In this paper, we analyze and optimize the energy efficiency of downlink cellular networks. With the aid of tools from stochastic geometry, we introduce a new closed-form analytical expression of the potential spectral efficiency (bit/sec/m$^2$). In the interference-limited regime for data transmission, unlike currently available mathematical frameworks, the proposed analytical formulation depends on the transmit power and deployment density of the base stations. This is obtained by generalizing the definition of coverage probability and by accounting for the sensitivity of the receiver not only during the decoding of information data, but during the cell association phase as well. Based on the new formulation of the potential spectral efficiency, the energy efficiency (bit/Joule) is given in a tractable closed-form formula. An optimization problem is formulated and is comprehensively studied. It is mathematically proved, in particular, that the energy efficiency is a unimodal and strictly pseudo-concave function in the transmit power, given the density of the base stations, and in the density of the base stations, given the transmit power. Under these assumptions, therefore, a unique transmit power and density of the base stations exist, which maximize the energy efficiency. Numerical results are illustrated in order to confirm the obtained findings and to prove the usefulness of the proposed framework for optimizing the network planning and deployment of cellular networks from the energy efficiency standpoint.