Since 1977 California has had the California Tax Postponement Program to protect low income seniors from the economy. If you're a low income senior and own your house in California you can defer property taxes indefinitely so that no matter what happens to housing prices you'll never feel pressured to move. Several states have something similar (eg. NJ senior freeze) but I think California is the most aggressive.
Prop 13 is wholly unnecessary. It could be removed entirely overnight and grandma wouldn't need to pay a dime.