metadata: authors: 0: author: Avner,Paolo
1: author: Hallegatte,Stephane
count: World
docna: 0: docna: Moral Hazard vs. Land Scarcity : Flood
            Management Policies for the Real World
docty: Policy Research Working Paper
abstracts: cdata!: This paper investigates the costs and
            benefits of three ex ante flood management strategies --
            risk-based insurance, zoning, and subsidized insurance -- in
            an urban economics framework that takes land scarcity into
            account. In a theoretical setting and in the absence of
            market failures, risk-based insurance perfectly internalizes
            flood risks and maximizes social welfare. However,
            risk-based insurance faces major technical, social, and
            political challenges and is not always realistic. Flood
            zoning and subsidized insurance are two second-best options
            that are easier to implement and less technically demanding.
            The paper explores analytically and with numerical
            simulations the welfare losses and distributional impacts
            with these second-best options, and demonstrates that total
            losses often remain small. Flood zoning is close to optimal
            when flood-prone areas are small, floods are frequent, and
            housing quality is low. Zoning keeps total land value
            unchanged but transfers wealth from landowners in
            flood-prone areas to landowners in safe locations.
            Subsidized insurance is close to optimal when a large
            fraction of a city is flood prone, floods are rare, and
            housing quality is high. And although it increases flood
            losses through the moral hazard effect, subsidized insurance
            encourages more construction, which reduces housing rents
            and benefits tenants regardless of where they live.
            Subsidized insurance transfers wealth from landowners in
            safe locations to landowners in flood-prone areas. When the
            implementation of risk-based insurance is unrealistic, as is
            often the case in developing countries, a combination of
            zoning in high-risk areas and subsidized insurance for
            low-risk areas might be a good alternative.
repnme: repnme: Moral Hazard vs. Land Scarcity : Flood
            Management Policies for the Real World
display_title: Moral Hazard vs. Land Scarcity :
            Flood Management Policies for the Real World
listing_relative_url: /research/2019/09/31405970/moral-hazard-vs-land-scarcity-flood-management-policies-real-world
publishtoextweb_dt: 2019-09-13T00:00:00Z
no_of_pages: 33
author_keys: 0: author_key: 1256219
1: author_key: 1103816
model_extractions: Policy Research Working Paper 9012 Moral Hazard vs. Land Scarcity Flood Management Policies for the Real World _Paolo Avner Stephane Hallegatte_ Global Facility for Disaster Reduction and Recovery September 2019
Policy Research Working Paper 9012

# **Abstract**

This paper investigates the costs and benefits of three ex ante flood management strategies—risk-based insurance, zoning, and subsidized insurance—in an urban economics framework that takes land scarcity into account. In a theoretical setting and in the absence of market failures, risk-based insurance perfectly internalizes flood risks and maximizes social welfare. However, risk-based insurance faces major technical, social, and political challenges and is not always realistic. Flood zoning and subsidized insurance are two second-best options that are easier to implement and less technically demanding. The paper explores analytically and with numerical simulations the welfare losses and distributional impacts with these second-best options, and demonstrates that total losses often remain small. Flood zoning is close to optimal when flood-prone areas are small, floods are frequent, and housing quality is low. Zoning keeps total land value unchanged but transfers wealth from landowners in flood-prone areas to landowners in safe locations. Subsidized insurance is close to optimal when a large fraction of a city is flood prone, floods are rare, and housing quality is high. And although it increases flood losses through the moral hazard effect, subsidized insurance encourages more construction, which reduces housing rents and benefits tenants regardless of where they live. Subsidized insurance transfers wealth from landowners in safe locations to landowners in flood-prone areas. When the implementation of risk-based insurance is unrealistic, as is often the case in developing countries, a combination of zoning in high-risk areas and subsidized insurance for low-risk areas might be a good alternative.

This paper is a product of the Global Facility for Disaster Reduction and Recovery. It is part of a larger effort by the World Bank to provide open access to its research and make a contribution to development policy discussions around the world. Policy Research Working Papers are also posted on the Web at http://www.worldbank.org/prwp. The authors may be contacted at pavner@worldbank.org.

_The Policy Research Working Paper Series disseminates the findings of work in progress to encourage the exchange of ideas about development issues. An objective of the series is to get the findings out quickly, even if the presentations are less than fully polished. The papers carry the names of the authors and should be cited accordingly. The findings, interpretations, and conclusions expressed in this paper are entirely those of the authors. They do not necessarily represent the views of the International Bank for Reconstruction and Development/World Bank and its affiliated organizations, or those of the Executive Directors of the World Bank or the governments they represent._ Produced by the Research Support Team
# **Moral Hazard vs. Land Scarcity: Flood Management Policies for the Real World**

Paolo Avner World Bank Stephane Hallegatte World Bank JEL Codes: R14, R11, R13, C63, H21, H23, Q54, Key words: Urban economics, urban floods, risk‐based insurance, land use zoning, subsidized insurance, moral hazard, land scarcity
# **1. Introduction**

Economic losses due to natural disasters – and especially floods – are increasing, mostly because of increases in population and wealth in urban areas (Pielke et al. 2008; Neumayer and Barthel 2011; Hallegatte 2017; Hoeppe 2016; Weinkle et al. 2018; Paprotny et al. 2018). Many have called for resolute action to reduce flood losses (e.g., Kunreuther, Michel‐Kerjan, and Doherty 2011). But what can governments do?

