If you are like most people, you probably think that your government takes too much of your money in taxes, yet you don't have any facts to back up that claim. There is also a common belief that when a government needs more money it can simply raise tax rates to generate more revenue. This page will show you that, yes, you are getting taxed way too much and it will show you the math leading to that conclusion. In addition, you will see just how little extra cash a government collects when it raises your taxes, but more importantly, you will see the devastating effects raising taxes has on the economy. It's not pretty!
To see the effects of taxation, we'll focus on a single economic event: you earn $1,000 for a week's worth of work at your job. We will follow the money as it works its way through the economy and into the government's coffers. We'll examine what happens when we vary a single variable which is the rate at which the government taxes your income. All other taxes will be ignored to keep it simple. Even this simple model will be very instructive.
Let's first look at what happens when tax rates are obscenely high at 90 percent. (See my explanation of working with percentages if you aren't comfortable with them.) You earned $1,000, so the government takes $900 of that, leaving you with only $100. You spend your $100 probably on food or some other essentials. The company where you spent your $100 uses your money to pay one of its own workers. Again, since the tax rate is 90 percent, the government takes $90 of their $100 salary, leaving the employee with only ten dollars to take home. That $10 doesn't get them much, but wherever they spent it, that company needs to pay a worker. Of the $10 wage earned, this employee gets to keep only $1, with the government taking $9.
Here is a table showing the amount of money spent in each transaction along with the amount of tax paid to the government. The last column shows a running total of the tax collected:
| Transaction | Money Spent | Tax Collected | Total Tax | |||
|---|---|---|---|---|---|---|
| − | − | $900 | $900 | |||
| 1 | $100 | $90 | $990 | |||
| 2 | $10 | $9 | $999 |
We can see from the table that the government has taken in a total of $999 so far, leaving only a single dollar of your pay still in circulation in the economy. Looking at the total amount of economic activity resulting from your getting paid, we see that it amounts to a mere $110.
As a second example, let's see what changes when the government takes half of your income. As before, you earned $1,000 for your hard work during the week; the only difference is the tax rate. The result is that you get to take home $500 with the government getting the other half. When you spend your $500, that money will work its way into the paychecks of the employees of the merchants you frequented. Those employees were paid the $500 you spent, so they take home $250 while the government takes its $250 share.
This process continues; each transaction is for half the amount of the previous one, because the government is taking half of what's left each time. Here again is a table showing the transactions that take place along with the tax collected:
| Transaction | Money Spent | Tax Collected | Total Tax | |||
|---|---|---|---|---|---|---|
| − | − | $500 | $500 | |||
| 1 | $500 | $250 | $750 | |||
| 2 | $250 | $125 | $875 | |||
| 3 | $125 | $62.50 | $937.50 | |||
| 4 | $62.50 | $31.25 | $968.75 | |||
| 5 | $31.25 | $15.63 | $984.37 | |||
| 6 | $15.63 | $7.81 | $992.19 | |||
| 7 | $7.81 | $3.91 | $996.09 | |||
| 8 | $3.91 | $1.95 | $998.05 | |||
| 9 | $1.95 | $0.98 | $999.02 |
We can see from the last column that the government has again ended up with all but about a dollar of your original $1,000 wage. It took a little longer, 9 transactions instead of 2, but it's clear that lowering the tax rate did not reduce the government's revenues at all.
Now take a look at the second column and add up the total amount spent in these nine transactions (hint: it's the same as the tax collected). The total amount of economic activity has risen from a mere $110 under the oppressive tax system to almost $1,000 with this system — the people are essentially nine times richer with taxes at 50 percent versus when they were taxed at 90 percent. And the government still took in the same amount of tax.
The 50 percent tax rate has some nice symmetry to it: you earned $1,000; the government collected $1,000 in tax; and $1,000 in economic activity was the result. Is this the best tax rate for the people and the government? It looks nice, but let's just suppose the tax rate were slashed to just 20 percent and see what results we find.
