Coordinating Supply Chains
Robert Akerlof and Richard Holden*†April 20, 2023

Abstract
Many complex final goods require a large number of inputs to come together in a timely and efficient manner for production to be successful. Notable examples include lithography machines used to make semiconductors,
airplanes, and lasers. We build a model to analyze this coordination problem
and show how coordination can be achieved. There is a manufacturer endowed with capital who needs an input from each of n suppliers to produce a
final good. The manufacturer may pay a markup to overcome supplier reluctance and achieve coordination. Coordination can also be achieved throughintegration, but integration inflates costs due to a lack of congruence between
manufacturer and supplier. We model integration and the associated cost
inflation along the lines of Aghion and Tirole (1997). We derive sharp predictions about firm structure and apply our model to a number of applications
including International Trade and Industry Policy.
Keywords: Coordination, Supply Chains, Theory of the Firm.

* Akerlof: University of Warwick and CEPR. r.akerlof@warwick.ac.uk.

Holden: UNSW Busi-

ness School. richard.holden@unsw.edu.au†
Preliminary and incomplete. Please do not circulate without permission.

1

1

Introduction

A crucial but underappreciated feature of supply chains is the need for coordination. Bringing together multiple suppliers to create complementary components
of a product that fit together in a timely and efficient manner is, at its core, a matter of coordination. Indeed, without effective coordination a supply chain does
not even exist.
To get a sense of the importance of coordination in supply chains, consider thefollowing example from the semiconductor industry, which produces the chips
used in a vast range of electronic devices from smartphones and computers, tocars and household appliances, to missiles and aircraft.
Dutch company ASML is the world’s leading manufacturer of photolithography machines used to fabricate advanced semi-conductors. Indeed, it has 100%
market share in extreme ultraviolet (EUV) lithography machines, which are essential for the production of the most advanced chips used in mobile phones andother devices.
The essence of producing chips is to carve transistors into silicon. The smaller
the transistor, the more transistors can be placed onto a given size piece of silicon,
and the more powerful and energy efficient the chip is.
The first semiconductors were made with a microscope, visible light, and photoresists (a light-sensitive material that produces a pattern). But as transistors
became smaller and smaller, chip producers ran into a problem. The wavelengthof visible light is several hundred nonometers (depending on the color of light).
To continue to make transistors smaller, manufacturers moved to ultraviolet light,
with wavelengths as small as 193nm. To make more efficient chips, manufacturers
needed to make use of EUV, which has a wavelength of 13.5nm.1
1

The interested reader may wonder how 7nm, 5nm, and even 3nm transistors can be producedwith EUV light having a wavelength of 13.5nm. In 1873 Ernst Abbe showed that the smallest
resolvable distance between two features of a sample–the optical resolution of a microscope–is proportional to the wavelength of light used to illuminate the sample and inversely proportional to
the refractive index of the medium between the lens and the sample. Denoting the wavelength as
λ and NA as the numerical aperture given by the sine of the half angle multiplied by the refractive
index of the medium, Abbe showed that d = λ/2N A. This principle applies to any light-based2But as Miller (2022) notes: one “can’t simply buy an EUV lightbulb. Producing
enough EUV light requires pulverizing a small ball of tin with a laser.” ASML’s
search for a supplier of EUV light focused on Cymer, founded and based in San
Diego, CA. ASML announced that they would acquire Cymer in October 2012
and the deal closed in May 2013.2 Cymer engineers devised a method to propel
a 1/30,000 mm wide ball of tin through a vacuum at 200 miles per hour. They
then fire a laser once at the tin to warm it up, and then a second time turn it intoa plasma 40-50 times hotter than the surface of the sun. This is repeated 50,000
times per second, to produce the required EUV light.
The lasers required to perform this task didn’t exist, so Cymer approached
German precision tooling company Trumpf to develop such a laser. Trumpf’s accomplishment in doing so is a story in and of itself, involving specialized gases in
the laser chamber, and building a laser which itself had 457,329 individual components. Yet there was another problem to overcome. The tin droplets reflected
light, which, if they shined back into the laser would wreak havoc.
To solve this problem, ASML approached German lens manufacturer Zeiss.
The central issue was that EUV is hard to reflect. The 13.5nm wavelength of EUV
is more like an X-ray than visible light. Just like with X-rays, many materials
absorb EUV rather than reflecting it. So Zeiss developed mirrors made of one
hundred alternating layers of molybdenum and silicon. Each of these layers wasaround 2nm thick.
As Miller (2002) puts it “building such a mirror with nanoscale precision proved
almost impossible. Ultimately, Zeiss created mirrors that were the smoothest objects ever made, with impurities that were almost imperceptibly small. If the mirrors in an EUV system were scaled to the size of Germany, the company said, theirbiggest irregularities would be a tenth of a millimeter. To direct EUV light with
precision, they must be held perfectly still, requiring mechanics and sensors so
exact that Zeiss boasted they could be used to aim a laser to hit a golf ball as far
projection system, including semiconductor photolithography.
2
https://www.asml.com/en/news/press-releases/2013/
asml-completes-acquisition-of-cymer.

3

away as the moon.”
In all, ASML uses more than 4,700 individual suppliers that go into the $200
million ASML machine used by semiconductor foundries like Taiwan Semiconductor Manufacturing Corporation (TSMC) and Intel to produce advanced chips.
There are a number of salient features from the above discussion of ASML and
EUV photolithography. First, many of their suppliers had to develop entirely new
products for ASML’s purposes. Second, multiple components were essential for
ASML. Trumpf’s new laser and Cymer’s method for creating EUV light in industrial quantities would have been useless to ASML without Zeiss’s new mirrors.
Third, ASML bought Cymer but not Trumpf or Zeiss.3 Across its supply chain
it is integrated with about 15% of its suppliers, but contracts with the remaining
suppliers (Miller, 2022). Fourth, by being the first to successfully coordinate these
suppliers–to build the requisite supply chain–ASML has, at least for now, a complete monopoly in the market for its advanced machines. In fact, ASML’s stock
price has risen from $154 in January 2019 to $546 at the end of 2022, or more than350%.
We analyze a model of the supply-chain coordination problem. There is a Manufacturer (M) endowed with capital who needs an input from each of n suppliers
to produce a final good. M makes a take-it-leave it offer to each supplier consisting of a price for the input and a fraction of M’s capital that is payable if the
supplier produces and M refuses to buy. Suppliers decide whether to accept thiscontract and produce. Finally M decides whether to buy the inputs produced or
pay the penalty for not purchasing. If M buys all the inputs then M produces final
goods.
3

Our model provides a rationale for why ASML bought Cymer but not Trumpf or Zeiss. The
value of Cymer’s business was largely tied to the successful development of EUV lithography, and
ASML had made a large bet on that prospect as well. In the context of our model they had high“congruence” (the parameter α in section 3). By contrast Trumpf made lasers more many otherpurposes and the lens Zeiss developed was a small part of its business. The two other acquisitions
ASML made in this period were also companies whose products were strongly tied to the successof EUV lithography. Wijdeven Motion (acquired in September 2012) essentially determines thesize of a nanometer; and Hermes Microvision (acquired June 16, 2016) tests the functioning ofadvanced chips.

