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| : ''Not to be confused with the "Multiplier effect" of a [[Transformer]].''
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| In economics, a '''multiplier''' is a factor of proportionality that measures how much an [[Endogenous (economics)|endogenous]] variable changes in response to a change in some [[exogenous]] variable.
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| For example, suppose variable ''x'' changes by 1 unit, which causes another variable ''y'' to change by ''M'' units. Then the multiplier is ''M''.
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| ==Common uses==
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| Two multipliers are commonly discussed in introductory [[macroeconomics]].
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| ===Money multiplier===
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| {{Main|Money multiplier}} {{See also|Fractional reserve banking}}
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| In monetary microeconomics and banking, the money multiplier'' measures how much the [[money supply]] increases in response to a change in the [[monetary base]].
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| The multiplier may vary across countries, and will also vary depending on what measures of money are considered. For example, consider [[Money supply#Empirical measures|M2]] as a measure of the U.S. money supply, and [[Money supply#Empirical measures|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.
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| ===Fiscal multipliers===
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| {{Main|Fiscal multiplier}}
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| Multipliers can be calculated to analyze the effects of [[fiscal policy]], or other exogenous changes in spending, on [[GDP|aggregate output]].
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| For example, if an increase in German government spending by €100, with no change in taxes, 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 tax]]es or [[proportional tax]]es).
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| ===Keynesian and Hansen-Samuelson multipliers===
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| [[Keynesian economics|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-LM|IS curve]] shifts left or right in response to an exogenous change in spending.)
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| 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:
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| :<math>C_{t} = C_{0} + cY_{t-1}</math>
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| :<math>1/(1-c(1-t)+m)</math>
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| 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:
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| :<math>I_{t} = I_{0} + I(r) + b (C_{t} - C_{t-1})</math>
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| 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 "[[Accelerator effect|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:
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| :<math>I_{t} = I_{0} + b (C_{t} - C_{t-1})</math>
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| Now, assuming away government and foreign sector, aggregate demand at time t is:
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| :<math>Ytd = C_{t} + I_{t} = C_{0} + I_{0} + cY_{t-1} + b (C_{t} - C_{t-1})</math>
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| assuming goods market equilibrium (so <math>Y_{t} = Ytd</math>), then in equilibrium:
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| :<math>Y_{t} = C_{0} + I_{0} + cY_{t-1} + b (C_{t} - C_{t-1})</math>
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| But we know the values of <math>C_{t}</math> and <math>C_{t-1}</math> are merely <math>C_{t} = C_{0} + cY_{t-1}</math> and <math>C_{t-1} = C_{0} + cY_{t-2}</math> respectively, then substituting these in:
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| :<math>Y_{t} = C_{0} + I_{0} + cY_{t-1} + b (C_{0} + cY_{t-1} - C_{0} - cY_{t-2})</math>
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| or, rearranging and rewriting as a second order linear [[difference equation]]:
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| :<math>Y_{t} - (1 + b )cY_{t-1} + b cY_{t-2} = (C_{0} + I_{0})</math>
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| The solution to this system then becomes elementary. The equilibrium level of Y (call it <math>Y_{p}</math>, the particular solution) is easily solved by letting <math>Y_{t} = Y_{t-1} = Y_{t-2} = Y_{p}</math>, or:
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| :<math>(1 - c - b c + b c)Y_{p} = (C_{0} + I_{0})</math>
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| so:
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| :<math>Y_{p} = (C_{0} + I_{0})/(1-c)</math>
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| The complementary function, <math>Y_{c}</math> is also easy to determine. Namely, we know that it will have the form <math>Y_{c} = A_{1}r_{1}t + A_{2}r_{2}t</math> where <math>A_{1}</math> and <math>A_{2}</math> are arbitrary constants to be defined and where <math>r_{1}</math> and <math>r_{2}</math> are the two [[Eigenvalues and eigenvectors|eigenvalues]] (characteristic roots) of the following characteristic equation:
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| :<math>r^{2} - (1+b )cr + b c = 0</math>
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| Thus, the entire solution is written as <math>Y = Y_{c} + Y_{p}</math>
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| 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. {{citation needed|date=June 2012}}
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| ==General method==
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| The general method for calculating long-run multipliers is called [[comparative statics]]. That is, comparative statics calculates how much one or more [[Endogenous (economics)|endogenous]] variables change in the long run, given a permanent change in one or more exogenous variables. The comparative statics method is an application of the [[Implicit Function Theorem]].
