Long-term relationships : Static gains and Dynamic
ine¢ ciencies
David
Hemous
Morten
Olseny
Septemb er 4, 2010
Abstract
This pap er form al izes the idea that rms can (partly) overcome the static costs asso ciated
with low contractibility by engaging in long-term relationships, but that the rigidity of these
relationships may lead to strong dynamic ine¢ ciencies. Sp eci cally, we consider the rep eated
interaction b etween a nal go o d pro ducer and an intermediate input sup plier. A pair pro du
cer/supplier can b e a go o d match or a bad match and the nature of the match do e s not
change over time. As a consequence, learning th at a m atch is go o d increases the value of the
relationship and creates the p otential for co op eration in general equilibrium. We then allow for
innovati on on the supplier side and show (i) that innovations need to b e larger for a new
supplier with sup erior technology to break up existing relationships than in either a set-up where
the input is contrac tible or w hen we preclude c o op eration in long-term relationships, an d (ii)
that th e rate of inn ovation is lower than in the contractible case, and may b e lower than when
co op eration is precluded in the noncontrac tible setting. We then show that, when innovation is
exogenous, co op e ration in long-term relationships may overcome most of the welfare loss asso
ciate d with non-contractibility, but that, once innovation is endogenized, welfare is much lower in
the noncontractible case than in the contrac ti ble case, and can even b e lower if co op eration o
ccurs in long-term relationships than if it do es not. However, we show that far from the techn
ologic al frontier, the establishment of long-term relationships can turn out to b e a go o d
substitute to high contractibility. An analysis of patents data in the US provide motivational
evidence to our work.
JEL. Keywords: contractibility, long-term relationships,
innovation.
H a r va r d U n
i ve r s i ty
yH a r va r d U n i ve r s i ty
1 Intro duction
Do contractual frictions still matter when rms engage in rep eated interac tion s? Although
recent work in the grow th literature has emphasized the central role that institutional features
like contractibility play in shaping income and growth di/erences across countries, at the same
time, there is widespread evidence that rms engage in imp licit contracting that could substitute
for formal contracting (for instance [Johnson et al., 2002] s hows in a survey across Eastern
Europ ean countries that after the rst year of a relationship the extent to which parties trust the
court has a very small impact on trust towards one another). At rst glance, co op eration in
long-term relationships could then overcome the ine¢ ciencies of a p o or contractual
environment. This pap er shows that implicit contracting is in fact a very p o or substitute for
formal contracts once dynamic asp ects are intro duced. Indeed, co op eration in long-term
relationships comes at a pri ce: stickiness in relationships; rms may prefer to stay with a known
partner wi th whom they are used to co op e rate rather than risking the switch to a new partner
with access to a more pro ductive technology. In fact, [Johnson et al., 2002] has shown that the
b elief in c ou rts e¢ ciency has a strong impact on the incentive for rms to try out ne w
suppliers. Rigidities in relations hips, in return, reduces the inc entive for entrants to innovate, up
to a p oint that the economy might even b e worse o/ when co op eration arises in long-term
relationships than w hen it do es not. Hence, the p ossibility of developing co op eration in
long-term relationships turns contractibility issues from a static problem of ine¢ cient allo cation
of resources into a dynamic problem of ine¢ cient developme nt of technologies. The idea that
long-term relationships can overcome institutional weaknesses but at the cost of
adding rigidities in the economy, is not new in the literature ([Johnson et al., 2002] evokes this
p ossibility), but, to our knowledge, it has not b een formally analyzed yet. In this pap er, we
provide a rst attempt at such an analysis, and we inve stigate some of the macro economic
consequences, fo cusing in particular on growth and innovation. However, rigidities in re
lationships could p otentially a/ect the economy in other dimensions, for instance the ability to
resp ond to macro economic sho ck s as well.
More sp eci cally, we consider an industry where pro duction requires that a nal go o d pro
ducer and an intermediate input supplier work together, but where the supplier is the only one
making an investment. We contrast the case where the provision of the input is contractible with
the case whe re it is not. In a non-rep eated framework, noncontractibility typically creates an ex
p ost hold-up situation leading to underinvestment by the supplier as in [Grossman and Hart,
1986]. The match b etween the pro ducer and the supplier can b e go o d or bad , where go o
d matches are characterized by a higher pro ductivity. The nature of the match is unknown to b
oth parties b efore they start working together and it do es not change over time. T herefore , if a
match turns out to b e go o d, the value of the relationship in the following p erio d is higher than
the exp ected value of a new relationship. In the noncontractible case, this
1
di/erence in values allows for an equilibrium where in a go o d match the suppli er invests more
than he would in a one s hot interaction, even though pro ducers are able to sw itch suppliers at
w ill and suppliers do not c o ordinate amongst thems elves to punish pro duce rs who switch
suppliers.
Having intro duced the p os sibility of rep e ate d interaction partly overcoming contractual
incompleteness, we go on to demonstrate the p ossible ine¢ ciencies of established
relationships. We allow a new supplier to enter the market with a b etter technology. Pro ducers
already engaged in a long-term relationship face a trade-o/: switching to the new supplier allows
them to have access to a more pro ductive technology, but at the risk of entering into a bad
match. Entering into a bad match always yields lower pro duction b ecause of the inherent lower
pro ductivity of the relationship; but, wh en the input is noncontractible and when suppliers co op
erate in go o d matches, bad matches are also characterized by more severe under-investment
than go o d matches. Hence, bad matches are even worse relative to go o d matches, this
worse bad match e/ ect is the principal force b ehind our result that co op eration in a we ak
contractible setting magni es rigidities in relationships. First, we show that, in order for an
innovator to capture a large share of the market, innovations have to b e more radical when i n
the noncontractible c as e with co op eration (which we lab el as the co op erative case ) than
in the contractible c as e or in the nonc ontractible case when co op eration is precluded in
long-term relationships (which we lab el as the Nash case ). As a cons equence, the mo del
predicts more technological di/erences across rms in countries with p o or contractibility institu
tion s (if coop eration aris es) than in countries with strong institutions. Then, we show that
innovation and even welfare can b e lower i n the co op erative case than in the Nash case. We
provide a numerical example where the establishment of co op eration in long-term relationships
allows to overcome most of the ine¢ ciencies of p o or c ontractibility when innovation is
exogenous, but is actually detrimental to welfare when innovation is endogenous.
