Size of Investment Multiplier

First Method. The basic income-expenditure equation is:
Y = C + I                                       Y = Income 
                                                      C = Consumption
Or   ΔY = ΔC + ΔI                        I = Investment

Dividing both sides by ΔY,

`(ΔY)/(ΔY)`= `(ΔC)/(ΔY)` = `(ΔI)/(ΔY)`

I = `(ΔC)/(ΔY)` + `(ΔI)/(ΔY)`

`(ΔI)/(ΔY)` = I = `(ΔC)/(ΔY)`

Taking reciprocals of both sides,

`(ΔY)/(ΔI)` = `1/(1-(ΔC)/(ΔY))``(ΔY)/(Δ1)` = K

K = `(1)/(1-c)`               `(ΔC)/(ΔY)` = c, marginal propensity to consume.

Second Method
K = `(ΔY)/(ΔI)`
ΔY = ΔC + ΔI
ΔI = ΔY - ΔC

Put the value in Eq. (i) from (ii), 

K = `(ΔY)/(ΔI)` = `(ΔY)/(ΔY - ΔC)`

Dividing Numerator and Denominator by ΔY, 

K = `1/((ΔY)/(ΔY)-(ΔC)/(ΔY))``1/(1-c)`

Thus, it is clear that there is a direct relation between MPC and multiplier. This can be explained with the help of following equations :

1. If MPC is zero, the size of the multiplier will be unity
   or If MPC = 0, then K = 1
   \[\mathbf{K}=\frac{1}{1-c}=\frac{1}{1-0}=\frac{1}{1}=1\]
   It means if MPC is equal to zero, the size of multiplier is equal to one.
2. Higher the MPC, higher will be the size of multiplier. If MPC = 0.8 i.e. too close to one, then the size of multiplier is 5.
   \[\mathbf{K}=\frac{1}{1-c}=\frac{1}{1-.8}=\frac{1}{.2}=\frac{1}{\frac{2}{10}}=\frac{10}{2}=5\]
   It reflects that if MPC is higher, the size of the multiplier is also higher.
3. Lower the MPC, lower the size of multiplier
   If MPC = 0.2 i.e. close to zero, then the size of multiplier is 1.25 or close to one
   \[\mathrm{K}=\frac{1}{1-c}=\frac{1}{1-0.2}=\frac{1}{8}=\frac{1}{\frac{8}{10}}=\frac{10}{8}=1.25\]
   It means if MPC is lower i.e. close to zero, the size of the multiplier is low i.e. close to one
4. If MPC is equal to one, the size of the multiplier will be infinity or If MPC = 1 then
   \[\mathrm{K}=\frac{1}{1-c}=\frac{1}{1-1}=\frac{1}{0}=\infty\]
   It depicts if MPC = 1, the size of multiplier is infinity.

Generally, MPC lies between zero and unity , therefore, the size of the multiplier will vary between unity and infinity.

The relation between MPC and multiplier can be shown with the help of Table 1. 

TABLE 1

Table 1 reflects that when MPC is zern, the size of multiplier is 1. When MPC is close to zero the size of multiplier will be low, say if MPC =,the size of multiplier is 1.33. Now MPC = 1, the size of multiplier is infinity. It means if MPC is near to one, the size of multiplier would be higher and vice versa. Thus it is a clear that MPC and multiplier are directly and positively related.
Relationship between MPC and multiplier can be depicted with the help of Fig. 1.

Fig. 1

In Fig. l(A), 45° line shows income is equal to consumption. It means C and Y increase in the same ratio. In other words, MPC = 1, so K = •. It is because

K = `(1)/(1-c)` = `(1)/(1-1)` = `1/0` = ∞

In Fig. l(B), CC curve is consumption curve. It is parallel to X-axis. It shows that change in income is not accompanied by any change in consumption. In this case MPC = 0 and therefore, multiplier is equal to one 

K = `(1)/(1-c)` = `(1)/(1-0)` = `1/1` = 1

In Fig. l(C), Y = C +I equilibrium line. CC is the consumption curve. It intersects the equilibrium line at point A, where income and consumption are equal. Change in consumption leads to change in income i.e. 11 Y. If MPC = `1/2`

K = `(1)/(1-c)` = `(1)/(1-1/2)` = `1/(1/2)` = 2

As we know that MPC and multiplier are positively related but MPS and K are inversely related. It is because MPC and MPS are inversely related. It means if MPC is higher, then MPS will be lower and vice-versa. The sum of MPC and MPS is always equal to one. That is,

MPC + MPS = 1
Since  Y = C + S                            or  ΔY = ΔC + ΔS

`(ΔY)/(ΔY)` = `(ΔC)/(ΔY)` = `(ΔS)/(ΔY)`          Where, `(ΔC)/(ΔY)` = MPC, `(ΔS)/(ΔY)` = MPS

1 = MPC + MPS
1 = c + s,             1 - c = s 

we know,  K = `(1)/(1-c)` = `1/s`

The derived equation states that there is an inverse relation between MPS and multiplier. The relation between MPS and multiplier can be shown with the help of certain equations :
(i) If MPS is equal to one, the size of multiplier is equal to one
or If MPS = 1,                             K `1/s` = `1/1` = 1

It shows that if MPS is equal to one, the size of multiplier is equal to one.
(ii) Lower the MPS, higher will be size of the multiplier
If MPS = 0.2 then K = 5                 K =`1/s` = `1/0.2` = `1/(2/10)` = 1.25

It shows if MPS is lower i.e. close to zero, the size of multiplier would be higher
(iii) Higher the MPS, lower the size of the multiplier
If MPS= 0 .8 then K = 1.25            K =`1/s` = `1/0.8` = `1/(8/10)` = 1.25

It shows if MPS is higher, the size of the multiplier will be low.
(iv) If MPS is equal to zero, the size of the multiplier will be infinity.

If MPS = 0 then                                  K =`1/s` = `1/0` = `1/0` = ∞

If MPS is equal to zero, the size of multiplier will be infinity.
Generally MPS lies between zero and unity, therefore, the size of the multiplier will vary between unity and infinity.

The relation between MPC, MPS and multiplier can be shown with the help of the Table 2:
TABLE 2

Table 2. depicts that when MPS is 1, the multiplier is equal to o·ne. If MPS is close to zero, the size of multiplier will be high. If MPS is equal to zero, the size of mutliplier would be infinity. In this way it is clear that MPS and K are inversely related.
Thus it is clear that as MPC increases or MPS falls, the size of multiplier is higher. It should also be noted that MPC cannot be negative and also cannot be greater than one. Therefore size of multiplier cannot be less than one, but it is always less than infinity. 

• The multiplier (K) depends on MPC and MPS.
• K = 1 / (1 − MPC) or K = 1 / MPS.
• Higher MPC → higher multiplier; higher MPS → lower multiplier.
• If MPC = 0, K = 1; if MPC = 1, K = ∞.