Table Of ContentEconometricaSupplementaryMaterial
SUPPLEMENTTO“ORGANIZINGTHEGLOBALVALUECHAIN”
(Econometrica,Vol.81,No.6,November2013,2127–2204)
BYPOLANTRÀSANDDAVINCHOR
This supplement documents several detailed proofs from Sections 2 and 3 of the
main text of the paper, that were omitted due to space constraints. It also contains
TablesS.I–S.XandFiguresS.2andS.3,whichwerementionedinthemaintextofthe
paper.
A. OPTIMALDIVISIONOFSURPLUS:SUFFICIENTCONDITION
INTHISSECTION,weshowthatthefunctionβ∗(j)inequation(15)ofthemain
text,characterizingtheoptimaldivisionofsurplusatstagej,indeedsatisfiesa
sufficientconditionfortheassociatedprofit-maximizationproblemofthefirm.
Ourapproachbuildsonrecastingthisasadynamicprogrammingproblem.
Rememberthatinthemaintext,wereducedtheproblemtothatoffinding
afunctionvthatmaximizes
(cid:2)
1(cid:3) (cid:4)
(A.1) π (v)=κ 1−v(cid:3)(j)(1−α)/α v(cid:3)(j)v(j)(ρ−α)/(α(1−ρ))dj(cid:7)
F
0
Asareminder,κ≡Aρ(1−ρ)(ρ−α)/(α(1−ρ))(ρ)ρ/(1−ρ) isapositiveconstant.
α 1−α c
DefinethevaluefunctionV(j(cid:8)v)associatedwiththisproblemas
(cid:2)
1(cid:3) (cid:4)
V(j(cid:8)v)=κsup 1−v(cid:3)(k)(1−α)/α v(cid:3)(k)v(k)(ρ−α)/(α(1−ρ))dk(cid:8)
v[(cid:3)j(cid:8)1] j
wherevintheargumentofthevaluefunctionsatisfiesv=v(j).TheHamilton–
Jacobi–Bellmanequationassociatedwiththisproblemis
(cid:5) (cid:3) (cid:3) (cid:4) (cid:4) (cid:6)
(A.2) −V (j(cid:8)v)=sup κ 1− v(cid:3) (1−α)/α v(cid:3)v(ρ−α)/(α(1−ρ))+V (j(cid:8)v)v(cid:3) (cid:8)
j v
v(cid:3)
with boundary condition V(1(cid:8)v)=0. The right-hand-side problem is strictly
concaveanddeliversauniquesolution:
(cid:7) (cid:8)
(cid:3) (cid:4)
α 1+ Vv(j(cid:8)v) = v(cid:3) (1−α)/α(cid:8)
κv(ρ−α)/(α(1−ρ))
whichwecanplugbackinto(A.2).Aftersomesimplifications,wehave
(cid:7) (cid:8)
α α/(1−α)
(A.3) −V (j(cid:8)v)=(1−α)
j κ
(cid:3) (cid:4)
× V (j(cid:8)v)+κv(ρ−α)/(α(1−ρ)) 1/(1−α)v(α−ρ)/((1−ρ)(1−α))(cid:7)
v
©2013TheEconometricSociety DOI:10.3982/ECTA10813
2 P.ANTRÀSANDD.CHOR
From well-known results (see, for instance, Bertsekas (2005, Proposi-
tion3.2.1),orLiberzon(2011,Section5.1.4)),ifthevaluefunctionassociated
with the solution v(cid:3) that satisfies the necessary conditions for optimality also
satisfiestheHJBequation(A.3),thenthatwouldbesufficienttoconcludethat
v(cid:3) (andthusβ∗)deliversamaximum.