In an extensions section, we derive additional predictions by mo difying the initial framework
in di/erent ways. First, we develop a q uick ex tension of the mo del where there is a xed
technological frontier that all suppliers can freely imitate on at the b eginning of the p erio d, in
this case co op eration in long-term relationships turns out to b e a go o d substitute to go o d
contractibility institutions, not only b ecause most of the growth o ccurs through imitation so that
the impact of innovation is reduce d but also b ecause imitation itself reinforces co op eration.
Second, we show how with a small mo di cation of the strate gies played in the co op erative
case in the original set-up, the co op erative case can easily feature multiple equilibria: some
with high levels of co op eration and reduced growth and some with low level of co op eration
and faster growth, and how innovation may tend to happ en through waves in the co op erative
case. Third, we show that co op eration in long-te rm relationships may incentivize innovators to
pro duce scarcer but larger innovations, and fourth, that the negative e/ect of co op eration
2
on innovation can b e reduced when patents are b etter enforced. Our pap er relates to two
main topics in the literature : the p ossi bility of building relation1ships
under imp erfect contractibility, and the impact of institutions, in particular contractibility,
on macro economic outcomes (growth or trade). A large b o dy of theoretical literature
addresses th e question of building relationships in the presence of contractual incompleteness:
the rep etition of the same interaction can give rise to equilibria of the Folk theorem typ e, where
parties co op erate and prov ide more e/ort (or investment) than they would in a one-shot inte
raction. [Kreps, 1996] gives a go o d overview of the theoretical work on this topic. A prominent
pap er in the eld is [Baker et al., 1994], the authors address the issue of formal versus implicit
contracts by considering the rep eated interaction b etwee n a rm and an employee. They
analyze a situation where rms rewards employees through two mechanisms: a wage that dep
ends on a contractible signal of the employee s p erformance, and a noncontractible b onus that
dep ends directly on the employee s unveri able p erformance. They demonstrate that
sometimes a signal b etter correlated to the actual p erformance (w hich can b e interpreted as a
proxy for more develop ed institutions) can prevent the formation of more e¢ cient implicit
contracts.
[Macaulay, 1963] is the rst pap er to show that interactions b etween rms in most markets are
rep eated and that rms engage in relational contracts. [Ellicks on , 1991] is the seminal
reference in the law literature arguing that, in close commun ities, p eople co op erate without
the he lp of the state, b ecause they are engaged in rep eated interactions. [Brown et al., 2004]
ran exp eriments showing the endogenous emergence of long-term relationships in the absence
of thi rd party enforcement, where low e/ort was punished by the termination of the relationship.
Furthermore, they showed that in successful relationships, e/ort was high from the very b
eginning. Our eq uilibrium shares these features. [Banerjee et al., 2000] show that in the Indian
software industry, reputation of rms matter for the kind of contrac ts they are o/ered. [Allen et
al., 2005], [Allen et al., 2006] and [Allen et al., 2008] show in related pap ers that in India and
China long-term relationships provide a success fu l way to nance rms. [MacLeo d, 2006]
provides a very interesting discussion of the di/erent mechanisms that allow the enforcement of
incomplete contracts, and comp ares the p erformance of informal enforcement with formal
enforceme nt. [Raiser et al., 2008] use World Bank Enterprise Survey (as we do) to asse ss the
determinants of trust in trans ition countries.
Since [North, 1981] a large literature has emerged on the impact of institutions on growth and
development. [La Porta et al., 1998] and [Acemoglu et al., 2001] are the se minal pap ers which
demonstrate empirically that variations i n the institutional foundations can have signi cant in uence
on economic outcomes. A nice theoretical treatment is given in [Acemoglu et al., 2006].
1I
n [ B a ke r e t a l . , 2 0 0 2 ] , t h i s m o d e l i s e x t e n d e d t o s t u d y t h e b o u n d a r i e s o f t h e r m . K l
e i n a n d L e › e r ( 1 9 8 1 ) s h ow t h a t r e p e a t e d p u r ch a s e s by c o n s u m e r s m ay i n c e nt i v i z e r
m s t o p r ov i d e h i g h q u a l i ty g o o d i n a n o ow c o nt r a c t i b l e s e t t i n g , b u t t h a t t h e p r i c e n e e
d s t o b e h i g h e r t h a n t h e c o m p e t i t i ve p r i c e . I n [ K l e i n , 1 9 9 6 ] , r m s e n g a g e d i n l o n g - t e
r m r e l a t i o n s h i p s , c o o p e r a t e w i t h i n a r a n g e o f p r i c e va r i a t i o n , b u t h o l d - u p o c c u r s o
utsidethisrange.
3
2Pro
ductivity increases b oth through imitation from the technological frontier and incremental
innovation, where the capacity of incremental innovation dep ends on the ability of th e
manager. Far from the technological frontier, rms pursue inves tment-based strategies
featuring long-term relationships b etween rms and their managers, but sacri ce selection of
managers, whereas close to the frontier, relations hips b ecome shorter and the selection of go
o d managers b ecomes more imp ortant. Institutions that favor the establishment of long-term
relationships are appropriate far from the frontier but turn out to b e a burde n close to it.
