Skip to main content


Log in

Is Risk a Limit or an Opportunity to Mitigate Greenhouse Gas Emissions? The Case of fertilization in Agriculture

  • Published:
Environmental Modeling & Assessment Aims and scope Submit manuscript


Risk can represent an important non-monetary barrier to the adoption of climate change mitigation practices. This paper deals with how risk aversion influences the fertilization behavior of farmers. We analytically show that a decreasing variance of yield along with nitrogen inputs encourages risk averse farmers to apply larger quantities of fertilizers. Data concerning three departments in France are then used to determine (i) crop yield response function to N fertilizer and (ii) farmers’ risk aversion behavior on the basis of their current fertilizer applications. We find that risk aversion is associated with an additional application of 29 kg/ha compared with risk neutral behavior, which represents an average loss of €76/ha. We show that the reduction of an abatement linked to risk aversion should appear only when crop yield variance is convex with respect to N fertilizer. Lastly, our results show that an insurance contract that covers yield variability may be an interesting tool to mitigate emissions.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Fig. 1

Similar content being viewed by others

Availability of Data and Material

The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.

Code Availability

This is not applicable.


  1. In France, nitrogen fertilization accounts for 44% of GHG emissions from agriculture and, alone, represented 8% of national net emissions in 2017 [74].

  2. Some examples of actions: setting a ceiling of 170 kg/ha spreadable not to be exceeded for the entire farm, setting up a regulatory schedule for spreading fertilizers specifying the dates authorized for the use of fertilizers (depending on their nature and the type of crop), buffer or protection perimeters moving the spreading limits away from sensitive areas, etc. These measures are included in a national action program. They can be reinforced (period of ban on spreading, etc.) or declined (soil cover, etc.) at the regional level.

  3. See the European Council agreement of October 2014 aimed at reducing European greenhouse gas emissions by 40% by 2030 relative to 1990 levels.

  4. In France, the current agricultural insurance system is characterized by public and private intervention that helps to cover damages caused by frost, hail, or drought. The public intervention is under the responsibility of the FNGRA (Fonds National de Gestion des Risques en Agriculture) and aims to cover farmers against uninsurable risks. Insurable risks are covered by the private market, alongside government intervention: farmers receive a grant representing, at most, 65% of the insurance premium.

  5. Such questioning also exists in the literature concerning the potential “risk-increasing” or “risk-decreasing” factor of biodiversity [75, 76]. For example, Finger and Buchman [75] proved empirically the risk-reducing effect of species diversity.

  6. The interested reader can refer to the review of Pannell [77] concerning potential explanations for over-application of nitrogen fertilizers by some farmers.

  7. The risk increasing, decreasing, or neutral nature of fertilization is not clear-cut in the literature (see Sect. 2). A quadratic specification for yields allows us to represent all these potential situations. In addition, it is consistent with Cerrato and Blackmer [72] and Bélanger et al. [73].

  8. We calculated a matrix of Pearson correlation coefficients regarding yield crops and implemented a significance test. In this test, a p value is calculated for each correlation coefficient and if it is lower than 0.05, we can conclude that the coefficient is significantly different than 0. In the matrix, every value was superior to zero which bears the assumption of no covariance between crops. The results of this analysis are available from the authors upon request.

  9. The results of parametric sensitivity analysis are available from the authors upon request.

  10. Our work focuses on the demand side of insurance, leaving the supply side aside. Indeed, the insurance contract considered may not be financially sustainable enough to be proposed by insurance companies. The analysis of the feasibility of the contract for the supply side of the insurance market will lead to further future research.

  11. Through contracting, defining such a benchmark is feasible. First, farmers monitor their quantity of fertilizer on farmlands for public authorities or cooperations. This is the case in particular in the framework of the Cahier d’épandage controlled by the Chamber of Agriculture in France. They could themselves define the \({x}_{i}^{*}\) at the level of crops. An alternate solution, could be to define an overall \({x}_{i}^{*}\) at the level of their farms.

  12. For example, the European FADN (Farm Accountancy Data Network) database gathers data from representative farms of a given territory but not the data of all farms. Moreover, it does not provide information about the amount of fertilizers but, instead, the cost linked to fertilization.

  13. Estimation is conducted with the MODEL procedure of the SAS statistical package (SAS INSTITUTE, 2012).

  14. Note that other articles also assess the risk aversion of French farmers like Bocquého et al. [65] and Bougherara et al. [66], but they assume the prospect theory as the theoretical framework, whereas we consider the expected utility framework.


  1. Galloway, J. N., Townsend, A. R., Erisman, J. W., Bekunda, M., Cai, Z., Freney, J. R., Martinelli, L. A., Seitzinger, S. P., & Sutton, M. A. (2008). Transformation of the nitrogen cycle: Recent trends, questions, and potential solutions. Science, 320(5878), 889–892.

