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

📚 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

First-order difference equations are recurrence relations that connect consecutive values of a sequence. In IB mathematics, the most important case is the linear equation yₙ₊₁ = a yₙ + 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。

2. Solving the Linear Equation | 线性差分方程的迭代求解

The fastest method for a single term is iteration: calculate y₁, y₂, y₃ step by step. However, to analyse long-term behaviour we need the closed-form solution. For yₙ₊₁ = a yₙ + b, the general solution is:

求某一项最快的方法是迭代：逐项算出 y₁

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