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Mathematical Proof that National Debt and the Deficit are Necessary (and therefore not 'Evil')

Citizens Oversight (2013-03-08) Ray Lutz

This Page: http://www.copswiki.org/Common/M1348
Media Link: http://youtu.be/Am0Na3lp-7c

Revised 2013-03-18

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The national economy, debts and deficits are largely totally misunderstood. People attempt to relate them to a family budget where you have to pay your bills and balance your budget. But the reality for the federal government is far different. Here, I will explain and "prove" that a balanced budget is not a reasonable goal, deficits are to be expected, and the national debt is largely not a concern.

The national economy is very complex. So we need to start with a simplified model and then gradually add complexity so you can hold the whole thing in your mind at one time.

We will initially consider a closed economy, one that has no trade deficits, no internal loans or debts -- transactions are all cash. There are two actors. The (federal) government, which by our constitution has the sole right to coin money, and the citizenry, who can trade with each other without limit. We will also ignore the Federal Reserve Bank for now, and assume that the government can create money when it needs it. The citizens are sufficient in number so they can provide almost anything in the economy that is needed, somewhat like a profile of our private sector. We will also assume the government must also fund certain activities, such as national defense, which the citizenry participate in providing and are paid to do so by the government.

Follow $100, closed economy.

First, let's consider $100, which is spent by the government to provide those services. To be simple, we will assume there is a flat tax rate of 30%, and taxes are levied instantly, and each transaction results in "income" that can be taxed.

On day 1, the government spends $100 to provide needed national defense services and the citizenry pays income taxes of 30% on that purchase. That leaves $70 circulating in the economy. In the next transaction $70 are spent and earned for services and products provided, and income taxes of 30% are paid on that leaving 0.7*70 = $49. And the circulation continues. We can abbreviate the process in the following table:

Step Govt Spends Govt Bank $ in Circulation Taxes
    0 0  
0 100 -100.00 100.00 0.00
1 0 -70.00 70.00 30.00
2 0 -49.00 49.00 21.00
3 0 -34.30 34.30 14.70
4 0 -24.01 24.01 10.29
5 0 -16.81 16.81 7.20
6 0 -11.76 11.76 5.04
7 0 -8.24 8.24 3.53
8 0 -5.76 5.76 2.47
9 0 -4.04 4.04 1.73
10 0 -2.82 2.82 1.21
11 0 -1.98 1.98 0.85
12 0 -1.38 1.38 0.59
13 0 -0.97 0.97 0.42
14 0 -0.68 0.68 0.29
15 0 -0.47 0.47 0.20
16 0 -0.33 0.33 0.14
17 0 -0.23 0.23 0.10
18 0 -0.16 0.16 0.07
19 0 -0.11 0.11 0.05
20 0 -0.08 0.08 0.03
21 0 -0.06 0.06 0.02
22 0 -0.04 0.04 0.02
23 0 -0.03 0.03 0.01
24 0 -0.02 0.02 0.01
25 0 -0.01 0.01 0.01

Of course, mathematically speaking, it is impossible to ever get all of it back because even if there is only a tiny bit left, taxing at 30% leaves an even tinier portion, etc., even though it is much less than a penny after 25 cycles or so. But if you spend a lot more, like a trillion instead of $100, then the portion that is less than a penny in this example is still significant. If you carry unlimited fractional amounts, you can never get it down to zero. Let me say that again. YOU CAN NEVER TAX IT ALL BACK! That's simply a mathematical fact.

Constant spending in a closed economy with constant average tax rate

Let's do this again but this time, let's have the govt consistently spent $100 in each period. I added another column, Tax Rate, so we can experiment with changing the tax rate.

