It depends on whether you have the intention to upgrade in the future.
When I first took my 2b licence, I was already aiming for class 2 bikes, so I bought my first bike which is a second hand which total about $1600 (RG125) for me to last about a year plus before I take class 2A which I also used about $1600 (GSXR400-Old mother hen) before I switched to a class 2.
All amount stated are OTR prices cos I didn't wanna spend too much since I already plan to upgrade.
I know of people who are very happy with their 2B bikes because it's low maintenance, low petrol consumption and easy to squeeze through traffic. Upgrading never cross their mind.
Oh well, it all depends on you. $2000 is definitely enough to get your hands on a second hand bike with relatively good condition but you'll require some patience to get the best deal. If in doubt, bring an experience friend along to view the bike.