The problem is what would be the effect on the width of the depletion region of an unbiased PN junction on the doping levels of the P and N sides
This is a conceptual question. There is no equation involved.
The Attempt at a Solution
Increase in doping level implies more free carriers (electrons in the n side and holes in the p side) available for diffusion. Thus the greater number of such available free carriers (as compared to the number of free carriers available for a lesser doping level) would neutralise the charges on a greater number of ions (positive and negative) in the depletion region. Hence the net effect would be to reduce the width of the depletion region.
I would request for a ratification of the correctness of this understanding. Thanks.