Integration of demand and continuous utility functions (Q1082230)

The starting point of the utility approach to demand theory is a continuous (or upper semi-continuous) utility function defined on \({\mathbb{R}}_+^{\ell}\); in that theory, the utility functions associated with demand functions are typically defined only on the range of the given demand function (which may be a strict subset of \({\mathbb{R}}_+^{\ell}).\) This paper presents conditions which assure that a given demand function has been generated by an upper semi-continuous or a continuous utility function defined on all of the commodity space. In the first part of this paper the author presents integrability type conditions sufficient to assure that a given demand function is generated by a utility function which is defined and upper semi-continuous on all of \({\mathbb{R}}_+^{\ell}.\) In the second part the author studies necessary conditions to assure that a given demand function is generated by a utility function defined and continuous on all of \({\mathbb{R}}_+^{\ell}\); these conditions are less restrictive than those used when this function is only defined in the range of the demand function.