We present electronic structure calculations of bulk properties of (U, Pu)O2, in the whole Pu content range for which only very few experimental data are available. We use DFT+U and the vdW-DF functional in order to take into account the strong 5f electron correlations and nonlocal correlations. We investigate structural, elastic, electronic, and energetic properties of (U, Pu)O2 in the approximation of the ideal solid solution as described by the special quasirandom structures (SQS) method. The results on electronic properties highlight the narrowing of the band gap due to the mixing of UO2 and PuO2. Results on the mixing enthalpy are used to describe the phase stability of (U, Pu)O2 solid solutions, using both SQS configurations and a parametric method. In particular, the issue of an ideal solid solution on a limited supercell size is discussed.