Journal of Natural Disaster Science, Volume 31, Number 1, 2009, pp11-21

Natural Disaster Risk Bearing Ability of Governments:
Consequences of kinked utility
Stefan HOCHRAINER*
Georg PFLUG***International Institute for Applied Systems Analysis (IIASA).
Schlossplatz 1, A-2361 Laxenburg, Austria.
Tel.: +43 2236 807-517 Email: hochrain@iiasa.ac.at**International Institute for Applied Systems Analysis (IIASA)
and Department of Statistics and Decision Support Systems.
Universitätsstraße 5, A-1010 Vienna, Austria.
(Received for 26 Feb., 2008 and in revised from 19 Jan., 2009)
ABSTRACTIn this paper the risk neutrality paradigm for government stocks and investments ischallenged within the context of catastrophe risk. We focus on government’s ability to
spread its natural disaster risk. Based on the classical approach of Arrow and Lind, thepaper shows the weaknesses of and reformulates the risk neutral assumption for government
decisions under uncertainty. The rationale that governments have kinked utility functions,which can arise from natural disaster events, is given through a network example and its
implications explained considering also risk aversion and the benefits of different types ofrisk management strategies.
Keyword: Catastrophe risk, kinked utility, government, risk bearing ability,
natural disasters, networks.
agement strategies available to them include risk reduction, risk transfer and risk spreading mechanisms.
However, their risk management strategies are often
a function of their perception of the risk they are exposed to (Loefstedt and Frewer, 1998; Slovic, 2000).
Low probability but high consequence events are especially neglected for a number of reasons. For example, people may set (subjective) threshold levels forprobability such that an event below the threshold is
not considered possible. As a consequence, this event
is not incorporated in their decision making process(Kunreuther, 1996). If the risk is however realized,
the high losses can result in the inability to (efficiently)
cope with the event and may have serious negative
consequences in the future, e.g. high indebtedness
over a long time period which limits development op-

1. Introduction
The magnitude of economic and insured losses fromnatural disaster events seem to exhibit a rising trend(Munich Re, 2007). As a consequence proactive risk
management strategies are becoming increasingly
important for reducing potential disaster losses andlessening their long term impacts (Gurenko, 2004).
Different stakeholders can or may use different sets ofoptions for managing their risk. On the country level,risk bearers can be grouped into the government, thedomestic private sector and international institutions(Miller and Keipi, 2005). The private sector can befurther distinguished as property owners (households
and businesses), insurers, reinsurers and the capital
market (Grossi and Kunreuther, 2005). The risk man11

