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 Author Topic: Math problem: When would the freicoin money supply converge?  (Read 7051 times)
jtimon
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 April 26, 2011, 10:36:41 AM

Imagine there's a bitcoin-like network with the following difference:

-Freicoins have an annualized demurrage of 3%. Whenever a transaction is made, the demurrage is charged proportionally to the number of blocks added to the chain since the last move (it is not charged annually, but for each block generated).
-The Freicoins issued with each block is constant through time (forever).
-The first freicoin block is issued at the beginning of 2012 (01/01/2012 00:00:00)

Case 1)
-1 new freicoin is issued with each block.

When will the amount of freicoins destroyed by demurrage equal the 1 btc generated per block? [2 btc]
That is, when the total supply of freicoins will converge? [5 btc]
What will be the total amount of freicoins then?

Case 2)
-The total amount of freicoins will converge in 21 millions at the same time that the bitcoin network does (¿2033?).

How many freicoins have to be issued with each block to achieve this? [5 btc]

I don't have the calculus skills to do it myself, but I understand usually mathematical demonstrations. Don't answer with just the number, please, explain how you've get there.

Feel free to ask whatever you don't understand or you think needs clarification.

http://www.freicoin.org/

2 different forms of free-money: Freicoin (free of basic interest because it's perishable), Mutual credit (no interest because it's abundant)
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jtimon
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 April 26, 2011, 10:38:13 AM

I forgot to mention that the time target is 10 minutes between blocks, just as in bitcoin.

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Lemmih
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 April 26, 2011, 02:17:25 PM

Case 1)
When will the amount of freicoins destroyed by demurrage equal the 1 btc generated per block? [2 btc]
That is, when the total supply of freicoins will converge? [5 btc]

With 52560 new coins per year from mining and a 3% demurrage, the growth of freicoins is defined as such: y' = 52560 * e^(-0.03x)

The total number of freicoins for a given year must then be: y = integrate(y', 0, x)

Using those definitions we can calculate that 50% of the coins will be generated in a little over 23 years, 90% of the coins will be generated in nearly 77 years and it will take 153 years to generate 99% of the coins.

What will be the total amount of freicoins then?

The number of coins produced approaches 1,752,000 as time approaches infinity. To calculate the limit, use this equation: 52560 blocks per year * coins per block / demurrage.

Case 2)
-The total amount of freicoins will converge in 21 millions at the same time that the bitcoin network does (¿2033?).

How many freicoins have to be issued with each block to achieve this? [5 btc]

You can't change both the coin limit (21 million) and when the limit is reached (2033, 21 years from 2012) by only changing how many coins are issued.

By issuing around 25.6 coins per block you will hit 21 million in 21 years but the max number of freicoins will be ~44,900,000. However, with a demurrage of 17% and 70 coins per block, you'll hit 21 million in 21 years with a limit of 21.6 million coins.

PM me if you feel that I deserve the reward(s).
jtimon
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 April 26, 2011, 04:00:50 PM

It seems that you deserve the reward. Although I must admit that I'm not really sure.
If it doesn't bother you, I'll make you some more request.
I'll change the second case to this:

case 2) How long would take to converge at 21 millions with 3% demurrage?

case 3) How many freicoins per block and in total for the convergence to occur at the same time of bitcoin's money supply?

What would be the general formula?

y' = 52560 * e^(-0.03x)

I assume
x = time in years
and 52560 = blocks per year

y = integrate(y', 0, x)

Sorry, I don't understand that notation.

I'd like the formula to have in it as variables:

-The coins per block
-The demurrage
-The time of convergence
-The total supply in convergence

Of course, this is outside the deal, you could reject to answer all this.
I don't have any BTCs now because I was dumb enough to sell them just before the last huge rise, but I'll buy them and pay you when you gime an address.

Thank you

2 different forms of free-money: Freicoin (free of basic interest because it's perishable), Mutual credit (no interest because it's abundant)
Meni Rosenfeld
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 April 26, 2011, 06:10:00 PM

You can't converge to the limit in a finite amount of time unless you're rounding, in either Bitcoin or Freicoin. I think it will take 128 years for the Bitcoin block reward to be rounded to 0.

If y'=52560 exp(-0.03x) (how accurate this is depends on your exact specification) then
y = 1752000 (1 - exp(-0.03x))
x is indeed the time in years since the beginning.

If B is the coins per block, Y is the number of blocks per year (which is closer to 52595 on average), and the demurrage per block is r/Y, then the total number of coins in convergence is C = B*Y/r. This is easy to see; at convergence, the coins demurred each block, which is C*r/Y, must equal the coins generated in the new block, which is B, so C*r/Y=B and C=B*Y/r.

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Lemmih
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 April 26, 2011, 06:48:34 PM

It seems that you deserve the reward. Although I must admit that I'm not really sure.
If it doesn't bother you, I'll make you some more request.
I'll change the second case to this:

case 2) How long would take to converge at 21 millions with 3% demurrage?

Well, it will never hit 21 millions. Like cutting something in half will never make it disappear altogether.

If you want a limit of 21,000,000 and a demurrage of 3%, the number of bitcoins added per block should be: 875/73 or ~12. This can be calculated with the equation I posted previously: max number of coins = blocks per year * coins per block / demurrage.
We can now use our CAS software to figure out how many years it'll take to generate X% of the coins: 50% is generated in 23.1 years, 90% in 76.8 years, 99% in 153.5 years, 99.9% in 230.2 years, and so on.

case 3) How many freicoins per block and in total for the convergence to occur at the same time of bitcoin's money supply?
What would be the general formula?

y' = 52560 * e^(-0.03x)

I assume
x = time in years
and 52560 = blocks per year

y = integrate(y', 0, x)

Sorry, I don't understand that notation.

