Contracts and Technology Adoption

By DARON ACEMOGLU,POL ANTRAS` , AND ELHANAN HELPMAN*

We develop a tractable framework for the analysis of the relationship between contractual incompleteness, technological complementarities, and technology adoption. In our model, a ﬁrm chooses its technology and investment levels in contractible activities by suppliers of intermediate inputs. Suppliers then choose investments in noncontractible activities, anticipating payoffs 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 differences across countries with different contracting institutions, and that differences in contracting institutions lead to endogenous comparative advantage differences. (JEL D86, O33)

There is widespread agreement that differ- ductivity differences across firms, industries, ences in technology are a major source of pro- and nations.1 Despite this widespread agreement, we are far from an established framework for the analysis of technology choices of firms. * Acemoglu: Department of Economics, Massachusetts In- In this paper, we take a step in this direction and stitute of Technology, E52-380b, Cambridge, MA 02139, Na- develop a simple model to study the impact of tional Bureau of Economic Research, Centre for Economic Policy Research, and Canadian Institute for Advanced Re- contracting institutions, which regulate the research (e-mail: [email protected]); Antra`s: Department of Eco- lationship between the firm and its suppliers, on nomics, Harvard University, Littauer 319, Cambridge, MA technology choices. 02138, NBER, and CEPR (e-mail: [email protected]); Our model combines two well-established Helpman: Department of Economics, Harvard University, Lit- tauer 229, Cambridge, MA 02138, NBER, CEPR, and CIAR approaches. The first is the representation of (e-mail: [email protected]). A previous version of this technology as the range of intermediate in- paper was circulated under the title “Contracts and the Division puts used by firms; a greater range of interme- of Labor.” We thank Gene Grossman, Oliver Hart, Kalina diate inputs increases productivity by allowing Manova, Damiań Migueles, Giacomo Ponzetto, Richard Rog- greater specialization and thus corresponds to erson, Rani Spiegler, two anonymous referees, participants in 2 the Canadian Institute for Advanced Research and Minnesota more “advanced” technology. The second is Workshop in Macroeconomic Theory conferences, and semi- Sanford J. Grossman and Oliver D. Hart’s nar participants at Harvard University, MIT, Tel Aviv Univer- sity, Universitat Pompeu Fabra, University of Houston, University of California, San Diego, Southern Methodist University, Stockholm School of Economics, Stockholm 1 Among others, see Peter J. Klenow and Andre´s Rodri- University (IIES), ECARES, University of Colorado–Boul- guez-Clare (1997), Robert E. Hall and Charles I. Jones der, the Kellogg School, New York University, University (1999), or Francesco Caselli (2005) for countries, and Tor J. of British Columbia, Georgetown University, the World Klette (1996), Zvi Griliches (1998), John Sutton (1998), or Bank, University of Montreal, Brandeis University, Univer- Klette and Samuel S. Kortum (2004) for firms. sity of Wisconsin, UC–Berkeley, Haifa University, Syra- 2 See, among others, Wilfred J. Ethier (1982), Paul M. cuse University, Universitat Auto`noma de Barcelona, Duke Romer (1990), and Gene M. Grossman and Helpman (1991) University, Columbia University, and Cornell University for previous uses of this representation. See also the text- for useful comments. We also thank Davin Chor, Alexandre book treatment in Robert J. Barro and Xavier Sala-i-Martin Debs, and Ran Melamed for excellent research assistance. (2003). There is a natural relationship between this view of Acemoglu and Helpman thank the National Science Foun- technology and the division of labor within a firm. We dation for financial support. Much of Helpman’s work for investigated this link in the earlier version of the current this paper was done when he was Sackler Visiting Professor paper, Acemoglu, Antra`s, and Helpman (2005), and do not at Tel Aviv University. elaborate on it here. 916 VOL. 97 NO. 3 ACEMOGLU ET AL.: CONTRACTS AND TECHNOLOGY ADOPTION 917

(1986) and Hart and John H. Moore’s (1990) the noncontractible activities. Since she is not approach to incomplete-contracting models of the full residual claimant of the output gains the firm. We study technology choice of firms derived from her investments, she tends to un- under incomplete contracts, and extend Hart derinvest. Greater contractual incompleteness and Moore’s framework by allowing contracts thus reduces supplier investments, making more to be partially incomplete. This combination advanced technologies less profitable. Further- enables us to investigate how the degree of more, a greater degree of technological comple- contractual incompleteness and the extent of mentarity reduces the incentive to choose more technological complementarities between inter- advanced technologies. Although greater tech- mediate inputs affect the choice of technology. nological complementarity increases equilib- In our baseline model, a firm decides on rium ex post payoffs to every supplier, it also technology (on the range of specialized inter- makes their payoffs less sensitive to their non- mediate goods), recognizing that a more ad- contractible investments, discouraging invest- vanced technology is more productive, but also ments and, via this channel, depressing the entails a variety of costs. In addition to the profitability of more advanced technologies. direct pecuniary costs of engaging more suppli- An advantage of our framework is its relative ers (corresponding to the greater range of inter- tractability. The equilibrium of our model can mediate inputs), a more advanced technology be represented by a reduced-form profit func- necessitates contracting with more suppliers. tion for firms given by All of the activities that suppliers undertake are ͑ ͒ Ϫ ͑ ͒ Ϫ relationship-specific, and a fraction of those is (1) AZF N C N w0 N, ex ante contractible, while the rest, as in the work by Grossman-Hart-Moore, are nonverifi- where N represents the technology level, C(N) able and noncontractible. The fraction of con- is the cost of technology N, A is a measure of tractible activities is our measure of the quality aggregate demand or the scale of the market, 3 of contracting institutions. Suppliers are con- and w0N corresponds to the value of the N tractually obliged to perform their duties in the suppliers’ outside options. F(N) is an increasing contractible activities, but they are free to function that captures the positive effect of choose their investments in noncontractible ac- choosing more advanced technologies on reve- tivities and to withhold their services in these nue. The effects of contractual incompleteness activities from the firm. This combination of and technological complementarity are summa- noncontractible investments and relationship- rized by the variable Z. This variable affects specificity leads to an ex post multilateral bar- productivity and is decreasing in the degree gaining problem. As in Hart and Moore (1990), of contract incompleteness and technological we use the Shapley value to determine the di- complementarity. Moreover, the elasticity of Z vision of ex post surplus between the firm with respect to the quality of contracting insti- and its suppliers. We derive an explicit solution tutions is higher when there is greater comple- for this division of surplus, which enables us mentarity between intermediate inputs. This last to develop a simple characterization of the result has important implications for equilib- equilibrium. rium industry structure and the patterns of com- A supplier’s expected payoff in the bargain- parative advantage, because it implies that ing game determines her willingness to invest in sectors (firms) with greater complementarities between inputs are more “contract dependent.” We use this framework to show that the 3 Eric S. Maskin and Jean Tirole (1999) question combination of contractual imperfections and whether the presence of nonverifiable actions and unforeseen contingencies necessarily leads to incomplete con- technology choice (or adoption) may have im- tracts. Their argument is not central to our analysis portant implications for cross-country income because it assumes the presence of “strong” contracting differences, equilibrium organizational forms, institutions (which, for example, allow contracts to spec- and patterns of international specialization and ify sophisticated mechanisms), while in our model a fraction of activities may be noncontractible, not because trade. of “technological” reasons but because of weak contract- First, we show that our baseline model can ing institutions. generate sizable differences in productivity from 918 THE AMERICAN ECONOMIC REVIEW JUNE 2007 differences in the quality of contracting institu- Zwiebel (1996a, b) and Yannis Bakos and Erik tions. In particular, using a range of reasonable Brynjolfsson (1993) are most closely related to values for the key parameters of the model, we ours. Stole and Zwiebel consider a relationship find that when the degree of technological between a firm and a number of workers whose complementarity is sufficiently high, relatively wages are determined by ex post bargaining modest changes in the fraction of activities that according to the Shapley value. They show how are contractible can lead to large changes in the firm may overemploy in order to reduce productivity. the bargaining power of the workers, and Second, we derive a range of implications discuss the implications of this framework for about the equilibrium organizational form. a number of organizational design issues. Our results here show that the combination of Stole and Zwiebel’s framework does not have weak contracting institutions and credit mar- relationship-specific investments, however, ket imperfections may encourage greater ver- which is at the core of our approach, and they tical integration. do not discuss the effects of the degree of Third, we present a number of general equi- contractual incompleteness and the degree of librium applications of this framework. The complementarity between inputs on the equi- general equilibrium interactions result from the librium technology choice.4 fact that an improvement in contracting institu- Finally, our paper is also related to the liter- tions does not increase the choice of technology ature on the macroeconomic implications of in all sectors (firms). Instead, more advanced contractual imperfections. A number of papers, technologies are chosen in the more contract- most notably Erwan Quintin (2001), Pedro S. dependent sectors. We show that in the context Amaral and Quintin (2005), Andre´s Erosa and of an open economy, this feature leads to an Ana Hidalgo (2005), and Rui Castro, Gian Luca endogenous structure of comparative advan- Clementi, and Glenn MacDonald (2004), inves- tage. In particular, among countries with iden- tigate the quantitative impact of contractual and tical technological opportunities (production capital market imperfections on aggregate pro- possibility sets), those with better contracting ductivity.5 Our approach differs from these pa- institutions specialize in sectors with greater pers since we focus on technology choice and complementarities among inputs. These predic- relationship-specific investments, and because tions are consistent with the recent empirical we develop a tractable framework that can be results presented in Nathan Nunn (2006) and applied in a range of problems. Although we do Andrei A. Levchenko (2003). not undertake a detailed calibration exercise as As mentioned above, our work is related to in these papers, the results in Section IVA sug- two strands of the literature. The first investi- gest that the economic mechanism developed in gates the determinants of firm-level technology this paper can lead to quantitatively large ef- (including the division of labor) and includes, fects. Finally, Levchenko (2003), Arnaud Costi- among others, Gary S. Becker and Kevin M. not (2004), Nunn (2006), and Antra`s (2005) Murphy (1992) and Xiaokai Yang and Jeff Bor- also generate endogenous comparative advan- land (1991) on the impact of the extent of the tage across countries from differences in con- market on the division of labor, and Romer tractual environments. Among these papers, (1990) and Grossman and Helpman (1991) Costinot (2004) is most closely related, since he on endogenous technological change. None also develops a model of endogenous compar- of these studies investigates the effects of contracting institutions on technology choice. The 4 second literature deals with the internal organi- Another related paper is Olivier J. Blanchard and Mi- chael Kremer (1997), which studies the impact of inefficient zation of the firm. It includes the papers by sequential contracting between a firm and its suppliers in a Grossman-Hart-Moore discussed above, as well model where output is a Leontief aggregate of the inputs of as those of Benjamin Klein, Robert G. Craw- the suppliers. ford, and Armen A. Alchian (1978) and Oliver 5 Acemoglu and Fabrizio Zilibotti (1999), David Marti- mort and Thierry Verdier (2000, 2004), and Patrick Fran- E. Williamson (1975, 1985), who emphasize cois and Joanne K. Roberts (2003) study the qualitative incomplete contracts and hold-up problems. impact of changes in the internal organization of firms on Here, the papers by Lars A. Stole and Jeffrey economic growth. VOL. 97 NO. 3 ACEMOGLU ET AL.: CONTRACTS AND TECHNOLOGY ADOPTION 919 ative advantage based on specialization, though (1998) in introducing the term N␬ϩ1Ϫ1/␣ in his approach is more reduced-form than our front of the integral, which allows us to sepa- model. rately control the elasticity of substitution be- The rest of the paper is organized as fol- tween inputs and the elasticity of output with lows. Section I introduces the basic environ- respect to the level of the technology. To see ment. Section II characterizes the equilibrium this, consider the case where X(j) ϭ X for all j. with complete contracts. Section III introduces Then the output of technology N is q ϭ N␬ϩ1X. incomplete contracts into the framework of Sec- Consequently, both output and productivity (de- tion I, characterizes the equilibrium, and derives fined as either q/(NX)orq/N) are independent of the major comparative static results. Section IV ␣ and depend positively on N, with elasticity considers a number of applications of our basic determined by the parameter ␬.6 framework. Section V concludes. Proofs of the There is a large number of profit-maximizing main results are provided in the Appendix. suppliers that can produce the necessary intermediate goods, each with the same outside op- I. Technology and Payoffs tion w0. For now, w0 is taken as given, but it will be endogenized in Section IVC. We assume that Consider a profit-maximizing firm facing a each intermediate input needs to be produced by demand function q ϭ ApϪ1/(1Ϫ␤) for its final a different supplier with whom the firm needs to product, where q denotes quantity and p denotes contract.7 price. The parameter ␤ ʦ (0, 1) determines the A supplier assigned to the production of an elasticity of demand, while A Ͼ 0 determines intermediate input needs to undertake relation- the level of demand or the “market size.” The ship-specific investments in a unit measure of firm treats the demand level A as exogenous. (symmetric) activities. There is a constant mar- 8 This form of demand can be derived from a ginal cost of investment cx for each activity. constant elasticity of substitution preference The production function of intermediate inputs structure for differentiated products (see Sec- is Cobb-Douglas and symmetric in the activi- tion IVC) and it generates a revenue function ties:

