Solving models with jump discontinuities in policy functions

Görtz, Christoph; Mirza, Afrasiab Working Paper Solving models with jump discontinuities in policy functions Birmingham Business School Discussion Paper Series, No. 2016-07 Provided in Cooperation with: Birmingham Business School, University of Birmingham Suggested Citation: Görtz, Christoph; Mirza, Afrasiab (2016) : Solving models with jump discontinuities in policy functions, Birmingham Business School Discussion Paper Series, No. 2016-07, University of Birmingham, Birmingham Business School, Birmingham, http://epapers.bham.ac.uk/2163/ This Version is available at: https://hdl.handle.net/10419/202671 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by-nc-sa/2.5/ BIRMINGHAM BUSINESS SCHOOL Birmingham Business School Discussion Paper Series Solving Models with Jump Discontinuities in Policy Functions Christoph Görtz Afrasiab Mirza 2016-07 *** This discussion paper is copyright of the University and the author. In addition, parts of the paper may feature content whose copyright is owned by a third party, but which has been used either by permission or under the Fair Dealing provisions. The intellectual property rights in respect of this work are as defined by the terms of any licence that is attached to the paper. Where no licence is associated with the work, any subsequent use is subject to the terms of The Copyright Designs and Patents Act 1988 (or as modified by any successor legislation). Any reproduction of the whole or part of this paper must be in accordance with the licence or the Act (whichever is applicable) and must be properly acknowledged. For non-commercial research and for private study purposes, copies of the paper may be made/distributed and quotations used with due attribution. Commercial distribution or reproduction in any format is prohibited without the permission of the copyright holders. *** Solving Models with Jump Discontinuities in Policy Functions.∗ Christoph G¨ortz†and Afrasiab Mirza‡ This version: November 2015. Abstract We show that the Value Function Iteration (VFI) algorithm has difficulties approximating models with jump discontinuities in policy functions. We find that 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 value function iteration with a local interpolation step (VFI-INT) is sufficient to obtain accurate approximations. Differences between policy functions generated by VFI and these alternative methods are economically significant. We highlight that these differences across methods cannot be identified using Euler equation errors as these are not a sufficient measure of accuracy for models with jump discontinuities in policy functions. As a result, speed comparisons across methods that rely on Euler equation errors as a measure for accuracy can be misleading. The combination of computational speed, relatively easy implementation and adaptability make VFI-INT especially suitable for approximating models with jump discontinuities in policy functions. Keywords: Dynamic Equilibrium Economies, Non-Convex Capital Adjustment Costs, Computational Methods, Nonlinear Solution Methods, Euler equation errors. JEL Classification: C63, C68, E37. ∗We thank Christian Bayer, Andrew Clausen, Wouter den Haan, Giulio Fella, John Fender, Jesus Fernandez-Villaverde, Tom Holden, Julia Iori, Paul Levine, Kaushik Mitra, Christopher Otrok, Morten Ravn, Pontus Rendahl, Peter Sinclair, Carlo Strub, Konstantinos Theodoridis, H˚akon Tretvoll, John Tsoukalas, Fabio Verona and participants at the Society of Computational Economics 2013 Conference and the Royal Economic Society Annual Meeting 2014 for useful comments and suggestions. All remaining errors are our own. †University of Birmingham, Department of Economics, J.G. Smith Building, Edgbaston, Birmingham, B15 2TT. Email: [email protected]. ‡University of Birmingham, Department of Economics, J.G. Smith Building, Edgbaston, Birmingham, B15 2TT. Email: [email protected]. 1 Introduction We examine differences in the answers produced by global approximation methods for solving 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 decisions by comparing the option values associated with different adjustments. Fixed adjustment 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 policy functions are continuous (e.g. McGrattan (1996), Santos (2000), Aruoba et al. (2006), Santos and Peralta-Alva (2012)), the literature provides 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 intersections of these values and leads to an imprecise determination of the jump discontinuity. Sufficient mitigation of this problem requires very fine grids that are infeasible in many applications due to the curse of dimensionality. To our knowledge the problem VFI exhibits for models with jump-discontinuities has 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 et al. (1999), Nilsen and Schiantarelli (2003) and 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)). 