The HPM has also been applied to estimate the impact of hazards on property values. For instance, Bernknopf, Brookshire, and Thayer (1990) and Beron et al. (1997) focus on earthquake and volcanic hazards; the effect of hazardous waste and superfund sites is analysed by authors such as Clark and Allison (1999) and McCluskey and Rausser (2001); wildfire hazard is considered by Donovan, Champ, and Butry (2007).
MacDonald, Murdoch, and White (1987) were probably the first authors to provide a theoretical framework for the application of HPM to estimate the WTP for a reduction in the probability of flooding. Their model is based on the application of the expected utility framework to the HPM by Brookshire et al. (1985) and the option price literature on supply uncertainty described by Smith (1985). The rational consumer will choose to live in a location which maximizes his expected utility. Flood risk is considered a characteristic of properties, and when individuals decide a location where to live this decision often includes the level of risk they face (Bin, Kruse, and Landry, 2008; MacDonald, Murdoch, and White, 1987). Since there is a potential loss associated with flood risk, individuals will incorporate this into his choice.
The model we describe in this section is based on MacDonald, Murdoch, and White (1987), Bin et al. (2008), Kousky (2010) and Bin and Landry (2013). Formally, we can redefine the HPF (1) to consider explicitly the property risk factors of the house. Following Hallstrom and Smith (2005), let the subjective probability of flooding, i.e. the homeowner’s subjective assessment of flood risk, be a function 𝑝(𝑖, 𝑟) of the set of information, 𝑖, the individual holds about flood risk in the location of the property, and 𝑟 which represents the site attributes related to flood risk, this could be locational
characteristics such as proximity to water bodies or elevation. It is important to distinguish the subjective assessment of the probability of flooding, 𝑝, from the objective measure of flood risk, 𝜋. This distinction implies three important things; first, that the perceived risk is not necessarily equal to the objective risk; second, that changes in the objective risk are not necessarily perceived; and third, that changes in the perceived risk do not necessarily arise from changes in the objective risk. In areas where flood risk disclosure is mandatory or public information about flood risk is available, the set of information, 𝑖, might include the objective probability of flooding, 𝜋.
Let the HPF be given by the following equation:
𝑃 = 𝑃(𝑍, 𝑟, 𝑝(𝑖, 𝑟)) (5)
Therefore, 𝑃 is exogenous to individual buyers and sellers, but reflects subjective risk perception 𝑝(𝑖, 𝑟). Following Brookshire et al. (1985) the model uses an expected utility framework that incorporates risk factors associated with a property. The utility function of individuals is given by equation (2), and the household’s decision is modeled using the following state dependent utility function:
𝐸𝑈 = 𝑝(𝑖, 𝑟) ∙ 𝑈𝐹[𝑍, 𝑟, 𝑄] + (1 − 𝑝(𝑖, 𝑟)) ∙ 𝑈𝑁𝐹[𝑍, 𝑟, 𝑄] (6)
𝑈𝐹(∙) is the utility of the homeowner in a state where a flood occurs and 𝑈𝑁𝐹(∙) is the
utility of the homeowner when there is no-flood. The budget constraint for the individual in state 𝐹 (with perceived probability 𝑝(𝑖, 𝑟)) and 𝑁𝐹 (with perceived probability (1 − 𝑝(𝑖, 𝑟))) is given by equations (7) and (8), respectively.
𝐹: 𝑀 = 𝑃(𝑍, 𝑟, 𝑝(𝑖, 𝑟)) + 𝑄 + 𝐿(𝑟) (7)
Note from equations (7) and (8) that the level of consumption of 𝑄 is different across states, in particular 𝑄𝐹 < 𝑄𝑁𝐹. Both, the level of utility and the marginal utility of income
may change with the state. The conditional loss 𝐿(𝑟) ∈ (0, 𝑆̅), is a function of the locational risk characteristics of the house, 𝑟, and reflects the magnitude of the loss should state 𝐹 occurs; 𝑆̅ represents the structure replacement cost of the property. Notice that budget constraint (8) is the same as (2) where no flood risk is considered. Thus, the occurrence of a flood is associated with a potential monetary loss 𝐿(𝑟).