Step Govt Spends Govt Bank $ in Circulation Avg Tax Rate Tax Revenue Budget Deficit
    0 0    
0 100 -100.00 100.00 30% 0.00  
1 100 -170.00 170.00 30% 30.00 70.00%
2 100 -219.00 219.00 30% 51.00 49.00%
3 100 -253.30 253.30 30% 65.70 34.30%
4 100 -277.31 277.31 30% 75.99 24.01%
5 100 -294.12 294.12 30% 83.19 16.81%
6 100 -305.88 305.88 30% 88.24 11.76%
7 100 -314.12 314.12 30% 91.76 8.24%
8 100 -319.88 319.88 30% 94.24 5.76%
9 100 -323.92 323.92 30% 95.96 4.04%
10 100 -326.74 326.74 30% 97.18 2.82%
11 100 -328.72 328.72 30% 98.02 1.98%
12 100 -330.10 330.10 30% 98.62 1.38%
13 100 -331.07 331.07 30% 99.03 0.97%
14 100 -331.75 331.75 30% 99.32 0.68%
15 100 -332.23 332.23 30% 99.53 0.47%
16 100 -332.56 332.56 30% 99.67 0.33%
17 100 -332.79 332.79 30% 99.77 0.23%
18 100 -332.95 332.95 30% 99.84 0.16%
19 100 -333.07 333.07 30% 99.89 0.11%
20 100 -333.15 333.15 30% 99.92 0.08%
21 100 -333.20 333.20 30% 99.94 0.06%
22 100 -333.24 333.24 30% 99.96 0.04%
23 100 -333.27 333.27 30% 99.97 0.03%
24 100 -333.29 333.29 30% 99.98 0.02%
25 100 -333.30 333.30 30% 99.99 0.01%

After a number of periods, the economic situation will stabilize, and almost will balance, but will never do so, just as in the case of a single spending period. The total govt debt equals the amount of money in circulation, which is (nearly) the amount the govt spends in any period divided by the tax rate. Thus, $100/.30 = $333.33. Please note that if we consider all money entered into the economy as the govt debt, then it is impossible to "pay off the debt," or we will have no money at all.

Stimulus: Budget can balance if spending goes up then down

If spending increases, and then returns to a stable figure, the budget can just barely balance. Note that long term, the amount of money in the economy is based on the average tax rate, but the short-term stimulus does help the economy balance IF it returns to a set value.

Step Govt Spends Govt Bank $ in Circulation Avg Tax Rate Tax Revenue Budget Deficit
    0 0    
0 100 -100.00 100.00 30% 0.00  
1 100 -170.00 170.00 30% 30.00 70.00%
2 100 -219.00 219.00 30% 51.00 49.00%
3 100 -253.30 253.30 30% 65.70 34.30%
4 100 -277.31 277.31 30% 75.99 24.01%
5 100 -294.12 294.12 30% 83.19 16.81%
6 100 -305.88 305.88 30% 88.24 11.76%
7 100 -314.12 314.12 30% 91.76 8.24%
8 100 -319.88 319.88 30% 94.24 5.76%
9 100 -323.92 323.92 30% 95.96 4.04%
10 110 -336.74 336.74 30% 97.18 11.66%
11 100 -335.72 335.72 30% 101.02 -1.02%
12 100 -335.00 335.00 30% 100.72 -0.72%
13 100 -334.50 334.50 30% 100.50 -0.50%
14 100 -334.15 334.15 30% 100.35 -0.35%
15 100 -333.91 333.91 30% 100.25 -0.25%
16 100 -333.73 333.73 30% 100.17 -0.17%
17 100 -333.61 333.61 30% 100.12 -0.12%
18 100 -333.53 333.53 30% 100.08 -0.08%
19 100 -333.47 333.47 30% 100.06 -0.06%
20 100 -333.43 333.43 30% 100.04 -0.04%
21 100 -333.40 333.40 30% 100.03 -0.03%
22 100 -333.38 333.38 30% 100.02 -0.02%
23 100 -333.37 333.37 30% 100.01 -0.01%
24 100 -333.36 333.36 30% 100.01 -0.01%
25 100 -333.35 333.35 30% 100.01 -0.01%

Notice above that in period 10, govt spending increased by 10%, then went back to the set value. As a result, there is a bit more money in circulation, which allows the budget to be in slight surplus for a while. When the system stabilizes, govt debt is the same.

