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Bivys are originally tents without a structure, designed to be used in places where you can't pitch one. They are every bit like a tent. Just roll yourself into a tent and you'll see it works just as well, minus room and plus claustrophobia.
Now modern bivy bags kinda have drifted away from that.
Some became more like a tent, and began to offer similar features like a mosquito net or a structure so the cloth doesn't stick to your face. They're comfy but incredibly expensive, and about as large to pack as a one person tent.
Others became smaller to pack, lighter and thus simpler, with less features but more durable, and eventually became just sacks with a zipper or cord. They're overall cheaper, but still not really "cheap".
The fabric is key. It needs to be waterproof, obviously, but also breathable, because it's going to be single layered and condensation will happen; an opening at the feet can be good for that, though an opening means water getting in if it rains.
It also needs to be durable, as durable as a ground cloth.
If you want something cheap but great, get a Terra Nova MoonLight. About 100€, but worth its price. It's properly waterproof- and know that very few bivys can actually withstand rain- and breathable. It's also ~220g and very small to pack.
But considering the prices, best thing would be to buy the fabric and stitch one yourself. It's simple enough for that: waterproof yet breathable top, tough non-breathable bottom, stitch three sides and apply nylon tape to stiches for waterproofing, and leave the top opening with a cord to close it... it's incredibly simple to do for anyone who knows how to use a sewing machine. I wish I knew.
If your concern is the weight and size, and not the fact that you have to pitch it, better use a tarp: you won't get weight, and it can cover you enough to leave it open on both ends for aeration.
