The Function of Government Spending

Why you pay for it by spending the money

Any proposed government policy is challenged with the same question: “how are you going to pay for it”.

The answer is: “by spending the money”.

Which may sound counter intuitive, but we can show how by using a bit of mathematics.

Any government outlay causes somebody somewhere to receive some income they didn’t have previously in return for supplying some good or service. They then pay some tax, and decide how much of that income to spend on other goods and services in the economy and how much to save for a rainy day. That process then continues with the money bouncing around between people causing transactions.

We can represent the initital government outlay with the following definitions.

X is the initial government outlay on transaction 0, and because it is the government it pays no tax and does no saving on that transaction.

From that you can define the outlay of any transaction in the spending chain with this function definition.

The amount spent at transaction step t is the initial government outlay less the sum of the tax and saving that happened up to that point.

This is the function of government spending and describes a geometric series over a sequence of transactions.

We can now add a few restrictions. First, that the tax rate for the transaction is positive and leaves some income

that you can only save your disposable income

and for those who worry about these sort of things:

Once you have all that in place, and you understand that a money system has a material limit (ie there is a unit that cannot be further subdivided, e.g a penny) such that, expressed in that unit:

you can then say that there is a finite number of transactions n such that

And that is why you pay for government expenditure by spending the money. The outlay (X) will be matched by taxation and excess saving to the penny after n transactions.

Expressing it using mathematics allows you to see what changing taxation rates attempts to do. It is trying to increase and decrease the magnitude of n — the number of transactions induced by the outlay. It has nothing to do with the monetary amount.

The changing tax rates may change the amount of saving undertaken, or it may cause more fully relieved transactions in the business sector. It is by no means certain that any change will bring the expected outcome. And since the sum of the saving amounts determines the government deficit by accounting identity changing tax rates has no direct influence on it — short of confiscating savings directly.

To learn more about how the government spending multiplier works in the aggregate, read these two excellent articles by Andrew Berkeley

Thanks to Michael Berks and Larry Brownstein on the MMT UK Google Group for ironing out the kinks in the maths, Alex Douglas and Brendan Mahoney for corrections, and to Andrew Berkeley for the tweets that shortened this article by several minutes.

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