Perturbation Solution and Welfare Costs of Business Cycles in DSGE Models

Perturbation Solution and Welfare Costs of Business Cycles in DSGE Models

Journal Pre-proof

Perturbation Solution and Welfare Costs of Business Cycles in DSGE Models Christopher Heiberger, Alfred Maußner PII: DOI: Reference:

S0165-1889(19)30214-3 https://doi.org/10.1016/j.jedc.2019.103819 DYNCON 103819

To appear in:

Journal of Economic Dynamics & Control

Received date: Revised date: Accepted date:

6 August 2019 20 November 2019 3 December 2019

Please cite this article as: Christopher Heiberger, Alfred Maußner, Perturbation Solution and Welfare Costs of Business Cycles in DSGE Models, Journal of Economic Dynamics & Control (2019), doi: https://doi.org/10.1016/j.jedc.2019.103819

This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Elsevier B.V. All rights reserved.

Perturbation Solution and Welfare Costs of Business Cycles in DSGE Models Christopher Heibergera,c,d and Alfred Maußnerb,c,d a

University of Augsburg, Department of Economics, Universitätsstraße 16, D-86159 Augsburg, Germany, [email protected] b Corresponding author, University of Augsburg, Department of Economics, Universitätsstraße 16, D-86159 Augsburg, Germany, [email protected] c We would like to thank two anonymous referees for helpful comments and suggestions. All remaining errors are ours. d We wish to confirm that there are no known conflicts of interest associated with this publication and there has been no significant financial support for this work that could have influenced its outcome.

This Version: November 14, 2019 JEL classification: C63, D60, E32 Keywords: Business cycles, Mean effect, Second order solution, Risk aversion, Welfare costs Abstract Lucas (1987) argues that the removal of cyclical fluctuations would barely improve economic welfare. He considers a risk-averse consumer valuing exogenously given streams 2 of consumption C t = C e g t e−σ /2+σε t driven by an iid sequence of draws ε t from a standard normal distribution. This allows him to change the size σ2 of the fluctuations without changing the mean of the process. In production economies, too, uncertainty is typically introduced in form of multiplicative, log-normally distributed shocks so that mean-preserving spreads can be analyzed in an analogous way. However, only few stochastic dynamic general equilibrium (DSGE) models admit an analytic solution. The most prevalent method to solve these models, perturbation methods, obtain an approximation to the stochastic model by perturbing the solution of the model’s deterministic counterpart with respect to the uncertainty parameter σ. Yet, widely available formulae to compute perturbation solutions are based only on the perturbation of σε t and do not adequately capture the additional deviation −σ2 /2 in the mean between the stochastic model and its deterministic version. We show within a model admitting an analytical solution that a second-order approximation of the welfare criterion also requires to perturb the mean. Thus, welfare measures based on the standard procedures for second-order solutions are (seriously) biased by a purely exogenous mean effect. We develop a general procedure of computing second-order accurate approximations of welfare gains or losses in the canonical DSGE model by extending the computation of second-order solutions pioneered by Schmitt-Grohé and Uribe (JEDC, 2004) to allow for mean preserving increases in the size of shocks. We apply our method to the model considered by Cho, Cooley, and Kim (RED, 2015) and show that different from the results reported by these authors removing the cycle is always welfare improving. Welfare measures computed from weighted residuals methods confirm the logic behind our perturbation approach and verify the accuracy of our estimates.

1 Introduction Would economic agents benefit, if the business cycle could be removed? And if so, by how much? Lucas (1987) started this discussion by arguing these benefits are negligible. He considers a representative, risk-averse consumer facing a stochastic consumption stream. The consumer values this stream according to an additively separable intertemporal utility function with iso-elastic period utility. The expected mean of the stream grows at a constant rate and the fluctuations around this trend match the variance of trend deviations of quarterly real U.S. consumption. For plausible values of the coefficient of relative risk-aversion (his Table 2 considers values between 1 and 20) he estimates a welfare gain not exceeding 0.1 percent of annual consumption, i.e., about $ 8.5 in 1983. Since then, many researchers have estimated the welfare costs of business cycles in more elaborate models.1 These include departures from the specification of preferences and of the consumption process (as, e.g., Obstfeld (1994), Dolmas (1998), Tallarini (2000), and Barro (2009)), models with uninsurable idiosyncratic risk (as, e.g.,˙Imrohoro˘ glu (1989), Krebs (2003), De Santis (2007), Heathcote et al. (2008), and Krusell et al. (2009)), the consideration of nominal frictions (as, e.g., Cho et al. (1997) and Galí et al. (2007)), endogenous growth (as, e.g., Barlevy (2004) and Heer and Maußner (2015)), and model uncertainty (Barillas et al. (2009)). The range of estimates is wide, from being close to Lucas’ 0.1 percent to several orders of magnitude beyond. For instance, ˙Imrohoro˘ glu (1989), p.1378 estimates 0.3 percent of average consumption, Krebs (2003), p. 862 finds 7.48 percent, Tallarini (2000), Table 3 calculates costs between 2.1 and 12.6 percent, and in the model of Barro (2009) the society would be willing to reduce GDP by about 20% to eliminate rare disasters. The focus of this paper is on welfare computations within dynamic stochastic general equilibrium (DSGE) models. This class of models encompasses real business cycle models as presented, e.g., in King, Plosser, and Rebelo (1988a,b) as well as New Keynesian monetary models in the spirit of Christiano, Eichenbaum, and Evans (2005) and Smets and Wouters (2003, 2007). Since its origins in the 1980s DSGE models have become the workhorse of macroeconomic research. The most basic source of uncertainty and the driving force of business cycles in these models are shocks to total factor productivity (TFP), say A t . Commonly, the log of TFP is assumed to be normally distributed, i.e. ln A t ∼ N (µ, σ2 ).2 Since σ2 provides a measure for the degree of uncertainty in the model, the most natural way to assess the effects of uncertain shocks on the welfare of economic agents is to compare the agents’ welfare for different values of σ2 . Moreover, an adequate estimate of the costs of risk should leave the mean of A t unchanged when varying σ2 , i.e. for an adequate estimate of the costs of business cycles the TFP shock must have the mean preserving spread (MPS) property. Since the mean of the log-normally distributed random variable A t is increasing in σ2 , i.e. E(A t ) = 2 eµ+σ /2 , a MPS analysis demands that µ = −σ2 /2. Different from the example of Lucas (1987), DSGE models usually admit no closed form 1 2

See Barlevy (2005) for a survey of the literature. The standard assumption is that log TFP follows an AR(1)-process, ln A t+1 = ρA ln A t + ε t+1 , ε t ∼ iid N (0, σε2 ), so that ln A t ∼ N (µ, σ2 ) with µ = 0 and σ2 =

2

σε2 . 1−ρA2

solution. The most wide-spread approximation technique for DSGE models are perturbation methods. From the perspective of the user, they are easy to apply and able to handle even large scale models. Perturbation methods start with the solution of the model’s deterministic version. The deterministic solution is subsequently perturbed with respect to the uncertainty parameter σ and an approximation to the stochastic model, which additionally captures the effects of σ up to an arbitrary order, is constructed. Formulae for perturbation solutions to DSGE models are widely available, see e.g. Schmitt-Grohé and Uribe (2004b), Gomme and Klein (2011), Andreasen (2012), and Binning (2013). However, the canonical framework on which they rest, assumes that only the variance of the model’s shocks depends on the perturbed parameter σ. While shocks with non-zero means are covered under this framework, the crux of MPS is not that the mean µ is non-zero but the fact that it is coupled to the perturbation parameter σ, e.g. by µ = −σ2 /2. When the effect of σ on the model solution is derived, e.g. by taking derivatives of the model’s equilibrium conditions with respect to σ, the effect of a simultaneously changing mean is neglected, i.e. the derivative dµ/dσ is not taken into account. The same is true for higher-order effects as well as for models with multiple shocks to other variables. Hence, the MPS property is not covered under the canonical framework and, to the best of our knowledge, no formulae which consider the additional effects are available. In consequence, welfare measures which rest on available formulae are (potentially seriously) biased. The contribution of our paper, therefore, is first and foremost methodological. We extend the canonical DSGE model of Schmitt-Grohé and Uribe (2004b) to allow for stochastic processes that have the MPS property. For this extended model we derive the second-order perturbation solution and provide Matlab code that implements this solution.3 Our solution appropriately adjusts the level effect of the perturbation parameter. Thus, it also shifts the agent’s value function required for welfare comparisons. We distinguish between conditional and unconditional welfare measures. The former compares two economies, A and B, say, which start at the same economic state. Economy A remains in this state forever while economy B is hit by exogenous shocks that drive the business cycle. The unconditional measure integrates out the effect of a specific starting point. Even when the model solution is computed accurately under the MPS property, the total effect of shocks on the welfare of economic agents within a production economy can be decomposed into two conceptually different components. Cho, Cooley, and Kim (2015) argue that on the one hand, risk-averse agents, having concave utility functions, dislike fluctuations: according to Jensen’s inequality the expected utility of a lottery does not exceed the utility obtained from the expected outcome of the lottery. Therefore, these agents will benefit, if the lottery is replaced by a certain stream of consumption and leisure equal to expected consumption and leisure from the lottery. On the other hand, in a production economy, expected consumption and leisure are not exogenously given even when the mean of shocks is fixed to a specified level. For example, in a two-country model with perfect capital markets and where only the TFP of the domestic country is subject to shocks, a social planner can optimally shift capital between the two countries in response to shocks to domestic TFP. In consequence, one can expect that in a world with shocks mean total output exceeds 3

