ABSTRACT

We study a mean–variance acreage model, $A=\alpha \left(E\right[p],V[p\left]\right)$, 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 $p=\pi \left(Ay\right)$ 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 $p=\pi \left(Ay\right)$ 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.

摘要

我们研究了一个平均方差种植面积模型，$\mathrm{一种}=\alpha \left(乙\right[p],五[p\left]\right)$，其中p是收获时的价格，E 和 V 是基于种植时已知信息的期望和方差算子。在假设下$p=\pi \left(\mathrm{一种}\mathrm{是的}\right)$其中产量 y 在种植时是随机且未知的，我们将研究这个不动点问题的存在性、唯一性和收敛性以及均值方差模型的一致性。众所周知，除非初始近似值足够接近真解，否则牛顿法不能保证其收敛性。理论上，变量/随机性越多，就越难找到一个好的初始猜测。具体来说，我们关注逆需求函数的情况$p=\pi \left(\mathrm{一种}\mathrm{是的}\right)$是隐式定义的。我们将通过牛顿法求解随机非线性方程，并研究为牛顿法选择随机初始值的最优且稳健的方法。稳健的初始值将使我们能够研究价格支持计划将如何影响消费者价格、农产品价格和政府支出以及它们的可变性。希望求解非线性随机方程能够为牛顿法初始值的选择提供一些启示。