I’m going to answer this question, but I also want to submit that the true answer is unknowable and it doesn’t matter in the grand scheme of things.
I went to Columbia for graduate school. I’d say from my observations Columbia students (especially in my department, psychology, but I’ve seen it in others), in aggregate have perhaps slightly better chances of getting into graduate programs at Columbia, assuming that they are otherwise excellent candidates. This really depends on the individual department, though. Two caveats.
One, do remember that while Barnard is affiliated with Columbia and is officially one of its four undergraduate colleges…it is also a separate institution, and it has its own economics department with its own faculty. You’ll have the opportunity to take economics classes across both departments, and to do research with Columbia professors - but you’ll most likely take most of your classes with the Barnard economics faculty.
Two, you can’t necessarily apply the high-level probability to your individual probability - that doesn’t mean that if you go to Barnard that you, personally, will have higher or lower chances of admission to any of Columbia’s programs. That depends on a whole variety of other factors - for example, a student who would thrive at another college might not mesh well with the atmosphere at Barnard.