This code is available from the authors upon request.

3

mean total output from a world without shocks even when TFP shocks have the MPS property. Hence, in production economies uncertainty may produce an additional, endogenous mean effect from the optimal adjustment of agents and this mean effect may very well be positive.4 Yet, with the available formulae for perturbation solutions which lack correction for the MPS, this mean effects will be systematically overestimated so that the costs of business cycles are systematically underestimated. We apply our techniques to a standard real business cycle model and its extension to a one-good, two-country model with perfect international capital markets. We calibrate the first model as in Cho, Cooley, and Kim (2015) so that we can compare our results to those presented in their Figure 2.5 We show that overestimation of the mean effect under the available standard perturbation solution is misleading and suggests positive welfare effects from business cycles, whereas removing the business cycle turns out always beneficial under our proposed solution. We confirm the logic behind our method both analytically and numerically. Our analytic derivation is limited to a toy model that admits a closed form solution. Numerically we compare the results from our perturbation method with the results obtained from two different weighted residuals methods where the MPS can be implemented trivially. In the two-country model, the extended potential for risk-sharing implies negative welfare effects of removing the cycle for low degrees of risk-aversion. However, the standard perturbation solution still systematically underestimates the welfare costs of uncertainty while the accuracy of our proposed method is again confirmed by the weighted residuals methods. From here we proceed with a sketch of the toy model in Section 2. In addition, we briefly describe the benchmark real business cycle model and its two-country extension. These models serve as examples of the canonical DSGE model presented in Section 3.1 and as a framework to illustrate the computation of conditional and unconditional welfare measures in Section 3.2. In Section 3.3 we sketch the solution of the one-country model via two weighted residuals methods. Section 4 presents our quantitative results. In particular, we provide conditional and unconditional welfare gains from removing fluctuations from the models of Section 2. Section 5 concludes. The Appendix covers additional material. In particular, it presents the full sets of equations behind our models, covers the derivation of our second-order perturbation solution, describes in more detail our weighted residuals methods, and supplies the tables that underly our graphical presentation of the results. 4

A similar trade-off between the costs of uncertainty and aggregate productivity is present in the heterogenous agent economy studied by Heathcote et al. (2008), where increasing wage dispersion raises aggregate productivity. 5 Cho, Cooley, and Kim (2015) present their results in terms of graphs that display the welfare measure as a function of the coefficient of risk-aversion and of the variance of the TFP shock.

4

2 Models 2.1 A toy model We will illustrate our approach and the related concepts with the following toy model. A representative agent chooses consumption c t and hours n t to maximize6 Vt := E t

∞ X

βs

s=0

subject to

u(c t+s − ωn t+s )1−η , β ∈ (0, 1), ω > 0, η ∈ R>0 /{1} 1−η

c t = A t n1−θ , z t := ln A t , z t iid N (− 21 τ2 , τ2 ), θ ∈ (0, 1). t

Note the the process for total factor productivity A t has the MPS property, since E(A t ) = 1 for all τ ≥ 0. The analytic solution for expected life-time utility conditional on the realization of z t is given by  ‹ (1−θ )(1−η)  ‹ θ (1−η−θ )(1−η)τ2 1−η β θ 1−η 1 − θ zt 2 θ 2θ + e e . Vt = 1−η ω 1−β

(1)

Unconditional expected life-time utility, therefore, is equal to u

V := E(Vt ) =

 ‹ (1−θ )(1−η) θ (1−η−θ )(1−η)τ2 1−θ θ 1−η 2θ 2 e . (1 − η)(1 − β) ω

Note that both Vt and V u are decreasing in τ iff θ + η > 1. Otherwise the endogenous mean effect dominates the effect of uncertainty. It is straight forward to compute the second-order approximation of (1) at the point (z t , τ) = (0, 0). The result is Bβ −η−1 Bθ 1−η + Bθ −η z t + B(1 − η)θ −η−1 z t2 + θ (1 − η − θ )τ2 , (1 − η)(1 − β) 1−⠁ ‹ (1−θ )(1−η) θ 1−θ , B= ω

Vt '

(2)

since both Vτ (0, 0) and Vzτ (0, 0) are equal to zero. The perturbation approach of Schmitt-Grohé and Uribe (2004b) approximates the function Vt by a quadratic in the state variable z t and the perturbation parameter σ, where σ = 0 indicates the deterministic model and σ = 1 the stochastic model. The respective formula is:  ‹• ˜ V Vzσ z t V (z t , σ) ' V (0, 0) + Vz z t + Vσ σ + [z t , σ] zz , Vσz Vσσ σ 6

We also treat the case η = 1. See Appendix A for this case and the detailed derivation of all results presented in this subsection.

5

where all partial derivatives are evaluated at the point (z t , σ) = (0, 0). We will now demonstrate that we will get the solution in (2) from this approach only, if we also perturb the mean of the process for the log of the TFP shock. Let z t := µ(σ) + στε t , ε t iid N (0, 1), µ(σ) = − 12 (στ)2

so that the mean µ of z t is a function of the perturbation parameter. We can then derive the partial derivatives from the recursive definition  ‹ (1−θ )(1−η) θ 1−η θ 1−η 1 − θ V (z, σ) := e θ z t + βE t V (µ(σ) + στε t+1 , σ). {z } | 1−η ω =z t+1

We focus on Vσσ and relegate the computation of the other coefficients to Appendix A. Differentiating V (z, σ) twice and evaluating the result at the point σ = 0, and, hence z t = 0 yields   Vσσ (0, 0) = β µ00 (0)Vz (0, 0) + (µ0 (0)2 + τ2 )Vzz (0, 0) + Vσσ (0, 0) . (3) Using µ00 (0) = −τ2

(4)

and solving for Vσσ finally yields the fourth term on the right-hand side of (2). The standard approach, however, ignores (4) and sets this term equal to zero so that (3) yields )(1−η) ‹ (1−θ1−η  β −η−1 1 − θ (1 − η)τ2 , θ Vσσ (0, 0) = 1−β ω

which obviously differs from the correct solution presented in (2). 2.2 One-country model We consider a standard real business cycle model similar to the one presented by Hansen (1985). Our nomenclature and parameter choices follows Cho, Cooley, and Kim (2015). Output y t is produced from capital k t and labor n t according to the production function y t = A t kθt n1−θ . t

(5)

The natural log of total factor productivity, z t := ln A t , is governed by an AR(1)-process  ‹ τ2 z t+1 = ρz t + ε t+1 , ε t ∼ iid N − , τ2 . (6) 2(1 + ρ) The innovations in the process ε t are iid normal. Its mean is chosen so that the unconditional expectation of A t is equal to unity and independent of the standard deviation τ.The

6

current-period utility function u is of the Cobb-Douglas type and depends on consumption c t and leisure 1 − n t : ( 1   α 1−α 1−η for η 6= 1, 1−η c t (1 − n t ) u(c t , 1 − n t ) := (7) α ln c t + (1 − α) ln(1 − n t ) for η = 1.

