Structural propagation in a production network with restoring substitution elasticities
SStructural propagation in a production network withrestoring substitution elasticities
Satoshi Nakano a , Kazuhiko Nishimura b, ∗ a The Japan Institute for Labour Policy and Training, Tokyo 177-0044, Japan b Faculty of Economics, Nihon Fukushi University, Aichi 477-0031, Japan
Abstract
We model an economy-wide production network by cascading binary compounding functions, based on the sequential processingnature of the production activities. As we observe a hierarchy among the intermediate processes spanning the empirical input–output transactions, we utilize a stylized sequence of processes for modeling the intra-sectoral production activities. Under theproductivity growth that we measure jointly with the state-restoring elasticity parameters for each sectoral activity, the network ofproduction completely replicates the records of multi-sectoral general equilibrium prices and shares for all factor inputs observedin two temporally distant states. Thereupon, we study propagation of a small exogenous productivity shock onto the structure ofproduction networks by way of hierarchical clustering.
Keywords:
Elasticity of Substitution, Productivity Growth, Linked Input-Output Tables, Hierarchical Clustering
1. Introduction
Given the technological interdependencies among industrialactivities, innovation (in terms of productivity shock) in one in-dustry may well produce a propagative feedback effect on theperformance of economy-wide production. Previous literaturepertaining to the study of innovation propagation has based itstheory upon the non-substitution theorem (e.g., Georgescu-Roegen,1951) that allows one to study under a fixed technological structurewhile restricting the analyses to changes in the net outputs (e.g.,Contreras and Fagiolo, 2014; Tsekeris, 2017). Otherwise, Ace-moglu et al. (2012) assume Cobb-Douglas production (i.e., unitsubstitution elasticity) with which the structural transformationis restricted to the extent that the cost-share structures are pre-served. To study propagation in regard to potential technologi-cal substitutions, however, a potential range of alternative tech-nologies must be known in advance. Nonetheless, empiricalestimation of the substitution elasticities (e.g., Dixon and Jor-genson, 2013) is elusive, and to this end, the dimension of theworking variables has been significantly limited.In contrast, this study is concerned with the economy-widepropagation of innovation that involves structural transforma-tion with regard to the potential range of technologies amonginput variables of large (385) dimension. In so doing we modelmultiple-input production activity by serially nesting (i.e., cas-cading) binary-input production functions of different substitu-tion elasticities. For each industry (or sector of an economy), theelasticity parameters of the production function are measuredjointly with the productivity changes so that the economy-wide ∗ Corresponding author
Email addresses: [email protected] (Satoshi Nakano), [email protected] (Kazuhiko Nishimura ) production system completely replicates monetary and physi-cal inputs in all sectors for two temporally distant equilibriumstates. These elasticity parameters are hence called as restoringelasticities.In the following, we give our rationale for modeling produc-tion by nesting binary compounding functions. Consider, say,the manufacture of a pair of jeans. For this case, one needs asufficient amount of denim fabric, a pair of scissors, a sewingmachine, a ball of yarn, some electricity, and a tailor. We knowthat a pair of jeans will not fall into place all at once but ratherit is made in a step-by-step fashion: using a pair of scissors anda sewing machine, the fabric is first cut into pieces, then theyare sewn together using the yarn with some help from electricpower. Production generally involves a series of processes thatcombine the output of a previous process with another input ofproduction, before handing the output over to the next process.A production activity can be configured as a tree diagramsuch as the one shown in Fig. 1 (left). In this example, theproduction system comprises a series of six inputs ( x , x , x , x , x , x ) , processed in a hierarchical manner, producing fiveintermediate outputs, ( X , X , X , X , X ) , by five processesthat are nested serially. Naturally, one may be concerned thatthe denim fabric, for example, is produced by another (satel-lite) system, and therefore the extended system is inclusive ofa parallel nest. Suppose, for simplicity, that denim fabric isproduced by a serially nested process of two factor inputs, say,the indigo-dyed wrap and the plain weft threads, ( x , x ) . No-tice that we may always re-configure a production system intoa sequence by decomposing the satellite process. In this case,denim fabric is decomposed into a sequence of two inputs (wrapand weft) and a set of cut fabric (i.e., X ) is, presumably, pro-duced by the sequence (wrap, weft, tailor, scissors), in whichcase, the sequence of the inputs of the extended system becomes a r X i v : . [ q -f i n . E C ] A p r arnElectricitySewing MachineDenim FabricScissorsTailor x x x x x x X X X X X I npu t s Outputs X X X X X x x x x x x ProcessJeansInputs
Figure 1: Serially nested configuration of a production system (left) and thecorresponding incidence matrix (right) spanning direct and indirect inputs andintermediate outputs. ( x , x , x , x , x , x , x ) . A cascaded configuration of processes can be transcribedinto a triangular incidence matrix, as shown in Fig. 1. The shadedintersections represent the direct and indirect inputs and the dis-tribution of outputs, while indirect feedbacks are ruled out forsimplicity. A notable feature of this configuration is that everyintermediate process constitutes a part in the overall sequence ofprocesses. For instance, the intermediate process that producesoutput X consists of four direct and indirect inputs, namely, ( x , x , x , x ) , that are processed in this sequence. The over-all sequence of processes is of the final output X , which is ( x , x , x , x , x , x ) , so the intermediate process X obvi-ously constitutes a part in the overall sequence. Thus, the se-quence of every intermediate process is knowable by investigat-ing the overall sequence of the triangular incidence matrix.Our sector-level modeling of serially nested production ac-tivities requires the identification of the sequence of inputs (intra-sectoral processes) for all sectors constituting the economy, andfor that matter, we utilize the economy-wide inter-sectoral trans-actions recorded in an input–output table. We note that, if theincidence matrix transcribed from the input–output table is com-pletely triangular, every sector-level sequence of processes be-comes known from the sequence of intermediate processes styl-ized in the triangulated incidence matrix. We hereafter referto this sequence as the universal processing sequence (UPS).We will find that the input–output incidence matrix of the 2005Japanese economy is not completely yet not too far from beingtriangular. From an empirical perspective, we exploit the uni-versal sequence of processes observed through the triangulationof the input–output incidence matrix. Square matrix triangula-tion (e.g., Chaovalitwongse et al., 2011; Ceberio et al., 2015)refers to a generic technique for finding a simultaneous permu-tation of the rows and columns such that the sum of the entriesabove the main diagonal is maximized.Given the hierarchical sequence of the inputs, we shall pro-ceed to set up a production function that reflects the actual range Of course, the weaving process of the threads requires a weaver. Then, thesame kind of input (labor) is put into process at different stages (weaving andtailoring). By allowing indirect inputs we merge these inputs into the lowerstage. Indirect feedback may be, for example, the case where X is fed back into x . We rule this case out because there will be no way of producing the firstpair of Jeans. Table 1: Notations of variables and observations in different states (observationsare highlighted). Variable Reference Current ProjectedPrice w p π Cost share s a b m
Productivity t θ θz of potential technologies. In this study, we use the constantelasticity of substitution (CES) function Arrow et al. (1961) formodeling the binary compounding process. CES has been ap-plied exensively (e.g., Ramcharran, 2001). Nesting CES func-tions creates a Cascaded CES function whose nest elasticitiesare the main subject of estimation. As we show later on, therestoring nest elasticities will be resolved through calibrationof the sectoral productivities, in order that the two temporallydistant equilibrium states, regarding prices and shares of inputsspanning all sectoral productions, are replicated. The calibrated multi-sectoral Cascaded CES general equi-librium model is used to simulate the production networks trans-formation ex post of some external productivity shock, and ac-count for its economy-wide influences in terms of welfare mea-sured by the gains in the final demand. Finally, we perform hi-erarchical cluster analysis upon the networks of production indifferent equilibria to study the potential structural propagationof the external productivity shock. The clustering of productionnetworks is studied in various ways (e.g., Blöchl et al., 2011;Hu et al., 2017; Sun et al., 2018) In this study, we base our clus-ter evaluation on the Leontief inverse which is particular case ofKatz-Bonacich centrality (Carvalho, 2014). Upon performinghierarchical cluster analysis (e.g., Newman and Girvan, 2004;McNerney et al., 2013) we measure sectoral distances based onPearson’s correlation between sectoral multipliers of the chang-ing production networks.In section 2, we explain how the elasticity parameters andthe productivity change of a Cascaded CES function are cali-brated, given the universal processing sequence. In section 3,we demonstrate that the equilibrium structures for both refer-ence and current states are replicated, and how the productionnetwork is transformed by an exogenous productivity stimulus.Section 4 provides concluding remarks. Note that superscriptsare hereafter reserved for exponents, while subscripts are for in-dicating variety of inputs, nests, and industries, but not for par-tial derivatives. The notation for key variables in different statesis summarized in Table 1.
