries are vulnerable to natural hazards. Large damages in New York City and New
Jersey â€” with a preliminary estimate of direct damages in excess of $50 billion â€” raise
questions on the level of coastal protection in these areas and on the rapid coastal devel-
opments that have driven so many households and so much capital and infrastructure in
risky areas. In spite of large ï¬?nancial and technical resources, the economic vulnerabil-
ity to hurricanes remain high in the richest country of the world, and statistical analyses
suggest it has been growing in the last hundred years (Pielke et al., 2008).
1
The World Bank, Sustainable Development Network, Ofï¬?ce of the Chief Economist, Washington D.C.,
USA (shallegatte@worldbank.org). This paper is a background paper for the 2014 World Development
Report â€œManaging risks for development.â€? The author wants to thank Paolo Avner, Michael Chaitkin,
Patrice Dumas, Marianne Fay, Mushï¬?q Mobarak, Antonin Pottier, David C. Popp, and Adrien Vogt-Schilb
for useful discussions on the content of this paper. The remaining errors are the authorâ€™s. The views
expressed in this paper are the sole responsibility of the author. They do not necessarily reï¬‚ect the views of
the World Bank, its executive directors, or the countries they represent.
2
This is not an isolated situation. Increasing investments in disaster risk reduction have
lead to a signiï¬?cant reduction in human casualties (Kahn, 2005; Kellenberg and Mobarak,
2008), but economic losses from natural disasters have been growing as fast or even faster
than economic growth in many countries; see for instance statistical analyses in Barredo
(2009), Miller et al. (2008), Neumayer and Barthel (2010), Nordhaus (2010), Pielke et
al. (2008), and Bouwer et al. (2007). Climate change is not responsible of this evolution
(Schmidt et al., 2009; Neumayer and Barthel, 2010; Bouwer, 2011; IPCC, 2011). The
trend in hurricane losses relative to wealth in the US is for instance fully explained by
migrations toward hurricane-prone areas and increasing population wealth (Pielke et al.,
2008). Globally, there is a trend toward more risk taking: between 1970 and 2010, global
population grew by 87%, but the population living in ï¬‚ood plains increased by 114% and
in cyclone prone coastlines by 192%. The GDP exposed to tropical cyclones increased
from 3.6 percent of total GDP to 4.3 percent over the same period (UN-ISDR, 2011).
This global trend in economic disaster losses is a major concern in many countries.
Disaster losses represent an increasing burden on economies and public ï¬?nances (e.g.,
Benson and Clay, 2004; Loayza et al., 2009; Fomby et al. 2011; Strobl 2011), and their
trend threatens the functioning of insurance and reinsurance markets (e.g., Michel-Kerjan,
2010). Cross-country studies have suggested that economic disaster losses are increas-
ing less rapidly than income, making them easier to manage (Toya and Skidmore, 2007;
Mendelsohn et al., 2011). But the trend in economic losses in the US and many other
countries suggests that the relationship between economic growth and disaster vulnera-
bility is more complex than these studies suggest, and that the economic vulnerability to
disasters may not always decrease with income (Schumacher and Strobl, 2011).
This paper investigates the interlinkages between economic disaster losses and de-
velopment. It considers the fact that higher income makes it possible to invest in better
protections against disasters, but may also lead to higher investment in at risk areas. It also
considers the role of investments in at risk areas in economic growth, for instance because
development of coastal areas is critical for export-led growth or because agglomeration
externality and urbanization (often in ï¬‚ood plains) are major drivers of development (Ci-
ccone et al., 1996; Ciccone, 2002; World Bank 2008). Compared with previous investiga-
tions of trends in disaster economic losses (e.g., Lewis and Nickerson, 1989; Schumacher
and Strobl, 2011), this analysis proposes an explicit modeling of risk taking (how much
is invested in risky areas?) and of protection investment (how much is invested in protec-
tion?), to characterize the two-way link between economic growth and disaster losses. It
does so in a more general framework than previous studies, and it conï¬?rms the robustness
of its results by investigating the role of risk aversion and different forms of behavioral
bias.
Within this framework, it is found that â€” even with no change in climate condi-
tions and hazard characteristics, and assuming rational decision-making and the absence
3
of wrong incentives from subsidized insurance and post-disaster support â€” risk-taking
generally increases with development. Natural disasters are likely to become more de-
structive in the future, even relative to income, and even in the presence of risk aversion.
Reciprocally, increased risk-taking reinforces economic growth, suggesting that risk tak-
ing should not be indiscriminately suppressed. Risk-taking is both a driver and a conse-
quence of economic development. In this context, average annual losses from disasters
grow with income, and they grow faster than income at low levels of development and
slower than income at high levels of development.
The next section describes the most generic model and the conditions under which
disaster losses increase with economic growth. Section 3 presents more detailed results
for speciï¬?c speciï¬?cation of protection costs, and Section 4 investigates special cases of
production functions. Section 5 applies this model to New Orleans, demonstrating that
reasonable parameter values lead to a situation where economic growth incentivizes risk
taking and increases disaster losses. Section 6 shows that these results are robust to the
introduction of risk aversion, and Section 7 that they are robust to imperfect decision-
making, through the introduction of systematic biases in risk perception and myopic ex-
pectations. Section 8 concludes.
2 Development and natural risks
It is generally accepted that richer populations invest more to protect themselves from
natural hazards. A richer population, however, may also invest more in at-risk areas,
increasing exposure to natural hazards. These two trends have opposite impacts on risk,
and the resulting trend in risk is thus ambiguous. This trend is investigated in this section
with a simple model.
2.1 A general economic growth framework
We assume there are two categories of capital. Capital R represents the capital related
to activities that need to be (or beneï¬?t from being) located in areas that are potentially at
risk of ï¬‚ooding. There are two categories of such capital. First, some activities directly
depend on being in risky areas, such as ports that are located in coastal areas or river
ï¬‚ood plains. Second, positive concentration externality is making it proï¬?table to invest
in at-risk areas even in sectors that are not directly dependent on being at risk â€” such
as ï¬?nance in New York City or manufacturing in Shenzhen â€” to beneï¬?t from spill-overs
from the industry that need to be located in ï¬‚ood-prone areas (such as ports): (i) lower
long-distance transport cost; (ii) infrastructure for local transportation, water, energy, and
communication; (iii) large labor market with access to skilled workers; and (iv) access to
public services and amenities (art and culture, schools, university, etc.).
2.1 A general economic growth framework 4
Capital S represents the rest of economy, which can be located in safe locations with-
out loss of productivity. These two capitals are inputs in the production function:
Y = eÎ³t F (R, S ) (1)
where t is time, F is a production function and Î³ is the exogenous growth in to-
tal factor productivity. Classically, we assume that âˆ‚R F (R, S ) > 0; âˆ‚S F (R, S ) > 0;
2 2
âˆ‚R F (R, S ) < 0 and âˆ‚S F (R, S ) < 0 (decreasing returns).
The capital R can be affected by hazards, like ï¬‚oods and windstorms. If a hazard
is strong enough, it causes damages to the capital installed in at-risk areas, and can be
labeled as a disaster. We assume that in that case, a fraction X of capital R is destroyed.
It is assumed that this is the only consequence of disasters.2
These disasters (i.e. hazards that lead to capital destruction) have a probability p0
of occurring every year, except if protection investments reduce this probability. These
protection investments take many forms, depending on which hazard is considered. Flood
protections include dikes and seawalls, but also drainage systems to cope with heavy
precipitations in urban areas. Windstorm protections consist mainly in building retroï¬?ts
and stricter building norms, to ensure that old and new buildings can resist stronger winds.
It is assumed that better defenses reduce the probability of disasters, but do not reduce
their consequences.3 This is consistent with many types of defenses. For instance, sea-
walls can protect an area up to a design standard of protection but often fail totally if this
standard is exceeded; building norms allow houses to resist up to a certain wind speed,
but when this wind speed is exceeded, houses are completely damaged and require total
rebuilding. This modeling choice is made without loss of generality, if there is no risk
aversion.4 In that case, indeed, reducing the probability of a disaster or the consequences
in case of disaster is equivalent.
