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.
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.
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.