  Wing flexibility governs the flying performance of flapping wing flyers.
Here, we use a self-propelled flapping-wing model mounted on a ``merry go
round'' to investigate the effect of wing compliance on the propulsive
efficiency of the system. Our measurements show that the elastic nature of the
wings can lead not only to a substantial reduction in the consumed power, but
also to an increment of the propulsive force. A scaling analysis using a
flexible plate model for the wings points out that, for flapping flyers in air,
the time-dependent shape of the elastic bending wing is governed by the wing
inertia. Based on this prediction, we define the ratio of the inertial forces
deforming the wing to the elastic restoring force that limits the deformation
as the \emph{elasto-inertial number} $\mathcal{N}_{ei}$. Our measurements with
the self-propelled model confirm that it is the appropriate structural
parameter to describe flapping flyers with flexible-wings.
