# Understanding the Use of Multiplier in Macroeconomics

#### 1- Concept of Multiplier

The concept of the multiplier, which plays an important role in the modern theory of income and employment in economics, was first propounded by R.H. Kahn, the eminent economist of Cambridge University early in the 1930s. The concept of Kan's multiplier is known as the employment multiplier but Keynes used this concept of the multiplier as an investment multiplier in economics.

According to Keynes' perception, the concept of multiplier relates to a change in income as a result of a change in investment. An economic fact is that when investment increases, the income also increases many more times than it increases in investment. Therefore, the number of times income increases as a result of an increase in investment is called the multiplier. In other words, the multiplier is the ratio of change in income due to a change in investment.
In the word of Samuelson, "The multiplier is the number by which the change in investment must be multiplied to determine the resulting change in total income."
Likewise, Kurihara defines, "Multiplier is the ratio of a change in income to the change in investment."
The above definition also justifies that the multiplier is the ratio of a change in income to the change in investment and it is expressed symbolically as follows.

Multiplier K=\frac{ΔY}{ΔI}
Where;
K = Multiplier,
ΔY = Change in income,
ΔI = Change in investment,

The concept of multiplier has also been explained with the help of the following diagram.

In the above diagram, the initial equilibrium point is E where its corresponding income level is equal to OY. When there is an increase in investment expenditure by ΔI, the aggregate demand also increases. As a result, the AD curve shifts upward where it intersects the AS curve at point E1. This is the new point of equilibrium that determines the OY1 level of income where income increases by ΔY, as a result of an increase in investment expenditure by ΔI. The number of times the income increase resulting in an increase in investment is the investment multiplier. So the multiplier is K=ΔY/ΔI.

#### 2- How Multiplier Functions

As mentioned in the above description, multiplier shows the number of times there is an increase in income as a result of an increase in investment. Hence, an increase in investment expenditure brings an increase in the level of income multiple times. For instance, let us suppose that there is an increase in autonomous investment expenditure by $10 million. This makes the first round of income generation and thereafter with the consumption expenditure, other rounds of income generation go on recurring itself as shown in the table.  Rounds Consumption Expenditure Income Generation First Round ---------------$10 Second Round $8 Million$8 Third Round $6.4 Million$6.4 Fourth Round $5.11 Million$5.12
If marginal propensity to consume is given as equal to 0.8, the additional consumption expenditure △Y×MPC will be $10×0.8=$8 million. The consumption expenditure further generates income as equal to consumption expenditure as shown in the second round. It is because the expenditure of an individual is the income of the other one. The additional consumption expenditure in the third round is equal to \$8×0.8=6.4 and the income again generates as equal to consumption expenditure, as shown in the third round. This is how the additional consumption expenditure, as well as income generation, goes on various rounds until it tends to zero. This the way the multiplier process functions.

#### 3- Algebraic Derivation of Investment Multiplier

The investment multiplier can be derived algebraically as follows.
The equilibrium condition of a two-sector economy can be written as;
Y=C+I.........(i)
Let us suppose that there is a change in investment by △I and it results in a change in income by △Y which brings a change in consumption expenditure by △C as well. Now, the new equilibrium identity can be written as follows.

Y+△Y=C+△C+I+△I.............(ii)
Subtracting equation (i) from (ii), we get;
△Y=△C+△I.............(iii)
The normal consumption function is C=a+bY.......(iv) but when income level changes as a result of a change in investment, the consumption function also gets changed as follows.

C+△C=a+bY+b△Y.............(v)
By subtracting equation (iv) from (v), we get;
△C=b△Y.............(vi)
By substituting equation (iv) for △C in the equation (iii), we get;
△Y=b△Y+△I
Or,\△Y-b△Y=△I
Or,\△Y(1-b)=△I
Or,\frac{△Y}{△I}\=\frac{1}{(1-b)}
Or,\frac{△Y}{△I}\=K=\frac{1}{(1-b)}
Thus, the term 1/(1-b) represents the investment multiplier. It tells that due to an increase in investment by some amount the income level gets increased by some more times than an increase in investment.

