Solution Methods for Microeconomic Dynamic Stochastic Optimization Problems

Note: The GitHub repo SolvingMicroDSOPs repo associated with this document contains python code that produces all results, from scratch, except for the last section on indirect inference. The numerical results have been conﬁrmed by showing that the answers that the raw python produces correspond to the answers produced by tools available in the Econ-ARK toolkit, more speciﬁcally those in the HARK which has full documentation. The MSM results at the end have have been superseded by tools in the EstimatingMicroDSOPs repo.

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Abstract
These notes describe tools for solving microeconomic dynamic stochastic optimization problems, and show how to use those tools for eﬃciently estimating a standard life cycle consumption/saving model using microeconomic data. No attempt is made at a systematic overview of the many possible technical choices; instead, I present a speciﬁc set of methods that have proven useful in my own work (and explain why other popular methods, such as value function iteration, are a bad idea). Paired with these notes is Mathematica, Matlab, and Python software that solves the problems described in the text.

Keywords

Dynamic Stochastic Optimization, Method of Simulated Moments, Structural Estimation, Indirect Inference
JEL codes

E21, F41

¹Carroll: Department of Economics, Johns Hopkins University, Baltimore, MD, ccarroll@jhu.edu The notes were originally written for my Advanced Topics in Macroeconomic Theory class at Johns Hopkins University; instructors elsewhere are welcome to use them for teaching purposes. Relative to earlier drafts, this version incorporates several improvements related to new results in the paper “Theoretical Foundations of Buﬀer Stock Saving” (especially tools for approximating the consumption and value functions). Like the last major draft, it also builds on material in “The Method of Endogenous Gridpoints for Solving Dynamic Stochastic Optimization Problems” published in Economics Letters, available at http://www.econ2.jhu.edu/people/ccarroll/EndogenousArchive.zip, and by including sample code for a method of simulated moments estimation of the life cycle model a la Gourinchas and Parker (2002) and Cagetti (2003). Background derivations, notation, and related subjects are treated in my class notes for ﬁrst year macro, available at http://www.econ2.jhu.edu/people/ccarroll/public/lecturenotes/consumption. I am grateful to several generations of graduate students in helping me to reﬁne these notes, to Marc Chan for help in updating the text and software to be consistent with Carroll (2006), to Kiichi Tokuoka for drafting the section on structural estimation, to Damiano Sandri for exceptionally insightful help in revising and updating the method of simulated moments estimation section, and to Weifeng Wu and Metin Uyanik for revising to be consistent with the ‘method of moderation’ and other improvements. All errors are my own. This document can be cited as Carroll (2023) in the references.

1 Introduction

These lecture notes provide a gentle introduction to a particular set of solution tools for the canonical consumption-saving/portfolio allocation problem. Speciﬁcally, the notes describe and solve optimization problems for a consumer facing uninsurable idiosyncratic risk to nonﬁnancial income (e.g., labor or transfer income), ﬁrst without and then with optimal portfolio choice,¹ with detailed intuitive discussion of the various mathematical and computational techniques that, together, speed the solution by many orders of magnitude compared to “brute force” methods. The problem is solved with and without liquidity constraints, and the inﬁnite horizon solution is obtained as the limit of the ﬁnite horizon solution. After the basic consumption/saving problem with a deterministic interest rate is described and solved, an extension with portfolio choice between a riskless and a risky asset is also solved. Finally, a simple example shows how to use these methods (via the statistical ‘method of simulated moments’ (‘MSM’) to estimate structural parameters like the coeﬃcient of relative risk aversion (a la Gourinchas and Parker (2002) and Cagetti (2003)).

2 The Problem

The usual analysis of dynamic stochastic programming problems packs a great many events (intertemporal choice, stochastic shocks, intertemporal returns, income growth, the taking of expectations, and more) into a single step in which the agent makes an optimal choice taking account of all of these elements. For the detailed analysis here, we will be careful to disarticulate everything that happens in the problem explicitly into separate steps so that each element can be scrutinized and understood in isolation.

We are interested in the behavior a consumer who begins period $t$ with a certain amount of ‘capital’ $kt$ , which is immediately rewarded by a return factor $Rt$ with the proceeds deposited in a bank account balance:

References

Cagetti, Marco (2003): “Wealth Accumulation Over the Life Cycle and Precautionary Savings,” Journal of Business and Economic Statistics, 21(3), 339–353.

Carroll, Christopher D. (2006): “The Method of Endogenous Gridpoints for Solving Dynamic Stochastic Optimization Problems,” Economics Letters, 91(3), 312–320, https://www.econ2.jhu.edu/people/ccarroll/EndogenousGridpoints.pdf.

__________ (2023): “Solving Microeconomic Dynamic Stochastic Optimization Problems,” Econ-ARK REMARK.

Gourinchas, Pierre-Olivier, and Jonathan Parker (2002): “Consumption Over the Life Cycle,” Econometrica, 70(1), 47–89.

Merton, Robert C. (1969): “Lifetime Portfolio Selection under Uncertainty: The Continuous Time Case,” Review of Economics and Statistics, 51, 247–257.

Samuelson, Paul A. (1969): “Lifetime Portfolio Selection by Dynamic Stochastic Programming,” Review of Economics and Statistics, 51, 239–46.