Thus, for given values of α ∈ (0, 1) the coefficient of relative risk aversion, CRRA := −

∂ 2 u/∂ c 2 c = 1 + α(η − 1), ∂ u/∂ c

increases with η. Capital depreciates at the constant rate δ ∈ (0, 1], and the representative agent discounts future utilities at the rate β ∈ (0, 1). This agent chooses consumption, leisure, and the future stock of capital to maximize the expected life-time utility ¨∞ « X s Vt := E t β u(c t+s , 1 − n t+s ) , β ∈ (0, 1) s=0

subject to the resource constraints y t+s ≥ c t+s + i t+s ,

k t+s+1 = (1 − δ)k t+s + i t+s ,

 

 0 ≤ c t+s , k t+s+1 , and n t+s ∈ (0, 1)

for s = 0, 1, . . .

and a given initial stock of capital k t . Expectations E t are conditioned on information available at time t . We provide the equilibrium conditions of this model in Appendix B. Our numeric results in Section 4 employ the parameter values presented in Table 1. Table 1: Calibration of the one-country model

Parameter Description α β δ θ ρ

Value

utility weight of consumption discount factor rate of capital depreciation capital share in production autocorrelation of log of TFP shock

0.35 0.99 0.025 0.36 0.95

2.3 Two-country model Vis-à-vis autarky the representative households living in a two-country world with perfect capital markets have more opportunities for risk sharing. One would, therefore, expect that

7

for small degrees of risk-aversion the removal of TFP shocks would decrease welfare. To test this prediction we will apply our method to a standard international business cycle model sketched next. Let the index j ∈ {d, f } denote variables of the domestic (foreign) country. The allocation of resources solves the problem max Vt := E t

∞ X

βs

1



1 2 u(cd t+s , nd t+s ) + 2 u(c f t+s , n f t+s )

s=0

subject to the constraints y j t+s = A j t+s kθj t+s n1−θ , j t+s

j ∈ {d, f },

k = (1 − δ)k j t+s + Ψ(i j t+s /k j t+s )k j t+s , j ∈ {d, f } X j t+s+1 X X y j t+s ≥ c j t+s + i j t+s ,

j∈{d, f }

j∈{d, f }

j∈{d, f }

0 ≤ c j t+s , k j t+s+1 , and n j t+s ∈ (0, 1),

j ∈ {d, f },

z t+s+1 = Rz t+s + ε t+s+1 , s = 0, 1, . . . , • ˜ • ˜ • ˜ z ln(Ad t ) ε z t := d t := , ε t := d t . zf t εf t ln(A f t )

For both countries we parameterize the current-period utility function u(·) as in (7) and the adjustment cost function Ψ(x) as in Jermann (1998): a1 1−ζ x + a2 . Ψ(x) := 1−ζ

We choose the parameters a1 and a2 so that Ψ(δ) = δ and Ψ 0 (δ) = 1. This ensures that adjustment costs have no impact on the stationary equilibrium of the model. Accordingly, in each country the stationary allocation of resources is the same as in the one-country model. The interested reader will find the equilibrium conditions of this model in Appendix C. We consider two specifications of the shock process. The first one assumes that shocks originate only in the home country so that ‹  τ2 , τ2 and z f t+s ≡ 0∀s = 0, 1, . . . . zd t+1 = ρzd t + εd t+1 , εd t ∼ iid N − (8) 2(1 + ρ)

If, in addition, we set ζ = 0, the model reduces to the two-country model of Cho, Cooley, and Kim (2015). The second case follows the literature on international business cycles and assumes • ˜ • ˜ • ˜ • ˜ ρ 0 εd t+1 εd t ρε 2 1 : z t+1 = z + , ∼ iid N (µε , Σε ) , Σε = τ , (9) 0 ρ t ε f t+1 εf t ρε 1 and where the specification of µε follows from the MPS property E[ezd t ] = E[ez f t ] = 1 as will be laid out in Section 3. We set ρ to the value given in Table 2 and for the correlation between the innovations we follow Dmitriev and Roberts (2012), Table 1 and choose ρε = 0.25.

8

As these authors, we choose the value of the parameter ζ so that for η = 2 and τ = 0.007 the model implies a standard deviation of domestic investment relative to the standard deviation of domestic output equal to 2.88. Table 2 summarizes our calibration. Table 2: Calibration of the two-country model

Parameter Description α β δ θ ζ ρ ρε

Value

utility weight of consumption discount factor rate of capital depreciation capital share in production elasticity of i/k wrt Tobin’s q autocorrelation of log of TFP shock correlation of TFP innovations

0.35 0.99 0.025 0.36 0.08 0.95 0.25

3 Second-order approximations of welfare effects in DSGE models This section draws on the work of Stephanie Schmitt-Grohé and Martin Uribe. In SchmittGrohé and Uribe (2004b) they propose a canonical framework for DSGE models and develop formulas for a second-order approximate solution of this class of models. In Schmitt-Grohé and Uribe (2006, 2007) they develop measures for welfare comparisons. 3.1 Canonical DSGE model Variables and equilibrium conditions. Let x t ∈ Rn(x) denote a vector of n(x) endogenous state variables, i.e., variables whose period t value is given at the beginning of period t but evolves endogenously from period to period. The vector y t ∈ Rn( y) collects n( y) endogenous variables being determined within period t . The driving forces of the model are n(z) purely exogenous variables gathered in the vector z t ∈ Rn(z) . The equilibrium conditions are given by 0(n(x)+n( y))×1 = E t g (x t+1 , z t+1 , y t+1 , x t , z t , y t ) ,

(10a)

where E t denotes mathematical expectations as of time t and g : R2(n(x)+n( y)+n(z)) → Rn(x)+n( y) .  T We denote the vector of all state variables by w t := x Tt z Tt ∈ Rn(w) where n(w) = n(x) + n(z). Driving process. Schmitt-Grohé and Uribe (2004b) specify the driving process of the exogenous variables as z t+1 = Rz t + σΩν t+1 .

9

The matrix R has all its eigenvalues within the unit circle. The parameter σ ≥ 0 is an arbitrary scalar factored out from the matrix Ω and plays the role of the perturbation parameter. Without loss of generality we will fix σ = 1 for the stochastic model in the following. The vector ν t+1 ∈ Rn(z) is distributed independently and identically (iid) with mean 0n(z)×1 and variance I n(z) .7 The matrix Ω determines the covariance of the innovations ε t+1 := σΩν t+1 ,

since E t (ε t+1 ε Tt+1 ) = σ2 ΩΩ T . For σ = 0 the process becomes deterministic and approaches the zero vector: lim t→∞ z t = 0n(z)×1 . If the deterministic model has a stable solution at the  T point x T 01×n(z) y T solving
0(n(x)+n( y))×1 = g x, 0n(z)×1 , y, x, 0n(z)×1 , y ,

one can invoke the implicit function theorem to approximate the solution for a nearby8 stochastic model σ = 1. To motivate our extension let us revert for the moment to the one-shock model in Section 2.2, where z t = [ln(A t )]. Note, if we assume ε t+1 ∼ iid N (−τ2 /(2(1 + ρ)), τ2 ), as in (6), an equivalent formulation is given by ε t+1 := −

τ2 ˜ + τν t+1 = µ(τ) + τν t+1 , ν t+1 ∼ iid N (0, 1), 2(1 + ρ) 2

τ ˜ where µ(τ) = − 2(1+ρ) . Setting further R := ρ and Ω := τ as well as σ = 1 for the stochastic model, the AR(1) specification (6) for log productivity implies that

(10b)

˜ (σΩ) + σΩν t+1 . z t+1 = Rz t + µ

Most importantly, note already here that this specification guarantees not only that the mean preserving spread property is met for the stochastic model with σ = 1, but also that it remains throughout valid when perturbing and taking derivatives with respect to σ. ˜ was non-zero but independent of σΩ, one could simply convert the previous Now, if µ system to
−1 ¯z t+1 = R¯z t + σΩν t+1 where ¯z t := z t − I n(z) − R µ ˜, (11)

being equivalent to the process of Schmitt-Grohé and Uribe (2004b). Since the required transformation would then be independent of σΩ and therefore in particular of the perturbation parameter σ, the second-order approximation derived in Schmitt-Grohé and Uribe (2004b) remains valid. ˜ (σΩ) is non-constant as it is the case in the problem at hand. However, this is different if µ In order to adequately account for the effect of uncertainty in the model, i.e. in order to adequately cancel out any shift of the mean in productivity in the perturbation approach, the 7 8

I n denotes the identity matrix of dimension n. I.e. if the elements on the diagonal of ΩΩ T are not too large.

