Contracts and Technology Adoption

28 Pages·2007·0.63 MB·English

Contracts and Technology Adoption
By DARON ACEMOGLU, POL ANTRA`S, AND ELHANAN HELPMAN*
We develop a tractable framework for the analysis of the relationship between
contractualincompleteness,technologicalcomplementarities,andtechnologyadop-
tion.Inourmodel,afirmchoosesitstechnologyandinvestmentlevelsincontract-
ibleactivitiesbysuppliersofintermediateinputs.Suppliersthenchooseinvestments
innoncontractibleactivities,anticipatingpayoffsfromanexpostbargaininggame.
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 agree-
ment,wearefarfromanestablishedframework
for the analysis of technology choices of firms.
*Acemoglu:DepartmentofEconomics,MassachusettsIn- Inthispaper,wetakeastepinthisdirectionand
stituteofTechnology,E52-380b,Cambridge,MA02139,Na- develop a simple model to study the impact of
tional Bureau of Economic Research, Centre for Economic
contracting institutions, which regulate the re-
Policy Research, and Canadian Institute for Advanced Re-
search(e-mail:[email protected]);Antra`s:DepartmentofEco- lationshipbetweenthefirmanditssuppliers,on
nomics, Harvard University, Littauer 319, Cambridge, MA technology choices.
02138,NBER,andCEPR(e-mail:[email protected]); Our model combines two well-established
Helpman:DepartmentofEconomics,HarvardUniversity,Lit-
approaches. The first is the representation of
tauer229,Cambridge,MA02138,NBER,CEPR,andCIAR
technology as the range of intermediate in-
(e-mail:[email protected]).Apreviousversionofthis
paperwascirculatedunderthetitle“ContractsandtheDivision 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,Damia´nMigueles,GiacomoPonzetto,RichardRog- greater specialization and thus corresponds to
erson,RaniSpiegler,twoanonymousreferees,participantsin
more “advanced” technology.2 The second is
theCanadianInstituteforAdvancedResearchandMinnesota
WorkshopinMacroeconomicTheoryconferences,andsemi- Sanford J. Grossman and Oliver D. Hart’s
narparticipantsatHarvardUniversity,MIT,TelAvivUniver-
sity, Universitat Pompeu Fabra, University of Houston,
University of California, San Diego, Southern Methodist
University, Stockholm School of Economics, Stockholm 1Amongothers,seePeterJ.KlenowandAndre´sRodri-
University(IIES),ECARES,UniversityofColorado–Boul- guez-Clare (1997), Robert E. Hall and Charles I. Jones
der,theKelloggSchool,NewYorkUniversity,University (1999),orFrancescoCaselli(2005)forcountries,andTorJ.
of British Columbia, Georgetown University, the World Klette(1996),ZviGriliches(1998),JohnSutton(1998),or
Bank,UniversityofMontreal,BrandeisUniversity,Univer- KletteandSamuelS.Kortum(2004)forfirms.
sity of Wisconsin, UC–Berkeley, Haifa University, Syra- 2See,amongothers,WilfredJ.Ethier(1982),PaulM.
cuseUniversity,UniversitatAuto`nomadeBarcelona,Duke Romer(1990),andGeneM.GrossmanandHelpman(1991)
University, Columbia University, and Cornell University forprevioususesofthisrepresentation.Seealsothetext-
forusefulcomments.WealsothankDavinChor,Alexandre booktreatmentinRobertJ.BarroandXavierSala-i-Martin
Debs,andRanMelamedforexcellentresearchassistance. (2003).Thereisanaturalrelationshipbetweenthisviewof
AcemogluandHelpmanthanktheNationalScienceFoun- technology and the division of labor within a firm. We
dationforfinancialsupport.MuchofHelpman’sworkfor investigated this link in the earlier version of the current
thispaperwasdonewhenhewasSacklerVisitingProfessor paper,Acemoglu,Antra`s,andHelpman(2005),anddonot
atTelAvivUniversity. elaborateonithere.
916
VOL.97NO.3 ACEMOGLUETAL.:CONTRACTSANDTECHNOLOGYADOPTION 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 thusreducessupplierinvestments,makingmore
to be partially incomplete. This combination advanced technologies less profitable. Further-
enables us to investigate how the degree of more,agreaterdegreeoftechnologicalcomple-
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-
vancedtechnologyismoreproductive,butalso ments and, via this channel, depressing the
entails a variety of costs. In addition to the profitability of more advanced technologies.
directpecuniarycostsofengagingmoresuppli- Anadvantageofourframeworkisitsrelative
ers(correspondingtothegreaterrangeofinter- 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
Alloftheactivitiesthatsuppliersundertakeare
relationship-specific, and a fraction of those is (1) AZF(cid:1)N(cid:2)(cid:1)C(cid:1)N(cid:2)(cid:1)w N,
0
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,
of contracting institutions.3 Suppliers are con- and w N corresponds to the value of the N
0
tractually obliged to perform their duties in the suppliers’outsideoptions.F(N)isanincreasing
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 andtechnologicalcomplementarityaresumma-
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
gainingproblem.AsinHartandMoore(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-
anditssuppliers.Wederiveanexplicitsolution tutions is higher when there is greater comple-
for this division of surplus, which enables us mentaritybetweenintermediateinputs.Thislast
to develop a simple characterization of the result has important implications for equilib-
equilibrium. riumindustrystructureandthepatternsofcom-
A supplier’s expected payoff in the bargain- parative advantage, because it implies that
inggamedeterminesherwillingnesstoinvestin sectors (firms) with greater complementarities
between inputs are more “contract dependent.”
