@Poster
I upgraded from 145/80/12 to 165/70/12 on my khyber
and the reason for this upgrade was to achieve a better grip as my dad had an accident and the main reason were the tyres as 145 is quite a thin tyre for khyber
now here is the difference
the total diameter of the wheel (the rim+ the rubber part) is the same as it was before (a negligible difference of 0.5mm has occured) and it resulted in better braking
Im not sure but I think that the same must be the case with 165/80/R13 and 185/70/R13
I would suggest you to go for 185/70/13 as it will not affect your fuel economy that much and will obviously result in better braking
Btw here is how you differenciate it
I hope you get the idea
145/80/R12
165/70/R12
Keeping in mind the above example,
If you upgrade from 165/80/13 to 185/70/13 then the dia i-e the red arrow will remain the same
and as Rim size i-e 13" will also be retained so the blue length will also remain same
the thing that will change will be the width i-e highlighted with the green arrow
I hope now the difference will be clear between the two sizes