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In macroeconomics, a multiplier is a factor of proportionality that measures how much an endogenous variable changes in response to a change in some exogenous variable.

For example, suppose variable x changes by k units, which causes another variable y to change by M × k units. Then the multiplier is M.

Common uses

Two multipliers are commonly discussed in introductory macroeconomics.

Commercial banks create money, especially under the fractional-reserve banking system used throughout the world. In this system, money is created whenever a bank gives out a new loan. This is because the loan, when drawn on and spent, mostly finishes up as a deposit back in the banking system and is counted as part of money supply. After putting aside a part of these deposits as mandated bank reserves, the balance is available for the making of further loans by the bank. This process continues multiple times, and is called the multiplier effect.

The multiplier may vary across countries, and will also vary depending on what measures of money are being considered. For example, consider M2 as a measure of the U.S. money supply, and M0 as a measure of the U.S. monetary base. If a $1 increase in M0 by the Federal Reserve causes M2 to increase by $10, then the money multiplier is 10.

Fiscal multipliers

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 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:

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

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

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:

${\displaystyle (1-<!--\; -\; -->c-<!--\; -\; -->bc+bc)Y_{}}$Zdroj:https://en.wikipedia.org?pojem=Multiplier_(economics)
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