Rensselaer. Working Papers in Economics

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1 Rensselaer Working apers in Economics Department of Economics, Rensselaer olytechnic Institute, 8th Street, Troy, NY, 8-59, US. Tel: ; Fax: ; URL: Tracking Global Factor Inputs, Factor Earnings, an Emissions ssociate with Consumption in a Worl Moeling Framework Faye Duchin Rensselaer olytechnic Institute Stephen H. Levine Tufts University Number 74 November 7

2 Tracking Global Factor Inputs, Factor Earnings, an Emissions ssociate with Consumption in a Worl Moeling Framework by Faye Duchin Department of Economics Rensselaer olytechnic Institute Stephen H. Levine Department of Civil an Environmental Engineering Tufts University bstract. This paper presents a new approach for estimating the amount of carbon emboie in a prouct consume in a given economy, taking account of where the inputs to that prouct were extracte an processe all along the supply chain. The metho is generalie to apply to all factor inputs, incluing materials an energy, as well as pollutant emissions an can track not only the flows of factors an goos as imports an exports along the global supply chain but also the payments for these inputs mae by ultimate consumers along the global value chain. The new metho makes use of absorbing Markov chains that track ownstream an upstream flows. These chains are first escribe in terms of the mathematics of a one-region input-output moel an then generalie to the global framework of a multiregional worl economy. The paper also escribes the stanar way of solving this problem, which we call the Big metho, an inicates the main avantages of the Markov chain approach, namely that it is implemente without loss of information using a more compact atabase an can aress a wier range of questions, especially ones relate to the recycling of materials. Finally, the paper iscusses the parameter requirements istinguishing this type of ex-post analysis from moel-base exploration of alternative scenarios about the future an makes the case for combining the two. C6, C67, F8, Q56, Q57

3 Tracking Global Factor Inputs, Factor Earnings, an Emissions ssociate with Consumption in a Worl Moeling Framework by Faye Duchin Department of Economics Rensselaer olytechnic Institute Stephen H. Levine Department of Civil an Environmental Engineering Tufts University. Introuction Economists often portray an economy as a circular money flow, where proucers pay salaries to workers who in turn use their incomes to purchase consumption goos an services from the proucers. Goos are assume to move aroun the circle in the opposite irection from the money flows. While the circular flow concept has been use to escribe iniviual economies, it is equally applicable to the more complex system of the worl economy. This image accurately conveys the uality between the physical flows an the money flows that constitute any economy. However, while the money flows are entirely containe within the economic system, this is not true of the physical flows. rouction requires inputs of resources that are extracte from the environment; an uring both prouction an consumption, wastes are ischarge into the environment. While some ischarges o re-enter the prouction process, reuse of proucts an recycling of materials are outsie the scope of this paper. Thus, a typical ton of iron ore will follow a path through the system, ening up emboie in a range of consumer items, incluing househol appliances an vehicles. Starting from the other en, a typical family car traces its ultimate roots to not only the iron require in the course of its fabrication but also other materials an energy as well as labor an capital an is associate also with the carbon ischarge at each stage along the way. This paper presents a metho for quantifying where that ton of ore ens up, where that car came from, an where the carbon was emitte. It also tracks the associate money flows from consumption outlays to factor payments. The approach taken escribes these paths through an economy as absorbing Markov chains an shows, both symbolically an using a numerical example, how they are implemente using inputoutput ata. This framework supplements the mathematics of input-output moeling an is a fully general treatment for associating all factor inputs an all wastes generate with specific components of final consumption, in both physical quantities an money values.

4 4 The ability to track the supply chain for consumer goos has been recognie as vital for climate change policy. party to the Kyoto rotocol now commits to targets for the carbon emitte on its territory. mechanism consiere both more effective for reucing global emissions an fairer than imposing proucer responsibility is to target emissions emboie in a country s consumption, or its consumer responsibility. To calculate the latter quantity, it must be possible to compute the amount of carbon, an where in the worl it is emitte, associate with a given final eman. Over the past several years a literature has accumulate on the use of input-output moels to calculate a country s carbon emissions uner both consumer responsibility an proucer responsibility. The carbon associate with prouction is relatively straightforwar to estimate, but quantifying the carbon emboie in consumption is more problematic because it involves ientifying the countries of origin for imports an the arrangements surrouning prouction in these countries, incluing their own imports, then the arrangements surrouning prouction in the countries from which they in turn import, an so on. Techniques have been evelope for estimating the total carbon emboiment for given countries using multi-regional input-output analyses (see Lenen, ae, an Munksgaar 4; eters an Hertwich, forthcoming. Further simplifications permit estimates of emboie carbon, incluing for imports, base on ata for only the one or several given countries (see, for example, eters an Hertwich, 6. The main objective of these stuies has been to compare, for iniviual countries, the carbon emboie in their consumption an in their prouction. This paper evelops path-base methos base on Markov chain analysis to aress a relate family of questions for all countries simultaneously an for all factor inputs an pollutant ischarges, while also accounting for irect an inirect imports. Other kins of path analysis have been applie to ecomposing input-output matrices an multipliers (Defourny an Thorbecke 984; Khan an Thorbecke 989; Sonis an Hewings, 998; eters an Hertwich 6; Lenen 7, but none has aresse the question pose here or mae use of the absorbing Markov chains. Bailey et al. (4 a,b use environ analysis, a metho rawing on input-output an to a lesser egree Markov chain analysis an previously applie to ecosystems (atten an Finn, 979, to perform path analysis, ultimately focusing on system wie measures such as average path length in the entire system, system throughput, an measures of recycling. Markov chains have been use in the analysis of energy an biomass flows in ecosystems, incluing etermining a measure of recycling (Barber, 976. We know of one pioneering effort to formally use Markov chains to stuy the recycling of materials in inustrial systems (Yamaa et al. 6; Matsuno et al. 7. The Bailey, Yamaa, an Matsuno papers are narrowly focuse on the flows of specific factors, such as aluminum, an not on their emboiment in proucts an consumption goos. They o not aress the broaer question of relating prouction to consumption as o many input-output moels. This paper, by contrast, aresses just these broaer issues an in oing so provies a brige between an input-output analysis of the entire economy an a promising approach to stuying recycling.

