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The following is a very simplified explanation of the Savings/Investment identity, ignoring imports, exports and government surpluses and deficits. It relies on equating two different ways of Gross Domestic Product: as Total Income and Total Spending. For a deeper explanation, search "Macroeconomics", "Gross Domestic Product" and "Supply and Demand for Loanable Funds".

Savings is just what people earn minus what they spend and what they pay in taxes. Lets call Savings (S), "what they earn" Income (Y), "what they spend" Consumption (C) and "what they pay in taxes" Taxes (T). So now:

S = Y - C - T (Equation 1)

Looking at the economy as a whole, the income of a nation (Y) is either spent by people (C), spent by government (G) or spent by businesses as investment (I). Now:

Y = C + G + I (Equation 2)

If we assume that the government doesn't spend more or less than it taxes, then G = T, or:

G - T = 0 (Equation 3)

Substituting the right side of Equation 2 into Equation 1, we have:

S = Y - C - T = (C + G + I) - C - T = I + (G - T) (Equation 4)

Finally, substituting the right side of Equation 3 into Equation 4, we have:

S = I + (G - T) = I + 0 = I.

Therefore, Savings = Investment.

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Q: Why must the total value of saving in the economy equal the total value of investment?

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