We use this framework to show that the
3Eric S. Maskin and Jean Tirole (1999) question combination of contractual imperfections and
whetherthepresenceofnonverifiableactionsandunfore-
technology choice (or adoption) may have im-
seen contingencies necessarily leads to incomplete con-
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,forexample,allowcontractstospec- and patterns of international specialization and
ify sophisticated mechanisms), while in our model a
trade.
fractionofactivitiesmaybenoncontractible,notbecause
First, we show that our baseline model can
of“technological”reasonsbutbecauseofweakcontract-
inginstitutions. generate sizable differences in productivity from
918 THEAMERICANECONOMICREVIEW JUNE2007
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 betweenafirmandanumberofworkerswhose
complementarity is sufficiently high, relatively wages are determined by ex post bargaining
modestchangesinthefractionofactivitiesthat accordingtotheShapleyvalue.Theyshowhow
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 discusstheimplicationsofthisframeworkfor
about the equilibrium organizational form. a number of organizational design issues.
Ourresultshereshowthatthecombinationof StoleandZwiebel’sframeworkdoesnothave
weak contracting institutions and credit mar- relationship-specific investments, however,
ket imperfections may encourage greater ver- whichisatthecoreofourapproach,andthey
tical integration. do not discuss the effects of the degree of
Third,wepresentanumberofgeneralequi- 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-
tionsdoesnotincreasethechoiceoftechnology 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 AnaHidalgo(2005),andRuiCastro,GianLuca
endogenous structure of comparative advan- Clementi,andGlennMacDonald(2004),inves-
tage. In particular, among countries with iden- tigatethequantitativeimpactofcontractualand
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
complementaritiesamonginputs.Thesepredic- 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 appliedinarangeofproblems.Althoughwedo
Andrei A. Levchenko (2003). not undertake a detailed calibration exercise as
As mentioned above, our work is related to in thesepapers,the results inSectionIVAsug-
two strands of the literature. The first investi- gestthattheeconomicmechanismdevelopedin
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),ArnaudCosti-
among others, Gary S. Becker and Kevin M. not (2004), Nunn (2006), and Antra`s (2005)
Murphy(1992)andXiaokaiYangandJeffBor- 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)ismostcloselyrelated,sincehe
on endogenous technological change. None also develops a model of endogenous compar-
of these studies investigates the effects of con-
tracting institutions on technology choice. The
second literature deals with the internal organi- 4AnotherrelatedpaperisOlivierJ.BlanchardandMi-
chaelKremer(1997),whichstudiestheimpactofinefficient
zation of the firm. It includes the papers by
sequentialcontractingbetweenafirmanditssuppliersina
Grossman-Hart-Moorediscussedabove,aswell modelwhereoutputisaLeontiefaggregateoftheinputsof
as those of Benjamin Klein, Robert G. Craw- thesuppliers.
ford, and Armen A. Alchian (1978) and Oliver 5AcemogluandFabrizioZilibotti(1999),DavidMarti-
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.
impactofchangesintheinternalorganizationoffirmson
Here, the papers by Lars A. Stole and Jeffrey economicgrowth.
VOL.97NO.3 ACEMOGLUETAL.:CONTRACTSANDTECHNOLOGYADOPTION 919
ativeadvantagebasedonspecialization,though (1998) in introducing the term N(cid:4)(cid:6)1(cid:4)1/(cid:5) 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.SectionIIcharacterizestheequilibrium this,considerthecasewhereX(j)(cid:3)Xforallj.
with complete contracts. Section III introduces ThentheoutputoftechnologyNisq(cid:3)N(cid:4)(cid:6)1X.
incompletecontractsintotheframeworkofSec- Consequently,bothoutputandproductivity(de-
tionI,characterizestheequilibrium,andderives finedaseitherq/(NX)orq/N)areindependentof
themajorcomparativestaticresults.SectionIV (cid:5)and depend positively on N, with elasticity
considersanumberofapplicationsofourbasic determined by the parameter (cid:4).6
framework. Section V concludes. Proofs of the Thereisalargenumberofprofit-maximizing
main results are provided in the Appendix. suppliers that can produce the necessary inter-
mediate goods, each with the same outside op-
I. TechnologyandPayoffs tionw .Fornow,w istakenasgiven,butitwill
0 0
beendogenizedinSectionIVC.Weassumethat
Consider a profit-maximizing firm facing a eachintermediateinputneedstobeproducedby
demand function q (cid:3) Ap(cid:4)1/(1(cid:4)(cid:2)) for its final adifferentsupplierwithwhomthefirmneedsto
product,whereqdenotesquantityandpdenotes contract.7
price. The parameter (cid:2)(cid:1) (0, 1) determines the A supplier assigned to the production of an
elasticity of demand, while A (cid:5) 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.Thereisaconstantmar-
This form of demand can be derived from a ginal cost of investment c for each activity.8
x
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:
(cid:1) (cid:3)
(cid:2)
(2) R(cid:3)A1(cid:4)(cid:2)q(cid:2).
1
(4) X(cid:1)j(cid:2)(cid:3)exp lnx(i,j)di ,
Productiondependsonthetechnologychoiceof
0
the firm. More advanced (more productive)
technologiesinvolveagreaterrangeofinterme- wherex(i,j)isthelevelofinvestmentinactivity
diategoodsandthusahigherdegreeofspecial- i performed by the supplier of input j. This
ization. The technology level of the firm is formulationwillallowatractableparameteriza-
denoted by N (cid:1) (cid:1) , and for each j (cid:1) [0, N], tion of contractual incompleteness in Section
(cid:6)
X(j) is the quantity of intermediate input j. III, whereby a subset of the investments neces-
GiventechnologyN,theproductionfunctionof sary for production will be nonverifiable and
the firm is thus noncontractible.
