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FUNDA ¸ C ˜ AO GETULIO VARGAS ESCOLA DE P ´ OS-GRADUA ¸ C ˜ AO EM ECONOMIA PEDRO LUIS ACCIOLI NOBRE BRETAN ESSAYS ON INDUSTRIAL ORGANIZATION Rio de Janeiro 2010

PEDRO LUIS ACCIOLI NOBRE BRETAN ESSAYS ON INDUSTRIAL

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FUNDACAO GETULIO VARGAS

ESCOLA DE POS-GRADUACAO EM ECONOMIA

PEDRO LUIS ACCIOLI NOBRE BRETAN

ESSAYS ON INDUSTRIAL ORGANIZATION

Rio de Janeiro

2010

PEDRO LUIS ACCIOLI NOBRE BRETAN

ESSAYS ON INDUSTRIAL ORGANIZATION

Tese submetida a Escola de Pos-Graduacao em Economia

da Fundacao Getulio Vargas como requisito para obtencao do

Tıtulo de Doutor em Economia

Orientadora: Maria Cristina Terra

Area de concentracao: Economia Industrial

Rio de Janeiro

2010

PEDRO LUIS ACCIOLI NOBRE BRETAN

ESSAYS ON INDUSTRIAL ORGANIZATION

Tese submetida a Escola de Pos-Graduacao em Economia

da Fundacao Getulio Vargas como requisito para obtencao do

Tıtulo de Doutor em Economia

Aprovado em 23/12/2010

pela comissao organizadora

Maria Cristina Terra (Orientadora - EPGE/FGV)

Humberto Moreira (EPGE/FGV)

German Pupato (EPGE/FGV)

Adriana Hernandez-Perez (ITAU-UNIBANCO)

Maurıcio Canedo (IBRE)

Rio de Janeiro

2010

Agradeco,

Aos meus pais e minha irma, pelo amor e apoio incondicional;

A minha orientadora, Maria Cristina Terra, por acreditar no meu projeto e guiar me em

seu seu desenvolvimento;

Aos amigos da EPGE e da FEA, que me apoiaram e me ”aguentaram” por todos esses anos.

A Gabi, pelo apoio incrıvel que me impulsionou no momento mais difıcil desta jornada.

Resumo

Esta tese compoe-se de tres ensaios que versam sobre o formato, em termos de estrutura de

governanca, das industrias e como a estrutura interna das mesmas influenciam esses formatos

e seus produtos. O primeiro capıtulo apresenta um modelo de como a qualidade institucional,

confianca e incompleteza contratual afetam as decisioes das firmas sobre a melhor forma de se

organizarem internacionalmente. O segundo capıtulo vai na mesma direcao de explicar a orga-

nizacao das industrias, mas com foco no efeito da transmissao de informacao entre as unidades

constituintes das organizacoes sobre o formato otimo das mesmas. Ambos trabalhos usam

modelos dinamicos. O terceiro capıtulo se utiliza da modelagem de organizacao industrial para

mostrar como a estrutura interna de uma industria influencia no risco de credito consignado.

Palavras-chave: integracao, terceirizacao, organizacao industrial, risco, credito consignado, in-

formacao, confianca, qualidade institucional.

Abstract

This thesis is comprised of three essays with a common goal: expand the theories about what

drives the shape of an industry and its consequences. All three are theoretical and applied,

in the sense of being detailed studies about the inner workings of industries and their impact

on the shape of the latter. In the first chapter I model how the interplay among institutional

quality, trust and contract incompleteness affects firms’ decisions about their international mode

of organization. Second second chapter also concerns industries’ ownership choices, but in a

complete different perspective, focusing in information transmission between the different parts

that constitute a supply chain. Finally, third chapter uses industrial organization modeling to

show how the internal structure of an industry influence the risk of payroll-backed loans.

Keywords: integration, outsourcing, industrial organization, risk, payroll-backed credit, infor-

mation, trust, institutional quality.

Sumario

Introducao - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 1

Capıtulo 1: Trust, Institutions and the choice between out-

sourcing and FDI - - - - - - - - - - - - - - - - - - - - - - - - - - 2

Capıtulo 2: Information and Ownership Choice - - - - - - - - 36

Capıtulo 3: An economic model of the risk of payroll-backed

loans - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -71

Trust, Institutions and the

choice between Outsourcing and FDI

Abstract

I model how the interplay among institutional quality, trust and contract incompleteness

affects firms’ decisions about their international mode of organization. A producer wants

to form a partnership with a foreign distributor, through either Outsourcing (O) or Foreign

Direct Investment (FDI), but is uncertain over his commitment. A non-committed partner

will try to engage in illicit activities: for FDI, he can steal business plans, blueprints, etc; for

Outsourcing, the distributor collects the revenue and may default on his payments.

In a dynamic model with Bayesian learning, one-sided incomplete information and non-

observable investments, I show that reputation-building may induce a change in the long

run from FDI to Outsourcing. In accordance with the empirical literature, I show that for

weak institutions, entry happens via FDI when uncertainty over the commitment is high

and via Outsourcing when it is low. Conversely, with sound institutions entry is always via

Outsourcing. If entry took place via FDI, I show that after some time Outsourcing may

become preferable to FDI. The level of trust at which this change happens is decreasing in

the institutional quality.

Finally, I study how institutional reforms affect producers’ expectations about the evo-

lution of the partnership to FDI. Results indicate an institutional trap where the lower the

institutional framework, the smaller the impact from the reforms over expectations.

3

1 Introduction

Trust between business associates is an important element in successful and lasting partner-

ships1. Although in our times legal contracts are the primary way to establish the terms of an

agreement to trade, trust interplays with them in a two-way manner such that one attempts

to fill the gaps the other, for whatever reasons, is not capable to perform.2 For instance, when

institutions are not good enough to enforce contracts, a business partnership built on trust may

be capable to thrive. This is the first way. The other is the standard: in the absence of trust,

well-written contracts are necessary to make sure the trade obligations will be honored.

A big problem with the latter, though, is that it relies on the existence of well developed

institutions capable of enforcing the contracts. Therefore, if there is no trust between partners

and enforcement is poor, the possibilities of trade developing are dim. The former is not free

of its problems either: although, intuitively, trust can substitute for inadequate enforcement,

in general trust develops through a slow process of confidence building that in its course will

provide the society with an inefficient provision of the good or service in question.

Enforcement difficulties are particularly acute in the area of international relations: if one of

the parties reneges on a contract, the other must look for assistance in the courts and, depending

on the specifics of the market in question, the judicial system may be incapable of enforcing the

agreed contract. Poor enforcement is more likely in an international business relationship since

the costs for one of the parties to sue the other increase due to differences in the law system

and country-specific judicial procedures, besides the costs related to the distance per se. These

increased costs, in turn, make it easier for the courts to stand by their fellow citizens.

These problems inspired work by Araujo and Ornelas (2008). In their model, the interplay

between trust and enforcement is analyzed in a setting where an exporter needs a distributor

in a foreign country but cannot differentiate between the two types available there: myopic

and patients. The latter is interested in building a long-term partnership with the exporter,

whereas the former is shortsighted and thus is going to default on the contract whenever is

possible. Trust is defined as the value the exporter assigns to the probability that his distributor

is myopic. Enforcement shows up through an exogenous probability that a myopic distributor

has in finding a corrupt legal officer to back him in his unlawful plans. They show that for a given

enforcement level the incomplete information about the type of the distributor depresses trade

flows, but trade increases with the passage of time as the producer learns about the distributor

type. Improvement in enforcement also increases trade flows.

The present work builds on their model but, whereas they were concerned with the effects

of the interplay of trust and enforcement in trade flows, the focus here is on the effects of these

1My use of trust for business partnerships follows the definition of Mayer et al. (1995). They define it as: ”thewillingness of a party to be vulnerable to the actions of another party based on the expectations that the otherwill perform a particular action important to the trustor, irrespective of the ability to monitor or control thatother party”.

2See Egan and Mody (1990, p. 327).For example, at page 326-327, they state that ”Trust implies a moralcontract and long-term commitment which ”reflects a condition of mutual dependency where both client andsubcontractor are in a position to influence the other by their behavior”.

4

variables on the decision of multinational firms about the best way to structure their divisions

internationally. In accordance with the discussion in the second paragraph, the lack of good

enforcement disrupts business plans and distorts the optimal organization of production. This

paper shows that the above-mentioned “first way” of the trust-contract interaction may lead in

the long-run to the optimal organization, through a trust-building process. On the other hand,

as was mentioned in the second paragraph, this process is a long-run enterprise and does not

eliminate the distortions caused by bad institutions in the meantime.

I find that, broadly, the better the institutions, the more likely the foreign firm will enter

the market via Outsourcing; however, if she enters via FDI, with some probability a change to

Outsourcing may happen in the future. Finally, I determine the effects of institutional reforms

over the expectations of producers of a future change to FDI, with some interesting results.

Although the interplay of trust, institutions and ownership choice is widely discussed in

the management and economics literature, there are no theoretical models that study the joint

dynamics of them, and very few empirical papers, specially considering international aspects of

alliances. Among these few, Gulati (1995) studies 2400 alliances in various industrial sectors,

finding that equity sharing in a partnership is negatively related the duration of a repeated

relationship. Moreover, he finds that that international alliances are more likely to be equity

based than domestic alliances. From this, he concludes that trust affects the structure of interfirm

partnerships. Also, the franchising literature presents us cases that go in the direction of this

paper’s findings: for instance, Ko (2008) shows that both Mcdonalds and KFC decided to enter

the Chinese market via FDI given the unexistence of adequate laws at that time. In 2007, 17

years after entry, Mcdonalds still used mostly integrated structures to do business in China

(just 1 franchised outlet), although in the rest of the world non-integrated operational modes

accounted for 70% of its outlets.

Other authors also have studied the relationship between trust and ownership forms:

Brouthers and Brouthers (2003) find that industrial firms’ propensity to entry via integrated

FDI has a negative correlation with trust levels. The first study of forward integration into

distribution, by John and Weitz (1988), found that both environmental uncertainty and behav-

ioral uncertainty (related to trust) increase the probability of integration. Johnson et al. (2002),

studying post-communist countries, find first that courts are more important when relationships

are weaker, mostly in market-based transactions. Second, that in an established relationship,

after the development of trust, is the relationship itself which determines the cooperation, re-

gardless of the efficiency of the enforcement system. Another paper in the literature that studies

the effects of trust and ownership structure is Bottazzi et al. (2008): he finds that the lack of

trust undermines the use of sophisticated contracts.

This paper is also related to the literature on relational contracts and the theory of the

firm, specially the papers by Baker et al. (2002a,b). They examine relational contracts and

ownership structure in a very broad way. While these author link the relational aspect of

interfirm partnerships to the possibility of non-enforceable promises between the partners, here

I abstract from this element and focus on a different aspect: the learning process associated with

5

a relationship and the effect of institutional changes over it.

Thus, based on these evidences, I setup a two-country model where a producer wants to form

a partnership with a distributor in the other country to sell her goods, and faces two alternative

operational modes to establish this enterprise: she may start through Outsourcing or through a

Foreign Direct Investment.

In this paper trust between business partners is captured via the uncertainty the producers

have about the distributors, similar to Araujo and Ornelas (2008)’s. That is because distributors

can be of two types, one commited to the joint enterprise and another opportunist, and the

producer does not possess enough information to tell one apart from the other. Together with

the degree of institutional quality of the country, they are responsible for the potentials pitfalls

facing the producer as she makes business in Foreign. If she enters via Foreign Direct Investment,

the local distributor is part of the integrated firm. At this position, he has greater access to

classified information that is valuable to competitors and, depending on his type, he will obtain

and sell them, bringing harm to his former employer in terms of smaller future profits. Foreign

Direct Investment is also subject to a fixed cost related to all the hurdles and expenses that

come with integration.

On the other hand, if the entry choice was Outsourcing, the pitfall is of another nature:

now the proceeds of the sale are collected by the distributor and, depending on his type, he

may decide to run away with them. In both cases, Outsourcing or FDI, in order to complete

the unlawful action the distributor must find a corrupt legal agent, which is not certain. The

uncertainty revolving around the distributor’s possibilities to find this agent is this paper’s

measure of institutional quality: the better the enforcement, the more difficult is to evade the

law.

This story is modeled as repeated-game where a distributor and a producer meet period after

period to reassess their partnership. This way, in both operational modes one agent discovers

more about the other with the passage of time: as trade happens and if there is no record of

misbehavior by the part of the distributor, the producer slowly learns his partner’s true type,

through Bayes Rule.

The above-mentioned learning, when coupled with incomplete contracts, are responsible for a

relevant distinction between Outsourcing and FDI in this model: in the latter, the compensation

for the distributor is fixed, whereas in the former it may vary with time. At every period the

partners trade, there is some non-contractible investment to be done by the distributor in behalf

of the partnership, and after this work is finished the goods are produced, shipped and a bargain

takes place to split the surplus from the trade. In the case of FDI the splitting share is assumed

to be fixed, while in the case of Outsourcing it is subject to the uncertainty regarding the

type of the distributor. Therefore, in case this incomplete information were severe, the bargain

power shifts to the producer, who will get more of the surplus in exchange for the risk. The

downside of this, of course, are diminished incentives for the distributor to promote the good in

the country. At the end, the incomplete information causes the total profit from the operation

via Outsourcing to decrease, in comparison to FDI. Learning changes, over time, the severity of

6

the uncertainty over the distributor’s type, increasing the ex-post share of the distributor and

his investments, improving the outlook of Outsourcing via-ı¿12 -vis FDI.

This approach to reflect the risk of an arm’s-length operation is novel and describes well

how foreign investors react facing an uncertainty. Better institutions can mitigate the impact

of the incomplete information through enforcement, diminishing the risk and improving the

surplus sharing between the agents. Even if there is near-certainty over the opportunism of the

distributor, which would prompt almost total bargain power to the producer as an insurance

against this enormous risk, there is some critical level of enforcement such that beyond it the

bargain power of the producer starts to diminish. Intuitively, if the police are perfect, the

honesty of your business partner is almost irrelevant: legal deterrence at its best.

Before describing the results concerning optimal entry under incomplete information, a

benchmark is needed in order to jump from analytical results to any policy recommendation.

In this work, under complete information the preferred mode of operation would be Outsourc-

ing, given that in it producer and distributor share equally the ex-post surplus from trade and

investments are greater than in the case of FDI. This benchmark, of course, depends on the

assumption that just the distributor invests on the partnership. I decided to take this approach,

and not one with both investing or just the producer, due to three facts: first, to explore a

neglected side of the literature, which mostly focuses on investments upstream; second, because

the emphasis in the distributor’s investment allows me to focus on the effects of uncertainty

over the results of the partnership in the foreign country; third, and the most important, when

entering a new final-good market usually a good deal of the work that is necessary for the success

of the enterprise is done overseas, periodically. The emphasis is important: although one-shot

investments in production and distribution are relevant, here the focus are on the recurrent

investments necessary to maintain the correct course of a business enterprise.

In a environment with incomplete information, I show how institutional quality influence the

decision of the producer between entry via Foreign Direct Investment or Outsourcing: for lower

levels of enforcement, entry takes place via Foreign Direct Investment when the prior probability

that the distributor is an opportunist is high and via Outsourcing when it is low. However,

for some high enough enforcement level entry happens directly via Foreign Direct Investment,

regardless of the probability that the distributor is committed to the firm. Conditional on

entering via FDI, I also show that there is a unique critical trust level such that at it a change

of operational mode takes place.

The linking of compensation to trust directly through the bargaining process gives novel

results for the dynamics of outsourcing that are absent in Araujo and Ornelas’ work. There the

exporter is passive in his negotiations with the distributor, whatever his beliefs about the latter’s

type. Therefore, in case the exporter had near-certainty that his partner was myopic, the authors

assume that profits would be negative, regardless of the enforcement level. Here this is not the

case: near-perfect enforcement brings maximum profits for the producer, since in this case the

reputation of the distributor will not be relevant to the business. Another difference is that

in their model the increase in profits with the passage of time is due to increased trade flows.

7

Here I use revenue functions, and profits increase due to greater investments that happen as

the compensation of the distributor augments, but they do not come necessarily from increased

sales, although they could be interpreted this way.

Moreover, notice that there is effectively two kinds of contract incompleteness here: the

unobservable-investment one and the incompleteness brought by the imperfect enforcement.

Both contribute to underinvestment, and the latter affects directly the former: the weaker the

enforcement, the greater the hold-up problem since it affects directly the share of the trade

surplus that the distributor obtains.

Finally, a relevant policy question is how institutional improvements affect producers’ expec-

tations about doing business in the foreign country via FDI. I find that there is an institutional

trap effect, such that expectations improve if trust and institutional levels are already high, and

decrease otherwise.

1.1 Other related literature

Besides the empirical and theoretical works cited in the beginning of the section, this paper

is related to a large economics literature on Foreign Direct Investment and institutional quality,

most of it empirical. Studying expropriation, a risk not considered here, Eaton and Gersovitz

(1984) analyzed the relation between the type of FDI and political risk, coming with the idea

that the ownership of intangible assets diminishes the incentives to expropriation. However,

their model considered just FDI.

Yet Albuquerque (2003), in a setting with imperfect enforcement, allows for different types

of investment. He concludes, first, that the higher the political risk of the host country, the

greater is its share of FDI; second, that FDI is a more stable type of investment compared to

other types.

In a similar vein, Hausmann et al. (2000), in a empirical study over the types of flows coming

into Latin America, concludes that

“the share of FDI in total flows tends to be larger in countries that are riskier, more

distant, resource rich, financially underdeveloped, institutionally weak and suffering

from original sin.”

Thus, one of the conclusions of this paper in in line with these empirical findings of Hausmann,

that is, that institutional quality is inversely related to relative FDI flows.

Finally, Schnitzer (2002) analyzes the trade-off between FDI and debt finds that FDI is more

likely to be chosen the riskier the project, the greater the efficiency of the foreign investor and

the better its outside option.

None of these papers, though, consider how the interplay of trust, reputation and institutions

affect organizational choices in the international arena.

The effects of learning in the choice between operational modes were also analyzed by Kotseva

and Vettas (2005). However, in their model the learning is over levels of demand, given that the

agents are uncertain about the its true level.

8

Probably the work that more closely resembles this is from Straub (2008). He analyzes

the trade-off between FDI and debt in a model with incomplete contracts due to institutional

constraints in the host countries, such as problems of commitment and corruption. Two modes

of corruption are studied: petty bureaucratic corruption and high-level political corruption. In

line with the findings of this paper, his model predicts that multinational firms prefer FDI the

weaker the commitment of the host country, while both types of corruption favor more flexible

operational modes. To reach these conclusions, he models an environment with incomplete

information and assumes the parties split the gains from trade through a bargaining process.

The incomplete information, however, is over variables different from the ones in this work; thus,

is not directly comparable. In addition, the model is static and does not consider learning or

the possibility of a change from one mode to another due to institutional advances.

The paper proceeds as follows. Section 2 describes the economic background. Section 3

presents the learning environment. Sections 4 and 5 present the economic environment and

solve the producer’s problem of a choice between outsourcing and FDI. Section 6 solves the

distributor’s problem. At Section 7 I show that the producer and distributor’s strategies and

beliefs obtained in the previous sections constitute an equilibrium. Section 8 shows the effect

of changes in enforcement over, among other things, the time necessary to move from FDI to

Outsourcing and over the probability that this event will be attained. Section 9 concludes.

