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Comparison of Different Water Supply Risk Management Tools for Irrigators: Option Contracts and Insurance


Irrigators must cope with the risk of not having enough water to meet crop demands. There are different tools for managing this risk, including water market mechanisms and insurance. Given the choice, farmers will opt for the tool that offers the greatest positive change in expected utility. This paper presents a theoretical assessment of farmers’ expected utility for two different water option contracts and a drought insurance policy. We analyze the conditions that determine farmers’ preferences for these instruments and perform a numerical application to a water-stressed Spanish region. Results show that farmers’ willingness to pay for the considered risk management tools are greater than the preliminary estimates of these instruments costs. This suggests that option contracts and insurance may help farmers manage water supply availability risks.

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  1. A MGF of a random variable is a specification of its probability distribution, which is a convenient means of collecting together all the moments of a random variable into a single power series.

  2. Our assessment takes into account only the changes in farmers’ expected utility caused by differences in water availability (due to an option contract or an insurance).

  3. \(b=c-P_{w}; c\) is the marginal profit of water use and \(P_w \) is the water tariff.

  4. This applies to farmers relying on inter-basin transfers, where, because of area-of-origin preferences, no volume is transferred unless minimum water volumes are stored in the region whence the transfer is derived. Any other condition can be established as a trigger for the option contract in its place.

  5. A farmer exercising the water supply option contract will pay\(P_w \) plus \(P_e\) for the optioned volume \(P_e\), defined as a surcharge on top of the price paid for the normal source of water supply \((P_w) \). If the exercise price agreed in the option contract were lower than the price paid for the normal source of water supply, \(P_e \) would then be negative. This is not a very common situation, but it can occur when the contract is established between water users who have very different water productivities. In order to simplify the presentation of this approach, unless otherwise stated, only positive \(P_e\) values are considered in the analysis. An example of an inter-basin exchange with a lower exercise price than \(P_w\) that took place in the Spanish water market is presented in Sect. 5.

  6. \({ MGF}_{{w}} \left( {-{rb}} \right) ={ UIMGF}_{{w}} \left( {-{rb}} \right) +{ LIMGF}_{{w}} \left( {-{rb}} \right) \). UIMGF and the LIMGF are calculated in the same way, the only difference being the value of the integral limits (the expression of \({ UIMGF}_{{w}} \left( {-{rb}} \right) \) is given in “Appendix 3”).

    $$\begin{aligned} { LIMGF}_{{w}} \left( {-{rb}} \right) = {MGF}_{{w}} \left( {-{rb}} \right) \left[ {-{Q}\left( {{\upalpha },\left( {{\lambda }+{rb}} \right) {w}_{{g}} } \right) +1} \right] \end{aligned}$$

    Q (.) is a regularized gamma function.

  7. \({ LIMGF}_{{w}} \left( {-{ rP}_{{e}} } \right) ={\mathop {\int }\nolimits _{0}^{{{w}}_{{g}}}} {e}^{-{{ rP}}_{{e}} {w}}{f}\left( {w} \right) { dw}\). As \({P}_{{e}} =0\); then \(\mathop \int \nolimits _{0}^{{w}_{{g}}} {e}^{0}{f}\left( {w} \right) { dw}=\mathop {\int } \nolimits _0^{{w}_{{g}} } {f}\left( {w} \right) {dw}={\gamma }\).

  8. Equation (17) can be rewritten as \({\mathop {\int } \nolimits _0^{w_{g}}} (1-e^{-rP_e \left( {w-w_g } \right) })f\left( w \right) dw<0,\) where \({\mathop {\int }\nolimits _0^{w_g }} (1-e^{-rP_e \left( {w-w_g } \right) })f\left( w \right) dw\) is the expected utility of \(\left( {-P_e \left( {w-w_g } \right) } \right) \), i.e., the expected disutility of the increase in the cost of water due to obtaining it through the option contract instead of from the usual water source. If \(P_e <0\), this expected utility would be positive and thus \(R_{ins} <R_{opt_b }\).

  9. Obviously, irrigators will only sign the option contract if their WTP (risk premium, \(R\)) is greater than the price that they have to pay for the contract \((P); R>P\).

