Themultiplier–accelerator model (also known asHansen–Samuelson model) is amacroeconomic model which analyzes thebusiness cycle.^[1] This model was developed byPaul Samuelson, who creditedAlvin Hansen for the inspiration.^[1][2][3] This model is based on the Keynesianmultiplier, which is a consequence of assuming that consumption intentions depend on the level of economic activity, and theaccelerator theory of investment, which assumes that investment intentions depend on the pace of growth in economic activity.

The multiplier–accelerator model can be stated for a closed economy as follows:^[3] First, the market-clearing level of economic activity is defined as that at which production exactly matches the total of government spending intentions, households' consumption intentions and firms' investing intentions.

${\displaystyle Y_t=g_t+C_t+I_t}$;

then an equation to express the idea that households' consumption intentions depend upon some measure of economic activity, possibly with a lag:

${\displaystyle C_t=\alpha Y_{t-1}}$;

then an equation that makes firms' investment intentions react to the pace of change of economic activity:

${\displaystyle I_t=\beta [C_t-C_{t-1}]}$;

and finally a statement that government spending intentions are not influenced by any of the other variables in the model. For example, the level of government spending could be used as the unit of account:

${\displaystyle g_t=1}$

where${\displaystyle Y_t}$ is national income,${\displaystyle g_t}$ is government expenditure,${\displaystyle C_t}$ is consumption expenditure,${\displaystyle I_t}$ is induced private investment, and the subscript${\displaystyle t}$ is time. Here we can rearrange these equations and rewrite them as a second-order lineardifference equation:^[3][4][5]

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

Samuelson demonstrated that there are several kinds of solution path for national income to be derived from this second order linear difference equation.^[3][4] This solution path changes its form, depending on the values of the roots of the equation or the relationships between the parameter${\displaystyle \alpha }$ and${\displaystyle \beta }$.^[3][4]

Jay Wright Forrester argues^[6] that the Accelerator-Multiplier Theory cannot create the assumed business cycle but instead is a major contributor to the economiclong wave.

• Bratt, E. C. (1961)."Multiplier-Accelerator Models".Business Cycles and Forecasting (Fifth ed.). Homewood: Irwin. pp. 188–211.
• Estey, J. A. (1956)."The Multiplier-Accelerator Interaction".Business Cycles: Their Nature, Cause, and Control (Third ed.). Englewood Cliffs: Prentice-Hall. pp. 275–287.
• Fellner, W. J. (1956).Trends and Cycles in Economic Activity. New York: Henry Holt. pp. 308–338.

• Business cycle theories
• Keynesian economics

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