Is money we make completely taken away by taxes?

In my opinion, any money we make will be completely taken away by taxes. Here is my logic.

When I earn some money (x), I have to give some as tax to the government. For example, let it be 10% of my income. After taxes, I have х - х * 10% = 0.9x of the money I earned. I will spend this money on my needs, and it will become someone's profit. Therefore, this person will also pay 10% to the government. In this way, the initial amount will remain 0,9х - 0,9х*10%=0,81x. Here is my calculation table:

As we can see, after 30 steps, only 4% of the initial amount will remain from this money. And this happens with any salary. In the next tax period, I will earn new money and the whole cycle will repeat itself.

My friends told me that I am wrong, and taxes don't take 100% of our money because after taxes we still have 90% of the money we earn. In my opinion, such a view is wrong, because it reflects only 1 present moment, not taking time into account. The money that remains in the hands of a specific person is just a temporary advance.

I also want to pay attention to the fact that our salary was also someone's income, therefore, compared to the employer, we are several steps down when it comes to money movement. In any case, the government with the help of taxes will take all 100% (close to 99.99% to be very precise) of our money, just not immediately, but a few steps later. So tax rate shows the speed with which the money will be withdrawn. For example, when tax rate is 30%, then money completely disappears in 10 moves.

Mathematically, everything seems to be true, but is everything like that in reality? People told me that I am wrong, but don't explain why, so where am I making a mistake?

• If you and your friends have money, and people can afford to buy stuff in general, your view must be wrong.... Feb 24 at 6:10
• By that reasoning, money is completely spent on goods/services as well. After all, the 90% you keep, you spend on goods/services. The shopkeeper who gets your money then spends 90% of that on goods and services. Furthermore, the taxes (10% by you, 10% by the shopkeep, etc) go to the government, and the government then also spends this money on goods and services! You could make a similar table showing that, over time, goods and services take 99.9999% of our money... Feb 24 at 10:19
• The question is based on an opinion that any money we make will be completely taken away by taxes. Unless no one has any money, the conclusion must be that it is a flawed opinion. Feb 24 at 13:16
• @AKdemy Let's make it simpler. I work in an orchard. On day one I pick an apple. Every day after first I pick one more apple. Every evening on day two onwards I have to give yesterdays apple away to my boss. When I die the apple that's left is taken by my boss. In this case every evening I have one apple. But every apple I ever pick will be taken away by my boss. See it now? Just because I have something at a point in time doesn't mean it doesn't eventually end up being taken by someone.
– DRF
Feb 24 at 13:25
• I showed this to my kids to instill the fear of going through life without a high school diploma.
– user43583
Feb 25 at 12:37

No, this will not happen.

Government does not tax people just for 'shits and giggles'. Governments do not hoard money at some pile as a some sort of dragon from a fantasy novel. Taxes are levied so government can spend that money on some public goods or to use them as transfers to alleviate poverty or inequality. As you correctly point in your question, your spending is someone's else income. Government spending works the same way. When government spends money on a new bridge it creates profits for the contractor, wages for the workers, rents for landowners and interest to capital owners. When it comes to transfers, transfers are not income per se but then spending of the people who receive transfers will create incomes for whatever business and its employees and investors the people getting transfers choose to patronize.

Sure if the government would keep all the money they tax, and if money supply would be kept constant by central bank, then at some point all the money would be eventually siphoned out of the economy and government would get 'all the money', while the economy would be in ruin because people would have to revert to barter and there would be no provision of basic public goods such as police etc. However, in real life governments rarely if ever avoid spending money they tax. In fact most real life governments are running relatively large debts (e.g. see OECD statistics) showing that most governments spend usually more than they even tax.

• I think what this demonstrates is yes, the government does eventually "take all money", but they also give it back, so it's more like: all money cycles through the government regularly. Feb 24 at 22:55
• @MooingDuck the question specifically refers to money completely disappearing (second to last para) so OP clearly uses "take all money" not in a sense that all money pass through government but simply end stuck there
– 1muflon1
Feb 25 at 10:16
• And here's the paradox: if we all spend 10% of every paycheck on food, eventually the grocery store gets all the money. So now the government and the grocery store each have all our money, which of course cannot be true. Feb 25 at 19:00

Wind the clock back in the other direction: where did this money come from in the first place? The government issued it. Render unto Caesar that which is Caesar's, etc.

