Why do Gods take avatar?
This is indeed a very good question. But unfortunately no specific reason can be pointed out as the sole cause for God's incarnations:
हरि अवतार हेतु जेहि होई। इदमित्थं कहि जाइ न सोई।। [Ramcharit Manas - 1.121]
However, from scriptures and whatever sages have said three primary reasons can be pointed out,
1. Establishment of Dharma 2. Deliverance of saintly people 3. Destruction of the wicked people
Shri Krishna has Himself told this to Arjuna as we know from the Gita:
paritranaya sadhunam vinasaya ca duskrtam
dharma-samsthapanarthaya sambhavami yuge yuge [BG - 4.8]
The same reason has also been stated for Rama:
जब जब होइ धरम कै हानी। बाढहिं असुर अधम अभिमानी।।
करहिं अनीति जाइ नहिं बरनी। सीदहिं बिप्र धेनु सुर धरनी।।
तब तब प्रभु धरि बिबिध सरीरा। हरहि कृपानिधि सज्जन पीरा।।
असुर मारि थापहिं सुरन्ह राखहिं निज श्रुति सेतु।
जग बिस्तारहिं बिसद जस राम जन्म कर हेतु।। [Ramcharit Manas - 1.121]
Whenever dharma declines and the demons, wicked and the haughty increase,
when they do such immoral work that they can not be described, and when Brahman, cow, gods, and earth are in danger,
In those times God taking different kinds of bodies take away the pain and suffering of the saintly persons,
Killing the demons He reinstates the gods and preserves the order of His own Vedas,
He spreads His glory throughout the world, for Rama's birth know these are the cause.
But then the question is why the almighty God, by whose mere thinking thousands and thousands of universes get annihilated, would take birth to kill mere demons and wicked humans who are just part of the universe itself? Can't He just do it by His will without taking so much trouble of taking birth here in this mortal world?
Hence, the above reason alone is not satisfactory. So Shrimad Bhagavatam states an even more appropriate reason for the cause of God's descend as a human. God takes avatar as a human not just to kill the demons but to teach us the way of life by practically observing it Himself:
martyāvatāras tv iha martya-śikṣaṇaṁ
rakṣo-vadhāyaiva na kevalaṁ vibhoḥ [SB 5.19.5]
This reason seems sensible as from Shri Rams's life we learn how to be a good son, good brother, good king, etc. But then the question arises, what is the necessity of God to take avatar and teach us Himself when prophets, saints and sages already do the job of teaching by living saintly life themselves? Again, if we take Shri Krishna's birth as example, then there are also many activities by Him which we should never try to do it ourselves. Hence, this can't be the sole reason. So Bhagavatam says, "However we may argue, there can be no other reason than amusement for which He who has no birth takes birth here":
na te ’bhavasyeśa bhavasya kāraṇaṁ
vinā vinodaṁ bata tarkayāmahe [SB - 10.2.39]
That is, God takes birth here for His amusement or entertainment. But this can not be the actual reason because He who Himself is always blissful (sadchidanda) would require no other thing for His amusement. Only they who are not happy, tired or stressed require and depend upon other things for happiness and amusement. But God being always blissful why would He require anything else for His pleasure or amusement? Hence, Bhagavatam states the most appropriate reason for God's descend is to bestow the highest benefit upon humankind by manifesting His pastime here:
vyaktir bhagavato nṛpa [SB - 10.29.14]
This reason seems sensible because without the incarnations of God, the world would be bereft of His sweet pastimes. Without which there would be no scriptures like Ramayana, Bhagavata, etc. and the devotees would have nothing to think and talk about. Only the concept of avyakta (unmanifested) nirakara (formless) God would have remained which is hard to comprehend and practice. So one of the primary reasons for God's descend as a human is for bestowing the mercy of devotion to people and even to the transcendentalists who worship the unmanifested formless absolute (avyakta nirākara brahma):
tathā paramahaṁsānāṁ munīnām amalātmanām
bhakti-yoga-vidhānārthaṁ kathaṁ paśyema hi striyaḥ [SB - 1.8.20]
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