    CAS  Google Scholar 

  2. Blottnitz, H. V., Rabl, A., Boiadjiev, D., Taylor, T., & Arnold, S. (2006). Damage costs of nitrogen fertilizer in Europe and their internalization. Journal of Environmental Planning and Management, 49, 413–433.

    Google Scholar 

  3. Dalmas, D., Moreau, R., Quévremont, P., & Frey, V. (2010). Elaboration d’un plan de lutte contre les algues vertes. Ministère de l’alimentation, de l’agriculture et de la pêche, 154.

  4. Ellerman, A. D., Convery, F. J., & Perthuis, C. D. (2010). Pricing emission: The European Union emissions trading scheme. Cambridge University Press.

  5. De Cara, S., & Jayet, P. A. (2011). Marginal abatement costs of greenhouse gas emissions from European agriculture, cost effectiveness, and the EU non-ETS burden sharing agreement. Ecological Economics, 70, 1680–1690.

    Google Scholar 

  6. Bourgeois, C., Fradj, N. B., & Jayet, P. A. (2014). How cost-effective is a mixed policy targeting the management of three agricultural N-pollutants? Environmental Modeling and Assessment, 19, 389–405.

    Google Scholar 

  7. Dequiedt, B., & Moran, D. (2015). The cost of emission mitigation by legume crops in French agriculture. Ecological Economics, 110, 51–60.

    Google Scholar 

  8. Isbasoiu, A., Jayet, P. A., & De Cara, S. (2021). Increasing food production and mitigating agricultural greenhouse gas emissions in the European Union: Impacts of carbon pricing and calorie production targeting. Environmental Economics and Policy Studies, 23, 409–440.

    Google Scholar 

  9. Bamière, L., Camuel, A., De Cara, S., Delame, N., Dequiedt, B., Lapierre, A., & Lévêque, B. (2017). Analyse des freins et des mesures de déploiement des actions d’atténuation à coût négatif dans le secteur agricole : Couplage de modélisation économique et d’enquêtes de terrain – Rapport final. 79.

  10. Marques, G., De Oliveira Silva, R., Gustavo Barioni, L., Hall, J. A. J., Fossaert, C., Tedeschi, L. O., Garcia-Launay, F., & Moran, D. (2021). Evaluating environmental and economic trade-offs in cattle feed strategies using multiobjective optimization. Agricultural systems, 195, 103308. Menapace, L., Colson, G., Raffaelli, R. (2013). Risk aversion, subjective beliefs, and farmer risk management strategies. American Journal of Agricultural Economics, 95, 384–389.

    Google Scholar 

  11. Pellerin, S., Bamière, L., Angers, D., Béline, F., Benoit, M., Butault, J.-P., Chenu, C., Colnenne-David, C., De Cara, S., Delame, N., Doreau, M., Dupraz, P., Faverdin, P., Garcia-Launay, F., Hassouna, M., Hénault, C., Jeuffroy, M.-H., Klumpp, K., Metay, A., Moran, D., Recous, S., Samson, E., Savini, I., Pardon, L., Chemineau, P. (2017). Identifying cost-competitive greenhouse gas mitigation potential of French agriculture. Environmental Science and Policy, 77, 130–139.

  12. Chavas, J. P., & Holt, M. T. (1996). Economic behavior under uncertainty: A joint analysis of risk preferences and technology. The Review of Economics and Statistics, 78(2), 329–335.

    Google Scholar 

  13. Bontemps, C., Bougherara, D., & Nauges, C. (2021). Do risk preferences really matter? The case of pesticide use in agriculture. Environmental Modeling and Assessment, 26, 609–630.

    Google Scholar 

  14. Khor, L. Y., Ufer, S., Nielsen, T., & Zeller, M. (2018). Impact of risk aversion on fertiliser use: Evidence from Vietnam. Oxford Development Studies, 46(4), 483–496.

    Google Scholar 

  15. Meyer-Aurich, A., & Karatay, Y. N. (2019). Effects of uncertainty and farmers’ risk aversion on optimal N fertilizer supply in wheat production in Germany. Agricultural Systems, 173(C), 130–139.

  16. Nauges, C., Bougherara, D., & Koussoubé, E. (2021). Fertilizer use and risk: New evidence from Sub-Saharan Africa. TSE Working Papers 21–1266. Toulouse School of Economics.

  17. Dury, J. (2011). The cropping-plan decision-making: A farm level modelling and simulation approach. Institut National Polytechnique de Toulouse (INP Toulouse). INPT.

  18. Berentsen, P., Kovacs, K., & Asseldonk, M. V. (2012). Comparing risk in conventional and organic dairy farming in the Netherlands: An empirical analysis. Journal of Dairy Science, 95, 3803–3811.

