Scale-invariant equal sacrifice in taxation and conditional functional equations (Q1124062)

The purpose of this paper, is, first, to call the attention of those working in functional equations to this interesting application to taxation. Second, we show that equal absolute sacrifice taxation (whether scale-invariant or not) leads, for small incomes, either to unbounded negative utility values or to negative after tax income. Third, we demonstrate how this inconvenience can be avoided, at least in determining scale-invariant equal sacrifice utility functions, through some modifications of the arguments of \textit{H. P. Young} [Progressive taxation and the equal sacrifice principle, J. Publ. Econom. 32 (1987)] by considering only incomes not smaller than a positive lower bound (which seems quite appropriate for taxation purposes). Mathematically this leads to conditional functional equations. We determine all solutions of these restricted domains by solving such equations. Our regularity suppositions will be weaker (continuity and non-constancy rather than monotonicity) in much of the paper. For completeness sake we repeat some arguments of H. P. Young with appropriate modifications. We indicate also a second method of proof and make some remarks on the interpretation of scale-invariance and other aspects of taxation and utility.