1 Ϫ ␤ ␤ (2) R ϭ A q . 1 (4) X͑ j͒ ϭ expͫ͵ ln x(i, j) diͬ, Production depends on the technology choice of 0 the firm. More advanced (more productive) technologies involve a greater range of interme- where x(i, j) is the level of investment in activity diate goods and thus a higher degree of special- i performed by the supplier of input j. This ization. The technology level of the firm is formulation will allow a tractable parameteriza- denoted by N ʦ ޒϩ, and for each j ʦ [0, N], tion of contractual incompleteness in Section X(j) is the quantity of intermediate input j. III, whereby a subset of the investments neces- Given technology N, the production function of sary for production will be nonverifiable and the firm is thus noncontractible. Finally, we assume that adopting (and using) 1/␣ N a technology N involves costs C(N), and im- (3) q ϭ N␬ ϩ 1Ϫ1/␣ͫ͵ X( j)␣ djͬ , pose: 0 6 In contrast, with the standard specification of the CES 0 Ͻ ␣ Ͻ 1, ␬ Ͼ 0. production function, without the term N␬ϩ1Ϫ1/␣ in front (i.e., ␬ ϭ 1/␣ Ϫ 1), total output is q ϭ N1/␣X, and the two A number of features of this production func- elasticities are governed by the same parameter, ␣. tion are worth noting. First, ␣ determines the 7 A previous version of the paper, Acemoglu, Antra`s, degree of complementarity between inputs; and Helpman (2005), also endogenized the allocation of ␣ ʦ inputs to suppliers using an augmented model with addi- since (0, 1), the elasticity of substitution tional diseconomies of scope. between them, 1/(1 Ϫ ␣), is always greater than 8 One can think of cx as the marginal cost of effort; see one. Second, we follow Jean-Pascal Benassy the formulation of this effort in utility terms in Section IVC. 920 THE AMERICAN ECONOMIC REVIEW JUNE 2007

ASSUMPTION 1: where the firm chooses a technology level N and makes a contract offer [{x(i, j)}iʦ[0,1],{s(j), (a) For all N Ͼ 0, C(N) is twice continuously ␶(j)}] for every input j ʦ [0, N]. If a supplier differentiable, with CЈ(N) Ͼ 0 and CЉ(N) Ն accepts this contract for input j, she is obliged to 0. supply {x(i, j)}iʦ[0,1] as stipulated in the con- Ͼ Љ Ј ϩ Ͼ ␶ (b) For all N 0, NC (N)/[C (N) w0] tract in exchange for the payments {s(j), (j)}. [␤(␬ ϩ 1) Ϫ 1]/(1 Ϫ ␤). A subgame perfect equilibrium of this game is a strategy combination for the firm and the suppliers such that suppliers maximize (5) and the The first part of this assumption is standard; firm maximizes (6). An equilibrium can be al- costs are increasing and convex. The second ternatively represented as a solution to the fol- part will ensure a finite and positive choice of N. lowing maximization problem: Let the payment to supplier j consist of two ␶ ʦ ޒ parts: an ex ante payment (j) before the (8) investment levels x(i, j) take place, and a payment s(j) after the investments. Then, the payoff N to supplier j, also taking account of her outside max R Ϫ ͵ ͓␶͑j͒ ϩ s͑j͔͒ dj Ϫ C͑N͒ option, is N,͕x͑i,j͖͒i j ,͕s͑ j͒,␶ ͑j͖͒j , 0

(5) subject to (7) and the suppliers’ participation

1 constraint, ␲ ͑j͒ ϭ max ␶(j) ϩ s(j) Ϫ ͵ c x(i, j) di, w . x ͭ x 0ͮ 1 0 (9) s͑ j͒ ϩ ␶ ͑j͒ Ϫ c ͵ x͑i, j͒ di x 0 Similarly, the payoff to the firm is Ն ʦ ͓ ͔ w0 for all j 0, N . N (6) ␲ ϭ R Ϫ ͵ ͓␶͑j͒ ϩ s͑j͔͒ dj Ϫ C͑N͒, 0 Since the firm 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 where R is revenue and the other two terms on firm’s objective function, (8), is (jointly) concave the right-hand side represent costs. Substituting in the investment levels x(i, j), and these invest- (3) and (4) into (2), revenue can be expressed as ments are all equally costly, the firm chooses the same investment level x for all activities in all (7) R ϭ A1 Ϫ ␤N␤͑␬ ϩ 1Ϫ1/␣͒ intermediate inputs. Now, substituting for (9) in (8), we obtain the following simpler unconstrained ␣ ␤/␣ N 1 maximization problem for the firm: ϫ ͫ͵ ͩexpͩ͵ ln x(i, j) diͪͪ djͬ . (10) 0 0 1 Ϫ ␤ ␤͑␬ ϩ 1͒ ␤ Ϫ Ϫ ͑ ͒ Ϫ max A N x cx Nx C N w0 N. II. Equilibrium under Complete Contracts N,x As a benchmark, consider the case of complete contracts (the “first best” from the viewpoint of From the first-order conditions of this problem, the firm). With complete contracts, the firm has we obtain full control over all investments and pays each supplier her outside option. In analogy to our 9 With complete contracts, ␶(j) and s(j) are perfect sub- treatment below of technology adoption under stitutes, so that only the sum s(j) ϩ ␶(j) matters. This will incomplete contracts, consider a game form not be the case when contracts are incomplete. VOL. 97 NO. 3 ACEMOGLU ET AL.: CONTRACTS AND TECHNOLOGY ADOPTION 921

͑ ͒͑␤͑␬ ϩ 1͒Ϫ1͒/͑1 Ϫ ␤͒ ␬␤1/͑1 Ϫ ␤͒ Ϫ ␤/͑1 Ϫ ␤͒ (11) N* A cx In the case of complete contracts, the size of the market (as parameterized by the demand ϭ Ј͑ ͒ ϩ C N* w0 , level A) has a positive effect on investments by suppliers of intermediate inputs and productiv- Ј͑ ͒ ϩ ity. The other noteworthy implication of this C N* w0 (12) x* ϭ . proposition is that, under complete contracts, ␬c x the level of technology and thus productivity does not depend on the elasticity of substitution Equations (11) and (12) can be solved recur- between intermediate inputs, 1/(1 Ϫ ␣). sively. Given Assumption 1, equation (11) yields a unique solution for N*, which, together III. Equilibrium under Incomplete Contracts with (12), yields a unique solution for x*.10 When all the investment levels are identical A. Incomplete Contracts and equal to x, output equals q ϭ N␬ ϩ 1x. Since NX ϭ Nx inputs are used in the pro- We now consider the same environment un- duction process, we can define productivity as der incomplete contracts. We model the imper- output divided by total input use, P ϭ N␬.In fection of the contracting institutions by the case of complete contracts, this produc- assuming that there exists a ␮ ʦ [0, 1] such tivity level is that, for every intermediate input j, investments in activities 0 Յ i Յ ␮ are observable and ␬ verifiable and therefore contractible, while in- (13) P* ϭ ͑N*͒ , vestments in activities ␮ Ͻ i Յ 1 are not contractible. Consequently, a contract stipulates which is increasing in the level of technology. investment levels x(i, j) for the ␮ contractible In the next section we compare this to equilib- activities, but does not specify the investment rium productivity under incomplete contracts.11 levels in the remaining 1 Ϫ ␮ noncontractible The next proposition describes the key activities. Instead, suppliers choose their invest- properties of the equilibrium (proof in the ments in noncontractible activities in anticipa- Appendix). tion of the ex post distribution of revenue, and may decide to withhold their services in these PROPOSITION 1: Suppose that Assumption 1 activities from the firm. We follow the incom- holds. Then with complete contracts there exists plete contracts literature and assume that the ex a unique equilibrium with technology and in- post distribution of revenue is governed by mul- vestment levels N* Ͼ 0 and x* Ͼ 0 given by tilateral bargaining, and, as in Hart and Moore (11) and (12). Furthermore, this equilibrium (1990), we adopt the Shapley value as the so- satisfies lution concept for this multilateral bargaining game (more on this below).12 ѨN* Ѩ x* ѨN* Ѩ x* The timing of events is as follows: Ͼ 0, Ն 0, ϭ ϭ 0. ѨA ѨA Ѩ␣ Ѩ␣ ● The firm adopts a technology N and offers a ␮ ␶ contract [{xc(i, j)}iϭ0, (j)] for every inter- 10 ʦ We show in the Appendix that the second-order con- mediate input j [0, N], where xc(i, j)isan ditions are satisfied under Assumption 1. investment level in a contractible activity and 11 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 pecu- 12 The incomplete contracts literature also assumes that niary. For this reason, in the previous version we considered revenues are noncontractible. As is well known, with bilat- another definition of productivity: output divided by the eral contracting or with multilateral contracting and a bud- number of suppliers, P ϭ q/N ϭ (N)␬x. The ranking of get breaker (e.g., Bengt R. Holmstro¨m 1982), contracting on productivity levels between complete and incomplete con- revenues would improve incentives. In our setting, we do tracts is the same under both definitions, and the quantitative not need this assumption, since each firm has a continuum effects of changes in contractibility on productivity are of suppliers, and contracts on total revenues would not smaller with the definition used here. provide additional investment incentives to suppliers. 922 THE AMERICAN ECONOMIC REVIEW JUNE 2007

␶(j) is an upfront payment to supplier j. The suppliers and the firm will engage in multilat- payment ␶(j) can be positive or negative. eral Shapley bargaining. Denote the Shapley ● Potential suppliers decide whether to apply value of supplier j under these circumstances by ៮ Ϫ for the contracts. Then the firm chooses N sx[N, xc, xn( j), xn(j)]. We derive an explicit suppliers, one for each intermediate input j. formula for this value in the next subsection. ● All suppliers j ʦ [0, N] simultaneously For now, note that optimal investment by sup- ʦ choose investment levels x(i, j) for all i [0, plier j implies that xn(j) is chosen to maximize ʦ ␮ ៮ Ϫ 1]. In the contractible activities i [0, ] sx[N, xc, xn( j), xn(j)] minus the cost of invest- ϭ Ϫ ␮ they invest x(i, j) xc(i, j). ment in noncontractible activities, (1 )cxxn(j). ● ϭ The suppliers and the firm bargain over the In a symmetric equilibrium, we need xn(j) Ϫ division of revenue, and at this stage, suppli- xn( j) or, in other words, xn needs to be a ers can withhold their services in noncon- fixed-point given by14 tractible activities. ● Output is produced and sold, and the revenue (14) xn ϭ arg max s៮x ͓N, xc , xn , xn ͑j͔͒ ͑ ͒ R is distributed according to the bargaining xn j agreement. Ϫ͑1 Ϫ ␮͒cx xn ͑j͒. We will characterize a symmetric subgame perfect equilibrium (SSPE for short) of this Equation (14) can be thought of as an “incentive game, where bargaining outcomes in all sub- compatibility constraint,” with the additional games are determined by Shapley values. symmetry requirement. In a symmetric equilibrium with technology N, with investment in contractible activities B. Definition of Equilibrium and given by xc and with investment in noncontract- Preliminaries ible activities equal to xn, the revenue of the ϭ 1Ϫ␤ ␬ϩ1 ␮ 1Ϫ␮ ␤ firm is given by R A (N xc xn ) . ϭ ៮ Behavior along the SSPE can be described by Moreover, let sx(N, xc, xn) sx(N, xc, xn, xn); ˜ ␶ ˜ a tuple {N, x˜c, xñ, ˜} in which N represents the then the Shapley value of the firm is obtained as level of technology, x˜c the investment in con- a residual: tractible activities, xñ the investment in noncontractible activities, and ␶˜ the upfront payment to Ϫ ␤ ␬ ϩ ␮ Ϫ ␮ ␤ s ͑N, x , x ͒ ϭ A1 ͑N 1x x1 ͒ every supplier. That is, for every j ʦ [0, N˜ ], the q c n c n ␶ ϭ ␶ upfront payment is (j) ˜, and the investment Ϫ ͑ ͒ ϭ ʦ ␮ ϭ Nsx N, xc , xn . levels are x(i, j) x˜c for i [0, ] and x(i, j) ʦ ␮ xñ for i ( , 1]. With a slight abuse of termi- ˜ nology, we will denote the SSPE by {N, x˜c, xñ}. Now consider the stage in which the firm chooses The SSPE can be characterized by backward N suppliers from a pool of applicants. If sup- induction. First, consider the penultimate stage pliers expect to receive less than their outside of the game, with N as the level of technology option, w0, this pool is empty. Therefore, for and xc as the level of investment in contractible production to take place, the final-good pro- activities. Suppose also that each supplier other ducer has to offer a contract that satisfies the than j has chosen a level of investment in non- participation constraint of suppliers under in- Ϫ contractible activities equal to xn( j) (these are complete contracts, i.e., all the same, because we are constructing a symmetric equilibrium), while the investment level in every noncontractible activity by sup- than others. It is straightforward to show, however, that the 13 plier j is xn(j). Given these investments, 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. 13 More generally, we would need to consider a distri- 14 This equation should be written with “ʦ” instead of ϭ bution of investment levels, {xn(i, j)}iʦ(␮,1] for supplier j, “ .” However, we show below that the fixed point xn in where some of the activities may receive more investment (14) is unique, justifying our use of “ϭ.” VOL. 97 NO. 3 ACEMOGLU ET AL.: CONTRACTS AND TECHNOLOGY ADOPTION 923

(15) s៮x ͑N, xc , xn , xn ͒ ϩ ␶ Ն ␮cx xc Osborne and Ariel Rubinstein 1994). In a bargaining game with a ﬁnite number of players, ϩ ͑ Ϫ ␮͒ ϩ 1 cx xn w0 each player’s Shapley value is the average of her contributions to all coalitions that consist of

for xn that satisfies ͑14͒. players ordered below her in all feasible permutations. More explicitly, in a game with M ϩ 1 players, let g ϭ {g(0), g(1), ... , g(M)} be a ␶ In other words, given N and (xc, ), each sup- permutation of 0, 1, 2, ... , M, where player 0 is plier j ʦ [0, N] should expect her Shapley value the ﬁrm and players 1, 2, ... , M are the suppli- j ϭ Ј͉ Ͼ Ј plus the upfront payment to cover the cost of ers, and let zg {j g(j) g(j )} be the set of investment in contractible and noncontractible players ordered below j in the permutation g. activities and the value of her outside option. We denote by G the set of feasible permutations The maximization problem of the ﬁrm can and by v:G 3 ޒ the value of the coalition then be written as consisting of any subset of the M ϩ 1 players.16 Then the Shapley value of player j is max sq ͑N, xc , xn ͒ Ϫ N␶ Ϫ C͑N͒ ␶ N,xc ,xn , 1 s ϭ ͸ ͓v͑zj ഫ j͒ Ϫ v͑zj ͔͒. j ͑M ϩ 1͒! g g subject to ͑14͒ and ͑15͒. gʦG