1 not been documented in the literature. We explore its implications and show that a Finite Element Method (FEM) and an adaptation of the Endogenous Grid Method (EGM) can overcome this problem.2This is essentially 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 extending VFI 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 well established in the literature and is based on Cooper and Haltiwanger (2006). Their paper provides widely used parameter estimates and statistics on the importance of capital adjustment costs and relies on VFI as an 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 make any 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, even for fine grids VFI yields multiple thresholds located across a wide range of capital values. 2Given that we consider non-concave problems, we focus on piece-wise linear approximations and do not implement methods that involve higher order polynomial approximations. 2 We also highlight that relying on the use of Euler equation errors alone is insufficient to assess the accuracy of approximation for models with jump discontinuities in policy functions. We show that standard measures in the literature such as average or maximum Euler equation errors fail to indicate how well the threshold is approximated. To assess how well the four different methods approximate the threshold and the size of the jump discontinuity we conduct a simulation exercise and focus on two key statistics: the size of investment spikes and firm’s average capital stock. These are very sensitive to the location and size of the discontinuity and are also frequently reported as key statistics in models with (S, s) behavior.3 We find that VFI generates statistics that are noticeably different from the “true” investment spike size and average capital stock.4This is in stark contrast to the performance of EGM, FEM and VFI-INT which deliver statistics very close to the true ones. Crucially, the differences between these methods and VFI are economically significant. For example the maximum percentage deviations across shocks are much higher using VFI: for a particular comparable grid, VFI implies that a firm’s investment spike size (mean capital stock) deviates up to 16% (8%) from the true size. In contrast VFI-INT only implies a maximum deviation of 4% (2%). The limited informativeness of Euler equation errors in determining the accuracy of solutions for models with jump discontinuities in policy functions, implies that the conventional speed comparisons across methods – that use Euler equation errors to benchmark accuracy – can be misleading. We provide a first indication on the relative speed of VFI, FEM, EGM and VFI-INT for models with jump discontinuities in policy functions without relying on Euler equation errors alone. We compare speed by benchmarking accuracy based on the 3Statistics resulting from simulations have been used in the literature as an alternative to Euler equation errors to assess accuracy of approximations across methods, see for example Heer and Maußner (2008). 4We define the ”true” statistics as the mean of those generated by EGM and FEM for very fine grids (defined in detail in Section 5) as the model does not have an analytical solution. 3 statistics introduced above. Surprisingly, we find – in contrast to the literature benchmarking on Euler equation errors – VFI is much slower than FEM. For the other methods, FEM is the slowest method followed by VFI-INT while EGM is the fastest. However, EGM is also by far the most complex method to implement as it requires a number of adaptations to be applicable to our model. The original EGM algorithm introduced by Carroll (2006) is limited to smooth models with at most one control and one endogenous state variable. The literature has proposed numerous extensions to accommodate more complex models as the applicability of EGM is context dependent. A number of extensions have been developed recently to allow EGM to be applied to more complex models.5The implementation of EGM for non-smooth and non-concave problems such as ours adds a significant layer of complexity. Fella (2014) shows how to extend EGM to such settings using a consumption model that involves fixed adjustment costs for durable goods. We adapt the algorithm to our model of a plant with fixed capital adjustment costs. This problem involves an endogenous continuous choice variable that is subject to fixed adjustment costs, unlike Fella (2014), where this choice is discrete. FEM and VFI-INT are of similar implementation complexity and are far less complex to implement for non-concave problems than EGM.6Both are general purpose methods that require only minimal changes to handle more complex models. However, a key drawback of FEM is that it is far more expensive in terms of computation time than VFI-INT. The combination of computational speed and easy implementation and adaptation make VFIINT ideal for approximating models with jump discontinuities in the policy functions. 5These extensions often combine EGM with VFI. Barillas and Fernandez-Villaverde (2007) show how to introduce additional control variables; Hintermaier and Koeniger (2010) demonstrate how to introduce additional endogenous state variables in a durable goods model and Ludwig and Sch¨on (2013) show how to accommodate additional endogenous state variables in a human capital model. 