This paper proves the Welfare Theorems and the existence of a competitive equilibrium for the club economies with private information in Prescott and Townsend (2005). The proofs cover lottery economies with a finite number of goods and without free disposal. A mapping based on Negishi (1960) is used

The relationship between our general equilibrium model with multi- member households and club models with multiple private goods is investigated. The main distinction in the definitions consists of the equilibrium concepts. As a rule, competitive equilibria among households where no group of...