  10. See “Appendix 6”, showing the remaining comparisons between the proposed tools.

  11. A comparative statics analysis has been carried out in order to determine the influence of the main parameters on the value of the risk premium for each instrument. This material is available from the authors upon request.


  13. As the \(p\) value approaches one, we have no basis to reject the hypothesis that the fitted distribution actually generated our data set (Source: @Risk Manual).

  14. Wealth data sourced from the Spanish Farm Accountancy Data Network (RECAN), published by the Spanish Ministry of Agriculture, Food and Environment, MAGRAMA,

  15. We are aware of agreements between water users in the Tagus (sellers) and Segura basins (buyers) to sell water at a price of . If there is a drought period and they are exempted from paying the aqueduct tariff, the final price of this water would be lower than the usual water price.

  16. .

  17. Nevertheless, in practice, the costs of insurance are usually very high, reaching levels sometimes unaffordable for potential customers. That is why agricultural insurance policies are subsidized in most countries. However, insuring water shortages based on clearly objective and transparent measures (such as those governing the Tagus–Segura Aqueduct and transfers) would perhaps be offered at reduced administrative costs, because there is no need to adjust losses in the fields. They could even be attached, as an optional guarantee, to already offered insurance policies covering crop losses. In this case, no matter whether the policy is subsidized, the administration and commercial cost of the premium may be reduced.



Constant absolute risk aversion


Decreasing absolute risk aversion


Moment generating function


Probability density function


Coefficient of variation


Willingness to pay


  • Antón J, Kimura S (2011) Risk management in agriculture in Spain. OECD Food, Agriculture and Fisheries Papers No. 43, OECD Publishing. doi:10.1787/5kgj0d57w0wd-en Cited 22 Apr 2015

  • Bielza M, Conte C, Dittmann C, Gallego J, Stroblmair J (2008) Agricultural insurance schemes. Directorate General, JRC. European Commission, Ispra

  • Bjornlund H (2006) Can water assist irrigators managing increased supply risk? Some Australian experiences. Water Int 31(2):221–232

    Article  Google Scholar 

  • Black F, Scholes MS (1973) The pricing of options and corporate liabilities. J Polit Econ 81:637–654

    Article  Google Scholar 

  • Brown C, Carriquiry M (2007) Managing hydroclimatological risk to water supply with option contracts and reservoir index insurance. Water Resour Res 43:W11423

    Google Scholar 

  • Calatrava J, Garrido A (2005a) Spot water markets and risk in water supply. Agric Econ 33:131–143

    Article  Google Scholar 

  • Calatrava J, Garrido A (2005b) Modelling water markets under uncertain water supply. Eur Rev Agric Econ 32(2):119–142

    Article  Google Scholar 

  • Calatrava J, Martínez-Granados D (2012) El valor del agua en el regadío de la cuenca del Segura y en las zonas regables del trasvase Tajo-Segura. Economía Agraria y Recursos Naturales 12(1):5–31

    Google Scholar 

  • Cerdá E and Quiroga S (2008) Cost-loss decision models with risk aversion. ICEI Working Papers, WP 01/08

  • Cheng W-Ch, Hsu N-S, Cheng W-M, Yeh W-G (2011) Optimization of European call options considering physical delivery network and reservoir operation rules. Water Resour Res 47:W10501

    Google Scholar 

  • Collender RN, Zilberman D (1985) Land allocation under uncertainty for alternative specifications of return distributions. Am J Agric Econ 67(4):779–786

    Article  Google Scholar 

  • Cubillo F (2010) Looking for efficiency through integrated water management between agriculture and urban uses. Water Sci Technol Water Supply 10(4):584–590

    Article  Google Scholar 

  • Cui J, Schreider S (2009) Modelling of pricing and market impacts for water options. J Hydrol 371:31–41

    Article  Google Scholar 

  • Cummins T, Thompson C (2002) Anticipating the next level of sophistication in water markets. Connections 4:4–9

    Google Scholar 

  • FAO (2003) Unlocking the water potential of agriculture. Natural Resources Management and Environment Department, FAO, Rome

  • Fleming E, Villano R, Williamson B (2013) Structuring exotic options contracts on water to improve the efficiency of resource allocation in the Australian water market. Australasian Agribusiness Perspectives, paper 96