Better defenses are also more expensive, and the annual cost of defenses C increases
when the remaining disaster probability p decrease.5 The function C (p, R) is assumed
twice differentiable, C (p0 , R) = 0 (the probability of occurrence is p0 in the absence
of protections), âˆ‚R C (p, R) â‰¥ 0 (it is as or more expensive to protect more capital),
2
Disaster fatalities and casualties are not considered in this simple model, assuming that early warning,
evacuation and emergency services can avoid them, which is consistent with the observation that disaster
deaths decrease with income, at least above a certain income level (Kahn, 2005; Kellenberg and Mobarak,
2008). Human losses could be taken into account if it is assumed that fatalities and casualties can be
measured by an equivalent economic loss, which is highly controversial; see a discussion in Viscusi and
Aldy (2003). The analysis remains valid with indirect losses, but only if the latter increase linearly with
direct losses (which is inconsistent with some analyses, see Hallegatte 2008 and Henriet et al. 2012).
3
This is equivalent to the self-protection of Ehrlich and Becker (1972).
4
The role of risk aversion is investigated in Section 6.
5
The probability p here includes both the probability that an event exceeds protection capacities, and the
defense failure probability, even for weaker events.
2.2 Optimal choice of p, R, and S 5
2
âˆ‚p C (p, R) â‰¤ 0 (the cost increase when the probability decrease), âˆ‚p C (p, R) â‰¥ 0 and
C (0, R) = +âˆž (the marginal cost is increasing and it is impossible to reduce the prob-
2
ability to zero) and âˆ‚pR C (p, R) â‰¤ 0 (the cost of protecting more capital increases when
the probability decreases).
Any given year, the economic surplus Ï€ is given by:
Ï€ = eÎ³t F (R, S ) âˆ’ C (p, R) âˆ’ L âˆ’ r (R + S ) (2)
where r is the interest rate, L is the losses from disasters, and is given by a random
draw with probability p. If a disaster occurs, losses are equal to XR, i.e. a fraction X of
the capital located in the risky area is destroyed. Any given year, the expected loss E[L]
is equal to pXR and the expected output is equal to:
E[Ï€ ] = eÎ³t F (R, S ) âˆ’ C (p, R) âˆ’ (r + pX )R âˆ’ rS (3)
Note that in this equation, disaster losses appear as an additional cost of capital at risk,
in addition to the interest rate r .
We also deï¬?ne the risk-free situation as a situation in which there is no risk, either
because p0 = 0 (no hazard), because X = 0 (no vulnerability), or because C (p, R) = 0
(costless protections). In the risk-free situation, there are still two capitals R and S , but
none of them is at risk.
2.2 Optimal choice of p, R, and S
We assume that a social planner â€” or an equivalent decentralized decision-making pro-
cess â€” decides which amounts of capital R and S are to be located in the risky and safe
areas, and the level of protection (p and C (p, R)) that is to be built. Its program is:6
maxp,R,S E[Ï€ ]
(4)
s.t.0 â‰¤ p â‰¤ p0
We assume ï¬?rst that there is no risk aversion and we assume that the expected surplus
is maximized. From the social plannerâ€™s perspective, doing so is acceptable if disaster
losses remain small compared with aggregated income, consistently with the Arrow-Lind
theorem for public investment decisions (Arrow and Lind, 1970). As discussed in Mahul
and Ghesquiere (2007), this theorem holds only if some conditions are met, including if
disaster losses can be pooled among a large enough population (e.g., a large country), and
6
This model is more general than the model of Schumacher and Strobl (2011). In the latter, the only
decision concerns protection investments that mitigate disaster consequences, and there is no beneï¬?t from
taking risks and thus no trade-off between safety and income. This model also differs from Hallegatte
(2011) in that it is more general on the shape of production and protection cost functions, and it introduces
the interest rate to account for the consumptionâ€“investment trade-off.
2.2 Optimal choice of p, R, and S 6
with many other uncorrelated risks, i.e. in the presence of comprehensive insurance cov-
erage or post-disaster government support, or if disaster losses can be smoothed over time
thanks to savings and borrowing (i.e. self-insurance) or reinsurance. In other terms, the
optimal pathways determined by this analysis are valid assuming that the social planner
ensures that individual losses remain small thanks to temporal smoothing and redistribu-
tion or insurance across individuals. Section 6 investigates the case with risk aversion.
If p = p0 , then there is no protection in place â€” because protections are too expensive
â€” and the situation is highly simpliï¬?ed: disaster risk reduces by a ï¬?xed fraction the
productivity of the capital at risk. Classically, this reduces the amount of such capital
without inï¬‚uencing its growth rate on the balanced growth pathway. In the following, we
assume that p < p0 .
First order conditions lead to the optimal values of p, R, and S :
eÎ³t âˆ‚R F (R, S ) âˆ’ âˆ‚R C âˆ’ (pX + r ) = 0 (5)
eÎ³t âˆ‚S F (R, S ) âˆ’ r = 0 (6)
âˆ‚p C = âˆ’XR (7)
While the marginal production of capital S is r , the marginal production of capital R
is âˆ‚R C + pX + r , i.e. the cost of capital r plus the capital losses due to disasters pX plus
the incremental cost of protection âˆ‚R C . The term âˆ‚R C + pX + r is what we deï¬?ne as the
risk-adjusted cost of capital, and it is larger than the risk-free cost of capital, to account
for natural risks.7
Since pX > 0, âˆ‚R C > 0, and âˆ‚R F is decreasing, the ï¬?rst equation shows that the
presence of risk (X > 0 and C > 0) leads to a reduction in R.
Proposition 1 The presence of risk and the possibility to protect lead to a reduction in
the capital that is located in the risky area, compared with the risk-free situation.
Taking the derivative of the three equations with respect to time t, one gets:
2 2 2 2
Î³eÎ³t âˆ‚R F + [eÎ³t âˆ‚R F âˆ’ âˆ‚R C ]âˆ‚t R + eÎ³t âˆ‚RS F âˆ‚t S = (X + âˆ‚pR C )âˆ‚t p (8)
Î³t 2 2
Î³e âˆ‚S F + eÎ³t âˆ‚S F âˆ‚t S + eÎ³t âˆ‚RS F âˆ‚t R = 0 (9)
2 2
âˆ‚t pâˆ‚p C = âˆ’(X + âˆ‚pR C )âˆ‚t R (10)
2
Equation (10) shows that if (X + âˆ‚pR C ) < 0, then an increase of at-risk capital
(everything else being unchanged) leads to a decrease in protection (i.e. an increase in
7
Equivalently, one can deï¬?ne the risk-adjusted marginal productivity of capital as the marginal produc-
tivity of capital reduced by the cost of protection and disaster capital losses: eÎ³t âˆ‚R F (R, S ) âˆ’ âˆ‚R C âˆ’ pX .
2.2 Optimal choice of p, R, and S 7
the probability of occurrence), because the cost of protection then increases more rapidly
2
with R and p than the avoided disaster losses. If X + âˆ‚pR C = 0, then the probability of
occurrence is independent of R, and thus constant over time even in presence of economic
growth.
If we have
2
(X + âˆ‚pR C) > 0 , (11)
then having more capital at risk leads to an increase in protection, and the probability
of occurrence p and the amount of capital at risk R evolve in opposite direction. In the
following, we assume that this condition is met.
Replacing âˆ‚t p in Eq. (8) and replacing eÎ³t âˆ‚R F by (r + pX + âˆ‚R C ) gives:
2 F
âˆ‚RS
Î³ (r (1 âˆ’ 2F )
âˆ‚S
+ pX + âˆ‚R C )
âˆ‚t R = 2 F )2 2 C )2
(X +âˆ‚pR
(12)
2 (âˆ‚RS 2
âˆ’eÎ³t (âˆ‚R Fâˆ’ 2F
âˆ‚S
) + âˆ‚R Câˆ’ 2
âˆ‚p C
2 F
âˆ‚RS
Since 2F
âˆ‚S
< 1, the capital at risk increases over time when:
2
Î³t 2
2
(âˆ‚RS F )2 2
(X + âˆ‚pR C )2
âˆ’e (âˆ‚R F âˆ’ 2
) > âˆ’âˆ‚R C + 2C
(13)
âˆ‚S F âˆ‚p
If this inequality is not satisï¬?ed, then development and growth leads to a reduction in
capital at risk. It probably corresponds to locations that are too costly to protect against
natural disasters and where the optimal solution is â€œstrategic retreatâ€?, i.e. the abandonment
of human settlements.
To interpret this inequality, we can disregard for now the interactions between R and
2
S (i.e. assuming that âˆ‚RS F = 0) and assume that protection costs are independent of R
2 2
(i.e. âˆ‚R C = 0 and âˆ‚pR C = 0). In this case, the capital at risk increases over time when:
2 X2
âˆ’eÎ³t âˆ‚R F > 2C
(14)
âˆ‚p
This inequality is veriï¬?ed if the marginal productivity of the capital at risk decreases
more rapidly with R than the marginal cost of protection increases with p. In this same
situation, the probability of disaster decreases with economic growth.