2. Model
Suppose that UPS of n +1 inputs is known. Then, a cascaded(i.e., serially nested) production function of n + 1 inputs for an Previous models are calibrated at one point Rutherford (1999), while ourmodel is subject to two-point calibration. j is omitted) can be described as follows: y = tF ( x , x , · · · , x n ) = tF ( x , x )= tX n +1 ( x n , X n ( x n − , · · · X ( x , x ) · · · )) Here, y ≥ is the output, x i ≥ is the i th input, and t > isthe productivity level. There are n nests, each consisting of onefactor input and a compound from the lower level nest, exceptfor the primary nest, which includes two non-compound inputs.Note that we allow t to decrease (i.e., production activity maylose output performance instead of gaining), as may be observedin reality. We define X = x , for convenience.The CES production function for the i + 1 th compound out-put processed at the i th nest is of the following form: X i +1 ( x i , X i ) = (cid:18) λ σi i x σi − σi i + Λ σi i X σi − σi i (cid:19) σiσi − (1)The nest production function (1) holds for i = 1 , · · · , n , sincethere are n nests in a production activity. Here, λ i = 1 − Λ i ∈ [0 , is the share parameter of the i th nest, X i is the compoundoutput from the i − th nest, and σ i is the elasticity of substitutionbetween i th input x i and the compound input from the lowerlevel nest X i . We assume that (1) is homogeneous of degreeone.The following is the dual (or unit cost) function of the i +1 thcompound output given in (1): W i +1 ( w i , W i ) = (cid:0) λ i w − σ i i + Λ i W − σ i i (cid:1) − σi (2)Here, w i is the price of the i th input, and W i is the price of thecompound input from the lower level nest. We may verify that(2) is a dual function of (1) by the following exposition. First, aswe presume zero profit in the nest process, the following identitymust hold: W i +1 X i +1 = w i x i + W i X i (3)Then, by virtue of (1) and (3), we have ∂X i +1 ∂x i / ∂X i +1 ∂X i = (cid:18) λ i Λ i X i x i (cid:19) /σ i = w i W i (4)Alternatively, by virtue of (2) and (3), we have ∂W i +1 ∂w i / ∂W i +1 ∂W i = λ i Λ i (cid:18) W i w i (cid:19) σ i = x i X i (5)Hence, it is safe to say that (2) is the unit cost function of (1),since (4) and (5) are equivalent. A Cascaded CES unit cost func-tion can then be created by nesting (2) serially as follows: c = t − H ( w , w , · · · , w n ) = t − H ( w , w )= t − W n +1 ( w n , W n ( w n − , · · · W ( w , w ) · · · )) (6)where c indicates the unit cost of the sector concerned. For con-venience, we may define that W = w . A worked example of the calibration procedure that we presentin this section for a two-stage Cascaded CES function is givenin Appendix A. To begin with, let us apply Euler’s homoge-neous function theorem and the no-arbitrage (i.e., zero profit)condition to the unit cost function: c = (cid:88) ∂c∂w i w i , c = (cid:88) w i x i y By recursively taking the derivatives for (2), we arrive at thefollowing identities for i = 0 , , · · · , n : ∂W i +1 ∂w i = λ i (cid:18) W i +1 w i (cid:19) σ i , ∂W i +1 ∂W i = Λ i (cid:18) W i +1 W i (cid:19) σ i Then, the cost share of the input of the k th nest, counting fromthe outermost of the n nests, which we denote by s n − k , can bederived as a function of the prices of the compound inputs, asfollows: s n − k = x n − k w n − k yc = ∂c∂w n − k w n − k c = ∂W n +1 ∂W n · · · ∂W n − k +2 ∂W n − k +1 ∂W n − k +1 ∂w n − k w n − k W n +1 = λ n − k w − σ n − k n − k W σ n − n +1 k − (cid:89) l =0 Λ n − l W σ n − l − − σ n − l n − l (7)Our task here is to solve for σ i for all i and the change in t , using the shares and prices observed in two different states,namely, the current and the reference. First, we standardize allprices by those of the inputs (and outputs) of the reference stateand set them at unity, i.e., ( w , w ) = (1 , ) , while denotingthe reference-standardized prices of the current state by p =( p , p , · · · , p n ) . The productivity level for the reference statemust also be standardized and set to unity, i.e., t = 1 , since weknow from (2) that W i +1 = w i = 1 for all i . We also let p be given outside of the system, because the primary input (i.e.,numéraire good or labor) is not produced industrially.We denote the observed cost share of the i th input of the ref-erence state by a i and that of the current state by b i . For laterconvenience, we introduce χ i = { a i , b i , p i } , a set of observ-ables in the two states for the i th input. Further, we note that χ n +1 = { p } and χ = { p } for convenience. Applying refer-ence state values into (7) yields the following: a n − k = λ n − k k − (cid:89) l =0 Λ n − l (8)The following modification may not be so obvious; nonetheless,we may see the equivalence by applying all possible k recur-sively into (8). By taking Λ i = 1 − λ i into account, Further (8)can be modified to calibrate the share parameters: λ n − k = a n − k − (cid:80) k − l =0 a n − l (9)3pplying current state values to (7) we have: b n − k = a n − k p − σ n − k n − k W σ n − n +1 k − (cid:89) l =0 W − σ n − l + σ n − l − n − l By rearranging terms, we obtain the following: σ n − k = ln b n − k a n − k + (cid:80) k − l =0 σ n − l ln W n − l W n − l +1 + ln W n +1 p n − k ln W n − k +1 p n − k We write the above in terms of its entries (unknowns and knownsseparated by a semi-colon) for convenience: σ n − k ( W n − k +1 , · · · , W n +1 , σ n − k +1 , · · · , σ n ; χ n − k ) (10)Also, (2) is evaluated at the current state: W n − k +1 = (cid:32) W − σ n − k +1 n − k +2 − b n − k +1 p − σ n − k +1 n − k +1 − b n − k +1 (cid:33) σn − k +1 Again, we write the above in terms of its entries: W n − k +1 ( W n − k +2 , σ n − k +1 ; χ n − k +1 ) (11)Now, let us work on (10) and (11) step by step from the outerlayer of the nests, i.e., k = 0 . For this particular layer, we mayresolve the unknowns, given t , viz., σ n ( t ; χ n +1 , χ n ) = ln b n W n +1 a n p n ln W n +1 p n , W n +1 ( t ; χ n +1 ) = tp Note that the second equation is restating (6) at the current state c = p . Using the above terms the compound price W n is evalu-ated as follows: W n = ( tp ) ln an − ln bn ln tp − ln pn − b n p ln an − ln bn ln tp − ln pn n − b n ln tp − ln pn ln an − ln bn = W n ( t ; χ n +1 , χ n ) We may work on a few layers below: W n ( W n +1 , σ n ; χ n ) ⇒ W n ( t ; χ n +1 , χ n ) σ n − ( W n , W n +1 , σ n ; χ n − ) ⇒ σ n − ( t ; χ n +1 , χ n , χ n − ) W n − ( W n , σ n − ; χ n − ) ⇒ W n − ( t ; χ n +1 , χ n , χ n − ) σ n − ( W n − , W n , W n +1 , σ n − , σ n ; χ n − ) ⇒ σ n − ( t ; χ n +1 , χ n , χ n − , χ n − ) We repeat this procedure and obtain the following series: W n − k +1 ( t ; χ n +1 , χ n , · · · , χ n − k +1 ) σ n − k ( t ; χ n +1 , χ n , · · · , χ n − k ) Thus, t can be calibrated by way of the condition of the final stage k = n at the current state, i.e., W ( t ; χ n +1 , χ n , · · · , χ ) = p (12)and the restoring elasticity parameters σ n − k can all be resolvedfor k = 0 , , · · · , n − , using the solution t of (12), which wehereafter denote by θ .
3. Empirical Analysis
The degree to which a macroscopic production structure agreeswith the hierarchical order of processing sequences is the linear-ity . In a perfectly linear structure, the processing sequences willonly cascade from upstream to downstream; if this is the case,then we may arrange the rows and columns of the input–outputmatrix according to the hierarchical order to obtain a triangularmatrix, and hence, the universal processing sequence.More specifically, for an n sector output system with n − intermediate inputs (excluding self-input), the furthest upstream(headwaters) sector has no intermediate input and n − outputdestinations (i.e., zero column and n − row entries) whereas thefurthest downstream sector has n − inputs with no intermedi-ate output destination (i.e., n − column and zero row entries).Let us denote a reordering of n sectors whose initial order is (1 , , · · · , n ) by a permutation mapping φ : ( φ (1) , φ (2) , · · · , φ ( n )) .Further, let k ( φ ) = { l | k = φ ( l ) } designate the inverse, so thata φ -permuted version of a matrix U = { u ij } can be specifiedas follows: U ( φ ) = { u ( φ ) ij } = (cid:8) u i ( φ ) j ( φ ) (cid:9) For later discussion, let us work on a discretized square input–output matrix U (i.e., input–output incidence matrix) whose el-ements are binary, specifically, u ij = 1 if transaction x ij (cid:54) = 0 ,and u ij = 0 if x ij = 0 . The linearity (cid:96) of a matrix U withpermutation φ is defined as follows: (cid:96) = (cid:80) i 1) + ∇ H ( , I − = a I + − AI = [ a + ] I = = Note that the last part of the equality is due to the nature ofshares: (cid:80) ni =0 s ij = 1 . Likewise, similar result is obtainablefor the current state, as follows: H ( p , p ) (cid:104) θ (cid:105) − = [ p ∇ H ( p , p ) + p ∇ H ( p , p )] (cid:104) θ (cid:105) − = b (cid:104) p (cid:105) + p (cid:104) p (cid:105) − B (cid:104) p (cid:105) = [ b + ] (cid:104) p (cid:105) = (cid:104) p (cid:105) = p We hereafter recognize H as the production networks compris-ing the entire potential alternative technologies, or the meta struc- ture, while ∇ H representing the equilibrium production net-work in terms of shares of the inputs for all sectors (i.e., input–output coefficient matrix). In order to describe the production network in different equi-librium states, as well as to make comparisons between them,we perform cluster analysis. Technological similarity of sectorsin a production network has been studied by various methodspertaining to the measurement of distances between a pair ofsectors McNerney et al. (2013); Blöchl et al. (2011); Newmanand Girvan (2004). In the present study, we measure clusterdistance based on the correlation between the net multipliers oftwo sectors. The net multiplier µ j of sector j measures the indi-rect requirements from all sectors needed to deliver a unit outputto final demand from sector j . Specifically, µ j is an n -columnvector of the following identity: [ µ , · · · , µ n ] = [ I − S ] − − I = S + S + S + · · · where S represents the input–output coefficient matrix of a cer-tain state. In other words, µ j portrays sector j ’s characteristicpattern of unit output propagation.In Fig. 7, we display the heatmap of the correlations betweenall possible pairs of the net multipliers ( µ j , µ k ) for the currentstate. Note that we convert Pearson correlations into the follow-ing scaled Euclidean distance: d jk = (cid:113) − Corr ( µ j , µ k ) Thus, a perfect correlation has zero distance, whereas a zero cor-relation has a distance of and a perfect negative correlation hasa distance of √ . The greatest distance shown in Fig. 7 is indi-cated by the darkest color. Notice that clustering is observablein the original order of the sector classification. This is suppos-edly because the sectoral classification of an input–output tableis based fundamentally on Colin Clark’s three-sector theory, andthe sectors are disaggregated while placed in the neighborhood.On the other hand, the clusters are homogenized more or less inthe case of stream order as far as Fig. 7 is concerned. We further examine the changes in the networking clustersin different states. However, we found that the changes in themultiplier correlations between different equilibrium states werevisually undetectable by way of a heatmap. Hence, we exam-ine the transformation of the networking clusters by way of thechanges in the correlation distances between all possible pairsof the multipliers. In Fig. 8, we show the results of hierarchi-cal clustering using dendrograms for the reference and currentdistance metrics of the correlations between the net multipliers.In Fig. 9, we display a histogram of the differences of distancebetween the two states. Overall, the