First-Order Difference Equations and the Cobweb Model | 一阶差分方程与蛛网模型精讲

2026-08-30 17:54 · 本文部分内容由AI辅助创作

📚 First-Order Difference Equations and the Cobweb Model | 一阶差分方程与蛛网模型精讲

In discrete-time modelling, many real-world quantities change at fixed intervals: prices adjust each market day, populations reproduce each season, or loans grow each month. A first-order difference equation describes how a variable at one time step, yₙ₊₁, is related to its value at the previous step, yₙ. The cobweb model in economics is a classic IB application, combining linear demand and supply functions with price dynamics to explain whether markets converge to equilibrium.

离散时间模型中的许多实际量按固定间隔变化：价格逐日调整、种群逐季繁殖、贷款逐月增长。一阶差分方程描述变量在相邻时间步之间的关系，即后一项 yₙ₊₁ 由前一项 yₙ 决定。蛛网模型是经济学中的经典应用，也是在 IB 数学考试中常见的考点，它将线性需求与供给函数和价格动态结合起来，用于判断市场是否能趋向均衡。

1. What is a First-Order Difference Equation? | 一阶差分方程入门

+ b, where a and b are constants. Given an initial value y₀, the entire sequence is determined by repeated substitution.

一阶差分方程是联系数列相邻两项的递推关系。在 IB 数学中，最重要的类型是线性方程 yₙ₊₁ = a yₙ + b，其中 a、b 为常数。给定初值 y₀，整串数列就可以通过反复代入而唯一确定。

Example: yₙ₊₁ = 2yₙ + 1 with y₀ = 0 gives y₁ = 1, y₂ = 3, y₃ = 7.
例如：yₙ₊₁ = 2yₙ + 1 且 y₀ = 0，则 y₁ = 1、y₂ = 3、y₃ = 7。
The order of the equation is determined by the maximum time lag. Here only one lag appears, so it is first-order.
方程的阶数由最大时间滞后决定。这里只出现一项滞后，因此是一阶方程。