The Bitcoin Price Model – Large Time Calculations of the Bitcoin Price



Just for a toy model, to start us off, I used excel to insist the log data is strictly linear, implying the price follows a perfect exponential curve.  We observe bitcoin hits $1000 on precisely….ok lets not get carried away.  The Price cannot be a pure exponential function of time – that seems to violate some economic conservation laws.  But these are still fun plots.  The source is an excel sheet which you can find here

A somewhat sophistocated model for the price of bitcoin based on diffusion

In a previous post I suggested that the log linear behavior of bitcoin was a trend that will continue and proposed a model by which this would operate.  Here is a gogle drive containing the mathematica code I wrote to investigate the matter.  These equations do support the long term log linear behavior and produce price curves which seem to match the behavior we have seen from bitcoin.

You will be able to manipulate a parameter controlling the effective bitcoin supply.

The rules I used to derive the formulas were:

1) the Bitcoin demand D(t) is proportional to the probability that somebody on earth has learned of Bitcoins existence.  This function is an exponential in t for small times

D(t) ~ exp[t / to]

However the probability question has vastly distinct long time behavior.  The function that really represents the long term demand of bitcoin over time is closely related to the hyperbolic tangent 


This family of functions has the form

D(t) = 1 / ( 1 + exp[- t / t0] )

2) The Bitcoin supply is known in the short term to be proportional to t, but since the number of bitcoins generated per unit time halves every fixed amount of years, dictated by the protocol, I said we should model the supply as

S(t) = t * 2^(-t / t’)

Where t’ is in general a parameter that can be adjusted to dictate the effective supply of bitcoin as a function of time.

This incredibly simplistic model was in fact successful at generating the family of curves that the bitcoin price and its log seem to belong to.

Perhaps in the future more sophisticated models for the supply curve can be employed, since it seems clear the supply curve is responsible for the volatility.  See tables beneath:

Anyone who has mathematica ready to can run this code.  An animation can be generated as well as a dynamic plot you may manipulate.  A simple table of outputs for the price and log of the price is given in the above google drive as the supply parameter is swept.  You can observe the supply in the model has a strong influence on the price – many orders of magnitude in fact.  Below are some excerpts from the PDF and interactive mathematica program provided in the google drive, as well as the generating mathematica code.

Demand[t_] := 1/(1 + Exp[-t/tp])
 Supply[t_] := t*2^(-t/th)
Slider[Dynamic[tp], {0, 29, .1}, ImageSize -> 1300], Dynamic[tp],
 "= propogation time tp" ,
 Slider[Dynamic[th], {0, 4, .1}], ImageSize -> 1300, Dynamic[th],
 "= effective bitcoin supply"}
 Plot[Demand[t]/Supply[t], {t, .0000001, 5}, ImageSize -> 500]]
 Plot[Log[Demand[t]/Supply[t]], {t, .0000001, 5}, ImageSize -> 500]]
 {Slider[Dynamic[tp], {0, 29, .1}, ImageSize -> 1300], Dynamic[tp],
 "= propogation time tp", Slider[Dynamic[th], {0, 2, .001}],
 ImageSize -> 1300, Dynamic[th], "= effective bitcoin supply"}
 Plot[Demand[t]/Supply[t], {t, .0000001, 5}, ImageSize -> 500]],
 Dynamic[Plot[Log[Demand[t]/Supply[t]], {t, .0000001, 5},
 ImageSize -> 500]]}*)








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Bitcoin Bubble? Not this time. A qualitative analysis of Bitcoin’s price cycle and unusual economy of scale

Edit: If you find my claims speculative and full of muck, here is the math, and a google drive to some plots the source code in mathematica.  

This isn’t a bubble

Bitcoin is experiencing some rapid growth at the moment that is once again turning heads in the media.  Everyone wants to know when to sell their Bitcoin just before the bubble pops – that is clear.  While the exact timing of the bubble pop is difficult to predict, it is easy to use some key metrics to show that what we are experiencing now is in fact long overdue growth as the past months represented the fallout of the last bubble.

Don’t get me wrong.  There will be another Bitcoin bubble someday. And likely another after that.  It is the natural cycle that bitcoin lives on, which is illuminated plainly by taking the logarithm of the price history.