(cid:1)(cid:2) (cid:3) Finally,weassumethatadopting(andusing)
1/(cid:5) a technology N involves costs C(N), and im-
N
(3) q(cid:3)N(cid:4)(cid:6)1(cid:4)1/(cid:5) X(j)(cid:5)dj , pose:
0
6Incontrast,withthestandardspecificationoftheCES
0(cid:6)(cid:5)(cid:6)1, (cid:4)(cid:7)0. production function, without the term N(cid:4)(cid:6)1(cid:4)1/(cid:5) in front
(i.e.,(cid:4)(cid:3)1/(cid:5)(cid:4)1),totaloutputisq(cid:3)N1/(cid:5)X,andthetwo
A number of features of this production func- elasticitiesaregovernedbythesameparameter,(cid:5).
tion are worth noting. First, (cid:5)determines the 7A previous version of the paper, Acemoglu, Antra`s,
degree of complementarity between inputs; and Helpman (2005), also endogenized the allocation of
since (cid:5)(cid:1) (0, 1), the elasticity of substitution inputs to suppliers using an augmented model with addi-
tionaldiseconomiesofscope.
betweenthem,1/(1(cid:4)(cid:5)),isalwaysgreaterthan 8Onecanthinkofc asthemarginalcostofeffort;see
x
one. Second, we follow Jean-Pascal Benassy theformulationofthiseffortinutilitytermsinSectionIVC.
920 THEAMERICANECONOMICREVIEW JUNE2007
ASSUMPTION 1: wherethefirmchoosesatechnologylevelNand
makes a contract offer [{x(i, j)} , {s(j),
i(cid:1)[0,1]
(a) For all N (cid:5) 0, C(N) is twice continuously (cid:9)(j)}] for every input j (cid:1) [0, N]. If a supplier
differentiable,withC(cid:7)(N)(cid:5)0andC(cid:8)(N)(cid:8) acceptsthiscontractforinputj,sheisobligedto
0. supply {x(i, j)} as stipulated in the con-
i(cid:1)[0,1]
(b) For all N (cid:5) 0, NC(cid:8)(N)/[C(cid:7)(N) (cid:6) w ] (cid:5) tract in exchange for the payments {s(j), (cid:9)(j)}.
0
[(cid:2)((cid:4)(cid:6) 1) (cid:4) 1]/(1 (cid:4) (cid:2)). Asubgameperfectequilibriumofthisgameisa
strategy combination for the firm and the sup-
pliers 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-
partwillensureafiniteandpositivechoiceofN. lowing maximization problem:
Let the payment to supplier j consist of two
parts: an ex ante payment (cid:9)(j) (cid:1) (cid:1) before the
(8)
investment levels x(i, j) take place, and a pay-
(cid:2)
ments(j)aftertheinvestments.Then,thepayoff
N
to supplier j, also taking account of her outside max R(cid:1) (cid:9)(cid:9)(cid:1)j(cid:2)(cid:11)s(cid:1)j(cid:2)(cid:10)dj(cid:1)C(cid:1)N(cid:2)
option, is
N,(cid:12)x(cid:1)i,j(cid:2)(cid:13)i,j,(cid:12)s(cid:1)j(cid:2),(cid:9)(cid:1)j(cid:2)(cid:13)j 0
(5)
(cid:4) (cid:5) subject to (7) and the suppliers’ participation
(cid:2)
constraint,
1
(cid:10)(cid:1)j(cid:2)(cid:3)max (cid:9)(j)(cid:11)s(j)(cid:1) c x(i,j)di,w . (cid:2)
x x 0
1
0 (9) s(cid:1)j(cid:2)(cid:11)(cid:9)(cid:1)j(cid:2)(cid:1)c x(cid:1)i, j(cid:2) di
x
0
Similarly, the payoff to the firm is
(cid:2) (cid:8)w forallj(cid:1)(cid:9)0,N(cid:10).
0
N
(6) (cid:10)(cid:3)R(cid:1) (cid:9)(cid:9)(cid:1)j(cid:2)(cid:11)s(cid:1)j(cid:2)(cid:10)dj(cid:1)C(cid:1)N(cid:2),
Sincethefirmhasnoreasontoproviderentstothe
0 suppliers, it chooses payments s(j) and (cid:9)(j) that
satisfy (9) with equality.9 Moreover, since the
where R is revenue and the other two terms on firm’sobjectivefunction,(8),is(jointly)concave
theright-handsiderepresentcosts.Substituting in the investment levels x(i, j), and these invest-
(3)and(4)into(2),revenuecanbeexpressedas mentsareallequallycostly,thefirmchoosesthe
same investment level x for all activities in all
(7) R(cid:3)A1(cid:4)(cid:2)N(cid:2)(cid:1)(cid:4)(cid:6)1(cid:4)1/(cid:5)(cid:2) intermediate inputs. Now, substituting for (9) in
(cid:1)(cid:2) (cid:6) (cid:6)(cid:2) (cid:7)(cid:7) (cid:3) (8),weobtainthefollowingsimplerunconstrained
(cid:5) (cid:2)/(cid:5) maximizationproblemforthefirm:
N 1
(cid:11) exp lnx(i,j)di dj .
(10)
0 0
maxA1(cid:4)(cid:2)N(cid:2)(cid:1)(cid:4)(cid:6)1(cid:2)x(cid:2)(cid:1)c Nx(cid:1)C(cid:1)N(cid:2)(cid:1)w N.
