Multiplier (economics)

Multipliers can be calculated to analyze the effects of fiscal policy, or other exogenous changes in spending, on aggregate output.

For example, if an increase in German government spending by €100, with no change in tax rates, causes German GDP to increase by €150, then the spending multiplier is 1.5. Other types of fiscal multipliers can also be calculated, like multipliers that describe the effects of changing taxes (such as lump-sum taxes or proportional taxes).

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:

${\displaystyle C_{t}=C_{0}+cY_{t-1}}$

${\displaystyle 1/(1-c(1-t)+m)}$

so present consumption is a function of past income (with c as the marginal propensity to consume). Here, t is the tax rate and m is the ratio of imports to GDP. Investment, in turn, is assumed to be composed of three parts:

${\displaystyle I_{t}=I_{0}+I(r)+b(C_{t}-C_{t-1})}$

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 b > 0. As we are concentrating on the income-expenditure side, let us assume I(r) = 0 (or alternatively, constant interest), so that:

${\displaystyle I_{t}=I_{0}+b(C_{t}-C_{t-1})}$

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

${\displaystyle Ytd=C_{t}+I_{t}=C_{0}+I_{0}+cY_{t-1}+b(C_{t}-C_{t-1})}$

assuming goods market equilibrium (so ${\displaystyle Y_{t}=Ytd}$), then in equilibrium:

${\displaystyle Y_{t}=C_{0}+I_{0}+cY_{t-1}+b(C_{t}-C_{t-1})}$

But we know the values of ${\displaystyle C_{t}}$ and ${\displaystyle C_{t-1}}$ are merely ${\displaystyle C_{t}=C_{0}+cY_{t-1}}$ and ${\displaystyle C_{t-1}=C_{0}+cY_{t-2}}$ respectively, then substituting these in:

${\displaystyle Y_{t}=C_{0}+I_{0}+cY_{t-1}+b(C_{0}+cY_{t-1}-C_{0}-cY_{t-2})}$

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

${\displaystyle Y_{t}-(1+b)cY_{t-1}+bcY_{t-2}=(C_{0}+I_{0})}$

The solution to this system then becomes elementary. The equilibrium level of Y (call it ${\displaystyle Y_{p}}$, the particular solution) is easily solved by letting ${\displaystyle Y_{t}=Y_{t-1}=Y_{t-2}=Y_{p}}$, or:

${\displaystyle (1-c-bc+bc)Y_{p}=(C_{0}+I_{0})}$

so:

${\displaystyle Y_{p}=(C_{0}+I_{0})/(1-c)}$

The complementary function, ${\displaystyle Y_{c}}$ is also easy to determine. Namely, we know that it will have the form ${\displaystyle Y_{c}=A_{1}r_{1}t+A_{2}r_{2}t}$ where ${\displaystyle A_{1}}$ and ${\displaystyle A_{2}}$ are arbitrary constants to be defined and where ${\displaystyle r_{1}}$ and ${\displaystyle r_{2}}$ are the two eigenvalues (characteristic roots) of the following characteristic equation:

${\displaystyle r^{2}-(1+b)cr+bc=0}$

Thus, the entire solution is written as ${\displaystyle Y=Y_{c}+Y_{p}}$

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.