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National product accounting identities imply $S=I$. Since these are flow quantities, everything saved in a particular period is invested, thus there is never an accumulated stock of savings.
Is this reasoning correct? If so, where do household savings during a particular year (say bank account balances) go?
The accounting identity assumes that anything saved is an investment, i.e. something will be done with that saving in the future. It's a bit circular, I know. It's not a lagged model, i.e. there's no explicit indexing by time of the variables, unlike (e.g.) in the Hansen–Samuelson model, which uses discrete time. There are also macroeconomic models with continuous time, e.g. Ramsey–Cass–Koopmans model.
Actually, there's a little more to be said here (as I may have committed a fallacy myself above). German economist Fabian Lindner takes issue with those (including economics textbooks authors) who use a one-good model, like Ramsey's to model financial savings:
The fallacies loanable funds theory commits might be explainable by the mis-application of some ideas and concepts of neoclassical growth models – especially the Ramsey (1928), Solow (1956) and Diamond (1965) models – to the sphere of money and finance. Those models are routinely taught in contemporary graduate economics classes (Blanchard and Fischer, 1989; Romer, 1996). The Ramsey and Solow models are models of real investment only. Financial markets, financial assets and financial saving do not play any role in those models. There is only one good which, for simplicity, will be called “corn”. Corn has three functions: it can be consumed, invested and used as a means of payment since wages and interest payments are made with it. Full employment is assumed.
Without money and other financial assets, the only way units can save is to increase their tangible assets, i.e. to invest. Given that full employment is assumed, corn production is always at its maximum in each period. If the corn is consumed, it cannot be invested; if it is invested, it cannot be consumed. There is thus a real trade-off between consumption and investment.
Only under this assumption does it indeed make sense to talk about a limited saving fund which is increased when it is not consumed. However, this trade-off between consumption and investment is not a finance constraint, but a resource constraint. [...]
In contrast to Solow’s and Ramsey’s model, Diamond’s (1965) full employment corn economy allows units to lend and borrow. However, they do not borrow and lend money but again the one good, corn. They face a triple trade-off: they can eat (=consume), plant (=invest) or lend their corn. Here, consumption are not consumption expenditures but the actual eating of the corn. When a unit wants to lend its corn, it can of course not eat it so that it has to restrain from consumption to be able to lend it. [...]
But since in the real world money is normally not eaten or planted and keeps circulating in the economy when it is spent or lent, those models cannot be any guide for the analysis of a monetary economy. Specifically, what is true in a one good economy – units have to consume less to lend and invest more – is fundamentally wrong in a monetary economy.
