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This paper proposes a non-monetary traffic demand management scheme, named CARMA, as a fair solution to the morning commute congestion. We consider heterogeneous commuters traveling through a single bottleneck that differ in both the desired arrival time and Value of Time (VOT). We consider a generalized notion of VOT by allowing it to vary dynamically on each day (e.g., according to trip purpose and urgency), rather than being a static characteristic of each individual. In our CARMA scheme, the bottleneck is divided into a fast lane that is kept in free flow and a slow lane that is subject to congestion. We introduce a non-tradable mobility credit, named karma, that is used by commuters to bid for access to the fast lane. Commuters who get outbid or do not participate in the CARMA scheme instead use the slow lane. At the end of each day, karma collected from the bidders is redistributed, and the process repeats day by day. We model the collective commuter behaviors under CARMA as a Dynamic Population Game (DPG), in which a Stationary Nash Equilibrium (SNE) is guaranteed to exist. Unlike existing monetary schemes, CARMA is demonstrated, both analytically and numerically, to achieve a) an equitable traffic assignment with respect to heterogeneous income classes and b) a strong Pareto improvement in the long-term average travel disutility with respect to no policy intervention. With extensive numerical analysis, we show that CARMA is able to retain the same congestion reduction as an optimal monetary tolling scheme under uniform karma redistribution and even outperform tolling under a well-designed redistribution scheme. We also highlight the privacy-preserving feature of CARMA, i.e., its ability to tailor to the private preferences of commuters without centrally collecting the information.