4

We then characterize the set of equilibria of this game, using Introspective Equilibrium as our solution concept (Kets and Sandroni (2021)). Introspective equilibrium is based upon level-k thinking. Each player has an exogenously-given“impulse” which determines how they react at level 0; at level k > 0, players reactaccording to the belief that other players are acting at level k − 1. Introspective
equilibrium is defined as the limit of this process as k → ∞.
We show that M uses its capital to provide insurance to the m suppliers who
are most reticent to produce. These suppliers are paid marginal cost, while theremaining suppliers receive a markup. We offer an explicit expression for themarkup as a function of marginal costs and impulses. If the size of the aggregate
markups exceeds the surplus generated from final good production then coordination is not possible.
Armed with this we consider the decision for M to make input i in-house or
buy it from a supplier. We use an approach in the spirit of Aghion and Tirole (1997)
where M and supplier i decide ex ante on the allocation of formal authority. If M
has formal authority then we describe them as being “integrated.”
In the general case with n suppliers we show that supplier i is more likely
to be integrated when the extent to which the cost increase under M-authority islower, or their impulse is lower. The basic tradeoff is that M can pay a markup
to overcome supplier reluctance and achieve coordination under non-integration.
Alternatively coordination can be achieved through integration, but this increases
costs through lack of congruence between manufacturer and supplier. The three
key factors mentioned above affect this tradeoff.
One important point of comparison to our paper is Property Rights Theory(Grossman and Hart (1986), Hart and Moore (1990)). Since hold-up problems are
not the focus of our analysis, we assume that investments are contractible. But
it is useful to point out that our model makes two starkly different predictions
than what PRT would within our framework. First, since investments are madeby suppliers, PRT would predict that suppliers should own manufacturers. In
our model it is only ever optimal for manufacturers to own suppliers, although
non-integration may certainly be optimal. Second, integration can only ever be5optimal in our model with two or more suppliers since, with only one supplier
there is nothing to coordinate.
Scholars of international trade have, since the 1980s, focused on what havebecome known as “Global Value Chains” (GVCs). This literature concerns thesize of trade flows within GVCs and how different countries and industries are
positioned within GVCs. For an excellent survey see Antràs and Chor (2022).
Within this literature it is now customary to distinguish between two broadtypes of value chains: “snakes” and “spiders”.4 Snakes involve a purely sequential production process where at each stage along the value chain a supplier produces an input which is then used by a different supplier at the next stage. This
proceeds stage-by-stage until a manufacturer produces a final good. Following
the earlier trade literature, beginning with Antràs (2003), models of snakes thatspeak to the organization of the production process typically take a PRT approach.
Sociologists such as Gereffi (1999) have long-emphasized the importance of
coordination in supply (or “commodity”) chains. They also distinguish between“Producer-driven commodity chains” (essentially spiders) and “Buyer-driven commodity chains” (essentially snakes). As Gereffi (1999) puts it: “Producer-driven
commodity chains are those in which large, usually transnational, manufacturers
play the central roles in coordinating production networks.” Notable examples include “automobiles, aircraft, computers, semiconductors and heavy machinery.”
Our paper focuses squarely on spiders as we are principally concerned withthe coordination-aspect of supply chains. We do not address sequential production processes, which are undoubtedly important in many industries, and we donot model the final goods market and thus cannot speak to issues driven by elasticities, which is the focus of much of the international trade literature. On theother hand, we put coordination of suppliers at the heart of the analysis. Indeed,
we take a complete contracts approach, but focus on the coordination problem.
This is consistent with our motivating example, and our results on which suppliers are integrated is consistent with the empirical evidence on “spiders” (see, forinstance, Gereffi (1999))
4

Baldwin and Venables (2013).

6

To get a more concrete sense of how incomplete-contracting models of snakes
works, consider the pioneering work of Antràs and Chor (2013). They build on theProperty Rights Theory (PRT), and follow a now venerable tradition of applying
PRT to international trade, which includes Antràs (2003), Antràs (2005), Antràsand Helpman (2004), Acemoglu et al. (2007), and Antràs and Helpman (2008).
Antràs and Chor (2013) analyze a model in which final goods production requires a continuum of production stages, each performed by a different supplier.
Each supplier makes a relationship-specific investment which affects compatibility with other components, but contracts cannot be written on compatibility.
Importantly, production is necessarily sequential so that one stage must be performed before the next stage can commence, and there is a specific technologicallydetermined order in which this takes place.
The authors show that if final-good demand is sufficiently elastic, integrationin production is determined by a cutoff in the production process. Early stages
are non-integrated and later stages are integrated. Conversely, when demand in
sufficiently inelastic early stages are integrated and later stages are not. This is a
striking result, and the authors find empirical support for it in data on the shareof intrafirm imports of total imports in U.S. data.5
Another strand of the international trade literature on value chains concerns“spiders”, where production is non-sequential. Inputs from many suppliers areassembled by a manufacturer to produce a final good. For instance, an aircraft
consists of many components (fuselage, wing assembly, engines, and so on) thatare assembled by a manufacturer such a Boeing or Airbus. An illuminating incompletecontracting model of spiders is offered by Chor and Ma (2021), and a completecontracting approach by Antras et al. (2017).
Antras et al. (2017) interpret international trade flows as the “legs” of a spider. Firms that assemble inputs from the legs decide from which countries theysource those inputs. A key ingredient in this framework is that countries differ in
5

Alfaro et al. (2019) consider an environment where choices of organizational form havespillovers along the value chain. In particular relationship-specific investments upstream affectthe incentives for specific investments by downstream suppliers. The authors show that the central results of Antràs and Chor (2013) go through in this richer setting.

