A Guide on Solving Non-convex Consumption-Saving Models
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A Guide on Solving Non-convex Consumption-Saving Models Jeppe Druedahl1 Accepted: 4 September 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract Consumption-saving models with adjustment costs or discrete choices are typically hard to solve numerically due to the presence of non-convexities. This paper provides a number of tools to speed up the solution of such models. Firstly, I use that many consumption models have a nesting structure implying that the continuation value can be efficiently pre-computed and the consumption choice solved separately before the remaining choices. Secondly, I use that an endogenous grid method extended with an upper envelope step can be used to solve efficiently for the consumption choice. Thirdly, I use that the required pre-computations can be optimized by a novel loop reordering when interpolating the next-period value function. As an illustrative example, I solve a model with non-durable consumption and durable consumption subject to adjustment costs. Combining the provided tools, the model is solved almost 50 times faster than with standard value function iteration for a given level of accuracy. Software is provided in both Python and C++. Keywords Endogenous grid method · Post-decision states · Stochastic dynamic programming · Continuous and discrete choices · Occasionally binding constraints JEL Classification C6 · D91 · E21
1 Introduction Multi-dimensional consumption-saving models with adjustment costs or discrete choices are typically hard to solve numerically due to the presence of non-convexities. Starting from value function iteration (VFI), which is simple and straightforward, but
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Jeppe Druedahl [email protected] CEBI, Department of Economics, University of Copenhagen, Øster Farimagsgade 5, Building 26, 1353 Copenhagen K, Denmark
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J. Druedahl
inherently slow, this paper provides a guide on reducing computational time in such models using three layers of optimization. In the first layer, I use that many consumption saving models have a nesting structure such that the continuation value can be efficiently pre-computed and the consumption choice solved separately before the remaining choices. I refer to this as the nested value function iteration (NVFI). In the second layer, I use that the nested consumption problem under weak assumptions can be solved efficiently by using an extension of the endogenous grid method (EGM) originally developed by Carroll (2006) for one dimensional models.1 I refer to this as the nested endogenous grid method (NEGM). This step relies on a onedimensional version of the multi-dimensional upper envelope algorithm developed in Druedahl and Jørgensen (2017). In the third layer of optimization, I use that the pre-computation of the continuation value needed in both NVFI and NEGM can be computed efficiently by introducing a novel loop reordering reducing the number of computations and improving memory access. I refer to the improved solution methods as NVFI+ and NEGM+. To study the implications f
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