S. HOCHRAINER, G. PFLUG

tions. This situation is not restricted to the household
level or business sector, but can also apply for the
public sector (Hochrainer, 2006).
The ability to deal with risks can be describedas risk-preference. This human characteristic can bemodelled through the use of utility functions. While
there are now a variety of other approaches to analyze
decisions under risk, e.g. prospect theory (see Kahneman and Tversky, 1979), in this paper we use theclassical approach to show and reformulate from atheoretical as well as practical standpoint some of the
weaknesses of the risk neutral assumption for government decisions under uncertainty. Hence, our main
focus is on the government and its ability to spread
its risk from natural disaster events. From a theoretical point of view, one must consider the Arrow-Lind
theorem (1970), which states that governments should
behave risk-neutrally and therefore evaluate theirinvestments and current stock under uncertainty only
through the expected present (social) value (Little andMirrless, 1974). The paper discusses and reformulates its weaknesses in practical situations motivatedby concrete examples and gives a proof for a special
case of utility functions, namely with kinks. That such
kinked functions appear also in practice is shown
through a network example. Furthermore, the problem
of contingent liabilities is discussed within this example for showing the marginal benefits of different risk
management strategies. While the treatment of kinked
utility functions for decision makers was alreadyanalyzed by others (see Segal and Spivak, 1990), we
restate and reformulate it from a government perspective.
The paper is organized in two parts: a theoreticalsection which discusses the Arrow-Lind theorem andrelated topics, and a more practical part where a simple network example should underline the arguments
made before, and implications of these are discussed.
In detail, section 2 starts with a reformulation of theArrow-Lind theorem and gives a short literature review. Section 3 summarizes the weaknesses of the assumptions from a practical perspective, and in section
4 a network example is given to motivate the argument that kinked utility functions may also be present
for governments. In section 5 a formal proof of why
governments should behave risk aversely in such situations is given. Section 6 then discusses possible risk
management strategies the government may use be-cause of its inability to spread risk and discusses thebenefits of mitigation and loss financing strategies.
2. Arrow-Lind Theorem: Discussion
Kenneth J. Arrow and Robert C. Lind investigatedthe question of how governments should treat uncertainty in the evaluation of public investment decisions (Arrow and Lind, 1970). They argue that if therisks associated with a public investment are publicly
borne (e.g. through taxation), the total cost of riskbearing is insignificant, and therefore governments
should ignore uncertainty. The reason behind this isthat governments can efficiently transfer risk throughtheir ability to distribute losses over such a large
population that the per capita loss for each individualis negligible. Besides this risk spreading ability, governments are also able to pool a large number of risks.
The pool ideally consists of a large number of independent public infrastructure assets in various regionsof the country. Thus the government’s risk portfolio isdiversified and risk, therefore, can be neglected. Consequently, governments should behave risk-neutrally
and evaluate their investments only through the expected net present (social) value (Little and Mirrless,1974).
In more detail, when the government undertakes
an investment, each taxpayer has a small share ofthat investment with the returns being paid throughchanges in tax levels. Assuming risk aversion for
taxpayers, the value of the asset to the individual isless than its expected return. Hence, to compute thevalue of the investment to the individual taxpayer, arisk-bearing cost (factor) must be subtracted from theexpected return. To prove that the total of the costs ofrisk-bearing goes to zero as the population of taxpayers becomes large, Arrow and Lind assumed a utility
function U which is bounded, continuous, strictlyincreasing and differentiable. Furthermore, all individuals are identical, have the same preferences, theirincomes are identically distributed variables and they
are subjected to the same tax rates. The assumptions
about the taxpayers were made to simplify the proof,
but they are not essential to the argument. Their proposition can be stated in mathematical terms as:
Proposition 1. If U is a twice differentiable andstrictly monotonic utility function, satisfying |U(u)|≤12

NATURAL DISASTER RISK BEARING ABILITY OF GOVERNMENTS

K(1 + u2) and X is a random variable with E[X]=0, E[X2] < ∞, then for all a(1)

dt of T . He concluded that only for small government risks is the total cost of risk bearing necessarily
insignificant. The main result, being the ability of thegovernment to efficiently spread risk, is still controversial today. For example Blake and Burrows (2001)
used the risk spreading argument to conclude that in
the context of mortality risk, the government should
issue survivor bonds. Contrary to this, Dowd (2003)
argued that capital markets are better suited than any
government to bear and share risks, because this istheir original purpose (see for example Priest (2003)
for a general discussion about public and private insurance in the context of the Arrow-Lind theorem).
In this paper, a closer look at the ability of thegovernment to spread risk in the special case of catastrophic events is made. Catastrophic events form aclass of their own and have therefore to be analysed
separately. In the next section, some of the problems
in this context are presented from a practical point ofview, to further motivate the issues.

lim n[a − U −1 (E[U (a + X/n)])] = 0n→∞The quantity
πn = [a − U −1 (E[U (a + X/n)]

is called the individual risk premium.
The technical assumption that U is bounded by someother function, e.g. |U(u)|≤ K(1 + u2), is needed for
applying the dominated convergence theorem, i.e. tointerchange limit and integral operations within the
proof. πn is introduced using the certainty equivalent,
i.e. E[U(a + X/n)] = U(a + πn).
The Arrow-Lind proposition supports the argumentthat governments should behave risk-neutrally, because dividing risk X by n individuals drives the collective risk premium nπn (i.e. the sum of all individual
risk premiums) to zero as n increases. Summarizing,
the collective risk premium nπn is close to zero, if (i)n is large, (ii) Var(X) is small and (iii) U"(a) is small.
Section 3 will reformulate these mathematical reasonsin more qualitative ones.
A lively and fruitful discussion was started (andcontinues presently) after Arrow and Lind publishedtheir paper. Criticisms on the assumptions made by
Arrow and Lind were raised (Mishan, 1972; McKean
and Moore, 1972; Nichols, 1972; Wellington, 1972).
The Arrow Lind setup was further investigated in
the case of environmental externalities where public
investments produce uncertain environmental side effects which must be taken into account in evaluating
public investment decisions (Fisher, 1973). Further
analyses, which made the roles of the various assumptions more transparent in order to bring out the implications of a more realistic specification of the fiscal
system in which public sector investments are embedded, were performed by Foldes and Rees (1977). They
concluded that the conditions set by Arrow and Lind
must be more stringent than originally defined, andthat the circumstances in which the conclusions of
the theorem apply are very restricted. Gardner (1979)
investigated the Arrow-Lind theorem in a continuum
economy were the taxpayers belong to the set T = [0,1] were a individual agent is an infinitesimal subset