5 5 Within the framework of input-output moels, the Leontief inverse matrix plays a privilege role. This is the matrix (I - that is the efining feature of the basic static input-output moel, (I x = y or x = (I - y, (- where y is a vector of final eman, is a matrix of input-output coefficients, an x is the vector of output require to satisfy y. The power of the Leontief inverse is that it captures not only the irect but also the inirect input requirements. Thus, if F is the matrix of factor requirements per unit of output, the Leontief inverse makes it possible to etermine the vector of factor inputs,, require irectly an inirectly to satisfy y: = Fx = F(I - y. (- If we substitute for F the matrix C measuring pollution generate, or emissions, per unit of output, then c = Cx quantifies emissions. (Note that this is the stanar representation for emissions in an input-output framework. While in fact F an C shoul be conceptually relate, this is a workable first approximation. To simplify the notation, we use F to represent either factor inputs or emissions. The paths joining final eman an factor inputs (or emissions, represente both in physical units an in money values, are the subject of this paper. If goos prices are applie to y an factor prices to, the associate paths of money flows implie by Eq. (-, i.e., those between factor incomes (or value ae an outlays for final eman, can also be tracke. These paths can be sai to escribe the supply chain an the value chain, respectively. Some specific instances of the general question to be aresse are: How much carbon is emitte in other economies to satisfy consumer eman in the US? Which countries consumption accounts for the carbon emitte in China? How much carbon is associate with the transport of goos? How much Mile Eastern oil is associate irectly an inirectly with consumption in the US? How much of the money outlays for total consumption in the US goes to paying royalties on this oil? What are the factor input requirements for refrigerators sol in the EU, an where o the factors originate? Of the money pai for these refrigerators in the EU, where oes it en up in factor payments? How much Chinese labor is emboie in consumption of specific other countries? What portion of labor income in China is pai by consumers in these other countries? If one writes the equivalent of Eq. (- for the worl economy an analyes the chain beginning with each factor in each region an the chain ening in an element in each region s final eman, one can solve this problem for all regions simultaneously. It is stanar to o this using what we will efine as the Big metho. However, it will be seen that there are avantages for formulating the problem instea as an absorbing Markov chain.

6 6 The remainer of this paper is organie as follows. In Section, we efine ownstream flows an upstream flows in the supply chain an the value chain for each prouct, taking account of the webs of interepenence among sectors; these concepts are formalie an quantifie in subsequent sections. bsorbing Markov chains that track ownstream an upstream flows in a single economy are introuce in Section, where they are escribe in relation to the mathematics of the one-region input-output moel. Section 4 moves on to a global framework an presents the algorithms for a similar analysis of the ownstream an upstream flows in a multiregional worl economy. numerical example for tracking the supply chain an value chain, both ownstream an upstream, in the global framework is provie in Section 5. Section 6 escribes the stanar way of solving the problem, namely the Big metho, an inicates the main avantages of the Markov chain approach. The funamental istinction between this analysis of one given outcome an a moel for analying alternative scenarios is iscusse in Section 7, which escribes a global multiregional moel that can provie the inputs for the kin of analysis aresse in this paper. The final section provies conclusions, an an appenix elaborates on some of the mathematical analysis.. Downstream Flows an Upstream Flows in rouction Chains The supply chain for a particular goo can be sai to begin with the extraction of raw materials an mobiliation of labor an other factors of prouction, continues through various stages of processing an fabrication, an ens with elivery of the finishe goo to consumers. From any point mistream in the chain, prouct output flows ownstream in the irection of the consumer, while the activities consiere upstream are those proucing intermeiate inputs an ultimately the factor inputs. proucer is concerne with securing upstream suppliers an ownstream customers. From the point of view not of iniviual proucers but of the economy as a whole, the terms have a similar significance: we are concerne with tracking consumer goos upstream to the factor inputs require for their prouction an with tracking factors of prouction ownstream to the consumer goos in which they are eventually emboie. While iniviual proucers are concerne mainly with securing their own inputs, in fact factors of prouction an other inputs are also require at every intermeiate stage in the upstream prouction chain. n input-output framework is neee to escribe an quantify all of the inirect relationships that join the chains in a web-like structure. In the global context we will be intereste in tracking a goo consume in one economy upstream to the factor inputs that were utilie in proucing it in the same an other economies. We will also wish to track a factor input extracte an use in a given economy ownstream to the consumption goos in the same an other economies in which it is embee, both irectly an inirectly. Money flows in the opposite irection from the material goos. Outlays by consumers for a particular goo in one country flow upstream to the owners of the constituent

