The Scale of Universalization in Multiplicative Kantian Equilibrium

Multiplicative Kantian equilibrium requires agents to evaluate whether they would prefer all players to change their actions by a common proportional factor. This note argues that such a counterfactual is not fully specified by the material payoff environment alone: one must also specify the scale in which proportional universalization is performed. Building on the Sher-Roemer discussion of strategic non-invariance, we characterize how a smooth monotone reparametrization of actions changes the multiplicative Kantian first-order condition. We then illustrate the issue in a simple two-country abatement problem. Universalizing physical abatement and universalizing abatement expenditures are both natural interpretations of "doing one's part," but they lead to different Kantian outcomes, even though they describe the same material environment. The note also identifies special cases in which the interior first-order condition is representation-invariant, including common-exponent power transformations and symmetric profiles under identical transformations. The conclusion is constructive: applying multiplicative Kantian equilibrium to an economic environment requires specifying both the use of the Kantian criterion and the represented scale in which the Kantian counterfactual is evaluated.