Optimal pattern of technology adoptions under embodiment: A multi‐stage optimal control approach

Abstract

SUMMARYBy deriving the necessary conditions for a multi-stage optimal control problem where the endogenous switching instants appear as an argument of the state equation, we analyze the optimal pattern of technology adoptions under embodiment with a finite planning horizon. We show that the optimal pattern of technology adoptions depends crucially on how the growth rate advantage compares to the adjustment and the obsolescence costs inherent to embodiment. We obtain non-stationary lifetimes for the adopted technologies due to finite planning horizon. We analyze numerically the effects of planning horizon, speed of adjustment to the new technology, growth rate of technology and the impatience rate on the optimal pattern.

Technological vs. Ecological Switch and the Environmental Kuznets Curve

“…In particular, one involves a threshold level for the state variable and the other no. Problems with multiple timing have already been tackled in the literature (see Saglam 2011) but in the latter literature only technological switching is considered.…”

Optimal timing of regime switching in optimal growth models: A Sobolev space approach

“…Some early contributions to the optimal regime switching problems have proposed multi-stage optimal control techniques that recall the Pontryagin maximum principle from a dynamic programming perspective (see Tomiyama, 1985;Tomiyama and Rossana, 1989;Makris, 2001;Saglam, 2010). The main idea is to reduce a two-stage problem into a standard one with a dynamic programming approach, first by solving the post-switch problem and then attaching its value function to the pre-switch one with the Pontryagin maximum principle concluding at the intermediate steps.…”

Capital Accumulation and Embodied Technological Progress

“…From [2,4] we know that the co-state variable is continuous at the technology adoption time T . This gives the transversality condition…”

“…Following [4,5], we introduce K(t) to be the capital stock measured in productivity units being valid at time 0. Before any switch in technology has occurred, an investment of one unit leads to an increase of K(t) with one unit.…”

“…Alternatively, in [4] it is assumed that q(t) = e γ t , where γ is a positive constant, while in [5] there are just two technologies with different values of q available from the start.…”