Does he have any ‘right to exert his authority, inside the limits that have been delineated for him’?

Somebody who has real authority enjoys a certain degree of autonomy, if not outright independence. ‘Authority’ is almost never clearly delineated, there is always a gray area where the discretion of the individual in charge is the one that calls the shots.
More over if we, the ‘subjects’, consider that he has ‘the right’ to exercise that authority then it’s us who are in deep trouble.
‘Exertion of authority’ ‘smacks’ of the situation  when the ‘authority man’ had conquered his position against the wish of his subjects – like the emperors of the old. (Or like the communist dictators of not so long ago, only they pretended to exercise their authority for the benefit of the people while the emperors of the old were more straightforward and declared themselves ‘gods’)
Nowadays, at least in the democratic states, authority is, theoretically, used as a tool, towards the accomplishment of what the person in charge is supposed to achieve, not as a right enjoyed by that person.
In fact the notion of a right to exert authority inside some limits is akin to what has been described as ‘feudalism’, a social arrangement not that different from the Athenian democracy. The people were divided in two categories, just as in the previous situation – the ‘imperiums’ of the Antiquity, the difference being that in an imperium the top class was inhabited by a single individual – the emperor/dictator, while in feudalism/Athenian democracy the top class was inhabited by the free people, whose authority/freedom extended only as far as it started to encroach the authority of the equivalent individuals. I have to remark here that in many circumstances feudalism has very quickly degenerated back to imperium – for instance in absolutist France, ‘L’etat c’est moi’, or in tsarist Russia, while England successfully avoided that due to the spirit enshrined in Magna Charta.
The difference between feudalism/Athenian democracy and the modern democracy being that currently we can no longer speak of individual authority simply because nowadays no one has the “right” to own slaves – as the Athenian ‘democrats’ had, nor even enjoy extensive authority (bar the right of life and death) over other people – the serfs, as the feudal barons did not so long ago.