Input-output interactions in a DSGE framework Sára Bisová1, Eva Javorská2, Kristýna Vltavská3, Jan Zouhar4 Abstract. Despite their (heavily advertised) microeconomic foundations, DSGE models are usually implemented for economic analyses using countrylevel aggregate data. However, national statistical offices of the EU members create national accounts that contain detailed information about individual industries, such as the Input-Output tables. In this paper, we present a preliminary study of a multi-sector extension of a canonical RBC model that allows to use data on the input-output structure with multiple sectors. We formulate a simple baseline model that allows for an arbitrary number of sectors with an arbitrary input-output structure. The practical obstacle to using the model in practice lies in the need to find approximate steady-state values of the variables. In the general case, finding a solution to the steady-state problem is diffucult; however, we do provide an analytic solution to a special symmetric case, which can sometimes provide a close enough approximation for non-symmetric problems as well. Keywords: DSGE models, Input-Output tables, National Accounts JEL classification: E16, E17 AMS classification: 91B51
1
Introduction
In today’s macroeconomic analyses, Vector Autoregressions (VAR), Vector Error Correction models (VEC) or DSGE models are typically used, with DSGE models being the newest of the three approaches, and one that is becoming increasingly popular with researchers and regulators alike. For instance, the Czech National Bank has developed a DSGE model “g3” [1], which is used for both the interpretation of current economic phenomena and for economic forecasting. A key feature that makes DSGE models so favourable is the fact that they are built on microeconomic foundations, and thus should be resistant to the Lucas critique. In our opinion, however, the idea of DSGE models’ microfoundations is at odds with the aggregate nature of data that are typically used for model calibration; country-level data are almost invariably used, even though industry level-data are available in all EU member states’ national accounts and presented e.g. in the Input-Output tables [4]. It has to be noted that some of the existing models do in fact divide the firms sector into a small number of “industries”, typically distinguishing either between manufacturers and retailers, or, as in case of the Hubert model that has been developed and used by the Ministry of Finance of the Czech Republic (see [5], [6]), between manufacturers, retailers, and importers. Nevertheless, in such settings, (i) the model comprises a very small number of industries, (ii) the distinction between the individual industries does not follow the structure of national accounts, and (iii) the input-output structure is limited to a one-sided relationship between retailers and manufacturers. The aim of this paper is to formulate a simple baseline model that allows for an arbitrary number of sectors with an arbitrary input-output structure. To the best of our knowledge, the only existing studies that develop a similar type of a model are the works of Bouakez et al [3], [2]. Even though we were undoubtedly inspired by these works, our model differs from that of Boukaez et al in several aspects. Firstly, their model is built to utilize the structure of the U.S. national accounts – which differs from that in both the Czech Republic and the EU as a whole; most importantly, the U.S. national accounts collect data on the capital flows among 1 Univ.
of Economics, Prague, Dept. of Econometrics, nám. W. Churchilla 4, 130 67 Praha 3,
[email protected] of Economics, Prague, Dept. of Economic Statistics, nám. W. Churchilla 4, 130 67 Praha 3,
[email protected] 3 Univ. of Economics, Prague, Dept. of Economic Statistics, nám. W. Churchilla 4, 130 67 Praha 3,
[email protected] 4 Univ. of Economics, Prague, Dept. of Econometrics, nám. W. Churchilla 4, 130 67 Praha 3,
[email protected] 2 Univ.
the individual sectors (the Capital Flow Tables), which allow for a simple incorporation of the capital formation proces into the DSGE model. As we aim to apply our model to Czech data in future, we had to resort to a different, less straightforward method of combining the input-output processes and capital stock creation, which proved to complicate the equilibrium derivation. Secondly, while the models of Bouakez et al take on the New Keynesian perspective, ours is mostly in line with the real business cycle (RBC) literature.
