Redemption, as seen through the framework of Covenant Theology, requires at least two Persons. As Paul puts it:
For there is one God, and there is one mediator between God and men, the man Christ Jesus, who gave himself as a ransom for all, which is the testimony given at the proper time.—1st Timothy 2:5-6 (ESV)
The Third Person is necessary to transmit the benefit of Christ's ransom to the world:
Great indeed, we confess, is the mystery of godliness:
He was manifested in the flesh,
vindicated by the Spirit,
seen by angels,
proclaimed among the nations,
believed on in the world,
taken up in glory.
—1st Timothy 3:16 (ESV)
Jonathan Edwards' Economic Trinity
Jonathan Edwards theory of the Immanent Trinity characterizes the Spirit as the very love between the Father and the Son. He transferred this idea to the Economic Trinity. He writes:
So that in this verse [1st Corinthians 1: 30] is shown our dependence on each person in the Trinity for all our good. We are dependent on Christ the Son of God, as he is our wisdom, righteousness, sanctification, and redemption. We are dependent on the Father, who has given us Christ, and made him to be these things to us. We are dependent on the Holy Ghost, for it is of him that we are in Christ Jesus; it is the Spirit of God that gives faith in him, whereby we receive him, and close with him.
The saints have both their spiritual excellency and blessedness by the gift of the Holy Ghost, and his dwelling in them. They are not only caused by the Holy Ghost, but are in him as their principle. The Holy Spirit becoming an inhabitant, is a vital principle in the soul. He, acting in, upon, and with the soul, becomes a fountain of true holiness and joy, as a spring is of water, by the exertion and diffusion of itself.
There are three roles1 in the act of redemption that the Trinity fills:
The Father: The giver of beneﬁt.
The Son: The purchaser of beneﬁt.
The Spirit: The beneﬁt purchased.
Edwards finds support for this construction (among other places) in Paul:
Oh, the depth of the riches and wisdom and knowledge of God! How unsearchable are his judgments and how inscrutable his ways!
“For who has known the mind of the Lord,
or who has been his counselor?”
“Or who has given a gift to him
that he might be repaid?”
For from him and through him and to him2 are all things. To him be glory forever. Amen.
—Romans 11:33-36 (ESV)
We depend on there being three persons in the Trinity to have equal honor in the task of redemption:
And each person of the Trinity is equally glorified in this work: there is an absolute dependence of the creature on every one for all: all is of the Father, all through the Son, and all in the Holy Ghost. Thus God appears in the work of redemption as all in all. It is fit that he who is, and there is none else, should be the Alpha and Omega, the first and the last, the all and the only, in this work.—Jonathan Edwards, "God Glorified in Man's Dependence"
I'm grateful to Richard M. Weber's paper "The Trinitarian Theology Of Jonathan Edwards: An Investigation Of Charges Against Its Orthodoxy" [PDF] for this way of expressing the roles.
Edwards translated the preposition in him. I've asked a question on whether Edwards' translation is supportable here.