2 Background

Consider a world economy with two countries, Home and Foreign. There is a continuum

of identical producers and differentiated goods in Home and Foreign, with one producer per

good, and aggregate demand of any differentiated product, every period, is equal to one in both

nations. Producers sell their goods at Home and can access the Foreign market through one

of two possible operational modes: Outsourcing (O) or Foreign Direct Investment (I), which

generate revenue equal to Ri, i ∈ {O, I}. Whichever mode of entry is chosen, the producer

needs a distributor in Foreign to handle the whole sales operation: in the case of Outsourcing,

the distributor is contracted arm’s length; in the case of Foreign Direct Investment (henceforth

FDI), he is vertically merged in the organization, that is, the distribution business is a division

of a larger firm. Producers discount time at a rate β and I assume that the good is manufactured

with a bulk investment equal to k, thus the marginal cost is zero. Hereafter I will use female

pronouns when referring to the producer and male ones to the distributor.

What is the role of the distributor in the partnership? I say he has superior knowledge of

local conditions and thus is important for the enterprise to be successful. Indeed, he is the one

in charge of the recurrent investments that allow the product to be known and accessible to the

public, marketing for example.

There are two types of distributors in Foreign: either they can be opportunists (P ) or com-

mitted (and honest) (C), and each one knows his own type, which is his private information.

The producer, thus, is uncertain over the type of any distributor, but has enough information

9

to form a subjective belief concerning the type’s distribution, that is, she thinks they are op-

portunists with probability θ or committed with probability 1 − θ. Opportunistic distributors

discount the future at a rate equal to zero and thus are interested only in short-term gains from

a partnership, whereas committed ones desire to be part on a long-time business relationship

with the producer.3

Whichever operational mode the producer chooses, I assume she always offers a short-term

contract to the distributor. I will use the uppercase letter T when referring to one of this contract

periods (one year, for example), whereas the subperiods inside a generic T will be referred to

through lowercase letters, t.

In the case of FDI, the distributor is part of the integrated firm and his correct rank in

this context would be middle manager, that is, an agent in charge of a relevant division inside

the organization; thus, he has access to deep levels of information. Because of this, if he is

an opportunist just before the end of the period he searches for a corrupt legal officer; with

probability (1 − λ) he finds it and betrays his parent firm stealing, at a cost l, classified infor-

mation to be sold to competing firms. In this case I assume the producer loses forever a fraction

1 − ρ ∈ (0, 1) of future profits. The classified information can be anything from designs and

blueprints to distribution contacts and marketing actions that may be valuable to competitors.

Operation via FDI is also subject to a specific liability: a fixed setup cost f must be paid at en-

try, representing the various legal hurdles, expanded management operations and other specific

foreign-investment-related expenses.

When the operational mode is Outsourcing, the illicit opportunities for the distributor are of

another nature. Now, as the head of an independent firm, he is directly in charge of the proceeds

of the sale and, if an opportunist, may just run away with them before making his payments

to the producer. If this indeed happens, I assume that next period, T + 1, the producer stops

outsourcing for Foreign. Again, in order to do that he must find a corrupt legal office, which

happens with probability (1 − λ). Behind this story of default lies an important assumption,

namely, that the contracting parties use trade credit in their relationship. This means that all

monetary payments are done some time after the sales occurred. This gap, in turn, gives the

distributor the precious opportunity to run away with the sales revenue 4 .

Finally, the contract that binds the two firms is incomplete, that is, investments and product

quality are not observed by a third party, although the contracting parties do observe them.

Relationship-specific investments coupled with incomplete contracts means both parties bargain

over the division of the ex-post gains from trade, with the clearing happening at the end of each

contract period.

In case no unlawful action was taken by the distributor, in both operational modes the pro-

ducer updates his belief θ about the distributor and a new one-period partnership is contracted

for period T + 1. Before renewing, though, the producer checks which of the two operational

3This assumption is explained further in section 6.24I restrict each operational mode to those illicit opportunities since they are the more possible in each case.

However, in reality either mode could suffer from either type of illicit act.

10

modes is more profitable for him for the future: that is what I call the Producer’s Problem.

After the choice, the story repeats itself as described.

What about the Distributor’s Problem? It is exactly to take an unlawful action at the

end of each period (the opportunistic ones, of course). Next I present the exact timeline of each

contract period T .

Timelines under a generic contract period T

At t = 0 of T , the producer solves his problem, that is, whether to enter the market via

Outsourcing or Foreign Direct Investment and, after this choice, she makes a business offer to

the distributor, who may accept or reject it. If accepted, next period the distributor makes

investments E in the enterprise. These investments (marketing, retail, establishing distribution

centers, etc) have a positive effect on demand. At t = 2 goods are manufactured.

From t = 2 onwards the timeline for FDI is slightly different from the one for Outsourcing.

In the latter, at t = 3 uncertainty is resolved and distributor and producer bargain over the

price and, in the case of an agreement, the product is shipped. At t = 4 goods are sold. At

t = 5 the distributor decides his problem, that is, whether to steal the revenue; if not, at t = 6,

payment is done. Figure 1 below presents the timeline

Figure 1: Outsourcing - timing of events in a generic contract period T.

When the operational mode is FDI, at t = 3 uncertainty is resolved, distributor and producer

bargain over the price and, if there is an agreement, goods are shipped. Sales and payment

happen at t = 4. At t = 5 the distributor decides his problem, that is, whether to steal classified

information. The timeline is presented below in Figure 2.

Figure 2: FDI - Timing of events in a generic contract period T.

11

3 The learning mechanism

This section starts with the learning mechanism that will be used to connect producer and

distributor’s decisions across periods; it is based on the structure defined by Araujo and Ornelas.

Consider an ongoing partnership between a producer and a distributor at date T . As this

relationship evolves, the producer updates his belief with respect to the type of the distributor

through his experience. If the distributor defaulted (in the case of Outsourcing) or betrayed (in

the case of FDI) the producer last period, she forms a posterior equal to 1, that is, concludes the

distributor is an opportunist, and the partnership ends. Otherwise, if the distributor played by

the rules and did not default or betray, the producer updates his belief about his type according

to the Bayes rule:

θT+1 ≡ Pr(opportunist|no default ∩ θT ) =λθT

λθT + 1− θT< θT . (1)

Therefore, each period that the distributor did not behave unlawfully causes the producer

to decrease his belief that he is an opportunist.

In order to construct the dynamic strategies, let 0 and 1 indicate respectively a record of

good behavior and one of misbehavior in a given experience. Then, define θT (Y ) as the belief

that the distributor is an opportunist given an experience yT with size T and cardinality Y ,

where Y =∑T

i=1 yi, yi ∈ {0, 1}. Therefore, if there is no record of misbehavior, the producer

belief is given by the Bayes formula above; if misbehavior happened anytime, she updates her

beliefs to 1. Formally,

θT (Y ) =

λT θ0λT θ0+1−θ0 if Y = 0

1 if Y 6= 0(2)

It is clear that if the distributor is period after period committed to the partnership, in the

long-run the producer eventually becomes convinced (almost surely) that he is committed.

4 Producer’s Problem: Environment

In this section I show the economic structure that will support the producer’s problem of

choosing between FDI and Outsourcing, to be addressed in the subsequent section.

Assume throughout this subsection that the producer has two beliefs about the distributors:

1. that opportunistic distributors default or betray whether they find a corrupt legal officer,

and committed ones do not.

2. That both opportunistic and committed distributors make the same investment decisions

in a partnership with the producer, for each operational mode.

Section 6 will show that these beliefs are sequentially rational since they are dominant strate-

gies for the distributors.

12

4.1 Foreign Direct Investment

Consider the problems facing a producer when the operational mode in effect is Foreign

Direct Investment. In this operational mode the producer, as the sole owner of the firm, has

some residual rights over its assets. I start with the bargain.

The Bargain

At t = 3, producer and distributor bargain over the gains from trade. Bargain is over

incomplete information since the producer does not know the type of her partner; however,

remember that the distributor’s unlawfull actions in FDI do not affect current period profits.

Thus, incomplete information regarding the type of the distributor is innocuous here.

Therefore, I will model the bargain as the standard Nash one. If there is an accord, producer’s

gains from trade are equal to S = RI − pI where pI is the pric of the good in this operational

mode; if the bargain breaks down, she can always fire the distributor and still obtain a share

of RI , so her ex-post utility is a proportion γ ∈ (1/2, 1) of RI . Therefore, there is always some

potential loss when the distributor is fired before the end of a contract period.

Investments

At t = 1 the producer expects the distributor (whichever type, see beginning of this section)

to make investments EI that affect demand positively, that is:

EI = arg maxE

(1− γ)RI(E)− E, (3)

where RI(·) is increasing and concave.

Assuming that RI(E) = A√E, then the distributor’s profit is given by5

πDI =A2

4

(1− γ

)2.

Accordingly, producer’s profit is

πPI =A2

2γ(1− γ)− k + TI ,

where TI is the ex-ante transfer which is equal to the distributor’s ex-post profit. Finally, profit

for the integrated firm (and thus ex-ante profit for the producer) is:

ΠI =A2

4

(1− γ2

)− k. (4)

4.2 Outsourcing

Now, consider the case when the operational mode in effect is Outsourcing. Now, the use of

trade credit implies that there is an opportunity for him to literally run away with the proceeds

5The use of a specific functional form allows a clear presentation of the paper’s motivation and results.

13

of the sale before paying the producer. As done in the previous section for FDI, I start with the

bargain that happens at subperiod t = 3.

The Bargain

At t = 3, distributor and producer bargain over the gains from trade. Contrary to the

bargain in FDI, here the incomplete information about the type of the distributor has a profound

influence over producer’s decisions, since his misdeeds do affect current period profits. Incomplete

information basically changes the bargain powers between the two players; actually, the game

happens as if there were 3 players bargaining: the producer and both types of distributors. I

find Harsanyi and Selten’s (1972) approach, essentially a generalization of the Nash Bargain for

asymmetric bargaining powers, the most suitable for the problems in this paper. Observe that

the incomplete information affects the bargaining position insofar as the possibility of a default

directly affects current period profits.

Given that investment costs are bygones at the time of the bargain, the ex-post expected

utility of the producer (U) is pO (1− θ(1−λ)) (pO is the price of the good to be cleared through

the bargain), the expected price to be received for the sale of one unit of the good, and her

outside option is zero. An opportunistic distributor has utility Up = λ(RO − pO) + (1 − λ)RO

from the relationship and outside option equal to Uoutp = 0. A committed distributor has utility

UC = RO − pO and outside option UoutC = 0.

At the bargain, I assume that the producer and distributor call for an arbitrator to decide

their dispute. This arbitrator will receive all the necessary information from both players and

will choose an allocation that shares the gains from trade between the partners. For example, he

would receive from the producer her information about the potential types of the distributor and

her beliefs about them, θ; also, the arbitrator knows that in order to default on the payments

the distributor needs to find an officer, which happens with probability 1 − λ. With these

informations, the arbitrator makes a take-or-leave-it offer to both agents, which they both accept,

by construction.

Thus, the arbitrator chooses an offer, a price pO for the good, obtained from the maximization

of the following Generalized Nash Product (NP ):

pO = arg maxp

(p (1− θ(1− λ))

)(RO − p

)1−θ(λ(RO − p) + (1− λ)RO

)θs.t.

RO − p ≥ 0

λ(RO − p) + (1− λ)RO ≥ 0

(5)

I obtain that pO = Ψ(λ, θ)RO. See the Appendix A for the exact formula for pO. Then,

UC = (1−Ψ((λ, θ))R and UP = (1− λΨ(λ, θ))RO. The function Ψ(λ, θ) gives the ex-post share

of each partner over the revenue from the sale of the goods. It is the Outsourcing’s twin for

FDI’s γ parameter ; however, here the impact of the incomplete information makes the sharing

14

an interesting multivariate function.

There is a point worth mentioning about the choice of the above mechanism for the bargain.

There is no reason to expect any kind of separating equilibrium here because in no way the

distributor would reveal his own type, if an opportunist, since I am assuming that he would go

to jail in this case and would prefer not to reveal. Consequently, the arbitrator does not ask

the distributor for his type. The Nash Problem of the arbitrator is trying to balance out the

wishes of the partners taking into account the uncertainty from the uninformed side and the

institutional elements of the country, which are known to all agents. Mathematically, that is a

price that distributes the gain from trade between the parties through the maximization of a

subjective risk weighted welfare function.

Comparative statics over the bargain price derived from (5) reveal that

∂pO∂θ

> 0;∂pO∂λ≤ 0. (6)

See Appendix A for the proofs. These results are intuitive: the producer price increases

when her beliefs about the distributor being an opportunist are greater, and decreases with

improvements in law-enforcement. That is, when faced with increased uncertainty, if the bargain

is successful the resulting price insures the producer accordingly, since she is the one bearing

all the default risk. For a given uncertainty over the distributor’s type, increased enforcement

allows the bargain price to decrease, that is, the business is less risky for the producer and thus

she gives up more of the surplus to the distributor. Also,

limθ→0

Ψ(θ, λ) = 1/2 , lim(θ,λ)→(0,0)

Ψ(θ, λ) = 1/2 , lim(θ,λ)→(0,1)

Ψ(θ, λ) = 1/2, (7)

lim(θ,λ)→(1,1)

Ψ(θ, λ) = 1/2 and lim(θ,λ)→(1,0)

Ψ(θ, λ) = 1.

That is, when the producer is near-certain the distributor is committed, the resulting price

is the traditional Nash result with an equal split of the surplus, regardless of the enforcement

level; when there is perfect enforcement, Nash result is obtained again since there is no risk that

the distributor will default. Finally, when the producer is near-certain that the distributor is an

opportunist and enforcement is very poor, the price share goes to 1, representing the enormous

risk facing her.

There are two different graphics with these results. First, take a look at Figure B.1, in

the appendix, showing in a 3D setting the graph of Ψ(θ, λ). Observe that for a given θ, Ψ is

non-increasing in λ, and for a given λ, Ψ is increasing in θ. Figure 3, plotted below, shows the

same results in a composite two-dimensional graphic.

The limit of Ψ(θ, λ) when θ → 1 for arbitrary levels of λ can be seen graphically in Figure

3. The graphic shows an interesting property of the Ψ function: observe that for λ = 0.3,

Ψ(1, 0.3) = 1, whereas Ψ(1, 0.6) < 1. Therefore, for a sufficient level of enforcement the producer

share in the bargain is below 1, even if the producer beliefs about the distributor being an

opportunist are almost sure. In this case, enough enforcement causes the producer to let go

15

Figure 3: Relation between Ψ, θ and λ.

some of the insurance in favor of the distributor. Therefore, when θ = 1 the function Ψ is kinked

at a λ = 1/2:6

Ψ(1, λ) =

1, if λ ≤ λ1

2λ , if λ > λ.(8)

Appendix A shows this kinked function in detail; it is also visible in Figure B.1. This result

is important considering that it will force me to split each forgoing analysis among several λ

intervals, as will be clear afterward. Now, continuing backwards, I will detail the producer’s

expected profit.

Investments

As before, at t = 1 the producer expects the distributor to make investments EO such that

EO = arg maxE

(1−Ψ(θ, λ))RO(E)− E, (9)

where RO(·) is increasing and concave.

For RO(E) = A√E, the distributor’s ex-ante profits are thus given by

πDO =A2

4

(1−Ψ(θ, λ)

)2. (10)

6In another version of the paper, I considered the possibility of an exogenous cost of corruption, that is, theopportunistic distributor would have to pay a bribe to the corrupt legal officer in order to obtain his help in thecourts. In this case, λ would be negatively correlated with the bribe, reflecting the fact that if corruption is toocostly, the real amount of enforcement is higher. Consequently, λ diminishes since it is possible to share some ofthe surplus to the distributor at lower levels of enforcement.

16

Accordingly, Producer’s profit is

ΠPO = (1− θ + θλ)

[A2

2Ψ(θ, λ)

(1−Ψ(θ, λ)

)+ TO

]− k,

where TO are the ex-ante transfers which are equal to the distributor expected ex-ante profits.

Observe that these transfers are determined at the beginning of the contract period, but in the

case of Outsourcing they are paid at t = 5, together with the payment for the goods.7 Therefore,

joint (and producer) expected profits are:

ΠO(θ, λ) ≡ ΠO(θ, λ) = (1− θ + θλ)

[A2

4

(1−Ψ(θ, λ)2

)]− k. (11)

Notice that ∂ΠO(θ,λ)∂θ ≤ 0 and ∂ΠO(θ,λ)

∂λ ≥ 0.

5 Producer’s Problem: choosing between Outsourcing and FDI

This section deals with the determination of the producer choice between Outsourcing ver-

sus FDI. In order to that, the producer must compare the expected future profits from either

operational mode. I start showing the value function from the point of view of Outsourcing.

5.1 Outsourcing

At the beginning of a period of contract (t = 0 in the timeline from figure 1), if the distributor

did not default the producer must decide whether to continue working with him. Again, there

is just one variable connecting the current period with later ones, the distributor’s reputation θ,

and I can write the producer’s expected aggregate future profits in terms of a value function as

follows

VO(θT+1) = max{0, ΠO(θT ) + β Pr(0|θT )VO(θT+1(0)}, (12)

where Pr(0|θ) = 1 − θT + λθT is the probability of no default, given that the distributor is

perceived as an opportunist with probability θT . This notation follows the one introduced at

section 3. The reason I am not considering the option to change from Outsourcing to FDI will

be clear later. Therefore, when starting an arm’s-length partnership at date T = 0 the producer

expected present-valued profit is

VO(θ0) = ΠO(θ0) + β Pr(0|θ0)VO(θ1(0)). (13)

7If the transfer were equal to the distributor expected profit including his illicit possibilities, them only oppor-tunistic distributors would show up to be contacted. Knowing that, the producer would automatically update hisprior to 1 and would be out. A better strategy is to charge ex-ante from anybody the value given by (10): withprobability 1 − θ + θλ the producer earns exactly this amount, and with probability θ(1 − λ) the producer earnsnothing.

17

Substituting back VO(θ1(0)) in (13) and subsequently, I obtain that

VO(θ0) = ΠO(θ0) +∞∑j=1

βjΠO(θj)

(j−1∏i=0

(1− θi(0) + λθi(0))

). (14)

Notice that VO(θ0) is decreasing in θ0, since both 1−θi(0)+λθi(0) and ΠO(θj) are decreasing

in θ, which is increasing in θ0 by (2).

5.2 Foreign Direct Investment

Now, consider the problem facing a producer when the partnership is via FDI. At t = 0, if

the distributor did not betray the producer, she must decide whether to continue working with

him via Outsourcing, via FDI or to quit the foreign market. If he betrayed, then I assume the

producer continue to operate forever with imported managerial labor 8. Again, the distributor’s

reputation θ is the unique variable connecting the current period with later ones. So, I can write

the producer expected future profits in terms of a value function as this

VI(θ) = ΠI

(1 +

βρ(1− Pr(0|θ))1− β

)+ Pr(0|θ)β ×max

{0, VI(θ|0, θ), VO(θ|0, θ)

}, (15)

where the first element in the right-hand side is the expected discounted producer’s future profit

in case the distributor betrayed in the former period, and Pr(0|θ) = 1− θ+ θλ is the probability

of no betrayal by the distributor. The producer proposes a partnership to the distributor if

VI(θ0) ≥ 0, where

VI(θ0) = ΠI

(1 +

βρ(1− Pr(0|θ0))

1− β

)+ Pr(0|θ0)β ×max

{0, VI(θ1(0)), VO(θ1(0))

}− f (16)

Building on the structure defined until here, next I presents two important results: first, I

show that in case there was no record of betray from the distributor, a change from FDI to

Outsourcing may take place, depending on the institutional quality of the economy. Obviously,

for this change to occur entry via FDI must have happened, so the second result are the factors

that determine the initial choice between FDI or Outsourcing.

