Endogenous Grids in Higher Dimensions: Delaunay Interpolation and Hybrid Methods

Abstract

Many quantitative economic models have to be solved with numerical methods. This is also true for many household models, e.g., standard models of consumption and savings. With an increasing number of variables that are of relevance for a household's decision (=state variables), this may become very costly (in terms of computational time). One reason for this complexity is that household decisions are characterized as functions of state variables which have to be approximated on grids. The solution of the economic decision problem has to be computed numerically for each gridpoint. State variables are, e.g., financial assets and nonfinancial assets such as housing, educational background or age of the household.The numerical solution of such a prototypical model is basically characterized by two computationally demanding numerical operations: the solution of a non-linear system of equations and interpolation of functions. Typically, the solution of the non-linear system of equations does not have a closed form. Recently, a method that has received considerable attention in the literature is the method of endogenous gridpoints (ENDGM). It is mainly applied to one-dimensional problems. In contrast to a standard method with exogenous grids (EXOGM), a smart redefinition of state variables in ENDGM may make it possible to solve the non-linear system of equations analytically. This greatly enhances speed of computations. However, we highlight that there exists a trade-off in higher dimensions: while the solution of the non-linear system of equations in ENDGM may still have a closed form, interpolation becomes much more complex. It is not clear how this trade-off is resolved vis-a-vis a standard EXOGM method.Against this background, we compare three numerical methods: The standard exogenous grid method (EXOGM), the method of endogenous gridpoints (ENDGM), and a hybrid method (HYBGM) as a combination of the former two. We do this comparison by solving a dynamic model with two continuous state variables (financial assets and health of the household) and occasionally binding borrowing constraints. Evaluation is based on speed and accuracy of the methods. Our conclusion is that HYBGM and ENDGM both dominate EXOGM. In dynastic models where representative households have long or infinite horizons, ENDGM also always dominates HYBGM. In a finite horizon model, the choice between HYBGM and ENDGM depends on the number of gridpoints in each dimension. For a standard choice of gridpoints, ENDGM is always faster. These insights will be very useful in applications of economic models with more than one continuous state variable. Such models include, amongst others, models with an explicit notion of portfolio choice decisions with respect to financial wealth and housing investments over a household's life-cycle, models in health economics such as our application in this paper or human capital models. Enhancing the speed of computations in such models will simplify their use for policy analysis. Endogenous Grids in Higher Dimensions

The endogenous grid method for discrete-continuous dynamic choice models with (or without) taste shocks

“…Since then the idea became widespread in economics. Further generalizations of EGM include Barillas and Fernández-Villaverde (2007), Hintermaier and Koeniger (2010), Ludwig and Schön (2013), Fella (2014), Iskhakov (2015). Jørgensen (2013) compares the performance of EGM to mathematical programming with equilibrium constraints (MPEC).…”

The method of endogenous gridpoints in theory and practice

“…4 Ludwig and Schön (2014) show that Delaunay interpolation can be used with this irregular array. However, the computational gains from ENDG are partly eroded by the serial construction of the Delaunay triangulation in each period.…”

“…For ease of reference, the model is a slight extension of Ludwig and Schön (2014), adding additional interperiod risk. …”

A general endogenous grid method for multi-dimensional models with non-convexities and constraints

“…In the multi-dimensional case, however, there are no simple algorithm for finding the neighboring points in a fully irregular grid. Ludwig and Schön (2014) therefore suggest to use a Delaunay-triangulation to divide a two dimensional irregular grid into triangles (or into simplexes in higher dimensions). A so-called visibility walk can then be used to find the triangle containing the point to be interpolated.…”

“…Interpolation on irregular grids have previously been shown to be the key bottleneck for the performance of multi-dimensional EGM (without non-convexities). Ludwig and Schön (2014) show that using Delaunay-triangulations and so-called visibility walks can result in the EGM being slower than time iterations in a two-dimensional model without non-convexities. White (2015) shows how assuming a specific form of monotonicity can be used to avoid the triangulation step, but his method still requires expensive visibility walks, and cannot handle non-convexities.…”