Vertical Production Chain and Global Sourcing (Preliminary Draft) Christian Schwarz∗ Abstract I extend the “global sourcing” framework by Antràs and Helpman (2004, JPE) by introducing a continuum of intermediate inputs in the production function. This allows to study input specific sourcing strategies. I find new effects that can exclusively emerge in a setting with strictly more than two inputs. First, I find that the optimal rev- enue share contributed to a supplier systematically varies in the input intensities of the other suppliers. It increases if the production process becomes more unequal in the inputs of the other suppliers. Second, the introduction of a vertical production chain leads to fact that a higher position in the vertical production chain favors integration of that the intermediate input. This finding is perfectly in line with re- cent empirical evidence on the role of a vertical production chain in the organization of production. JEL-Classification: F23, L23 Keywords: Multinational Firms, Organization of Production, Outsourcing, Vertical Production Chain Version: 12.08.2008 ∗Christian Schwarz, Department of Managerial Economics, University of Duisburg-Essen, Lotharstraße 65, LB 319b, 47057 Duisburg, Germany. Email: [email protected]. 1 Introduction Increasing fragmentation of production due to technological progress has been a crucial trend in the last decades. Multinational enterprises can nowa- days choose from a richer pattern of sourcing strategies than ever before. For a multitude of intermediate inputs firms face the classical “make or buy” decision (i.e. outsource vs. integrate) and may also opt to offshore the in- termediate input production to a foreign country. Various authors provide empirical evidence and examples from the business press. A recent study by Linden, Kraemer and Dedrick (2007) reports that Apple has completely outsourced the production of the 451 mostly generic inputs that are needed to fabricate the newest iPod. Only final assembly is done by Apple itself. Alfaro and Charlton (2007) find for General Motors Corporations that out of 2,248 entities 455 subsidiaries are located outside the United States. The theory of the firm predicts that the decision to outsource or offshore intermediate input production relates to the characteristics of the countries and the characteristics of the products being produced. In their seminal con- tribution “global sourcing” Antràs and Helpman (2004) introduce a North- South model of international trade where firms choose from a variety of organizational forms, depending on their individual productivity and sec- tor characteristics. Their framework, which combines firm heterogeneity in spirit of Melitz (2003) with organizational structures as in Antràs (2003), is especially helpful for coming to grips with newly emerged empirical facts about arm’s length outsourcing and intra-firm trade. The main result is that firms in headquarter-intensive sectors are more likely to choose integration strategies, whereas component-intensive sectors solely focus on outsourcing strategies. As it is common in the outsourcing and offshoring literature Antràs and Helpman (2004) study a production process with two intermediate inputs. The firm’s decision whether to integrate or outsource intermediate input production is modeled with respect to one intermediate input. In the light of increasing fragmentation it is however straightforward to search for new effectsthatexclusivelyemergeinasettingwithstrictlymorethantwoinputs. 1 My model builds on “global sourcing” by Antràs and Helpman and ex- tends their framework by introducing a continuum of intermediate inputs in the Cobb-Douglas type production function. This approach allows to dif- ferentiate between intermediate inputs on the one hand and on the other hand to explore input specific sourcing strategies. The production process in Antràs and Helpman can be regarded as a two-stage game. In the first stage a final-good producer distributes revenue shares from the potential sale of the final-goods. In the second stage intermediate input suppliers take their revenue shares, which were assigned in the first stage, as given and maximize their individual profits by providing optimal input contributions. Various new effects appear exclusively in a setting with strictly more than two inter- mediate inputs. First, I derive the suppliers’ profit maximizing intermediate input con- tribution in the second stage. I find that the optimal input relation of two representative suppliers is independent of other suppliers. Second, I derive the profit maximizing distribution