This paper presents an empirical framework for matching with imperfect competition, analyzing the sources of wage inequality in the labor market. The authors develop a model that incorporates strategic interactions in wage setting and two-sided heterogeneity among workers and firms. They provide a tractable characterization of the model equilibrium, demonstrating its existence and uniqueness. This characterization allows for the derivation of comparative statics and the assessment of the relative contributions of worker skill, preference for amenities, and strategic interactions to wage inequality. Using instrumental variables, the authors identify labor demand and supply parameters and estimate them using matched employer-employee data from Denmark. They perform a series of counterfactual analyses to evaluate the main sources of wage inequality in Denmark. The model features strategic interactions in wage setting ("oligopsony") and two-sided heterogeneity, allowing for a richer understanding of wage inequality. The authors consider a static labor market with a large population of individuals divided into K finite categories. Each individual chooses to work at a firm or be non-employed, and each firm chooses wages associated with each worker type. Workers have heterogeneous preferences over firms, and firms have production functions that allow for strategic interactions in wage setting. The authors derive a set of comparative statics, showing that firms' strategic interactions in wage setting amplify the pass-through effect of a firm-specific productivity shock on equilibrium wages. They also consider a firm-specific amenities shock, showing that the equilibrium effect on the firm's wage is ambiguous. The authors introduce a social welfare function and study its properties, establishing a connection between the social welfare function and the generalized concentration index (GCI). They show that under certain conditions, increases in market concentration lower social welfare. The authors consider an empirical model that imposes parametric assumptions and establish identification. They use an instrumental variables strategy to identify labor supply parameters and construct labor supply elasticities and deterministic preferences for amenities. The authors exploit firm optimization to derive an estimating equation for the relative labor demand between worker types. They use this equation to identify firm-level production functions that feature heterogeneous labor inputs, flexible asymmetric substitution elasticities, match-specific labor productivity, and imperfect competition in labor markets. The authors describe their data and report their empirical results, finding that wages are marked down roughly 17 percent below the marginal revenue product of labor. They find significant heterogeneity in the distribution of labor supply elasticities across establishments and workers. The authors perform a series of counterfactual experiments to quantitatively examine the role of labor supply and demand factors in driving wage inequality, labor market concentration, and welfare. They find that all the primary channels in their model drive wage inequality, with some mechanisms always increasing inequality and others always decreasing it. The paper contributes to the literature on imperfect competition in labor markets, identification of production functions, and matching models. It provides a formal identification strategy for the structural labor supply elasticity that remains valid in the presence of strategic interactions. The authors also contribute to the literature on identification of production functions and the search-and-matching literature.This paper presents an empirical framework for matching with imperfect competition, analyzing the sources of wage inequality in the labor market. The authors develop a model that incorporates strategic interactions in wage setting and two-sided heterogeneity among workers and firms. They provide a tractable characterization of the model equilibrium, demonstrating its existence and uniqueness. This characterization allows for the derivation of comparative statics and the assessment of the relative contributions of worker skill, preference for amenities, and strategic interactions to wage inequality. Using instrumental variables, the authors identify labor demand and supply parameters and estimate them using matched employer-employee data from Denmark. They perform a series of counterfactual analyses to evaluate the main sources of wage inequality in Denmark. The model features strategic interactions in wage setting ("oligopsony") and two-sided heterogeneity, allowing for a richer understanding of wage inequality. The authors consider a static labor market with a large population of individuals divided into K finite categories. Each individual chooses to work at a firm or be non-employed, and each firm chooses wages associated with each worker type. Workers have heterogeneous preferences over firms, and firms have production functions that allow for strategic interactions in wage setting. The authors derive a set of comparative statics, showing that firms' strategic interactions in wage setting amplify the pass-through effect of a firm-specific productivity shock on equilibrium wages. They also consider a firm-specific amenities shock, showing that the equilibrium effect on the firm's wage is ambiguous. The authors introduce a social welfare function and study its properties, establishing a connection between the social welfare function and the generalized concentration index (GCI). They show that under certain conditions, increases in market concentration lower social welfare. The authors consider an empirical model that imposes parametric assumptions and establish identification. They use an instrumental variables strategy to identify labor supply parameters and construct labor supply elasticities and deterministic preferences for amenities. The authors exploit firm optimization to derive an estimating equation for the relative labor demand between worker types. They use this equation to identify firm-level production functions that feature heterogeneous labor inputs, flexible asymmetric substitution elasticities, match-specific labor productivity, and imperfect competition in labor markets. The authors describe their data and report their empirical results, finding that wages are marked down roughly 17 percent below the marginal revenue product of labor. They find significant heterogeneity in the distribution of labor supply elasticities across establishments and workers. The authors perform a series of counterfactual experiments to quantitatively examine the role of labor supply and demand factors in driving wage inequality, labor market concentration, and welfare. They find that all the primary channels in their model drive wage inequality, with some mechanisms always increasing inequality and others always decreasing it. The paper contributes to the literature on imperfect competition in labor markets, identification of production functions, and matching models. It provides a formal identification strategy for the structural labor supply elasticity that remains valid in the presence of strategic interactions. The authors also contribute to the literature on identification of production functions and the search-and-matching literature.