II. EquilibriumunderCompleteContracts x 0
N,x
As a benchmark, consider the case of complete
Fromthefirst-orderconditionsofthisproblem,
contracts(the“firstbest”fromtheviewpointof
we obtain
thefirm).Withcompletecontracts,thefirmhas
full control over all investments and pays each
supplier her outside option. In analogy to our
9Withcompletecontracts,(cid:9)(j)ands(j)areperfectsub-
treatment below of technology adoption under stitutes,sothatonlythesums(j)(cid:6)(cid:9)(j)matters.Thiswill
incomplete contracts, consider a game form notbethecasewhencontractsareincomplete.
VOL.97NO.3 ACEMOGLUETAL.:CONTRACTSANDTECHNOLOGYADOPTION 921
(11) (cid:1)N*(cid:2)(cid:1)(cid:2)(cid:1)(cid:4)(cid:6)1(cid:2)(cid:4)1(cid:2)/(cid:1)1(cid:4)(cid:2)(cid:2)A(cid:4)(cid:2)1/(cid:1)1(cid:4)(cid:2)(cid:2)c(cid:4)(cid:2)/(cid:1)1(cid:4)(cid:2)(cid:2) In the case of complete contracts, the size of
x
the market (as parameterized by the demand
(cid:3)C(cid:7)(cid:1)N*(cid:2)(cid:11)w , levelA)hasapositiveeffectoninvestmentsby
0
suppliers of intermediate inputs and productiv-
C(cid:7)(cid:1)N*(cid:2)(cid:11)w ity. The other noteworthy implication of this
(12) x*(cid:3) 0. proposition is that, under complete contracts,
(cid:4)c
x the level of technology and thus productivity
doesnotdependontheelasticityofsubstitution
Equations (11) and (12) can be solved recur- between intermediate inputs, 1/(1 (cid:4) (cid:5)).
sively. Given Assumption 1, equation (11)
yieldsauniquesolutionforN*,which,together III. EquilibriumunderIncompleteContracts
with (12), yields a unique solution for x*.10
Whenalltheinvestmentlevelsareidentical A. Incomplete Contracts
and equal to x, output equals q (cid:3) N(cid:4)(cid:6)1x.
Since NX (cid:3) Nx inputs are used in the pro- We now consider the same environment un-
ductionprocess,wecandefineproductivityas der incomplete contracts. We model the imper-
output divided by total input use, P (cid:3) N(cid:4). In fection of the contracting institutions by
the case of complete contracts, this produc- assuming that there exists a (cid:13)(cid:1) [0, 1] such
tivity level is that,foreveryintermediateinputj,investments
in activities 0 (cid:14) i (cid:14) (cid:13) are observable and
verifiable and therefore contractible, while in-
(13) P*(cid:3)(cid:1)N*(cid:2)(cid:4),
vestments in activities (cid:13)(cid:14) i (cid:14) 1 are not con-
tractible. Consequently, a contract stipulates
which is increasing in the level of technology. investment levels x(i, j) for the (cid:13)contractible
In the next section we compare this to equilib- activities, but does not specify the investment
riumproductivityunderincompletecontracts.11 levels in the remaining 1 (cid:4) (cid:13)noncontractible
The next proposition describes the key activities.Instead,supplierschoosetheirinvest-
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.Thenwithcompletecontractsthereexists pletecontractsliteratureandassumethattheex
a unique equilibrium with technology and in- postdistributionofrevenueisgovernedbymul-
vestment levels N* (cid:5) 0 and x* (cid:5) 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
(cid:12)N* (cid:12)x* (cid:12)N* (cid:12)x* The timing of events is as follows:
(cid:7)0, (cid:8)0, (cid:3) (cid:3)0.
(cid:12)A (cid:12)A (cid:12)(cid:5) (cid:12)(cid:5)
● The firm adopts a technology N and offers a
contract [{x (i, j)}(cid:13) , (cid:9)(j)] for every inter-
c i(cid:3)0
10WeshowintheAppendixthatthesecond-ordercon- mediate input j (cid:1) [0, N], where xc(i, j) is an
ditionsaresatisfiedunderAssumption1. investmentlevelinacontractibleactivityand
11Thismeasureofproductivityimplicitlyassumesthat
all the investments x(i, j) are measured accurately. There
may be some tension between this assumption and the
assumptionthatthecostoftheseinvestmentsisnotpecu- 12Theincompletecontractsliteraturealsoassumesthat
niary.Forthisreason,inthepreviousversionweconsidered revenuesarenoncontractible.Asiswellknown,withbilat-
another definition of productivity: output divided by the eralcontractingorwithmultilateralcontractingandabud-
number of suppliers, P (cid:3) q/N (cid:3) (N)(cid:4)x. The ranking of getbreaker(e.g.,BengtR.Holmstro¨m1982),contractingon
productivitylevelsbetweencompleteandincompletecon- revenues would improve incentives. In our setting, we do
tractsisthesameunderbothdefinitions,andthequantitative notneedthisassumption,sinceeachfirmhasacontinuum
effects of changes in contractibility on productivity are of suppliers, and contracts on total revenues would not
smallerwiththedefinitionusedhere. provideadditionalinvestmentincentivestosuppliers.
922 THEAMERICANECONOMICREVIEW JUNE2007
(cid:9)(j) is an upfront payment to supplier j. The suppliers and the firm will engage in multilat-
payment (cid:9)(j) can be positive or negative. eral Shapley bargaining. Denote the Shapley
● Potential suppliers decide whether to apply valueofsupplierjunderthesecircumstancesby
for the contracts. Then the firm chooses N s(cid:1) [N, x , x ((cid:4)j), x (j)]. We derive an explicit
x c n n
suppliers, one for each intermediate input j. formula for this value in the next subsection.
