Can union and firm market power counteract each other? What are the output and welfare effects of employer and union labor market power? Using data from French manufacturing firms, we leverage mass layoff shocks to competitors to identify a negative effect of employment concentration on wages. In line with the reduced form evidence and the French institutional setting, we develop and estimate a multi-sector bargaining model that incorporates employer market power. We find that, in the absence of unions, output decreases by 0.48 percent because they partially counteract distortions coming from oligopsony power. The reallocation of employment across space is key to realize the output gains from unions.