3. Weaknesses and Limitations in PracticeFirst, some examples of when the theorem seems towork and when not are provided. Arrow and Lind argue that the risk-bearing cost will be small if the sizeof the share borne by each taxpayer is negligible incomparison to his/her income. It appears reasonable
that this indicates also that the total cost of risk bearing is small. This is because under such circumstances
the investment is also small with respect to the total
wealth of the taxpayers. In the case of a federally
sponsored investment, the number of taxpayers is not
only large, but also the investment represents only asmall fraction of the national income and therefore
the risk-bearing costs are small even when the investment itself appears large in absolute terms. Suchsituations seem to be the case in developed countries
with wealthy and stable economies. For example, inGermany the losses resulting from the 2002 flooding
were dramatically high, with direct losses of approximately 9.2 billion Euro (Munich Re, 2003). But
mainly from taxation (Sonderfonds Aufbauhilfe),
more money for financing than needed was available(Werner et al., 2003). However, in developing countries one often finds a different picture. For example,
Hurricane Mitch caused direct damages in Honduras
equivalent to 2 billion dollars (Mechler, 2004a). This
13

S. HOCHRAINER, G. PFLUG

is much lower than the 2002 flood losses in Germany,
but recognizing that more than 50 percent of the 6.2
million people in Honduras live below the poverty
line, a roughly estimated cost of 320 US$ per person
if the losses are spread over the entire population indicates that additional taxation would not be feasible.
Hence, with a limited number of taxpayers the cost to
individuals depends upon the absolute size of the total
cost to be spread and the actual number and wealth of
the taxpayers. Additionally, as developing countries
usually undertake just a few large projects such thatrisk pooling is not an appropriate consideration (Brent,1998), the situation is for them even more problematic. Hence, the theorem can be quite misleading inpractice if it is taken for granted in all situations.
Let us now return to proposition 1. We alreadystated that the collective risk premium approaches
zero only under some certain conditions. Stated differently, the proposition is not applicable, if
・countries have relatively small populations (ncannot increase very much)・the risk is relatively large (e.g. the variance of
X is large),
・even small per capita risk is not negligible,
・U is very nonlinear or nondifferentiable.

・Countries subject to high natural hazard exposure・Countries subject to high economic vulnerability
・Countries with few large infrastructural assets
and high geographical correlation between
those assets
・Countries with concentrated economic activity
centers exposed to natural hazards.
Additional to these points above, there is also anotherimportant reason why the risk neutral argument does
not apply in the case of natural catastrophes, having
mainly to do with the assumptions of the utility function. The implications are interesting from a theoretical point of view and also have practical implications.
We start with a simple network example for showingthat kinked utility functions are quite natural and need
to be considered in catastrophic events, also from agovernment’s perspective.
4. Non-optimal destruction of Networks
Rationales for using utility functions with a kink can
be found in behavioral frameworks such as prospect
theory to distinguish people’s different preferenceswith respect to gains and with respect to losses relative to a reference point (Kahnemann and Tver-sky,
1979,1992; Starmer 2000) or similarly in first-orderrisk aversion models (with kinked utility) where
losses more heavily weighted than gains in the investor’s utility (Epstein and Stanley, 1990; Gul, 1991) as
well as in insurance type of analysis, e.g. to determine
the appropriateness of unregulated markets for humanwealth and liability risks (Sinn, 1982). In this section
it is illustrated how kinked utility functions can arise
in a natural way for catastrophe risk. While utility
functions are typically considered as artefacts, which
are unobservable and can only be indirectly guessedby observing preference decisions, if available, concrete benefit functions as they appear in various OR
applications are preferable as utility functions.
In the following, a network example is constructed, as for instance a road system, for showing
how kinked utility functions may appear. Here, thecosts are the investments in the arcs (the roads), while
the utility is the total throughput through the system.
For simplicity, we consider a directed network and