7their ability to reduce marginal costs for the assembling firm, but also in the fixed
costs paid to important from a country. This implies that–in contrast to standard
models of exports–whether it is optimal to import from one country depends on
the other countries from which a firm is sourcing its inputs. That is, there is interdependence across suppliers. In fact, depending on elasticities of demand in
the final goods market, different input-source countries can be complements or
substitutes. The authors show how, despite this complexity, such a model can be
structurally estimated. They show focus on how a change in the benefits of sourcing from a country like China affects the sourcing decisions of US firms and how
this varies depending on firm productivity and the other countries from which
they were initially sourcing.
Chor and Ma (2021) analyze a state-of-the-art multi-country, multi-industry
model of sourcing inputs. Final good producers source a continuum of inputs
from different suppliers. Both suppliers and the headquarters make relationshipspecific investments, leading to the possibility of bilateral holdup. The key contribution of the paper is that for each input variety the firm chooses two things. One,
from which country to source the input; and two, whether to vertically integratewith the supplier. In this environment, integration can be the optimal organizational form because of hold-up between firms and suppliers. These integration
decisions can be made on a dyad-by-dyad basis since there are no relevant interactions among suppliers that affect integration decisions. The authors show how
these firm-level decisions can be aggregated into a gravity equation by industry
and organizational form. This has direct implications for the share of intra-firmtrade. The authors also provide an expression for the welfare gains from trade.
Finally, a number of papers have emphasized the importance of coordination
in investment decisions. Among these, Akerlof and Holden (2016) and Akerlof
and Holden (2019) focus on the role of coordination in investment decisions, as
does the literature on “catalytic finance” such as Corsetti et al. (2006) and Morrisand Shin (2006).
The remainder of the paper proceeds as follows. Section 2 states the model ofsupply-chain coordination, and characterizes its equilibrium. Section 3 considers
8the manufacturer’s make-or-buy decision and analyzes which suppliers will optimally be integrated. Section 4 provides some applications of our theory. Section 5
concludes by putting our “coordination-based theory of the firm”, analyzed here,
into context. Proofs not offered in the main text are contained in the appendix.

2

The Model

A manufacturer of widgets (M) is endowed with capital k. Selling widgets generates revenue R. There are n suppliers of production inputs. To make widgets, M
needs an input from each supplier. M has no production costs aside from the costof obtaining inputs. It costs supplier i (Si for short) ci to produce the input neededP
by M. The surplus from making widgets is strictly positive: S ≡ R − i ci > 0.
The contracting environment is one where M makes take-it-or-leave-it offers to
suppliers (and thus has all of the bargaining power). For reasons that will become
clear momentarily, it makes sense for M to make offers to suppliers before theydecide whether to produce. A contract between M and Si takes the form of a pricepi that M offers to pay for input i, plus a claim over a fraction θi of M’s capital if
P
Si produces the input and M fails to pay (with i θi ≤ 1). We assume that the
suppliers have limited liability, so in the event they are unable to deliver inputs, it
is not possible to penalize them.6
The timing of events is as follows:
1. M makes take-it-or-leave-it offers to each supplier i of the form (pi , θi ).
2. Suppliers simultaneously decide whether to accept the offers and produce
inputs.
3. For each supplier i who produces, M decides whether to buy their input or
pay the penalty θi k. If M buys all inputs, M produces widgets and generates
6

If liability was not limited, the problem of coordinating suppliers could be solved by offering
contracts where suppliers receive a penalty for failing to produce if and only if all other suppliers
sign a contract.

9

revenue R. Otherwise, M produces no widgets and generates zero revenue.
Notice that if M contracted with suppliers after they produce inputs—rather than
before—suppliers would receive zero for their inputs since M has all of the bargaining power. Suppliers would, consequently, choose not to produce in the firstplace. M solves this hold-up problem by contracting with suppliers ex ante rather
than ex post, as we have assumed.

2.1

Analysis

For ease of exposition, we will assume that suppliers accept M’s offer when otherwise indifferent, and both M and the suppliers produce when otherwise indifferent.
We can analyze the game by backward induction. At time 3, there are two
possibilities: either all suppliers produced inputs, in which case M can produce
widgets, or M is unable to produce widgets. In the former case, we can assume
that M will pay pi to each supplier and produce widgets (if not, M would haveP
offered different contracts at time 1). M’s payoff is R − i pi and Si ’s payoff ispi − ci . In the event that M cannot produce widgets, M obtains zero revenue andpays each supplier i that produced either pi or θi k (whichever is smaller). Noticethat the penalty θi k is only used if θi k ≤ pi . Thus, it makes sense to choose θiinitially so that θi k ≤ pi . Hence, we can assume that M pays the penalty rather
than pi .
Now, consider time 2. Suppliers are weakly better off if they accept M’s offer
since, if they accept, they can always choose not to produce (there is no penalty
for failing to do so). Thus, suppliers will always accept M’s offer. Let ai denote
Si ’s decision whether to produce. Based on our analysis of time 3, we conclude
that Si ’s payoff is:



0,


πSi (a1 , ...an ) = pi − ci ,θ k − ciif ai = 0
if a1 = a2 = ... = an = 1i10otherwiseIn order to get the suppliers to produce, M must set pi ≥ ci for all i, so let us focus
Pon this case. If k ≥ i ci , M can set θi = cki for all i; this makes ai = 1 a weakly
dominant strategy and ensures that all suppliers produce. In this case, moreover,P
M can set pi = ci , which gives M all of the surplus (R − i ci ).
P
If instead k <i ci , it is not possible to make ai = 1 a weakly dominant
strategy for all suppliers: θi k − ci < 0 for some i. In this case, the suppliers face acoordination game with multiple Nash equilibria: there is a Nash equilibrium inwhich all suppliers produce (a1 = a2 = ... = an = 1) and a Nash equilibrium inwhich no supplier produces (a1 = a2 = ... = an = 0).
In order to refine the set of time-2 equilibria, we will apply an equilibrium concept developed by Kets and Sandroni (2021) called introspective equilibrium. Introspective equilibrium is based upon level-k thinking (see Crawford et al. (2013)
for a review). Each player has an exogenously-given “impulse” which determines
how they react at level 0; at level k > 0, players react according to the belief that
other players are acting at level k − 1. Introspective equilibrium is defined as thelimit of this process as k → ∞.
As we will see, the introspective equilibrium depends critically upon players’
impulses, or level-0 thinking. We make the following assumption regarding suppliers’ impulses.
Assumption 1. Let x0i ∈ {0, 1} denote the impulse of supplier i: x0i = 1 means that
supplier i’s impulse is to produce and x0i = 0 means that supplier i’s impulse is to not
produce. We assume that x0i = 1 with probability γi . The x0i ’s are mutually independent.
x0i is private information of supplier i.
The parameter γi captures supplier i’s perceived willingness to produce. We
formally define introspective equilibrium in our context as follows.
Definition 1 (Introspective Equilibrium for Supply Chains). An introspective equilibrium (a∗1 , a∗2 , ..., a∗n ) is constructed as follows:
1. Supplier i’s choice at level k > 0, denoted xki , is obtained by letting each supplier11

best respond to the belief that other suppliers are at level k − 1:1, if E [(Π xk−1 )(p − c ) + (1 − Π xk−1 )(θ k − c )] ≥ 0i
j̸=i jii
j̸=i jiixki =0, otherwise.
2. An introspective equilibrium is the limit as k → ∞:a∗i = lim xki .
k→∞

The following lemma characterizes the introspective equilibria of the game.
Lemma 1. The introspective equilibrium is unique and:1. All suppliers produce in equilibrium if condition 1 holds for all i:pi Γ−i + θi k(1 − Γ−i ) ≥ ci ,where Γ−i =Q(1)

j̸=i γj .