A first option is to do nothing, leaving affected people to pay for their own losses after a flood through self‐insurance or a market insurance (Michel‐Kerjan and Kunreuther 2011). In that case, a national or local government has nothing to do, beyond the provision of information about flood risks (Camerer and Kunreuther 1989) and the regulation of insurance markets to ensure fair competition and acceptable insurer bankruptcy risks. In theory, landowners and tenants would then include the cost of floods into their decisions regarding the quality and robustness of building construction, the level of rents, their localization choices, and the purchase of insurance. In the absence of market failures, this solution leads to an optimal level of construction and of flood losses.

The real world is far from this ideal situation. A first issue is the low penetration of flood insurance that is usually observed. This has two consequences. First, the insurance premium is one of the main channels through which the risk of floods is internalized in housing prices (Speyrer and Ragas 1991; Beltrán, Maddison, and Elliott 2018; Trond and Hofkes 2015; Bin, Kruse, and Landry 2008). In the absence of insurance, housing prices tend to ignore flood risks. Second, in the absence of insurance, affected people are usually in deep financial trouble after floods. In such a context, governments usually have to step in and support the recovery of the population, for two main reasons. The first reason is the widely accepted responsibility of governments to support people in times of crisis – it is impossible for a government to ignore the situation of people who have lost their home or job in a large disaster. Governments face a time consistency issue and a lack of ability to commit, as is well known in monetary and fiscal policy (Kydland and Prescott 1977). The second reason is the externalities linked to recovery and reconstruction. The inability of one family to recover and rebuild their home has implications for the community: to ensure the sustainability of infrastructure networks and the functioning of a community, people need to come back to the affected areas and rebuild, an issue illustrated by the difficulty of rebuilding New Orleans’ neighborhoods after hurricane Katrina.

When governments provide post‐disaster support without having the systems in place before the event, they often use ad hoc targeting and delivery mechanisms, which are expensive and lack predictability, making it difficult for people to take into account this support in their decision making. Also ad hoc post‐disaster support systems have been accused of creating a moral hazard, which in turn reduces the take‐up of insurance contracts.

Another option for government is to mandate insurance against natural disasters, ensuring that penetration reaches 100 percent. This option is however politically difficult, mostly because of technical and political issues. The technical issues relate first to the difficulty in assessing the risk level and therefore the appropriate risk‐based premium and second relate to enforcement issues, especially in countries with weak tenure systems and administrative capacities. Political issues relate to the heterogeneity in risk‐based premium and affordability issues. It is difficult for people to accept a mandatory system that makes them pay an insurance premium that varies widely across neighborhoods, based on models with large uncertainties. And the implementation of mandatory 2
insurance can have a strong impact on households who settled in flood‐prone areas (especially the poorest who may have chosen this localization because this is where land prices are the lowest) (Lall and Deichmann 2010; Michel‐Kerjan and Kunreuther 2011). While complementary policies or tax breaks to help households cope with insurance costs have been proposed in rich countries (Kunreuther (1996) and Picard (2008)), mandatory insurance remains politically challenging.

If neither the laissez‐faire approach nor the mandatory insurance system is desirable or possible, then what are the other options? This paper explores two other second‐best policy instruments that are implemented: flood zoning, i.e. a ban of construction in flood‐prone areas, and a subsidized insurance system in which people have insurance but pay a premium that is independent of the risk level (with cross‐subsidies from low‐risk to high‐risk households).

These options are not costless. Flood zoning diminishes the land available for construction, and interacts with urbanization and land‐use management (Burby 2001; Burby et al. 2001; Lall and Deichmann 2010). In a situation of land scarcity, which is the case in many cities and coastal areas, it increases real estate pressure, possibly leading to increased rents and smaller dwellings for all inhabitants.

A subsidized, mandatory insurance system can be financed from a local tax whose rate is independent of the individual flood risk level. Such a scheme exists in France with the CAT‐NAT system (Grislain‐ Letrémy and Peinturier 2010). This option is also costly, as it creates moral hazard: households who decide to settle in risk‐free areas have to pay for those who take more risks and benefit from these risks (e.g., through an easier access to infrastructure or through amenities). This system thus subsidizes risk‐taking behavior and may lead to larger flood losses (Laffont 1995).<sup>1</sup> This paper investigates flood risks in an idealized city prone to flooding and provides a framework for analyzing the relative costs and advantages of flood zoning, subsidized insurance, and risk‐based market insurance, taking into account the impact of these policies on rent and housing affordability. It builds on (Frame 2001) in that it combines a classical urban economic model (Alonso 1964; Mills 1967; Muth 1969; Fujita 1989) with a simple representation of flood risks to investigate the efficiency of first‐ best and second‐best options in different contexts and for different floods. This paper adds to Frame (2001) by investigating a fuller set of flood management policies, including in particular flood zoning in the policy portfolio, and by modeling decisions regarding housing investments and density in flood and safe areas.<sup>2</sup> And this paper goes beyond the demonstration that subsidies can theoretically make sense by quantifying the trade‐offs between efficiency and political acceptability and identifying conditions that make the various policies particularly attractive in a specific city and to manage specific events.