Sticking with the same example, changing only the tax rate to 20 percent, your $1,000 wage provides you with $800 in take−home pay. The government gets only $200 of it. This just has to mean the government gets less revenue with such a low tax rate, right? Let's make a table again as we did for the other tax rates and see what we find; the first 20 transactions are shown below:
| Transaction | Money Spent | Tax Collected | Total Tax | |||
|---|---|---|---|---|---|---|
| − | − | $200 | $200 | |||
| 1 | $800 | $160 | $360 | |||
| 2 | $640 | $128 | $488 | |||
| 3 | $512 | $102.40 | $590.40 | |||
| 4 | $409.60 | $81.92 | $672.32 | |||
| 5 | $327.68 | $65.54 | $737.86 | |||
| 6 | $262.14 | $52.43 | $790.28 | |||
| 7 | $209.72 | $41.94 | $832.23 | |||
| 8 | $167.77 | $33.55 | $865.78 | |||
| 9 | $134.22 | $26.84 | $892.63 | |||
| 10 | $107.37 | $21.47 | $914.10 | |||
| 11 | $85.90 | $17.18 | $931.28 | |||
| 12 | $68.72 | $13.74 | $945.02 | |||
| 13 | $54.98 | $11.00 | $956.02 | |||
| 14 | $43.98 | $8.80 | $964.82 | |||
| 15 | $35.18 | $7.04 | $971.85 | |||
| 16 | $28.15 | $5.63 | $977.48 | |||
| 17 | $22.52 | $4.50 | $981.99 | |||
| 18 | $18.01 | $3.60 | $985.59 | |||
| 19 | $14.41 | $2.88 | $988.47 | |||
| 20 | $11.53 | $2.31 | $990.78 |
Just as we saw when the tax rate was lowered from 90 percent to 50 percent, the government still makes the same amount of money in tax revenue when the rate is lowered all the way to just 20 percent (10 more transactions are needed to reach $999 in revenue). It just takes more time since more transactions need to take place. But again take a look at the second column (money spent) and add up the total amount of economic activity that resulted from your earning $1,000. Actually I can just tell you that it approaches $4,000!
That is a significant finding — it is as if money is created out of thin air. You got paid only $1,000 (and only got to keep $800 of it), and yet four times as much economic activity took place. This could be called the "Cold Fusion" of economics. Is there any limit to this? Could we lower taxes even further and get an even bigger multiplier effect? The answer is a qualified yes. The cost of gaining this multiplier effect is a lengthening of the time it takes for the government to collect all of the tax. The good news is that we can come up with some equations that put a limit on the number of transactions to see just how much we can lower taxes with no significant negative impact on government revenue.
We'll start by defining the variables that will make up our equations. First we need to represent the tax rate, which we'll call r. To aid in understanding, we'll also use real numbers along with the variables, so we'll set r to 20 percent:
| tax rate = r = 0.2 |
The government takes 20 percent of your income in taxes, so you get to keep 80 percent, which is equal to:
| (1−r) = 0.8 |
Your original $1,000 income gets multiplied by r to determine how much the government gets in taxes and by (1−r) to determine how much you get to spend. As we saw above, the numbers are $200 tax and $800 take-home pay. This is before any purchases are made (transactions).
The only other variable needed for our equations is the number of transactions that take place, which we'll call N. If we set N = 1, then only one transaction occurs which involves you spending your $800. That $800 becomes other people's salary: $800 · r goes to the government and $800 · (1−r) is paid to the employees.
Let's build a table that illustrates what's happening, but we'll keep the entries in terms of the tax rate r. The initial economic event is your earning $1,000 and the government taking its share. So far no purchases have been made, so the table looks like this:
| Transaction | Money Spent | Tax Collected | Take-Home Pay | |||
|---|---|---|---|---|---|---|
| − | − | 1000 · r | 1000 · (1−r) |
The first transaction consists of you spending your take-home pay; that money ends up as wages for employees, so the government takes its cut. The following table adds this transaction to the record:
| Transaction | Money Spent | Tax Collected | Take-Home Pay | |||
|---|---|---|---|---|---|---|
| − | − | 1000 · r | 1000 · (1−r) | |||
| 1 | 1000 · (1−r) | 1000 · (1−r) · r | 1000 · (1−r)2 | |||
| [= $800] | [= $160] | [= $640] |
I realize that this looks complicated, but there is already a pattern emerging: the money spent in a transaction (second column) is just the take-home pay from the previous line. The tax collected is the previous line's tax multiplied by 80 percent (1−r). The take-home pay of the next round of employees is also 80 percent of the previous line.