The Bitcoin Price-Epochs 

You can detect Bitcoin bubbles by realizing a few facts:

1) on average, Bitcoin price increases about 1000% / yr

2) short term deviations from this behavior represent recessions and bubbles

3) Bitcoin’s natural mode (for the moment, because it is young and had not reached equilibrium with other currencies) is to exhibit incredibly fast growth – it’s worth repeating: Bitcoin has gained a staggering 1000% year average growth since its inception.  That rate of increase is so massive, that a recession superimposed over it has the appearance of price stability or even small, slow growth.



Shown above is the entire price history of Bitcoin.  There was a “little” $11 bubble followed by a BIG bubble in April…and now we’re gearing up for an even BIGGER BUBBLE! Right?

Wrong.  Here’s the log plot, which effectively filters out  Bitcoins automatic repeated doubling every few months.


This is a very illuminating chart.  It really makes me happy as a clam, because almost everything about it falls in line with the basic tenets linear impulse response. Bitcoin is quite well behaved in fact.  Notice:

• The truly daunting growth rate of bitcoin is to attain about 10 times its price the previous year (I know, I’ve mentioned this now three times.  I just can’t get over it).

• The bubble in 2011 was actually a much greater deviation from this standard behavior than the bubble in April 2013.  This is entirely expected.  Mark my words – the next bubble will appear quite large on the linear axis, but indeed be smaller than the last in the log linear sense

• Because the 2011 Bitcoin Bubble was so much bigger than the one in April 2013, the associated recession was quite long.  The recession following the 2013 bubble was shorter because it was a smaller bubble.

• The recession periods are in proportion to the size of the bubble.  This is to be anticipated, as just about anything in the universe has a response proportionate in amplitude to that of the impulse which disturbed it.

• The price today is actually the natural expected log linear price – Bitcoin has stayed true to this line since it began.  It cannot rise in this manner forever, but there is no reason to think Bitcoin has left its rapid growth state yet – the currency is still brand new and relatively few people in the world have interacted with it.  The upside will continue until of course it cannot sustain log linear growth.  But it is, to put it lightly, kind for Bitcoin investors that the day has not yet come.

Economy of Scale

Bitcoin will probably continue to follow this pattern for quite some time because of the extremely low friction within the Bitcoin system.  The principle is known as economy of scale.  The simple fact is, there is very little stopping Bitcoin from growing.  Unlike a company, which feels considerable growing pains as its stock price shoots up, and eventually levels off when growth is impossible due to diminishing returns, Bitcoin doesn’t experience any backlash for attracting more users, more excitement, more of everything.  Bitcoin doesn’t feel growing pains. Nothing works like that – not usually anyway, which is why Bitcoin has fooled the silver haired finance veterans, who, despite their vast experience in markets, have gone their entire lives never having met a beast that has nothing to stop it from gaining velocity.

Conclusion:  Bitcoin has a truly unique economy of scale, limited only by the flow of information and technology around the world, and the robustness of the Bitcoin protocol itself.

If I lost you on the last part, watch this TED talk by a particularly captivating scientist who researches economies of scale in both biological and economic systems:

“Geoffrey West: The surprising math of cities and corporations.”

You will be reminded of the Bitcoin price when he characterizes the economies of scale in organisms and likens this to the development and life cycle of an entire city – its people, its culture, its structural map, and its entire social being.

It is not very difficult to swallow, for me, that this really is the way the Bitcoin economy works.  A city is nothing if not a set of connections – the social and economic influences, and the relationships between people in close proximity.  In 2013, where “close proximity” has been redefined to mean “within arms reach of a keyboard” it is no wonder we see the topological structures inherent in the growth of economic centers emerging in the cloud.

You’ll also (hopefully) be entertained by his breakdown of economies of scale, and almost certainly will be enlightened.

Thesis: hold your coins, and watch the log plot.


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When should I sell my Bitcoin Mining hardware? Bitcoin Mining Hardware Resale Value vs Projected Return

If you have any questions about this process, you can contact me with the form below!  Happy mining. 

In general, the profit margin for miners becomes slimmer through time as the competition to break into bitcoin mining increases and the barriers to entry are higher than ever.  Customers have in general had poor experiences with ASIC manufacturers who have delivered the mining products late, making the hardware purchase a net loss for the investor.  The situation is not so grim for the miner, however, because the bitcoin price and price velocity are as high as they have ever been, and are not showing signs of flinching.

