Abstract | Punishment is a cost imposed on a target, in response to an un- desirable action. Yet choosing to punish also reveals information about the authority’s own motives and values. We propose that observers jointly infer the wrongness of the action and the authority’s motivations. Using hypothetical scenarios in un- familiar societies, we experimentally manipulated observers’ prior beliefs and measured human observers’ inferences after observing punishment. These inferences were recapitulated in a formal model that inverts an intuitive causal model of authorities who make rational choices about punishment by weighing its costs and benefits (i.e. utilities). An essential component of this model, driving these inferences, is that legitimate authorities consider the utility of a proportional response to harmful actions, which depends on the balance between the wrongness of the act and the severity of the punishment.