Toproveso,letusbeginbydefining
(cid:2)
(cid:3) (cid:4) 1(cid:3) (cid:3) (cid:4) (cid:4) (cid:3) (cid:4)
J j(cid:8)v(cid:8)v(cid:3) =κ 1− v(cid:3)(k) (1−α)/α v(cid:3)(k) v(k) (ρ−α)/(α(1−ρ))dk(cid:8)
[j(cid:8)1]
j
whichisthefunctionalinthemaintext,butwiththelowerlimitoftheintegral
startingatj∈[0(cid:8)1].Theoptimizationproblemforthisfunctionalisanalogous
totheprobleminourBenchmarkModelexceptforthelowerlimitoftheinte-
gral.AsshownintheAppendixofthemaintext,wemusthave
(cid:7) (cid:8)
(1−α)C (1−ρ)/(1−α)
v(k)= 1(k−C ) (cid:8) and
1−ρ 2
(cid:7) (cid:8)
(1−α)C (α−ρ)/(1−α)
v(cid:3)(k)= 1(k−C ) C (cid:7)
1−ρ 2 1
Here, C and C are the associated constants of integration. The key dif-
1 2
ference here from our Benchmark Model is in the initial condition, which
is now v(j)=v, while the transversality condition continues to be given by
v(cid:3)(1)(1−α)/α=α.Usingthesetwoconditions,wefindthatC andC areimplic-
1 2
itlydefinedby
(cid:7) (cid:8)
(1−α)C (1−ρ)/(1−α)
(A.4) 1(j−C ) =v(cid:8) and
1−ρ 2
1−ρ v
(A.5) (1−C )(α−ρ)/(1−α) =αα/(1−α)(cid:7)
1−α 2 (j−C )(1−ρ)/(1−α)
2
Note that C and C are therefore functions of j and v. Moreover, since we
1 2
must have C < j <1 in order for v to be greater than 0, the constants of
2
integrationC andC arecontinuouslydifferentiableinj andv.
1 2
ThevaluefunctionV(j(cid:8)v)isthen
(cid:2)
1(cid:3) (cid:3) (cid:4) (cid:4) (cid:3) (cid:4)
V(j(cid:8)v)=κsup 1− v(cid:3)(k) (1−α)/α v(cid:3)(k) v(k) (ρ−α)/(α(1−ρ))dk
v[(cid:3)j(cid:8)1] j
(cid:2) (cid:7) (cid:7)(cid:7) (cid:8) (cid:8) (cid:8)
1 (1−α)C (α−ρ)/(1−α) (1−α)/α
=κ 1− 1(k−C ) C
1−ρ 2 1
j
ORGANIZINGTHEGLOBALVALUECHAIN 3
(cid:7) (cid:8)
(1−α)C (α−ρ)/(1−α)
× 1(k−C )
1−ρ 2
(cid:7) (cid:8)
(1−α)C (ρ−α)/(α(1−α))
×C 1(k−C ) dk(cid:8)
1 1−ρ 2
whichcanbesimplifiedto
(cid:2) (cid:7) (cid:7) (cid:8) (cid:8)
1 1−ρ (1−α)C ρ/α
V(j(cid:8)v)=κ 1 (k−C )(ρ−α)/α−C1/α dk(cid:7)
1−α 1−ρ 2 1
j
Evaluatingtheintegral,wehave
(cid:9) (cid:7) (cid:8)
1−ρα (1−α)C ρ/α(cid:10) (cid:11)
V(j(cid:8)v)=κ 1 (1−C )ρ/α−(j−C )ρ/α
1−αρ 1−ρ 2 2
(cid:12)
−C1/α(1−j) (cid:8)
1
which,usingequations(A.4)and(A.5),canbereducedto
(cid:9)
1−ρα(cid:10) (cid:11)
(A.6) V(j(cid:8)v)=κ Cρ(1−α)/(α(ρ−α))αρ/(α−ρ)−vρ(1−α)/(α(1−ρ))
1−αρ 1
(cid:12)
−C1/α(1−j) (cid:7)
1
ByeliminatingC fromequations(A.4)and(A.5),onecanseethatC itselfis
2 1
givenimplicitlyby
(1−α)C
(A.7) 1(1−j)+v(1−α)/(1−ρ)=αα/(α−ρ)(C )(1−α)/(ρ−α)(cid:7)
1−ρ 1
Ourfinalstepistoshowthatthevaluefunctiondefinedbyequations(A.6)
and (A.7) indeed satisfies the Hamilton–Jacobi–Bellman equation in (A.3).