The rational consumer will choose to live in a location which maximizes his expected utility subject to the budget constraint. If a property is subject to frequently substantial flooding, the owner may incur substantial repair costs and additional associated losses; alleviating strategies include constructing flood-proofing structures and engaging in environmental flood control practices. All these future costs might easily exceed the cost of buying an equivalent property outside the flood risk area (Bin and Kruse, 2006; Lamond, 2012; MacDonald, Murdoch, and White, 1987; Zimmerman, 1979). Consumers will locate within a floodplain if they are compensated for accepting the potential loss (MacDonald, Murdoch, and White, 1987). Intuitively this means that flood risk is capitalised into property prices.
Formally, maximizing expected utility (6), with respect to the subjective probability of flooding, 𝑝, subject to the homeowner’s budget constraint, and dividing by the expected marginal utility of income yields:
𝜕𝑃
𝜕𝑝 =
𝑈𝐹− 𝑈𝑁𝐹
𝑝(𝑖, 𝑟)𝜕𝑈𝜕𝑄 + (1 − 𝑝(𝑖, 𝑟))𝐹 𝜕𝑈𝜕𝑄𝑁𝐹 (9)
Equation (9) is the coefficient on the risk variables estimated in hedonic regressions. It indicates that the marginal implicit hedonic price for flood risk reflects the incremental utility difference across states; dividing by the expected marginal utility of income produces a measure of marginal WTP. This implicit price can be used to estimate welfare effects of marginal changes in the independent variable (Bin et al., 2008; Kousky, 2010; MacDonald, Murdoch, and White, 1987).
To see why, following MacDonald, Murdoch, and White (1987) and Smith (1985), consider 𝜎 to be a reduction to 𝑝. The Option Price (𝑂𝑃) is defined as the maximum WTP for an improvement in the chance of the desirable state, 𝑁𝐹, holding expected utility constant, and can be expressed as:
(𝑝(𝑖, 𝑟) − 𝜎) ∙ 𝑈𝐹[𝑍, 𝑟, 𝑄 − 𝑂𝑃] + (1 − 𝑝(𝑖, 𝑟) + 𝜎) ∙ 𝑈𝑁𝐹[𝑍, 𝑟, 𝑄 − 𝑂𝑃]
= 𝑝(𝑖, 𝑟) ∙ 𝑈𝐹[𝑍, 𝑟, 𝑄] + (1 − 𝑝(𝑖, 𝑟)) ∙ 𝑈𝑁𝐹[𝑍, 𝑟, 𝑄] (10)
Therefore, the marginal WTP for reducing the probability of a flood can be expressed as the change in 𝑂𝑃 due to a reduction in the probability of flooding (𝜎). From equation (10) and assuming constant expected utility we get:
𝜕𝑂𝑃
𝜕𝜎 =
𝑈𝑁𝐹− 𝑈𝐹
(𝑝(𝑖, 𝑟) − 𝜎) 𝜕𝑈𝜕𝑄 + (1 − 𝑝(𝑖, 𝑟) + 𝜎)𝐹 𝜕𝑈𝜕𝑄𝑁𝐹 (11)
Thus, in a housing market with flood risk, the locations that improve the chances of state 𝑁𝐹 will get bid up, ceteris paribus. Notice from equations (9) and (11) that:
𝜕𝑃
𝜕𝑝 = −
𝜕𝑂𝑃
𝜕𝜎 , for 𝜎 = 0 (12)
Therefore, the marginal WTP for a reduction of 𝑝 whilst remaining indifferent is captured by the sales price differential resulting in housing markets as consumers bid for locations with lower 𝑝. This justifies interpreting the coefficients from hedonic regressions as estimates of the amount of compensation a homeowner requires, through a lower property price, to move into a riskier area. Some examples of applications of the hedonic price model to valuation of flood hazard include MacDonald, Murdoch, and White (1987), Speyrer and Ragas (1991), Harrison, Smersh, and Schwartz (2001) and Bin and Kruse (2006).