10

variance-covariance matrix and the mean vector in (10b) have to be perturbed simultaneously with σ. To stress this point even further, note the following. If we held fixed µ = −τ2 /2(1+ρ) for different values of τ in the model of Section 2.2, we would compare different deterministic stationary states: in the transformed model, ln A t approaches µ/(1 − ρ) = −τ2 /(2(1 − ρ 2 )) so that A = exp{µ/(1 − ρ)} is decreasing in τ. Only if we let µ continuously vary with the perturbation parameter σ, can we secure not to jump from one deterministic solution to another. Consequently, the second-order Taylor approximation for the stochastic model constructed around σ = 0 differs from the standard results in Schmitt-Grohé and Uribe (2004b), if the ˜ gradient or Hessian matrix of µ(σ) := µ(σΩ) at σ = 0 are non-trivial. Obviously, for the σ2 τ2 we get present case where µ(σ) = − 2(1+ρ) τ2 σ 0 = 0, µ (0) = − 1 + ρ σ=0 τ2 µ00 (0) = − < 0. (1 + ρ) This motivates our extension of the shock process as part of the canonical DSGE model:
0n(z)×1 = z t+1 − Rz t − µ(σ) − σΩν t+1 , ν t+1 i.i.d. N 0n(z)×1 , I n(z) , (10c) µ(σ = 0) = 0n(z)×1 , (10d) µσ (σ = 0) = 0n(z)×1 , (10e) µσσ (σ = 0) 6= 0n(z)×1 . (10f) The MPS property in the vector-valued case. (10c) is given by

The unconditional mean of the process

µz = (I n(z) − R)−1 µ(σ)

and its covariance matrix σ2 Σz solves the discrete Lyapunov equation9 σ2 Σz = Rσ2 Σz R T + σ2 Σε , Σε = ΩΩ T .

Therefore, each element zi t , i = 1, 2, . . . , n(z) of the vector z t is normally distributed with mean µzi and variance equal to the i th diagonal element of the matrix σ2 Σz , σ2 τii , say. Accordingly, a MPS property for all Zi := ezi , i = 1, . . . , n(z), demands that E(Zi ) = E(ezi ) = 2 eµzi +σ τii /2 = 1 and and therefore requires µzi = −σ2 τii /2. The proper specification of the mean vector µ(σ), therefore, is given by µ(σ) = − 21 σ2 (I n(z) − R)diag(Σz )

so that

µσσ = −(I n(z) − R)diag(Σz ),

where diag(Σz ) denotes the vector of diagonal elements of the matrix Σz . 9

See, e.g., Lütkepohl (2005), p. 27.

11

Approximate solution. tion of

The stationary point of the deterministic model is found as solu-


0(n(x)+n( y))×1 = g x, 0n(z)×1 , y, x, 0n(z)×1 , y .

The solution of the perturbed model are vector valued functions h x : Rn(x)+n(z)+1 → Rn(x) and h y : Rn(x)+n(z)+1 → Rn( y) given by x t+1 = h x (w t , σ),

(12a) (12b)

y

y t = h (w t , σ).

Schmitt-Grohé and Uribe (2004b) show how to compute a second-order approximation of  T these functions at the point of expansion x T , 01×n(z) , 0 , i.e. at the stationary point of the deterministic model. The form of this solution reads
x x (w t − w) + 12 Hσσ σ2 , (13a) h x (x t , z t , σ) = x + H wx (w t − w) + 21 I n(x) ⊗ (w t − w) T H ww
1 1 y y h y (x t , z t , σ) = y + H wy (w t − w) + 2 I n( y) ⊗ (w t − w) T H ww (w t − w) + 2 Hσσ σ2 . (13b)

Setting σ = 1 in these equations delivers the solution of the stochastic model. The Matlab code provided by Schmitt-Grohé and Uribe (2004b) for the computation of x y x y the matrices H ww , H ww , Hσσ , and Hσσ rests on tensor formulas. In the Appendix D we apply a chain rule for the second derivative of a vector-valued composite function as proposed by Gomme and Klein (2011) and extend the computation to the case of perturbed means as specified in (10c)-(10f). The results are summarized in the following proposition. Proposition 1 For the extended canonical DSGE model with shock process as specified in (10c)-(10f): 1. the first-order solution H wx , H wy and, in particular, Hσx = 0, Hσy = 0, remain unchanged compared to the standard solution. y x remain unchanged compared to the standard solu, H ww 2. the second-order terms H ww tion. y x and Hσσ are determined as the solution of the linear system 3. the second-order terms Hσσ





I n(x)+n( y) ⊗ N T gss N ΩΩ T − g y 0 trm I n( y) ⊗ (ΩΩ T ) Hzzy
− gz 0 + g y 0 Hzy µσσ   •H y ˜ σσ , = g y + g y 0 g x 0 + g y 0 H xy x Hσσ   0n(x)×n(z) • y ˜  y y H x x Hzyx I n(z)   y y where N =  , H = H , H , H = .  y w x z ww Hzy H xz Hzzy 0[n(w)+n( y)]×n(z) − trm

12

(14)

and where gi , i ∈ {x 0 , y 0 , z 0 , y, z} denotes the Jacobian matrix of g(x0 , z0 , y0 , x, z, y) with respect to i , gss is the Hessian matrix of g(·), and trm(·) denotes the matrix trace operator. y Since the solutions for the matrices H wy and H ww are not affected from our extension, this system differs from the system in equation (51) on p. 610 of Gomme and Klein (2011) only by the term in the second line. This term distinguishes the standard solution from our extended version of the canonical DSGE model in (10).

3.2 Welfare measures Conditional measure. Suppose we have two different equilibrium time paths i ∈ {a, r} of the model of Section 2.2.10 Let « ¨∞ X 1 α(1−η) (1−α)(1−η) s (1 − ni t+s ) (15) Vi t := E t c β 1 − η i t+s s=0

denote the associated life-time utility, and note that it is conditioned on the given initial capital stock k t and the known initial realization of the TFP shock ln A t . Let i = r denote our reference equilibrium to which we want to compare an alternative equilibrium i = a. Schmitt-Grohé and Uribe (2004a), p. 17f define the welfare measure as the fraction of consumption which the representative agent would be willing to forgo in equilibrium r to be equally well-off as in the alternative equilibrium a. Here we follow Cho, Cooley, and Kim (2015) and define λc as the fraction of consumption which has to be given to the representative agent in equilibrium r in order to be as well-off as in equilibrium a, i.e., « ¨∞ X 1 c α(1−η) (1−α)(1−η) s [(1 + λ )c r t+s ] (1 − n r t+s ) = (1 + λc )α(1−η) Vr t . Vat = E t β 1 − η s=0 Solving for λc yields 1 • ˜ α(1−η) Vat c λ = − 1. Vr t

(16)

The superscript c shall remind the reader that this measure is conditioned on the initial point (k t , ln A t ). In order to evaluate λc , we will use the second-order approximation of Vi t at the stationary solution: Vi Vi Vi t ' Vi + H wVi (w t − w) + 12 (w t − w) T H ww (w t − w) + 21 Hσσ .

(17)

To get this solution, we must add an equation for the variable Vi t to the system (10) and define an initial condition (w t − w). The first requirement is easily met, because the infinite sum (15) has a recursive representation: 1 α(1−η) Vi t = (18) c (1 − n t )(1−α)(1−η) + βE t Vi t+1 . 1−η t 10

The extension to the two-country model of Section 2.3 follows in an obvious manner so that we add only remarks where necessary.

