Endogenous directed technical change for energy transition

The finite nature of certain fossil fuels, which are essential inputs in production, imposes a substantial constraint on the economy's long-term growth potential. This paper develops an endogenous directed technical change model under uncertainty regarding the timing of a potential backstop technology discovery. The model directs R&D towards increasing the probability of a technological breakthrough, triggering thereby a transition from a fossil fuel-dependent economy to one powered by cleaner, renewable energy sources. We derive analytical solutions using a general optimal control framework allowing to establish the Pontryagin (maximum principle) first-order optimality conditions, complemented with an adequate numerical assessment to a model of the optimal allocation of national economic resources between consumption, investment, and R&D. We identify the optimal time paths for consumption in both the pre- and post-backstop adoption economies, the depletion rate of fossil fuels, investment in R&D, and the utilization rate of the substitute, under two scenarios: when energy input and physical capital are substitutes in production, and when they are complements.