1. 41]), we have v (p; h; f ) having increasing di�erences in (k; f ) for each (K; z; h): By Topkis ([85], Theorem 2.8.1), the optimal solutions a (p; h; f ) are therefore ascending in the strong set order on F;K + the subset of positive linear functionals in K: Then by the complete characterization of monotone Lipschitz functions in Jeyakumer, Luc, and Schaible
2. Corollary 3), for any limiting distribution (f ) 2 ' J (f ) associated with a Markovian equilibrium in ' T (f 0 ) 2 E 0 , there is a ?xed point (f 0 ) 2 ' J (f 0 ) associated with a Markovian equilibrium in economy ' T (f 0 ) 2 E 0 such that (f 0 ) (f ) in stochastic dominance. A similar argument shows the for any the limiting distribution at (f ) 2 ' J (f ); there exists a Markovian equilibrium in ' T (f 0 ) such that its limiting distribution (f 0 ) (f ): The existence of a monotone selection then follows from Smithson ([79], corollary 1.8) noting that for example;For (ii), ?x 2 E 0 : Then the Sweak set relation monotone comparative statics result for the limiting distribution for the ?xed point correspondence ' J (f ) follows noting ?rst that by proposition 8(b) due to Topkis [85], the operator T h has ?xed points that strong set order compatiable
3. A theorem on partially order sets with applications to ?xed point theorems;S Abian;Canadian Journal of Mathematics,1962