Contracts and Technology Adoption

Transcription

1 Contracts and Technology Adoption The Harvard community has made this article openly available. Please share how this access benefits you. Your story matters Citation Acemoglu, Daron, Pol Antras, and Elhanan Helpman. 27. Contracts and technology adoption. American Economic Review 97(3): Published Version Citable link Terms of Use This article was downloaded from Harvard University s DASH repository, and is made available under the terms and conditions applicable to Other Posted Material, as set forth at nrs.harvard.edu/urn-3:hul.instrepos:dash.current.terms-ofuse#laa

2 Contracts and Technology Adoption Daron Acemoglu Department of Economics, MIT Pol Antràs Department of Economics, Harvard University Elhanan Helpman Department of Economics, Harvard University November 28, 26 Abstract We develop a tractable framework for the analysis of the relationship between contractual incompleteness, technological complementarities, and technology adoption. In our model a rm chooses its technology and investment levels in contractible activities by suppliers of intermediate inputs. Suppliers then choose investments in noncontractible activities, anticipating payo s from an ex post bargaining game. We show that greater contractual incompleteness leads to the adoption of less advanced technologies and that the impact of contractual incompleteness is more pronounced when there is greater complementary among the intermediate inputs. We study a number of applications of the main framework and show that the mechanism proposed in the paper can generate sizable productivity di erences across countries with di erent contracting institutions and that di erences in contracting institutions lead to endogenous comparative advantage di erences. Keywords: comparative advantage, economic growth, incomplete contracts, technology choice, theory of the rm. JEL Classi cation: D23, F, L23, O3. A previous version of this paper was circulated under the title Contracts and the Division of Labor. We thank Gene Grossman, Oliver Hart, Kalina Manova, Damián Migueles, Giacomo Ponzetto, Richard Rogerson, Rani Spiegler, two anonymous referees, and participants in the Canadian Institute for Advanced Research and Minnesota Workshop in Macroeconomic Theory conferences, and seminar participants at Harvard, MIT, Tel Aviv, Universitat Pompeu Fabra, Houston, UCSD, Southern Methodist, Stockholm School of Economics, Stockholm University (IIES), ECARES, Colorado-Boulder, Kellogg, NYU, UBC, Georgetown, World Bank, Montreal, Brandeis, Wisconsin, UC Berkeley, Haifa, Syracuse, and Autònoma-Barcelona for useful comments. We also thank Davin Chor, Alexandre Debs and Ran Melamed for excellent research assistance. Acemoglu and Helpman thank the National Science Foundation for nancial support. Much of Helpman s work for this paper was done when he was Sackler Visiting Professor at Tel Aviv University.

3 Introduction There is widespread agreement that di erences in technology are a major source of productivity di erences across rms, industries and nations. Despite this widespread agreement, we are far from an established framework for the analysis of technology choices of rms. In this paper, we take a step in this direction and develop a simple model to study the impact of contracting institutions, which regulate the relationship between the rm and its suppliers, on technology choices. Our model combines two well-established approaches. The rst is the representation of technology as the range of intermediate inputs used by rms; a greater range of intermediate inputs increases productivity by allowing greater specialization and thus corresponds to more advanced technology. 2 The second is Grossman and Hart s (986) and Hart and Moore s (99) approach to incomplete-contracting models of the rm. We study technology choice of rms under incomplete contracts, and extend Hart and Moore s framework, by allowing contracts to be partially incomplete. This combination enables us to investigate how the degree of contractual incompleteness and the extent of technological complementarities between intermediate inputs a ect the choice of technology. In our baseline model a rm decides on technology (on the range of specialized intermediate goods), recognizing that a more advanced technology is more productive, but also entails a variety of costs. In addition to the direct pecuniary costs of engaging more suppliers (corresponding to the greater range of intermediate inputs), a more advanced technology necessitates contracting with more suppliers. All of the activities that suppliers undertake are relationship-speci c, and a fraction of those is ex ante contractible, while the rest, as in the work by Grossman-Hart-Moore, are nonveri able and noncontractible. The fraction of contractible activities is our measure of the quality of contracting institutions. 3 Suppliers are contractually obliged to perform their duties in the contractible activities, but they are free to choose their investments in noncontractible activities and to withhold their services in these activities from the rm. This combination of noncontractible investments and relationship-speci city leads to an ex post multilateral bargaining problem. As in Hart and Moore (99), we use the Shapley value to determine the division of ex post surplus between the rm and its suppliers. We derive an explicit solution for this division of surplus, which enables us to develop a simple characterization of the equilibrium. A supplier s expected payo in the bargaining game determines her willingness to invest in the noncontractible activities. Since she is not the full residual claimant of the output gains derived from Among others, see Klenow and Rodriguez (997), Hall and Jones (999) or Caselli (24) for countries, and Klette (996), Griliches (998), Sutton (998) or Klette and Kortum (24) for rms. 2 See, among others, Ethier (982), Romer (99) and Grossman and Helpman (99) for previous uses of this representation. See also the textbook treatment in Barro and Sala-i-Martin (23). There is a natural relationship between this view of technology and the division of labor within a rm. We investigated this link in the earlier version of the current paper, Acemoglu, Antràs and Helpman (25), and do not elaborate on it here. 3 Maskin and Tirole (999) question whether the presence of nonveri able actions and unforeseen contingencies necessarily lead to incomplete contracts. Their argument is not central to our analysis because it assumes the presence of strong contracting institutions (which, for example, allow contracts to specify sophisticated mechanisms), while in our model a fraction of activities may be noncontractible not because of technological reasons but because of weak contracting institutions.