2
The model
The model economy consists of a single infinitely-lived household and J firms, representing one sector each. The household is the sole labour supplier to all firms, and the firm’s production can be either consumed by the household, used as a material input in other sectors, or used in the form of a capital investment. In our treatment, we use the social-planner formulation of the model, which should give identical equilibrium conditions as the competitive equilibrium formulation, but with a lower notational burden, see [7]. The social planner’s aim is to maximize the expected sum of households (discounted) utilities, i.e. to maximize E
∞ X
t
β U (Ct , Ht ) = E
t=0
∞ X
β t log Ct − φ log Ht ,
t=0
where U is the (instantaneous) utility function, Ct and Ht are the consumption and labour-time bundle in period t and φ is a non-negative parameter (labour disutility). The consumption bundle is defined as a weighted Dixit-Stiglitz aggregate of sector-specific consumptions: hP i1/θ θ Ct ≡ , j γj Cjt where Cjt is the quantity consumed of goods from sector j in period t, γj is the relative weight of the goods from sector j in the consumption bundle (γj are nonnegative and sum to one), and θ is a parameter greater than 1, such that θ/(θ − 1) is the elasticity of substition among the sectors’ production. Similarly, the labour time bundle is defined as hP i1/υ υ Ht ≡ , j Hjt where Hjt are hours worked in sector j and period t and υ is a parameter less than 1 representing the substition among hours spent working in individual sectors. Note that in this setting the household prefers diversity in labour time. The production function in sector j has the form of a Cobb-Douglas function with constant returns to scale: α αjK α Yjt = (eZt Hjt ) jH Kj,t−1 MjtjM , where Zt is an exogenous aggregate labour productivity shock, Kj,t−1 is the capital stock in sector j at the beginning of period t (i.e. capital stock determined by the end of period t − 1), Mjt is a bundle of all material flows to sector j from other sectors, and the α’s are sector-specific output elasticities that sum to one. The material bundle is defined as hP i1/θ θ Mjt ≡ µ M , ij ijt i where Mijt is the quantity of sector i’s output being supplied (as an input) to sector j, and µij is the weight of sector i’s output in sector j’s material (input) bundle. The labor productivity shock is assumed to follow the standard AR(1) process, Zt = ρZt−1 + εt ,
εt ∼ N (0, σ 2 ), i.i.d.
The budget constraint is given as Yjt = Cjt + Ijt +
P
k
Mjkt ,
where Ijt denotes capital investment in sector j and period t defined as Ijt ≡ Kjt − (1 − δ)Kj,t−1 , where δ is the depreciation rate.
(1)
Comments on alternative model specifications and calibration. The original sector-specific data (and, most notably, Input-Output tables) presented by the Czech Statistical Office are divided into 81 industries. Technically, such a number of sectors would probably prove too high for the resulting model to be computationally tractable. Therefore, it seems necessary to aggregate these industries in order to reduce the number of sectors to an acceptable level. Following the CZ-NACE classification, the 81 industries can be aggregated into as few as 19 sectors. As for calibration, the µij parameters are supposed to be taken directly from Input-Output tables, and the γj parameters will be calibrated from the consumer price index. Moreover, we need the parameters of the Cobb-Douglas production function; we intend to use estimates based on the index number approach. The θ parameter can be estimated as the ratio of gross value added in current prices and compensation of employees. These data are published in the national accounts.
3 3.1
Solving the model Equilibrium conditions.
First, we form the Lagrangian as2 h i P∞ P P L = E t=0 β t log Ct − φ log Ht − j λjt − Yjt + Cjt + Ijt + k Mjkt h P P i P∞ P = E t=0 β t log Ct − φ log Ht + j λjt − Yjt + Cjt + Kjt − (1 − δ)Kj,t−1 − i j λit Mijt . The first-order conditions are obtained by setting the partial derivatives of the Lagrangian (w.r.t. the decision variables and Lagrange multipliers) to zero. After the elimination of Lagrange multipliers and some obvious algebraic manipulation, we can list the equilibrium conditions as follows: θ−2 −θ αjH γj Yjt Cjt Ct = φ (Hjt /Ht )υ
for all j,
θ−2 1−θ θ−2 −θ γi Cit Mijt = γj αjM µij Yjt Cjt Mjt h i θ−2 −θ θ−2 −θ Cjt Ct = β E t Cj,t+1 Ct+1 αjK (Yj,t+1 /Kjt ) + 1 − δ
Yjt = (eZt Hjt )
αjH
Yjt = Cjt + Ijt +
αjK Kj,t−1
P
k
α MjtjM
Mjkt
for all i, j, for all j, for all j, for all j,
Zt = ρZt−1 + εt , εt ∼ N (0, σ 2 ), i.i.d. Altogether, this gives (J 2 + 4J + 1) equations. Note that if one wants to use a software package like Dynare, symbolic expressions containing loops and sums are not supported in the code. Even for J as low as 3, writing down these equations in hand is a very tedious task, and typing errors are likely to creep into the code. For these reasons, we put up a Matlab script that creates the appropriate Dynare code for an arbitrary number of sectors; we will gladly provide the Matlab script to anyone interested at an email request. 3.2
Steady state conditions
In the analysis of equilibrium dynamics, one has to find the steady state first. The steady state conditions are easily obtained from the equilibrium conditions by dropping the time subscript (t) and the error term
2 We deliberately omitted the constraint (1) and its Lagrange multiplier in order to save space; note that Z is an t exogenous random process, so it does not enter any first-order conditions w.r.t. the decision variables. The equilibrium conditions given below, however, do include the constraint on Zt .