Implicitdifferentiationof(A.7)produces
dC C
1 =− 1 (cid:8) and
dj αα/(α−ρ)1−ρC(1−ρ)/(ρ−α)−(1−j)
ρ−α 1
dC v(ρ−α)/(1−ρ)
1 = (cid:7)
dv αα/(α−ρ)1−ρC(1−ρ)/(ρ−α)−(1−j)
ρ−α 1
4 P.ANTRÀSANDD.CHOR
Usingtheseexpressionsto(totally)differentiateV(j(cid:8)v)in(A.6)withrespect
toj andv,andsimplifying,weobtain
1−α
V (j(cid:8)v)=−κ C1/α(cid:8) and
j α 1
(cid:7) (cid:8)
1
V (j(cid:8)v)=κ C(1−α)/αv(ρ−α)/(1−ρ)−v(ρ−α)/(α(1−ρ)) (cid:7)
v α 1
PluggingtheaboveexpressionsforV (j(cid:8)v)andV (j(cid:8)v)into(A.3),itisstraight-
j v
forward to verify that the Hamilton–Jacobi–Bellman equation in (A.3) is in-
deedsatisfied.Thisconfirmsthatthefunctionβ∗(j)satisfiesthesufficientcon-
ditionforamaximum.Note,finally,thatbecausewehaveonlyonecandidate
solution for a maximum that satisfies the Euler–Lagrange equation, we can
comfortablystatethatβ∗(j)istheglobalmaximizerwithinthesetofpiecewise
continuouslydifferentiablereal-valuedfunctions.
B. OPTIMALDIVISIONOFSURPLUS:CONSTRAINEDPROBLEM
Insolvingfortheoptimaldivisionofsurplusateachstageinthemainpaper,
we have not constrained the optimal bargaining share β∗(m) to be nonnega-
tive or no larger than 1. The latter assumption is without loss of generality,
since the solution to the problem satisfies β∗(m)=1−αm(α−ρ)/α ≤1. Note,
however, that in the sequential complements case (ρ > α), when m is sufi-
ciently small we necessarily have β∗(m) < 0. As argued in the main text, a
negative β∗(m) can be justified by appealing to the fact that the firm might
finditoptimaltocompensatecertainsupplierswithapayoffthatexceedstheir
marginal contribution. Still, it is worth exploring how the optimal division of
surplus is affected by imposing the constraint β∗(m)≥0. It might seem nat-
ural that the modified solution in the complements case would be given by
β∗(m)=max{1−αm(α−ρ)/α(cid:8)0}, but we will show below that this would be an
incorrectguess.
Theproblemthatweseektosolvecanbewrittenas
(cid:2)
1(cid:3) (cid:4)
max 1−v(cid:3)(j)(1−α)/α v(cid:3)(j)v(j)(ρ−α)/(α(1−ρ))dj
{v(cid:3)(j)}j∈[0(cid:8)1] 0
s.t. 0≤v(cid:3)(j)≤1(cid:8)
with initial condition v(0)=0. Remember that this formulation follows from
defining
(cid:2)
j(cid:3) (cid:4)
v(j)≡ 1−β(k) α/(1−α)dk(cid:8)
0
whichinturnimpliesβ(j)=1−v(cid:3)(j)(1−α)/α.
ORGANIZINGTHEGLOBALVALUECHAIN 5
Observe first that in the sequential substitutes case (ρ<α), the solution to
theunconstrainedproblemdoesnotviolatetheconstraint0≤v(cid:3)(j)≤1,since
0≤αj(α−ρ)/α <1. Thus, the solution obtained from solving the unconstrained
problemisnecessarilyalsothatwhichyieldsthemaximumfortheconstrained
problem.
