The role of insurance companies in a risky economy

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Risk prevention in cities prone to natural hazards

Abstract

Cities located in regions prone to natural hazards such as flooding are not uniformly exposed to risks because of sub-city local characteristics (e.g. topog-raphy). Spatial heterogeneity thus raises the issue of how these cities have spread and should continue to develop. The current paper investigates these questions by using an urban model in which each location is characterized by a transport cost to the city center and a risk exposure. Riskier areas are developed nearer to the city center than further away. Investment in building resilience leads to more compact cities. At a given distance to the city center, riskier areas have lower land prices and get lower household density and higher building resilience. Actuarially fair insurance generates optimal density and resilience. An increase of insurance subsidization leads to an increase of density in the riskiest areas and a general decrease of resilience. In this case density restrictions and building codes have to be enforced to limit risk over-exposure.
Keywords: natural disaster risks, city development, insurance, prevention, ur-ban density, building resilience.

Introduction

In October 2012, hurricane Sandy hit the East Coast of the USA, killing 54 people and generating more than 50 billion dollars of losses.1 The damage was tremendous in Greater New York: 17% of the city was flooded and 150,000 homes were dam-aged (The Economist, 2012, 2013). Insurance indemnities were paid to affected households that were insured, and relief had to be organized for those that were not covered. People whose houses were destroyed wondered if they should abandon or rebuild them, and if so, how high they should elevate their new homes. Govern-ments wondered if they should authorize development in risky areas like Oakwood Beach on Staten Island, and if so, according to which building codes. Sandy is only one example of extreme meteorological events that have caused large flood-ing damages in the world in the recent years. Among those, Xynthia superstorm affected the European coast in February 2010, hurricane Katrina struck the New Orleans region in the USA in August 2005 and Maharashtra heavy rains flooded the area of Mumbai in India in July 2005. Each time, these events and their devastating losses have raised the same questions about the necessity of better managing urban development in areas prone to natural hazards.
Most risk-prone regions were initially urbanized because of the many advan-tages they offered to communities. In particular, many cities are located near seas and/or rivers, as they can provide natural resources and transport facilities. Nowadays, many industries and services rely on these specificities, and agglom-eration forces continue to drive urbanization at these locations (Fujita & Thisse, 2002). However, these locations are often double-edged because of exposure to flooding in the case of extreme meteorological events. Natural hazards coupled with expanding urbanization have already increased losses in the last few decades, and these are expected to escalate with the rising sea level and more severe rainfall patterns due to climate change (IPCC, 2014). At a sub-urban scale in risk-prone cities, locations are differentiated not only by their distance to valuable amenities such as the city center but also by exposure to risk due to local characteristics (e.g. topography for flooding risks). The sub-city spatial heterogeneity raises the essential question of how these cities have spread until now and how they should continue to develop in the future.
The paper investigates these issues by using an urban model in which each location is characterized by a transport cost to the city center (or to other valu-able amenities) and a risk of natural hazard (such as flooding). It focuses on the impacts of risk spatial variation and insurance subsidization on city development.2 My results are the following. Riskier areas are developed nearer to the city center than further away. Investment in building resilience leads to more compact cities. At a given distance to the city center, riskier areas have lower land prices and get lower household density and higher building resilience. Actuarially fair insurance promotes the optimal development of the city in terms of risk prevention, with optimal household density and optimal building resilience. I analyze how an in-crease of insurance subsidization affects the city development. If the subsidy is financed by households in the city, it leads to an increase/decrease of density in the riskiest/safest areas. If the subsidy is financed by households in the country, it leads to a general increase of density in the city because it attracts households from other cities. Moreover, in any case, an increase of insurance subsidization leads to a general decrease of building resilience in the city. These results show that density and zoning restrictions as well as building codes have to be enforced in the city to limit risk over-exposure when insurance is subsidized.3
