Is there an equilibrium between privately owned and publicly listed companies? Private equity funds are increasingly taking formerly public companies off the exchanges. Is there an economic reason for any public companies to exist and what is the equilibrium share of public companies in the total corporate 'population'?
I don't know exactly how the numbers would work out, but you would expect an equilibrium but it will depend on many factors.

A complication is that you have public companies taken private and private companies that never go public.  It may be there is some equilibrium between these as they are unlikely to be equivalent. Ex public companies are often highly levered,  while never public are often debt free and cash rich.

Particular important factors would be interest rates and credit spreads.  This is because it boils down to the distribution of capital.  If there is lots of money available to lend then LBOs become attractive and companies can be taken private. If there is strong demand for equity then IPOs become easy.

I think an economy with a shortage of public companies would show overvalued equities and hence IPOs are encouraged (think dot com bubble - shortage of public web companies vs demand). If debt is cheap and there are too many public companies then you will see either take overs or LBOs.

I suspect the effect is most visible within sectors rather than the whole economy.