3Amongst
these institutional features, the extent to which one can enforce contracts is of
particular interest, as it varies widely even across develop ed countries. [Acemoglu et al., 2007]
have shown that countries with weaker legal institutions adopt inferior technologies and develop
a comparative advantage in sectors where there is more substitutability across inputs.4Although
[Acemoglu and Johnson, 2005] downplays the imp ortance of contractibility compared to prop
erty rights institutions, [Cowan and Neut, 2007] show empirically, that pro ductivity is relatively
larger in countries with go o d legal enforceme nt in sectors with a more complex intermediate
structure, and, similarly, [Nunn, 2007] show that the se countries develop a comparative
advantage in sectors that rely more on relation-sp eci c investments.[Dixit, 2004] is closely
related to our pap er as it analyzes the typ e of informal institutions that emerge when the jud
iciary system of a country is not very develop ed yet (a famous example based on rep eated
interactions is [Greif, 1993] s Maghribi traders).
Starting with [Aghion and T irole , 1996], an imp ortant b o dy of literature analyzes the
organization of R&D in an incomplete contracts framework (see [Aghion and Howitt, 1998] for
an interesting analysis of th ese issues), in contrast, our pap er ignores the issue of the exact
organization of the R&D pro cess and fo cuses on the impact of the organization of the pro
duction pro cess on innovation. The literature on incomplete contracts and macro economics
also include: [Francois et al., 2003], who study the impact of growth on contractual
arrangements (some of our results p oint towards feedback e /ects whe re the frequenc y and
the typ e of innovation a/ect in return the extent of co op eration b etween business partners);
and [Caballero and Ham mour, 1998] who relate incomplete contracts with macro economics
sho ck s ampli cation. Finally, a related idea that long-term relationships b etween pro ducers
and suppliers can b e barriers to entry was already formalized in [Aghion and Bol ton, 1987],
who show that when an incumb ent fac es entry by p otential comp etitors with sup erior
technology, she will sign long-term contract that reduces the risk of entry. In our set-up,
however, the relationship
2[
B o n g l i o l i e t a l . , 2 0 0 9 ] p r e s e nt a s i m i l a r t r a d e - o /. I n t h e i r m o d e l , r i g i d l o n g - t e r m c
o nt r a c t s t h a t f avo r i nve s t m e nt by m a n a g e r s , c a n b e o p t i m a l a t e a r l y s t a g e o f d e ve l o p
m e nt b u t a s c a p i t a l a c c u mu l a t e s , t h e r e t u r n t o t a l e nt i n c r e a s e s a n d e x i b l e c o nt r a c t
saremoreappropriate.
3[ O t t av i a n o , 2 0 0 7 ] s t u d i e s h ow c o nt r a c t e n f o r c e m e nt s i mu l t a n e o u s l y a /e c t s t h e l o c
a t i o n o f R & D a c t i v i t i e s a n d w h e t h e r p r o d u c t i o n a n d R & D a r e i nt e g r a t e d o r n o t , a n d h
ow i n r e t u r n t h i s l a s t d e c i s i o n a /e c t s t h e r e t u r n s f r o m R & D .
4I n t h e t r a d e l i t e r a t u r e , [ R a u ch , 1 9 9 9 ] s h ow s t h e i m p o r t a n c e o f n e two r k s i n s h a p i n g t
r a d e , e s p e c i a l l y f o r m o r e d i /e r e nt i a t e d p r o d u c t s .
4
is of a di/erent nature as the contract is implicit and we rule out e xplicit contracts that would last
more th an a single p erio d.
Section 2 pres ents the motivational e vidence. Section 3 presents the mo de l in a static
framework, and shows how long-term relationships bring static gains. Sec ti on 4 intro duces
innovation in this framework, and contains our main results. Section 5 extends the mo de l in
di/erent directions and contains additional predictions. Finally section 6 concludes an d suggests
future research.
2 Motivational evidence
Our motivational e vidence are based on the dis tribution of fore ign patents recorded at the
United States Patents O¢ ce (USPT O) b etween 1992 and 2006. We show that countries with
b etter contracting institutions tend to in novate relatively more in s ectors that are more
contract intensive. More sp eci cally, we run regressions of the form:
c
= Lc
Cs + X Xc Cs + s + c + t
+ "c;s;t; (1)
and t
Yc;s;t
s
,
c
6
where Yc;s;t
s
c
5
5
2.1 Data
7
5
6
7
is a measure of the amount of innovation by rms from country c, in sector s, in year
t, Lis a measure of the quality of contracting institutions in country c, Xis a vector of country-sp
eci c controls, Cis a measure of contractual intensity in sector s, and
are sector-sp eci c, country-sp eci c and year-sp eci c xed e/ects.
Patents data have b een w idely used as a proxy for innovation. The NBER dataset, describ ed
in [Hall et al., 2001], contains all patents rec orde d at the USPTO from 1963 to 2006 and the
numb er of citations made to each patent until 2006 adjusted for the age of the patent. We
complete the dataset available online with the dataset on the assignees of the patents from [Lai
et al., 2009], and we fo cus on the 2064233 patents granted after 1992. We are able to identify
the country of the rst assignee for 1927079 patents, dropping the 1023038 patents from US
origin, patents from countries with less than 50 patents total in the p erio d 1992 - 2006, patents
from scal havensand patents attributed to the USSR, Czechoslovakia or Yugoslavia, we are
left with a total of 889208 patents. Each patent is given (at least) one technological co de
according to the IPC classi cation.Using a concordance b etween th e IPC co de and the ISIC
classi cation,we are able to match each observation with a sp eci c sector (at the two or three
digit level), and then each patent to a sp eci c sector by randomly selecting a single sector
We d r o p p a t e nt s f r o m B a h a m a s , B a r b a d o s , B e r mu d a , C ay m a n i s l a n d s , C o o k i s l a
n d s , N e t h e r l a n d s a nt i l l e s , S t . K i t t s a n d N e v i s a n d V i r g i n ( B r i t i s h ) i s l a n d s .
T h e r e i s a t o t a l o f 9 2 4 9 1 4 o b s e r va t i o n s f o r o n l y 9 0 0 3 4 6 p a t e nt s .
