In this paper, the performance of a visible light communication (VLC) network exploiting power control is considered. The achievable rate region of two VLC communication pairs is characterized. The system employs optical orthogonal frequency division multIPlexing (O-OFDM) with power control performed at the transmitters. The optical signal clIPping effect is taken into consideration which results from the physical limitations at the transmitters. Also, the system is constrained by the desired illumination power for each transmitter. We prove that the achievable rate region is defined as the union of two regions where the boundary of each of these regions can be obtained by a single variable optimization problem. Our numerical results show the effects of the system parameters including direct current (DC) bias, the desired illumination power, clIPping noise and the relative positions of the receivers relative to the LED transmitters on the achievable rate region of the two users. We show the enhancement in the performance resuLTEd from using power control compared to orthogonal resource allocation techniques.