A Test of the Expected Utility Model: Evidence from Earthquake Risks

The purposes of this paper are twofold. The first is to demonstrate that the expected utility hypothesis is a reasonable description of behavior for consumers who face a low-probability, high-loss natural hazard event, given that they have adequate information. The second is to demonstrate that in California information on earthquake hazards was generated by a 1974 state law that created a market for safe housing that previously did not exist.


I. Introduction
In a recent survey article on expected utility theory, Schoemaker (1982) describes the theory as "the major paradigm in decision mak-consumers account for this information in their decision making. In our theoretical development, consumers are assumed to be informed about relatively safe and unsafe locations. The information may be attained by visual inspection, word of mouth, or a government program that delineates relatively unsafe housing locations for home buyers. The empirical results in Section III not only support the contention that information is available and considered in home purchase decisions, they shed light on the source of the information.
The consumer's problem is to maximize expected utility over two states of the world: the earthquake state and no earthquake state, which occur with probabilities p and 1p, respectively. The consumer pays p(a, s) for a house where a = (al, . . ., a,,) is a vector of n attributes and s is the safety attribute. Specifically, s is the monetary loss that the consumer perceives would be sustained during an earthquake. The function p(a, s) is assumed to be twice continuously differentiable in all arguments with first partial derivatives positive for i = 1, . . . , i. This implies that the n attributes are all desirable; if, for instance, neighborhood crime is considered, the attribute is the absence of crime. The partial derivative of the hedonic price equation with respect to the safety attribute is necessarily negative as shown below.
Expected utility is written as

V = pU[W(a) p(a, s)s] + (1p)U[W(a) p(a, s)],
where U has continuous first and second partial derivatives. The function VV(a) is the wealth equivalent of the bundle of attributes the consumer has in the two states and is also assumed to be twice continuously differentiable. The safety attribute (or the amount of selfinsurance) appears in both states as a reduction in the price of the house but appears again in the earthquake state as a damage loss. The optimum choice of attributes is characterized by the following first-order conditions; a: pUf(W1 -Upi) + (1p)U'(W, -pi) = 0, i =1 * . . , n; ( where subscripts on W and p denote partial derivatives and the e subscript on U denotes evaluation in the earthquake state. Assuming nonsatiation (Us, U' > 0), condition (2) implies that the ith attribute is chosen where W, = pi, or its marginal value to the consumer equals its marginal cost in the market. Condition (3) indicates that at the optimum the ratio of marginal utilities in the two states must equal the price ratio of self-insurance where the prices are weighted by the state of the world probabilities.3 Note also that -p1 <,p < 0, or an additional dollar spent on safety must decrease damages by more than a dollar. Assuming second-order conditions are satisfied, optimum values of a and s solve conditions (2) and (3). Either risk neutrality or risk aversion is compatible with second-order sufficient conditions1 Equation (3) forms the basis for testing the expected utility model. That is, given values for the unknown parameters in equation (3) one can determine whether or not individuals act in accordance with expected utility theory. The empirical analysis presented in the next section is directed at determining both the existence of a price gradient with respect to relative earthquake safety and the magnitude of any price differential (p5). In addition, the source of this location information is examined. In Section IV the estimated price differential is combined with probability and expected damage estimates to analyze the expected utility model.

III. Empirical Analysis: Hedonic Housing Equations
In the theoretical model it was hypothesized that individuals, acting on hazard information and possessing varying levels of risk aversion, would locate along a hedonic price gradient, with relatively safer homes commanding higher prices, everything else equal. In this section, a methodology that enables this hypothesis to be tested is described. Empirical tests are conducted for both Los Angeles County and the San Francisco Bay Area counties-Alameda, Contra Costa, and San Mateo. Also included is a description of the data base and the test results.
