Are there public functions that might benefit from more competition, including competition from private firms?

Yes there are. One example would be Government involvement in taking care of illegal immigration.
Illegal immigrants are not entitled for the minimum wages that are set by the Government. So, the companies often agressively seeking for them so they can pay them cheaply for low-level tasks,and give the company a decrease in total expense. (which will give a competitive advantages toward companies that use legal workkers)

The business cycle is the periodic but irregular up-and-down movement in economic activity, measured by fluctuations in real gross domestic product (gdp) and other macroeconomic variables. a business cycle is typically characterized by four phases—recession, recovery, growth, and decline—that repeat themselves over time. economists note, however, that complete business cycles vary in length. the duration of business cycles can be anywhere from about two to twelve years, with most cycles averaging six years in length. some business analysts use the business cycle model and terminology to study and explain fluctuations in business inventory and other individual elements of corporate operations. but the term "business cycle" is still primarily associated with larger (industry-wide, regional, national, or even international) business trends.