4 her investments, she tends to underinvest. Greater contractual incompleteness thus reduces supplier investments, making more advanced technologies less pro table. Furthermore, a greater degree of technological complementarity reduces the incentive to choose more advanced technologies; though greater technological complementarity increases equilibrium ex post payo s to every supplier, it also makes their payo s less sensitive to their noncontractible investments, discouraging investments and, via this channel, depressing the pro tability of more advanced technologies. An advantage of our framework is its relative tractability. The equilibrium of our model can be represented by a reduced-form pro t function for rms given by AZF (N) C (N) w N; () where N represents the technology level, C (N) is the cost of technology N, A is a measure of aggregate demand or the scale of the market, and w N corresponds to the value of the N suppliers outside options. F (N) is an increasing function that captures the positive e ect of choosing more advanced technologies on revenue. The e ects of contractual incompleteness and technological complementarity are summarized by the variable Z. This variable a ects productivity and is decreasing in the degree of contract incompleteness and technological complementarity. Moreover, the elasticity of Z with respect to the quality of contracting institutions is higher when there is greater complementarity between intermediate inputs. This last result has important implications for equilibrium industry structure and the patterns of comparative advantage, because it implies that sectors ( rms) with greater complementarities between inputs are more contract dependent. We use this framework to show that the combination of contractual imperfections and technology choice (or adoption) may have important implications for cross-country income di erences, equilibrium organizational forms, and patterns of international specialization and trade. First, we show that our baseline model can generate sizable di erences in productivity from di erences in the quality of contracting institutions. In particular, using a range of reasonable values for the key parameters of the model, we nd that when the degree of technological complementarity is su ciently high, relatively modest changes in the fraction of activities that are contractible can lead to large changes in productivity. Second, we derive a range of implications about the equilibrium organizational form. Our results here show that the combination of weak contracting institutions and credit market imperfections may encourage greater vertical integration. Third, we present a number of general equilibrium applications of this framework. The general equilibrium interactions result from the fact that an improvement in contracting institutions does not increase the choice of technology in all sectors ( rms). Instead, more advanced technologies are chosen in the more contract-dependent sectors. We show that in the context of an open economy, this feature leads to an endogenous structure of comparative advantage. In particular, among countries with identical technological opportunities (production possibility sets), those with better contracting institutions specialize in sectors with greater complementarities among inputs. These predictions are consistent with the recent empirical results presented in Nunn (25) and Levchenko 2

5 (23). As mentioned above, our work is related to two strands of the literature. The rst investigates the determinants of rm-level technology (including the division of labor) and includes, among others, Becker and Murphy (992) and Yang and Borland (99) on the impact of the extent of the market on the division of labor and Romer (99) and Grossman and Helpman (99) on endogenous technological change. None of these studies investigate the e ects of contracting institutions on technology choice. The second literature deals with the internal organization of the rm. It includes the papers by Grossman-Hart-Moore discussed above, as well as Klein, Crawford and Alchian (978) and Williamson (975, 985), who emphasize incomplete contracts and hold-up problems. Here, the papers by Stole and Zwiebel (996a,b) and Bakos and Brynjolfsson (993) are most closely related to ours. Stole and Zwiebel consider a relationship between a rm and a number of workers whose wages are determined by ex post bargaining according to the Shapley value. They show how the rm may overemploy in order to reduce the bargaining power of the workers and discuss the implications of this framework for a number of organizational design issues. Stole and Zwiebel s framework does not have relationship-speci c investments, however, which is at the core of our approach, and they do not discuss the e ects of the degree of contractual incompleteness and the degree of complementarity between inputs on the equilibrium technology choice. 4 Finally, our paper is also related to the literature on the macroeconomic implications of contractual imperfections. A number of papers, most notably Quintin (2), Amaral and Quintin (25), Erosa and Hidalgo (25) and Castro, Clementi and MacDonald (24), investigate the quantitative impact of contractual and capital market imperfections on aggregate productivity. 5 approach di ers from these papers since we focus on technology choice and relationship-speci c investments, and because we develop a tractable framework that can be applied in a range of problems. Although we do not undertake a detailed calibration exercise as in these papers, the results in subsection 5. suggest that the economic mechanism developed in this paper can lead to quantitatively large e ects. Finally, Levchenko (23), Costinot (24), Nunn (25) and Antràs (25) also generate endogenous comparative advantage across countries from di erences in contractual environments. Among these papers, Costinot (24) is most closely related, since he also develops a model of endogenous comparative advantage based on specialization, though his approach is more reduced-form than our model. The rest of the paper is organized as follows. Our Section 2 introduces the basic environment. Section 3 characterizes the equilibrium with complete contracts. Section 4 introduces incomplete contracts into the framework of Section 2, characterizes the equilibrium, and derives the major comparative static results. Section 5 considers a number of applications of our basic framework. Section 6 concludes. Proofs of the main results are provided in the Appendix. 4 Another related paper is Blanchard and Kremer (997), which studies the impact of ine cient sequential contracting between a rm and its suppliers in a model where output is a Leontief aggregate of the inputs of the suppliers. 5 Acemoglu and Zilibotti (999), Martimort and Verdier (2, 24) and Francois and Roberts (23) study the qualitative impact of changes in the internal organization of rms on economic growth. 3