The second transaction is exactly the same as the previous one, but with an added factor of (1−r) in each table entry. This is because only 80 percent of the money from the previous transaction is still available for purchases; the rest has been collected as tax by the government:
| Transaction | Money Spent | Tax Collected | Take-Home Pay | |||
|---|---|---|---|---|---|---|
| − | − | 1000 · r | 1000 · (1−r) | |||
| 1 | 1000 · (1−r) | 1000 · (1−r) · r | 1000 · (1−r)2 | |||
| 2 | 1000 · (1−r)2 | 1000 · (1−r)2 · r | 1000 · (1−r)3 | |||
| [= $640] | [= $128] | [= $512] |
Just to be sure, let's add one more transaction to the table:
| Transaction | Money Spent | Tax Collected | Take-Home Pay | |||
|---|---|---|---|---|---|---|
| − | − | 1000 · r | 1000 · (1−r) | |||
| 1 | 1000 · (1−r) | 1000 · (1−r) · r | 1000 · (1−r)2 | |||
| 2 | 1000 · (1−r)2 | 1000 · (1−r)2 · r | 1000 · (1−r)3 | |||
| 3 | 1000 · (1−r)3 | 1000 · (1−r)3 · r | 1000 · (1−r)4 |
Now we come to the fun part; we need to add up the two middle columns to determine the total amount of money spent (economic activity) and total tax collected, given the tax rate r and the number of transactions (N). The total economic activity is the sum of the second column:
| Economic Activity = 1000 · [(1−r) + (1−r)2 + ... + (1−r)N] |
There is a well-known compact equation for finding the sum of increasing powers of a number ((1−r) in this case), which I won't show here, but if you are interested in seeing how it is derived, see my page on the sum of powers. Skipping to the answer we have:
| Economic Activity = 1000 · | (1−r) | · | [1 − (1−r)N] |
| r |
Let's plug in the numbers to make sure that this is indeed the equation we're looking for:
| Economic Activity = 1000 · | 0.8 | · | [1 − (0.8)N] |
| 0.2 |
When N=1, the economic activity calculates to $800, and when N=2, it is $1440, which is exactly as we expect. An important thing to notice is that the factor (0.8)N gets smaller as N increases. In fact for any arbitrarily small value greater than zero, you can find a value for N such that (0.8)N is less than that value. In math lingo we say that (0.8)N approaches zero as N gets larger, though for any finite value of N it is never quite zero.
To express this observation in mathematical notation, we write:
| lim | (0.8)N = 0 |
| N → ∞ |
This equation is read, "the limit as N approaches infinity of the expression is zero." Using this information, we can see that for large values of N, the amount of economic activity approaches:
| Economic Activity → 1000 · | 1−r | = 1,000 · | 0.8 | = $4,000 |
| r | 0.2 |
(1−r)/r is called a multiplier since it amplifies the amount of economic activity. Here is a table showing the multiplier in action. The second column shows the limit of the amount of economic activity that can occur when you earn $1,000 for a variety of tax rates. The final column is the inverse of the multiplier which we'll call the effective tax rate:
| Tax Rate | Impact of $1,000 | Effective Tax Rate | ||
|---|---|---|---|---|
| 75% | $333.33 | 300% | ||
| 60% | $666.67 | 150% | ||
| 50% | $1,000.00 | 100% | ||
| 40% | $1,500.00 | 67% | ||
| 35% | $1,857.14 | 53.8% | ||
| 30% | $2,333.33 | 42.9% | ||
| 25% | $3,000.00 | 33.3% | ||
| 20% | $4,000.00 | 25% | ||
| 15% | $5,666.67 | 17.6% | ||
| 10% | $9,000.00 | 11.1% | ||
| 5% | $19,000.00 | 5.26% | ||
| 1% | $99,000.00 | 1.01% |
The last column shows the tax rate as a proportion of economic activity. You may have noticed that in our example the government is always taking money out of the economy and not replenishing it. If that continued indefinitely, there would eventually be no money left in circulation, and transactions could not take place. It is this last column that shows how much money the government needs to put back into the economy, as a proportion of the total activity.