distances have contractedin the current state relative to the reference state.Given the meta structure H , we may project the equilibriumstate, given an exogenous productivity shock z = ( z , · · · , z n ) .7 O ff i c e s upp li e s S t ea m and ho t w a t e r s upp l y I ndu s t r i a l w a t e r s upp l y N on - li f e i n s u r an c e C a r r en t a l and l ea s i ng R ea l e s t a t e r en t a l s e r v i c e S e w age d i s po s a l ** M i sc e ll aneou s edu c a t i ona l and t r a i n i ng i n s t i t u t i on s ( p r o f i t - m a k i ng ) R ea l e s t a t e agen c i e s and m anage r s F a c ili t y s e r v i c e f o r r oad t r an s po r t M ob il e t e l e c o mm un i c a t i on s H i r ed c a r and t a x i t r an s po r t W o r k e r d i s pa t c h i ng s e r v i c e s B u s t r an s po r t s e r v i c e B u il d i ng m a i n t enan c e s e r v i c e s C l ean i ng P o s t a l s e r v i c e s and m a il de li v e r y P ub li c b r oad c a s t i ng C on s i gned f r e i gh t f o r w a r d i ng G a s s upp l y M i sc e ll aneou s t e l e c o mm un i c a t i on s I n t e r ne t ba s ed s e r v i c e s P r i v a t e non - p r o f i t i n s t i t u t i on s s e r v i ng en t e r p r i s e s H a r bo r t r an s po r t s e r v i c e J ud i c i a l , f i nan c i a l and a cc oun t i ng s e r v i c e s W a s t e m anage m en t s e r v i c e s ( i ndu s t r y ) P e t r o l eu m r e f i ne r y p r odu c t s ( i n c l ud i ng g r ea s e s ) W a t e r s upp l y I n f o r m a t i on s e r v i c e s E l e c t r i c i t y N e w s pape r F i nan c i a l s e r v i c e R a il w a y t r an s po r t ( f r e i gh t ) T i r e s and i nne r t ube s W a s t e m anage m en t s e r v i c e s ( pub li c ) ** F i x ed t e l e c o mm un i c a t i on s C oa s t a l and i n l and w a t e r t r an s po r t A d v e r t i s i ng s e r v i c e s R oad f r e i gh t t r an s po r t ( e xc ep t s e l f - t r an s po r t ) S t o r age f a c ili t y s e r v i c e C i v il eng i nee r i ng and c on s t r u c t i on s e r v i c e s P ub li c a t i on A i r t r an s po r t R e s ea r c h i n s t i t u t e s f o r c u l t u r a l and s o c i a l sc i en c e ( p r o f i t - m a k i ng ) P ho t og r aph i c s t ud i o s W o v en f ab r i c appa r e l K n i tt ed appa r e l M i sc e ll aneou s w ea r i ng appa r e l and c l o t h i ng a cc e ss o r i e s R e s ea r c h i n s t i t u t e s f o r na t u r a l sc i en c e s ( p r o f i t - m a k i ng ) G ood s r en t a l and l ea s i ng ( e xc ep t c a r r en t a l ) R a il w a y t r an s po r t ( pa ss enge r s ) M o t o r v eh i c l e m a i n t enan c e s e r v i c e s C o rr uga t ed c a r d boa r d bo x e s R e t a il t r ade P r i v a t e po w e r gene r a t i on W ho l e s a l e t r ade E l e c t r i c bu l b s C o m p r e ss ed ga s and li que f i ed ga s M i sc e ll aneou s pe r s ona l s e r v i c e s M i sc e ll aneou s r epa i r s , n . e . c . B aggage , handbag s , s m a ll l ea t he r c a s e s and m i sc e ll aneou s l ea t he r p r odu c t s M i sc e ll aneou s bu s i ne ss s e r v i c e s O il and f a t p r odu c t s , s oap , sy n t he t i c de t e r gen t s and s u r f a c e a c t i v e agen t s P r i n t i ng , p l a t e m a k i ng and boo k b i nd i ng M e t a lli c f u r n i t u r e W a t c he s and c l o cks P a ck i ng s e r v i c e V i deo p i c t u r e , s ound i n f o r m a t i on , c ha r a c t e r i n f o r m a t i on p r odu c t i on R e s ea r c h and de v e l op m en t ( i n t r a - en t e r p r i s e ) M i sc e ll aneou s i ndu s t r i a l i no r gan i c c he m i c a l s I ndu s t r i a l s oda c he m i c a l s B a tt e r i e s R o ll ed and d r a w n a l u m i nu mM i sc e ll aneou s r ubbe r p r odu c t s R epa i r o f m a c h i ne R epa i r o f c on s t r u c t i on M i sc e ll aneou s non - f e rr ou s m e t a l p r odu c t s M i sc e ll aneou s pape r c on t a i ne r s W ooden f u r n i t u r e M i sc e ll aneou s g l a ss p r odu c t s M i sc e ll aneou s f u r n i t u r e and f i x t u r e s C e ll u l a r phone s P l u m be r ' s s upp li e s , po w de r m e t a ll u r g y p r odu c t s and t oo l s M i sc e ll aneou s r ead y - m ade t e x t il e p r odu c t s M i sc e ll aneou s i r on o r s t ee l p r odu c t s M i sc e ll aneou s pu l p , pape r and p r o c e ss ed pape r p r odu c t s M i sc e ll aneou s m e t a l p r odu c t s C oa l p r odu c t s R o ll ed and d r a w n c oppe r and c oppe r a ll o ys P l a s t i c p r odu c t s M i sc e ll aneou s f i na l c he m i c a l p r odu c t s C oa t ed pape r and bu il d i ng ( c on s t r u c t i on ) pape r R e s ea r c h i n s t i t u t e s f o r na t u r a l sc i en c e ( pub i c ) ** M i sc e ll aneou s f ab r i c a t ed t e x t il e p r odu c t s S y n t he t i c d y e s and o r gan i c p i g m en t s Lead and z i n c ( i n c l ud i ng r egene r a t ed l ead ) C oa l m i n i ng , c r ude pe t r o l eu m and na t u r a l ga s M i sc e ll aneou s i ndu s t r i a l o r gan i c c he m i c a l s M i sc e ll aneou s f ab r i cs R ubbe r and p l a s t i c f oo t w ea r C o l d - f i n i s hed s t ee l M o t o r v eh i c l e pa r t s and a cc e ss o r i e s M agne t i c t ape s and d i scs H o t r o ll ed s t ee l Lea t he r f oo t w ea r T i m be r I r on and s t ee l s hea r i ng and s li tt i ng G e l a t i n and adhe s i v e s A c t i v i t i e s no t e l s e w he r e c l a ss i f i ed P ape r B o l t s , nu t s , r i v e t s and s p r i ng s M i sc e ll aneou s c e r a m i c , s t one and c l a y p r odu c t s M e t a l c on t a i ne r s , f ab r i c a t ed p l a t e and s hee t m e t a l S pe c i a l f o r e s t p r odu c t s ( i n c l ud i ng hun t i ng ) C o rr uga t ed c a r dboa r d P a i n t and v a r n i s he s S a l t C oa t ed s t ee l C a s t and f o r ged s t ee l M i sc e ll aneou s ed i b l e c r op s M a c h i n i s t s ' p r e c i s i on t oo l s M i sc e ll aneou s w