of revenue, i.e. the optimal revenue shares that should ideally be assigned to each intermedi- ate input supplier in the first stage. I find that the optimal revenue share contributed to a supplier systematically varies in the input intensities of the other suppliers, even if the own input intensity remains constant. It increases if the production process becomes more unequal in the inputs of the other suppliers, i.e. if the input intensity dispersion increases. Similar joint profits and revenues increase in the input intensity dispersion. Third, I differentiate between intermediate inputs by considering the proximity to the final good, i.e. I introduce a vertical production chain. If, for example, technology-services are higher in the vertical production chain than labor services, I find that it is more likely that firms will integrate the production of technology-services. This theoretical prediction is perfectly in line with new empirical evidence by Alfaro and Charlton (2007). They find that multinationals tend to own the stages of production proximate to their final-good. Therefore, the novelty of my approach is that it introduces a new determinant of sourcing strategies, namely a vertical production chain. My extension of the Antràs and Helpman (2004) framework is comple- 2 mentarytothetheoryofthefirmandinparticularthestrandoftheliterature that investigates the ownership structure of firms. The ownership decision of firms is a lively field of both theoretical and empirical research as the sur- vey by Helpman (2006) illustrates. Various aspects are under investigation. Grossman and Helpman (2002) address the choice between outsourcing and integration within a one-input production function. McLaren (2000) and Grossmann and Helpman (2003) focus on the matching between a final-good producer and unaffiliated intermediate input suppliers. Antràs (2003) uses a setting of incomplete contracts to study the ownership decision of firms with respect to the input intensities of the production process. The impact of legal contractibility with respect to specific inputs is addressed by Antràs and Helpman (2006). To study the organizational choice of individual firms, new theories have to account for the empirical fact of within-in sectoral heterogeneity. The seminal contribution by Melitz (2003) allows productivity to differ across firms and has become the cornerstone in the “new” new trade theory. In this framework, the interaction between productivity differences across firms and fixed costs of exporting leads to the prediction that only the most pro- ductive firms export. However, final-goods production can also be offshored to a foreign country. Helpman, Melitz and Yeaple (2004) extend the Melitz (2003) model by incorporating the horizontal motive of FDI, i.e. transport cost savings. They show that among firms that serve foreign markets, the less productive firms export while the more productive firms engage in hori- zontal FDI. Unlike horizontal FDI the classical motive for vertical FDI is the firms’ attempt to take advantage of cross-border factor cost differences like in Helpman (1984) and Helpman and Krugman (1985). The first ones who combine firm heterogeneity in spirit of Melitz (2003) with the organization of firms as in Antràs (2003) are Antràs and Helpman (2004). They introduce a North-South model of international trade where firms choose from a variety of organizational forms, depending on their indi- vidual productivity and sector characteristics. Their main result is that in a sector that is component intensive, i.e. the input provided by the component supplier is relatively more important for the production process than the one 3 of the final-good producer, firms solely focus on domestic and international outsourcing. In case the sector is intensive in the input provided by the final-good producer, i.e. headquarter-services, integration and outsourcing can coexist. However, all these contributions focus on one single intermediate input that can either be integrated or outsourced. This strand of the literature often neglects that final good production is accomplished by combining var- ious inputs. The major drawback of this approach is that it does not allow to explore interdependencies between intermediate input specific sourcing strategies. Shy and Stenbacka (2005) study for example a Cournot model with a continuum of intermediate inputs. They derive the equilibrium frac- tion of outsourced intermediate inputs, which is a result that can exclusively be derived in a setting with a continuum of intermediate inputs. My exten- sion of the Antràs and Helpman (2004) framework can therefore be seen as a new impulse to study the interdependencies between input specific sourcing strategies. 