We therefore concentrate below on the sequential complements case
(ρ>α). As mentioned above for the unconstrained problem, we necessar-
ily have that v(cid:3)(j) > 0 (or β(j) < 1) for all j > 0. As we will show below,
the same will be true for the solution to the constrained problem (i.e., when
imposing v(cid:3)(j) ≤ 1), and thus, for the time being, we ignore the constraint
v(cid:3)(j)≥0.
To solve the constrained problem, it is simplest to write down the Hamilto-
nianassociatedwiththeproblem,where,forsimplicity,wedroptheargument
j anddefineu≡v(cid:3):
(cid:3) (cid:4)
H= 1−u(1−α)/α uv(ρ−α)/(α(1−ρ))+λu+θ(1−u)(cid:7)
Here, λ is the costate variable and θ is the multiplier associated with the
constraint u ≤ 1. The first-order conditions associated with this problem
are
(cid:7) (cid:8)
1
(B.1) H = 1− u(1−α)/α v(ρ−α)/(α(1−ρ))+λ−θ=0(cid:8) and
u α
(cid:3) (cid:4) ρ−α
(B.2) H = 1−u(1−α)/α u v(ρ−α)/(α(1−ρ))−1=−λ(cid:3)(cid:7)
v α(1−ρ)
Combiningthesetoeliminateλandλ(cid:3),wehave
1−α ρ−α
u1/α v(ρ−α)/(α(1−ρ))−1
α2 (1−ρ)
1−α
=− u(1−α)/α−1u(cid:3)v(ρ−α)/(α(1−ρ))−θ(cid:3)(cid:7)
α2
Note that when the constraint u ≤ 1 (or β ≥ 0) does not bind, we have
θ(cid:3)=θ=0,andthisreducesto
(cid:13) (cid:14)
ρ−αu2
v(ρ−α)/(α(1−ρ))u(1−α)/α−1 +u(cid:3) =0(cid:8)
1−ρ v
which is identical to equation (14) in the main text. For reasons analogous to
thoseintheunconstrainedproblem,theprofit-maximizingfunctionumustset
theterminthesquarebracketsto0,whichimplies
u=C v−(ρ−α)/(1−ρ)(cid:8)
1
6 P.ANTRÀSANDD.CHOR
where C is a strictly positive constant. Note that for v sufficiently small (in
1
particular,intheneighborhoodofj=0),andgivenρ>α,wenecessarilyhave
thatu>1,sothattheconstraintu≤1willhavetobind,implyingθ>0.Notice
then thatequation(B.2)implies that λ(cid:3)=0,which, inlightofequation (B.1),
in turn implies that θ is a monotonically decreasing function of j as long as
theconstraintbinds.Asaresult,iftheconstraintbindsatsomejˆ∈(0(cid:8)1),then
θ>0forallj<jˆandsotheconstraintmustbindaswellforallj<jˆ,implying
u(j)=1forallj≤jˆ.
