This note shows that a standard real　business cycle model with a speciﬁc parameterrange can weakly generate a hump-shaped　output response output to productivity　shocks. This result requires only that the technology shocks are nearly random walk

Shirai Daichi

Hosei University Repository

A note on hump-shaped output in the RBC model著者 Shirai Daichi出版者 Institute of Comparative Economic Studies,Hosei Universityjournal orpublication title比較経済研究所ワーキングペーパーvolume 188page range 1-12year 2014-11-17URL http://hdl.handle.net/10114/9641A note on hump-shaped output in the RBC model¤Daichi ShiraiySeptember 25, 2014AbstractThis note shows that a standard real business cycle model with a speciﬁc parameterrange can weakly generate a hump-shaped output response output to productivityshocks. This result requires only that the technology shocks are nearly random walk.Keywords: RBC model; Hump-shaped outputJEL Classiﬁcation numbers: E10, E321 IntroductionThe hump-shaped responses of macroeconomic variables to exogenous shocks are one ofthe most important features of observed business cycles and ﬁnancial crises.1 Many studiesfocus on reproducing this feature by introducing some friction, for example, investmentadjustment costs, labor adjustment costs, and agency costs. This note shows that astandard real business cycle (RBC) model without additional friction can weakly generatea hump-shaped output response to persistent productivity shocks.Previously, it was thought that the output response to productivity shocks in thestandard RBC model displays no hump shape, except for a case of complete depreciation of¤I thank Ryoji Hiraguchi, Masaru Inaba, and Keiichiro Kobayashi for helpful comments and discussions.All remaining errors are my own.yThe Canon Institute for Global Studies. Email: shirai.daichi@canon-igs.org1Kobayashi and Shirai (2014) summarize these stylized facts.1capital. However, we ﬁnd that even in a case of partial depreciation of capital, a standardRBC model can weakly reproduce hump-shaped responses. To reproduce this feature,technology shocks have to be suﬃciently persistent, that is, almost a random walk. Kingand Rebelo (1999) point out that the standard RBC model requires a persistent exogenousshock to replicate the main statistical features of business cycles. In addition, a persistentproductivity shock can generate hump-shaped output.Standard RBC models have two kinds of propagation mechanisms: capital accumula-tion and intertemporal substitution. We emphasize that capital accumulation is an im-portant propagation mechanism to reproduce the hump-shaped output. When a change incapital stock is larger than a change in productivity shock, then output displays a humpshape. This situation can be generated from a relatively high investment rate, which prop-agates capital accumulation and makes the output response countercyclical during someperiods.The remainder of this note is organized as follows. Section 2 presents the standardRBC model used in this note. Section 3 analyzes the condition of the hump-shaped outputwith complete capital depreciation. Section 4 shows the impulse response function (IRF)for output to a productivity shock with partial depreciation. Section 5 concludes.2 ModelIn this section, we describe a standard RBC model with a representative household andﬁrm. All markets are competitive.2.1 SettingsThe representative household maximizes expected discounted lifetime utility deﬁned overconsumption, C, and hours of work, L:U =1Xt=0¯t [ln(Ct) + ° ln(1¡ Lt)] ;subject to a budget constraint:Ct + It = wtLt + rtKt;2where ¯ is the subjective discount rate, such that 0 < ¯ < 1, It is investment, wt is thewage, rt is the rental rate of capital, and Kt is the capital stock, which obeys the usuallaw of motion:Kt+1 = (1¡ ±)Kt + It;where ± is the depreciation rate.The representative ﬁrm produces output according to a Cobb–Douglas productionfunction:Yt = ztK®t L1¡®t ; 0 · ® · 1;where zt is productivity. In this note, we assume that the economy was initially in asteady state and the technology shock makes the initial value of z1 lower (greater) thanthe steady state value, z1 = !z¤; ! > 0, in such a way that productivity at time 1 islower (greater) than its steady state value by 1%, ! = 0:99 (! = 1:01). Productivity isassumed to follow the process:zt+1 = ½zt + (1¡ ½)z¤; 0 < ½ < 1; t ¸ 1:This assumption implies that the evolution of productivity can be represented as fol-lows:zt = (1¡ ½t¡1(1¡ !))z¤; t ¸ 1: (1)3 Mechanism of hump-shaped outputIn this section we provide an intuitive explanation of the mechanism of the hump-shapedoutput using a special case of parameter setting, that is, complete depreciation of capitalwithin a one period, or ± = 1. It is well known that exact analytical solutions can beobtained in RBC models with log utility, Cobb–Doglas production functions, and completedepreciation of capital. 