Trouble at differentiating a consumption function 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 &quot;Macroeconomia&quot;, by Mario Henrique Simonsen, there is the statement of this consumption function:
$(1)$ $C = aA + bY_d$   with $a&gt;0$ and $b&gt;0$
(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.
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
$$
Y_d - C = S = I = \frac{dA}{dt}
$$