2 Model Consider an economy with a set of sectors S. Consumers have identical preferences which are represented by Z 1 U = x + X(k)µdk (1) 0 µ S with 0 < µ < 1 and where x is consumption of a homogeneous good. 0 Aggregate consumption X in sector s is a constant elasticity of substitution s function (cid:20)Z (cid:21)1 α X = x (k)α (2) s s Vs with 0 < α < 1 and where V is the set of final-good varieties in sector s s. The elasticity of substitution between any two final-good varieties in a given sector is 1/(1−α). For simplicity, I assume that the final-good varieties x are constant. Aggregate consumption simplifies to X = x d1/α s s s s 4 and rises in the number of final-good varieties d . Consumers find it more s easy to substitute varieties within a sector than across sectors if α > µ holds. Consumer preferences lead to an inverse demand function for each variety x s of p = Xµ−αxα−1. (3) s s s Each final-good variety x is accomplished by combining a continuum of s intermediate inputs m . Each intermediate input m (i) is provided by a s s unique intermediate input supplier A(i). Output of a final-good variety x s is given by a sector specific Cobb-Douglas type production function of the form (cid:20)Z (cid:18) (cid:19) (cid:21) m (k) s x (θ,m (i)) = θexp e (k)ln dk (4) s s s e (k) Is s with 0 < e (i) < 1. The productivity parameter θ is firm specific, whereas s the parameters e are sector specific. The larger e (i) the more intensive is s s the production of the final-good variety x in the intermediate input m (i). s s As in Antràs (2003) I call e (i) the input intensity of the intermediate input s m (i) provided by the intermediate input supplier A(i). Total revenue R = s s x p of a final-good variety is given by s s (cid:20) Z (cid:18) (cid:19) (cid:21) m (k) R = d(µ−α)/αθµexp µ e (k)ln s dk . (5) s s s e (k) Is s Production of intermediate inputs requires only one factor of production, named labor. An intermediate input supplier A(i) faces a perfectly inelastic supply of workers at the wage rate w(i). Intermediate input production involves variable costs of w(i) and fixed costs f (i). Final-good production can be seen as a two stage production process. In the following I drop the sectorindexsforconvenience. Nowconsidertheproductionprocessindetail: 1st Stage: Distribution of Revenue In the first stage intermediate input suppliers distribute the potential fu- ture revenue from the sale of final goods varieties. Each agent’s revenue R share is given by b(i) with b(k)dk = 1. One of the intermediate input Is suppliers is the final-producer A(h) that provides headquarter services. The 5 final-good producer offers a contract to each other intermediate-input sup- plier A(i), h 6= i, that involves a share of revenue b(i) and a participation fee. The participation fee has to be paid by the other intermediate-input sup- pliers to the final-good producer. The final-good producer has an incentive to raise the participation fee as much as possible, as long as the participation constraint for the intermediate input supplier is satisfied. Once a relation- ship between a final-good producer and an intermediate-input supplier is formed the participation fee has no further effects on the outcomes. As a result, the final-good producer is interested in the distribution of revenue (cid:16) (cid:17) b∗ ≡ b∗ ,···,b∗ that maximizes the potential revenue from the sale of (1) (n) the final goods. 2nd Stage: Production Stage In the second stage intermediate input suppliers take their revenue share b(i) assigned in the first stage as given and maximize their individual profit π(i). The intermediate input suppliers’ individual profit at the second stage is given by the share of revenue less the cost of producing the intermediate input and fixed costs. The maximization problem of a supplier A(i) is maxπ(i) = b(i)R−w(i)m(i)−f (i). (6) m(i) I solve this two-step production process via backward induction. Solution 2nd Stage The maximization problem of an agent A(i) in the 2nd stage leads to the first order condition ∂π(i) e(i) = b(i)Rµ −w(i) = 0. (7) ∂m(i) m(i) 6 Solving the system of first order conditions leads to the optimal input con- tribution m∗(i) of an agent A(i) and is given by e(i)b(i) (cid:16) (cid:17) 1 (cid:20) µ Z (cid:18) b(k) (cid:19) (cid:21) m∗(i) = µθµdµ−α 1−µ exp e(k)ln dk . (8) α w(i) 1−µ w(k) I Proof see Appendix. Proposition 1 Two intermediate input suppliers A(i) and A(i) contribute optimal inputs m∗(i) and m∗(j) if and only