对加权蛛网模型稳定性的探讨

在传统的蛛网模型中引入权值,用以反映生产者对未来市场的预测和决策,并利用动力系统知识研究了需求函数y=f(x)和供应函数x=g(y)交点(即平衡点)的稳定性条件.当用近两周期价格加权平均预测下一周期商品的价格,从而确定生产数量时,即x_(k+1)=g(λy_k+y_(k-1))/(1+λ)),对任意的λ∈(0,+∞),证明了当λ=2时,平衡点的稳定区域最大(αβ3),其中-α,β分别为需求函数和供应函数在平衡点处的斜率;当用近三个周期价格加权平均预测下一周期商品的价格,从而确定生产数量时,即xk+1=g(λy_k+μy_(k-1)+y_(k-1))/(1+λ+μ),证明了当λ=5,μ=4时,平衡点的稳定区域为αβ5.与传统的蛛网模型相比,平衡点的稳定区域扩大了,从而更有利于经济趋于稳定.

The Discussion of Stability on the Weighted Cobweb Model

In this paper,weight is introduced in the traditional cobweb model to reflect the producer's forecasting of future market and decision-making.Moreover,we use the knowledge of dynamical systems to study the stability condition of demand and supply function in the point of intersection(which is called the equilibrium point).When we use weighted average of the prices of the past two cycles to forecast the next cyclical price of the commodity in order to determine the amount of production,that isx_(k+1) = g(λy_k+y_(k-1)/1+λ),to willfullyλ∈(0,+∞),this paper proved that whenλ= 2,the stable region of the equilibrium point is the largest(αβ<3),of which—α,βis the slope of the demand and supply functions in the equilibrium point respectively.when we use weighted average of the prices of the last three cycles to forecast the next cyclical price of the commodity in order to determine the amount of production,that isx_(k+1)= g(λy_k+μy_(k-1)+y_(k-2)/1+λ+μ),this paper proved that whenλ= 5,μ=4,the stable region of the equilibrium point isαβ<5.Comparing with the traditional cobweb model,the stable region of the equilibrium point is expanded,thus it is advantageous for the economy tending to be stable.