Supplementary Material
Appendix A. Tables
Table A1 - Test for Discontinuity at Thresholds 1000 and 1500 (Bandwidth of 20).
Threshold 1000
(Bandwidth 20)
mean
α
1.0584
-0.39
140.6507
-61.663
12.9578
-9.313
13.2676
-3.054
13.4887
-2.501
Variable
Active Share
Flood Risk
Payroll
Income
Housing Value
Population
Density
4400
-3900
∗∗∗p < 0.01, ∗∗p < 0.05, ∗p < 0.1.
p
0***
0***
0***
0***
0***
Threshold 1500
(Bandwidth 20)
mean
α
0.773
-0.096
97.3842
-5.335
11.2467
2.35
9.8025
0.63
10.4656
1.12
p
0.14
0.611
0.029**
0***
0***
0***
426.0833
0.753
156.6
Table A2 - Test for Discontinuity at Thresholds 1000 and 1500 (Bandwidth of 30).
Threshold 1000
(Bandwidth 30)
mean
α
0.9461
-0.288
86.214
-3.455
17.0471
-4.627
10.2649
-0.016
11.7061
-0.572
Variable
Active Share
Flood Risk
Payroll
Income
Housing Value
Population
Density
2800
-2200
∗∗∗p < 0.01, ∗∗p < 0.05, ∗p < 0.1.
p
0***
0.532
0.001***
0.912
0.005***
Threshold 1500
(Bandwidth 30)
mean
α
0.6829
-0.002
94.0945
-3.653
9.6834
4.04
10.0526
0.365
10.6591
0.904
p
0.977
0.709
0***
0.005***
0.001***
0***
557.5361
0.871
80.43
1
Table A3 - Falsification Test: Arbitrarily Determined Threshold of 723 (Bandwidths 20, 25, 30).
Variable
Active
Share
Flood Risk
Threshold 723
(Bandwidth 20)
mean
α
p
Threshold 723
(Bandwidth 25)
mean
α
0.409
p
Threshold 723
(Bandwidth 30)
mean
α
p
0.254
0.004***
0.507
0.141
0.102
0.5805
0.043
0.595
95.064
0.499
0.912
103.172
-4.916
0.612
96.6333
-0.016
0.998
Payroll
12.961
-2.092
0.365
12.647
-1.383
0.501
13.128
-1.803
0.327
Income
10.049
0.177
0.326
10.027
0.231
0.222
10.1832
0.028
0.893
11.022
0.052
0.891
10.866
0.217
0.608
11.1843
-0.153
0.716
980.925
38.602
0.974
1300
-230.091
0.809
1100
-358.119
0.714
Housing
Value
Population
Density
∗∗∗p < 0.01, ∗∗p < 0.05, ∗p < 0.1.
Appendix B: Tiered Incentive Schemes.
Theoretical Model
We assume that all communities have an equilibrium level of (public) flood risk management
activity, Y*. Most communities have Y*=0. For those communities with Y*>0, we suppose that
the (political) process generating Y* draws from a smooth distribution of positive Y values, such
as a lognormal distribution or chi-square. In other words, we remain agnostic about the process
for determining Y* except that positive values of Y* be distributed continuously and that
increasing net benefits to increasing Y will not diminish Y*.1 We further consider that Y is
observable, while a latent index of optimal flood management effort, Z, exists but is no directly
observable. Z*, unlike Y*, can take negative values in equilibrium.
Under the CRS, the federal government subsidizes Y. This results in a new distribution
of flood risk management activity, Y**. We expect that Y** Y* at all levels. The cumulative
distribution function (CDF) for Y* is entirely at or to the left of the CDF for Y**, or Y** first-order
stochastically dominates Y*, as some communities will not be affected by the CRS discount, but
others will take the encouragement of CRS to upgrade their Y attainment.
In practice, we do not observe Y*. We only observe Y**. Under our theory, where CRS
never leads a community to do even less flood management, we have Y*= Y** for all CRS
nonparticipants. But for all participants, we observe Y** and never Y*. Estimating the effect of
CRS on effort Y is thus problematic because the counterfactual (Y*) is not observed for the
communities that volunteer to be “treated” and neither the actual (Y**) and the counterfactual (Y*)
are observed for the untreated nonparticipants. Being unable to observe the counterfactual of
what communities would have done in the absence of CRS constrains our evaluation of the CRS
program.
