# Multiplier (economics)

In economics, a multiplier is a factor that indicates the extent to which an original economic impulse (independent variable) affects a variable (dependent variable) to be explained. If the rates of change of the independent and dependent variables correspond, the multiplier is equal to one. Mostly, however, the concept explains causal relationships in which the dependent variable changes by a multiple of the independent variable (multiplier is greater than one).

The term multiplier is a technical term and is mainly used in the context of Keynesianism . Together with the accelerator process , the multiplier process or the multiplier effect is an essential mechanism by which there is a self-reinforcement of economic impulses. The multiplier principle or multiplier theory is used in economic policy mainly in the context of the question of stimulating overall economic demand and monetary policy.

Example of a negative multiplier effect (negative competition)

## Multipliers in the context of overall economic demand

The Keynesian theory uses the multiplier concept to explain various relationships in the context of overall economic demand . For example, if government spending is increased by a certain amount, it does not increase total demand by the same amount (then the multiplier would be one), but by an even larger amount.

The reason for this is that the multiplier process takes place in several stages. In the first stage, the increase in government spending (primary impulse) leads to an equal increase in income. With a consumption quota c > 0, part of the increased income is used for additional consumer spending, which also increases the income of other economic agents at the next level. The expansion process continues, albeit with a decreasing effect, since the households each save part of the additional income. The savings are withdrawn from the expenditure-income cycle and lead to so-called “seepage effects” in the expansion process until this comes to a complete standstill in the new equilibrium. Its overall strength and duration are greater (smaller) the smaller (greater) the savings rate.
In total, the total demand is increased by a multiple of the original impulse.

Arithmetic determination of the multiplier effect

The arithmetic determination of the multiplier effect can be seen in the example in the figure on the right.

All components of overall economic demand are used as a multiplier in the context of promoting demand

${\ displaystyle \! \ Y = C + I + G + X-Im}$

in question. In general, however, the state demand G, which can be directly influenced, is in the foreground.

### Government expenditure multiplier

The government expenditure multiplier indicates the multiple by which gross domestic product increases when government expenditure is increased. The basic assumption here is the existence of a closed economy .

The multiplier can be derived from the formula for overall economic demand: using a simple calculation . ${\ displaystyle \! \ Y = C + I + G}$

• Y: aggregate demand, GDP
• C: consumer demand
• I: investments
• G: government spending
• c: marginal consumption rate, which indicates how much of an additional euro of income Y is consumed
• s: marginal savings rate (1 - c ), which accordingly indicates how much of an additional euro of income Y is saved

If one also considers GDP as an indicator of total income, on which consumer spending depends, the consumption function can be shown as follows:

${\ displaystyle \! \ C = C_ {0} + c \ cdot Y}$

Consumer demand consists of an autonomous (income-independent) consumption C 0 and a variable, income-dependent consumption . ${\ displaystyle c \ cdot Y}$

Substituting the consumption function into the first formula gives:

${\ displaystyle \! \ Y = C_ {0} + c \ cdot Y + I + G}$

If you bring cY to the left and then exclude Y , you get:

${\ displaystyle \! \ (1-c) \ cdot Y = C_ {0} + I + G}$

Furthermore, it is assumed that investments I and autonomous consumption expenditure C 0 are initially constant. In this case, the change in equilibrium income depends only on the change in government spending. Solving for Y gives the solution:

${\ displaystyle \ Delta Y = {\ frac {\ Delta G} {1-c}} = {\ frac {\ Delta G} {s}}}$

or as a geometric series :

${\ displaystyle \ Delta Y = \ sum \ limits _ {t = 0} ^ {\ infty} c ^ {t} \ cdot \ Delta G}$.

The multiplier is the reciprocal of the savings rate:

${\ displaystyle M = {\ frac {1} {1-c}} = {\ frac {1} {s}}}$.

This changes the income according to the formula:

${\ displaystyle \ Delta Y = M \ cdot \ Delta G}$.

