The Lorenz system was derived on the basis of a model of convective atmospheric motions and may serve as a paradigmatic model for considering a complex climate system. In this study we formulated the thermodynamic efficiency of convective atmospheric motions governed by the Lorenz system by treating it as a nonequilibrium thermodynamic system. Based on the fluid conservation equations under the Oberbeck-Boussinesq approximation, the work necessary to maintain atmospheric motion and heat fluxes at the boundaries were calculated. Using these calculations, the thermodynamic efficiency was formulated for stationary and chaotic dynamics. The numerical results show that for both stationary and chaotic dynamics, the efficiency tends to increase as the atmospheric motion is driven out of thermodynamic equilibrium when the Rayleigh number increases. However, it is shown that the efficiency is upper bounded by the maximum efficiency, which is expressed in terms of the parameters characterizing the fluid and the convective system. The analysis of the entropy generation rate was also performed for elucidating the difference between the thermodynamic efficiency of conventional heat engines and the present atmospheric heat engine. It is also found that there exists an abrupt drop in efficiency at the critical Hopf bifurcation point, where the dynamics change from stationary to chaotic. These properties are similar to those found previously in the Malkus-Lorenz waterwheel system.