6Our FEM code approximates the value function using piece-wise linear functions with weights updated via iteration on the Bellman operator rather than minimization of the Galerkin residual as in McGrattan (1996) and Aruoba et al. (2006). The latter approach has been shown to work well for smooth problems while for our context with jump discontinuities in the policy function we find that this approach is problematic as results are highly dependent on the algorithm’s start values. 4 The rest of the paper is organized as follows. The next section presents the model we use to illustrate our results. We then provide descriptions of the methods we use to solve the model. Section 4 discusses the model parameterization. Section 5 analyses the differences in the solutions across methods. The final section concludes. 2 The Model We consider a general class of models where in every period the agent makes both a continuous and discrete choice (c0, d0) based on the state (c, d) consisting of previous period’s choices. The set of possible states is denoted by Ω. The agent’s choice set is constrained as follows: (c, c0, d, d0)∈Γ, where Γ is R4 +.7Importantly, this specification of the constraints includes the case where c or dare subject to non-convex adjustment costs. The agent solves the following dynamic programming problem: V(c, d, A) = sup (c0,d0)∈Γ(c,·;d,·;A) u(c, c0;d, d0;A) + βX A0∈A π(A0|A)V(c0, d0, A0) where Ais the set of all possible shock realizations A∈ A,πis the corresponding transition matrix, the domain of Vis Ω × A, the per-period utility function of the agent is u, and the discount-factor is β. We assume that u(·, c0;d, d0;A) and u(c, ;d, d0;A) are differentiable on int(Γ(·, c0;d, d0;A)) and int(Γ(c, ·;d, d0;A)), respectively. Importantly, the value function Vis non-concave in the presence of non-convex adjustment costs to cor d. As a result, the agent compares the option values associated with choices of c0and d0. A kink in the value function arises at the point of indifference between these options and implies a jump discontinuity in 7We define particular subsets of Γ as follows: Γ(c, ·;d, ·) = {(c0, d0) : (c, c0, d, d0)∈Γ}, Γ(c, ·;d, d0) = {c0: (c, c0, d, d0)∈Γ}, Γ(·, c0;d, d0) = {c: (c, c0, d, d0)∈Γ}. 5 where ˆ Vij(K, A) =        V0 ij(K, A) + V0 i+1j−V0 ij Ki+1−Ki(K−Ki) if K∈[Ki, Ki+1] 0 otherwise (9) so that we effectively use a piece-wise linear approximation for each value of productivity. We then apply the Bellman operator (8) using ˆ V(K0, A0) as our guess for tomorrow’s value function and update our initial guess V0(K, A) on the grid points. Finally, we iterate to convergence on ˆ V(K, A). The key difference between FEM and VFI is that with FEM tomorrow’s value function can be evaluated at any point in the state space. Crucially, this implies that the optimal choice of tomorrow’s capital is not restricted to be on the exogenous grid [K1,...Kn]. Therefore, the optimization step in the Bellman operator can be carried out using a standard constrained optimization routine that enforces the irreversible investment constraint I≥0. Hence, FEM permits an additional degree of freedom above VFI but it comes at a cost as we are forced to employ the computational expensive constrained optimization routine repeatedly. Note that the policy function generated by this procedure is also a piece-wise linear function akin to (9). Our algorithm for FEM is no more difficult to implement than VFI given that we do not rely on Galerkin weighting and use standard methods for implementing root-finding such as Golden Section Search.12 3.4 Endogenous Grid Method EGM as introduced by Carroll (2006) suggests assigning an exogenous grid over the control variable K0rather than the state variable K. Then, using the following first-order condition allows us to determine an endogenous grid over K, given the exogenous grid K0 12For our case of jump discontinuities in the policy function we find that Galerkin weighting is not suitable as it leads to results that are highly dependent on start values for the algorithm. 12 and the derivative of the value function with respect to K0,VK0(K0, A0),13 pI+γK0−(1 −δ)K K=βEA0|AVK0(K0, A0).(10) Interpolating the solution on the endogenous grid, to evaluate it on the exogenous grid, we can obtain a set of optimal control and state pairs that can then be used to approximate the value function. Crucially, this procedure requires a unique solution to the first-order condition (10) for every K0when solving for the endogenous grid over K. As shown in Figure 1, for our class of models fixed costs introduce kink(s) in the value function resulting in jump discontinuities in the (otherwise smooth and decreasing) slope of the value function, VK0(K0, A0). As a result, the first-order condition (10) does not imply a unique endogenous grid over K.14 Therefore, EGM as introduced by Carroll (2006) is not directly applicable. Instead, we employ a modification of EGM proposed by Fella (2014) and implement the following steps for our case with fixed capital adjustment costs: 1. We begin by assigning an (exogenous) grid on K0and an initial guess for VK0(K0, A0). 