Abstract
Indexbased insurances offer promising opportunities for climaterisk investments in developing countries. Indeed, contracts conditional on, e.g., weather or livestock indexes can be cheaper to set up than conventional indemnitybased insurances, while offering a safety net to vulnerable households, allowing them to eventually escape poverty traps. Moreover, transaction costs by insurance companies may be additionally reduced if contracts, instead of arranged with single households, are endorsed by collectives of households that bear the responsibility of managing the division of the insurance coverage by its members whenever the index is surpassed, allowing for additional flexibility in what concerns risksharing and also allowing insurance companies to avoid the costs associated with moral hazard. Here we resort to a population dynamics framework to investigate under which conditions household collectives may find collective index insurances attractive, when compared with individual index insurances. We assume risk sharing among the participants of each collective, and model collective action in terms of an Nperson threshold game. Compared to less affordable individual index insurances, we show how collective index insurances lead to a coordination problem in which the adoption of index insurances may become the optimal decision, spreading index insurance coverage to the entire population. We further investigate the role of riskaverse and riskprone behaviors, as well as the role of partial correlation between insurance coverage and actual loss of crops, and in which way these affect the original coordination thresholds.
This is a preview of subscription content, log in to check access.
References
Alaton P, Djehiche B, Stillberger D (2002) On modelling and pricing weather derivatives. Appl Math Finance 9(1):1–20
Arrow K (1963) Aspects of the theory of risk bearing, YRJO Jahnsson lectures, also in 1971. Essays in the theory of risk bearing. Markham, Chicago
Barnett BJ, Black JR, Hu Y, Skees JR (2005) Is area yield insurance competitive with farm yield insurance? J Agric Resour Econ 30(2):285–301
Barrett CB, McPeak JG (2006) Poverty traps and safety nets. Springer, Berlin
Carter MR, Galarza F, Boucher S (2007) Underwriting areabased yield insurance to crowdin credit supply and demand. Sav Dev 31(3):335–362
Clarke DJ (2011) A theory of rational demand for index insurance. University of Oxford, Discussion paper Series, Number, Department of Economics 572
Clarke D, Kalani G (2011) Microinsurance decisions: evidence from Ethiopia, University of Oxford, PhD Thesis
Clarke D, Das N, de Nicola F, Hill RV, Kumar N, Mehta P (2012a) The value of customized insurance for farmers in rural Bangladesh. Research paper, International Food Policy Research Institute (IFPRI)
Clarke DJO, Mahul K, Rao N, Verma N (2012b) Weather based crop insurance in India. World Bank Policy Research, Working Paper 5985
Cole S, Giné X, Tobacman J, Townsend R, Topalova P, Vickery J (2013) Barriers to household risk management: evidence from India. Am Econ J Appl Econ 5(1):104
De Bock O, Gelade W (2012) The demand for microinsurance: a literature review. ILO Microinsurance Innovation Facility Research Paper, Number 26
De Janvry A, Dequiedt V, Sadoulet E (2014) The demand for insurance against common shocks. J Dev Econ 106:227–238
Deng X, Barnett BJ, Vedenov DV (2007) Is there a viable market for AreaBased Crop Insurance? Am J Agric Econ 89(2):508–519
Dercon S, Hill RV, Clarke D, OutesLeon I, Seyoum Taffesse A (2014) Offering rainfall insurance to informal insurance groups: evidence from a field experiment in Ethiopia. J Dev Econ 106:132–143
Dixit AK, Levin SA, Rubenstein DI (2013) Reciprocal insurance among Kenyan pastoralists. Theor Ecol 6(2):173–187
Gaurav S, Cole S, Tobacman J (2011) Marketing complex financial products in emerging markets: evidence from rainfall insurance in India. J Mark Res 48(SPL):S150S162
Giné X, Townsend R, Vickery J (2007) Statistical analysis of rainfall insurance payouts in southern India. Am J Agric Econ 89(5):1248–1254
Hazell PB (1992) The appropriate role of agricultural insurance in developing countries. J Int Dev 4(6):567–581
Hess U, Skees J, Stoppa A, Barnett B, Nash J (2005) Managing agricultural production risk: innovations in developing countries. Agriculture and Rural Development (ARD) Department Report (32727GLB)
Hossain S (2013) Problems and prospects of weather index based crop insurance for rural farmers in Bangladesh. Dev Ctry Stud 3(12):208–220
Keswell M, Carter MR (2014) Poverty and land redistribution. J Dev Econ 110:250–261
Norton M, Osgood D, Madajewicz M, Holthaus E, Peterson N, Diro R, Mullally C, Teh TL, Gebremichael M (2014) Evidence of demand for index insurance: experimental games and commercial transactions in Ethiopia. J Dev Stud 50(5):1–19
Osgood DE, McLaurin M, Carriquiry M, Mishra A, Fiondella F, Hansen JW, Peterson N, Ward MN (2007) Designing weather insurance contracts for farmers in Malawi, Tanzania and Kenya: final report to the Commodity Risk Management Group. Agriculture and Rural Development, World Bank
Pacheco JM, Santos FC, Souza MO, Skyrms B (2009) Evolutionary dynamics of collective action in Nperson stag hunt dilemmas. Proc R Soc B 276(1655):315–321
Santos FC, Pacheco JM (2011) Risk of collective failure provides an escape from the tragedy of the commons. Proc Natl Acad Sci USA 108(26):10421–10425
Souza MO, Pacheco JM, Santos FC (2009) Evolution of cooperation under Nperson snowdrift games. J Theor Biol 260(4):581–588
Vasconcelos VV, Santos FC, Pacheco JM, Levin SA (2014) Climate policies under wealth inequality. Proc Natl Acad Sci USA 111(6):2212–2216
Acknowledgments
The authors would like to dedicate this paper to Mats Gyllenberg on the occasion of his 60th birthday. We also thank Avinash Dixit for his insightful comments. J.M.P. and F.C.S. acknowledge financial support from FCTPortugal through Grants EXPL/EEISII/2556/2013, PTDC/MATSTA/3358/2014, and PTDC/EEISII/5081/2014, and by multiannual funding of CBMAUM and INESCID (under the Projects UID/BIA/04050/2013 and UID/CEC/50021/2013) provided by FCTPortugal. S.A.L. acknowledges financial support from National Science Foundation Grants EF1137894, GEO1211972 and Project “Green Growth Based on Marine Resources: Ecological and SocioEconomic Constraints (GreenMAR)”, funded by Nordforsk.
Author information
Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare no competing financial interests.
Appendix: When actual loss and insurance coverage do not fully overlap
Appendix: When actual loss and insurance coverage do not fully overlap
Here we follow (Clarke 2011) and investigate the effect of introducing two different probabilities instead of the risk r employed in the main text: p, the probability that harvest is lost; q, the probability that any index insurance is activated (as defined previously already in terms of the risk r).
Depending on which one of the four possible states we are in, namely :