    CAS  Google Scholar 

  19. Stuart, D., Schewe, R., & McDermott, M. (2014). Reducing nitrogen fertilizer application as a climate change mitigation strategy: Understanding farmer decision-making and potential barriers to change in the US. Land Use Policy, 36, 210–218.

    Google Scholar 

  20. Smit, B., & Skinner, M. W. (2002). Adaptation options in agriculture to climate change: A typology. Mitigation and Adaptation Strategies for Global Change, 7, 85–114.

    Google Scholar 

  21. Huang, W. Y., Heifner, R. G., Taylor, H., & Uri, N. D. (2001). Using insurance to enhance nitrogen fertilizer application to reduce nitrogen losses to the environment. Environmental Monitoring and Assessment, 68, 209–233.

    CAS  Google Scholar 

  22. Agence de l’eau Loire-Bretagne. (2016). 10ième programme de l’agence de l’eau Loire-Bretagne – 2016–2018. Fiche 10.

  23. Lambert, D. K. (1990). Risk considerations in the reduction of nitrogen fertilizer use in agricultural production. Western Journal of Agricultural Economics, 15(2), 234–244.

    Google Scholar 

  24. Leathers, H. D., & Quiggin, J. C. (1991). Interactions between agricultural and resource policy: The importance of attitudes toward risk. American Journal of Agricultural Economics, 73, 757–764.

    Google Scholar 

  25. Just, R. E., & Pope, R. D. (1979). Production function estimation and related risk considerations. American Journal of Agricultural Economics, 61(2), 276–284.

    Google Scholar 

  26. Monjardino, M., McBeath, T., Ouzman, J., Llewellyn, R., & Jones, B. (2015). Farmer risk-aversion limits closure of yield and profit gaps: A study of nitrogen management in the southern Australian wheatbelt. Agricultural Systems, 137, 108–118.

    Google Scholar 

  27. Rajsic, P., Weersink, A., & Gandorfer, M. (2009). Risk and nitrogen application levels. Canadian Journal of Agricultural Economics/Revue Canadienne d’Agroéconomie, 57, 223–239.

    Google Scholar 

  28. Babcock, B. A. (1992). The effects of uncertainty on optimal nitrogen applications. Review of Agricultural Economics, 14(2), 271–280.

    Google Scholar 

  29. Finger, R. (2012). Nitrogen use and the effects of nitrogen taxation under consideration of production and price risks. Agricultural Systems, 107, 13–20.

    Google Scholar 

  30. Sriramaratnam, S., Bessler, D. A., Rister, M. E., Matocha, J. E., & Novak, J. (1987). Fertilisation under uncertainty: An analysis based on producer yield expectations. American Journal of Agricultural Economics, 69, 349–357.

    Google Scholar 

  31. COMIFER. (2011). Calcul de la fertilisation azotée. Comifer.

  32. Gandorfer, M., Pannell, D., & Meyer-Aurich, A. (2011). Analyzing the effects of risk and uncertainty on optimal tillage and nitrogen fertilizer intensity for field crops in Germany. Agricultural Systems, 104, 615–622.

    Google Scholar 

  33. Regev, U., Gotsch, N., & Rieder, P. (1997). Are fungicides, nitrogen and plant growth regulators risk-reducing? Empirical evidence from Swiss wheat production. Journal of Agricultural Economics, 48, 167–178.

    Google Scholar 

  34. Antle, J. M. (2010). Asymmetry, partial moments, and production risk. American Journal of Agricultural Economics, 92, 1294–1309.

    Google Scholar 

  35. Bougherara, D. (2014). Rôle de l’aversion au risque des agriculteurs dans l’utilisation de pesticides et implications pour la régulation. Rapport Scientifique du Projet AVERSIONRISK.

  36. Sheriff, G. (2005). Efficient waste? Why farmers over-apply nutrients and the implications for policy design. Applied Economic Perspectives and Policy, 27, 542–557.

    Google Scholar 

  37. Ramaswami, B. (1993). Supply response to agricultural insurance: Risk reduction and moral hazard effects. American Journal of Agricultural Economics, 75(4), 914–925.

    Google Scholar 

  38. Horowitz, J. K., & Lichtenberg, E. (1993). Insurance, moral hazard, and chemical use in agriculture. American Journal of Agricultural Economics, 75, 926–935.

    Google Scholar 

  39. Smith, V. H., & Goodwin, B. K. (1996). Crop insurance, moral hazard, and agricultural chemical use. American Journal of Agricultural Economics, 78, 428–438.

    Google Scholar 

  40. Neumann, L. J., & Morgenstern, O. (1947). Theory of games and economic behavior. Princeton University Press Princeton.

