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