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This note reports and reemphasizes the importance of a balancedness condition for production sets which—if combined with otherwise standard assumptions—guarantees the non-emptiness of the core. Balancedness neither requires nor implies any property related to returns to input scale. It is argued that the emphasis laid by Scarf and others on a particular kind of increasing returns to scale is not essential for the existence result in general. It is shown, however, that Scarf's distributivity assumption implies balancedness. Finally, it is pointed out that earlier results indicate that equilibria in the core require a specific profit distribution to consumers rather than a particular behavioral rule of producers even in the presence of nonconvexities.