This muddles three distinct ideas - cost effectiveness, productivity and efficiency.  When economists talk about productive efficiency, they usually mean the fourth definition, "improved ratios of output per unit per unit cost of input".  That's more or less the definition used in  R Jacobs, P Smith, A Street, Measuring efficiency in health care, Cambridge UP 2006,  which defines efficiency as "the ratio of a weighted sum of outputs to a weighted sum of inputs". 

 

As a general proposition, private provision tends to be more efficient in this sense than public provision.  The reason is simple: the private sector is able to reduce average costs per unit by adverse selection (not doing work that pushes up the average cost), and the public services don't have that option.  Public services have to aim for cost-effectiveness, which is about achieving stated goals at the lowest possible cost; it follows that public services are intended to be inefficient, because they are trying to do something else instead.  The attached graph shows the difference in theory. That also implies that while the private sector should be able to deliver lower costs per unit, it doesn't necessarily add up to lower costs overall - efficient is not the same as cheap, and efficient in part is not the same as cost-effective on the whole. 

 

There's a full-blooded discussion of what measuring efficiency might involve in M Stone, 'How not to measure the efficiency of public services (and how one might)' Journal of the Royal Statistics Society A 2002 165(3) 405-434.  It's about the relative efficiency of police forces, but that shouldn't get in the way of the principles.  Then some of the reservations to be made about political positions and valuations are in  Smith P, Street A, 2005, Measuring the efficiency of public services, Journal of the Royal Statistical Society Series A 168(2) 401-417.