The accelerator model of investment leads to a difference equation of the form

Y_t = C₀ + C₁Y_t-1 + C₂Y_t-2
 

where usually C₁ is positive and C₂ is negative.

With the coefficients of the difference equation and the first two values of Y; i.e.,Y₁ and Y₂ one can determine Y_t for all subsequent values of t. Enter the values of the coefficients in the table below and the initial values of Y in the table which follows it. When the "Iterate" button is clicked upon the values of Y for t=3 to t=10 are computed.

From simulations, such as the above, some of the dynamics of an economy described by the difference equation specified. But often simulations provide only an imperfect understanding of the nature of the economy. The equilibrium can be found only approximately using simulation. An analytic solution is a better approach. For example, the equilibrium level can be determined precisely through analysis.

The equilibrium level Y is given by:

Y = C₀/(1 - C₁ - C₂).
 

Thus the homogenous difference equation for the deviations from equilibrium (y_t = Y_t - Y) is:

y_t = C₁y_t-1 + C₂y_t-2.
 

For the previously specified simulation the analytical solution of the homogeneous difference equation for the deviations from the equilibrium can be found.

The solution can be represented as being equal to the equilibrium solution plus terms that represent the deviations from the equilibrium. In the case of negative or complex roots the deviations from the equilibrium are cyclical. The cycle period can be found by dividing the angle of the roots into 360 degrees.