I am not sure I understand the problem. Marginal costs are the Lagrange multiplier associated with the production function in the firms’ maximization problem. They naturally show up there, regardless of whether it’s an RBC model or a NK model.
Right. I just want to make sure that the marginal cost in RBC is a constant.
Your FOC should simply imply that
mc_real=(epsilon-1)/epsilon;
Definitely. Thank you for confirming it.