Thesolutionfortheoptimalshareforj>jˆthussolvesthedifferentialequa-
tion
(B.3) v(cid:3)=C v−(ρ−α)/(1−ρ)(cid:8)
1
withtheboundaryconditionv(cid:3)(jˆ)=1andthetransversalitycondition v(cid:3)(1)=
αα/(1−α).Asintheunconstrainedproblem,equation(B.3)implies
(cid:7) (cid:8)
(1−α)C (1−ρ)/(1−α)
(B.4) v(j)= 1(j−C ) (cid:8) and
1−ρ 2
(cid:7) (cid:8)
(1−α)C (α−ρ)/(1−α)
(B.5) v(cid:3)(j)=C 1(j−C ) (cid:8)
1 1−ρ 2
butnowC andC followfromsolving
1 2
(cid:7) (cid:8)
(1−α)C (α−ρ)/(1−α)
C 1(jˆ−C ) =1(cid:8) and
1 1−ρ 2
(cid:7) (cid:8)
(1−α)C (α−ρ)/(1−α)
C 1(1−C ) =αα/(1−α)(cid:7)
1 1−ρ 2
Thissystemyields
(cid:7) (cid:7) (cid:8)(cid:8)
1−α 1−jˆ (ρ−α)/(1−ρ)
(B.6) C =αα/(1−ρ) (cid:8) and
1 1−ρ 1−αα/(ρ−α)
jˆ−αα/(ρ−α)
(B.7) C = (cid:7)
2 1−αα/(ρ−α)
Note,however,thatatjˆ,wemustalsohave
(cid:2)
jˆ
v(jˆ)≡ udk=jˆ(cid:7)
0
Plugging(B.6)and(B.7)into(B.4)thenyields
(1−α)αα/(ρ−α)
jˆ= (cid:7)
(1−ρ)(1−αα/(ρ−α))+(1−α)αα/(ρ−α)
ORGANIZINGTHEGLOBALVALUECHAIN 7
Finally, substituting this expression for jˆ, together with (B.6) and (B.7), into
(B.5)produces
(cid:7) (cid:8)
(ρ−α)αα/(ρ−α) (α−ρ)/(1−α)
v(cid:3)(j)=αα/(1−α) j−(1−j) (cid:8)
1−ρ
which,givenβ(j)≡1−v(cid:3)(j)(1−α)/α,finallyimplies
(cid:7) (cid:8)
(ρ−α)αα/(ρ−α) (α−ρ)/α
β∗(j)=1−α j−(1−j) for j>jˆ(cid:7)
1−ρ
Notethatwecanthensummarizethesolutiontotheconstrainedproblemas
⎧
⎪1−αj(α−ρ)/α(cid:8) ifα>ρ(cid:8)
⎪
⎨ (cid:9) (cid:7) (cid:8) (cid:12)
β∗(j)=⎪max 1−α j−(1−j)(ρ−α)αα/(ρ−α) (α−ρ)/α(cid:8)0 (cid:8)
⎪⎩ 1−ρ
ifρ>α(cid:8)
forallj∈[0(cid:8)1](cid:7)
In the accompanying Figure S.1, we plot the abovesolution (thick curve)and
compare it to that which solves the unconstrained problem (thin curve). Ob-
viously, in the sequential substitutes case, the two solutions coincide. Inter-
estingly, for all j >jˆ, we find that the optimal bargaining share received by
the firm in the unconstrained problem is higher than its bargaining share un-
dertheconstrainedproblem.Intuitively,intheconstrainedproblem,thefirm
wouldhavepreferredtoincentivizeupstreamsuppliersbyofferingthemapay-
off exceeding their marginal contribution, but when it is not able to do so, it
attempts to alleviate upstream investment inefficiencies by offering their full
FIGURES.1.—Profit-maximizingdivisionofsurplusforstagem.
8 P.ANTRÀSANDD.CHOR
marginalcontributiontoalargermeasureofsuppliersandbyofferingahigher
shareoftheirmarginalcontributiontotheremainingsuppliers.
Despite these differences, notice that the optimal bargaining share β∗(j)
continues to be (weakly) increasing in the sequential complements case and
strictly decreasing in the sequential substitutes case. Hence, the statement in
Proposition 1 of the paper remains valid except for the fact that β∗(j) is now
onlyweaklyincreasinginmwhenρ>α.