13

The answer to the second question derives from our ultimate goal. We want to compare an economy without uncertainty and, thus, without TFP shocks to one with a given amount of risk as specified by the choice of τ2 (or τii in the vector-valued cases). Therefore, we may assume that both economies start at the stationary solution and that the one without shocks stays there for ever. This gives Vr =

1 c α(1−η) (1 − n)(1−α)(1−η) (1 − β)(1 − η)

for the reference solution. The second-order approximation of Vat follows from (13b) for w t − w = 0n(w) and is given by Va Vat ' Vr + 12 Hσσ ,

Va y where Hσσ is the element of the vector Hσσ which refers to the variable Vat . Note however, that the MPS property was constructed for the unconditional distribution of the model’s shocks, i.e. so that E[ezi t ] = 0 for all i = 1, . . . , n(z) and t ∈ N. Yet, if the state in which the economy starts in period t is specified, a construction of a MPS for the conditional distribution is in general not possible. Consider, for example, the univariate case

z t+s+1 = ρz t+s + µ + τν t+s+1 , ν t+s+1 ∼ iid N (0, 1).

The conditional mean and variance of z t+s are given by E t [z t+s ] = ρ s z t + µ

so that E t [e

z t+s

1 − ρ 2s 1 − ρs and var[z t+s ] = τ2 1−ρ 1 − ρ2



 2s 1 − ρs 2 1−ρ ] = exp ρ z t + µ +τ . 1−ρ 2(1 − ρ 2 ) s

Dependence on the step ahead s obstructs any specification µ that yields E t [ezt+s ] = 1 for all s ∈ N. In consequence, unconditional welfare measures, except for the case of uncorrelated shocks ρ = 0, suffer from a bias. Unconditional measure. Schmitt-Grohé and Uribe (2006) also develop a measure of unconditional welfare gains or losses. We start by integrating both sides of equation (15) with respect to the distribution of the state variables w t . This gives ™« ¨ –∞ X 1 α(1−η) ci t+s (1 − ni t+s )(1−α)(1−η) , (19) Viu := E E t βs 1 − η s=0

where E denotes unconditional expectations. As before, let λu (u for unconditional) denote the fraction of consumption c r t+s the representative household would require to be as welloff in equilibrium r as in equilibrium a: ¨ –∞ ™« X 1 α(1−η) Vau = E E t βs ((1 + λu )c r t+s ) (1 − n r t+s )(1−α)(1−η) , 1 − η s=0 = (1 + λu )α(1−η) Vru .

14

Accordingly, λu is given by  u

λ =

Vau Vru

1  α(1−η)

(20)

− 1.

In order to obtain the unconditional moments, we take unconditional expectations E on both sides of equation (13b):   y y ¯ Tt H ww ¯ t + 12 Hσσ ¯ t + 12 E I n( y) ⊗ w w , E¯ y t = H wy Ew

¯ t := where the bar denotes deviations from the stationary solution, i.e., y¯ t := y t − y and w w t − w. Employing the trace operator to the second term on the right-hand side of this equation and letting Γ w denote the covariance matrix of the states gives   y w  1 Γ tr H ww . 1  + 1Hy . .. ¯t + 2 (21) E¯ y t = H wy Ew 2 σσ yn( y) w tr {H Γ }

The unconditional expectation of the states follows from stacking equations (13a) and (10c),   w ˜ w ˜w + ν ˜ ww ¯ t+1 = H ¯ Tt+1 H ¯ + 12 H ¯ + 12 I n(w) ⊗ w ˜ t+1 , w ww t σσ w t • ˜ • ˜ • ˜ x H wx H ww 0n(x)×1 ˜ w := ˜ w := : ˜ H , H , ν = , t+1 w ww 0n(z)×n(x) R 0n(z)n(w)×n(w) µ + Ων t+1

where µ = µ(σ = 1) and taking expectations on both sides. The result is a linear system in ¯t the unknown vector Ew   w w  ˜ 1Γ tr H ww   .. 1  + 1H ˜w ˜ w Ew ¯ (22) I n(w) − H = E˜ ν + t t+1 2 2 σσ . w  w.n(w) w ˜ ww Γ tr H In the last step we determine the covariance matrix Γ w from the linearized solution for the vector of states: ˜ ww ¯ t+1 = H ¯ +ν ˜ t+1 . w w t

The matrix Γ w solves the discrete Lyapunov equation11 • ˜
˜ wΓ w H ˜ w T + Σν˜ , Σν˜ = 0n(x)×n(x) 0n(x)×n(z) Γw = H . w w 0n(z)×n(x) ΩΩ T

(23)

Note that the Matlab program of Schmitt-Grohé and Uribe (2004b) does not consider the term E˜ ν t+1 = [01×n(x) , µ T ] T , so that the unconditional measure λu is biased by i) disregard˜w . ing this effect and by ii) disregarding the effect of µσσ on the vector H σσ 11

For the derivation of this equation see, e.g., Hamilton (1994), p. 264f or Lütkepohl (2005) p. 26f.

15

Mean and Fluctuations Effect. Here we follow Cho, Cooley, and Kim (2015) and decompose the welfare effect into two components. First, even under a MPS where the mean of exogenous shocks is held constant, the mean of the endogenous variables may vary with the standard deviations of shocks. This mean effect ωm reflects the optimal response of labor supply and capital accumulation to the model’s shock(s). It is defined as the fraction of consumption required by agents living in the stationary environment to be equally well-off as agents who enjoy the expected value of consumption and leisure obtained from living in the stochastic environment. Let ˜c and n˜ denote the expected values of consumption and hours obtained from the solution of equation (21). Then, the mean effect is defined by (1 + ωm )α(1−η) Vr = V˜ :=

∞ X

βs

s=0

1 α(1−η) ˜c ˜)(1−α)(1−η) . (1 − n 1−η

(24)

The mean effect may very well be positive and increasing in the standard deviations of the shocks. The second component ωu , the fluctuations effect, captures risk-aversion. Accordingly, it solves the equation Vau = (1 + ω f )α(1−η) V˜

(25)

so that agents in the stochastic environment enjoy the same unconditional expected lifetime utility as those agents being provided with a steady stream of consumption equal to ˜ hours. The fluctuations effect, thus, will (1 + ω f )˜c and working a constant fraction of n always be negative for risk-averse agents. Combining Vau = (1 + λu )α(1−η) Vr

with the previous two equations gives (1 + ωm )(1 + ω f ) = (1 + λu ),

so that the welfare effect λu is approximately equal to the sum of the mean and the fluctuations effect. 3.3 Mean Weighted Residuals Methods In order to check the logic underlying our assumptions in (10c)-(10f) and the accuracy of our perturbation solution in Proposition 1, we also solve the model with two variants of a mean weighted residuals method (see Appendix E for a detailed description). In particular, we use a finite element method (FEM)12 and a Chebyshev-Galerkin method (CGM)13 to approximate the solutions n t = h(k t , z t ) for hours and Vt = V (k t , z t ) for the value function in the one country model, or nd t = hd (kd t , k f t , zd t , z f t ) for domestic hours, n f t = h f (kd t , k f t , zd t , z f t ) 12 13

See, e.g., McGrattan (1995). See, e.g., Judd (1992) and for a textbook presentation Heer and Maußner (2009), Chapter 6.

16

for foreign hours and Vt = V (kd t , k f t , zd t , z f t ) for the value function in the two country model. In both of these methods it is easy to implement the MPS assumption on the mean of the innovations in equation (6) and (8) or (9), respectively. To see this, note that both methods require integration over the probability density function of the variable ε t+1 in order to evaluate the residual of two, or three in case of the two country model, functional equations. These equations are the Euler equation for consumption and the recursive definition of the value function, equations (B.1f) and (B.1g) in Appendix B and (C.1.12), (C.1.13) and (C.1.14) in Appendix C, respectively. For instance, the residual function for the former condition in the one-country model is given by (see equation (E.7)): ! Z τ2 ε t+1 + 2(1+ρ) 1 rhs1 (k t , z t , ε t+1 , h) p exp − dε t+1 , R1 (k t , z t , h) := 1 − 2τ2 2πτ2 R  ‹ λ t+1 y t+1 : rhs1 (k t , z t , ε t+1 , h) = β θ + 1 − δ ≡ 0, λt k t+1 where λ t is the marginal utility of consumption which can be, as all other variables, derived analytically from the policy n t = h(k t , z t ). In the two-country model for j ∈ {d, f }, Z j j R1 (kd t , k f t , zd t , z f t , hd , h f ) := q j t − rhs1 (kd t , k f t , zd t , z f t , ε t+1 , hd , h f ) R2  ‹ 1 1 T −1 exp − (ε t+1 − µε ) Σε (ε t+1 − µε ) dε t+1 ≡ 0, p 2 2π det(Σε )   € i j t+1 Š y j t+1 i j t+1 λ t+1 j ) , rhs1 (kd t , k f t , zd t , z f t , ε t+1 , hd , h f ) := β θ − + q j t+1 (1 − δ + Ψ λt k j t+1 k j t+1 k j t+1 We evaluate this expectation by means of the Gauss-Hermite quadrature. The finite element method employs a grid Γ ⊂ R2 or Γ ⊂ R4 , respectively, over the space of the state variables. Between the grid points cubic C 2 splines approximate the respective function. We determine the function values at the grid points such that the residuals from the two (or three in the case of the two-country model) functional equations vanish (see (E.8) for the definition of the residual of the value function). The Chebyshev-Galerkin method employs a product base of Chebyshev polynomials to approximate the solution for hours and the value function. For instance, in the one country model, the solution for hours is approximated by ˆh(k t , z t ) :=