2
Proposition 2 If X + âˆ‚pR C > 0 and if the concavity of the production function is high
enough â€” or if the convexity of the protection costs is high enough â€” then economic
growth leads to an increase in capital at risk (i.e. an increase in losses when a disas-
ter occurs), and a decrease in the probability of occurrence. If the concavity is lower,
then economic growth leads to a decrease in capital at risk (and an increase in disaster
probability).
2.3 Trend in average annual losses 8
It is interesting to note that â€” counter-intuitively â€” the capital at risk increases over
time when the concavity is high (i.e. returns of capital at risk are rapidly decreasing), and
decreases otherwise.
Note that if development leads to a reduction in capital at risk R and an increase in p,
then at one point the economy reaches a situation where there is no protection and p = p0 .
In such a situation, as already stated, capital at risk R grows at the same rate than in the
risk-free situation.
The capital R in the risk-free situation is referred to as Rs , and its evolution is:
2 F
âˆ‚RS
Î³ (r (1 âˆ’ 2 F ))
âˆ‚S
âˆ‚t Rs = 2 F )2
(âˆ‚RS
(15)
2
âˆ’eÎ³t (âˆ‚R Fâˆ’ 2F
âˆ‚S
)
Assuming that the capital R increases with economic growth, i.e. that condition (13) is
veriï¬?ed, then the comparison of Eqs. (12) and (15) explains how the protection inï¬‚uences
the evolution of capital:
â€¢ The term pX + âˆ‚R C in the numerator is the impact of risk and protection marginal
costs on marginal productivity; it increases the numerator and accelerates the abso-
lute growth in R.
2
â€¢ The term âˆ‚R C in the denominator is the decreasing or increasing return on protec-
tion; if the returns on protection are constant (e.g. C (p, R) = RC (p) or C (p, R) =
C (p)), then this term does not exist; if the returns are decreasing (i.e. costs are
2
convex and âˆ‚R C > 0), then this term increases the denominator and slows down
the growth in R; if the returns are increasing, then the growth in R is accelerated.
(X +âˆ‚ 2 C )2
â€¢ The term âˆ’ âˆ‚ 2pR in the denominator is the impact of the change in protection
pC
that is provided if more capital is installed in at risk areas (if the probability of
2
occurrence is ï¬?xed, this term does not appear). Since âˆ‚p C > 0, this term is negative
and reduces the denominator and thus accelerate the growth in R.
2.3 Trend in average annual losses
The average economic losses due to disasters are equal to E[L] = pXR.
âˆ‚t E[L] = XRâˆ‚t p + pXâˆ‚t R (16)
It can be rewritten:
ï£® ï£¹
2
pâˆ‚p C
âˆ‚t E[L] = âˆ‚t p ï£°âˆ‚p C âˆ’ 2 C
âˆ‚pR
ï£» (17)
1+ X
9
2
Since âˆ‚p C < 0 and âˆ‚p C > 0, and under condition (11), then E[L] and p evolve in
opposite directions.
2
Proposition 3 If X + âˆ‚pR C > 0, then average annual disaster losses increase when the
probability of occurrence decreases over time.
This result highlights the need to consider the combination of exposure (R) and prob-
ability (p) to investigate risks. In particular, a reduction in the probability of occurrence
does not mean that average losses decrease; on the opposite, this general analysis suggests
that under mild conditions a decrease in the probability of occurrence leads to an increase
in average losses, because of the increase in capital at risk.
Importantly, this analysis illustrates that protection reduces the hazard (the probabil-
ity of occurrence of an event), but its impact on risk is more complex, because it also
increases exposure (here, the capital at risk R). As a result, protection transfers part of the
risk from one kind of risk (frequent and low-cost events) to another kind (exceptionnal
and high-impact events), a process already stressed in Etkin (1999).
3 Special cases for protection costs
We already made assumptions on the shape of C (p, R), but it is useful to explore two
extreme cases for the dependence of C (p, R) to R.
In a ï¬?rst case, we can consider a coast or a river, where additional capital investments
are done at a ï¬?xed density and are thus using additional land, which in turn requires
additional protection. In such a case, the protected areas increases proportionally with
the invested capital in the risky zone, and C (p, R) = RC Ëœ (p). This is the case in Florida,
or in the south of France, where population density is low and ï¬‚ood exposure increases
mainly through the construction of individuals houses, at low density. This situation can
be labeled â€œhorizontalâ€? or â€œarea-increasingâ€? accumulation.
In a second case, we consider a given risky areas, which is protected against coastal
ï¬‚oods and where investment takes place. In such a case, the risky and protected area does
not increase with investments, and the cost of protection is independent of the amount of
protected capital: C (p, R) = C Ëœ (p). This is notably the case where additional investments
take place through higher concentration and density, on a given area. Examples of such
places are the Netherlands, New Orleans, or Manhattan in New York City. This situation
can be labeled â€œverticalâ€? or â€œdensity-increasingâ€? accumulation.
Let us explore the consequence of these two patterns on ï¬‚ood risks.
3.1 Horizontal accumulation 10
3.1 Horizontal accumulation
In that case, we can assume that the protection cost function has the form C (p, R) =
RCËœ (p). The marginal cost and beneï¬?t of protection are equal (Eq. (7)), which means
Râˆ‚p CËœ (p) = âˆ’XR. Therefore, p is independent of R and constant over time.
2
Deriving the previous equation with respect to R gives X + âˆ‚pR C = 0, and we can
rewrite Eq. (5) as:
râ€²
Ëœ (p) + pX + r
eÎ³t âˆ‚R F (R, S ) = C (18)
Since C Ëœ (p) is constant and positive, the risk-adjusted marginal productivity r â€² is also a
constant, larger than r . In this case, R evolves like a capital without risk, but with a larger
interest rate r â€² instead of r . Since marginal productivity needs to be larger, the amount of
capital is lower in presence of decreasing returns (R < Rs ), i.e. risk leads to a reduction
in capital R. With classical production functions (CES or Cobb-Douglas) and neutral
technological change, the capital at risk R increases at the same rate as economic growth.
Since p is constant, âˆ‚t E[L] = pXâˆ‚t R, and average annual losses grow at the same rate as
capital at risk and as risk-free economic growth.
Proposition 4 In horizontal-accumulation locations â€” i.e. where ï¬‚ood exposure in-
creases because the developed area at risk is expanded and where protection costs in-
crease therefore proportionally with protected capital â€” rational decision-making leads
to annual ï¬‚ood losses growing at the same rate as economic growth, with a constant
ï¬‚ood probability, regardless of how protection costs vary with the residual probability of
occurrence.
3.2 Vertical accumulation
In that case, we can assume that C (p, R) = Î¾pâˆ’Î½ + C0 . Assuming p < p0 , we can use
Eq. (7) to ï¬?nd:
1
âˆ’ 1+
RX Î½
p= (19)
Î½Î¾
2 2
We have âˆ‚pR C = 0, and thus X + âˆ‚pR C > 0 if there is risk, so that condition (11) is
always veriï¬?ed. As a consequence, we know that p and R evolve in opposite directions:
if capital increases (resp. decreases), the protection is improved (resp. weakened) and
disaster probability decreases (resp. decreases). We are in the situation where p decreases
and R increases when condition (13) is veriï¬?ed, and it can be rewritten:
2
2 (âˆ‚RS F )2 X 2 pÎ½ +2
âˆ’eÎ³t (âˆ‚R Fâˆ’ 2
) > (20)
âˆ‚S F Î½ (Î½ + 1)Î¾
11
Also, we know from the general analysis that âˆ‚t E[L] is positive and average disaster
losses are increasing over time.
Proposition 5 In locations where ï¬‚ood exposure rises as a result of increased density in a
given protected area (e.g., New Orleans), rational decision-making results in a continuous
increase in annual ï¬‚ood losses, when ï¬‚ood probability decreases over time.
To know whether capital at risk and ï¬‚ood losses grow more or less rapidly than eco-
nomic growth, assumptions are needed on the shape of the production function. This is
what is investigated in the next section.
4 Special cases for the production function
In this section, we keep the â€œvertical-accumulationâ€? assumption, i.e. C (p, R) = Î¾pâˆ’Î½ +
C0 . All qualitative results are however valid if C (p, R) = Î¾Rc pâˆ’Î½ + C0 , where 0 â‰¤ c < 1.
We will assume ï¬?rst that capital at risk R and safe capital S are separable inputs in
the production function, and then show that results can be generalized to other production
functions.
2
If F (R, S ) = f (R) + g (S ), then âˆ‚RS F = 0, and all equations can be simpliï¬?ed.
The evolution of the capital at risk is:
Î³ (r + pX + âˆ‚R C )
âˆ‚t R = 2 C )2
(X +âˆ‚pR
(21)
2 2
âˆ’eÎ³t âˆ‚R f (R) + âˆ‚R Câˆ’ 2C
âˆ‚p
In absence of risk and protection, the evolution would be:
Î³r
âˆ‚t Rs = 2
(22)
âˆ’eÎ³t âˆ‚R f (R s )
We can also calculate âˆ‚t S as:
Î³r
âˆ‚t S = 2
(23)
âˆ’eÎ³t âˆ‚S g (S )
If we now assume that f (R) = Î»RÂµ and g (S ) = Î±Î»S Âµ , we can solve Eq. (6):
1 Î³t âˆ’1
S = (Î±Î»Âµ) 1âˆ’Âµ e 1âˆ’Âµ r 1âˆ’Âµ (24)
and we have:
âˆ‚t S âˆ‚t RS Î³
= = (25)
S RS 1âˆ’Âµ
So, with this shape of production function, a productivity growth at rate Î³ leads to a
Î³ Î³
risk-free economic growth at rate 1âˆ’ Âµ
(i.e. Rs and S grow at rate 1âˆ’ Âµ
).
4.1 Low development level 12
1
âˆ’ 1+ Î½
RX
With p = Î½Î¾
, we can rewrite Eq. (5):
1
âˆ’ 1+
X Î½ 1
eÎ³t Î»ÂµRÂµâˆ’1 = r + pX = r + X Râˆ’ 1+Î½ (26)
Î½Î¾
This equation cannot be solved analytically, but its behavior in special cases can be
analyzed.
4.1 Low development level
If productivity is low, capital is also limited (R is small) and thus the probability of occur-
rence of a disaster is large (p is large).8 In such an extreme situation, the capital interest
rate r is small in the at-risk area compared with the ï¬‚ood-related capital losses pX , and
Eq.(26) can be simpliï¬?ed by removing r , leading to the solution:
Î³
Î½ t
R(t) = R0 e 1+Î½ âˆ’Âµ (27)
Î½
If Âµ > 1+ Î½
, R decreases and p increases over time. It is likely that this situation
corresponds to locations that are abandoned over time, since they are too expensive to
protect against natural disasters.
Î½
In the situation in which protection improves over time, Âµ < 1+ Î½
, the growth rate
Î³ Î³
of capital at risk is Î½ âˆ’Âµ , which is always larger than 1âˆ’Âµ . In this case, therefore, the
1+Î½
capital at risk grows more rapidly than the safe capital and risk-free income. The relative
vulnerability of the economy can be measured by the amount of disaster losses when a
disaster occurs divided by income or by the â€œfraction at riskâ€?, i.e. the share of capital at
risk R in total capital R + S . This vulnerability is increasing over time, as shown in Fig.1
with the fraction at risk. Interestingly, the growth in capital at risk is more rapid when Î½ is
smaller, i.e. when the convexity of protection costs is lower and protection costs increase
slowly with the protection level.
Average losses pXR have a growth rate equal to:
Î³
Î³L = (28)
1 âˆ’ Âµ 1+
Î½
Î½
Î½
In the situation in which protection improves over time, Âµ < 1+ Î½
, average losses
Î³
increase over time, and they increase more rapidly than 1âˆ’Âµ and thus more rapidly than
8
We assume here that p0 is large, i.e. that the area-at-risk has a large ï¬‚ood probability in absence of
protection, and that p < p0 . If productivity is so low that p = p0 , then pX can be replaced by p0 X , which
is independent of R. Then, Eq. (26) can be simpliï¬?ed by replacing its right-hand-side by r + p0 X = râ€² .
In this situation, the capital at risk is lower than in absence of risk (R < Rs ), but it grows at the same
rate. When development increases productivity, there is a time when protection is such that p < p0 , and the
following calculation holds if p0 is large enough.
4.1 Low development level 13
30
25
Fraction of capital at risk (R/(R+S)) (%)
20
15
10
5
0 âˆ’1 0 1 2 3
10 10 10 10 10
Aggregated income (billion USD)
Figure 1: Evolution of the â€œfraction at riskâ€?, i.e. the share of capital at risk R in total
capital R + S , as a function of time. The fraction at risk increases with development, until
it stabilizes at high development level. Calculations using numerical values from New
Orleans (see Section 5) and Î± = 2.
4.2 High development level 14
risk-free economic growth, i.e. the growth rate of Rs and S (in other terms, the income
elasticity of average annual losses is larger than one).
So, in this case, annual disaster losses grow more rapidly than risk-free economic
growth at low levels of development, when disaster losses dominate the interest rate in the
assessment of the cost of capital, and when the returns on capital at risk are decreasing
rapidly, more rapidly than a limit value deï¬?ned by the shape of the production costs.
In that case, the economic surplus (generated by the at-risk capital) is
E[Ï€ ] = eÎ³t F (R, S ) âˆ’ C (p, R) âˆ’ (r + pX )R
Î³
and is growing at the same rate as losses, i.e. 1âˆ’Âµ 1+Î½ . It means that the process of
Î½
increasing risk-taking leads to a growth in economic surplus that is more rapid than risk-
free economic growth. So, increasing risk-taking is also a driver of economic growth
(even though the presence of risk leads to a lower output, see proposition 1).
Note that this case is equivalent to the case explored in Hallegatte (2011) where the
total amount of capital (R + S ) is ï¬?xed at an exogenous level K , and is independent of
the risk level (which is equivalent to r = 0 provided that R â‰¤ K ). It is a situation in
which there is no consumptionâ€“investment trade-off, and in which the capital at risk can
keep increasing more rapidly than growth, until all the capital is located in the risky area.
What follows shows that accounting for the consumptionâ€“investment trade-off changes
signiï¬?cantly the results at high level of development.
4.2 High development level
At a high development level, capital productivity is large, then the amount of capital
at risk R is large.9 As a consequence, the probability of occurrence (p) is small. In
such an extreme situation, the capital interest rate r is large compared with ï¬‚ood-related
capital losses pX , and Eq.(26) can be simpliï¬?ed by removing pX , leading to the risk-free
solution:
Î³
R(t) = R0 e 1âˆ’Âµ t (29)
At high level of development, when disaster probability is very low, the capital at risk
grows at the same rate than economic growth. This is why the fraction at risk stabilizes at
high income, as shown in Fig.1.
When productivity is high, the average losses pXR has a growth rate equal to:
9
Using a reductio ad absurdum argument, it is easy to show from Eq. (26) that R tends toward inï¬?nity
when productivity grows. Assume that R is bounded when t â†’ +âˆž. In this case, the left-hand-side of
Eq. (26) tends toward inï¬?nity when t increases, so the right-hand-side has to do the same. In that case, r
becomes negligible over time, and the solution of Eq. (26) tends toward Eq. (27), which is not bounded
when t â†’ +âˆž. This is in contradiction with our initial hypothesis.
4.2 High development level 15
Î³ Î½
Î³L âˆ’â†’ (30)
tâ†’+âˆž 1 âˆ’ Âµ (1 + Î½ )
Average annual losses are thus growing less rapidly than economic growth at high
level of development, but they never decrease in absolute terms. At high development
level, the growth rate in annual disaster losses is equal to economic growth multiplied by
a â€œprotection factorâ€? (Î½/(1+ Î½ )), which depends only on the shape of the protection costs
and is lower than one.
The protection factor is also the income elasticity of average disaster losses (in a given
region, for a given hazard). It should not be confused with the income elasticity of disaster
damages (when a disaster occurs), which is larger than one.
In that case, the different terms of the economic surplus from at-risk capital (E[Ï€ ] =
Î³t
e F (R, S ) âˆ’ C (p, R) âˆ’ (r + pX )R) are growing at different rates. The production
Î³
eÎ³t F (R, S ) and capital cost rR are growing at a rate 1âˆ’ Âµ
. The protection costs and average
Î³ Î½
losses pXR are growing at a rate 1âˆ’Âµ 1+Î½ , i.e. more slowly than production. It means that
Î³
when productivity tends to the inï¬?nity, the economic surplus is growing at the rate 1âˆ’ Âµ
,
i.e. at the rate of risk-free economic growth.