I'd like the formula to have in it as variables:

-The coins per block
-The demurrage
-The time of convergence
-The total supply in convergence

Yes, 'x' is time in years and 52560 is indeed the number of blocks per year (which is not quite accurate as Holy-Fire points out).
Y is the total supply of coins for a given year and is defined (it is the antiderivative of y') as: y = 1752000*(1-e^(-0.03x))
jtimon
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 April 27, 2011, 07:56:10 AM

Thank you very much.
Lemmih, if it's ok with you, I think I should share out the bounty. As you answered first and you were right, what about 8 for you and 4 for holy-fire (6/6 if you feel generous)?

2 different forms of free-money: Freicoin (free of basic interest because it's perishable), Mutual credit (no interest because it's abundant)
Lemmih
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 April 27, 2011, 12:15:38 PM

I'm totally fine with a 6/6 split. My address is 1JMrMgWsWVKuSxESAg8VSXvhjCvBN3NCBD.
jtimon
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 April 27, 2011, 01:38:52 PM

Uups...I don't know what happened. I told mtgox to buy 12 but it did just bought 11.932. I will owe you 0.07 btc until I add more funds. I cannot set a buy order with 9.318 usd

04/27/11 13:29   Bought BTC   9.55 for 1.9123   9.548   -18.259   11.932   9.318
04/27/11 13:29   Bought BTC   2.38 for 1.912   2.375   -4.541   2.384   27.577

I just can believe what I did last week on Monday.
I'll send you the coins when I'm home. Can't do it from work.

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jtimon
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 April 27, 2011, 10:49:01 PM

lemmith, your 6 btc are sent.

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Lemmih
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 April 28, 2011, 01:46:15 AM

Meni Rosenfeld
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 April 28, 2011, 04:37:35 AM

lemmith, your 6 btc are sent.
I'm ok with receiving any amount from 0 to 21M. Whatever amount you choose to pay me, you can send it to the address in my signature.

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jtimon
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 April 28, 2011, 04:48:42 PM

Paid 5.93

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jtimon
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 April 28, 2011, 04:51:20 PM

By the way, what do you think about those freicoins?
Feel free to criticize them. Nobody seem to like demurrage here.

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Lemmih
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 April 28, 2011, 06:22:48 PM

Providing extra-market incentives for people to spend rather than save - to consume now rather than later - will lead to inefficiencies and misallocations. Purchasing power naturally changes over time and the market rate of the currency accurately represents this. Hence, any monetary grain associated with deferring consumption reflects a genuine increase in the quantity and/or quality of produced goods/services and is not merely an artifact of the currency's non-depreciating nature.

Since the temptation to consume now approaches infinity as the savings rate goes up, there is no feedback loop.

Price stability is desirable in many situations and should be bought at the market rate.

This is the Austrian position as I understand it.
Meni Rosenfeld
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 April 28, 2011, 06:23:01 PM

By the way, what do you think about those freicoins?
Feel free to criticize them. Nobody seem to like demurrage here.
Demurrage is equivalent to simply increasing the block reward by the same percentage each year. I don't feel strongly about this either way, but I think that since we can't divine what the future growth rate of the economy will be, it's better to make the currency non-inflative rather than setting an arbitrary inflation rate.

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jtimon
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 May 04, 2011, 11:40:17 PM

@Lemmih
Hoarding is not the only way of saving. You can lend the money or store things that you will consume in the future (like Robinson saving fish). You could also save by selling your freicoins for bitcoins and hoarding them.

@Holy-Fire
I don't understand why you call it inflationary if the monetary base is constant. The point is to charge hoarding to make transactions cheaper. I must admit that I cannot think of a demurrage rate/units generated per block that is not arbitrary.

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Meni Rosenfeld
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 May 05, 2011, 03:46:33 AM

@Holy-Fire
I don't understand why you call it inflationary if the monetary base is constant. The point is to charge hoarding to make transactions cheaper. I must admit that I cannot think of a demurrage rate/units generated per block that is not arbitrary.
Because what matters is the amount of money you store relative to the total money in existence. This varies in the same way whether new money is generated or your own money decays.

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jtimon
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 May 05, 2011, 08:00:49 AM

@Holy-Fire
I don't understand why you call it inflationary if the monetary base is constant. The point is to charge hoarding to make transactions cheaper. I must admit that I cannot think of a demurrage rate/units generated per block that is not arbitrary.
Because what matters is the amount of money you store relative to the total money in existence. This varies in the same way whether new money is generated or your own money decays.

But with a fixed monetary base, there is not likely to be price inflation. If the growth (which leads to price deflation) equals the demurrage, the purchasing power of your money would remain constant. If the demurrage were less than the growth, your purchasing power would increase with time.

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Meni Rosenfeld
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 May 05, 2011, 08:57:47 AM

@Holy-Fire
I don't understand why you call it inflationary if the monetary base is constant. The point is to charge hoarding to make transactions cheaper. I must admit that I cannot think of a demurrage rate/units generated per block that is not arbitrary.
Because what matters is the amount of money you store relative to the total money in existence. This varies in the same way whether new money is generated or your own money decays.

But with a fixed monetary base, there is not likely to be price inflation. If the growth (which leads to price deflation) equals the demurrage, the purchasing power of your money would remain constant. If the demurrage were less than the growth, your purchasing power would increase with time.
The same will happen if the growth equals the inflation. Hence, demurrage is equivalent to inflating the block reward.

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