2. No supplier produces in equilibrium otherwise, except in the following special casewhere no equilibrium exist: (i) there are exactly two suppliers and (ii) condition 1
holds for one supplier and is violated for the other.
Proof. First, consider supplier i’s leve1-1 choice (x1i ). At level 1, supplier i assumes
that other suppliers are at level 0. The expected return to supplier i from producing is pi Γ−i + θi k(1 − Γ−i ) (since Γ−i is the probability that all of the other suppliers
produce). Hence, supplier i produces at level 1 (x1i = 1) if and only if equation 1
holds. Now, consider supplier i’s level-2 choice (x2i ). Supplier i knows that othersuppliers produce at level 1 if and only if equation 1 holds for all j ̸= i. Hence,
supplier i produces at level 2 if and only if equation 1 holds for all j ̸= i.
Let us split our analysis of level 3 into two cases: (i) more than two suppliersand (ii) two suppliers. In case (i), supplier i produces at level 3 if and only ifequation 1 holds for all suppliers (including i). To see this, take any two suppliers
12besides i. These two suppliers both produce at level 2 if and only if equation 1
holds for all suppliers. From level 3, it is easy to iterate up. It is clear that, at any
level k ≥ 3, supplier i produces if and only if equation 1 holds for all suppliers.
This establishes the lemma for case (i).
Now consider case (ii). In the case where there are two suppliers only, supplieri produces at level 3 if and only if equation 1 holds for supplier i. The reason isthat supplier j produces at level 2 if and only if equation 1 holds for supplier i.
More generally, at odd levels, supplier i produces if and only if equation 1 holds
for supplier i and, at even levels, supplier i produces if and only if equation 1
holds for supplier j. Hence, for all levels k ≥ 1, both suppliers produce (neither
supplier produces) if and only if equation 1 holds (is violated) for both suppliers.
If equation 1 holds for one supplier and is violated for the other, there is cycling
and no convergence, so no equilibrium exists. This establishes the lemma for case(ii).
Suppose the manufacturer wishes to produce widgets rather than not. The
P
manufacturer’s problem is to minimize costs ( i pi ) subject to the incentive compatibility condition for of the suppliers (equation 1) and the incentive compatibility constraint of the manufacturer (θi k ≤ pi ). It clearly makes sense to choose
pi such that the suppliers’ incentive compatibility constraints bind, which implies−i ). Substituting for pi , we can rewrite the incentive compatibilthat pi = ci −θi k(1−ΓΓ−iity constraint for the manufacturer as θi k ≤ ci . We can thus rewrite the manufacturer’s problem as:
X ci − θi k(1 − Γ−i )
min
θi
Γ−i
i
subject to: θi k ≤ ci for all i
Without loss of generality, assume that Γ−1 ≤ Γ−2 ≤ ... ≤ Γ−n . This means that
it is hardest to convince supplier 1 to produce and easiest to convince supplier n.
Note that, with this indexing, supplier 1 also has the greatest impulse to produce:
γ1 ≥ γ2 ≥ ... ≥ γn . Intuitively, despite supplier 1’s high impulse, it is hard toconvince supplier 1 because of the low impulse of other suppliers.
13

Additionally, let us define m to be the number of suppliers who can be paidP
off using capital k. More precisely, let m be the integer such that m
i=1 ci ≤ k <Pm+1Pn
i=1 ci (or, in the event thati=1 ci ≤ k, let m = n). Note that there might besome capital remaining after paying off the first k suppliers that could be used
to partially pay off the m + 1-st supplier. For the sake of simplifying exposition,
we will assume that there is zero capital remaining after paying off the first m
suppliers.
Assumption 2. There is no capital remaining after paying off the first m suppliers:Pm
i=1 ci = k.
Proposition 1. If M produces widgets:
1. M fully insures the first m suppliers and does not insure other suppliers:• θi = cki for i ≤ m.
• θi = 0 for i > m.
2. M pays the first m suppliers marginal cost and pays a markup to other suppliers:• pi = ci for i ≤ m.
1 − Γ−ii= ci + ci ·• pi = Γc−i
for i > m.
Γ
| {z −i }markup paid to i

M produces widgets if the total markup does not exceed the surplus:
X
i>m

|

1 − Γ−i
≤S
Γ−i
{z
}

ci ·

total markup

Otherwise, M does not produce widgets and offers suppliers nothing for their inputs (pi =0 for all i).

143

Make or Buy

We now extend the model to consider whether it makes sense for M to make input
i in-house or buy the input from another firm. Our approach to the make-or-buy
problem is based on Aghion and Tirole (1997). As in Aghion and Tirole (1997),
M and Si decide ex ante on an allocation of formal authority. If M has formal
authority, we will say that M and Si are integrated; otherwise, we will say thatthey are separate firms. The party with formal authority has the right to maketwo decisions: (1) whether input i is produced and (2) which production methods
are used. The timing is similar to the baseline model:
1. M makes take-it-or-leave-it offers to each supplier of the form (pi , θi , Ai ),where Ai ∈ {M, Si } is an allocation of formal authority.
2. Suppliers decide whether to accept the offers.
3. Suppliers exert effort at generating cost-cutting production methods. The
party with authority then chooses among the production methods.
4. The parties with authority simultaneously decide whether to produce inputs.
5. For each input i that is produced, M decides whether to use the input andpay pi to Si , or pay a penalty θi k to Si . If M uses all of the inputs, M produces
widgets and generates revenue R. Otherwise, M produces no widgets andgenerates zero revenue.
In Section 3.1, we will carefully spell out what happens at time 3. As a preview,the main finding is that M-authority raises the cost of producing input i to (1 +βi )ci . In Section 3.2, we will show that, while M-authority increases production
costs, it also lowers the costs of coordination. This is the main tradeoff governing the
make-or-buy decision.