d1 X d2 X

φi, j Ti−1 (ψ1 (k t ))T j−1 (ψ2 (z t )),

i=1 j=1

where Tl denotes the l -th order Chebychev polynomial, d1 , d2 ∈ N are the degrees of approximation, and ψi , i = 1, 2 are bijections between [k, ¯k] and [z, z¯], respectively, into the domain of Chebyshev polynomials [−1, 1]. The d1 d2 coefficients φi, j solve a system of nonlinear equations which requires that weighted sums of the residuals vanish.

17

4 Results In this section we present the welfare gains and losses computed for the models from Section 2 and discuss the accuracy of these measures. Our results rest on the parameter values presented in Table 1. In order to study the effects of different degrees of risk-aversion and uncertainty, we follow Cho, Cooley, and Kim (2015) and solve the model for ten different values of the parameter η and five different values of the parameter τ and report the mea˜ := λ(c/ y), where c and y denote, sures in percent of income, i.e., instead of λ we report λ respectively, the stationary value of consumption and income. 4.1 One-country model Our results for the one-country model are summarized in Table F.1 in Appendix F and illustrated in Figure 1. Welfare measures. First, consider Figure 1. In its panel (a) it displays the conditional and in its panel (b) the unconditional welfare measures from our extension of the perturbation solution for models with the MPS property. Panels (c) and (d) show the results from the available standard perturbation formulae and are close to the results reported in Figure 2 of Cho, Cooley, and Kim (2015) (henceforth CCK). Finally, panel (e) displays the conditional measure from our Chebshev-Galerkin solution (CGM) while panel (f) shows the unconditional measure from our finite elements solution (FEM). First, note that both the conditional and the unconditional measures from our method indicate welfare gains from removing the business cycle for all combinations of risk-aversion and the amount of risk as parameterized by η and τ, respectively. Hence, the fluctuations effect always dominates the mean effect, so that the representative agent living in the stationary and certain environment would not want to live in the stochastic economy, i.e., ˜ i < 0. λ The results from our perturbation method are confirmed by the weighted residual methods pictured in panels (e) and (f). In particular, both methods yield negligible differences for the conditional welfare measure. The unconditional measures introduce additional sources of errors. These additional errors can not be ascribed to the method used to approximate the model solution but result from the fact that further approximations must be made in the subsequent computation of unconditional moments given the approximated model solution. First, computation of the unconditional moments from the perturbation solution in (22) involves the covariance matrix Γ ω of state variables. However, Γ ω is computed only from the linearized model solution in (23). Second, the unconditional moments from the solution of weighted residual methods can only be derived by Monte-Carlo simulations and therefore are subject to simulation error. Despite these additional errors which are not touched by our extended perturbation solution, a comparison of panels (b) and (f) shows that unconditional welfare measures from both methods are still sufficiently in line. However, the results are quite different if one uses the available standard perturbation solution. In panels (c) and (d) we show the results if we only adjust the TFP shock according to equation (11) and, hence, ignore both the effect of µσσ in equation (14) and the non-zero

18

Figure 1: Welfare measures (a) Conditional measure pert-MPS

(b) Unconditional measure pert-MPS

0

0

-0.05

-0.05

-0.1 -0.1

-0.15 -0.15

-0.2

-0.2 -0.25

-0.25 -0.3

-0.3

-0.35

-0.4 1

2

3

4

5

6

7

8

9

10

-0.35 1

(c) Conditional measure pert

2

3

4

5

6

7

8

9

10

8

9

10

(d) Unconditional measure pert

0.15

0.2

0.1 0.15

0.05

0.1 0

-0.05

0.05

-0.1 0

-0.15

-0.05 -0.2

-0.25 1

2

3

4

5

6

7

8

9

10

-0.1 1

(e) Conditional measure CGM

2

3

4

5

6

7

(f) Unconditional measure FEM

0

0

-0.05

-0.05 -0.1

-0.15 -0.1

-0.2

-0.15 -0.25

-0.3 -0.2

-0.35

-0.4 1

2

3

4

5

6

7

8

9

10

19

-0.25 1

2

3

4

5

6

7

8

9

10

mean µ in equation (22). The graphs show that both measures are positive for values of η smaller than (about) 4 (Panel (c)) and 5.5 (Panel (d)). In Figure 2 of CCK the graphs cut the abscissa at a value of η slightly larger than 5. In addition, the starting and endpoints of each graph in Panel (d) are close to those in Figure 2 of CCK, so that we suspect that their Figure 2 reports the unconditional welfare measure from the standard perturbation solution with mean adjustment as in equation (11). Thus, as a first result, we note that a proper implementation of the MPS property requires to perturb both the mean and the variance of the innovations. Otherwise one overstates the mean relative to the fluctuations effect. This can be seen by comparing the lines labeled ωm and ω0m in Table F.1 in Appendix F. They show, respectively, the mean effect computed from our proposed method and the mean effect, if we disregard the effect of µσσ on Hσσ . The latter overstates the former by at least 56.7 percent and at most by 111.7 percent. Let us turn now to the size of the effects. As is to be expected, the welfare loss increases with increasing risk-aversion, measured by the parameter η, and with increasing uncertainty, measured by the parameter τ. The welfare gains from removing fluctuation are small. According to Table F.1, they range from approximately 0.001 percent to 0.36 percent of income for the conditional measure and from 0.0011 percent to 0.31 percent for the unconditional measure. For the latter and in terms of U.S. real per capita income in 2018 of about 57 thousand Dollars (see https://fred.stlouisfed.org) the representative household would be willing to give up between 63 cents and 176.5 Dollars of annual consumption in order to stay within the certain environment. For the common choice of τ = 0.007 and an intermediate value of η = 5 the figure would be 15 Dollars of annual consumption. This is not too far from Lucas’s $ 8.5. Thus, as a second result, we confirm CCK and find small welfare gains from removing the cycle in the benchmark real business cycle model for plausible values of the risk-aversion parameter and the standard deviation of the TFP shock. However, the discrepancy between ˜ i and the standard solution λ ˜ i is large. The percentage deviation our extended solution λ e s i i i i ˜ −λ ˜ )/λ ˜ | lies between 41.8 percent and almost 404 percent for the conditional ∆ := |(λ s e e welfare measure (i = c ) and between 68.6 percent and 475 percent for the unconditional measure (i = u). ˜ m in Table F.1 report the size of the mean effect as computed from The lines labeled ω equation (24). The size of this effect is increasing in both the risk aversion parameter η and the standard deviation τ and ranges between 0.05 percent of income and 0.38 percent. Relative to the absolute value of the overall effect, the mean effect is large for η = 1 (log preferences) and about 4.25 times the size of the overall effect, irrespective of the value of τ. For η = 2 it is still about twice the size of the overall effect. For the remaining values of η between 3 and 10 the mean effect is between 1.6 and 1.17 of the size of the overall effect. Accuracy. In order to add further proof for the accuracy of our method to implement the mean preserving spread property we gauge the accuracy of our extended perturbation solution vis-à-vis the standard perturbation solution with Euler equation residuals. For both functional equations (E.7) and (E.8) defined in Appendix E we compute the percentage increase in consumption that would be required to equate the left-hand side to the right-

