# Multiplier accelerator

Proposed by English economist Roy Harrod (1900-1978) and American economist Paul Samuelson (1915- ) as an extension of the work of English economists John Maynard Keynes (1883-1946) and Richard Kahn (1905-1989), multiplier-accelerator is a model analyzing economic fluctuations through the effects of the accelerator and multiplier models.

An increase in government expenditure may lead to an increase in consumer incomes which (through the multiplier effect) leads to an increase in output which in turn (through the accelerator process) raises investment. This process tends to work on a cyclical basis.

Also see: Kondratieff cycles, sunspot theory, product life-cycle theory, acceleration principle, fine-tuning, political business cycle, trade cycle

Source:
R F Harrod, The Trade Cycle (Oxford, 1936);
P Samuelson, ‘Interactions Between the Multiplier Analysis and Principles of Acceleration’, Review of Economic Statistics 21, 2 (May, 1939), 75-78

The multiplier–accelerator model (also known as Hansen–Samuelson model) is a macroeconomic model which analyzes the business cycle.[1] This model was developed by Paul Samuelson, who credited Alvin Hansen for the inspiration.[1][2][3] This model is based on the Keynesian multiplier, which is a consequence of assuming that consumption intentions depend on the level of economic activity, and the accelerator theory of investment, which assumes that investment intentions depend on the pace of growth in economic activity.

## Model

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 linear difference 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 }