With no restrictions on ␶, the participation In the Appendix, we derive the asymptotic constraint (15) will be satisfied with equality; Shapley value of Robert J. Aumann and Shap- otherwise the firm could reduce ␶ without vio- ley (1974) by considering the limit of this ex- lating (15) and increase its profits. We can pression as the number of players goes to therefore solve ␶ from this constraint, substitute infinity.17 Leaving the formal derivation to the the solution into the firm’s objective function, Appendix, here we provide a heuristic deriva- and obtain the simpler maximization problem:15 tion of this Shapley value. Suppose the firm has adopted technology N, ͑ ͒ ϩ ͓៮ ͑ ͒ (16) max sq N, xc , xn N sx N, xc , xn , xn all suppliers provide an amount xc of every N,xc ,xn contractible activity, and all suppliers other than j invest x (Ϫj) in every noncontractible activity, Ϫ ␮ Ϫ ͑ Ϫ ␮͒ ͔ Ϫ ͑ ͒ Ϫ n cx xc 1 cx xn C N w0 N while supplier j invests xn(j). To compute the Shapley value for supplier j, first note that the ͑ ͒ subject to 14 . firm is an essential player in this bargaining game. (If a coalition does not include the firm, ˜ The SSPE {N, x˜c, xñ} solves this problem, and then its output equals zero regardless of its the corresponding upfront payment satisfies size.) Consequently, the supplier j’s marginal contribution is equal to zero when a coalition ␶ ϭ ␮ ϩ ͑ Ϫ ␮͒ ϩ (17) ˜ cxx˜ c 1 cxx˜ n w0 does not include the firm. When it does include the firm and a measure n of suppliers, the mar- ϭ Ϫ s៮x ͑N˜ , x˜c , xñ , xñ ͒. ginal contribution of supplier j is m(j, n) ѨR៮ /Ѩn, where C. Bargaining

We now derive the Shapley values in this game (see Lloyd S. Shapley 1953, or Martin J. 16 In our game, the value of a coalition equals the amount of revenue this coalition can generate. 17 More formally, we divide the interval [0, N] into M 15 Note that, as in the case with complete contracts, the ﬁrm equally spaced subintervals with all the intermediate inputs chooses its technology and investment levels to maximize sale in each subinterval of length N/M performed by a single revenues net of total costs. The key difference is that with supplier. We then solve for the Shapley value and take the incomplete contracts, this maximization problem is con- limit of this solution as M 3 ϱ. See Aumann and Shapley strained by the “incentive compatibility” condition (14). (1974) or Stole and Zwiebel (1996b). 924 THE AMERICAN ECONOMIC REVIEW JUNE 2007

៮ 1 Ϫ ␤ ␤͑␬ ϩ 1Ϫ1/␣͒ Ϫ R ϭ A N other suppliers invest xn( j) in their noncontractible activities, every supplier invests xc in ␣ ␤/␣ n 1 her contractible activities, and the level of tech- ϫ ͫ͵ ͩexpͩ͵ ln x(i, k) diͪͪ dkͬ nology is N. Then the Shapley value of supplier jis 0 0 is the revenue derived from the employment of (20) s៮x ͓N, xc , xn ͑Ϫj͒, xn ͑j͔͒ n inputs with technology N, and the last input ϭ ϭ ͑1 Ϫ ␮͒␣ k n is provided by supplier j. Since x(i, k) xn ͑j͒ Յ Յ ␮ Յ Յ ϭ ϭ͑1 Ϫ ␥͒A1 Ϫ ␤ͫ ͬ xc for all 0 i and all 0 k n, x(i, k) x ͑Ϫj͒ Ϫ ␮ Ͻ Յ Յ Ͻ n xn( j) for all i 1 and all 0 k n, and ϭ ␮ Ͻ Յ ϭ ␤␮ ␤͑ Ϫ ␮͒ ␤͑␬ ϩ ͒Ϫ x(i, k) xn(j) for all i 1 and k n, ϫ x x ͑Ϫj͒ 1 N 1 1, evaluating the previous expression enables us to c n write the marginal contribution of supplier j as where ␥ is defined in (19). ␤ ͑ ͒ ϭ 1 Ϫ ␤ ␤͑␬ ϩ 1Ϫ1/␣͒ (18) m j, n ␣ A N A number of features of (20) are worth noting. First, in equilibrium, all suppliers invest x ͑j͒ ͑1 Ϫ ␮͒␣ equally in all the noncontractible activities, i.e., ϫ ͫ n ͬ ␤␮ ͑Ϫ ͒␤͑1 Ϫ ␮͒ ͑␤ Ϫ ␣͒/␣ x (j) ϭ x (Ϫj) ϭ x , and so xc xn j n . n n n xn ͑Ϫj͒ (21) s ͑N, x , x ͒ ϭ s៮ ͑N, x , x , x ͒ The Shapley value of supplier j is the average of x c n x c n n her marginal contributions to coalitions that con- Ϫ ␤ ␤␮ ␤͑ Ϫ ␮͒ ␤͑␬ ϩ ͒Ϫ ϭ͑ Ϫ ␥͒A1 x x 1 N 1 1 sist of players ordered below her in all feasible 1 c n orderings. A supplier who has a measure n of R players ordered below her has a marginal contri- ϭ͑1 Ϫ ␥͒ , bution of m(j, n) if the firm is ordered below her N (probability n/N), and 0 otherwise (probability Ϫ ϭ 1Ϫ␤ ␤␮ ␤(1Ϫ␮) ␤(␬ϩ1) 1 n/N). Averaging over all possible orderings of where R A xc xn N is the total the players and using (18), we obtain revenue of the firm. Thus, the joint Shapley value of the suppliers, Nsx(N, xc, xn), equals the Ϫ ␥ 1 N n fraction 1 of the revenue, and the firm s៮ ͓N, x , x ͑Ϫj͒, x ͑j͔͒ ϭ ͵ ͩ ͪm͑j, n͒ dn receives the remaining fraction ␥, i.e., x c n n N N 0 ͑ ͒ ϭ ␥ 1 Ϫ ␤ ␤␮ ␤͑1 Ϫ ␮͒ ␤͑␬ ϩ 1͒ (22) sq N, xc , xn A xc xn N x ͑j͒ ͑1 Ϫ ␮͒␣ ϭ͑ Ϫ ␥͒ 1 Ϫ ␤ n 1 A ͫ ͬ ϭ ␥ xn ͑Ϫj͒ R.

ϫ ␤␮ ͑Ϫ ͒␤͑1 Ϫ ␮͒ ␤͑␬ ϩ 1͒Ϫ1 xc xn j N , This is a relatively simple rule for the division of revenue between the firm and its suppliers. where Second, the derived parameter ␥ ϵ ␣/(␣ ϩ ␤) represents the bargaining power of the firm; it is ␣ increasing in ␣ and decreasing in ␤. A higher ␥ ϵ (19) ␣ ϩ ␤ . elasticity of substitution between intermediate inputs, i.e., a higher ␣, raises the firm’s bargain- We therefore obtain the following lemma (see ing power, because it makes every supplier less the Appendix for the formal proof): essential in production and therefore raises the share of revenue appropriated by the firm. In LEMMA 1: Suppose that supplier j invests contrast, a higher elasticity of demand for the ␤ xn(j) in her noncontractible activities, all the final good, i.e., higher , reduces the firm’s VOL. 97 NO. 3 ACEMOGLU ET AL.: CONTRACTS AND TECHNOLOGY ADOPTION 925

bargaining power, because, for any coalition, it investment in noncontractible activities and reduces the marginal contribution of the firm to thus underinvests in these activities. Second, as the coalition’s payoff as a fraction of revenue.18 discussed above, multilateral bargaining distorts ␣ ៮ Furthermore, when is smaller, sx[N, xc, the perceived concavity of the private return Ϫ xn( j), xn(j)] is more concave with respect to relative to the social return. Using the first-order xn(j), because greater complementary between condition of this problem and solving for the ϭ the intermediate inputs implies that a given fixed point by substituting xn(j) xn yields a change in the relative employment of two inputs unique xn: has a larger impact on their relative marginal products. The impact of ␣ on the concavity of (23) xn ϭ x៮ n ͑N, xc ͒ ϵ ͓␣͑1 Ϫ ␥͒ ٪ ៮ sx will play an important role in the following ␤ ϫ͑ ͒Ϫ1 ␤␮ 1 Ϫ ␤ ␤͑␬ ϩ 1͒Ϫ1͔1/͓1 Ϫ ␤͑1 Ϫ ␮͔͒ results. The parameter , on the other hand, cx xc A N . affects the concavity of revenue in output (see ៮ ៮ (2)) but has no effect on the concavity of sx with Note that xn(N, xc) is increasing in xc; since the respect to xn(j), because with a continuum of marginal productivity of an activity rises with suppliers, a single supplier has an infinitesimal investment in other activities, investments in effect on output. contractible and noncontractible activities are complements.19 Another implication of (23) is D. Equilibrium that investment in noncontractible activities is increasing in ␣. Mathematically, this follows To characterize an SSPE, we first derive the from the fact that ␣(1 Ϫ ␥) ϭ ␣␤/(␣ ϩ ␤)is incentive compatibility constraint using (14) increasing in ␣. The economics of this relation- and (20): ship is the outcome of two opposing forces. The share of the suppliers in revenue, (1 Ϫ ␥), is x ͑j͒ ͑1 Ϫ ␮͒␣ decreasing in ␣, because greater substitution ϭ ͑ Ϫ ␥͒ 1 Ϫ ␤ͫ n ͬ xn arg max 1 A between the intermediate inputs reduces the ͑ ͒ xn xn j suppliers’ ex post bargaining power. But a ␣ ϫ ␤␮ ␤͑1 Ϫ ␮͒ ␤͑␬ ϩ 1͒Ϫ1 Ϫ ͑ Ϫ ␮͒ ͑ ͒ greater level of also reduces the concavity of ٪ xc xn N cx 1 xn j . ៮ sx in xn, increasing the marginal reward from investing further in noncontractible activities. Relative to the producer’s first-best choice char- Because the latter effect dominates, xn is in- acterized above, we see two differences. First, creasing in ␣. the term (1 Ϫ ␥) implies that the supplier is not Now, using (21), (22), and (23), the firm’s the full residual claimant of the return from her optimization problem (16) can be expressed as

1 Ϫ ␤͓ ␮៮ ͑ ͒1 Ϫ ␮͔␤ ␤͑␬ ϩ 1͒ (24) max A xc xn N, xc N 18 To clarify the effects of ␣ and ␤ on ␥, let us return to N,xc the derivation in the text and note that the marginal contribution of the ﬁrm to a coalition of n suppliers, each one Ϫ ␮ Ϫ Ϫ␮ ៮ cx N xc cx N(1 )xn (N, xc ) investing xc in contractible activities and xn in noncontractible activities, can be expressed as Ϫ ͑ ͒ Ϫ C N w0 N, n ␤/␣ n ␤/␣ m ͑n͒ ϭ A1 Ϫ ␤N␤͑␬ ϩ 1͒x␤␮x␤͑1 Ϫ ␮͒ͩ ͪ ϭ ͩ ͪ R, q c n N N 19 The effect of N on x is ambiguous, since investment ϭ 1Ϫ␤ ␤(␬ϩ1) ␤␮ ␤(1Ϫ␮) n where R A N xc xn is the equilibrium level in noncontractible activities declines with the level of tech- ␤/␣ of the revenue (when n ϭ N). The expression (n/N) is nology when ␤(␬ ϩ 1) Ͻ 1 and increases with N when decreasing in ␤ and increasing in ␣ for all n Ͻ N. The ␤(␬ ϩ 1) Ͼ 1. This is because an increase in N has two ␤/␣ parameter ␥ is given by the average of the (n/N) terms, opposite effects on a supplier’s incentives to invest: a greater number of inputs increases the marginal product of 1 N n ␤/␣ ␣ investment due to the “love for variety” embodied in the ␥ ϭ ͵ ͩ ͪ dn ϭ , N N ␣ ϩ ␤ technology, but at the same time, the bargaining share of a 0 supplier, (1 Ϫ ␥)/N, declines with N. For large values of ␬, the former effect dominates, while for small values of ␬, the and is also decreasing in ␤ and increasing in ␣. latter dominates. 926 THE AMERICAN ECONOMIC REVIEW JUNE 2007

៮ where xn(N, xc) is defined in (23). Substituting below the level of investment in contractible (23) into (24) and differentiating with respect to activities, and this “underinvestment” reduces N and xc results in two first-order conditions, the profitability of technologies with high N.As ˜ 20 ␮ 3 which yield a unique solution (N, x˜c) to (24): 1 (and contractual imperfections disap- pear), both these bracketed terms on the left-hand ˜ ͑␤͑␬ ϩ 1͒Ϫ1͒/͑1 Ϫ ␤͒ ␬␤1/͑1 Ϫ ␤͒ Ϫ ␤/͑1 Ϫ ␤͒ ˜ 3 21 (25) N A cx side of (25) go to 1 and (N, x˜c) (N*, x*). The impact of incomplete contracts on produc- 1 Ϫ ␣͑1 Ϫ ␥͒͑1 Ϫ ␮͒ ͑1 Ϫ ␤͑1 Ϫ ␮͒͒/͑1 Ϫ ␤͒ tivity follows directly from their effect on the ϫ ͫ ͬ choice of technology. In particular, productivity 1 Ϫ ␤͑1 Ϫ ␮͒ ␬ under incomplete contracts, P˜ ϭ N˜ , is always Ϫ ␤͑ Ϫ ␮͒ ͑ Ϫ ␤͒ lower than P* as given in (13), since N˜ Ͻ N*. ϫ͓␤ 1␣͑1 Ϫ ␥͔͒ 1 / 1