4.2.1 Development and disaster trends
Appendix A shows that these results are unchanged if R and S are two production factors
in a Cobb-Douglas function. This analysis leads to three conclusions, concerning the
trends in capital at risk, average annual losses, and economic surplus.
Proposition 6 If (i) capital at risk and safe capital can be separated in the production
function or are two factors in a Cobb-Douglas function, (ii) protection costs grow less
than proportionnally with the amount of capital to protect, and (iii) capital returns are
decreasing more rapidly than a threshold that depends on the convexity of protection cost
(i.e. if Âµ < Î½/(1 + Î½ )), then economic losses in case of disaster (R) grow more rapidly
than risk-free economic growth, but their rate of growth converges toward the rate of risk-
free economic growth as development proceeds. The relative vulnerability of the economy
(the â€œfraction at riskâ€?) is thus increasing over time.
Proposition 7 The average annual disaster losses follow a bell-shaped curve in relative
terms with income: disaster losses are growing more rapidly than risk-free economic
growth at low stages of development and then keep growing in absolute terms but more
slowly than risk-free economic growth at higher productivity levels. At high productiv-
ity, the growth rate of annual losses is the risk-free economic growth rate reduced by a
â€œprotection factorâ€? that depends only on the convexity of protection costs.
16
Proposition 8 The presence of risk reduces the economic surplus. But because of in-
creasing risk-taking, the growth rate of economic surplus is larger than risk-free eco-
nomic growth at low development level, and it tends to risk-free economic growth at high
development level.
As already stated, this result can be generalized to cases where C (p, R) = Î¾Rc pâˆ’Î½ +
C0 and 0 â‰¤ c < 1. If c = 1, then protection is constant and average losses and capital at
risk grow at the same rate than economic growth.
Importantly, we do not need capital to be â€œmore productiveâ€? in at-risk areas to have
capital at risk increasing more rapidly than risk-free economic growth and the capital
located in safe areas. We only need a separable production function with two categories
of capital (R and S ), which are imperfectly substituable and have decreasing returns, and
R related to activities located in risky areas. In that case, there is an incentive to invest in
at-risk areas to beneï¬?t from high marginal returns at low capital levels. Results are then
independent of the relative productivity Î± of the two capitals. The productivity ratio (Î±)
determines the ratio between the two capitals (S/R) at high development level.
5 Numerical application to New Orleans
We apply these formulations to the case of New Orleans, using the following illustrative
assumptions:
â€¢ The capitals are separable, and we consider only the capital R in the ï¬‚ood-prone
area of New Orleans. the capital S is the capital located in safe areas in the rest of
the region or the country. The interest rate is r = 5 %.
â€¢ The area is ï¬?xed, and the protection costs depend only on the probability of occur-
rence p, not on the amount of capital to protect: C (p, R) = Î¾pâˆ’Î½ + C0 . In that case,
condition (11) is always veriï¬?ed, and p and R evolve in opposite directions.
â€¢ In absence of protection, the city would be ï¬‚ooded every year (p0 = 1).
â€¢ The cost of protecting New Orleans against category-3 storms is about $3 billion
in investment, and we assume a 10% per year maintenance cost; the probability of
such a storm is one out of 50 years. The annualized protection cost is C (1/50) =
$450 million per year, taking into account the cost of capital and maintenance costs.
â€¢ Protecting New Orleans against category-5 hurricane ï¬‚oods would cost about $30
billion, with also a 10% per year maintenance cost, and the probability of such a
storm is one out of 200 years, so that C (1/200) = $4.5 billion per year.10
10
State ofï¬?cials estimated the cost of Category 5 protection between $2.5 and $32 billion (Carter, 2005;
17
Even though each of these assumptions can be discussed, they provide an order of
magnitude for the cost of protecting the city. Using these assumptions, we have:
C ( p) = Î¾ pâˆ’ Î½ âˆ’ pâˆ’
0
Î½
, (31)
with p0 = 1, Î½ = 1.66, and Î¾ = 6.8 Â· 10âˆ’4.
For New Orleans, we assume that 50% of capital at-risk is lost in case of ï¬‚oods.
This large fraction translates the facts that part of the city is under normal sea level and
stayed ï¬‚ooded for weeks after Katrina hit the city, amplifying damages to houses and
buildings, and that ï¬‚oods occur through dike failure, leading to high water velocity and
large damages (RMS, 2005).
The production function is Y = eÎ³t F (K ) = Î»eÎ³t RÂµ , with Âµ = 0.3 and Î³ = 0.015
(total factor productivity grows by 1.5% per year). The risk-free growth rate, i.e. the
Î³
growth in the capital S located in safe areas in the rest of the country, is 1âˆ’ Âµ
â‰ˆ 2.1% per
year.
The variable Y is the local GDP in the ï¬‚ood-prone areas of New Orleans. Using
a New Orleans exposed population of 500,000 people, and the GDP per capita of New
Orleans ($24,000 per year in 2009), we have a city GDP of $12 billion. To estimate Î»,
we solve numerically Eq. (26) to ï¬?nd R as a function of Î», and we chose Î» so that the
income in absence of disaster is Y = Î»RÂµ = $12 billion (in economic data, protection
expenditures are included in income). The value is Î» = 3.53. We ï¬?nd that the optimal
capital at risk in New Orleans is R = $59 billion, i.e. about 5 times the local income.
Losses in case of ï¬‚ood would be about $30 billion, which is consistent with data for the
ï¬‚ood due to Katrina (removing losses due to wind) (RMS, 2005). The optimal probability
of ï¬‚ood is found at 2.2% per year, i.e. a return period of 45 years (which is also close to
the return period of a category-3 hurricane, i.e. the current protection level in the city).
Then, we can solve numerically Eq. (26) for a series of t, to investigate the dependence
of risk to income. We can calculate the trend in R(t) and in average annual loss. We ï¬?nd
that R is growing at a rate 2.4% against 2.1% for the growth without risk. Average annual
losses due to ï¬‚oods are growing at a rate 1.5%, slower than economic growth. Thanks to
increased risk taking, the economic surplus Ï€ is growing at a rate 2.2%, i.e. more rapidly
than risk-free economic growth.
In the case of New Orleans, therefore, a rational decision-maker would make average
disaster losses increase less rapidly than economic growth (1.5 vs. 2.1%), but would
increase capital at risk more rapidly than risk-free growth (2.4 vs 2.1%). The consequence
is that the average cost of disasters is decreasing relative to regional or national wealth,
but the consequences when a ï¬‚ood occurs increase, even relatively to wealth. It means
that the New Orleans region evolves toward fewer disasters with consequences that are
Revkin and Drew, 2005; Schwartz, 2005). More recent and detailed estimates by Louisiana Coastal Protec-
tion and Restoration (LACPR, led by the U.S. Army Corps of Engineers) reach even larger values.
18
4
Current local income
24 billion USD
3.5
Annual growth rate of
capital at risk R
3
Growth rate (%)
2.5
Surplus growth rate
2 riskâˆ’free
economic
growth
1.5 Annual growth rate of
average annual losses
1 0 1 2 3
10 10 10 10
Aggregated income (billion USD)
Figure 2: Annual growth rate of capital at risk R, average annual disaster losses, and
economic surplus growth rate, as a function of local income. The horizontal line is the
rate of risk-free economic growth. Growths in capital at risk and in economic surplus
are more rapid than economic growth at early development stage, and these growth rates
converge toward economic growth rate over time. Growth in average annual losses is
faster than risk-free economic growth at low income and slower at high income. The
vertical dashed line shows the current income in New Orleans.
19
0.9
0.8
0.7
Annual probability of occurrence
0.6
0.5
0.4
0.3
0.2
0.1
0 âˆ’1 0 1 2 3
10 10 10 10 10
Aggregated income (billion USD)
Figure 3: Optimal disaster probability of occurrence in New Orleans, as a function of
local income.
growing relative to income, leading to an increased need for recovery and reconstruction
support.
One can investigate how this result depends on the local income in New Orleans (and
thus on total factor productivity), assuming that everything else remains unchanged (in-
cluding protection costs C (p) and the fraction of capital at risk lost in case of ï¬‚ood X ).
