**Keynesian and Hansen-Samuelson Multipliers**

Keynesian economists often calculate multipliers that measure the effect on aggregate demand only. (To be precise, the usual *Keynesian multiplier* formulas measure how much the IS curve shifts left or right in response to an exogenous change in spending.)

American Economist Paul Samuelson credited Alvin Hansen for the inspiration behind his seminal 1939 contribution. The original Samuelson multiplier-accelerator model (or, as he belatedly baptised it, the "Hansen-Samuelson" model) relies on a multiplier mechanism that is based on a simple Keynesian consumption function with a Robertsonian lag:

so present consumption is a function of past income (with c as the marginal propensity to consume). Investment, in turn, is assumed to be composed of three parts:

The first part is autonomous investment, the second is investment induced by interest rates and the final part is investment induced by changes in consumption demand (the "acceleration" principle). It is assumed that 0 < b . As we are concentrating on the income-expenditure side, let us assume I(r) = 0 (or alternatively, constant interest), so that:

Now, assuming away government and foreign sector, aggregate demand at time t is:

assuming goods market equilibrium (so ), then in equilibrium:

But we know the values of and are merely and respectively, then substituting these in:

or, rearranging and rewriting as a second order linear difference equation:

The solution to this system then becomes elementary. The equilibrium level of Y (call it, the particular solution) is easily solved by letting, or:

so:

The complementary function, is also easy to determine. Namely, we know that it will have the form where and are arbitrary constants to be defined and where and are the two eigenvalues (characteristic roots) of the following characteristic equation:

Thus, the entire solution is written as

Opponents of Keynesianism have sometimes argued that Keynesian multiplier calculations are misleading; for example, according to the theory of Ricardian equivalence, it is impossible to calculate the effect of deficit-financed government spending on demand without specifying how people expect the deficit to be paid off in the future.

Read more about this topic: Multiplier (economics), Common Uses