How do Jesus' authority and power relate to his purpose?
There's alot going on in Luke 4:31-44. Luke highlights the BIG IDEA that the power of Jesus is linked to the identity of Jesus, and that Jesus is the Christ, the Son of God. He demonstrates this by Jesus' ability to cast demons out of people and to heal people from sickness. But before he gets to these power stories, note what how this section starts:
And he went down to Capernaum, a city of Galilee. And he was teaching them on the Sabbath, and they were astonished at his teaching, for his word possessed authority. (Luke 4:31-32)
The thing that made the teaching of Jesus unique was its authority. Mark adds this clarification: 'for he taught them as one who had authority, and not as the scribes.'. The scribes would only recount what other, earlier, famous scribes had said about scripture. But Jesus would teach with 'But I say to you'.
In teaching the Law and the Prophets of the Old Testament, as the Word incarnate, Jesus was teaching with authority the Word He had spoken. So more than the ideas of others, Jesus was giving the definitive interpretation of the word.
The idea of Jesus' authority is important. His power is encouraging, because He is stronger than darkness and sickness. But his authority is what gives him the right to do what he does.
In short, power is the might to do something, authority is the right to do something. And anything Jesus did, he did it with both - power and authority.
Another way of thinking about this is that Jesus appropriated the Godness of God in his ministry. And that's appropriate, because He is God. When I say that he appropriated God's God-ness, I mean that Jesus appropriately defined truth and right and wrong. His teaching - or defining of truth - was appropriate because - it's God's prerogative to define truth. Jesus unasashamedly and authoritatively defined reality.
The common Christian confession of the early church is that Jesus is Lord. This means he is the Sovereign Master who is in control. It also means he has the authority in our lives to tell us what to believe, how to live, what is true, and what is right and wrong.
The bad news is that we can deviate from what Jesus wants in two ways. First, we can deny his Lordship, and resist the idea that He has the authority to be God in our lives. And it is God's prerogative to define truth, morality, and what He expects from us. Second, we can acknowledge his Lordship in principle, but find a clever way to disagree with what he actually wants. That is, humans are geniuses at finding ways of calling bad good and ascribing it to God's endorsement.
The good news is that Jesus is Lord, and He is a good shepherd. If we submit to him, we can trust that He will bring us to a good place. We may not be completely comfortable with the place. But if it is the place of his choosing - whether related to our situation in life, what we believe, how we live, what we think is right or wrong - then we can embrace it as reflective of God's Godness.
The first order principle of all reality is that God is God, and there is no other. An intrinsic component of God's Godness is his authority. So even though we may not like the word authority - it tends to rub our permissive and relativistic society the wrong way - it's important, for in lies our salvation.
In short, God can save us becase He has both the authority and the power to save us. That means our salvation is effective (it works, because God is powerful), and it is right - because God has the authority to do it. God's purpose of bringing us to himself is accomplished through his authority and power.