It does matter when you label them the exact same thing. If t=0 and t=T, they're all zero. t=0 means t is zero. It doesn't mean anything else. t=T means T is the same thing as t, which is also zero.
This is how I meant it to be:
http://i.imgur.com/FmpcmeO.png
But you can't use
x to refer both to "what you invest" and "what you would have if you didn't invest", if
x increases in value if you invest and does not increase in value if you don't. They're two different quantities.
I have 10 in my pocket now. Let 10 = x. If I invest 10, I invest x. If I didn't invest the 10, how much would I have? 10, which is x.
What I invest = 10
What would I have if I didn't invest? the 10 in my pocket.
Is 10 = 10 not the same quantity? Is there something wrong with this assumption? Sorry I'm not getting where you're coming from.
Not if the real value of the currency increases. In that case, the 8 you get by investing can be used to buy 16 goods (because the currency increased in value as a result of the investment), as opposed to the 10 you get by not investing which can only buy 10 goods (because, by not investing, the currency did not increase in value). 16 goods are better than 10, last time I checked.
Even if you didn't invest, population grows (unreasonable to assume we can stop this), your 10 is now worth more in real terms of distribution of income. In this case, the 8 I used to hire 16 singers in the next period to sing to me, is most certainly less than the number I can hire if I kept the 10.
Appreciate the comments 