  • Garrido A (2007) Water markets design and evidence from experimental economics. Environ Resour Econ 38:311–330

    Article  Google Scholar 

  • Garrido A, Zilberman D (2008) Revisiting the demand of agricultural insurance: the case of Spain. Agric Finance Rev 68:43–66

    Article  Google Scholar 

  • Garrido A, Calatrava J (2009) Agricultural water pricing: EU and Mexico. OECD, Paris

    Google Scholar 

  • Garrido A, Gómez-Ramos A (2009) Propuesta para la implementación de un centro de intercambio basado en contratos de opción. In: Gómez-Limón JA, Calatrava J, Garrido A, Sáez FJ and Xabadia A (eds) La economía del agua de riego en España. Fundación Cajamar, pp. 321–341

  • Garrido A, Llamas MR, Varela-Ortega C, Novo P, Rodríguez-Casado R, Aldaya MM (2010) Water footprint and virtual water trade in Spain. Springer, New York

    Book  Google Scholar 

  • Garrido A, Bielza M, Rey D, Mínguez MI, Ruiz-Ramos M (2012a) Insurance as an adaptation to climate variability in agriculture. In: Mendelsohn R, Dinar A (eds) Handbook on climate change and agriculture. Edward Elgar, Arnold, pp 420–445

    Google Scholar 

  • Garrido A, Rey D, Calatrava J (2012b) Water trading in Spain. In: de Stefano L, Llamas MR (eds) Water, agriculture and the environment in Spain: Can we square the circle?. CRC Press, Botín Foundation, pp 205–216

    Chapter  Google Scholar 

  • Gil M, Garrido A, Gómez-Ramos A (2009) Análisis de la productividad de la tierra y del agua en el regadío español. In: Gómez-Limón JA, Calatrava J, Garrido A, Sáez FJ, Xabadia A (eds) La economía del agua de riego en España. Fundación Cajamar, Almeria, pp 95–114

    Google Scholar 

  • Gómez-Limón JA, Riesgo L, Arriaza M (2002) Agricultural risk aversion revisited: a multicriteria decision-making approach. In: Paper prepared for presentation at the Xth EAAE Congress ‘Exploring Diversity in the European AgriFood System’, Zaragoza (Spain), 28–31 August 2002

  • Gómez-Limón JA, Arriaza M, Riesgo L (2003) An MCDM analysis of agricultural risk aversion. Eur J Oper Res 151:569–585

    Article  Google Scholar 

  • Gómez-Ramos A, Garrido A (2004) Formal risk-transfer mechanisms for allocating uncertain water resources: the case of option contracts. Water Resour Res 40:W12302

    Article  Google Scholar 

  • Gómez-Ramos A (2013) Drought management, uncertainty and option contracts. In: Maestu J (ed) Water trading and global water scarcity: international experiences. RFF Press, Routledge, pp 286–297

    Google Scholar 

  • Hansen K, Howitt R, Williams J (2006) Implementing option markets in california to manage water supply uncertainty. In: Paper presented at the American Agricultural Economics Association annual meeting, Long Beach, California, July 23–36

  • Hansen K, Howitt R, Williams J (2013) Water trades in the western United States. In: Maestu J (ed) Water trading and global water scarcity: international experiences. RFF Press, Almeria, pp 56–67

    Google Scholar 

  • Hardaker JB (2000) Some issues in dealing with risk in agriculture. In: Working papers in agricultural and resources economics. Armidale, N.S.W. School of Economic Studies. University of New England

  • Hardaker JB, Huirne RBM, Anderson JR, Lien G (2004) Coping with risk in agriculture, 2nd edn. CABI publishing, Wallingford

    Book  Google Scholar 

  • Just DR, Peterson H (2010) Is expected utility theory applicable? A revealed preference test. Am J Agric Econ 92(1):16–27

    Article  Google Scholar 

  • Kahneman D, Tversky A (1979) Prospect theory: an analysis of decision under risk. Econometrica 47:263–291

    Article  Google Scholar 

  • Kandel S, Stambaugh RF (1991) Asset returns and inter-temporal preferences. J Monet Econ 27(1):39–71

    Article  Google Scholar 

  • Lefebvre M (2011) Irrigation water allocation mechanisms and drought risk management in agriculture. Doctoral Thesis, University of Montpellier I