You've also chosen to stop your analysis at the point where you pay into government. The government will promptly pay a lot of that straight out again as salaries, benefits, bond repayments, purchases of everything from office supplies to weapons.

Or you could make the same argument for some other provider of essential services: if everyone spends (say) 10% of their income on energy bills to keep the lights on, at some point the energy provider gets all the money. Yes .. but they spend it again?

It's a bit like arguing that eventually all the water in the world will have passed through the urinary tract of some animal: it may or may not be true, but it's not as exciting as it sounds either way.

• It's worth noting that many governments (e.g. USA, Japan) actually spend quite lot more than they collect in taxes. Feb 24 at 17:21
• Also worth nothing is the corruption that happens in government aka people in power. The Govt pay more than needed, for access, for privilege, for influence. While the sellers also increase their prices for similar reasons, because its the govt with seemingly bottomless coffers and little oversight. Mar 3 at 5:52

Your calculation is fundamentally wrong because you are forgetting an important factor - the money that government spends.

In your calculation you seem to think that the 10% the government takes in taxes vanishes into a black hole. But what actually happens is that it get paid to a teacher, or a soldier, or a road construction firm, and it enters the economy again.

So if you track all the individual dollars: yes, every one will eventually become a tax dollar and go to the government. But every one will also come out of the government and go to a person and start the cycle again.

(Incidentally you could make exactly the same calculation for Jeff Bezos - some fraction of all spending goes to Amazon and some of that gets taken as profit for Jeff Bezos; and so as money keeps circulating eventually all money will go to Jeff Bezos - although of course it will also be spent by him.)

• It's less clear that Bezos has any material desires left to spend money on. He could be satisfied with spending his first 100 billion for the rest of his life and hoard any more money he gets without spending it. Although he of course his net wealth is not generally made up of cash.
– bdsl
Feb 25 at 12:13

A distinction must be made, as also other answers suggest, between money and income.

If you speak of money, meaning the amount of money you have at some instant of time, and with which you pay taxes and buy consumption goods, giving this money to other people, you are right. Calculations similar to yours can say that at the end of the process all the money would come back to government as taxes.

We can show it formally.

Suppose that you have a dollar coming, for example, from a helicopter drop: a helicopter drops $$1$$ dollar and you take it. Suppose that the tax rate is $$10 \%$$.

You pay $$\frac{1}{10}$$ of taxes and spend $$\frac{9}{10}$$, giving it to another person to buy goods. This person has now $$\frac {9}{10}$$ dollars, pays $$\frac{1}{10} \frac{9}{10}$$ taxes and buys $$\frac{9}{10}\frac{9}{10}$$ of goods. The process continues infinitely, the government at each step takes $$\frac{1}{10}$$ of taxes from each person. The process of levying taxes can be described as follows:

$$T= \frac{1}{10} +\frac{1}{10} \frac{9}{10}+ \frac{1}{10} (\frac{9}{10})^2+...+ \frac{1}{10} (\frac{9}{10})^n+...=$$

$$= \frac{1}{10} [ 1+\frac{9}{10}+ (\frac{9}{10})^2+ (\frac{9}{10})^n+...]=$$

$$= \frac{1}{10} \sum_ 0^\infty (\frac{9}{10})^n \qquad (1)$$

where $$T$$ is the total amount of taxes the government levies.

The sum in formula $$(1)$$ is a geometric series of ratio $$\frac{9}{10}$$, and the theory of series says us that it converges to $$\frac{1}{1- 9/10}$$.

Therefore we can write $$(1)$$ as:

$$T =\frac{1}{10}\sum_ 0 ^\infty (\frac{9}{10})^n = \frac{1}{10} \frac {1}{1-9/10}=$$

$$=\frac{1}{10} \frac {1}{1/10}= \frac {10}{10}=1$$.