The findings of the paper are as follows. First, as expected in the absence of market failures, (actuarially‐fair) risk‐based premium insurance dominates other flood management strategies in terms of household welfare irrespective of the characteristics of the floods (return period, flood prone area and vulnerability of buildings and infrastructure). However, for most types of floods, the additional cost of second‐best options (land use zoning or subsidized insurance) compared with the optimum is

> 1 Because of moral hazard, subsidized insurance is often completed with risk reduction measures (see Michel‐ Kerjan and Kunreuther, 2011, on the National Flood Insurance Program in the U.S.).

2 Frame (2001) assumes that people consume land directly, without having to invest in building of different height and density. As a result, he cannot explore the effect of moral hazards when investments in building in flood zones are incentivized by subsidized insurance and is thus missing an essential component of the problem.

3
small, representing in most instances less than half a percentage point of income and in many cases less than 0.2%. In particular, moral hazard does increase flood losses, but the effect remains limited (and it could be limited further with appropriate deductibles). The difference between first and second‐ best options is smaller when floods are rare, flood affected areas are limited, and damages are low. This distance increases for more frequent, larger in extent, and more damaging floods. If the subsidized system has lower transaction costs than the risk‐based system (for instance because of the need to assess risk at the plot level and to provide heterogeneous pricing), then the subsidized system could dominate, especially in low‐risk environments.

Second, the preferred second‐best option (zoning or subsidized insurance) depends on the characteristics of the floods. For frequent floods affecting a small share of the urban area, land use zoning yields lower welfare losses than subsidized insurance. This is because the loss of land remains moderate and does not significantly affect rents and housing affordability, while the effect of moral hazard for frequent floods is large. For floods with larger extent, the impact of zoning on housing affordability becomes significant and subsidized insurance dominates. Zoning also becomes preferable when buildings and infrastructure are more vulnerable, which may be the case due to low quality housing or due to flood characteristics (e.g., in places with flash floods that are particularly destructive). This paper only considers economic losses, but the inclusion of (non‐insurable) human losses would only reinforce these findings.

Finally, as already noted in (Frame 2001), the urban land markets play a key role in distributing the costs and benefits of flood management policies across the population. The costs of zoning are concentrated on the owners of the land that cannot be constructed, but an additional cost is imposed to all tenants through increased rents (which benefit landowners in safe areas). And the benefits from the subsidies in a subsidized insurance system are not fully captured by people living (or owning land) in risky areas: the increased construction reduces rents, which benefits all tenants (at the expense of people owning land in safe places). These distributed costs are small per capita and largely invisible (and therefore disregarded in policy discussions), but the aggregate numbers can be large and play a key role in explaining which policy dominates.

Taken together, these findings highlight the large distributional impacts of flood management, which explain why implementation of such policies is challenging (World Bank 2013). They also suggest that while risk‐based premium insurance is the preferable option if transaction costs are not too large, there are other options that are easier to implement and more politically acceptable, and incur only a small incremental aggregate cost. A combination of flood zoning (in high‐frequency low‐scale floods and high‐vulnerability areas) and subsidized insurance (to manage rarer but large extent events) is close to the optimal solution and much easier to implement both in political and practical terms. At the core of these findings lies the trade‐off between limiting damages by reducing the housing capital exposed to floods and reducing housing rents by increasing housing supply.

The paper is organized as follows. Section 2 presents the urban economy theory and our approach to include flood risks. Then, Section 3 introduces the two insurance systems as well as the land use zoning policy and provides analytical expressions of households’ welfare (and its income equivalent), the costs of each policy and the impact on land rents. Section 4 provides a numerical assessment and ranks the three systems for different characteristics of flood risks in the city. Section 5 concludes on the practical application of these results.

4
# **2. Urban economic theory**

## **2.1. Principles**

In this section, we present the principal features and equations of the standard urban economic modeling framework. We do so briefly, because it has been extensively described in many papers and books (e.g., (Brueckner 2011; Fujita 1989)). We use a model inspired by von Thünen (1826), adapted by Alonso (1964), Mills (1967), and Muth (1969) and comprehensively described in Fujita (1989). Assuming that a city is defined by a number of jobs located in a single location, the CBD, this static model is based on two trades‐offs, made by two categories of economic actors:

- Households choose their housing location in the agglomeration by arbitrating between larger and cheaper dwellings further from the city center, and increased commuting costs to the city center where all jobs are assumed to be located.

- Landowners choose where and how much to invest (i.e., what buildings to construct), as a function of expected rents at each location.

These two trades‐offs characterize a static equilibrium in the urban system.

**Table 1: Nomenclature**

|**Symbol**|**Signification**|
|---|---|
|R|Housing rents (or, equivalently, annualized real estate<br>price per square meter)|
|R�|Agricultural rent|
|L|Available land for construction|
|K|Invested capital|
|ρ|Depreciation rate of capital|
|i|Interest rate : cost of capital|
|δ|Fraction of damaged capital when a flood occurs|
|τ|Return time of a flood (i.e. the inverse of frequency)|
|p|Unitary transport cost/km|
|θ|Share of flood prone areas in an agglomeration|
|n|Household density|
|q<br>|Dwelling size|
|N|Population in the agglomeration|
|r�|Limit of the city,maximum radius to the center|