Before, though, some assumptions to keep the problem framed and interesting.

5.3 Assumptions

Assumption 2a: A is high enough and k low enough such that when Ψ = 1/2, ΠO(·) > 0.

Using the results from (7) and Assumption 2a, I obtain the asymptotic behavior of ΠO(θ, λ):

8This begs the question of why foreign labor is not used since the beginning. Foreign labor is more expensiveand does not have the local knowledge. Thus, it is used just as the last resource.

18

limθ→0

ΠO(θ, λ) > 0, lim(θ,λ)→(0,0)

ΠO(θ, λ) > 0, lim(θ,λ)→(0,1)

ΠO(θ, λ) > 0, (17)

lim(θ,λ)→(1,1)

ΠO(θ, λ) > 0 and lim(θ,λ)→(1,0)

ΠO(θ, λ) < 0.

As was the case in (7), there is a limit missing from the above list, the one when θ → 1 for

an arbitrary λ. In order to determine it, it is necessary to use the results given by condition (8).

It allows me to define a unique critical value λ such that9

limθ→1

ΠO(θ, λ) = 0. (18)

Thus,

limθ→1

ΠO(θ, λ)

≥ 0 if λ ≥ λ,

< 0 otherwise.(19)

Figure B.2 in the Appendix B shows the graphic of ΠO when A = 1 and k = 0.05. A line

indicates the plane where ΠO(θ, λ) = 0 and the value of λ is also displayed in the figure.

Assumption 2b: A is high enough and k low enough such that when γ = 1/2,ΠI(1/2) > 0.

Assumption 3: f < f = limγ→1/2

(1−β(1−ρ))1−β ΠI .

This last assumption says that when incentives are at the maximum (and thus profits), even

if the distributor betrayed the producer immediately after entry it is still worthwhile to enter

via FDI, that is, expected discounted profits net of the fixed cost f are positive.

Remember that ∂ΠI∂γ < 0 and lim

γ→1ΠI < 0. Together with Assumption 2b, these imply that

there exists a value γ such that

f =(1− β(1− ρ))

1− βΠI(γ). (20)

In order to keep the model interesting such that a choice between FDI and Outsourcing could

be considered, I will assume that it is always profitable to enter via FDI, given a fixed costs f

satisfying Assumption 3. This lead us to

Assumption 4: γ ∈ (1/2, γ).

5.4 A Change from FDI to Outsourcing

Assuming entry took place via FDI, why and when a change of operational mode may

happen? Remember that the mechanism of learning produces decreases in the belief parameter

θ with the passage of time, which in turn increases the value of current period’s expected

9Notice that when Ψ equal one, profit is negative (see equation 11); according to hyphotesis 2a, profit is positivewhen Ψ equals 0.5. Equation (8) guarantees that there is an intermediate λ such that profit is zero.

19

outsourcing profits. After enough time, future prospects from Outsourcing may look better than

the ones from FDI and a change of operational mode takes place. The next Lemma formalizes

this idea.

Lemma 1. Suppose the producer entered the market in Foreign via Foreign Direct Investment.

Then, for λ ∈ [0, λc], λc > 1/2, there is a unique value θ∗ such that at this value the producer

decides to switch the operational mode. Additionally, θ∗ is increasing in λ.

Proof: Appendix A.

This Lemma assures the existence of a unique threshold such that, after some time learning

about the distributor’s type, the producer decides to change the operational mode. The condition

is valid just for λ ∈ [0, λc], since in according with the discussion in Section 4.2, for values above

this the enforcement is good enough and the bargain price is low enough such that for any θ

the per-period expected profit obtained from Outsourcing is always greater than that obtained

from FDI, and therefore there is no change to be decided since there is no way FDI would ever

be chosen from the beginning.

5.5 Producer’s problem: Mode of entry

The results presented in Lemma 1 rested on the assumption that entry had already taken

place via FDI, but this may not be the case. A relevant question to ask, though, is whether is

more profitable for the producer to enter via FDI, with a possible change later to Outsourcing,

or directly via the latter. Next proposition answers this:

Proposition 1. Suppose the producer is considering entering the market in Foreign, and that

whichever operational mode he chooses, the distributor enters the partnership if invited to. Then,

entry occurs according to the following pattern:

1. For λ ∈ (0, λ), there is a unique θ∗0 ∈ (θ∗, θ0], such that at it the producer is indifferent

between entry via FDI or Outsourcing. For θ0 > θ∗0, entry is via FDI; otherwise, is via

Outsourcing.

2. For λ ∈ [λ, λ), there is a unique θ∗0 ∈ (θ∗, 1], such that at it the producer is indifferent

between entry via FDI or Outsourcing. For θ0 > θ∗0, entry is via FDI; otherwise, is via

Outsourcing.

3. For λ ∈ [λ, 1), entry is always via Outsourcing.

Proof: See Appendix A. λ, λ are defined, respectively, in equations (A.13) and (A.14).

Therefore, a producer who entered with great uncertainty (a high θ0) over the commitment

of her business partner chooses a mode of entry which gives her greater control. Eventually, with

experience, she might become convinced of the true type of the distributor, and in this process

at some point she will change from one operational mode to the other.

20

However, this story depends on the institutional quality λ, as Proposition 1 shows. If en-

forcement is good enough, the producer enters directly via Outsourcing, since the probability

of default is low enough (due to the good institutions) to make her give up some rent in the

bargain and therefore the incentives to invest are greater than the case of FDI. If it is low, entry

happens via FDI as an insurance against the risk and the change only occurs if the partnership

lasts enough to bring θ close to θ∗.

6 Distributor’s Behavior

This section deals with the determination of the opportunistic distributor’s problem of

whether to steal the classified information, in the case of FDI, or the revenue, in the case

of outsourcing.

6.1 Foreign Direct Investment

I now solve the distributor’s problem when the operational mode is FDI. Consider first the

problem faced by an opportunistic distributor. By definition, he does not care much about the

future, thus is less interested in building a good reputation. Also, he does not betray unless

he is able to find a corrupt legal agent, since otherwise he would be forced by the courts to

compensate the producer and pay a penalty fee Z > 0. When the distributor does find a corrupt

agent, he must spend an effort l to discover the producer’s plans. If the distributor does not

betray, he simply receives a fraction 1− γ of the revenue.

Thus, upon finding a corrupt agent, the distributor chooses to betray as long as

α(1− ρ)ΠI

1− β− l ≥ 0, (21)

=⇒ ΠI ≥l(1− β)

α(1− ρ). (22)

The right hand side of (21) gives the payoff from selling the confidential information to

another producer: is equal to the losses to be incurred by the producer, brought to the present

by the producers’ discount factor β, multiplied by the share to be appropriated by the distributor,

α.10 I assume henceforth that this inequality is always satisfied, that is, l and ρ are small enough

and α and β high enough such that (22) is valid.

6.2 Outsourcing

In the case of Outsourcing, obviously he always accepts the producer’s offer since he never

loses from doing it. If he is committed, he does not default.11 If he is an opportunist, first he

10I assume there is a bargain between the distributor and a potential local producer.11Instead, suppose the committed distributor is not intrinsically honest as I assumed. Then, if his profits from

the operation were some positive value (due to search costs for example), there exists some discount factor suchthat for values above it, a distributor prefers not to default, as show Araujo and Ornelas in their model. In orderto simplify the model, I assumed perfect competition in the distributor market, preventing them from obtaining

21

must decide whether to default at the end of the period. In order to do this, he must find a

corrupt legal officer. So, upon finding this agent, he defaults only if he finds profitable to do so:

A2

2

(1−Ψ(θ, λ)

)>A2

2

(1−Ψ(θ, λ)

)2m

Ψ(θ, λ) > 0.

(23)

Therefore, if a corrupt agent is found, an opportunistic distributor always defaults, for any

revenue. Given this behavior, the distributor, either type, invest EO in the partnership, given

by (9). Notice that if the investments were different the distributor type would be revealed

and, case he were an opportunist, he knows the producer would immediately update his prior

to θ = 1 and the bargain would break-down, leaving him with nothing. Therefore, either type

of distributor invests the amount indicated.

7 Equilibrium

Now I solve for the equilibrium strategies and beliefs of the game played by the producer

and the distributor. In order to do that, I show that the distributor’s strategy described in this

section, together with the producer strategies described in Proposition 1, are part of an squential

equilibrium12

Proposition 2. There is a sequential equilibrium where a Producer with a initial belief θ0 enters

according to Proposition 1. The distributor always accepts a partnership offer and, if an oppor-

tunist, always defaults; if committed, never does. If entry was via Foreign Direct Investment,

then with probabilityT ∗0−1∏j=0

1− θj(θ0) + λθj(θ0)

there is a change of operational mode to Outsourcing, at belief θ∗ and time T ∗0 .

Proof. Proposition 1 gives the producer’s entry strategies, contingent on the distributor default-

ing or betraying if an opportunist (and if a corrupt officer is found). But defaulting/betraying

is a dominant strategy for the opportunistic distributor, therefore the producer strategy is a

best-reply to the distributor’s one. These strategies are also sequentially rational, since both

agents are optimizing over their choice space. Given these strategies, the sequence of beliefs

{θi}∞i=1 given by equation 2 is consistent, since are formed by the Bayes Rule. T ∗0 is given by

(A.20) .

some profit.12See Kreps and Wilson (1982) and Araujo and Ornelas (2008).

22

8 Institutional Reforms, Expectations and the Attraction of FDI

As I show now, this model gives interesting results about the effects of institutional improve-

ments in the Foreign country over producers’ expectations over the evolution of the partnership

to the point of a change to FDI. The focus on FDI is due to the importance of this measure to

developing countries, with the consequence that these nations compete fiercely to attract this

kind of investment, implementing policies aimed directly at them.

In short, an institutional change affects two core variables related to producers’ expectations:

first, the probability that the partnership will last until change a from FDI to outsourcing; second,

the time span until the attainment, by the partnership, of this point of change, conditional on

it being reached. Basically, I identify an ”institutional trap” where improvements augment

expectations in”developed” areas and diminish them in ”underdeveloped” ones. An in-depth

analysis follows.

8.1 The time span until the change of operational mode

Consider an ongoing relationship between a producer and his distributor. Suppose the change

of operational mode, if possible, happens at some θ∗ < θj , with due time T ∗j , where j represents

their current trust level (from some initial θ0). What would be the effect of the unexpected

change of λ on the time span between today and the change of operational mode, conditional on

it being attained? Section 5.4’s Lemma says that θ∗ increases with λ, reducing the time span.

However, the improvement also reduces the future reputation of the distributor, that is, not just

θ∗ is affected but also all the future values of θ that must be attained before they reach the value

θ∗; for a given θ∗, this increases the time span.

The impact of the change over the time span is given by

∂T ∗j∂λ0

=

∂θ∗

∂λ0log λ0

(1−θ∗)θ∗ −log

((1−θj)θ

θj(1−θ∗)

)λ0

(log λ0)2. (24)

Therefore, an improvement in the enforcement of contracts in Foreign causes a decrease in

the time span between entry via FDI and the change to Outsourcing if and only if

∂T ∗j∂λ

< 0⇐⇒ ∂θ∗

∂λ>T ∗jλθ∗(1− θ∗). (25)

Rewriting the inequality in terms of elasticities,

∂T ∗j∂λ

< 0⇐⇒ η(θ∗, λ) > T ∗j (1− θ∗), (26)

where η(θ∗, λ) is the elasticity of θ∗ with respect to λ.

Therefore, given that T ∗j is equal or greater than one, the impact of an institutional improve-

ment in θ∗ must be high enough to overcome the “bad” effect in the reputation. This is more

23

likely to happen when enforcement already attained some respectable level. In order to show

this, first I parametrize the model with reasonable values: β = 0.8, ρ = 0.75, A = 1, γ = 0.75

and (to simplify) j = 0.13 For these values, λ = 0.879655, that is, above this level entry is

direct via Outsourcing, thus with no room for a change of the operational mode. Given these

parameters, it is possible to obtain a (rather complicated) function of θ∗ depending on λ.

Figure B.3, in the Appendix B, shows the graph of T as a function of λ and θj . It was

constructed based on equation (A.20), substituting θ∗ for the function obtained from (A.6) and

taking into account that, for a given λ < λ, the θjs suitable for the analysis are those such that

θ∗0(λ) < θj(λ); for the other values of θj , T was set to Null, corresponding to the empty areas in

the figure.

Notice that the partial derivative of T with respect to λ is increasing for low values of λ

and becomes negative for higher values. Therefore, consider the case of a country with poor

institutions (low λ) and suppose that the prior belief of the producer, θ0, reflects this fact, for

instance θ0 > 0.5. In this case, an institutional improvement increases the time span necessary to

reach the trust level related to the change from FDI to Outsourcing, that is, the loss in learning

capabilities more than offset the increase in θ∗ that would allow the change to happen earlier.

Summarizing, for the given parameters an institutional improvement diminishes the time

necessary to attain the trust level related to the change of operational modes only if the country

already has a high level of enforcement. When enforcement is already high, any additional

tightening of it has a greater impact on θ∗ than on reputations, since the latter role is limited

if institutions are already good. There is a kind of institutional trap in this result, since the

improvements make more difficult for the agents to proceed towards the more efficient mode of

operation.

8.2 Probability of survival

The preceding analysis was made considering the effect of an institutional improvement over

the time span until the change from fdi to outsourcing, conditional on it being reached. But

the institutional improvement also effects the chance of survival of the partnership, that is, the

probability of that attaining θ∗. Therefore, starting from point θj , j ≥ 0, the probability that

the partnership will last until the attainment of θ∗ is

Pr(partnership lasts T ∗j periods) =

T ∗j −1∏i=j

(1− θi(θj) + λθi(θj)

). (27)

Changes in λ affect this probability through two channels: via the chance of not having a

betrayal or default in each of the preceding periods, and via the number of periods necessary to

reach the target event, that is, it affects T , the number of terms inside the productory. Figure

B.4 at the appendix, based on the same parameter values of the previous figure, shows the

13Calculations and figure were constructed with the software Mathematica 6.0.

24

graph of equation (27), in terms of the enforcement level λ and initial trust level θj ,with θ∗ as

the reference event in the future.

Notice that for a given θj , the probability is flat or increasing for the smaller values of λ, but

increases much faster for the larger ones. This pattern is caused by the two channels discussed

just above. The first one causes an small increase in the probability, given that every period

the chance of betrayal diminishes. Notice also some ripples in the figure: these are produced by

the second effect, the one of λ over T ∗j . Previous section showed that in general T ∗j is increasing

in λ for its lower values (due to the lock-in effect) and decreasing in it for the higher ones.

Therefore, starting from a given θj , j ≥ 0, as one moves along the lambda axis there will be

discrete changes in T ∗j , since in the calculation of (27) the time must be considered exactly as

in equation (A.20), that is, non-negative integers. For lower enforcement levels, the increase in

time adds another term to the product, counterbalancing the positive effect from the decrease

in each period’s probability of default it; the opposite happens when the time decreases with

higher enforcement levels, causing a huge increase in the probability.

9 Conclusion

The interplay between trust and contracts and their effects on the international organization

of production were analyzed here in a model with some novel features.

In a dynamic setting, I show that for lower levels of enforcement entry happens via Foreign

Direct Investment when the prior probability that the distributor is an opportunist is high and

via Outsourcing when it is low. However, for some high enough enforcement level entry is directly

via Outsourcing, regardless of the belief about the distributor. Conditional on entering via FDI,

I also show that there is a unique critical trust level such that at it a change of operational mode

takes place.

Finally, I study how institutional reforms affect producers’ expectations about doing busi-

ness in the foreign country via FDI. Results indicate an institutional trap where the lower the

institutional framework, the smaller the impact from the reforms over expectations.

The model generates some testable predictions: first, whether institutions (from trade fairs

to government promoted international business missions) that facilitate the exchange of infor-

mation across business partners promote the entry of multinational firms with a more flexible

organizational structure. Second, if different levels in the quality of enforcement of contracts

change the preferred way to enter a foreign country. Third, the effects of reputation-building on

the way multinational corporations organize their production internationally. That is, to test

whether after some time making business in a country with poor institutions (usually a devel-

oping one) the multinational decides to change from Foreign Direct Investment to Outsourcing

or some less rigid organizational structure due to improved confidence in the local partners.

25

References

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direct investment,” Journal of International Economics, 2003, 61 (2), 353–383.

Araujo, L. and E. Ornelas, “Trust-Based Trade,” Working Paper, 2008.

Baker, G., R. Gibbons, and K.J. Murphy, “Relational Contracts and the Theory of the

Firm*,” Quarterly Journal of Economics, 2002, 117 (1), 39–84.

, , and , “Relational Contracts in Strategic Alliances,” 2002. mimeo.

Bottazzi, Laura, Marco Da Rin, and Thomas F. Hellmann, “The Importance of Trust

for Investment: Evidence from Venture Capital,” SSRN eLibrary, 2008.

Brouthers, K.D. and L.E. Brouthers, “Why Service and Manufacturing Entry Mode

Choices Differ: The Influence of Transaction Cost Factors, Risk and Trust*,” Journal of

Management Studies, 2003, 40 (5), 1179–1204.

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Choice in Alliances,” Academy of Management Journal, 1995, 38, 85–85.

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Games with Incomplete Information,” Management Science, 1972, 18 (5), 80–106.

Hausmann, R., Research Dept, and Inter-American Development Bank, Foreign Di-

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355.

Johnson, S., J. McMillan, and C. Woodruff, “Courts and Relational Contracts,” The

Journal of Law, Economics, & Organization, 2002, 18 (1), 221–277.

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http://hbr.org/product/mcdonald-s-is-china-lovin-it/an/HKU802-PDF-ENG.

Kotseva, R. and N. Vettas, “Foreign Direct Investment and Exports Dynamics with Learning

about Demand,” Working Paper, 2005.

Kreps, D.M. and R. Wilson, “Sequential equilibria,” Econometrica: Journal of the Econo-

metric Society, 1982, pp. 863–894.

26

Mayer, R.C., J.H. Davis, and F.D. Schoorman, “An integrative model of organizational

trust,” The Academy of Management Review, 1995, 20 (3), 709–734.

Schnitzer, M., “Debt v. Foreign Direct Investment: The Impact of Sovereign Risk on the

Structure of International Capital Flows,” Economica, 2002, 69 (273), 41–67.

Straub, Stephane, “Opportunism, corruption and the multinational firm’s mode of entry,”

Journal of International Economics, 2008, 74 (2), 245–263.

27

Appendix A

Formula for the bargain price

From equation (5), I obtain

pO =RO.Ψ(θ, λ, b)

= RO ·

(2− θ + λ(1 + θ)−

√(2− θ + λ(1 + θ))2 − 8λ

)(A.1)

Partial derivatives of the bargain price

From (A.1), define s(θ, λ) = 2 + λ+ θ(λ− 1). First, some results I will use later:

s(θ, λ) > 0

s(θ, λ) ≥ λ+ 2

s(θ, λ)2 − 8λ > 0

The last is less obvious: s(θ, λ)2 = ((1 + 2λ) + (1− θ)(1− λ))2 > (1 + 2λ)2 > 8λ.

