Large-scale extraction of carbon dioxide (CO2) from Earth's atmosphere ("negative emissions") is important for stringent climate change mitigation scenarios, and we examine optimal (i.e. least-cost) pathways of negative emissions in the presence of learning by doing ("endogenous learning"). Optimal pathways solve a variational problem involving minimization of discounted costs subject to a constraint on total negative emissions across time. A minimum pathway exists if the marginal cost curve of negative emissions is increasing with annual rate of emissions reduction. In the absence of endogenous learning, the optimal pathway has annual negative emissions increasing with time: with more rapid increase in emissions rate occurring in case of large discount rate and slower increase of the cost curve. Endogenous learning can have contrary effects depending on how it is included in models. This paper identifies a basic distinction, between additive and multiplicative effects on marginal costs of endogenous learning, which governs its qualitative effects in such models. If endogenous learning is best modeled as a negative addition to the cost function, shifting the cost curve downward, the optimal pathway has higher emissions rate early on when compared to the no-learning case, however with emissions increasing with time. In contrast if endogenous learning is a multiplicative effect, scaling down marginal cost uniformly, then benefits of learning are slowly manifest as marginal cost rises and the optimal pathway begins at lower emissions rates that increase more rapidly as compared to if endogenous learning were absent.
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