Almost all predictions in this forum are biased and its mostly what everyone wishes to happen.
http://bitcoincharts.com/charts/chart.png?width=940&m=mtgoxUSD&SubmitButton=Draw&r=&i=&c=0&s=&e=&Prev=&Next=&t=S&b=&a1=&m1=10&a2=&m2=25&x=0&i1=&i2=&i3=&i4=&v=0&cv=0&ps=0&l=1&p=0&All I am pointing out is that this data for historical price log linear on the time scale for the entire life of the currency. It is not an uncommon relationship in physics and economics. This chart actually shocked me when I realized it but sure as shit - it really does show roughly 10x increase yearly.
I am impressed at the fast doubling time of the price, but the exponential character doesn't surprise me at all because it is what you obtain from diffusion of the signal (knowledge, or news of bitcoin) through the human race. If you model the demand for bitcoin being proportional to the number of people who have ever even heard of it, since this is still a small fraction of the population on earth, the demand will have the exponential form for early times. It is only when a significant part of the populous has already learned of bitcoin and decided they don't like it that the demand will cease to be exponential.
if you don't believe me, draw on a piece of paper 50 to 100 dots. Suppose at t = 0 only one dot is special. Connect that dot to another dot which represents two nodes in communication. The let each of those nodes "tell" another dot at random. There is now a 99.5% (check this?) chance that there will be 4 connected nodes now, because chances are they will be connecting with another unoccupied dot. Even at this stage, if we repeat the step, we will most likely be left with 8 nodes connected.
You get this chart for N, the number of nodes the signal has propagated to
t |0 | 1 | 2 | 3 | 4 | 5 |
N |1 | 2 |4 | 8 | ? | ? |
In the last two bins we might not get 16 and 32...there is some non trivial probability at this point that one of the nodes will connect to another which already is connected. it is around this time when the propogation of news through a network ceases to be exponential as some nodes receiving word may "already know."
What this does establish is good to prove my point so I will say that now... for small times (e.g. not too many folks know about bitcoin) the demand for the currency D(t) will go like
D(t) ~ et/to for small t
The supply of bitcoin is, at any moment in the present, a linear function in t. But the supply function is not actually ~ t because we know it has the limiting form S(t) -> S
final for large t. In fact we know that it roughly have the form
S(t) ~ t /2t/t'
The math speaks for itself. If we model the price simply as P(t) = k D(t) / S(t) what do you find? It;s pretty marvelous actually! Although for intermediate values of t the demand will begin to deviate from exponential behavior, the diminishing supply having the exponential in its denominator serves to offset this effect and in a sense create a roughly exponential demand price as a function of time even after the demand is no longer increasing in such a manner.