● All suppliers j (cid:1) [0, N] simultaneously For now, note that optimal investment by sup-
chooseinvestmentlevelsx(i,j)foralli(cid:1)[0, plier j implies that x (j) is chosen to maximize
n
1]. In the contractible activities i (cid:1) [0, (cid:13)] s(cid:1) [N,x ,x ((cid:4)j),x (j)]minusthecostofinvest-
x c n n
they invest x(i, j) (cid:3) x (i, j). mentinnoncontractibleactivities,(1(cid:4)(cid:13))cx (j).
c x n
● The suppliers and the firm bargain over the In a symmetric equilibrium, we need x (j) (cid:3)
n
divisionofrevenue,andatthisstage,suppli- x ((cid:4)j) or, in other words, x needs to be a
n n
ers can withhold their services in noncon- fixed-point given by14
tractible activities.
● Outputisproducedandsold,andtherevenue (14) x (cid:3)argmaxs(cid:1) (cid:9)N,x ,x ,x (cid:1)j(cid:2)(cid:10)
n x c n n
R is distributed according to the bargaining xn(cid:1)j(cid:2)
agreement.
(cid:4)(cid:1)1(cid:1)(cid:13)(cid:2)c x (cid:1)j(cid:2).
x n
We will characterize a symmetric subgame
perfect equilibrium (SSPE for short) of this Equation(14)canbethoughtofasan“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 givenbyx andwithinvestmentinnoncontract-
c
Preliminaries ible activities equal to x , the revenue of the
n
firm is given by R (cid:3) A1(cid:4)(cid:2)(N(cid:4)(cid:6)1x(cid:13)x1(cid:4)(cid:13))(cid:2).
c n
BehavioralongtheSSPEcanbedescribedby Moreover, let s (N, x , x ) (cid:3) s(cid:1) (N, x , x , x );
x c n x c n n
a tuple {N˜, x˜ , x˜ , (cid:9)˜} in which N˜ represents the thentheShapleyvalueofthefirmisobtainedas
c n
level of technology, x˜ the investment in con- a residual:
c
tractibleactivities,x˜ theinvestmentinnoncon-
n
tractibleactivities,and(cid:9)˜ theupfrontpaymentto
s (cid:1)N, x , x (cid:2)(cid:3)A1(cid:4)(cid:2)(cid:1)N(cid:4)(cid:6)1x(cid:13)x1(cid:4)(cid:13)(cid:2)(cid:2)
everysupplier.Thatis,foreveryj(cid:1)[0,N˜],the q c n c n
upfrontpaymentis(cid:9)(j)(cid:3)(cid:9)˜,andtheinvestment
(cid:4)Ns (cid:1)N,x ,x (cid:2).
levelsarex(i,j)(cid:3)x˜ fori(cid:1)[0,(cid:13)]andx(i,j)(cid:3) x c n
c
x˜ for i (cid:1) ((cid:13), 1]. With a slight abuse of termi-
n
nology,wewilldenotetheSSPEby{N˜,x˜ ,x˜ }. Nowconsiderthestageinwhichthefirmchooses
c n
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, w , this pool is empty. Therefore, for
0
andx asthelevelofinvestmentincontractible production to take place, the final-good pro-
c
activities.Supposealsothateachsupplierother 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-
contractibleactivitiesequaltox ((cid:4)j)(theseare complete contracts, i.e.,
n
all the same, because we are constructing a
symmetric equilibrium), while the investment
level in every noncontractible activity by sup-
thanothers.Itisstraightforwardtoshow,however,thatthe
plier j is xn(j).13 Given these investments, the bestdeviationforasupplieristochoosethesamelevelof
investmentinallnoncontractibleactivities.Forthisreason
we save on notation and restrict attention to only such
deviations.
13More generally, we would need to consider a distri- 14Thisequationshouldbewrittenwith“(cid:1)”insteadof
butionofinvestmentlevels,{x(i,j)} forsupplierj, “(cid:3).” However, we show below that the fixed point x in
n i(cid:1)((cid:13),1] n
wheresomeoftheactivitiesmayreceivemoreinvestment (14)isunique,justifyingouruseof“(cid:3).”
VOL.97NO.3 ACEMOGLUETAL.:CONTRACTSANDTECHNOLOGYADOPTION 923
(15) s(cid:1) (cid:1)N, x , x , x (cid:2)(cid:11)(cid:9)(cid:8)(cid:13)c x Osborne and Ariel Rubinstein 1994). In a bar-
x c n n x c
gaining game with a finite number of players,
(cid:11)(cid:1)1(cid:1)(cid:13)(cid:2)c x (cid:11)w each player’s Shapley value is the average of
x n 0
hercontributionstoallcoalitionsthatconsistof
forx thatsatisfies(cid:1)14(cid:2). playersorderedbelowherinallfeasiblepermu-
n
tations. More explicitly, in a game with M (cid:6) 1
players, let g (cid:3) {g(0), g(1),..., g(M)} be a
In other words, given N and (x , (cid:9)), each sup- permutationof0,1,2,...,M,whereplayer0is
c
plierj(cid:1)[0,N]shouldexpectherShapleyvalue the firm and players 1, 2,..., M are the suppli-
plus the upfront payment to cover the cost of ers, and let zj (cid:3) {j(cid:7)(cid:8)g(j) (cid:5) g(j(cid:7))} be the set of
g
investment in contractible and noncontractible players ordered below j in the permutation g.
