The Economic Trinity is the doctrine concerning how the Father, Son, and Holy Spirit relate to each other and the world. The word economic is used from the Greek oikonomikos, which means relating to arrangement of activities. Each person has different roles within the Godhead, and each has different roles in relationship to the world (some roles overlap).
The Father sent the Son (John 6:44, 8:18). The Son came down from Heaven not to do His own will but the will of the Father (John 6:38). The Father gave the Son (John 3:16), who is the only begotten (John 3:16), to perform the redemptive work (2 Cor. 5:21, 1 Pet. 2:24). The Father and Son sent the Holy Spirit. The Father, who chose us before the foundation of the world (Eph. 1:4), predestined us (Eph. 1:5, Rom. 8:29) and gave the elect to the Son (John 6:39).
It was not the Son who sent the Father. The Father was not sent to do the will of the Son. The Son did not give the Father, nor was the Father called the only begotten. The Father did not perform the redemptive work. The Holy Spirit did not send the Father and Son. It is not said that the Son or the Holy Spirit chose us, predestined us, and gave us to the Father.
For a single verse that shows differences in roles, see 1 Pet. 1:2, "According to the foreknowledge of God the Father, by the sanctifying work of the Spirit, that you may obey Jesus Christ and be sprinkled with His blood," You can see that the Father foreknows. The Son became man and sacrificed Himself. The Holy Spirit sanctifies the church.
To be overly simplistic, we could say that the Ontological Trinity deals with what God is and the Economic Trinity deals with what God does.
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