# Multiplier (economics)

Last updated

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.

## Contents

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:

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

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. [ citation needed ]

### Multiplier formula in an open economy

The three most known multiplier formula are as depicted, where:

Multipliers in an open economy
Keynesian multiplierFiscal multiplier driven by an increase in taxFiscal multiplier driven by tax cuts
The multiplier's characteristics
${\displaystyle \Delta G>0\$ ;\Delta T=0}
${\displaystyle \Delta G>0\$ ;\Delta G=\Delta T}
${\displaystyle \Delta T<0\$ ;\Delta G=0}
Formula
${\displaystyle \Delta Y={1 \over s+m}\cdot (\Delta I+\Delta G+\Delta X)}$
${\displaystyle \Delta Y={s \over s+m}\cdot \Delta G+{1 \over s+m}\cdot (\Delta I+\Delta X)}$
${\displaystyle \Delta Y={c \over s+m}\cdot \Delta (-T)+{1 \over s+m}(\Delta I+\Delta G+\Delta X)}$
Multiplier given
${\displaystyle {1 \over s+m}}$
${\displaystyle {s \over s+m}}$
${\displaystyle {c \over s+m}}$

## General method

The general method for calculating short-run multipliers is called comparative statics. That is, comparative statics calculates how much one or more endogenous variables change in the short run, given a 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. [1] A graph showing the impact on some endogenous variable, over time (that is, the multipliers for times t, t+1, t+2, etc.), is called an impulse-response function. [2] The general method for calculating impulse response functions is sometimes called comparative dynamics.

## History

The Tableau économique (Economic Table) of François Quesnay (1758), which laid the foundation of the Physiocrat school of economics is credited as the "first precise formulation" of interdependent systems in economics and the origin of multiplier theory. [3] 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, [4] 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. [5]

## Related Research Articles

Keynesian economics are the various macroeconomic theories and models of how aggregate demand strongly influences economic output and inflation. In the Keynesian view, aggregate demand does not necessarily equal the productive capacity of the economy. It is influenced by a host of factors that sometimes behave erratically and impact production, employment, and inflation.

The IS–LM model, or Hicks–Hansen model, is a two-dimensional macroeconomic model which is used as a pedagogical tool in macroeconomic teaching. The IS–LM model shows the relationship between interest rates and output in the short run in a closed economy. The intersection of the "investment–saving" (IS) and "liquidity preference–money supply" (LM) curves illustrates a "general equilibrium" where supposed simultaneous equilibria occur in both the goods and the money markets. The IS–LM model shows the importance of various demand shocks on output and consequently offers an explanation of changes in national income in the short run when prices are fixed or sticky. Hence, the model can be used as a tool to suggest potential levels for appropriate stabilisation policies. It is also used as a building block for the demand side of the economy in more comprehensive models like the AD–AS model.

In economics, the fiscal multiplier is the ratio of change in national income arising from a change in government spending. More generally, the exogenous spending multiplier is the ratio of change in national income arising from any autonomous change in spending. When this multiplier exceeds one, the enhanced effect on national income may be called the multiplier effect. The mechanism that can give rise to a multiplier effect is that an initial incremental amount of spending can lead to increased income and hence increased consumption spending, increasing income further and hence further increasing consumption, etc., resulting in an overall increase in national income greater than the initial incremental amount of spending. In other words, an initial change in aggregate demand may cause a change in aggregate output that is a multiple of the initial change.

Nicholas Kaldor, Baron Kaldor, born Káldor Miklós, was a Hungarian economist. He developed the "compensation" criteria called Kaldor–Hicks efficiency for welfare comparisons (1939), derived the cobweb model, and argued for certain regularities observable in economic growth, which are called Kaldor's growth laws. Kaldor worked alongside Gunnar Myrdal to develop the key concept Circular Cumulative Causation, a multicausal approach where the core variables and their linkages are delineated.

The General Theory of Employment, Interest and Money is a book by English economist John Maynard Keynes published in February 1936. It caused a profound shift in economic thought, giving macroeconomics a central place in economic theory and contributing much of its terminology – the "Keynesian Revolution". It had equally powerful consequences in economic policy, being interpreted as providing theoretical support for government spending in general, and for budgetary deficits, monetary intervention and counter-cyclical policies in particular. It is pervaded with an air of mistrust for the rationality of free-market decision making.

Endogenous growth theory holds that economic growth is primarily the result of endogenous and not external forces. Endogenous growth theory holds that investment in human capital, innovation, and knowledge are significant contributors to economic growth. The theory also focuses on positive externalities and spillover effects of a knowledge-based economy which will lead to economic development. The endogenous growth theory primarily holds that the long run growth rate of an economy depends on policy measures. For example, subsidies for research and development or education increase the growth rate in some endogenous growth models by increasing the incentive for innovation.

The marginal propensity to save (MPS) is the fraction of an increase in income that is not spent and instead used for saving. It is the slope of the line plotting saving against income. For example, if a household earns one extra dollar, and the marginal propensity to save is 0.35, then of that dollar, the household will spend 65 cents and save 35 cents. Likewise, it is the fractional decrease in saving that results from a decrease in income.

In economics, the marginal propensity to consume (MPC) is a metric that quantifies induced consumption, the concept that the increase in personal consumer spending (consumption) occurs with an increase in disposable income. The proportion of disposable income which individuals spend on consumption is known as propensity to consume. MPC is the proportion of additional income that an individual consumes. For example, if a household earns one extra dollar of disposable income, and the marginal propensity to consume is 0.65, then of that dollar, the household will spend 65 cents and save 35 cents. Obviously, the household cannot spend more than the extra dollar. If the extra money accessed by the individual gives more economic confidence, then the MPC of the individual may well exceed 1, as they may borrow or utilise savings.

