Cobweb Model as an Application of Difference Equation

Let the demand and supply function as

\(X_{dt}\; =a\; +\; bp_{t}\; \) and

\(X_{st}\; =A\; +\; Bp_{t-1}\; \)

According to the above equations the demand of the commodity depends on the current price of the commodity and the supply depends on the prices of previous period denoted by \(p_{t\; }\; and\; p_{t-1}\; \) respectively.

As we know at equilibrium,

\(X_{dt}\; =\; X_{st}\)

or \(a\; +\; bp_{t}\; =\; A\; +\; Bp_{t-1}\)

or \(bp_{t}\; -\; Bp_{t-1}\; =\; A\; -\; a\) ——————- (A)

This (A) is the basic equation of the difference equation.

Now the complete solution comprises of two solutions :

  1. Particular solution
  2. Complementary solution

For particular solution,

Let \(p_{t}\; =\; p_{t-1}\; =\; \overline{p}\)

The above equation will take the form of the equation

\(b\overline{p} -\; B\overline{p} =\; A\; -\; a\),

\(\overline{p}\left( b\; -\; B \right)\; =\; A\; -\; a\),

\(\overline{p}=\; \; \frac{A\; -\; a}{b\; -\; B}\),

The above solution is the particular solution.

For complementary solution,

Let \(p_{t}\; =\; \alpha \beta ^{t}\; and\; p_{t-1}\; =\; \alpha \beta ^{t-1}\; and\; R.H.\mbox{S}\; =\; 0\) in equation (A)

\(b\alpha \beta ^{t}\; -\; B\alpha \beta ^{t-1}\; =\; 0\),

\(\alpha \beta ^{t}\left\{ b\; -\; B\beta ^{-1} \right\}\; =\; 0\; \),

As \(\alpha \beta ^{t}\; \neq \; 0\),

So \(b\; -\; \frac{B}{\beta }\; =\; 0\),

\(b\; =\; \frac{B}{\beta }\),

\(\beta \; =\; \frac{B}{b}\),

So the complementary solution can be written as \(p_{t}\; =\; \alpha \; \left( \frac{B}{b} \right)^{t}\)

The general solution can be written as 

\(p_{t}\; =\frac{A\; -\; a}{b\; -\; B} + \alpha \; \left( \frac{B}{b} \right)^{t}\)

Now taking initial condition i.e., \(t\; =\; 0\) , so the above equation can be rewritten as 

\(p_{0}\; =\; \frac{A\; -\; a}{b\; -\; B}\; +\; \alpha \),

\(p_{0}\; -\; \frac{A\; -\; a}{b\; -\; B}\; \; =\; \alpha \)

The complete equation can be written as,

\(p_{t}\; =\frac{A\; -\; a}{b\; -\; B} + \left( p_{0}\; -\; \frac{A\; -\; a}{b\; -\; B} \right) \; \left( \frac{B}{b} \right)^{t}\),

Where “t” shows the time path of the equation.

 

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