Scott Farrow and Michael Scott
UMBC, Salisbury University
Prepared for the AERE Summer workshop
June, 2011
Abstract
Floods provide a useful example of risky events that span a range from small to catastrophic. In such circumstances the economically preferred measure of the benefits is based on an ex-ante measure that takes into account a person’s willingness to pay to avoid variability in their income or wealth. An analytical structure is developed to estimate the ex-ante value based on an adjustment factor to the estimated damage should an event occur which other research has shown can be large for individual outcomes. The probability and damage from events are analyzed and expected values are estimated for the case of Baltimore City using a flood model developed by the Federal Emergency Management Agency.
An apparently new derivation of the transformed probability of an event based on exceedance probability estimates is derived. The results suggest that an ex-ante adjustment in this case leads to moderate but not dramatic increases in willingness to pay. The ex-post expected value is about $320 million per year to completely avoid flood damage. The values chosen for the base case ex-ante alternative increase willingness to pay by about 40 percent, a value which can be increased with higher risk aversion, reduced by a higher level of wealth, and is essentially unaltered by extending the upper limits of integration.
Appreciation is extended to Thomas Wallace, Michele Stegman, and Chava Carter for research assistance; to seminar and conference participants at the Society for Benefit-
Cost Analysis Annual Meeting, and Carnegie Mellon University. Appreciation is also extended to the John D. and Catherine T. Macarthur Foundation for partial funding.
Results are preliminary and should be cited with caution.
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I.
Introduction
Damages caused by flooding are the subject of frequent stories in the popular press and often form a first approximation to how much people are willing to pay to avoid such damages. Such estimates are very location and event specific. A more consistent approach to information about willingness to pay could be useful to inform benefit-cost analyses of structural improvements, such as levees, and for insurance programs, such as the National Flood Insurance Program. However, the theoretically preferred benefit-cost analysis of investments in flood prevention and insurance depend on the ex-ante, not the ex-post, willingness to pay.
An existing framework in the literature for the difference between the ex-post and the exante values is used to estimate the ex-ante adjustment given knowledge about conditional, or ex-post, damages. While many prior analyses focus on floods with an expected return period of 100 years (the 100 year flood) due to its importance in the National Flood
Insurance Program, this analysis evaluates the expected ex-post and the ex-ante willingness to pay for the entire distribution of return periods from 1 to 500 years.
Estimates of ex-post damages for varying flood heights are obtained from the HAZUS damage model used by the Federal Emergency Management Administration (FEMA,
2009). Results from Baltimore City are used as the pilot example as data are still being generated for other counties in the Maryland. Analyzing natural hazards such as floods,
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which have a wide range of possible outcomes, may also inform the analysis in related areas such as terrorism.
The paper proceeds in Section II by developing the theoretical difference between an ex-post and an ex-ante measure of willingness to pay and a forecasting equation for that difference. Estimation of the conditional (ex-post) forecast damages from various flood heights (return periods) is developed in Section III. Also developed in section III are the empirical details to generate the ex-ante adjustment to the ex-post value and the probability of each outcome in order to define the expected values of the measures.
Section IV presents the preliminary empirical results and sensitivity tests with Section V summarizes the ambiguous impact of behavioral adjustments to the expected utility model while Section VI concludes.
II. Ex-ante and ex-post values for reducing consequences
Consider two economic models of consumer behavior based on Freeman (1989). One, essentially based on a risk neutral consumer probabilistically weighting the value of a certain outcome, assumes knowledge of the utility impact of an adverse outcome, A*, conditional on its occurring and knowledge of its probability, q. Policies may alter either outcomes or probabilities.
A second approach is based on an expected utility framework to model how much an individual would pay to achieve the same utility as the random occurrence of the event
3

