Structure of Optimal Trajectories in a Nonlinear Dynamic Model with Endogenous Delay

One of the modern applications of integral equations is the replacement of capital equipment under technological change. Corresponding models are known as vintage capital models (VCMs), see [1, 2, 3, 4, 5, 13, 14, 17]. They focus on optimization of the equipment lifetime and can be expressed via special Volterra integral equations with delay (e.g., Corduneanu [5]). Existing results about endogenous equipment lifetime in VCMs include mainly the case of constant lifetime. This paper is devoted to the construction of exact solutions to an optimization problem (OP) with endogenous equipment lifetime in the well-known Solow VCM [16]. The authors investigated the integral models with endogenous delay for various applied problems of economics, ecology, and engineering (see [6, 7, 8, 9, 10, 12, 19, 20] and the references therein). They provided an asymptotic analysis of the OP under study and discovered turnpike properties of the optimal equipment lifetime in [21] (see also [7, 8]). More complicated models with many inputs and outputs were investigated in [7, 10, 19]. The paper is organized as follows. The OP for the Solow model with endogenous capital lifetime is formulated in Section 2. Section 3 exposes preliminary results such as the condition for an extremum, gradient of the OP, and arising auxiliary nonlinear integral equation. In Section 4, the exact structure of optimal trajectories is established.