# Why is GDP equal to the sum of revenues?

National Income can be expressed in two ways :

• In can be derived from the GDP, as, roughly, a sum of "money spent" : GDP is equal to C + G + I + NX, where C, G, I and NX are houshold consumption, government expenditures, investment, and net exports. National Income is then obtained by adding net income from abroad and removing consumption of fixed capital.
• It can be calculated as the sum of incomes (details here) : mainly pre-tax employee compensation, pre-tax corporate profits, and taxes on production and imports.

The fact that the two ways give the same results (let alone the "statistical discrepancy") is usually taken for granted.

Of course the principle makes sense, but I am looking for a rigorous approach / mathematical proof.

• One could say that whatever agents earn, they spend. But what if they spare some money ?
• One could say that whatever is spent, is earned by some agent. But I tried this and couldn't get rid of the 'I' in the first formula.

Could somebody enlighten me ?

• Actually I think I've seen that investments are not charges, e.g. if the profit of a company is 20 dollars and it spends 12 dollars in an investment, the profit is still 20 dollars. Only its loss of value ("Consumption of fixed capital") is counted as charges. That would explain why I couldn't get rid of the investment from the formula : investments are in the profits. – yannick1976 Jul 24 '15 at 20:39
• Im not sure what you problem is. But in the first calculation the savings are equal to the investments. Therefore you can say, that $Y=C+S+G+NX$. All the savings are being reinvested. – callculus Jul 25 '15 at 14:52
• @calculus if I earn 100, spend 90, and put 10 in the bank, the 10 go to the 'I' in the formula ? – yannick1976 Jul 25 '15 at 20:27

It is right that the whole savings are invested.

You can use the (simplified) national income and the national output to show the equality.

The national income is $NI=S+I$

The national product is equal to the national income: $NP=NI$

The national product is equal to the sum of consumption and investment: $NP=C+I$

By using this three equations it can be shown, that $I=S$.

This presumption is made for the IS-Curve at the IS-LM-Model.

• Thanks for this answer, but actually this is not what I wanted to prove. Maybe what i want to prove is more clear from my answer ? And actually maybe the proof is what you said in your answer, that national product is equal to national income ? But this is not clear to me yet. – yannick1976 Jul 26 '15 at 14:52

Ok so I think I'm starting to get it. But I'll still be glad if somebody confirms this + corrects the flaws there still are.

The reasoning is that everything that is spent (GDP) goes to somebody. If we remove Govt employees salaries, VTA and imports from G, C and I, and add exports, we get companies sales. So :

Companies sales = G + C + I - Govt Salaries - VTA - Imports + Exports
=> Companies sales = G + C + I + NX - Govt Salaries - VTA
=> GDP = G + C + I + NX = Companies sales + Govt salaries + VTA

Companies sales then can be divided into salaries and benefits. Charges disappear in this aggregated view (but there still is a problem with imports in think).

So : GDP = Benefits + Private companies salaries + Profits + Govt salaries + VTA
=> GDP = (pre-tax) employee compensation + (pre-tax) profits + taxes on production, which is what I wanted to prove.

• Where is capital expenditure? Capex? – luchonacho Apr 18 '17 at 9:48