Results are reproduced in Fig. 2 and 3. Figure 2 shows the annual growth rate of capital
at risk R, of average annual losses (pXR), and of the economic surplus, as a function of
local income Y . The vertical line shows the current income in New Orleans, and the hor-
izontal line the rate of risk-free economic growth. At very low productivity, the growth
in capital at risk R would be 4.6% per year, i.e. more than twice the rate of risk-free
economic growth. In this situation, the annual probability of ï¬‚ood would exceed 80%
(see Fig. 3). This growth then converges toward the rate of risk-free economic growth
(the horizontal dashed line) as development proceeds. The economic surplus growth rate
is also larger than the risk-free economic growth rate at low development level, and it
converges toward it as development proceeds.
Figure 1 shows how the fraction at risk â€” i.e. the fraction of capital located in at
risk area â€” increases with income, assuming Î± = 2 (i.e., that in the absence of risk, this
fraction would be 38 percent, according to Eq. (24)).
For average annual disaster losses, the growth rate is 2.8% per year at low development
level, i.e. 40% faster than economic growth. At an income of 40 million USD per year in
20
the city, the growth rate of annual losses is equal to the rate of economic growth (2.1%),
and this growth rate keeps decreasing until 1.3% per year, which is economic growth
(2.1%) corrected by the â€œprotection factorâ€?, equal to 0.62 in the case of New Orleans.
In this case, therefore, the income elasticity of average disaster losses would be equal
to 0.62 (i.e. a 1% growth in the US would lead to a 0.62% growth in average annual losses)
and the income elastricity of disaster losses (when a disaster occurs) would be equal to
1.1 (i.e. a 1% growth in the US would lead a 1.1%-increase in the size of disasters when
they occur).
6 Taking into account risk aversion
The present analysis does not include risk aversion, following Arrow and Lind (1970). It
thus assumes that the social planner who determines the appropriate level of risk ensures
that (i) aggregate losses remain limited compared with national income; (ii) there is risk
sharing across individuals in the country to avoid large individual losses11 ; and (iii) there
is temporal smoothing of disaster losses, through savings and borrowing (self-insurance)
or reinsurance. In absence of these elements, risk aversion needs to be taken into account.
This is the case, for instance, in small countries where the entire economy can be affected
(as in Grenada after hurricane Ivan in 2004 where losses reached 200% of GDP) and
where the risk-free level of capital R (Rs ) would be large compared with the rest of the
economy.
To take into account risk aversion, we need to introduce an utility function, which we
assume to depend on the economic surplus u(Ï€ ) and to have decreasing returns uâ€² (Ï€ ) >
0 and uâ€²â€² (Ï€ ) < 0. The utility cost of disasters can be approximated by the insurance
premium Î´ that the region would be ready to pay to avoid all losses, which is deï¬?ned by:
u(Ï€0 âˆ’ Î´ ) = pu(Ï€0 âˆ’ L) + (1 âˆ’ p)u(Ï€0 ) (32)
where Ï€0 is the surplus in absence of disaster and is equal to eÎ³t F (R, S ) âˆ’ C (p, R) âˆ’
r (R + S ). This equation deï¬?nes a function Î´ (p, R, S, t), which replaces pXR in Eq. (3)
when risk aversion is accounted for.
Ï€
Î³t
E[u] = e F (R, S ) âˆ’ C (p, R) âˆ’ r (R + S ) âˆ’ Î´ (p, R, S, t) (33)
And the maximization program becomes:
11
Since fatalities and casualties cannot be shared, it means that forecasts and early warning systems
reduce human losses, as is observed in most developed countries where economic losses have increased
while human losses have decreased.
21
maxp,R,S Ï€
(34)
s.t.0 â‰¤ p â‰¤ p0
We use a constant relative risk aversion (CRRA) utility function under the form u =
Ï€ 1âˆ’Ï?
1âˆ’Ï?
, then we have:
1âˆ’Ï? ( 1âˆ’
1
Ï?)
Î´ = Ï€0 âˆ’ p(Ï€0 âˆ’ RX )1âˆ’Ï? + (1 âˆ’ p)Ï€0 (35)
Since Î´ > pRX in presence of positive risk aversion, the taking into account of risk
aversion makes perceived risk larger and creates a non-linearity between R and risk. To
go further, the optimization program can be solved numerically with the parameters and
functional forms from Section 5 on New Orleans, and using the same methodology to
calibrate Î». Since risk aversion introduces total income in the equations of R and p, it
creates a link between R and S even when the two capitals are separable in the production
function. It means that the value of S (i.e. Î±) also needs to be calibrated. In practice, the
value of S depend on how disaster risks in New Orleans are shared with risk-free capital
(or capital that is subject to a risk independent of hurricane risk). As an illustration,
equations are solved assuming that Î± = 2.
Results for the fraction at risk are presented in Fig. 4, for a risk aversion Ï? = 2. It
shows that risk aversion leads to locating less capital in at risk areas, at all development
levels.
Figure 5 shows that risk aversion has an ambiguous impact on the probability of occur-
rence: at low development level, the capital at risk is so much smaller with risk aversion
that it is optimal to increase the probability of occurrence; at higher development level,
risk aversion leads to better protection and to a decrease in the probability of occurrence.
Finally, Fig. 6 shows that capital at risk still grow more rapidly than income at all devel-
opment levels, and the growth rate converges toward the risk-free growth rate. At high
development level and in this simulation, the capital at risk is lower but increases more
rapidly with risk aversion than without risk aversion. At high development level, average
annual losses grow at a lower rate than risk-free economic growth, like in the case without
risk aversion.
Numerical simulations suggest therefore that the qualitative results in the case without
risk aversion remain valid with risk aversion. An exploration of various values of risk
aversion (Ï?) and of various risk sharing level (modeled through Î± here) conï¬?rms that
results are robust to the presence of risk aversion.
7 Taking into account biases in risk perception
The assumption of rational and perfectly informed decision-making is clearly an over-
simpliï¬?cation of reality. The analysis of disaster protection and risk in such a framework
22
30
25
Fraction of capital at risk (R/(R+S)) (%)
No risk aversion
20
15
Risk aversion (Ï? = 2)
10
5
0 0 1 2 3
10 10 10 10
Aggregated income (billion USD)
Figure 4: Evolution of the â€œfraction at riskâ€?, i.e. the share of capital at risk R in total
capital R + S , as a function of time, with and without risk aversion. Risk aversion reduces
the fraction at risk at all development levels. Calculations using numerical values from
New Orleans (see Section 5) and Î± = 2.
23
0.16
0.14
Annual probability of occurrence
0.12
Risk
aversion
0.1 (Ï?=2)
0.08
0.06
0.04 No risk aversion
0.02
0 0 1 2 3
10 10 10 10
Aggregated income (billion USD)
Figure 5: Evolution of the annual probability of occurrence, without risk aversion and
with risk aversion (Ï? = 2).
suggests that â€” in a normative context â€” it is optimal from an economic perspective
to increase risk taking with development (provided that fatalities can be avoided) and
disasters should become rarer and more costly as income grows.
But interpreting real-world disaster loss data series would require a descriptive ap-
proach, where realistic characteristics of decision-making are considered.12 To do so, the
analysis needs to include behavior imperfections.
First, there are information and transaction costs. Since the information on natural
hazards and risk is not always easily available, households and businesses may decide not
to spend the time, money and effort to collect them, and disregard this information in their
decision-making process (Camerer and Kunreuther, 1989; and Hogarth and Kunreuther,
1995).
Second, risk perceptions are sometimes biased by cognitive failures. Indeed, individ-
uals do not always react rationally when confronted to small probability risks, and they
defer choosing between ambiguous choices (Tversky and Shaï¬?r 1992; Trope and Lieber-
man, 2003). Moreover, private and public investment decisions do not always adequately
take long and very long-term consequences into account (Kunreuther et al. 1978; Thaler,
1999). Kahneman and Tversky (1979) proposed the prospect theory to better explain
12
Other features may be needed to fully explain the current trends in disaster losses, such as the role of
externality and moral hazard (e.g., Laffont 1995).
24
3.5
Capital at risk (no risk aversion)
Average losses (no risk aversion)
3 Income in riskâˆ’free situation
Capital at risk (Ï? = 2)
Average losses (Ï? = 2)
Growth rate (%)
2.5
2
1.5
1
1 Current income 100 1 000
Aggregated income (billion USD)
Figure 6: Evolution of capital at risk R and average annual disaster losses as a function
of time, without risk aversion and with risk aversion (Ï? = 2).
25
individual behaviors, taking into account observed behaviors and experimental results.
In the following, we modify the model to account for some of these effects. We
assume that capital investment decisions are made with imperfect knowledge, or with
biases in risk perception or behavior. This assumption is consistent with the observa-
tion that most capital investment decisions are not made using all available disaster risk
information.