ooden p r odu c t s M e t hane de r i v a t i v e s R e s ea r c h i n s t i t u t e s f o r c u l t u r a l and s o c i a l sc i en c e ( p r i v a t e , non - p r o f i t ) * P o tt e r y , c h i na and ea r t hen w a r e S t ee l p i pe s and t ube s H en egg s M i sc e ll aneou s m anu f a c t u r i ng p r odu c t s A b r a s i v e S t a r c h P l y w ood , g l ued l a m i na t ed t i m be r I no r gan i c p i g m en t S hee t g l a ss and s a f e t y g l a ss A li pha t i c i n t e r m ed i a t e sJ e w e l r y and ado r n m en t s H ea l t h and h y g i ene ( p r o f i t - m a k i ng ) B edd i ng C a r bon and g r aph i t e p r odu c t s N on - f e rr ou s m e t a l c a s t i ng s and f o r g i ng s P u l s e s M i sc e ll aneou s o r e s E l e c t r i c w i r e s and c ab l e s M i sc e ll aneou s li v e s t o ck P l a s t i c i z e r s " T a t a m i " ( s t r a w m a tt i ng ) and s t r a w p r odu c t s G l a ss f i be r and g l a ss f i be r p r odu c t s , n . e . c . R e s ea r c h i n s t i t u t e s f o r na t u r a l sc i en c e s ( p r i v a t e , non - p r o f i t ) * P e t r o c he m i c a l a r o m a t i c p r odu c t s ( e xc ep t sy n t he t i c r e s i n ) C a s t and f o r ged m a t e r i a l s ( i r on ) T he r m op l a s t i cs r e s i n s C he m i c a l f e r t ili z e r G r a i n m illi ng E l e c t r i c li gh t i ng f i x t u r e s and appa r a t u s M i sc e ll aneou s sy n t he t i c r e s i n s P o t a t oe s and s w ee t po t a t oe s F i be r y a r n s C o tt on and s t ap l e f i be r f ab r i cs ( i n c l ud i ng f ab r i cs o f sy n t he t i c s pun f i be r s ) M i sc e ll aneou s gene r a l - pu r po s e m a c h i ne r y I n t e r na t i ona l s h i pp i ng M ea t F l o w e r s and p l an t s M i sc e ll aneou s i ned i b l e c r op s H i gh f un c t i on r e s i n s S uga r T he r m o - s e tt i ng r e s i n s A l u m i nu m ( i n c l ud i ng r egene r a t ed a l u m i nu m ) W i r i ng de v i c e s and s upp li e s M i sc e ll aneou s non - f e rr ou s m e t a l s S il k and a r t i f i c i a l s il k f ab r i cs ( i n c l ud i ng f ab r i cs o f sy n t he t i c f il a m en t f i be r s ) B ea r i ng s F l ou r and m i sc e ll aneou s g r a i n m ill ed p r odu c t s D e x t r o s e , sy r up and i s o m e r i z ed s uga r M i sc e ll aneou s s e r v i c e s r e l a t i ng t o c o mm un i c a t i on C yc li c i n t e r m ed i a t e s R i c e F r u i t s M i sc e ll aneou s e l e c t r i c a l de v i c e s and pa r t s V ege t ab l e s A ud i o and v i deo r e c o r d s , o t he r i n f o r m a t i on r e c o r d i ng m ed i a F r o z en f i s h and s he ll f i s h C oppe r P ape r boa r d M i sc e ll aneou s e l e c t r on i c c o m ponen t s M ea s u r i ng i n s t r u m en t s Log s A g r i c u l t u r a l c he m i c a l s R e s ea r c h i n s t i t u t e s f o r c u l t u r a l and s o c i a l sc i en c e ( pub li c ) ** M anu f a c t u r ed i c e A n i m a l o il and f a t s , v ege t ab l e o il and m ea l P r i n t i ng i n k E l e c t r i c a l equ i p m en t f o r i n t e r na l c o m bu s t i on eng i ne s P ho t og r aph i c s en s i t i v e m a t e r i a l s P ape r t e x t il e f o r m ed i c a l u s eLea t he r and f u r sk i n s S y n t he t i c r ubbe r R o t a t i ng e l e c t r i c a l equ i p m en t M e t a l p r odu c t s f o r a r c h i t e c t u r e C a r pe t s and f l oo r m a t s C e m en t C h i ck en s D a i r y f a r m p r odu c t s C ab l e b r oad c a s t i ng E l e c t r i c m ea s u r i ng i n s t r u m en t s S e m i c ondu c t o r de v i c e s I n t e r na l c o m bu s t i on eng i ne s f o r m o t o r v eh i c l e s P e t r o c he m i c a l ba s i c p r odu c t s S y n t he t i c f i be r s M a r i ne f i s he r y M ed i c a m en t s L i qu i d c r ys t a l pane l M i sc e ll aneou s s t r u c t u r a l c l a y p r odu c t s T r an s f o r m e r s and r ea c t o r s B ee f c a tt l e G a s and o il app li an c e s and hea t i ng and c oo k i ng appa r a t u s W ooden c h i p s M i sc e ll aneou s f ood s P u m p s and c o m p r e ss o r s I n t eg r a t ed c i r c u i t s C l a y r e f r a c t o r i e s B o tt l ed o r c anned v ege t ab l e s and f r u i t s T ea and r oa s t ed c o ff ee W hea t, ba r l e y and t he li k e C ond i m en t s and s ea s on i ng s F eed and f o r age c r op s O p t i c a l f i be r c ab l e s F eed s D a i r y c a tt l e f a r m i ng P r e s e r v ed ag r i c u l t u r a l f ood s t u ff s ( e xc ep t bo tt l ed o r c anned ) C a s t i r on p i pe s and t ube s S eed s and s eed li ng s H og s R a y on and a c e t a t e I n l and w a t e r f i s he r y and aqua c u l t u r e B o tt l ed o r c anned s ea f ood M a r i ne aqua c u l t u r e M i sc e ll aneou s i ndu s t r i a l e l e c t r i c a l de v i c e s and pa r t s M i sc e ll aneou s p r o c e ss ed s ea f ood M e t a lli c o r e s R e l a y s w i t c he s and s w i t c hboa r d s S a l t ed , d r i ed o r s m o k ed s ea f ood C o s m e t i cs , t o il e t p r epa r a t i on s and den t i f r i c e s A g r i c u l t u r a l s e r v i c e s ( e xc ep t v e t e r i na r y s e r v i c e ) C r u s hed s t one s R ead y m i x ed c on c r e t e W ooden f i x t u r e s G r a v e l and qua rr y i ng P e r f o r m an c e s ( e xc ep t m o v i e t hea t e r s ) , t hea t r i c a l c o m r an i e s M u s i c a l i n s t r u m en t s P r o c e ss ed m ea t p r odu c t s S o ft d r i n ks T r a v e l agen cy and m i sc e ll aneou s s e r v i c e s r e l a t i ng t o t r an s po r t E l e c t r i c aud i o equ i p m en t O r gan i c f e r t ili z e r s , n . e . c . R ad i o c o mm un i c a t i on equ i p m en t ( e xc ep t c e ll u l a r phone s ) C e m en t p r odu c t s B o tt l ed o r c anned m ea t p r odu c t s O p t i c a l i n s t r u m en t s and l en s e s S po r t i ng and a t h l e t i c good s F i s h pa s t e A pp li ed e l e c