  • Leiva AJ, Skees J (2008) Using irrigation insurance to improve water usage of the Rio Mayo irrigation system in northwestern Mexico. World Dev 36(12):2663–2678

    Article  Google Scholar 

  • Lorenzo-Lacruz J, Vicente-Serrano SM, López-Moreno JI, Beguería S, García-Ruiz JM, Cuadrat JM (2010) The impact of droughts and water management on various hydrological systems in the headwaters of the Tagus River (central Spain). J Hydrol 386:13–26

    Article  Google Scholar 

  • Merton RC (1973) Theory of rational option pricing. Bell J Econ Manag Sci 4:141–183

    Article  Google Scholar 

  • Mesa-Jurado MA, Martín-Ortega J, Ruto E, Berbel J (2012) The economic value of guaranteed water supply for irrigation under scarcity conditions. Agric Water Manag 113:10–18

    Article  Google Scholar 

  • Mongin P (1997) Expected utility theory. In: Davis J, Hands W, Maki U (eds) Handbook of economic methodology. Edward Elgar, London, pp 342–350

    Google Scholar 

  • OECD (2009) Managing risk in agriculture: a holistic approach. OECD Publishing, Paris

    Google Scholar 

  • Pérez-Blanco CD, Gómez CM (2012) Design of optimum private insurance schemes as a means to reduce water overexploitation during drought events. A case study in Campo de Cartagena (Segura River Basin, Spain). Paper presented at 86th Annual Conference of the Agricultural Economics Society, University of Warwick, United Kingdom, April

  • Pérez-Blanco CD, Gómez CM (2013) Designing optimum insurance schemes to reduce water overexploitation during drought events: a case study of La Campiña, Guadalquivir River Basin, Spain. J Environ Econ Policy 2(1):1–15

    Article  Google Scholar 

  • Quiroga S, Garrote L, Fernandez-Haddad Z, Iglesias A (2011) Valuing drought information for irrigation farmers: potential development of a hydrological risk insurance in Spain. Spanish J Agric Res 9(4):1059–1075

    Article  Google Scholar 

  • Rigby D, Alcon F, Burton M (2010) Supply uncertainty and the economic value of irrigation water. Eur Rev Agric Econ 37(1):97–117

    Article  Google Scholar 

  • Ruiz J, Bielza M., Garrido A, Iglesias A (2014) Managing drought economic effects through insurance schemes based on local water availability. Manuscript

  • Sivakumar MVK, Motha RP (2007) Managing weather and climate risks in agriculture. Springer, Berlin

    Book  Google Scholar 

  • Tobarra MA (2008) Gestión del recurso natural agua en situaciones de información asimétrica, racionamiento e incertidumbre. Doctoral Thesis, Universidad Politécnica de Cartagena, Murcia, Spain

  • Tomkins CD, Weber TA (2010) Option contracting in the California water market. J Regul Econ 37:107–141

    Article  Google Scholar 

  • Varela D (2008) Recargos de las primas: funciones y criterios de cálculo. In: Presented at the Course Marketing del seguro Aspectos técnicos del cálculo de las primas de seguros agro-pecuarios. Madrid, CEIGRAM, 18–19th Nov 2008

  • Williamson B, Villano R, Fleming E (2008) Structuring exotic option contracts on water to improve the efficiency of resource allocation in the water spot water. In: Paper presented at AARES 52nd annual conference, Feb 2008

  • Zeuli KA, Skees JR (2005) Rainfall insurance: a promising tool for drought management. Water Res Dev 21(4):663–675

    Article  Google Scholar 

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This study has been developed in the framework of the European project ‘Water Markets Scenarios for Southern Europe: new solutions for coping with water scarcity and drought risk?’ WATER CAP & TRADE (ERA-Net, P100220C-631).

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Correspondence to Dolores Rey.