Thus, the idea that a balanced budget is "good" is really immaterial in this case because the overall govt debt is the same. The only way we see a surplus is because there is a big additional deficit in the year of increased spending.

NOTICE THAT LONG TERM, the govt debt must equal spending divided by the average tax rate, and the deficit will follow to make it so.

Double spending, Tax rate the same

It turns out that even if spending is doubled, if it is consistent, the deficit will approach (but never reach) zero. But the amount of money in circulation doubles, which pumps up the economy.

Step Govt Spends Govt Bank $ in Circulation Avg Tax Rate Tax Revenue Budget Deficit
    0 0    
0 200 -200.00 200.00 30% 0.00  
1 200 -340.00 340.00 30% 60.00 70.00%
2 200 -438.00 438.00 30% 102.00 49.00%
3 200 -506.60 506.60 30% 131.40 34.30%
4 200 -554.62 554.62 30% 151.98 24.01%
5 200 -588.23 588.23 30% 166.39 16.81%
6 200 -611.76 611.76 30% 176.47 11.76%
7 200 -628.23 628.23 30% 183.53 8.24%
8 200 -639.76 639.76 30% 188.47 5.76%
9 200 -647.83 647.83 30% 191.93 4.04%
10 200 -653.48 653.48 30% 194.35 2.82%
11 200 -657.44 657.44 30% 196.05 1.98%
12 200 -660.21 660.21 30% 197.23 1.38%
13 200 -662.15 662.15 30% 198.06 0.97%
14 200 -663.50 663.50 30% 198.64 0.68%
15 200 -664.45 664.45 30% 199.05 0.47%
16 200 -665.12 665.12 30% 199.34 0.33%
17 200 -665.58 665.58 30% 199.53 0.23%
18 200 -665.91 665.91 30% 199.67 0.16%
19 200 -666.13 666.13 30% 199.77 0.11%
20 200 -666.29 666.29 30% 199.84 0.08%
21 200 -666.41 666.41 30% 199.89 0.06%
22 200 -666.48 666.48 30% 199.92 0.04%
23 200 -666.54 666.54 30% 199.95 0.03%
24 200 -666.58 666.58 30% 199.96 0.02%
25 200 -666.60 666.60 30% 199.97 0.01%

Half the tax rate, standard spending

Let's go back to fixed $100 spending per period and decrease the tax rate.

Step Govt Spends Govt Bank $ in Circulation Avg Tax Rate Tax Revenue Budget Deficit
    0 0    
0 100 -100.00 100.00 15% 0.00  
1 100 -185.00 185.00 15% 15.00 85.00%
2 100 -257.25 257.25 15% 27.75 72.25%
3 100 -318.66 318.66 15% 38.59 61.41%
4 100 -370.86 370.86 15% 47.80 52.20%
5 100 -415.23 415.23 15% 55.63 44.37%
6 100 -452.95 452.95 15% 62.29 37.71%
7 100 -485.01 485.01 15% 67.94 32.06%
8 100 -512.26 512.26 15% 72.75 27.25%
9 100 -535.42 535.42 15% 76.84 23.16%
10 100 -555.10 555.10 15% 80.31 19.69%
11 100 -571.84 571.84 15% 83.27 16.73%
12 100 -586.06 586.06 15% 85.78 14.22%
13 100 -598.15 598.15 15% 87.91 12.09%
14 100 -608.43 608.43 15% 89.72 10.28%
15 100 -617.17 617.17 15% 91.26 8.74%
16 100 -624.59 624.59 15% 92.57 7.43%
17 100 -630.90 630.90 15% 93.69 6.31%
18 100 -636.27 636.27 15% 94.64 5.36%
19 100 -640.83 640.83 15% 95.44 4.56%
20 100 -644.70 644.70 15% 96.12 3.88%
21 100 -648.00 648.00 15% 96.71 3.29%
22 100 -650.80 650.80 15% 97.20 2.80%
23 100 -653.18 653.18 15% 97.62 2.38%
24 100 -655.20 655.20 15% 97.98 2.02%
25 100 -656.92 656.92 15% 98.28 1.72%

