Choose an appropriate bitrate when going up in encoding resolution

[tufffta]

If I encode one video to two versions:
*1280x720, 2000kbps
*1920x1080, then:
1) if I choose 2000 kbps for 1920x1080 version, will it look worse than 1280x720 on the same screen (which native resolution is 1366x768), becase every pixel gets less bits, so is of worse quality then? Or could it actually be better?

2) Should I increase the bitrate proportionally to the increase in resolution to keep the same quality (so that every pixel got the same amount of bits, hence the quality was the same)? In this case: 1920/1280=1,5; therefore 2000kbps * 1,5 = 3000kbps for 1920*1080? Or is it more complicated and the quality, when increasing resolution and bitrate, change is determined by some other coefficient?

In essense I am asking whether bits per pixel (measured by resolution x*y/(bitrate/8) if I get the formula right) is the factor that I should look at when I'm deciding what should be an increase in bitrate when increasing the resolution?

[Warperus]

1) If you original video is 1920*1080, your higher resolution encoding will use some bits for unneseccary details. As bits count is the same you'll end up with overall worse quality. It's not going to be ultimately worse in your situation as presentation device will make some blur, but still it's a tendency.

2) If you want similar bits per pixel, you need to include both dimensions into equation: 1920*1080/1280/720=2,25 so 2000kbps*2,25=4500kbps. I'd say you need a bit less, probably you'll be ok with 4000kbps.

[nibus]

Try out CRF encoding. It takes a lot of this guesswork out of the equation, and automatically scales bitrate for the video source and resolution. Just experiment with CRF values and find one that looks good to you. Try anywhere from CRF 18 to 24 and see what you think.

I've found that one bitrate for a smaller resolution usually gives more texture; while increasing the resolution at the same bitrate may add some sharpness at the expense of this texture.

[tufffta]

Thank you for the explanation and suggestions, guys!