ϭ Ј͑ ˜ ͒ ϩ C N w0 , E. Implications of Incomplete Contracts Ј͑ ˜ ͒ ϩ C N w0 We now provide a number of comparative (26) x˜ c ϭ . ␬cx static results on the SSPE under incomplete contracts, and compare the incomplete- As in the complete contracts case, these two contracts equilibrium technology and invest- conditions determine the equilibrium recur- ment levels to the equilibrium under complete sively. First, (25) gives N˜ and then, given N˜ , contracts. The comparative static results are fa- (26) yields x˜c. Moreover, using (23), (25), and cilitated by the block-recursive structure of the (26) gives the level of investment in noncon- equilibrium; any change in A, ␮,or␣ that tractible activities as increases the left-hand side of (25) also in- ˜ creases N, and the effect on x˜c and xñ can then ␣͑1 Ϫ ␥͓͒1 Ϫ ␤͑1 Ϫ ␮͔͒ be obtained from (26) and (27). The main re- (27) x˜ ϭ n ␤͓1 Ϫ ␣͑1 Ϫ ␥͒͑1 Ϫ ␮͔͒ sults are provided in the next proposition (proof in the Appendix). CЈ͑N˜ ͒ ϩ w ϫ 0 ͩ ␬ ͪ. PROPOSITION 2: Suppose that Assumption 1 cx holds. Then there exists a unique SSPE under ˜ incomplete contracts, {N, x˜c, xñ}, characterized ˜ Comparing (12) to (26), we see that, for a given by (25), (26), and (27). Furthermore, {N, x˜c, xñ} ˜ Ͼ N, the implied level of investment in contract- satisfies N, x˜c, xñ 0, ible activities under incomplete contracts, x˜c,is identical to the investment level in contractible x˜ n Ͻ x˜ c , activities under complete contracts, x*. This

highlights the fact that differences in the invest- ѨN˜ Ѩx˜ c Ѩx˜ n Ͼ 0, Ն 0, Ն 0, ment in contractible activities between these ѨA ѨA ѨA economic environments result only from differences in technology adoption. In fact, compar- ѨN˜ Ѩx˜ c Ѩ͑x˜ n /x˜ c ͒ ing (11) with (25), we see that N˜ and N* differ Ͼ 0, Ն 0, Ͼ 0, only because of the two bracketed terms on the Ѩ␮ Ѩ␮ Ѩ␮ left-hand side of (25). These represent the dis- ѨN˜ Ѩx˜ Ѩ͑x˜ /x˜ ͒ tortions created by bargaining between the firm Ͼ c Ն n c Ͼ and its suppliers. Intuitively, technology adop- Ѩ␣ 0, Ѩ␣ 0, Ѩ␣ 0. tion is distorted because incomplete contracts reduce investment in noncontractible activities 21 ␮ 3 Note that as 1 the investment level xñ does not converge to x*, because the effect of distortions on the noncontractible activities does not go to zero. What goes to 20 See the Appendix for a more detailed derivation and zero, however, is the importance of noncontractible activi- for the second-order conditions. ties in the production of final goods. VOL. 97 NO. 3 ACEMOGLU ET AL.: CONTRACTS AND TECHNOLOGY ADOPTION 927

The main results in this proposition are intu- ogy choices and investments. The reason is related itive. Suppliers invest less in noncontractible to the discussion in the previous subsection where activities than in contractible activities, in par- it was shown that a higher ␣ reduces the share of ٪ ៮ ticular, each supplier but also makes sx less concave. Because the latter effect dominates, a lower degree of complementarity increases supplier invest- x˜ ␣͑1 Ϫ ␥͓͒1 Ϫ ␤͑1 Ϫ ␮͔͒ n ϭ Ͻ ments and makes the adoption of more advanced (28) ␤͓ Ϫ ␣͑ Ϫ ␥͒͑ Ϫ ␮͔͒ 1, x˜c 1 1 1 technologies more profitable. The reduced-form profit function depicted which follows from equations (26) and (27) and in equation (1) can also be derived at this from the fact that ␣(1 Ϫ ␥) ϭ ␣␤/(␣ ϩ ␤) Ͻ ␤ point. Combining (23), (24), and the condi- (recall (19)). Intuitively, the firm is the full tion for the firm’s optimal choice of xc, the residual claimant of the return to investments in firm’s payoff can be expressed as (see the contractible activities, and it dictates these invest- Appendix): ments in the contract. In contrast, investments in noncontractible activities are decided by the sup- ϩ͑␤͑␬ ϩ ͒Ϫ ͒ ͑ Ϫ ␤͒ (29) ␲ ϭ AZ͑␣, ␮͒N1 1 1 / 1 pliers, who are not the full residual claimants of the returns generated by these investments (recall Ϫ C͑N͒ Ϫ w N, (21)) and thus underinvest in these activities. 0 In addition, the level of technology and investments in both contractible and noncontract- where ible activities is increasing in the size of the market, in the fraction of contractible activities (30) Z͑␣, ␮͒ ϵ ͑1 Ϫ ␤͒␤␤␮/͑1 Ϫ ␤͒ (quality of contracting institutions), and in the elasticity of substitution between intermediate ϫ͓␣͑1 Ϫ ␥͔͒␤͑1 Ϫ ␮͒/͑1 Ϫ ␤͒ inputs.22 The impact of the size of the market is intuitive; a greater A makes production more 1 Ϫ ␣͑1 Ϫ ␥͒͑1 Ϫ ␮͒ ͑1 Ϫ ␤͑1 Ϫ ␮͒͒/͑1 Ϫ ␤͒ ϫ ͫ ͬ profitable and thus increases investments and 1 Ϫ ␤͑1 Ϫ ␮͒ equilibrium technology. Better contracting institutions, on the other hand, imply that a greater ϫ͑c ͒Ϫ␤/͑1 Ϫ ␤͒ fraction of activities receives the higher invest- x Ͻ ment level x˜c rather than xñ x˜c. This makes the choice of a more advanced technology more represents a measure of “derived efficiency” profitable. A higher N, in turn, increases the and captures the distortions arising from incom- profitability of further investments in x˜c and xñ. plete contracts. The extent of these distortions Better contracting institutions also close the depends on the model’s parameters as shown in (proportional) gap between x˜c and xñ because Proposition 2. In addition, we have the follow- with a higher fraction of contractible activities, ing lemma (proof in the Appendix): the marginal return to investment in noncontractible activities is also higher. A higher ␣, i.e., lower complementarity be- LEMMA 2: Suppose that Assumption 1 holds. tween intermediate inputs, also increases technol- Let ␨␮(␣, ␮) ϵ [␮ ϫ ѨZ(␣, ␮)/Ѩ␮]/Z(␣, ␮) be the elasticity of Z(␣, ␮) with respect to ␮ and let ␨␣(␣, ␮) ϵ [␣ ϫ ѨZ(␣, ␮)/Ѩ␣]/Z(␣, ␮) be the 22 Equation (13) above implies that the measure of pro- elasticity of Z(␣, ␮) with respect to ␣. Then, we ductivity, P ϭ q/(NX), has the same comparative statics as technology. The same results also apply if we were to define have that productivity as P ϭ q/N, that is, as output divided by the number of suppliers. Yet, in this case firms operating under (a) ␨␮(␣, ␮) Ͼ 0 and ␨␣(␣, ␮) Ͼ 0; and different contracting institutions would have different pro- (b) Ѩ␨␮(␣, ␮)/Ѩ␣ Ͻ 0 and Ѩ␨␣(␣, ␮)/Ѩ␮ Ͻ 0. ductivity levels, not only because they choose different levels of technology, but also because they have different investment levels. See Acemoglu, Antra`s, and Helpman Part (a) of this lemma implies that better (2005) for more details on this point. contracting institutions and greater substitutability 928 THE AMERICAN ECONOMIC REVIEW JUNE 2007

between intermediate inputs lead to higher levels applications emphasize both the potential quan- of Z(␣, ␮). Part (b), on the other hand, implies that titative effects of our main mechanism and its the proportional increase in Z(␣, ␮) in response to implications for equilibrium organizational an improvement in contracting institutions is choices and endogenous patterns of compara- greater when there is more technological comple- tive advantage. mentarity between intermediate inputs. The intu- ition is that contract incompleteness is more A. Quantitative Implications damaging to technologies with greater complementarities, because there are more significant We first investigate whether our baseline investment distortions in this case. This last model can generate significant productivity dif- result implies that sectors with greater comple- ferences from cross-country variation in con- mentarities are more contract dependent, a fea- tracting institutions. Our purpose here is not to ture that will play an important role in the undertake a full-fledged calibration, but to give general equilibrium analysis in Section IVC. a sense of the empirical implications of our Finally, to compare the complete and incom- baseline model for plausible parameter values. plete contracts equilibria, recall that they both To obtain closed-form solutions for the equilib- lead to the same allocation as ␮ 3 1. Together rium value of N and for the productivity mea- with Proposition 2, this implies (proof in the sure P ϭ N␬, we assume that the function C(N) Appendix): is linear in N, C(N) ϭ ␾N. Equation (25) then implies24 PROPOSITION 3: Suppose that Assumption 1 ˜ holds. Let {N, x˜c, xñ} be the unique SSPE with incomplete contracts, and let {N*, x*} be the A␬ ␬͑1 Ϫ ␤͒/͑1 Ϫ ␤͑␬ ϩ 1͒͒ P˜ ϭ ͩ ͪ unique equilibrium with complete contracts. ␾ ϩ w Then 0 ϫ ␤␬/͑1 Ϫ ␤͑␬ ϩ 1͒͒c Ϫ ␤␬/͑1 Ϫ ␤͑␬ ϩ 1͒͒ N˜ Ͻ N* and xñ Ͻ x˜c Յ x*. x 1 Ϫ ␣͑1 Ϫ ␥͒͑1 Ϫ ␮͒ ␬͑1 Ϫ ␤͑1 Ϫ ␮͒͒/͑1 Ϫ ␤͑␬ ϩ 1͒͒ This proposition implies that since incomplete ϫ ͫ Ϫ ␤͑ Ϫ ␮͒ ͬ contracts lead to the choice of less advanced 1 1 (lower N) technologies,23 they also reduce pro- ␬␤͑1 Ϫ ␮͒/͑1 Ϫ ␤͑␬ ϩ 1͒͒ ductivity and investments in contractible and ␣͑1 Ϫ ␥͒ ϫ ͫ ͬ . noncontractible activities. ␤

IV. Applications Our focus is on the impact of the quality of contracting institutions, ␮, on ﬁrm-level pro- We next discuss a number of applications of ductivity. For this purpose, consider the ratio the basic framework developed so far. These of productivity in two economies with the ␮ fraction of contractible tasks given by 1 and ␮ Ͻ ␮ : 23 It is useful to contrast this result with the overemploy- 0 1 ment result in Stole and Zwiebel (1996a, b). There are two important differences between our model and theirs. First, our model features investments in noncontractible activities, which are absent in Stole and Zwiebel. Second, Stole and 24 Recall from the discussion in footnote 11 that there Zwiebel assume that if a worker is not in the coalition of the are alternative ways of measuring productivity in the bargaining game, she receives no payment. Since in our model, and which one of these is more appropriate will model investment in noncontractible activities is not veriﬁ- depend on how productivity is measured in practice. If, able, a supplier receives the upfront payment ␶ indepen- for example, we were to measure productivity as output dently of whether she is in the coalition, and she receives divided by the number of suppliers, the impact of im-

only the rest of the payment, i.e., sx, through bargaining. provements in contracting institutions on productivity The treatment of the outside option in Stole and Zwiebel is would be even larger, because such improvements raise essential for their overemployment result (see Catherine C. investment in contractible and noncontractible activities de Fontenay and Joshua S. Gans 2003). as well. VOL. 97 NO. 3 ACEMOGLU ET AL.: CONTRACTS AND TECHNOLOGY ADOPTION 929

␬͑1Ϫ␤͑1Ϫ␮1 ͒͒ Ϫ ␣͑ Ϫ ␥͒͑ Ϫ ␮ ͒ Ϫ␤͑␬ϩ ͒ 1 1 1 1 1 1 ˜͑␮ ͒ ͫ Ϫ ␤͑ Ϫ ␮ ͒ ͬ P 1 1 1 1 (31) ϭ ␬͑ Ϫ␤͑ Ϫ␮ ͒͒ P˜͑␮ ͒ 1 1 0 0 Ϫ ␣͑ Ϫ ␥͒͑ Ϫ ␮ ͒ Ϫ␤͑␬ϩ ͒ 1 1 1 0 1 1 ͫ Ϫ ␤͑ Ϫ ␮ ͒ ͬ 1 1 0