Despite the apparent victory for the miner when the price rises, it is a common misconception that a positive return of investment can be saved for a poor mining hardware purchase if the bitcoin price rises enough.  This is not actually true – well, not exactly.  The truth is, if the only way to make ROI is if the price of bitcoin rises, then you should have simply bought bitcoins, rather than the miner.

The question of when a “good” time to sell your miner is complicated.  What you have is a machine that prints money.  But it prints money at an ever decreasing rate.  When do you sell it?  You need to first make a few back of the envelope calculations.

Calculate your miner’s daily performance today

First you find the number of bitcoins your miner will earn you per day at the present time.  There are many online calculators for this.  You can do it by hand in a number of ways; my favorite way is the following:

• Get an estimate online of the total network hashrate (at the time of writing it is about 4200 TH/s).

• Figure out what fraction of that total hashrate is your contribution; this is

Fraction = (your hasrate)/(network hashrate)

• That’s the fraction of all bitcoins per day which will belong to you (it’s a generous estimate because of mining pool fees which we disregard).

• The total number of bitcoins generated per day is about:

(25 BTC) / (1 block) * (1 block) / (10 minutes) * (60 minutes) / (1 hour) * (24 hours) / (1 day) = 3600 BTC per day

• Your expected Bitcoins/day is then just

My BTC/day = Fraction * 3600 BTC/day

For example: if you own a 576 GH/s KNCMiner Jupiter…

• Your share of the network hashrate is 576/(4200*1000) = .00013714

• This is the fraction of daily bitcoins produced by everyone, which is always 3600.  You generate about 3600*.00013714 = .4937 BTC/day

• This estimate is roughly true when taken for the period of 2 weeks over which the difficulty is adjusted, and is a central tendency.  You actual daily rake will differ day to day due to the nature of Bitcoin.  However this way of estimating is surprisingly accurate, if you get a good Hashrate estimate.

Calculate the expected return of your miner from now on

This can be in principle extremely difficult.  I will present an extremely over simplified model that can be used to roughly calculate the expected return of your miner given market trends by invoking the approximation of a geometric series for the difficulty.

• Divide the last difficulty by the current difficulty.

• This is the value r or the common ratio.  r must be between 0 and 1, or you have have a mistake.

• If you can, find r for the previous diff change or previous 2 or 3 and average them.

Once you have the r you want to use, the number of bitcoins your miner will EVER produce is approximately

Bitcoins from now to eternity = (number of bitcoins I’ll produce this 2 weeks period)/(1 – r)

For today the difficulty is 510 million.  Prior it was 390 million.  That gives an r = .76 which we will proceed with, though you can go back a few steps and average the result if you like by checking a chart like this. 

Anyway, once we have r, we find

Bitcoins ever = (number of bitcoins I’ll produce this 2 weeks period)/(1 – r)

= (14  days)* (.4397 BTC / day)/ (1 – .76)

= 25.6 BTC

Decide when to sell 

Snce I love to disregard complications, let’s assume you can liquidate your Miner instantly, without any problems/costs/risks for a “market price.”  You should sell your miner precisely when the market price of the miner will allow you to buy more bitcoins than the total number left your miner will ever create.

Let’s do this analysis again with KNCMiner (for no real reason other than consistency at this point) as an example.  Here is some data from on sales of “knc miner” equipment over the past 7 days:

Item Title Sold Format Start Price

End Price

Bids End Date
KnCMiner Jupiter **In Hand** 500+ GH/s, Asic Bitcoin Miner Yes $9,999.99 $9,999.99 1 2013-10-28
In Hand KNCminer Jupiter ASIC Bitcoin Miner – 500 GH/s – FREE 2 Day Shipping Yes Bid $5,000.00 $8,100.00 24 2013-10-28
535 GH/s+ KnCMiner Jupiter bitcoin miner – In Hand – Ships Immediately Yes Bid $300.00 $8,100.00 39 2013-11-03
KNCminer Jupiter 550GHash Bitcoin with 1200W Power Supply Yes Bid $5,000.00 $8,050.00 9 2013-10-28
KNC 550+ GH/S Bitcoin miner. KnCminer. Jupiter in Hand Hashing with power supply Yes Bid $5,000.00 $8,000.00 15 2013-10-31
KNCminer Jupiter 550GHash Bitcoin with 1200W Power Supply IN HAND Yes Bid $7,850.00 $7,950.00 2 2013-10-29
KnCMiner Jupiter 550GH/s Asic Bitcoin Miner with SeaSonic 1000W PSU **in-hand** Yes Bid $5,000.00 $7,890.00 1 2013-10-28
KnCMiner Jupiter ASIC Bitcoin Miner 500+GH/s *In-Hand* (Will Sell Now for $7200) Yes Bid $4,995.00 $7,200.00 4 2013-11-03
KnCMiner SATURN ASIC Bitcoin Miner 285+ GH/s IN HAND! FREE Overnight w/ BuyNow Yes Bid $3,800.00 $4,970.00 1 2013-10-29
KnCMiner Jupiter **In Hand** 550+ GH/s, Asic Bitcoin Miner Yes $4,900.00 $4,900.00 1 2013-11-01
KnCMiner Saturn **In Hand** 275+ Gh/s, ASIC Bitcoin Miner Yes Bid $1.00 $3,995.00 14 2013-11-03