20

hand side, if we insert the approximate instead of the true policy function.14 We compute the residuals on an equal grid G of 100 × 100 points over the square p spaced rectangular p [0.9k, 1.1k] × [−2τ/ 1 − ρ 2 , 2τ/ 1 − ρ 2 ]. To prevent the results from being driven by the least reliable residuals at the grid borders, we report the mean absolute values of the residuals over the grid in Table F.7 in Appendix F. The residuals for the Euler equation (E.7) from the standard perturbation solution are between 0.0026 percentage points (for η = 1 and τ = 0.003) and 0.0584 percentage points (for η = 10 and τ = 0.019) larger than those from our extended solution. This difference is even more pronounced for the value function (E.8), where the Euler residuals from the standard solution are between 0.0054 (η = 1, τ = 0.003) and 0.2168 (η = 10, τ = 0.019) percentage points larger. These results further validate our method to implement the MPS property in DSGE models. Finally, we also compare the accuracy of our extended perturbation solution for the MPS property with the solutions from the two global methods. Tables F.5 and F.6 in Appendix F display the maximum absolute value of the Euler residuals over the grid G for the Euler equation (E.7) and the value function (E.8), respectively. In terms of this measure, and as is well-known from, e.g., Aruoba et al. (2006) and Heer and Maußner (2008), the perturbation solution performs worst. Its Euler residuals for the functional equation (E.8) (which determines the value function required in equation (16)) range from 0.018 percent for η = 1 and τ = 0.003 to 0.58 percent for η = 10 and τ = 0.019. For the finite element method they range from 0.000289 percent for η = 1 and τ = 0.011 to 0.002481 percent for η = 1 and τ = 0.019. Even more accurate is the Chebyshev-Galerkin method (CGM) with a minimum Euler equation residual of 6.88×10−9 percent for η = 1 and τ = 0.003 and a maximum value of 5.8 × 10−6 for η = 10 and τ = 0.019. For this reason we also present the conditional welfare measure computed from this solution in the lines labeled proj in column S of Table F.1 and displayed in panel (e) of Figure 1. The absolute difference between the conditional welfare measure from our perturbation solution and the CGM solution lies between 1.6 × 10−8 percentage points for η = 3 and τ = 0.003 and 1.2 × 10−3 percentage points for η = 10 and τ = 0.019. For the latter this difference is equal to about 68 cents of real per-capital consumption in 2018. Hence, even though the perturbation solution is quite less accurate outside a small neighborhood of the stationary solution, this imprecision is of almost negligible consequence for the conditional welfare measure. In order to compare unconditional measures from the perturbation approach to those obtained from the weighted residuals methods we have to resort to simulation, since the distribution of the state variables w t := (k t , ln A t ) T is unknown. Our estimate of the unconditional mean of Vau from equation (19) is Vau =

T 1X Vˆ (k t+s , ln A t+s ), T s=0

where Vˆ (k t , ln A t ) denotes the approximate FEM solution of the value function. Since the welfare effects for small values of η and τ are of the order of magnitude between 10−4 and 14

See Christiano and Fisher (2000) for this kind of Euler residuals. The original concept is introduced in Judd and Guu (1997).

21

10−5 , we must choose T very large. To see this, note that the variance of the sample mean z¯ of log TFP is equal to •
˜ 1 τ2 2 2 T +1 var(¯ z) = 2 T + . (T − 1)ρ − T ρ + ρ T 1 − ρ2 (1 − ρ)2

For T = 5 × 108 this is still as large as 1.7 × 10−5 . Table F.1 displays the means from 15 simulations of size T = 5 × 108 each. Compared to the results from the perturbation solution the FEM solution yields smaller effects being close to those from the perturbation solution for τ ≤ 0.011. For τ = 0.015 the FEM measures are between 7 and 13 percent smaller than those from the perturbation solution. For the even larger value of τ = 0.019, the FEM measure is between 27 and 58 percent smaller. Yet, in absolute terms, this latter difference is equal to 0.026 percentage points or 14.8 U.S. dollars of per-capital consumption in 2018. We trace this difference to the fluctuations effect, since the FEM measure of the unconditional mean effect is close to the mean effect from the perturbation solution. At the maximum, the former exceeds the latter by 2.4 percent for η = 2 and τ = 0.030 and is 4.3 percent below the latter for η = 8 and τ = 0.019. We, thus, conclude that the perturbation approach overestimates the unconditional fluctuations effect for large TFP shocks. As a third result, we therefore note that our perturbation method delivers reliable estimates of the conditional welfare measure and also for the unconditional measure if the amount of uncertainty in the model as measured by τ is of moderate size. 4.2 Two-country model Figure 2 graphs the welfare measures presented in Table F.2 in Appendix F for the twocountry model of Section 2.3 without adjustment costs of capital and TFP shocks in the home country only. The values of the parameters that characterize production and preferences in both countries are those presented in 2. The Š TFP shock zd t := ln(Ad t ) follows € Table τ2 2 , τ . the process (8) with ρd = 0.95 and εd t iid N − 2(1+ρ d) Figure 2: Welfare measures: two-country, one-shock model (b) Unconditional measure pert-MPS

(a) Conditional measure pert-MPS

(c) Conditional measure pert

(d) Unconditional measure pert

The graphs in panels (a) and (b) confirm the intuition that the greater potential for risksharing is welfare increasing. The conditional welfare measure shows that only for high

22

degrees of risk-aversion, i.e., η ∈ {9, 10}, is the removal of the cycle welfare-increasing. According to the unconditional measure, shown in panel (b), the effect of uncertainty never dominates the mean effect. Panels (c) and (d) display the welfare measures derived from the standard perturbation solution, where we have only adjusted the mean of the TFP shock according to equation (11). Obviously, both the conditional and the unconditional measures indicate larger gains from removing the cycle than does our extended solution. The conditional measure overstates the effect from our solution between 0.0019 (η = 4, τ = 0.003) and 0.0781 (η = 1, τ = 0.019) percentage points, the unconditional measure between 0.0027 (η = 4, τ = 0.003) and 0.1089 (η = 1,τ = 0.019) percentage points (see Table F.2 in Appendix F).15 Figure 3 illustrates the welfare effects of the business cycle in the two-country model with adjustment costs of capital and correlated TFP shocks in both countries. The parameter values for this model are presented in Table 2 and the numerical results in Table F.3 in Appendix F. Figure 3: Welfare measures: two-country, two-shock model (b) Unconditional measure pert-MPS

(a) Conditional measure pert-MPS

(c) Conditional CGM

All three measures show that for η > 2 removing the cycle is welfare increasing. There are three forces behind this outcome vis-à-vis the two-country, one-shock model. First, the additional TFP shock increases uncertainty. Second, it also increases the potential to risk sharing so that the mean effect is larger than in the one-shock model. Third, adjustment costs of capital accumulation lower the degree to which this potential can be exploited. Obviously, the first and the third effect dominate the second one. We illustrate the welfare decreasing effect of adjustment costs in Table F.4, which presents the welfare measures for the two-country, two-shock model without adjustment costs. Instead of η > 2 it requires risk aversions of η = 6 (η = 7 for the unconditional measure) and more for the effect of uncertainty to dominate. As in the one-country model, the difference between the perturbation solution and the weighted residuals method are small. The absolute difference between the perturbation 15

Note that neither the graph in Panel (c) nor the graph in Panel (d) comes close to Figure 6 of Cho, Cooley, and Kim (2015).

23

solution and the Chebyshev-Galerkin method (CGM) lies between 1.09 × 10−6 for η = 1 and τ = 0.003 and 4.0 × 10−3 for η = 10 and τ = 0.019 percentage points.