    Google Scholar 

  41. Polomé, P., Harmignie, O., & de Frahan, B. H. (2006). Farm-level acreage allocation under risk. Annual meeting, July 23–26, Long Beach. American Agricultural Economics Association.

  42. Agreste. (2014). Pratiques culturales sur les grandes cultures et prairies portant sur l’année 2011.

  43. RICA. Réseau d’information agricole.

  44. Agreste. (2011). Les Dossiers n° 12 - Juillet.

  45. Finger, R., & Schmid, S. (2008). Modeling agricultural production risk and the adaptation to climate change. Agricultural Finance Review, 68, 25–41.

    Google Scholar 

  46. Saha, A., Shumway, C. R., & Talpaz, H. (1994). Joint estimation of risk preference structure and technology using expo-power utility. American Journal of Agricultural Economics, 76, 173–184.

    Google Scholar 

  47. Pope, R. D., LaFrance, J. T., & Just, R. E. (2011). Agricultural arbitrage and risk preferences. Journal of Econometrics, 162, 35–43.

    Google Scholar 

  48. Reynaud, A., & Couture, S. (2012). Stability of risk preference measures: Results from a field experiment on french farmers. Theory and Decision, 73, 203–221.

    Google Scholar 

  49. Tevenart, C., & Brunette, M. (2021). Role of farmers’ risk and ambiguity preferences on fertilization decisions: An experiment. Sustainability, 13, 9802.

    CAS  Google Scholar 

  50. De Cara, S., Jacquet, F., Reynaud, A., Goulevant, G., Jeuffroy, M.-H., Montfort, F., & Lucas, P. (2011). Economic analysis of summer fallow management to reduce take-all disease and N leaching in a wheat crop rotation. Environmental Modeling and Assessment, 16, 91–105.

    Google Scholar 

  51. Bigman, D. (1996). Safety-first criteria and their measures of risk. American Journal of Agricultural Economics, 78(1), 225–235.

    Google Scholar 

  52. Kennedy, J. O. S., & Francisco, E. M. (1974). On the formulation of risk constraints for linear programming. Journal of Agricultural Economics, 25(2), 129–145.

    Google Scholar 

  53. Mitter, H., Heumesser, C., & Schmid, E. (2015). Spatial modeling of robust crop production portfolios to assess agricultural vulnerability and adaptation to climate change. Land Use Policy, 46, 75–90.

    Google Scholar 

  54. Alexander, P., Moran, D., Smith, P., Hastings, A., Wang, S., Sünnenberg, G., Lovett, A., Tallis, M. J., Casella, E., Taylor, G., Finch, J., & Cisowska, I. (2014). Estimating UK perennial energy crop supply using farm-scale models with spatially disaggregated data. GCB Bioenergy, 6, 142–155.

    Google Scholar 

  55. Coletti, G., van der Gaag, L. C., Petturiti, D., & Vantaggi, B. (2020). Detecting correlation between extreme probability events. International Journal of General Systems, 49(1), 64–87.

    Google Scholar 

  56. Isik, M., & Devadoss, S. (2006). An analysis of the impact of climate change on crop yields and yield variability. Applied Economics, 38(7), 835–844.

    Google Scholar 

  57. Ritter, C., Dicke, D., Weis, M., Oebel, H., Piepho, H. P., Büchse, A., & Gerhards, R. (2008). An on-farm approach to quantify yield variation and to derive decision rules for site-specific weed management. Precision Agriculture, 9, 133–146.

    Google Scholar 

  58. Borgne, É. (2021). Les mauvais rendements font les mauvais résultats de l'agriculture normande, Le bilan économique 2020 – Insee Conjoncture Normandie n°27, juillet 2021.

  59. Agreste. (2023). Synthèses conjoncturelles. Grandes cultures, n°400, Février 2023.

  60. Su, Y., Gabrielle, B., & Makowski, D. (2021). The impact of climate change on the productivity of conservation agriculture. Nature Climate Change, 11, 628–633.

    Google Scholar 

  61. Solow, R. (1956). A contribution to the theory of economic growth. The Quarterly Journal of Economics, 70(1), 65–94.

    Google Scholar 

  62. Ingersoll, J. E. (1987). Theory of financial decision making. Rowman & Littlefield. 3.

  63. Tanaka, T., Camerer, C. F., & Nguyen, Q. (2010). Risk and time preferences: Linking experimental and household survey data from Vietnam. The American Economic Review, 100, 557–571.

    Google Scholar 

  64. Kahneman, D., & Tversky, A. (1979). Prospect theory: An analysis of decision under risk. Econometrica, 47(2), 263–291.