(εt ). After a slight algebraic manipulation3 we obtain a system of non-linear equations αjH γj Yj Cjθ−2 C −θ = φ (Hj /Ht )υ 1−θ γi Ciθ−2 Mij = γj αjM µij Yj Cjθ−2 Mj−θ 1 = β αjK (Yj /Kj ) + 1 − δ
Yj =
α Hj jH
α Kj jK
Yj = Cj + Ij + 3.3
α Mj jM
P
k
Mjk
for all j,
(2)
for all i, j,
(3)
for all j,
(4)
for all j,
(5)
for all j.
(6)
Steady state – symmetric case
Consider now a special symmetric case where for all j, αjK = αK , αjH = αH , αjM = αM , γj = γ, and µij = µ for all i 6= j, µij = 0 otherwise. In this case, the analytic steady-state solution has the form given in the proposition below. Proposition 1. Let R=
αK , 1/β − 1 + δ
S = µ1/θ αM (J − 1)(1−θ)/θ ,
α
T =R
− αK H
α
S
− αM H
.
Then, in the steady state of the symmetric problem, for all j, φ 2 C , αH j αM Yj , Mij = J −1 Yj =
(7)
Kj = R Y j ,
(8)
Cj =
αH , φ(1 − δR − αM )
(9)
Hj = S Y j . Proof. It is useful to derive the following identities first: P H υ = j Hjυ = J Hjυ , and so P C θ = j γCjθ = γJ Cjθ , and so P θ θ Mjθ = i6=j µMij = µ(J − 1)Mij , and so
(10)
(Hj /H)υ = 1/J ,
(11)
θ
(12)
θ
(13)
(Cj /C) = 1/(γJ) , (Mij /Mj ) = 1/[µ(J − 1)] ,
the last equation holds for all i 6= j. Now, plugging (11) and (12) into (2) immediately yields αH Yj = φCj2 , a simple rearrangement of (7). In (3), the terms γi , γj and Ci , Cj can be cancelled due to symmetry, and αM plugging in (13) gives Mij = J−1 Yj . Note that summing this formula across the second subscript yields P P αM (14) k Mjk = k6=j J−1 Yk = αM Yj . The equation Kj = R Yj follows immediately from (4). Next, note that θ i hX i1/θ h 1/θ αM θ Mj = µMij = µ(J − 1) Yj = S Yj . J −1 i The equation (9) can be obtained by plugging (7), (8) and (14) into (6). From (5) and the previous results, we have Yj = HjαH (R Yj )αK (S Yj )αM , which gives (10), and completes the proof. It is easily seen that the equations in Proposition 1 follow a recursive scheme in the decision variables – Cj is expressed in terms of the model parameters only, Yj is in terms of Cj , and the rest is given in terms of Yj . 3 To be specific, we cancelled equal terms on both sides of the third equation, and noted that in the equilibrium, Z = ρZ, and hence Z = 0.
4
Conclusions
As we pointed out in the introduction, this paper was primarily meant to show that DSGE models can be formulated to work with a disaggregated structure of the economy, and start the academic discussion that will hopefully lead to creating DSGE models that explain some of the real-life phenomena that were not captured by the extant models. We carried out first steps in this direction by devising a simple RBC-based DSGE model that can accomodate an arbitrary number of sectors, the interactions of which are described by the input-output tables produced by the national accounts along the accepted EU accounting standards. In chapter 3, we derived the equilibrium conditions that can be e.g. entered into the widely-used Dynare software package. However, even in this simple setting, there are some practical obstacles that need to be tackled before the model can be used with real-life data. Concretely, efficient numeric solutions of the steady state are required. As of yet, we have only obtained some preliminary results, namely the characterization of the steady state in the special case of a symmetric problem, described in Proposition 1. In future, we aim to test to what extent can these results be used to find initial guesses for the numeric algorithms that solve the problem in the general case. Other goals for future reseach include calibrating the model on Czech data and extending the baseline models to a more elaborate setting; these extensions might be as follows: • more sector-specific parameters (such as the depreciation rate, technology shocks etc.), • the introduction of the government and foreign sectors, • explicit treatment of the labour market and its imperfections, • incorporating possible price, labour, and capital adjustment costs.
Acknowledgements This research was supported by the Internal Grant Agency of the University of Economics, Prague, project No. 18/2013, “The usage of DSGE models in national accounts”.
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