Obtaining estimates of pesticide productivity is an economic response to the growing public concern about the steady increase of pesticide use in the United States. This type of research indicates the cost of limiting pesticide use in terms of foregone output. Previous empirical studies give a “snap-shot”, or “average”, look at pesticide productivity. This research effort employs a random coefficient model to determine the trend of the marginal value product of pesticides in agriculture in the United States. Results show a distinct downward trend in two states, Iowa and Texas. California, however, shows no evidence of a downward trend.

This paper discusses analytical and numerical issues related toelliptic equations with random coefficients which are generallynonlinear functions of white noise. Singularity issues are avoidedby using the Itô-Skorohod calculus to interpret the interactionsbetween the coefficients and the solution. The solution is constructedby means of the Wiener Chaos (Cameron-Martin) expansions. Theexistence and uniqueness of the solutions are established underrather weak assumptions, the main of which requires only that theexpectation of the highest order (differential) operator is anon-degenerate elliptic operator. The deterministic coefficientsof the Wiener Chaos expansion of the solution solve a lower-triangularsystem of linear elliptic equations (the propagator). This structureof the propagator insures linear complexity of the related numericalalgorithms. Using the lower triangular structure and linearity of thepropagator, the rate of convergence is derived for a spectral/hp finiteelement approximation. The results of related numerical experiments arepresented.