Thus,∂pO∂θ

= (1− λ)(s(θ, λ)−

√s(θ, λ)2 − 8λ

)> 0 (A.2)

∂pO∂λ

= (2− θ)(s(θ, λ)−

√s(θ, λ)2 − 8λ

)− 4λ (A.3)

Notice that(s(θ, λ)−

√s(θ, λ)2 − 8λ

)− 2λ = (s(θ, λ)− 2λ)−

√s(θ, λ)2 − 8λ

= (s(θ, λ)− 2λ)−√

(s(θ, λ)− 2λ)2 + 4λs(θ, λ)− 4λ(λ+ 2)

= (s(θ, λ)− 2λ)−√

(s(θ, λ)− 2λ)2 + 4λ (s(θ, λ)− (λ+ 2)) ≤ 0,

(A.4)

since it is equal to zero for some values of λ when θ = 1. Therefore, ∂pO∂λ ≤ 0.

Derivation of the critical value λ

When evaluated at θ = 1, pO from (A.1) becomes

pO =RO.Ψ(1, λ)

=R ·(

1 + 2λ− |1− 2λ|4λ

) (A.5)

28

Notice that Ψ(1, λ, b) equals 1 whenever what is inside the absolute is positive. Therefore,

the critical value λ is the λ such that |1− 2λ| = 0, =⇒ λ = 12 .

Proof of Lemma.

Suppose that there was entry via FDI and the distributor did not betray the producer. Thus,

it is possible that after some time a operational mode change takes place. At this point, which is

related to a trust level θ∗, the producer would be indifferent between making FDI for one more

period and outsourcing from there on or starting outsourcing right at that moment:

ΠI

(1 +

βρ(1− Pr(0|θ∗))1− β

)+ Pr(0|θ∗)βVO(θ1(θ∗(0))) = VO(θ∗)

ΠI

(1 +

βρ(1− Pr(0|θ∗))1− β

)= VO(θ∗)− Pr(0|θ∗)βVO(θ1(0))

= ΠO(θ∗)

(A.6)

The right-hand side is decreasing in θ, whereas the left-hand side is increasing. Call RHS

the right hand side of (A.6), and LHS the left hand side. Then, using (7) and Assumption 4,

limθ→0

LHS < limθ0→0

RHS (A.7)

The limits approaching 1 are again more difficult. But it is possible to show that there exists

a unique λc such that

limθ→1

LHS = limθ0→1

RHS (A.8)

⇒ ΠI

(1 +

βρ(1− λc)1− β

)= ΠO(1;λc) (A.9)

Notice that λc ≥ λ = 1/2. This cutoff exists: first,the right hand side of (A.6) is increasing

in λ for any θ whereas the left hand side is non-increasing and linear. Second, from (7) and

Assumption 4,

lim(θ,λ)→(1,1)

LHS = Π(γ) < lim(θ,λ)→(1,1)

RHS = ΠO(1; 1)

lim(θ,λ)→(1,0)

LHS = Π(γ)

(1 +

βρ

1− β

)> 0 > lim

(θ,λ)→(1,0)RHS = ΠO(1; 0)

Thus, for λ ∈ (0, λc], limθ→1

LHS > limθ0→1

RHS and θ∗ is unique. Finally, given that both

ΠO(θ∗, λ) and Pr(0|θ∗) are decreasing in θ and increasing in λ, θ∗ is increasing in λ.

Proof of Proposition 1.

29

The proof follows the following procedure: first I define a hypothetical indifference condi-

tion (between FDI and Outsourcing), followed by convexity and asymptotic properties of this

condition

Consider the value functions from entry via Outsourcing and from entry via FDI, presented

respectively in sections 4.2 and 4.1. Notice that

limθ0→0

VO(θ0) =ΠO(0;λ)

1− β, (A.10)

and

limθ0→0

VI(θ0) = ΠI + limθ0→0

(Pr(0|θ0)×max

{βVI(θ1(0)), βVO(θ1(0))

})− f

≤ ΠI +βΠO(0;λ)

1− β− f

<ΠO(0;λ)

1− β,

(A.11)

since the maximum expected value possible for future profits is βΠO(0;λ)1−β , and ΠI < ΠO(0;λ),

by Assumption 4 and equation (7). Therefore, for prior beliefs close to 0, that is, when the

producer is near-certain that the distributor is committed, entry will be via Outsourcing.

Now, the limits when θ0 approaches 1. First, see that

limθ0→1

VI(θ0) = ΠI

(1 +

βρ(1− λ)

1− β

)+ limθ0→1

λmax{βv(θ1(0)), βVO(θ1(0))

}− f > 0, (A.12)

by Assumption 4.

In order to calculate the limit of VO(θ;λ) when θ approaches 1, I am forced to split the

analysis among some cases. In order to do that two cutoffs must be defined. Based on (8) and

(17) I define a cutoff λ such that

VO(1; λ) =ΠO(1; λ)

1− λβ= 0. (A.13)

For the second, define λ such that when θ = 1 the producer is indifferent between outsourcing

or FDI:

ΠI

(1− β + βρ(1− λ)

(1− β)(1− λβ)

)− f = VO(1; λ). (A.14)

This cutoff exists, since the right hand side is non-decreasing in λ (equation (8)), the left

hand side is increasing in λ, lim(θ,λ)→(1,1) ΠO(θ, λ) > ΠI and lim(θ,λ)→(1,0) ΠO(θ, λ) < 0 <

ΠI

(1 + βρ

1−β

)− f (equation (7) and Assumption 4). Notice that λ < λc.

With these cutoffs, the limit of VO(θ0;λ) when θ0 approaches 1 is immediately split among

three cases:

30

Case 1: λ ∈ (0, λ).

limθ0→1

VO(θ0;λ) =ΠO(1;λ)

1− λβ< 0. (A.15)

Case 2: λ ∈ [λ, λ).

0 ≤ limθ0→1

VO(θ0;λ) < ΠI

(1− β + βρ(1− λ)

(1− β)(1− λβ)

)− f. (A.16)

Case 3: λ ∈ [λ, 1).

limθ0→1

VO(θ0;λ) ≥ ΠI

(1− β + βρ(1− λ)

(1− β)(1− λβ)

)− f. (A.17)

Now, notice that when there is ex-ante certainty that the distributor is committed, entry via

Outsourcing is preferable, since by equation (6),

ΠI

1− β− f < ΠO(0)

1− β(A.18)

Besides this special case, the choice between FDI and Outsourcing rests on the values of the

parameter θ0, that is, on the initial reputation of the distributor.

From (A.10) and (A.11), it is clear that for prior beliefs close to 0, that is, when the producer

is near-certain that the distributor is committed, entry will be via Outsourcing, regardless of

the λ value, confirming (A.18). But as (A.12), (A.15),(A.16) and (A.17) show, the result is not

so straight for θ close to 1. Therefore, it is instructive to analyze the entry decision in some

separate cases, which in turn are determined exactly by the λ intervals defined in equations

(A.15)-(A.17).

Case 1: λ ∈ (0, λ)

For λ in this range, there is a value θ0 such that, for θ0 > θ0, VO(θ0) < 0; otherwise, is equal

or greater than zero. First, note that

limθ0→0

VO(θ0;λ) =ΠO(0;λ)

1− β> 0 >

ΠO(1;λ)

1− λβ= lim

θ0→1VO(θ0;λ).

Also, remember that ΠO is increasing in λ and VO(θ0) decreasing in θ0. Applying these

results to (A.13), the result follows.

Therefore, for θ0 > θ0, entry is via FDI. From (A.10), (A.11) and (A.12), for θ0 close to 0,

entry happens via Outsourcing. Thus, there may be a threshold θ∗0 such that at it the producer

is indifferent between entering via FDI or via Outsourcing:

VI(θ∗0) = VO(θ∗0)

31

Π(γ)z0 + x0z1βΠ(γ) + x0x1z2β2Π(γ) + · · ·+

(T−1∏i=0

x(θi)

)βTVO(θ∗T (0))− f =

ΠO(θ0)+βx(θ0)ΠO(θ1) + · · ·+ βT

(T−1∏i=0

x(θi)

)VO(θ∗T (0))

(A.19)

where xi = 1− θi +λθi and zi = (1 + βρ(1−Pr(θi|0))1−β ). Notice that θ∗T is the one obtained in (A.6),

where the producer is indifferent between the two operational modes, and T is the size of the

history preceding this event. T is endogenous for θ ∈ [θ∗, 1], and can be represented as:

T ∗0 ≡ T (θ0|θ∗) =

⌈ln (1− θ0)θ∗ − ln(1− θ∗)θ0

lnλ

⌉(A.20)

Therefore, (A.19) actually represents the equality of two finite series:

B︷ ︸︸ ︷Π(γ;A)

z(θ0) +

T ∗0−1∑j=1

βjzj

(j−1∏i=0

x(θi)

)− f =

C︷ ︸︸ ︷ΠO(θ0) +

T ∗0−1∑j=1

βjΠO(θj)

(j−1∏i=0

x(θi)

)

⇓(1 +

βρ

1− β

)Π(γ;A)

1 +

T ∗0−1∑j=1

βj

(j−1∏i=0

x(θi)

)︸ ︷︷ ︸

D

−f = ΠO(θ0) +

T ∗0−1∑j=1

βjΠO(θj)

(j−1∏i=0

x(θi)

)︸ ︷︷ ︸

E

+

βρ

1− βΠ(γ;A)

T ∗0−1∑j=1

βjx(θj)

(j−1∏i=0

x(θi)

)+βρΠ(γ;A)

1− βx(θ0)︸ ︷︷ ︸

F

(A.21)

Both D, E and F are decreasing in θ0, since ΠO(θk) and x(θk) are decreasing in θk, for any

k, and θ(k) in turn is increasing in θ0 (see equation (2)). Finally, whereas the second derivative

of E and F is not clear, D is linear, since the productory is linear in θ0. Notice that if f were

zero, for θ ∈ [θ∗, 1] entry is via FDI, otherwise is via Outsourcing ; that is, θ∗ from Lemma is

also a threshold for entry choice.

Also,

limθ0→θ∗+

B < limθ0→θ∗+

C (A.22)

m(1 +

βρ(1− x(θ∗))

1− β

)ΠI − f < ΠO(θ∗) (A.23)

32

which is true from (A.6), and

limθ0→θ−0

B > limθ0→θ−0

C (A.24)

m

Π(γ;A)

z(θ0) +

T ∗0−1∑j=1

βjzj(θj)

(j−1∏i=0

x(θi)

)− f > ΠO(θ0) +

T ∗0−1∑j=1

βjΠX(θj)

(j−1∏i=0

x(θi)

)(A.25)

Rearranging,

f < Π(γ;A)

z(θ0) +

T ∗0−1∑j=1

βjzj(θj)

(j−1∏i=0

x(θi)

)−ΠO(θ0)−T ∗0−1∑j=1

βjΠX(θj)

(j−1∏i=0

x(θi)

)

= Π(γ;A)

z(θ0) +

T ∗0−1∑j=1

βjzj(θj)

(j−1∏i=0

x(θi)

)−ΠO(θ0)−T ∗0−1∑j=1

βjΠX(θj)

(j−1∏i=0

x(θi)

)

+ βT

(T−1∏i=0

x(θi)

)VO(θ∗T (0))− βT

(T−1∏i=0

x(θi)

)VO(θ∗T (0))

= VI(θ0)− VO(θ0) = VI(θ0)

(A.26)

since VO(θ∗0) is equal to zero at θ0. By Assumption 3, the above inequality is valid. Therefore,

in this λ interval, there is a unique θ∗0 ∈ (θ∗, θ0], such that the producer is indifferent between

entry via FDI or Outsourcing. For θ0 > θ∗0, entry is via FDI; otherwise, is via Outsourcing.

Case 2: λ ∈ [λ, λ)

In this case, the only difference from the previous case are the limits approaching 1:

limθ0→θ∗+

B < limθ0→θ∗+

C (A.27)

m(1 +

βρ(1− x(θ∗)

1− β

)ΠI − f < ΠO(θ∗) (A.28)

which is true again from (A.6), and

limθ0→1

C = ΠI

(1− β + βρ(1− λ)

(1− β)(1− βλ)

)− f (A.29)

limθ0→1

C =ΠO(1;λ)

1− βλ≥ 0 (A.30)

where limθ0→1

T ∗0 = ∞. Both B and C are positive, with B > C: from (A.15) and (A.16), when

33

λ→ λ+, B > C; when λ→ λ−, B > C.

Case 3: λ ∈ [λ, 1)

limθ0→θ∗+

B < limθ0→θ∗+

C (A.31)

m(1 +

βρ(1− x(θ∗)

1− β

)ΠI − f < ΠO(θ∗) (A.32)

which is true again from (A.6), and

limθ0→1

A < limθ0→1

B (A.33)

m (A.34)

ΠI

(1− β + βρ(1− λ)

(1− β)(1− βλ)

)− f < ΠO(1;λ)

1− βλ(A.35)

which is true by definition (see (A.17)).

Appendix B

Figure B.1: Relation between Ψ, θ and λ.

34

Figure B.2: ΠO(θ, λ), showing λ. The red line indicates the grid line corresponding to ΠO(θ, λ) =0.

Figure B.3: Relation between T ∗k , θ0 and λ.

35

Figure B.4: Probability of the partnership lasting until trust level θ∗, in T ∗k periods.

Information and ownership choice

Abstract

This paper shows that a transaction-costs model embedded with interfirm information

exchange and learning provide new results about the optimal vertical organizational form in

a supply chain.

The model is applied to study the effects of shocks over the organizational decisions of

firms, with a new type of vertical fdi described, without reference to factor costs: shocks

prompts an input supplier to acquire his buyer and quit the former business. I exemplify this

effect with documented actions and characteristics of two leading Brazilian meat-exporting

firms.

37

1 Introduction

The literature in optimal vertical organizational form in Industrial Organization has ad-

vanced remarkably since Williamson (1971, 1975, 1979, 1985) and Grossman and Hart (1986)’s

seminal works. In the present decade there was a renewed interest in this topic insofar the incom-

plete contract theory gave an useful foundation to explain much of the changes that happened

internationally with the globalization process1.

At the same time, an emerging literature in Organization Theory2 studies several different

aspects of the organization of the firm, both in terms of the separate units that constitute an

integrated firm and inside these units. This literature aims to explain the determinants and

consequences of Delegation, Centralization, Hierarchies, and other different kinds of governance

structures that exist. An important message of these works is the relevance of understanding the

appropriate ways communication flows between productive units in a economy, be these units

simple agents or larger structures, and how these flows contributes to give the firm a shape.

This paper intends to bring these two literatures a little closer, investigating the consequences

of information flows and knowledge exchange for the choice of optimal vertical organizational

form among productive units in a multi-layered supply chain. In other words, how information

and knowledge affect the decision each owner of a productive unit in a supply chain must

take: the best way to maximize profits in a supply chain is integrate or outsource with the

business partners? This is done through a transaction costs dynamic model that departs from

the traditional setup: there are no investment-related incentives and the productive process is

shaped by several persistent economic structures.

In a nutshell, I obtain that the decision of two units to integrate or outsource will depend on

the interplay among some factors: first, there are governance fixed costs related to integration;

second, integration may provide interfirm externalities that improve the joint profit of the two

units, vis-A -vis outsourcing; third, all firms in the supply chain are subject to random shocks

that prompt them to exchange information with their business partners. This exchange implies

that the firms learn about each other business and develop some kind of partnership-specific

knowledge capital. Fourth, both the strength of the externalities and of the governance costs

depend on the size of this knowledge capital. Fifth, in turn, the process of building the knowledge

capital depends on three key characteristics of the firms: their quality, a proxy for the dexterity

of the firm’s labor force in handling information knowledge; their size (number of inputs); the

degree of relationship between the firms in the supply chain, what I call connectivity.

An important message of this model is that the interconnected nature of supply chains

matters to explain their shape. Specially, the study of interfirm knowledge sharing is relevant,

since vertical decisions are not just about minimizing factor costs but also about appropriating

1For example, Grossman and Helpman (2003, 2005), Antras and Helpman (2004), Helpman (2006), amongothers.

2See Bolton and Dewatripont (1994), Puga and Trefler (2002), Dessein and Santos (2006), Cremer et al.(2007), Hart and Moore (2005), Rajan and Zingales (2001), Radner (1992), Radner (1993), Harris and Raviv(2002), Garicano (2000).

38

knowledge, as Antras and Rossi-Hansberg (2008) stress when discussing future directions for the

literature. In their landmark work, Ethier and Markusen (1996) recognized the importance of

knowledge-based capital for the optimal structure of an organization. However, their focus was

on specific technologies, embodied in products. Here, I focus on partnership-specific knowledge,

obtained only with the passage of time.

This model ends with some advantages compared to the workhorse of the literature, the

Property Right Theory (PRT, henceforth). First, is based on Transaction Costs Theory (TC

henceforth), which as stressed by Lafontaine and Slade (2007)’s survey, has a much better

empirical record than PRT. Second, it provides linkages with the Theory of Organization without

sacrificing the study of the supply chain, as in usual analysis of bilateral relationships.

Third, the local and economy-wide nature of the two empirical studies cited above make

models based on PRT poor fitted to offer an explanation for them. One reason is the dependence

of PRT on investment-related hold-ups which, as stressed by Hart and Holmstrom (2008), are

still ill-defined concepts; thus, the power of PRT to explain the shape of the supply chain is

more useful with static problems related to the initial establishment of innovative industrial

firms, specially concerning R&D issues, than to explain the broad structural elements behind

the development of a generic supply chain, which is a dynamic problem. Moreover, PRT is

more useful to explain settings in which large differences in factor costs between locations allow

an optimal arrangement of the spatial distribution of firms in a supply chain, to maximize the

return of their different types of investments. This is the success story of PRT in explaining

the globalization process, through the papers cited in the first paragraph. However, is not clear

why these elements would play such an important role into national economies, which are more

homogeneous in factor costs.

I use a novel term, connectivity, to measure the degree of a relationship between inputs

used in productive units. Connectivity is a measure of the number of channels for exchange of

information that exists between participants in some productive process.

Next, I show in detail the main elements of the model, followed also by a detailed description

of the results.

Characteristics of the model

First, I embed interfirm learning dynamics into the buyer-supplier relationships in supply

chains: units are subject period after period to shocks that prompt them to adapt their products

and managerial practices. In order to do that, these units exchange information with their buyer

and suppliers, given that these must also make changes in their own units. Also, given that the

buyers and suppliers participate in the readaptation effort, their own buyers and suppliers are

also called on: any shock reverberates upwards and downwards through all the production chain,

and at any time any unit may receive information not just from its suppliers but also from its

buyer, developing a knowledge stock about each of them.

Second, based on information theory, I show the information one unit absorbs with relation

39

to the another at any point in time depends on two elements: reception and comprehension of

the informational flow, which are functions of the units’ dexterity in handling information flows.

Third, units may be associate through one of two possible operational modes: outsourcing or

integration. The latter internalizes the externalities between the units, decreasing ordinary costs

relative to Outsourcing, but is subject to governance costs. These governance costs represent

costs that are absent only in a partnership that lasts for some time, like the other unit’s manage-

rial culture, knowledge of and relationship with the partner’s buyers and suppliers, preferential

agreements, detailed information about the partner’s production process, and so on.

