Anticipation echoes in vintage capital models

The paper analyzes the structure of optimal trajectories in the one-sector vintage capital model that optimizes the endogenous lifetime of age-structured capital under technological change. A new repetition pattern (anticipation echoes) is demonstrated in both optimal investment and optimal capital lifetime in the case of finite-horizon optimization. The anticipation echoes are caused by expectation of the future “no-investment” policy at the end of the planning horizon. It is shown that the anticipation echoes represent a general structural property of dynamic systems with endogenous delay. Mathematically, the problem under study is the optimal control of non-linear integral equations with unknowns in the integration limits.