Some prominent economists have begun to analyze formally the market for a privately issued outside money that they associate with Bitcoin. Rodney Garratt and Neil Wallace (2018) (ungated version here) model the relative values of (exchange rates between) "Bitcoin 1" and other hypothetical cryptoassets ("Bitcoin 2," etc.). Linda Schilling and Harald Uhlig (2018) take a related approach to the exchange rate between "Bitcoin" and "the US dollar." I use quotation marks here to indicate that the authors' subjects are modeling entities, named after but not the real things. Their correspondence to the real things should not be taken for granted.
Both pairs of authors draw on a well-known theoretical result by Kareken and Wallace (1981): when two fiat currencies are perfect substitutes, the equilibrium exchange rate between them is indeterminate. To get the intuition, imagine that any payment made in US dollars can equally be made in Canadian dollars valued according to the going exchange rate. Nobody then has a reason to swap one currency for an equivalent amount of the other. No matter the level of the exchange rate, there is no pressure for it to change. Only the combined real money supply and demand matter, and any exchange rate is consistent with combined real combined real supply equaling combined real money demand. The combined real money stock could be 99% US dollars (at an exchange rate making one USD worth many CAD) or 99% Canadian dollars. Either rate is compatible with monetary equilibrium.
It easy to see, of course, that in the real-world fiat currencies are not perfect substitutes in all uses. Most obviously, legal tender laws and tax-payment restrictions imposed by nation-states make any two fiat currencies non-interchangeable. Canadians cannot pay their Canadian-dollar-denominated debts or taxes with equivalent amounts of US dollars; they need Canadian dollars. Garratt and Wallace (p. 1887) are aware of this objection to the relevance of the Kareken-Wallace (hereafter KW) result. But, they assert, no such objection can be made in the case of Bitcoin and coins that are identical to Bitcoin in all respects but brand name:
Bitcoin and its actual and potential rivals—in the title intentionally mislabeled bitcoin 1, bitcoin 2, … in order to indicate that there could be many of them—do seem to satisfy all the assumptions that Kareken and Wallace made to get exchange-rate indeterminacy.
Consequently, they write (p. 1896):
Much of the uncertainty in the value of bitcoin comes from the ease of creating perfect substitutes. It is easy to clone bitcoin and the creation of very close substitutes makes the value of bitcoin rest on beliefs that may be hard to pin down.
Without questioning the Garratt-Wallace analysis given their assumptions, I want to point out that a key assumption – that perfect substitutes are easy to create – is factually false with respect to real-world cryptoassets. Bitcoin and its actual and potential rivals are in fact not perfect substitutes and thus do not satisfy the KW assumptions. The reason does not come from legal tender or tax laws, but primarily from network economies in monetary systems that apply to cryptocurrency systems. The size of the Bitcoin network matters, and is not easy or cheap to replicate.
The concern that newer cryptoassets will be close to perfect substitutes for Bitcoin, and that because very cheap to produce they will drive Bitcoin's value toward zero, has also been expressed less formally in blog posts by John Cochrane and much earlier by Brad DeLong. DeLong wisely included the caveat that Bitcoin could retain its value if it could remain differentiated from its rivals, but he doubted that it can do so, wrongly suggesting that its differentiation is tenuous because it rests on nothing but its being the oldest cryptoasset.
To grasp the relevance of network effects, it is important to note the basic fact that each coin has its own blockchain. New cryptocurrencies do not provide access to the established Bitcoin system, only to their own much smaller systems. Bitcoin's thousands of validation nodes are not their validation nodes, which number many fewer. Newer coins are not as widely accepted in payments. They do not offer an equivalent ecosystem of wallets, retail payment processors like Bitpay, and other ancillary service providers. They are not traded on as many exchanges, or in similar volumes, and consequently have larger bid-ask spreads. Given the network advantages of a larger system with wider acceptance and established robustness, new coins that merely clone the Bitcoin code are not in fact close substitutes for Bitcoin.
Economic theory predicts that the marginal me-too coin, with a near-zero cost of production, will have a market cap of close to zero. Such me-too coins will not bring down the market cap of Bitcoin (or other established cryptoassets) when potential investors and users have no reason to prefer them. In general, to attract users and thereby attain a positive market cap, a new coin has to offer improvements over the Bitcoin system. The cost of making significant technical improvements is of course not negligible, so new cryptoassets providing them will never be easy to create or superabundant.
The cryptoassets that have gained positive value competing against Bitcoin have done so not by cloning it but by offering new-and-better features. The most prominent improvements have been in four areas: greater speed in payment validation (e.g. Ripple, Stellar, Bitcoin Cash, Litecoin), greater privacy (Monero, Dash, Zcash), greater security against 51 percent attacks (NEO, Peercoin, Decred), and better infrastructure for smart contracts and applications (Ethereum, EOS). Stability of value may be considered a fifth area, or so-called stablecoin projects may alternatively be considered an entirely different proposition.
New-and-improved coin projects continue to be launched. The publicity for new coin projects characteristically emphasizes how their technology differs from and improves on the Bitcoin protocol. It never says: we are merely a clone of Bitcoin. For a recent example, on the 3rd of January 2019, the same day as the tenth anniversary of Bitcoin's launch, a new coin called Beam was launched. Beam implements a next-generation blockchain technology ("Mimblewimble") that enables greater privacy (less information about transactions is publicly revealed on its blockchain) and faster validation.
To say that a particular cryptoasset is distinct, and not a perfect substitute for Bitcoin, is not to claim that its market value in Bitcoins or in dollars is stable or readily predictable. Nor is it to deny that its value can go to zero, if the market completely abandons it as no longer a plausible contender. But it is to claim that news that raises or lowers the odds of the coin's wider future adoption will drive changes in its relative value. Thus measured price correlations between other cryptoassets and Bitcoin are significantly less than one, and as low as 0.2 in the case of Ripple. If two coins with predetermined quantity paths were perfect substitutes, by contrast, it would be hard to explain why any news should affect their relative values. That news does affect the relative values of real-world cryptoassets is evidence that they do not fit the assumptions of the KW indeterminacy result.
Philosophers distinguish a merely valid argument – the conclusion follows from the premises – from a sound argument, which is both valid and has true premises. Arguments that assume perfect substitutability among cryptoassets may be valid, but they are not sound.
 I am an advisor to Beam.
 Charles Bovaird writes of Bitcoin and Ripple (XRP) in an article for Coindesk: "The two currencies are quite distinct, a situation noted by cryptocurrency fund manager Jacob Eliosoff, and this situation might help explain their weak price relationship. Bitcoin and litecoin have differing value propositions and target separate audiences from XRP."