6 2 Technology and Payo s Consider a pro t-maximizing rm facing a demand function q = Ap =( ) for its nal product, where q denotes quantity and p denotes price. The parameter 2 (; ) determines the elasticity of demand while A > determines the level of demand or the market size. The rm treats the demand level A as exogenous. This form of demand can be derived from a constant elasticity of substitution preference structure for di erentiated products (see Section 5.3) and it generates a revenue function R = A q : (2) Production depends on the technology choice of the rm. More advanced (more productive) technologies involve a greater range of intermediate goods and thus a higher degree of specialization. The technology level of the rm is denoted by N 2 R + and for each j 2 [; N], X (j) is the quantity of intermediate input j. Given technology N, the production function of the rm is q = N + = Z N X (j) dj =, < < ; > : (3) A number of features of this production function are worth noting. First, determines the degree of complementarity between inputs; since 2 (; ), the elasticity of substitution between them, = ( ), is always greater than one. Second, we follow Benassy (998) in introducing the term N + = in front of the integral, which allows us to separately control the elasticity of substitution between inputs and the elasticity of output with respect to the level of the technology. To see this, consider the case where X (j) = X for all j. Then the output of technology N is q = N + X. Consequently, both output and productivity (de ned as either q= (N X) or q=n) are independent of and depend positively on N, with elasticity determined by the parameter. 6 There is a large number of pro t-maximizing suppliers that can produce the necessary intermediate goods, each with the same outside option w. For now, w is taken as given, but it will be endogenized in Section 5.3. We assume that each intermediate input needs to be produced by a di erent supplier with whom the rm needs to contract. 7 A supplier assigned to the production of an intermediate input needs to undertake relationshipspeci c investments in a unit measure of (symmetric) activities. There is a constant marginal cost of investment c x for each activity. 8 The production function of intermediate inputs is Cobb-Douglas and symmetric in the activities: Z X (j) = exp ln x (i; j) di, (4) 6 In contrast, with the standard speci cation of the CES production function, without the term N + = in front (i.e., = = ), total output is q = N = X, and the two elasticities are governed by the same parameter,. 7 A previous version of the paper, Acemoglu, Antràs and Helpman (25), also endogenized the allocation of inputs to suppliers using an augmented model with additional diseconomies of scope. 8 One can think of c x as the marginal cost of e ort; see the formulation of this e ort in utility terms in Section

7 where x (i; j) is the level of investment in activity i performed by the supplier of input j. This formulation will allow a tractable parameterization of contractual incompleteness in Section 4, whereby a subset of the investments necessary for production will be nonveri able and thus noncontractible. Finally, we assume that adopting (and using) a technology N involves costs C (N), and impose Assumption (i) For all N >, C (N) is twice continuously di erentiable, with C (N) > and C (N). (ii) For all N >, NC (N) = [C (N) + w ] > [ ( + ) ] = ( ). The rst part of this assumption is standard; costs are increasing and convex. The second part will ensure a nite and positive choice of N. Let the payment to supplier j consist of two parts: an ex ante payment (j) 2 R before the investment levels x (i; j) take place, and a payment s (j) after the investments. Then, the payo to supplier j, also taking account of her outside option, is Similarly, the payo to the rm is x (j) = max (j) + s (j) Z c x x (i; j) di; w : (5) = R Z N [ (j) + s (j)] dj C (N) ; (6) where R is revenue and the other two terms on the right-hand side represent costs. Substituting (3) and (4) into (2), revenue can be expressed as R = A N (+ =) Z N 3 Equilibrium under Complete Contracts Z = exp ln x (i; j) di dj. (7) As a benchmark, consider the case of complete contracts (the rst best from the viewpoint of the rm). With complete contracts, the rm has full control over all investments and pays each supplier her outside option. In analogy to our treatment below of technology adoption under incomplete contracts, h consider a game form where i the rm chooses a technology level N and makes a contract o er fx (i; j)g i2[;] ; fs (j) ; (j)g for every input j 2 [; N]. If a supplier accepts this contract for input j, she is obliged to supply fx (i; j)g i2[;] as stipulated in the contract in exchange for the payments fs (j) ; (j)g. A subgame perfect equilibrium of this game is a strategy combination for the rm and the suppliers such that suppliers maximize (5) and the rm maximizes (6). An equilibrium can be alternatively represented as a solution to the following maximization problem: max N;fx(i;j)g i;j ;fs(j);(j)g j R Z N [ (j) + s (j)] dj C (N) (8) 5

8 subject to (7) and the suppliers participation constraint, Z s (j) + (j) c x x (i; j) di w for all j 2 [; N] : (9) Since the rm has no reason to provide rents to the suppliers, it chooses payments s (j) and (j) that satisfy (9) with equality. 9 Moreover, since the rm s objective function, (8), is (jointly) concave in the investment levels x (i; j) and these investments are all equally costly, the rm chooses the same investment level x for all activities in all intermediate inputs. Now, substituting for (9) in (8), we obtain the following simpler unconstrained maximization problem for the rm: max N;x A N (+) x c x Nx C (N) w N: () >From the rst-order conditions of this problem, we obtain: (N ) (+) A =( ) =( ) cx = C (N ) + w ; () x = C (N ) + w c x : (2) Equations () and (2) can be solved recursively. Given Assumption, equation () yields a unique solution for N, which, together with (2), yields a unique solution for x. When all the investment levels are identical and equal to x, output equals q = N + x. Since NX = Nx inputs are used in the production process, we can de ne productivity as output divided by total input use, P = N. In the case of complete contracts this productivity level is P = (N ) ; (3) which is increasing in the level of technology. In the next section we compare this to equilibrium productivity under incomplete contracts. The next proposition describes the key properties of the equilibrium (proof in the Appendix). Proposition Suppose that Assumption holds. Then with complete contracts there exists a unique equilibrium with technology and investment levels N > and x > given by () and (2). Furthermore, this equilibrium = : 9 With complete contracts, (j) and s (j) are perfect substitutes, so that only the sum s (j) + (j) matters. This will not be the case when contracts are incomplete. We show in the Appendix that the second-order conditions are satis ed under Assumption. This measure of productivity implicitly assumes that all the investments x (i; j) are measured accurately. There may be some tension between this assumption and the assumption that the cost of these investments is not pecuniary. For this reason in the previous version, we also considered another de nition of productivity: output divided by the number of suppliers, P = q=n = (N) x. The ranking of productivity levels between complete and incomplete contracts is the same under both de nitions and the quantitative e ects of changes in on productivity are smaller with our main de nition. 6

