A Community-Based Flood Insurance Option (2015)

Appendix B

Technical Discussion of the Responsibility for Insurance Is Irrelevant (RII) Proposition

For the sake of discussion and illustration, assume there are two actors, the community, c, and its residents, r. The analysis will be conducted using two residents, r₁ and r₂. The community takes actions, a, that diminish expected flood damages to the residents, d₁ and d₂, and to the community’s infrastructure, d_c. Such actions could be building a dike or restoring a marsh. The community could also restrict residents from building or living in certain areas, or impose restrictions on residences, such as requiring that houses be built with pilings. For now, leave aside risk aversion for the individuals, as well as concerns about aggregate risk for the community and the insurer. Posit actuarially fair insurance, which enables one to expect flood damages as a measure of loss. The goal of these simplifications is to enable us to get our thinking straight in the simplest case.

The individuals take measures, m₁ and m₂, which reduce their personal expected flood losses but not any other party’s losses. Posit that both the action of the community and the measures of the residents are calibrated in dollar terms. The objective is to have the combination of a and m₁ and m₂ that minimizes total flood costs (TFCs), as comprised by expenditures by both the residents and the community plus the damages to the residents and the community’s infrastructure.

¹ If the community were to restrict locations or impose building requirements, then the tally of “damages” would also include the costs to the individuals in direct dollars or willingness-to-pay for meeting those impositions.

Posit, as is usually assumed and is highly likely to be true, that there are diminishing returns to a and to m₁ and m₂ in reducing damages over the relevant range of values.² Damages to a resident are limited by her measure plus the action of the community.³ Damages to the community infrastructure are limited solely by the community’s expenditure. Thus,

The different damage functions for residents 1 and 2 arise because they reside in different places and have different structures.

The goal, to reiterate, is to find those values a*, m₁*, and m₂* that minimize (1) given the production functions (2). Thus, substitute the values in (2) for d₁, d₂, and d_c and then substitute them into equation (1). Then take derivatives with respect to each of the three choice variables, a, m₁ and m₂, and set them equal to 0. Let the derivatives of the f, g, and h functions with respect to their arguments be denoted by subscripts of those arguments. The efficiency conditions are

For a

Subtracting 1 from each side of the three equations in (3) gives us more easily interpreted conditions:

For a

The interpretation of the efficiency conditions in (4) is straightforward. The community should continue to spend dollars until the sum of the reduction in expected damages to the two residents and itself is just $1. The residents should only be concerned with themselves and should spend so $1 of m reduces damages by $1. As long as the community and the residents adjust to each other’s actions, they need not coordinate in any way.

The outcome above will be achieved if the community takes its residents’ damages fully into account, as it should, and if the residents optimize for themselves.

Now consider insurance provided by the federal government to the

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² There may be increasing returns over some range. For example, building half a dike may not reduce risks by much. However, the relevant range is beyond such a point; it begins where returns start to diminish.

³ Consider a one-period model, but implicitly are taking account of multiple periods. Thus, one damage-reducing measure might be locating in an area with lesser flood risk.

residents on an actuarially fair basis. The insurance cost, k, for expected damages, d, will just equal d. Thus, substituting k for d in the analysis above, everything goes through as before. (Moreover, risk aversion is ruled out as a concern, given insurance.) The requirement for efficiency is that the community now has to take the residents’ insurance costs into account. The residents themselves have to raise their m until the sum of m + k, what is now their total costs, is minimized.

If the residents are fully insured, then they no longer suffer financially from damages. Thus, the efficiency condition for the community is that it take account of the residents’ insurance costs, as it should. If insurance is only partial, then resident i will pick m_i to minimize m_i + k_i + d_i, the community will pick a to minimize the sum d_c + [k₁ + d₁] + [m₂ + k₂ + d₂], and everything goes through as before.

Given the assumptions listed here in Appendix B, it matters not whether the community or the residents pay insurance premiums.

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