This is the third post in a series of posts looking at the fiscal multiplier. Previously, we have examined how the circular flow of money interacts with government spending and taxation (as well as private saving) by considering a mathematical structure called a geometric series. This interpretation of the fiscal multiplier is based around the concept of "spending rounds" which represent successive events in which income received previously is spent onwards, creating new income which is spent in the next round, and so on. Each spending round involves a successively smaller amount of circulating money because a fraction of all income is collected in tax (or saved). Eventually, all of the money has been withdrawn from circulation via taxation (and saving) and the spending stops. In the interim period, the circulation of the ever-reducing money stock produces a total, cumulative amount of income.
This approach is an intuitive way of thinking about sequences of spending. It enables us to conceive of how the money initially introduced by government spending is passed around the economy and what the implications of taxation and saving are. But whilst it arguably does a good job of describing how individual acts of spending follow the receipt of income, it should be recognised that in real economies collective spending does not precede collective income in discrete, ordered stages. Spending and the receipt of incomes arise via an incredibly complex network of millions of overlapping transactions occuring continuously.
The concept of the spending round also leads to questions such as how long it takes for a single spending round to occur, or the number of rounds that should be considered. It seems reasonable, for example, that the number of spending rounds included in any analysis would depend on the time period under consideration. But it is not really clear how spending rounds relate to absolute time. However, it turns out that these concerns can be adequately side-stepped by using a coherent accounting framework. This post attempts to tighten up our understanding of the fiscal multiplier and presents an alternative mathematical derivation which is more conducive to inclusion in more complex models.