The vice presidency has been the butt of jokes for generations. But politicians beg for the job, and with the right chemistry and personal dynamic, it can be a powerful position. We talk with the former chief White House reporter for The New York Times, Peter Baker, who has written a book about Vice President Dick Cheney’s relationship with President George W. Bush, “Days of Fire: Bush and Cheney in the White House.” We also talk with Amy Chozick and Jonathan Martin, political reporters for The New York Times, about what we know about Senator Tim Kaine and Governor Mike Pence, the current candidates for the job and whether, once elected, Hillary Clinton or Donald J. Trump is prepared to hand over real responsibility and stature to either of them.