Fourth, the exchange of information between units with the passage of time augments the

knowledge capital of one with relation to another. Consequently, it diminishes the governance

costs of integration and improves the strength of externalities in integration; Thus, this effect

sheds light on the findings of Kellogg (2008), the unique empirical paper in my knowledge

on interfirm relationship dynamics. This author shows that there are considerable productivity

gains from interfirm relationships, specially on non-technical dimensions. Acemoglu et al. (2004)

also shows that integration is correlated with the age of the firms, which hints at the importance

of dynamic considerations in the vertical decisions of the them.

Main results of the model

Based on this whole structure, I analyze the choice betweens Integration versus Outsourcing

in two different times: in a static environment with instantaneous information sharing, and in a

dynamic one where learning happens with time.

Under the static, instantaneous information model, I obtain the following results:

• first, the unit of the highest quality (dexterity in handling information flows) has just lower-

quality integrated supplier units to itself, no outsourced ones. In turn, these integrated

units may have some integrated units to themselves, and some outsourced, depending on

this: the lower the difference in quality of the parent unit relative to the supplier unit,

and the lower the quality of the parent unit, the greater the chance that this supplier unit

will be outsourced. Therefore, from the highest level of the supply chain to the lowest,

firms have increasingly more outsourced units and less integrated ones. This fits well with

the stylized fact that the most important managerial units are not outsourced or kept in

a separated entity, as discussed by Hortacsu and Syverson (2009). Integration is always

downward;

• low-quality units are small and numerous, whereas top-quality units are large and inte-

grated and the medium-quality units have intermediate configurations;

• based on the above results, this model sheds light on two conflicting empirical results, be-

tween Hortacsu and Syverson (2009)’s results of a positive relationship between integration

and suppliers productivity and Acemoglu et al. (2004)’s results on a inverse relationship

between supplier’s R&D and downward integration.

40

Under the dynamic model, I show that downward integration is more likely, among other

determinants, the higher is the the buyer’s quality, the lower the supplier’s quality relative to

the buyer’s one. However, the higher the difference in quality, the slower the learning process

for the partner and thus integration takes more time to come, although this is a second order

effect.

Contrary to the static model, here forward integration is viable, and also is more likely the

higher is the the buyer’s quality, the lower the supplier’s quality relative to the buyer’s one

and, as a first-order effect, the higher the connectivity of the supplier. However, the larger

the difference in quality, the more time it takes to reach it, here as first-order effect. Thus,

integration comes sooner for top quality firms close but not equal in quality, and the higher the

connectivity of the supplier. Either case, the position of the firms in the chain influences the

gathering of information and the probability of the merger.

These results hint to some reasons forward integration is less likely, as shown in data: first,

due to the learning advantages of buyers relative to suppliers; second, due to the first-order effect

of the difference in quality for forward information gathering, related to the second-order in the

case of downward.

Finally, the dynamic model is applied to study the effects of regulatory shocks over the

organizational decisions of units, with a new vertical foreign-direct investment effect described:

shocks prompts an input supplier to acquires his buyer and quit the former business. The model’s

prediction about the production chain characteristics that enhance the likelihood of this effect is

confirmed with documented actions and characteristics of two leading Brazilian meat-exporting

firms.

Findings with relation to the Literature

This paper is related to two strands of the literature. First, there is a vast number of papers

that study the design of internal organizations, that is, from inside the firm. The literature is

huge, for instance: Williamson (1967) studied hierarchical structures from the side of produc-

tivity enhancement and span of control, Rosen (1982), Harris and Raviv (2002) and Garicano

(2000) from the view of task assignment and coordination, Radner (1992, 1993) from the view

of information processing, Aghion and Tirole (1997), Baker et al. (1999), Rajan and Zingales

(2001) and Hart and Moore (2005) from the view of authority, developing the concept of delega-

tion. Second, there are those papers that look through the external view of the firm, like Antras

and Helpman (2004), Hart and Holmstrom (2002) and Acemoglu et al. (2004).

What is the difference between this paper and these others? This is the first paper to consider

the effect of an heterogeneous supply chain with two-way communication between firms in the

organizational mode. Albeit the results of this paper may look similar to Antras and Helpman

(2004) and Acemoglu et al. (2004) results, here I do not use property-rights theory, actually

incentives do not play any role3; I also endogenize the size of each unit, their informational

3Contrary also to the works by Acemoglu et al. (2007) and Hart and Holmstrom (2002).

41

channels and show with details the possibilities on which not just downward but also forward

integration happens, which is absent in the literature and allowed me to explain and describe

the occurrence of an new type of FDI.

Second, the dynamics of learning bring new results to the effect of the decision to integrate

versus outsourcing: contrary to Hart and Holmstrom (2002), outsourcing does not comes from

a deliberate behavior of agents in ignoring externalities, but is the result of a difficult process of

learning that takes time to develop.

The paper proceeds as follows. Section 2 describes the economic environment. Section 3

determines the fundamental, building blocks of the model. Section 4 defines the informational

environment and at Section 5 the learning mechanism, based on the informational environment,

is explained. Section 6 integrates all elements to build the economic components of the model:

revenue and costs functions. Section 7 uses the whole previous structure to analyze the choice

between integration versus outsourcing at the static, instantaneous information model. Section

8 discusses a puzzle in the literature and shows how the model helps understanding it. Section

9 shows the dynamics of integration versus outsourcing. Section 10 applies the dynamic model

to the behavior of two large Brazilian units facing regulatory shocks, showing a different kind of

vertical foreign-direct investment. Section 11 concludes.

2 Background

Suppose there is a final good in the economy that must be produced: thus, a supply chain

must be structured, but what is a supply chain?

A supply chain represents the production process behind the final good. Different from the

traditional view of a production function, it gives attention to the specifics behind the relationship

between all the elements that make part in the various processes that constitute the final good;

thus, it presents the various underlying production processes of the production function.

The aim of this paper is evaluate how specific characteristics of an arbitrary supply chain

affect the decision of firms in a economy to integrate or outsource. In order to start, this section

introduces the background economic environment related to the structure of this arbitrary supply

chain.

The economy where the supply chain exists is a discrete-time one. In this economy, units

are the standard types of structure in the productive process and I assume the whole economy

is composed of a infinitely large number of them. Thus, supply chains are made of units.

Definition 1 (Units).

A unit is an element of the productive process, such that:

• There are some individuals in the economy that own it;

• There is an homogeneous output produced at it;

42

• At least one individual works at it.

Definition 2 (Inputs).

Each unit in the supply chain makes one unit of a output each period. This output is produced

from a combination of some intermediate inputs or labor. Intermediate inputs are the outputs

from other units; labor inputs are those from workers inside each unit.

The economy is populated by many supply chains, each made of different number of units.

It is not necessary for this paper to take some specific supply chain as reference. Thus, when I

refer to the supply chain, I am referring to an arbitrary one with at least 2 units belonging to it

(otherwise there is no chain).

In this supply chain, each supplier sells just to one buyer, but each buyer can have multiple

suppliers. All buyer-supplier unit pairs in this supply chain have a business partnership that

lasts indefinitely.

Consider a long-term relationship between two units of this supply chain: an forward unit

(supplier) and a downward one (buyer). The governance structure of this partnership is of two

possible forms: the units can be either Integrated, or Outsourced with relation to each other. As I

show later, at Integrated units information flows between units is faster compared to outsourced

units; however, there are governance costs of integration. This decision is evaluated at the

beginning of each period t of their partnership.

At any period t, following the definition of the ownership structure of the units, there is the

realization of a shock over some units in the supply chain. This temporary shock prompts some

units to change their outputs or internal environments. After a unit receives the call to change,

she arranges a meeting with her suppliers and her buyer to address the necessary changes the

partners must also accomplish. Summarizing, any shock reverberates not just upward in the

production chain but also downwards, and a output related to several different inputs expects to

be subject to a greater number of shocks with the passage of time, directly or indirectly. After

the shock, the units that received it make the necessary adjustments.

Thus, following the realization of the adaptations, the buyer-supplier pairs exchange proto-

types and based on this they meet to discuss the terms of trade, that is, the prices.

After that, the first unit in the supply chain produces its output and sends it to its buying

unit, who receives it along other inputs, and produces its own output, and so on, until the final

output is made. Figure 1 next page summarizes this story.

43

Figure 1: Timeline.

Figure 2 below presents an example tree of a four stage supply chain. This example will

also be used next in this section and in others to facilitate the understanding of the model. The

vertical dimension of the supply chain in the example is 4. In turn, each stage is made of 1 to 3

inputs.

Figure 2: Example of a Supply Chain Tree.

As the figure shows, supply chains are complex. However, I will use a simple notation when

referring to units in the chain: I am going to consider one buyer unit (i) and her forward

suppliers, called indistinctly by (k). Figure 3 next page shows this easier notation.

Next section I explain in detail exactly the properties associate with units that constitute

the supply chain (and the economy). After that, sections 4 and 5 explain the shocks and

informational flows along the supply chain, and the mechanism used by units to learn from each

other.

44

Figure 3: Notation to be used throughout the paper.

3 Structure of the Units

This section presents the structural properties endowed to the units in the supply chain; as

I show along the paper, these properties have a great impact on the way units relate to each

other.

3.1 Characteristics

Each unit in the economy is born with three characteristics associated with. The three char-

acteristics are: quality, related to the dexterity of the unit’s workers in handling informational

flows from other units.; size, related to the number of intermediate inputs necessary to the pro-

duction of the output of the unit; and connectivity, a measure of the way the inputs of some unit

are related to each other. At birth, each characteristic is drawn from appropriate distributions.

Notice that the model’s findings are dependent neither on any particular combination of these

characteristics.

Next, I define these three in detail and explain their role in the paper.

Definition 3 (Size). Size refers to the number of intermediate inputs some unit uses in the

production of its output. Formally:

Each unit (i) in the supply chain makes one unit of a output each period from the combination

of some inputs. There are two types of inputs: intermediate inputs, and one labor input. Define

• |Mi| = mi as the number of intermediate inputs that are used by unit (i);

As I show in section 5, the greater the size of the business partners of some unit, the greater

the expected amount of information this unit is going to receive and learn from these other units.

This effect is important for choice between integration versus outsourcing in a dynamic model,

at section 9.

The definition of connectivity is made through the development of two concepts, presented

in sequence.

Concept 1 (Messages). Consider two inputs k, k′of unit (i) (either labor or intermediate).

Communication between them is done through a message Mk,k′, which is a informative content,

45

an information for short, about k that reaches k′. If the message Mk,k′ is totally absorbed by k′,

then k′ is capable to reproduce it; consequently, is able to reproduce part of k.

Concept 2 (Connections). A connection between two inputs is the existence of a channel for

transmission of information between them. Formally:

A connection between inputs k and k′ is a directed and oriented graph4, that is it, an

ordered pair (Vk,k′ , Ak,k′), where Vk,k′ = {k, k′} is a set of vertexes and Ak,k′ an oriented ordered

pair (k, k′) (called an arrow), such that

• k is the the sending vertex;

• k′ is the the receiving vertex;

• Unlimited messages Mk,k′ are allowed to flow between k and k′.

Figure 4 presents a directed and oriented graph:

Figure 4: A connection between inputs k and k’: a direct oriented graph.

In terms of the example at figure 3, there are two conections: from i to k and vice-versa.

Definition 4 (Connectivity). Connectivity is a measure of the interaction of some input with

respect to all the necessary parts that the output is made of. Formally:

Let Gi be a set containing the connections, in the sense of Concept 2, among all its in-

puts that participate into the production process of the output of unit (i). Formally, Gi =

{(Vk,k′ , Ak,k′) ∀ k, k′ ∈Mi}. This set has the following properties:

• Each element in Gi is a connection as defined in Concepts 1 and 2;

• Gi(k) ⊂ Gi, k ∈ Mi, is the set of connections where input k is present as a sender or as

a receiver;

• The connectivity of an input k ∈Mi is si(k) = |Gi(k)||Gi| ∈ [0, 1];

In terms of figure 3, set Gi has dimension equal to 1 (there is just one input overall), and

the connectivity of input k is equal to 1 since i is made entirely from k.

As I show in sections 4 and 5, ceteris paribus a supplier with greater connectivity to his

buyer will learn more from it, but does not affect the buyer learning from the supplier. This is

important when I analyze the difference between downward and forward integration in a dynamic

setting, at section 9.

4Graphic demonstration here: http://mathworld.wolfram.com/OrientedGraph.html

46

Definition 5 (Quality).

Quality of some unit (i), represented by qi ∈ [0, 1], is a measure of this unit’s labor dexterity

in handling informational flows from other units. The greater the quality, the greater the units

capacity to receive, comprehend and execute information transmitted from other units. Thus, it

is a measure of productivity in dealing with information flows.

The effect of quality is as follows: If some message is transmitted to some unit (k), it receives

just the fraction qk of it, comprehends a fraction qk of the information available at the received

message and, depending on the ownership mode, executes it with dexterity determined by its

quality. The next three sections are dedicated to explain those effects in greater detail.

4 Shocks and information flows

This section is concerned with the definition of the way information affects and flows between

units in the supply chain. I start with the concept of a shock.

4.1 The Shock

Units are always tweaking their internal elements to address a variety of demands. Regu-

latory norms, preference changes, cost-cutting necessities, innovations, all present a unit with

the necessity to change its way of producing and internal structure. Therefore, every begin of

a period t there is the possibility of a temporary shock that prompts some unit to modify its

product or internal environment. This shock is placed on some final or intermediate input, thus

in some unit itself.

After some unit (i) received a call to change, she arranges a meeting with her suppliers and

her buyer to address the necessary changes the partners must also accomplish. That is, the unit

that received the shock sends messages, as defined in Concept 1, to the buyer and suppliers.

These messages contain informative elements about the changes that the suppliers must do in

their outputs. However, the content of the message includes informative elements about the

other inputs that make part in (i)’s output.

In terms of the example from figure 3, if unit (i) receives the shock, she talks with unit (k)

about possible changes. If the shock were over (k), she would talk with (i).

Summarizing, any shock reverberates not just upward in the production chain but also down-

wards, and a product made of several different units expects to be subject to a greater number

of shocks with the passage of time, directly or indirectly.

It is important to be able to pinpoint the unit that will do the initial studies concerning the

necessary modifications. For example, suppose the original shock was placed on the final good

producing unit. However, actually the shock was entirely reflected in one specific input of this

final product. Then, this input unit is the one who has actually received the shock first.

Definition 6. The unit responsible for the initial redesign is the first one, from the top of the

47

supply chain to the bottom, that had more than one of its suppliers affected (since, in this case,

this firm must lead the redesign in order to manage the simultaneous changes in all its suppliers).

Timing of a Shock

The signal that shows which units will be affected by a shock at any instant of time comes

from a Poisson Process. Thus, the probability of receiving a signal u periods from today is

independent from the last time the unit got it, and is given by γe−γu.

Target and Size of a Shock

After unit (i) received the shock, she compiles a list of her direct intermediate inputs (from

Mi) that must be adapted. Let Λti be the set of (i)′s intermediate inputs that were affect by the

shock.

4.2 The Propagation of the Information

After unit (i) received the shock and evaluated the necessary changes, it will contact her

partners to discuss the necessary changes they must also accomplish. In order to do so, she must

send messages to the other units, downward and upward. How are the messages distributed with

the suppliers, and what is their content?

4.2.1 Downwards

An useful proxy is that the share of each input in the redevelopment process is a function of

its connectivity. Thus, if (i) is hit by a shock, the messages sent to supplier k areMti,k = si(k)Λti.

But suppose this input supplier also had 2 external inputs. Call one of them k. Thus, ks share

will be sk(k)si(k)× Λti, and so on as long as there are suppliers.

Therefore, in the sense of Concept 1, the messages allow the receptor to replicate the content,

thus to learn part of the business of the unit that sent the message.

4.2.2 Upwards

Now, if unit (k) was hit by a shock, his buyer, some other unit (i), will too be called to

make changes in his productive process due to the supplier’s modifications. Moreover, the buyer

will share the information received from this supplier with his other suppliers, according to the

following share: each of the other k′ input suppliers of (k)’s will receive messages

Mtk,k′ = si(k

′)× si(k)× Λtk.

However, the buyer receives the whole information from her supplier, that is, Mtk,i = Λtk

. Next figure shows the flow of raw information between partners in the same supply chain

48

considered in figure 2. Notice that unit on which the shock was placed does not learn about her

partners.

Figure 5: Distribution of the information through a supply chain tree.

49

5 Absorptive capacity and the Learning Mechanism

As explained last section, after the shock the flow of information is bidirectional, from the

one who received it to the suppliers and buyers. But how this information is absorbed by the

units? In other words, how do they receive and comprehend this information so that it can be

used? This section proposes a structure to model this question.

Reception of information

In order to model the reception of information, I propose an structure inspired by the work

of Claude Shannon, on Information Theory(Shannon, 1984). Shannon derived a concept called

Information Capacity, which is the maximum amount of information that can pass through

a channel without error, for a given amount of noise in the system5. Another definition of

Information Capacity is a measure of the units’ mutual information, that is, the information

the units share with each other. Thus, the greater the noise in the communication between two

units, the lower is their expected mutual information and channel capacity.

I model reception of information inspired on this concept, with some differences accounting

to the economic environment.

First, I define noise as the difficult in transmitting some given amount of information in a

language such that it can be understood by both receiver and sender. For example, consider

two people from different divisions in a company discussing a mutual problem. Since they have

different environments and internal languages, there may be some noise in their communication.

However, noise is asymmetric: the unit with greater ability to handle informational flows is

able to receive proportionally more of the messages transmitted from other unit. Thus, I model

this asymmetric noise considering that there is noise just when the receiver is of less quality than

the sender.

Thus, based on the reasoning explained above, the formula to model the reception of infor-

mation by unit (k) from unit (i) is (1

1 + max{qi, qk} − qk

), (1)

The denominator is a function of the distance between two units, when the sender is of greater

quality than the receiver (the max operator).

Comprehension of the information

Besides problems related to the transmission of information, there are also some problems

related to the comprehension of the information during an exchange. That is, even if information

is error-free and fully transmitted, the information transmitted must be understood6. In the

simplest way, I model the comprehension threshold of a unit as its own quality.

5See Sims (2003)6See Boisot and Canals (2004) for a discussion over the distinction between knowledge and information in

economics. Also, see the introduction in Shannon (1948).

50

Absorptive Capacity

Thus, bringing both together, the absorptive capacity of information by some unit (i)

from other unbit (k) is: (qi

1 + max{qi, qk} − qk

)(2)

The informational stock

Finally, I present the formula that represents the way the informational stock of one unit

about other is formed.

Let Hti (k) represent the set of information of unit (i) with respect to her supplier. Thus,

unit (i) learns according to this function:

Hti (k) = Ht−1

i (k) ∪(

qi1 + max{qi, qk} − qi

)×Mt

k,i \Ht−1i (k), (3)

where H0i (k) = ∅. Its algebraic analog, hti(k), is given by equation (A.5).

What is the intuition behind this formula? First, learning is incremental, thus it starts at the

previous period’s learning stock, Ht−1i (k). Second, each new shock brings messagens that contain

new information, which changes the knowledge base only if adds something to it, represented by

the termMtk,i \H

t−1i (k). Third, the term inside parenthesis is the absorptive capacity explained

just before.

Accordingly, (k)’s learning as a supplier to (i) is

Htk(i) = Ht−1

k (i) ∪(

qk1 + max{qk, qi} − qk

)×Mt

i,k \Ht−1k (i), (4)

with analogous properties to (3) and algebraic version htk(i) given by (A.4).

The following proposition summarizes the results from these learning equations.

