How Does Resistance Vary with Wire Cross-Sectional Area Changes?

[Sheneron]

Homework Statement

A current-carrying ohmic metal wire has a cross-sectional area that gradually becomes smaller. The current has the same value for each section of the wire. How does the resistance per unit length vary? Increase, decrease or constant?

Here is what I think: Since the wire is a conductor then the potential at one point will be the same as another point, so the voltage will be constant. And since the current is constant then the resistance is constant. Is that wrong?

 

[Sheneron]

Or wait is this it:

Since the current density is proportional to the electric field and current density is defined as I/A then the as A decreases the current density will increase, which will increase the electric field which will increase the voltage.