but without the variation in outcomes, an ex-ante approach
1
. Each approach is modeled in more detail below based on Freeman (1989).
Formally, the ex-post willingness to pay seeks to identify the amount of payment, CS, in the conditional world such that, given wealth M and indirect utility function V:
(1) V(M,A*)=V(M-CS,0) .
The expected ex-post willingness to pay for total avoidance is then q*CS if q is the probability per period (here assumed constant). Freeman works more intensely with a marginal value of a small change in A* and the amount of money one would give up to maintain utility. After totally differentiating equation 1, setting it equal to zero, and weighting by probability, one obtains the ex-post marginal willingness to pay:
(2 ) WPA*=dM/dA* = -qV
A*
/V
M*
The willingness to pay, ex-post, is seen to be a function of the marginal utility value, V
A* converted to monetary value by the marginal utility of money given that the event occurs.
The ex-ante approach seeks a similar monetary payment to maintain utility, but one that takes into account the variation in outcomes in the determination of a state independent payment. Formally the consumer seeks to find a value, X, that in other settings has been
1 Behavioral adjustments to the expected utility model are considered in Section V.
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called an option price, which generates the same utility with and without the payments and so seeks to solve:
(3) qV(M,A*) + (1-q)V(M,0)= qV(M-X,0) + (1-q)V(M-X,0)=V(M-X,0)
A marginal expression similar to equation 2 can be derived such that
(4) WAA = dM/dA* = dM dA
 
* [ qV
M * qV
A *
  q V
M
0
]
Where M
0
and M* is the level of income in the world with the event and without. When equations 2 and 4 are combined, an expression can be derived for the difference between the ex-ante and ex-post measures as in equation 5:
(5) Proportional difference=
WAA *

