A mean–variance acreage model

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.

KEYWORDS:

• Mean–variance analysis
• rational expectations
• two-period model
• fixed point problem
• Newton's method

AMS CLASSIFICATIONS:

• Primary: 65H05
• 65D32
• 65C05
• 47H10
• 91B84
• 91G60
• Secondary: 91G10
• 90-08

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 Since market demand is often defined as the sum of individual demands, a lot of market demand can be defined this way. The isoelastic function, $D\left(p\right)=k{p}^r$, is also called constant price elasticity demand function in microeconomics. For most goods the elasticity r (the responsiveness of quantity demanded to price) is negative. Elasticity can be described as elastic (or very responsive r<−1), unit elastic (r = −1), or inelastic (not very responsive, $-1<r<0$).

2 By Central Limit Theorem, Monte Carlo estimator converges with rate $\frac{\sigma }{\sqrt{n}}$. For Gauss–Hermite quadrature[51, p.890], the error term is (1) $E=\frac{n!\sqrt{\pi }}{2^n\left(2n\right)!}{f}^{\left(2n\right)}\left(\eta \right).$(1)