# Who Produces For Whom In The World Economy?

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Figure 1: Three ways to look at the same export flowsThe left side of the figure shows total export flows as they appear in standard trade statistics (forshorthand, “standard trade”). First, country A exports 10 cars without windshields to country B. Then,country B installs the windshields and exports the finished goods to C to be consumed. The exported goodsare classified as exports of manufacturing goods (motor vehicles and parts), but they contain 100 hours ofmaintenance services.The top right figure shows the calculation of vertical exports. “Cars without windshields” are countedtwice in standard trade statistics: once when they are exported from A to become inputs in B and once whenthey are exported from B for consumption in C. These flows indicate the integration of the productionprocess between countries A and B.The bottom right figure shows “value-added” exports as it appears in our data. The value-added exportflows shows that no final user in B utilizes goods from A. B is just used as an intermediate (or transit)country for A. All the final users of country A’s exports are in country C. So the only value-added exportsthat B realizes with C are the windshields that are produced in B and used as a final product in C. That meansthat country A’s exports do not depend on the final demand of country B, but on the final demand of C. They5

reconciled input-output and trade data and the GTAP is the best source of such information, as thecommunity of compatible general equilibrium economists recognizes. One can only hope better quality datawill arise in time.2.2 Theoretical foundation of the calculation162.2.1 A closed economyIn a closed economy with n industries and products, equilibrium between output and final demandrequires that for each product, output is equal to the sum of intermediate deliveries and of final demand:P A P FD(1)in which P is a n x 1 vector of output, FD is a n x 1 vector of final demand and A is a n x n matrix ofinput coefficients taken from the intermediate deliveries matrix. This matrix is composed of elements aij,defined as the amount of product i required for the production of one unit of product j.Assuming that the matrix A is fixed17, this yields the following relation:P (I-A)-1 FD(2)In which I is the n x n identity matrix and (I-A)-1 the Leontief inverse matrix derived from the inputcoefficients matrix A. This relation is a well-known result in input-output analysis that links the final demandof each product and the production. It gives the total direct and indirect effects of an increase in final demandon the production of each industry.18Furthermore, each output vector P is itself associated with a value-added vector VA:VA P - diag(P) A’ I(3)where diag(P) is the n x n matrix having the elements of P on its diagonal and 0 elsewhere, A’ is thetranspose of matrix A and I is the identity column vector filled by 1s. Each element of diag(P) A’ I gives the16 This is discussed in greater depth in Daudin, Rifflart, Schweisguth and Veroni (2006).17 This assumption is needed to derive the I-O model. It means that all inputs are required in a fixed proportion of theoutput. This implies a production function with constant returns to scale.18 (I-A)-1 is derived from the expression I A1 A2 A3 , where I is used to calculate the direct effect linked to theincrease in final demand, A1 is the first-order matrix used to calculate the production of inputs required by the increaseof final demand, A2 is the second-order matrix used to calculate the production of inputs required by this increase ofinput demand, A3, etc. The sufficient condition to derive the A matrix is that all the leading principal minors be positive(Hawkins Simon Conditions). A minor of a matrix is the value of a determinant. The principal leading minors of an n xn matrix are evaluated on what is left after the last m rows and columns are deleted, where m runs from (n-1) down to 0.11

value of the required input to produce each element of P.19Hence, substituting (1) into (2), the value-added vector VA associated with the final demand vector FDis equal to:VA (I – A)-1 FD – diag((I – A)-1 FD) A’ I(4)2.2.2 Opening up a single economyBefore moving to the study of value-added trade, let us compute VS (the imported inputs used inexports).Let a n x 1 vector Im give the imports used for the production of each sector.Let X be a n x 1 vector of exports (part of final demand). Following (2), the production of all the inputsfor X requires a total domestic production of (I-A)-1X. VS, the amount of imports required to produce X canhence be defined as :VS Im((I-A)-1X)2.2.3 Value-added tradeWe can transpose this writing to the world as a whole using an inter-country input-output table. To dothis, the world is considered in the same way as a single closed economy with c countries, each onecontaining n sectors. We assume that each sector in each country produces a very specific product, which isproduced nowhere else and which is not substitutable with any other product. The construction of theappropriate matrixes is detailed in the appendix and discussed in Hoen (2002).Applied to the world, the previous equations (1), (2) and (3) can be rewritten as:P* (I-A*)-1 FD*VA* P* – diag(P*) A*’ I(5)VA* (I – A*)-1 FD* – diag((I – A*)-1 FD*) A*’ I(6)WhereP* is the nc x 1 vector of output by country and by industry,FD* is the nc x 1 vector of final demand by country and by industry,19 In contrast, A P gives the vector of output used as inputs for further production.12

A* is the nc x nc input coefficients matrix showing all the inter-industrial and inter-regional trade ofinputs all over the world (See Technical Appendix for its construction). A* being a fixed matrix, A* can betransposed into A*’,and VA* is the nc x 1 vector of value added, provided by country and by industry.Equation (6) allows the computation of the value-added production required by the consumption orinvestment of any final product. It is possible to use this formula to compute the effects of an increase infinal demand on the value-added sector: VA* (I – A*)-1 FD* – diag((I – A*)-1 FD*) A*’ I(7)Assuming that fd ia and vabj are items of the vectors FD* and VA*, with a and b for the countries and iand j for the products or industries, a one unit change in a final product i0 of a country a0, fdia0 will generate0an increase in the value added va bj ,,ia0 of any sector j of any country b.0If b a0, va bj ,,ia0 will be considered as a value-added export from the sector j of the country b to the0sector i0 of the country a0. The total export in value added

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