First, how is public sector productivity defined?
"Most public services are provided free at the point of use, and so there are no prices with which to value their outputs. This also makes it more difficult to place a value on the outcomes produced by those services. Ideally, we might value a hospital admission by the amount by which it improved the patient’s health, or a year of education by the benefits it has for a child’s development and future life trajectory. In practice, this is rarely feasible." (The Institute for Fiscal Studies)
Is the Fire and Rescue Service less productive if it puts out fewer fires, or more productive if its work prevents more fires?