WPA *
WAA *
    q )
V
M
V
M
*
0
The difference is seen to hinge on the difference in the marginal utility of wealth in the different states of the world. If that utility is state independent or results in a very small change in wealth, then the measures are equal or nearly equal. The measures can differ in important ways that are unclear empirically in the case of either natural hazards or terrorism.
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The key behavioral assumption is the relation between the marginal utility of wealth in the damaged compared to the undamaged state of the world. In what may be a common assumption, the marginal utility of wealth in the “no event” state of the world, V
M
0
is assumed smaller than the marginal utility after the event. In that case the ex-ante measure will be larger than the ex-post measure. But some types of events may exhibit
“irreplaceable” preferences such that the marginal utility of wealth is larger before the event than after (Freeman, 1989; Cook and Graham, 1977). In such cases the ex-ante amount may be smaller than the ex-post amount.
Focusing on what may, but need not be case, Freeman investigated several different risk averse specification of V and concluded that it was precisely where probabilities were low but potential losses were high that the difference between ex-ante and ex-post is largest. For instance, an event leading to a loss of 50 percent of wealth with a probability of .001 yielded a percentage difference between the ex-ante and ex-post values that was ten times higher than that with a probability of .9; rising from an error of 6 percent in the latter probability case to about a 65 percent error in the former case.
Instead of specifying a particular utility function, the approach taken here is to fit both a second order linear regression and a log-log regression with independent variables probability (q), damage (CS), wealth (M), and level of risk aversion (r ) to the output data from Freeman’s various utility functions 2
. The resulting response surface demonstrates his findings within the range of damages that he estimated; a large proportional damage with a small probability can lead to the ex-ante measure exceeding the ex-post by more
2 The regression is based on work by Michelle Stegman and Chava Carter.
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than a factor of two as illustrated in Figure 1 using the second order linear model. Note that care should be used in the use of fitted forms or extrapolating outside of estimated areas. When the surfaced used for the above is extended beyond the bounds used by
Freeman, some ex-ante values are less than ex-post values. To date, the empirical results to be reported later are also sensitive to whether a second order or log-linear equation is fit to the data. Use of the log-linear form led to convergence problems and so the second order form is used here.
Figure 1: Response surface: Ex-ante multiple of ex-post for varying risk and
damage levels, risk aversion equal to 2
WAA* multiple of WPA*
Risk, q
Damage as proportion of wealth
The results for the second order specification used in the numerical estimates are reported in Appendix B. The forecast regression results fit the data reasonably well with an adjusted R
2
of .85.
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The coefficients can be used to predict the adjustment factor to convert an ex-post measure of damages to an ex-ante measure. Equation 5 can be manipulated to define the ex-ante measure as a function of the adjustment factor and the ex-post value conditional on probability, level of risk aversion, and damages as a proportion of wealth. While one could simply work from a parent utility function with the specified risk aversion, we are using the response surface both to consider values not estimated by Freeman (but which could be estimated independently) and in the future consideration of integrating uncertainty in the forecast of the ex-ante adjustment which can be facilitated by information of the variance-covariance matrix of the regression results.
Given the potential for natural hazards (and terrorism) to be low probability and high consequence events, it is possible that potentially large adjustments between ex-ante and ex-post values could alter the results of standard benefit-cost analysis. That potential conclusion will be investigated in the context of floods in the following sections.
III. Implementation and data to estimate the ex-ante adjustment
There are challenges in adapting the ex-ante valuation approach to an applied setting.
With regard to Table 1, the data needed to adjust an ex-post measure are conditional damages, probability, wealth, and the degree of risk aversion. Some portions of the following discussion provide general results; others are targeted at the narrowing of the case study to the city of Baltimore for an illustrative, empirical case study. Figure 2
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illustrates the flow of analysis, where the prior section provides an estimate of the ex-ant adjustment to the ex-post value.
Figure 2: Estimation sequence
Estimation of conditional damages
3
The data sought are the damages resulting from different flood heights, which in common usage are related to different return period, such as the 100 year flood. The Federal
Emergency Management Agency has developed a national level model, HAZUS-MH, that is designed to model flooding (and in a related portion, earthquakes) at the census block level (FEMA, 2009; Scawthorn, et al., 2006a, 2006b).
3 The following description is based on Joyce and Scott (2005).
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To carry-out flood analysis, FEMA developed software and data components to support
