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Investment multiplier mathematically reviewed

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7/15/2015 7:49:17 PM
Posted: 2 years ago
Investment multiplier formula can be expressed as:

accumulated income from government investment = initial investment/savings rate

Please reconstruct the spreadsheets on your computer to understand my two spreadsheets. This board has hard-to-crack constraints. It is more important that it may look to you.

Let's set savings rate = 10% and initial investment = 1,000 and see how many turns of money is necessary to result in what the formula means.

Column 1: turn of money (money flow from hands to hands)
Column 2: income at that turn of money
Column 3: savings 10%
Column 4: consumption 90%
Column 5: accumulated income through that turn.

1st 1,000 100.0000 900 1,000
2nd 900 90.0000 810 1,900
3rd 810 81.0000 729 2,710
58th 2 0.2465 2 9,978
59th 2 0.2219 2 9,980
60th 2 0.1997 2 9,982

As shown above, it takes about 60 turns of money.
Now, let's set savings rate = 1% and initial investment = 100 to produce the same result.

Column 1: turn of money (money flow from hands to hands)
Column 2: income at that turn of money
Column 3: savings 1%
Column 4: consumption 99%
Column 5: accumulated income through that turn.

1st 100.00 1.0000 99.00 100
2nd 99.00 0.9900 98.01 199
3rd 98.01 0.9801 97.03 297
58th 56.39 0.5639 55.83 4,417
59th 55.83 0.5583 55.27 4,473
60th 55.27 0.5527 54.72 4,528
598th 0.25 0.0025 0.25 9,975
599th 0.25 0.0025 0.24 9,976
600th 0.24 0.0024 0.24 9,976

As shown, it takes 600 turns of money to result in what the same formula means.

These two calculations signify that the formula means differently, in terms of the number of turns of money required, according to the rate of savings. Every turn of money requires time. Therefore, it can be rightfully said that the investment formula does not have time factor.

If the savings rate is zero, the investment multiplier is infinite. It means that a government investment creates infinite amount of income no matter what the amount of investment is. It is true, according to the formula itself, because infinite number any amount becomes infinite. A hundred dollars/year times 1,000 years and a thousand dollars/year times 100 years are the same 100,000 dollars.

Therefore, investment multiplier means production for a long time is multiple times of production for a short period.

Time is not ignorable in the real world because the government debt grows by interest over time. Maximization of investment multiplier requires minimization of savings. People must spend all the money after tax. The government must spend all the taxes collected. There is no way to repay the principal and interest of government debt, if we are to maximize the investment multiplier.

Time is not ignorable also because consumption is negation of value while income is addition of value. The kind of income that is a part of investment multiplier is getting created every day without government spending. From the beginning of human existence, we created value constantly. We forgot all of that because we consumed most of it. The only thing within our touch is what is remaining, which is savings in another term. Keynesian income is like walking without proceeding, because savings is cursed.

Let's ask one serious question. Is it possible that Keynes considered time factor while formulating the multiplier but deliberately ignored it because he thought it was not important?

One time, somebody argued that Keynes was concentrating on short-term economic matters only and that long-term prospects should be considered. Keynes replied, "In the long run, we all die." This conversation means that Keynes was simply not interested in time factor.

When time is forgotten, one can easily be trapped by the famous Zeno's Paradox of Achilles and the Turtle. If time does not flow enough for Achilles to catch up with the Turtle, then Achilles cannot catch up with the Turtle. Investment multiplier can mean something to us only if time flows unequally: if stops for negation of value during consumption, stops for interest accrual, stops for due date of principal repayment, and flows freely for income. The multiplier is a variation of Zeno's Paradox.