We r s t u s e a c o n c o r d a n c e I P C - N AC E r e v 1 a n d t h e n N AC E r e v 1 - I S I C r e v 3 .
when a patent falls in several sectors. To ge t our rst measure of innovation, we aggregate the
numb er of patents at the country - s ector - year level, add 1 and take the logarithm. However,
each patent do es not h ave the same economic value: some innovations are worth much more
than others, a proxy commonly used for the size of an innovation asso ciated with a patent is
the numb er of citations made to the patent (see for in stance [Tra jtenb erg, 1990]). For our
second measure of innovation we then aggregate the numb er of patents at the country - sector
- year level with the numb er of citations m ade to these patents, add 1 and take the logarithm.
For our third measure, we replace the numb er of citation as measure of the size of one
innovation by the logarithm of 1 plus the numb er of citations. For our second and third
measures, we restrict attention to patent granted b efore 2003 ([Hall et al., 2001] advise to
ignore the last three years of data when using citations).
The measures of contracting institutions that we use are from the World Bank ([Bank, 2004])
and have b een used in several pap ers including [Acemoglu and Johnson, 2005]. Following the
metho dology of [Djankov et al., 2003], the World Bank collected data on the pro cedures involve
d when a rm tries to collect a commercial debt contract worth 50 p ercent of the coun try s
annual income p er capita from a reluctant buyer. They computed two measures: the total numb
er of pro c edures and an index (on a scale from 1 to 10) of the degree of legal formalism
involved. The underlying assumption is that contracts are easier (and l ess exp ensive) to
enforce when the numb er of pro cedures and the degree of legal formalism is limited. The
control variables that we use are GDP p er capita in 1992 (from the World Bank) and the
measure of protection against expropriation by government averaged over 1985 - 1995 from
Political Risk Serv ices (this measure is used for instance in [Knack and Keefer, 1995] and in
[Acemoglu and Johnson, 2005]).
8Our measure of contractual intensity is based on the latest version of the classi cation of go o
ds elab orated by [Rauch, 1999]. Rauch classi e s go o ds (at the 4 digit SITC re v2 level) dep
ending on whethe r they are sold on organized exch an ge , reference priced in catalogs or
neither. Presumably, go o ds sold on an organized exchange market are quite homogenous with
a thick marke t, while go o ds that are not even reference p ric ed are more di/erentiated and sp
eci c to a limited numb er of buyers. The scop e for hold-up is then larger for the latter go o ds
than for the former, and as se veral other studies have done (for instance [Nunn, 2007]), we can
use this classi cation to proxy for the contractual intensity of a sector. More sp eci cally, we build
an index taking the value 0 when the go o d is sold on an organized exchanged market, 0.5
when it is reference priced and 1 when it is neither sold on an organized market nor reference
priced. We use concordance tables to compute such a measure at the ISIC rev3 2 digits and 3
digits levels.Rauch gives a conservative and a lib eral c las si cation, we will use b oth in our
empirical analysis.
8We
useconcordancetablesgoingfromSITCrev2toSITCrev3andthenS
ITCrev3toISICrev3.
6
In App endix Z, we rep ort a table giving the distribution of patents across countries together
with the m easure of contracting ins titutions in the country, and we rep ort a table describing the
distribution of patents. Because a higher value of Lindicates worse contracting institutions and a
higher value of Csca higher level of contractual intensity, we exp ect to nd < 0: innovation
should b e relatively larger in countries with go o d contracting institutions in sectors with a high
level of contractual intensity.
2.2 Results
In our baseline regression we have observations for 46 countries across 33 sectors and in 15
di/erent years. However, many observations at the country - sector - year levels are 0 (47% of
the observations in our baseline case), we then run separately our regression on the intensive
margin (whether there is at least one patent in a given country - sector - year) and on the
extensive margin.
t ht hFor
the intensive margin, we adopt a logit mo del. The results are rep orted in table 1
(standard errors are clustered at the country-industry leve l). Column s (1) shows that in
countries where the numb er of pro cedures to collect a check is higher, the probability of
innovation is reduced in sectors where contractual intensity is higher (when one uses the lib eral
classi cation); the e/ect is signi cant at the 1% level. Column (3) shows that the same holds
when one uses the c on servative Rauch classi cation and column (2) shows that the formalism
index has the same negative e/ect (signi cant at the 5% level when one uses the lib eral
classi cation as in column (2), it is signi cant at the 1% level when one uses the conservative
classi cation). An obv ious alternative ex planation is that sectors with a higher contractual
intens ity may also b e more "c omplex" (although a quick lo ok at table 0.2 shows that there
may not b e a p erfect correlation), and the re fore only develop ed countries are susceptible to
innovate in thes e s ectors. To take into account this p ossibility, we include the interaction b
etween GNI p er capita in 1992 with the contractual intensity of the sector as a control variable.
Even though, this control is highly e ndogenous, the e/ect of worse contractibility institutions do
e s not disapp ear: it remains signi cant at the 10% level when one uses the lib eral
classi cation, it is signi cant at the 1% level with the numb er of pro cedures and the
conservative classi cation (and at the 5% level w ith the formal index and the conservative
classi cation in a non rep orted regress ion). In column (8), we als o control for the risk of
expropriation; in many pap ers, th e measu re of the quality of the judicial s ystem used mixes
the quality of contracting institutions with property rights protection, here we see that the prop
erty rights variable has no explanatory p ower. Finally, the e/ect is economically signi cant, in
table 1, the o dd s ratio is 0.926, so that if a country s numb er of pro cedures moves from the
75p ercentile (22, e.g.Germany) to the 25
t hp ercentile (16, e.g. Ireland), the o dds ratio increases by a factor 1.14 more in the sector in
the 75p ercentile of contractual intensity (1, e.g. Manuf of general purp ose machinery ) than in
7
the sector in the 25t h p ercentile (0.71, e.g. Manuf of other electrical equipment n.e.c.).