t r on i c equ i p m en t V i deo equ i p m en t and d i g i t a l c a m e r a V e t e r i na r y s e r v i c e R e f r i ge r a t o r s and a i r c ond i t i on i ng appa r a t u s N ood l e s E l e c t r on t ube s R e t o r t f ood s K n i tt i ng f ab r i cs R e f i ned s a k e E ng i ne s B r ead P a v i ng m a t e r i a l s Y a r n and f ab r i c d y e i ng and f i n i s h i ng ( p r o c e ss i ng on c o mm i ss i on on l y ) M i sc e ll aneou s c o mm un i c a t i on equ i p m en t T o ys and ga m e s F e rr o a ll o ys R ad i o and t e l e v i s i on s e t s P r epa r ed f r o z en f ood s C on f e c t i one r y C r op s f o r be v e r age s M i sc e ll aneou s a m u s e m en t and r e c r ea t i on s e r v i c e s P u l p W i r ed c o mm un i c a t i on equ i p m en t M i sc e ll aneou s c l ean i ng , ba r be r s hop s , beau t y s hop s and pub li c ba t h s H ou s eho l d e l e c t r i c app li an c e s ( e xc ep t a i r- c ond i t i one r s ) M e t a l p r odu c t s f o r c on s t r u c t i on D i s he s , s u s h i and l un c h bo x e s M ed i c a l i n s t r u m en t s M i sc e ll aneou s li quo r s B o il e r s S e r v i c e s r e l a t i ng t o w a t e r t r an s po r t C on v e y o r s T u r b i ne s R epa i r o f s h i p s P o r t and w a t e r t r a ff i c c on t r o l ** E l e c t r on i c c o m pu t i ng equ i p m en t ( a cc e ss o r y equ i p m en t ) M a l t li quo r s W h i sk e y and b r and y B i cyc l e s C r ude s t ee l ( e l e c t r i c f u r na c e s ) H ea l t h and h y g i ene ( pub li c ) ** P e r s ona l C o m pu t e r s C r ude s t ee l ( c on v e r t e r s ) B eau t y s hop s S t a t i one r y R epa i r o f a i r c r a ft s M i sc e ll aneou s o ff i c e m a c h i ne s P r i v a t e b r oad c a s t i ng E l e c t r on i c c o m pu t i ng equ i p m en t ( e xc ep t pe r s ona l c o m pu t e r s ) E a t i ng and d r i n k i ng s e r v i c e s I n t e r na l c o m bu s t i on eng i ne s f o r v e ss e l s S uga r c r op s P i g i r on H ou s eho l d a i r- c ond i t i one r s M a c h i ne r y f o r s e r v i c e i ndu s t r y A i r c r a ft s C he m i c a l m a c h i ne r y M i sc e ll aneou s t r an s po r t equ i p m en t S il v i c u l t u r e O r dnan c e C op y m a c h i ne T r u cks , bu s e s and m i sc e ll aneou s c a r s M ed i c a l s e r v i c e ( m i sc e ll aneou s m ed i c a l s e r v i c e ) A i r po r t and a i r t r a ff i c c on t r o l ( i ndu s t r i a l ) R obo t s M i sc e ll aneou s p r odu c t i on m a c h i ne r y M e t a l m a c h i ne t oo l s M e t a l p r o c e ss i ng m a c h i ne r y T e x t il e m a c h i ne r y R epa i r o f r o lli ng s t o ck M a c h i ne r y f o r ag r i c u l t u r a l u s e M a c h i ne r y and equ i p m en t f o r c on s t r u c t i on and m i n i ng T w o - w hee l m o t o r v eh i c l e s M e t a l m o l d s T oba cc o S e r v i c e s r e l a t i ng t o a i r t r an s po r t R o lli ng s t o ck M i sc e ll aneou s S h i p s ( e xc ep t s t ee l s h i p s ) S e m i c ondu c t o r m a k i ng equ i p m en t S upp l e m en t a r y t u t o r i a l sc hoo l s , i n s t r u c t i on s e r v i c e s f o r a r t s , c u l t u r e and t e c hn i c a l sk ill s A i r po r t and a i r t r a ff i c c on t r o l ( pub li c ) ** S po r t f a c ili t y s e r v i c e , pub li c ga r den s and a m u s e m en t pa r ks R a il w a y c on s t r u c t i on E l e c t r i c po w e r f a c ili t i e s c on s t r u c t i on C e r e m on i a l o cc a s i on s S t ee l s h i p s M i sc e ll aneou s c i v il eng i nee r i ng and c on s t r u c t i on P ub li c c on s t r u c t i on o f r i v e r s , d r a i nage s and m i sc e ll aneou s pub li c c on s t r u c t i on P a ss enge r m o t o r c a r s P ub li c ad m i n i s t r a t i on ( l o c a l ) ** A g r i c u l t u r a l pub li c c on s t r u c t i on T e l e c o mm un i c a t i on f a c ili t i e s c on s t r u c t i on P ub li c c on s t r u c t i on o f r oad s H o t e l s P ub li c ad m i n i s t r a t i on ( c en t r a l ) ** S c hoo l l un c h ( pub li c ) ** S c hoo l l un c h ( p r i v a t e ) * N u c l ea r f ue l s R e s i den t i a l c on s t r u c t i on ( w ooden ) R e s i den t i a l c on s t r u c t i on ( non - w ooden ) N on -r e s i den t i a l c on s t r u c t i on ( w ooden ) N on -r e s i den t i a l c on s t r u c t i on ( non - w ooden ) L i f e i n s u r an c e H ou s e r en t H ou s e r en t ( i m pu t ed hou s e r en t ) S c hoo l edu c a t i on ( pub li c ) ** S c hoo l edu c a t i on ( p r i v a t e ) * S o c i a l edu c a t i on ( pub li c ) ** S o c i a l edu c a t i on ( p r i v a t e , non - p r o f i t ) * M i sc e ll aneou s edu c a t i ona l and t r a i n i ng i n s t i t u t i on s ( pub li c ) ** M ed i c a l s e r v i c e ( ho s p i t a li z a t i on ) M ed i c a l s e r v i c e ( e xc ep t ho s p i t a li z a t i on ) M ed i c a l s e r v i c e ( den t i s t r y ) M ed i c a l s e r v i c e ( pha r m a cy d i s pen s i ng ) S o c i a l i n s u r an c e ** S o c i a l w e l f a r e ( pub li c ) ** S o c i a l w e l f a r e ( p r i v a t e , non - p r o f i t ) * S o c i a l w e l f a r e ( p r o f i t - m a k i ng ) N u r s i ng c a r e ( f a c ili t y s e r v i c e s ) N u r s i ng c a r e ( e xc ep t f a c ili t y s e r v i c e s ) P r i v a t e non - p r o f i t i n s t i t u t i on s s e r v i ng hou s eho l d s , n . e . c . * B a r be r s hop s P ub li c ba t h s M o v i e t hea t e r s S t ad i u m s and c o m pan i e s o f b i cyc l e , ho r s e , m o t o r c a r and m o t o r boa t r a c e s A m u s e m en t and r e c r ea t i on f a c ili t i e s P r odu c t i v i t y G r o w t h Figure 6: Sectoral productivity growths measured under Cascaded CES functions, displayed in stream order. Sectors with positive productivity growths includeMiscellaneous amusement and recreation services ( . ), Tobacco ( . ), and Metallic ores (0.219). Sectors with negative productivity growths include Whiskeyand brandy ( − . ), Personal Computers ( − . ), and Electronic computing equipment (except personal computers) ( − . ). These numbers are almostidentical to the log of Törnqvist indices obtainable via (14). Figure 7: Scaled Euclidean distances d jk between net multipliers of industries ( µ j , µ k ) in original order (left) and in stream order (right), for the current state. The projected equilibrium price π = ( π , · · · , π n ) is the solu-tion to the following system of equations: π = H ( π , p ) (cid:104) θ (cid:105) − (cid:104) z (cid:105) − The solution can be obtained by iteration, if the given produc-tivity shock is enhancing, i.e., z j ≥ for all j , since the do-main will be contracting during this iteration process. The iter-ation begins with the current equilibrium price; thus, the initial ( k = 0) guess of the equilibrium price is w (0) = p . The recur-rence formula for w can be described as follows: w ( k +1) = H (cid:16) w ( k ) , p (cid:17) (cid:104) θ (cid:105) − (cid:104) z (cid:105) − (16)Specifically, the domain ( , p ) will be contracted p ≥ p (cid:104) z (cid:105) − = H ( p , p ) (cid:104) θ (cid:105) − (cid:104) z (cid:105) − ≥ H ( , p ) (cid:104) θ (cid:105) − (cid:104) z (cid:105) − > Hence, (16) is a contraction mapping that converges monoton-ically to the equilibrium such that p ≥ w ( k ) → π > . Notethat we only used the monotonicity of H to show the conver- gence of the process (16) for the enhancing case (i.e., z ≥ ).For a non-enhancing case, we need a guarantee that an equilib-rium exists, which is provided if H is a concave mapping Kras-nosel’ski˘ı (1964); we may then take any initial guess to arrive atthe equilibrium via (16).The social benefit of innovation (in terms of enhanced pro-ductivity z > ) relative to the current state must be the reducedprojected price of commodities π < p . In this study, we evalu-ate such social benefit by the increase in earnable final demandfor the same total amount of the primary input. In other words,we evaluate innovation in terms of how much more net outputcan be provided from the same total amount of primary input.For later convenience, we introduce the projected input–outputcoefficients ex post of z over current state as follows: m = p ∇ H ( π , p ) (cid:104) π (cid:105) − (cid:104) θ (cid:105) − (cid:104) z (cid:105) − M = (cid:104) π (cid:105) ∇ H ( π , p ) (cid:104) π (cid:105) − (cid:104) θ (cid:105) − (cid:104) z (cid:105) − Then, our innovation assessment problem can be specified asthe following: max δ δ s.t. m [ I − M ] − (cid:104) π (cid:105) f δ ≤ b [ I − B ] − f where f is the current-state net output (or final demand) ob-served in the form of a column vector, while δ is the scalar tobe maximized. Note that the right-hand side of the constraintis the total of primary inputs of the current state, and the objec-tive term is the total earnable final demand at the projected statewhose commodity-wise proportion is fixed at the current state.The solution of the problem can be obtained by the followingcalculation: δ ∗ = b [ I − B ] − fm [ I − M ] − (cid:104) π (cid:105) f Further, we may examine the distribution of the primary inputs8 .8 0.6 0.4 0.2 0.0 0.0 0.2 0.4 0.6 0.8 Figure 8: Hierarchical clustering dendrograms by the scaled Euclidean distances d jk between the net multipliers of sectors for the reference (left) and the current(right) states. Each leaf of a tree corresponds to a sector. Network transforma-tion regarding d jk can be monitored by the difference in the order of sectorsconfigured by the dendrogram. The same leaves (sectors) of the two trees areconnected by a gray line. Distance Change F r equen cy Figure 9: Distribution of changes in scaled Euclidean distances between the netmultipliers of A and B . in the current and projected states, as follows: ∆ v = b [ I − B ] − (cid:104) f (cid:105) − m [ I − M ] − (cid:104) π (cid:105) (cid:104) f δ ∗ (cid:105) Here, ∆ v denotes the redistribution of the primary inputs, whoseentries will sum to zero. Moreover, we may calculate the eco-nomic welfare gain provided by z , as the gain in the final de-mand, as follows: ∆ f = ∆ f = ( δ ∗ − Note that is the aggregated demand, which equals the GDPof the current state economy. Figure 10: Hierarchical clustering dendrograms by d jk between the net mul-tiplier of sectors for the current state (left) and the projected state given byRMC110 (right). The left tree is identical with the right tree of Fig. 8 Here, we examine the structural propagation effect of anRMC110 innovation (where “RMC” refers to the “Ready mixedconcrete” sector and “110” indicates a 10% productivity increase)injected into the current state economy. RMC110 is specificallythe following vector: z = (1 , · · · , , z RMC , , · · · , , z RMC = 1 . (17)Note that RMC appears 145th in the original order and 244thin the (upstream-first) stream order. The empirically observedTörnqvist index value is approximately . for the interval be-tween the reference and current states. By recursive means, π isobtained via (16) under (17), and accordingly, the ex post equi-librium structure ( m , M ) is obtained as well. We calculate thescaled Euclidean distances between the net multipliers of M andcompare them with those of B in Fig. 11. Since RMC110 is avery small injected productivity shock , the scaled Euclideandistances of the output multipliers have changed just slightly to-wards sparsity. In Fig. 10, we show the tanglegram compris-ing the results of hierarchical clustering for the current and pro-jected distance metrics of the correlations between the multipli-ers. Note that the right-hand tree of Fig. 8 is identical to theleft-hand tree of Fig. 10. The clusters have transformed to a cer-tain extent between the reference and current states, as well asslightly between the current state and the projected state