In a socialist market economy, businesses are cooperatively owned. In a capitalist market economy, businesses are generally privately owned. If someone owns a means of production, but does all of the labor to create and sell the goods using those means of production is it private property, social property, or something else? Is it possibly personal property because it isn't owned by a group of workers but isn't being used to exploit workers by extraction of surplus value? Do market socialist accept this form of property?
Man, you're gonna have a hard
Man, you're gonna have a hard time finding any market socialists on this site to answer this question...
If it's market socialism, any
If it's market socialism, any goods produced would need to belong to the firm (cooperative), because a market requires a system of alienable property rights (i.e. unless coops own their products, they can't trade them).
FS98 wrote: In a socialist
FS98
They are socially owned rather than owned by the co-operative -- they are, however, run by the associates who are in the cooperative. That is important because otherwise the associates could hire workers and so reintroduce wage-labour. See:
I.3.3 What does socialisation mean?
and:
Proudhon, Property and Possession
FS98
Private ownership ignores the key point -- does the owner hire workers or not to toil in their property. A peasant or an artisan may "own" their land and tools but it does not make them capitalists...
FS98
In answer to your question, may I suggest:
I.3.7 What about people who do not want to join a syndicate?
No genuine socialist would force people to become communists -- if a workers wants to labour away on their own, then let them. I would expect that over time they would see the benefits of (libertarian) communism.