As time approaches infinity, TimeCoin is, I suspect, superior to BitCoin and Expo/BernankeCoin in that its money supply is mathematically guaranteed to be constant and thus have neither inflation nor deflation and avoid both inflationary and deflationary spirals.
The best analogy to the money supply in *Coin is to consider a vessel into which water is being poured (via money creation) at irregular intervals (but the pour occurs instantaneously) and evaporating (e.g. lost wallets in the block chain currencies & sufficiently damaged notes in fiat currencies). If, as I suspect, the percentage of the water in the vessel that evaporates is connected to a variable that has no correlation to time, volume of water in the vessel, or the amount of water being poured (and can thus be effectively considered to be its long-run average over t=(0,inf)), then (where
v(
t)=volume of water in vessel at time
t,
e=percentage of water at time
t that evaporates between
t and
t+1,
c(
t)=water vapor that condenses in the vessel between
t and
t+1 (analogous to lost wallets being "recovered") and
f(
t)=volume of water poured into the vessel instantaneously at
t):
v(
t+1)=
ev(
t)+
f(
t)+
cFor BitCoin: lim(
t->inf,
f(
t))=0 and d
f/d
t: never positive
For TimeCoin: d
f/d
t=0 =>
f(
t)=
x (
x: positive & real)
For Expo/BernankeCoin: d
f/d
t=(
r+
e)
t (
r=the desired inflation rate... actually the better term is arguably
KingCoin, given that the BoE theoretically has only an inflation target unlike the Fed)
In the case of fiat currencies, I suspect that
c is zero (though I suppose that
c could just stand for counterfeiting...). For the block chain based currencies,
c is vanishingly small (basically being the chance that someone randomly creating secret keys finds one that's already been used and lost) and likely to have a rather large variance (it may be years between successes even if the mining collectives turn their computing power to finding keys to harvest [hopefully lost] coins): it can essentially be ignored (and will have the basically same effect in any block chain currency (the effect on confidence in block chain currencies if a secret key is duplicated not being considered here)).
Assuming positive
e:
For BitCoin, as
t->inf,
f(
t)-
ev(
t-1) approaches zero from below thus lim(
t->inf,
v(
t))=0
For TimeCoin, lim(
t->inf,
f(
t)-
ev(
t-1))=0 thus lim(
t->inf,
v(
t))=
y (
y: positive & real)
For Expo/BernankeCoin,
f(
t)>
ev(
t-1) thus lim(
t->inf,
v(
t))=inf
e is, it should be noted, unknowable (since it's impossible to tell if a given key is lost or merely not circulating).
I also suspect, however, that there is little incentive for TimeCoin adoption prior to
t being arbitrarily large...