In the five years since they began their life together in the desert sprawl of greater Phoenix, Devon Lawrence and Eren Mendoza have bounced from one itinerant home to another.

They have camped alongside a freeway off-ramp, using a gas station sink as their bath and a plastic tarp as their refuge from the relentless sun. They have slept on an air mattress in a friend’s living room. For the last two years, they have crammed into rooms at motels, paying as much as $650 a week.

Ms. Mendoza and Mr. Lawrence are both 32, and both have jobs. She works at a supermarket deli counter. He stocks shelves at a convenience store. Together, they earn about $3,500 a month. Yet they have been stymied in their reach for a modest dream: They cannot find an affordable home in a safe neighborhood in Phoenix, where rents have roughly doubled over the last decade.

“These prices are just wild,” Ms. Mendoza said. “It’s pretty much all anybody talks about. The fact that a dual income can’t support us is insanity.”

The impossible arithmetic of housing is a potent source of economic anxiety in Phoenix, and in many major American cities — a reality that could influence control of the White House.

Arizona is one of six battleground states likely to determine the result of the presidential election. Its unemployment rate was only 3.7 percent in February, lower than the 3.9 percent national rate. Inflation has slowed. In the Phoenix area, optimism is buoyed by $60 billion in investments in factories that make advanced computer chips — a Biden administration talking point.