## 2.1.1. Households

Each household is composed of one representative worker living at a distance r from the city business district (CBD) where all jobs are located. Each worker has to commute once a day to the CBD, and this commuting entails transportation costs T�r� . All households are supposed to earn the same income Y and – at the equilibrium – they have the same level of satisfaction described by a utility function U that depends on both the level of consumption of a composite good z and of the size of their dwelling q . The level of rents per square meter (or, equivalently, the annualized real estate price per square meter) is R�r� _,_ at each location in the city. Each household maximizes his or her utility function under 5
rzq U(z,q) st. z+RQ)qt+T) <Yt+L a a K(r) H(r) H(r) = F(K(r),L(r)) f R(t) = Ra m(r) = [R(r)F(K(r), L(r)) — G+ p)K@)] K(r) = Argmax[R(r)F(L(r), K(r)) —(9 + i)K(r)] ~=0

-

pr ~~ez~~ N U(z,q) =z* ® wherea,B>Oandat+fPp=1
F�L, K��AL<sup>�</sup> K<sup>�</sup> where a, b�0 and a�b�1 (5) We also make the following simplifying assumptions:

- Transport costs increase linearly with distance from the city center, T�r��pr , where _p_ is the generalized cost of transportation including the cost of time.

- The agricultural rent R� is supposed to be null. This simplification is not necessary but it makes most of the calculations much easier without altering results.

- At each distance r from the city center, and assuming a perfectly circular city, we can define the available land as the following increasing function of r: L�r��lr where l =2 π

This modeling framework is particularly well suited to study the costs and benefits of different flood‐ management strategies. Indeed, by informing on household utility, it enables us to quantify both the monetary cost of floods and the disutility from reduced land availability, increased rents and smaller dwelling sizes. It thus constitutes a comprehensive, although schematic, framework for flood management policy evaluation.

# **2.3 Introducing floods in the urban economy model**

We introduce floods in the traditional urban economy model, assuming that the only impact of floods is through capital destruction. All other consequences (e.g., casualties and fatalities) are disregarded, assuming they can be avoided with relevant tools, e.g., early warning and evacuation.<sup>3</sup> We consider that there are two zones in the agglomeration. One zone is flood prone and represents a fraction θ of total available land; one zone is safe and represents a fraction 1 �θ . The flood‐prone zones are not confined to a specific area in the city; instead, they are assumed to be distributed homogeneously in terms of distance to the CBD throughout the agglomeration. One could picture this situation as a disk, a slice of which is flood prone, such that flood prone areas and non‐flood prone areas are in proportions θ and 1 �θ (see Figure 1). In flood‐prone zones, the invested capital at each location in the city is noted K������r� . Conversely, the invested capital in other parts of the city is K�����r� .

> 3 The fact that economic losses increase with income while the number of fatalities from natural disasters decreases with income (Kahn 2005) suggest that human losses can be reduced with other tools than economic losses.

7
**Figure 1: Representation of flood‐prone and safe areas in the urban area. Flood‐prone zones are not confined to a location but are distributed homogeneously in terms of distance to the CBD throughout the urban area.**

In addition to the fraction of flood prone areas in the agglomeration ( θ ), floods are characterized by two parameters: the return time of a flood ( τ ), which is the inverse of its frequency; and the percentage of damage that it causes when it occurs ( δ ), as a proportion of the invested capital. In case of flood, occurring with an annual probability 1/τ , flood losses are equal to δ�K������r�dr . The average annual loss per capita is thus equal to: We assume that these flood risk characteristics are known with certainty, and that all economic actors are perfectly rational in the way they manage risks and make decisions. Also, we take the number of inhabitants in the city as a given, and floods and flood management policies are assumed not to affect demographics. Finally, no risk aversion is considered, which does not affect results since outcomes with second‐best policies are compared with the full‐insurance case, in which there is no residual risk.<sup>4</sup>

# **2.4 Introducing flood management options**

This subsection investigates three different policy options that can be implemented: land use zoning, risk‐based‐premium insurance and subsidized insurance. Investments in flood risk mitigation – either with dikes, pumping station, improved drainage, or nature‐based solutions – are not considered here, and we focus on the management of the residual risk once mitigation measures have been implemented. We will get back to the role of flood protections in the conclusion.<sup>5</sup>

> 4 Risk aversion would be critical to compare welfare in the first‐ and second‐best scenarios with a baseline with no insurance, or to estimate the expected take‐up of a non‐mandatory insurance system.

> 5 All flood characteristics are in fact open to policy influence. Indeed, investing in flood defenses, for example, could lead to less frequent floods (impacting τ ), or to less widespread floods (impacting θ ), or to less severe events (impacting δ ).

8
Z = (1—@)L z= (1 -— 0)Argmax[RzF(L, Kz) — (p + 1) Kz] s = Argmax[RsF(L, Ks) —(p + i) Ks] Flood = 9 Ks Safe = (1 - 6)Ks

> __5 ~~=—]~~ 0Tf“Krtooas(r dr = ~~9—~~ |“0fKs(r)

