Technical appendix: Business cycle accounting for the Japanese economy in a deterministic way Masaru Inaba

∗

November 26, 2007

Introduction. This paper is a technical appendix for Kobayashi and Inaba (2006) which apply business cycle accounting to the Japanese economy in a deterministic way. The idea of BCA developed by Chari, Kehoe and McGrattan (2002, 2004, 2007) is to assess which wedge is important for the fluctuation of an economy which is assumed to be described as a prototype model with time-varying wedges. These wedges resemble productivity, labor and investment taxes, and government consumption. Since these wedges are measured using the production function and first order conditions to fit the actual macroeconomic data, this method can be interpreted as a generalization of growth accounting. While Chari et al. (2004, 2007) assume the dynamic stochastic general equilibrium model in the prototype economy when applying BCA, Kobayashi and Inaba (2006) assume perfect foresight when applying BCA to the Japanese economy. The assumption of perfect foresight enables us to avoid complicated arguments and calculations concerning the stochastic process of wedges, which Chari et al. (2004) discuss in detail. Since the perfect foresight version in Chari et al. (2002a) provides identical implications for the Great Depression as the stochastic version in Chari et al. (2004), we adopt this simplification in this paper1 .

∗ Research

Institute of Economy, Trade, and Industry. Email: [email protected] (2004) apply BCA to the Japanese economy using a log-linearized dynamic stochastic general equilibrium model. The simulation result on the investment wedge is somewhat different from Kobayashi and Inaba (2006). Inaba(2007) apply BCA using the parameterized expectation algorithm to the Japanese economy to solve a non-linear dynamic stochastic general equilibrium model. I find that the result of BCA using PEA is similar to the result of the perfect foresight BCA by Kobayashi and Inaba. Therefore, we can conclude that the causes of the difference in the results between Chakraborty and Kobayashi-Inaba must be in data constructions, data sources and log-linearization. In cases where the economy is far away from the steady state or highly non-linear, the approximation error may be large. Therefore, taking account of non-linearities may be important. 1 Chakraborty

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The prototype model This section describes the prototype model with time-varying wedges: the efficiency wedge At , the labor wedge 1 − τl,t , the investment wedge 1/(1 + τx,t ), and the government wedge gt . The household maximizes: [∞ ] ∑ t max E0 β U (ct , lt )Nt ct ,kt+1 ,lt

subject to

{

ct + (1 + τx,t )

Nt+1 kt+1 − kt Nt

t=0

} = (1 − τl,t )wt lt + rt kt + Tt , 0 < β < 1,

where ct denotes consumption, lt employment, Nt population, kt capital stock, wt the wage rate, rt the rental rate on capital, Tt the lump-sum taxes per capita. All quantities written in lower case letters denote per-capita quantities except for Tt . The firm maximizes max At F (kt , γ t lt ) − {rt + (1 + τx,t )δ}kt − wt lt , kt ,lt

where δ denotes the depreciation of capital stock and γ the balanced growth rate of technical progress. The resource constraint is ct + xt + gt = yt ,

(1)

where xt is investment, gt the government consumption and yt the per-capita output. The law of motion for capital stock is Nt+1 kt+1 = (1 − δ)kt + xt . Nt

(2)

The equilibrium is summarized by the resource constraint (1), the law of motion for capital (2), the production function, yt = At F (kt , γ t lt ),

(3)

and the first-order condtions, −

Ul,t = (1 − τl,t )At γ t Fl,t , Uc,t

Uc,t (1 + τx,t ) = βEt Uc,t+1 [At+1 Fk,t+1 + (1 − δ)(1 + τx,t+1 )] ,

(4) (5)

where Uct , Ult , Flt and Fkt denote the derivatives of the utility function and the production function with respect to their arguments. The functional form of the utility function is given by U (c, l) = ln c + ϕ ln(1 − l), where ϕ > 0 is a parameter. Also the functional form of the production function is given by F (k, l) = k α l1−α . 2

