Exploiting Model Structure In Numerically Solving Macrodynamics

This paper is concerned with the development of computationally efficient algorithms for the solution of the dynamics of macroeconomic models. The paper focuses on a particular continuous-time macroeconomic model. In the paper we exploit the model structure to improve the efficiency of solving the model's dynamics.The paper uses a well-known representative agent model (Matsuyama, 1987), which has a number of important dynamic properties. These properties have significant implications, common to a range of macroeconomic models, for computing the model solution. Firstly the model has a number of stable and unstable trajectories so that it is likely to be complicated to solve the model for a stable solution. The economy is initially at a stable steady-state equilibrium, and when shocked by, say, an exogenous change in world interest rates, then it moves to a stable trajectory leading to a new steady-state equilibrium. The movement to the new equilibrium is assumed to come about as a consequence of optimising behaviour of the agents in the model. In the model, certain variables jump instantaneously after the shock, and force the economy onto the trajectory leading to the stable equilibrium.A second property of the model is that it is nonlinear with nonlinearities arising as a direct consequence of optimising behaviour by the representative agents. The usual approach is to linearise the model in the neighbourhood of the steady-state and then solve the linearised model. Using a linear approximation rather than the true nonlinear model can be particularly unreliable if the jumps required to bring the economy back onto a stable path are particularly large.These properties, especially the property of jumps to the stable path, introduce some interesting challenges to developing computationally efficient algorithms for solving the model.To generate the macrodynamics resulting from an exogenous shock it is necessary to find that manifold which defines the stable solution to the model. When shocked the model jumps from the pre-shock steady-state onto this manifold and then evolves to the new (post-shock) steady-state. For an algorithm to calculate these dynamics it has information about the pre-shock steady-state and the post-shock steady-state as well as the differential equations of the model. That is, it has all the terminal conditions and the initial conditions for the non-jumping variables. The problem is to find the initial conditions of the jumping variables that place the model on the stable manifold.This is an example of a class of problem known as two-point boundary value problems. It can be solved by a forward shooting algorithm where a numerical search procedure is used to find the unknown initial conditions that, when used with the differential equations, generate the known terminal conditions.It is possible to take advantage of certain characteristics of the model in computing a solution. Firstly, the model is autonomous so that calendar time plays no role in the model solution. Thus the model can be solved in reverse time. Secondly, the model has a block recursive structure and can be divided into two sub-models. The first sub-model can be solved independently of the second sub-model. This first sub-model has a saddle structure, and the algorithm exploits the separatrix property of saddles to generate the stable arm of the sub-model with one sub-model solution in reverse time. A single interpolation then gives the initial condition of the jumping variable. The stable arm for the first sub-model can be treated as an exogenous nonlinear forcing function in the second sub-model. Taking the solutions to the first sub-model as exogenous and given, the second sub-model is linear in its endogenous variables and the algorithm exploits these properties in generating a solution.By exploiting the model structure through the use of these particular properties of the model, the paper demonstrates how a computationally superior algorithm can be developed for solving model dynamics.