A Continuous Extension that Preserves Concavity, Monotonicity and Lipschitz Continuity

Borradores de Economia
Number: 
230
Published: 
Classification JEL: 
C02, C20
Keywords: 
Human capital agglomeration, Social returns, Private returns, Externalities, Uncertainty, Fiscal policy
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The following is proven here: let W : X × C ? R, where X is convex, be a continuous and bounded function such that for each y?C, the function W (·,y) : X ? R is concave (resp. strongly concave; resp. Lipschitzian with constant M; resp. monotone; resp. strictly monotone) and let Y?C. If C is compact, then there exists a continuous extension of W, U : X × Y ? [infX×C W,supX×C W], such that for each y?Y, the function U(·,y) : X ? R is concave (resp. strongly concave; resp. Lipschitzian with constant My; resp. monotone; resp. strictly monotone).