Proximity to earthquake-related hazards is the important variable under study. Relatively hazardous areas have been delineated through research programs conducted by the U.S. Geological Survey and the California Division of Mines and Geology. The outcome of these efforts was the Alquist-Priolo Special Studies Zone Act passed by the California legislature in 1972 and amended in 1974, 1975, and 1976. This act represents an attempt to provide society with information concerning relative earthquake-associated risk.
Special Studies Zones (SSZs) are designated areas of elevated relative risk determined by potentially and recently active earthquake fault traces (surface displacement has occurred in Holocene time, i.e., over the last 11,000 years). The evidence of faults may be directly observable (ruptured streets, crooked fences, etc.) or inferred (i.e., geomorphic shapes). The length of an SSZ coincides with the fault length whereas the width is generally one-eighth of a mile on each side of the fault.
Within California, the total number of SSZs designated through January 1979 was 251. There are two important ways in which consumers become aware of these. First, when an SSZ is designated, property owners in the zone are notified. Second, consumers selling property in an SSZ are required to notify prospective buyers that the property is in a zone (Alquist-Priolo Special Studies Zones Act 1974). This latter requirement has been implemented by the Department of Real Estate by having agents disclose the information via an addendum to the purchase contract. The buyer is then granted a period to collect additional information or to cancel the sale.
The potential effects of the Alquist-Priolo Act form the basis of a testable hypothesis. The null hypothesis is that consumers respond to the awareness of hazards associated with SSZs with the alternative being that they do not.

Data Specifics
The study areas are Los Angeles County and the San Francisco Bay Area counties, and observations are confined to single family residences. Thus, we do not consider the impact of hazard location on other structures (multiple family dwellings, mobile homes, commercial, etc.) or other ownership types (rental, leasing, etc.). Therefore, within our sample, this research asks if Los Angeles and San Francisco Bay Area households will pay a premium in the form of higher housing values for homes located outside an SSZ and what is the magnitude of that willingness to pay.
The data base was constructed so that hypotheses concerning the impact of SSZ location differences on housing sale price could be tested. The dependent variable in the entire analysis is the sale price of owner-occupied single family residences.5 The independent variable set consists of variables that correspond to three levels of aggregation: house, neighborhood, and community. The Appendix describes further the data employed in the study.
The housing characteristic data, obtained from the Market Data Center (a computerized appraisal service centered in Los Angeles), 5Note that sale price or the discounted present value of the How of rents rather than actual rent is used as the dependent variable. The two are interchangeable given the appropriate discount rate. pertain to houses sold in 1978 and contain information on nearly every important structural and/or quality attribute. The Appendix provides summary statistics for the housing, neighborhood, and community characteristics used in the hedonic analysis. It should be emphasized that housing data of such quality (e.g., micro level of detail) are rarely available for studies of this nature. Usually outdated data that are overly aggregate (for instance, census tract averages) are employed. These data yield functions relevant for the "census tract" household but are only marginally relevant at the household (micro) level.
The Market Data Center provided computer data tapes listing all houses sold in Los Angeles County and the San Francisco Bay Area counties during the period specified. The number of entries was unmanageably large, so the data set was reduced as follows. First, a data set was constructed that contained houses within SSZs.J' This was accomplished by first searching the tape for all houses located in census tracts that were wholly or partly in an SSZ. This list was further reduced through a random number matching system. The addresses of the remaining entries were then checked against a detailed map to select those clearly within an SSZ. The numbers of' valid Los Angeles County and the San Francisco Bay Area SSZ data points were 292 and 745, respectively.
Second, data sets were constructed that included houses not located in hazard areas. After deletion of incomplete data entries, a random number matching system was utilized to choose sample sizes of approximately five thousand observations in each study area. The safety variable is then represented by a dummy variable that takes on the value one for houses in an SSZ and zero otherwise.
In addition to the immediate characteristics of a home, other variables that could significantly affect its sale price are those that reflect the condition of' the neighborhood and community in which it is located. That is, school quality, ethnic composition, proximity to employment centers (and in Los Angeles County, distance to the beach), and measures of the ambient air quality have a substantial effect on sale price. In order to capture these impacts and to isolate the independent influence of location vis-A-vis the SSZs, these variables were included in the econometric modeling.
