It is actually an old ladies myth that the space elevator cannot be built with current technology. It would just be extraordinarily more expensive with current materials.
The elevator needs to be built from geostationary orbit downward. This means that all the material needs to be pulled up by conventional propulsion, this is the principal factor affecting cost. The conventional approach to build an elevator without materials satisfying the strong tensile requirements that are required for the single-tether design would have to be an exponential structure; a single tether connected to earth, supported by two tether, supported themselves by 4 tethers,.. until reaching above geo orbit, where we'll have to attach $2^n$ tethers to the counterbalancecounterweight, where $n$ is given by the tensile properties of the chosen material.
If you have already one space elevator, then it is easier/cheaper to build a second one, because the expensive step of moving the mass up has already been bootstrapped. In the case of mars, it would make more economical sense to actually send the building materials from earth than to uplift them from mars surface, assuming of course we already have a working elevator on earth