15

3.1

Cost cutting

Let us consider how production methods are chosen at time 3. We will focus on
input i and we will drop i subscripts to reduce notation.
There are P aspects of the production process and a production method forp)each aspect. Let mp denote the production method used for aspect p and let c(mPPdenote the associated cost. The overall cost of production is P1 Pp=1 c(mp ).
For each aspect of production, there is a default method with a cost cH and twocheaper methods (m1 and m2) that might be uncovered. One of the cheaper methods has an associated cost cL < cH and the other method has an associated costcL −∆ < cL . The cost of production is borne by the supplier, so the supplier prefers
the cheapest possible method. The manager receives a private benefit of negligible size from one of the two cheaper methods. The manager prefers whichever ofthe cheaper methods generates this negligible private benefit. The supplier’s preferred method is the same as the manager’s preferred method with probability α.
Parameter α captures the degree of preference congruence of M and the supplier.
The supplier uncovers the cheaper methods for process p with probability ηS >0 and the manufacturer uncovers the cheaper methods with probability ηM > 0.
The chance of uncovering cheaper methods is independent across players andprocesses. After methods have been uncovered, the party with formal authority
chooses among the production methods they know or, alternatively, delegates the
choice to the other party.
Analysis
Let us now examine how the allocation of authority affects the cost of production.
M-authority. If the manager has formal authority, they choose the method themselves whenever they uncover the cheap methods and they delegate the choice tothe supplier otherwise. The expected cost to the supplier of producing the input

16

is:
C M −auth = cH − ηM (cH − cL + α · ∆) −|{z}
M uncovers the cheap methods(1 − ηM )ηS (cH − cL + ∆)
{z
}
|

.

M delegates and S uncovers the cheap methods= cH − (ηM + ηS + ηM ηS )(cH − cL ) − (ηS + ηM (α − ηS ))∆Supplier-authority. If the supplier has formal authority, they choose the method
themselves whenever they uncover the cheap methods and they delegate to the
manager otherwise. The expected cost to the supplier of producing the input is:
C S−auth = cH −η (c − c + ∆)}| S H {z L−S uncovers the cheap methods(1 − ηS )ηM (cH − cL + α · ∆)
|
{z
}

.

S delegates and M uncovers the cheap methods= cH − (ηM + ηS + ηM ηS )(cH − cL ) − (ηS + ηM (1 − ηS )α)∆Notice that C S−auth < C M −auth when α < 1. Intuitively, α < 1 means that the
manufacturer is not sufficiently concerned with cost-cutting and allocating formal
authority to them leads to inflated costs.
Because it is probabilistic whether the manufacturer and the supplier uncover
the cheaper methods, the cost may differ from the expected cost. However, as
the number of aspects of production P becomes large, the cost converges to the
expected cost (due to the law of large numbers). Since it simplifies the exposition
to deal with a fixed cost rather than a random cost, we will focus on the caseM −authwhere P → ∞. Moreover, we will let ci ≡ C S−auth and βi ≡ C ci −ci , so that(1+βi )ci = C M −auth . Parameter βi , which is increasing in preference incongruence,
captures the extent to which costs increase under M-authority.

3.2

Authority allocation

Let us now examine how authority will be allocated at time 1. To aid intuition, let
us begin by considering a simple case where there are just two suppliers and the17manager has zero capital (k = 0). We will then examine the general case.
3.2.1

Two suppliers and k = 0

If neither supplier is integrated, the problem is the same as in Section 2. In orderto produce widgets, the manager must pay a markup of:ΦN I = c1 ·1 − γ2
1 − γ1+ c2 ·γ2
γ1

Now suppose that both suppliers are integrated. At time 4, there is no longer acoordination problem since M has the authority to produce both types of inputs.
Hence, there is no need to pay a markup because of the coordination problem.
However, there is a markup in this case arising from the added production costsof M authority. At time 1, M must offer supplier i a contract where pi = (1 + βi )ci .
Consequently, there is a markup of βi · ci on input i, and the total markup is:Φ12 = c1 · β1 + c2 · β2
Finally, consider the case where only supplier i is integrated. At time 4, theparties deciding whether to producing inputs are M and supplier j. The coordination game is played, in other words, between M and j rather than i and j. To
analyze this case, we will assume that M has an impulse just as suppliers do.
Assumption 3. The manufacturer has an impulse to produce, denoted x0M ∈ {0, 1}, justlike suppliers. We assume that x0M = 1 with probability γM . x0M is private information of
the manufacturer.
In order to get supplier j to produce at time 4, the supplier must pay a markup
Mof cj · 1−γ. Because M is the second party in the game, the only party that must be
γMpaid a markup is supplier j; however, there is an incentive compatibility constraintthat must be satisfied in order for M to choose to produce input i. The incentive
compatibility constraint for M follows immediately from Lemma 1:

18

= S − Φi ≥S − (total markup)
|
{z
}

1
ci (1 + βi )γj(2)payment to M for producing input i≡pM

While supplier i is not part of the time-4 game, and thus does not need to be
paid a coordination markup, the allocation of authority to M increases the cost of
producing input i by ci · βi .
Thus, the total markup that is paid in this case is:Φi = ci · βi + cj ·1 − γMγM

To summarize, the manufacturer will choose an authority structure based uponwhich minimizes the markup; however, the authority structure where supplier i
is integrated but not j is only available if the IC constraint for the manufacturer(Condition 2) is satisfied. The following proposition immediately follows.
Proposition 2. Suppose there are two suppliers.
1. Integrating supplier i is preferred to no integration (Φi ≤ ΦN I ) if and only if:cj ·M − γiγγM · γi
1 − γj ≥ ci β i −γj(3)

2. Integrating both suppliers is preferred to integrating i only (Φ12 ≤ Φi ) if and onlyif:
1 − γM≥ βj(4)γM
Equation 3 shows what makes it is preferable to integrate supplier i rather
than have no integration. Whether it is preferable depends in part upon is how
the markup on input i changes when supplier i’s authority is taken away (βi −1−γj). Additionally, it depends upon whether supplier i has a low impulse toγjproduce compared to the manufacturer (γM − γi ). Equation 4 shows what makes
it is preferable to integrate both suppliers rather than supplier i only. What matters19

in this case is how the markup on input j changes when supplier j’s authority isMtaken away (βj − 1−γ).
γM
Notice that in the special case where γ1 = γ2 = γM = γ and β1 = β2 = β, full
≥ β and non-integration is preferred otherwise.
integration is preferred when 1−γγ
In other words, integration occurs when the cost of giving authority to M (β) is
low relative to the costs of coordinating suppliers (γ).
3.2.2

