Phil Appenzeller, the longtime chief executive officer of Munsch Hardt Kopf & Harr, said his firm increasingly outsources office services such as hospitality, mail room work and copying. But when it comes to higher-level functions such as accounting and information technology, he has never considered taking the work to outside companies.

“My [chief financial officer] and I have this debate all the time. I’m kind of old school. I want them to be our employees,” he said.