1
Note that this allows for a discontinuity to occur when Y*=0 and for Y* to remain constant even if net benefits
from raising Y* also increase.
2
Looking at the CRS program’s arbitrary thresholds is a way around this problem. A
dominant feature of the CRS program – and many other nonlinear incentive schemes – is the
tiered nature of the subsidy. Every additional 500 credit points raise the subsidy by a discrete
5% increment. This “step function” introduces a particular set of incentives that revolve around
those thresholds. Contrasting it with a linear incentive scheme, where the subsidy is a constant
proportion of credit score (e.g., credits/100 = % discount), the reward for investing additional
effort in Y is generally continuous rather than “lumpy” or discontinuous at the thresholds.
To sharpen this comparison, consider two alternative incentive schemes: a linear and a
tiered system. Figure A1 depicts the subsidy rate (s, a discount on flood insurance premiums
here) on the vertical axis and the credit score (Y) on the horizontal axis. Hold equal the area
under the two alternative subsidy lines in order to keep total program spending comparable
between the alternatives.2 Under the linear system, suppose that a community achieves some
level of Y = Y’. This value Y’ is either “near” a threshold for the tiered system (e.g., 500, 1000)
yet below that threshold or it is not. If the incentive scheme was switched from linear to tiered,
let the new credit scores be Y’’. Communities with values of Y’ near a threshold from below
may find it worthwhile to invest in additional Y in order to obtain the discrete jump in s at the
next threshold. Other communities will reduce their Y’’ to the minimal Y in their tier as long as
costs rise in Y and they seek lower costs. Notice that for either type of community, the resulting
Y’’ value is necessarily at a tier threshold. For some communities, that means Y’’ < Y’ as they
just barely cross the threshold of Y’. For other communities, that means Y’’ > Y’ as they upgrade
their Y to obtain the jump in subsidy. What constitutes “near” or how large the ranges are where
it is worthwhile to upgrade Y to take advantage of the nonlinear jump in subsidy depends on the
communities’ decision making process – likely weighing the costs of the marginal upgrade in Y
against the benefits of the much larger subsidy. On net, whether a linear subsidy induces more
effort than a tiered subsidy (i.e., Y’ > Y’’) depends on many contextual factors and remains an
empirical question.
2
Keeping the areas under the curve equal is the simplest way to highlight the effect of nonlinear incentives without
confounding it with different subsidy levels as well. In this set-up, the subsidy lines cross at the midpoint of each
tier. Of course, the linear subsidy line could be drawn to intersect the upper corners of the tiered subsidy line –
along the lines apparently envisioned by Zahran et al. (2009) – but such a comparison is an implicitly more generous
subsidy program (i.e., that linear subsidy amount is always as great or greater for every Y) and implies a positive
subsidy at Y=0.
3
Fig. B1 - Linear and Tiered Subsidy Incentives.
s
linear system
10
tiered system
above
5
“near” and below
500
1000
1500
Y
While Figure A1 helps clarify the effect of switching from a linear to a tiered incentive
system, the effect of introducing a tiered subsidy when no previous subsidy existed is more
involved. The basic logic remains. Suppose that Y* is the initial credit score in the absence of
the CRS subsidy. Some initial Y* values will be near-yet-below a CRS threshold, while others
will be above a threshold. As before, those communities near-yet-below a threshold will upgrade
to a Y** > Y* just above the threshold. The other communities, however, will not reduce their Y**
to the lower threshold value because Y* was optimal in the absence of a subsidy. Introducing a
subsidy will not induce a Y**<Y*. We should expect to see many communities’ optimal Y values
remain unchanged with the tiered subsidy simply because they subsidy amount is insufficient to
induce a jump to the next tier. (For these communities, the subsidy is not affecting behavior and
amounts to a wealth transfer.) Those communities “near” a threshold from below might upgrade
to the next tier. How large the “near” neighborhood is depends again on relative costs of
upgrading and the decision processes for communities. Regardless, Y**  Y* for all communities,
and Y** > Y* for some. Except for some communities whose optimal Y* in the absence of the
CRS happened to be at a CRS threshold, the bulk of the communities achieving Y values at the
thresholds can be seen as communities responsive to the CRS (as Y** > Y). Finally, if the CRS
subsidy were linear, it might be seen to change the Y behavior of more communities, but the
tiered system’s larger effect on fewer, “near” communities may amount to more or less of a
cumulative impact on Y.3
3
Aside from the theoretical ambiguity in comparing the power of linear or tiered subsidies to alter behavior, there
are some compelling theoretical predictions that follow from the tiered system. The threshold induces upgrading
behavior to just above the next threshold and never downgrading behavior. If the CRS subsidy were linear, we
would not have these threshold effects which allow us to identify the communities responsive to the subsidy.
