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The standard approaches for calculating GDP are: $$ \text{GDP} = \text{Consumer Spending + Capital Investments + Govt. Spending + Net Exports} $$ And $$ \text{GDP} = \text{Total wages + Profits + Taxes + Return on Capital Investment} $$ I have an issue in seeing the equivalence of these two approaches in different cases. Say we pay the government taxes. In the second approach that gets counted in "taxes", but in the first approach it doesn't get directly counted. Yes, the government spending usually comes out of tax revenue, but not all of tax revenue goes to government spending. So how does the "taxes" in the second approach get divided in the first approach. A good chunk of it goes to government spending, but what about the rest? Are we making an assumption that it's all spent? What if in a simple "two good economy", we have an individual/firm paying taxes but we don't add government spending. Does the tax money go "waste" here in our GDP calculation?

Secondly, we don't count wages in "capital investments" since we assume that the wages paid by firms (counted in the second approach) become consumer spending; but are we assuming that consumers spend ALL their income? How do we account for savings and simply models in which wages do not necessarily equal spending?

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Yes the two approaches are equivalent. However, that does not mean that the respective individual variables are equivalent. Taxes are not equivalent to government spending but the both equations equally capture all Government output. For example in the first approach:

$$GDP = C+ I + G +NX$$

Where $C$ is consumer spending, $I$ is investment and $G$ government spending, $I$ is also equal to private and public saving $I=S + T-G$, so if government spend less than what it collect in taxes the public savings and public portion of investment increases, and when it runs budget deficit vice versa, so the possibility of running either surplus or deficit or balanced budget is already accounted for there. No assumption that its all spent is necessary.

Second, any income, whether it is wage or not, can be only consumed or saved. The consumption part is explicitly included in the formula as $C$ the saving part is the $S$ which represent the private savings from which investment is created. So if a person spend less their income investment increases thanks to private saving, and if a person lives on debt investment decreases to reflect that.

Also, both formulas are just accounting formulas. By itself they have no behavioral assumptions/spending assumptions. They are national equivalent of the kind of accounting that you could see in firms book just track the sources of income and spending.

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