This paper introduces a novel application of the Kaya identity to assess the roles of technological change and consumption behaviour in shaping global greenhouse gas emissions. Drawing on numerical insights from counterfactual emission reduction scenarios, we quantify the adjustments in technology and consumption required to remain within the carbon budget by 2050 and explore their distributional implications. Building on this analysis, we develop a simple analytical model that formalizes the resulting carbon budget trilemma: under binding ecological constraints, rising consumption, technological progress, and widening inequality cannot sustainably coexist. We place these transformations in historical perspective by examining a set of precedents---from wars and epidemics to economic collapses and episodes of rapid technical upgrading---that provide comparative magnitudes for the scale of change implied by a binding carbon budget. Our conclusions unveil a race between technological innovation and consumption sobriety to reach planetary sustainability, in which global inequality acts as a boundary constraint. Given past technological progress, and current levels of global inequality, it is unlikely that sustained reductions in average consumption can be avoided if we are to respect ecological constraints.