This study considers time-inconsistent output subsidy/tax policies in free-entry mixed markets under different competition modes. In a committed regime where the subsidy is determined before entry, the optimal rate is zero in either Cournot or Stackelberg game with private leadership, while it is negative in Stackelberg game with public leadership. In a non-committed regime where the subsidy is determined after entry, the optimal rate is always positive. We also show that private leadership is the best for social welfare regardless of the timing of subsidy policies, but public leadership might be equilibrium if the timing of entry decisions is endogenous.