We study monetary policy in an environment where price and wage Phillips curves exhibit true curvature. To this end, we propose a New Keynesian model with endogenous adjustment of price and wage setting frequencies, moving beyond the quasi-linear structure of standard nonlinear NK Phillips curves. Using euro area data from 1999Q1 to 2024Q4, we estimate and simulate the non-linear model, analyzing the recent inflation surge and the implications of state-dependent prices and wages for monetary policy. Unlike conventional models, our framework does not attribute inflation dynamics primarily to exogenous supply shocks. Instead, the impact of shocks is asymmetric and depends on their timing, size, and the business cycle. Consequently, the inflation–output stabilization trade-off is state-dependent: monetary policy is more effective in curbing inflation, and supply shocks have larger effects during periods of high inflation.