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There are two examples, one for a bandit, the another one is for the government.

Suppose we have a simple economy, a bandit and a teacher. The teacher can teach the bandit, but the bandit can't offer anything of value to the teacher.

Suppose at the start the bandit has \$100, while the teacher has \$0. The bandit pays for lessons $100 and the teacher consequently teaches them. Now the teacher has all the money and the bandit has nothing but their gun. If we will stop here GDP will be \$100 because Consumption=\$100, while everything else is zero.

But our bandit being bandit uses their gun to take all money from the teacher. Here we go again, the bandit has \$100, while the teacher has zero. And again, our bandit pays \$100 for lessons. If we will stop here our GDP will increase by \$100, totaling to \$200. Conclusion #1: We will be able to increase our GDP with these robbing-spending cycles, each cycle increasing GDP by \$100.

Now let's replace our bandit with the government. Our bandit declares themselves to be the Sovereign over these lands, with God given divine right to take people's money by force, (i.e. to tax them). The teacher recognizes them as their Sovereign. We replace our robbing-spending cycle with taxing-spending cycle. The Sovereign pays for lessons, the teacher enjoys sitting on the money for a while and then they get taxed of all of their money. This time Consumption=0, but Government Expenditures increase by \$100 for each iteration of the cycle ("The State is me", as our Sovereign would say). Conclusion #2: We will be able to increase our GDP with these taxing-spending cycles, each cycle increasing GDP by \$100.

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Well this is trivially true because GDP measures output. Here the only producer in the whole economy is the teacher, if the bandit or government can make teacher work more then of course GDP increases.

Note the GDP increase here because the teacher actually produces 200\$ worth of services not because money ‘circulate’. You can calculate GDP even for barter economy where there is no money. What matters is that this bandit or government managed to ‘convince’ the teacher to produce 200\$ of services.

Even in a totalitarian society where everyone is forced to do things at the end of the barrel of a gun GDP won’t be necessary zero.

However, this being said it is not realistic to assume that if government set taxes to 100% people will still provide goods and services assuming government can’t use forced labor.

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This seems like a critique of GDP as a measure of economic activity, disguised as a critique of taxation.

Conclusion #1: We will be able to increase our GDP with these robbing-spending cycles, each cycle increasing GDP by $100.

This really only makes sense if the bandit forgets the lesson each time. Otherwise the teacher will eventually have nothing to teach and the bandit is just paying for conversation - something the bandit could just as easily offer the teacher.

You're also skirting the issue of cycle definition - GDP refers to a single cycle. In such an economy, does it make sense to define the cycle as a year? Or as a course of lessons?

The robber-teacher example forces one to ask what, precisely, GDP is really measuring, and leads to the conclusion that the answer is highly context-dependent.

Conclusion #2: We will be able to increase our GDP with these taxing-spending cycles, each cycle increasing GDP by $100.

So far, it appears that society has not yet run out of things with material public benefit to build and maintain. It has not "run out of things to teach", and the activities financed through public spending operate on a significantly wide range of time scales that the choice of a one-year cycle is as good as many other choices.

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