As mentioned in the above description, the term 1/(1-b) shows the value of the investment multiplier K. Let us recall that b=MPC and 1-MPC=MPS. Therefore multiplier K can also be expressed as follows.
K=\frac{△Y}{△I}
Or,\K=\frac{1}{(1-b)}
Or,\K=\frac{1}{(1-MPC)}
Or,\K=\frac{1}{(MPS)}

 AlternativelyThe initial equilibrium identity of a simple economy before a change in investment is expressed as follows.Y_1=C+I.........(i)Substituting consumption function a+bY_1 for C in equation (i) we get;Y_1=a+bY_1+IOr,\Y_1-bY_1=a+IOr,\Y_1(1-b)=a+IOr,\Y_1=\frac{1}{(1-b)}\(a+I)..........(ii)When there is an increase in investment by △I, the equilibrium identity gets changed as follows;Y_2=C+I+△I..........(iii)Substituting new consumption function a+bY_2 for C in equation (iii) we get;Y_2=a+bY_2+I+△IOr,\Y_2-bY_2=a+I+△IOr,\Y_2(1-b)=a+I+△IOr,\Y_2=\frac{1}{(1-b)}\(a+I+△I)........(iv)Now, subtracting the equation (ii) from equation (iv), we get;Y_2-Y_1=\frac{1}{(1-b)}\(a+I+△I)-\frac{1}{(1-b)}\(a+I)Or,\△Y=\frac{1}{(1-b)}\△IOr,\frac{△Y}{△I}=K=\frac{1}{(1-b)}Thus, the investment multiplier K=1/(1-b)This shows that the increase in income level by △Y is 1/(1-b) times more than the increase in investment.

#### 4- Multiplier in a Three-sector Economy Model

The three-sector economy model consists of household, business, and government sectors. The inclusion of government affects the income level depending on the multiplier effect of its fiscal operations. Under this model, we will analyze the government expenditure multiplier (G-Multiplier) and tax multiplier (T-Multiplier).

#### 4.1- Government Expenditure Multiplier (G-Multiplier)

To derive a government expenditure multiplier, let us assume that;
1- The government makes expenditures on goods and services.
2- There is no provision for transfer payment.
3- C, I & G are constant and,
4- The consumption function is given.

To derive the G-multiplier, let us recall the equation of equilibrium income determination in a three-sector economy.
Y=\frac{1}{(1-b)}\(a-bT+I+G)...........(i)

Now, let us suppose that government expenditure increases by △G and all other factors are given. This △G causes an increase in aggregate demand and consequently, there is a rise in income level by △Y. The equilibrium income determination identity (i) with △G can be expressed modifying as follows.

Y+△Y=\frac{1}{(1-b)}\(a-bT+I+G+△G)..........(ii)
Substracting equation (i) from equation (ii), we get;

Y+△Y-Y=\frac{1}{(1-b)}\(a-bT+I+G+△G)-{\frac{1}{(1-b)}\(a-bT+I+G)}
Or,\△Y=\frac{1}{(1-b)}\△G
Or,\frac{△Y}{△G}=\frac{1}{(1-b)}
Thus, the government expenditure multiplier G_m can be expressed as;
G_m\or\frac{△Y}{△G}=\frac{1}{(1-b)}

#### 4-2 The Tax Multiplier (T-Multiplier)

A tax is a withdrawal from the income stream of the economy and it has a negative impact on the equilibrium level of national income. To explain the impact of a change in tax (T-Multiplier) on income level, we include here only two types of the tax system.
1- Lump-sum tax,
2- Proportional income tax,

1- Lump-sum tax
In order to find out the impact of a change in lump-sum tax on income level, let us include △T in the equilibrium equation. Let us again recall the equilibrium equation.

Y=\frac{1}{(1-b)}\(a-bT+I+G)...........(iii)
When there is a change (increase) in tax by △T, it will bring a change (fall) in national income by △Y. When △T and △Y are included in the above equation, it gets changed in the following form of the equation.

Y+△Y=\frac{1}{(1-b)}\[a-b(T+△T)+I+G]
Y+△Y=\frac{1}{(1-b)}\(a-bT-b△T+I+G)........(iv)
The effect of a change in tax △T or T_m on equilibrium national income can be obtained by subtracting the equation (iii) from the equation (iv) as follows.

Y+△Y-Y=\frac{1}{(1-b)}\(a-bT-b△T+I+G)-{\frac{1}{(1-b)}(a-bT+I+G)}
Or,\△Y=\frac{1}{(1-b)}\(-b△T)
Or,\frac{△Y}{△T}=\frac{-b}{(1-b)}
Or,\frac{△Y}{△T}=\T_m=\frac{-b}{(1-b)}

Note that, an increase in tax has a negative impact on the equilibrium of national income. A rise in tax by △T causes a fall in the equilibrium of national income by a multiple of △T, and on the other hand, a tax release by △T results in a rise in the equilibrium level of national income.