    Google Scholar 

  65. Bocquého, G., Jacquet, F., & Reynaud, A. (2014). Expected utility or prospect theory maximisers? Assessing farmers’ risk behaviour from field-experiment data. European Review of Agricultural Economics, 41, 135–172.

    Google Scholar 

  66. Bougherara, D., Gassmann, X., Piet, L., & Reynaud, A. (2017). Structural estimation of farmers’ risk and ambiguity preferences: A field experiment. European Review of Agricultural Economics, l44(5), 782–80.

  67. Dubois, P. (2006). Efficacité des contrats agricoles: le cas de la production de blé en Midi-Pyrénées. INRA Sciences Sociales, 2006, 1–4, April.

  68. Barnett, B. J., & Coble, K. H. (1999). Understanding crop insurance principles: A primer for farm leaders. Research Reports 15784, Mississippi State University, Department of Agricultural Economics.

  69. Collier, S. M., Ruark, M. D., Oates, L. G., Jokela, W. E., & Dell, C. J. (2014). Measurement of greenhouse gas flux from agricultural soils using static chambers. Journal of Visualized Experiments, 90, e52110.

    Google Scholar 

  70. Gilliot, J. M., Vaudour, E., Michelin, J., & Houot, S. (2017). Estimation des teneurs en émission organique des sols agricoles par télédétection par drone. Revue Française de Photogrammétrie et de Télédétection, 213–214, 105–115.

    Google Scholar 

  71. Cayatte, J. L. (2004). Introduction à l'économie de l'incertitude. De Boeck Supérieur.

  72. Cerrato, M., & Blackmer, A. (1990). Comparison of models for describing; corn yield response to nitrogen fertilizer. Agronomy Journal, 82, 138–143.

    Google Scholar 

  73. Bélanger, G., Walsh, J. R., Richards, J. E., Milburn, P. H., & Ziadi, N. (2000). Comparison of three statistical models describing potato yield response to nitrogen fertilizer. Agronomy Journal, 92, 902–908.

    Google Scholar 

  74. (2019). CITEPA, Rapport CCNUCC.

  75. Finger, R., & Buchmann, N. (2015). An ecological economic assessment of risk-reducing effects of species diversity in managed grasslands. Ecological Economics, 110, 89–97.

    Google Scholar 

  76. Mouysset, L., Doyen, L., & Jiguet, F. (2013). How does economic risk aversion affect biodiversity? Ecological Applications, 23(1), 96–109.

    CAS  Google Scholar 

  77. Pannell, D. J. (2017). Economic perspectives on nitrogen in farming systems: Managing trade-offs between production, risk and the environment. Soil Research, 55(6), 473–478.

    Google Scholar 

Download references


Warm thanks go to Stéphane De Cara, Shahbano Soomro, and Caroline Orset for their reading and advice. This research is part of the Agriculture and Forestry research program by the Climate Economics Chair.


This research is part of the Agriculture and Forestry research program by the Climate Economics Chair. The authors want to thank the Climate Economics Chair for financial support. The UMR BETA is supported by a grant overseen by the French National Research Agency (ANR) as part of the “Investissements d’Avenir” program (ANR-11-LABX-0002–01, Lab of Excellence ARBRE).

Author information

Authors and Affiliations



All authors contributed to the study conception and design. Material preparation, data collection, and analysis were performed by Benjamin Dequiedt and Emmanuel Servonnat. The chapter thesis was written by Benjamin Dequiedt, and the corresponding first draft of the manuscript was written by Marielle Brunette and Philippe Delacote. All authors commented on previous versions of the manuscript. Marielle Brunette and Benjamin Dequiedt realized the revision of the article. All authors read and approved the final manuscript.

Corresponding author

Correspondence to Marielle Brunette.

Ethics declarations

Conflict of Interest

The authors declare no conflict of interests.

Ethics Approval

This research does not involve humans or animal subjects.

Consent to Participate

This research does not involve human subjects.

Consent to Publish

This research does not involve human subjects and consequently, we do not need consent for publication from the participants.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.


Appendix 1: Modeling of the CARA Utility Function

Let us consider one farmer with an initial profit \({\pi }_{0}\) and facing a stochastic profit \(\widetilde{\pi }.\) Under CARA, her utility function is defined as \(u\left(\widetilde{\pi }+{\pi }_{0}\right)={-e}^{-\alpha (\widetilde{\pi }+{\pi }_{0})}\), with a risky profit \(\widetilde{\pi }\) corresponding to the sum of profit on its different crops: \(\widetilde{\pi }= {\sum }_{i=1}^{n}{\widetilde{\pi }}_{i}\)

Consequently, we have \(u\left(\widetilde{\pi }+{\pi }_{0}\right)={-e}^{-\alpha ({\pi }_{0}+{\sum }_{i=1}^{n}{\widetilde{\pi }}_{i})}\)