9 In the case of complete contracts, the size of the market (as parameterized by the demand level A) has a positive e ect on investments by suppliers of intermediate inputs and productivity. The other noteworthy implication of this proposition is that under complete contracts, the level of technology and thus productivity do not depend on the elasticity of substitution between intermediate inputs, = ( ). 4 Equilibrium under Incomplete Contracts 4. Incomplete Contracts We now consider the same environment under incomplete contracts. We model the imperfection of the contracting institutions by assuming that there exists a 2 [; ] such that, for every intermediate input j, investments in activities i are observable and veri able and therefore contractible, while investments in activities < i are not contractible. Consequently, a contract stipulates investment levels x (i; j) for the contractible activities, but does not specify the investment levels in the remaining noncontractible activities. Instead, suppliers choose their investments in noncontractible activities in anticipation of the ex post distribution of revenue, and may decide to withhold their services in these activities from the rm. We follow the incomplete contracts literature and assume that the ex post distribution of revenue is governed by multilateral bargaining, and, as in Hart and Moore (99), we adopt the Shapley value as the solution concept for this multilateral bargaining game (more on this below). 2 The timing of events is as follows: The rm adopts a technology N and o ers a contract [fx c (i; j)g i= ; (j)] for every intermediate input j 2 [; N], where x c (i; j) is an investment level in a contractible activity and (j) is an upfront payment to supplier j. The payment (j) can be positive or negative. Potential suppliers decide whether to apply for the contracts. suppliers, one for each intermediate input j. Then the rm chooses N All suppliers j 2 [; N] simultaneously choose investment levels x (i; j) for all i 2 [; ]. In the contractible activities i 2 [; ] they invest x (i; j) = x c (i; j). The suppliers and the rm bargain over the division of revenue, and at this stage, suppliers can withhold their services in noncontractible activities. Output is produced and sold, and the revenue R is distributed according to the bargaining agreement. We will characterize a symmetric subgame perfect equilibrium (SSPE for short) of this game, where bargaining outcomes in all subgames are determined by Shapley values. 2 The incomplete contracts literature also assumes that revenues are noncontractible. As is well known, with bilateral contracting or with multilateral contracting and a budget breaker (e.g., Holmström, 982), contracting on revenues would improve incentives. In our setting, we do not need this assumption, since each rm has a continuum of suppliers and contracts on total revenues would not provide additional investment incentives to suppliers. 7

10 4.2 De nition of Equilibrium and Preliminaries n Behavior along the SSPE can be described by a tuple ~N; ~xc ; ~x n ; ~o in which N ~ represents the level of technology, ~x c the investment in contractible activities, ~x n the investmenthin noncontractible activities, and ~ the upfront payment to every supplier. That is, for every j 2 ; N ~ i the upfront payment is (j) = ~, and the investment levels are x (i; j) = ~x c for i 2 [; n ] and x (i; oj) = ~x n for i 2 (; ]. With a slight abuse of terminology, we will denote the SSPE by ~N; ~xc ; ~x n. The SSPE can be characterized by backward induction. First, consider the penultimate stage of the game, with N as the level of technology, x c as the level of investment in contractible activities. Suppose also that each supplier other than j has chosen a level of investment in noncontractible activities equal to x n ( j) (these are all the same, because we are constructing a symmetric equilibrium), while the investment level in every noncontractible activity by supplier j is x n (j). 3 Given these investments, the suppliers and the rm will engage in multilateral Shapley bargaining. Denote the Shapley value of supplier j under these circumstances by s x [N; x c ; x n ( j) ; x n (j)]. We derive an explicit formula for this value in the next subsection. For now, note that optimal investment by supplier j implies that x n (j) is chosen to maximize s x [N; x c ; x n ( of investment in noncontractible activities, ( j) ; x n (j)] minus the cost ) c x x n (j). In a symmetric equilibrium, we need x n (j) = x n ( j), or in other words, x n needs to be a xed-point given by: 4 x n = arg max x n(j) s x [N; x c ; x n ; x n (j)] ( ) c x x n (j) : (4) Equation (4) can be thought of as an incentive compatibility constraint, with the additional symmetry requirement. In a symmetric equilibrium with technology N, with investment in contractible activities given by x c and with investment in noncontractible activities equal to x n, the revenue of the rm is given. by R = A N + x c x Moreover, let sx (N; x c ; x n ) = s x (N; x c ; x n ; x n ), then the Shapley value of the rm is obtained as a residual: n s q (N; x c ; x n ) = A N + x c xn Ns x (N; x c ; x n ) : Now consider the stage in which the rm chooses N suppliers from a pool of applicants. If suppliers expect to receive less than their outside option, w, this pool is empty. Therefore, for production to take place, the nal-good producer has to o er a contract that satis es the participation constraint of suppliers under incomplete contracts, i.e., s x (N; x c ; x n ; x n ) + c x x c + ( ) c x x n + w for x n that satis es (4). (5) 3 More generally, we would need to consider a distribution of investment levels, fx n (i; j)g i2(;] for supplier j, where some of the activities may receive more investment than others. It is straightforward to show, however, that the best deviation for a supplier is to choose the same level of investment in all noncontractible activities. For this reason we save on notation and restrict attention to only such deviations. 4 This equation should be written with 2 instead of =. However, we show below that the xed point x n in (4) is unique, justifying our use of =. 8

