This paper extends the classical Samuelson multiplier-accelerator model for national economy. Actually, this new modeling structure removes the basic shortcoming of the original model producing stable business cycles when realistic values of the parameters (multiplier, accelerator) are entered into the system of equations. Under this new approach, we introduce some kind of randomness and memory into the system. We assume that consumption, private investment and governmental expenditure depend upon the national income values of the last n (n 1) years and further assume that multiplier and accelerator factors are stochastic variables. Then stochastic delayed difference equations of higher order are employed to describe the model, while the respective solutions of higher order polynomials for the expectation of national income variables correspond to the typical observed business cycles of real economy. Stability and controllability conditions are investigated while numerical examples provide further insight and better understanding as regards the control actions, system design, and produced business cycles.
Samuelson model Stochastic Difference equations Stability Control State feedback