Appendix 1: Farmers’ Expected Utility with No Risk Management Tool

$$\begin{aligned} EU_0 \left( {\widetilde{\uppi }} \right)= & {} \mathop \int \nolimits _0^{\overline{w}} U\left( {\widetilde{\pi } _0 }\right) f\left( w \right) dw\nonumber \\= & {} \mathop \int \nolimits _0^{\overline{w}}\left[ {1-e^{-r\left( {a+bw} \right) }} \right] f\left( w \right) dw\nonumber \\= & {} \mathop \int \nolimits _0^{\overline{w}} f\left( w \right) dw - \mathop \int \nolimits _0^{\overline{w}} e^{-ra} e^{-rbw} f\left( w \right) dw \nonumber \\= & {} 1-e^{-ra}\mathop \int \nolimits _0^{\overline{w}} e^{-rbw }f\left( w \right) dw \nonumber \\= & {} 1-e^{-ra}{\textit{MGF}}_w \left( {-rb} \right) \end{aligned}$$

Appendix 2: Expected Utility and Risk Premium with Option Contract (a)

$$\begin{aligned} EU_{opt_a } \left( {\widetilde{\pi } } \right)= & {} Z{\mathop {\int }\nolimits _{0}^{w_{g}}} \left[ {1-e^{-r\left( {a+bw_g -P_{opt_a } -P_e \left( {w_g -w} \right) } \right) }} \right] f\left( w \right) dw\nonumber \\&\quad +\,\left( {1-Z} \right) {\mathop {\int }\nolimits _{0}^{w_{g}}} \left[ {1-e^{-r\left( {a+bw-P_{opt_a } } \right) }} \right] f\left( w \right) dw\nonumber \\&\quad +\,{\mathop {\int }\nolimits _{w_{g}}^{\bar{w}}} \left[ {1-e^{-r\left( {a+bw-P_{opt_a } } \right) }} \right] f\left( w \right) dw\nonumber \\= & {} Z\gamma - Ze^{{ - r(a + bw_{g} - P_{{opt_{a} }} - P_{e} w_{g} )}} {\textit{LIMGF}}_{w} ( - rP_{e} )\nonumber \\&\quad +\,\gamma - e^{{ - r(a - P_{{opt_{a} }} )}} {\textit{LIMGF}}_{w} ( - rb) - Z\gamma \nonumber \\&\quad +\,Ze^{{ - r(a - P_{{opt_{a} }} )}} {\textit{LIMGF}}_{w} ( - rb) + (1 - \gamma ) - e^{{ - r(a - P_{{opt_{a} }} )}} {\textit{UIMGF}}_{w} ( - rb) \nonumber \\= & {} 1 - e^{{ - r(a - P_{{opt_{a} }} )}} ({\textit{UIMGF}}_{w} ( - rb) + {\textit{LIMGF}}_{w} ( - rb)) \nonumber \\&\quad -\,Ze^{{ - r(a + bw_{g} - P_{{opt_{a} }} - P_{e} w_{g} )}} {\textit{LIMGF}}_{w} (-rP_{e} )\nonumber \\&\quad +\,Ze^{{ - r(a - P_{{opt_{a} }} )}} {\textit{LIMGF}}_{w} (- rb) =1-e^{-r\left( {a-P_{opt_a } } \right) }{\textit{MGF}}_w \left( {-rb} \right) \nonumber \\&\quad -\,Ze^{-r\left( {a+bw_g -P_{opt_a } -P_e w_g } \right) }{\textit{LIMGF}}_w \left( {-rP_e } \right) \nonumber \\&\quad +\,Ze^{-r\left( {a-P_{opt_a } } \right) }{\textit{LIMGF}}_w \left( {-rb} \right) \end{aligned}$$
$$\begin{aligned} EU_0 \left( {\widetilde{\pi } } \right)= & {} EU_{opt_a } \left( {\widetilde{\pi } } \right) \\ 1-e^{-ra}{\textit{MGF}}_w \left( {-rb} \right)= & {} 1-e^{-r\left( {a-R_{opt_a } } \right) }{\textit{MGF}}_w \left( {-rb} \right) \end{aligned}$$
$$\begin{aligned}&\quad -Ze^{-r\left( {a+bw_g -R_{opt_a } -P_e w_g } \right) }{\textit{LIMGF}}_w \left( {-rP_e } \right) +\,Ze^{-r\left( {a-R_{opt_a } } \right) }{\textit{LIMGF}}_w \left( {-rb} \right) \nonumber \\&\quad -\,e^{-ra}{\textit{MGF}}_w \left( {-rb} \right) = -e^{-ra}e^{rR_{opt_a } }\left[ {\textit{MGF}}_w \left( {-rb} \right) +Ze^{-r\left( {b-P_e } \right) w_g }{\textit{LIMGF}}_w \left( {-rP_e } \right) \right] \nonumber \\&\quad -\,{\textit{ZLIMGF}}_w \left( {-rb} \right) \nonumber \\&R_{opt_a } = \frac{1}{r} ln\left( {\frac{{\textit{MGF}}_w \left( {{-}rb} \right) }{\left( {1{-}Z} \right) {\textit{MGF}}_w \left( {{-}rb} \right) +\hbox {Z}\left[ {e^{{-}r\left( {b{-}P_e } \right) w_g }{\textit{LIMGF}}_w \left( {{-}rP_e } \right) +{\textit{UIMGF}}_w \left( {{-}rb} \right) } \right] }} \right) \nonumber \\ \end{aligned}$$