The accelerator effect in economics is a positive effect on private fixed investment of the growth of the market economy. Rising GDP implies that businesses in general see rising profits, increased sales and cash flow, and greater use of existing capacity. This usually implies that profit expectations and business confidence rise, encouraging businesses to build more factories and other buildings and to install more machinery. This may lead to further growth of the economy through the stimulation of consumer incomes and purchases, i.e., via the multiplier effect.

In economics, the consumption function describes a relationship between consumption and disposable income. The concept is believed to have been introduced into macroeconomics by John Maynard Keynes in 1936, who used it to develop the notion of a government spending multiplier.

In economics, comparative statics is the comparison of two different economic outcomes, before and after a change in some underlying exogenous parameter.

In monetary economics, the money multiplier is the ratio of the money supply to the monetary base. If the money multiplier is stable, it implies that the central bank can control the money supply by determining the monetary base.

A balanced budget is a budget in which revenues are equal to expenditures. Thus, neither a budget deficit nor a budget surplus exists. More generally, it is a budget that has no budget deficit, but could possibly have a budget surplus. A cyclically balanced budget is a budget that is not necessarily balanced year-to-year but is balanced over the economic cycle, running a surplus in boom years and running a deficit in lean years, with these offsetting over time.

The permanent income hypothesis (PIH) is a model in the field of economics to explain the formation of consumption patterns. It suggests consumption patterns are formed from future expectations and consumption smoothing. The theory was developed by Milton Friedman and published in his A Theory of Consumption Function, published in 1957 and subsequently formalized by Robert Hall in a rational expectations model. Originally applied to consumption and income, the process of future expectations is thought to influence other phenomena. In its simplest form, the hypothesis states changes in permanent income, rather than changes in temporary income, are what drive changes in consumption.

Foundations of Economic Analysis is a book by Paul A. Samuelson published in 1947 by Harvard University Press. It is based on Samuelson's 1941 doctoral dissertation at Harvard University. The book sought to demonstrate a common mathematical structure underlying multiple branches of economics from two basic principles: maximizing behavior of agents and stability of equilibrium as to economic systems. Among other contributions, it advanced the theory of index numbers and generalized welfare economics. It is especially known for definitively stating and formalizing qualitative and quantitative versions of the "comparative statics" method for calculating how a change in any parameter affects an economic system. One of its key insights about comparative statics, called the correspondence principle, states that stability of equilibrium implies testable predictions about how the equilibrium changes when parameters are changed.

Luigi L. Pasinetti was an Italian economist of the post-Keynesian school. Pasinetti was considered the heir of the "Cambridge Keynesians" and a student of Piero Sraffa and Richard Kahn. Along with them, as well as Joan Robinson, he was one of the prominent members on the "Cambridge, UK" side of the Cambridge capital controversy. His contributions to economics include developing the analytical foundations of neo-Ricardian economics, including the theory of value and distribution, as well as work in the line of Kaldorian theory of growth and income distribution. He also developed the theory of structural change and economic growth, structural economic dynamics and uneven sectoral development.

The multiplier–accelerator model is a macroeconomic model which analyzes the business cycle. This model was developed by Paul Samuelson, who credited Alvin Hansen for the inspiration. This model is based on the Keynesian multiplier, which is a consequence of assuming that consumption intentions depend on the level of economic activity, and the accelerator theory of investment, which assumes that investment intentions depend on the pace of growth in economic activity.

The Cambridge capital controversy, sometimes called "the capital controversy" or "the two Cambridges debate", was a dispute between proponents of two differing theoretical and mathematical positions in economics that started in the 1950s and lasted well into the 1960s. The debate concerned the nature and role of capital goods and a critique of the neoclassical vision of aggregate production and distribution. The name arises from the location of the principals involved in the controversy: the debate was largely between economists such as Joan Robinson and Piero Sraffa at the University of Cambridge in England and economists such as Paul Samuelson and Robert Solow at the Massachusetts Institute of Technology, in Cambridge, Massachusetts, United States.

The wage unit is a unit of measurement for monetary quantities introduced by Keynes in his 1936 book The General Theory of Employment, Interest and Money. A value expressed in wage units is equal to its price in money units divided by the wage of a man-hour of labour.

The monetary/fiscal policy debate, otherwise known as the Ando–Modigliani/Friedman–Meiselman debate, was the exchange of viewpoints about the comparative efficiency of monetary policies and fiscal policies that originated with a work co-authored by Milton Friedman and David I. Meiselman and first published in 1963, as part of studies submitted to the Commission on Money and Credit.

## References

1. James Hamilton (1994), Time Series Analysis, Chapter 1, page 2. Princeton University Press.
2. Helmut Lütkepohl (2008), 'Impulse response function'. The New Palgrave Dictionary of Economics, 2nd. ed.
3. The multiplier theory, by Hugo Hegeland, 1954, p. 1
4. The Economic record, by the Economic Society of Australia and New Zealand, 1962, p. 74 Donald Markwell, Keynes and Australia, Reserve Bank of Australia, 2000, pages 34-7. http://www.rba.gov.au/publications/rdp/2000/pdf/rdp2000-04.pdf
5. The origins of the Keynesian revolution, by Robert William Dimand, p. 117