4.3- Balanced Budget Multiplier (BBm)
Now let us examine the balanced budget policy of the government on income level. When the government adopts a balanced budget policy, it spends as much as it collects through taxation. It means T=G and △T=△G. The effect of a balanced budget policy on national income is measured through the balanced budget multiplier which is always equal to one.
Let us again recall the equilibrium identity of the national income determination of a three-sector economy. It is as given as;
Y=\frac{1}{(1-b)}\(a-bT+I+G)........(vi)
In order to find out the balance budget multiplier, let us include △T and △G in the equation (vi) and we will get;

Y+△Y=\frac{1}{(1-b)}\(a-b(T+△T)+I+G+△G).........(vii)
By subtracting equation (vi) fram the equation (vii) we get;

Y+△Y-Y=\frac{1}{(1-b)}\(a-b(T+△T)+I+G+△G)-{\frac{1}{(1-b)}\(a-bT+I+G)}
Or,\△Y=\frac{1}{(1-b)}\(-b△T+△G)......(viii)
Since, △T=△G in a balanced budget policy, by substituting △G for △T in the equation (viii) we have the following form of the equation.
△Y=\frac{1}{1-b}\(-b△G+△G)
Or,\△Y=\frac{1}{1-b}\(△G-b△G)
Or,\△Y=\frac{1}{1-b}\{△G(1-b)}
Or,\frac{△Y}{△G}=\frac{(1-b)}{(1-b)}
Or,\frac{△Y}{△G}=1
Or,\frac{△Y}{△G}=BB_m=1
Thus, BB_m=1

#### 5- Leakages of Multiplier

Leakages are the withdrawals of income from the income stream which tend to weaken the multiplier effect of new investment. Given the MPC, the income propagation gradually becomes smaller and smaller in the successive rounds due to leakages. There are some important leakages as follows.

5.1- Saving
Saving is the most important leakage of the multiplier process. As MPC is less than one, it means the entire income is not spent on consumption. It implies that a portion of the entire income has been saved and that remains out of the income stream. It causes a fall in income in the successive rounds. So the more the saving, the higher will be the leakage, and it will result in a smaller size of the multiplier, and vice-versa.

5.2 The high liquidity preferences
When people hoard a portion of the increased income due to their high liquidity preferences for the purpose of the transaction, precautionary, or speculative motives, it is a leakage. This habit of people affects the multiplier process adversely.

5.3 Purchase of stock and security
If a part of the increased income is spent on the purchase of old stock and securities instead of buying consumer goods, consumption expenditure falls. Consequently, the size of the multiplier also gets smaller than before.

5.4- Debt cancellation
Debt cancellation or repayment of debt causes leakage of income from the income stream and it reduces the size of the multiplier. It is because, when a part of increased income is used for the debt cancellation instead of spending for consumption, there is less possibility of re-injection of that income into the income stream.

5.5- Inflation
As the increase in investment leads to inflation, the amount of expenditure on consumer goods increases due to the absorption of high prices, and there is a fall in real consumption. As a result the multiplier effect on income gradually becomes weaker and weaker.

5.6- Net Import
When a portion of increased income is spent on imported goods, it causes the income to leak out of the domestic income stream. As a result, there is a fall in the consumption of domestic goods and it reduces the multiplier effect on income.

5.7- Undistributed profit
When the entire income of joint-stock companies in the economy is not distributed to their shareholders but stored in the reserve fund, it constitutes a leakage from the income stream. It tends to reduce the income and then consumption expenditure thereby weakening the multiplier process.

5.8- Imposition of Tax
Imposition direct or indirect of tax weakens the multiplier process. The progressive tax lowers the disposable income of tax-payers and commodity tax tends to raise prices of goods and a part of increased income is absorbed income by the high price. Thus, the imposition of tax lowers the consumption expenditure and then income level by weakening the size of the multiplier.

#### 6- Importance or uses of multiplier

The importance of multiplier has been shortly explained as follows.

6.1- Investment
The multiplier theory highlights the importance of investment. Consumption is somewhat stable in the short-run and the change in income and employment is the result of the change in investment. A fall in investment brings about a cumulative decline in income and employment by the multiplier process and vice-versa. The regain of income and employment by the multiplier process, it requires an increase in investment. Thus it highlights how important is an investment.

The multiplier process evaluates the different phases of the trade cycle. It points at, a fall in investment, causes a fall in income and employment in a cumulative manner leading a recession and then depression. Conversely, an increase in investment leads to revival and then a boom in the economy.

6.3- Deficit financing
It highlights the importance of deficit financing and also focuses on how to accelerate the process of economic expansion.

6.4- Government development expenditure
The study of multiplier highlights the importance of government expenditure on development function and resultant achievement of employment and growth rate.

6.5- Saving and investment equality
Multiplier helps to equalize the saving and investment. If there is a divergence between these variables, an increase in investment brings about an increase in income level through a multiplier process by more than the rise in investment. As a result, the income increases, and with that, the saving also rises, which equals the investment.

6.6- Achieving full employment
It is the multiplier theory that highlights an initial increase in investment leads to a rise in income and employment by multiple times, increase in investment. if a single dose of investment fails to uplift the level of employment, the state can inject regular doses of investment until full employment is achieved.