On the other hand, we model protection decisions as made with perfect knowledge
of natural risks and assuming (wrongly) that capital investment decisions will then also
be made optimally and with perfect knowledge. There is thus an inconsistency in the
model between protection decisions and capital investment decisions. This hypothesis
is justiï¬?ed by the fact that (public and private) decisions concerning disaster protections
(e.g. the design of a dike system) are most of time designed through sophisticated risk
analyses, taking into account all available information and assuming optimal behaviors.
We borrow from the prospect theory (Kahneman and Tversky, 1979) the idea that
people weight different outcomes of a decision not using the probability of the outcome
(here, the probability of occurrence p), but a â€œdecision weithâ€? Ï€ . In a classical decision-
making framework â€” based on expected utility maximization â€” Ï€ can be interpreted
as the perceived probability of disaster occurrence, and the difference between p and
Ï€ is the bias in risk perception. In a prospect theory framework, the decision weights
â€œshould not be interpreted as measure of beliefâ€? but they can also be inï¬‚uenced by other
factors such as ambiguity (Kahneman and Tversky, 1979). In this case, there are thus
two distinct issues: the misestimation of probability of occurrence (due for instance to
biased risk perception) and over- or under-weighting of some possible outcomes (which
is a preference, not a mistake).
In the model, Ï€ is used by investors to decide of the amount of capital to install in at-
risk areas. In this section, we assume however that potential losses remain small enough
for the utility function (in an expected utility maximization framework) or the value func-
tion (in a prospect theory context) to remain linear.13 Equation (5) therefore becomes:
eÎ³t âˆ‚R F (R, S ) âˆ’ âˆ‚R C âˆ’ (Ï€X + r ) = 0 (36)
We investigate two ways of modeling the decision weight Ï€ . The ï¬?rst model assumes a
static relationship between the actual probability of occurrence p and the decision weight
Ï€ . The second model takes a dynamic view on risk perception and introduces myopic
adaptive expectations.
13
And the reference point used by decision-makers to assess a situation is a situation with zero economic
surplus.
7.1 Systematic perception bias or decision weighting 26
7.1 Systematic perception bias or decision weighting
A ï¬?rst model can be proposed where the relationship between p, the real probability of
occurrence, and Ï€ , the decision weight, can be represented as:
Ï€ = B ( p) (37)
Kahneman and Tversky (1979) propose that (i) the decision weight is higher than
probability for low probabilities; (ii) â€œsubcertaintyâ€? means that Ï€ is less sensitive to
change in probabilities than perceived probability; (iii) Ï€ changes abruptly near the end-
points, with Ï€ (0) = 0 and Ï€ (1) = 1. One function that satisï¬?es these conditions is:
0 if p < pmin
Ï€ ( p) = 1 if p > pmax (38)
Î²
pb + p otherwise
Events with a probability below pmin are considered impossible; low-probability events
with probability higher than pmin are overweighted; and non-certain higher-probability
events are underweighted.
Calculation from Section 4 can be redone with decision weight, leading to replace
Eq. (26) by:
eÎ³t Î»ÂµRÂµâˆ’1 = r + Ï€X (39)
At very high development level, protection is so high that p < pmin and disasters are
considered impossible. This may be the situation in the Netherlands for most decision-
makers. In that case, as in previous cases, the capital at risk R grows as fast as risk-free
economic growth. At higher development level, p > pmin and we have:
Î²
âˆ’ 1+
Î³t Âµâˆ’1 X Î½ Î²
e Î»ÂµR = r + Xpb + X Râˆ’ 1+Î½ (40)
Î½Î¾
In that case, a high productivity still leads to a situation where the ï¬?xed term r + Xpb
dominates the right hand side of Eq. (40), and the capital at risk R still grows as fast as
risk-free economic growth.
So at high development level, risk perception bias or a prospect theory decision frame-
work still leads to a growth rate in capital at risk that is as fast as risk-free economic
growth, and thus to a growth rate in average annual losses that is lower than risk-free eco-
nomic growth (namely, the risk-free rate reduced by the protection factor, like in previous
cases).
Bias in risk perception or different decision-making however changes results at lower
development level. If the ï¬?xed term r + Xpb is dominated in the right hand side of
Eq. (40), then the growth rate in capital at risk R becomes:
7.2 Myopic expectations 27
Î³
Î³R = Î²
(41)
1âˆ’Âµâˆ’ 1+Î½
in locations where capital at risk R and the protection level would be increasing with
economic growth (i.e., where the probability of occurrence p decreases), the presence of
a large underestimation (or underweighting) of risk can lead to the opposite outcome, that
is a decrease of capital at risk and protection level over time.
Î²
If the growth rate of R is positive (Âµ < 1 âˆ’ 1+ Âµ
), then this growth rate increases with
Î² . It means that capital at risk increases more rapidly if hazards are more under-weighted.
Regardless of risk perception, however, the growth in capital at risk is larger than capital
in a risk-free situation, and the economy evolves toward more risk taking.
Average losses pXR have a growth rate equal to:
Î³
Î³L = 1âˆ’Î²
(42)
1âˆ’ Âµ 1+
Î½
Î½
+ Î½
Î½
In the situation in which protection improves over time, Âµ < 1+ Î½
, and in presence of
risk under-weighting, average losses increase over time, and they increase more rapidly
than with perfect information (Î² = 1), and thus more rapidly than risk-free economic
growth, i.e. the growth rate of Rs and S . In short, the introduction of a systematic bias
in risk perception (or of decision weights instead of probabilities) does not change the
main conclusion of this paper, namely that development leads to more risk taking (capital
at risk increases more rapidly than risk-free economic growth). Risk under-weighting or
under-estimation can however amplify this effect.
7.2 Myopic expectations
This modeling is a simpliï¬?cation, especially because it disregards the dynamics of risk
perception. In practice, it is likely that perceived risk is higher than actual risk during the
years following an event, and lower after some times and when the memory of disaster
losses has lost its acuteness. A second model can be proposed where decisions on the
amount of capital to install in the risky area are based on a disaster probability that is
estimated empirically, based on previous disasters (Hallegatte 2011). The empirically
estimated disaster probability is Ï€ and is given by:
u=t
1 t âˆ’u
Ï€ (t) = eâˆ’ Ï„ F (u)du (43)
Ï„ u=âˆ’âˆž
Where F (u) is equal to one if a disaster occurred at time u, and zero otherwise.
This modeling corresponds to backward-looking adaptive expectation, in which past
events have an exponentially decreasing weight (with time scale Ï„ ). In other terms, agents
assess future disaster risks from past events, with a memory characteristic time Ï„ . The
7.2 Myopic expectations 28
consequence is that the estimated disaster probability is higher than the real one just after
a disaster, and lower than the real one when no disaster has occurred for a while. This
behavior appears consistent with many observations (e.g., Kunreuther and Slovic, 1978;
Tol et al., 1998).
The efï¬?ciency of this empirical process depends on the disaster probability. If there
are many disasters over a period Ï„ (i.e. if 1/p << Ï„ ), the estimated probability remains
close to the real one. If the memory is too short, i.e. if Ï„ is too low, then the estimated
probability will often be different from the real one.
Here, we are interested in the dynamics between two disasters.14 Assuming that the
last disaster occurs at time t0 , we have F (t) = 0 for t > t0 , and:
u=t0 u=t
1 t âˆ’u 1 t âˆ’u t âˆ’t 0
Ï€ (t) = eâˆ’ Ï„ F (u)du + eâˆ’ Ï„ F (u)du = Ï€ (t0 )eâˆ’ Ï„ (44)
Ï„ u=âˆ’âˆž Ï„ u=t0
Replacing p by Ï€ in Eq. (26) gives:
t âˆ’t 0
eÎ³t Î»ÂµRÂµâˆ’1 = r + Ï€X = r + XÏ€ (t0 )eâˆ’ Ï„ (45)
At high development level and if economic actors know that disaster probability is low,
then Ï€X is very small compared with r , and the introduction of myopic expectations does
not change anything: capital at risk increases as fast as economic growth, like in all other
cases. This is also a case in which periods between disasters are the longest, and therefore
the risk is most underestimated. Fortunately, it corresponds to situation where risk is a
minor component in the cost of capital, and the consequences of this underestimation are
minor.
But at low development levels, and where risk perception is high (Ï€ is large), then
Eq. (45) can be simpliï¬?ed, and the growth rate in capital at risk R is equal to:
1
Î³+Ï„ 1
Î³R = = Î³S + (46)
1âˆ’Âµ Ï„ (1 âˆ’ Âµ)
In that case, therefore, capital at risk increases faster than the risk-free economic
growth (Î³S ), like in other cases. But in this case, the growth rate is independent of the
shape of the protection cost function (Î½ ), and only depends on the expectation timescale
Ï„.