7 7 factors of prouction in the same an other economies, such that the total outlay of all consumers purchasing a given prouct is equal to the incomes receive by the owners of the factors emboie in the prouct. Moving in the other irection, the earnings of a particular factor in a given economy are pai by all ownstream users of the factor an ultimately by consumers of the goos in which the factor is embee. In the case where we are tracking not factors of prouction but wastes like carbon emissions, this waste ischarge may or may not be price (e.g., taxe. In the event, for example, of a carbon tax, the logic of the last paragraph hols. This means that the outlays of consumers can be trace upstream to factor payments an carbon taxes associate with all or part of a final bill of goos, an the earnings from a carbon tax can be trace ownstream to all the ultimate consumers whose outlays contribute to paying it. If the carbon emissions are not taxe, they simply have a ero price an can be hanle formally like any price factor or waste.. The Input-Output Moel an bsorbing Markov Chains Before consiering the multiregional global framework, we escribe the simpler context of a one-region economy with n sectors, each proucing a characteristic goo, an k factors of prouction. ll variables an coefficients are measure in relevant physical units, such as tons of steel or kwh of electricity per ton of steel. Money values are, of course, a special case of physical unit. Our examples are in general physical units, an unit prices are subsequently introuce explicitly into the analysis. In this section factors are associate with final eman first using the familiar inputoutput equations. Then we introuce an absorbing Markov chain moel an solve the same problem. Markov chains can be use to moel systems where the selection of the next state on a path is a function only of the present state an the transition probabilities associate with the branches available at the present state. In an economic system ifferent factors, an ifferent units of a given factor, take ifferent paths through the system. Thus a proportion of each factor is use irectly in sector, another proportion in sector, an so on. These proportions, of course, a to., an it is this property that a factor (or an intermeiate goo is entirely istribute among a clearly efine set of irect uses -- that allows us to use a Markov chain moel for this system. We interpret the Markovian transition probabilities as proportions of factors an goos, an it is in terms of these proportions that we will work. The section ens with a formal escription of the Markov moel an its relationship to the input-output moel in the case of a oneregion system. The Input-Output Moel Starting from Eqs. (- an (-, we istinguish the requirements for a particular factor, say i, corresponing to iniviual components of final eman by replacing the vector y by the iagonal matrix ŷ. If we call the resulting matrix Φ, then

8 8 Φ = F(I - ŷ, (- where each row of the k x n matrix Φ quantifies the istribution of one factor among the n components of final eman, while the columns escribe the requirements of all factors to satisfy each component of final eman. For a numerical example, consier an economy escribe by, F, an y: F y 8 9 Then one can etermine x, = , an Φ Reaing across the first row of the Φ matrix shows the ways in which the require 8.5 units of the first factor are istribute:. units are require to eliver the first goo to consumers, 6.7 to eliver the secon goo, an 8.5 to eliver the thir goo. The first column of the same matrix shows that elivering the first goo to consumers requires. units of the first factor an.6 units of the secon factor. Thus the Φ matrix provies the solution to our problem. The bsorbing Markov Chain Moel Next consier an analysis base on representing this economic system as a Markov chain. The system structure of our example economy is shown in Figure -, where the network noes represent the states of the system. The noes labele F refer to factors of prouction, X to outputs, an Y to final emans.

9 9 x ij = a ij x j - flow from sector i to sector j x Y y i - final eman for sector i rj = f rj x j - flow of factor r to sector j X y x x x x F F x x X X x x Y y y Figure -: Flows of Goos an Factors in a Three-Sector, Two-Factor Economy Y The objective is to follow each unit of a factor of prouction as it progresses through the system shown in Figure -, eventually ening by satisfying final eman. Because all utilie factors eventually en up embee in one or more components of final eman, the state of being part of final eman is referre to as an absorbing state an the Markov chain is referre to as an absorbing chain. (We remin the reaer that wastes an recycling are not consiere in this paper. The other states, corresponing to factors an goos, are referre to as transient states. Since the Markov chain moel quantifies each flow out of a state as a proportion of the total flow out of that state, we nee to etermine exactly what proportion of factor r is use irectly by sector j. The number of units of factor r use by sector j is rj, an the total use of factor r is r = r + r + r, so the proportion in question will be: f rj = rj / r (- This is the rj th term in the k x n matrix F. We now efine so-calle irect transition matrices of the form (a,b, where a an b are vectors representing two states, an the ij th entry of (a,b is the portion of flow out of state a i going irectly to state b j. By this efinition, (,x = F, the matrix of proportions of irect flows of factors to sectoral outputs x. Next we consier the proportional flows from the total supply of proucts, x, to final eman, y, taking into account that an intermeiate goo can flow from one sector to another any number of times before being absorbe in final eman. In the absence of