activities and the value of her outside option. WedenotebyGthesetoffeasiblepermutations
The maximization problem of the firm can and by v:G 3 (cid:1) the value of the coalition
then be written as consistingofanysubsetoftheM(cid:6)1players.16
Then the Shapley value of player j is
max s (cid:1)N,x ,x (cid:2)(cid:1)N(cid:9)(cid:1)C(cid:1)N(cid:2)
q c n (cid:9)
N,xc,xn,(cid:9) 1
s (cid:3) (cid:9)v(cid:1)zj (cid:2)j(cid:2)(cid:1)v(cid:1)zj(cid:2)(cid:10).
subjectto(cid:1)14(cid:2)and(cid:1)15(cid:2). j (cid:1)M(cid:11)1(cid:2)! g(cid:1)G g g
With no restrictions on (cid:9), 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 (cid:9)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
thereforesolve(cid:9)fromthisconstraint,substitute infinity.17 Leaving the formal derivation to the
the solution into the firm’s objective function, Appendix, here we provide a heuristic deriva-
andobtainthesimplermaximizationproblem:15 tion of this Shapley value.
Suppose the firm has adopted technology N,
(16) maxs (cid:1)N,x ,x (cid:2)(cid:11)N(cid:9)s(cid:1) (cid:1)N,x ,x ,x (cid:2) all suppliers provide an amount x of every
q c n x c n n c
N,xc,xn contractibleactivity,andallsuppliersotherthan
jinvestx ((cid:4)j)ineverynoncontractibleactivity,
(cid:1)(cid:13)c x (cid:1)(cid:1)1(cid:1)(cid:13)(cid:2)c x (cid:10)(cid:1)C(cid:1)N(cid:2)(cid:1)w N n
x c x n 0 while supplier j invests x (j). To compute the
n
Shapley value for supplier j, first note that the
subjectto(cid:1)14(cid:2).
firm is an essential player in this bargaining
game. (If a coalition does not include the firm,
The SSPE {N˜, x˜ , x˜ } solves this problem, and then its output equals zero regardless of its
c n
the corresponding upfront payment satisfies size.) Consequently, the supplier j’s marginal
contribution is equal to zero when a coalition
(17) (cid:9)˜ (cid:3)(cid:13)c x˜ (cid:11)(cid:1)1(cid:1)(cid:13)(cid:2)c x˜ (cid:11)w doesnotincludethefirm.Whenitdoesinclude
x c x n 0
the firm and a measure n of suppliers, the mar-
(cid:4)s(cid:1) (cid:1)N˜,x˜ ,x˜ ,x˜ (cid:2). ginal contribution of supplier j is m(j, n) (cid:3)
x c n n (cid:12)R(cid:1)/(cid:12)n, where
C. Bargaining
We now derive the Shapley values in this
game (see Lloyd S. Shapley 1953, or Martin J. 16Inourgame,thevalueofacoalitionequalstheamount
ofrevenuethiscoalitioncangenerate.
17More formally, we divide the interval [0, N] into M
15Notethat,asinthecasewithcompletecontracts,thefirm equallyspacedsubintervalswithalltheintermediateinputs
choosesitstechnologyandinvestmentlevelstomaximizesale in each subinterval of length N/M performed by a single
revenues net of total costs. The key difference is that with supplier.WethensolvefortheShapleyvalueandtakethe
incomplete contracts, this maximization problem is con- limitofthissolutionasM3(cid:15).SeeAumannandShapley
strainedbythe“incentivecompatibility”condition(14). (1974)orStoleandZwiebel(1996b).
924 THEAMERICANECONOMICREVIEW JUNE2007
R(cid:1) (cid:3)A1(cid:4)(cid:2)N(cid:2)(cid:1)(cid:4)(cid:6)1(cid:4)1/(cid:5)(cid:2) other suppliers invest x ((cid:4)j) in their noncon-
n
(cid:1)(cid:2) (cid:6) (cid:6)(cid:2) (cid:7)(cid:7) (cid:3) tractible activities, every supplier invests xc in
(cid:5) (cid:2)/(cid:5) hercontractibleactivities,andtheleveloftech-
n 1
(cid:11) exp lnx(i,k)di dk nologyisN.ThentheShapleyvalueofsupplier
j is
0 0
is the revenue derived from the employment of (20) s(cid:1) (cid:9)N, x , x (cid:1)(cid:4)j(cid:2),x (cid:1)j(cid:2)(cid:10)
x c n n
n inputs with technology N, and the last input (cid:1) (cid:3)
k (cid:3) n is provided by supplier j. Since x(i, k) (cid:3) x (cid:1)j(cid:2) (cid:1)1(cid:4)(cid:13)(cid:2)(cid:5)
xcforall0(cid:14)i(cid:14)(cid:13)andall0(cid:14)k(cid:14)n,x(i,k)(cid:3) (cid:3)(cid:1)1(cid:1)(cid:15)(cid:2)A1(cid:4)(cid:2) x (cid:1)n(cid:4)j(cid:2)
x ((cid:4)j)forall(cid:13)(cid:14)i(cid:14)1andall0(cid:14)k(cid:14)n,and n
n
x(i, k) (cid:3) xn(j) for all (cid:13)(cid:14) i (cid:14) 1 and k (cid:3) n, (cid:11)x(cid:2)(cid:13)x (cid:1)(cid:4)j(cid:2)(cid:2)(cid:1)1(cid:4)(cid:13)(cid:2)N(cid:2)(cid:1)(cid:4)(cid:6)1(cid:2)(cid:4)1,
evaluatingthepreviousexpressionenablesusto c n
write the marginal contribution of supplier j as
where (cid:15)is defined in (19).