Appendix 3: Upper Incomplete Moment Generation Function (UIMGF)

We consider that variable \(\widetilde{w}\) follows a gamma distribution \(f(w)\):

$$\begin{aligned} f\left( w \right)= & {} \frac{\lambda ^{\alpha }}{{\Gamma }\left( \alpha \right) }w^{\alpha -1}e^{-\lambda w}\nonumber \\ {\textit{UIMGF}}_w \left( {-rb} \right)= & {} \mathop \int \nolimits _{w_g }^{\overline{w}} e^{-rbw}\frac{\lambda ^{\alpha }}{{\Gamma }\left( \alpha \right) }w^{\alpha -1}e^{-\lambda w} dw= \left[ {-\frac{\lambda ^{\alpha }w^{\alpha }E_{1-\alpha } \left( {\left( {\lambda +rb} \right) w} \right) }{{\Gamma }\left( \alpha \right) }} \right] _{w_g }^{\overline{w}} \end{aligned}$$

\(E\) is an exponential integral function.

$$\begin{aligned} E_n \left( z \right)= & {} z^{n-1}{\Gamma }\left( {1-n,z} \right) \\ E_{1-\alpha } \left( {\left( {\lambda +rb} \right) w} \right)= & {} \left( {\left( {\lambda +rb} \right) w} \right) ^{-\alpha }{\Gamma }\left( {\alpha ,\left( {\lambda +rb} \right) w} \right) \end{aligned}$$

So, the expression of \({{\textit{UIMGF}}}_{w}(-{rb})\) is

$$\begin{aligned} {\textit{UIMGF}}_w \left( {-rb} \right)= & {} -\frac{\lambda ^{\alpha }}{{\Gamma }\left( \alpha \right) }\left[ {w^{\alpha }\left( {w\left( {\lambda +rb} \right) } \right) ^{-\alpha }{\Gamma }\left( {\alpha ,\left( {\lambda +rb} \right) w} \right) } \right] _{w_g }^{\overline{w}}\nonumber \\= & {} -\frac{\lambda ^{\alpha }}{{\Gamma }\left( \alpha \right) }\left[ {\left( {\lambda +rb} \right) ^{-\alpha }{\Gamma }\left( {\alpha ,\left( {\lambda +br} \right) w_g } \right) } \right] _{w_g }^{\overline{w}}\nonumber \\= & {} \frac{\lambda ^{\alpha }}{\left( {\lambda +rb} \right) ^{\alpha }}\left[ {\frac{-{\Gamma }\left( {\alpha ,\left( {\lambda +rb} \right) w_g } \right) }{{\Gamma }\left( \alpha \right) }} \right] _{w_g }^{\overline{w}}\nonumber \\= & {} {\textit{MGF}}_w \left( {-rb} \right) \left( {\left[ {Q\left( {\alpha ,\left( {\lambda +rb} \right) w_g } \right) } \right] -\left[ {Q\left( {\alpha ,\left( {\lambda +rb} \right) {\overline{w}}} \right) } \right] } \right) \end{aligned}$$

\(Q\)(.) is the regularized gamma function, whose domain is [0,1].

