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Wages for the vast majority of workers have stagnated since the 1980s while, productivity has grown. We investigate two coexisting explanations based on rising market power: (1) monopsony, where dominant firms exploit the limited mobility of their own workers to pay lower wages; and (2) monopoly, where dominant firms charge too high prices for what they sell, which lowers production and the demand for labor, and hence equilibrium wages economy-wide. Using establishment data from the US Census Bureau between 1997 and 2016, we find evidence of both monopoly and monopsony, where the former is rising over this period and the latter is stable. Both contribute to the decoupling of productivity and wage growth, with monopoly being the primary determinant: In 2016, monopoly accounts for 75% of wage stagnation, monopsony for 25%.