C. THEBENCHMARKMODELWITHEXANTETRANSFERS
In Section 3.1.1 of the paper, we argue that introducing ex ante lump-sum
transfersbetweenthefirmandthesuppliershasverylittleimpactonourmain
results. Because these ex ante transfers have no effect on ex post decisions
made after agents are locked in by the contracts, investment levels continue
to be characterized by equation (10) in our main text. The key implication of
introducing ex ante transfers is that the objective function of the firm is no
longertheirexpostpayoff(asinequation(11)ofthemainpaper),butrather
thejointsurpluscreatedalongthevaluechain,or
(cid:7)(cid:2) (cid:8) (cid:2)
1 ρ/α 1
(C.1) π =A1−ρθρ x(j)αdj − cx(j)dj(cid:7)
T
0 0
This might reflect, as in Antràs (2003) and Antràs and Helpman (2004), the
fact that the firm has full bargaining power ex ante, in the sense that it can
make take-it-or-leave-it offers to suppliers that include an initial transfer to
the firm. With a perfectly elastic supply of potential suppliers, each with an
ex ante outside option equal to 0, these ex ante transfers would thus be set
in a way that allows the firm to appropriate all the surplus created along the
valuechain.Alternatively,evenwhenboththefirmandsuppliershavesomeex
ante bargaining power (perhaps because the number of potential suppliers is
limited),thefactthatagentshaveaccesstoameanstotransferutilityexante
inadistortionarymannerimplies,bytheCoasetheorem,thattheorganization
ofproductionalongthevaluechain(i.e.,whichstagesgetintegratedandwhich
getoutsourced)willbedecidedefficiently,namely,inajoint-profit-maximizing
manner.
Notefromequation(6)inthemaintextthatcx(j)=α(1−β(j))r(cid:3)(j)forall
j∈[0(cid:8)1].Pluggingthisinto(C.1),wehave
(cid:2) (cid:2)
1(cid:3) (cid:4) 1(cid:3) (cid:3) (cid:4)(cid:4)
π =r(1)−α 1−β(j) r(cid:3)(j)dj= 1−α 1−β(j) r(cid:3)(j)dj(cid:7)
T
0 0
After substituting in the expressions from equations (8) and (9) of the main
paper,wefindthat
(cid:2)
1(cid:3) (cid:3) (cid:4)(cid:4)(cid:3) (cid:4)
π =κ 1−α 1−β(j) 1−β(j) α/(1−α)
T
0
ORGANIZINGTHEGLOBALVALUECHAIN 9
(cid:13)(cid:2) (cid:14)
j(cid:3) (cid:4) (ρ−α)/(α(1−ρ))
× 1−β(k) α/(1−α)dk dj(cid:8)
0
whereκ≡Aρ(1−ρ)(ρ−α)/(α(1−ρ))(ρθ)ρ/(1−ρ) isapositiveconstant.
α 1−α c
Defining
(cid:2)
j(cid:3) (cid:4)
v(j)≡ 1−β(k) α/(1−α)dk(cid:8)
0
wecanwriteπ as
T
(cid:2)
1(cid:3) (cid:4) (cid:10) (cid:11)
(C.2) π =κ 1−αv(cid:3)(j)(1−α)/α v(cid:3)(j) v(j) (ρ−α)/(α(1−ρ))dj(cid:7)
T
0
The Euler–Lagrange equation associated with choosing the function v(cid:3)(j)
from the set of piecewise continuously differentiable real-valued functions to
maximize (C.2) can be derived in a manner analogous to the case without ex
antetransfers.ThisEuler–Lagrangeequationreducesto
(cid:13) (cid:14)
1−α ρ−αv(cid:3)(j)2
v(j)(ρ−α)/(α(1−ρ))v(cid:3)(j)(1−α)/α−1 +v(cid:3)(cid:3) =0(cid:8)
α 1−ρ v(j)
which,asinthecasewithoutexantetransfers,implies
(cid:7) (cid:8)
(1−α)C (1−ρ)/(1−α)
v(j)= 1(j−C ) (cid:8) and
1−ρ 2
(cid:7) (cid:8)
(1−α)C (α−ρ)/(1−α)
v(cid:3)(j)=C 1(j−C ) (cid:8)
1 1−ρ 2
with initial condition v(0)=0. The main difference relative to the case with-
out ex ante transfersis that the transversality condition is now v(cid:3)(1)(1−α)/α=1
(insteadofv(cid:3)(1)(1−α)/α=α),whichimpliesthattheoptimaldivisionofsurplus
isnowgiveninsteadby
β∗(j)=1−v(cid:3)(j)(1−α)/α=1−j(α−ρ)/α(cid:7)
T
Notethattheslopeofβ∗(j)withrespecttoj continuestobecruciallyshaped
T
by whether ρ is higher or lower than α, just as in our Benchmark Model. It
followsthenthatProposition1,whichwereproducebelow,continuestohold
inthesetupwithexantetransfers.