Appendix 4: Expected Utility and Risk Premium with Option Contract (b)

If we assume that irrigators will always exercise the option at the maturity date when their water allotment is below \(w_{g}\), their profit function is

$$\begin{aligned} \widetilde{\pi } _{opt_b } \left( w \right)= & {} a+b\widetilde{w}-P_{opt_b } \quad if\quad \widetilde{w}\ge w_g\nonumber \\ \widetilde{\pi } _{opt_b } \left( w \right)= & {} a+b\hbox {w}_g -P_{opt_b } -P_e \left( {w_g -\widetilde{w}} \right) \quad if \quad \widetilde{w}<w_g \end{aligned}$$
$$\begin{aligned} EU_{opt_b } \left( {\widetilde{\pi } } \right)= & {} \mathop \int \nolimits _0^{w_g } \left[ {1-e^{-r\left( {a+bw_g -P_{opt_b } -P_e \left( {w_g -w} \right) } \right) }} \right] f\left( w \right) dw\nonumber \\&+\,\mathop \int \nolimits _{w_g }^{\overline{w}} \left[ {1-e^{-r\left( {a+bw-P_{opt_b } } \right) }} \right] f\left( w \right) dw\nonumber \\= & {} \gamma -e^{-r\left( {a+bw_g -P_{opt_b } -P_e w_g } \right) }\mathop \int \nolimits _0^{w_g } e^{-rP_e w}f\left( w \right) dw\nonumber \\&+\,\left( {1-\gamma } \right) -e^{-r\left( {a-P_{opt_b } } \right) }\mathop \int \nolimits _{w_g }^{\overline{w}} e^{-rbw} f\left( w \right) dw\nonumber \\= & {} 1-e^{-r\left( {a+bw_g -P_{opt_b } -P_e w_g } \right) }{\textit{LIMGF}}_w \left( {-rP_e } \right) -e^{-r\left( {a-P_{opt_b } } \right) }{\textit{UIMGF}}_w \left( {-rb} \right) \nonumber \\ \end{aligned}$$
$$\begin{aligned} EU_0 \left( {\widetilde{\pi } } \right)= & {} EU_{opt_b } \left( {\widetilde{\pi } } \right) \nonumber \\ 1-e^{-ra}{\textit{MGF}}_w \left( {-rb} \right)= & {} 1-e^{-r\left( {a+bw_g -R_{opt_b } -P_e w_g } \right) }{\textit{LIMGF}}_w \left( {-rP_e } \right) \nonumber \\&\quad -e^{-r\left( {a-R_{opt_b } } \right) }{\textit{UIMGF}}_w \left( {-rb} \right) \nonumber \\ -e^{-ra}{\textit{MGF}}_w \left( {-rb} \right)= & {} -e^{-r\left( {a+bw_g -R_{opt_b } -P_e w_g } \right) }{\textit{LIMGF}}_w \left( {-rP_e } \right) \nonumber \\&\quad -e^{-r\left( {a-R_{opt_b } } \right) }{\textit{UIMGF}}_w \left( {-rb} \right) \nonumber \\ {\textit{MGF}}_w \left( {-rb} \right)= & {} e^{rR_{opt_b } }\left[ {e^{-r\left( {b-P_e } \right) w_g }{\textit{LIMGF}}_w \left( {-rP_e } \right) +{\textit{UIMGF}}_w \left( {-rb} \right) } \right] \nonumber \\ R_{opt_b }= & {} \frac{1}{r}\ln \left( {\frac{{\textit{MGF}}_w \left( {-rb} \right) }{e^{-r\left( {b-P_e } \right) w_g }{\textit{LIMGF}}_w \left( {-rP_e } \right) +{\textit{UIMGF}}_w \left( {-rb} \right) }} \right) \end{aligned}$$