The General CaseLet us turn now to the general case where there are n suppliers and the manufacturer has capital k. The following lemma gives a formula for the markup.
Lemma 2. Suppose the manufacturer integrates a set of suppliers I and their capital isused to fully ensure a set of suppliers K (i.e. θi = 1 for i ∈ K and θi = 0 for i ∈/ K).
When at least one supplier is integrated,
1. The total markup is:XΦI (K) =ci · βi

+

i∈I

|

Q

ci ·

i∈I∪K
/

{z

}

non-congruence costs

where Γ−i−I =X|

1 − Γ−i−I · γM,
Γ−i−I · γM{z}coordination costs

γj .

j ∈{i}∪I/

2. The incentive compatibility constraint for the manufacturer is:
S − ΦI (K)| {z }≥

payment to M for producing inputs I≡pM

1 X
ci (1 + βi ),Γ−I i∈I

where ΦI (K) denotes the total markup and Γ−I =

Q

γj .

j ∈I
/

The first term of the markup formula is the added cost of producing inputs
when suppliers are integrated. This cost is lower when the integrated suppliers’
preferences are more congruent with those of M. The second term of the markup
20

formula is the cost of coordinating the non-integrated suppliers. The difference
relative to the formula in Lemma 1 is that manufacturer M takes the place of theintegrated suppliers. As in the two-supplier case, there is an incentive compatibility constraint for the manufacturer since the manufacturer must be properly
incentivized to produce at time 4. This incentive compatibility constraint can bederived from Lemma 1.
The following corollary is a consequence of Lemma 2.
Corollary 1.
1. The manufacturer does not insure suppliers who are integrated (θi = 0 for i ∈ I).
2. Among the non-integrated suppliers, the manufacturer insures those who are mostreluctant to produce (i.e. those of lowest index).
Point 1 of the corollary can be seen from the markup formula. The reason to
insure a supplier is to reduce the coordination markup; but there is zero coordination markup for integrated suppliers, and hence no reason to insure them. Point
2 can also be seen from the markup formula and the logic is akin to that in Proposition 1. The low-index suppliers are the most reluctant to invest. Insuring these
suppliers has the greatest bang for the buck in terms of reducing the total markup.
The next proposition examines when a given supplier is more or less likely to
be integrated. The proof is given in the Appendix.
Proposition 3. Lowering βi or γi makes it more likely supplier i will be integrated. Moreprecisely, let I ∗ (βi , γi ) denote the firms that are integrated as a function of βi and γi . Ifi ∈ I ∗ (β, γ, c):
1. i ∈ I ∗ (β ′ , γ, c) for all β ′ < β.
2. i ∈ I ∗ (β, γ ′ , c) for all γ ′ < γ.
Intuitively, lowering βi reduces the cost of integrating supplier i, and hence
raises the chance that supplier i will be integrated. When γi is low and supplier21

i is not integrated, it is hard to coordinate the other suppliers. Thus, lowering
γi raises the benefit of integrating supplier i, and hence raises the chance that
supplier i will be integrated.
Proposition 4 gives some additional results (the proof is given in the Appendix).
Proposition 4.
1. If βi = 0 for all suppliers, all suppliers are integrated.
2. If suppliers differ only in their β’s (ci = c and γi = γ for all i), the suppliers whoare integrated are those for whom βi is lowest.
3. If suppliers differ only in their γ’s (ci = c and βi = β for all i), the suppliers whoare integrated are those for whom γi is lowest.
4. If suppliers are homogeneous (ci = c, βi = β, and γi = γ for all i), it is possible thatsome suppliers are integrated and some are not.
Point 1 follows from the observation that there is no cost to integration—only
benefit—when betai = 0 for all i. Point 2 follows from the observation that the costof integration for the most congruent suppliers (i.e. the suppliers with the lowest
betai ’s). Point 3 follows from the observation that the coordination benefits of integration are greatest for the suppliers with the lowest impulses (i.e. the supplierswith the lowest γi ’s).
Point 4 considers the case where all suppliers are homogeneous. In this case,cost of integrating an additional firm is β. The marginal benefit of integration
is decreasing, by contrast. Consequently, the point where the marginal cost of
integration equates with the marginal benefit may be interior. Point 4 has the interesting implication that initially identical suppliers may be treated differently by
the manufacturer; moreover, their ultimate costs of production may be different(c vs. (1 + β)c). This result speaks to the growing literature on persistent performance differences between firms (see, for instance, Gibbons and Henderson(2012), Chassang (2010), Ellison and Holden (2014)).
22

4Applications

Supplier Investments
Suppose the inputs the manufacturer needs to produce a widget are generic andhave a low marginal cost of production. In such a case, the manufacturer is likelyPhave sufficient capital to purchase all of the inputs (k ≥ i ci ), in which case no
coordination problem arises.
There are two circumstances where coordination problems are more likely toarise: (i) the inputs are generic but have a high marginal cost of production, or (ii)the inputs are non-generic. In both cases, a coordination problem is likely because
the cost of producing inputs (ci ) is high. In the latter case, the cost of producing inputs is high because suppliers need to make investments to produce them.
This was a key reason for ASML’s coordination problem: its main suppliers (e.g.
Cymer, Trumpf, and Zeiss) needed to make expensive, years-long investments.
The idea that coordination problems arise when suppliers make expensive investments suggests an interesting extension to the model. Consider a dynamic
version of the model where there are several potential manufacturers of widgets.
At time 1, it is hard to produce widgets because input suppliers have not made therequisite investments. If suppliers make investments at time 1, the coordination
problem disappears at time 2. Such a model suggests that a given manufacturer
M might prime the pump at time 1. If M solves the coordination problem at time 1,
they create a supply chain that other manufacturers can exploit at time 2.
One consideration that arises here is akin to the classic dilemma in the patent
literature, where manufacturer M might be disincentivized from creating the supply chain at time 1 if they lose market power at time 2. Hence, preserving market
power at time 2 becomes an important consideration for manufacturer M. This
desire to preserve market power is arguably another important reason why wemight see integration. By integrating suppliers at time 1, M obtains market powerin the input market at time 2.

