  Consider a causal structure with endogeneity (i.e., unobserved
confoundedness) in empirical data, where an instrumental variable is available.
In this setting, we show that the mean social welfare function can be
identified and represented via the marginal treatment effect (MTE, Bjorklund
and Moffitt, 1987) as the operator kernel. This representation result can be
applied to a variety of statistical decision rules for treatment choice,
including plug-in rules, Bayes rules, and empirical welfare maximization (EWM)
rules as in Hirano and Porter (2020, Section 2.3). Focusing on the application
to the EWM framework of Kitagawa and Tetenov (2018), we provide convergence
rates of the worst case average welfare loss (regret) in the spirit of Manski
(2004).