Y+Ls—Ds
P p = (1—6)Argmax|RpF(L, Ksarep) — (0 + Ksarep| + 6 Argmax [ReF(L, Krtoodp) - (6 +it ~~-)~~ Kewooap|

q(r) N _ N (v={[0"fn(r)dr = I0"f H(r) ~~—~~ 4(r) _ 'f v= ~~-~~ |0 nu(r)dr n(r) H(r) H(r) = F(K,L) 1 ~~1~~ _ [pp ~~oa”~~ TY No ya+y)yt? 2 ~_ ~~y_~~ N yt+1 b _ B= ~~(SS)(B)\~~ <u>a</u> ~~+~~ <u>f</u> and y=_ ~~1ap~~

f=1
[0,1] y= (1 — 0)Argmax [Ru,F(L, Ksare) — (p + i)Ksare| + 6 Argmax [RupPC, Kriooa) — (6 +it ~~—)~~ Kewooa|

f f=0 -¢=Ks f=1 ;-=K, fff (1-f) N I n(r)dr i <u>q(r)</u> _ "F f= ~~-~~ | m(r)dr — F=A-f ~~O—]~~ (0)'fKyiooalr) y ~~s~~ <u>r( (W;y+2+</u> 1) <u>Y</u> " nn <u>ceo)</u> ~~a~~ F ~~ave~~ (y + 1)(W; +1) a _ * ~~f~~ rae <u>(W; + 1)</u>

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<!-- Start of picture text --> P 1 ity<br>— =CpY(Ws +1) ¥<br>Ss<br>~> 1<br>us<br>1<br>pY<br>4 1+y 4 1+y<br>== 4 ( -) 4 Y (Ws +1) — 7 = () Y (Ws +)atTYY<br><!-- End of picture text --> s Zz s(¥%) = Uz(¥z) = Up(Y) = up $ s(¥%) = up ~~=—~~ s = YCpit7(Ws + 1) Z z(Vz) = up ~~a~~ 1(, ~~2~~ Zzp\1+v
# **4. Numerical results**

To explore how these theoretical results can inform decision‐making, we now rely on numerical exercises. We calibrate an idealized city on the characteristics of the Paris agglomeration (in terms of transport speed and cost, urban density, population, income, etc. see Appendix C) but a similar exercise could easily be performed for different cities.

Then, we explore how the different policies perform with various values of flood parameters. In particular, we consider cases in which the idealized city is exposed to frequent or rare floods, looking at return periods ranging from one year to 50 years<sup>9</sup> (for instance because the city has flood protection to the corresponding standard). We also explore cases where the flood zones represent a small or a large fraction of land, with flood‐prone areas ranging from 1 percent to 100 percent of the city area. Finally, we explore for varying degrees of damage inflicted by floods. This is expressed through the percentage of buildings construction value destroyed in flood prone areas when a flood occurs. We represent cases where floods are nearly harmless, for example because they are only constituted by a couple centimeters of water, to very damaging floods. The range of damage is 1% to 100% of buildings’ construction value.<sup>10</sup> The real Paris agglomeration would be a situation with rare floods (with a return period of 50 to 100 years) and a relatively small area at risk (around 20 percent). A city like New Orleans would be exposed to rare floods (the current protection system is supposed to protect the city against the 100‐year hurricane at least), but with a large fraction of the city at risk (Katrina flooded 80 percent of the city). And a city like Dar es Salam or Mumbai would be cases in which floods are very frequent (these cities experience floods almost every year), but with a relatively small flooded area.

With the model presented in the previous section, we explore how risk‐based insurance, subsidized insurance, and flood zoning perform in these different situations (keeping the socioeconomic characteristics of the city constant).

## **4.1 Trade‐off between damages from floods and housing costs**

The most common argument against subsidized insurance is that it creates moral hazard by incentivizing developments in risky areas and thereby increasing flood damages. Indeed, Table 2shows the annual damages to buildings per household (annually destroyed capital) expressed in percentage of household income in each of the three flood management strategies and for various flood characteristics (flood prone areas, θ , and return period of the flood, τ ). As in the rest of the document, S refers to the subsidized insurance, R refers to the risk‐based premium insurance and Z refers to land use zoning.

> 9 We explore return periods ranging from 1 to 100 years but only present the results up to 50 years as differences are small beyond that.

> 10 In this section, we mainly present results when damages to flooded buildings correspond to 40% ( δ ) of the buildings’ construction value. Note that the range of damage is itself open to policy influence as the enforcement of more resilient construction standards would reduce the damage on structures (albeit at a cost equivalent to increased construction costs).

16
Flood zoning is associated to zero damages as the policy simply forbids construction in flood prone areas. But subsidized insurance is associated with (much) higher annual damages per household than the risk‐based premium insurance. This is because landowners and developers do not bear the cost of flood risks and therefore build much more in the flood‐prone area, leading to large losses. Even with relatively rare floods – with a 20‐year return period – flood losses are more than doubled in the subsidized insurance case, compared with risk‐based insurance. The increase is even larger for more frequent floods. These results confirm the importance of the moral hazard effect in the model.<sup>11</sup> Note that while annual damages per household decrease with subsidized insurance as the return period of floods increases, they increase with risk‐based premium insurance. This is because when floods occur less frequently, developers faced with risk‐based premium insurance invest more in dwellings so that when a flood occurs, it causes higher damages. Conversely, when floods are frequent, investments in flood prone areas are very limited with risk‐based insurance (see right panel of Figure 2). With the subsidized insurance, construction decisions are only indirectly affected by flood characteristics so that the invested capital does not vary much but damages are lower when floods are less frequent.

