Curiously, the three systems of economic organizations that Seth Ackerman (2012) investigates in The Red and the Black all feature something called "prices". Prices are uncontroversially (I hope!) a key part of standard market capitalism, but Ackerman reports that both the alternative systems he examines, presumably Albert and Hahnel's (1991, 1992a, 1992b, 2002)* "participatory economics" as well as Soviet-style central planning, also have prices. According to Ackerman (2012), Albert and Hahnel's plan uses a democratic process to set prices; similarly, the economic task in the former Soviet Union after 1989 was to "Get Prices Right": not to introduce a price system, but to correct an already existing price system (which they found to be already correct). Thus, it is clearly not the case that prices per se are the critical factor; every economic system seems to have "prices." Differences in economic systems consist not of whether or not some system has "prices," but of what prices represent or do, and how these prices interact with human behavior.
*It is perhaps the fault of Jacobin (the online version of the magazine is very difficult to correctly cite), but Ackerman infuriatingly fails to completely cite many of his sources. Hopefully the citations here will assist future scholars.
An economic system consists of a collection of institutions that has three primary functions. First, a quantity and mix of stuff (goods and services), constrained by the available labor and natural resources, must be produced and distributed. Second, the institutions themselves must be reproduced: the human beings who participate in the institutions, and the quantity and mix of stuff will change over time. I use the passive voice above purposefully, because the third and most important function of an economic system is designating which people in which institutions have how much and which control over the production, distribution, and consumption of stuff, the supply of labor offered by the people, and the reproductive future of the institutions.
In On Communism and Markets, I think Matthijs Krul (2013) gives these last two functions special emphasis. It might (or might not) be more "efficient" in some sense to have some system of distribution, but so long as that system reproduces capitalist relations of production, it's not socialism.
However, I think a stronger argument is possible. For Marx, a necessary presupposition for a socialist revolution is that capitalist relations have become fetters on human productive powers. If that is indeed the case, then preserving any kind of specifically capitalist relations is at best dodgy, and requires a much stronger positive case than Ackerman (2012) makes. It is insufficient to argue that capitalism has kept the grocery stores stocked, capitalism uses some unspecified price system, therefore we should preserve this unspecified price system to keep the grocery stores stocked.
These issues are certainly important, but the technical problem of producing and distributing the right quantity and mix of stuff under labor and resource constraints still remains.
I want to construct a minimal definition of price. All three of the systems have something called a "price": to understand how they differ, I want to understand what is common between them all. In the general form, a price is a number attached to some definite, objectively determinable quantity of stuff. The number must be directly observable by and exactly the same for every relevant actor. The number must directly affect the actors' economic decisions: if the number were nontrivially different, the actors would make different economic decisions just because of that difference in price.
Thus the differences between different economic systems consists of how a society attaches these observable numbers to definite quantities of stuff, how these numbers affect people's economic behavior, and how the institutions and individuals reproduce these mechanisms of attachment and effect. In addition to briefly describing the capitalist theory of prices, I will examine each of the systems Ackerman (2012) and Krul (2013) discuss: market socialism, Albert and Hahnel's (1991 etc.) participatory economics, and "Soviet-style" central planning.
Ackerman, Seth. December 2012. The red and the black. Jacobin 8. Retrieved August 13, 2016 from jacobinmag.com/2012/12/the-red-and-the-black
Albert, Michael, and Robin Hahnel. 1991. The political economy of participatory economics. Princeton University Press.
---. 1992a. Participatory planning. Science & Society 56.1: 39-59.
---. 1992b. Socialism as it was always meant to be. Review of Radical Political Economics 24.3-4: 46-66.
---. 2002. In defense of participatory economics. Science & Society 66.1: 7-28.
Krul, Matthijs. 2013. On Communism and Markets. Notes & Commentaries. Retrieved August 14, 2016 from http://mccaine.org/2013/01/30/on-communism-and-markets-a-reply-to-seth-ackerman/