HAZUS-MH. Specifically, flood vulnerability analysis requires block-level census data containing building stock, employment profiles, and population counts, stream gauge locations and flow volumes, and lifeline locations, all provided as part of the
HAZUS-MH program. In addition, users must download the 30-meter digital elevation model (DEM) data from the US Geological Survey. Both riverine and coastal flooding can be evaluated in HAZUS. Baltimore City is subject to both types of flooding. flooding only. The spatial level of detail must also be selected. The minimum stream drainage area size essentially functions as a “resolution” setting in the model. The potential range of area size runs from 0.25 sq miles (local scale) to 400 sq miles (regional scale). As with any resolution, the trade-off is size versus detail. A small stream drainage area size will encompass many small streams and creeks and examine them for their flooding potential. However, it is easy to overwhelm the software with too much detail, making a model run difficult to finish. On the other hand, choosing a large stream drainage area size ensures the model will finish easily, as there may be only a few drainage areas that meet the size criteria. Unfortunately if the area size is too large, many important streams reaches may be omitted, thus underreporting the level of vulnerability.
Through empirical testing, a computable number of reaches appeared to be between 60 and 80. In Baltimore city this means about 1 square mile as the drainage area to be modeled .
The major steps involved in the hydrologic analysis included automatically delineating the drainage area for each stream reach, determining the stream gauges that are either
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upstream or downstream of each reach, and finally calculation the flow volume for the entire set of stream reaches in the study case. Characterization of the shoreline involves picking the type of coast and the flooding characteristics for each coastline segment. The type of coast consists of both the degree of wave exposure (from sheltered to full exposure) and the shoreline morphology (from rocky to small dunes to large dunes to flood protection structure). The flooding characteristics involve recording the 10-year,
50-year, 100-year, and 500-year flood height, plus any wave heights (if available).
The calculation of the riverine and coastal flood hazard are accomplished in separate processes. For the riverine flood hazard, a hydraulics analysis is completed (FEMA, 2009;
Scawthorn, et al., 2006a, 2006b ). To summarize the important methodological steps of the riverine hydraulics analysis: the model approximates the floodplain associated with a stream reach, finds the upstream and downstream limits of that approximated floodplain, generates a set of cross-sections, and associates those cross-sections with flood elevations and discharge values. The result of both the coastal and the riverine flood hazard determination are two flood depth grids for a particular recurrence interval that can be used to overlay the demographic data to estimate loss. When examining both coastal and riverine flooding in the same county, the model picks the “predominant” flooding mechanism and its associated flood depth when intersecting the flood zone with the demographic data.
Socio-economic data embedded in HAZUS include census data to the census block level and supplemental data on the commercial building inventory based on Department of
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Energy data (FEMA, 2009). The degree of damage to structures is based on case studies of flood events and engineering damage functions. Several measures of relatively direct loss are available including the cost of repair and replacement of buildings damaged and destroyed, the cost of damage to building contents, losses of building inventory involving contents related to business activities, relocation expense for businesses and institutions, a measure of the loss of services or sales, wage loss consistent with income loss, and rental income loss to building owners along with counts and characteristics of buildings damaged (FEMA, 2009; Joyce and Scott, 2005).
These analyses intersect the census block data with the flood depth information to create an estimate of amount and degree of damage to buildings as well as the resulting potential direct economic losses. The usual caveats apply as for any spatial interpolation process. Most importantly however, the location of buildings within a census block is not known. Therefore, any block that is not completely contained in the flood zone is assumed to have its characteristics evenly distributed throughout it – which may be relatively reasonable in a dense urban area but less accurate in rural areas with larger census blocks. The program takes from a day to a day and one-half to generate one observation for one county using a desk-top computer.
Table 1 presents data from Joyce and Scott (2005) for several Baltimore counties and
Baltimore city for the 100 year flood. Costs related to the capital stock, such as building damage and inventory loss, are seen to be major components but other elements directly related to income are also important. These measures do not include secondary effects or
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non-use values and are point estimates. For instance, for the City of Baltimore, the case studied in more detail here, the total estimated damage in Joyce and Scott (2005) from a
100 year flood was $526,653,000. The comparable result used here, based on an updated version of HAZUS as well as being valued in 2008 dollars, is $837,183,000 where the context is a total building exposure in Baltimore city of approximately $97 billion of which about $1 billion is within the 100 year floodplain. Only that fraction of damage related to building damage and contents, if the latter coverage is purchased separately, is covered by the National Flood Insurance Program.