10 imports, the omestic supply is simply total omestic output, x. The proportion of supply that flows from the i th sector irectly to the i th component of final eman is simply the ratio y i /x i, an no output flows irectly from the i th sector to any other component of final eman; that is, the latter proportion is. In matrix form these proportions are elements of the n x n iagonal matrix, (x,y = x - ŷ ( = ŷ x -. Next we examine the intermeiate use of the total supply of output of each goo. The proportion of total supply, or output, flowing irectly from sector i to sector j is a ij = x ij /x i. This is the ij th term in the n x n matrix, where of course = (x,x. fter this first roun of irect flows among sectors, the intermeiate goos may be elivere either to final eman or to other sectors, an so on for all subsequent rouns. The matrix of proportional flows from x to y, incluing both irect an inirect flows, is therefore (x,y = (x,y + (x,x (x,y + (x,x (x,x (x,y + = ŷ x - + ŷ x - + ŷ x - + = (I ŷ x - = (I - ŷ x -. (- The evelopment of Eq. (- is base on the same logic as the representation of the Leontief inverse (I - as the matrix power series (I + + +, an the relation between an is spelle out below. Combining Eqs. (- an (- leas to the matrix of overall proportional flows from factors to final eman (,y = (, x(x,y = F (I - ŷ x -. (-4 The rows of this matrix istribute proportions of each factor over all components of final eman; it is reaily seen that each row total equals. since (,y is the prouct of two matrices that both have this property. Finally, to calculate the actual amounts of factors, we calculate the matrix Θ = (,y = F (I - ŷ x -. (-5 Evaluating Eq. (-5, we get.4 (,x = ,.455

11 (x, y , (,y = ,.46 an finally Θ It shoul come as no surprise that this Θ = Φ, the result we obtaine by the familiar input-output metho. It shoul be pointe out that we can express Eq. (-5 in terms of the input-output matrices an F by recogniing that f rj = f rj (x j / r an a ij = a ij (x j /x i. Therefore where X is the intersectoral flow matrix. F = - F x, an (-6a = x - x = x - X (-6b The Formal Markov Chain Moel The familiar input-output manipulations on a atabase measure in a common unit (namely money values or mass are a special case of the more general Markov chain analysis. When the flows are measure in mixe units, traitional input-output moels are not Markov chains. In this case the input-output moel can be solve provie that the matrix satisfies the Simon-Hawkins conitions, meaning that the economy oes not operate at a loss. The Markov chain moel in mixe units can be analye even for an economy that epens upon subsiies. We conclue this section by presenting a formal absorbing Markov chain analysis of this system to emonstrate its aitional properties. In Section 4, we will exploit the greater generality of the Markov metho to link factors to final eman in the more complex system of the global multiregional economy. In a Markov chain the system is represente by a number of states an the one-step transition probabilities, what we are referring to as the irect proportional flows, from

12 one state to another. These irect transition probabilities are represente for the entire system in what is calle the systemwie transition matrix, enote as M. Thus m ij is the probability that state i will transition in the next step to state j. In an absorbing Markov chain analysis, the states are partitione into absorbing states an transient states, an the transition matrix is put into canonical form by orering the states starting with the absorbing states first. Then M can be partitione as follows, I M. (-7 R Q The absorbing states have the property that once entere they can never be exite. Thus if state i is an absorbing state m ii =. (since % of the flow remains in the same state an m ij = for j i. The ientity matrix I an the matrix, the latter being rectangular, reflect these properties. The irect proportional flows among the transient states are represente by the matrix esignate Q an the irect proportional flows from transient to absorbing states by the matrix esignate R. The absorbing chain is sai to have a funamental matrix (Kemeny an Snell, 976, efine as N = I + Q + Q + = (I Q -. (-8 This matrix contains information about the average number of times a unit of a factor of prouction or a sectoral output passes through each transient state before reaching an absorbing state. Since R contains the irect proportional flows from those transient states to the absorbing states, the matrix B = NR (-9 contains the proportion of the output of each transient state that ultimately reaches each absorbing state, either irectly or inirectly. The B matrix is therefore what we sought in the analysis an the numerical example above. In our example the final emans (y correspon to the absorbing states while the transient states are the factors ( an the sectoral outputs (x. Noting that, an making use of the irect transition matrix notation, (y,y = I, the partitione M matrix is

13 y x F I (- (x,x (x,y x ( (y,y M, Therefore, F Q (- an the funamental matrix is (I (I F I N. (- The matrices in the first row an column of N correspon to the factors while the matrices in the secon row an column correspon to the sectoral outputs. Finally, y x R (- an y x (I y x (I F B. (-4

14 4 The matrix constituting the first block of B correspons to the proportional flows from the factors to final eman. We will efine this as B = F (I - x - ŷ. (-5 The secon block matrix of B correspons to the proportional flows from the sectors to the final emans an is efine as B x = (I - x - ŷ. (-6 B is the matrix of interest, an we see that in fact it is exactly what we previously labele (,y. The other component of B, B x, correspons to (x,y. In this section we have shown how the absorbing Markov chain analysis replicates the results of the more traitional input-output analysis. However, by virtue of Eq. (- being a funamental matrix of an absorbing chain, we can erive aitional results escribing the paths taken by the factors as they flow through the system. s one example, the row sums in N corresponing to the factors are the average path lengths taken by those factors before being absorbe as final eman. Similarly, the row sums corresponing to the sector outputs are the average path lengths taken by those outputs. In the form presente in Eq. (- this is an average across all the final emans, but the funamental matrix can also be moifie to escribe the average path length until absorption by a specific component of final eman, that is, a specific absorbing state. Note that while (I - has the property that its row sums are average path lengths, the same is not true for the row sums (or the column sums when flows are measure in physical units of (I The Global Supply an Value Chains as bsorbing Markov Chains In this section we generalie the concepts presente in the one-region moel of Section in orer to apply them to a worl economy consisting of m regions, each escribe in terms of n sectors an k factors of prouction. In this global framework, we explicitly escribe the role of imports an exports in each region s economy. The inter-regional transport of trae flows itself calls for factor inputs an generates emissions, an these will be explicitly aresse. Note that this transport is require per unit of trae flow between specific regions, not per unit of one region s output, so its representation requires special treatment in an input-output framework. We treat the inustries proviing international transport as sectors in each regional economy an for now represent the eman for their output as part of the final eman of importing regions. Later the treatment of this istinctive sector will be further elaborate. Figure 4- below is a network moel illustrating the flows of goos an services, measure in physical units, in a -region system.