The data base assembled for the housing value study is appropriate to test the hypotheses outlined above for two reasons. First, the housing characteristic data are extremely detailed at the household level of' aggregation and extensive in that a relatively large number of' observations are considered. Second, a variety of neighborhood and com-' See Hart (1977) for the location of SSZs.

Empirical Results
The underlying structure of the hypothesis test is a single-equation empirical model that attempts to explain the variation in sale prices of houses located in Los Angeles County and the San Francisco counties.7 The estimated coefficients of these hedonic equations specify the effect a change in a particular independent variable has on sale price. In reference to the SSZ location variable, this procedure allows one to focus on its significance while separating out the influence of other extraneous variables. Therefore, this analysis yields two outputs concerning the relationship of hazard location differentials to housing price. First, the relative significance of location variations is determined and, second, the estimated coefficient pertaining to location implicitly measures its monetary value. The estimated Los Angeles and San Francisco hedonic gradients that provide the best fit of the data are presented in table 1.8 A number of aspects of the equations are worth noting. First, as measured by R2, the nonlinear form is a significant improvement over linear specifications. In addition, a comparison of the log of the likelihood values (semilog to the linear) indicated that the semilog form was a significant improvement at the 1 percent level (see Judge et al. 1980). As Rosen (1974) pointed out, this is to be expected since consumers cannot always arbitrage by dividing and repackaging bundles of housing attributes. Thus, on both theoretical and empirical grounds the semilog specification proved to be a better functional form.
Second, in the semilog equations all coefficients have the expected sign and are significantly different from zero at the 1 percent level. The SSZ dichotomous location variable has the a priori expected relationship to home sale price and is significant at the 1 percent level. This result is invariant with respect to various sample sizes, model formulations (various independent variable sets were tested), and estimated functional form.9 These results indicate that individuals are 7See Freeman (1979) and Mdler (1977) for a review of estimates of hedonic housing equations. 8 The main difference between the Los Angeles and Bay Area analyses is the locational variables. In the Bay Area distance to beach (ocean) is unimportant due to the presence of the bay. In addition, the three Bay Area counties were assigned dichotomous variables to account for county differences. San Mateo County is the excluded group and therefore is included in the constant term. 9 Since the SSZ location variable is a zero-one variable then our choice set over functional forms was essentially restricted to the linear and semilog forms. Thus, possi-acting on hazard information when making locational choices, and this action is translated into a measurable hedonic gradient.
Regarding the monetary impact on housing sale price, the nonlinear specification does not allow straightforward interpretation since the effect of any independent variable depends on the level of' all other variables. However, the Los Angeles County (Bay Area) results indicate that if all other variables are assigned their mean values, then living outside of an SSZ causes an increase in home value of approximately $4,650 ($2,490) over an identical home located in an SSZ. In relative terms the magnitude has approximately one-half' the impact of a swimming pool or one-third the value of a view.
In the next section, these monetary figures are used to test the expected utility model. But before proceeding to this analysis, we can confirm the source of the hazard information used by home buyers. As indicated above, the Alquist-Priolo Act was enacted in 1974. Therefore, a pre-1974 analysis of the housing market would yield insight concerning the importance of the act in providing consumers relative risk information.
Housing data for the 1972 time period are used in the test of the Alquist-Priolo Act. Successful enhancement of consumers' awareness by the Alquist-Priolo disclosure provisions would require a change in the hedonic rent gradient over time. This change could take one of' two forms: (i) an SSZ location would be an insignificant housing characteristic in 1972 yet significant in 1978; or (ii) the location variable would be significant in both years but its relative magnitude would increase over time. The first type of' change could be considered a strong test of the impact of the Alquist-Priolo Act since the act would have filled an existing information void. Thus evidence of' a direct market effect would be available. The magnitude change of the SSZ variable would imply a weaker response since it would be evident that consumers had hazard location information from some other source and were already acting on it before passage of the Alquist-Priolo Act.
