Basic question on S=I Consider the identity:
$$
Y=C+I+G+X-M
$$
 where $Y$ is defined either as value-added output, or income, or
total expenditure. Now, consider private individuals, who decide to
save money, as opposed to spending it. Aggregate demand decreases,
and so output follows, due to the Keyensian cross logic. However,
income that is not consumed, is saved. If savings increase, so does
investment. As such, why is it not the case that if consumption goes
down, investment goes up by the same amount? And if so, aggregate
output should remain unchanged, regardless of the decision of the
consumers after they receive their income?
The reason for that is that if you check carefully the model an important assumption is that investment is fixed $I=\bar{I}$ once you relax this assumption and allow $I$ be some variable function of interest the paradox of thrift disappears in long run.
This does not meant that fixing $I$ in the short run is incorrect assumptions as during recession you will get liquidity trap in a short run. Banks are scared to borrow money because they can’t distinguish good lenders from bad. So in short run all the excess savings during recession get absorbed into banks hoarding liquidity and no matter how much you save $I$ is fixed. Even without recession another way of thinking about it is that in short run it’s hard for companies to change their investment profile.
However, in long run once the $I$ is some variable function of interest rate and of course it has to equal public and private saving  $I=S+ T-G$ the paradox of thrift disappears ($I=S$ only if you assume government runs balanced budget $T=G$ which is fine but I wanted to point out that you should not forget about public saving).
Also I basically took the answer to this one from Blanchard et al. Macroeconomics an European Perspective. If you want to understand this deeper I really recommend checking the book, it’s in my opinion the best undergraduate macro book on the market. If you google hard enough you can even find a free pdf version...