if m∗(i) b(i) e(i) w(j) = (9) m∗(j) b(j)e(j) w(i) holds. The optimal input relation solely depends on the relative distribution of revenue b(i)/b(j), the relative input intensities e(i)/e(j) and the inverse wage differential w(j)/w(i) of the intermediate input suppliers. Note that it does neither depend on the aggregate consumption in the economy X nor on factors that are associated with other suppliers A(k), k 6= j 6= i, involved in the final-goods production. Whether two representative input suppliers provide an optimal contribution relation is therefore independent from the supplement of other suppliers. The economic intuition is the following. Assume that there are no inter- mediate input wage differences and the inputs are both equally important for the production process. In this case intermediate-input suppliers are only willing to contribute relative quantities equal to the relative revenue shares they capture from the relation. If one allows for differences in the input intensities the intermediate-input supplier with the more important input also invests relatively more in order to avoid underinvestment. If the wage gap between the two intermediate-input suppliers increases, the supplier of the relatively more expensive input decreases the own contribution. This is intuitive since the input suppliers have to bear an increase in the variable 7 costs of intermediate-input production for their own. After having solved the second stage I now proceed with the first stage of the production process. Solution 1st Stage R The total value of the final-good production is given by π = π(i)dk. I If all agents contribute optimal inputs m∗ the total value of the final-good production is given by Z π∗ = dαµ−−ααµθ1−µµψ − f (k)dk (10) I with (cid:20) Z (cid:21) (cid:20) Z (cid:18) (cid:19) (cid:21) µ b(k) ψ = 1−µ e(k)b(k)dk exp e(k)ln dk . (11) 1−µ w I I Proof see Appendix. One of the intermediate input suppliers is the final-producer A(h) that provides headquarter services. This agent offers a contract to each other intermediate-input supplier A(i), h 6= i, that involves a share of revenue b(i) and a participation fee t(i). The participation fee has to be paid by an intermediate input supplier and is captured by the final-good producer. The profit of an intermediate input supplier A(i) at the 1st stage is then π(i) = b(i)R−w(i)m(i)−f (i)−t(i). The final-good producer A(h) has an incentive to raise the participation fee t(i) as much as possible, as long as the participation constraint π(i) ≥ 0 for each other intermediate input supplier A(i) is satisfied. It is important to note that if the participation constraint is satisfied the participation fee has no further effects on the outcomes and the intermediate input supplier will contribute optimal inputs m∗(i) as derived in (8). As a result the equi- librium value of t(i) satisfies π(i) = 0 for all intermediate-input suppliers A(i) except the final-good producer A(h). The final-good producer captures 8 the total value of the final-good production, i.e. π(h) = π, due to the sum of participation fees. The final-good producer is therefore interested in the ex-post distribution of revenue that maximizes potential revenue from the sale of the final-goods. (cid:16) (cid:17) In the following, I derive the distribution of revenue b∗ = b∗ ,···,b∗ (1) (n) that maximizes the potential revenue from the sale of the final goods. Po- tential revenue is maximized if ψ is maximal. A necessary condition for a maximum of ψ with respect to the optimal distribution of revenue b∗ is dψ | = 0. (12) b∗ db(i) Proposition 2 The distribution of revenue b∗ that maximizes the potential revenue from the sale of the final-goods is implicitly given by Z e(i)−b(i)−(1−µ)e(i)b(i)+[b(i)−µe(i)] e(k)b(k)dk = 0 (13) I due to the fact that it provides a solution to (12). Proof see Appendix. An explicit form of b∗ is possible to derive for the two intermediate in- put cases (headquarter-services and manufactured components) Antràs and Helpman (2004) examine1. For other cases with strictly more than two in- termediate inputs numerical methods help to provide b∗. However, a basic economic intuition can be established analytically from proposition 2. If an intermediate input supplier provides an input that is not required for the final-good production the optimal assigned revenue share is zero, i.e. b(i) = 0 ⇔ e(i) = 0. Furthermore if the production process solely relies on only one single input, the intermediate-input supplier completely captures the revenue, i.e. e(i) = 1 ⇒ b(i) = 1. 1Iprovidethederivationofb∗ fortheAntràsandHelpman(2004)caseintheAppendix. 9