15 5 Region Final Deman Region Factors of rouction u = y + t Region Final Deman Region Factors of rouction u = y + t Region,F e e Region,F e e e e Region,F Region Factors of rouction u = y + t Region Final Deman y consumer eman t inter-regional transport eman e import/export technical matrix T inter-regional transport matrix F factor matrix t = T e + T e t = T e + T e t = T e + T e Figure 4-: Flows of Goos an Factors in -Region Economic System Consumer eman an inter-regional transport eman are combine into one vector, u, an referre as final eman. The u s an e s are n-element vectors, one element for each sector, an the s are k-element vectors, one for each factor. The noes representing the regions are more complex than the others because they involve the economic structure characterie by the matrices of intermeiate inputs an of factors of prouction, an F, consiere in Section. Figure 4- illustrates the internal structure of the noe for region ; the noes for regions an are similarly organie. The intermeiate an total output vectors are w an x, respectively, while is the vector of total omestic supply incluing both omestic prouction an imports.

16 6, F x w + u e e e e Figure 4-: Details of Region Noe for Flows of Goos an Factors s in the previous section we will analye the system in Figure 4- as an absorbing Markov chain, with the final eman noes treate as absorbing states. Since each noe in Figure 4- represents a set of noes, one for each economic sector or factor of prouction, the flows associate with the branches are all vectors. We first evelop the irect transition matrices, (a,b, an then incorporate them into a formal absorbing chain moel. aralleling the escription in Section, we begin with the flow in region g from g to x g. The proportion of the r th factor flowing to the j th sector is grj f grj n. (4- s This is the rj th term in the k x n matrix F g = ( g,x g. The proportion of the i th sector s intermeiate output flowing irectly to the j th sector is x gij /w gi, where w gi is the total intermeiate use of the output of sector i. This is the ij th term of the n x n matrix (w g,x g = where X g is the intersectoral flow matrix for region g. grs ŵ - X g g, (4-

17 7 The proportion of x g flowing irectly to g, total omestic supply, is clearly (x g, g = I, (4- the n x n ientity matrix, since each component of x g is irectly an completely incorporate in the corresponing component of g. The same is true for (e hg, g for h g. The proportion of g flowing irectly to w g is the proportion of each unit of supply of g that is incorporate into w g, an is therefore ( g,w g = g ẑ - ŵ. (4-4 g The same relationship hols for g to e gh, h g, as well as g to u g (recalling that u g = y g + t g. Flows from region g to region h are represente as Finally, ( g, h = ( g,e gh (e gh, h ẑ - ê I = g gh = ẑ - g gh ( g,x g = ( g,w g (w g,x g ẑ - ŵ ŵ - X g = g ê. (4-5 g g = ẑ - X g g = g, (4-6 where x gij a gik. (4-7 gi is, as in Section (see Eq. (-6b, the irect istribution of output to ifferent sectors as a proportion of total omestic supply. In the multiregional case, omestic supply inclues imports as well as omestic output: x e. (4-8 g g h g hg Noes, the final emans, contain the n absorbing states. For each of these noes there is only one branch leaving the noe an it returns into the same noe. (These branches, not shown in Figure 4-, are shown in Figure 4-. The matrix of proportions of flow associate with this single branch, (u,u, is the n x n ientity matrix I.

18 8 We now have etermine all the matrices require for the absorbing Markov chain. To put the system in canonical form, the absorbing states (i.e., the final emans, are labele noes -; an the transient states are noes 4. We recast the network moel in Figure 4- as a Markov chain, shown in Figure 4-. Each region is now represente by two noes corresponing to the vectors x an, the latter incluing imports. Vector States u in Region u in Region - u in Region 4 - in Region 5 - in Region 6 - in Region (,u (u,u (x, 8 (,x 7 x in Region 8 - x in Region 9 - x in Region in Region in Region in Region (,x 5 (, (, (, ( (u,u (,, (,u (,u (u,u (, (,x (x, (,x 7 (x, 4 (,x (,x 9 6 Figure 4- bsorbing Markov Chain Moel for -Region System Showing the Flows of Goos an Factors. Each (, is a Direct roportional Flow Matrix The canonical form for the matrix of proportions of irect flows of an absorbing chain, as escribe in Section, is shown in Figure 4-4. For the multiregional system, the imension of the I matrix is (nm x nm, the matrix is (nm x m(k+n, the R matrix is (m(k+n x nm, an the Q matrix is (m(k+n x m(k+n. Recall that the R matrix escribes the irect proportions of flows from the transient states (the s, s an x s to the absorbing states (the u s, an the Q matrix escribes the irect proportions of flows between the transient states. ll row sums of M equal.: this is evient for the first nm rows, an also hols for the row sums of R plus Q, since R contains the portions of omestic supply of goos ( istribute irectly to final eman an Q contains the portions use as intermeiate inputs an the portions exporte.