Because s (or c ) lies in the interval , the multiplier is always greater than or equal to 1 . This induces self-reinforcing, sustainable growth. In addition, it becomes clear that the multiplier is higher the lower the savings rate or the higher the propensity to consume in the respective country. ${\ displaystyle] 0.1 [}$${\ displaystyle {\ frac {1} {s}}}$${\ displaystyle (M \ geq 1)}$

#### Expanded government expenditure multiplier for taxes

An income tax results in a lower multiplier effect and the equation changes to:

${\ displaystyle Y = C_ {0} + c \ cdot Y ^ {v} + I + G}$

In the equation equals "Disposable Income" at a tax rate of . If you replace and dissolve after , the result is: ${\ displaystyle Y ^ {v} = (1-t) * Y}$${\ displaystyle t}$${\ displaystyle Y ^ {v}}$${\ displaystyle Y}$

${\ displaystyle Y = {\ frac {C_ {0}} {(1-c + c \ cdot t)}} + {\ frac {I} {(1-c + c \ cdot t)}} + {\ frac {G} {(1-c + c \ cdot t)}}}$

Assuming that investments and autonomous consumption initially remain constant, the multiplier effect for taxes results:

${\ displaystyle M_ {tax} = {\ frac {1} {1-c + c \ cdot t}}}$

#### Extended government expenditure multiplier for taxes and crowding out (investments not constant)

Assuming that government spending is financed by credit, this additional demand for money leads to an increase in interest rates. The rising interest rates lead to a decrease in investments.

${\ displaystyle z_ {i}}$: Interest rate elasticity of investments

${\ displaystyle m_ {i}}$: Interest rate elasticity of money demand

${\ displaystyle M_ {Tax + CrowdingOut} = {\ frac {1} {1-c + c \ cdot t + {\ frac {z_ {i}} {m_ {i}}}}}}}$

### Investment multiplier

According to the investment multiplier, an exogenous increase in investment demand leads to an increase in demand to the same extent as an expansion of government spending. The derivation can be carried out in the same way as the government expenditure multiplier. Like the government expenditure multiplier, the model is based on the assumption of a closed economy.

The increase in overall economic demand is calculated as follows:

${\ displaystyle \ Delta Y = {\ frac {\ Delta I} {1-c}} = {\ frac {\ Delta I} {s}}}$

or as a geometric series:

${\ displaystyle \ Delta Y = \ sum \ limits _ {t = 0} ^ {\ infty} c ^ {t} \ cdot \ Delta I}$.

It should be noted that the principle of the investment multiplier only works if there are permanent changes in the autonomous investments.

### Export multiplier

In an open economy it has to be taken into account that part of the increasing demand is satisfied by imports from abroad. The value of the multiplier becomes smaller, the greater the import propensity m is.

In the case of an open economy with autonomous (i.e. given independently of income Y ) net investment, autonomous export X, income-dependent consumption

${\ displaystyle C = c \ cdot Y}$

and also income-related import

${\ displaystyle Im = m \ cdot Y}$

if the economic activity of the state is neglected:

${\ displaystyle \ Delta Y = {\ frac {\ Delta X} {1-c + m}} = {\ frac {\ Delta X} {s + m}}}$

respectively as a geometric series

${\ displaystyle \ Delta Y = \ sum \ limits _ {t = 0} ^ {\ infty} c ^ {t} \ cdot \ Delta X}$.

So the multiplier is:

${\ displaystyle {\ frac {1} {1-c + m}} = {\ frac {1} {s + m}}}$.

If economic activity receives an impetus from additional exports, income Y increases the less it “seeps into” savings or the less the additional demand is satisfied by imports Im from abroad, i.e. the smaller s, the marginal savings rate and the import propensity is m . In more comprehensive models, repercussions on other countries are taken into account. The demand benefiting from abroad is also partly satisfied there by imports, which can increase domestic exports.

### Tax multiplier

State intervention can stimulate economic development in different ways. On the one hand, government spending can be increased and, on the other hand, taxes can be reduced, thereby increasing the disposable income of private households and thus increasing income-related consumer demand.

The tax multiplier indicates how much national income changes when a change in direct taxes is made.

In a closed economy with active fiscal policy, a distinction is made between national income and disposable income. The following applies to the disposable income of private households:

${\ displaystyle \! \ Y_ {V} = Y-T _ {\ mathrm {dir}} + Tr}$

With

T dir : direct taxes,
Tr: transfer payments.