A

Measuring the wedges

We take the government wedge τg directly from the data. To obtain the values of the other wedges, we use the data for yt , lt , xt , gt and Nt , together with a series on kt constructed from xt by (2). The efficiency wedge and the labor wedge are directly calculated from (3) and (4). To solve (5), we need to posit a strict assumption on the values of the wedges for the time period after the target period of business cycle accounting. Denoting the target period of BCA by t = 0, 1, 2, · · · , T , we assume that At = A∗ = AT , gt = g ∗ = gT , and τl,T = τl∗ = τl,T for t ≥ T + 1. The growth rate of the population is assumed to be constant for t ≥ T + 1. We also assume that τx,t is an unknown constant τx∗ for t ≥ T . Under these assumptions, given that kT +1 is constructed from the data xt (t ≤ T ), we pick a value for τx∗ and calculate the equilibrium path of ct ,kt (t ≥ T + 1) which converges to the balanced growth path with constant wedges. Since the equilibrium path of ct (and kt ) is uniquely determined for a given value of τx∗ , we can choose the ”true” value of τx∗ such that τx,T = τx,T +1 = τx∗ and the initial consumption cT +1 (τx∗ ) satisfy (5) at t = T , given cT and kT +1 . Once τx∗ = τx,T is determined by this method, tx,t for t = 0, 1, 2, · · · , T , are obtained by solving (5) backward. Suppose ct , yt , lt , xt , gt and kt are detrended by γ = (1 + gz ). We assume that the utility function is ln(ct ) + ϕ ln(1 − lt ) and the production function is yt = At ktα lt1−α . The algorithm in detail is as follows:

Nested shooting method for measuring the wedges • Initialization: Assume that the target period is t = 1, · · · , T . For example, the target period for Japan is from 1981 to 2005. Set a stopping parameter ϵ > 0. Assume that after period T all wedges are the values at T which remain constant as the steady state values. • Step 1: Fix the lower bound value of τx∗ at τ ∗x and the upper value of τx∗ at τ¯x∗ . • Step 2: Set an initial value of τx,T = τx∗ = (¯ τx∗ − τ ∗x )/2. Given kT +1 , simulate the prototype model by the shooting method2 after T + 1 to find the values of cT +1 and lT +1 which satisfy the transversality conditions. Note that all wedges remain constant at the steady state values. • Step 3: If (1 + τx∗ )(1 + gz )

} 3 1 1 { 1−α >β (1 + τx∗ )(1 − δ) + αA∗ kTα−1 +1 lT +1 , cT cT +1

set τ¯x∗ = τx∗ and go to Step 2; 2 For

the details, see King and Rebelo (1995) values of cT and kT +1 are from actual data. The values of cT +1 and lT +1 are derived from the shooting method in Step 3. 3 The

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Else if (1 + τx∗ )(1 + gz )

} 1 { 1 1−α <β (1 + τx∗ )(1 − δ) + αA∗ kTα−1 +1 lT +1 , cT cT +1

set τ ∗x = τx∗ and go to Step 2; Else if ∥ (1 + τx∗ )(1 + gz )

} 1 1 { 1−α −β (1 + τx∗ )(1 − δ) + αA∗ kTα−1 +1 lT +1 ∥< ϵ, cT cT +1

go to next step. • Step 5: τx,t for t = 0, 1, 2, · · · , T , are obtained by solving (5) backward.

References [1] Chari, V., P. Kehoe, and E. McGrattan. (2002) ”Accounting for the great depression ”American Economic Review 92, 22-27. [2] Chari, V., P. Kehoe, and E. McGrattan. (2004) ”Business cycle accounting” Federal Reserve Bank of Minneapolis Research Department Starff Report 328. [3] Chari, V., P. Kehoe, and E. McGrattan. (2007) ”Business cycle accounting” Econometrica 75, 781-836. [4] Inaba, M. (2007) ”Business cycle accounting for the Japanese economy using the parameterized expectations algorithm” mimeo. [5] King, R., S. Rebelo (1995) ”Transitional Dynamics and Economic Growth in the Neoclassical Model” NBER working paper No. 3185. [6] Kobayashi, K. and M. Inaba. (2005) ”Data appendix: business cycle accounting for the Japanese economy.” http://www.rieti.go.jp/en/publications/dp/05e023. [7] Kobayashi, K. and M. Inaba. (2006) ”Business cycle accounting for the Japanese economy” Japan and the World Economy 18, 418-440.

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