Implementability under Monotonic Transformations in Differences

Consider a social choice setting in which agents have quasilinear utilities over monetary transfers. A domain D of admissible valuation functions of an agent is called a revenue monotonicity domain if every 2-cycle monotone allocation rule is truthfully implementable (in dominant strategies) and satisfies revenue equivalence. We introduce the notions of monotonic transformations in differences, which can be interpreted as extensions of Maskin's monotonic transformations to quasilinear environments, and show that if D admits these transformations then it is a revenue monotonicity domain. Our proof is elementary and does not rely on strenuous additional machinery. We show that various economic domains, with countable or uncountable allocation sets, admit monotonic transformations in differences. Our applications include public and private supply of divisible public goods, multi-unit auction-like environments with increasing valuations, allocation problems with single-peaked valuations, and allocation problems with externalities.