Retarded-advanced Differential Equation in Optimal Economic Growth Models

We analyse the dynamics of simple class of neoclassical growth models with time-to-build. Time-to-build comes from the difference between the investment decisions and delivery of finished capital goods, as it was proposed by Tinbergen and Kalecki. This kind of delay in production of capital goods influence the optimal path of consumption of infinitely living economy. The optimal saving and consumption of households is chosen by the social planner in the way of solving the optimization problem with delay. Due to Kolmanovskii and Myshkis (1999) the classical Pontryagin maximum principle of dynamical optimization can be extended on the class of systems with time delay. The Hamiltonian for such systems can be simple constructed and the optimality condition can be derived. As a result we obtain a forward-looking Euler type equation. We compare the dynamics of economic systems with delay with the dynamics of their counterparts without the delay to show that both admit saddle type solutions. The paper points out the importance of the retarded-advanced dynamical systems in the economic theoretical investigations.