In this paper, we apply two optimization methods to solve an optimal control problem of a linear neutral differential equation (NDE) arising in economics. The first one is a variational method, the second follows a dynamic programming approach. Due to the infinite dimensionality of the NDE, the second method requires the reformulation of the latter as an ordinary differential equation in an appropriate abstract space. It is shown that the resulting HJB equation admits a closed-form solution, allowing for a much finer characterization of the optimal dynamics compared to the alternative variational method. The latter is clearly limited by the nontrivial nature of asymptotic analysis of NDEs.