the aggregate recreational visit function in Equation (4) estimated across all households would represent the willingness to pay, or demand, of these households for recreational fishing visits to the lake in response to changes in the fee rate f. Changes in environmental quality of the lake would therefore cause this demand curve to “shift,” and the welfare consequences, or value, of this change in environmental quality would be measured by changes in consumer surplus from this shift in the demand for fishing visits.

Instead of z being a desirable commodity such as recreational visits, it could alternatively be “bad,” such as the incidence of waterborne disease from use of a microbially polluted aquatic system as a source of domestic water supply. This implies that ∂U / ∂z < 0 in the utility function from Equation (1). The household may not be able to allocate its labor time to affect the incidence of the disease, but it may be able to allocate expenditures p^vv that would mitigate the adverse effects of z or reduce its occurrence. For example, these could be purchases of marketed goods (e.g., bottled water, water filters, medical treatment) or payment for access to public services (e.g., improved sewage treatment or water supply). In addition, any improvements in water quality q may also mitigate the incidence of disease. As a result, Equation (2) is now modified to

(5)

where ∂z/v∂ < 0 and ∂z/∂q < 0. By assuming that the household’s allocation of its labor time is not relevant to this simplified problem, the budget constraint in Equation (3) is now

(6)

where M is total household income, including any labor income. Maximizing the utility function of Equation (1) with respect to Equations (5) and (6) yields the optimal demand for any mitigating good or service purchased v^*, as a function of prices p^x and p^v; household income M; and water quality q. By substituting latter demand for v^* into the disease incidence function of Equation (5), totally differentiating, and rearranging, one can obtain an estimable reduced form relationship between disease incidence z^* and levels of water quality q.