19 9 M = I O R Q I irect proportional flows from between absorbing states (ientity matrix irect proportional flows from absorbing states to transient states (ero matrix R irect proportional flows from transient states to absorbing states Q irect proportional flows from transient states to transient states Figure 4-4: Canonical Form of bsorbing Chain Direct roportional Flow Matrix For our -region system the M matrix is shown in Figure 4-5. The first of these matrices is the irect representation of Figure 4- an is partitione in accorance with its canonical form into the I,, R an Q matrices, with the last three further partitione. The matrix M is sparse: a number of the partitions are matrices, an others are iagonal matrices.

20 M =, (, ( x, (,u (, (, ( x,,u (, (, ( x, (,u (, (x, (x, (x x, ( x, ( x, (,u (u,u (u,u (u = e e u e e u e e u I I I F F F I I I Figure 4 5: -Region roportional Flow Matrix in Canonical Form for the Flow of Goos an Factors

21 s in Section, we wish to etermine the absorption matrix B = NR. To exploit the wellefine structure an sparsity of M, we write: R u, Q F I e an B B B x. B Then we are able to express B as follows: B = (I Q - R (I QB = R B QB = R. Solving for B we obtain: This leas to B F B x =, B x - B =, an B B x ẑ - ê B = ẑ - û. B = F (I - ẑ - ê - ẑ - û, an B x = B = (I - ẑ - ê - ẑ - û. (4-9a (4-9b (4-9c (4-a (4-b We note in Figure 4-5 that is a block-iagonal matrix while the submatrix we have calle ẑ - ê has eroes in the iagonal blocks. If we efine à = + ẑ - ê, then à is the mn x mn matrix with the proportional istributions for regional prouction matrices own the iagonal an the information on exports in off-iagonal blocks. We can rewrite the key equations as follows: B = F (I à - ẑ - û, an B x = B = (I à - ẑ - û. (4-a (4-b These equations generalie Eq. (-5 an (-6 to the case of multiple regions that trae among themselves. To calculate the amounts, an not just the proportions, of factors associate with each component of final eman, we construct the augmente vector of multiregional factor use

22 an then, parallel to Eq. (-5, Φ = B = F (I Ã - ẑ - û. (4- Finally we return to the important fact that factors of prouction are require, an carbon an other pollutants emitte, in carrying internationally trae goos. These requirements have been accounte for but combine with consumer eman in the u vector. We can o this without istortion or loss of information since there is ero consumer eman for the output of the inter-regional transport sectors. However, as our goal is to associate all factor inputs an pollutant emissions with consumer eman for the goos-proucing sectors, it will be necessary to appropriately reallocate them to final eman for the outputs of these other sectors. This can be one by appropriately moifying the Φ matrix using a proceure that will be escribe an illustrate in the numerical example of Section Region Numerical Example In this section we emonstrate the absorbing chain approach through a -region example. Each region s economy is escribe in terms of four sectors: ( agriculture, ( manufacturing, ( oil extraction, an (4 inter-regional transportation. There are three factors of prouction: ( labor, ( crue oil, an ( lan. The illustrative ata are such that region is a stylie representation of a inustrialie economy, region of an agricultural economy, an region of a non-inustrialie economy that is well-enowe with oil. It is assume that not only the parameters but also the values of all variables are known: the implications of this assumption are the subject of Section 6. The specifics of the example follow, starting with the matrices of intermeiate an factor inputs per unit of output with all quantities, unless otherwise specifie, measure in physical units: F F F Final eman an output in each region are:

23 y 5 y 6 8 y an x 75 x 98.7 x. The vectors of factor use are: , 6, an Trae goos are transporte between regions with the requirements for transport of each unit of imports to region j from region i escribe by entries in the rows of the matrix T ij (shown below corresponing to ifferent transport sectors a single such sector in the example below. Each entry in that row (the 4 th row, measure in ton-kilometers, is the prouct of the mass of the goo transporte multiplie by the istance between the origin region an the estination region. Thus for each unit of agricultural goos importe from region to, or region to,. ton-km of inter-regional transport is require. For heavier manufacture goos, the corresponing figure is.5 ton-km T T.5..4 T T...48 T T The trae vectors quantify the flows of goos an services among the three regions: 5.5 e e e 67.8 e 8 e 7.5 e. The unit prices of goos, which reflect ifferences in transport costs, are:

24 4 p p p, an the per-unit factor prices an scarcity rents are π 5 π.5 5 π an r r. 5 r, respectively. In this case, there is a scarcity rent only on lan in region. When there are no scarcity rents, or when they are presume to be inclue in the factor prices, the π vectors alone suffice. (The reason for incluing explicit scarcity rents becomes clearer in Section 7, where the Worl Trae Moel is iscusse. The matrices B, B x, an B are compute as escribe in Section 4. The row sums of B are equal to., except for those rows corresponing to factors that were not utilie, since each row shows the proportions of flows, both irect an inirect, from a given factor in one region to each component of final eman in all regions. B The columns of B correspon to the components of the final eman vectors u, u, an u, each with four components: agriculture, manufacturing, oil extraction, an international transportation. The nine rows correspon to the components of the vectors,, an, each with three components for labor, crue oil, an lan. Thus, the first row shows that 44% of the labor in region (the sum of the first 4 entries is require to satisfy its own final eman both irectly an inirectly. That means that 56% of the region s labor is emboie in the consumption goos of the other regions. Note that in this very aggregate example, regions an extract no crue oil an the oil-rich region uses no lan for agricultural prouction, as reflecte by the rows of eroes in B. Note

