This is from a class I'm teaching. Some will know that the expression translated "faith in Christ" in Greek is more usually, "the faith of Jesus" or "the faith of Christ." European scholarship has tended to follow the traditional "faith in Christ" translation (in this corner, James Dunn as the type). American scholarship has tended to follow the "faithfulness of Christ" translation (in the other corner, Richard Hays as the type).
I wrestled with this issue for fifteen years before I had any real sense of what I think. Here's where I'm at.
1. It's clear that Jesus' "obedience to death" (Phil. 2:8) was a significant part of the equation. Hebrews 5:7 speaks of Jesus praying to the one who could save him out of death and being heard because of his godliness. Romans 5:19 speaks of the importance of Christ's one act of obedience and how many will be justified because of it. So it clearly would fit with Paul's theology that we are justified through the faithfulness of Jesus in some way. The question is whether Paul actually used the expression "faith of Jesus Christ" in this way.
2. That Paul could think of having faith upon Christ in some way also seems clear to me from Romans 9:33, "The one who has faith upon him will not be ashamed." The object of faith here is clearly the rock of stumbling, namely Jesus the Lord. We know this from 10:11, which repeats this same Scripture from Isaiah 28:16. The having faith upon him is parallel there to calling upon the name of the Lord, namely, "Jesus is Lord," as in 10:9. Having faith upon Jesus in this context is thus having faith that he is Lord which involves having faith that God raised him from the dead for our justification (4:25).
3. If "faith upon him" in 9:33 broadly means faith on Christ, meaning faith that God has raised Jesus from the dead and established Jesus as Lord, then we can backtrack to 9:32, where Paul says that Israel did not reach the "rule of righteousness" because they did not pursue it "on the basis of faith but on the basis of works." The that the following verse, which speaks of putting "faith on him," refers to faith on Christ implies that the expression "on the basis of faith" here refers to human faith, not Christ's faith. But since this expression is a formula Paul uses over and over, we must conclude that human faith is a major part of what Paul means by the expression ek pisteos, "on the basis of faith."
This would imply that, at best, Hays is only half right to see the expression in relation to Christ's faith. It would seem that Paul uses this expression in relation to a general principle, namely, that God justifies on the basis of faith. Perhaps this is true in relation to Christ as well--who is justified or found not guilty on the basis of a sinless faith rather than, as in our case, a faith in the midst of sinfulness. But at this point I conclude that Dunn is more correct when it comes to the expression ek pisteos, "on the basis of faith," in both Galatians and Romans.
4. The clincher for me on the specific expression "faith of Jesus" was 2 Corinthians 4:13. Here Paul quotes Psalm 115:1 (LXX): "I have had faith, therefore I have spoken. What is Paul talking about? What Paul speaks is that the one who raised Jesus will also raise us (4:14). Here is my interpretation of this passage. Paul is comparing his sufferings to those of Jesus. Here we remember Hebrews 5:7: "Jesus prayed to the one able to save him out of death and he was heard because of his godliness." My argument is that Paul likely read Psalm 115/116 as the words of Christ, as he and other Christians did so many psalms. Thus he sees in Psalm 115:1 Jesus expressing his faith that God would raise him.
My argument is somewhat involved (it comes out in CBQ sometime over the course of the next year) but basically I believe Paul is saying in this verse that just as Jesus had faith (in the God who could raise him from the dead) so also Paul has such faith. This verse was for me the straw that tipped the "faith of Jesus" debate Hays' way in terms of the specific expression, "the faith of Jesus." I believe that the idea of the "faith of Jesus" was thus not only a part of Paul's vocabulary, but even more likely a commonplace in Christian circles. This would explain how Paul can use an expression that seems so ambiguous to us without feeling the need to explain it.
5. So in conclusion, I believe that Paul begins several theological discussions with a commonly understood phrase, "the faith of Jesus" (Rom. 3:22; Gal. 2:16). Note that Jesus is the immediate word following faith in these particular expressions (see also Rom. 3:26). But then Paul balances out the faith of Jesus with the importance of humans having faith (and Paul primarily sees God as the object of such faith, see Romans 4, although he can loosely speak of having faith on him, Christ, as in 9:33). When he finally gets to expressions like "the faith of Christ," then, he has established a general principle that applies to everyone, both Christ and humans. The expression, "the faith of Christ" thus becomes a sort of double entrendre referring to both Christ's faith and our faith in Christ. From then on, the expression "on the basis of faith" comes to refer primarily to human faith in what God has done in Christ.
We might say, therefore, that Paul argues "ek" Hays "eis" Dunn.