2. We generate an endogenous grid for Kusing the first-order condition (10).15 3. We then split our endogenous state space into two regions: one where the value function is concave (VK0(K0, A0) is smooth) and another where the value function is not concave (VK0(K0, A0) exhibits jump discontinuities). (a) We apply Carroll (2006)’s algorithm in the concave region. (b) We apply VFI in the non-concave region to identify and retain only global optima. 13The derivation of the first-order condition is shown in the Appendix. 14Clausen and Strub (2012) show that at the optimum the first-order condition holds and the envelope condition is valid. 15Note that this implies that the endogenous grid changes in every iteration. 13 4. We then proceed to interpolate over the endogenous state space and construct optimal (K, K0) pairs in both the active and inactive cases. 5. We use these pairs to construct an approximation of the values to being active and inactive and thereby the overall value function. 6. We use the slope of this value function to construct the endogenous grid as in step 2. Steps 2-5 are repeated until the value function is deemed to have converged. The computationally most demanding task of this algorithm is the interpolation step which is far less expensive than the maximization/optimization steps in VFI or FEM. Note however that the applicability of EGM is context dependent, for example it cannot necessarily accommodate additional variables. The reason is that EGM’s applicability rests on finding a unique solution to the first-order condition(s) for the endogenous grids. This limits EGM to models with only one continuous choice variable. The literature shows how to accommodate additional variables for specific classes of models by often combining EGM with VFI steps (see for example Barillas and Fernandez-Villaverde (2007), Hintermaier and Koeniger (2010), Fella (2014), and Ludwig and Sch¨on (2013)). In comparison to smooth and convex problems, implementation complexity increases even more for models with jump discontinuities in the policy function due to the need to identify and handle the non-concave region of the value function (Step 3) separately. 4 Parameterization Our choice of the model parameters is based on estimates by Cooper and Haltiwanger (2006). Using annual plant level data of the Longitudinal Research Database, they estimate the above model with convex and non-convex capital adjustment costs and find that a 14 combination of these fits the data well.16 Importantly, they find evidence for substantial fixed adjustment costs of roughly 4% of the average plant-level capital stock. We approximate the stochastic process for productivity by a ten-state Markov chain using the method proposed by Rouwenhorst (1995) and calibrate the Markov chain to match the standard deviation σε= 0.03 and persistence, ρ= 0.885. All estimates of Cooper and Haltiwanger (2006), that we use to calibrate the model, are summarized in Table 1. We approximate the value function for all solution methods over the same state space for capital.17 As a baseline scenario we use 700 (VFI), 97 (EGM), 95 (FEM), and 385 (VFI-INT) capital grid points, which is representative for many practical applications. For VFI-INT we use 35 interpolation points on each side of the optimal grid point identified by the value function iteration algorithm. This capital grid choice for the baseline scenario generates comparable average log-absolute Euler equation errors across methods (see Table 2).18 As conventional in studies which consider the performance of different approximation methods we use an equally spaced grid for capital. Table 1: Model Parameters (based on Cooper and Haltiwanger (2006)) β0.95 discount factor δ0.069 capital depreciation rate pI1 price to buy capital α0.592 returns of capital ρ0.885 persistence of plant specific shock σε0.03 standard deviation of plant specific shock γ0.049 convex adjustment costs F0.039 fixed adjustment costs 16For the sake of simplicity of exposition we do not include the possibility of selling capital considered by Cooper and Haltiwanger (2006). Selling plant’s capital stock at a price smaller than pIwould introduce an additional kink in the value function. The solution methods can be adjusted to accommodate the additional choice, but as our findings can be generalized to these additional kinks we assume irreversibility of capital for ease of exposition. 17The state space is chosen so that capital does not hit any boundaries during our simulations. Convergence is evaluated by considering the largest absolute distance between corresponding points of the value function of two consecutive iterations. If this absolute distance falls below 10−4the algorithm is deemed to have converged. 18As the Euler equation is a necessary but not a sufficient condition in our setup, Euler equation error statistics are calculated for policy functions across all shocks solely in the area of the state space in which all approximation methods imply positive investment. 