This study empirically examines the public and social policy question: Do state restrictive abortion laws affect the likelihood that women use more highly effective contraceptive methods? Using contraceptive use data from the Behavioral Risk Factor Surveillance System 2002 survey, the empirical results show that Medicaid Funding Restrictions, Informed Consent Laws, and Two-Visit Laws have no significant impact on adult women’s (ages 18-44, 18-24, 25-34, 35-44) use of highly effective contraceptive methods.
I wouldn't imagine that these laws would have any effect on contraception use.
What would be even more interesting is if they examined abortion access and its affect on contraception use. If you know that you can't get an abortion easily, will you be more careful?
A state’s antiabortion attitudes, which likely contribute to the enactment of restrictive abortion laws in a state, are a major factor in inducing greater use of highly effective contraceptive methods by adult women at-risk of an unintended pregnancy. The empirical findings remain robust for various population subgroups of adult women (i.e., married, single, employed, unemployed, with children, no children and college educated).
And how does this affect actual rates of pregnancy and abortion? That would be an interesting question.