␬␤͑␮0Ϫ␮1 ͒ Ϫ1 ϫ͓␤ ␣͑1 Ϫ ␥͔͒ 1Ϫ␤͑␬ϩ1͒ , where, recall, ␥ ϵ ␣/(␣ ϩ ␤). Equation (31) shows that the proportional impact of ␮ on aggregate productivity depends only on the val- 25 ues of the parameters ␣, ␤, and ␬. FIGURE 1. RELATIVE PRODUCTIVITY FOR ␤ ϭ 0.75 AND The parameter ␤ governs the elasticity of ␬ ϭ 0.25 substitution between final-good varieties, and also determines the markup charged by final- good producers. In our benchmark simulation, we set this parameter equal to 0.75. This implies and also that this variety growth was the main an elasticity of substitution between final-good source of productivity (TFP) growth in the US varieties equal to four, which corresponds to the economy, we can use our model to back out a mean elasticity estimated by Christian Broda value of ␬. We take the rate of TFP growth and David E. Weinstein (2006) using US import per annum to be 0.5 percent and use the fact data. This value of ␤ also implies a markup over that productivity is given by P ϭ N␬. This ␤Ϫ1 Ϫ Ӎ ␬ Ϫ ␬ ␬ ϭ marginal cost equal to 1 0.33, which implies that (Ntϩ1 Nt )/Nt 0.005 and is comfortably within the range of available generates a value of ␬ ϭ ln(1.005)/ln(1.02) Ӎ estimates.26 0.25.27 We choose the parameter ␬ to match Mark The value of ␣ is harder to pin down. Since Bils and Klenow’s (2001) estimates of variety the main theoretical ideas in the paper relate to growth in the US economy from Bureau of complementarity between different inputs, a Labor Statistics (BLS) data on expenditures low value of ␣ may be natural. Nevertheless, on different types of goods. Their findings since we have no direct way of relating ␣ to indicate that the growth of the number of aggregate data, we show the implications of varieties in the US economy was around 2 changes in the quality of contracting institutions percent per annum between 1959 and 1999. If for different values of ␣. we assume that the variety of intermediate Figure 1 depicts the value of the ratio in (31) goods grew at the same rate over this period, for different values of ␣ when ␤ ϭ 0.75 and ␬ ϭ 0.25. We consider three experiments. The middle curve depicts a shift from ␮ ϭ 0.25 to 25 In deriving equation (31), we assume that the remain- 0 ing parameters are held fixed when ␮ changes. We should thus interpret our results as reflecting the partial-equilibrium response of productivity to changes in contractibility. 27 A somewhat different number is obtained from 26 On the lower side, Catherine J. Morrison (1992) finds Broda and Weinstein’s (2006) estimate that the number markups ranging from 0.1 in 1961 to 0.29 in 1970, while on of varieties in US imports between 1972 and 2001 grew the higher side Julio J. Rotemberg and Michael Woodford at an annual rate of 3.7 percent. Using the same formula, (1991) use an estimate of 0.6. See Susanto Basu (1995) for this would imply a value of 0.135 for ␬. Note, however, a discussion of the sensitivity of the estimated markups to that this is likely to be an overestimate of variety growth model specification and calibration. It is interesting to note and thus an underestimate of ␬ because Broda and Wein- that Morrison (1995) finds markups between 0.1 and 0.39 in stein consider the same good imported from different Japan and 0.1 and 0.2 in Canada, Mark J. Roberts (1996) countries as different varieties, and the number of ex- finds average markups between 0.22 and 0.3 in Colombia, porting countries to the United States increased over and Jean-Marie Grether (1996) finds average markups be- time. For this reason, we choose ␬ ϭ 0.25 as our bench- tween 0.34 and 0.37 in Mexico. mark parameter value. 930 THE AMERICAN ECONOMIC REVIEW JUNE 2007

˜ ˜ FIGURE 2. RELATIVE PRODUCTIVITY P(0.75)/P(0.25) FOR FIGURE 3. RELATIVE PRODUCTIVITY P˜ (0.75)/P˜ (0.25) FOR ␬ ϭ ␤ 0.25 AND ALTERNATIVE ’S ␤ ϭ 0.75 AND ALTERNATIVE ␬’S

␮ ϭ 1 0.75, the lowest curve depicts a more ␮ ϭ 1 ␮ ϭ 2 modest increase from 0 ⁄3 to 1 ⁄3 , 0.7 reduces the effect of improved contractibil- while the highest curve illustrates the most ex- ity on productivity, but for ␣ ϭ 0.25 we still ␮ ϭ ␮ ϭ treme shift from 0 0to 1 1. find that the experiment increases productivity The patterns in Figure 1 confirm the results in by a factor of 1.8. Lemma 2 and show that the impact of an in- Finally, Figure 3 shows the results for alter- crease in ␮ on productivity is larger for lower native values of ␬ (␬ ϭ 0.2 and ␬ ϭ 0.3) while values of ␣. More importantly, the figure shows holding ␤ at 0.75. When ␬ ϭ 0.2, the estimated that the quantitative effects can be sizable for a effects are somewhat smaller than in our bench- large range of values of ␣. For example, when mark simulation, but still sizable; with low val- ␣ ϭ 0.25, productivity increases by a factor of ues of ␣, the impact of an improvement in the 2.5, 4.1, and 19.7 in the three experiments. As ␣ quality of contracting institutions on productiv- increases and intermediate inputs become more ity is still large. For example, with ␣ ϭ 0.25, substitutable, however, the magnitude of the productivity increases by a factor of 2.0. On the quantitative effects diminishes. For example, other hand, greater values of ␬ lead to more when ␣ ϭ 0.75, productivity increases by a significant responses of productivity to in- factor of 1.4, 1.7, and 3.2 in the three cases. creases in ␮. Even for a very high value of ␣ Though smaller, these are still sizable effects. such as 0.75, productivity increases by a factor We next discuss the sensitivity of these quan- of 4.8 as ␮ increases from 0.25 to 0.75. titative results to alternative parameter values. Overall, this simple quantitative evaluation While markup estimates are consistent with a suggests that the mechanism highlighted in our value of ␤ ϭ 0.75, other evidence suggests model is capable of generating quantitatively somewhat higher or lower values of ␤. Figure sizable effects from relatively modest variations 2 shows the results of an increase in ␮ from in contracting institutions. 0.25 to 0.75 when ␤ ϭ 0.78 and when ␤ ϭ 0.7, while leaving ␬ at 0.25.28 The figure shows that B. Choice of Organizational Forms the quantitative effects are considerably larger in the case in which ␤ ϭ 0.78, even for large An important insight of the incomplete con- values of ␣. For example, with ␣ ϭ 0.75, the tracts literature, and especially of Grossman and improvement in contracting institutions leads to Hart (1986) and Hart and Moore (1990), is to a proportional increase in productivity of 4.4, consider organizational forms (and the owner- which is a very sizable effect. Setting ␤ equal to ship of assets) as a choice variable affecting ex ante investments. Our framework is tractable enough to allow these considerations and can be 28 Only values of ␤ less than 0.8 are consistent with used to discuss issues of vertical integration Assumption 1 combined with ␬ ϭ 0.25. versus outsourcing, and how the employment VOL. 97 NO. 3 ACEMOGLU ET AL.: CONTRACTS AND TECHNOLOGY ADOPTION 931 relationship between the firm and its suppliers choice of the upfront payment ␶ is not restricted, should be organized. We now briefly discuss it can be shown that the firm always prefers how this can be done. A more detailed treatment outsourcing to vertical integration.30 In contrast, is presented in our working paper version, Ace- when suppliers face credit constraints (in the moglu, Antra`s, and Helpman (2005). sense that the upfront payment ␶ is restricted to We have thus far assumed that the threat be nonnegative), integration may be preferable point of a supplier is not to deliver the noncon- to outsourcing. In particular, when credit contractible activities, which—in view of the Cobb- straints are present and ␣ ϩ ␤ Ͻ 1, there exists Douglas structure of the production function of ␦៮ ʦ (0, 1) such that vertical integration is pre- the intermediate input in (4)—is equivalent to ferred by the firm for ␦ Ͻ ␦៮ and outsourcing is assuming that the supplier does not deliver X(j) preferred for ␦ Ͼ ␦៮. Furthermore, ␦៮ is decreas- at all. Assume, instead, that the threat point of a ing in ␣, so that integration is more likely when supplier is not to deliver a fraction 1 Ϫ ␦ of her there is greater complementarity between inter- X(j), where 0 Յ ␦ Յ 1. The magnitude of ␦ mediate inputs. This result implies that vertical depends, among other things, on whether the integration is more likely when both contractual supplier is an employee of the firm or an outside frictions and credit market imperfections are contractor. Our analysis above corresponds to present.31 the case ␦ ϭ 0. The following lemma general- izes the Shapley value to the case where ␦ Ͼ 0 C. General Equilibrium (proof in the Appendix): We now discuss how the technology choice LEMMA 3: Suppose that supplier j invests can be embedded in a general equilibrium ver- xn(j) in her noncontractible activities, all the sion of our model. Besides verifying that gen- Ϫ other suppliers invest xn( j) in their noncon- eral equilibrium interactions do not reverse our tractible activities, every supplier invests xc in partial equilibrium results, this analysis is useful her contractible activities, and the level of tech- as a preparation for the results in the next nology is N. Then, the Shapley value of supplier subsection, where we investigate endogenous j is given by (20), where patterns of comparative advantage across countries. We start with an equilibrium with a given ␣ ϩ ␤ number of producers (final goods) and then ␣͑1 Ϫ ␦ ͒ (32) ␥ ϵ ␣ . endogenize this with free entry. In this and the ͑␣ ϩ ␤͒͑1 Ϫ ␦ ͒ next subsection, we focus on the case where 0 Ͻ ␮ Ͻ 1, and ␦ ϭ 0, as in the baseline model. Lemma 3 implies that the formula for the Assume that there exists a continuum of final Shapley value is the same as before, except that goods q(z), with z ʦ [0, Q], where Q represents now the firm’s share in the bargaining game, ␥, the number (measure) of final goods. All con- depends on ␦. Clearly, ␥ in (32) equals ␥ in (19) sumers have identical preferences, when ␦ ϭ 0, but is greater when ␦ Ͼ 0. This is natural, since, with ␦ Ͼ 0, the bargaining game is more advantageous to the firm. In the limit, as ␦ ␥ a supplier can do is not to cooperate in the use of her goes to 1, the firm’s share also goes to 1. intermediate input, which will reduce the efficiency with Using this lemma, the working paper version which the firm can employ this input, but may not reduce demonstrated that all the results in Propositions this efficiency to zero. See Grossman and Hart (1986). 2 and 3 hold for any ␦ ʦ [0, 1) and employed 30 This is because the firm does not undertake any rela- this generalization to analyze the choice be- tionship-specific investments. Since suppliers are the only agents undertaking noncontractible relationship-specific in- tween integration and outsourcing. Briefly, sup- vestments, it is efficient to give them as much bargaining pose that for an integrated firm, we have ␦ Ͼ 0, power as possible. This will not necessarily be the case if while with outsourcing ␦ ϭ 0.29 Then, when the the firm were to also make relationship-specific investments. See, for example, Hart and Moore (1990), Antra`s (2003, 2005), and Antra`s and Helpman (2004). 31 The positive effect of complementarity on the integra- 29 An integrated firm has ␦ Ͼ 0 because in this case the tion decision has been derived before in the property-rights firm owns all the intermediate inputs. In this event, the most literature (see Hart and Moore 1990). 932 THE AMERICAN ECONOMIC REVIEW JUNE 2007