15 5 Results This section first documents two specific issues when approximating models with jump discontinuities: (i) VFI fails to accurately approximate such models and (ii) Euler equation errors are not a suitable measure for algorithm accuracy. Then we show in Section 5.2 via a simulation exercise that the inaccuracies resulting from (i) are economically significant. This exercise also shows that VFI-INT, EGM and FEM can address the shortcomings of VFI. In light of (ii), we provide in Section 5.3 a comparison of methods with respect to speed and implementation complexity. 5.1 Specific Problems in Models with Jump Discontinuities Problems with VFI. As noted in Section 2, theory predicts that the policy function exhibits a jump discontinuity at the threshold separating the active (positive investment) and inactive (no investment) regions.19 Figure 2 shows the policy functions for tomorrow’s capital generated by VFI, VFI-INT, EGM and FEM for the baseline scenario. This figure highlights that VFI-INT, EGM and FEM all produce similar policy functions. However, these differ substantially from the one produced by VFI in two important aspects. First, VFI does not uniquely determine the threshold separating the active and inactive regions. Moreover, VFI does not approximate the shape of the active region accurately. To more clearly see the problems that arise in the determination of the threshold with VFI, we show in the bottom panel of Figure 3 the values to the plant of being active and inactive for increasingly finer capital grids. The intersection of these values determines the capital threshold below which the plant is active and above which the plant is inactive. While 19Theory also predicts additional jump discontinuities in the policy function in the active region due to the interaction between fixed and convex variable adjustment costs (see e.g. Clausen and Strub (2012)). The variable costs penalize the plant for making large adjustments while fixed costs penalize the plant for making small and frequent investments. The result is that the active region of the policy function consists of concave parts that are separated by jump discontinuities. 16 theory predicts a single intersection of these functions, VFI generates multiple intersections as a result of approximating these values using step functions. The reason for these steps is that only a finite set of points can be used to approximate the values of being active and inactive because VFI limits the choices for both the values of the endogenous state and the control variable to a fixed grid.20 VFI’s inaccurate determination of the threshold can also be seen in the corresponding policy functions for tomorrow’s capital which are shown in the top panel of Figure 3. From there it is evident that even a very fine grid using 3000 points does not deliver a unique intersection of the option values. In addition to this illustration, we provide a more comprehensive overview about the inaccurate determination of the threshold: we consider the percentage difference between the value for today’s capital implied by the grid point min(Vi> V a) and today’s capital implied by the grid point max(Vi< V a). Table 2 (column 7) shows that for the VFI baseline scenario (grid scenario 6) the mean across all shocks of this measure is 8.56, i.e. the capital stock to the right of the last intersection of the option values Viand Vais 8.56% higher than the capital stock to the left of the first intersection of the option values. This is equivalent to an average of 17.3 capital grid points across all shocks (column 9). The standard deviation across shocks of the percentage difference between the two capital stocks is 5.58%. This indicates that even using 700 capital grid points (baseline scenario) the threshold is determined very imprecisely across all shocks. While the percentage difference decreases with finer grids, even for grids as fine as 1900 points (grid scenario 10) the imprecise determination of the threshold is still apparent as the mean capital stock to the right of the last intersection of the option values is 3.20% higher than the capital stock to the left of the first intersection, with a standard deviation across shocks of 1.42%. Essentially, when 20Such approximations are particularly prone to error when the slope of the underlying function is steep. In our problem, the slope of the value to being inactive is much larger than the slope of the value to being active. Hence, as shown in Figure 3, the approximation of the value of being inactive is much worse than the approximation of the value of being active. 