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As Strauss demonstrated with inescapable lucidity many decades ago, the two nativity stories of Matthew and Luke disagree at almost every point, one exception being the location of Jesus' birth in Bethlehem. [...] Matthew assumes Jesus was born in the home of Mary and Joseph in Bethlehem, and that they only relocated to Nazareth in Galilee after taking off for Egypt to avoid Herod the Great's persecution. Luke knows nothing of this but instead presupposes that Mary and Joseph lived in Galilee and "happened" to be in Bethlehem when the hour struck for Jesus' birth because the Holy Couple had to be there to register for a Roman taxation census. [...] For the moment, my point is to suggest that Luke and Matthew both seem to have been winging it, just as they did with their genealogies. They began with an assumption and tried to connect the dots. This time, their common assumption was that Jesus was born in Bethlehem. Whence this assumption? Was there historical memory that Jesus was born there? Hardly; if there had been, we cannot account for Mark's utter lack of knowledge of the fact. No, it seems much more natural, much less contrived, to suggest that Matthew and Luke alike simply inferred from their belief in Jesus' Davidic lineage that he must have been born in Bethlehem. [...] Matthew and Luke both placed the birth of Jesus in Bethlehem because they mistakenly thought prophecy demanded it. They went to work trying to connect the dots with narrative or historical verisimilitude, but with limited success.
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Robert M. Price (The Incredible Shrinking Son of Man: How Reliable is the Gospel Tradition?)