In standard models of labor market monopsony, the profits derived from firm monopsony power depends on the firm's labor supply elasticity. There are two puzzles facing these standard models. First, different standard approaches to estimating labor supply elasticities produce dramatically different estimates and hence measures of profits from monopsony power. Second, commonly used low labor supply elasticities imply profit shares of aggregate income that are too high after accounting for price markups and capital income. This paper argues that both of these issues arise from the same limitation - that firms can increase employment only by raising wages. To address this, we develop a tractable model where firms use both higher wages and costly recruiting expenditures to attract workers. Firms have wage setting power due both to search frictions and workers' heterogeneous preferences over workplaces. We show that whether firms profit from their wage setting power depends on the shape of firms' recruiting cost function, and the rents acquired by firms from wage setting power can be dissipated by recruiting costs. In a calibrated quantitative model that also accounts for the strategic behavior of a large firm, profits from wage setting power account for 6% of labor market-wide marginal product and 5% of output. Our findings suggest that wage setting power alone does not imply profits for firms that exploit this power.