2 In this case, the evolution of capital stock is governed by thefollowing:Kt+1 = ®¯ztK®t L1¡®: (2)2For example, see McCallum (1989), Ljungqvist and Sargent (2012) and Stokey, Lucas and Prescott(1989).3In addition, all variables are obtained analytically:Ct = (1¡ ®¯)ztK®t L1¡®;L =X1 +X;where X ´ (1¡®)°(1¡®¯) . In this special case, labor inputs are always constant even when theeconomy is in the transition toward the steady state because of complete balancing ofincome and substitution eﬀects. In addition, the investment rate, st = Yt¡CtYt , is constantand equal to ®¯.A humped-shape variable responds countercyclically to a shock during some periods.Most studies usually judge graphically whether the response of a variable is hump-shaped.In this note, we deﬁne “hump-shaped” explicitly for descriptive purposes.Deﬁnition 1. After revelation of a negative (positive) technology shock z1 at the end ofperiod 0, the hump-shaped response of a variable in period 2 is the same as or decreases(increases) more than its value in period 1.The following proposition provides a simple characterization of the hump-shaped re-sponse of output to the technology shock.Proposition 1. Assume the log utility, Cobb–Douglas production, and complete capitalstock depreciation in one period. Then the output displays a hump-shaped response totechnology shocks when the following condition is satisﬁed:½ ¸ 1¡ !1¡®1¡ ! : (3)Proof. Because the hump-shaped response is determined by the growth rate in period 2,as per Deﬁnition 1, the output is the same or decreases (increases) more than its value inperiod 1 in response to the negative (positive) technology shock. The output growth ratein period 2 is as follows:Y2 ¡ Y1Y1=z2K®2 L1¡®z1K®1 L1¡® ¡ 1:Equation (1) and (2) imply thatY2 ¡ Y1Y1=1¡ ½(1¡ !)!!® ¡ 1: (4)4When output in period 2 equals its value in period 1, the left side of equation (4) equalszero, and ﬁnally, we can obtain equation (3).Proposition 1 implies that the condition (3) satisﬁes the case of large values of (½; ®).3In other words, when the technology shock displays persistence to some extent or theinvestment rate is high, standard RBC models are capable of producing hump-shapedoutput responses to technology shocks. The intuition of Proposition 1 can be understoodfrom equation (4), which shows that output growth rate is decomposed into rate of changeof productivity, 1¡½(1¡!)! , and rate of change of capital stock, !®. On the one hand,productivity converges to the steady state monotonically and the response of capital stockis always countercyclical and hump-shaped during some periods. When the capital growthrate is higher than the productivity growth rate, that is, when ½ and ® are large, the outputresponse is also hump-shaped. After the impact period of a productivity shock, investmentleads to a large change in the relatively high investment rate and the next period’s outputchanges substantially. This eﬀect creates the propagation mechanism. Therefore, whenthe propagation mechanism of capital accumulation is larger than the productivity growthrate, then the output response is hump-shaped.Figure 1 depicts the output response to a 1% technology shock, that is, ! = 0:99. Weshow ﬁve cases corresponding to ½ = 0:5; 0:701; 0:8; 0:9; 0:99 and other parameters areset standard values: ® = 1=3; ¯ = 0:98; ° = 1:8; ± = 1; z¤ = 1.4 The result for ½ = 0:5,output converges to the steady state monotonically. The result for ½ = 1¡!1¡®1¡! = 0:701which strictly satisﬁes the condition, equation (4), output in period 2 is the same level inperiod 1 and converges to the steady state after period 3. In case of a large ½, that is,more than 0.701, output is conspicuously hump-shaped.In Figure 2, we set the value of ½ equal to 0.9 and show ﬁve cases corresponding to® = 0:25; ® = 0:3; ® = 1=3; ® = 0:35; ® = 0:4. This ﬁgure shows that the higher the3In addition, McCallum (1989) and Romer (2011) show that this special case of the RBC model cangenerate hump-shaped responses. However, they do not explain explicitly the condition that can generatehump-shaped responses.4In addition, we can consider a large value for ® because we interpret output, including remainingcapital stock.50 5 10 15 20 25 30−1.4−1.2−1−0.8−0.6−0.4−0.20                                                              t%Yt  ρ=0.5ρ=0.701ρ=0.8ρ=0.9ρ=0.99Figure 1: Response to