Jeff Allyn, left, and David Barenholtz in Rancho Mirage, Calif., where they recently bought a new home. The couple had about $1.5 million to spend. Philip Cheung for The New York Times

This past spring, David Barenholtz and Jeff Allyn decided it was time to leave Texas. The couple, both self-employed business consultants, had lived in Austin for eight years after moving from Los Angeles. They were still formulating a plan when their four-bedroom house in Austin’s Barton Hills neighborhood was unexpectedly snapped up by an eager buyer.