Identifying the communities responsive to a linear subsidy would require knowing both the counterfactual Y* and the
actual Y’ values. The tiered system makes it easier to identify the responsive communities (although there will be
fewer of them and knowledge of Y* is still needed in order to estimate the magnitude of the responses).
4
To see this graphically, consider a simplified situation in Figure A2. This frequency
distribution shown with a solid line represents the PDF for Y* (in a shape resembling a loglogistic distribution). We conjecture this simple counterfactual distribution (i.e., how Y would be
distributed in the absence of the CRS) to reflect diminishing net returns to investment in Y or the
increasing rareness of communities that heavily invest in flood mitigation. For simplicity, only
two thresholds are included, at 500 and 1000. The dashed lines represent a possible frequency
distribution of Y** values that results from a tiered CRS-style subsidy. Areas where the density
of Y* is greater than for Y**, indicated with an Ai in Figure A2, represent the credit score regions
where there is temptation for some communities to upgrade their Y scores in order to obtain the
CRS subsidy. The figure shows some of the density shifting from Ai to the region Bi, just above
the next threshold. The size of the “near and below” range depends on the context and the
community in question (i.e., for two communities with Y* = 950, only one might find it
worthwhile to upgrade to 1000). The distributions in the Bi ranges also reflect how some
communities will upgrade to slightly more than the next threshold’s minimum value. This might
be due to lumpiness in how points are awarded, uncertainty in how FEMA will audit scores, or
other reasons. Throughout, the logic involves communities upgrading from A1 to B1 (and maybe
even B2) and from A2 to B2, because they have no incentive to downgrade in the presence of a
subsidy. The areas under the baseline and the tiered system lines in Figure A2 might be the same
or, if the subsidy induces some communities with Y*=0 to start flood mitigation, the area under
the dashed line might be greater.
Fig. B1 - Stylized Shift in Distribution for Tiered Subsidy System.
frequency
baseline
(Y*)
tiered system
(Y**)
B1
“near” and below
A1
B2
A2
500
1000
1500
Y
The threshold discontinuity under the tiered subsidy of the CRS program invites the
exploration of regression discontinuity approaches. First, we assume that communities
participating in CRS cannot fully control their final CRS score (Y**). For instance, we can
5
assume that the final scores that communities receive from doing a particular CRS activity are
determined by FEMA auditors. Based on this assumption, we then suppose that communities on
either side of the threshold (to the next CRS tier) are effectively randomly assigned to their
treatment class. Under these assumptions, with the help of a Regression Discontinuity Design
(RDD), we claim to have found evidence of a causal effect of the subsidy discount. We can say
things like “a jump from class 1 to class 2 causes more Z,” where Z is some measure of an
outcome, such as the proportion of housing units in the floodplain. The use of RDD does not
require control variables. An advantage to the RDD is that the assumption that the thresholds
work to randomize the treatment (in this case, the CRS discount) can be tested, because under the
assumption of random assignment we expect that communities just above the threshold should
look identical to those just below the threshold in all kinds of non-outcome (non-Z) variables, X
(e.g., property values, population densities, flood risks, income, education, form of government,
etc.). We can then test whether X is different above and below thresholds. Our hypothesis is that
we will reject the randomization assumption that communities above and below thresholds will
look markedly different. Based on the literature on CRS, we expect that even around the
thresholds, communities participating in CRS will differ based on their educational levels,
financial resources, flood risks, demographic characteristics, and political (e.g., power) or
economic (e.g., private property values) reasons.
The basic theory is that Y** scores just above the threshold (Zahran et al., 2010 used 50
CRS points from the threshold) will tend to be strongly influenced by the discount, while Y**
scores just below the threshold will tend to have weaker or no influence from additional
discount. Those “just above” the threshold will tend to be there because they responded to the
subsidy’s incentives; those just below are at their optimal Y regardless of the subsidy.
6