A mean–variance acreage model. Issue 4 (4th March 2022) - Fang, Ming; Perng, Cherng-tiao
ABSTRACT: We study a mean–variance acreage model, A = α ( E [ p ], V [ p ] ), where p is price at harvest time and E and V are the expectation and variance operators conditional on information known at planting time. Under the assumption that p = π ( A y ) where yield y is random and unknown at planting time, we will investigate the existence, uniqueness, and convergence of this fixed point problem as well as the coherence of the mean–variance model. As is well known, Newton's method can not guarantee its convergence unless the initial approximation is sufficiently close to a true solution. In theory, the more variables/randomness one has, the harder it is to find a good initial guess. Specifically we focus on the case when the inverse demand function p = π ( A y ) is implicitly defined. We will solve the random nonlinear equations by Newton's method and investigate the optimal and robust way to choose random initial values for Newton's method. The robust initial value will allow us to study how the price support program will affect consumer prices, farm prices, and government expenditures as well as their variabilities. Hopefully solving nonlinear random equations will shed some light on the choice of initial values for Newton's method.