11 In other words, given N and (x c ; ), each supplier j 2 [; N] should expect her Shapley value plus the upfront payment to cover the cost of investment in contractible and noncontractible activities and the value of her outside option. The maximization problem of the rm can then be written as: max s q (N; x c ; x n ) N C (N) subject to (4) and (5). N;x c;x n; With no restrictions on, the participation constraint (5) will be satis ed with equality; otherwise the rm could reduce without violating (5) and increase its pro ts. We can therefore solve from this constraint, substitute the solution into the rm s objective function and obtain the simpler maximization problem: 5 max s q (N; x c ; x n ) + N [s x (N; x c ; x n ; x n ) c x x c ( ) c x x n ] C (N) w N subject to (4). N;x c;x n n o The SSPE ~N; ~xc ; ~x n solves this problem, and the corresponding upfront payment satis es (6) ~ = c x ~x c + ( ) c x ~x n + w s x ~N; ~xc ; ~x n ; ~x n : (7) 4.3 Bargaining We now derive the Shapley values in this game (see Shapley, 953, or Osborne and Rubinstein, 994). In a bargaining game with a nite number of players, each player s Shapley value is the average of her contributions to all coalitions that consist of players ordered below her in all feasible permutations. More explicitly, in a game with M + players, let g = fg () ; g () ; :::; g (M)g be a permutation of ; ; 2; :::; M, where player is the rm and players ; 2; :::; M are the suppliers, and let z j g = fj j g (j) > g (j )g be the set of players ordered below j in the permutation g. We denote by G the set of feasible permutations and by v : G! R the value of the coalition consisting of any subset of the M + players. 6 Then the Shapley value of player j is s j = (M + )! X v z j g [ j g2g v zg j : In the Appendix, we derive the asymptotic Shapley value of Aumann and Shapley (974), by considering the limit of this expression as the number of players goes to in nity. 7 Leaving the formal derivation to the Appendix, here we provide a heuristic derivation of this Shapley value. Suppose the rm has adopted technology N, all suppliers provide an amount x c of every con- 5 Note that, as in the case with complete contracts, the rm chooses its technology and investment levels to maximize sale revenues net of total costs. The key di erence is that with incomplete contracts, this maximization problem is constrained by the incentive compatibility condition (4). 6 In our game, the value of a coalition equals the amount of revenue this coalition can generate. 7 More formally, we divide the interval [; N] into M equally spaced subintervals with all the intermediate inputs in each subinterval of length N=M performed by a single supplier. We then solve for the Shapley value and take the limit of this solution as M!. See Aumann and Shapley (974) or Stole and Zwiebel (996b). 9

12 tractible activity, and all suppliers other than j invest x n ( j) in every noncontractible activity, while supplier j invests x n (j). To compute the Shapley value for supplier j, rst note that the rm is an essential player in this bargaining game (if a coalition does not include the rm, then its output equals zero regardless of its size). Consequently, the supplier j s marginal contribution is equal to zero when a coalition does not include the rm. When it does include the rm and a measure n of suppliers, the marginal contribution of supplier j is m (j; n) R=@n, where R = A N (+ =) Z n Z = exp ln x (i; k) di dk is the revenue derived from the employment of n inputs with technology N, and the last input k = n is provided by supplier j. Since x (i; k) = x c for all i and all k n, x (i; k) = x n ( j) for all < i and all k < n, and x (i; k) = x n (j) for all < i and k = n, evaluating the previous expression enables us to write the marginal contribution of supplier j as m (j; n) = A N (+ =) xn (j) ( x n ( j) ) x c x n ( j) ( ) n ( )=. (8) The Shapley value of supplier j is the average of her marginal contributions to coalitions that consist of players ordered below her in all feasible orderings. A supplier that has a measure n of players ordered below her has a marginal contribution of m (j; n) if the rm is ordered below her (probability n=n) and otherwise (probability n=n). Averaging over all possible orderings of the players and using (8), we obtain: s x [N; x c ; x n ( j) ; x n (j)] = N Z N n N m (j; n) dn = ( ) A xn (j) x n ( j) ( ) x c x n ( j) ( ) N (+), where + : (9) We therefore obtain the following lemma (see the Appendix for the formal proof): Lemma Suppose that supplier j invests x n (j) in her noncontractible activities, all the other suppliers invest x n ( j) in their noncontractible activities, every supplier invests x c in her contractible activities, and the level of technology is N. Then the Shapley value of supplier j is s x [N; x c ; x n ( j) ; x n (j)] = ( ) A xn (j) ( x n ( j) ) x c x n ( j) ( ) N (+), (2) where is de ned in (9). A number of features of (2) are worth noting. First, in equilibrium, all suppliers invest equally

