An integral equation for the identification of causal effects in nonlinear models

When the causal relationship between X and Y is specified by a structural equation, the causal effect of X on Y is the expected rate of change of Y with respect to changes in X, when all other variables are kept fixed. This causal effect is not identifiable from the distribution of (X,Y). We give conditions under which this causal effect is identified as the solution of an integral equation based on the distributions of (X,Z) and (Y,Z), where Z is an instrumental variable.