In this case, it takes much longer to stabilize. There is still 2x funds circulating, comparable to doubling spending, but the deficit is much larger. If spending is fixed, the system will eventually stabilize at "nearly balanced" no matter what the spending level or tax rate, but it takes LONGER to stabilize if spending is too low or tax rate is too low.

Thus, if you care about the deficit, you will opt for increased spending rather than lower taxes, and you must expect that the budget will never quite balance, and this is if there is NO EXPANSION, which is not realistic.

Expanding Economy

As the population increases, there will be a requirement for an expanding economy. On average, the federal debt increases at about 6.3% a year.

Plot of total public and private debt with federal debt 1915 to 1985.jpg:
plot_of_total_public_and_private_debt_with_federal_debt_1915_to_1985.jpg

Now we will add some realistic changes to the economy. First, we will assume that the economy is expanding, and spending increases at a steady rate. We will set the expansion at a set 3% per period.

Step Govt Spends Govt Bank $ in Circulation Avg Tax Rate Tax Revenue Budget Deficit
    0 0    
0 100.00 -100.00 100.00 30% 0.00  
1 103.00 -173.00 173.00 30% 30.00 70.87%
2 106.09 -227.19 227.19 30% 51.90 51.08%
3 109.27 -268.31 268.31 30% 68.16 37.63%
4 112.55 -300.36 300.36 30% 80.49 28.48%
5 115.93 -326.18 326.18 30% 90.11 22.27%
6 119.41 -347.73 347.73 30% 97.85 18.05%
7 122.99 -366.40 366.40 30% 104.32 15.18%
8 126.68 -383.16 383.16 30% 109.92 13.23%
9 130.48 -398.69 398.69 30% 114.95 11.90%
10 134.39 -413.47 413.47 30% 119.61 11.00%
11 138.42 -427.85 427.85 30% 124.04 10.39%
12 142.58 -442.07 442.07 30% 128.36 9.97%
13 146.85 -456.31 456.31 30% 132.62 9.69%
14 151.26 -470.67 470.67 30% 136.89 9.50%
15 155.80 -485.27 485.27 30% 141.20 9.37%
16 160.47 -500.16 500.16 30% 145.58 9.28%
17 165.28 -515.40 515.40 30% 150.05 9.22%
18 170.24 -531.02 531.02 30% 154.62 9.18%
19 175.35 -547.06 547.06 30% 159.31 9.15%
20 180.61 -563.56 563.56 30% 164.12 9.13%
21 186.03 -580.52 580.52 30% 169.07 9.12%
22 191.61 -597.97 597.97 30% 174.16 9.11%
23 197.36 -615.94 615.94 30% 179.39 9.10%
24 203.28 -634.44 634.44 30% 184.78 9.10%
25 209.38 -653.48 653.48 30% 190.33 9.10%

You will note that the budget stabilizes with a consistent deficit. There is no way to fix this if the economy is expanding, even if tax rates are increased. Let's double the tax rates to try to get rid of the deficit.