Table 1: Direct Economic Losses
Source: Joyce and Scott (2005)
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Multiple flood return periods
There is a significant policy focus on the event termed a 100 year flood due to policies in the National Flood Insurance Program for which that flood defines a bright line between required insurance in some cases and no requirement (Kousky, Luttmer and Zeckhauser,
2006). However, broader programmatic analyses are likely to be concerned about damages from the entire distribution of potential floods and with evaluating measures such as the statistically expected losses. Consequently, the HAZUS model was rerun for nine different flood periods; 10, 30, 50, 75, 100, 150, 200, 350, and 500 years. Flood return periods are a frequently used measure of the magnitude of a flood based on a transformation of data such a water flow rates.
Figure 3 provides a visual display of the estimated direct damages from a 100 year flood, for the city of Baltimore based on the output of the HAZUS model.
Figure 3: Direct Economic Loss, 100 year flood, City of Baltimore
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A parsimonious representation of the distribution of conditional damages was estimated by a logarithmic regression of damages on the flood period. As other spatial units become available, more complicated econometrics can be expected to involve the building stock exposure and other information, but given the single site case study such information is included in the constant term in the regression summarized in Table 2.
The return period is highly significant but with diminishing marginal damages based on the elasticity of damages with respect to return period of .25. This equation forms the basis both the evaluation of the ex-post damages, and as the base for an adjustment to estimate the ex-ante damages. It is important to note the case specific role of the damage function. Here the estimate is of increasing but diminishing marginal damages. In other contexts such as homeland security or perhaps for the largest floods, the damage function may increase at an increasing rate as systematic linkages among damaged parts of the area could change the shape of the damage function.
Table 2: Prediction equation for conditional (ex-post) damages: Baltimore City
Dependent Var.
Ln Total Damage
Constant
Ln Flood Return
Period
Observations 9
Adj. R
2
= .97
Root MSE= .04976
Coefficient
12.3899
.2537
Standard Error
.0672
.014 t Value
184.48
17.72
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Probability of an individual event and the expected value of the distribution of damages
The probability of a flood event is central to estimating the ex-ante value based on the expost damages using the coefficients in Table 1. In addition, benefit-cost analyses generally focus on the expected value of the appropriate valuation measure, also requiring the probability for each possible outcome.
If the probability of a specific event is estimated, then the expected value calculation is straightforward. However, as in the case of flooding and some catastrophic analysis, the preferred probabilistic statement is typically that of the exceedance probability, the probability of an event such as stream flow, x , being equal to or greater than some specific flood value, P(x>x o
). This probability is a statement about the inverse or complementary cumulative distribution function, CCDF that equals 1-F(x) where F(x) is the usual cumulative distribution function. (Grossi and Kunreuther, 2005; Chin, 2000;
Scawthorn, et al., 2006a,b).
Hydrologists analyze estimated exceedance probabilities but typically focus on a transformation of the flood size defined as the inverse of the exceedance probability,
1/CCDF. One statistical interpretation of this measure is the expected number of time periods, T, until a certain flood size, x
0
, is exceeded. (Chin, 2000; Prakash, 2004). As
Chin (p. 257) states however, “it is more common to describe an event by its return period than its exceedance probability”.
It is asserted here that this common practice defines a transformation of the underlying flood random variable, x, into another random variable, R(x), defined with the same transformation, 1/CCDF, as the return period. The HAZUS program uses the return period in this latter way to correspond to a given flood event, x
0.
When used in this way, the probability density function of R(x) can be derived from that of x. An intuitive derivation is provided here, Appendix A contains a more detailed derivation with the same result.
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The intuitive derivation asserts that the probability of the chance of exceedance, 1-F(x), should equal the same chance of exceedance when measured in terms of R(x), such that
1-F(x) is equal to 1-F(R(x)). In words, if there is a five percent chance of a flood exceeding a size x
0
, then there should also be a five percent chance of a flood exceeding the transformed variable R(x
0
). In that case the density function of R can be immediately derived by substitution and the first fundamental theorem of calculus. Then:
1-F(x) = R(x)
-1
by definition
1-F(R) = R
-1
by assumption and substitution, then: 𝑑(1 − 𝐹(𝑅)
= 𝑑𝑅 𝑑𝑅 −1 𝑑𝑅
−𝑓(𝑅) = −𝑅 −2 𝑓(𝑅) = 𝑅 −2
This density function is used both in the adjustment calculation for ex-ante damages, and for the estimation of the expected value of either the ex-ante or ex-post distributions. A derivation based on integration by substitution is in Appendix A.
Given the density function of R, expected values can now be computed. Defining the conditional damage from Table 2 as D(R), then the expected values for the ex-post damage estimate and the ex-ante estimates are in equations 6 and 7 where