Table 1: Intensive margin
(1) (2) (3) (4) (5) (6) (7) (8) Num pro
c 0.076*** 0.080** 0.049* 0.059*
*Cont int lib (0.028) (0.032) (0.028) (0.032) 0.926 0.923 0.952 0.943
Form index 0.398** 0.296* *Cont int lib (0.163) (0.157)
0.672 0.744 Num pro c 0.121*** 0.093***
*Cont int con (0.034) (0.034) 0.886 0.911
GNI p er cap 0.706*** 0.734*** 0.782*** *Cont int lib (0.167) (0.163) (0.205)
2.025 2.083 2.185 GNI
p er cap 0.611***
*Cont int con (0.203) 1.842
Prot. Exprop 0.175 *Cont int lib (0.291)
20.839 pseudo R0.56 0.56 0.56 0.49 0.54 0.54 0.54 0.53 Obs 22275 22275 22275
14355 20295 20295 20295 18810 Clusters 1485 1485 1485 957 1353 1353 1353 1254
Note: Logit mo del. Table re p orts co e¢ cient, standard errors clustere d by country-indus try in
parentheses and o dds ratio. All regressions include country, sector and year xed e/ects.
Column 4 restricts attention to countries with more than 500 patents in total. * p <0.1, ** p<0.05,
*** p<0.01
We now deal with the extensive margin, that is the numb er of patents
conditional on a least one patent b eing granted in the country - se ctor year. We run a simple OLS regress ion on the sample of country - industry year with at least one patent. The res ults are rep orted in table 2 (standard
errors are clustered at the country-industry level). In all columns except
column 4 the dep endent variable is the logarithm of the total numb er of
patents, in column 4, the dep endent variable is the logarithm of the total
numb er of patents plus the total numb er of citations made to these
patents. In columns (1), (2) and (3) we nd a negative and signi cant at 1%
level e/ect of a higher numb er of pro cedures or more formal legal syste m
on the numb er of patents in more contractual intensive sectors. Column (4)
shows that the e/ect carries on when one takes into account the size of
inventions as proxied by the numb er of citations made to those patents.
Column (5) restrict attention to countries with at least 500 patents in total. In
columns (6) and (7) we control for initial GNI p er capita, th e e/ect of the
numb er of pro cedures remains signi cant at the 10% level when one u ses
the lib eral classi cation, but the e/ect of the
8
t
h
f
o
r
m
a
l
i
s
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1
1 (1
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U
= clarity we w ill drop the subscripts t
where denotes the di scount rate of the c on sumers. For
when this do es not lead to c on fu sion.
We assume that the p ro ductivity of the di/erentiated go o d (to b e sp eci ed b e low) is
low enough that the outside go o d always remain active. Each variety is pro duced monop
olistically
10
X
Table 3: Tobit mo del
(1) (2) (3) (4) (5) Dep var: log(1+Pat)
Num pro c 0.064*** 0.030** *Cont int lib (0.015) (0.015)
Form index 0.257*** 0.099 *Cont int lib (0.075) (0.070)
Num pro c 0.084*** *Cont int con (0.019)
GNI p er cap 0.504*** 0.531*** *Cont int lib (0.083) (0.081)
pseudo R20.47 0.47 0.47 0.47 0.47 Observations 22770 22770 22770 20790
20790 Clusters 1518 1518 1518 1386 1386
Note: Tobit mo del. Table rep orts co e¢ cient an d standard errors clustered by
countryindustry in parentheses. All regressions include c ountry, sector and year xed
e/ects . * p <0.1, ** p<0.05, *** p<0.01
3 Sustaining long-term relationships and contractibility
In this section we deve lop a mo del of rep eated interaction b etween nal go o d pro ducers
and intermediate input s uppliers in general equilibrium, where some matches pro duce r/s
upplier are exogenously b etter than others. We rst show that when the prov ision of the
input is noncontractible, the classic hold-up problem arises and suppliers have an incentive to
underinvest in an one-shot interaction. We then let the game b e rep eated, and sh ow that
the prosp ect of continuin g the re lation ship in the following p erio d provides suppliers in a
go o d match with an incentive to co op erate with the pro ducer, by investing more than they
would in the one-shot interaction.
3.1 Preferences and pro duction
We consider a quasi-general equilibrium mo del where consumers consume only two typ es
of nal go o ds: a set of di/erentiated go o ds (denoted c) of measure 1, and a homogenous
outside go o d (denoted Coi). Aggregate p re fe re nces are given by a representative agent
with utility function:
1Z
C
by a nal go o d pro ducer. The demand for a variety j (c) and the quantity of variety j pro
duced ( qjj), can then b e written as a function solely of its own price :9
qj = cj = p j: (3)
1
Every p erio d a mass
j
D
=
10
k jk
Ak)
D
1 1
qj
jkAk
b
=
g
jk
jk
jk
.RThe outside sector pro duces a homogen
evenues of the pro ducer of variety jcan then b e expressed
one
as q
with lab or and we normalize its price to 1, suc
to 1. All the action in the mo del takes place in
duction of each nal go o d jin the di/erentiate
provided by an intermediate input supplier. Int
quality, and, only go o d quality in puts have v
is supplied by only one intermediate input sup
can s upply any numb er of nal go o d pro du
is a mass 1 of intermediate input suppliers.
of existing nal g
ones. Intermediate input suppliers are in nitely
= ( jkAk)1 1X; (4) where is a match sp eci c an
the pro ductivity of the inte rm ediate input sup
inputs of go o d quality provided by the suppli
ductivity, but, throughout the pap er we make
ductivity). The match sp eci c level of pro duct
matches, and <1 in bad matches . T he qua
and the pro ducer only once they start working
probability bthat a random pair pro ducer/supp
one unit of go o d quality intermediate input re
quality inputs can b e pro duced costless ly).
Throughout the pap er we will normalize th
the supplier by the pro ductivity of the relation
can the n expres s revenues as jkAkjkAk 1 ), a
normalized revenues ( R(x) x 1 (x) where (x
9T
h e f u n c t i o n a l f o r m o f t h e u t i l i ty f u n c t i
m e /e c t s g o i n g t h r o u g h t h e wa g e s ( t h a n k s t o t h e p r e s e n e d w i t h t h e p r i c e e l a s t i c i ty o f t h e C E S a g
c e o f t h e h o m o g e n o u s g o o d ) o r t h e p r i c e i n d e x ( a s t h e e l e a t u r e s wo u l d c o m p l i c a t e t h e a n a l y s i s
a s t i c i ty o f s u b s t i t u t i o n b e twe e n t h e va r i e t i e s i s e q u a l i z t s .