5 Conclusion Our paper is inspired by the contributions of Lucas (1987) and Cho, Cooley, and Kim (2015). The former ranked exogenously given stochastic streams of consumption with the mean preserving spread (MPS) property according to a standard additively separable intertemporal utility function and found that removing the business cycle implies negligible benefits for the representative consumer. The latter argue convincingly that the welfare effects of economic fluctuations in production economies have two conceptually different sources. The fluctuations effect captures the fact that risk-averse economic agents would prefer living in a certain environment. The mean effect stems from the optimal adjustment of input factors. Economic agents may benefit from this effect, if, for example, the reduced form aggregate production function is convex in the shocks that drive the model. Measuring both effects without distortion also requires that the shocks have the MPS property. Introducing mean preserving spreads into a model requires to consider their means as functions of their variance. As a consequence, approximations based on perturbing the variance should adequately and simultaneously also perturb the mean. Otherwise, by comparing different levels of uncertainty, the researcher exogenously introduces shifts of the means and mixes level effects with effects from removing uncertainty. Yet, available perturbation methods and code assumes that the mean of innovations in the vector-autoregressive process for shocks is not perturbed. As a consequence, the impact of the mean preserving spreads is missing in the second-order approximation of the value function, which in turn is required to compute the conditional and unconditional welfare measure. Furthermore, the non-zero means also bias the computation of unconditional moments required to compute the unconditional welfare measure. The main contribution of this paper, therefore, is to extend the canonical stochastic dynamic equilibrium model to allow for mean preserving spreads. In a model with closed form solution we demonstrate the need to also perturb the mean of the processes that drive the model. We derive formulas for the computation of second-order approximations of the policy functions and provide the respective Matlab code. We find that our second-order solution differs from the standard case and interpret this term as the effect of unwanted shifts in means. We then compute the welfare effects of removing fluctuations in two standard real business cycle models. The first model is the model that underlies Figure 2 in Cho, Cooley, and Kim (2015). Different form the results reported there, the mean effect never dominates the fluctuations effect. In the two-country version of this model, the effect of uncertainty dominates the mean effect for values of the parameter of relative risk aversion greater than η = 2. Finally, we solve both models with the aid of two weighted residuals methods and confirm both our method and our results. We, therefore, are confident that our extension of the perturbation method is the proper way to approximate the welfare effects of economic fluctuations in DSGE models.

24

References Andreasen, M. M. (2012). On the effects of rare disasters and uncertainty shocks for risk premia in non-linear dsge models. Review of Economic Dynamics 15, 295–316. Aruoba, S. B., J. Fernández-Villaverde, and J. F. Rubio-Ramírez (2006). Comparing solution methods for dynamic equilibrium economies. Journal of Economic Dynamics and Control 30, 2477–2508. Barillas, F., L. P. Hansen, and T. J. Sargent (2009). Doubts or variability? Journal of Economic Theory 144(6), 2388 – 2418. Barlevy, G. (2004, September). The cost of business cycles under endogenous growth. American Economic Review 04(4), 964–990. Barlevy, G. (2005). The costs of business cycles and the benefits of stabilization. Economic Perspectives 33(1), 32–49. Barro, R. J. (2009). Rare disasters, asset prices, and welfare costs. American Economic Review 99(1), 243–264. Binning, A. (2013). Third-order approximation of dynamic models without the use of tensors. Working Paper 2013/13, Norges Bank. Cho, J.-O., T. F. Cooley, and H. S. K. Kim (2015). Business cycle uncertainty and economic welfare. Review of Economic Dynamics 18, 185–200. Cho, J.-O., T. F. Cooley, and L. Phaneuf (1997). The welfare cost of nominal wage contracting. The Review of Economic Studies 64(3), 465–484. Christiano, L. J., M. Eichenbaum, and C. L. Evans (2005). Nominal rigidities and the dynamic effects of a shock to monetary policy. Journal of Political Economy 113, 1–45. Christiano, L. J. and J. D. Fisher (2000). Algorithms for solving dynamic models with occasionally binding constraints. Journal of Economic Dynamics and Control 24, 1179–1232. De Santis, M. (2007). Individual consumption risk and the welfare costs of business cycles. American Economic Review 97(4), 1488–1506. Dmitriev, A. and I. Roberts (2012). International business cycles with complete markets. Journal of Economic Dynamics and Control 36, 862–875. Dolmas, J. (1998). Risk preferences and the welfare cost of business cycles. Review of Economic Dynamics 1(3), 646 – 676. Galí, J., M. Gertler, and J. D. López-Salido (2007). Markups, gaps, and the welfare costs of business fluctuations. Review of Economics and Statistics 89(1), 44–59.

25

Gomme, P. and P. Klein (2011). Second-order approximation of dynamic models without the use of tensors. Journal of Economic Dynamics and Control 35, 604–615. Hamilton, J. D. (1994). Time Series Analysis. Princeton, NJ: Princeton University Press. Hansen, G. D. (1985). Indivisible labor and the business cycle. Journal of Monetary Economics 16, 309–327. Heathcote, J., K. Storesletten, and G. L. Violante (2008). Insuarance and opportunities: A welfare analysis of labor market risk. Journal of Monetary Economics 55(3), 501/525. Heer, B. and A. Maußner (2008). Computation of business cycle models: A comparison of numerical methods. Macroeconomic Dynamics 12, 641–663. Heer, B. and A. Maußner (2009). Dynamic General Equilibrium Modelling. 2nd Edition. Berlin: Springer. Heer, B. and A. Maußner (2015). The cash-in-advance constraint in monetary growth models with labor market search. Macroeconomic Dynamics 19, 144–166. ˙Imrohoro˘ glu, A. (1989). Cost of business cycles with indivisibilities and liquidity constraints. Journal of Political Economy 97(6), 1364–1383. Jermann, U. J. (1998). Asset pricing in production economies. Journal of Monetary Economics 41, 257–275. Judd, K. L. (1992). Projection methods for solving aggregate growth models. Journal of Economic Theory 58, 410–452. Judd, K. L. and S.-M. Guu (1997). Asymptotic methods for aggregate growth models. Journal of Economic Dynamics and Control 21, 1025–1042. Kågström, B. and P. Poromaa (1994). Lapack-style algorithms and software for solving the generalized sylvester equation and estimating the separation between regular matrix pairs. Working Notes UT, CS-94-237, LAPACK. King, R. G., C. I. Plosser, and S. Rebelo (1988a). Production, growth and business cycles I. The basic neoclassical model. Journal of Monetary Economics 21, 195–232. King, R. G., C. I. Plosser, and S. Rebelo (1988b). Production, growth and business cycles II. New directions. Journal of Monetary Economics 21, 309–341. Krebs, T. (2003). Growth and welfare effects of business cycles in economies with idiosyncratic human capital risk. Review of Economic Dynamics 6(4), 846 – 868. Krusell, P., T. Mukoyama, A. ¸ Sahin, and A. A. Smith (2009). Revisiting the welfare effects of eliminating business cycles. Review of Economic Dynamics 12(3), 393 – 404. Lucas, Jr, R. E. (1987). Models of Business Cycles. Oxford: Basil Blackwell.

26

Lütkepohl, H. (2005). New Introduction to Multiple Time Series Analysis. Berlin: Springer. Magnus, J. R. and H. Neudecker (1999). Matrix Differential Calculus with Applications in Statistics and Econometrics (3rd ed.). Chichester: John Wiley &Sons. McGrattan, E. R. (1995). Solving the stochastic growth model with a finite element method. Journal of Economic Dynamics and Control 20, 19–42. Obstfeld, M. (1994). Evaluating risky consumption paths: The role of intertemporal substitutability. European Economic Review 38(7), 1471 – 1486. Schmitt-Grohé, S. and M. Uribe (2004a). Optimal simple and implementable monetary and fiscal rules. Working Paper W10253, National Bureau of Economic Research (NBER). Schmitt-Grohé, S. and M. Uribe (2004b). Solving dynamic general equilibrium models using a second-order approximation to the policy function. Journal of Economic Dynamics and Control 28, 755–775. Schmitt-Grohé, S. and M. Uribe (2006). Optimal simple and implementable monetary and fiscal policy rules: Expanded version. Working Paper W12402, National Bureau of Economic Research (NBER). Schmitt-Grohé, S. and M. Uribe (2007). Optimal simple and implementable monetary and fiscal rules. Journal of Monetary Economics 54, 1702–1725. Smets, F. and R. Wouters (2003). An estimated dynamic stochastic general equilibrium model of the euro area. Journal of the European Economic Association 1(5), 1123–1175. Smets, F. and R. Wouters (2007). Shocks and frictions in u.s. business cycles: A bayesian dsge approach. American Economic Review 97, 586–606. Sydsæter, K., A. Strøm, and P. Berck (1999). Economists’ Mathematical Manual, Third Edition. Berlin: Springer. Tallarini, T. D. (2000). Risk-sensitive real business cycles. nomics 45(3), 507 – 532.

27

Journal of Monetary Eco-