Carbon budget Concept, its Deviation and the Equation: Climate Economics Perspective

Abstract. The carbon budget concept is based on discoveries made at the end of the 2000s, claiming that global mean temperature (GMT) increase is roughly linearly dependent on cumulative emissions, with a proportionality metric named transient climate response to cumulative carbon emissions (TCRE). Since its emergence in natural science, the carbon budget concept has gained prominence as a tool for policymakers and climate economists alike. However, its usage in economic assessments has been critiqued due to its inability to capture time-delay effects and TCRE changing with climatic conditions.

In this paper, we define the "perfect budget'' equation as a purely linear relationship between GMT and cumulative emissions. Hereby, we distinguish two sources of deviations from the perfect carbon budget: emission scenario and climate state dependence. The former stems from the scenario choice under the fixed cumulative emissions, the latter from the change in TCRE with increased cumulative emissions. Introducing Green's function formalism in the context of the temperature response to an emission pulse (pulse response), we provide reasoning behind time delay and possible scenario-dependent deviations and connect Green's formalism with the carbon budget equation. Using an optimization program, we show that extreme scenario-dependent deviations are relatively small compared to the overall GMT increase and even smaller when there are no more emissions after the year of optimization. We explain the behaviour of scenario-dependent deviations by the shape of the pulse response. Additionally, we show that the pulse response changes with climatic conditions, through which we further explain carbon budget state dependency. Finally, we derive a generalized analytical carbon budget equation, which captures the state-dependent change of TCRE through a weak exponential GMT increase dependency on cumulative emissions. The analysis is done in FAIRv2.0.0, a simple climate model that includes climate feedback modifying the carbon cycle.