the technology shock (zt shock) corresponding to each ½Note: This ﬁgure depicts the nonlinear dynamic response to a 1% technological shock in terms of percentagedeviation (log deviation times 100) from the steady state. Parameter settings are ® = 1=3; ¯ = 0:98; ° =1:8; ± = 1 and z¤ = 1value of ®, the stronger is the propagation mechanism.4 Quantitative experimentsThe assumption of complete capital stock depreciation in one period allows us to under-stand the hump-shaped mechanism analytically. In this section, we relax this assumptionto introduce capital depreciation partially, 0 < ± < 1. In this section, we calculate impulseresponse functions, and some parameter settings can generate hump-shaped responses.First, we set the depreciation rate of capital, ±, as 0.025 and compute the impulse re-sponses to the 1% negative technology shock corresponding to ® = 0:25; 0:3; 1=3; 0:35; 0:37and ½ = 0:96; 0:97; 0:98; 0:99; 0:993.5 Table 1 shows the diﬀerence between percentage5These impulse responses are calculated with Dynare. In addition, we solve the model using the forwardshooting algorithm. This algorithm can solve nonlinear system of equations globally. However, there is nodiﬀerence between standard impulse response function and forward shooting algorithm. Thus, we do not60 5 10 15 20 25 30−1.7−1.6−1.5−1.4−1.3−1.2−1.1−1−0.9                                                                                       t%Yt  α=0.25α=0.3α=1/3α=0.35α=0.4Figure 2: Response to the technology shock (zt shock) corresponding to each ®Note: This ﬁgure depicts the nonlinear dynamic response to a 1% technological shock in terms of percentagedeviation (log deviation times 100) from the steady state. Parameter settings are ¯ = 0:98; ° = 1:8; ± =1; ½ = 0:9; and z¤ = 17deviation from the steady state in period 2 and period 1 corresponding to each ® and½. If the response of output to the negative technology shock is hump-shaped, the diﬀer-ence of percentage deviation between periods 2 and 1 is negative by Deﬁnition 1. Table1 reports that a RBC model can weakly generate hump-shaped responses in the case ofgreater than or equal to ½ = 0:99. In addition, Figure 3 depicts the impulse responsefunction: ® = 1=3; ± = 0:025, and corresponding to each ½. When the technology shock isstrongly persistent or nearly random walk, the output response is hump-shaped even witha standard parameter setting. Usually, standard RBC models require that the technologyshocks are persistent to ﬁt with observed business cycle data and many studies frequentlyassume that the technology evolves according to a random walk, for example, Chang,Gomes and Schorfheide (2002). This is not a special case, even in empirical studies. Nel-son and Plosser (1982) show that productivity contains a unit root. In addition, Table 2shows that DSGE models estimate that technology shocks have a persistently large value.Table 1: Diﬀerence between percentage deviation from the steady state in periods 2 and1 corresponding to each ® and ½®0.25 0.3 1/3 0.35 0.370.96 0.0516 0.0452 0.0415 0.0399 0.03800.97 0.0358 0.0300 0.0267 0.0252 0.0235½ 0.98 0.0207 0.0156 0.0127 0.0113 0.00990.99 0.0069 0.0024 -0.0001 -0.0012 -0.00250.993 0.0030 -0.0011 -0.0035 -0.0046 -0.0058Notes: These values are based on impulse reponses to the 1 % technology shock under each parametersetting.However, the RBC model with standard parameter values cannot produce a suﬃcientlystrong hump-shaped response that is not quantitatively similar to observed business cycledata. Therefore, many studies introduce additional mechanisms, such as, the specializedreport the results of the forward shooting algorithm.80 5 10−1.4−1.35−1.3−1.25−1.2                            t%Yt  ρ=0.980 5 10−1.32−1.315−1.31−1.305−1.3                            t%Yt  ρ=0.990 10 20 30−1.35−1.3−1.25−1.2                            t%Yt  ρ=0.993Figure 3: Impulse Responses corresponding to each ½Notes: Each panel depicts the impulse response to a 1 standard deviation technology shock in terms ofpercentage deviation from the steady state. Parameter settings are ® = 1=3; ¯ = 0:98; ± = 0:025; ° =1:8 and z¤ = 1Table 2: Estimation results of persistence of technology shockvalue dataSmets and Wouters (2003) 0.811 Euro AreaOnatski and Williams (2010) 0.954 Euro AreaLevin, Onatski, Williams and Williams (2006) 0.964 U.S.Sugo and Ueda (2008) 0.949 JapanIiboshi, Nishiyama and Watanabe (2006) 0.818 Japanadjustment costs of investments, (e.g., Smets and Wouters, 2003; Christiano, Eichen-baum and Evans, 2005), agency costs (e.g., Carlstrom and Fuerst, 1997), learning-by-doing mechanisms, (e.g., Chang et