We already discussed this matter in the past (
here, for your reading pleasure)
Someone has come up with exactly the same reason there, namely, Thiers' Law. However, Gresham's Law, or its proper extension, still persists if we consider two currencies, which are freely exchangeable at market rates. Thiers' Law basically describes a situation when one currency fails as money, i.e. no one wants to accept it. In that case people simply refuse to transact in it, and thus no free exchange is possible. Obviously, this is not the case here. Bitcoin is a worthy speculative asset, and it is not a good idea overall to spend your capital on everyday needs, Gresham's Law or otherwise (read, it is Thiers' Law which is not applicable here)
But Gresham’s law cannot be applied to “two currencies, which are freely exchangeable at market rates”. That’s is an incorrect interpretation of the law’s setting. Gresham’s law can be applied to two variations of commodity money that are enforced to be accepted at the same exchange rate by a legal tender law. At the same time, one of the assets has more commodity value (good money) than the other (bad money). It can be also extended to similar occasions where there are two identical assets of the same nominal value but differing in intrinsic value, which is clearly not our case.
Lets consider a simple example. Say 10 grams of gold cost $1. We have $1 coins that weight 10 grams and are pure gold. That’s our good money. The government melts a fraction of these coins and creates new $1 coins that also weight 10 gram but consist of 50% gold and 50% copper. The newly minted bad money has an obviously lower intrinsic value, but the government, using legal tender, forces merchants to accept in on par with good money. Good money starts being hoarded, because people will apparently prefer to get rid of a less valuable asset in the first place, which drives good money out of the circulation. In time, the price of gold starts to grow tending to $2 for 10 grams (because dollar’s commodity value was reduced by half) and people start melting good money to sell it as a commodity, because in this case they get more value than the face value of a coin. That’s how bad money inevitably destroy good money.
It’s easy to see why we cannot apply that law to Bitcoin: it’s “face” value is determined by an open market and it has no intrinsic value to be melted for.
If we create a setting where two currencies (BTC and a national fiat) are widely accepted as a medium of exchange (in other words, are equally liquid), BTC will destroy the fiat currency, which happens according to Thiers' Law. I actually modelled that scenario in Part 1, you must have missed it.
The premise that “Bitcoin is a worthy speculative asset, and it is not a good idea overall to spend your capital on everyday needs” shows where you took the wrong path. Indeed, Bitcoin is a good speculative asset, which is why it is not equally liquid to fiat, which is why it is not a currency, which is why neither Gresham’s, nor Thiers' Law can be applied to the pair Bitcoin-fiat under normal conditions. These are not two different money representations (good and bad) but a money and a speculative asset. It’s the same as applying Gresham’s law to bonds or securities. The premise that Bitcoin will be prefered for savings as a better value storage is correct, but it has nothing to do with Gresham’s law. In fact, Gresham himself likely new nothing about inflation and fiat money, as he lived in 16th century.
BTW, I’ve just checked out your link. You seem to worry about fractional reserve and the appearance of derivatives that will inevitably screw everything up. I mostly share your concern and I believe that a fractional reserve system shouldn’t be built on top of Bitcoin, at least on the scale of currently available systems. I also addressed that matter in Part 1. Furthermore, I suppose that a blockchain platform that is meant to be an actual currency should not support Turing-complete scripts.