The Use of Extremal Vector Field Analysis to Study Debt Dynamics

Willi Semmler - New School for Social Research and Malte Sieveking - University of Frankfurt

This paper introduces a new method to study the dynamics of the state equations in intertemporal models. More specifically, for a dynamic growth model with n capital stocks and one equation for the evolution of external debt we study the dynamics of the state equations by applying vector field analysis. We show that the usual transversality conditions are not sufficient to provide the existence of solutions. Extremal vector field analysis shows the existence of the solutions and tracks the dynamic paths for (1) a discrete time version, (2) continuous time version discretized through the Euler procedure, (3) an example from growth theory with borrowing of economic agents. We demonstrate that intertemporal models with borrowing and lending are typically state constrained. Extremal vector field analysis shows analytically and numerically the region in which the borrowers remain creditworthy. Moreover, we show the relation of extremal vector field analysis to the Hamilton-Jacobi-Bellman (HJB)optimality equations. As it turns out extremal vector field analysis is another way to compute the HJB equations. Discretization error bounds are also provided. By using this new method a number of problems in growth theory such as the dynamics of the state variables for the multivariate case, the numerical computation of the borrowing capacity of the borrower, the role of debt ceilings in lending and borrowing behavior as well as the affects of interest rates which include a default risk can be discussed. Although the problems are studied in the context of deterministic models it is indicated how the results are relevant for stochastic models.

Scheduled for Session 5.3 Techniques In Economic Dynamics