<p>Saving is there <em>implicitly</em> through investment, taxes and government spending. You can't include it explicitly in that equation as you would be counting saving twice.</p>
<p>By definition (see Blanchard et al Macroeconomics) <em>private</em> saving in macroeconomics is defined as:</p>
<p><span class="math-container">$$S=Y-T-C(Y-T) \implies S+T = Y-C(Y-T) \tag{1}$$</span></p>
<p>Next we can substitute the saving into the AD equation:</p>
<p><span class="math-container">$$Y=C(Y–T)+I(r)+G \implies Y- C(Y–T) = I(r) +G\tag{2}$$</span></p>
<p>The substitution will get us:</p>
<p><span class="math-container">$$ S + T = I(r) +G \implies S=I(r) +G-T\tag{3}$$</span></p>
<p>Now solving equation 3 for investment we get:</p>
<p><span class="math-container">$$ I(r)=S + (T-G)$$</span></p>
<p>which shows that investment will be equal to private saving <span class="math-container">$S$</span> and public saving <span class="math-container">$T-G$</span>. Also I did not written it explicitly but saving is function of interest rate so you could write the last equation also as: <span class="math-container">$I(r)=S(r) + (T-G)$</span></p>
<p>Hence saving is already included in the formula <span class="math-container">$Y=C(Y–T)+I(r)+G $</span> through investment. As a consequence you cannot explicitly include the saving there again as you would be double counting.</p>
<p>You can either use:</p>
<p><span class="math-container">$$Y=C(Y–T)+I(r)+G $$</span></p>
<p>or in case you wanna include saving explicitly use</p>
<p><span class="math-container">$$Y=C(Y–T)+S(r) +T $$</span></p>
<p>but you cannot do both at the same time as that would be double counting the same variable twice.</p>

CC BY-SA 4.0
2022-09-09T08:26:41.757Z
2022-09-09T08:26:41.757Z
1muflon1
<p>Savings are included in <span class="math-container">$I$</span>, 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 <span class="math-container">$C$</span>. (Thanks to <a href="https://economics.stackexchange.com/questions/52647/why-doesnt-the-function-y%cc%85-cy%cc%85-t%cc%85irg%cc%85-include-the-savings-variable/52648?noredirect=1#comment85684_52648">1muflon1 for the correction</a>.)</p>
<p>The demand function merely describes who is using up how many units of the output.</p>
<blockquote>
<p>The function Y̅=C(Y̅–T̅)+I(r)+G̅ can be altered to: Supply of Y= Demand of Y.</p>
</blockquote>
<p>This is not an alteration. The notation in a lot of textbooks is quite poor IMO. <span class="math-container">$Y=C(Y–T)+I(r)+G$</span> is the <em>definition</em> of the demand function; instead of <span class="math-container">$Y$</span> one could write <span class="math-container">$D_Y$</span>. <br>
In <em>equilibrium</em> Supply of <span class="math-container">$Y$</span> = Demand of <span class="math-container">$Y$</span>. This is not an alteration, but a use of the demand function.</p>

CC BY-SA 4.0
2022-09-09T07:35:01.660Z
2022-09-09T08:44:59.607Z
2022-09-09T08:44:59.607Z
Giskard
<p>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 ?</p>
<p>What is my reasoning behind this:</p>
<p>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.</p>
<p>We know that:</p>
<ul>
<li><p>The consumption function is the amount of money one can choose to either consume or save. So, why not removing the Savings option?</p>
</li>
<li><p>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?</p>
</li>
</ul>
<p>So, do we write: Y̅=C(Y̅–T̅)+I(r)+G̅, instead of Y̅=C(Y̅–T̅)-S+I(r)+G̅, because S&lt;I(r)? How could we even compare savings to the interest &quot;RATES&quot;?</p>
<p>Where am I wrong?</p>

CC BY-SA 4.0
2022-09-09T07:37:11.370Z
I could not parse what you mean by '*How could we even compare savings to the interest "RATES"?*'
Giskard
<macroeconomics>
Why doesn't the function Y̅=C(Y̅–T̅)+I(r)+G̅ include the 'savings' variable?
student
# Why doesn't the function Y̅=C(Y̅–T̅)+I(r)+G̅ include the 'savings' variable?

(Asked by: username_0 on 2022-09-09)

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

Where am I wrong?

**Tags:** macroeconomics
> **Comments:**
> **username_1**: I could not parse what you mean by '*How could we even compare savings to the interest "RATES"?*'  

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## 2 answers

### Answer by username_2 on 2022-09-09. Score: 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.


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### Answer by username_1 on 2022-09-09. Score: 1

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](https://economics.stackexchange.com/questions/52647/why-doesnt-the-function-y%cc%85-cy%cc%85-t%cc%85irg%cc%85-include-the-savings-variable/52648?noredirect=1#comment85684_52648).)

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.


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