1 0We c o u l d e q u a l l y we l l h ave a s s u m e d t h a t t h e i nt e r m e d i a t e g o o d s u p p l i e.
r s d i e w i t h p r o b a b i l i ty
11
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er
,
a
di
/e
re
nt
st
at
e
of
th
e
w
or
ld
is
dr
a
w
n
e
v
er
y
p
er
io
d
af
te
r
th
e
pr
o
d
u
c
er
h
a
s
c
h
o
s
e
n
hi
s
s
u
p
pl
ie
r.
T
o
b
e
of
g
o
o
d
q
u
al
it
y,
a
n
in
p
ut
m
u
st
b
e
ta
il
or
e
d
to
th
e
s
p
e
ci
c
st
at
e
of
th
e
w
or
ld
.
F
ur
th
er
a
n
in
p
ut
is
s
p
e
ci
c
to
a
p
ar
ti
c
ul
ar
pr
o
d
u
c
er
a
n
d
is
u
s
el
e
s
s
to
a
n
y
ot
h
er
a
g
e
nt
in
th
e
e
c
o
n
o
m
y,
a
n
d,
af
te
r
th
e
st
at
e
of
th
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w
or
ld
h
a
s
b
e
e
n
re
v
e
al
e
d,
a
pr
o
d
u
c
er
c
a
n
n
ot
n
d
a
n
ot
h
er
s
u
p
pl
ie
r;
th
er
ef
or
e
th
e
s
et
-u
p
b
e
c
o
m
e
s
o
n
e
of
bi
la
te
ra
l
m
o
n
o
p
ol
y.
If
th
e
in
p
ut
i
s
c
o
nt
ra
ct
ib
le
,
th
e
pr
o
d
u
c
er
a
n
d
th
e
s
u
p
pl
ie
r
c
a
n
in
iti
al
ly
si
g
n
a
c
o
nt
ra
ct
th
at
fu
lly
s
p
e
ci
e
s
th
e
c
h
ar
a
ct
er
is
ti
c
s
of
th
e
in
p
ut
fo
r
e
a
c
h
p
o
s
si
bl
e
st
at
e
of
th
e
w
or
ld
,
w
h
er
e
a
s
th
is
in
p
o
s
si
bl
e
if
th
e
in
p
ut
is
n
o
n
c
o
nt
ra
ct
ib
le
,
th
at
is
,
if
a
c
o
ur
t
c
a
n
n
ot
v
er
if
y
w
h
et
h
er
th
e
in
p
ut
is
of
g
o
o
d
or
of
b
a
d
q
u
al
it
y.
N
o
n
c
o
nt
ra
ct
ib
ilit
y
le
a
d
s
to
a
d
o
u
bl
e
h
ol
du
p
pr
o
bl
e
m
:
th
e
pr
o
d
u
c
er
c
a
n
h
ol
du
p
th
e
s
u
p
pl
ie
r
b
y
cl
ai
m
in
g
th
at
th
e
in
p
ut
is
of
b
a
d
q
u
al
it
y,
w
hi
le
th
e
s
u
p
pl
ie
r
c
a
n
h
ol
du
p
th
e
pr
o
d
u
c
er
b
y
d
el
iv
er
in
g
a
b
a
d
q
u
al
it
y
in
p
ut
,
a
n
d,
n
o
c
o
ur
t
w
o
ul
d
b
e
a
bl
e
to
c
h
e
c
k
w
hi
c
h
p
ar
ty
is
w
ro
n
g.
T
h
er
ef
or
e,
a
n
y
c
o
nt
ra
ct
s
p
e
ci
fy
in
g
th
e
a
m
o
u
nt
of
in
p
ut
s
of
g
o
o
d
q
u
al
it
y
to
b
e
pr
o
vi
d
e
d
is
w
or
th
le
s
s.
W
e
c
o
n
si
d
er
th
at
re
v
e
n
u
e
s
ar
e
s
h
ar
e
d
th
ro
u
g
h
e
x
-p
o
st
N
a
s
h
B
ar
g
ai
ni
n
g
(a
ft
er
th
e
st
at
e
of
th
e
w
or
ld
h
a
s
b
e
e
n
re
al
iz
e
d
th
e
n)
,
a
n
d
w
e
d
e
n
ot
e
b
y
2
(0
;1
)
th
e
b
ar
g
ai
ni
n
g
p
o
w
er
of
th
e
s
u
p
pl
ie
r.
T
hr
o
u
g
h
o
ut
th
e
p
a
p
er
w
e
wi
ll
c
o
m
p
ar
e
th
e
si
tu
at
io
n
w
h
er
e
th
e
in
p
ut
is
fu
lly
c
o
nt
ra
ct
ib
le
to
th
e
o
n
e
w
h
er
e
it
is
n
ot
.
3
.
3
S
t
r
u
c
t
u
r
e
o
f
t
h
e
g
a
m
e
a
n
d
o
n
e
s
h
o
t
v
e
r
s
i
o
n
.
and a
mass
D
11
12
In every p erio d, the timeline is then the follow ing:
. Final go o d pro ducers die with
probability D
of new nal go o d pro ducers is b orn.
2. Every supplier makes a take it or leave it o/er of an exWhen the inpu
1 t is contractible, they can also contrac t on
input to b e provided, give n the nature of the match.
3. Pro ducers cho ose their sup plier, and the transfer tfro
paid.
We m a ke t h e c l a s s i c a s s u m p t i o n t h a t r e ve nu e s a r e o b
n o n ve r i a b l e , a n d s o c a n n o t b e p a r t o f a c o nt r a c t .
