This paper presents a framework for developing efficient and interpretable carbon-cycle emulators (CCEs) as part of climate emulators in Integrated Assessment Models, enabling economists to custom-build CCEs accurately calibrated to advanced climate science. We propose a generalized multi-reservoir linear box-model CCE that preserves key physical quantities and can be use-case tailored for specific use cases. Three CCEs are presented for illustration: the 3SR model (replicating DICE-2016), the 4PR model (including the land biosphere), and the 4PR-X model (accounting for dynamic land-use changes like deforestation that impact the reservoir's storage capacity). Evaluation of these models within the DICE framework shows that land-use changes in the 4PR-X model significantly impact atmospheric carbon and temperatures -- emphasizing the importance of using tailored climate emulators. By providing a transparent and flexible tool for policy analysis, our framework allows economists to assess the economic impacts of climate policies more accurately.