The formula for equilibrium national income is:

${\ displaystyle \! \ Y = C \ cdot (Y-T _ {\ mathrm {dir}} + Tr) + I + G}$

Direct taxes T dir are levied in proportion to income:

${\ displaystyle T _ {\ mathrm {dir}} = t \ cdot Y}$.

The following applies to consumer demand C , taking into account the tax aspect:

${\ displaystyle \! \ C = C_ {0} + c \ cdot (Y- (t \ cdot Y) + Tr)}$.

Here it becomes clear that higher taxes reduce the propensity of households to consume.

C 0 = C a : autonomous consumption by private households
G: government demand
I: Investments by entrepreneurs
t: tax rate
c: marginal consumption rate

A change in national income due to the change in direct taxes is expressed in the following relationship:

${\ displaystyle \ Delta Y = - {\ frac {c \ cdot \ Delta T _ {\ mathrm {dir}}} {1-c}} = - {\ frac {c \ cdot \ Delta T _ {\ mathrm {dir} }} {s}}}$

respectively as a geometric series:

${\ displaystyle \ Delta Y = \ sum \ limits _ {t = 0} ^ {\ infty} c ^ {t} \ cdot \ Delta T _ {\ mathrm {dir}}}$.

This simplified multiplier differs from the other multipliers only in the sign and in the primary effect of the expansion. It is negative because income falls when taxes rise and vice versa. The expression says that, for example, the amount of a tax cut is only partially spent (depending on the propensity to consume c ), the other part is saved. Thus the primary effect here has an indirect (dampened) effect on demand, which in turn induces further secondary effects and increases the national income several times over. ${\ displaystyle (c \ cdot \ Delta T _ {\ mathrm {dir}})}$

### Calculation using differential calculus

The multiplier can also be calculated by differentiating the goods market equilibrium condition:${\ displaystyle \! \ Y = C + I = cY + I}$

The total differential is:

${\ displaystyle \ Delta Y = {\ frac {\ partial C} {\ partial Y}} \ cdot \ Delta Y + \ Delta I}$.

After reshaping and inserting : ${\ displaystyle {\ frac {\ partial C} {\ partial Y}} \ equiv c}$

${\ displaystyle \! \ (1-c) \ cdot \ Delta Y = \ Delta I}$

Ultimately, the following is resolved for the unknown endogenous quantity : ${\ displaystyle \ Delta Y}$

${\ displaystyle \ Delta Y = {\ frac {\ Delta I} {1-c}} = {\ frac {\ Delta I} {s}}}$.

You get the already known multiplier . ${\ displaystyle {\ frac {1} {1-c}}}$

### Haavelmo theorem

The Haavelmo theorem states that an increase in government spending increases the equilibrium income in the goods market even if it is financed by the tax increase.

Thus: . ${\ displaystyle {\ frac {\ mathrm {d} Y} {\ mathrm {d} G}} = 1}$

## Money creation multiplier

The money creation multiplier is defined analogously . It indicates how much the money supply increases when the central bank gives a monetary policy impulse. The higher the minimum reserve and the liquidity reserve ratio , the lower it is.

## Critical consideration of the multiplier theory

The Keynesian multiplier model is based on a number of assumptions and assumes a greatly simplified picture of macroeconomic structures. The question of the applicability of this theoretical model in practice depends on the individual case and cannot be answered across the board.

In the following example, one of the points of criticism regarding the mode of operation of the multiplier is to be taken up.

As already stated, the multiplier effect depends, among other things, on the marginal propensity to consume of households c . The greater the propensity to consume, the greater the multiplier effect. For example, a multiplier of and a multiplier of${\ displaystyle c = 0 {,} 8}$${\ displaystyle M = {\ frac {1} {1-0 {,} 8}} = 5}$${\ displaystyle c = 0 {,} 5}$${\ displaystyle M = {\ frac {1} {1-0 {,} 5}} = 2}$

• The multiplier effect lasts 3 income distribution periods (1 period = an average of 2 months),
• For additional primary investment is: .${\ displaystyle I = 100}$

For and we get the series:${\ displaystyle M = 5}$${\ displaystyle c = 0 {,} 8}$${\ displaystyle I + I \ cdot c + I \ cdot c ^ {2} + I \ cdot c ^ {3} = 100 + 100 \ cdot 0 {,} 8 + 100 \ cdot 0 {,} 8 ^ {2 } +100 \ cdot 0 {,} 8 ^ {3} = 295 {,} 20 = 59 \ \% \ {\ text {of}} \ 500}$

