We develop a New Keynesian model with staggered price and wage setting where downward nominal wage rigidity (DNWR) arises endogenously through the wage bargaining institutions. It is shown that the optimal (discretionary) monetary policy response to changing economic conditions then becomes asymmetric. Interestingly, we find that the welfare loss is actually slightly smaller in an economy with DNWR. This is due to that DNWR is not an additional constraint on the monetary policy problem. Instead, it is a constraint that changes the choice set and opens up for potential welfare gains due to lower wage variability. Another finding is that the Taylor rule provides a fairly good approximation of optimal policy under DNWR. In contrast, this result does not hold in the unconstrained case. In fact, under the Taylor rule, agents would clearly prefer an economy with DNWR before an unconstrained economy ex ante.