This paper addresses the problem of the firms operating on cross-border or inter-regional platforms that are subject to the enforcement of each local government's carbon emissions regulatory policy, thus causing an imbalance in the sharing of the burden of the greening of the total supply chain. We introduce the concept of equity as the incentive mechanism to coordinate this green supply chain which is a function of the carbon emission permits and the revenue generated by the firms. Due to the complexity and imbalance in the original incentive mechanism to this problem, we provide a new equivalent supply chain network equilibrium model under elastic demand based on user equilibrium theory. We state the user equilibrium conditions and provide the equivalent formulation. We show the trade-offs under various carbon emissions regulatory policies. A product with higher price elasticity and carbon emission intensity not only hampers the firm from gaining a higher revenue, but it also reduces the equity of the system under an invariant emission regulatory policy.