Implementing an Approximation of Cumulative Prospect Theory into Mixed Linear Programming - an Application to Bio-Economic Modelling at Farm-Scale Considering Crop Insurance

Many empirical studies have found Cumulative Prospect Theory (CPT) superior in depicting risk behavior compared to the expected utility approach and literature now offers also CPT related parameter estimates for European farmers. CPT combines two segments of utility functions, a convex, risk loving one for losses and a concave, risk averse one for gains, and assigns subjective weights to the pay-offs according to their cumulative probabilities. So far, no implementation of CPT into constrained optimization problems exists, allowing for instance, the simulation of risk management under CPT in farm-scale programming models. To close this gap, we propose to combine endogenous sorting of the pay-offs based on integer variables with a piece-wise linear approximation of the value function using SOS2 (Special Ordered Sets of Type 2) variables. The SOS2 variables are required to deal with the convexity of the loss segment of the utility function. The integer sorting assigns the weights to the pay-offs according to their cumulative probabilities, it requires that all pay-offs are equally likely. Simulating optimal uptake levels of variants of a hypothetical crop insurance product with an evolved bio-economic model at farm-scale serves a proof of concept. The model considers adjustments in the cropping plan and allows for partial insurance coverage, in opposite to existing studies which evaluate the uptake of crop insurance at fixed crop choices and depict coverage as a yes-no decision. The approximation error of the approach is found as negligible small and the numerical burden compared to optimization under risk neutrality as still acceptable. The proposed approximation approach is quite general and applicable for any utility function increasing in the pay-off value and does not require its differentiability. It can also be applied without probability weighting. The empirical application underlines that the approach generates the expected behavior when a risk reducing strategy, here crop insurance, is considered under CPT. Insured acreage generally increases with higher strike levels where more frequently occurring but lower crop damages are covered, and with reduced cost of the insurance products. Using crop insurance as a risk management strategy is found to interact with other measures such as adjustments in cropping shares. This underlines the usefulness of an approach which allows to optimize interacting risk management strategies at farm-scale under CPT, considering resource and other relevant constraints.