## A mathematical theory of pure exchange economies without the no-critical-point hypothesis |

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admissible allocations assume Chapter consequently consider contains no disastrous convexity defined definite negative denoted dense set derivative dimensional submanifold disastrous allocations Du^x Dui(xi easily seen Edgeworth box equations equilibria ex cr g X,x,p GELDROP Given Global analysis Hence it follows iel(z implicit function theorem implies LEMMA Let u e Let x e Lyapunov stability m-tuple manifold Math matrix Morse function open and dense open neighbourhood open subset pair w,u parametrized Pareto optima point z e points x,p preference relation product topology pure exchange economy regular economy regular pair regular value satisfying Section set E r,u Smale smooth functions smooth map solution strict Pareto optimum submanifold of dimension tangent space total resources trade curves utility functions utility tuples vector field Whitney C topology