An Expanded Multiplier-Accelerator Model

This paper revisits the standard multiplier-accelerator model, as advanced by Samuelson. While borrowing on the main assumptions of the multiplier-accelerator, we check the validity of Keynesian theory. Using higher-order difference equations and advanced-level mathematical techniques we solve the tax-augmented multiplier-accelerator model, as well as the open economy one. We find that the values of equilibrium national income are identical to the simple national-income model in the absence of the accelerator. We solve the simple multiplier-accelerator model both in present terms and with prolonged consumption. We solve for equilibrium consumption, tax, and imports which are unaffected by the accelerator. All results conform to Keynesian theory where investment, government spending and exports have a favorable multiplying effect on national income through their respective multipliers. The accelerator coefficient affects neither those multipliers, nor the income and the non-income tax multipliers. Expanding the multiplier-accelerator by the volume of foreign trade, taxation or both does not change the values of Keynesian variables. Adding an accelerator leaves optimal values unaffected but, more importantly, reinforces Keynesian theory.