17 0 10 20 30 40 50 0 20 40 60 80 (a) Value Function Iteration capital today capital tomorrow 0 10 20 30 40 50 0 20 40 60 80 (b) Value Function Iteration with interpolation capital today capital tomorrow 0 10 20 30 40 50 0 20 40 60 80 (c) Finite Element Method capital today capital tomorrow 0 10 20 30 40 50 0 20 40 60 80 (d) Endogenous Grid Method capital today capital tomorrow 0 10 20 30 40 50 0 20 40 60 80 (e) All Methods (a)−(d) capital today capital tomorrow Figure 2: Policy Function for capital implied by different approximation methods. The baseline grid generates comparable average log-absolute Euler equation errors across methods. Subplots are shown for shock value 7. The blue dashed line in each subplot indicates the no investment decision (1 −δ)K. For better visibility we do not show part of the state space to the right of the threshold where the policy function is equal to the no-investment line. 18 using VFI to approximate models with jump discontinuities in the policy function, extremely fine grids are required to determine the threshold relatively precisely. Such fine grids are typically infeasible in most applications due to the curse of dimensionality.21 0 10 20 30 40 50 0 10 20 30 40 50 60 70 80 (1a) VFI policy function − 350 grid points capital today capital tomorrow 0 10 20 30 40 50 0 10 20 30 40 50 60 70 80 (1b) VFI policy function − baseline, 700 grid points capital today capital tomorrow 0 10 20 30 40 50 0 10 20 30 40 50 60 70 80 (1c) VFI policy function − 3000 grid points capital today capital tomorrow 31 32 33 34 35 36 37 38 39 40 41 116 117 118 119 120 121 122 123 124 125 126 127 (2a) VFI value functions − 350 grid points capital today option values 33 34 35 36 37 38 39 116 117 118 119 120 121 122 123 (2b) VFI value functions − baseline, 700 grid points capital today option values 34.4 34.6 34.8 35 35.2 35.4 35.6 35.8 116.4 116.6 116.8 117 117.2 117.4 117.6 117.8 118 capital today option values (2c) VFI value functions − 3000 grid points active investment inactive investment Figure 3: Approximation using Value Function Iteration. Top panel: Policy functions for capital for different grid sizes for a particular productivity level. Bottom panel: Option values to the plant of being active (red dashed) and inactive (blue solid) for different capital grids (zoomed in to show multiple intersections). Subplots are shown for shock 7. Table 2 reports the same statistics also for VFI-INT, FEM and EGM. The grid scenarios 1-12 in this table are comparable across methods in terms of average log-absolute Euler equation errors. For the baseline scenario, VFI approximates the threshold with an imprecision that is up to 10 times larger than for other methods – 8.56% versus 0.85% (VFI-INT), 3.48% (FEM), 3.45% (EGM). The three alternative methods deliver – in line with the predictions by theory – a single intersection of Viand Va(the mean number of grid points across shocks is exactly unity). These methods are therefore much more suitable than VFI for approxi- 21For similar reasons, VFI is also unable to correctly approximate the jump discontinuities and concave parts of the policy function in the active region. However, while the threshold is crucial for the dynamics of the model, the poor approximation of the active region is only of larger importance when the persistence of the technology shock is low. 19 mating models with jump discontinuities in policy functions. The percentage differences in capital stocks reflect here only the distance between two adjacent grid points to the left and right of the threshold. The relatively large numbers for coarse grids highlight that in order to approximate the location of the jump discontinuity precisely, these three methods require finer grids relatively to grids that are sufficient to approximate smooth and concave models (i.e. convex choice sets). Limited Informativeness of Euler Equation Errors. It is striking from Table 2 that for comparable average Euler equation errors, VFI and the other three methods deliver very different policy functions in terms of (at least) the determination of the threshold. Euler equation errors are often employed as a measure of accuracy and for comparisons across methods. However, it has so far been overlooked in the literature that the information about the accuracy of approximation provided by average and maximum Euler equation errors for problems with jump discontinuities in the policy functions is limited. This becomes evident when considering the recursive problem outlined in Section 2: Euler equation errors measure only how well the active region is approximated, i.e. the accuracy of the decision in equation (6), using information about the slope of the value function.22 So they do provide an indication about the accuracy of approximation of the policy function in the active region. However, Euler equation errors fail to measure the accuracy of the binary decision to be active or inactive shown in equation (5), determining the location of the threshold requires information about the slope and the level of the value function. As a result, Euler equation errors alone are not a sufficient measure of accuracy when policy functions exhibit jump discontinuities. We also compute Euler equation errors near the threshold. These are generated by 22Clausen and Strub (2012) show that an Euler equation holds in the active region for our class of models. They do not hold in the inactive region. 20 simulating the model’s (S, s) behavior for each shock value individually and calculating the mean of absolute log-Euler equation errors across all shocks for observations with positive investment.23 These are reported in Table 2 (column 6) and it is evident that – as one would expect from the discussion above – also the Euler equation errors at the threshold provide limited guidance on accuracy as they are very similar across grid scenarios. 