Palestinian Jews and Diaspora Jews

Bart Erhman's basic objection to the traditional authorship of Matthew is the improbability that a Palestinian Jew could write literary Greek. This raises several issues:

i) For many Jews, Greek was their native tongue. Indeed, that was so widespread that it necessitated Greek translations of the OT like the LXX. 

ii) "Palestinian Jew" is ambiguous. The fact that Matthew was living in Palestine at the time Jesus summoned him doesn't imply that Matthew was a native of Palestine. As the religious capital of Judaism, Jerusalem was a magnet for Diaspora Jews. There's no presumption that Matthew was born and raised in Palestine just because he happened to be there as an adult when Jesus summoned him. 

A textbook example is St. Paul, a bilingual Diaspora Jew who took up residence in Jerusalem–as did his sister (Acts 23:16). Barnabas is another example of a Diaspora Jew living in Palestine (Acts 4:36). 

iii) Likewise, Matthew's job as a minor gov't employee doesn't tell us much about his background, aside from the fact that he needed to be bilingual to communicate with Greek-speaking Roman officials (his employers) and Aramaic-speaking Jews.

Paul was a tent-maker. That gives you absolutely no indication regarding Paul's social class or education. 

Unless you were an aristocrat, or you were born rich, you had to take what you could get to support yourself. 

iv) There are different levels of proficiency in a language. An ability to understand the spoken word. An ability to speak it. Read it. And/or write it.

Suppose Matthew lacked the educational background to compose Greek. He could still dictate to a scribe.

Paul used scribes even though he had the educational background to do his own writing if he wanted to. The fact, moreover, that both Peter (1 Pet 5:12) and Paul used scribes tells you something about the availability of Christian scribes to assist early church leaders.