The first and second points are usually fulfilled bysmall island states and most developing countries.
Contrary to common health or injury risks, natural
catastrophe risks are geographically correlated. Thismeans that for small countries there is a higher probability of larger losses relative to population. For developing countries with less developed early warning
systems or adherence to construction codes, relative
losses will be even larger. Furthermore, in developing
countries with widespread poverty, even small damages or small income shocks may not be negligible.
Concerning the third point, when even small per capita risk is not negligible, the individual’s utility function could be interpreted as high curved, implying that
the Arrow-Pratt risk aversion coefficient (Pratt, 1964)
is high. The reason for this is that the wealth of these
individuals is very low, so that also small capita risk
are seen as not negligible. Mechler (2004b) summarized and also reformulated the qualitative conditions
when the Arrow-Lind theorem should not or can not
be used to assume risk neutrality for the government:

14

NATURAL DISASTER RISK BEARING ABILITY OF GOVERNMENTS

The catastrophic event chooses now an arc j at random decreases its capacity, which was kj(cj) to kj(cj− D), where D denotes the amount of damage. We
assume for simplicity that all other capacities remainunchanged. We assume further that after the catastrophic event the network flows will adjust to the
new situation and equal the solution of the max-flow
problem under the reduced capacity. Let the expected
maximal flow after the event be U_(B0, D) (say). Evidently,

just one origin-destination throughput. The difference
between network construction by planned investment
and network destruction by natural catastrophes becomes evident in this example: While infrastructure
construction is typically based on an optimal costbenefit decision, the infrastructure destruction by
natural catastrophes occurs in an erratic manner. This
leads in a natural way to different marginal utility increases for construction and decreases for destruction.
To be more concrete, consider the following directed
network (Figure 1):

U_(B0, D) ≤ U(B0 − D),
k3

F

@

@ k7
@
@
R
@
@
k1
k4@
k8 @ k11
@
@
R
@
@
R

@
@
k2@ k5
k
k
9
12
@
@
@
@
R
R
@
@

k6@
k10
@
@
R

F

because the random shock does not take away somemarginal capacity at each arc, but strikes the systemmuch harder by unsystematically hitting fully just one
component as illustrated in Figure 3.

-

Figure 1. A network example: The flow F represents
the return. Investments can be made to increase the arc capacities ki.
The government as investor may invest in the capacities of the arcs. The capacity ki of arc i is a function
ki(ci) of the amount ci invested in this arc. The utilityof the investment is the maximal flow F through thisnetwork. For a given budget B, the optimal investment strategy maximizes total network flow underthe budget constraint. Denote this maximal flow by
U(B). Figure 4 shows this function as a dotted line.
The concrete specifications of this example are given
in Appendix B. Suppose that the actual network is the
optimal system for a budget of B0. As illustration, theoptimal investment of a budget of B0 = 10 is shownin Figure 2, where the width of the arcs represents thecapacity.

Figure 3. Left: The catastrophic event reduces for
instance the capacity of only arc 12. This
network can be built with a budget of B0 − D.
Right: with the same amount of budget B0− D a more effective network could be built
as is shown here. The utility of this network
is much higher than the one left over by thecatastrophe.

The marginal loss in utility caused by the catastrophic
event is higher than the marginal increase of utility bya marginal increase of the budget B. Thus consideringa basic budget of B0 and a return (utility) function Ū,
which equals U(B) for B > B0 and U_(B0, B0 − B) for
B < B0, this function has a kink at B0 (figure 4).

Figure 2. The optimal capacities of the links for agiven budget of B0 = 10.
15

S. HOCHRAINER, G. PFLUG

parison to b and Z (and therefore w.r.t. c and X), we
parameterize the quantities c and X by a small parameter , i.e. we consider

5.5
5
4.5
4
3.5

B0 + c + X.

