Equivalence Between the Two-Dimensional Value Model and the Pareto Weight Model

IN THIS APPENDIX, we establish an equivalence between (i) our “two-dimensional” model, in which the designer maximizes total value (VAL) over feasible mechanisms according to Definition 3 and (ii) a “one-dimensional” model in which agents only report their rates of substitution r and the designer maximizes weighted surplus (with Pareto weights λj) according to (VAL’). While we only need one direction of the equivalence to justify our derivation of optimal mechanisms in Section 4, we demonstrate the full equivalence to show that we could just as well start our analysis with the one-dimensional model (with Pareto weights given as a primitive of the model) and our conclusions would remain identical. To simplify notation, we use (X̄B X̄S T̄B T̄S) to denote a generic (direct) mechanism eliciting (v v) and (XB XS TB TS) to denote a generic (direct) mechanism eliciting r. Formally, a mechanism (XB XS TB TS) is feasible in the one-dimensional model if, for all r r̂,