5.2 Economic Significance We explore the economic relevance of the differences across methods documented above through a simulation exercise that focuses on two key statistics of the model: the size of investment spikes and the mean of capital. These two statistics are often used to calibrate models with (S, s) adjustment of capital to the data. Moreover, the mean of capital is a popular measure for firm size. These two statistics crucially depend on a precise determination of the location and size of the jump discontinuity. The following simulation exercise focuses on the effects near the threshold as this is most important for model dynamics. We evaluate the distance between the two key statistics implied by the different approximation methods and the “true” statistics. As the model does not have an analytical solution, we solve it using FEM and EGM — the two methods that allow by construction for the highest accuracy of threshold determination — for a large number of grid points and label the average statistics produced by the resulting policy functions to be the “true” statistics.24 We solve the model with the four approximation methods at the grid scenarios shown in Table 2. For each method and grid scenario, we 23To clearly identify the effects of imprecise threshold determination we determine in this (and the following) simulation exercises the model’s (S, s) adjustment behavior implied by a particular shock value at a time and report the mean of these exercises across shock values. We simulate the model for each shock value for 1050 periods and discard the first 50 periods to remove any impact of start values. 24In particular we use 4000 capital grid points for FEM, and 2500 for EGM. These deliver a unique intersection of Viand Va. The average absolute percentage deviation between the two sets of statistics across all shocks is 0.5% for the average capital stock and 0.6% for the average investment spike size. These differences are also consistent with results we obtain using alternative grids (2400 for FEM and 1500 for EGM). Our results also continue to hold using these alternative grids. 21 example Barillas and Fernandez-Villaverde (2007), Hintermaier and Koeniger (2010), Fella (2014) and Ludwig and Sch¨on (2013)). For models with jump discontinuities, demarcating the non-concave region adds substantial programming complexity, and using VFI-INT rather than VFI in extension to EGM is then more appropriate in light of our findings. VFI, VFI-INT and FEM are far simpler to implement and easily extend to additional state and control variables. The combination of computational speed and relatively easy implementation and adaptation make VFI-INT especially suitable for approximating models with jump discontinuities in the policy functions. 6 Conclusion Differences across approximation methods have been extensively studied for dynamic economies where policy functions are continuous. However, the literature provides little guidance about the adequacy and accuracy of computational methods for dynamic economies where agents face non-concave problems. This paper is a first attempt to fill this gap. We highlight that for models with jump discontinuities in policy functions (i) using Value Function Iteration (VFI) is problematic as it fails to accurately identify both the location and size of jump discontinuities; and (ii) Euler equation errors are not a sufficient measure for accuracy as they do not provide indications about how well the location of the discontinuity is approximated. We show that much more accurate approximations for this class of models are delivered by the Endogenous Grid Method (EGM), the Finite Element Method (FEM) and value function iteration when extended with a local interpolation step (VFIINT). 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A.1 Euler equation when plant is active When the plant is active (I > 0), the optimal investment strategy can be characterized by an Euler equation. Using equations (3) and (6) the plant’s problem in this case can be formulated as V(K, A) = max K0AKα−pI(K0−(1−δ)K)−FK −γ 2KK0−(1 −δ)K K2 +βEA0|AV(K0, A0). Following Proposition 1 of Clausen and Strub (2012), at the optimal choice of capital tomorrow the following first-order condition holds: pI+γK0−(1 −δ)K K=βEA0|AVK0(K0, A0) where VK0(·) denotes the function’s derivative with respect to K0. Then, the following Euler equation characterizes investment dynamics for an active plant pI+γI K=βEA0|A αA0(K0)α−1+pI(1 −δ)−F+γ 2I0 K02+γ(1 −δ)I0 K0!,(A.1) where I0=K00 −(1 −δ)K0. Given that the plant in not active for all possible values of the state variables, the above equation holds only when investment is strictly positive. A.2 Value Function Iteration We implement discrete value function iteration (VFI) as follows: 1. Fix the upper and lower bound for capital at [Kmin, Kmax]. 2. Generate an equally spaced capital grid of npoints GK={Ki}n i=1 on [Kmin, Kmax]. 