The phase behavior of a mixture of bilayer forming lipids and polymer-lipids (lipids with covalently attached polymer to their hydrophilic moieties) in excess water is studied theoretically. The mixture is predicted to exhibit complex phase behavior for polymer molecular weights 2000 and 5000, depending on the concentration (fraction) of polymer-lipids in the lipid mixture. The bilayer is characterized by a maximal concentration n(sat) (saturation Limit) of polymer-lipids that it can incorporate, as determined by its material properties (elastic modulus of area expansion and critical area expansion). At a different concentration ntr, which we call the thermodynamics crossover, micelle formation becomes energetically favorable over bilayer formation. We show that for DSPC and SOPC bilayers n(tr) < n(sat). Increase of the polymer-lipid concentration above n(tr) leads to a gradual transition from a bilayer to a micellar phase; bilayers and micelles can coexist. In the transition region the polymer-lipid concentration is higher in the micellar phase than in the coexisting bilayer.