For and we get the series:${\ displaystyle M = 2}$${\ displaystyle c = 0 {,} 5}$${\ displaystyle I + I \ cdot c + I \ cdot c ^ {2} + I \ cdot c ^ {3} = 100 + 100 \ cdot 0 {,} 5 + 100 \ cdot 0 {,} 5 ^ {2 } +100 \ cdot 0 {,} 5 ^ {3} = 187 {,} 50 = 94 \ \% \ {\ text {of}} \ 200}$

After 6 months, we get 94% of the "full" multiplier, but with only 59% of the "full" multiplier. So it has to be taken into account that a larger multiplier requires more time to develop its full effect. The problem is that the time lag in reality can differ significantly from that in the model. ${\ displaystyle M = 2}$${\ displaystyle M = 5}$

One of the main points of criticism is certainly the fact that the multiplier model is static and does not capture any dynamics. If one of the parameters, such as the marginal consumption rate, were to change in the short term, the results could be different.

Another negative aspect of the multiplier can be seen in government investments during a boom. If production capacities were fully utilized, additional investments would lead to a rise in the price level and thus increase the national income in nominal terms, but not at all or only slightly in real terms. This would lead to the devaluation of the money.

In addition, it must be noted that the multiplier shows its effect both “upwards” and “downwards”. For example, if government spending falls, private household consumer spending and corporate investment spending would also fall. This would trigger a multiplicative effect and lead to a disproportionate decline in national income. However, this is exactly what is currently being observed in the euro zone as a result of the austerity policy .

Since the exact determination of the duration of a multiplier process is problematic in practice, it cannot be ruled out that several multiplier processes overlap and thus reinforce or compensate each other in their effect. This makes an exact determination of the multiplier effect hardly possible.

The actual size of the multipliers of state financial policy in connection with the “austerity policy” is controversial.

## literature

• Klaus Rose, Karlhans Sauernheimer: Theory of foreign trade. 14th edition, Verlag Franz Vahlen, Munich 2006, ISBN 3-8006-3287-X . 675 pp.
• Peter Bofinger: Fundamentals of Economics. 3rd edition, Pearson Studium, Munich 2010, ISBN 3-8273-7354-9 . 618 pp.
• Nicholas Gregory Mankiw, Mark Peter Taylor; Adolf Wagner, Marco Herrmann (translator): Fundamentals of economics. (Original English title: Principles of Economics ) 4th edition, Schäffer-Poeschel Verlag, Stuttgart 2008, ISBN 3-7910-2787-5 . 989 pp.
• Paul Anthony Samuelson , William Dawbney North House : Economics. The international standard work on macro and microeconomics. 4th edition, mi-Wirtschaftsbuch (Münchner Verlagsgruppe), Munich 1998. ISBN 3-86880-089-1 . 1104 pp.

## Individual evidence

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2. A. Farner: Introduction to the basics of economics. 1992, p. 269 f.
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7. Klaus Rose, Karlhans Sauernheimer: Theory of foreign trade. 14th edition, Verlag Franz Vahlen, Munich 2006, ISBN 3-8006-3287-X , pp. 129-131 (675 pp.).
8. Renate Neubäumler, Brigitte Hewel: Fundamentals of economic theory and economic policy. 2nd edition, Gabler Verlag (Springer Science + Business Media), Wiesbaden 1998, ISBN 3-409-23474-8 , pp. 279-281 (667 pp.).
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12. a b c Dieter Dahl: Economics. 7th edition, Gabler Verlag (Springer Science + Business Media), Wiesbaden 1993, ISBN 3-409-60217-8 , pp. 465-467 (669 pp.).
13. Ralf Streck: Austerity increases the deficit instead of decreasing it , April 24, 2013.
14. Renate Neubäumler, Brigitte Hewel: Fundamentals of economic theory and economic policy. 2nd edition, Gabler Verlag (Springer Science + Business Media), Wiesbaden 1998, ISBN 3-409-23474-8 , p. 282 (667 p.).
15. Manfred Schäfers faz.net February 21, 2013: "Dissent about the austerity policy"