3
2.5

Introduce the risk premium π as the certainty equivalent, i.e. the solution of

2
1.5
1
0.5

0

2

4

6

8

10

12

U(B0 + c − π()) = E[U(B0 + c + X)],

14

Figure 4. A catastrophe makes the return on investment drop below the historic return curve(The basic stock is B0 = 10). The function
U(B) is shown as a dotted line, the function
Ū(B) as a solid line.

that isπ()= B0 + c − U−1E[U(B0 + c + X)].
The concavity of U implies that π() ≥ 0, i.e. the risk
premium is positive.
Recall that the Arrow-Lind Theorem states thatthe collective risk premium π()/ tends to zero with 
tending to zero (set  =1/n and use proposition 1.) The
following proposition shows that this effect disappears for kinked utilities.
Proposition 2. Assume that U is strongly left-andright-sided differentiable at B0 with left side derivative γ− and right side derivative γ+, i.e. there is a δ> 0
and a K> 0 such that for v> 0

In developing countries, where redundant systems areoften unaffordable, this effect is magnified because of
the threat that a failure anywhere can cause a failure
everywhere.
5. Kinked Utility functions for Governments andRisk Aversion
Given the example in section 4 we now prove one of
the consequences of such an situation for proposition
1, e.g. that one can not assume that the collective risk
premium is small if the marginal return on infrastructure repair after an event is larger than the marginal
return on infrastructure investment in a no-event scenario. As a consequence, risk spreading is not an option and therefore the government should behave risk
averse in such situations.
Assume that the actual stock value at the beginning of the reference year is B0 and that a budget of b
is foreseen for this year. A possible catastrophic eventof random size Z reduces the stocks (w.l.o.g. at the
beginning of every year). The stock value after the
event and the investment of b is|U(B0 + v) − U(B0) − vγ+| ≤ K|v|1+δ
and for v< 0
|U(B0 + v) − U(B0) − vγ−| ≤ K|v|1+δ ,with γ+ > 0. Assume further that E(|X|1+δ) < ∞.
++Case A. If γ−γ−γ
E ([X + c] ) >c, then
−

lim
↓0

π()
γ− − γ++E ([X + c] ).
=

γ−

+
+
Case B. If γ−γ−γ
E ([X + c] ) <c, then−B0 − Z + b = B0 + c + X
with c = b − E(Z) is the expected change in stocks
and X = −Z + E(Z) is the zero-mean risk part. We assume that c> 0, otherwise the budget would not be
sufficient to cover the expected annual damage. The
utility function U is assumed to be strictly monotonic
and concave. Since we assume that B0 is large in com-

lim
↓0

γ− − γ+π()(E ([X + c]−).
=γ+Here [a]+ = max(a, 0) and [a]− = − min(a, 0) are thepositive and negative part of a, respectively.
Proof. See Appendix A.
16

NATURAL DISASTER RISK BEARING ABILITY OF GOVERNMENTS

cal analysis of Arrow and Lind (1970). This position
can and has however been criticized from various
perspectives. Risk spreading ability is especiallycontroversial: how many people are needed so thatthe risk premium for each one approaches zero, whatmust the wealth of the taxpayers be in this case, what
does individual risk aversion have to do with it and is
the size of the country important? As discussed, therisk spreading ability can be limited dependent on theanswers to the questions above. This paper adds to thediscussion the implications of kinked utility functions
due to catastrophe events and challenges the risk neutrality paradigm from such an perspective.
Motivated by a network example which ideally
represents the infrastructure of a country several rationales that governments have kinked utility functions
were given. Basically, whereas the development of
infrastructure is a careful, return-maximizing process,natural catastrophes remove capital stock in a lessthan optimal way. For example, catastrophes may
damage a fraction of a power plant that disables theentire facility, destroy a segment of a road rendering
the entire road useless, or attack key telecommunications centers, rather than recent peripheral investments. As a consequence, they can seriously affectthe economy as a whole. For example, Honduras was
unprepared in terms of resources for rapid recovery
when hit by Hurricane Mitch. Many infrastructurereconstruction needs still remain mainly because of
insufficient domestic financial resources (Telford etal., 2004). Recognizing, that ”Infrastructure represents, if not the engine, then the ”wheels” of economic activity” (World Development Report, 1994),
negative economic effects can be prolonged by a lackof full infrastructure reconstruction. Governments
should therefore take these risks into account, at leastin their development planning processes.
From a government perspective the analysis
has important implications. Risk neutrality can not
be assumed and therefore risk management strategies should be considered. Generally speaking, risk
instruments to lessen the risk of large contingent liabilities for the government in the case of a cat event,can be separated into ex-post resources or pro-active
measures. While the former includes taxation, budget
diversion and outside assistance, the latter includes
insurance, reserve funds, structural mitigation measures and redundant systems. A discussion of all these