As you can see, the amount of taxes levied by government at the end will be $$1$$ dollar, exactly the dollar you had at the beginning.$$^1$$

$$\;$$

Things are different if we speak of income: at each step, when a person buys goods, this expenditure becomes the income of another person, and it is counted as income. At each step, therefore, the overall income increases.

Economic theory says that in this case the total income increases by $$\frac {1}{t}$$, $$^2$$ where $$t$$ is the tax rate, $$\frac{1}{10}$$ in our case.

So, at the end of the process the increase of overall income, generated by the initial dollar, call it $$\Delta Y$$, is

$$\Delta Y= \frac {1}{1/10}=10$$

Therefore, you can see that the total amount of taxes, $$1$$ dollar, is paid out of an overall income of $$10$$, so that the overall tax rate will be still $$10\%$$.

$$***$$

Obviously, all that holds ceteris paribus. If the government, at the same time, increases the money supply, through government spending, or the money supply increases through other means of money creation (including banks and foreign sector), it is of course possible that the total amount of money doesn’t decrease, it can remain the same or also increase.

$$^1$$ More rigorously, as we have an infinite process, we say that the sum converges to $$1$$ as the steps of the process approach infinity.

$$^2$$ This formula in economics is called the keynesian multiplier. More specificcally, in general the formula is $$\frac{1}{1- (1-t)c}$$, where $$c$$ is the rate of disposable income (income after taxes) we spend in consumption goods, that in our case is $$1$$, because we are supposing we don't save anything, we spend all our disposable income.

• You can prove with the same formula that all money becomes wages or company income at some point. It's meaningless. Feb 25 at 20:49
• This is only a formalization of the calculations of the OP, mathematics is an instrument that you can use for many purposes. The formula is an abstract formula of the theory of infinite series, that can be applied where it can be useful. It is as if you use additions or multiplications, you can apply it to many subjects. The argument of the OP is correct in mathematical terms, formalizing it more rigorously. If you want to prove in the same way other hypotheses, you can do it. Feb 25 at 21:05
• Without an interpretation of the result, a simple mathematical prove is meaningless. If you follow a given dollar, it may never reach the government. That's also true and seemingly contradicts your result, even though both are true. Interpretation is necessary. Feb 26 at 13:32
• Maybe you are right, but I can't see your point. The fact that I use a single dollar is to semplify the formalization, but you can do it with any amount of dollars. Or think as if $1$ dollar were the total amonut of money in the economy. That is the same reasoning of the keynesian multiplier, formally it is exactly the same, a geometric series that converges. Feb 26 at 13:43

I will be in partial opposition to other answers here, because I think that, technically, you are completely right!

Yes, government takes all your money... (back!)... The thing, which I find wrong in your reasoning is the statement that this money were even yours in the beginning. You actually do not own any money! You only temporarily hold it. Money is a government's product, which government lends to its people to make transactions (through Central Banks). Of course that, eventually, the government takes all this money back through taxes, because the government is the reason why money even exist in the first place (right now I mean those goverment's ones!). I would say it is like a central repository through which all the money flows and must flow.

If you have centralized money they will flow through the central.

That is my perspective of the problem.

• The money is among other things a store of value. If you didn't functionally own your money as if it were any other physical private property, then it wouldn't be useful as a store of value, and we'd all revert to barter. Maybe we can think of the money being on a kind of indefinite loan to private citizens for the purpose of storing their value and conducting transactions, but it's functionally equivalent to ownership. The question is less about who is entitled to "own" the money in the first place. Feb 24 at 17:23
• 1) Why do you think ownership is a necessary condition for having storage of value? Do you really own the trinket you have stored in your pockets during some war time? Or do you just hold it? I would vote for the latter. 2) In my opinion the whole conception of ownership is kinda problematic... For ownership to really exist there has to be some authority that issues the ownership. Other than that, ownership is in reality impossible to achieve. You can only temporarily hold stuff, determined by your power to hold it. Feb 24 at 17:29
• (I'm sure I'll get banned for daring to comment but) This is also true about everything, right? Such as shareholders. All money comes from the government and goes back to it, but also, all money comes from shareholders and goes back to them; all money comes from plumbers and goes back to plumbers; all money comes from farmers and goes back to farmers; etc. Feb 24 at 19:01

Let's look at this from a more theoretical point of view.