4. The typ e of the match is revealed if the two parties are interacting for the rst time (it is
already known otherwise) .
5. State of the world is realized.
6. Suppliers decide on how much go o d quality input to provide in the noncontractible case
(they are b ound to follow the contract in the contractible case).
7. Revenues are shared b etween pro ducers and suppliers through ex p ost Nash
Bargaining.
1 2We
res tric t attenti on to contracts sp eci ed by the game ab ove, so, contracts last for one p
erio d only. The fact that suppliers make take it or leave it o/er to pro ducers w hen deciding on
the ex-ante transfer, implies that they are engaged in Bertrand comp etition.As a consequence,
if all suppliers are symmetric they do not capture any rent from the relationshi p. However, if one
supplier has an advantage over the others, in the sense that the relationsh ip with that supplier
is of higher value than any other p otential relationship, the supplier can capture th e entire
surplus of that relationship over the oth er p otential relationships. Bertrand comp etition (instead
of ex-ante Nash bargaining) simpli es the exp osition but do es not a/ect our results , we discuss
what happ ens with ex-ante Nash bargaining or when pro duce rs make take-it or leave-it o/ers
at the end of the sec tion .
Before pro ceeding to describ e the equilibrium that we study, we consider a on e p erio d
version of the game ab ove to demonstrate the ine¢ ciencies that rep e ate d interaction can
overcome. Consider rst the case where the input is fully contractible. Suppliers, engaged in
Bertrand comp etition, o/er the contrac t that guarantees the highest value to the pro ducer,
while making them still break even. Therefore, they o/er to pro d uce the amount of go o d
quality inputs that maximize joint pro ts, and o/er an ex-ante transfer tsuch that the nal go o d
pro ducer captures the entire value of the relationship. Hen ce normalized go o d quality
inputs is at the rst b est level ( m) given by:
xm argmax
R(x) x= (( 1)= )
;
so that the amount of go o d quality inputs is min go o d matches and min bad matches. Now,
consider the situation in which no enforceable contract can b e written on the provision
of the input. The normalized investme nt mas found ab ove is no longer in the self-interest of the
supplier, as he b e ars the full marginal cost but only gets a share of the marginal b ene t.
Hence the supplier, once h e has b een chosen, would not commit to any announcement he
may have made ab out the quantity of go o d quality input he would provide, instead he will cho
ose an investment level that maximizes his ex-p os t pro ts. Ex-p ost pro ts are given by a
share
1 2T
heformalproofofthisisinAppen
dixA.
13
of the revenues minus the whole cost of the investment, so normalized amount of go o d
quality inputs provided by the supplier is at the "Nash" leve l ( n), give n by:
xn argmax x 1
x= m;
1 3and
the amount of go o d quality inputs in nin go o d matches and nin bad matches. It follow
s that n<m: as in any standard mo del, there is under investme nt. The ex-ante transfer re ects
the amount of go o d quality inputs eventually provided, so that t= (1 b+ b )( R(n) n), and
the nal go o d pro ducer still captures the entire b ene t of the relationship.
3.4 Sustaining co op eration
We now let the simple game of the previous subsec tion b e rep eated an in nite numb er of
times, and show that the rep eated interaction can incentivize a supplier in a go o d match to
invest more than what his short term self-interest would require. We take as given the structure
of the one shot interaction: ex-ante monetary transfers are paid just afte r the pro ducers have
chosen a supplier, suppliers undertake the investment, and revenues are shared according to
ex-p ost Nash bargaining. The contractible case remains very simple: it is a rep etition of the
one-shot interaction. Normalized investment is always at the rs t b est level m. A newb orn pro
ducer initially cho oses a supplier at random, and keeps switching supplier every p erio d until
he nds a go o d match. Once he has found a go o d match, he stick s with him, and, b ecause
of Bertrand comp etition, the go o d match supplier o/ers an ex-ante transfer that allows him to
capture the entire surplus of the ongoing relationship over any new relationship. In the
noncontractible case, the same SPNE exists but the s upplier s normalized investment is s et at
n, we will refer to this nonco op erative SPNE as the Nash case . We now derive a co op
erative SPNE.
We say that a supplie r co op erates with his pro ducer if he provides more (normalize d) go
o d quality inputs than he would in a one-shot interaction, that is if his normalized investment b
elongs to (n;m]. Of course, there is a profusion of SPNE featuring some level of co op eration,
and we are going to restrict attention to a particular class of equilibria. In every p erio d, the pro
ducer has to cho ose b etween continuing a relationship or switching to a new partner. The di¢
culties in generating co op eration when pro ducers can switch s uppliers at will are well known.
C las sic s olu tion s often involve some form of "collusive" b ehavior from the suppliers: for
instance, in the rst mo d el of [Kranton, 1996], parties agree to a limited level of co op eration at
the b egin of a relationship, this b ehavior is collusive b ecause in a new relationship, the
supplier has an incentive to agree with the pro ducer to b oth d eviate to strategies allowing for
full co op eration from the start of the relationship. In the context of innovation, which is an
activity undertaken by a rm - sometimes an entrant - in order to increase its market share, we
1 3N
o t e t h a t u n d e r i nve s t m e nt n o t o n l y d e c r e a s e s p r o t s b u t f u r t h e r r e d u c e s we l f a r e :
i n t h e f u l l c o nt r a c t i b i l i ty c a s e , t h e m o n o p o l y d i s t o r t i o n a l r e a d y l e a d s t o a p r o d u c t i
o n o f d i /e r e nt i a t e d g o o d s l owe r t h a n t h e we l f a r e m a x i m i z i n g l e ve l , i n c o m p l e t e c o nt r
a c t i b i l i ty a g g r ava t e s t h i s i n i t i a l m o n o p o l y d i s t o r t i o n .