13 in all the noncontractible activities, i.e., x n (j) = x n ( j) = x n, and so s x (N; x c ; x n ) = s x (N; x c ; x n ; x n ) = ( ) A x where R = A x ( ) c xn ( ) c x n N (+) = ( ) R N ; (2) N (+) is the total revenue of the rm. Thus, the joint Shapley value of the suppliers, Ns x (N; x c ; x n ), equals the fraction of the revenue, and the rm receives the remaining fraction, i.e., s q (N; x c ; x n ) = A x ( ) c x n N (+) = R: (22) This is a relatively simple rule for the division of revenue between the rm and its suppliers. Second, the derived parameter = ( + ) represents the bargaining power of the rm; it is increasing in and decreasing in. A higher elasticity of substitution between intermediate inputs, i.e., a higher, raises the rm s bargaining power, because it makes every supplier less essential in production and therefore raises the share of revenue appropriated by the rm. In contrast, a higher elasticity of demand for the nal good, i.e., higher, reduces the rm s bargaining power, because, for any coalition, it reduces the marginal contribution of the rm to the coalition s payo as a fraction of revenue. 8 Furthermore, when is smaller, s x [N; x c ; x n ( j) ; x n (j)] is more concave with respect to x n (j), because greater complementary between the intermediate inputs implies that a given change in the relative employment of two inputs has a larger impact on their relative marginal products. The impact of on the concavity of s x () will play an important role in the following results. The parameter, on the other hand, a ects the concavity of revenue in output (see (2)) but has no e ect on the concavity of s x with respect to x n (j), because with a continuum of suppliers, a single supplier has an in nitesimal e ect on output. 4.4 Equilibrium To characterize a SSPE, we rst derive the incentive compatibility constraint using (4) and (2): x n = arg max x n(j) ( ) A xn (j) ( x n ) x ( ) c x n N (+) c x ( ) x n (j). 8 To clarify the e ects of and on, let us return to the derivation in the text and note that the marginal contribution of the rm to a coalition of n suppliers, each one investing x c in contractible activities and x n in noncontractible activities, can be expressed as m q (n) = A ( ) N (+) x ( ) c xn n N = = n N = R; where R = A N (+) x c xn is the equilibrium level of the revenue (when n = N). The expression (n=n) = is decreasing in and increasing in for all n < N. The parameter is given by the average of the (n=n) = terms, = N and is also decreasing in and increasing in. Z N n N = dn = + ;

14 Relative to the producer s rst-best choice characterized above, we see two di erences. the term ( First, ) implies that the supplier is not the full residual claimant of the return from her investment in noncontractible activities and thus underinvests in these activities. Second, as discussed above, multilateral bargaining distorts the perceived concavity of the private return relative to the social return. Using the rst-order condition of this problem and solving for the xed point by substituting x n (j) = x n yields a unique x n : x n = x n (N; x c ) h ( ) (c x ) x c A N (+) i =[ ( )] : (23) Note that x n (N; x c ) is increasing in x c ; since the marginal productivity of an activity rises with investment in other activities, investments in contractible and noncontractible activities are complements. 9 Another implication of (23) is that investment in noncontractible activities is increasing in. Mathematically, this follows from the fact that ( ) = = ( + ) is increasing in. However, the economics of this relationship is the outcome of two opposing forces. The share of the suppliers in revenue, ( ), is decreasing in, because greater substitution between the intermediate inputs reduces the suppliers ex post bargaining power. But a greater level of also reduces the concavity of s x () in x n, increasing the marginal reward from investing further in noncontractible activities. Because the latter e ect dominates, x n is increasing in. Now, using (2), (22) and (23), the rm s optimization problem (6) can be expressed as h max A x c x n (N; x c ) i N (+) N;x c c x Nx c c x N ( ) x n (N; x c ) C (N) w N; (24) where x n (N; x c ) is de ned in (23). Substituting (23) into (24) and di erentiating with respect to N and x c results in two rst-order conditions, which yield a unique solution ~N; ~xc to (24): 2 ~N (+) A cx ( ) ( ) ( ) ( ) ( ) ( ) = C N ~ + w ; (25) C ~N + w ~x c = c x : (26) As in the complete contracts case, these two conditions determine the equilibrium recursively. First, (25) gives ~ N, and then given ~ N, (26) yields ~x c. Moreover, using (23), (25), and (26) gives 9 The e ect of N on x n is ambiguous, since investment in noncontractible activities declines with the level of technology when ( + ) < and increases with N when ( + ) >. This is because an increase in N has two opposite e ects on a supplier s incentives to invest; a greater number of inputs increases the marginal product of investment due to the love for variety embodied in the technology, but at the same time, the bargaining share of a supplier, ( ) =N, declines with N. For large values of the former e ect dominates, while for small values of the latter dominates. 2 See the Appendix for a more detailed derivation and for the second-order conditions. 2