Academics in insurance economics have shown much interest in natural dis-asters, in particular because of the numerous imperfections in natural disaster insurance markets (Kunreuther, 1984; Kunreuther & Michel-Kerjan, 2009). On the supply side (Charpentier & Le Maux, 2014; Jaffee & Russell, 1997), diver-sification issues lead private insurers to supply contracts at prices largely above actuarially fair rates. On the demand side (Botzen et al., 2015; Kunreuther et al., 2007; Raschky & Weck-Hannemann, 2007), households under-insure even if in-surance is fair, in particular because they under-estimate the risk or they expect free assistance (charity hazard). In this context, policy makers have implemented natural disaster public policies such as the National Flood Insurance Program (NFIP) in the USA and various programs in Europe like the CatNat in France (Bouwer et al., 2007; Kunreuther & Michel-Kerjan, 2009). To deal with diversifi-cation issues, public insurance/relief can complement the weak private insurance supply (e.g. in the USA) or public reinsurance can help private insurance to sup-ply contracts at lower prices (e.g. in France). However, these policies cannot solve the weak insurance demand issues without subsidizing insurance or/and making it mandatory. For instance, the NFIP in the USA subsidizes contracts in risky areas thanks to taxpayers, and insurance is requested for access to loans. Meanwhile, the CatNat Program in France subsidizes contracts in risky areas with the other contracts and insurance is mandatory to avoid adverse selection. If the advantage of subsidization is to improve insurance demand and risk sharing (Browne & Hoyt, 2000; Grace et al., 2004), the disadvantage is to lead to risk over-exposure because it does not provide the right incentives for individual risk prevention (Bagstad et al., 2007; Courbage et al., 2013; Picard, 2008).
Academics in urban economics have focused on natural disaster issues in the context of city development. As modeled first by Alonso (1964), households spread out in the space surrounding the city center to commute there for consumption or work, and those settled further away incurring higher transport costs are compen-sated by lower land rent, which explains the increasing housing lot sizes and the decreasing density with distance to the city center. Polinsky & Shavell (1976) and Scawthorn et al. (1982) add in their model the existence of a negative amenity such as exposure to natural hazard. These models show that, at a given distance from the city center, the land price decreases when the loss exposure increases. Many empirical studies have confirmed this effect for natural disaster risks, as summarized in the meta-analysis by Daniel et al. (2009). Because households do not want to incur too much transport cost or natural disaster cost, Frame (1998) demonstrates that riskier areas are developed nearer to the city center than further away, and some risky areas inside the city outer boundary may stay undeveloped. The tradeoff between transport cost and natural disaster cost has been observed empirically for instance by Smith (1993) and Atreya & Czajkowski (2014). Frame (1998) also points out that insurance subsidization decreases the land price dif-ference between risky areas and safe areas, as confirmed empirically by Shilling et al. (1989). Furthermore, Frame (2001) shows theoretically that risk aversion can lead households to under-develop risky areas. However, many empirical stud-ies, such as Browne & Hoyt (2000), Harrison et al. (2001) and Michel-Kerjan et al. (2012), suggest that households are more inclined to risk over-exposure because of insurance subsidization, risk under-estimation or charity hazard, than to risk under-exposure because of risk aversion.4 In this case, urban regulation should be enforced to limit over-exposure, in particular in terms of zoning/density restrictions and building codes (Bagstad et al., 2007; Kunreuther, 1996; Kunreuther & Michel-Kerjan, 2013). In an urban theoretical model with risk exposure but no transport costs, Grislain-Letrémy & Villeneuve (2014) show that zoning restrictions can be Pareto improving in the case of full insurance subsidization. In empirical analy-sis, Czajkowski & Simmons (2014) and McKenzie & Levendis (2010) respectively observe that investments in building resilience reduce natural disaster losses and increase housing values.
The present paper aims to further analyze the role of natural hazard exposure and insurance subsidization in the development of risk-prone cities with transport costs. Relative to the previously cited theoretical papers on urban economics, the present paper adds building resilience modeling and analyzes how densities and  resiliences are affected by natural hazard exposure and insurance subsidization. This analysis is essential from the perspective of implementing efficient urban regulation, in terms of zoning restrictions, density restrictions and building codes, for cities with transport costs and natural disaster risks. The rest of the paper is organized as follows. Section 2 sets up the model. Section 3 provides an analysis of city development. Section 4 provides an analysis of the impact of a change in insurance subsidization. Section 5 concludes.

