The mining of bitcoin would continue but transferring them and spending them becomes a problem since there would be no quoted price. The bitcoin protocol is about mining bitcoin not pricing bitcoin. There is nothing in the protocol about establishing a market price for bitcoin; you need a market for that, but what if all the exchange markets are shut down?
As a medium of exchange and barter, those looking to transact using bitcoin who name their price in individual transactions would continue to do so. And these transactions, in aggregate can be reported to give market participants a general idea of where the price per bitcoin generally is; like beaver pelts in early American history. But will this approach satisfy bitcoin’s aspirations to take on the Dollar, Yen, Pound and Euro? I don’t think so.
For this reason, I have suggested that some entity, possibly the Bitcoin Foundation (and this can be done on a non-profit basis), make a market in bitcoin and broadcast a current price for bitcoin (a peg) that various exchanges and merchants can use as a benchmark.
This market making activity should be done completely out of reach of regulators. Honesty on the part of the ‘peg’ maintainers would come by way of interacting with various exchanges (that are open) since maintaining this peg would require lots of buying and selling on the various exchanges to maintain an inventory (of bitcoin) that is necessary to maintain a peg and any resultant price gaps or arbitrage would be quickly closed by savvy traders. (By the way, if the VC community is serious about funding bitcoin start-ups they should pool their capital to fund such a mechanism).
With this added layer of price discovery in the bitcoin’s existence as a currency, the possibility of scaling up to the multi-hundred billion valuations necessary to get it on the first rung of the global currency market becomes a possibility. Without it, we’re talking about beaver pelts.