# THE CONCEPT OF MULTIPLIER

The Concept of Multiplier - Introduction
Equilibrium national income changes if injections and/ or leakages change.
Under this section we introduce you to the concept of Multiplier. This analyses
the magnifying effects of changes in leakages and/or injections on equilibrium
level of income. Multiplier can be defined as a process through which any
changes in the aggregate expenditure (spending), affect the equilibrium level of
income. It can also be said to be a scalar through which national income
parameter is multiply to give equilibrium level of income. The scalar is often
represented by which can be seen in RHS of equation 4.7 and 4.20. It should be noted that this is often refer to as Income, Government and Investment
multiplier. However, when tax is involved the multiplier becomes (i.e. tax multiplier) .

The multiplier principle
Consider the components of aggregate demand expressed as:
Y = C + I + G + En, where, as defined previously
Y = disposable income
C = consumption expenditure
I = investment expenditure
G = government expenditure
En = net exports, i.e. export less imports.
Any increase or decrease in the magnitude of any of the variables in the right�hand side of this equation will have some effects on disposable income. More
specifically, an increase in either consumption, investment, government
expenditure or net exports will have an expansionary effect on income and
employment. Conversely, any decrease in any of these variables will have a
contractionary effect on both income and employment. These effects are based
on what is referred to as the multiplier principle.
To sharpen our understanding of the multiplier concept, let us focus on
investment alone for the time being. In other words, let us study the meaning and
working of the investment multiplier. `is derived by differentiating the
classical and Keynesian model with respect to income, investment and
government expenditures. It can be illustrated as follows;
Y = C + I + G ...........................1
C = a + bY ................................2
Therefore, equation 1 becomes;
Y = a + bY + I + G......................3
Collect like terms, to have;
Y - bY = a + I + G ......................4
Y(1 - b) = a + I + G ......................5
Make Y the subject and differentiate w r t , I and G
Y = a + I + G / 1+ b ...........................6
.......................................7 equation 7 is achieved through first
derivative of equation 6 w r t I by holding G and a constant.
Similarly;
.....................................8 through first derivative of 6 w r t G holding a and I constant.

Self Assessment Exercise
i. Determine the income multiplier (k) given that Y= C + I + G and that I and G
are autonomous, while C = a + bY

### The Geometrical Illustration of Multiplier

Figure 4.2.1 depicts the working of the multiplier. The initial change was an
increase in investment expenditure depicted as (+∆I). This change caused an
increase in income shown as ∆Y. The increase in income (∆Y) gave rise to
increase in consumption expenditure depicted as ∆C and an increase in savings
not shown in the diagram. Since consumption expenditure is a component of AE
aggregate expenditure increased necessitating in income denoted as ∆Y. This sets in motion another chain of reactions until the economy converges to a new equilibrium income level depicted as Y* corresponding to aggregate expenditure AE2.

Self Assessment Exercise
i. Draw a diagram to illustrate the concept of multiplier

### Algebraic Determination of the Multiplier

We will employ equation for the illustration.
Y = 1 ( a – bT0 +I0 + G0 + X0 – M0) .............. 10
1– b( 1 – t) + m
Equation 10 has two main components. The expression for the multiplier 1
1+ b ( 1-t) +m
and the autonomous components (a – bT0 + 10+ G0 + X0 – M0 ).
From equation above, if any of the autonomous components changes for
example Investment, income will change by
∆Y = 1 (∆10) (11)
1 – b(1 – t) + m
From equation above the changes in Y with respect to I could be expressed as.
∆Y = 1
∆I 1 – b (1 – t) + m
Equation 1.16 is the investment spending multiplier. The value of equation 1.16
is the member of times by which a change in investment will be multiplied to
obtain the resultant change in income.
Alternative Explanation to multiplier concept
The multiplier (k) is a number by which an initial change in investment (ΔI) is
multiplier to obtain the final effect on the national income (Y).
Example 1:
Suppose an investment of N200 million brings about an increase in income of
N1,000 million, the multiplier is.
m = 5 = N1.000
200 This is because 5 x N200million = N1.000 million
Example 2:
If the increase in income, given the same level of investment, is N1,500 million,
the multiplier is 7.5 (i.e. N1,500/200).
The crucial question is, why does a given change in the level of investment bring
about a multiplier work? We shall examine this important question by the use of
numerical illustrations.
The working of the multiplier: A numerical illustration

#### To explain the working of the multiplier, let us make two simple assumptions

(a) that our small economy of Zamba decides to invest N100 million annually
(b) that the MPC of Zamba is 0.75, This means that for every hundred naira
received, 75 naira is spent and 25naira is saved.
let us assume that this economy has neither government nor external trade
transactions.
If N100 million is invested in the Zamban economy annually, it will give rise to
several rounds of expenditure, and each expenditure round will induce yet
another expenditure round. Let us trace the details of this process of induced
expenditure by examining Table 4.2.1 below;
It will be apparent from table 4.1 that the N100 million initial investment gives
rise directly to a N100 million increase in income in the first round. Part of this
income will be consumed and part of it saved. The part consumed is obtained by
multiplying the income by the MPC, i.e. N100 million x 0.75 = N75 million
(column 3). The part saved is N100 – 75 = N25million or N100 million x MPS =
(0.25), column 4.
The second period of expenditure is the income of N75 million obtained from
increased in consumption. Using the same process of multiplying this by the
MPC of 0.75 and MPS of 0.25, the actual values for consumption and savings,
respectively, will be found in Period 2, columns 3 and 4.
In general, each income period gives rise to an expenditure period as well as a
savings period. Finally, the process of spending and re-spending will yield an
income several times the initial amount. In our case, the ultimate amount will be
N400 million. This is the multiplier effect of the yearly expenditure of N100.
What is the value of the multiplier in this case?
Note that from the multiplier effect of N400 goes to savings. Thus, the total
reflects the size of both the MPC and the MPS. Note also that the final increment
in income (the multiplier effect) is the multiplier of the reciprocal of MPS. This
gives a clue on how to obtain the multiplier effect of any increase in investment.
Simply multiply the change in investment by the MPS turned upside down.
Using our case as an illustration, we have 100 x 4/1 = N400 million.

Self Assessment Exercise
i. Given the following equation determine the multiplier Y= C + I + G + X – M,
where C = a + bYd , Yd = Y – T, I = Io, T = To + tY, G = G0, X = Xo, M = Mo.

### Conclusion

This unit concludes that every economy has its definite marginal propencity to
consume which is the main factor that determine the multiplier process, the
larger the MPC the better

### Summary

This unit looked at the concept of multiplier both algebraically and graphically and used these economic tools to explain in a clear term how multiplier process works. in addition, we used numerical analysis for better understanding.

### Marked Assignment

i. Given the following equation determine the multiplier Y= C + I + G + X – M,
where C = a + bYd , Yd = Y – T, I = Io, T = To + tY, G = G0, X = Xo, M = Mo.
ii. Generate a multiplier process given that initial National income increase by
200m and that MPC for the economy is 0.9.
iii. graphically explain the concept of multiplier.
iv. Discuss the multiplier effect of continuous increase in tax revenue base
without a corresponding increase in the level of income

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