23Movers and Shakers
Our model considers three tools that the manufacturer can use to achieve coordination: (i) insuring suppliers, (ii) paying markups, and (iii) integrating suppliers.
Potentially, additional tools might be available to the manufacturer besides thesethree. Akerlof and Holden (2016) consider the role that “movers and shakers”
play in coordinating parties. They consider, for instance, the role that financial
institutions or real estate developers play in creating expectations that a projectwill succeed. Movers and shakers, according to this view, create a virtuous cycle
where each party’s confidence raises the confidence of the other parties.
In terms of the present model, we might view movers and shakers as changing
the impulses of suppliers (γi ). They might raise a supplier’s γi by working to sellthem on the project. Since raising γi reduces the overall markup that the manufacturer needs to pay, we might also expect movers and shakers to command large
rents for the role that they play.
Movers and shakers might also serve as a substitute for integration. We might,
more generally, think of movers and shakers as generating a sense of trust between
parties. In cases where trust between firms is strong, integration might be lower.
This may explain, for instance, the types of company relationships we see in Japanand Korea (i.e. keiretsu and chaebol).
International Trade
Suppose manufacturer M decides to obtain an input from overseas rather thandomestically. How might this affect the integration decision? One possibility istrust between the parties declines. We might think of this as a reduction in γi . At
the same time, the congruence of preferences between M and the supplier might
decline as well. We might think of this as an increase in βi . According to Proposition 3, a decline in γi pushes the manufacturer toward integrating the supplier
while an increases in βi pushes the manufacturer against integrating the supplier.
Hence, there are competing considerations. The model suggests that whether integration increases or decreases when production moves abroad depends upon24

which of these considerations dominates.
Industrial Policy
A number of rationales used to justify industrial policy in advanced economies
in the 1960s and 1970s have been thoroughly debunked. For instance, industrial
policy was sometimes justified on the grounds of job creation, exchange-rate orbalance-of-payments management, or sectoral externalities. These arguments did
not stand up to scrutiny (see, inter alia, Scitovsky (1954), Bhagwati et al. (1978)).
Our theory offers a different rationale for the potential efficacy of industrial
policy. If there are projects with positive surplus that are not occuring, a government can play a coordinating role. This could happen through the movingand-shaking channel we discussed above, or it could happen by the government
providing capital. By providing a manufacturer with additional capital, a government could facilitate coordination where it otherwise would not occur, or help it
be achieved at lower cost by reducing the need to pay markups.
Complex issues such as the green-energy transition exhibit obvious coordination problems, suggesting a potentially important role for “coordination-based”
industrial policy.

5

Concluding Remarks

We have provided a model of supply chains that emphasizes the importance of
coordination and speaks to relationships between a single manufacturer and multiple suppliers—or “spiders” as they have become known. Our model not only
provides stark predictions that are consistent with existing empirical evidence,but is tractable enough to facilitate the analysis of a number of applications. These
include important issues in International Trade and Industry Policy.
At the heart of our approach in this paper is a novel theory of the firm basedon the importance of coordination. Coase (1937) first raised the deep questionof why firms exist if markets are an efficient means of allocating resources. The
25

Transaction Costs theory of the firm that Coase suggested and Williamson developed (Williamson (1971), Williamson (1975)) was the first alternative to the rather
unsatisfying Neoclassical theory of the firm.
Grossman and Hart (1986) and Hart and Moore (1990) pioneered the Property
Rights Theory of the firm. PRT emphasizes the importance of assets ownership inalleviating hold-up problems and hence encouraging ex ante relationship-specificinvestments.
There is also an earlier Principal-Agent Approach which emphasizes the importance of agency problems for hiring employees our outsourcing production to independent contractors. See for instance (Alchian and Demsetz (1972) and Holmstrom (1982)).
A fourth major approach to theory of the firm is the Incentives Approach ofHolmstrom and Milgrom (1994). Those authors view the make-or-buy decisions
as two alternative systems for managing incentives. Gibbons (2005) provides an
excellent overview of these theories.
We offer a distinct approach—a Coordination-Based theory of the firm.7 Thisapproach emphasizes the role that a firm (the manufacturer in our model) plays
in coordinating the provision of inputs from a number of disparate suppliers. The
firm can encourage provision of an input either through contract—by providing
insurance to a supplier in the event that they supply the input but other suppliers
do not—or via control of decision rights, whereby they can compel provision of
the input from a given supplier. Our coordination-based theory provides sharp
predictions about when exchange is mediated by contract, and when integration
is preferable. We have showed that the coordination-based theory can provide
starkly different predictions than PRT.
Our view is that all five of these rationales for the existence of firms are important. But to date the coordinating role that firms play has received little formal
attention. We hope that our contribution has partially remedied this oversight.
7

Alonso et al. (2008) emphasize the role of coordination in a different way to us. They focus
on the need to adapt decisions to local conditions but also coordinate those decisions in a multidivisional firm. As such they focus on the internal structure of an organization whereas we focus
on how the boundaries of firms are determined–i.e. the make-or-buy decision.

26

References
Acemoglu, Daron, Pol Antràs, and Elhanan Helpman, “Contracts and technology adoption,” American Economic Review, 2007, 97 (3), 916–943.
Aghion, Philippe and Jean Tirole, “Formal and real authority in organizations,”
Journal of political economy, 1997, 105 (1), 1–29.
Akerlof, Robert and Richard Holden, “Movers and shakers,” Quarterly Journal of
Economics, 2016, 131 (4), 1849–1874.
and , “Capital assembly,” The Journal of Law, Economics, and Organization,
2019, 35 (3), 489–512.
Alchian, Armen A and Harold Demsetz, “Production, information costs, and economic organization,” The American economic review, 1972, 62 (5), 777–795.
Alfaro, Laura, Davin Chor, Pol Antras, and Paola Conconi, “Internalizing global
value chains: A firm-level analysis,” Journal of Political Economy, 2019, 127 (2),508–559.
Alonso, Ricardo, Wouter Dessein, and Niko Matouschek, “When does coordination require centralization?,” American Economic Review, 2008, 98 (1), 145–179.
Antràs, Pol, “Firms, contracts, and trade structure,” Quarterly Journal of Economics,
2003, 118 (4), 1375–1418.
, “Incomplete contracts and the product cycle,” American Economic Review, 2005,
95 (4), 1054–1073.
and Davin Chor, “Organizing the global value chain,” Econometrica, 2013, 81(6), 2127–2204.
and , “Global Value Chains,” Handbook of International Economics, 2022, 5,
297–376.
and Elhanan Helpman, “Global sourcing,” Journal of Political Economy, 2004,
112 (3), 552–580.
and, “Contractual Frictions and Global Sourcing,” The Organization of Firms
27