Consider the case of a tax rate of 50 percent. From the above table we see that the effective tax rate is 100 percent. What does that mean? It means that the government needs to put $1,000 back into the economy for every $1,000 of economic activity that takes place (since it removed $1,000 in tax). It does this by funding public services such as national defense and public education. In order for the economy to function, the size of the government needs to be as big as the entire economy!
And with the government controlling where every dollar is spent, it becomes difficult for small businesses to be created. Consider instead a tax rate of 20 percent (effectively 25 percent). The government steers $1,000 toward its programs, yet there is an extra $3,000 in residual economic activity left over for the people to decide how the money is spent. Lower taxes are essential for entrepreneurs who share the American Dream.
Whenever there is talk of starting a new government program, it is followed by talk of raising taxes to pay for it. If the program is worthwhile, the theory goes, then perhaps the people won't mind a tax increase. Here we will look at exactly how much extra tax revenue is collected when taxes are raised. Going back to the table of transactions and adding up the column of tax collections we find that the total tax collected is:
| Tax Collected = 1000 · r · [1 + (1−r) + (1−r)2 + ... + (1−r)N] |
This equation also contains a sum of successive powers which has a compact solution:
| Tax Collected = 1000 · [1 − (1−r)N+1] |
As we saw above, the term (1−r)N+1 gets smaller and smaller as the number of transactions, N, increases. Therefore the total tax collected has an upper limit of $1,000. Using the terminology from above, we would say that the total tax collected approaches $1,000 as N approaches infinity. This makes sense because only $1,000 was paid to you in weekly salary; there just isn't any more money available. Varying the tax rate only results in changing how fast the government collects all the money. And as we saw above, it has a profound effect on economic activity.
Here is a graph showing the tax revenue collected by the government at each tax rate. We start off with no transactions having occurred (N = 0) and see that tax collected is linearly increasing with the tax rate. It is this picture people have in mind that leads them to believe that increasing the tax rate leads to more government revenue.
| Tax Revenue | ||
| 0 | 50 | 100 |
| Tax Rate | ||
| Transactions: |
Now press the "Add Transaction" button to increase N by 1. The graph now shows the total amount of tax collected after you've spent your income. The lighter color represents the tax collected from the previous transaction. The total is the sum of the income tax you paid when you earned your $1,000 plus the tax paid by the employees of the merchants where you spent it.
Continue pressing the button to see how total tax revenue changes as the number of transactions increases. You will notice that the curve gets flatter and flatter at the top, meaning that raising the rate in that area actually yields little additional revenue.
When you reach 10 transactions, you will see that the government can extract more than 90 percent of the money from the economy when the tax rate is just 20 percent! And at that rate, the government collects all of the money after a total of only about 24 transactions.
Now we will put all of these equations to use to see what the effects are of changing tax rates. This first calculator shows the tax revenue collected and economic activity created as a result of your getting paid $1,000 for a week's work. You can vary the tax rates in the top row and also the number of transactions that occur in the left-hand column. If you are unclear what is meant by a transaction, please read the first part of this page which explains it fully.
This next calculator could more appropriately be labeled, "Raising Taxes: How to Kill a Vibrant Economy." Select values for the current tax rate, change in tax rate, and number of transactions. The calculator will then show you the increase or decrease in tax revenue collected by the government as well as the change in economic activity that will result from your earning $1,000 for a week's work.
Playing around with the numbers will show you that changing tax rates has very little effect on the amount of tax collected. However, you will see that economic activity is greatly affected. Increasing tax rates from 30 percent to 40 percent kills off more than a third of the economy! That would mean higher unemployment, more jobs moved overseas, etc. But on the other hand if the tax rate is lowered from 30 percent to 20 percent, the economy would get an adrenaline shot in the form of almost 70 percent more money flowing. The downside being that the government collects slightly less revenue (not even 1 percent less). Seventy percent upside with only one percent downside seems like a simple trade-off to make.
