The long-term impact of algorithmic decision making is shaped by the dynamics between the deployed decision rule and individuals' response. Focusing on settings where each individual desires a positive classification---including many important applications such as hiring and school admissions, we study a dynamic learning setting where individuals invest in a positive outcome based on their group's expected gain and the decision rule is updated to maximize institutional benefit. By characterizing the equilibria of these dynamics, we show that natural challenges to desirable long-term outcomes arise due to heterogeneity across groups and the lack of realizability. We consider two interventions, decoupling the decision rule by group and subsidizing the cost of investment. We show that decoupling achieves optimal outcomes in the realizable case but has discrepant effects that may depend on the initial conditions otherwise. In contrast, subsidizing the cost of investment is shown to create better equilibria for the disadvantaged group even in the absence of realizability.