Lack of divine coincidence in New Keynesian models

The literature has long agreed that the divine coincidence holds in standard New Keynesian models: the monetary authority is able to simultaneously stabilize inflation and output gap in response to preference and technology shocks. I show that the divine coincidence holds only when inflation is stabilized at exactly zero. Even small deviations from zero generate policy trade-offs. I demonstrate this result using the model׳s non-linear equilibrium conditions to avoid biases from log-linearization. When the model is log-linearized, a non-zero steady state level of inflation gives rise to what I call the endogenous trend inflation cost-push shock in the New -Keynesian Phillips curve.