# Saving identity

The saving identity or the saving-investment identity is a concept in national income accounting stating that the amount saved in an economy will be the amount invested in new physical machinery, new inventories, and the like. More specifically, in an open economy (an economy with foreign trade and capital flows), private saving plus governmental saving (the government budget surplus or the negative of the deficit) plus foreign investment domestically (capital inflows from abroad) must equal private physical investment. In other words, the flow variable investment must be financed by some combination of private domestic saving, government saving (surplus), and foreign saving (foreign capital inflows).

This is an "identity", meaning it is true by definition. This identity only holds true because investment here is defined as including inventory accumulation, both deliberate and unintended. Thus, should consumers decide to save more and spend less, the fall in demand would lead to an increase in business inventories. The change in inventories brings saving and investment into balance without any intention by business to increase investment. Also, the identity holds true because saving is defined to include private saving and "public saving" (actually public saving is positive when there is budget surplus, that is, public debt reduction).

As such, this does not imply that an increase in saving must lead directly to an increase in investment. Indeed, businesses may respond to increased inventories by decreasing both output and intended investment. Likewise, this reduction in output by business will reduce income, forcing an unintended reduction in saving. Even if the end result of this process is ultimately a lower level of investment, it will nonetheless remain true at any given point in time that the saving-investment identity holds.

## Algebraic statement Illustration of the saving identity with the three sectors, the computation of the surplus or deficit balances for each, and the flows between them.

In a closed economy with government, we have

$Y=C+I+G\to I=Y-C-G$ meaning that whatever of aggregate output ($Y$ ) is neither consumed by the private ($C$ ) nor the government ($G$ ) must be what is invested ($I$ ). But it is also true that

$Y=C+S+T\to S=Y-T-C$ where T is the amount of taxes levied; this equation says that saving is disposable income ($Y-T$ ) minus consumption. Combining both expressions (by solving for $Y-C$ on one side and equating) gives

$I+G=S+T\to I=\underbrace {S} _{\text{private saving}}+\underbrace {(T-G)} _{\text{public saving}}$ In an open economy, a similar expression can be found. The national income identity now is

$Y=C+I+G+(X-M)$ where $(X-M)$ is the balance of trade (exports minus imports). Private saving is still $S=Y-T-C$ , so again combining (by solving for $Y-C$ on one side and equating) gives

$I+G+(X-M)=S+T\to I=\underbrace {S} _{\text{private saving}}+\underbrace {(T-G)} _{\text{public saving}}+\underbrace {(M-X)} _{\text{capital inflow}}$ ### Intended and unintended investment

In the above equations, $I$ is total investment, both intended and unintended (with unintended investment being unintended accumulation of inventories). With this interpretation of $I$ , the above equations are identities: they automatically hold by definition regardless of the values of any exogenous variables.

If $I$ is redefined as only intended investment, then all of the above equations are no longer identities but rather are statements of equilibrium in the goods market.