Are there any indicator specifically designed to examine privatization? Do you think there should be different indicators to examine privatization in the developed and developing countries?
It depends on goals of SOE privatisation, e.g. state security guaranted by new owners connected with global powers, jobs preservation, profitability improvement, implementation of projects important for the country, and the like... You can devise huge number of indicators for each unclassified an classified goal...
It might be interesting to look at the expectations that politicians have when privatizing public services. These expectations usually range across three dimensions: from efficiency/technical performance, to altered relationships with clients and other stakeholders, to the organization itself.
With regard to the first, you could look at the quality, quantity, and value for money per unit of output, whatever that may be. But I think you should also look at the other two dimensions.
Regarding the second group, I see - broadly defined - the interests of citizens (i.e. satisfaction, trust, equity, participation, etc.), and accountability as the most important indicators.
The last group of indicators concern the organization itself: what happens to staff in the organization? Are they more satisfied, committed, motivated and/or productive?
Article Great Expectations of Public Service Delegation: A systematic review
I think the same criteria should be applied in developed and developing countries, asking questions such as whether the privatisation improved productivity and whether the amount the government received from the privatisation exceeds the net present value of the future income it would have received from the asset had it not been sold.
The best Australian writer on this topic in my opinion is Prof John Quiggin from UQ in books such as Does Privaitisation Pay? and Great Expectations.