Simple tax puzzle I recently saw some post on facebook whining about taxes. Simplifying it (and changing numbers, facts, etc.), this was saying:
For each dollar an employer wants to pay you:


*

*20% go in taxes that your employer pays

*out of the remaining 80%, you pay 40% of various taxes (from income taxes, to sales taxes etc.)

*48% of the initial amount remains for you and 52% has gone in taxes.
Now, assume that you spend all of your money. This amounts to spending everything in paying people's salary : whether it is directly a taxi driver, or the set of people that are required to bring your tomato to the supermarket: cashier, CEO, farmer etc. (the tomato itself does not cost money as the earth is not asking us to bury dollar bills each time we harvest a tomato).


So, out of these salaries, as an (indirect) "employer", the 48% you spent will partially go in employer taxes (20% in our scenario). The facebook post concludes that in fact, out of the initial money that was offered to you, a total of 61.6% have gone into taxes.
I want to go further and say that, actually, the people that I indirectly employed also paid their 40% share of "various taxes", and with their remaining 48% they also used their money to buy things (hence employ people, that will buy things...), etc.
You recognize here a converging series... that converges to 100%.
Solutions to the paradox:


*

*there is no paradox: if the government just absorbs taxes and does not employ itself people, there is indeed 100% of an initial amount of money that end up in taxes

*the problem is that we should stop without entering the recurrence : you have 48% of the initial amount, you spent it, and bought stuffs with that. That's the value you got, that got transformed into goods (or services).

*when I said a tomato costs nothing per se, and only work has a value in terms of salary, I was wrong. (why?).

*I made a mistake and the series does not converge to 100% (why??).


I actually can't find the actual solution to this paradox I imposed myself. No politics please ;)
 A: I'm not sure if this is on topic for this site, but no one has voted the question down yet so I'll give a shot at an answer.
Assume that the money supply is constant and assume that every monetary transaction is taxed at a constant rate (whenever an amount of money changes hands then the government takes a fixed proportion of that amount.)
If we assume that money held by the government does not circulate, and that money held by everyone else circulates at a constant rate (called the velocity of money) then your argument shows that the proportion of the money supply that ends up in the hands of the government approaches 100% as time goes on.
To be precise, the amount of money not held by the government at time $t$ forms a geometric series that approaches zero as $t \to \infty$.
However, this situation does not reflect reality. The government spends money too, rather than holding on to it forever.  For example, the government spends money on payroll, entitlement programs, and buying goods from the private sector.
Now if we assume instead that the government spends money like everyone else, a somewhat different thing happens. Every dollar keeps circulating forever, and is taxed an unlimited number of times, so eventually the total amount the government has collected will exceed the money supply, and indeed will eventually exceed any fixed amount.  (But of course it will not have all this money at that time, because it will have already spent much of it.)
Eventually every dollar will have been involved in many transactions: sometimes being part of taxable transactions within the private sector, sometimes being paid to the government in taxes, and sometimes being paid back by the government to the private sector.