PROPOSITION C.1: The (unconstrained) optimal bargaining share β∗(j) is a
T
weakly increasing function of j in the complements case (ρ>α), while it is a
weaklydecreasingfunctionofj inthesubstitutescase(ρ<α).
10 P.ANTRÀSANDD.CHOR
In sum, Proposition C.1 confirms that whether the incentive for the firm
to retain a larger surplus share increases or decreases along the value chain
continues to crucially depend on the relative size of the parameters ρ and α,
whichweviewasthecentralresultofourpaper.
The key difference from our Benchmark Model without ex ante transfers
relatestotheleveloftheshareβ∗(j).Inparticular,notethatinthesequential
T
complementscase(ρ>α),wenecessarilyhaveβ∗(j)≤0forallj∈[0(cid:8)1],and
thus the firm finds it optimal to outsource all production stages, as discussed
inthemaintext.Inthesequentialsubstitutescase(α>ρ),wehaveβ∗(0)=0
and β∗(1)=1, which necessarily implies that the most upstream stages will
necessarilybeintegrated,whilethemostdownstreamstageswillnecessarilybe
outsourced. As a result, the cutoff stage separating the upstream integrated
stages from the downstream outsourced stages necessarily lies strictly in the
interiorof(0(cid:8)1).
D. LINKAGESACROSSBARGAININGROUNDS
In this section, we include the details related to the variant of our model
outlined in Section 3.1.2, in which we allow suppliers to internalize the effect
of their investment levels and their negotiations with the firm on the subse-
quent negotiations between the firm and downstream suppliers. As argued in
thepaper,itnowbecomesimportanttospecifypreciselytheimplicationsofan
(off-the-equilibrium path) decision by a supplier to refuse to deliver its input
to the firm. The simplest case to study is one in which, once the production
processincorporatesanincompatibleinput(say,becauseasupplierrefusedto
trade with the firm), all downstream inputs are then necessarily incompatible
as well, and thus their marginal product is zero and firm revenue remains at
r(m) ifthedeviationhappenedatstagem.(Wewillbrieflydiscussalternative
assumptionsbelow.)
Forreasonsthatwillbecomeapparent,itisnecessarytodevelopourresults
withinadiscrete-playerversionofthegamebetweenthefirmandthesuppliers,
inwhicheachofM>0supplierscontrolsameasure1/M ofproductionstages.
WewilllaterrunthelimitasM →∞tocompareourresultswiththoseinthe
Benchmark Model in our paper. Assuming that each supplier sets a common
investmentlevelforalltheproductionstagesunderitscontrol(rememberthat
leavingasidethesequentialityofstages,theproductionfunctionissymmetric
ininvestments),revenuegenerateduptosupplierK<M isgivenby
(cid:19) (cid:21)
(cid:20)K 1 ρ/α
R(K)=A1−ρθρ X(k)α (cid:8)
M
k=1
if all the suppliers upstream of K have delivered compatible inputs before
supplierK makesitsowninvestmentdecision(thusrespectingthenaturalse-
quencing of the stages). We use uppercase letters to denote variables in the
Description:SUPPLEMENT TO “ORGANIZING THE GLOBAL VALUE CHAIN”. (Econometrica, Vol. 81, No. 6, November 2013, 2127–2204). BY POL ANTRÀS AND