Step Govt Spends Govt Bank $ in Circulation Avg Tax Rate Tax Revenue Budget Deficit
    0 0    
0 100.00 -100.00 100.00 60% 0.00  
1 103.00 -143.00 143.00 60% 60.00 41.75%
2 106.09 -163.29 163.29 60% 85.80 19.13%
3 109.27 -174.59 174.59 60% 97.97 10.34%
4 112.55 -182.39 182.39 60% 104.75 6.93%
5 115.93 -188.88 188.88 60% 109.43 5.60%
6 119.41 -194.96 194.96 60% 113.33 5.09%
7 122.99 -200.97 200.97 60% 116.97 4.89%
8 126.68 -207.07 207.07 60% 120.58 4.81%
9 130.48 -213.30 213.30 60% 124.24 4.78%
10 134.39 -219.71 219.71 60% 127.98 4.77%
11 138.42 -226.31 226.31 60% 131.83 4.76%
12 142.58 -233.10 233.10 60% 135.79 4.76%
13 146.85 -240.09 240.09 60% 139.86 4.76%
14 151.26 -247.30 247.30 60% 144.06 4.76%
15 155.80 -254.72 254.72 60% 148.38 4.76%
16 160.47 -262.36 262.36 60% 152.83 4.76%
17 165.28 -270.23 270.23 60% 157.41 4.76%
18 170.24 -278.33 278.33 60% 162.14 4.76%
19 175.35 -286.68 286.68 60% 167.00 4.76%
20 180.61 -295.28 295.28 60% 172.01 4.76%
21 186.03 -304.14 304.14 60% 177.17 4.76%
22 191.61 -313.27 313.27 60% 182.49 4.76%
23 197.36 -322.67 322.67 60% 187.96 4.76%
24 203.28 -332.35 332.35 60% 193.60 4.76%
25 209.38 -342.32 342.32 60% 199.41 4.76%

Thus, even with incredible 60% tax rates, the deficit continues to persist in an expanding economy. This is also the case with a TRADE DEFICIT, and the table will look about the same with a trade deficit as with an expanding economy. Note, however, that with the double tax rates, it cuts the amount of money in the economy by a factor of two.

Just for fun, let's reduce the tax rate to the lower 15% rate to see what happens.

Step Govt Spends Govt Bank $ in Circulation Avg Tax Rate Tax Revenue Budget Deficit
    0 0    
0 100.00 -100.00 100.00 15% 0.00  
1 103.00 -188.00 188.00 15% 15.00 85.44%
2 106.09 -265.89 265.89 15% 28.20 73.42%
3 109.27 -335.28 335.28 15% 39.88 63.50%
4 112.55 -397.54 397.54 15% 50.29 55.32%
5 115.93 -453.83 453.83 15% 59.63 48.56%
6 119.41 -505.16 505.16 15% 68.08 42.99%
7 122.99 -552.38 552.38 15% 75.77 38.39%
8 126.68 -596.20 596.20 15% 82.86 34.59%
9 130.48 -637.25 637.25 15% 89.43 31.46%
10 134.39 -676.05 676.05 15% 95.59 28.87%
11 138.42 -713.07 713.07 15% 101.41 26.74%
12 142.58 -748.68 748.68 15% 106.96 24.98%
13 146.85 -783.23 783.23 15% 112.30 23.53%
14 151.26 -817.01 817.01 15% 117.49 22.33%
15 155.80 -850.25 850.25 15% 122.55 21.34%
16 160.47 -883.19 883.19 15% 127.54 20.52%
17 165.28 -915.99 915.99 15% 132.48 19.85%
18 170.24 -948.84 948.84 15% 137.40 19.29%
19 175.35 -981.86 981.86 15% 142.33 18.83%
20 180.61 -1015.19 1015.19 15% 147.28 18.46%
21 186.03 -1048.94 1048.94 15% 152.28 18.14%
22 191.61 -1083.21 1083.21 15% 157.34 17.88%
23 197.36 -1118.09 1118.09 15% 162.48 17.67%
24 203.28 -1153.66 1153.66 15% 167.71 17.50%
25 209.38 -1189.99 1189.99 15% 173.05 17.35%

You can see that the amount of money in the economy is increased, as is the total govt debt, but it takes much longer for the economy to stabilize.

Lowering tax rates pumps up the amount of money in circulation but winds up with a higher deficit because it takes much longer to stabilize.