Since protection is done with perfect knowledge about the amount of installed capital,
1
âˆ’ 1+ Î½
RX
the actual probability of occurrence remains equal to p = Î½Î¾
, and we have:
Î³ Î½ 1 Î½
Î³L = + (47)
1 âˆ’ Âµ (1 + Î½ ) Ï„ (1 âˆ’ Âµ) (1 + Î½ )
14
A more complete dynamical analysis is made using a numerical model in Hallegatte (2011).
29
The last term of the equation is due to myopic expectations, and it leads to an increase
Î³
in the growth rate of average annual losses. Since the risk-free growth is equal to 1âˆ’ Âµ
, the
growth rate of annual losses can be either slower or faster than risk-free economic growth,
depending on the values of Î½ and Ï„ .
At low development level and with myopic expectations, annual average losses are
growing more rapidly than risk-free growth if:
Î½ > Î³Ï„ (1 âˆ’ Âµ) (48)
This is the case if protection costs increase rapidly with the desired safety standard (Î½
is large), if expectation are short-sighted (Ï„ is small), but also if economic growth is slow
(Î³ is small) or if the production function of capital at risk is close to constant return (Âµ is
close to one). Using parameter values from our New Orleans case study, this condition
is met if Ï„ is lower than 158 years, which is a very long timescale. It seems therefore
possible that at low development level, when natural risks dominate in the cost of capital,
a dynamic bias in risk perception leads to mean annual losses that increase more rapidly
than risk-free economic growth.
8 Conclusion and discussion
This paper proposes an economic analytical framework to analyze the trade-off between
disaster losses and investment returns in areas at risk from natural hazards. This trade-off
is analyzed under various assumptions on decision-making, including the presence of risk
aversion, biases in risk perception, and alternative decision theories such as the prospect
theory.
Under conditions that ensure that protection improves over time, the presence of risk
and the possibility to protect against disasters lead to a lower amount of capital in risky
area (compared with the risk-free situation), but it also increases the growth rate of capital
at risk where protection costs increase less rapidly than the amount of protected capital
(i.e. where investments are at least partly done by increasing capital density and concen-
tration).
By improving protection, economic development drives the economy toward more
risky behaviors (i.e. a growing share of capital is installed in at-risk areas). Protection
reduces the probability of occurrence of an event, but its impact on risk is more complex.
In particular, it transfers part of the risk from one kind of risk (frequent and low-cost
events) to another kind (exceptionnal and high-impact events).
Reciprocally, the increasing in risk-taking is found to accelerate economic growth.
Along an optimal growth pathway, increasing risk-taking is thus both a driver and a con-
sequence of economic development. This interlinkage between development and risk
30
taking suggests that risk should not be reduced at all cost.
By focusing on average losses that mix probability and loss (e.g., Toya and Skidmore,
2007; Rashky, 2008), econometric studies have left out of their analysis the potential in-
crease in damages when a disaster occur. Current trends in disaster losses appear however
consistent with the prediction of fewer but larger disasters (e.g., Etkin, 1999; Nordhaus,
2010; Bouwer et al., 2007; Pielke et al., 2008; Bouwer, 2011; Schumacher and Strobl,
2011). These results are also in line with UN-ISDR (2009), which observes that poor
countries suffer from frequent and low-cost events, while rich countries suffer from rare
but high-cost events.
The paper suggests that natural disasters will become less frequent but more costly
with development and economic growth, and this result has some policy-relevant con-
sequences. In particular, it means that development requires more resilience, i.e. an
improved ability to deal with and recover from rare events, which exceed the protec-
tion capacity. The Tohoku Paciï¬?c earthquake could thus be an illustration of the type of
events the world will have to deal with in the future. Such a trend toward larger disas-
ters translates into a strong and increasing need for crisis management and post-disaster
support, through (1) forecasts and early warning to mitigate human losses (e.g., Sub-
biah et al., 2008; Hallegatte 2012); (2) rainy-day funds and insurance and reinsurance
schemes to support reconstruction (e.g., Ghesquiere and Mahul, 2010; Jaffee et al., 2010;
Michel-Kerjan, 2010); and (3) new international instruments for post-disaster support and
solidarity (e.g., Linnerooth-Bayer et al. 2009). Finally, the growing role of exceptional
disasters, on which knowledge and data is the scarcest, call for decision-making processes
that are able to cope with deep uncertainty and the so-called â€œblack swansâ€? (Lempert and
Collins, 2007; PatÂ´ e-Cornell, 2012; Hallegatte et al., 2012). Such low-probability high-
impact events represent a challenge for the design of development policies.
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35
A Capital at risk and safe capital as substituable inputs
in a Cobb-Douglas function
If R and S are imperfectly substituable, we can assume that F (R, S ) = Î»RÂµ1 S Âµ2 . This
section demonstrates that this situation is similar to the situation where R and S are sep-
arable in the production function.
In this case, the marginal productivity of S gives us:
1
eÎ³t Î»Âµ2 1âˆ’Âµ2 Âµ1
S= R 1âˆ’Âµ2 (49)
r
1
âˆ’ 1+ Î½
RX
With p = Î½Î¾
, we have:
1
âˆ’ 1+
X Î½ 1
pX = X Râˆ’ 1+Î½ (50)
Î½Î¾
and the marginal productivity of R gives us:
Âµ2 1
âˆ’ 1+
Î»Âµ2 1âˆ’Âµ2 Î³t Âµ1 +Âµ2 âˆ’1 X Î½ 1
Î»Âµ1 e 1âˆ’Âµ2
R 1âˆ’Âµ2
=r+X Râˆ’ 1+Î½ (51)
r Î½Î¾
Here, we can use the same approach as before.
A.1 Low development level
At low level of development, and using the same assumptions on p0 , pX is larger than r ,
and the equation can be approximated by assuming that r << pX , which gives:
Î³
1âˆ’Âµ2 t
1âˆ’(Âµ1 +Âµ2 )âˆ’ 1+Î½
R(t) = R0 e (52)
So R is increasing if 1 âˆ’ (Âµ1 + Âµ2 ) > 11+
âˆ’Âµ2
Î½
, i.e. if Î½ > 1Âµ1 +2Âµ2
âˆ’Âµ1 âˆ’Âµ2
. Using classical values
for decreasing return (i.e. Âµ1 + Âµ2 â‰ˆ 0.3), and assuming that the capital at risk and the
safe capital have the same exponent, it leads to Î½ > 0.64, which is the case if protection
costs are convex.
Since economic growth in absence of risk would be 1âˆ’(ÂµÎ³ 1 +Âµ2 )
, the capital at risk in-
creases more rapidly than risk-free economic growth. Average losses E[L] = pXR are
growing at a rate:
Î³ Î½
Î³L = 1âˆ’Âµ2 (53)
1 âˆ’ (Âµ 1 + Âµ 2 ) âˆ’ 1+Î½
Î½+1
Average losses increase more rapidly than risk-free economic growth if:
A.2 High development level 36
Î³ Î½ Î³
1âˆ’Âµ2 > (54)
1 âˆ’ (Âµ 1 + Âµ 2 ) âˆ’ 1+Î½
Î½+1 1 âˆ’ (Âµ 1 + Âµ 2 )
If R is increasing, then the denominator is positive, and this inequality is always ver-
iï¬?ed. So, in this setting, at low level of development and under mild conditions insuring
that the probability of occurrence decreases with time, average disaster losses increase
more rapidly than risk-free economic growth.
A.2 High development level
At high level of development, pX is very small compared with r , and the equation can be
solved by assuming at pX = 0:
Î³
t
R(t) = R0 e 1âˆ’(Âµ1 +Âµ2 ) (55)
Which is also the rate of risk-free economic growth. Average losses E[L] = pXR are
then growing at a rate:
Î³ Î½
Î³L = (56)
1 âˆ’ (Âµ 1 + Âµ 2 ) Î½ + 1
In this case, the growth rate of disaster losses is lower than the rate of risk-free eco-
nomic growth. Indeed, the growth rate in annual disaster losses is equal to economic
growth multiplied by the same â€œprotection factorâ€? Î½/(1 + Î½ ) as in the case of a separable
production function.
Proposition 9 The case where R and S are substituable inputs in a Cobb-Douglas func-
tion is equivalent to the case where the production function is separable.