(D(R),risk, wealth, f(R)) is the adjustment factor derived from Equation 5 that is parameterized from the regression results in Appendix Table 1.
(6) E(ex-post WTP)= 
R

1
 
R

1
1
R
2
 b
R dR
 
R

1

R b

2 dR
 b


1
R b

1
R

1
(7) E(ex-ante WTP)=
R


1
1
 
( , , ) dR
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Once the damage function is estimated, the expected value of damages can be computed for a chosen maximum return period. Equation 6 admits of a closed form solution, equation 7 is less amenable to such a solution and was solved numerically using
Mathematica7
4
(Wolfram Research, 2010: Just, 1998).
Wealth and Risk Aversion
The definition of the wealth or income over which the individual is averse can significantly affect results (Meyer and Meyer, 2006). For instance, Freeman developed his approach using income although for the purposes of the flood case, wealth will be used. His maximum damage as a share of income was 50 percent. For major events such as floods or terrorism, some individuals may well suffer losses significantly exceeding 50 percent of annual wealth although some specific forms of utility functions become undefined if the loss exceeds total wealth. The measure of estimated wealth used here is adjusted from improvements in the 100 year flood plain and approximately adjusted for larger floods and other elements of total damages included in HAZUS such as contents and inventory loss. The resulting value used for the base case is three billion dollars approximated from the total exposed value of improvements in the 100 year flood plain of $641 million (2001 dollars; Joyce and Scott, 2005). Consequently, for Baltimore City, a value of $3 billion as the base case for total damages exposure in the 100 year flood plain (of which about $1 billion are structures), and $97 billion for the total city exposure
(FEMA, 2009). No systematic or general equilibrium risks, as might link flooding in multiple jurisdictions are considered.
4 The N[Integrate] procedure with the default integration rule was used. Program available from author.
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With regard to relative risk aversion, the Arrow-Pratt measure of risk aversion weighted by wealth, Freeman considered values of .5, 2, and 10. The functional forms used by
Freeman incorporate both decreasing absolute risk aversion and constant relative risk aversion. The base case used is relative risk aversion equal to 2, with sensitivity tests for
0 (risk neutrality) and 10.
IV: Results and sensitivity testing
The starting point for numerical results is the ex-post estimate of expected damages. In
Table 3, the ex-post value is reported in row 1 and reflects a risk aversion value of zero, exposed wealth of $3 billion (which doesn’t affect the calculation), and an upper limit of integration of the 500 year flood. The resulting point estimate for the expected ex-post flooding damages for Baltimore City is about $319 million while the expected return period is 6.2 years.
Rows 2 through 5 are all based on an ex-ante adjusted willingness to pay to avoid total losses which incorporates risk aversion as described above. The base ex-ante model uses the parameters from the ex-post model but has a risk aversion coefficient of 2. The expected value of the ex-ante measure is moderately higher, 25 percent, than the risk neutral case with an estimated willingness to pay of $399 million. While one might anticipate from Figure 1 that some adjustments would be large, for instance, the 500 year flood damage represents almost a 40 percent loss of the assumed wealth, the probability
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of such large floods is very small so that the expected ex-ante value represents a moderate but not dramatic adjustment. Although no formal sensitivity tests of a statistical difference are currently possible from the HAZUS output, the difference seems likely to be policy significant.
Table 3: Expected Value Results and Sensitivity Testing
Scenario Risk
Aversion
Exposure
(Wealth)
Upper limit
Integration
1. Ex-Post/
Risk neutral
0 3 B 500
2.Base-
Ex-ante
3. Upper limit of integration
4. High Wealth
2
2
2
3 B
3 B
97 B
500
1000
500
500 5. Very risk averse
10
Source: Author calculations (preliminary)
3 B
WTP Result
Million 2008 $
$ 319
$ 399
$ 402
$ 321
$ 562
The later rows of Table 3 report sensitivity tests which are taken in turn. Row 3 doubles the upper limit of integration to 1,000 which is twice the base upper limit and well beyond the data on which the damage equation is estimated. The result is less than a one percent increase over the ex-ante estimate based a 500 year limit of integration. This sensitivity test reinforces the sense that expected value calculation substantially reduces the impact of very low probability but high damage events. Another sensitivity test is to increase the exposed wealth in row 4 to the improved value of all of Baltimore City. The larger wealth reduces the ex-ante premium that people would be willing to pay such that the ex-ante value is only about one percent larger than the base ex-post value. Finally, relative risk aversion is increased to ten in Row 5 which substantially increases the ex-
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ante value by almost 40 percent over the base ex-ante case and almost 90 percent over the ex-post case.
V.
Behavioral considerations 5
A growing body of literature seeks to identify the behavioral determinants of willingnessto-pay for flood risk reduction. Much of this research is based around nonexpected utility models such as Kahneman and Tversky’s (1979) prospect theory model, which are based on the general premise that individual choices under risk are made based on perceived probability and asset valuation rather than on the absolute/objective values of these factors. The process of pinpointing specific heuristics and biases, as well as the degree and direction of their influence on decisions involving flood risk, is difficult. Only a limited portion of that literature is specific to flood risk reduction.
The several biases and heuristics that have been empirically shown to affect decisions involving flood risk are presented in Table 4. The affect heuristic refers to the role of feelings and emotions in decision-making. In the context of flood risk perception, it is likely that individuals who have experienced flooding will have stronger negative emotions toward flood risk - and a higher risk perception – compared to those who haven’t. Closely related is the availability heuristic, which increases the perceived risk of events which are easily recalled. This too might suggest that those who have experienced flooding would have a higher flood risk perception, and that the temporal nearness of the flood experience would influence the degree to which increased risk is perceived.
5 Section based on material by Chava Carter.
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Conversely, individuals who have experienced a low-probability event may believe that this reduces their likelihood of experiencing such an event again (Botzen et al, 2009).
The ambiguity affect describes a bias in which probabilities that are unknown are removed from consideration. Botzen et al (2009) found that a lack of understanding of the causes of flooding had a negative effect on the perception of flood risk, and Bell
(2007) found that self-rated knowledge of flood processes and hydrology increased the likelihood that an individual would take protective action against flood hazard. The bandwagon effect refers to the tendency of individuals to base their own decisions on the choices of those around them. Somewhat related is neglect of probability, that is, the disregard of objective probabilities in making decisions involving risk. Bell (2007) attempted to capture this effect by asking residents in a flood-prone area about the number and type of flood information sources which people sought before, during, and after a significant flood, and found a significant though small positive effect on perceived flood risk.
While the aforementioned biases alter the perception of probability, several others affect the perceived utility of an outcome. Loss aversion, one of the fundamental principles underlying prospect theory, describes a cognitive bias in which the disutility associated with losing an object is greater than the utility associated with acquiring it. It has been measured as the difference between what an individual is initially willing to pay for a certain object with the amount that the individual is willing to accept to surrender that object after it has been given to him (Kahneman & Tversky, 1979). Closely related to
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loss aversion is the pseudo-certainty effect, which refers to the tendency to exhibit riskaverse behavior toward gains and risk-seeking behavior toward losses. That is, people are generally willing to take higher risks in order to avoid a loss than they are to enable a gain; the choices individuals make regarding insurance coverage and deductibles have been offered as evidence of this bias. Hyperbolic discounting reflects time-inconsistency in preferences; for example, people will strongly prefer to receive $10 today over receiving $10 in five years, but will exhibit a considerably weaker preference for receiving $10 in five years over receiving $10 in ten years. Kunreuther, Onculer & Slovic
(1998) found that this effect was present in decisions regarding protective measures.
Heuristic
Affect heuristic
Measurement
Flood experience
Effect on WTP Affect prob. or utility?
Increase Probability
Ambiguity effect Understands flood risk Decrease
Availability heuristic
Bandwagon effect
Hyperbolic discounting
Temporal nearness of flood experience
Neighbors’ flood decisions
ΔWTP over time
Increase
Increase
Decrease
Loss aversion WTP vs. WTA values Increase
Pseudo-certainty effect
Neglect of probability
Insurance purchases
Flood info. sought
Increase or
Decrease
Decrease
Probability
Probability
Probability
Utility
Utility
Utility
Probability
The various effects identified in Table 4 indicate that behavioral responses may shift probability or utility in either direction, increasing or decreasing perceived probability or utility. Further, such shifts presumably would not be monotonic throughout the range of events if total probability is maintained. While this provides rich material for further
23

