The Durban Conference of Parties (COP17) has approved a deal to negotiate and arrange by 2015, a global commitment to reduce greenhouse gases (GHGs) starting from 2020 onwards. COP17 confirmed the Cancún (COP16) agreement concluding that future global warming should be limited to below 2°C post-industrial. This report investigates scenarios of gradually stringent remaining emissions quotas (REQ) resulting to increased probabilities to limit temperature rise below 2°C. REQ are applied as cumulative bounds in the combined TIMES-MACRO model of the USA with the MERGE Integrated Assessment Model both able to analyse technological change. The study summarises the main findings and conclusions of this parametric analysis where all world regions accept a binding protocol or Accord starting in 2020 mitigating global warming. The mathematical description of the combined model that integrates in one set of equations and one objective function two hybrid top-down and bottom-up models with complementary regional representation is described in this paper.