**Table 2: Annually destroyed capital per household expressed in percentage of income (€) in the three flood management options for various flood characteristics, assuming in all cases that when they occur, floods destroy 40% of buildings in flood prone area.**

|Flood<br>return<br>period<br>(years)||5 years||1|0 years||2|0 years||
|---|---|---|---|---|---|---|---|---|---|
|Flood<br>prone<br>area(%)|S|R|Z|S|R|Z|S|R|Z|
|10%|2.26%|0.24%|0|1.14%|0.3%|0|0.57%|0.27%|0|
|20%|4.42%|0.52%|0|2.26%|0.63%|0|1.14%|0.56%|0|
|30%|6.48%|0.86%|0|3.35%|1.02%|0|1.70%|0.89%|0|
|40%|8.46%|1.28%|0|4.42%|1.47%|0|2.26%|1.24%|0|

> 11 Even though as previously noted, the model does not include non‐monetary non‐insurable losses, from the personal risks and trauma caused by having one’s home flooded to the loss of personal items like photographs and books. Also, insurance claims take time to be processed, and often leave a deductible, which mitigate the moral hazard effect.

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<!-- Start of picture text --> 3000 Capital density in safe areas 2500 Capital density in flooded areas<br>—_——— Zoning —_——— Zoning<br>2500 ———= Subsidized Insurance ————= Subsidized Insurance<br>=e Risk-based Insurance 2000 == Risk-based Insurance<br><!-- End of picture text --> <!-- Start of picture text --> = = 1500<br>2 4<br>1500 Ss<br>= = 1000<br>w 1000 w<br><!-- End of picture text --> <!-- Start of picture text --> 00<br>020 40 60 0 20 40 60<br>Distance to the city center (km) Distance to the city center (km)<br><!-- End of picture text --> <!-- Start of picture text --> 25 Housing surfaces 10 Housing rents<br>— Zoning —_— Zoning<br>=a == Subsidized Insurance == Subsidized Insurance<br>2 2 —eennnn== Risk-based Insurance 8 enn Risk-based Insurance<br>2<br>E15t\ 6<br>7) x<br>3£<br>ow<br>—E 1 4<br>o<br>G<br>><br>9 0.5 2<br>0:0<br>020 40 60 0 20 40 60<br>Distance to the city center (km) Distance to the city center (km)<br><!-- End of picture text -->
households. At the core of this reframed question, is the acknowledgment that there is a trade‐off in urban areas between minimizing damages and lowering housing costs.

To illustrate this point, Table 3 compares subsidized insurance and risk‐based insurance for a set of floods, looking at the impact on land values or landowners’ profits (LO) and on rents for tenants (T). It shows that subsidized insurance makes a transfer of wealth from landowners in the safe areas to landowners in the flood‐prone area. This is the direct effect of the subsidy.

But the subsidy also results in a decrease in rent in the safe and flood‐prone areas of the city. The decrease is around 5 percent if 40 percent of the city is affected every 20 years, or 20 percent of the city is affected every 5 years. Importantly, this benefit goes toward all tenants in the city, not only those living in the flood‐prone area, since rents are not affected by floods.

**Table 3: Variation (%) between subsidized and risk‐based premium insurance for Landowners’s profits (LO) and Tenants’s rents (T) in flood prone and safe areas for various flood return periods and flood‐prone land fraction, assuming a 40% damage fraction**

||Flood<br>return<br>period<br>(years)|5 ye|ars|10 ye|ars|20 ye|ars|
|---|---|---|---|---|---|---|---|
|Flood<br>prone<br>area<br>(%)||LO|T|LO|T|LO|T|
|10%|Safe|‐9.9|21|‐7.0|18|‐4.6|125|
||Flood|306|‐.|130|‐.|59|‐.|
|20%|Safe|‐19.3|44|‐13.9|37|‐9.1|256|
||Flood|263|‐.|113|‐.|51|‐.|
|30%|Safe|‐28.3|69|‐20.7|57|‐13.5|39|
||Flood|223|‐.|96|‐.|44|‐.|
|40%|Safe|‐37.0|97|‐27.2|79|‐17.9|53|
||Flood|184|‐.|80|‐.|36|‐.|

# **4.2 Optimal strategy and distance to second best**

The optimal flood management that delivers the highest utility to households is unambiguously the risk‐based premium insurance. It dominates subsidized insurance and zoning for any type of flood, irrespective of the return period of the flood ( τ ), of the fraction of flood prone urban area ( θ ), and of the share of damage inflicted to buildings when it occurs ( δ ). Figure 4, shows how risk‐based premium insurance compares to other flood management options, for various metrics, and for an estimated damage to structures of 40% in case of floods. For all of these (utility and income equivalent utility differentials), risk‐based insurance compares favorably.

The distance between the 1<sup>st</sup> and 2<sup>nd</sup> best option however varies dramatically depending on how frequent the floods are ( τ ), how much of the urban area they impact ( θ ), and what share of the installed capital they destroy ( δ ). For frequent floods that affect a very large share of the urban area, the gaps are very substantial. Consider for example flood events that return every year and affect 99% (the near totality) of the urban area – graphically, at the bottom right of each of Figure 4 quadrants. In this extreme case, households’ utility would be 13% higher with risk‐based insurance rather than with the second best. In income equivalent terms, this would represent close to 7,800 euros or 13% of 19
households’ annual income.<sup>12</sup> But these results are obtained for very extreme flood events and are outliers in the heatmaps presented in Figure 4. In most other instances, the distance between the first and second‐best flood management option is much smaller. Consider for example a flood event that would occur every 50 years, affect 50% of the urbanized land and destroy 40% of the structures in flood affected areas. Despite being quite a dramatic event, choosing to implement the second best, rather than the risk‐based premium insurance scheme would result in a moderate 0.14% decrease in household utility, equivalent to an extra annual cost to households of 71.5 euros, 0.12% of their annual income.

