The“core Equal treatment in the core: if x is an allocation in the r-core of a given economy, then any two agents of the same type must receive the same bundle Proof:not:Letx=∑1x1,x=1∑ j So∑。x+∑ /=1MB= Ir That is x+XB=w+wB Every agent below the average will coalize to improve upon the allocation
The “core” • Equal treatment in the core: if x is an allocation in the r-core of a given economy, then any two agents of the same type must receive the same bundle. Proof: if not. Let , – So – That is – Every agent below the average will coalize to improve upon the allocation. 1 1 r A A r j x x = = 1 1 r B B r j x x = = 1 1 1 1 1 1 1 1 r r r r r r r r A B Aj Bj j j j j x x w x = = = = + = + A B A B x x w w + = +
The“core Shrinking core: there is a unique market equilibrium x" from initial endowment w if y is not the equilibrium there is some replication r, such that y is not in the r-core Proof: since y is not the equilibrium, there is another allocation g improve upon A(or B)at least. That means see the fig. 8=CWA+(1-aya for some 8>0 Let &=T/V(T and V are integers)
The “core” • Shrinking core: there is a unique market equilibrium x * from initial endowment w. if y is not the equilibrium, there is some replication r, such that y is not in the r-core. Proof: since y is not the equilibrium, there is another allocation g improve upon A(or B) at least. That means see the fig. – Let (T and V are integers) (1 ) for some 0 A A g w y = + − =T V/