The relative impact of hazard information independent of the Alquist-Priolo Act is also tested using the pre-and postdata sets; that is, if SSZ location remains a stable (no relative magnitude change), significant determinant of' housing price, then consumers are acting on some available information although their preferences have not been enhanced or changed by the public disclosure program. ble forms such as quadratic, log, inverse semilog, exponential, semilog exponential, and the Box-Cox transformation of the SSZ location variable are not available since they inevitably reduce to zero-one or cannot be estimated (e.g., log of zero). Further, a Box-Cox transformation of the dependent variable that is not equivalent to linear or semilog yields difficult to interpret results. Finally, the translog transformation is not available because the objective is to determine the separate influence of SSZ locations.
The 1972 time period results are presented in table 2. The senilog functional form provides the best fit of' the data, and all coefficients, with the exception of' SSZ location, are significant at the 1 percent level and related to home sale price as expected. However, the most noteworthy aspect of' the equations is that the SSZ location variable does not demonstrate significance in 1972, even at the 1() percent level. The combined 1972 and 1978 results indicate that the Alquist-Priolo Act has caused a structural change in the hedonic gradient over time. This is evidenced both by the significant monetary impact change over time and by the change in significance. Theref'Ore, in the study areas the Alquist-Priolo Act does pass a strong test of' effectiveness, suggesting that the act provided information that consumers used in their market decisions.

If' consumers behave as if' they maximize expected utility, then firstorder condition (3) must necessarily be satisfied. The terms in condition (3) include the probability of' an earthquake, marginal utilities of' income, marginal damage to a house, and the marginal change in the house price. Our approach is to solve equation (3) f`or this latter term by substituting in reasonable values of' all the former terms for the Los
Angeles region. This provides an analytical solution f'or the price difference between houses in and out of' SSZs. This price difference is then compared to the observed differencee in housing prices estimated in the previous section. The two differences are shown to be close, thereby supporting the expected utility paradigm.
In the empirical work, houses were described as either in or out of' unsafe areas so that the safety attribute was (liscrete. In equation (3) the attribute is continuous. Therefore, the partial derivative p, in (3) is approximated as np/As, where Ap is the total price differencee between safe and unsafe houses, and As is the total damage in dollars resulting from an earthquake. Equation (  perceive the role of inflation and keep their homes for a longer period, then use of the real rate of interest would be more appropriate in calculating the true cost differential for living outside of' an SSZ. From the early 1950s up until 1978 the real rate of interest on home mortgages averaged around 3 percent. If we use this rate of interest, we obtain a real cost differential of' $140 per year. These figures provide a range for comparison to Ap from equation (4) after substituting in values for p, As, and UO/U'.
First, consider a range of' values for Us /U'. As a lower bound, and to be consistent with second-order maximization conditions, we use risk neutrality where Us/U' = 1. For risk aversion, however, 1 < UM/U' < xc To establish an upper bound we appeal to recent work that employs cross-sectional data on household assets to establish properties of household utility functions. In particular, Cohn et al. (1975) found evidence that the coefficient of' relative risk aversion is slightly decreasing in wealth. Friend and Blume (1975) found that "if there is any tendency for increasing or decreasing proportional risk aversion, the tendency is so slight that for many purposes the assumption of constant proportional risk aversion is not a bad first approximation" (p. 915). More recently, Morin and Suarez (1983) found the coefficient to be slightly decreasing for wealth levels up to $100,000, after which it becomes approximately constant. Furthermore, Friend and Blume estimated the market price of risk to determine a value for the coefficient, which they argue is greater than one and may be as high as two. Since we are interested in the ratio of marginal utilities and not the coefficient of relative risk aversion, we cannot use these results directly; but we can explore the implications suggested by this work.
