First-Order Linear Difference Equation: Cobweb Model Analysis

Published on June 4, 2026

Word-Problem

Solve a first-order linear difference equation for an agricultural market's price path, analyzing convergence and sketching its behavior.

Question

please answer

Answer

To solve this problem, we will follow the steps outlined in the prompt.

Given Information:

Demand function:
Supply function:
Market equilibrium condition:

a. Formulate the first-order linear difference equation for the price in equilibrium

To find the difference equation, we set the demand equal to the supply:

Now, we rearrange the equation to solve for

in terms of

Divide both sides by

The first-order linear difference equation is:

b. Find the price time path that consists of complementary function and particular solution

The general solution is

1. Find the particular solution ():
The particular solution (steady state) occurs when

So,

2. Find the complementary function ():
The homogeneous part of the equation is

.
The characteristic equation is

, which gives

.
The complementary function is of the form

3. Combine them:

c. Solve for the time path of price given an initial price

We use the initial condition

to find the constant

Substituting

back into the general solution:

d. Determine if the time path converges or diverges and sketch the behavior

Convergence/Divergence:
The behavior of a first-order difference equation

depends on the absolute value of the coefficient

.
Here,

.
Since

, the time path converges to the steady state

General Behavior:
Because

is negative (

), the price will exhibit damped oscillations. This means the price will alternate above and below the equilibrium value (50), with the magnitude of the oscillations decreasing over time as it approaches 50.

(above 50)
(below 50)
(above 50)
(below 50)

Sketch Description:
The graph starts at

, drops to

, rises to

, drops to

, and continues to “zig-zag” with smaller and smaller steps, eventually flattening out at the horizontal line

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