When aggregated over the whole urban area, even these moderate differentials between first‐ and second‐best options do add up, as they are multiplied by the millions of households living in the urban area (4.9 million in the case of Paris). In the previously cited example of a flood event occurring every 50 years, affecting 50% of the urban area and destroying 40% of the installed housing stock in affected areas, the costs of adopting the second‐best option rather than the first would represent 350 million euros annually.

This additional cost with the subsidized insurance can inform the trade‐off between the efficiency of the flood management system and its feasibility and political acceptability. This cost can be first compared with the transaction costs linked to a risk‐based system that requires a precise evaluation of the risk level to determine the premium in each plot.<sup>13</sup> And the uncertainty on the level of risk, and thus on the insurance premium, is also a significant implementation obstacle.

Finally, if the insurance is supposed to be mandatory, the loss of efficiency estimated here can be compared to the political challenges of implementing a risk‐based insurance whose prices can vary by orders of magnitude over short distance and represent a large fraction of disposable income for poor households already installed in high‐risk areas. While a risk‐based insurance poses difficult affordability issues in some areas, the subsidized insurance spreads the risk broadly, resulting in small (and more acceptable) costs in most instances.

> 12 The model is calibrated for the urban area of Paris, France and households’ average annual income is approximately 57,500 euros (see Appendix C).

> 13 A comparison would also consider the practical and financial arrangement and the cost of capital needed to make the insurance system viable and credible. In many places, like in the French Cat‐Nat system, the cost of capital is kept acceptable thanks to public guarantees that reduce the need for insurers to maintain large reserves.

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<!-- Start of picture text --> Income equivalentq differential between second<br>.<br>best woe 5 .<br>and risk-based premium insurance (% of income)<br>50 | 0.0% | 0.0% | 0.1% | 0.1% | 0.1% | 0.1% | 0.1% | 0.1% | 0.1% | 0.1% | 0.1%<br>20<br>igS<br>ES<br>3 30 | 0.0% | 0.1% | 0.2% | 0.2% | 0.3% | 0.3% | 0.4% | 0.4% | 0.4% | 0.3% | 0.3% 15<br>2<br>3<br> *°<br>3Re)2 10) O08 | 04s | 00% | 188 | te “ “mee 10<br>5| 0.0% | 0.2% 0.9% 2.0% | 2.8% | 4.1% 7.4% 5<br>1 | 0.0% | 0.0% | 0.1% | 0.1% | 0.2% | 0.3% | 0.4% | 0.6% | 1.0% | 2.1%<br>0<br>1 10 20 30 40 50 60 70 80 90 99<br>Fraction of flood prone areas, (%)<br><!-- End of picture text -->

<!-- Start of picture text --> on . . . : :<br>Utility differential between risk-based insurance and other options<br>20<br>50 | 0.00% | 0.04% | 0.07% | 0.10% 0.14% | 0.15% 0.14% | 0.13% is<br>ogenom een fe mom om om m oe T<br>2 14<br>3 30 0.01% | 0.10% | 0.18% | 0.25% | 0.31% | 0.36% 0.40% | 0.40% | 0.37% | 0.34%<br>2 12<br>oO<br>2 °oe om om | |] mr %<br>no]Re)=3 ®soso | | a] on 4 8<br>m5 0.58% | 0.98% 2.14% Sem Beara Soe 4<br>2<br>1 | 0.00% | 0.03% | 0.07% | 0.12% | 0.19% | 0.28% 1.09%<br>0<br>1 10 20 30 40 50 60 70 80 90 99<br>Fraction of flood prone areas, (%)<br><!-- End of picture text -->

<!-- Start of picture text --> Income equivalent differential between second<br>best and risk-based. premium. .insurance (euros)<br>°° vw [news] na [ae ne | ne me] 12000<br>©5 “° voz se] on |e [an |e] |e 10000<br>2<br>ao 30 131.8 | 162.5 | 185.7 | 201.2 206.4 176.0<br>g 8000<br>=<br>‘5oO 20] 107 271.2 | 336.0 420.5 | 437.4 | 4349 | 411.1<br>aa 6000<br>=<br>€ 4000<br>a<br>“<br>° [sae] ar] a am a 2000<br>0<br>1 10 20 30 40 50 60 70 80 90 99<br>Fraction of flood prone areas, (%)<br><!-- End of picture text -->

