“At a certain point the school system becomes no longer viable,” said Grant Oliphant, the new president of the Pittsburgh Foundation, which is overseeing a program that provides college scholarships worth up to $40,000 for any student who has attended the city’s public schools since the ninth grade and graduates from high school with a grade point average of at least 2.0.

“The notion is to create an incentive to stay in school and graduate,” Mr. Oliphant said. “The second aspect is economic preservation  to create an incentive for people to keep their kids in school or move here with their kids  to keep enough taxpayers in town.”

Since 1980, the city’s population has plunged from 423,000 to about 312,000. Since 2000 alone, the metropolitan area has lost 60,000 people.

While natural decrease occurred in many Southern cities that were magnets for retirees, the overall population was replenished by the influx of younger migrants. But in Pittsburgh and other places outside the South, not only has the population aged in place, but also, to a lesser extent, the very old  often disabled and widowed  have returned to spend their last years with children and grandchildren and avail themselves of better medical treatment and transportation.

“You think of this as a rural or small-city phenomenon,” said Gordon F. De Jong, a sociologist at the University of Pittsburgh. “Here’s a large metropolitan area where it’s happening.”