A solid but timid response. After 2...e5 the game would be a reverse Philidor's. 2 d3 makes it easier for Black to achieve equality than 2 e5.
Hitting e4 and threatening a quick Queen trade which leaves White without castling rights and down a pawn. 2...d5 continues the Alekhine theme of attacking the White e4-pawn until it is captured or moves.
?! This waiting move is too slow. Better was 3 e5 when 3...Nd7 leaves Black with a more cramped position. White's first move advantage is essentially given away with 3 a3.
Threatening to win a pawn after 4...exd3 5 Bxd3 while if 5 Qxd3 then 5...Qxd3 6 Bxd3 leaves Black standing even better.
A decent move but somewhat slow. White can't avoid trading a pawn for a single developed piece after 4...exd3 5 Bxf6 6 gxf6 Bxd3. 4 Nc3 might have been better, allowing the dark-squared bishop to choose between f4 and g5 until later. 4 Bg5 does have the advantage of tying down Black's e-pawn since ...e6 and ...e5 are both met with dxe4.
I made this move to keep up the trading threats on the d-file in case White played Bxf6, which seemed likely. Winning a pawn with 4...exd3 seemed okay, but would leave me with doubled f-pawns and two tempi behind White due to Bd3 and having the e-pawn still on e7. 4...Bf5 keeps up the pressure on d3, which is the theme in the Scandanavian Variation of Alekhine's Defense.
Forcing doubled f-pawns but also setting a trap. If 5...gxf6 then 6 Qh5 Bg6 (6...e6 7 g4! winning a tempo) 7 Qb5+ Nd7 8 Nf3 exd3 9 Bxd3 and White's threat to b7 and his better development are good compensation for his pawn. 5 g4 would be met by 5...Bg6 6 h4 (6 f4 exf3 7 Nxf3 Nc6 =) exd3 7 Bxd3 (7 h5 dxe2!) Bxd3 when both 8 cxd3 and 8 Qxd3 Qxd3 9 cxd3 leave White with an isolated d-pawn which will quickly come under attack.
Preventing the trap which would be sprung after 5...gxf6, but creating a new problem: a pin on the e-file. If now 6 Qh5 then 6...g6 leaves Black ahead. 5...exf6 also saves a tempo by opening the a3-f8 diagonal to the dark-squared Bishop without loss of time.
Taking immediate advantage of the pin. 6 g4 is much weaker because any follow up on that attack will take two tempi which allows Black precious time needed to get his King to safety. White's threats include immediate capture on e4 as well as adding to the pressure with Nc3 followed by Rd1 or 0-0-0.