Risk-prone city model

I consider a static model of a city with commuting transport costs and natural hazard exposure, in the spirit of Frame (1998, 2001), Polinsky & Shavell (1976) and Scawthorn et al. (1982). The city is inhabited by N identical households.5 The sub-city scale grid is modeled by a two-dimensional continuous space with the coordinate system x = (x1, x2). Because of spatial heterogeneity due to transport costs and natural hazards, all variables potentially depend on location x. Moreover, each variable has a unique value at each location x because I consider identical households and identical housing developers. The city has a pre-established center located at x = (0, 0), also called the central business district where work and consumption activities are concentrated.
Households compete to spread out in the space around the city center and commute between their housing location and the city center. They choose their housing location x and the quantity of goods purchased in the city center, aggre-gated in a composite good denoted z(x). Besides composite good consumption, households value their housing good consumption, characterized by lot size, mea-sured in land area unit and denoted s(x).6 The utility function of each household, denoted v(.), depends on z(x) and s(x) and is classically supposed to be twice continuously differentiable, strictly increasing in each argument (with ∂zv(0, s) = ∞ and ∂sv(z, 0) = ∞) and globally concave. The composite good supplied in the city center is considered as the numéraire (i.e. price equal to 1 for one unit of good) and the housing good supplied by housing developers at location x has a housing unit price denoted ph(x) (i.e. price for one land area unit with housing). The composite good expenses and the housing rent for one household located at x are thus respectively z(x) and ph(x)s(x).
Besides composite good expenses and housing rent, households incur commut-ing transport costs and expenses related to natural hazards. One household settling at location x incurs the given transport cost t(x) because of commuting between its housing location and the city center (or potentially other valuable amenities). For example, a city located next to an estuary is depicted in figure 1.1.7 On the land, the darkness of the square units characterizes the commuting transport cost t(x) for each household located at x. Darker areas represent locations further from the city center with higher transport costs. In stylized models, transport costs are often considered to be proportional to the distance to the city center. However, real transport costs are more complex than this stylized form, in particular because of transport system complexity. Moreover, other potential amenities (e.g. the pos-itive amenities of being near the water-front) should be taken into account in the transport costs. Note also that transport costs should include different costs, in particular the direct transport cost but also the time opportunity cost.
One household settling at location x is also exposed to natural hazards (such as flooding), with the given probability of impact π(x). The level of the loss in case of impact, denoted l(.), depends on the housing lot size s(x) and on the building resilience, denoted b(x). The loss function l(.) is assumed to be twice continuously differentiable. It is decreasing with b at a decreasing rate because the most efficient resilience investments are made first. Besides, if it is reasonably assumed that more households on a land unit leads to more total losses on this land unit (for a given building resilience level), the loss function is such that ≥ ∂s∂l (s, b) for any s and b.8 Note that losses should include direct and indirect losses. The city depicted in figure 1.1 is also represented in figure 1.2 for natural hazard exposure. On the land, the darkness of the square units characterizes the probability π(x) of being affected by a natural hazard for each household located at x. The higher the risk, the darker the location. For flooding risks, locations at lower altitude are usually more subject to flooding and should be darker. The probability of being affected by a natural hazard can correspond for example to the probability that the water level reaches a threshold level that induces significant losses for households. Besides, I consider that insurance is supplied to households at or below fair prices because I do not consider any insurance transaction cost and I consider potential insurance subsidy. As households are risk-averse (i.e. their utility function is concave), they deliberately purchase full insurance coverage and bear a certain cost related to natural disaster risks, which is the insurance premium. With insurance subsidy corresponding to a fraction λ ∈ [0, 1] of expected losses, the premium paid by a household located at x is (1 − λ)π(x)l(s(x), b(x)). The higher λ, the higher the subsidy. The insurance subsidy can be financed either by the city through a lump-sum tax on household wealth or by another party outside the city. In the former case, the tax borne by each household in the city

Table of contents :

Introduction
1 Risk prevention in cities prone to natural hazards
1.1 Introduction
1.2 Risk-prone city model
1.3 Risk-prone city development
1.4 The impact of insurance subsidization
1.5 Conclusion
1.6 Appendix
2 The role of insurance companies in a risky economy
2.1 Introduction
2.2 The model of the risky economy
2.3 Pareto optimality and Arrow-Debreu economy
2.4 Decentralized equilibrium with insurance companies
2.5 Conclusion
3 Insurability of low-probability catastrophic risks
3.1 Introduction
3.2 A model with aggregate loss uncertainty
3.3 Insurability without aggregate uncertainty
3.4 Insurability with aggregate uncertainty
3.5 Conclusion
3.6 Appendix
4 Pooling natural disaster risks in a community
4.1 Introduction
4.2 Caribbean countries and natural disasters insurance
4.3 The model
4.4 Optimal insurance and reinsurance
4.5 Conclusion
4.6 Appendix
Conclusion
Bibliography

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