Habakkuk 2:2 And the Lord answered me, and said, Write the vision, and make it plain upon tables, that he may run that readeth it. 3 For the vision is yet for an appointed time, but at the end it shall speak, and not lie: though it tarry, wait for it; because it will surely come, it will not tarry. 4 Behold, his soul which is lifted up is not upright in him: but the just shall live by his faith.
Romans 1:17 ends with the words, The just shall live by faith. Paul takes this from the words we read from Habakkuk 2 above. We see in Habakkuk 2:3 the mention of an appointed time. We see that it will not tarry and it will surely come. What is Habakkuk speaking of here? This is a mention of the coming of Christ. The vision is telling those at that time the Messiah will surely come. He will come at the appointed time. Those for who Christ came, the just ones will live by His faith. How did Christ purchase eternal life for us? How are we redeemed? Life comes by that faith, that gospel of Christ. That is the faith by which we live. My faith does not give me life. My life gives me faith. Our life comes from that work that Christ completed and that is what Paul is telling us here. Christ earned and established righteousness by that faith, by His gospel. Life came by His obedience. God did not bestow life because of my faith, but because of His. To not see this is to miss the message of the gospel. Yes, we live lives of faith but we cannot say that our faith is perfect. It took that perfect faith, that finished word of Christ, to satisfy God and in turn give us life. If we were in Him, then we have that life that He earned. That is representation. We will see later in Romans 5:
Romans 5:18 Therefore as by the offence of one judgment came upon all men to condemnation; even so by the righteousness of one the free gift came upon all men unto justification of life.
As one brought judgment which equaled condemnation, the other brought justification which equaled life. That is what Christ was doing for us at and by the cross. It was His obedience unto death that saved us, that gave us life.