Gun deaths is a disingenuous term in the modern American political climate, because it includes suicides and assumes a lack of access to guns would eliminate the 60 percent of handgun deaths that are suicides, or guns used by criminals.
Gun ownership does not seem to lead to more murders, and murders are the real concern. There, we have a Handgun Paradox, where even in states like California, which make it more difficult to get a concealed weapons permit, gun ownership has boomed but murders have plummeted.
Yet if a trickle down effect is acceptable economics, it might also work in guns.