in a Global Economy, 2008.
Antras, Pol, Teresa C Fort, and Felix Tintelnot, “The margins of global sourcing:
Theory and evidence from us firms,” American Economic Review, 2017, 107 (9),2514–2564.
Baldwin, Richard and Anthony Venables, “Spiders and snakes: offshoring andagglomeration in the global economy,” Journal of International Economics, 2013,
90 (2), 245–254.
Bhagwati, Jagdish N et al., “Foreign trade regimes and economic development:
Anatomy and consequences of exchange control regimes,” NBER Books, 1978.
Chassang, Sylvain, “Building routines: Learning, cooperation, and the dynamics of incomplete relational contracts,” American Economic Review, 2010, 100 (1),448–465.
Chor, Davin and Lin Ma, “Contracting frictions in global sourcing: Implications
for welfare,” mimeo, 2021.
Coase, Ronald, “The Nature of the Firm,” Economica, 1937, 4 (16), 386–405.
Corsetti, Giancarlo, Bernardo Guimaraes, and Nouriel Roubini, “International
lending of last resort and moral hazard: A model of IMF’s catalytic finance,”
Journal of Monetary Economics, 2006, 53 (3), 441–471.
Crawford, Vincent P, Miguel A Costa-Gomes, and Nagore Iriberri, “Structuralmodels of nonequilibrium strategic thinking: Theory, evidence, and applications,” Journal of Economic Literature, 2013, 51 (1), 5–62.
Ellison, Glenn and Richard Holden, “A theory of rule development,” The Journalof Law, Economics, & Organization, 2014, 30 (4), 649–682.
Gereffi, Gary, “International trade and industrial upgrading in the apparel commodity chain,” Journal of International Economics, 1999, 48 (1), 37–70.
Gibbons, Robert, “Four formal (izable) theories of the firm?,” Journal of Economic
Behavior & Organization, 2005, 58 (2), 200–245.

28

and Rebecca Henderson, What do managers do?: Exploring persistent performancedifferences among seemingly similar enterprises, Citeseer, 2012.
Grossman, Sanford J and Oliver D Hart, “The costs and benefits of ownership:
A theory of vertical and lateral integration,” Journal of Political Economy, 1986,
94 (4), 691–719.
Hart, Oliver and John Moore, “Property Rights and the Nature of the Firm,”
Journal of Political Economy, 1990, 98 (6), 1119–1158.
Holmstrom, Bengt, “Moral hazard in teams,” The Bell journal of economics, 1982,
pp. 324–340.
and Paul Milgrom, “The firm as an incentive system,” The American economic
review, 1994, pp. 972–991.
Kets, Willemien and Alvaro Sandroni, “A theory of strategic uncertainty andcultural diversity,” The Review of Economic Studies, 2021, 88 (1), 287–333.
Miller, Chris, Chip War: The Fight for the World’s Most Critical Technology, Simonand Schuster, 2022.
Morris, Stephen and Hyun Song Shin, “Catalytic finance: When does it work?,”
Journal of International Economics, 2006, 70 (1), 161–177.
Scitovsky, Tibor, “Two concepts of external economies,” Journal of political Economy, 1954, 62 (2), 143–151.
Williamson, Oliver, Markets and Hierarchies: Analysis and Antitrust Implications,
New York: Free Press, 1975.
Williamson, Oliver E., “The Vertical Integration of Production: Market Failure
Considerations,” American Economic Review, 1971, 61 (2), 112–123.

29

6

Appendix

Proof of Proposition 3. Suppose that a set of suppliers I are integrated. Let us
consider how adding supplier i to set changes the total markup. Let ∆(I) denotethe change in the total markup from integrating i:∆(I) ≡ ΦI∪{i} (K ′ ) − ΦI (K),
where K is the optimal insurance choice when suppliers I are integrated and K ′is the optimal choice when suppliers I ∪ {i} are integrated. To prove the result,
it is sufficient to show that ∆(I) is weakly increasing in βi and γi . There are fourcases we need to consider:
1. I is nonempty and i ∈/ K.
2. I is nonempty and i ∈ K.
3. I is empty and i ∈/ K.
4. I is empty and i ∈ K.
In case 1, corollary 1 implies that K ′ = K. We find that:∆(I) = ci · [βi −1 − Γ−i−I · γM]−
Γ−i−I · γMXj ∈I∪{i}∪K/cj · [

1 − Γ−j−I · γM
1 − Γ−i−j−I · γM−]
Γ−j−I · γM
Γ−i−j−I · γM

It is clear that ∆(I) is increasing in βi . γi appears in the second term of the above
expression: a higher γi increases Γ−j−I , which increases ∆(I).
In case 2, integrating i frees up some capital to insure other suppliers. We have
K ′ = [K − {i}] ∪ A, where A denotes the additional insured suppliers. We find
that:∆(I) = ci ·βi −Xj ∈I∪K/cj ·[1 − Γ−i−j−I · γM
1 − Γ−j−I · γM 1 − Γ−i−j−I · γM X
−
]−
cj ·[
]
Γ−j−I · γM
Γ−i−j−I · γMΓ−i−j−I · γMj∈A30

It is clear that ∆(I) is increasing in βi . γi appears in the second term of the above
expression: a higher γi increases Γ−j−I , which increases ∆(I).
In case 3, corollary 1 implies that K ′ = K. We find that:∆(I) = ci · [βi −X
1 − Γ−j−i · γM
1 − Γ−i1 − Γ−j]−−]cj · [
Γ−i
Γ−j
Γ−j−i · γMj ∈{i}∪K/

It is clear that ∆(I) is increasing in βi . γi appears in the second term of the above
expression: a higher γi increases Γ−j , which increases ∆(I).
In case 4, integrating i frees up some capital to insure other suppliers. We have′
K = [K − {i}] ∪ A, where A denotes the additional insured suppliers. We find
that:∆(I) = ∆(I) = ci · βi −

X
j ∈K
/

cj · [

X
1 − Γ−j 1 − Γ−j−i · γM
1 − Γ−j−i · γM−]−
cj · []
Γ−j
Γ−j−i · γM
Γ−j−i · γMj∈A

It is clear that ∆(I) is increasing in βi . γi appears in the second term of the above
expression: a higher γi increases Γ−j , which increases ∆(I).
This completes the proof.
Proof of Proposition 4.
The reasoning behind Points 1-3 is straightforward and given in the text. To
establish Point 4, let us focus on the case where there is no supplier heterogeneity,k = 0, and γM = γ. In this case, the markup when g firms are integrated is:Φ(g) = g · β + (n − g)c(γ g−n − 1)
Differentiating with respect to g:
Φ′ (g) = β + c + c · γ g−n [(n − g) log γ − 1]
Observe that Φ′ (n) = β and Φ′ (0) = β + c + c · γ −n [n log γ − 1]. If β > 0, γ < 1,c > 0, and n is large, Φ′ (n) > 0 and Φ′ (0) < 0. Hence, the value of g for which the31markup is lowest is interior (0 < g ∗ < n). This establishes the result.

32