(cid:2)
(18) m(cid:1)j, n(cid:2)(cid:3)(cid:5)A1(cid:4)(cid:2)N(cid:2)(cid:1)(cid:4)(cid:6)1(cid:4)1/(cid:5)(cid:2) A number of features of (20) are worth not-
ing. First, in equilibrium, all suppliers invest
(cid:1) (cid:3)
x (cid:1)j(cid:2) (cid:1)1(cid:4)(cid:13)(cid:2)(cid:5) equallyinallthenoncontractibleactivities,i.e.,
(cid:11) n x(cid:2)(cid:13)x (cid:1)(cid:4)j(cid:2)(cid:2)(cid:1)1(cid:4)(cid:13)(cid:2)n(cid:1)(cid:2)(cid:4)(cid:5)(cid:2)/(cid:5). x (j) (cid:3) x ((cid:4)j) (cid:3) x , and so
x (cid:1)(cid:4)j(cid:2) c n n n n
n
(21) s (cid:1)N, x , x (cid:2)(cid:3)s(cid:1) (cid:1)N, x , x , x (cid:2)
TheShapleyvalueofsupplierjistheaverageof x c n x c n n
her marginal contributions to coalitions that con-
(cid:3)(cid:1)1(cid:1)(cid:15)(cid:2)A1(cid:4)(cid:2)x(cid:2)(cid:13)x(cid:2)(cid:1)1(cid:4)(cid:13)(cid:2)N(cid:2)(cid:1)(cid:4)(cid:6)1(cid:2)(cid:4)1
sist of players ordered below her in all feasible c n
orderings. A supplier who has a measure n of
R
players ordered below her has a marginal contri- (cid:3)(cid:1)1(cid:1)(cid:15)(cid:2) ,
butionofm(j,n)ifthefirmisorderedbelowher N
(probability n/N), and 0 otherwise (probability
1(cid:4)n/N).Averagingoverallpossibleorderingsof where R (cid:3) A1(cid:4)(cid:2)x(cid:2)(cid:13)x(cid:2)(1(cid:4)(cid:13))N(cid:2)((cid:4)(cid:6)1) is the total
c n
theplayersandusing(18),weobtain revenue of the firm. Thus, the joint Shapley
(cid:2) valueofthesuppliers,Ns (N,x ,x ),equalsthe
(cid:6) (cid:7) x c n
1 N n fraction 1 (cid:4) (cid:15) of the revenue, and the firm
s(cid:1) (cid:9)N, x , x (cid:1)(cid:4)j(cid:2),x (cid:1)j(cid:2)(cid:10)(cid:3) m(cid:1)j,n(cid:2)dn receives the remaining fraction (cid:15), i.e.,
x c n n N N
0
(cid:1) (cid:3) (22) s (cid:1)N,x ,x (cid:2)(cid:3)(cid:15)A1(cid:4)(cid:2)x(cid:2)(cid:13)x(cid:2)(cid:1)1(cid:4)(cid:13)(cid:2)N(cid:2)(cid:1)(cid:4)(cid:6)1(cid:2)
x (cid:1)j(cid:2) (cid:1)1(cid:4)(cid:13)(cid:2)(cid:5) q c n c n
(cid:3)(cid:1)1(cid:1)(cid:15)(cid:2)A1(cid:4)(cid:2) n
x (cid:1)(cid:4)j(cid:2) (cid:3)(cid:15)R.
n
(cid:11)x(cid:2)(cid:13)x (cid:1)(cid:4)j(cid:2)(cid:2)(cid:1)1(cid:4)(cid:13)(cid:2)N(cid:2)(cid:1)(cid:4)(cid:6)1(cid:2)(cid:4)1,
c n This is a relatively simple rule for the division
of revenue between the firm and its suppliers.
where Second,thederivedparameter(cid:15)(cid:16)(cid:5)/((cid:5)(cid:6)(cid:2))
representsthebargainingpowerofthefirm;itis
(cid:5) increasing in (cid:5)and decreasing in (cid:2). A higher
(19) (cid:15)(cid:10)(cid:5)(cid:11)(cid:2). elasticity of substitution between intermediate
inputs,i.e.,ahigher(cid:5),raisesthefirm’sbargain-
We therefore obtain the following lemma (see ingpower,becauseitmakeseverysupplierless
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
x (j) in her noncontractible activities, all the final good, i.e., higher (cid:2), reduces the firm’s
n
VOL.97NO.3 ACEMOGLUETAL.:CONTRACTSANDTECHNOLOGYADOPTION 925
bargaining power, because, for any coalition, it investment in noncontractible activities and
reducesthemarginalcontributionofthefirmto thusunderinvestsintheseactivities.Second,as
thecoalition’spayoffasafractionofrevenue.18 discussedabove,multilateralbargainingdistorts
Furthermore, when (cid:5) is smaller, s(cid:1) [N, x , the perceived concavity of the private return
x c
x ((cid:4)j), x (j)] is more concave with respect to relativetothesocialreturn.Usingthefirst-order
n n
x (j), because greater complementary between condition of this problem and solving for the
n
the intermediate inputs implies that a given fixed point by substituting x (j) (cid:3) x yields a
n n
changeintherelativeemploymentoftwoinputs unique x :
n
has a larger impact on their relative marginal
products. The impact of (cid:5)on the concavity of (23) x (cid:3)x(cid:1) (cid:1)N, x (cid:2)(cid:10)(cid:9)(cid:5)(cid:1)1(cid:1)(cid:15)(cid:2)
n n c
s(cid:1) (cid:2)willplayanimportantroleinthefollowing
x
results. The parameter (cid:2), on the other hand, (cid:11)(cid:1)c (cid:2)(cid:4)1x(cid:2)(cid:13)A1(cid:4)(cid:2)N(cid:2)(cid:1)(cid:4)(cid:6)1(cid:2)(cid:4)1(cid:10)1/(cid:9)1(cid:4)(cid:2)(cid:1)1(cid:4)(cid:13)(cid:2)(cid:10).