1/␤ Q The first-order condition of the maximization (33) u ϭ ͵ q(z)␤ dz Ϫ c e,0Ͻ ␤ Ͻ 1, of (29) yields ͩ ͪ x 0 ␤␬ (34) AZ͑␣, ␮͒N͑␤͑␬ ϩ 1͒Ϫ1͒/͑1 Ϫ ␤͒ 1 Ϫ ␤ where e is the total effort exerted by this individual, the elasticity of substitution between ϭ wCЈ ͑N͒ ϩ w , Ϫ ␤ L 0 final goods, 1/(1 ), is greater than 1, and cx represents the cost of effort in terms of real for all ␣ ʦ ͑0, 1͒. consumption. These preferences imply the demand function Since each individual can be employed at the wage w in the process of technology adoption, Ϫ1/͑1 Ϫ ␤͒ ϭ p͑z͒ S their outside option as a supplier is w0 w. q͑z͒ ϭ ͫ ͬ , pI pI Equation (34) then implies that the technology choice depends on a ϵ A/w, which is the in- where p(z) is the price of good z, S is the verse real cost of technology adoption (or the aggregate spending level, and “equilibrium market size”). Let the equilibrium technology choice of N as a function of a, ␣, Ϫ͑1 Ϫ ␤͒/␤ Q and ␮ implied by (34) be N(a, ␣, ␮). Defining pI ϵ ͫ͵ p(z)Ϫ␤/(1 Ϫ ␤) dzͬ total labor demand by a firm with technology N ϵ ϩ 0 as CT(N) CL(N) N, condition (34) implies that N(a, ␣, ␮) is implicitly defined by is the ideal price index, which we take to be the I ϭ ␤␬ numeraire, i.e., p 1. The implied demand ͑␤͑␬ ϩ ͒Ϫ ͒ ͑ Ϫ ␤͒ Ϫ1/(1Ϫ␤) (35) aZ͑␣, ␮͒N͑a, ␣, ␮͒ 1 1 / 1 function A[p(z)] for each firm is there- 1 Ϫ ␤ fore identical to the demand function used in the previous sections, with A ϭ S. ϭ CTЈ͓N͑a, ␣, ␮͔͒, Recall that each firm in this economy solves the maximization problem (24) and its reduced-form for all ␣ ʦ ͑0, 1͒. profit function is given by (29). We assume that the degree of technological complementarity, ␣, Since firms with higher elasticities of substitu- varies across firms (or sectors) with its support tion choose higher N (Proposition 2), this equa- given by a subset of (0, 1). We denote its cumu- tion implies that N(a, ␣, ␮) is increasing in ␣. lative distribution function by H(␣). This formu- The demand for labor by a firm choosing tech- ␣ ␮ ␣ ␮ lation implies that if Q products are available for nology N(a, , )isCT[N(a, , )], thus labor consumption, a fraction H(␣) of them are pro- market clearing can be expressed as32 duced with elasticities of substitution smaller than 1/(1 Ϫ ␣). 1 The key general equilibrium interaction re- (36) Q ͵ C ͓N͑a, ␣, ␮͔͒ dH͑␣͒ ϭ L, T sults from competition of producers for a scarce 0 resource, labor. Assume that labor is in fixed supply L. A firm that adopts technology N em- where the left-hand side is total demand for labor ␣ ␮ ploys N individuals as suppliers and CL(N) and L is labor supply. Since N(a, , ) is increas- workers in the process of technology adoption ing in a, this condition uniquely determines the (implementation, use, or creation of the tech- equilibrium value of a, i.e., our measure of the real nology). We assume that these are the only uses demand level. The relationship between a and the of labor (Assumption 1 now applies to CL(N)). model’s parameters, as embodied in this equation, Denoting the wage rate in terms of the numeraire by w, this implies that the total cost of ϭ adopting technology N is C(N) wCL(N). The 32 Since it is straightforward to verify that the wage is wage rate w is taken as given by each firm, but always strictly positive, (36) is written as an equality rather is endogenously determined in equilibrium. than in complementary slackness form. VOL. 97 NO. 3 ACEMOGLU ET AL.: CONTRACTS AND TECHNOLOGY ADOPTION 933 illustrates the key general equilibrium feedback in Equation (38) shows that the proportional our model. Proposition 2 implies that N(a, ␣, ␮)is change in N in response to ␮ˆ , Nˆ (a, ␣, ␮), can be increasing in ␮, so we may expect better contract- positive only for some ␣’s (again because of the ing institutions to encourage all firms to adopt resource constraint). Since Ѩ␨␮(␣, ␮)/Ѩ␣ Ͻ 0 more advanced technologies. The resource con- (from Lemma 2), the left-hand side of equation straint, (36), implies, however, that not all the (37) is decreasing in ␣. Consequently, there N(a, ␣, ␮)’s can increase with Q constant. Con- exists a critical value ␣␮ such that Nˆ (a, ␣, ␮) Ͼ sequently, the equilibrium value of a has to adjust 0 for all ␣ Ͻ ␣␮ and Nˆ (a, ␣, ␮) Ͻ 0 for all ␣ Ͼ to clear the labor market when ␮ rises. ␣␮. This implies that low ␣ firms, with greater To derive the implications of an increase in ␮ technological complementarity between inputs, on the cross-sectional distribution of technology are more contract dependent. This establishes choices, recall that the number of products, Q,is the following proposition (proof in the text): given, and differentiate the first-order condition (35) to obtain PROPOSITION 4: Suppose Assumption 1 holds. (37) Then there exists ␣␮ ʦ (0, 1) such that in the general equilibrium economy with Q constant, aˆ ϩ ␨␮ ͑␣, ␮͒␮ˆ ϭ ␭͓N͑a, ␣, ␮͔͒Nˆ ͑a, ␣, ␮͒, an increase in ␮ raises the level of technology N(a, ␣, ␮) in all firms with ␣ Ͻ ␣␮ and reduces where yˆ, defined as dy/y, represents the propor- it in all firms with ␣ Ͼ ␣␮. tional rate of change of variable y, ␨␮(␣, ␮)is the elasticity of Z(␣, ␮) with respect to ␮, and We next discuss how the number of products, ␭(N) is the elasticity of the marginal cost curve Q, can be endogenized with free entry. To do Ј ␤ ␬ ϩ Ϫ Ϫ ␤ CT(N) minus [ ( 1) 1]/(1 ), i.e., this in the simplest possible way, suppose that an entrant faces a fixed cost of entry wf, where C Љ ͑N͒N ␤͑␬ ϩ 1͒ Ϫ 1 f is the amount of labor required for entry. This ␭͑ ͒ ϵ T Ϫ N . cost is borne in addition to the cost of tech- CЈT ͑N͒ 1 Ϫ ␤ nology adoption. Moreover, in the spirit of Assumption 1 implies that ␭(N) Ͼ 0. Moreover, Hugo A. Hopenhayn (1992) and Marc J. Melitz as proved in Lemma 2, ␨␮(␣, ␮) Ͼ 0 and ␨␮(␣, ␮) (2003), suppose that an entrant does not know a is decreasing in ␣. Next, differentiating (36), we key parameter of the technology prior to entry. obtain a relationship between ␮ and a: While in their models the entrant does not know its own productivity, we assume instead that it 1 does not know ␣, but knows that ␣ is drawn (38) ͵ ␴ ͑␣͒Nˆ ͑a, ␣, ␮͒ dH͑␣͒ ϭ 0, from the cumulative distribution function L ␣ ␣ 0 H( ). Since the relationship between and productivity is determined in general equilib- ␴ ␣ ϵ Ј ␣ ␮ ␣ ␮ where L( ) CT[N(a, , )]N(a, , ). Sub- rium, the distribution of productivity is en- stituting (37) into (38) then yields dogenous. After entry, each firm learns its ␣ and maxi- 1 mizes the profit function (29). This maximiza- Ϫ1 ͵ ␴ ͑␣͒␨␮͑␣, ␮͒␭͓N͑a, ␣, ␮͔͒ dH͑␣͒ tion leads to the choice of technology as a L ␮ ␣ ␣ ␮ 0 function of a, , and , i.e., N(a, , ), in (35). aˆ ϭ Ϫ ␮ˆ . Let us define 1 ͵ ␴ ͑␣͒␭͓N͑a, ␣, ␮͔͒Ϫ1 dH͑␣͒ L ⌸͑ ␣ ␮͒ ϵ ͑␣ ␮͒ 0 a, , aZ , ϫ N͑a, ␣, ␮͒1 ϩ͑␤͑␬ ϩ 1͒Ϫ1͒/͑1 Ϫ ␤͒ Ϫ C ͓N͑a, ␣, ␮͔͒, Since the term in front of ␮ˆ on the right-hand T side of this equation is negative, an improvement in contracting institutions increases wages where Z(␣, ␮) is given by (30). This is an relative to expenditure and reduces a. indirect profit function, with profits measured in 934 THE AMERICAN ECONOMIC REVIEW JUNE 2007 units of labor. It is straightforward to verify that productivity and firm size. The only difference it is increasing in a, ␣, and ␮. is that the larger country has proportionately Free entry implies that expected profits must more final good producers. This result contrasts equal the entry cost wf,or with the case with an exogenous number of products, where differences in L would generate 34 1 comparative advantage. (39) ͵ ⌸͑a, ␣, ␮͒ dH͑␣͒ ϭ f. 0 D. Comparative Advantage This free-entry condition uniquely determines Perhaps the most interesting general equilib- the equilibrium value of a, without reference to rium application of our framework is to inter- the labor market clearing condition (36). Con- national trade. Consider a world consisting of sequently, given the equilibrium value of a, the two countries, indexed by ᐉ ϭ 1, 2. We now distribution of ␣ induces a distribution of pro- derive an endogenous pattern of comparative ductivity and firm size in the economy (from the advantage between these two countries from first-order condition (35)): firms with larger ␣’s differences in contracting institutions.35 Sup- choose more advanced technologies and they pose that there is a fixed number of products and are more productive.33 that every product is distinct not only from other Finally, the labor market clearing condition products produced in its own country, but also determines the number of entrants. Since labor from products produced in the foreign country. demand now includes individuals working in All products can be freely traded between the the founding of the firms, the market clearing two countries. condition (36) has to be replaced with Suppose also that the two countries are identical, except for their contracting institutions. In 1 2 1 2 1 1 particular, L ϭ L , Q ϭ Q , and H (␣) ϭ Q ͵ C [N(a, ␣, ␮)] dH(␣) ϩ f ϭ L. H2(␣) for all ␣ ʦ (0, 1), but the fraction of ͫ T ͬ ␮ᐉ 0 activities that are contractible differs across countries. Without loss of generality, we as- Together with N(a, ␣, ␮) from (35) and a from sume that ␮1 Ͼ ␮2, so that country 1 has better the free entry condition (39), this modified labor contracting institutions. market clearing condition determines the num- The equilibrium condition for technology ber of entrants Q. adoption (34) holds in both countries, with dif- An interesting implication of the general ferent wage rates, wᐉ, for the two countries (A is equilibrium with free entry is that an increase in the same for both countries and equal to world the supply of labor L does not create a scale expenditure). Defining aᐉ ϵ A/wᐉ and Nᐉ(␣) ϵ effect and is not a source of comparative advan- N(aᐉ, ␣, ␮ᐉ), we have tage. If two countries that differ only in L freely trade with each other, their wages are equalized, ␤␬ (40) aᐉZ͑␣, ␮ᐉ͒Nᐉ͑␣͒͑␤͑␬ ϩ 1͒Ϫ1͒/͑1 Ϫ ␤͒ their technology level is the same in every in- 1 Ϫ ␤ dustry ␣, and they have the same distribution of ᐉ ϭ CЈT ͓N ͑␣͔͒,

33 Note also that the degree of dispersion of productivity for all ␣ ʦ ͑0, 1͒, ᐉ ϭ 1, 2. depends on the degree of contract incompleteness; in countries with better contracting institutions there is less productivity dispersion and less size dispersion. This follows from the fact that small firms (i.e., low-␣ firms) are larger in 34 We have also worked out an endogenous growth model countries with better contracting institutions, while large with expanding product variety, where the long-run rate of firms (high-␣ firms) are smaller in those same countries. growth depends on the degree of contract incompleteness. We This prediction of the model is consistent with the empirical do not discuss this model in order to save space. evidence reported in James R. Tybout (2000), which indi- 35 The link between contracting institutions and endog- cates that there are significantly fewer medium-sized enter- enous comparative advantage was previously discussed by prises in less-developed economies, which typically have Levchenko (2003), Costinot (2004), Nunn (2006), and worse contracting institutions. Antra`s (2005). VOL. 97 NO. 3 ACEMOGLU ET AL.: CONTRACTS AND TECHNOLOGY ADOPTION 935

In addition, the labor market clearing condition 4 implies that there exists an ␣␮ such that 1 2 1 2 (36) holds in both countries. Consequently, the N (␣) Ͻ N (␣) for all ␣ Ͼ ␣␮ and N (␣) Ͼ N (␣) country with better contracting institutions, coun- for all ␣ Ͻ ␣␮. As a result, country 1, which has try 1, will have higher wages and a lower aᐉ.36 the better contracting institutions, tends to export The pattern of trade can now be determined low-␣ products and import high-␣ products. If, in by comparing the revenues of firms with the addition, ␭(N) is constant, the proportional change same value of ␣ in the two countries. We show in N(a, ␣, ␮) in response to an increase in ␮ is in the Appendix (see the proof of Proposition 2) always smaller when ␣ is greater (because the that the revenue of a producer with parameter ␣ partial Ѩ␨␮(␣, ␮)/Ѩ␣ is negative; see Lemma 2). In in country ᐉ is this case, R1(␣)/R2(␣) is everywhere decreasing in ␣ and there exists ␣៮ ␮ ʦ (0, 1) such that ᐉ͑␣͒ ϭ ͑␣ ␮ᐉ͒ ᐉ͑␣͒␤␬/͑1 Ϫ ␤͒ 1 ␣ 2 ␣ Ͼ ͐1 1 ␣ ␣ ͐1 2 ␣ ␣ (41) R AZ , N R ( )/R ( ) [ 0 R ( ) dH( )]/[ 0 R ( ) dH( )] for all ␣ Ͻ ␣៮ ␮, and the opposite inequality holds ᐉ 1 Ϫ ␤͑1 Ϫ ␮ ͒ for all ␣ Ͼ ␣៮ ␮. Consequently, country 1, with ϫ ␣ , the better contracting institutions, is a net ex- ͑ Ϫ ␤͒ Ϫ ␤͑ Ϫ ␮ᐉ͒ ␣ 1 ͫ1 1 ␣ ϩ ␤ͬ porter in low- sectors and a net importer in high-␣ sectors. This result is summarized in the which is increasing in total world expenditure, following (proof in the text): A,inZ(␣, ␮), and in the level of technology. But it is also directly affected by the parameters PROPOSITION 5: Suppose Assumption 1 holds through the last term on the right-hand side. and ␭(N) is constant. Then there exists ␣៮ ␮ ʦ (0, 1) If such that in the two-country world equilibrium, country 1 with ␮1 Ͼ ␮2 is a net exporter of 1͑␣͒ ͐1 1͑␣͒ ͑␣͒ ␣ Ͻ ␣៮ R 0 R dH products with ␮ and a net importer of Ͼ , 2͑␣͒ ͐1 2͑␣͒ ͑␣͒ products with ␣ Ͼ ␣៮ ␮. R 0 R dH The most important implication of this result then country 1 is a net exporter of goods with is that differences in contracting institutions cre- substitution parameter ␣. Consequently, we sim- ate endogenous comparative advantage. A country ply need to determine the distribution of R1(␣)/ with better contracting institutions gains a com- R2(␣) across different ␣’s. From (41) we have parative advantage in sectors that are more contract dependent, which, in our model, correspond R1͑␣͒ Z͑␣, ␮1͒ N1͑␣͒ ␤␬/͑1 Ϫ ␤͒ to the sectors with greater technological comple- (42) ϭ ͫ ͬ R2͑␣͒ Z͑␣, ␮2͒ N2͑␣͒ mentarities between inputs. The main implications in Proposition 5 re- ␣ ceive support from a number of recent empir- 1 Ϫ ␤͑1 Ϫ ␮2͒ ical papers. Nunn (2006) and Levchenko 1 Ϫ ␤͑1 Ϫ ␮1͒ ␣ ϩ ␤ ϫ . (2003) use international trade data and the 1 Ϫ ␤͑1 Ϫ ␮2͒ ␣ classification of industries into more and less Ϫ ␤͑ Ϫ ␮1͒ 1 1 ␣ ϩ ␤ contract-dependent groups. Consistent with Proposition 5, they find that countries with Both Z(␣, ␮1)/Z(␣, ␮2) and the last term in better contracting institutions specialize in the (42) are decreasing in ␣.37 Second, Proposition export of goods that are more contract dependent. Moreover, the impact of cross-country differences in the contracting institutions on 36 To obtain a contradiction, suppose that this is not the exports is quantitatively large. Nunn, for ex- case; then (40) implies N1(␣) Ͼ N2(␣) for all ␣’s (since ample, finds that the estimated impact of the Nᐉ(␣) is increasing in ␮ for given a), and so labor market quality of the legal system on the export of clearing cannot be satisfied in both countries. contract-dependent sectors is comparable to 37 Recall from Lemma 2 that Z(␣, ␮) is increasing in ␣ 1 2 the effect of human capital abundance on the and ␮, and Ѩ␨␣(␣, ␮)/Ѩ␮ Ͻ 0. Therefore, Z(␣, ␮ )/Z(␣, ␮ ) is decreasing in ␣ whenever ␮1 Ͼ ␮2. The fact that the last export of human capital–intensive products. term is decreasing in ␣ follows from ␮1 Ͼ ␮2. In other words, according to these estimates, 936 THE AMERICAN ECONOMIC REVIEW JUNE 2007 differences in legal systems are as important mechanism for endogenous comparative ad- as differences in human capital abundance for vantage. These implications are consistent explaining trade flows. with a range of recent empirical results in the literature. V. Conclusion It is also useful to note that while we have used a “love-for-variety” model of technol- In this paper, we developed a tractable frame- ogy, our results do not depend on this specific work for the analysis of the impact of contractual modeling approach. In equation (29), N can incompleteness and technological complemen- be interpreted as a general technological in- tarities on the equilibrium technology choice. In vestment, and the linkages between contracts our model, a firm chooses its technology corre- and technology adoption (or investment) sponding to the range of intermediate inputs highlighted in our analysis would apply inde- used in production, and offers contracts to sup- pendent of the specific assumptions regarding pliers, specifying the required investments in the nature of technology. contractible activities. Investments in the re- A number of areas are left for future remaining, noncontractible activities are then cho- search. These include, but are not limited to, sen by the suppliers in anticipation of the ex the following. The model assumes that all post bargaining payoffs. activities are symmetric; an important exten- We used the Shapley value to characterize sion is to see whether similar results hold with the division of surplus between the firm and a more general production function, where the its suppliers, and derived an explicit solution firm may wish to treat some suppliers of in- to these payoffs. Using this setup, we estab- termediate inputs differently from others, de- lished that greater contractual incompleteness pending, for example, on how essential they reduces investments in noncontractible and are for production. Another area for future contractible activities and depresses technol- study is an investigation of the simultaneous ogy choice. The impact of contractual incom- determination of the range of intermediate pleteness on technology adoption is greater in inputs used by the firm and the division of sectors with more complementary intermedi- labor among the suppliers. Finally, it is im- ate inputs. portant to investigate whether the relationship The key mechanism developed in the paper between contracting institutions, technologi- leads to potentially large differences in produc- cal complementarities, and the choice of tech- tivity in response to differences in contracting nology is fundamentally different when we institutions. We also derived implications of use alternative approaches to the theory of the this mechanism for differences in the internal firm, such as the managerial incentives ap- organization of the firm across societies with proach of Holmstro¨m and Paul R. Milgrom different contracting institutions and a new (1991).

