  The dynamic polarizabilities of the 4s, 3d and 4p states of Ca$^+$, are
calculated using a relativistic structure model. The wavelengths at which the
Stark shifts between different pairs of transitions are zero are computed.
Experimental determination of the magic wavelengths can be used to estimate the
ratio of the $f_{3d_{J}\to 4p_{J'}}$ and $f_{4s_{1/2} \to 4p_{J'}}$ oscillator
strengths. This could prove valuable in developing better atomic structure
models and in particular lead to improved values of the polarizabilities needed
in the evaluation of the blackbody radiation shift of the Ca$^+$ ion.
