What happens to the resistance as the conductor is made thinner?

The resistance of the wire will increase if we made the conductor thinner because R (resistance) = p (rho). l (length) / A (area of cross-section or thickness of wire).

 

Area is directly proportional to the radius of the wire.

So as you can see if area of cross-section ( or thickness) of wire increases, resistance decreases and more current flows through it.