To determine an upper bound, one approach is to examine Us/U' for various utility functions that exhibit the properties cited above. The largest upper bound is associated with a utility function exhibiting constant relative risk aversion equal to two; thus, we use U(A) -A-', where A is total wealth. The denominator of UsO/U' is evaluated at total wealth, while the numerator is evaluated at total wealth minus the dollar value of earthquake damage. Again, to determine the largest upper bound, we assume the maximum expected damage of about $20,000 developed below. To obtain total wealth we note from Friend and Blume's data (table 3, p. 908) that over their entire sample the market value of a house as a percentage of total wealth averaged 16 percent."' Since the average market value of houses in our sample is $83,153, we use as an estimate of total wealth A = $83,153/.16 = $519,706. Finally, using U(A) = -A-', we obtain UeolU' = 519,7062/499,7062 = 1.08 for the largest upper bound. This second approach gives an identical estimate to the first developed above and suggests that risk aversion plays a surprisingly small role in our analysis apparently due to the relatively small changes in lifetime wealth involved. To estimate the odds of an event in the Los Angeles area, we use two sources. First, Kunreuther et al. (1978) report results of a survey question among California residents on the subjective beliefs concerning the odds of an earthquake. The average perceived odds of an event from that survey are about 2 percent per year.12 To obtain a more objective estimate of the risk of an event we turn to a report '' Note that U"A/U' will be a negative number for risk-averse individuals. Thus, issued by the Federal Emergency Management Agency (FEMA 1980), which estimated the odds of a large earthquake to be from 2 percent to 5 percent per year for the Los Angeles area. The upper bound of that range, 5 percent, resulted from scientific concerns over the Palmdale bulge, a temporary uplifting of the desert floor north of' Los Angeles that occurred in the late 1970s. The lower bound estimate, which was widely publicized prior to the FEMA report, is based on the historical pattern of large earthquakes that have occurred in the Los Angeles area (Sieh 1978). For the relevant time period for our study, 1972-78, and for the Los Angeles area, there exists a remarkable coincidence between subjective and objective measures of' risk of an earthquake. The FEMA lower bound estimate, which is appropriate prior to the occurrence of' the Palmdale bulge, and the Kunreuther et al. estimate both imply p = .02 for estimating Ap in equation (4).

Another approach for estimating UI/U' is to use a linear approximation (first-order
Finally, we need to develop an estimate of earthquake losses or damages associated with residing in an SSZ as opposed to residing outside of an SSZ, defined as As in equation (4). Again, we can obtain a subjective estimate of about $20,000 from Kunreuther et al. (1978) for the average total damage people expect to occur to their homes if' an earthquake occurs.'3 As an alternative measure, engineering studies suggest that the average damage to a single-story frame house should a great earthquake occur near Los Angeles would be about 5 percent of the home's value (NOAA 1973). This implies a level of' damage for the average house in our property value sample (worth $83,153) of $4,158. However, homes in areas of maximum ground shaking, such as would occur in an SSZ if' the local fault ruptured, would suffer damage equal to about 25 percent of the home's value (NOAA 1973). For the average house in our sample, this implies damages of $20,788 (for a home in an area of' maximum ground shaking). These figures obviously span the Kunreuther et al. estimate, with the upper bound figure quite close, suggesting that households answering the Kunreuther survey may have perceived the question to imply that their home would be located in an area of maximum damage. Note, however, that As represents the difference in damages an individual would expect from living in versus outside of an SSZ should an earthquake occur. Thus, as an absolute upper bound, we will use a value of As of $20,000 consistent with a subjective assessment that homes outside of an SSZ will suffer no damage. As a lower bound we will take the difference in the objective engineering assess-'" Again, this average was obtained by weighting expected damage by frequency of occurrence among the survey respondents from fig. 5.6, p. 94, of Kunreuther et al. (1978). ments ($20,788 minus $4,158) of $16,630. Thus, the lower bound assumes homes in an SSZ will suffer the maximum level of ground shaking and homes outside an SSZ will suffer average levels of ground shaking.