<!-- Start of picture text --> Total urban income equivalent differential between second<br>best and risk-based. premium: :insurance (millionPTT euros) x104<br>°° fe fe |e ]aalo foe ae |e] a °<br>©5 “ [= je] [a mlm]ooo ae | °<br>2<br>o 30 796 1020 | 1011<br>g 4<br>=<br>‘5oO 20| 52 | 501 1328 | 1646 | 1892 | 2060 | 2142 | 2130 | 2013 | 1809<br>aa 3<br>=<br>€ 2<br>=I<br>“ ° Lo | | | ne | | |r ae i 1<br>0<br>1 10 20 30 40 50 60 70 80 90 99<br>Fraction of flood prone areas, (%)<br><!-- End of picture text -->
<!-- Start of picture text -->
- Highest household utility<br>[___] Subsidized insurance<br>45 [EEE Zoning (30% damage)<br>_ HE Zoning (50% damage)<br>2 40<br>oo<br>235<br>co}<br>8 30<br>=<br>w 25<br>°<br>o<br>€ 20<br>=<br>5 15<br>a<br>®<br>& 10<br>5<br>10 20 30 40 50 60 70 80 90<br>Increasing fraction of flood prone areas, (%)<br><!-- End of picture text -->
<!-- Start of picture text --> Utility differential between subsidized insurance and zoning<br>* a TM 30<br>20<br>‘@ 40 0.1% | 0.8% | 1.8% | 2.9% | 4.2% | 5.7% | 7.7% 13.8% 37.3%<br>io 10<br>—><br>3 * — °<br>fe}<br>|<br>2<br>a -10<br>Ss 20 | 0.0% 0.5% 1.1% 1.9% 2.9% | 4.2% 5.9% 8.3% | 11.6% 35.3%<br>ro} -20<br>2<br>—<br>2 ° _— *<br>c<br>—<br>2 -40<br>MO 5] 0.1% | -1.4% | -2.6% | -3.5% | -4.2% | -4.5% | -4.4% | -3.5% | -1.4%<br>-50<br>1 10 20 30 40 50 60 70 80 90 99<br>Fraction of flood prone areas, (%)<br><!-- End of picture text -->
For rarer floods, it is preferable to provide a subsidized insurance contract to people living in the flood zone than to prevent them from living here: the additional cost from floods is more than compensated by the benefits from reduced land scarcity.

# **5. Conclusions**

We use the standard urban economics model to characterize housing investment choices, utility throughout the city, rents and dwelling sizes, in an agglomeration where a fraction of land is flood prone. This theoretical framework is well suited for the study of flood costs and flood‐management policy costs in an urban context. It enables to capture flood capital losses (i.e., housing capital destruction), indirect costs (rent‐related losses and insurance costs), and the variation of landowners’ aggregated profits, assuming that other losses (e.g., human and cultural losses) can be avoided. Going beyond the sole direct costs has the advantage of providing more comprehensive insights from a social welfare perspective.

The results have relevant policy implications. First, except in extreme cases with large and frequent floods, the difference in efficiency between a perfect risk‐based insurance system and a subsidized insurance system with uniform premiums is small. Considering the challenges to implement a risk‐ based system, this result suggests that it makes sense in most places in the world to implement a subsidized uniform system, at least as a first step. The efficiency cost due to the subsidy is small because the moral hazard that causes an increase in flood losses is largely compensated by the benefits from reduced housing scarcity and lower housing costs.

However, the model also shows that subsidized insurance is dominated by flood zoning for floods that are too frequent or that cover a small fraction of city. Indeed, if flood‐prone areas are small, banning construction there does not affect housing prices significantly. And if floods are too frequent, then the cost of the moral hazard created by insurance subsidies becomes too high. This suggests that in a city which is affected by a variety of flood events, of varying return periods and severity, a mixed strategy of zoning and subsidized insurance may constitute the second‐best – zoning for small areas prone to frequent flooding complemented by subsidized insurance for the rest of the urban area.

The framework and model proposed here can be used in specific cases to inform policy making on flood management and quantify the trade‐off between political acceptability and policy efficiency. In cases like Paris or New Orleans where the flood probability of occurrence is below 1 percent per year, the efficiency cost of an insurance subsidy is estimated below 0.1 percent of local income, and less than 15<sup>14</sup> percent of flood losses in the optimal case. In a situation like Dar es Salam, where floods occur yearly and affect 10 to 20 percent of the city area, the additional cost from an insurance subsidy would approach 20 percent of local income and clearly be unacceptable. There, flood zoning – or investment to prevent floods in the first place – is the only alternative.

The model also stresses the distributional impacts of various flood management options. Flood zoning affects primarily the owner of land in flood‐prone areas: in these places, flood zoning reduces values to zero (unless other usages like agriculture are possible, but this is not considered in our study). But flood zoning also increases land value in the rest of the city and thus benefits other landowners, at the

> 14 This is for damages up to 60% of building values. For lower damages, this figure goes down substantially: 7% for damages worth 30% of building construction costs.

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expense of tenants who have to pay more per unit of housing. Subsidized insurance creates a transfer of rent from the owners of safe land to the owners of flood‐prone land. But it also creates a benefit for all tenants who see a decrease in housing costs due to increased housing supply.

The urban economy framework highlights how land and housing markets redistribute across space and individuals the cost of floods and the benefits of flood management policies and shows that focusing only on the direct beneficiaries of these policies and/or only on the flood‐prone area can be highly misleading.

# **Acknowledgments**

This research paper has benefited from financial support from Association pour la Promotion de la Recherche en Economie du Carbone (APREC). We wish to thank Mark Roberts for an in‐depth review of this paper which helped improve subsequent drafts and Vincent Viguié for useful comments. We also wish to thank Harris Selod for his help in the interpretation of the “equilibrium” vs “optimum” mathematical proofs in Fujita (1989). The views expressed in this paper are the sole responsibility of the authors. They do not necessarily reflect the views of the World Bank, its executive directors, or the countries they represent.

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