The behavior of the Fermi energy and the electron thermoelectric power in an asymmetric quantum well of the Morse potential type as a function of electron concentration are studied. It is shown that the Fermi energy is a piecewise-linear function, and the electron thermoelectric power is an oscillating function. The analysis shows that the nature of these oscillations differs significantly from that in films of a parabolic and rectangular profile, namely, the oscillation period is a nonmonotonic function and reaches a maximum at some value of the electron concentration. The possibility of estimating the parameters of the film profile (quantum well) is proposed. A comparison with the experiment shows a qualitative agreement.