# Why Saving is a Function of Income

Under the Keynesian consumption function, it has been mentioned that consumption expenditure increases when there is an increase in income but not by so much as there is an increase in income. It implies that a portion of disposable income remains unspent on consumption. The portion of income that is not spent on consumption or remains with consumers after their consumption expenditure is known as saving.
1. Saving Function
Saving is the counterpart of the consumption function. It states the relationship between disposable income and saving. Therefore, saving is the function of disposable income. The saving function is expressed symbolically as follows.
S = f(Yd) ............(i)

As we know that the sum of consumption expenditure and saving is equal to disposable income i.e. Yd = C + S. So, if one function is known, the other can be easily derived. Given the consumption function C = a + bYd, the Saving function can be written as follows.

Yd = C + S ..........(ii)
Or, Yd - C = S
Or, S = Yd - C ..........(iii)

Substituting a + bYd, for C we get;
S = Yd - (a + bYd)
= - a + Yd - bYd
= - a + (1 - b)Yd .......(iv)

Where;
- a    = Y-intercept or saving at zero income,
1 - b = Marginal propensity to save or slope of saving curve,
Yd     = Disposable income,
b       = MPC

Let us give a numerical example. Suppose the following consumption function is given.
C = 200 + 0.8Yd
S = Yd - C
Substituting the given consumption function for C we get,
S = Y- 200 - 0.8Yd
= - 200 + Yd - 0.8Y
= - 200 + (1 - 0.8)Yd
= - 200 + 0.2Yd
In the above equation, 0.2 is the marginal propensity to save. It follows from above that the sum of marginal propensity to consume and marginal propensity to save is equal to one ie. MPC + MPS = 1.

The saving function has also been explained with the aid of the following diagram.
In the above diagram, CC and SS are the consumption and saving curves respectively. The saving is negative until the income reaches Y1. As the consumption curve intersects the unity line at point A, the consumption expenditure equals disposable income. So, there is zero income at point B. Thereafter, with the increase in income, consumption expenditure increases but not by so much as the income increases. The gap between consumption expenditure and disposable income is equal to saving which is represented by the area after Y1, between X-axis and SS curve, as shown in the diagram. The triangled area SY1O equals autonomous expenditure equal to the area CAO. After Y1 saving goes on increasing as income increases. Hence, saving is also the function of income. Here C = EY2 and S = ED or Y2F.

2- Properties or Technical Attributes of Saving Function
There are two properties of saving functions. They are average propensity to save and marginal propensity to save.

2.1- Average Propensity to Save (APS)
The average propensity to save is the ratio saving (S) to any particular level of disposable income (Yd). It increases with an increase in disposable income. It can be calculated by diving the amount of saving by income. It is expressed as follows.

APS = S/Yd
Where;
APS = Average propensity to save,
S = Saving
Yd = Disposable income,
As the entire income is not spent on consumption, a portion of income is saved. So we have the following expression.

C + S = Y
Diving both sides by Yd we get,
Or, \frac{C}{Y_d}+\frac{S}{Y_d}=\frac{Y_d}{Y_d}
Or, \frac{C}{Y_d}+\frac{S}{Y_d}=\1
Here, \frac{C}{Y_d}=\APC and \frac{S}{Y_d}=\APS
So, APC + APS = 1 or the sum of APC and APS is equal to 1,
Or, APS = 1 - APC

2.2- Marginal Propensity to Save (MPS)
A marginal propensity to save is a ratio of change in saving to change in income. It can be found by dividing saving (ΔS) by a change in income (ΔYd). Symbolically, it is expressed as follows.
MPS = ΔS/ΔYd .......(i)

Where;
MPS = Marginal propensity to save,
ΔS    = Change in saving,
ΔYd  = Change in disposable income,

From C + S = Yd, it follows that any change in income (ΔYd) must induce either a change in consumption expenditure or a change in saving (ΔS).

Thus, ΔC + ΔS = ΔYd ......... (ii)
Dividing both sides by ΔYd, we get;
\frac{\triangleC}{\triangleY_d}+\frac{\triangleS}{\triangleY_d}=\frac{\triangleY_d}{\triangleY_d}

\frac{\triangleC}{\triangleY_d}+\frac{\triangleS}{\triangleY_d}=1

Here,\frac{\triangleC}{\triangleY_d}= MPC and\frac{\triangleS}{\triangleY_d}=MPS
MPC + MPS = 1 or the sum of MPC and MPS is equal to 1,
So, MPS = 1 - MPC,

3- Relation between APC and MPC, and APS and MPS
The relation between these variables has been mentioned as follows.
3.1 In the short-run, APC is greater than MPC and APS is less than MPS at all levels of income.
3.2- Since, APC + APS = 1. It implies that if APC falls steadily with an increase APS als0 must increase steadily.
3.3- If autonomous consumption is zero, APC, MPC, APS, and MPS remain constant.

The above relationship between the given variables has been shown in the following table as well.
 Y C S= Y-C APC=C/Y APS=S/Y MPC=ΔC/ΔY MPS=ΔS/ΔY 0 20 -20 ∝ ∝ - - 50 60 -10 1.2 -0.2 0.8 0.2 100 100 0 1 0 0.8 0.2 150 140 10 0.93 0.07 0.8 0.2 200 180 20 0.90 0.10 0.8 0.2

4- Paradox of Thrift (Saving is a vice, not a virtue)
There is a paradox among economists on the thrift (The quality of using money and other resources carefully) of human beings. Classical economists had focused that saving as a virtue and thought that when an individual saves a portion of his income, he can make his fortune by maximizing his earning. According to them, more saving leads to more investment and more investment generates more income. When all individuals of society save a portion of their income, there is an increase in the saving of society which is beneficial for all members and the society itself. Hence, according to the classical economist saving is a private as well as a social virtue.
Keynes denies the classical view and claims that saving is a vice, not a virtue. According to him, more saving lessens consumption expenditure which results in a fall in effective demand. A fall in effective demand causes a fall in output, employment, and income level. Hence, saving is a social vice, not a virtue.
With the help of the following diagram, Keynes has explained how saving is a social vice, not a virtue.
In the above diagram, it has been shown that saving and investment are equal at point E where the income level is equal to OY1. Now, let us suppose that there is an increase in aggregate saving which results in a shift in the saving curve upwards from  SS to S1S1. This upward shift in the saving curve results in a point of equality between these variables at point E1. Consequently, there is a fall in income from  OY1 to OY. This fall in income further will bring a fall in saving from EY1 to E1Y and also a fall in investment. This is how saving is a social vice, not a virtue. Such a paradox among economists is seen in relation to the human quality of using money and other resources carefully.