The Arrow-Lind theorem is contained as the special
case that U is smooth at a, i.e. γ+ = γ− and that c=0. In this case, one sets  =1/n, i.e. divides the risk
among n persons with identical utility and finds thatn · πn = n · (B0 − U−1(E[U(B0 + X/n))) → 0.
Proposition 2 is valid for small . In the special
case that the utility function is piecewise linear i.e.
U(x)= γ+[x − B0]+ − γ−[x − B0]− ,then no limit operation is necessary, since then for every π()= π(1)with

π(1) =⎧⎪⎨ γ− −γ+ E([X + c]+ )γ−⎪⎩ γ− −γ+ (E([X + c]− )γ+

if

γ− −γ+
E([X + c]+ ) > cγ−(Case A)ifγ− −γ+
E([X + c]+ ) < cγ−(Case B).

Notice that in the case of Proposition 2, π()/ =0
only if either γ− = γ+ (i.e. there is no kink) or Case B
is fulfilled and E([X +c]−)=0, which means that X ≥−cor equivalently Z ≤ b, i.e. that the maximally possible
catastrophe is such small that all its damage can be
repaired with the annual budget b. We remark that E([X + c]+)=0 cannot happen for c> 0.
Summarizing, governments should not behave
risk neutral when utility functions are kinked after acatastrophe, because marginal utility of reconstructionin the case of an catastrophic (cat-) event can be quite
higher than in the case of a no cat-event. This implies
that in case of no resources for reconstruction a catevent can have dramatic impacts on the economic
performance (as illustrated by the network flow example) while with resources the marginal utility of
the reconstruction process (e.g. repairing roads) canbe very high. In other words, beneath the direct losses
the indirect effects can be enormous and therefore
have to be included in the risk management decision
making process.
6. Discussion
The risk neutrality paradigm for governments withregards to catastrophe risk dates back to the theoreti17

S. HOCHRAINER, G. PFLUG

measures is not possible here (see Mechler, 2004b;
Hochrainer, 2006) and we restrict ourselves on a discussion of possible pro-active measures for the network example in section 4.
Mitigation measures like strengthening bridges
in key areas (important network knots) would decrease the marginal utility loss in a cat-event scenario.
If mitigation was enough to withstand a natural disaster with a certain impact, this cat event would have nofurther consequences. If mitigation lessens the impact,
the marginal utility loss in the cat-event would be lessthan the marginal utility loss in the cat-event with no
mitigation, but on the other hand, the absolute utility
in the cat event with mitigation measures would behigher than in the scenario without mitigation. Hence,
mitigation measures are efficient if the marginal investment utility is high. Redundant systems are similar to (physical) mitigation measures, but rather thanstrengthening constructions to lessen the impact of adisaster, redundant systems could reduce the risk of
total failure if they are sufficiently spatially uncorrelated with the risk, e.g. earthquakes are usually localized events and therefore redundant systems have to
be separated from such areas (it should be noted that
only a single event is assumed here, further researchin the future will take into account multi-event scenarios). In this case, to develop an effective and cheap
redundant system, risk based infrastructure interdependencies should be assessed and network improvement to counteract disastrous events analyzed (Brown
et al., 2004; Poorzahedy and Bushehri, 2005).
In contrast to structural mitigation measures proactive financial risk management instruments likeinsurance, catastrophe bonds or reserve fund arrangements do not lessen the physical impact of disasters,
but ensure that money is available without delay after
the cat-event so that a quick repair and return to thestatus quo is possible. Using the network example
above, financial instruments are preferable to mitigation measures when marginal utility without a cat
event is low, but absolute utility is high.
Further research includes the analysis of investment strategies in risk management instruments fordifferent marginal return rates as well as the incorporation of simultaneous multi event scenarios in network systems.