Proposition 1. Based on the learning structure defined above, I can say that the expected

accumulated information to be received by some unit (k) from her buyer (i) at some period t is

greater:

1. The smaller the noise through their communication, that is, their distance in quality.

2. The better the unit’s own quality, a proxy for her capacity to ”understand” the received

information;

3. The larger the unit’s connectivity, since there are more channels of information between

the units.

4. The greater is the size of her buyer’s other partners since this means more shocks and more

information shared.

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Proof: Appendix.

Accordingly,

Proposition 2. Based on the learning structure defined before, I can say that the accumulated

information received by some unit (i) from one of her suppliers, (k), at some period t is greater:

1. The better the unit’s own quality, a proxy for her capacity to ‘”understand” the received

information;

2. The greater is the size of her supplier’s other partners, since this means more shocks and

more information shared;

Proof: Appendix.

6 Revenue and Costs

This section uses the building blocks from the previous sections to build the units’ funda-

mental decision variables: revenue and costs function. Specifically, I show how the way the

information stock of one unit about others affect their main variables. I start with the determi-

nation of revenue. Since the units produce one unit of output per period, revenue of any unit is

given by its price.

6.1 Revenue

In the case of two vertically integrated units (i) and (k), input price pk is given by a measure

vk related to the market assessment of the value of the output.

That is just not the case if the units are outsourced in relation to each other. They determine

their trade price through a bargain process. As shown in figure 1, at t = 4 all parties meet to

put through their positions in the bargain.

If the bargain breaks down, unit k has outside option equal to vk − c where c is the cost

to bring the good to the market and vk the revenue from selling to the market ; otherwise, her

utility is Uk = pk. The buyer, unit (i), has utility Ui = pi−pk−P−ki if the bargain is successful,

and Uoi = pi− vk−P−ki if it is not, where P−ki is the price of the other intermediate inputs from

(i), excluding unit (k).

The bargain revenue, given by the Nash result, is given by pk = vk − 12 (c).

6.2 Costs

6.2.1 Adaptation Costs

A unit with quality qk called by external events to modify its product faces adaptation costs

that, as I show later, have a great impact over the choice of organizational form.

Adaptation costs are related to the redesign of the unit’s input and its delivery to the buyer;

in this case, adaptation by (k) to deliver to (i). In particular, there is the possibility that (k)

52

takes to too much time to research the adaptation. Since the shock comes in the beginning

of a period t, the unit must complete the research in adaptations until the half of this period

so that products sold in t + 1 already incorporate the necessary changes. With probability ρ

the unit completes the adaptation work in time; otherwise, sales happen normally without the

modification but there is a penalty cost K > 0.

There is a difference, though, between integration and outsourcing. For the latter, the greater

proximity of the units implies that their mutual stock of knowledge decreases the possibility

of delays, since there are few barriers between buyer and supplier. Hurdles are greater for

outsourcing, since the units are separate. There are several transaction costs that are diminished

in integration, usually those related to informational flows between adjacent units: a common

business culture, common regulations, same physical space make decisions faster, besides that

fact that there is no contract binding all actions. Thus, making adjustments is easier when

integrated.

Thus, the difference in ρ between integration and outsourcing is a measure of the dexterity

of the execution of the new information assimilated. I capture this effect in a simple way:

Let htk,i =htk(i)+h

ti(k)

2 the average bilateral stock of information; then I define the probability

of receiving a penalty cost K as:

ρk(i) = qihtk,i + qk(1− htk,i) if integration,

ρk(i) = qk if outsourcing.(5)

Thus, the bilateral stock of information of one unit with relation to another makes procedures

between these two units faster when their are integrated, compared to the case of outsourcing. I

use the bilateral stock since it reflects the bilateral exchange of information between units. The

greater the bilateral stock, the faster is communication in integration vis-a-vis outsourcing.

Consequently, for each mode of ownership, variable costs are given by

• Vertical Integration

Since is the forward firm that pays the price for the delays, its costs are given by

K(1− qihtk,i − qk(1− htk,i)

). (6)

• Outsourcing

Again, costs of the forward firm are given by

K(1− qk). (7)

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6.2.2 Fixed Costs of integration

Finally, I define the fixed governance costs related to integration: they are given by

F (htk(i)) for forward integration,

F (hti(k)) for downward integration.(8)

with F ′(·) < 0, F ′′(·) > 0, (htk(i) is the information stock of the supplier with relation to the

buyer, (hti(k) the information stock of the buyer with relation to the supplier.

Contrary to adaptation costs, governance costs are related to the unilateral stock of infor-

mation of the buying firm with respect to the acquired one. This is so because the acquirer, as

the new owner of the combined firm, must set the new governance structure and the more he

knows about the other unit, the better he is able to do it. Since the choices are prerrogative of

the owner, his information is the one more relevant.

7 Static, Instantaneous Information Model

In this section I present a model where information is readily observed by all firms at the

begin of times and is instantaneously shared. Thus, everything happens as if there was an

instantaneous shock of size 1 in all units in the economy.

The reason behind this radical assumption is to separate general properties of the model from

those specific for the dynamic model, present in Section 8. Also, this model presents results that

are not usual in the literature.

Since I assume information is readily observed by all firms, for a relationship between a buyer

and a supplier, their proportional information stock is given only by the absorptive capacity of

one with relation to the other.

With this background, I proceed to the steps necessary to deliver the problem of choosing

between integration and outsourcing.

7.1 Joint Profit of a Partnership

From the title, notice that I am defining the joint profit of a relationship, not the aggregate

one for the two partners. The latter includes the costs for both the forward and downward units

with relation to their respective buyer and suppliers, besides an explicit account of their profits

with these other relationships: the whole supply chain.

For the case of Integration, the joint profit of the partnership between units (i) and (k) at

period t is given by

pi −[K(1− qihtk,i − qk(1− htk,i)

)]− P−ki − Pk − ωk − ωi − F (·) (9)

where the form of F (·) depends whether integration is forward or downward (see (8)), Pk is the

sum of prices from the suppliers to unit (k) and both ωk and ωi are the wage of the respective

54

units. For the case of Outsourcing, aggregate profit between unit (k) and (i) is given by

pi − [K(1− qk)]− P−ki − Pk − ωk − ωi. (10)

7.2 Ranking of qualitites

Equilibrium properties of this model starts with a rank of the qualities from all units of the

supply chain. Notice that, regardless of the ownership structure, for any conceivable pair of units

that may participate in any stage of the production process, the quality of the unit downward

must be higher than that of the unit forward. Suppose this was not true and consider two units

qi, qk, k downward to i with qi > qk. If qi > qk, then by either (9) or (10) aggregate profits could

be increased if the firms switched position in the chain. In the case of outsourcing, at equation

(10) is clear cost are increasing in the quality of the supplier.

Results are not so straightforward with respect to integrated firms. However, notice that

costs again would decrease if the firms switched place, for the following reasons:

1. Governance fixed costs are decreasing in quality;

2. Adaptation costs are decreasing in quality since:

• From (9), cost is increasing in ρ. Notice that if both firms switched place, the average

informational stock would go up, because absoptive capacitites would increase:

qi +qk

1 + max{qi, qk} − qk− qk −

qi1 + max{qi, qk} − qi

> 0 (11)

• A greater average informational bilateral informational stock makes ρ smaller if the

downward quality is greater than the forward one, see (5).

Therefore, the following proposition summarizes the assortment property of the static model:

Proposition 3. Regardless of their owership structure, downward units are of greater quality

than forward ones in the static model.

7.3 Choice of Outsourcing versus Integration

When all information is expected to be readily shared between the units in the industries,

what drives the division of some activities between integrated units and outsourced ones?

It is fair to say that the reason lies entirely in the differences of quality. Explaining: for a

given quality, the greater the connectivity, the greater is the supplier’s accumulated information

about the parent unit. Therefore, the decision about downward integration versus outsourcing

is based on whether (9) is greater than (10), that is, whether the numerator is greater than the

denominator in the following ratio:

Θ(qi, qk) =pi −

[K(

1− qihtk,i − qk(1− htk,i))]− P−ki − Pk − ωk − ωi − F (·)

pi − [K(1− qk)]− P−ki − Pk − ωk − ωi(12)

55

where the form of F (·) depends whether integration is forward or downward (see (8). Based on

Θ(qi, qk), next proposition follows.

Proposition 4. Consider the ratio Θ(qi, qk) in (12) above. Based on it, I can say that

1. For a given supplier quality, qk, there is a threshold qi such that for values of qi above it,

integration between units qi and qk happens. Notice that

∂Θ(qi, qk)

∂qi> 0, (13)

and,∂qi(qk)

∂qk> 0.

Therefore, the unit of the highest quality has all his supplier units integrated to itself,

no outsourced ones. In turn, these integrated units may have some integrated units to

themselves, and some outsourced: the smaller the difference in quality of the parent unit

relative to the supplier unit, and the lower the quality of the buying unit, the greater the

chance that this supplier will be outsourced. Hence, from the highest level of the supply

chain to the lowest, firms have increasingly more outsourced units and less integrated ones.

2. There is never forward integration in this static model.

Proof at Appendix.

The intuition behind this results follows: the greater the difference in quality, the more

integration adds value to the partnership, since information flows are greater. Also, the greater

the quality of the downward unit, the lower are the governance fixed costs.

The fact that forward integrations are not possible in this static model is due to the static

nature of it, that gives the higher quality firm (the buyer) all the advantage since their learning,

instantaneous in this case, is always greater.

Notice that both size and connectivity do not play great roles here. This is because their

influence appears only in a dynamic setting. At Section 9 I show that in a dynamic model

connectivity and size have a great role in the ownership choice, and that forward integration is

possible.

8 Relation to findings of the literature

I show now how the model developed here sheds some light on the findings of Acemoglu et

al. (2004)’s empirical findings of an inversed relationship between integration and supplier R&D

intensity and Hortacsu and Syverson (2009)’s findings on an increasing relationship between

integration and supplier plant productivity.

Notice that the theory developed until here implies that

56

1. Units are more likely to be integrated the greater the distance in quality between them

and the buyer, the greater the quality of the buyer and the higher the connectivity of the

supplier.

2. Integrated structures are more productive than outsourced ones of the same industry, due

to the externalities.

This model speaks in terms of labor quality in an informational sense. If one assume that

labor quality is necessary for intense if R&D expenditures, the model gives that high-tech

units (those with higher R&D expenditures) must be of high-quality, which in turn are more

productive. However, but for a given buyer quality integration is decreasing in the quality of

the supplier, and for a given supplier quality, is increasing in the quality of the buyer. Thus,

the model shows Property Rights Theory is not necessary to explain Acemoglu et al.’s empirical

findings, but also these finds do not contradict Hortacsu and Syverson’s findings as well. Of

course, a new round of econometric evaluation is necessary, since the three models are very

different.

9 The Dynamic model

Previous section results were done over the assumption that information between firms is

instantaneously shared. Following the environment developed in section 6, now I consider the

dynamics of integration versus outsourcing, starting from moment zero, when there is no infor-

mation shared. In contrast with the previous section’s model, now the decision to integrate or

outsource must take into account the production shocks and the learning process that follows

from it, with very different results from the static model.

9.1 Outsourcing versus Integration with Learning

Consider the case of a supplying unit (k) and her buyer (i): their joint value functions in

each organizational mode in some specific period t are:

V V,tk,i = pi −K

(1− qihtk,i − qk(1− htk,i)

)− P−k

i − Pk − ωk − ωi + Et+1

(βV V,t+1

k,i

); (14)

V O,tk,i = pi −K (1− qk)− P−k

i − Pk − ωk − ωi + Et+1βmax {V V,t+1k,i − F t(·), V O,t+1

k,i }. (15)

where O index the outsourcing value function, and V the integration one. The form of F (·)depends whether integration is forward or downward (see (8)).

Notice that the above equations represent the functional form of the present-value joint profit

functions, not the solved value function with the solution for the trajectories. The existence of

a solution for each period is given just ahead in section 9.2 .

57

9.2 Ranking of qualities

Ranking in the dynamic model is no different than that of section 7.2: at each time t,

regardless of the ownership structure, for any conceivable pair of units that may participate in

any stage of the production process of the supply chain, the quality of the unit downward must

be higher than that of the unit forward.

The proof follows that from section 7.2 since, by (14) or (15), the only difference between

the static and the dynamic model is the dynamic movement of the value stock of information.

First, notice that if that some pair of firms kept their ownership structure intact for the

future, then the dynamic model is just a repetition of the static one and the same proof from

7.2 applies. This is always true, thus, for integrated units at birth.

However, for outsourced units, does the possibility of a change to integration make it possible

that the quality of a downward unit be smaller than the quality of a forward unit? Not likely,

since from 7.2 costs decrease when units switch places in both ownership modes.

Proposition 5. In the dynamic model, downward units are of greater quality than forward ones,

regardless of their ownership mode.

9.3 The choice between Integration and Outsourcing

Next series of propositions consider the interplay between interfirm learning dynamics and

supply chain characteristics over the choice between Integration and Outsourcing.

Lemma. Sufficient conditions for the existence of an knowledge threshold for integration are

high enough fixed costs and the buyer’s quality being sufficiently greater than the supplier’s.

Proposition 6. Consider the decision between two vertically adjacent units in a supply chain, (i)

and (k), of whether or not to integrate. Provided the conditions on Lemma 9.3 are met, the units

start their partnership via Outsourcing and change latter to Integration, at some information

threshold h.

Corollary 1. Assume the sufficient condition specified in lemma 9.3 is valid. Then, about

downward mergers, one can say that

1. Considering the value of the threshold necessary for a transition from outsourcing to inte-

gration:

it is decreasing in the quality of the buyer:

∂hi(k)

∂qi< 0, (16)

58

and is increasing with reference to the quality of the supplier:

(17)

∂hi(k)

∂qk> 0. (18)

2. Considering the accumulation of knowledge capital, downward mergers happen faster:

(a) the greater is the the buyer’s quality,

(b) The greater is the size of the supplier’s (and of his suppliers, and so on...), since this

means more shocks and more information shared;

Proof at Appendix.

The above proposition address two elements of importance when considering integration

versus outsourcing in a dynamic setting: first, it says that for a given supplier quality, a buyer

of greater quality need attain an information threshold smaller since the extra value obtained

from the integration is greater. Accordingly, for a given buyer, the smaller the quality of the

supplier, the lower the necessary threshold since, again, the extra value obtained from integration

is greater.

Second, at item 2, proposition says which elements make it faster the coming of the inte-

gration threshold. The greater quality of the buyer is due to increased learning from shocks.

Learning also is greater the larger the dimension supplier’s own supply chain, since there are

more shocks that may hit the supplier and prompt information sharing.

Now, let me show results concerning the existence of forward integration.

Corollary 2. Assume the sufficient condition specified in Corollary 9.3 is valid. Then, about

forward mergers, I can say that

1. Considering the value of the threshold necessary for a transition from outsourcing to inte-

gration:

it is decreasing in the quality of the buyer and is increasing in the quality of the supplier:

∂hk(i)

∂qk> 0, (19)

∂hk(i)

∂qi< 0. (20)

2. Considering the accumulation of knowledge capital, forward mergers happen faster:

(a) The smaller the noise through their communication, that is, their distance in quality,

a proxy for her capacity to receive the information;

(b) The better the unit’s own quality, a proxy for her capacity to ‘”understand” the infor-

mation;

59

(c) The larger the unit’s connectivity;

(d) The greater is the size of the buyer (and of the other partners of the buyer, and so

on...), since this means more shocks and more information shared.

3. Summarizing: two elements affect the time until the change of ownership mode. First,

there is the target level itself, which as shown in the last item, is increasing in the buyer’s

quality and decreasing in the supplier’s one. Second, the expect time to reach this level

depends on the elements described in item 2. Therefore, although with a larger difference

in quality the target level is smaller, the time necessary to reach it augments, since this

difference diminishes the accumulation of information.

Summing Up

• From the point of view of maximizing value, the higher the buyer’s quality in the supply

chain, the more likely is both types of integration.

– forward integration happens sooner between firms not too much apart in quality,

where the connectivity of the supplier is high and the firms are located at the top of the

supply chain. In this setup, an increased difference in quality increases the creation

of value but diminishes learning; location is also important: forward information

gathering increases with the distance from the top of the chain, since the frequency of

shocks is higher, but also decreases with it, since the size of shocks are more diluted.

The contrary happens to downward integration.

– downward integration happens sooner between firms apart in quality: an increased

difference in quality augments the creation of value; location is also important: down-

ward information gathering decreases with the distance from the top of the chain, since

the frequency of shocks is lower, but also augments with it, since the size of shocks

are more diluted.

• At any time, both downward and forward integration may happen, contrary to the model

from Section 7. Yet, from Proposition 6, Corollaries 1 and 2, is clear that if buyers are of

greater quality than suppliers, downward mergers are more likely than forward mergers,

since buyers in average absorb more information per unit of time. Also, as shown in the

items above, the difference in quality between units hinders the supplier’s gathering of

information, worsening the prospects of forward integration vis-ı¿12 -vis the downward one.

10 Case study

In this section I show some interesting application of the above theory, related to a real

case that happened with two Brazilian companies, namely, Sadia and Friboi. Brazil is one the

60

world’s greatest raw meat exporter, being the first in beef, the second in poultry, specially chicken

and the fourth in pork, as of 2008. Several meat packers export these products to all around

the globe. Besides raw inputs, some of the most successful Brazilian companies also produce

processed meat; among these, two are reference: Sadia7 and JBS-Friboi8, the latter being the

largest exporting and beef-processing unit in the world.

In the last few years, there were several outbreaks of foot-and-mouth disease in Brazil that

severely hampered Brazilian exports of raw and processed beef products. Although this disease

does not poses harm to consumers, it is used by importer countries and competitors as excuses

to lobby in order to displace Brazilian producers off the meat market.

This type of restriction also affected the poultry export market, which is Sadia’s main one:

twice in 2007 Russia, Brazil’s larger buyer, halted imports alleging sanitary concerns.

In the last four years, these two Brazilian units adopted similar strategies: internationaliza-

tion. Friboi bought in 2007 the American Swift Foods Company to access the American and

Argentinian markets; Sadia opened in 2008 its first foreign industrial plant, in Russia.

According to JBS-Friboi’s executive director of business, in an interview to one of Brazil’s

most prestigious business newspapers (Gazeta Mercantil, 2006a) at the time of the first foreign

acquisition (translation by the author):

”We are looking for diversification in foreign markets and also a ’sanitary hedge’.”

In the same vein, commenting for the same newspaper (Gazeta Mercantil, 2006b) the decision

to build a plant in Russia, Sadia’s CEO at that time said that:

“Nowadays, sanitary quality is our greatest problem in Brazil.”

Both companies exported raw and semi-processed products to these markets for at least 10

years, and after the devaluation of Brazil’s currency, the Real, in 1999, their exports increased

remarkably. The quotes above indicate that the sequence of regulatory shocks of the last 5 years

was one of the reasons that prompted them to build stronger positions in their main exporting

markets, in order to defend themselves.

These two stories fit within the model developed in the preceding sections. First, notice that

these companies were exporters for a long time and had already gained considerable information

about their foreign partners and market structure, after a series of small shocks into the daily

operation of their units. Second, they confronted large shocks in the last few years that, if not

addressed, could halt completely their exports and harm their relations with the buyers. These

shocks represented large costs, as in equation (7), thatif not addressedwould lead to great losses.

Thus, these companies perceived that with some great probability their presence in the foreign

market, built for so many time, would be lost. The alternative? Some merger or acquisition

that allowed the continuation of their business, from the perspective of the buyer.

