Define a market as an arrangement whereby people interact to exchange commodities (the term as used here includes all goods, services, intangibles such as "attention" etc) the "exchange rate" of which is broadly determined to the relative demand and supply of different commodities. With this definition, yes, the market is an adequate tool to express the value exchanged in this economy. Price quantitative evaluation will not, however, be sufficient to express value. So the way this tool is used has to differ from how it is used in the exchange of, say, shares.

    Barter is one way, and more specifically, implicit transactions in a "cooking-pot" market.

    From "Is reputation a convertible currency" [2]:

    There is, here, the first glimpse of a process of give and take, by which people do lots of work on their creations which are distributed not for nothing, but in exchange for things of value. People "put in" to the Internet because they realise that they "take out" from it. Although the connection between giving and taking seems tenuous at best, it is in fact crucial. Because whatever resources there are on the Net for you to take out, without payment, were all put in by others, also without payment; the Net's resources that you consume were produced by others for similar reasons - in exchange for what they consumed, and so on. Thus the economy of the Net begins to look like a vast tribal cooking-pot, surging with production to match consumption, simply because everyone understands - instinctively, perhaps - that trade need not occur in single transactions of barter, and that one product can be exchanged for millions at a time. The cooking-pot keeps boiling because people keep putting in things as they themselves, and others, take things out.
2 Rishab Aiyer Ghosh, 1996, "Is reputation a convertible currency?", published as part of "Cooking pot markets: an economic model for the trade in free goods and services on the Internet", First Monday, March 1998, http://www.firstmonday.dk/issues/issue3_3/ghosh/

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