Multiplier-accelerator model

The multiplier–accelerator model (also known as Hansen–Samuelson model) is a macroeconomic model which analyzes the business cycle. This model was developed by Paul Samuelson, who credited Alvin Hansen for the inspiration. This model is based on the Keynesian multiplier and the accelerator theory of investment.

Model

See also: Multiplier (economics) § Keynesian and Hansen–Samuelson multipliers

The multiplier–accelerator model can be stated as follows:

$Y_{t}=g_{t}+C_{t}+I_{t}$

$C_{t}=\alpha Y_{t-1}$

$I_{t}=\beta \mid C_{t}-C_{t-1}\mid$

$g_{t}=1$

where $Y_{t}$ is national income, $g_{t}$ is government expenditure, $C_{t}$ is consumption expenditure, $I_{t}$ is induced private investment, and the subscript $t$ is time. Here we can rearrange these equations and rewrite them as a second order linear difference equation:

$Y_{t}=1+\alpha (1+\beta )Y_{t-1}-\alpha \beta Y_{t-2}$

Samuelson demonstrated that there are several kinds of solution path for national income to be derived from this second order linear difference equation. This solution path changes its form, depending on the values of the roots of the equation or the relationships between the parameter $\alpha$ and $\beta$.

Further reading

• Fellner, W. J. (1956). Trends and Cycles in Economic Activity. New York: Henry Holt. pp. 308–338.