Existence of shadow prices in finite probability spaces

A shadow price is a process <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$${\widetilde{S}}$$</EquationSource> </InlineEquation> lying within the bid/ask prices <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$${\underline{S},\overline{S}}$$</EquationSource> </InlineEquation> of a market with proportional transaction costs, such that maximizing expected utility from consumption in the frictionless market with price process <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$${\widetilde{S}}$$</EquationSource> </InlineEquation> leads to the same maximal utility as in the original market with transaction costs. For finite probability spaces, this note provides an elementary proof for the existence of such a shadow price. Copyright Springer-Verlag 2011