3. Approximate the AR(1) process for the log of productivity using an m-state Markov chain. Denote the set of states by A. Productivity is drawn from the set GA≡eA. 4. Guess an initial value function V0(Ki, Aj) = AjKα iand policy function K00(Ki, Aj)=0 at each point [Ki, Aj] on the grid where Ki∈[Kmin, Kmax] and Aj∈GA. 34 5. Set the tolerance parameter tol = 10−4. This parameter is used to determine if the value function has converged. 6. For each level of capital Ki∈[Kmin, Kmax] and productivity Ajin GA: (a) compute the value of being inactive: Vina(Ki, Aj)≡AjKα i+βEA0|AjV0(Ki(1 −d), A0) (b) compute the value of being active: i. for each possible K0≥(1 −d)Ki∈[Kmin, Kmax] compute ˜ V(K0)≡AjKα i−p(K0−Ki(1 −δ)) −FKi−γKi 2K0−Ki(1 −δ) Ki2 +βEA0|AjV0(K0, A0) ii. the value of being active is Vact(Ki, Aj)≡maxK0˜ V (c) update the value and policy functions i. if Vina(Ki, Aj)≥Vact(Ki, Aj) then V1(Ki, Aj) = Vina(Ki, Aj), K01(Ki, Aj) = (1 −d)Ki ii. if Vina(Ki, Aj)< V act(Ki, Aj) then V1(Ki, Aj) = Vact(Ki, Aj), K01(Ki, Aj) = arg max ˜ V 7. Check if kV0−V1k∞< tol. If not set V0=V1,K00=K01and repeat Step 5. 8. Verify that Kmin < K0(Kmin, Aj)< K0(Kmax, Aj)< Kmax for all j. If not, enlarge grid and repeat Steps 1-6. A.3 Value Function Iteration with Local Interpolation We implement local interpolation within VFI as follows: 1. First follow Steps 1-6b) for VFI above. 2. Then, let (K∗ i−1, Aj), and (K∗ i+1, Aj) be the grid points adjacent (resp. to the left and right) to the optimal value of K0(Ki, Aj), denoted (K∗ i, Aj) found by VFI. 3. Generate 35 new capital grid points on the intervals [(K∗ i−1, Aj),(K∗ i, Aj)], and [(K∗ i+1, Aj),(K∗ i, Aj)]. 4. Compute the new values based on interpolation of being active, Va INT , and inactive, Vi INT , using these new points as additional possible values for tomorrow’s capital. 5. Update the optimal value at Ki, Ajagain as follows: V(Ki, Aj) = max{Va INT , V i INT }. 35 6. Update the policy function with the optimal capital value corresponding to the maximum found in the previous step. 7. Proceed with step 7 of the VFI algorithm. A.4 Finite Element Method We implement a Finite Element Method approximation to the value function via the following algorithm: 1. Fix the upper and lower bound for capital at [Kmin, Kmax]. 2. Generate an equally spaced capital grid of npoints GK={Ki}n i=1 on [Kmin, Kmax]. 3. Approximate the AR(1) process for the log of productivity using an m-state Markov chain. Denote the set of states by A. Productivity is drawn from the set GA≡eA. 4. Guess an initial value function {{V0 ij}n i=1}10 j=1 at each point [Ki, Aj] of the state space. We set V0 ij =AjKαfor all i, j. 5. Set the tolerance parameter tol = 10−4. 6. We approximate the value function V(K, A) as ˆ V(K, A), a piece-wise linear interpolation through the points {{V0 ij}n i=1}10 j=1 where ˆ V0 ij(K, A) = (V0(Ki, Aj) + V0 i+1−V0 i Ki+1−Ki(K−Ki) if K∈[Ki, Ki+1] 0 otherwise 7. For each point in the capital grid find the value of being inactive Vina(K, A) where Vina(K, A) = AKα+βEA0|Aˆ V0(K(1 −d), A0) 8. Find the value of being active, Vact(Ki, Aj): •first find K0(Ki, Aj) = arg maxK0≥Ki(1−d)˜ V(K, A)≡AKα i−p(K0−Ki(1 −δ)) − FKi−γKi 2K0−Ki(1−δ) Ki2+βEA0|Aˆ V0(K0, A0) •the value of being active is then Vact(Ki, Aj) = ˜ V(K0(Ki, Aj)). 9. Update the value and policy functions (a) if Vinv(Ki, Aj)≥Vact(Ki, Aj) then V1(Ki, Aj) = Vinv(Ki, Aj), K0(Ki, Aj) = (1 − d)Ki (b) if Vinv(Ki, Aj)< V act(Ki, Aj) then V1(Ki, Aj) = Vact(Ki, Aj), K0(Ki, Aj) = arg max ˜ V 36 10. Check if kV0−V1k∞< tol. If not set V0=V1and repeat the steps above. 11. Verify that Kmin < K0(Kmin, Aj)< K0(Kmax, Aj)< Kmax for all j. If not, enlarge grid and repeat Steps 1-12. A.5 Endogenous Grid Method We implement the Endogenous Grid Method as follows: 1. Fix the upper and lower bound for capital at [Kmin, Kmax]. 2. Generate an n-point equally spaced grid GK0={K0 i}n i=1 for tomorrow’s capital over [Kmin, Kmax]. 3. Approximate the AR(1) process for the log of productivity using an m-state Markov chain. Denote the set of states by A. Productivity is drawn from the set GA≡eA. 4. Guess an initial value for EV (K0, A0) at each point [Ki, Aj] of the state space and construct a corresponding guess for EVK0(K0, A0). 5. Set the tolerance parameter tol = 10−4. 6. Construct an endogenous grid of capital points {Kend i}n i=1 using the Euler equation Kend i=γIi βEA0|A αA0(K0 i)α−1+p(1 −δ)−F+γ 2I0 i K0 i2+γ(1 −δ)I0 i K0 i!−p to obtain nmatching pairs {Kend i, K0 i}n i=1. 7. Ensure capital is not reversible: if K0/Kend i<(1 −δ) then set Kend i=K0 i/(1 −δ). 8. Identify the non-concave region (a) identify the set of jumps in EVK0(K0, A0) by noting that around these jump points there are sharp changes in the slope of EVK0(K0, A0). (b) find the minimum and maximum of values of EVK0(K0, A0) at these jumps, and denote these by V , V . (c) the non-concave region for tomorrow’s productivity A0 jconsists of all pairs {Kend i, K0 i} where EVK0(K0 i, A0 j)∈[V,V]. 9. For each pair {Kend i, K0 i}inside the non-concave region (a) compute the value to being active for every K0 j6=K0 i (b) if the maximum does not occur at K0 ithen discard the pair {Kend i, K0 i} 37