7http://www.sadia.com/en/home/index.asp8http://www.jbs.com.br/defaultEn.aspx?PagId=FLGCORVM

61

Consider the model developed at Section 9, with a supply chain made up of just a supplier

indexed 1 and a buyer indexed 0, at some generic time t, and assume that a strong regulatory

shock happened. If the supplier does not make the costly adaptations C, he is out of the export

business. Yet, even making the adaptations, with probability 1−z they are useless in preventing

an outbreak of disease and the cost is even bigger, J . Than, the supplier-buyer pairs must

choose between three options: nothing, outsourcing with exports, outsourcing with domestic

production.

max

{0, z(V O − C) + (1− z)(V O − J), OV − F

}. (21)

The novelty is the third option, where the supplier acquires the buyer, does not invests in

sanitary measures and continue buying from another supplier.

Before moving to explain the decision of the Brazilian companies, one has to explain why

prior to the shock the units were operating in outsourcing mode. According to the model, that

would be the case if both had medium and close qualities such that integration externalities

would be negligible but interfirm learning would be medium for both.

The meatpacking business has characteristics that satisfy these descriptions: first, these firms

probably have close qualities since the skill requirements are similar in all countries9; Also, given

the commodity-related nature of good, the average quality of the labor in the meatpacking is

probably not high and and the small degree of specificity of this business diminishes the incentives

to integration.

The next proposition follows, a straightforward application of the theory developed through

this paper:

Proposition 7. Consider two firms operating via outsourcing, each in a different country, H

and F , where H ′s firm (supplier) exports to F ′s (buyer). After a regulatory shock, the supplier

will decide to quit its local operations, buy its downward buyer and outsource from F according

to the following:

• the greater the size of those buyer-related units;

• the lower the difference in quality between the two units;

• the larger the supplier’s connectivity;

• the greater the amount of time of their relationship.

Proof. Equation (21) and Corollary 2.

This definition applies very well to the Brazilian units quoted before:

9I am still short on data about the costs of these units and plants, but given the comparative advantagesof Brazil in labor and natural resources, probably Brazilian costs are smaller: public data on the non-profitresearch institute Agri Benchmark shows that Brazil has one of the lowest livestock prices of the world. Seehttp://www.agribenchmark.org/.

62

• Both produce inputs, almost-final good products or processed meat, that demand great

interconnection with the buyers, that is, their inputs’ connectivity is high. For two reasons:

technical, since the meat are the most important intermediate inputs that enters in the final

product, and constitute the larger fraction of it; quality concerns, that demand constant

interconnection;

• The supply chain’s dimensional is smaller in the vertical dimension relative to most other

industries (since the meat comes directly from livestock farms to those units), but the

horizontal dimension of the chain related to the buyer is large since it involves, downward,

several interconnected wholesale, distribution, marketing and retail operations.

• Some knowledge of operating in foreign countries and companies can be learned indepen-

dently of the market, since they are common to all of them:

– Sadia exports to 65 different markets and is the leading producer of processed poultry

in Brazil.

– JBS-FRiboi is the second largest exporter of meat in the world, the largest of processed

meat. Has 500 clients in 110 countries.

• The companies had at least 10 years of experience exporting to these markets.

• Due to the low-tech nature of the industry, the quality of the Brazilian firms probably is

close or superior to those of the acquired foreign firms, facilitating the interfirm learning.

Although the horizontal dimension forward is also large (numerous producers), nowadays the

shocks coming from these are usually small by the simple nature of their tasks and because there

is already considerable integration forward. The fact that forward integration came sooner in

Brazil is not difficult to understand: in the last decades the considerable disruption in the live-

stock production and slaughtering business caused many problems for an efficient management

of the supply chain. Product quality was specially important (Lawrence et al., 2001; Hayenga

et al., 2000). downward, however, the competitive aspect of the retailing market, the lack of

stringent technical and food safety regulations and simpler consumer preferences in terms of va-

rieties (due to smaller market sizes) all contributed to low product complexity and slow interfirm

learning from the downward side of the chain.

However, this changed in the last decade: increased consumption of meat by the emerging

markets, increased number of product varieties due to more diversified consumer preferences,

increased horizontal concentration in the retailing sector, non-food technical regulations, export-

ing market barriers, all pose great pressure in the units in the middle of the chain. Shocks come

specially from the retail segment, since in several countries there is extreme concentration of

power with few units (Taylor, 2006; Netland et al., 2008). Schulze et al. (2006) discuss these

new developments and the implication in line of transaction costs.

63

11 Conclusion

This paper shows that a transaction-costs model together embedded with interfirm infor-

mation exchange and learning provides an unified explanation to several recent unexplained

empirical findings about the optimal vertical organizational form in a diverse supply chain.

Each period firms in a multi-layered supply chain must reevaluate their organizational form.

Also, they may receive a shock, which induces changes in the firm’s output, compelling it to

exchange information with its partners, which learn about it in turn. The shock disseminates

through all the chain, according to each firm’s quality, size (number of intermediate inputs) and

connectivity (measure of the informational channels between inputs). The knowledge capital one

firm has about its partners diminish the size of the governance costs of integration and improves

interfirm externalities of integration.

First, in a static model, I show that from the highest level of the supply chain to the lowest,

integration is more likely the higher the quality of the buyer, and the difference in quality between

firms. Thus, firms from top to bottom firms have increasingly more outsourced units and less

integrated ones. Also, just downward integration happens. These results shed lights on two

apparently contradictory empirical results of the literature.

Second, in a dynamic model, I show the determinants of interfirm downward and forward

learning, the characteristics that causes forward and downward integration and why the latter is

more likely, as shown in data. Contrary to the static model, now forward integration happens.

Finally, the dynamic model is applied to study the effects of regulatory shocks over the

organizational decisions of firms, with a new type of vertical foreign-direct investment explained.

I exemplify this effect with documented actions and characteristics of two leading Brazilian meat-

exporting firms.

64

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Appendix

Proof of Propositions 1 and 2:

Remember that the buyer must share information with his other suppliers, so the shock prompts some level of

information sharing across all the units in the industry. Therefore, which is the accumulated information one unit

has from her buyer and supplier in some period t?

Notice that the learning from the suppliers comes from shocks happening just in the “tree” that starts with

the unit itself and goes through the suppliers own ramifications. Learning from the buyer side comes from shocks

originated in the rest of the tree.

Therefore, the accumulated information unit (k) obtains from her buyer i at time t, from time t− 1, is

Htk(i) = Ht−1

k (i) ∪(

qk1 + max{qk, qi} − qk

((At

i × si(k)

)\Ht−1

k (i)

), (A.1)

where Ati =⋃

Λti are all the messages sent from (i)’s10.

Accordingly, the accumulated information unit (i) obtains from her supplier k at time t, from time t− 1, is

Hti (k) = Ht−1

i (k) ∪(

qi1 + max{qi, qk} − qi

)×(At

k \Ht−1i (k)

), (A.2)

where analogously Atk =⋃

Λtk are all the messages sent from (k)’s.

It is going to be useful later to translate into some algebraic measure the information in the form of partitions

of both the shocks received from other units, Λ, as the accumulated information Ht one unit has about another.

Define the algebraic size of the shock, αti,as the proportion of inputs from Mi that must be adapted at time t:

αti =|Λti|mi

. (A.3)

Thus, the algebraic analogs of (A.1) and (A.2) are:

htk(i) = ht−1k (i) +

(qk

1 + max{qk, qi} − qk

)(ati × si(k)− ht−1

k (i))+, (A.4)

.

Accordingly,

hti(k) = ht−1i (k) +

(qi

1 + max{qi, qk} − qi

)(atk − ht−1

i (k))+. (A.5)

.

Proof of Proposition 4:

1. I start the proof considering a downward integration, that is, F (·) is F (qi).

First, check that limqi→1 Θ(qi, qk) > 1 ∀qk; limqi→0 Θ(qi, qk) < 1∀qk < qi; limqk→1 Θ(qi, qk) > 1∀qi > qk;

Now, the derivatives. Θ(·) is increasing in qi, since its sign rests on:

∂Θ(qi, qk)

∂qi=

(−F ′(qi) +K

∂ρk(i)

∂qi

)> 0, (A.6)

10To facilitate the notation, I hereafter will define the shocks affecting some unit (i) but that originated inanother unit as Λi(k), with k ∈ Zi; the same to their interval analog, αi.

68

considering that∂ρk(i)

∂qi= (qi − qk)

(((1 + qi − qk)2)− si(k)qk

2(1 + qi − qk)2

)+ h

ti,k > 0. (A.7)

Therefore, there exists a qi such that

− F (qi) +Kρk(i) = Kqk (A.8)

Notice that

∂qi∂qk

=K(∂ρk(i)∂qk

− 1)

−K ∂ρk(i)∂qi

+ F ′(qi)> 0, (A.9)

given that the denominator is negative from F (·) decreasing. The next figure shows the value of the

numerator (referred to by Φ) for arbitrary qi and qk, notice it is always negative:

Figure 6: Numerator of (A.9)

Now, consider the possibility of forward integration. Since there is an instantaneous shock, the information

received by the dowstream unit about the forward partner is always greater than the one received by the forward

about the downward partner. This happens because the downward unit has greater quality (see section 7.2) and

information received by the forward unit is proportional to the connectivity of this unit, which is smaller or equal

to 1. Thus, there is never forward integration.

Proof of Lemma and Proposition 6:

A threshold exists when there is an equilibrium with integration starting today (14) (LHS) equals outsourcing

today and integration tomorrow (15) (RHS). It is different whether integration is downward or forward. A

threshold exists if...

in the case of downward integration versus outsourcing:

69

• limhi(k)→0 LHS < limhi(k)→0 RHS

Valid for a high F (0) and qi close to qk;

• limhi(k)→1 LHS > limhk(i)→1 RHS.

Always valid.

Now, for forward integration versus outsourcing, conditions are the same (limits, in this case, are for hk(i)).

• ∂V Vk,i

∂hi(k)≥ 0, since the continuation payoff is never smaller - information accumulated does not depreciate.

Also, fixed costs get smaller.

• F (·) decreases with more information.

• ∆VO

∆hi(k)≥ 0. To see this, notice that next period’s value function is not smaller than current one, given that

the information does not depreciates, and the unit has the option to choose the best alternative.

Therefore, there is at one level of information that, if reached, prompts a change from outsourcing to integra-

tion. This level is hi(k) (for downward Integration), or hk(i) (forward integration).

Considering downward integration, the threshold is such that

Kqk − β EF(hi(k)

)= Kρ(hk(i))− F (hi(k)). (A.10)

For forward integration,

Kqk − β EF(hk(i)

)= Kρ(hk(i))− F (hk(i)). (A.11)

Proof of Corollary 1:

See that

∂hi(k)

∂qi=

−K ∂ρk(i)∂qi

k ∂ρ(hi(k))

∂hi(k)− F ′(hi(k)) + β EF ′

(hi(k)

) < 0, (A.12)

since the derivative in the numerator is negative as shown in (A.6) and the denominator is positive given that

F (·) is decreasing and convex and from (5).

Finally,

∂hi(k)

∂qk=

K(

1− ∂ρk(i)∂qk

)k ∂ρk(i)

∂hi(k)− F ′(hi(k)) + β EF ′ (hi(k))

> 0. (A.13)

The sign depends on the value of the numerator, which is determined by the following terms:(qi,

si(k)qk1 + qi − qk

, t,si(k)(1 + qi)(qi − qk)

(1 + qi − qk)2

). (A.14)

It turns out that the behavior of the numerator follows the inverse of the function showed in figure 6, thus it

is always positive.

Proof of Corollary 2:

∂hk(i)

∂qi=

−K ∂ρk(i)∂qi

k ∂ρk(i)

∂hk(i)− F ′(hk(i)) + β EF ′

(hk(i)

) < 0, (A.15)

since the derivative in the numerator is positive.

Finally,

∂hk(i)

∂qk=

K(

1− ∂ρk(i)∂qk

)k ∂ρk(i)

∂hk(i)− F ′(hk(i)) + β EF ′

(hk(i)

) > 0. (A.16)

70

An economic model of the risk of payroll-backed loans

Abstract

This work presents a theoretical evaluation of the main characteristics that drive the risk

of credit losses on a portfolio of consumer payroll-backed loans. Consumer credit through

payroll-backed loans is one of the fastest growing credit lines in Brazil. Under reasonable

assumptions, I construct a microeconomic-based model of an industry whose workers are

borrowers of payroll-backed loans. Then, I build a formula for the probability of default

and expected losses for both an individual loan and a portfolio of loans from the industry’s

workers. Finally, variables of the formula are subject to a sensibility analysis to determine

their influence on the risk of payroll-backed loans.

72

1 Introduction

Models for credit risk assessment were introduced by Merton (1974). In his approach the

stochastic behavior of the value of a firm’s assets is modeled and if the value becomes lower

than a threshold, the company is considered to be in default. This model is seminal and was

chosen as the foundation for the capital allocation formula proposed by the Bank of International

Settlements, the Basel II Accord; Merton’s model gave the base not just for corporate risk models

but also to modeling risk on consumer loans, as stressed by Thomas (2009).

However, due to the very different nature of the underlying economic structure behind cor-

porate and consumer credit, there is considerable skepticism about the applicability of Merton’s

framework to measure consumer credit risk, as highlighted by Thomas et al. (2005). Notwith-

standing, the Basel propose has highlighted that although there are a number of well established

credit risk models for portfolios of corporate loans which are widely used by financial organi-

zations, there are no such established consumer credit risk models for portfolios of consumer

loans. Work towards establishing this benchmark has begun with, among others, de Andrade

and Thomas (2007) and Thomas (2009). These two papers construct Merton-style option-based

models, substituting the concept of a consumer’s call option value on the value of its reputation

for the assumption, present in corporate models, of a call option on the assets of the firms.

In this paper, I intend on contributing to the literature on consumer credit risk models

through a microeconomic study disentangling the risk properties of a popular and important

kind of consumer credit that is thriving in Brazil: consumer payroll-backed loans. Payroll-

backed loans are a fast-growing business in Brazil and is one the credit products responsible for

the explosion on borrowing in recent years.

I build a microeconomic model of a industrial sector, whose workers take payroll-backed

loans. Then, this model is used to construct a formula for the probability of default and expected

losses on both an individual loan and on a portfolio of payroll-backed loans. This theoretical

risk formula is very tractable and allows me to study the effects of changes in variables on the

risk of this credit product, and thus on its supply by the financial institutions.

There are three variables responsible for the bulk of the behavior of the above-mentioned

equation for the probability of default: the quality of the industry’s labor, the size of the in-

dustry’s labor force and the exposure of the industrial sector to shocks at its prices. Those

three variables, in turn, are related to the main driver of default in payroll-backed loans: unem-

ployment. Therefore, at subsection 1.1 next I explain a little about the risk characteristics of

this credit product, with a focus on unemployment. Following, at subsection 1.2, I explain the

modeling strategy and its findings.

1.1 Unemployment and risk in payroll-backed loans

Basically, payroll-backed loans are contracts among workers from some industry, their union

and a financial institution. The union negotiate a rate with the latter and the workers can borrow

at this rate until a fixed proportion of their payroll income, now 20%. Since the loan’s payments

73

are automatically deduced each month from the borrowers payroll, credit risk is minimal for

those employed. That is why the biggest market for payroll-backed loans in Brazil are workers

of the public sector, that have job stability.

However, there is also a considerable market for payroll-backed loans to private sector work-

ers. Credit risk in this case is also usually thought to be small, but is that true?

As the recent world economic crisis has shown us, that is not always the case. The main

driver for risk in payroll-backed loans is unemployment. As stressed in several newspapers, some

banks in Brazil had considerable losses from loans linked to sectors affected with more severity

by the world crisis. For instance, because of the crisis, one of the leading banks in Brazil for

this type of credit, BMG, decided to stop operating payroll-backed loans with private firms; say

Ricardo Guimaraes, president of BMG (Estado de Minas, 2010):

”(...) we decided, in 2009, that payroll-backed loans to the private sector were not

our focus anymore. Risk and default rates are greater than those from public sector

workers and recipients of public pension funds.”

De facto, default rates for workers from the private sector are much greater than those from

the public sector. In a recent interview to the Brazilian business journal Brasil Economico

Pirelli, a director of Bradesco, Brazil’s third largest financial institution, said that at that time,

09/24/2010, default rates for private-sector workers were almost 4%, against 1.8% of public-

sector workers (Brasil Economico, 2010). However, in the same interview the bank’s president,

Luis Trabuco, reckons that the biggest growth in the future will come from the private sector.

Thus, payroll-backed loans are very sensitive to aggregate risk, that cause mass unemploy-

ment. Besides, the aggregate risk is sector-based: in this sense, it feels like a corporate-risk

model since the industrial sector must be modeled in order to extrapolate its characteristics to

the loans its workers have taken. Credit risk of payroll-backed loans is also linked to idiosyncratic

unemployment risk: workers’ mistakes that cause their layoff.

The importance of understanding workers’ behavior (related to idiosyncratic risk) and indus-

try performance (related to aggregate risk) to assess default risk in portfolios of payroll-backed

loans is already reckoned in the consumer credit industry. For instance, Both Itau Unibanco

and Santander, Brazil’s second and fourth largest banks respectively, offer insurance together

with the payroll-backed credit to assure the payment of part of the credit in case of default.1

They also demand some minimum employment time (six months at Itau Unibanco) to minimize

idiosyncratic unemployment risk.

In short, two elements make payroll-backed loans apart from other types of consumer loans:

1. Large and direct risk from aggregate unemployment shocks;

2. It is a consumer product with a considerable industry-based risk;

1See http://ww2.itau.com.br/creditoconsignado/func_servpub.htm.

74

Thus, although all consumer credit has its risk linked to unemployment in some way, in the

case of payroll-backed loans risk is directly linked to the performance of the sector; this allows

one to focus the study of credit risk on the sector’s behavior, besides workers’ behavior.

1.2 About the model and its results

In order to model both the industrial sectors’ macro behavior, related to the aggregate risk,

and the workers’ behavior, related to the idiosyncratic risk, this paper uses a simple and flexible

two-period model based on the work of Kremer (1993).

Kremer’s model allows one to simulate the characteristics of labor inside an industry with

a focus on the ”quality” of labor, that is, the chance each worker has in making a critical

mistake on the production. Thus, KLremer’s model allows me to link the risk from idiosyncratic

unemployment to the probability of a worker making a mistake and being fired.

Considering the modeling of aggregate risk, Kremer’s model also poses no hurdles. This risk

is realized through an ad hoc aggregate shock that happens every period and affects with random

severity the industry’s prices, at each one of them. Since the industry chooses the size of its

labor force each period, after the first one it must either hire new workers if the shock happened

and was positive (in the sense of allowing hiring) or dismiss some workers if happened and was

negative (in the sense of causing unemployment).

Analysis of default is made with reference to the second period. Thus, if a negative shock

happens and some workers are fired, there is the possibility of default. Otherwise, default is

possible due to idiosyncratic risk. Thus, I obtain a formula for the probability of default and

expected losses for both a single loan and the portfolio of payroll-backed loans. The formula is

tractable and allows me to perform a sensibility analysis of its core variables: the quality of the

workers, that is, their probability of doing some critical mistake (idiosyncratic risk); the initial

size of the labor force; and the exposure of the industrial sector to an aggregate shock.

