Differentiability of the reduced-form win-probability function

Prove differentiability of the ex-ante equilibrium hiring probability function P*(b, e; x) with respect to both the bid b and signaling effort e under the modeling assumptions of Section 5, where P*(b, e; x) denotes a worker of observable type x’s reduced-form win probability and signal production includes additive noise and employer utility includes an idiosyncratic taste shock. Establishing differentiability is required to justify the use of first-order conditions for identification and estimation of worker costs and abilities.

Background

In the identification of labor supply, the authors invert workers’ first-order conditions to recover costs and abilities from observed bids and efforts. This procedure presumes that the reduced-form win-probability P*(b, e; x) is differentiable in b and e so that partial derivatives exist and can be used in the first-order conditions.

The paper motivates differentiability by the presence of signal-production noise and employer taste shocks, but does not provide a formal proof. A proof would strengthen the theoretical foundation of the identification strategy that relies on Equations (7) and (8) and the subsequent inversion in Equations (9) and (10).