Let the Punishment Fit the Crime: Enforcement with Error

Given an externality function that implements a social objective, this paper examines the possibility of implementing the social objective when the action is observed with error. Provided that the signal is informative in the sense that it separates certain distributions of actions and agents are risk-neutral, the social objective remains implementable even with observational error. In addition, when errors are small, there is a closed form second-order approximation for the penalty function that depends only on first and second moments and two derivatives of the externality function. The formula is applicable when activity is measured reasonably accurately, which is necessary for an acceptable implementation. This formula is simple enough to lend itself to actual implementation.