Barth, after referring her to John Calvin responds in part by saying:
"Paul did not write all that in order to teach and to canonize a certain concept of the relation between man and woman; he took that relationship (which he considers the right one) to illustrate the relation between God and man, as it should exist in the Christian community. … bear in mind that not only Paul, but the whole Bible, assumes that the man-woman relationship on earth and in time is not a matriarchy but a patriarchy. That is a fact; just as it is a fact that the Chosen People (to whom Christ also belonged) were not the Carthaginians nor the Spartans but the Israelites. Or just as it is a fact that in his revelation (in human language, history, and philosophy-of-life) God did not choose the Middle Ages, nor the 20th century, but late antiquity, thus making that era the centre of all ages. These selections do not imply any recognition of special worth and excellence in male persons nor in the Israelite people nor in the people of late antiquity. But they indicate special dispositions made by God in his dealings with human beings. Their importance is only temporal. But as such they cannot be simply ignored, nor can they be disputed with personal arguments" (cf. 1 Corinthians l l.11).
"Whether or not we recognize them as right, whether we like them or not, we have to accept these dispositions in this life until the time of the New Heaven and the New Earth. We are, of course, at liberty to object to these dispositions; in so doing we may experience to some extent whether they are right or wrong. For instance, a great many Christians in Germany today object to the fact that Christ was a Jew, time will show whether their objections are salutary. And women can object to the fact that the Bible says "man is the woman's head." Time will show whether it is good to reverse this disposition or (as you would like to do) to neutralize it. I will refrain from saying anything about Paul's sociological, psychological, or physiological arguments. But I may point out that there is another possibility: not to oppose God's dispositions but to accept them without argument because they are, and then perhaps with time to realize that they are good. These dispositions are bound up with the fact of his revelation in Christ, who was a man, thus confirming the superiority of Adam. …The essential point, which cannot be too much stressed, is that Paul's concept of man's superiority is merely a means to an end, merely to illustrate God's superiority over human beings. Read this passage carefully from this approach, and you will see that it excludes all idea of mutualism! I think it should then be easier for you to understand basically what Paul says about the man-woman relationship (within the context of this higher truth)."
** A theological footnote: Keep in mind that Barth is here speaking of Adam’s superiority as rank or position, not unlike God the Father as superior to God the Son. Moreover, if it was the joy of Christ to become servant of all (Philippians 2.5), certainly, we ought not to reject the office he gives to us; this too is for His glory. The argument is often set forth in such a manner that “subordination equals denigration” and “equality equals indistinguishability.” However, we ask ourselves the question about the role of the Son of God, in what manner is He denigrated because He obeys the will of the Father? “Is the church denigrated by her subordination to her Lord?” What about submission to pastors and elders (Hebrews 13.17) or children submitting to their parents are these actions demeaning the worth or value of the individual?