I find that the greater the idiosyncratic risk, the higher the probability of default, everything

else constant. Also, the effect of the size of the labor force over the probability of default

is complex, because any change on this variable causes two opposing effects: an increase on it

raises revenue for a given total productivity, but also decreases productivity; however, I find that

the lower are both the labor force and the idiosyncratic risk, the more likely that an increase in

labor force will lead to an increase in the overall probability of default for an individual loan.

The exposure of the industry to aggregate shocks is studied via a mean-preserving spread of

the distribution of severity of the shock, which causes an overall increase in the probability of

default, everything else constant.

The structure of the paper is as follows: section 2 summarizes the institutional framework

governing payroll-backed loans in Brazil; section 3 presents the microeconomic model that gives

the foundation for the development of the risk formula; section 4 incorporates the preceding

model in a model of default risk and expected losses; section 5 presents the portfolio view of the

previous analysis. Section 6 presents a sensitivity analysis of the risk formula with respect to

75

important variables. Section 7 concludes.

2 The Brazilian laws regulating payroll-backed loans

In 2003, the Brazilian Congress passed a law (number 10820/2003) regulating the so called

emprestimo consignado, namely, payroll-backed loans. This law represented a watershed moment

for credit in Brazil: it gave both consumers and financial institutions the necessary tools to make

credit more affordable and less risky.

As stated in the introduction, according to the law a payroll-backed loan is a contract among

workers, firms, unions and banks. Firms must make the necessary arrangements to share all

necessary information about the workers to the financial institutions, and make the monthly

deductions of the loan parcels from the payroll. Financial institutions negotiate with the unions

on the available rates and maturities for the loans, which the worker/borrower may accept or not.

Finally, although the firm may itself arrange special terms with banks, these must be approved

by the labor union in behalf of the employees.

These special allowances to the unions gives them lots of leverage to negotiate the terms

of these loans with the financial institutions, especially when representing employees with poor

credit history which would have great difficult in getting good terms, or any credit at all. Thus,

payroll-backed loans gave cheaper credit for borrowers with good credit history, and gave those

with poor credit score an instrument to borrow again.

The picture next show the explosive growth for payroll-backed loans since the approval of

the law:

Figure 1: Payroll-backed Loans - Balance in Million R$.

76

3 The Microeconomic model

3.1 Background

The economy is composed of several industries, indexed by ω. Time is discrete, restricted to

two periods, 1 and 2. Workers from these industries are indexed by i and each one produces one

unit of output each period.

At each one of these two periods, the industrial sector is subject to aggregate risk, that

materializes via an industry-specific shock. This shock comes right at the beginning of each

period, and affects the industry’s price. It can either increase or decrease it.

Following the shock, the industry’s labor force is determine in period 1; accordingly, at period

2 after the same shock labor force is reevaluated. Thus this second shock, if smaller then the one

at the previous period, can cause unemployment; otherwise, the hiring of new workers. These

shocks are independent and identically disributed over time, happens in any of the two periods

and affects the industry’s price P (zt) with intensity zt, which is given by a distribution F (zt), t

indexing the period 1 or 2. Assume P ′(zt) > 0 and .

Unemployment is also caused by worker-specific, idiosyncratic risks, that materializes

via shocks that affect the workers directly. This shock I describe through the variable v ∈ [0, 1],

with the following meaning: with probability v any worker makes a mistake that hampers

production at his industry and he is fired. Thus, it is a proxy for the quality of the industry’s

labor force. Thus, the worker-specific shock happens with probability v and may happen once

per worker during each period 1 and 2.

This shock can be anything from the entrance of a new competitor, a global crisis, etc.

Figure below shows the timeline associate with one period:

Figure 2: Timeline of the model.

In the following section I solve for both periods’ optimal labor level. I start with a brief recall

of what happens in each period, following figure 2.

• Consider the economy at period 1. At first there is the realization of shock z1, z∗1 . Following,

labor n1 ≡ n1(z∗1) is hired.

• Consider period 2. At the beginning, shock z∗2 is realized; subsequently, labor n2 ≡ n2(z∗2)

is defined; thus, there is the hiring or laying-off of some workers from previous period.

77

3.2 Problem of the industry

Consider some industry ω. Recall that each worker of this industry has probability v of

making an critical error that hampers production; thus, with probability

(1− v)nt ,

no worker does any critical, unemployment-inducing error and each one produces one unit of

output, resulting in nt of output sold at price P (zt). Finally, each worker is paid the nominal

wage W .

Following the timeline presented above, to solve the problem of the industry, the problem of

the industry in each period, 1 and 2, is to choose its labor force nt. First period is the initial

hiring stage, whereas at the second one there is a reevaluation of the mature project.

Thus, in each period t, for a given realization of the respective shock zt, the problem of the

industry is to choose the respective labor force nt:

maxnt

P (zt)nt(1− v)nt − ntW. (1)

First-order condition at both periods is given by2:

(1− v)nt + nt(1− v)nt log(1− v) =W

P (z∗t ). (2)

Now, notice that I am interested not just in the value of the labor forces, but specifically

whether there are hires or layoffs at period 2 relative to period 1.

Thus, given that some labor n1 was chosen at period 1, for which values of shock z2 is there

dismissal of workers in period 2 relative to period 1?

Let me define a critical value of z2, z2, separating a second-period where more labor is hired

(relative to n1) from a second-period with layoffs. This critical value is the realized second-period

shock such that the second period’s market price makes the industry choose the same number

of workers chosen at period 1; therefore, z2 = z∗1 .

Therefore, at the second period if z∗2 > z2, it is optimal to hire people n2 ≡ n(z∗2) > n1;

hereafter I will call shocks that follow this inequality positive shocks.

Otherwise, if z∗2 < z2 then it is optimal to dismiss workers until labor force n2 ≡ n(z∗2);

accordingly, hereafter I will call shocks that follow this inequality negative shocks.

Notice that first period labor has no effect into the second period profit, since there are no

upfront cost associated with hiring or dismissing labor. Now that first and second period labor

forces were determined, the modeling of the risk of default of payroll-backed loans can start.

2Since marginal cost is positivem there is a solution just if marginal revenue is positive, which happens all{v, nt} such that 1 + nt log(1− v) ≥ 0.

78

4 Modeling the risk of payroll-backed loans

At this section, I use the microeconomic model developed to determine the probability of

default and expected losses at period 2 of an individual payroll-backed loan that some worker

hired by the industrial sector took at period 1. Therefore, is an assessment of the future expected

losses at period 2, from the perspective of period 1, after first-period labor force were determined.

As explained before, in order to model the risk of losses from the default of some worker’s

payroll-backed loan, first one must determine the chance of this particular worker being un-

employed. Unemployment is caused by 2 independent events: the worker’s industry received a

negative shock and he is laid-off; second, he does some individual layoff-inducing mistake.

However, being unemployed is neither necessary nor sufficient for a default: the dismissed

worker must also be short on collateral, liquid assets such that he is not able to bear the loan

payments.

In order to model the situations described above, first a description of the dynamics of

workers’ assets flows is necessary. Let Ai be the end-of-period value of the i-th borrower assets,

described by the processes

Ai2 = Ai1 + IisiW (3)

Ai1 = Ai0 + siW (4)

where si is the worker’s saving share of the income and Iit an indicator function equal to one

if the worker was not dismissed at the beginning of period 2, zero otherwise. Ai0 is the initial

asset status of each worker that was hired at period 1, before joining the industry.

First equation, (3), gives second period’s end-of-period asset stock, formed by first period’s

asset stock plus savings from the salaries received during the second period, conditional on this

worker being employed. Accordingly, second equation, (4), gives first period’s end-of-period

asset stock, formed by the workers’ initial asset stock plus savings from the salaries received

during the first period.

Notice that although I continue the assumption that the workers are identical in terms of

their probability v of making a layoff-inducing error, I do not assume that their initial assets Ai0

are the same.

A consumer defaults over his payroll-backed loan if the value Ai of his assets at period 2 falls

below the contractual present value Bi of its payments:

Pr[worker defaults in period 2] ≡ Pr[Ai2 < Bi

2] (5)

In order to solve the above equation, I begin with the following assumption

Assumption: if assets from borrower i at period 1 are greater than his liabilities, expected

79

loss is zero. Thus, I assume full expected recovery of assets in the event of a default. Formally,

Pr[loss|Ai1 ≥ Bi

2] = 0.

This assumption allows me to focus on the loans for which losses are certain if there is

unemployment.

Therefore, for those loans that do not match the above criteria, expected losses depend on

the default probability which, in turn, is a function of the probability of unemployment by

idiosyncratic and aggregate shocks. There are three cases to be considered here3:

a) If z∗2 < z, then a negative shock happened and default might come for those workers chosen

to be laid-off.

b) If z∗2 < z,negative shock happened but the worker is not chosen to be laid-off; then, default

may happens due to idiosyncratic risk events, that is, workers’ mistakes.

c) If z∗2 > z, positive shock happened; then, default may happens due to idiosyncratic risk

events.

Thus, the probability of default on an individual loan is:

F (z2)

[EG

(n1 − n2(z2)

n1

)︸ ︷︷ ︸

Case a)

+

(1− EG

(n1 − n2(z2)

n1

))v︸ ︷︷ ︸

Case b)

]+ (1− F (z2)) v︸ ︷︷ ︸

Case c)

,(6)

where distribution G is F (z2|z2 < z2).

First brace represents Case a) above, that is, it is the expected probability of being fired

given an unemployment-inducing aggregate shock. It is based on the idea that the probability

of being fired, for any worker, is the number of dismissals (that is, n1 minus n2) over the total

number of workers from period 1. The expectation is over the possible realizations of the number

of dismissals.

Second brace represents Case b) above, that is, shock happened but was not unemployment-

inducing for this worker. Default comes from the probability of idiosyncratic unemployment,

v.

Third brace represents the case of a positive shock. Again, default comes from the probability

of idiosyncratic unemployment, v.Now is possible to have a complete assessment of the probability of default:

Pr[Ai2 < Bi

2] =

0 if Ai

1 ≥ Bi2;

F (z2)

[EG

(n1 − n2(z2)

n1

)+

(1− EG

(n1 − n2(z2)

n1

))v

]+ (1− F (z2)) v,

if Ai1 < Bi

2.

(7)

3I do not address the case where the consumer is not fired but his income drops such that he is not able to paythe loan because in Brazil employers are not allowed to decrease payroll of the employees.

80

5 Portfolio Risk

The previous analysis considered credit risk on individual loans. Now, I will use the reasoning

developed there to determine the expected losses on a portfolio of payroll-backed loans.

Consider a portfolio consisting of k ≤ n1 payroll-backed loans in equal value amounts, with

the same term T . Assume one loan for worker, thus I am dealing with loans from k workers.

Let Li be the gross loss (before recoveries) on the i-th loan, so that Li = 1 if the i-th worker

defaults and Li = 0 otherwise. Let L be the portfolio percentage gross loss, that is,

L =1

k

k∑i=1

Li. (8)

Considering just the k∗ workers that have Ai1 < Bi

24,

L∗ =1

k∗

k∗∑i=1

L∗i . (9)

I am interested in the portfolio’s default. Thus, I must consider losses from the workers

affected by an aggregate shock and from those that were affect by idiosyncratic shocks.

Notice that all events are mutually exclusive, you can’t be unemployed more than once. Also,

the probability of a worker being chosen for dismissal is, ex-ante, independent of another worker.Thus, the probability of a default up to r% of the portfolio means up to k∗r

100 workers default

in their loans. Therefore, k∗×m100 workers either got the aggregate negative shock:

PDN = Pr[negative shock and some worker defaults]

= F (z2)

[EG

(n1 − n2(z2)

n1

)+

(1− EG

(n1 − n2(z2)

n1

))v

](10)

or a positive one

Pr[positive shock and some worker defaults]

= (1− F (z2)) v.(11)

Thus, folows the paper’s first proposition:

Proposition 1. Consider a portfolio of k payroll-backed loans in equal value amounts, with the

same term T . The expected loss of the portfolio is given as follows:

• Consider those borrowers/workers i such that Ai1 ≥ Bi

2; then, Expected Loss is zero.

4I will use the asterisk (*) to mark any function defined over the subset of workers satisfying this inequality.

81

• Otherwise, the Probability of a Portfolio Loss less or equal to a share r is given by

Pr[L∗ ≤ r] = Pr[ k∗r or less workers default ]

= F (z2)B

(k∗r; k∗,

PDN

F (z2)

)+ (1− F (z2))B

(k∗r; k∗,

v

(1− F (z2))

) (12)

where PDN is given respectively by (10) and (11). B(k∗r; k∗, PDN

F (z2)

)is the cumulative

distribution function at position k∗r of a Binomial distribution with parameters k∗ andPDNF (z2)

; accordingly, B(k∗r; k∗, v

(1−F (z2)

)is the cumulative distribution function at position

k∗r of a Binomial distribution with parameters k∗ and v(1−F (z2)

:

B(k∗r; k∗, PDN) =k∗r∑i=0

(k∗

i

)(PDN

F (z2)

)i(1− PDN

F (z2)

)k∗−i(13)

B(k∗r; k∗, v) =

k∗r∑i=0

(k∗

i

)(v

1− F (z2)

)i(1− v

1− F (z2)

)k∗−i. (14)

In both binomials in (12) the number of trials is represented by the size of the portfolio and

”success” by the probability of default.

The use of a binomial distribution is due to way I chose to analyze default here: probability

of default in a percentage of the portfolio. Thus I must consider, for instance, out of k∗ workers,

the probability of one worker fired because of idiosyncratic unemployment and k∗ − 1 not fired,

or two workers fired and k∗ − 2 not, and so on until k∗r workers.

6 Comparative Statics

Previous sections showed formulas giving the riskiness of individual loans and of a portfolio

made up from payroll-backed loans. Here, I go further to analyze the sensitivity of the underlying

forces that drive default. Unless otherwise noticed, I will base the analysis on the formula for

an individual loan, equation (7), since the individual view is enough to assess the effects of the

main drivers of risk.

Default probabilities are driven by three variables: v, n1 and F (z). I describe their effects in

this sequence.

Proposition 2. The effects over the default risk from changes in the variables v, n1, everything

else constant (for each one) are:

(a) An increase in the idiosyncratic probability of unemployment, v, everything else constant,

increases the probability of default;

(b) An increase in first-period labor, n1, has ambiguous results. However, the lower are both n1

and the idiosyncratic probability of default, the more likely that an increase in labor force

will lead to an increase in the overall probability of default for an individual loan.

82

(c) A mean-preserving spread increases the probability of default from an aggregate negative

shock, everything else constant.

Proof: Appendix.

The variable whose effect is more difficult to describe is the labor force at the initial period,

n1. This is due to two opposing effects caused by any change in the labor force: first, any rise in

n1 increases output for a given productivity; however, it diminishes productivity, but this effect

is smaller the lower is v.

To capture the impact of this effect, consider a specific economic scenario: how the risk of

an individual payroll-backed loan changes during some period of downturn in the sector.

In terms of the model, the equivalent question is this:

Under which conditions the ocurrence of a negative shock in period 1 decreases the

probability of default at period 2?

Based on the effects described in Proposition 2, given that the shock at period one lowers

this period’s labor force, according to equation (A.1) the lower is v the more likely that this

decrease in n1 will lead to a decrease in z2, in turn lowering the probability of a second negative

shock.

Intuition is this:

1. First, there is the direct effect from the fall in n1 over z2 and over F (z2):

i. a fall in n1 decreases output but increases total productivity (because if there are fewer

workers, there are less things that can go wrong, see (1)) for a given output;

ii. the lower is v, the lower the impact of the decrease in n1 over total productivity and

thus the less likely that the increase in productivity compensates the other effect,

resulting in lower overall revenue;

iii. However, the lower are v and n1 after the shock, the higher the marginal revenue;

iv. Therefore, at second period, the equivalent marginal cost to induce the same choice of

labor from the first-period is higher than in the same cenario with a higher v;

v. Finally, for the given nominal salary, this implies a lower price, and thus a lower

threshold z2 (P ′ > 0).

2. Finally, the lower n1 decreases the probability of any remaining worker being chosen for

layoff at second-period, if another negative shock happens.

Next proposition summarizes this discussion:

Proposition 3. Consider a scenario where a negative aggregate shock happens at period one.

Then, the lower is the idiosincratic probability of default and the labor force at period one, the

smaller is the total probability of default at period 2.

83

7 Conclusion

This work presents a theoretical evaluation of the main characteristics that drive the risk of

credit losses on a portfolio of consumer payroll-backed loans. Consumer credit through payroll-

backed loans is one of the fastest growing credit lines in Brazil. Under reasonable assumptions, I

construct a microeconomic-based model of an industry whose workers are borrowers of payroll-

backed loans. Then, I derive a formula for the probability of default and expected losses for

both an individual loan and a portfolio of loans from the industry’s workers. Finally, the core

variables of the formula are subject to a sensibility analysis to determine their influence on the

risk of payroll-backed loans.

84

References

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ber 24, 2010.

de Andrade, F.W.M. and L. Thomas, “Structural models in consumer credit,” European

Journal of Operational Research, 2007, 183 (3), 1569–1581.

Estado de Minas, “Credito consignado garante lucro recorde ao BMG,” http:

//www.em.com.br/app/noticia/economia/2010/01/26/internas_economia,145230/

credito-consignado-garante-lucro-recorde-ao-bmg.shtml. January 26, 2010.

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nomics, 1993, 108 (3), 551–575.

Merton, R.C., “On the pricing of corporate debt: The risk structure of interest rates,” Journal

of finance, 1974, pp. 449–470.

Thomas, L.C., “Modelling the Credit Risk for Portfolios of Consumer Loans: Analogies with

corporate loan models,” Mathematics and Computers in Simulation, 2009, 79 (8), 2525–2534.

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85

Appendix

Proof of Proposition 2:

Item (a): See (7).

Item (b):

The proof follows by showing the effect of a change in one variable over the default equation,

everything else constant.

I start with the effects an increase in n1 has over the individual elements of the default

equation. After that, I given an example to integrate the whole effect.

• From (7), EG

(n1−n2(z2)

n1

)increases with an increase in n1, everything else constant.

• The threshold z2 is affected by n1, that is, the threshold probability of a negative shock

given an aggregate shoc, everything else constant. I obtain that

∂z2∂n1

T 0⇐⇒ 2 + n1 log (1− v) T 0 (A.1)

• Finally, if the threshold z2 is affected, so are both F (z2) and expectations over the truncated

distribution function G(z) (G = F (z|z < z2). I obtain the following effects:

– An increase in z2 increases the cumulative probability given by F (z2);

– An increase in z2 increases EG(z).

Now, an example to make all these effects more clear.

For instance, suppose both n1 and v are large, such that 2 + n1 log (1− v) < 0; then, an

increase in n1...

• increases the default probability via the following channels:

– the chance of a particular worker being chosen for layoff increases, because EG

(n1−n2(z2)

n1

)increases for a given n2 and a given distribution G;

• decreases the default probability via the following channels:

– the probability of an idiosyncratic shock (F (z2)), since z2 decreases;

– the expectation of n2(z2) decreases due to the changes in the truncated distribution

G(z2), caused by the decrease in z2

Item (c): In order to assess the effect of changes in the distribution of severity, F (zt), let me

examine the effect of a mean-preserving spread on F (z2). Call the new distribution F ′(z2). A

mean-preserving spread makes tails fatter; therefore, E(G′(z)) decreases, that is, the expected

negative shock increases because distribution G is distribution F truncated below: since there is

more mass in the left tail, the mean decreases.