We propose a non-linear state-space model to examine the relationship between CO$_2$ emissions, energy sources, and macroeconomic activity, using data from 1971 to 2019. CO$_2$ emissions are modeled as a weighted sum of fossil fuel use, with emission conversion factors that evolve over time to reflect technological changes. GDP is expressed as the outcome of linearly increasing energy efficiency and total energy consumption. The model is estimated using CO$_2$ data from the Global Carbon Budget, GDP statistics from the World Bank, and energy data from the International Energy Agency (IEA). Projections for CO$_2$ emissions and GDP from 2020 to 2100 from the model are based on energy scenarios from the Shared Socioeconomic Pathways (SSP) and the IEA's Net Zero roadmap. Emissions projections from the model are consistent with these scenarios but predict lower GDP growth. An alternative model version, assuming exponential energy efficiency improvement, produces GDP growth rates more in line with the benchmark projections. Our results imply that if internationally agreed net-zero objectives are to be fulfilled and economic growth is to follow SSP or IEA scenarios, then drastic changes in energy efficiency, not consistent with historical trends, are needed.

翻译：我们提出了一种非线性状态空间模型，利用1971年至2019年的数据来考察CO$_2$排放、能源来源与宏观经济活动之间的关系。CO$_2$排放被建模为化石燃料使用的加权和，其排放转换系数随时间演变以反映技术变化。GDP则表示为线性增长的能源效率与总能源消耗的结果。该模型使用全球碳预算项目的CO$_2$数据、世界银行的GDP统计以及国际能源署（IEA）的能源数据进行估计。基于共享社会经济路径（SSP）和IEA净零路线图的能源情景，模型给出了2020年至2100年CO$_2$排放和GDP的预测。模型的排放预测与这些情景一致，但预测的GDP增长较低。另一种模型版本假设能源效率呈指数级提升，所产生的GDP增长率更接近基准预测。我们的研究结果表明，如果国际社会商定的净零目标要实现，且经济增长需遵循SSP或IEA情景，那么能源效率需要发生与历史趋势不一致的剧烈变革。