No, seriously...

Each person (individual) in the Godhead holds a unique office.

The Father holds all authority.

The Son is the kinsman redeemer of mankind and the creator of all things created in the beginning.

The Holy Spirit is the author of the Bible (2 Peter 1:20-21) and the revealer of truth / testifier of Christ / the Divine reporter.

As far as equality goes all three are equal. The Father is in authority which the Son and the Spirit submit as should we all.

But this does not make him better than the Son or the Spirit.

Walter Martin used to say the President is greater than I but he will tell you before anyone else that he is not better than I am...

Good example of authority and not being better / higher (as the OP says) than anyone else being considered.