There is an advantage for birds to fly in a V-Formation as stated by Portugal et al, 2013:
As a bird flaps, a rotating vortex of air rolls off each of its
wingtips. These vortices mean that the air immediately behind the bird
gets constantly pushed downwards (downwash), and the air behind it and
off to the sides gets pushed upwards (upwash). If another bird flies
in either of these upwash zones, it gets free lift. It can save energy
by mooching off the air flow created by its flock-mate.
There is no advantage to the lead bird though, only the birds behind will get these free lifts by being in the upwash zones.
A study on Ibises determined that there was no clear lead bird and birds cycled between positions. Flying in a V-formation is a collective effort, there would be no benefit to always being in front, so the birds take turns. I would assume the same aerodynamics and the cycling of lead bird apply to all birds that generally fly in V-formations such as Geese. Otherwise the lead bird would just exhaust itself and not have enough energy to complete the migration, it would be the same situation as if the bird was flying alone.
By carefully observing the flock, the investigators learned that the
amount of time a bird is leading a formation is strongly correlated
with the time it can itself profit from flying behind another bird.
By working together, a flock of birds is greater than the sum of its
parts. By not being selfish, individual birds can reap the collective
advantage of aerodynamic wash-up, while also allowing for larger
flocks, which result in yet more time for an individual to be in a
Portugal, Hubel, Fritz, Heese, Trobe, Voelk, Hailes, Wilson & Usherwood. 2013. Upwash exploitation and downwash avoidance by flap phasing in ibis formation flight. Nature http://dx.doi.org/10.1038/nature12939