There would be a net gravitational blue shift.
Work Step by Step
The energy of a photon is given by E = hf, where h is Planck's constant. If the energy of light is lowered, as in traveling upward in a gravitational field, its frequency is lowered (the light is said to be red-shifted). On the other hand, if the light's energy is increased, as in falling downward in a gravitational field, the frequency is raised (the light is blue-shifted). Light leaving the moon is red shifted (rising in the moon’s gravitational field), but then is blue-shifted even more (falling through Earth’s stronger field). There is a net gravitational blue shift.