al., 2002; Cooper and Johri, 2002), adjustment costsof labor, (e.g., Cogley and Nason, 1995), and redistribution (e.g., Kobayashi and Shirai,2014), which amplify propagation mechanisms. These additional mechanisms are partiallysuccessful at generating a small hump-shaped impulse response.This note stresses that the hump-shaped output in the RBC model is due to persistenttechnology shocks and high investment rates. These factors are important for replicatinghump-shaped output and have substantial eﬀects on the dynamics of capital stock. Many9studies introduceinvestment adjustment costs; on the other hand, capital adjustment costsmodels are less successful in replication because, as Cogley and Nason (1995) point out,the ﬂow of investment is very small relative to the capital stock in capital adjustmentmodels. Hence, one way to replicate the strong hump-shaped output is to introduce afactor that substantially aﬀects the dynamics of capital stock.5 ConclusionIn this note, we provided intuitive understanding of the hump-shaped output mechanism.In addition, we showed that a standard RBC model with a speciﬁc range of parameterscan weakly reproduce a hump-shaped output. This hump shape is very weak, althoughan additional mechanism is not necessary to reproduce this feature. However, as is wellknown, RBC models have weak internal propagation mechanisms and do not generate asuﬃciently strong hump shape.ReferencesCarlstrom, Charles T. and Timothy S. Fuerst (1997) “Agency Costs, Net Worth, and Busi-ness Fluctuations: A Computable General Equilibrium Analysis,” American EconomicReview, Vol. 87, pp. 893–910.Chang, Yongsung, Joao F. Gomes, and Frank Schorfheide (2002) “Learning-by-Doing asa Propagation Mechanism,” American Economic Review, Vol. 92, pp. 1498–1520.Christiano, Lawrence J., Martin Eichenbaum, and Charles L. Evans (2005) “NominalRigidities and the Dynamic Eﬀects of a Shock to Monetary Policy,” Journal of PoliticalEconomy, Vol. 113, pp. 1–45.Cogley, Timothy and James M Nason (1995) “Output Dynamics in Real-Business-CycleModels,” American Economic Review, Vol. 85, pp. 492–511.Cooper, Russell and Alok Johri (2002) “Learning-by-doing and aggregate ﬂuctuations,”Journal of Monetary Economics, Vol. 49, pp. 1539–1566.10Iiboshi, Hirokuni, Shin-Ichi Nishiyama, and Toshiaki Watanabe (2006) “An estimateddynamic stochastic general equilibrium model of the Japanese economy: A Bayesiananalysis,” March, mimeo.King, Robert G. and Sergio T. Rebelo (1999) “Resuscitating real business cycles,” inJohn B. Taylor and Michael Woodford eds. Handbook of Macroeconomics, Vol. 1B ofHandbook of Macroeconomics: Elsevier, Chap. 14, pp. 927–1007.Kobayashi, Keiichiro and Daichi Shirai (2014) “Heterogeneity and redistribution in ﬁ-nancial crises,” CIGS Working Paper Series, The Canon Institute for Global Studies14-004E, The Canon Institute for Global Studies.Levin, Andrew T., Alexei Onatski, John Williams, and Noah M. Williams (2006) “Mone-tary Policy Under Uncertainty in Micro-Founded Macroeconometric Models,” in MarkGertler and Kenneth Rogoﬀ eds. NBER Macroeconomics Annual 2005, Vol. 20, Cam-bridge, MA: MIT Press, Chap. 4, pp. 229–312.Ljungqvist, Lars and Thomas J. Sargent (2012) Recursive Macroeconomic Theory, Cam-bridge, MA: The MIT Press, 3rd edition.McCallum, Bennett T. (1989) “Real Business Cycle Models,” in Robert J. Barro ed.Modern business cycle theory, Cambridge, MA: Harvard University Press, pp. 16–50.Nelson, Charles R. and Charles I. Plosser (1982) “Trends and random walks in macroe-conmic time series : Some evidence and implications,” Journal of Monetary Economics,Elsevier, Vol. 10, pp. 139–162.Onatski, Alexei and Noah Williams (2010) “Empirical and policy performance of aforward-looking monetary model,” Journal of Applied Econometrics, John Wiley &Sons, Ltd., Vol. 25, pp. 145–176.Romer, David (2011) Advanced Macroeconomics, New York: McGraw-Hill, 4th edition.Smets, Frank and Raf Wouters (2003) “An Estimated Dynamic Stochastic General Equi-11librium Model of the Euro Area,” Journal of the European Economic Association, Vol.1, pp. 1123–1175.Stokey, Nancy L., Robert E. Lucas, Jr., and Edward C. Prescott (1989) Recursive Methodsin Economic Dynamics, Cambridge, MA: Harvard University Press.Sugo, Tomohiro and Kozo Ueda (2008) “Estimating a dynamic stochastic general equilib-rium model for Japan,” Journal of the Japanese and International Economies, Vol. 22,pp. 476–502.12

A note on hump-shaped output in the RBC model

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A note on hump-shaped output in the RBC model

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