Consistent with Polomé et al. [41], we assume that no covariance exists between crops, so that the yield distributions are independent. We can then write:

$$u\left(\widetilde{\pi }\right)={-e}^{-\alpha {\pi }_{0}-\alpha \widetilde{{\pi }}_{1}-\alpha \widetilde{{\pi }}_{2}-\alpha \widetilde{{\pi }}_{3}-... -\alpha \widetilde{{\pi }}_{n}}={-e}^{{-\pi }_{0}-\alpha \widetilde{{\pi }}_{1}}\times {e}^{-\alpha \widetilde{{\pi }}_{2}}{\times e}^{-\alpha \widetilde{{\pi }}_{3}}\times \dots \times {e}^{ -\alpha \widetilde{{\pi }}_{n}}$$

Profits between the different fields are independent. Consequently, the optimum in one field is not influenced by the profit in another field. Moreover, we know from the literature that the expected utility of CARA preferences can be represented by a mean variance Markowitz form as follows (see Cayatte [71] for a demonstration):

Assumption: \(\widetilde{{\pi }}_{i}\to N (m, {\sigma }^{2})\)

$$E[u\left(\widetilde{{\pi }}_{i}\right)]=E[\widetilde{{\pi }}_{i}]-c V[\widetilde{{\pi }}_{i}]$$

where \(c=\nicefrac{\alpha}{2}\), where the expected profit is as follows: \(E\left[\widetilde{{\pi }}_{i}\right]={l}_{i}({E[\widetilde{y}}_{i}]{p}_{i}-{x}_{i}w)\) and,

where \({l}_{i}\) is the area (ha), \({y}_{i}\) is the yield (q.ha−1), \({p}_{i}\) is the crop price (€.q−1), \({x}_{i}\) is the fertilizer applied (kgN.ha−1), and \(w\) is the fertilizer price (€.kgN−1).

Expected yield is specified as quadratic: \({E[y}_{i}]={\beta }_{1}+{\beta }_{2}{x}_{i}+{\beta }_{3}{{x}_{i}}^{2}\)

Profit variance is as follows: \(V\left[\widetilde{{\pi }}_{i}\right]=V[{l}_{i}({\widetilde{y}}_{i}{p}_{i}-{x}_{i}w)]={{l}_{i}}^{2}{{p}_{i}}^{2}V[{\widetilde{y}}_{i}]\)

Yield variance is specified as quadratic as well: \(V\left[{y}_{i}\right]={\rho }_{1}+{\rho }_{2}{x}_{i}+{\rho }_{3}{{x}_{i}}^{2}\)

In the literature, it is not clear-cut if fertilizer is a risk-increasing factor, a risk-decreasing factor, or a neutral one (see Sect. 2). A quadratic specification for yield allows us to represent all these potential situations. In addition, it is consistent with Cerrato and Blackmer [72] and Bélanger et al. [73].

The expected utility function can be written in the complete specified form as:

$$E[u\left(\widetilde{{\pi }}_{i}\right)]={l}_{i}(({\beta }_{1}+{\beta }_{2}{x}_{i}+{\beta }_{3}{{x}_{i}}^{2}){p}_{i}-{x}_{i}w)-c {{l}_{i}}^{2}{{p}_{i}}^{2}({\rho }_{1}+{\rho }_{2}{x}_{i}+{\rho }_{3}{{x}_{i}}^{2})$$

The first-order condition to find the optimal nitrogen application is as follows:

$$\frac{{dU}_{i}}{{dx}_{i}}=0\iff 0={l}_{i}\left(\left({\beta }_{2}+2{\beta }_{3}{{x}_{i}}^{*}\right){p}_{i}-w\right)-c{{l}_{i}}^{2}{{p}_{i}}^{2}({p}_{2}+{2p}_{3}{{x}_{i}}^{*})$$

where \({{x}_{i}}^{*}\) is such that \(0={l}_{i}(({\beta }_{2}+2{\beta }_{3}{{x}_{i}}^{*}){p}_{i}-w)-c {{l}_{i}}^{2}{{p}_{i}}^{2}({\rho }_{2}+2{\rho }_{3}{{x}_{i}}^{*})\) and

$$=\hspace{0.17em}\hspace{0.17em}>\hspace{0.17em}{{x}_{i}}^{*}=\frac{c{{p}_{i}}^{2}{l}_{i}{\rho }_{2}+w-{\beta }_{2}{p}_{i}}{(2{\beta }_{3}{p}_{i}-2{\rho }_{3}c{l}_{i}{{p}_{i}}^{2})}$$

Consequently, we observe that the optimal quantity of fertilizer is determined by the risk aversion coefficient. This value is used in Sect. 4.3 to determine the risk aversion coefficient per farmer. Note that in theory, this equation allows for a negative quantity of fertilizers; however, in practice (simulations), and in the rare cases where it happens, the quantity is limited to zero.