25 5 also that rows 4 an 6, corresponing to an, the istribution of its utilie lan an labor, are ientical because region prouces only one output. The same is true for rows 7 an 8. Utiliing Eq. (4-, we can calculate the Φ matrix in physical quantities. Note that the row sums in B that are correspon to the failure to use specific factors an therefore get multiplie by in this operation: Φ Each row of Φ shows how a particular factor in a given region is istribute among all components of final eman in all regions, while each column shows how many units of each factor originating in all regions are require to meet final eman for a specific goo in a particular region. Naturally, the vector of row sums is exactly equal to the vector. Columns can in general not be totale since factors may be measure in ifferent units. Now we examine the istribution of income in this multiregional system. The relevant matrices are etermine by multiplying each row of B, an each row of Φ, by the region-specific factor price, incluing any scarcity rents, that is, by the relevant component of (π i + r i for region i; we call the resulting matrices B W an Φ W : B W

26 6 Each row sum of B W is equal to the earnings of one unit of that factor, an the row shows the portion of the unit earnings pai by consumers of each goo in each region. Φ w Each row total of Φ W shows the total earnings of one factor in each region, an iniviual entries in the row show the source of the payment in the consumption of a particular final goo in some region. For example, the total of row is 7.7 money units, say thousans of ollars; this means that workers in region earn $7,7, of which $75,8 comes out of consumption outlays in the same region (the sum of the first 4 figures in the row. Most worker income in region is associate with manufacturing, amounting to over $, of the total. The largest factor earnings (i.e., the highest row total are for lan in region, which takes in $75,; this is not surprisingly the region that specialies in agriculture. Total income in region is $8,6, the sum of the first row sums. Now assume that the thir factor correspons to water pollution instea of lan inputs an that the ischarge of a unit of pollutant is taxe at ifferent rates in the ifferent regions. Then we see that total worl taxes amount to $76,5 (the sum of row-sums, 6 an 9, most of which are pai by region, while these ischarges are either insignificant in quantity, or not taxe, in region. Most of the tax is pai by consumers of agricultural goos, mainly the large number of them in region. Each column total of Φ W measures the outlays for all factors associate with the consumption of a particular goo in a given region, an the sum of all column totals for a given region is the total value of consumption outlays, p i T y i. Thus consumption of agricultural goos in region amounts to $, (column sum, most of which goes to pay lan owners in region ($6, an workers in region ($5,. The sum of all row totals for a given region is the total factor income, or the (supply-sie GD for that region. The sum of all column totals for each region is the value of total consumption outlays. The latter iffers from the former by the value of net exports. For example, region has total factor earnings of $84, but nees to lay out $475, for consumption; this eficit is explaine by the fact that it is a net importer, an the value of its trae eficit accounts for the ifference. By contrast region is a net exporter. Its

27 7 factor earnings are $,5,, but its outlays for consumption are only $9,7, with its trae surplus accounting for the ifference. Finally, we return to the question of the factor resources require by inter-regional transport of trae goos an reallocate these factors to the importe goos. Reallocation of Transport to Final Deman for Goos We wish to reallocate the factors associate in each region with the international transport of its imports to the final eman in the same an other regions where those imports are absorbe. To o this, the information neee, an its sources, are as follows: (a The amounts of factors to be reallocate: in the column of Φ for each region corresponing to the transportation sector(s. (b The proportion of each region s transport eman (in ton-km associate with the import of each goo: obtaine from T ij e ij. The factors are allocate among imports in these proportions. (c The proportions among components of final eman, by region an by sector, where each region s imports en up, which is the same as where each region s total omestic supply, (output plus imports, ens up: obtaine from the rows of B. The factors associate with each import are allocate to final eman in these proportions. See the appenix for a more etaile escription of the algorithm. The process may nee to be iterate but is assure to converge. In the -region example, this yiels (after iterations the matrix we call B R, to be compare with B : B R The bsorbing Markov Chain pproach Compare to the Big pproach s shown in Section, the upstream an ownstream chains in the one-region case can be etermine using either the stanar input-output approach or the absorbing Markov