APPENDIX Second-Order Conditions in the Complete Contracting Case

⌸ϭ 1Ϫ␤ ␤(␬ϩ1) ␤ Ϫ Ϫ Ϫ Let A N x cxNx C(N) w0N. Using the ﬁrst-order conditions (11) and (12), the matrix of the second-order conditions can be expressed as

Ѩ2⌸ Ѩ2⌸ Ϫ͑ Ϫ ␤͒␬ 2͑ Ј ϩ ͒Ϫ1 Ϫ ͓ Ϫ ␤͑ ϩ ␬͔͒ 1 Ncx C w0 cx 1 1 Ѩx2 ѨxѨN ϭ ΄ Ѩ2⌸ Ѩ2⌸ ΅ ΄ ΅. 2 Ϫ ͓ Ϫ ␤͑ ϩ ␬͔͒ Ϫ͑ ␬͒Ϫ1͓ Ϫ ␤͑ ϩ ␬͔͒͑␬ ϩ ͒͑ Ј ϩ ͒ Ϫ Љ ѨNѨx ѨN cx 1 1 N 1 1 1 C w0 C

The second-order conditions are satisfied if this matrix is negative definite, which requires its diagonal elements to be negative and its determinant to be positive. The first diagonal element is negative; the second diagonal element is negative if and only if VOL. 97 NO. 3 ACEMOGLU ET AL.: CONTRACTS AND TECHNOLOGY ADOPTION 937

CЉN ͓␤͑1 ϩ ␬͒ Ϫ 1͔͑␬ ϩ 1͒ Ͼ Ј ϩ ␬ , C w0 and the determinant is positive if and only if

CЉN ␤͑1 ϩ ␬͒ Ϫ 1 Ͼ Ј ϩ Ϫ ␤ . C w0 1

Note that when ␤(1 ϩ ␬) Ͻ 1, both of these conditions are satisfied, and when ␤(1 ϩ ␬) Ն 1, the second inequality implies the first inequality. Therefore Assumption 1 is necessary and sufficient for the second-order conditions to be satisfied.

PROOF OF PROPOSITION 1: The ﬁrst part of the proposition is a direct implication of Assumption 1. The comparative statics of N* follow from the implicit function theorem by noting that, except for ␤, N* increases in response to an increase in a parameter if and only if this parameter raises the left-hand side of (11). Using the results concerning the response of N* to changes in parameters together with (12) then implies the responses of x* to parameter changes.

PROOF OF LEMMA 1 AND RELATED RESULTS: We now provide a formal derivation of the Shapley value introduced in the text, building on the work of Aumann and Shapley (1974). Let there be M suppliers, each one controlling a range ␧ ϭ N/M of the continuum of intermediate inputs. Due to symmetry, all suppliers provide an amount xc of contractible activities. As for the noncontractible Ϫ activities, consider a situation in which a supplier j supplies an amount xn( j) per noncontractible activity, while the M Ϫ 1 remaining suppliers supply the same amount xn( j) (note that we are again appealing to symmetry). To compute the Shapley value for this particular supplier j, we need to determine the marginal contribution of this supplier to a given coalition of agents. A coalition of n suppliers and the ﬁrm yields a sales revenue of

͑ ␧͒ ϭ 1 Ϫ ␤ ␤͑␬ ϩ 1Ϫ1/␣͒ ␤␮͓͑ Ϫ ͒␧ ͑Ϫ ͒͑1 Ϫ ␮͒␣ ϩ␧ ͑ ͒͑1 Ϫ ␮͒␣͔␤/␣ (A1) FIN n, N; A N xc n 1 xn j xn j , when the supplier j is in the coalition, and a revenue

͑ ␧͒ ϭ 1 Ϫ ␤ ␤͑␬ ϩ 1Ϫ1/␣͒ ␤␮͓ ␧ ͑Ϫ ͒͑1 Ϫ ␮͒␣͔␤/␣ (A2) FOUT n, N; A N xc n xn j , when supplier j is not in the coalition. Notice that even when n Ͻ N, the term N␤(␬ ϩ 1Ϫ1/␣) remains in front, because it represents a feature of the technology, though productivity suffers because the term in square brackets is lower. Following the notation in the main text, the Shapley value of player j is

1 (A3) s ϭ ͸ ͓v͑ zj ഫ j͒ Ϫ v͑ zj ͔͒. j ͑M ϩ 1͒! g g gʦG

ϭ ϩ ϭ j ഫ ϭ j ϭ The fraction of permutations in which g(j) i is 1/(M 1) for every i.Ifg(j) 0 then v(zg j) v(zg) 0, because in this event the ﬁrm is necessarily ordered after j.Ifg(j) ϭ 1 then the ﬁrm is ordered before j with probability 1/M and after Ϫ j ഫ ϭ ␧ j ഫ ϭ j with probability 1 1/M. In the former case v(zg j) FIN(1, N; ), while in the latter case v(zg j) 0. Therefore the j ഫ ϭ ␧ conditional expected value of v(zg j), given g(j) 1, is (1/M)FIN(1, N; ). By similar reasoning, the conditional expected j ␧ ϭ Ͼ j value of v(zg)is(1/M)FOUT(0, N; ). Repeating the same argument for g(j) i, i 1, the conditional expected value of v(zg ഫ ϭ ␧ j Ϫ ␧ j), given g(j) i,is(1/M)FIN(i, N; ), and the conditional expected value of v(zg)is(i/M)FOUT(i 1, N; ). It follows from (A3) that

M M 1 1 s ϭ ͸ i͓F ͑i, N; ␧͒ Ϫ F ͑i Ϫ 1, N; ␧͔͒ ϭ ͸ i␧͓F ͑i, N; ␧͒ Ϫ F ͑i Ϫ 1, N; ␧͔͒␧. j ͑M ϩ 1͒ M IN OUT ͑N ϩ␧͒ N IN OUT i ϭ 1 i ϭ 1

Substituting for (A1) and (A2), 938 THE AMERICAN ECONOMIC REVIEW JUNE 2007

1 Ϫ ␤ ␤͑␬ ϩ 1Ϫ1/␣͒ ␤␮ M A N xc s ϭ ͸ i␧͕i␧x ͑Ϫj͒͑1 Ϫ ␮͒␣ ϩ␧͓ x ͑ j͒͑1 Ϫ ␮͒␣ Ϫ x ͑Ϫj͒͑1 Ϫ ␮͒␣͔͖␤/␣␧ j ͑N ϩ␧͒ N n n n i ϭ 1

1 Ϫ ␤ ␤͑␬ ϩ 1Ϫ1/␣͒ ␤␮ M A N xc Ϫ ͸ i␧͓i␧x ͑Ϫj͒͑1 Ϫ ␮͒␣ Ϫ␧x ͑Ϫj͒͑1 Ϫ ␮͒␣͔␤/␣␧. ͑N ϩ␧͒ N n n i ϭ 1

The ﬁrst-order Taylor expansion of the previous expression gives

1 Ϫ ␤ ␤͑␬ ϩ 1Ϫ1/␣͒ ␤␮͑␤ ␣͒␧ ͑ ͒͑1 Ϫ ␮͒␣ M A N xc / xn j s ϭ ͸ ͑i␧͓͒i␧x ͑Ϫj͒͑1 Ϫ ␮͒␣͔͑␤ Ϫ ␣͒/␣␧ϩo͑␧͒, j ͑N ϩ␧͒ N n i ϭ 1 or

x ͑ j͒ ͑1 Ϫ ␮͒␣ 1 Ϫ ␤ ␤͑␬ ϩ 1Ϫ1/␣͒ n ␤␮ ␤͑1 Ϫ ␮͒ ͑␤ ␣͒ͫ ͬ ͑Ϫ ͒ M A N / xc xn j sj xn ͑Ϫj͒ o͑␧͒ ϭ ͸ ͑i␧͒␤/␣␧ϩ . ␧ ͑N ϩ␧͒ N ␧ i ϭ 1

Now, taking the limit as M 3 ϱ, and therefore ␧ ϭ N/M 3 0, the sum on the right-hand side of this equation becomes a Riemann integral:

x ͑j͒ ͑1 Ϫ ␮͒␣ 1 Ϫ ␤ ␤͑␬ ϩ 1Ϫ1/␣͒͑␤ ␣͒ͫ n ͬ ␤␮ ͑Ϫ ͒␤͑1 Ϫ ␮͒ A N / xc xn j N sj xn ͑Ϫj͒ limͩ ͪ ϭ ͵ z␤/␣ dz. ␧ 2 M3ϱ N 0

Solving the integral delivers

x ͑j͒ ͑1 Ϫ ␮͒␣ ͑ ␧͒ ϭ ͑ Ϫ ␥͒ 1 Ϫ ␤ͫ n ͬ ␤␮ ͑Ϫ ͒␤͑1 Ϫ ␮͒ ␤͑␬ ϩ 1͒Ϫ1 lim sj / 1 A ͑Ϫ ͒ xc xn j N , M3ϱ xn j with ␥ ϭ ␣/(␣ ϩ ␤). This corresponds to equation (20) in the main text, and completes the proof of the lemma. ϭ Ϫ In addition, imposing symmetry, i.e., xn(j) xn( j), the ﬁrm’s payoff is

ϭ 1 Ϫ ␤ ␤͑␬ ϩ 1͒ ␤␮ ␤͑1 Ϫ ␮͒ Ϫ ϭ ␥ 1 Ϫ ␤ ␤␮ ␤͑1 Ϫ ␮͒ ␤͑␬ ϩ 1͒ s0 A N xc xn Nsj A xc xn N , as stated in equation (22) in the text.

PROOF OF PROPOSITION 2: First, we verify that the second-order conditions are again satisﬁed under Assumption 1. To see this, note that the ϭ ៮ problem in (14) is strictly concave and delivers a unique xn xn(N, xc), as given in (23). Plugging this expression into (24), we obtain

␲ ϭ ͑1 Ϫ ␤͒/͑1 Ϫ ␤͑1 Ϫ ␮͒͒⌿ 1 ϩ͓͑␤͑␬ ϩ 1͒Ϫ1͒/͑1 Ϫ ␤͑1 Ϫ ␮͔͒͒ ␤␮/͓1 Ϫ ␤͑1 Ϫ ␮͔͒ Ϫ ␮ Ϫ ͑ ͒ Ϫ (A4) A N xc cx xc N C N w0 N, where

Ϫ1 ␤͑1 Ϫ ␮͒/͓1 Ϫ ␤͑1 Ϫ ␮͔͒ ⌿ ϵ ͓͑cx ͒ ␣͑1 Ϫ ␥͔͒ ͓1 Ϫ ␣͑1 Ϫ ␮͒͑1 Ϫ ␥͔͒.