Acknowledgments
The work was sponsored by the “Österreichische
Nationalbank Jubiläumsfonds” (OeNB), project number 11127.
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Starmer, S. (2000). ’Developments in Non-Expected Utility

and planning for developing countries, London, UK:

Theory: The Hung for a Descriptive Theory of Choice

Heinemann Educational Books.

under Risk’, Journal of Economic Literature 38: 332-

Loefstedt, R. and Frewer, L. (1998) Risk & Modern Society,382.

London, UK: Earthscan Publications.

Telford, J., Arnold, M. and Harth, A. (2004) ’Learning Les-

McKean, R.N. and Moore, J.H. (1972) ’Uncertainty and thesons from Disaster Recovery’, Disaster Risk Manage-

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World Bank.

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Wellington, D. (1972) ’Uncertainty and the Evaluation ofDisasters’, in K. Demeter,

Public Investment Decisions: Comment’, The Ameri-

N.E. Erkan and A. Guener (eds.) The Role of Local Govern-

can Economic Review 62: 170.

ments in Reducing the Risk of Disasters, Washington,

World Development Report (1994) Infrastructure For De-

D.C.: The World Bank Institute, pp. 35-47.

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Appendix A: Proof of Proposition 2

able development department technical paper series.
Set Y = X + c. Notice that E(|X|1+δ) < ∞ implies that also

Washington, D.C.: IDB and World Bank.

E(|X|1+δ) < ∞. Recall that the risk premium π(E) is defined

Mishan, E.J. (1972) ’Uncertainty and the Evaluation of
Public Investment Decisions: Comment’, The Ameri-by the relation

can Economic Review 62: 161-64.
(2)

Munich Re (2003) Topics Geo -Annual Review: Natural Ca-

U(B0 + c − π()) = E[U(B0 + Y )]tastrophes 2002, Munich: Munich Reinsurance Group.
Munich Re (2007) Topics Geo -Annual Review: Natural Ca-

i.e.

tastrophes 2006, Munich: Munich Reinsurance Group.
(3)

Nichols, A. (1972) ’Uncertainty and the Evaluation of Pub-π()= B0 + c − U−1(E[U(B0 + Y )]).

lic Investment Decisions: Comment’, The American
Economic Review 62, 168-169.

We show first that there is a constant K1 such that

Poorzahedy, H. and Bushehri, S.N.S. (2005) ’Network per(4)formance improvement under stochastic events withπ() ≤ K1.

long-term effects’, Transportation 32: 65-85.
First, the definition implies that π() tends to 0 as  tend to 0.

Pratt, J.W. (1964) ’Risk Aversion in the Small and in the
Large’, Econometrica 32: 122-36.

Since U is strictly

19

S. HOCHRAINER, G. PFLUG

increasing and concave in a neighborhood of B0, its inverse|γ+E[Y ]+ − γ−E[Y ]− + γ−(π() − c)|≤ 1+δKE[|Y |1+δ]+ 

U−1 must satisfy

1+δ

K2.

Therefore, dividing by  and letting  go to zero, one obtains|U−1(y) − B0|≤ K0|y − U(B0)|(5)

for some K0 > 0 and for y in some neighborhood of U(B0).

(9)

The strong differentiability assumption implies that

γ+E([Y ]+) − γ−E([Y ]−)+ γ−−π() − c→ 0.


