Why doesn't the function Y̅=C(Y̅–T̅)+I(r)+G̅ include the 'savings' variable?

Question

Why isn't the function Y̅=C(Y̅–T̅)+I(r)+G̅, actually: Y̅=C(Y̅–T̅)-S+I(r)+G̅, or Y̅=C(Y̅–T̅)+G̅ ?

What is my reasoning behind this:

The consumption function includes all the money you are legally allowed to spend (ie: after paying taxes). But, as we know, a part of this money can be saved.

We know that:

• The consumption function is the amount of money one can choose to either consume or save. So, why not removing the Savings option?

• The function Y̅=C(Y̅–T̅)+I(r)+G̅ can be altered to: Supply of Y= Demand of Y. And I don't think that savings are part of the Demand of Y. That is why, we would have to deduct it from the equation?

So, do we write: Y̅=C(Y̅–T̅)+I(r)+G̅, instead of Y̅=C(Y̅–T̅)-S+I(r)+G̅, because S<I(r)? How could we even compare savings to the interest "RATES"?

Where am I wrong?

Answer 1

Saving is there implicitly through investment, taxes and government spending. You can't include it explicitly in that equation as you would be counting saving twice.

By definition (see Blanchard et al Macroeconomics) private saving in macroeconomics is defined as:

$$S=Y-T-C(Y-T) \implies S+T = Y-C(Y-T) \tag{1}$$

Next we can substitute the saving into the AD equation:

$$Y=C(Y–T)+I(r)+G \implies Y- C(Y–T) = I(r) +G\tag{2}$$

The substitution will get us:

$$ S + T = I(r) +G \implies S=I(r) +G-T\tag{3}$$

Now solving equation 3 for investment we get:

$$ I(r)=S + (T-G)$$

which shows that investment will be equal to private saving $S$ and public saving $T-G$. Also I did not written it explicitly but saving is function of interest rate so you could write the last equation also as: $I(r)=S(r) + (T-G)$

Hence saving is already included in the formula $Y=C(Y–T)+I(r)+G $ through investment. As a consequence you cannot explicitly include the saving there again as you would be double counting.

You can either use:

$$Y=C(Y–T)+I(r)+G $$

or in case you wanna include saving explicitly use

$$Y=C(Y–T)+S(r) +T $$

but you cannot do both at the same time as that would be double counting the same variable twice.

Answer 2

Savings are included in $I$, e.g.: storing money in the bank which is lent out to people making physical investments. Keeping cans of food at home or similar things that may be considered saving from a laymen's perspective is not considered savings from a macro point of view, and are counted in $C$. (Thanks to 1muflon1 for the correction.)

The demand function merely describes who is using up how many units of the output.

The function Y̅=C(Y̅–T̅)+I(r)+G̅ can be altered to: Supply of Y= Demand of Y.

This is not an alteration. The notation in a lot of textbooks is quite poor IMO. $Y=C(Y–T)+I(r)+G$ is the definition of the demand function; instead of $Y$ one could write $D_Y$.
In equilibrium Supply of $Y$ = Demand of $Y$. This is not an alteration, but a use of the demand function.