For example, suppose variable x changes by 1 unit, which causes another variable y to change by M units. Then the multiplier is M.
Common uses 
Two multipliers are commonly discussed in introductory macroeconomics.
Money multiplier 
The multiplier may vary across countries, and will also vary depending on what measures of money are 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 
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 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:
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:
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:
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:
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.
General method 
The general method for calculating long-run multipliers is called comparative statics. That is, comparative statics calculates how much one or more 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.
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. 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 function. The general method for calculating impulse response functions is sometimes called comparative dynamics.
The tableau économique (Economic Table) of François Quesnay (1759), which lay the foundation of the Physiocrats economic theory, is credited as the "first precise formulation" of interdependent systems in economics and the origin of multiplier theory. 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.
The modern theory of the multiplier was developed in the 1930s, by Kahn, Keynes, Giblin, and others, 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.
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.
See also 
- James Hamilton (1994), Time Series Analysis, Chapter 1, page 2. Princeton University Press.
- Helmut Lütkepohl (2008), 'Impulse response function'. The New Palgrave Dictionary of Economics, 2nd. ed.
- The multiplier theory, by Hugo Hegeland, 1954, p. 1
- The Economic record, by the Economic Society of Australia and New Zealand, 1962, p. 74
- The origins of the Keynesian revolution, by Robert William Dimand, p. 117
- Cassidy, John (10 October 2011). "The Demand Doctor". The New Yorker.