Effect of the Fractional Reserve Banking System

The fractional reserve banking system allows banks to lend more money than they have as deposits. Thus, if they have $100 deposited, then they can loan x times that amount, depending on the required reserve amount. Assuming a 10% reserve is required, they could lend $1000, and thereby magically create $900 of "IOU Money" based on the promise of the borrower. That 'new money' circulates through the economy and some people will earn that money as their income, and therefore they pay income taxes, and this can contribute to the overall tax revenue. So indeed, whenever the reserve amount is reduced and there is new "IOU money" created, then it can appear that much more money exists in the economy than was created by the govt. However, as those loans are repaid, the money "disappears." The banks can then make new loans up to the reserve limit, and the IOU money is recreated. So in a stable economy, the net result of the fractional reserve system is zero, in terms of its contribution to the ability of the govt to recover any new money spent.

If reserve requirements are decreased, say from 10% to 4%, this can provide what appears to be more money in the economy, and it may be possible to balance the budget for a short time using this "IOU money" to fund it. If you reduce it later, say from 4% to 10% reserve requirement, this will reduce the amount of IOU money, and the economy must go into a larger deficit situation.

In the financial crash of 2008, the financial industry had pushed "subprime" loans, and then they packaged those up into bundles that could be traded on the stock market. The reality is that real estate is not very liquid, and so it it is not compatible with the computerized trading schemes used in the stock market today. But these bundles are compatible. Speculation pushed home prices up to unbelievable levels because of the demand of subprime loans the the traders that were eager to package them up. Eventually, the bubble did burst, and it was discovered that many of the loans would never be paid off. When that happens, the IOU money instantly disappears, and thus the crash and recession.

The bottom line of the Fractional Reserve System is that it cannot help to balance the budget for any length of time UNLESS the total private debt is going up at the same time either through a steady reduction in the reserve requirement or far more loans made that will may look good at first but probably will not be paid off. So for purposes of considering the balanced budget, we have to ignore the impact of the fractional reserve system.

Sequestration

Sequestration and other austerity measures are absolutely the wrong thing to do and will not stimulate the economy or help the deficit situation over the long term. Cutting government spending will not move us toward a balanced budget as this will remove money from the economy at exactly the time when we need more spending. The mindset that the federal govt should balance the budget will drive the economy toward recession, and a "balanced approach" means that we run a deficit of about 7% of GDP.

CONCLUSION

A balanced budget will only work if you have steady and somewhat declining economy. The principles of austerity being promoting by many governments so as to try to achieve a balanced budget can only be achieved with a steady or declining budget. An expanding economy REQUIRES a deficit, no matter what level of taxation you use. Similarly, if you have a TRADE DEFICIT, you will wind up with a situation similar to steady expansion.

Furthermore, in the U.S., the government has delegated the creation of new money to the Federal Reserve Bank, which loans the money to the govt with interest. As you can see, there is no way that the overall debt can ever be paid off, and there is no source for the interest, and it will simply add the the overall govt debt.

This debt is meaningless, will never be paid off, is not shouldered by our children, etc. And it does not mean that "we are broke." Anyone who says that is trying to fool you.

There are two ways to increase the amount of money in circulation, cutting tax rates or increasing spending. Both have the same effect if allowed to stabilize. But higher spending with modest tax rates results in lower deficits faster.


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Title Mathematical Proof that National Debt and the Deficit are Necessary (and therefore not 'Evil')
Publisher Citizens Oversight
Author Ray Lutz
Pub Date 2013-03-08
Media Link http://youtu.be/Am0Na3lp-7c
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Forum Link https://www.facebook.com/raylutz/posts/382746195156609
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Keywords Budget And Taxation
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plot_of_total_public_and_private_debt_with_federal_debt_1915_to_1985.jpgjpg plot_of_total_public_and_private_debt_with_federal_debt_1915_to_1985.jpg manage 34.0 K 2014-06-26 - 17:23 Raymond Lutz plot of total public and private debt with federal debt 1915 to 1985.jpg
Topic revision: r12 - 2016-05-16, UnknownUser
 

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