Let us now study the impact of an emission price on the optimal amount of fertilizer:

  • Expected utility with a tax on emissions:

$${U}_{i}={l}_{i}(({\beta }_{1}+{\beta }_{2}{x}_{i}+{\beta }_{3}{{x}_{i}}^{2}){p}_{i}-{x}_{i}w-{x}_{i}{f}_{tax})-c {{l}_{i}}^{2}{{p}_{i}}^{2}({\rho }_{1}+{\rho }_{2}{x}_{i}+{\rho }_{3}{{x}_{i}}^{2})$$

where tax is the emission price and \(f\)the emission factor.

The first-order condition becomes.

$$\frac{d{U}_{i}}{d{x}_{i}}={l}_{i}\left(\left({\beta }_{2}+2{\beta }_{3}{{x}_{i}}^{*}\right){p}_{i}-w-{f}_{tax}\right)-c {{l}_{i}}^{2}{{p}_{i}}^{2}\left({\rho }_{2}+2{\rho }_{3}{{x}_{i}}^{*}\right)=0$$


$${{x}_{i}}^{*}=\frac{c{{p}_{i}}^{2}{l}_{i}{\rho }_{2}+w+{f}_{tax}-{\beta }_{2}{p}_{i}}{(2{\beta }_{3}{P}_{i}-2{\rho }_{3}c{l}_{i}{{P}_{i}}^{2})}$$

The second-order condition is:

$$\frac{d{{U}_{i}}^{2}}{{d}^{2}{x}_{i}}=2{l}_{i}{\beta }_{3}{p}_{i}-\alpha {{l}_{i}}^{2}{{p}_{i}}^{2}{\rho }_{3}<0$$
  • Impact of an emission price on fertilizer reduction:

$$\frac{{{dx}_{i}}^{*}}{d\mathrm{tax}}=\frac{f}{2{\beta }_{3pi}-{2}_{\rho 3}{{cl}_{i}p_{i}}^{2}}\Leftrightarrow \frac{{{dx}_{i}}^{*}}{dtax}=\frac{{l}_{i}f}{\frac{{d}^{2}E[\tilde{\pi} (x_i)]}{{{dx}_{i}}^{2}}-c\frac{{d}^{2}V[\tilde{\pi} (x_i)]}{{{dx}_{i}}^{2}}}$$

If \({\rho }_{3}\ne 0\) then the emission reduction will be influenced by the risk aversion coefficient.

Appendix 2: Impact of Risk Aversion on \({{x}_{i}}^{*}\)

Let us consider condition (7): \(f(x,c)=\frac{dE[\widetilde{{\pi }}_{i}({{x}_{i}}^{*})]}{d{x}_{i}}-c \frac{dV\left[\widetilde{{\pi }}_{i}\left({{x}_{i}}^{*}\right)\right]}{d{x}_{i}}=0\) and a point (\({{x}_{i}}^{*}\),\(c\)) that satisfies \(f({{x}_{i}}^{*}\),\(c\)) = 0. Then, according to the implicit function theorem, we have:

$$\frac{\partial {x}_{i}^{*}}{\partial c}= - \frac{\frac{\partial f}{\partial c}}{\frac{\partial f}{\partial {x}_{i}^{*}}}=\frac{\frac{dV\left[\widetilde{{\pi }}_{i}\left({{x}_{i}}^{*}\right)\right]}{d{x}_{i}}}{\frac{{d}^{2}E[\widetilde{{\pi }}_{i}({{x}_{i}}^{*})]}{{d{x}_{i}}^{2}} -c \frac{{d}^{2}V\left[\widetilde{{\pi }}_{i}\left({{x}_{i}}^{*}\right)\right]}{{d{x}_{i}}^{2}}}$$

Looking at the effect of risk aversion on \({{x}_{i}}^{*}\) and using Eq. (6) and the result of the implicit function theorem below, we find that:

$$\frac{\partial {x}_{i}^{*}}{\partial c}=\frac{dV\left[\widetilde{{\pi }}_{i}\left({x}_{i}^{*}\right)\right]}{d{x}_{i}}\times {\left(\frac{{d}^{2}E[\widetilde{\pi }({x}_{i})]}{{d{x}_{i}}^{2}}-c \frac{{d}^{2}V\left[\widetilde{{\pi }}_{i}\left({x}_{i}\right)\right]}{{d{x}_{i}}^{2}}\right)}^{-1}$$

The sign of this condition depends on the specifications of the profit function and the variance of profit. We consider four different profit function specifications and five profit variance ones, leading to 20 scenarios. The following Table 8 summarizes the impact of risk aversion on the optimal amount of fertilizer application for all of the scenarios.