Consider Government G, Company C and Yourself Y.

Company C is a Monopolist for everything (or an aggregate of all companies, same thing), so the only company in existence.

Y is employed by C. (Y is doing all work or an aggregate of all workers)

Y receives 1 per period from C for providing work. Y spends all money on taxes and necessities (let's ignore savings). Say taxes are 10%.

C receives 0.9 from Y (necessities) and pays 0.09 in taxes.

G receives 0.19 in taxes from C and Y.

However, G also pays C for doing some work. As G may also not generate savings, G pays 0.19 * 1,11... to C and recovers 0,11... in taxes.

What you got now is an equilibrium. Now, you may ask, if G doesn't spend money, wouldn't all money be collected by G eventually and therefore all money go to taxes?

Yes!

Same if C or Y doesn't spend any money. Over time, they'll accrue all money in the system. But: none of C, G and Y can actually do that. Spending is necessary.

Therefore, the government doesn't have any more possibilities of gathering all money than you do. Actually, look around, companies and individuals are better at gathering money ;).

P.s. if you add savings you get the same thing but with an equilibrium over several periods unless someone saves forever, in which money actually does leave the cycle (money sink). However, that money sink can be anyone, not just the government.

• The text of your answer did not enlighten me and your C,Y notation is very unconventional, but +1 for the drawing!! Feb 26 at 19:54
• @Giskard it's a CGY drawing, the same stuff they use in Hollywood! Feb 26 at 20:10

TL;DR The fact that all money "gets taxed away in some sense" is uninteresting. How fast that process happens is interesting and whether you want it to be fast or slow depends on how much you believe that what the government does is value building.

I think pjc50's answer is really good in that it sums up

1. all (fiat) money "belongs" to the government (in as much as that is a thing). They create it so that they can use it to collect tax and than use it to buy things.
2. a similar argument can be made for any essential commodity (food/energy) if every person needs the commodity at some percentage then as the money gets passed between people you get "all" the money being used for the commodity.
3. the government doesn't hoard it, it just goes ahead and gives it back to "the people" in the form of paychecks, infrastructure, defense, healthcare, etc.

But there is a little more to it than that. For one taxes are much more ubiquitous, many/most transfers of money between private and even state actors are taxed in some way. Which does make the government a little more interesting.

Secondly the taxes are distributed based on the governments interests.

This is the real crux. Because money as such, isn't particularly useful. I can have many million Deutsche mark but their value will be rather negligible now they can't be converted. What is useful is the perceived value of the money by other people. Money is used to obviate the need to write a piece of code for your grocer when you want to buy two apples.

And the perceived valued of money is something that the government can and does influence based on how it spends the money it gets. If it creates things that are useless or destroys stuff, we all get poorer. If it uses the money to create and support useful things we might even get richer.

Since governments are universally pretty bad at managing money, how fast the money they give out gets back to them, is interesting and does have an effect on the economy and the "value of money".

So the fact that all money "gets taxed away in some sense" is uninteresting. How fast that process happens is interesting, and whether you want it to be fast or slow, depends on how much you believe that what the government does is value building.

PS: Value building in all kinds of sense by the way, maybe you want pro-abortion advocacy or you want healthcare for everyone or you want lot's of guns, or good family values, or even all of that. It might be lowering the actual amount of food you can afford, but if you've always been able to eat way more than you needed it to, you're happy now your neighbour can also afford to eat.

You are correctly calculating how much of the money from your paycheck will eventually flow to the government. But your interpretation, "any money we make will be completely taken away by taxes", is not the right conclusion.

Think of money like water flowing around in pipes.

