SAVINGS AND CONSUMPTION FUNCTION, MPS AND MPC, APC AND APS

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Savings is the part of the national income that is not consumed.

Consumption is the part of national income that is consumed.

If we assume that all national income is disposable income, then national income is the sum of consumption and savings.

$$Y=S+C$$

But, what effect will changes in national income have on savings and consumption?

This can be analyzed using MPS and MPC.

The marginal propensity to save (MPS) is the change in savings that result from changes in national income.

It is the fraction of the rise in national income that is saved.

It is the derivative of the savings function in re the national income

The marginal propensity to consume (MPC) is the change in consumption that results from changes in national income.

It is the fraction of the rise in national income that is consumed.

It is important to note that the sum of MPC and MPS is equal to one

$$1=MPC+MPS$$

Example 1

Given that the consumption function is given as $C=150+0.8Y$. Determine

1. the Savings function

2. MPS 

3. MPC

Solution:

1. Earlier, we noted that 

$Y=S+C$

Making S the subject of the formula

$S=Y-C$

By substitution

$S=Y-(150+0.8Y)$

$S=Y-150-0.8Y$

$S=-150+0.2Y$

2. The MPS is the first derivative of the savings function.

$S=-150+0.2Y$

Using power rule of differentiation

$MPS=0.2$

3. The MPC is the first derivative of the Consumption function.

$C=150+0.8Y$

$MPC=0.8$

Example 2

If the saving function is $S=0.02Y^2-Y+100$. Determine the 

A. Consumption function

B. The MPC when y=30

C. The MPS when y=30

Solution:

1. $C=Y-S$

$C=Y-(0.02Y^2-Y+100)$

$C=Y-0.02y^2+Y-100$

$C=-0.02Y^2+2Y-100$

2. MPC is the derivative of the consumption function

$C=-0.02Y^2+2Y-100$

$MPC=-0.04Y+2$

When Y=30

$MPC=-0.04(30)+2$

$MPC=-1.2+2=0.8$

3. The MPS is the derivative of the savings function

$S=0.02y^2-y+100$

$MPS=0.04Y-1$

When Y=30

$MPS=0.04(30)-1$

$MPS=0.2$

Example 3

If the consumption function is $C=0.01Y^2+0.8Y+100$. Determine the total savings where $Y=10$

Solution:

To get the total savings where $Y=10$, we have to get the savings

$S=Y-C$

$S=Y-(0.01Y^2+0.8Y+100)$

$S=Y-0.01Y^2-0.8Y-100$

$S=-0.01Y^2+0.2Y-100$

When $Y=10$

$S=-0.01(10^2)+0.2(10)-100$

$S=-1+2-100=-99$

In addition to MPC and MPS, economists also solve for APC and APS.

The average propensity to consume (APC) is defined as the ratio of consumption to national income  

It is computed as follows:

$$APC=\frac{C}{Y}$$

On the other hand, the average propensity to save (APS) is defined as the ratio of savings to national income.

$$APS=\frac{S}{Y}$$

It is computed as follows:

It is important to note that the sum of APC and APS will always equal one.

$$1=APC+APS$$

Example 4

If the APC of an economy is 0.6, what is the total savings when Y=N1000?

Solution:

$1=APC+APS$

$APS=1-APC$

Since APC=0.6

$APS=1-0.6=0.4$

Earlier, we noted that

$APS=\frac{S}{Y}$, and we are told that Y=1000,

$0.4=\frac{S}{1000}$

$S=400$

Example 5

In an economy, the ratio of APC and APS is 5:3. The level is 12,000. What is the total consumption

Solution:

$APC=\frac{C}{Y}$

$\frac{5}{8}=\frac{C}{12,000}$

$60,000=8C$

$C=7500$

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