The Causal Shadow Price: Identification, Doubly Robust Estimation, and Semiparametric Efficiency for the Lagrange Multiplier of Interventional Fairness Constraints

In constrained fairness optimisation, the Lagrange multiplier μ* is treated as a tunable hyperparameter with no principled relationship to the underlying data-generating process. This opacity impedes interpretability and auditing of fairness-constrained models. Contributions. We establish a new class of gradient-space causal functionals induced by optimisation duality. Specifically: (i) Identification. When fairness is defined via the interventional effect under the backdoor criterion and overlap, the KKT stationarity condition combined with inverse probability weighting (IPW) yields an explicit functional expression for μ* in terms of propensity-weighted gradient geometry—the causal shadow price. (ii) Doubly robust representation. Replacing IPW with the augmented IPW (AIPW) estimator produces a doubly robust version of the shadow price that remains consistent when either the propensity score or the outcome regression model is correctly specified, but not necessarily both. (iii) Asymptotic normality and plug‑in validity. Treating θ* as fixed, the plug-in AIPW estimator μ̂* satisfies a central limit theorem with an explicit influence function; if θ* is replaced by a √n-consistent estimator θ̂, the influence function is preserved up to op(n^{-1/2}) under approximate KKT conditions. (iv) Semiparametric efficiency. The AIPW-based estimator achieves the nonparametric Cramér–Rao lower bound for estimating this functional under a nonparametric model for (X,A,Z). Interpretation. The shadow price μ* = G(P,f_θ*) is an emergent, model-class dependent property of the optimisation geometry under causal constraints—neither a structural causal parameter nor a free tuning knob. Its value is the negative projection of expected loss sensitivity onto the propensity-weighted causal gradient, scaled by the sign of the interventional discrepancy. Deviations between the optimisation-derived and estimator-derived values provide a practical diagnostic for causal model misspecification.