Only two Gospels, Matthew & Luke, contain birth narratives.
Matthew's focus is on Joseph and includes accounts of the Magi, the slaughter of infants and the flight to Egypt.
Luke focuses on Mary and includes the accounts of John the Baptist’s birth, the shepherds, the inn/manger and three poems/songs of Mary, Zechariah and Simeon.
The accounts in Matthew and Luke are similar in that they approach Jesus as a saviour who was born during the reign of Herod as a fulfilment of prophecy.
There are many differences between these two accounts in terms of characters, locations and perspectives.
Luke focuses heavily on the birth of John the Baptist, and later in the Gospel clarifies how John the Baptist was a forerunner to Jesus.
Matthew focuses on Jesus’ Jewish ancestry and on him as a fulfilment of Israel’s hopes.
One event in Matthew that could potentially have been widely known, the slaughter of infants, is not mentioned in Luke.
Redaction criticism is the science of examining how source material has been edited (redacted) by editors to reflect their point of view.
When we report a common story, it will inevitably reflect something of our concerns, pet issues and biases.
Redaction critics of the birth narratives suppose that both Matthew and Luke shared a common story, but told it in ways that reflected their theological concerns and audiences.
Redaction critics note the many references to the Hebrew scriptures in Matthew’s account inferring he was a Jewish believer writing for Jews who needed to be convinced that Jesus was the fulfilment of Jewish prophecy.
By contrast, Luke wrote for a non-Jewish audience, thus emphasizing Jesus as a universal saviour (‘a light to lighten the gentiles’).
The doctrine of the incarnation includes the concept of the ‘hypostatic union’: Jesus was a union of human and divine, fully God and fully human.
One way of understanding incarnation is the substantial presence model which says that Jesus possessed all the attributes of God – that is he was omniscient, omnipotent, etc at every moment.
A different way of understanding the incarnation is the kenotic model (Greek for ‘emptying’); in being born, Jesus surrendered some of these attributes.
The birth narratives could be seen as kenotic in terms of the fragility of Jesus or the danger he faced.
Or, as reflecting the ‘substantial presence’ model (the miraculous good fortune around the birth of Jesus).