“Consumption lowers investment” mankiw explanation

Question

I am teaching myself macroeconomics and I am stuck on this reading Mankiw:

Because the economy’s output is fixed by the factors of production and the level of government purchases is fixed by the government, the increase in consumption must be met by a decrease in investment. For investment to fall, the interest rate must rise. Hence, a reduction in taxes, like an increase in government purchases, crowds out investment and raises the interest rate

Before that, we have the fined the following identity:

Y - C - G = I

Output - Consumption - Gov. spending = Investment.

Looking at the identity, it’s easy to see why the bolded part is right, but I can’t intuitively understand why an increase in consumption would decrease investment.

Answer 1

This is because investment is equal to private and public savings (but assuming balanced budget public saving drops out so it would only be private savings I=S). This is because you can only invest what you save.

Moreover saving and consumption are mutually exclusive. From your income you can either consume or save there is no third option economically speaking (actually technically saving is postponed consumption but I don’t want to confuse you too much so I won’t go further into that).

Hence it’s quite intuitive that the more you consume, the less savings available and the higher the interest rate due to low savings and so on.

So to sum up consumption must decrease investment as the more proportion of your income you consume the less you can save. If you consume 70% of your income you can only save 30%, if you consume 90% you can only save 10% and so on. If you consume 100% of your income your saving and thus investment is 0.

Answer 2

Mankiw makes the Big Assumption that total output $Y$ is fixed. Informally and intuitively, we can imagine this as arising because everything is running at full capacity. Factories and workers are working as long and as hard as they can.

So, every \$100 increase in consumption (e.g. production of bread, clothing) can only possibly occur with a corresponding \$100 decrease in investment (e.g. production of machinery, housing), because we'd have to get those factories and workers that are now producing bread and clothing to instead produce machinery and housing.

In the real world, Mankiw's Big Assumption does not generally hold: Total output is not usually fixed. And so, a \$100 increase in consumption does not logically and necessarily lead to a \$100 decrease in investment.