There is money flowing from your employer to you, flowing from you to the government, flowing from you to the stores you shop in, and so on. The money flows around and around these pipes. This is how the economy works. Even the government is both receiving and spending money, so all the money flowing into the government is also flowing back out in other pipes.

We can also think about savings accounts as water tanks, and think of banks as more complicated tanks that can make loans, etc., but for now let's ignore the tanks and just think about the pipes, like your calculation, which also assumed nobody is putting any of their money into savings. And even if we include all these water tanks, water is still flowing all around the system if we take a long-term view where money flows both into and out of the savings accounts.

Does all water flow to the government?

Now, if we look at a molecule of water in this giant pipe network, what is the chance that it will flow to the government? This is what you are calculating. After you receive it, there is a 10% chance it will continue directly to the government, and a 90% chance it will flow somewhere else. But of course, after flowing somewhere else, there is still a 10% chance of flowing to the government, and so on. As you correctly see from your calculations, it is basically inevitable that sooner or later it will flow to the government.

Of course, it doesn't stop when it reaches the government. It will keep flowing around and around the economy, and sooner or later it will visit every part of the economy. It will even eventually come back to where it started (but it won't stop, it will just keep flowing around and around).

A negative view of taxes

If the government were a giant sinkhole where money would just disappear (so the water would just empty out of the pipes and not return), then your conclusion would be right: all the water would eventually drain out of the network because of taxes. But of course that is not what happens. The government spends the money (on salaries, army weapons, etc.), so the water keeps flowing around the economy.

A more positive view

For a more positive perspective, remember that as long as you are receiving and spending money, all the water in the system will eventually flow through your bank account, because it is part of the giant pipe network. So if you believe that "any money we make will be completely taken away by taxes", you should also believe that "all the money anybody makes will be completely paid into your personal bank account". It will just take a much much longer time for it to find its way there, because your bank account is much much smaller than the government's!

Money will not be completely taken by the government as long as government spends the money. Basic macroeconomic identity is GDP equation:

$$Y=C+I+G+NX$$

Where $$Y$$ is aggregate income, $$C$$ consumption spending, $$I$$ investment spending, $$G$$ government spending and $$NX$$ spending on net exports. Consumption spending is further function of disposable (after tax) income $$C(Y-T)$$. What is important here is that aggregate income also depends on government spending $$G$$, if government spends all money it gets $$T=G$$, then aggregate income won't change.

In step 1, you made \$1 as income. In step 2, you paid a tax of 10% and kept \$0.9

Now that money is yours. You don't pay taxes on that \$0.9 again. Eventually there is a cycle, if you see for some time.... where the governments directly and indirectly pump the money back to the people and then folks pay them back and the governments again get a percentage off that. Like you said governments are going to keep getting the money more and more, and people lose money. People would be having lesser money to buy stuff. This means governments keep getting lesser chunks. They want people to make more, so that they get more (as a percentage). They print more money and pay folks even more, so that people pay even more to the government back. (I am not an expert. This is just my take on this.) • Of course I dont pay tax again. But I will buy something from you for this 0.9, and it will be your income, so then you will pay tax on this money, and you will have 0.81 left, and everything will be as in the table. In the end, after all the exchanges, the government will receive the entire 1$ anyway Feb 24 at 3:15
• @TamilaAmbeon And aren't you leaving out the step where the government in turn passes on the money to someone else? So the money continues to circulate. This way the government actually receives more than "the entire $\$1$" because it can receive "the same money" over and over again. Based on some hypothetical initial$\$1$, the government can receive an arbitrarily large sum of tax from it, not just $\$1$. And lots of other people also receive this money along the way. Feb 24 at 13:37 What I think you're missing is that businesses are taxed on their net income, not their entire revenue. (I'm simplifying greatly.) So if you buy something for \$100, their cost to make that product would probably have been at least \$90, and their net income would be less than \$10. Then they pay less than \$1 in tax on that. As you keep going further along the supply chain, the amounts of taxes that come from your original payment get smaller and smaller. They'll never add up to the full \$100 you paid.