How is productivity measured in public sector? How do they measure productivity in the public sector? Companies just say this is how much money we make per input but what about the public sector?
The BLS has a manual by Katharine G. Abraham (1998) on the subject.
https://www.bls.gov/lpc/iprprody.pdf
page 17 and page 21;  covers outputs and inputs respectively.

Measuring output:

“The specification and measurement of output is the single most troublesome problem
in computing productivity. It is a more serious problem for government service
than for private sector organizations. Some of the problems are endemic to a specific
government activity; others are more general in nature.”
…
“For most public sector output measures, physical quantities are used to calculate the
outputs.”

Measuring inputs:

“Two labor measures that are commonly used to measure private sector productivity are the number of hours and the number of persons. These two measures impart very different information and for this reason both are calculated. The preferred measure is the number of hours because this measure reflects the time worked to produce the outputs.

Hours are the preferred measure for government too, but governmental units do not usually collect these data. Instead, most governments collect and use the number of full-time-equivalent employees.”

The OECD has put out an article on the subject but at the country of Finland
http://www.oecd.org/sdd/na/2666079.pdf