28 8 chain approach. In fact, the same is true in the case of the multiregional worl economy, for which the absorbing Markov chain approach was escribe in Sections 4 an 5. Here we escribe the stanar input-output approach to the multiregional problem. Following the input-output notation of Section, the solution to the -region, -factor, 4- goo problem coul be written as Φ = F(I Å - ŷ, where ŷ is the x iagonal matrix of final eman for the regions, F is the 9 x block-iagonal matrix with all regions F matrices (each of imension x 4 arrange own the iagonal, an, which we will call the Big matrix, is efine as follows:. (6- In the Big matrix, each region s matrix, i, is ivie column-wise into parts, one corresponing to omestically supplie inputs per unit of its output ( ii an the others to imports from each other region, j, per unit of region i s output ( ji. Thus, taking region as an example, = + +. It is reaily seen that the Big matrix satisfies the Hawkins-Simon conitions for an acceptable input-output matrix if the matrices i o. This Big matrix has become familiar because the multiregional input-output moel use by regional economists generally takes the following form (see, for example, (eters an Hertwich forthcoming: (I x = y + e, or I I I x x x = y y y + e e e, (6- which can be solve for x, given y (here efine as final eman for only those goos prouce an consume in a region an the vectors of exports to final eman in other regions, e i. Using the matrices F, ij, an, Φ can be calculate from Eq. (-. Thus we have expressions for Φ: Eq. (6- shows the comparable Big an path-base expressions, an Eqs. (6-4 an (6-5 evelop them to highlight the similarities an ifferences. In Eq. (6- an (6-5, is a block-iagonal matrix, an each block escribes factor flows ( ij = f ij x j in a region. Φ F(I ~ y (I y (6-

29 9 y y y I I I F F F Φ (6-4 y y y I e e e I e e e I Φ (6-5 Formulating the problem in terms of the Big matrix has the avantage that manipulating the Big matrix is alreay familiar to regional input-output economists: it is hanle like a one-region input-output matrix, an its Leontief inverse is compute. The properties of the Leontief inverse are sufficiently attractive that analysts are incline to put all extensions to the basic static input-output moel in the form of Eq. (-a. Eq. (6- is a multi-regional moel written in that form, an the ynamic input-output moel an the moel close for househols have also been written an solve in a Big form, among other formulations (but naturally with a ifferent Big matrix, for this reason. From a practical point of view, statistical offices have begun to publish the breakown of a country s input-output table into a omestic input-output table an an import table -- although the latter are not now further broken own to istinguish iniviual country origins of imports in response to the eman of analysts. However, the solution by the Big metho of Eq. (6-4 has several rawbacks. First, the numerous off-iagonal components of the Big matrix are not available in publishe ata. Even when a regionally-aggregate import matrix is available, it oes not contribute aitional information content over the smaller quantity of ata require for the Markov metho, as import matrices are generally erive on the assumption that the omestic an import shares for a given goo are the same in all uses of that goo. That is, ij = i r ji ŝ i, where i r is the iagonal matrix showing the share of imports in omestic supply in region i an ji ŝ shows the share of all imports to region i from region j. There are many categories of ata that analysts woul like from statistical offices, but arguably better ij matrices for benchmark years base on the collection of more etaile ata an final eman vectors istinguishing omestically prouce from importe goos - shoul not be accore high priority. While matrices are a goo choice of parameters because they change slowly an in ways that can be unerstoo in terms of technological change, this is not the case for ij matrices where a proucer s choice between a omestically prouce input an its similarly-price importe counterpart, or between an import of a particular goo from one region rather than another region at similar prices is circumstantial an volatile. By contrast, the off-iagonal blocks in the path-base approach are iagonal matrices with exports as a share of omestic supply own the iagonal.

30 secon rawback of the Big approach is conceptual. It is natural, an generally effective, to approach new research problems by making use of concepts an techniques that worke well for relate problems in the past; this is the reason the Leontief inverse an its constituent multipliers are put in the service of new problems. However, this practice also has the effect of slowing the aoption of new approaches able to aress a broaer set of questions by generaliing existing concepts an methos. Duchin an Steenge (7 provie a critique of the Big strategy. The path-base approach set out to follow the physical flows rather than to erive an extene Leontief inverse. The analysis in Sections 4 an 5 provies a metho to quantify the factors (an emissions emboie in a region s consumption, taking full account of its imports, both irect an inirect, an their transport. Resources use in transporting imports are reallocate to the importe goos. The T matrices require to internalie these inputs o not fit into a Big matrix because they are multiplie not by an output vector but by a region s imports from each istinct origin. Distinguishing the transport associate with trae flows requires a more general representation. Finally, the absorbing Markov chain analysis offers aitional information about the input-output problem that is not available using stanar input-output methos, that has not been exploite in the present paper. In particular, the N matrix, erive from the canonical Markov matrix, measures the number of times a noe is visite. This property is use by Yamaa et al. (6 an Matsuno et al. (7 to examine the paths of recycle a material through an economy an count the average number of times it is recycle before ening up in a lanfill. Similarly, it was utilie by Levine (98 to erive a measure of trophic position within an ecosystem. The approach can profitably be generalie to a global analysis such as that reporte here. The Big metho has the avantage of familiarity among input-output analysts, by contrast with the absorbing Markov chain analysis propose in the paper. Our claim is that the approaches are complementary, enjoying areas of overlap where they provie results that are ientical, an areas of non-overlap where the Markov results require minimal information to answer familiar questions an provie consistent answers to a larger set of emerging questions. The basic logic of the Markov approach is natural an intuitive, as illustrate by the iagrams of physical flows shown in Figures., 4., an 4.. The efining characteristic of the metho is to associate each flow of a factor or goo with a set of shares aing to. that inicate the estination (or origin in the case of pollution of that flow. Once this simple logic is internalie, the mathematics of the Markov metho is as straightforwar as that for manipulating an input-output moel. However, both the Big metho an the Markov chain analysis escribe in this paper are ex-post approaches to analying historical ata or ata generate by a worl moel; neither is a substitute for a moel capable of analying alternative scenarios.