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Trying to understand idea of savings = investments. Private savings formula is (for closed economy): $$S=Y(GDP) - C - T$$ which means "income - private spendings".
But households don't spend money they earned from Y , they spend money they earned from $$Y_{y-1}$$ where y is current year. What do I miss here?

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Your statement is not correct. Households spend current $Y$. Past $Y_{t-1}$ was already spent so it can't be spent twice. Saving in all forms is also considered spending in macroeconomics, so even when households save money from previous years they are 'spending' that money using terminology in macroeconomics.

Moreover, since definitionally someones spending is someone else's income, when we talk about GDP it always has to be all spent in the year when it is created.

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