15 the level of investment in noncontractible activities as ~N ( ) [ ( )] ~x n C + w A : (27) [ ( ) ( )] c x Comparing (2) to (26), we see that for a given N the implied level of investment in contractible activities under incomplete contracts, ~x c, is identical to the investment level in contractible activities under complete contracts, x. This highlights the fact that di erences in the investment in contractible activities between these economic environments only result from di erences in technology adoption. In fact, comparing () with (25), we see that ~ N and N di er only because of the two bracketed terms on the left-hand side of (25). These represent the distortions created by bargaining between the rm and its suppliers. Intuitively, technology adoption is distorted because incomplete contracts reduce investment in noncontractible activities below the level of investment in contractible activities and this underinvestment reduces the pro tability of technologies with high N. As! (and contractual imperfections disappear), both of these bracketed terms on the left-hand side of (25) go to and ~N; ~xc! (N ; x ). 2 The impact of incomplete contracts on productivity follows directly from their e ect on the choice of technology. In particular, productivity under incomplete contracts, ~ P = ~ N, is always lower than P as given in (3), since ~ N < N. 4.5 Implications of Incomplete Contracts We now provide a number of comparative static results on the SSPE under incomplete contracts, and compare the incomplete-contracts equilibrium technology and investment levels to the equilibrium under complete contracts. The comparative static results are facilitated by the block-recursive structure of the equilibrium; any change in A, or that increases the left-hand side of (25) also increase ~ N, and the e ect on ~x c and ~x n can then be obtained from (26) and (27). The main results are provided in the next proposition (proof in the Appendix). Proposition 2 Suppose n that oassumption holds. Then there exists a unique SSPEn under incomplete contracts, ~N; ~xc ; ~x n ; characterized by (25), (26) and (27). Furthermore, ~N; ~xc ; ~x o n satis es ~ N; ~x c ; ~x n >, ~x n < ~x c ; 2 Note that as! the investment level ~x n does not converge to x, because the e ect of distortions on the noncontractible activities does not go to zero. What goes to zero, however, is the importance of noncontractible activities in the production of nal goods. 3

16 @ N > N > (~x n =~x c ) N > (~x n =~x c ) > The main results in this proposition are intuitive. activities than in contractible activities, in particular, Suppliers invest less in noncontractible ~x n ( ) [ ( )] = <, (28) ~x c [ ( ) ( )] which follows from equations (26) and (27) and from the fact that ( (recall (9)). ) = = ( + ) < Intuitively, the rm is the full residual claimant of the return to investments in contractible activities and it dictates these investments in the contract. In contrast, investments in noncontractible activities are decided by the suppliers, who are not the full residual claimants of the returns generated by these investments (recall (2)) and thus underinvest in these activities. In addition, the level of technology and investments in both contractible and noncontractible activities are increasing in the size of the market, in the fraction of contractible activities (quality of contracting institutions), and in the elasticity of substitution between intermediate inputs. 22 impact of the size of the market is intuitive; a greater A makes production more pro table and thus increases investments and equilibrium technology. Better contracting institutions, on the other hand, imply that a greater fraction of activities receive the higher investment level ~x c rather than ~x n < ~x c. This makes the choice of a more advanced technology more pro table. A higher N, in turn, increases the pro tability of further investments in ~x c and ~x n. Better contracting institutions also close the (proportional) gap between ~x c and ~x n because with a higher fraction of contractible activities, the marginal return to investment in noncontractible activities is also higher. A higher, i.e., lower complementarity between intermediate inputs, also increases technology choices and investments. The reason is related to the discussion in the previous subsection where it was shown that a higher reduces the share of each supplier but also makes s x () less concave. Because the latter e ect dominates, a lower degree of complementarity increases supplier investments and makes the adoption of more advanced technologies more pro table. The reduced-form pro t function depicted in equation () in the Introduction can also be derived at this point. Combining (23), (24), and the condition for the rm s optimal choice of x c, the rm s payo can be expressed as (see the Appendix): The (+) + = AZ (; ) N C (N) w N; (29) 22 Equation (3) above implies that the measure of productivity, P = q= (NX), has the same comparative statics as technology. The same results also apply if we were to de ne productivity as P = q=n, that is, as output divided by the number of suppliers. Yet, in this case rms operating under di erent contracting institutions would have di erent productivity levels not only because they choose di erent levels of technology, but also because they have di erent investment levels. See Acemoglu, Antràs and Helpman (25) for more details on this point. 4

17 where Z (; ) ( ) ( ) [ ( )] ( ) ( ) ( ) ( ) (c x ) (3) represents a measure of derived e ciency and captures the distortions arising from incomplete contracts. The extent of these distortions depends on the model s parameters as shown in Proposition 2. In addition, we have the following lemma (proof in the Appendix): Lemma 2 Suppose that Assumption holds. Let (; ) (; ) =@) =Z (; ) be the elasticity of Z (; ) with respect to and let let (; ) (; ) =@) =Z (; ) be the elasticity of Z (; ) with respect to. Then, we have that. (; ) > and (; ) > ; and (; ) =@ < (; ) =@ <. Part of this lemma implies that better contracting institutions and greater substitutability between intermediate inputs lead to higher levels of Z (; ). Part 2, on the other hand, implies that the proportional increase in Z (; ) in response to an improvement in contracting institutions is greater when there is more technological complementarity between intermediate inputs. The intuition is that contract incompleteness is more damaging to technologies with greater complementarities, because there are more signi cant investment distortions in this case. This last result implies that sectors with greater complementarities are more contract dependent and will play an important role in the general equilibrium analysis in Section 5.3. Finally, to compare the complete and incomplete contracts equilibria, recall that they both lead to the same allocation as!. Together with Proposition 2, this implies (proof in the Appendix): n o Proposition 3 Suppose that Assumption holds. Let ~N; ~xc ; ~x n be the unique SSPE with incomplete contracts and let fn ; x g be the unique equilibrium with complete contracts. Then ~N < N and ~x n < ~x c x. This proposition implies that since incomplete contracts lead to the choice of less advanced (lower N) technologies, 23 they also reduce productivity and investments in contractible and noncontractible activities. 23 It is useful to contrast this result with the overemployment result in Stole and Zwiebel (996a,b). There are two important di erences between our model and theirs. First, our model features investments in noncontractible activities, which are absent in Stole and Zwiebel. Second, Stole and Zwiebel assume that if a worker is not in the coalition of the bargaining game, she receives no payment whatsoever. Since in our model investment in noncontractible activities is not veri able, a supplier receives the upfront payment independently of whether she is in the coalition, and she receives only the rest of the payment, i.e., s x, through bargaining. The treatment of the outside option in Stole and Zwiebel is essential for their overemployment result (see de Fontenay and Gans, 23). 5

