What He "gave up" was not any part of the divine essence. He "gave up" by taking ON the form of a servant.
The context of the Carmen Christi is Paul's admonition for each of as equals to treat each other as better.
Phi 2:3 Let nothing be done through strife or vainglory; but in lowliness of mind let each esteem other better than themselves.
This is spot on!

We seem to have duplicated several threads on this topic, so I hope the mods will forgive me for repeating a post I've just made on a different thread.
I am not an expert at Greek as some people are on this board, but here is my own attempt at translating Philippians 2:6, preserving the Greek word order to some degree:
'Who, in the form of God existing, not as a harpagmos he reckoned being equal to God, but He emptied Himself, the form of a slave taking......' So the $64 question is, what is a
harpagmos (ἁρπαγμὸs)? The problem is that it is a
hapax legomenon, a word that occurs only here in the whole NT, so there has been a difficulty in establishing exactly what is does mean. The verb
harpazo has to do with 'grasping' or 'seizing,' but what does the noun mean? Well, recently a guy (mentioned by SATS Prof somewhere) has trawled all through all the occurrences of
harpagmos in ancient and koine Greek, and he has proved to most people's satisfaction that it means 'Something held to one's advantage,' like a 'Get out of Jail Free' card when you're playing
Monopoly. The new NIV gives this meaning.
So the Lord Jesus did not regard being in the form of God something to be held to His advantage. He did not say, "Well I am God, and I am jolly well not going to go down to earth and subject Myself to the most horrendous punishments and agonies on behalf of these miserable human beings!" No, He emptied Himself- of what? Not His deity, but His prestige and glory. He
'made Himself of no reputation.' So far from hanging on to His privileges as God, He relinquished them and took on the nature of a slave. I think it is worth looking at
Exodus 21:1-6 and the person who gives up his freedom because he loves his master. His ear is pierced through. Now consider Psalm 40:6-8. The KJV says,
'My ears You have opened,' but the Hebrew can certainly mean
'pierced.' Christ is the One who has given up His position and His freedom to become a slave to the Father; to live the life of perfect obedience to His will that we cannot live, and to take the punishment that we deserve to receive.