Appendix 5: Expected Utility and Risk Premium with Insurance

The farmers’ profit function in this case is

$$\begin{aligned} \widetilde{\pi } _{ins} \left( w \right)= & {} a+b\widetilde{w} -P_{ins}\quad if\quad \widetilde{w}\ge w_g\\ \widetilde{\pi } _{ins} \left( w \right)= & {} {a+b\hbox {w}_{g}} -{P}_{ins} \quad if \quad \widetilde{w}<w_g \end{aligned}$$
$$\begin{aligned} EU_{ins} \left( {\widetilde{\pi } } \right)= & {} \mathop \int \nolimits _0^{w_g } 1-e^{-r\left( {a+bw_g -P_{ins} } \right) }f\left( w \right) dw\nonumber \\&+\,\mathop \int \nolimits _{w_g }^{\overline{w}} 1-e^{-r\left( {a+bw-P_{ins} } \right) }f\left( w \right) dw\nonumber \\= & {} \gamma -\gamma e^{-r\left( {a+bw_g -P_{ins} } \right) }+\left( {1-\gamma } \right) -e^{-r\left( {a-P_{ins} } \right) }{\textit{UIMGF}}_w \left( {-rb}\right) \nonumber \\= & {} 1-\gamma e^{-r\left( {a+bw_g -P_{ins} } \right) }-e^{-r\left( {a-P_{ins} } \right) }{\textit{UIMGF}}_w \left( {-rb} \right) \end{aligned}$$
$$\begin{aligned} EU_0 \left( {\widetilde{\pi } } \right)= & {} EU_{ins} \left( {\widetilde{\pi } } \right) \nonumber \\ 1-e^{-ra}{\textit{MGF}}_w \left( {-rb} \right)= & {} 1-\gamma e^{-r\left( {a+bw_g -R_{ins} } \right) }-e^{-r\left( {a-R_{ins} } \right) }{\textit{UIMGF}}_w \left( {-rb} \right) \nonumber \\ R_{ins}= & {} \frac{1}{ r} ln \left( {\frac{{\textit{MGF}}_w \left( {-rb} \right) }{\gamma e^{-rbw_g }+ {\textit{UIMGF}}_w \left( {-rb} \right) }} \right) \end{aligned}$$

Appendix 6: Comparison of Instruments

1.1 Comparison Between the Two Option Contracts (a) and (b)

We first compare the risk premiums and then assess the conditions that make one instrument more attractive to the farmer than the other. \(R_{opt_b}\) will be greater than \(R_{opt_a}\) for all cases. Intuitively, the conclusion is the same, as option contract (b) offers more guarantees than contract (a), and the probability of the farmer being able to purchase the optioned volume at the maturity date is greater.

If \(R_{opt_b } >R_{opt_a } \), then

$$\begin{aligned}&e^{-r\left( {b-P_e } \right) w_g }{\textit{LIMGF}}_w \left( {-rP_e } \right) + {\textit{UIMGF}}_w \left( {-rb} \right) \nonumber \\&\quad <\left( {1-Z} \right) {\textit{MGF}}_w \left( {-rb} \right) +\hbox {Z }\left[ {e^{-r\left( {b-P_e } \right) w_g }{\textit{LIMGF}}_w \left( {-rP_e } \right) +{\textit{UIMGF}}_w \left( {-rb} \right) } \right] \nonumber \\&e^{-r\left( {b-P_e } \right) w_g }{\textit{LIMGF}}_w \left( {-rP_e } \right) + {\textit{UIMGF}}_w \left( {-rb} \right) < {\textit{MGF}}_w \left( {-rb} \right) \end{aligned}$$

For \(R_{opt_b } \) to be positive, the above expression must hold (as the numerator of the logarithm on the right side of the expression has to be greater than the denominator on the left side).

We calculate the conditions that determine whether farmers will take out an option contract or insurance thus:

$$\begin{aligned} P_{opt_b } - P_{opt_a } <\frac{1}{r} ln\left( {\frac{D_{opt_a } }{D_{opt_b } }} \right) \end{aligned}$$

If \(P_{opt_b } <P_{opt_a } +\frac{1}{r} ln\left( {\frac{D_{opt_a } }{D_{opt_b } }} \right) \), they would choose the option contract (b); and if \(P_{opt_b } >P_{opt_a } +\frac{1}{r }\ln \left( {\frac{D_{opt_a } }{D_{opt_b } }} \right) \), they would purchase the option contract (a) \((D_{opt_a } \) is always higher than \(D_{opt_b } )\). The farmer would be indifferent to the two if \( P_{opt_b } =P_{opt_a } +\frac{1}{r} ln\left( {\frac{D_{opt_a } }{D_{opt_b } }} \right) \).

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Rey, D., Garrido, A. & Calatrava, J. Comparison of Different Water Supply Risk Management Tools for Irrigators: Option Contracts and Insurance. Environ Resource Econ 65, 415–439 (2016).

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