18 5 Applications We next discuss a number of applications of the basic framework developed so far. These applications emphasize both the potential quantitative e ects of our main mechanism and its implications for equilibrium organizational choices and endogenous patterns of comparative advantage. 5. Quantitative Implications We rst investigate whether our baseline model can generate signi cant productivity di erences from cross-country variation in contracting institutions. Our purpose here is not to undertake a full- edged calibration, but to give a sense of the empirical implications of our baseline model for plausible parameter values. To obtain closed-form solutions for the equilibrium value of N and for the productivity measure P = N, we assume that the function C (N) is linear in N, C (N) = N. Equation (25) then implies: 24 ( ) A (+) ~P = + w (+) cx (+) ( ) ( ) ( ) ( ( )) (+) ( ) ( ) (+). Our focus is on the impact of the quality of contracting institutions,, on rm-level productivity. For this purpose, consider the ratio of productivity in two economies with the fraction of contractible tasks given by and <, ~P ( ) ~P ( ) = where recall that = ( + ). h i ( ( )) ( )( ) (+) ( ) h ( )( ) ( ) i ( ( )) (+) ( ) ( ) (+), (3) Equation (3) shows that the proportional impact of on aggregate productivity only depends on the values of the parameters,, and. 25 The parameter governs the elasticity of substitution between nal-good varieties and also determines the markup charged by nal-good producers. In our benchmark simulation, we set this parameter equal to :75. This implies an elasticity of substitution between nal-good varieties equal to 4, which corresponds to the mean elasticity estimated by Broda and Weinstein (26) using U.S. import data. This value of also implies a markup over marginal cost equal to ' :33, which is comfortably within the range of available estimates Recall from the discussion in footnote that there are alternative ways of measuring productivity in the model, and which one of these is more appropriate will depend on how productivity is measured in practice. If, for example, we were to measure productivity as output divided by the number of suppliers, the impact of improvements in contracting institutions on productivity would be even larger, because such improvements raise investment in contractible and noncontractible activities as well. 25 In deriving equation (3), we assume that the remaining parameters are held xed when changes. We should thus interpret our results as re ecting the partial-equilibrium response of productivity to changes in contractibility. 26 On the lower side, Morrison (992) nds markups ranging from : in 96 to :29 in 97, while on the higher side Rotemberg and Woodford (99) use an estimate of :6. See Basu (995) for a discussion of the sensitivity of the estimated markups to model speci cation and calibration. It is interesting to note that Morrison (995) nds markups between : and :39 in Japan and : and :2 in Canada, Roberts (996) nds average markups between 6

19 ~ P(.75) ~ P(.25) ~ P ~ () P() 2 8 ~ P(2 / 3) ~ P(/ 3) α Figure : Relative productivity for = :75 and = :25 We choose the parameter to match Bils and Klenow s (2) estimates of variety growth in the U.S. economy from BLS data on expenditures on di erent types of goods. Their ndings indicate that the growth of the number of varieties in the U.S. economy was around 2% per annum between 959 and 999. If we assume that the variety of intermediate goods grew at the same rate over this period and also that this variety growth was the main source of productivity (TFP) growth in the U.S. economy, then we can use our model to back out a value of. We take the rate of TFP growth per annum to be.5% and use the fact that productivity is given by P = N. This implies that Nt+ Nt =N t = :5 and generates a value of = ln (:5) = ln (:2) ' : The value of is harder to pin down. Since the main theoretical ideas in the paper relate to complementarity between di erent inputs, a low value of may be natural. Nevertheless, since we have no direct way of relating to aggregate data, we show the implications of changes in the quality of contracting institutions for di erent values of. Figure depicts the value of the ratio in (3) for di erent values of when = :75 and = :25. We consider three experiments. The middle curve depicts a shift from = :25 to = :75, the lowest curve depicts a more modest increase from = =3 to = 2=3, while the highest curve illustrates the most extreme shift from = to =. The patterns in Figure con rm the results in Lemma 2 and show that the impact of an increase in on productivity is larger for lower values of. More importantly, the gure shows that the :22 and :3 in Colombia, and Grether (996) nds average markups between :34 and :37 in Mexico. 27 A somewhat di erent number is obtained from Broda and Weinstein s (26) estimate that the number of varieties in U.S. imports between 972 and 2 grew at an annual rate of 3.7%. Using the same formula, this would imply a value of :35 for. Note, however, that this is likely to be an overestimate of variety growth and thus an underestimate of because Broda and Weinstein consider the same good imported from di erent countries as di erent varieties, and the number of exporting countries to the U.S. increased over time. For this reason, we choose = :25 as our benchmark parameter value. 7