(0, 0)—(no Loss, no insurance activated)

(0, I)—(no Loss, insurance activated)

(L, 0)—(Loss, no insurance activated)

(L, I)—(Loss, insurance activated)
we can write down the following payoff tables for Cs and Ds:
Ds  

State  Probability  Payoff 
(0,0)  \(p_{0,0}\)  b 
(0,I)  \(p_{0,1}\)  b 
(L,0)  \(p_{1,0}\)  0 
(L,I)  \(p_{1,1}\)  0 
Cs  

State  Probability  Payoff  
\(n_C \ge M\)  \(n_C <M\)  
(0,0)  \(p_{0,0}\)  \(bc_c \)  \(\gamma (bc)+(1\gamma )b\) 
(0, I)  \(p_{0,1}\)  \(b(1+\alpha )c_c\)  \(\gamma \left[ {b(1+\alpha )c} \right] +(1\gamma )b\) 
(L, 0)  \(p_{1,0}\)  \(c_c \)  \(\gamma (c)\) 
(L, I)  \(p_{1,1}\)  \(\alpha bc_c \)  \(\gamma (\alpha bc)\) 
In the simplest scenario, we may assume that the probability p of occurrence of a loss is statistically independent from the probability q that the index I of the II is activated. Then we can write for the probabilities of occurrence of each of the 4 states:
Such a statistical independence is unlikely, however. Thus, the joint probability distribution will not, in general, be the product of the 2 marginal probability distributions. Following (Clarke 2011), we investigate a symmetric joint probability distribution, defined in terms of a new parameter u, which corresponds to the probability that an individual will incur a Loss but without the II be activated (\(p_{1,0}\)). In terms of u, the probabilities now read:
Again, we can write for the fitness difference between Cs and Ds
with \(\Delta =\gamma c\big (\frac{p_{0,1} +p_{1,1} }{m}1\big )\)and \(\Omega \Delta =c \big (\gamma +\frac{p_{0,1} +p_{1,1} }{m}s_C \big )\).
The expressions for \(\Delta \) and \(\Omega \Delta \)depend only on the combination \(p_{0,1} +p_{1,1} =q\), despite the fact that, individually, both \(p_{0,1} \) and \(p_{1,1}\) depend on p, q and u. Since \(p_{0,1} +p_{1,1} =q\), irrespective of whether the p and q distributions are independent or not, this means that the results we obtain for the original model also hold for this “extended model” provided that the original risk r is replaced by the probability q.
\(\square \)
Rights and permissions
About this article
Cite this article
Pacheco, J.M., Santos, F.C. & Levin, S.A. Evolutionary dynamics of collective index insurance. J. Math. Biol. 72, 997–1010 (2016). https://doi.org/10.1007/s0028501509393
Received:
Revised:
Published:
Issue Date:
Keywords
 Index insurance
 Collective action
 Evolutionary game theory
 Nonlinear returns
Mathematics Subject Classification
 91XX
 91A22
 91A06
 91A40
 91B18
 91B30