Because there is some variance here and not too many data points, let’s calculate some quantities for only one of these examples – the one on 11/3 which went for $8100.

On 11/3 let’s assume the expected return for the Miner was 25.6 BTC just as it is today. The price of bitcoin – that is, the bitcoin to USD conversion – is a far (perhaps infinitely) more volatile quantity than the expected return of Mining hardware when returns are stated in BTC.  So a quick search of data on for example shows the price that day was, on average, 218.24 USD.  So if this seller collected precisely 8100 USD for this sale (he didn’t because of ebay and paypal fees) he could have turned around about bought

BTC earned by selling = (8100 USD)/(218.24 USD/BTC) = 37 BTC

That’s a pretty good sell, IF the seller turned around with his money and bought BTC immediately.  This is true regardless of the future price of bitcoin because this is a bitcoin to bitcoin comparison (in general, by staying within bitcoin when calculating, it is easier to extract information from the market data).

Limitations of this model

This model is completely invalid if the difficulty is not increasing in a pretty particular manner which it seems to be for now.  Specifically, this model will completely fail if the difficulty is increasing but in smaller increments each time.  This model will fail even more massivly if the difficulty is stable or decreasing, which is not the case presently, but has happened before.  A stable difficulty will also almost certainly occur one day again – but I will not digress as to why this is the case in this post.

Furthermore this model assumes bluntly that the network difficulty takes geometric steps upward, which would be the result of a purely exponential hashrate plot. Simply put, the network difficulty is not geometric in general, but only approximates geometric behavior in a sufficiently short period of time. In fact, it is a mathematical certainty that in an appropriate time scale, any relationship at all will trace an exponential curve quite closely and it is this universal principle that I have exploited in this model.

But the difficulty really cannot really be geometric – not only because the ratio of two consecutive difficulties is pseudorandom, but rather because the network hashrate displays kinks, corresponding to generations of ASIC technology..  However, it can be seem empirically that the difficulty function is  approximately geometric within 3 to 5 difficulty increases, over which the ratio r does not deviate strongly from a central value.  Luckily, 4  increases of difficulty amounts to 8 weeks.  Since the advent of ASIC architecture, 8 weeks has been, perhaps consequently, the approximate lifetime of a mining device (some devices has an even shorter lifetime).

This is congruent with the intuition which says if your mining income rate falls off faster than a geometric series, you don’t have much time left before your mining income will be quite literally zero; though an old pentium 4 CPU can “mine” a bit, there comes a point when you will simply die before you mine an amount of bitcoin that exceeds standard transactions fees.  Whether this is the case today is left as an exercise for the reader.

Because of the probabilistic nature of the the bitcoin network’s protocol, there is a fundamental limit on the accuracy one can hope to have predicting short term trends.  bitcoin is also a financial instrument, and being such lends itself poorly to any model with long term accuracy.  Thus we must be sure of the time scale over which a particular model has any validity.  This empirical rule of geometric increases in difficulty is merely a model, so use it with caution.  If you plan on doing such a calculation, it would be wise to trace the difficulty back a few steps and see if something else is going on in the bitcoin economy.  If that is the case, it is time to make a new model.

This week I will make a detailed mathematical analysis of the geometric approximation to network changes.  This will come with a post to generalize the fit to longer times through the use of more complicated models resembling statistical mechanics of physics.  If I forget to do this remind me, seriously.

– Altoidnerd

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