The second-order conditions can be checked, analogously to the case with complete contracts, by computing the Hessian and checking that it is negative deﬁnite. VOL. 97 NO. 3 ACEMOGLU ET AL.: CONTRACTS AND TECHNOLOGY ADOPTION 939

Here, we present an alternative proof, which also serves to illustrate how the reduced-form proﬁt function (29) in the main ␤␮ Ͻ Ϫ ␤ ϩ ␤␮ text is derived. In particular, notice that for a given level of N, the problem of choosing xc is convex (since 1 ) and delivers a unique solution:

␤ Ϫ 1⌿ ͓1 Ϫ ␤͑1 Ϫ ␮͔͒/͑1 Ϫ ␤͒ cx (A5) x ϭ Aͫ ͬ N͑␤͑␬ ϩ 1͒Ϫ1͒/͑1 Ϫ ␤͒. c 1 Ϫ ␤͑1 Ϫ ␮͒

Plugging this solution in (A4) then delivers

␲ ϭ 1 ϩ͓͑␤͑␬ ϩ 1͒Ϫ1͒/͑1 Ϫ ␤͔͒ Ϫ ͑ ͒ Ϫ AZN C N w0 N, where

␤ Ϫ 1 ␤␮/͑1 Ϫ ␤͒ cx 1 Ϫ ␤ Z ϵ ͫ ͬ ⌿͓1 Ϫ ␤͑1 Ϫ ␮͔͒/͑1 Ϫ ␤͒ ͩ ͪ , 1 Ϫ ␤͑1 Ϫ ␮͒ 1 Ϫ ␤͑1 Ϫ ␮͒ which simplifies to equation (30) in the text. This reduced-form expression of the profit function immediately implies that Ѩ2⌸/ѨN2 Ͻ 0 if and only if the second part of Assumption 1 holds. Finally, from equation (A5) we also ␮ ϭ 1 ϩ [(␤(␬ ϩ 1)Ϫ 1)/(1Ϫ ␤)]␤␮ Ϫ ␤ have cxxcN AZN /(1 ), which combined with (19) and (28) implies that the firm’s revenues are

1 Ϫ ␤͑1 Ϫ ␮͒ ϭ ␲ ϩ ͑ ͒ ϩ ϩ ␮ ϩ ͑ Ϫ ␮͒ ϭ 1 ϩ͑␤͑␬ ϩ 1͒Ϫ1͒/͑1 Ϫ ␤͒ R C N w0 N cx xc N cx xn N 1 AZN ␣ , ΂ ͑ Ϫ ␤͒ Ϫ ␤͑ Ϫ ␮͒ ΃ 1 ͫ 1 1 ␣ ϩ ␤ͬ which will be used in Section IVC (see equation (41)). Next, the comparative static results follow from the implicit function theorem as in the proof of Proposition 1. First, ѨN˜ /ѨA Ͼ 0 follows immediately. To show that ѨN˜ /Ѩ␣ Ͼ 0 and ѨN˜ /Ѩ␮ Ͼ 0, let us take logarithms of both sides of (25) to obtain

␤͑␬ ϩ 1͒ Ϫ 1 ln N˜ ϩ ln͑ A␬␤1/͑1 Ϫ ␤͒c Ϫ ␤/͑1 Ϫ ␤͒͒ ϩ F͑␽, ␮͒ ϭ ln͑CЈ͑N˜ ͒ ϩ w ͒, 1 Ϫ ␤ x 0 where

1 Ϫ ␤͑1 Ϫ ␮͒ 1 Ϫ ␽ ͑1 Ϫ ␮͒ ␤͑1 Ϫ ␮͒ (A6) F͑␽, ␮͒ ϭ lnͩ ͪ ϩ ln͑␽␤Ϫ1͒, 1 Ϫ ␤ 1 Ϫ ␤͑1 Ϫ ␮͒ 1 Ϫ ␤ and ␽ ϵ ␣(1 Ϫ ␥) Ͻ ␤ is monotonically increasing in ␣. Simple differentiation delivers

ѨF͑␽, ␮͒ ͑1 Ϫ ␮͒͑␤ Ϫ ␽͒ ϭ , Ѩ␽ ͑1 Ϫ ␤͒␽ ͑1 Ϫ ␽ ͑1 Ϫ ␮͒͒ which implies ѨF/Ѩ␣ Ͼ 0, and establishes that ѨN˜ /Ѩ␣ Ͼ 0. Furthermore,

1 Ϫ ␽͑1 Ϫ ␮͒ ␤ ␽ Ϫ ␤ ϩ ␤͑1 Ϫ ␽ ͑1 Ϫ ␮͒͒ͩ lnͩ ͪ ϩ lnͩ ͪͪ ѨF͑␽, ␮͒ 1 Ϫ ␤͑1 Ϫ ␮͒ ␽ ϭ Ѩ␮ ͑1 Ϫ ␽͑1 Ϫ ␮͒͒͑1 Ϫ ␤͒ and 940 THE AMERICAN ECONOMIC REVIEW JUNE 2007

Ѩ2F͑␽, ␮͒ ͑␤ Ϫ ␽͒2 ϭ Ϫ Ͻ 0. Ѩ␮2 ͑1 Ϫ ␽͑1 Ϫ ␮͒͒2͑1 Ϫ ␤͑1 Ϫ ␮͒͒͑1 Ϫ ␤͒

Thus, ѨF(␽, ␮)/Ѩ␮ reaches its minimum over the set ␮ ʦ [0, 1] at ␮ ϭ 1, in which case it equals

␽ Ϫ ␤ ϩ ␤ ln͑␤/␽͒ Ͼ 0. 1 Ϫ ␤

This inequality follows from the fact that ␽ Ϫ ␤ ϩ ␤ ln(␤/␽) is decreasing in ␽ for ␽ Ͻ ␤, and is equal to 0 at ␽ ϭ ␤.We thus have shown that ѨF(␽, ␮)/Ѩ␮ Ͼ 0 for all ␮, so that ѨN˜ /Ѩ␮ Ͼ 0. Ѩ Ѩ ϭ Ѩ Ѩ␣ Ͼ Ѩ Ѩ␮ Ͼ Finally, note that straightforward differentiation of (28) delivers (xñ/x˜c)/ A 0, (xñ/x˜c)/ 0 and (xñ/x˜c)/ 0. ␣ ␮ ˜ In light of equation (26), the effects of A, , and on x˜c follow directly from those on N, where the inequalities become strict .whenever CЉ٪ Ͼ 0

PROOF OF LEMMA 2: Comparison of (30) with (A6) implies that

Z͑␣, ␮͒ ϵ ␸eF͑␽,␮͒, where

␣␤ ␽ ϵ ␣͑ Ϫ ␥͒ ϭ 1 ␣ ϩ ␤

␸ ␤ ␽ ␮ and is a parameter that depends only on and cx. We have that F( , ) is increasing in both its arguments, and that ␽ is increasing in ␣, which establish part (a) of the lemma. To prove part (b), ﬁrst note that

Ѩ2F͑␽, ␮͒ Ϫ͑␤ Ϫ ␽͒ ϭ Ͻ 0, Ѩ␽Ѩ␮ ͑1 Ϫ ␽ ͑1 Ϫ ␮͒͒2␽ ͑1 Ϫ ␤͒ because ␽ Ͻ ␤. Moreover,

␮ ѨZ͑␣, ␮͒ ѨF͑␽, ␮͒ ␨␮ ͑␣, ␮͒ ϵ ϭ ␮ , Z Ѩ␮ Ѩ␮ and therefore

2 Ѩ␨␮ ͑␣, ␮͒ Ѩ F͑␽, ␮͒ Ѩ␽ ϭ ␮ ϫ Ͻ Ѩ␣ Ѩ␽Ѩ␮ Ѩ␣ 0.

Similarly,

␣ ѨZ͑␣, ␮͒ ѨF͑␽, ␮͒ Ѩ␽ ␨␣ ͑␣, ␮͒ ϵ ϭ ␣ ϫ , Z Ѩ␣ Ѩ␽ Ѩ␣ and therefore

2 Ѩ␨␣ ͑␣, ␮͒ Ѩ F͑␽, ␮͒ Ѩ␽ ϭ ␣ ϫ Ͻ Ѩ␮ Ѩ␽Ѩ␮ Ѩ␣ 0.

PROOF OF PROPOSITION 3: ˜ ␮ 3 ˜ The proof follows from Proposition 2, since {N, x˜c} converge to {N*, x*} as 1, and N, x˜c, and x˜n are all increasing in ␮ (nondecreasing in ␮, in the case of the investment levels). VOL. 97 NO. 3 ACEMOGLU ET AL.: CONTRACTS AND TECHNOLOGY ADOPTION 941

PROOF OF LEMMA 3: The proof is similar to that of Lemma 1. A coalition of n suppliers and the ﬁrm yields a sales revenue of

͑ ␧͒ ϭ 1 Ϫ ␤ ␤͑␬ ϩ 1Ϫ1/␣͒ ␤␮͓͑ Ϫ ͒␧ ͑Ϫ ͒͑1 Ϫ ␮͒␣ ϩ␧ ͑ ͒͑1 Ϫ ␮͒␣ ϩ ͑ Ϫ ␧͒␦␣ ͑Ϫ ͒͑1 Ϫ ␮͒␣͔␤/␣ (A7) FIN n, N; A N xc n 1 xn j xn j N n xn j ,

when the supplier j is in the coalition, and a sales revenue

͑ ␧͒ ϭ 1 Ϫ ␤ ␤͑␬ ϩ 1Ϫ1/␣͒ ␤␮͓ ␧ ͑Ϫ ͒͑1 Ϫ ␮͒␣ ϩ␧␦␣ ͑ ͒͑1 Ϫ ␮͒␣ ϩ ͑ Ϫ ͑ ϩ ͒␧͒␦␣ ͑Ϫ ͒͑1 Ϫ ␮͒␣͔␤/␣ (A8) FOUT n, N; A N xc n xn j xn j N n 1 xn j ,

when supplier j is not in the coalition. As in the case with ␦ ϭ 0, the Shapley value of supplier j can be written as

M 1 s ϭ ͸ i␧͓F ͑i, N; ␧͒ Ϫ F ͑i Ϫ 1, N; ␧͔͒␧. j ͑N ϩ␧͒ N IN OUT i ϭ 1

␧ ␧ Plugging the new formulas for FIN(n, N; ) and FOUT(n, N; ) in (A7) and (A8) then delivers

1 Ϫ ␤ ␤͑␬ ϩ 1Ϫ1/␣͒ ␤␮ M A N xc s ϭ ͸ i␧͕i␧͑1 Ϫ ␦␣͒ x ͑Ϫj͒͑1 Ϫ ␮͒␣ ϩ␧͓x ͑j͒͑1 Ϫ ␮͒␣ Ϫ x ͑Ϫj͒͑1 Ϫ ␮͒␣͔ ϩ N␦␣x ͑Ϫj͒͑1 Ϫ ␮͒␣͖␤/␣␧ j ͑N ϩ␧͒ N n n n n i ϭ 1

1 Ϫ ␤ ␤͑␬ ϩ 1Ϫ1/␣͒ ␤␮ M A N xc Ϫ ͸ i␧͓i␧͑1 Ϫ ␦␣͒x ͑Ϫj͒͑1 Ϫ ␮͒␣ ϩ␧͓␦␣x ͑j͒͑1 Ϫ ␮͒␣ Ϫ x ͑Ϫj͒͑1 Ϫ ␮͒␣͔ ϩ N␦␣x ͑Ϫj͒͑1 Ϫ ␮͒␣͔␤/␣␧. ͑N ϩ␧͒N n n n n i ϭ 1

The ﬁrst-order Taylor expansion of the previous expression gives

x ͑ j͒ ͑1 Ϫ ␮͒␣ 1 Ϫ ␤ ␤͑␬ ϩ 1Ϫ1/␣͒ ␣ n ␤␮ ␤͑1 Ϫ ␮͒ ͑␤ ␣͒͑ Ϫ ␦ ͒ͫ ͬ ͑Ϫ ͒ M A N / 1 xc xn j sj xn ͑Ϫj͒ o͑␧͒ ϭ ͸ ͑i␧͓͒͑i␧͒͑1 Ϫ ␦␣͒ ϩ N␦␣͔͑␤ Ϫ ␣͒/␣␧ϩ . ␧ ͑N ϩ␧͒N ␧ i ϭ 1

Taking the limit as M 3 ϱ now yields

x ͑j͒ ͑1 Ϫ ␮͒␣ 1 Ϫ ␤ ␤͑␬ ϩ 1Ϫ1/␣͒͑␤ ␣͒͑ Ϫ ␦␣͒ͫ n ͬ ␤␮ ͑Ϫ ͒␤͑1 Ϫ ␮͒ A N / 1 xc xn j N sj xn ͑Ϫj͒ limͩ ͪ ϭ ͵ z͓z͑1 Ϫ ␦␣͒ ϩ N␦␣͔͑␤ Ϫ ␣͒/␣ dz. ␧ 2 M3ϱ N 0

Finally, integrating the last term by parts delivers

x ͑j͒ ͑1 Ϫ ␮͒␣ ͑ ␧͒ ϭ ͑ Ϫ ␥͒ 1 Ϫ ␤ͫ n ͬ ␤␮ ͑Ϫ ͒␤͑1 Ϫ ␮͒ ␤͑␬ ϩ 1͒Ϫ1 lim sj / 1 A ͑Ϫ ͒ xc xn j N , M3ϱ xn j

where

␣͑1 Ϫ ␦␣ ϩ ␤͒ ␥ ϵ ␣ , ͑␣ ϩ ␤͒͑1 Ϫ ␦ ͒

as claimed in the lemma. 942 THE AMERICAN ECONOMIC REVIEW JUNE 2007

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