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.