The reason why Michelin wants this is to make the sport easier to access for other tire manufacturers. Many of them make 18 inch tires for endurance racing and it would it take a lot less effort to adopt those for F1 than to design 13 inch tires from scratch, which then can only be used in F1. For Michelin the point of racing really is to show what they are worth on the track and being sole supplier really does not help. They can provide the teams with square tires and still win every race. Unfortunately this early demand from Michelin has not been well received and they have been forced to tone it down to be even considered. I do see this happening in the future because for reasons already mentioned above, running 13 inch just makes no sense anymore.
If you should see a man walking down a crowded street talking aloud to himself, don't run in the opposite direction, but run towards him, because he's a poet. You have nothing to fear from the poet - but the truth.
(Ted Joans)