givenby RMC110.Figure 12 displays the log-absolute difference between cur-rent and projected primary inputs ln (cid:107) ∆ v j (cid:107) in stream order, un- Japan’s GDP of the current state was , BJPY, while the output of theRMC sector was , BJPY. Thus, 10% of the current RMC output amountsto 0.0243% of the current GDP. e+011e+031e+05 -0.2 0.0 0.2 0.4 0.6 0.8 Distance Change F r equen cy [ l og sc a l e ] Figure 11: Distribution of changes in scaled Euclidean distances between thenet multipliers of B and M .Table 2: Evaluation of structural propagation of RMC110 (unit: millionJapanese yen). y RMC is the gross output of the RMC sector, which is the sumof final demand and the intermediate inputs total. v RMC is the value added ofthe RMC sector. Overall, non-structural propagation (Leontief) gains twice the10% amount of the current gross output of RMC sector (i.e., , MJPY),whereas structural propagation (with Cascaded CES) gains thrice this amount. unit: [MJPY] Current Projected(Leontief) Projected(CCES) y RMC , , 144 1 , , 056 6 , , v RMC , 835 379 , 682 2 , , f ± , 452 +334 , der Cascaded CES (open dots) and Leontief (filled dots) net-works. Note that Leontief network eliminates any sort of tech-nology substitution (i.e., assuming σ ij = 0 for all i and j ).Structural propagation under Cascaded CES network clearly out-performs non-structural propagation under Leontief meta struc-ture.Table 2 summarizes the welfare calculation with regard tostructural propagation of RMC110 applied to the current state.Here, y RMC and v RMC indicate gross output and value added,respectively, of the RMC sector. The breakdown of v RMC indi-cates that primary inputs are to be redistributed to sectors such asEngine, Ready mixed concrete, and Wholesale trade, from sec-tors such as Public construction of roads, Miscellaneous civilengineering, and Civil engineering and construction services,in order to gain an economic welfare equivalent to +334 , MJPY, given by RMC110. Further, note that a crude evaluationof RMC110 is to note first that 10% of the current state outputof RMC amounts to +113 , MJPY. This is the amount thatthe RMC sector is able to gain initially from RMC110. In con-trast, the propagation effect that pertains to the total amount ofwelfare gain, even without technological substitution (Leontief),can be almost two-fold larger ( +206 , ), and would be three-fold larger ( +334 , ) if we were to consider the full propaga-tion effect, inclusive of the potential structural change. 4. Concluding Remarks Unlike utility, production is a step-by-step practice. Accord-ingly, we took into account the configuration of the processes Sectors (stream order) P r opaga t i on Figure 12: Each dot represents the sectoral propagation of RMC110 in termsof log-absolute difference between current and projected primary inputs i.e., ln (cid:107) ∆ v j (cid:107) . Open dots correspond to structural propagation under Cascaded CESwhereas filled dots correspond to non-structural propagation under Leontief net-works. that underlie a production activity. We broke down the config-uration of production into activities that comprised binary andnested processes. For empirical purposes, we applied a CESfunction for each nest process, resulting in a Cascaded CESfunction for modeling industrial production. Moreover, we usedthe sequence of inputs obtained by the triangulated input–outputincidence matrix for modeling the intra-sectoral sequence ofprocesses, as we observed a stylized hierarchy among the inter-mediate processes spanning the empirical input–output transac-tions.By using linked input–output tables as the two-point datasource, the elasticity parameters of Cascaded CES functions wereresolved synchronously though the calibration of productivity.At the same time, substantial negative nest elasticity parametersand productivity growths were observed. These amounts beingnegative may be attributed to bias in price measurement in thepresence of qualitative changes. Still, these parameters causedno problems, as far as the general equilibrium analysis was con-cerned. The credibility of the calibrated system is validated bythe complete replication of the two observed states portrayed inthe linked input–output tables. Naturally, the analysis becomesmore decisive when we advance the study to work more on cap-ital and growth, quality considerations, and international trade,all of which remain for further investigation. Appendix A. Two-stage Cascaded CES calibration We start with a following two-stage Cascaded CES function,where W = w . c = t − W W = (cid:0) λ w − σ + Λ W − σ (cid:1) − σ W = (cid:0) λ w − σ + Λ W − σ (cid:1) − σ able A.3: Sample data (shaded values correspond to χ , χ , χ and p ) andthe calibrated parameters. a b p σ i θ Törnqvistoutput 0.8 . 946 0 . input 2 0.3 0.2 1.2 1.88input 1 0.5 0.7 0.6 3.54input 0 0.2 0.1 0.9 ! p0 Figure A.13: Equation (A.1) is solved at t = 0 . ≡ θ . This is the two-stage ( n = 2 ) version of the function defined in(2) and (6). As regards (9), the share parameters are equalt tothe cost shares at the reference state: λ = a , λ = a − a The following exposition shows the procedure for parametercalibration via backward induction: W = tp = W ( t ; χ ) σ = ln b a + ln W p ln W p = σ ( t ; χ , χ ) W = (cid:18) W − σ − a p − σ − a (cid:19) − σ = W ( t ; χ , χ ) σ = ln b a + σ ln W W + ln W p ln W p = σ ( t ; χ , χ , χ ) W = (cid:18) W − σ − a p − σ − a (cid:19) − σ = W ( t ; χ , χ , χ ) Hence, we can calibrate t for the final condition, which is thetwo-stage version of (12): W ( t ; χ , χ , χ ) = p (A.1)For demonstration purposes, we calibrate t on the data given inTable A.3. We use the data, i.e., χ i = { a i , b i , p i } and p forcalibrating t . Note that a and b are not used in the calculation,because the shares are degenerate, i.e., (cid:80) i s i = 1 . 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