Notice that c = E(Y )= E[Y ]+ − E[Y ]−. Replacing E[Y ]− in (9)|U(B0 + Y ) − U(B0)| ≤  · max(γ+, γ−)|Y | + 1+δ · K|Y |1+δ ,by E[Y ]+ − c one gets finally in Case Aπ()γ− − γ++→
γ− E[Y ] .

i.e. by taking expectations and for  ≤ 1,
|E[U(B0 + Y )] − U(B0)| ≤ C · ,

Notice that in turn, if Case A is fulfilled, then  c < π( )eventually.

where C = max(γ+, γ−)E|Y | + KE|Y |1+δ

. Setting y = E[U(B0Case B. Assume now that c > π(). Then+ Y )] in (5) one obtains for sufficiently small (10) |E[U(B0 + c − π())] − U(B0) − γ+(c − π())|≤ K[c +|U−1(E[(B0 + Y )]) − B0|≤ K0|E[U(B0 + Y )] − U(B0)|≤ CK0π()]1+δ ≤ 1+δK2.

and substituting c − π() for U−1(E[(B0 + Y )]) − B0 one gets

Putting (7) and (10) together, using (6) and the identity (3)one obtains that|c − π()|≤ CK

(6)|γ+E[Y]+ − γ−E[Y]− + γ+(π() − c)| ≤ 1+δKE[|Y|1+δ]+ 1+δK1.
which implies (4) with K1 = CK0 + c.
Therefore, dividing by  and letting  go to zero, one obtains

Now we will approximate both sides of (2). First, theright hand side of (2) is approximated. From the strong dif-π() − c→ 0.
ferentiability one gets with [Y ]+ = max(Y, 0) and(11) γ+E([Y ]+) − γ−E([Y ]−)+ γ+[Y ]− = − min(Y, 0) = [Y ]+ − Y

Replacing E[Y ]+ in (11) by E[Y ]− + c on gets finally in Case Bπ()γ− − γ +→E[Y ]− .
γ+|U(B0 + Y ) − U(B0) − γ+E[Y ]+ + γ−[Y ]−| ≤ K1+δ|Y |1+δAgain, in turn, in Case B eventually c > π().

and therefore, by taking the expectation on both sides, onegets further

Appendix B: Specification of the Network example(7)|E[U(B0 + Y )] − U(B0) − γ+E[Y ]+ + γ−E[Y ]−|≤ K1+δE[|Y |1+δ].

The costs ci as functions of the capacities ki for the example shown in Figure 1 were specified as

For the approximation of the left side of (3) one has toci(ki)= αiki + βiki2,

distinguish Case A and Case B.
Case A. If c < π(), thenfor the twelve arcs, i =1,..., 12, where the constants α and β
were chosen as(8)|E[U(B0 + c − π())] − U(B0) − γ−(c − π())|≤ K[c + π()]1+δ ≤ 1+δK2,

α = [0.1, 0.09, 0.12, 0.11, 0.13, 0.08, 0.1, 0.1, 0.09, 0.11,
0.12, 0.11]

where K2 = K(c + K1)1+. Putting (7) and (8) together, using

β = [0.3, 0.31, 0.27, 0.26, 0.26, 0.33, 0.31, 0.29, 0.3, 0.3,
0.31, 0.28](6) and the identity (3) one obtains that20

NATURAL DISASTER RISK BEARING ABILITY OF GOVERNMENTS

Thus the marginal costs increase with increasing capacity.

f9 + f10 = f12

The capacities as functions of the costs arefi ≤ kiki (ci) =i =1,..., 122i iαiki + β k = ci i =1,..., 12

α2
ci
α
+ −
.
4β 2 β
2β

12

i=1

ci ≤ BFor every fixed budget B, the optimal investment into

The decision variables are the flows fi and the provided

the arcs was calculated. Denote by fi the flow on arc i. The

capacities ki. For every budget B in the range 1 ≤ B ≤ 13,

max-flow problem under budget constraints is:the optimal value U(B) of the maximization problem wascalculated. Figure 4 shows the function B → U(B) as a dot-

Maximize F := f1 + f2

ted line. It was then assumed that the optimal network forunder the constraints thatbudget B0 = 10 is affected in such a way that an arc is cho-

f1 = f3 + f4

sen at random with equal probability 1/12 and its capacity

f2 = f5 + f6

is dropped form ki to ki' in such a way that αiki'+ βiki'2 = αiki

f3 = f7+ βiki2 − D. The expected maximal flow for the partly de-f4 + f5 = f8 + f9

stroyed network as a function of the damage D is depicted

f6 = f10

as a solid line in Figure 4.

f7 + f8 = f1121