The optimized effective potential (OEP) method allows for calculation of the local, effective single particle potential of density functional theory for explicitly orbital-dependent approximations to the exchange-correlation energy functional. In the present work the OEP method is used together with the approximation due to Krieger, Li, and Iafrate (KLI). We present the first application of this method to polymers. KLI calculations have been performed for the insulating polyethylene and the results have been compared to those from other orbital-dependent potentials. Various properties of the band structure are also calculated. The single-particle band gap strongly depends on the basis set with larger basis sets yielding narrow gaps. For certain physical quantities such as the total energy and the exchange energy, the various orbital-dependent Kohn-Sham exchange-only and Hartree-Fock results differ only slightly. For the highest occupied orbital energy the difference is more significant than expected. In order to get the right band gap in OEP the exchange contribution to the derivative discontinuity is calculated and added to the Kohn-Sham gap. The corrected gap obtained by the KLI approach is 12.8 eV compared with the Hartree-Fock and experimental values of 16.6 and 8.8 eV, respectively. We observe, however, the strong dependence of the derivative discontinuity on the basis set.