I don't know what you consider not very expensive, but AFAIK the absolute cheapest you can get a 37" 1080P TV is to buy a Vizio refurb off of ebay. Search for the VO37LF model and make sure to buy from an authorized Vizio reseller. Make sure that it's refurbished by Vizio and comes with Vizio's 90-day warranty. Vizio refurbs are supposed to be in like-new condition, and they have a pixel-perfect warranty, so they'll fix/replace the TV for even one dead/stuck/bright pixel.
I recommend 1080P, because 1366x768 is way to low of a resolution for a computer monitor, and some TVs won't even accept that resolution and will only display 1360x768. If you choose buy it now and use both Bing and ebates cashback, then you can get 11% cashback from ebay (I think). I've seen the VO37LF go for as low as $485 shipped after cashback.