This paper documents the short run and long run behavior of the search and matching model with staggered Nash wage bargaining. It turns out that there is a strong tradeoff inherent in assuming that previously bargained sticky wages apply to new hires. If sticky wages apply to new hires, then the staggered Nash bargaining model can generate realistic volatility in labor input, but it predicts a strong counterfactually negative long run relationship between inflation and unemployment. This finding is robust to including a microeconomically realistic degree of indexation of wages to inflation. The lack of a negative long run relationship between trend inflation and unemployment provides indirect evidence against the proposed mechanism that high inflation systematically makes new hiring more profitable by depressing the real wages of new hires.