Solving Models with Jump Discontinuities in Policy Functions

DOIhttp://doi.org/10.1111/obes.12203
Date01 April 2018
Published date01 April 2018
434
©2017 The Department of Economics, University of Oxford and JohnWiley & Sons Ltd.
OXFORD BULLETIN OF ECONOMICSAND STATISTICS, 80, 2 (2018) 0305–9049
doi: 10.1111/obes.12203
Solving Models with Jump Discontinuities in Policy
Functions*
Christoph G ¨
ortz and Afrasiab Mirza
Department of Economics, University of Birmingham, Edgbaston, Birmingham, UK
(e-mail: c.g.gortz@bham.ac.uk), (e-mail: a.mirza@bham.ac.uk)
Abstract
We compare global methods for solving models with jump discontinuities in the policy
function. We find that differences betweenvalue function iteration (VFI) and other methods
are economically significant and Euler equation errors fail to be a sufficient measure of
accuracy in such models. VFI fails to accurately identify both the location and size of
jump discontinuities, while the endogenous grid method (EGM) and the finite element
method (FEM) are much better at approximating this class of models. We further show that
combining VFI with a local interpolation step (VFI-INT) is sufficient to obtain accurate
approximations. The combination of computational speed, relatively easy implementation
and adaptability make VFI-INT especially suitable for approximating models with jump
discontinuities in policy functions: while EGM is the fastest method, it is relativelycomplex
to implement; implementation of VFI-INT is relatively straightforward and it is much faster
than FEM.
I. Introduction
Weexamine differences in the answers produced byglobal approximation methods for solv-
ing dynamic economies where agents face non-concave problems (i.e. non-convex choice
sets). Non-concave problems can result from the inclusion of fixed adjustment costs that
are empirically relevant in many circumstances.1In such problems, agents make discrete
JEL Classification numbers: C63, C68, E37.
*We thank Francesco Zanetti and two anonymous referees for their constructive comments, which significantly
improved the paper.We further thank Christian Bayer,Andrew Clausen, Wouterden Haan, Giulio Fella, John Fender,
Jesus Fernandez-Villaverde,TomHolden, Julia Iori, Kaushik Mitra, Christopher Otrok, Morten Ravn, PontusRendahl,
Peter Sinclair, Carlo Strub, KonstantinosTheodoridis, H˚akonTretvoll, JohnTsoukalas, Fabio Veronaand participants
at the Society of Computational Economics 2013 Conference and the Royal Economic SocietyAnnual Meeting 2014
for useful comments and suggestions. All remaining errors are our own. G¨ortz acknowledges support from a British
Academy Small Grant.
1The relevance of fixed adjustment cost is highlighted for example in studies of investment (e.g. Caballero et al.,
1995; Doms and Dunne, 1998; Power,1998; Cooper, Haltiwanger and Power, 1999; Nilsen and Schiantarelli, 2003;
Cooper and Haltiwanger, 2006; Whited, 2006; Bayer,2006; Khan and Thomas, 2008; Bloom, 2009; Wang and Wen,
2012), consumer-durables choice (e.g. Jose Luengo-Prado, 2006; Bajari et al., 2013), portfolio choice models with
transaction costs and asset prices (e.g. Vayanos, 1998), costly technology adoption (e.g. Khan and Ravikumar,2002)
and optimal dynamic capital structure choice (e.g. Hennessy and Whited, 2005).
Solving models with jumps in policy functions 435
decisions by comparing the option values associated with different adjustments. Fixed ad-
justment costs generate kink(s) in the value function at the intersection of these option
values and imply jump discontinuities in the policy function. While differences across
approximation methods have been extensively studied for dynamic economies where pol-
icy functions are continuous (e.g. McGrattan, 1996; Santos, 2000; Aruoba, Fernandez-
Villaverde and Rubio-Ramirez, 2006; Santos and Peralta-Alva, 2012), the literature pro-
vides little guidance about the adequacy and accuracy of computational methods when
policy functions exhibit jump discontinuities. The goal of this paper is to fill this gap.
We document that the exact intersection of the option values – and thereby the location
of a jump discontinuity in the policy function – is difficult to determine using discretized
value function iteration (VFI). The use of a finite grid on state and control variables limits
VFI to approximating the option values as step functions. This results in multiple inter-
sections of these values and leads to an imprecise determination of the jump discontinuity.
Such imprecision may lead to economically significant approximation errors. Sufficient
mitigation of this problem requires very fine grids that are infeasible in many applications
due to the curse of dimensionality.
Toour knowledge the problemVFI exhibits for models with jump-discontinuities has not
been documented in the literature. We explore its implications and showthat a finite element
method (FEM) and an adaptation of the endogenous grid method (EGM) can overcome
this problem.2This is essential because both methods approximate the option values over
the entire state space using piece-wise linear functions – effectively approximating these
values using an infinite set of points – leading to a single intersection of option values
and therefore a unique determination of the jump discontinuity in the policy function. We
also show that extendingVFI to allow the option values to be approximated locally around
each grid point using piece-wise linear functions (VFI-INT) is sufficient to obtain a unique
intersection and precise solutions.
We illustrate differences across methods for non-concave problems using a partial
equilibrium model of a Plant, where investment is subject to both variable and fixed capital
adjustment costs. This model is wellestablished in the literature and is based on Cooper and
Haltiwanger (2006).3Their paper provides widely used parameter estimates and statistics
on the importance of capital adjustment costs and relies on VFI as the approximation
method. In this model, in the presence of fixed costs the plant determines its investment
strategy each period by comparing the option value of remaining inactive (not investing)
with the option value of becoming active (investing). The optimal investment strategy
follows an (S,s) adjustment process whereby the plant does not makeany investment until
capital depreciates below a threshold level at which point the plant makes a substantial
investment to re-build its capital stock (investment spike). The threshold is determined by
the intersection of the plant’s option values.To correctly capture the dynamics of investment
it is crucial to determine this threshold accurately. We show that EGM, FEM and VFI-INT
yield a unique threshold, while in contrast, evenfor fine g ridsVFI yields multiple thresholds
located across a wide range of capital values.
2Given that we consider non-concaveproblems, we focus on piece-wise linear approximations and do not imple-
ment methods that involve higher order polynomial approximations.
3We illustrate these differencesalso in a general equilibrium setting in Appendix S1.
©2017 The Department of Economics, University of Oxford and JohnWiley & Sons Ltd

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