[FootballMuse]

Define "save for a rainy day". Are you talking about permanently putting it under your matress, leaving it in your checking account, or investing it? It will eventually be spent, and the only thing that is changing is the velocity. Either way, that is not necessarily correct.

GDP has multiple different calculations:

Y = C + I + G + NX

or

GDP = Consumption + Investment + Government Spending + Net Exports

Another calculation is

Y = C + S + T

GDP = Consumption + Saving + Net Taxes

Saving either through investment and cash hoarding both count toward GDP. The percentage of GDP that is Saving is the Gross National Savings or National Savings Rate.

tldr; In isolation, scenario 1, Investment is $10 as inventory and Savings is $10. Both add toward GDP

https://en.wikipedia.org/wiki/National_saving