It really depends one what you mean by "bandwidth."
If you mean, "how much data can I send per second," then it's pretty clear that you can send much more information by modulating a 1 GHz carrier than you can by modulating a 1KHz carrier. As an example, say you wanted to transmit audio. The highest frequency you need to modulate is 20KHz. On a 1MHz carrier, you could do this by shifting the center frequency by +- 10KHz, or 1%, but on a 1GHz carrier, you only need to shift the center frequency by .0001% to send the same data. So, higher frequencies are capable of carrying much more data per octave.