Optimization of a Spatially Variable Resource: An Illustration for Irrigated Crops

Abstract

Economic optimization of a resource which is space dependent and displays irregular spatial variation was investigated. The resource (yield) of an entire field is regarded as a random variable which depends on a random function on the one hand, and a control variable on the other hand. The study focuses on the question of how economically efficient levels of relevant control variables may vary as a function of random spatial variability of resources, taking into account the decision maker's attitude to risk. A general approach to cope with the above question is developed, and the effects of uncertainty on the levels of the optimal control variables are investigated by comparing a deterministic case with two uncertainty levels of decision making: risk neutral and risk averse. Following the general methodology, expectation value and variance of average yield are first approximated by Taylor series expansion and then closed‐form solutions are obtained for a specific, exponential, yield production function. Subsequently, we illustrate the general analysis for the example of irrigated corn, the only control variable being water quantity, and with corp yield a spatial random function in the plane. The results suggest that the impact of random soil properties on optimum level of water application might be substantial. Hence it is generally worthwhile to account for spatial variability in optimizing level of irrigation or similar control variables.