Table 8 Impact of risk aversion on optimal fertilizer application

The following four pages provide an illustration of the impact of risk aversion on the fertilizer application function of the profit function (four specifications), the variance of profit (five specifications), and farmers’ preferences towards risk (three possibilities: risk neutrality, risk aversion, and risk loving).

The legend to understand the different graphs is as follows:

figure a
figure b
figure c
figure d
figure e

Appendix 3: Impact of Risk Aversion on Emission Tax Incentive

Let us consider condition (7): \(f(x,\mathrm{tax})=\frac{dE[\widetilde{{\pi }}_{i}({{x}_{i}}^{*},tax)]}{d{x}_{i}}-c \frac{dV\left[\widetilde{{\pi }}_{i}\left({{x}_{i}}^{*}\right)\right]}{d{x}_{i}}=0\), and a point (\({{x}_{i}}^{*}\),\(\mathrm{tax}\)) that satisfies \(f({{x}_{i}}^{*}\),\(\mathrm{tax}\)) = 0. Then, according to the implicit function theorem:

$$\frac{\partial {x}_{i}^{*}}{\partial tax}= - \frac{\frac{\partial f}{\partial tax}}{\frac{\partial f}{\partial {x}_{i}^{*}}}=\frac{- \frac{{\partial }^{2}E\left[\widetilde{{\pi }}_{i}\left({{x}_{i}}^{*},tax\right)\right]}{\partial tax \partial {x}_{i}}}{\frac{{\partial }^{2}E\left[\widetilde{{\pi }}_{i}\left({{x}_{i}}^{*},tax\right)\right]}{{\partial {x}_{i}}^{2}} -c \frac{{\partial }^{2}V\left[\widetilde{{\pi }}_{i}\left({{x}_{i}}^{*}\right)\right]}{{\partial {x}_{i}}^{2}}}=\frac{f{l}_{i}}{\frac{{\partial }^{2}E[\widetilde{{\pi }}_{i}({{x}_{i}}^{*},tax)]}{{\partial {x}_{i}}^{2}} -c \frac{{\partial }^{2}V\left[\widetilde{{\pi }}_{i}\left({{x}_{i}}^{*}\right)\right]}{{\partial {x}_{i}}^{2}}}$$

The impact of risk aversion on emission reductions triggered by emission prices is given by

$$\frac{{\partial }^{2}{x}_{i}^{*}}{\partial tax \partial c}= - f{l}_{i} \frac{{d}^{2}V\left[\widetilde{{\pi }}_{i}\left({x}_{i}\right)\right]}{{d{x}_{i}}^{2}}\times {\left(\frac{{d}^{2}E[\widetilde{\pi }({x}_{i})]}{{d{x}_{i}}^{2}}-c \frac{{d}^{2}V\left[\widetilde{{\pi }}_{i}\left({x}_{i}\right)\right]}{{d{x}_{i}}^{2}}\right)}^{-2}$$

Condition (11) depends on the specification used for the variance of profit. When the variance is convex, Eq. (11) < 0; when it is concave, Eq. (11) > 0; and when it is linearly decreasing, Eq. (11) = 0.

Appendix 4: Illustration of the Impact of an Insurance Policy on Nitrogen Quantities for One Crop

figure f

Appendix 5: Database Treatment

figure g

This figure details the data treatment. In Step 1, the Epicles database provides information about 325,048 plots corresponding to 6397 farms. We removed those that were not relevant for our analysis, leading to the 24,729 plots corresponding to the 2301 farms used in Step 2. In this step, we wanted to obtain a balanced sample to estimate risk-aversion coefficients in Step 3. To do this, we once again removed some unrelevant plots, leaving 10,990 plots (from 795 farms) at the beginning of Step 3. Risk aversion parameters are estimated on this sample. Finally, to simulate the two policy instruments (emission price and insurance), we considered the initial fertilization levels of the 2013 season, which represent 2774 plots (from 684 farms).

Appendix 6: Accuracy of Risk Aversion Scenario with Observed Emissions


Aversion coefficient

Type of coefficient

Risk averse


Risk loving


All risk attitudes




 − 7.0

 − 12.4



 −  6.9


Uniform: 7.5.10 –7


 − 25.9

 − 12.4



 −  11.7


Uniform: 0.0075










 − 25.9

 − 12.4



 −  11.7

  1. Average distance in % per ha between observed and estimated nitrogen application (the categorization of attitude toward risk is based on the CARA 1 scenario)

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Dequiedt, B., Brunette, M., Delacote, P. et al. Is Risk a Limit or an Opportunity to Mitigate Greenhouse Gas Emissions? The Case of fertilization in Agriculture. Environ Model Assess 28, 735–759 (2023).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: