Editing Mechanism design (section)

====Sufficiency====

Mechanism design papers usually make two assumptions to ensure implementability:

<math>\frac{\partial}{\partial \theta} \frac{\partial u / \partial x_k}{\left|\partial u / \partial t\right|} > 0 \ \forall k</math>

This is known by several names: the [[single-crossing condition]], the sorting condition and the Spence–Mirrlees condition. It means the utility function is of such a shape that the agent's [[Marginal rate of substitution|MRS]] is increasing in type.{{Clarification|date=July 2024}}

<math>\exists K_0, K_1 \text{ such that } \left| \frac{\partial u / \partial x_k}{\partial u / \partial t} \right| \leq K_0 + K_1 |t|</math>

This is a technical condition bounding the rate of growth of the MRS.

These assumptions are sufficient to provide that any monotonic <math>x(\theta)</math> is implementable (a <math>t(\theta)</math> exists that can implement it). In addition, in the single-good setting the single-crossing condition is sufficient to provide that only a monotonic <math>x(\theta)</math> is implementable, so the designer can confine his search to a monotonic <math>x(\theta)</math>.