x c
affects the concavity of revenue in output (see
(2))buthasnoeffectontheconcavityofs(cid:1) with Notethatx(cid:1) (N,x )isincreasinginx ;sincethe
x n c c
respect to x (j), because with a continuum of marginal productivity of an activity rises with
n
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 (cid:5). Mathematically, this follows
To characterize an SSPE, we first derive the from the fact that (cid:5)(1 (cid:4) (cid:15)) (cid:3) (cid:5)(cid:2)/((cid:5)(cid:6) (cid:2)) is
incentive compatibility constraint using (14) increasingin(cid:5).Theeconomicsofthisrelation-
and (20): shipistheoutcomeoftwoopposingforces.The
(cid:1) (cid:3) share of the suppliers in revenue, (1 (cid:4) (cid:15)), is
x (cid:1)j(cid:2) (cid:1)1(cid:4)(cid:13)(cid:2)(cid:5) decreasing in (cid:5), because greater substitution
xn(cid:3)argmax(cid:1)1(cid:1)(cid:15)(cid:2)A1(cid:4)(cid:2) nx between the intermediate inputs reduces the
xn(cid:1)j(cid:2) n suppliers’ ex post bargaining power. But a
greater level of (cid:5)also reduces the concavity of
(cid:11)xc(cid:2)(cid:13)xn(cid:2)(cid:1)1(cid:4)(cid:13)(cid:2)N(cid:2)(cid:1)(cid:4)(cid:6)1(cid:2)(cid:4)1(cid:1)cx(cid:1)1(cid:1)(cid:13)(cid:2)xn(cid:1)j(cid:2). s(cid:1) (cid:2)inx ,increasingthemarginalrewardfrom
x n
investing further in noncontractible activities.
Relativetotheproducer’sfirst-bestchoicechar- Because the latter effect dominates, x is in-
n
acterized above, we see two differences. First, creasing in (cid:5).
theterm(1(cid:4)(cid:15))impliesthatthesupplierisnot Now, using (21), (22), and (23), the firm’s
thefullresidualclaimantofthereturnfromher optimization problem (16) can be expressed as
(24) maxA1(cid:4)(cid:2)(cid:9)x(cid:13)x(cid:1) (cid:1)N,x (cid:2)1(cid:4)(cid:13)(cid:10)(cid:2)N(cid:2)(cid:1)(cid:4)(cid:6)1(cid:2)
c n c
18Toclarifytheeffectsof(cid:5)and(cid:2)on(cid:15),letusreturnto N,xc
thederivationinthetextandnotethatthemarginalcontri-
bution of the firm to a coalition of n suppliers, each one (cid:4)c N(cid:13)x(cid:4)c N(1(cid:4)(cid:13))x(cid:1) (N,x )
x c x n c
investingx incontractibleactivitiesandx innoncontract-
c n
ibleactivities,canbeexpressedas (cid:4)C(cid:1)N(cid:2)(cid:1)w N,
(cid:6) (cid:7) (cid:6) (cid:7) 0
n (cid:2)/(cid:5) n (cid:2)/(cid:5)
m (cid:1)n(cid:2)(cid:3)A1(cid:4)(cid:2)N(cid:2)(cid:1)(cid:4)(cid:6)1(cid:2)x(cid:2)(cid:13)x(cid:2)(cid:1)1(cid:4)(cid:13)(cid:2) (cid:3) R,
q c n N N
19TheeffectofNonx isambiguous,sinceinvestment
owfhethreeRre(cid:3)venAu1e(cid:4)((cid:2)wNh(cid:2)e((cid:4)n(cid:6)n1)x(cid:3)c(cid:2)(cid:13)Nxn(cid:2))(.1(cid:4)T(cid:13)h)eisexthpereesqsiuoinlib(rniu/Nm)(cid:2)l/e(cid:5)veisl innolnoognycownhtreancti(cid:2)b(l(cid:4)ea(cid:6)ctiv1i)ti(cid:14)ensd1ecalnindesinwcriethastehsewleivthelNofwtehcehn-
decreasing in (cid:2)and increasing in (cid:5)for all n (cid:14) N. The (cid:2)((cid:4)(cid:6) 1) (cid:5) 1. This is because an increase in N has two
parameter(cid:15)isgivenbytheaverageofthe(n/N)(cid:2)/(cid:5)terms, opposite effects on a supplier’s incentives to invest: a
(cid:2) (cid:6) (cid:7) greaternumberofinputsincreasesthemarginalproductof
1 N n (cid:2)/(cid:5) (cid:5) investment due to the “love for variety” embodied in the
(cid:15)(cid:3)N N dn(cid:3)(cid:5)(cid:11)(cid:2), technology,butatthesametime,thebargainingshareofa
0 supplier,(1(cid:4)(cid:15))/N,declineswithN.Forlargevaluesof(cid:4),
theformereffectdominates,whileforsmallvaluesof(cid:4),the
andisalsodecreasingin(cid:2)andincreasingin(cid:5). latterdominates.