research, it may indicate that the results presented here based on the expected utility model provide an appropriate starting place for analysis.
VI.
Conclusion
The results are an intriguing departure from the casual implications of Figure 1 where the ex-ante adjustment for small probability, high consequence events could be more than a factor of two and so economically and policy significant. The preliminary results, which may be sensitive to the functional form of the regression summarizing utility, suggest that an ex-ante adjustment in this case leads to differences that may be economically and policy relevant but the high potential adjustment for a single rare event is attenuated by their low probabilities. Extending the limits of integration appear to have little effect in this case although a larger effect is found when risk aversion is very substantially increased. When exposed wealth is much larger, the ex-ante adjustment is substantially reduced and is essentially equivalent to the ex-post value. Importantly, this preliminary result is depends on the case specific damage function. A function that instead increases at an increasing rate for a “catastrophic” event may well lead to a different conclusion.
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Appendix A: Alternative derivation of the density function for R(x)
Define x: flood measure (height or flow, a non-negative value);
F(x) cumulative distribution function of x with density function f(x)
R(x) = 1/(1-F(x)) which is a monotonic transformation of x given the properties of
F(x). Since F(x) is increasing in x, R(x) is increasing in x.
Apply integration by substitution to R(x). Then
∫ 𝑥𝑚𝑎𝑥 𝑥𝑚𝑖𝑛 𝑓(𝑅) 𝑑𝑅 𝑑𝑥 𝑑𝑥 = ∫
𝑅(𝑥𝑚𝑎𝑥_ 𝑓(𝑅)𝑑𝑅
𝑅(𝑥𝑚𝑖𝑛)
Substituting in dR/dx equal to R
2 f(x) from above, then
∫ 𝑥𝑚𝑎𝑥 𝑥𝑚𝑖𝑛 𝑓(𝑅)𝑅 2 𝑓(𝑥)𝑑𝑥 = ∫
𝑅𝑚𝑎𝑥 𝑓(𝑅)𝑑𝑅
𝑅𝑚𝑖𝑛
Consequently, 𝑓(𝑅)𝑅 2 𝐹(𝑥) = 𝐹(𝑅) 𝑓(𝑅) =
𝐹(𝑅)
𝑅 −2
𝐹(𝑥)
The density function of R is then seen to be equal to R -2 if the cumulative distribution function F(R) equals F(x) which is asserted to be the intent of the transformation. This derivation provides a further clarification of the role of the equivalent cumulative distribution functions which was used in the more intuitive derivation in the text.
25