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| Dynamic multipliers can also be calculated. That is, one can ask how a change in some exogenous variable in year ''t'' affects endogenous variables in year ''t'', in year ''t+1'', in year ''t+2'', and so forth.<ref>James Hamilton (1994), ''Time Series Analysis'', Chapter 1, page 2. Princeton University Press.</ref> A graph showing the impact on some endogenous variable, over time (that is, the multipliers for times ''t'', ''t+1'', ''t+2'', etcetera), is called an [[Impulse response|impulse-response function.]]<ref>Helmut Lütkepohl (2008), 'Impulse response function'. ''The New Palgrave Dictionary of Economics'', 2nd. ed.</ref> The general method for calculating impulse response functions is sometimes called [[comparative dynamics]].
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| == History ==
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| [[File:Quesnay Tableau.jpg|thumb|240px|Illustration of the original visualisation of the [[tableau économique]] by [[François Quesnay|Quesnay]], 1758.]]
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| The [[Tableau économique]] (Economic Table) of [[François Quesnay]] (1758), which laid the foundation of the [[Physiocrats|Physiocrat]] school of economics is credited as the "first precise formulation" of interdependent systems in economics and the origin of multiplier theory.<ref>[http://books.google.com/books?id=YAIeAAAAIAAJ The multiplier theory], by Hugo Hegeland, 1954, [http://books.google.com/books?id=YAIeAAAAIAAJ&q=%2B%22tableau+economique%22&pgis=1#search_anchor p. 1]</ref> In the tableau économique, one sees variables in one period (time ''t'') feeding into variables in the next period (time ''t+1''), and a constant rate of flow yields geometric series, which computes a multiplier.
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| The modern theory of the multiplier was developed in the 1930s, by [[Richard Kahn, Baron Kahn|Kahn]], [[Keynes]], [[Lyndhurst Giblin|Giblin]], and others,<ref>The Economic record, by the Economic Society of Australia and New Zealand, 1962, [http://books.google.com/books?id=8aAKAAAAIAAJ&q=%2B%22tableau+economique%22+multiplier&dq=%2B%22tableau+economique%22+multiplier&p=74&pgis=1 p. 74]</ref> following earlier work in the 1890s by the Australian economist Alfred De Lissa, the Danish economist Julius Wulff, and the German-American economist [[N. A. J. L. Johannsen]].<ref>[http://books.google.com/books?id=cYirAAAAIAAJ&pg=PA117 The origins of the Keynesian revolution], by Robert William Dimand, [http://books.google.com/books?id=cYirAAAAIAAJ&pg=PA117 p. 117]</ref>
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| ==Critiques==
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| Economist [[Robert Barro]] believes that the [[Keynesian multiplier]] is close to zero. For every dollar the government ''borrows'' and spends, spending elsewhere in the economy falls by almost the same amount.<ref>{{cite news|url=http://www.newyorker.com/reporting/2011/10/10/111010fa_fact_cassidy|last=Cassidy|first=John|title=The Demand Doctor|accessdate=9 October 2011|newspaper=The New Yorker|date=10 October 2011}}</ref>
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| ==See also==
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| *[[Complex multiplier]]
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| *[[Multiplier uncertainty]]
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| ==References==
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| <references />
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| [[Category:Macroeconomics]]
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| [[nl:Multiplier (economie)]]
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