If you have not read it yet, you must read Peter Leithart’s review of Khaled Anatolios’ defense of “person” language, and what that entails, in the doctrine of the Trinity:
I have touched upon this issue in the past. In particular, I am fascinated by where Barth would stand in the current discussion. It is not easy to say. As I have blogged before — Barth and the “fellowship in the Trinity — he can be appropriated by both sides, those for and against social or personalist models of the Trinity. I also discussed these matters here: “In God, subordination is not deprivation,” which is one of my favorite posts.
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Image: Khaled Anatolios (source: Boston College)
dogmatics.wordpress.com 
For Barth the Lord God is a single “person” or acting Subject who exists in, and only in, three ways or “modes” of being.” These three ways of being are therefore also necessarily personal. For Barth, the mystery of the Holy Trinity thus transcends the false dichotomy between “social” and “personalist” models, while incorporating elements of each. See my entry in The Oxford Handbook on the Trinity.
Thanks, Professor, for stopping by and recommending the resource. I’ll soon check our seminary library for said Handbook.
By the way, for any readers who may see this, I was using “personalist” as interchangeable with “social,” because that is how Leithart uses it. It could, however, be used as indicating a singular “person” or “subject,” and therefore opposed to “social.”