Appendix B: Regression result: Second order regression for ex-ante adjustment
Dependent variable: proportional difference between ex-ante and ex-post measure
Independent variables: 1) q, probability; 2) CS_M, ratio of damages to wealth; 3) r, relative risk aversion and second order and interaction terms.
Source | SS df MS Number of obs = 27
-------------+------------------------------ F( 9, 17) = 18.83
Model | 1.49950562 9 .166611735 Prob > F = 0.0000
Residual | .150437988 17 .008849293 R-squared = 0.9088
-------------+------------------------------ Adj R-squared = 0.8606
Total | 1.64994361 26 .063459369 Root MSE = .09407
------------------------------------------------------------------------------
D | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
q | .0972039 .1898152 0.51 0.615 -.303271 .4976789
CS_M | 2.063893 .5666262 3.64 0.002 .8684157 3.259369
r | .0748124 .0387012 1.93 0.070 -.00684 .1564649
rsq | -.0039643 .003441 -1.15 0.265 -.0112242 .0032956
qr | -.041162 .0118053 -3.49 0.003 -.0660689 -.016255
qCS_M | -1.162338 .2282071 -5.09 0.000 -1.643813 -.6808631
qsq | -.0024656 .1927943 -0.01 0.990 -.409226 .4042948
rCS_M | .0501882 .0201258 2.49 0.023 .0077265 .09265
CS_Msq | -2.099173 1.026932 -2.04 0.057 -4.26581 .067464
_cons | -.1378926 .0686641 -2.01 0.061 -.2827613 .006976
------------------------------------------------------------------------------
26

27

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