The social economy: Unlocking value and productivity through social technologies http://bit.ly/2m5cL1e via @McKinsey_MGI

While this report is a little old (2012) it does state "By using social technologies, companies can raise the productivity of knowledge workers by 20 to 25 percent." 

The thing is, I found this report only after measuring a project of using internal social at my old company and we got a similar measure of 25% increase in employer efficiency.

Looking on Wikipedia,  my old company has 137,000 employees, just think, 34,250 employees for free!

That is a pretty good business case inself.

I'm CEO of DLA Ignite dlaignite.com and I cannot measure it, as of day 1, we are a social company.  I tell my staff that I don't answer emails between 09:00 AM and 17:00, if they want me during the day then they can get me on social.

Back to my previous company ...

If 25% more people in your company isn't a business case then how about?

95% accurate sales forecast and increasing the efficiency of the sales function, for example getting sales and non sales people collaborating.

Increased employee experience by supporting returners, getting new starters contributing quicker, getting new salespeople contributing quicker

Business is afterall about communication and social media provides a friction less way to communicate.  Better still, there is no end-user training as we all use social products already.  We all come to work now expecting an "Facebook" user experience with the IT systems used.  So why not give it to them.

PS: We use Slack to run the business, but as this article points out there are plenty to of other social platforms.