Fluctuations and Limitations of a Multi-Sector Economic Model with Delays

A general multi-sector model of the economy is investigated. A sector's input to production, labor, evolves according to a jump Markov process. Labor jumps between sectors to balance supply and demand, where each sector differs by its productivity. The jump model captures the intrinsic noise of the micro agents on the macro level, which is represented by the random timing of labor jumps. Quantifying this noise is a central theme of this paper. An operator theoretic approach is utilized to capture the fluctuations of a linearized jump system exactly. As an illustrative example two sector and three sector economies are studied. In each case the optimal aggressiveness, gain, of a sector is determined for minimal variance. Delays are then introduced into the model. It is shown that the presence of a delay creates a limit on the minimum variance achievable and that high gain is destabilizing. For both the two and three sector models the nonlinear jump systems are simulated. It is shown that the operator theoretic approach is an appropriate method for quantifying the second moments.