"higher responsibility" is correct and can be used in written English.
For example, "He has taken on a higher responsibility in the company since his promotion."
"It's a higher responsibility".
I have a higher responsibility than that," he replied.
"We have no higher responsibility than stopping terrorist attacks," he added.
Mr. Boyd said they thought the theft was justified by a higher responsibility to report wrongdoing.
It's a form of self-indulgence and abdication of a higher responsibility to family".
To the legal responsibily of the profession was to be added the higher responsibility of man.
Or does leadership carry with it higher responsibilities?
Or maybe it would be wiser to extend his contract and appoint him to higher responsibilities.
It's precisely in the service to these supposed higher responsibilities that they often let more basic ones slip away.
After Conroy's departure, Shorten suddenly reappeared at right faction meetings, which he hadn't been attending, given his higher responsibilities.
The mobilisation of white British South Africa Police officers for military service led to black male and white female constables taking on higher responsibilities.