To obtain an upper bound estimate for the annual value of living outside of an SSZ to an expected-utility-maximizing household, we substitute values of UJIU' = 1.08, p = .02, and As = $20,000 into equation (4). These figures are consistent with the highest observed coefficient of relative risk aversion of' 2 and the subjective evidence obtained by Kunreuther et al. on earthquake risk and damages. To obtain a lower bound estimate we assume risk neutrality so U/IU' = 1 and use scientific-engineering evidence for p = .02 and As = $16,630.
These assumptions yield a range for Ap of from $333 to $431 per year. In contrast, from the estimated property value equation, the perceived annual cost of living outside of an SSZ ranges from $140 to $440 depending on use of real or nominal interest rates. This evidence suggests that the estimated property value equation for Los Angeles is consistent with utility-maximizing behavior with respect to earthquake risks.

V. Conclusion
Schoemaker (1982, p. 552) summarizes the problems of' expected utility theory as follows: "As a descriptive model seeking insight into how decisions are made, EU [expected utility] theory fails on at least three counts. First, people do not structure problems as holistically and comprehensively as EU theory suggests. Second they do not process information, especially probabilities, according to the EU rule. Finally, EU theory, as an 'as if' model, poorly predicts choice behavior in laboratory situations. Hence, it is doubtful that the EU theory should or could serve as a general descriptive model." Our analysis provides only indirect evidence with respect to Schoemaker's first point. However, having demonstrated consistency between our property value market results and the expected utility model for Los Angeles, we can strengthen the argument considerably by briefly considering the San Francisco case.
For San Francisco, home sale prices, damage to homes should an earthquake occur, and, presumably, risk preferences are all similar to the Los Angeles case analyzed in the previous section. However, the probability of a damaging earthquake is considerably less according to available scientific evidence. For example, the FEMA report (1980, p. 3) states: "the current estimated probability . . . is smaller [than for Los Angeles] but significant," and later gives annual odds for a great earthquake on the San Andreas fault near San Francisco as 1 percent.
These are half the odds given for a great earthquake in the Los Angeles area in the same report. Thus, from equation (4) of the previous section one would predict, on the basis of expected utility theory, that the property value differential for houses in SSZs in the Bay Area should be about half that observed in Los Angeles. From the two property value studies the differentials are $2,490 and $4,650, respectively. This successful "prediction" suggests both that individual households process probability information in a reasonably rational and accurate way and that, at least in a market situation with a well-defined institutional mechanism, the expected utility model may perform well in predicting behavior. It should be pointed out that through the decade of the 1970s, the media in California carried an average of two stories per week relating to local earthquake events, actual or possible damages, and probabilities (see, e.g., Los Angeles Times, April 7, 1975; April 4, 1976; April 22, 1978). Possible earthquake events are a topic of considerable interest within the state, and the level of awareness among state residents is very high (Turner et al. 1979). The scientific evidence summarized in the 1980 FEMA study used in our calculations was widely publicized throughout the 1970s and may well be responsible for the similarity between the Kunreuther et al. (1978) subjective probability estimates of earthquake risk and more objective scientific assessments.
In summary, the property value studies make a strong case for selfinsuring behavior consistent with maximization of expected utility. Further support of this result can be found by comparing the property value studies with surveys (Brookshire et al. 1982). In our survey of homeowners located in SSZs in Los Angeles (Brookshire et al. 1980), when asked how much more they would pay to purchase the same home outside of an SSZ, only 26 percent of respondents were willing to pay anything more. However, the average of all responses (including zero bids) was $5,920, very close to the average sale price differential of $4,650 from the Los Angeles property value study.'4 Efficient prices should convey information to consumers. We have shown that the property value markets for both Los Angeles and San Francisco convey hedonic price differentials to consumers that correspond closely to expected earthquake damages for particular homes located in SSZs. Although the information provided by the SSZ program is by no means perfect, our results suggest that programs to provide consumers with hazard information may well be effective.