<p>This does not seem to have anything to do with calculus.</p>
<p>The idea is that the income not consumed <span class="math-container">$Y_d - C$</span> is saved (usually denoted by <span class="math-container">$S$</span>).</p>
<p>This saving is then lent out to companies (via banks) who invest it (usually denoted by <span class="math-container">$I$</span>), and the accumulated capital is used in production. I am guessing this is denoted by <span class="math-container">$A$</span>?</p>
<p>And the change of <span class="math-container">$A$</span> in time is the investment (assuming there is no amortization).</p>
<p>Hence
<span class="math-container">$$
Y_d - C = S = I = \frac{dA}{dt}
$$</span></p>

CC BY-SA 4.0
2021-05-02T18:43:23.160Z
I thought that it was required some mathematical foundation behind the intuition that $A = Y_d - C$, but your logic seems right. I think I overthought about it and waste some time trying to face it with a calculus approach.
PedPg
CC BY-SA 4.0
2021-05-02T17:42:27.110Z
2021-05-02T17:42:27.110Z
Giskard
<p>everyone.</p>
<p>This is kind of a basic question, but I have a weak background on calculus and I already tried to figure this out by myself for a few hours without success. So, I am finally appealing on you - I even feel embarrassed about asking such a basic question among all of these complex questions in the forum.</p>
<p>Let's get to it: in the middle of a chapter about aggregate demand and IS curve in the textbook &quot;Macroeconomia&quot;, by Mario Henrique Simonsen, there is the statement of this consumption function:</p>
<p><span class="math-container">$(1)$</span> <span class="math-container">$C = aA + bY_d$</span>   with <span class="math-container">$a&gt;0$</span> and <span class="math-container">$b&gt;0$</span></p>
<p>(C represents consumption, A represents non-human wealth and Yd represents disposable income)</p>
<p>Then, as keynesian theory takes A as given in the short-run (so it is treated as a constant), there is this differentiation of A in respect of t:</p>
<p><span class="math-container">$(2)$</span> <span class="math-container">$\frac{dA}{dt} = Y_d - C = (1-b)Y_d - aA$</span></p>
<p>And there is my problem. Mathematically, I don't know how <span class="math-container">$\frac{dA}{dt} = Y_d - C$</span>.</p>
<p>My train of thought is to differentiate the consumption function like this:</p>
<p><span class="math-container">$\frac{dC}{dt} = a \frac{dA}{dt} + b \frac{dY_d}{dt}$</span></p>
<p>Then, as <span class="math-container">$\frac{dA}{dt} = 0$</span>:</p>
<p><span class="math-container">$0 = b \frac{dY_d}{dt} - \frac{dC}{dt}$</span></p>
<p>But it is pretty different than equation (2). What are my mistakes? Thank you.</p>

CC BY-SA 4.0
2021-05-02T17:39:35.193Z
$A$ is non-human wealth? Is it owned by lizardmen?
Giskard
CC BY-SA 4.0
2021-05-02T18:38:50.457Z
Well, at least in Blanchard's Macroeconomics, non-human wealth is the portion of wealth provided by someone's assets, such as financial assets. Human wealth is the portion provided by labor at present value after taxes. Therefore, I don't see any particular reason for sarcasm here.
PedPg
CC BY-SA 4.0
2021-05-02T19:17:37.647Z
I see, thanks! I guess I should have read more macro in English, I was unfamiliar with the terminology. Thanks agian.
Giskard
<macroeconomics><mathematical-economics><aggregate-demand>
Trouble at differentiating a consumption function
PedPg
# Trouble at differentiating a consumption function

(Asked by: username_0 on 2021-05-02)

everyone.

This is kind of a basic question, but I have a weak background on calculus and I already tried to figure this out by myself for a few hours without success. So, I am finally appealing on you - I even feel embarrassed about asking such a basic question among all of these complex questions in the forum.

Let's get to it: in the middle of a chapter about aggregate demand and IS curve in the textbook "Macroeconomia", by Mario Henrique Simonsen, there is the statement of this consumption function:

$(1)$ $C = aA + bY\_d$ with $a>0$ and $b>0$

(C represents consumption, A represents non-human wealth and Yd represents disposable income)

Then, as keynesian theory takes A as given in the short-run (so it is treated as a constant), there is this differentiation of A in respect of t:

$(2)$ $\frac{dA}{dt} = Y\_d - C = (1-b)Y\_d - aA$

And there is my problem. Mathematically, I don't know how $\frac{dA}{dt} = Y\_d - C$.

My train of thought is to differentiate the consumption function like this:

$\frac{dC}{dt} = a \frac{dA}{dt} + b \frac{dY\_d}{dt}$

Then, as $\frac{dA}{dt} = 0$:

$0 = b \frac{dY\_d}{dt} - \frac{dC}{dt}$

But it is pretty different than equation (2). What are my mistakes? Thank you.

**Tags:** macroeconomics, mathematical-economics, aggregate-demand

> **Comments:**
> **username_1**: $A$ is non-human wealth? Is it owned by lizardmen?  
> **username_0**: Well, at least in Blanchard's Macroeconomics, non-human wealth is the portion of wealth provided by someone's assets, such as financial assets. Human wealth is the portion provided by labor at present value after taxes. Therefore, I don't see any particular reason for sarcasm here.  
> **username_1**: I see, thanks! I guess I should have read more macro in English, I was unfamiliar with the terminology. Thanks agian.  

---

## 1 answers

### Answer by username_1 on 2021-05-02. Score: 5 | Accepted answer

This does not seem to have anything to do with calculus.

The idea is that the income not consumed $Y\_d - C$ is saved (usually denoted by $S$).

This saving is then lent out to companies (via banks) who invest it (usually denoted by $I$), and the accumulated capital is used in production. I am guessing this is denoted by $A$?

And the change of $A$ in time is the investment (assuming there is no amortization).

Hence
$$
Y\_d - C = S = I = \frac{dA}{dt}
$$

> **Comments:**
> **username_0**: I thought that it was required some mathematical foundation behind the intuition that $A = Y_d - C$, but your logic seems right. I think I overthought about it and waste some time trying to face it with a calculus approach.  

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