Multidimensional Economic-Growth Models with an Integral Utility Function

An $$n$$ -dimensional economic model is considered that has a Cobb–Douglas production function on the infinite planning horizon such that the utility function is an integral-type functional with a discount and a logarithm-type integrant. It is assumed that all of the model’s amortization factors are equal to one another. The constructed optimum control contains $$n-1$$ special segments that are described analytically. A special sweep procedure for consecutively solving two Cauchy problems on each segment is developed to find moments of switching between segments and the shape of the optimum trajectory. On the last segment, the optimum trajectory lies along a special ray; from the viewpoint of economy, this ray can be interpreted as the mode of equilibrium growth. A Pontrjagin maximum-principle problem with a special transversality condition is used to construct the optimum solution. Optimality is confirmed using the Kiselev theorem on sufficient conditions. Moving to problems of large dimensions considerably increases the number of technical difficulties. The description of the optimality verification procedure is therefore presented in detail for methodological reasons.