The identity of savings investment is
1. From the perspective of expenditure, gross domestic product (GDP)= consumption C+ investment I+ government purchase G+ net export (X-M).
2. Two departments: consumers (residents) and enterprises, I = s;; In other words, government savings (T-G) will be removed on the basis of three departments.
3. Three departments: consumers (residents), enterprises and government departments, I = S+(T-G); Excluding foreign sector savings (M-X) on the basis of four sectors.
4. Four departments: consumers (residents), enterprises, government departments and foreign departments, I=S+(T-G)+(M-X).
Macroeconomics studies the whole social and economic activities. The whole social and economic activities are mainly measured and expressed by the concept of national output. It includes gross national product (gnp), gross national product (nnp), national income (ni), personal income (pi) and disposable personal income (dpi), among which gnp is the core concept.
Only the family and enterprise sectors have c+i≡c+s. Because the gross national product is y=c+i from the perspective of expenditure and y=c+s from the perspective of income, there is C+I ≡ C+S, that is, S ≡ I. It should be noted that this identity is derived from the definitions of savings and investment. Including family, enterprise and government, the expenditure is y=c+i+g, and the income is Y = C+S+T. (Here, S is residents' savings, which can also be written as sp, G is government purchase expenditure, and T is government tax. )
Therefore, there is c+i+g≡c+s+t, that is, I ≡ S+T _ G. T _ G here can be regarded as the savings of government revenue MINUS expenditure, or it can be written as SG ≡ T _ G. So i=sp+sg. The four-sector economy, including family, enterprise, government and foreign countries, is y=c+i+g+x from the perspective of expenditure and y=c+s+t+m from the perspective of income (where x stands for export and m stands for import). So there is c+i+g+x≡c+s+t+m, that is, I ≡ S+T-G+M _ X, where M _ X can be regarded as the savings of foreign departments, because M is export, that is, income, and X is import, that is, expenditure. So m_x is the balance of foreign income minus expenditure, which can be regarded as the savings of foreign departments and can be expressed by sr, so there is I ≡ SP+SG+SR, which shows that no matter how many sectors of the economy, savings and investment are easy to establish identities.