14
think that it is particularly i mp ortant to avoid relying on such collusive equilibria, and to let pairs
co op erate as much as they can from the b eginning of the relationship. T he presence of
heterogenous matches in our mo del is what allows to sustain co op eration without resorting
to collus ive b ehavior by the suppliers. In fact, in her s econd mo del, [Kranton, 1996]
considered an equilibrium robust to pairwis e deviation where some suppliers had a higher
discount rate, in our setting relationship-sp eci c bad matches play the same role but avoid the
intro duction of adverse selection.
More sp eci cally, every rep etition of the game is comp osed of three moves: in phase 2
suppliers make their o/er for the ex-ante transfer, in phase 3 pro d ucers cho ose a supplier, and
in phase 6 s uppliers unde rtake the investment. We say that a go o d match supplier su/ered a
"terminal deviation" with a pro ducer if, either the supplier has at least once inves ted less than
he was sup p osed to in a relationship with the same pro ducer, or if the pro ducer has already
worked with another go o d match supplier since the rst time he worked with the supplier. We
denote by Ht(j;k), the set of his torie s of the game after trep etitions just b efore phase 6 where
pro ducer jhas chosen to work for the p erio d twith supplier k. We then denote by Hd t(j;k) the s
ubset of histories Ht(j;k) where the supplier kis a go o d match for pro ducer j, but where a
terminal d eviation has o ccurred b etween pro duc er jand supplier k, by Hb t(j;k) the subset of
histories Ht(j;k), where the supplier kis a bad match for the pro ducer j, and by Hg t(j;k) the
subset of histories Ht(j;k) where the supplier kis a go o d match for pro ducer kand no terminal
deviation within the pair has o ccurred. We then restrict attention to symm etric SPNE with the
following characteristics:
1. strategies of s uppliers towards a pro ducer dep end only on their p ersonal history with the
pro duc er and on w hether the pro ducer knows another supplier who is a go o d match and
on whether that other supplier is one with whom a terminal de viation has o ccurred or not.
Strategies of pro ducers are indep endent on the history of the game with other pro ducers
(condition C1);
2. normalized investment with pro ducer jby supplier kin a history b elonging to Hd t(j;k) is
given by n(condition C2);
3. normalized investment with pro ducer jby supplier kin a history b elonging to Hb t(j;k) is
given by n(condition C3);
k (j;k), the strategies played by the pro ducer j, jjht
2Hg t
;
kjht0
j
t
=
t
0 jjht
0 kjht; 0jh
j
k
0jht)+uk
0jht)
>uj jhtt+ is als o a SPNE (
satis es C4 for any ht+
kjhtalso
k 2Hg
t+ (
jht
t
. at any history ht
, and the supplier k, , must b e such that there is no pair 4of strategies for the pro ducer
and the supplier where jhdenotes the strategies of the other suppliers), and
where (j;k) and >0 following history h, such that u( ( ), where uis the value of the pro
ducer and u( jh)+uof the supplier.(condition C4).
15
1 4 1 5As
is common in co op erating games where pro ducers can start a new relationship,
condition C1 restricts attention to strategies that dep end on the "minimal" amount of information
on the history of the game . S trategie s played with di/erent pro ducers are entirely dep endent
of each other, and the only information that a supplier uses in a relationship with a pro ducer is
his own p ersonal history with the pro ducer and whether the pro ducer knows a go o d match
with whom no terminal deviation has o ccurred, a go o d match with whom a terminal deviation
has o ccurred or no go o d matches. This is, in general, an essential information as it a/ects the
outside option of the pro ducer, and cannot b e ignored (however strategies can b e indep end
ent on whether the pro ducer knows more than one othe r go o d match sup plier). C ondition C2
means that within a pair, players are using trigger strategies: a deviation is punished by
reverting to the strategies of the one-shot game. It is a natural assumption in this context, and
features the sharp est punishment within the pair ([Baker et al., 1994] or [Francois et al., 2003]
use trigger strategies in s imilar contexts). An interpretation of C2 is that trust in the pair pro
ducer/supplier is forever broken if a deviation o ccurs: if a supplier deviates once, the pro ducer
is afraid the supplier will cheat again, therefore he no longer accepts an ex-ante trans fe r
leaving any rent to the supplier, who, in return, keeps investing n; similarly, if a pro ducer do es
not keep his go o d match and nds a new go o d match, the su pplier is afraid the pro ducer will
keep reverting to his new go o d match in the ne xt p erio d and is not will ing to invest more
than n.
Condition C3 rule s out co op erati on in bad matches : pro ducers in a bad match w ill lo ok
for a new supplier in the following p erio d, which, in turn, deters co op eration. In App endix A,
we justify condition C3 by showing, rst, that there is no SPNE satisfy ing C1, C2 and C4 at any
no de in Ht(j;k)nHd t1 6(j;k), that is it is imp ossible to extend the equilibrium to bad matches as
well; and, second, that when bad matches are su¢ ciently bad, there is no equilibrium where a
single supplier would b e willing to co op erate in a bad match. Note that even if bad matches
were not to involve an actual loss in pro ductivi ty the equilibrium we describ e would still h ol d,
what is esse ntial, is that for some reason, co op eration is imp os sible in some matche s. The
pro ductivity loss provides a re as on for why this might b e the case.
Finally, condition C 4 means that strategies are designed such that parties in a go o d match
and who have not exp erienced a terminal deviation, achieve the highest level of co op eration
p ossible, knowing that in subsequent p erio ds their strategies will also lead to the highest
level of co op eration unle ss a terminal deviation has o ccurred. Condition C4 is the one that
rules out situations where co op eration builds up progressively, or where all suppliers co
ordinate on
t i s o bv i o u s l y i m p o s s i b l e t o g e n e r a t e c o o p e r a t i o n i f c o n d i t i o n C 4 we r e t (j;k):
o
a l s o a p p l y a t h i s t o r i e s Hb t
1 4I
15
16
16
H e r e a g o o d m a t ch s u p p l i e r f o r g i ve s a p r o d u c e r w h o s w i t ch e s , b u t o n l y n d s o u t a b
a d m a t ch . T h a t i s , a g o o d m a t ch s u p p l i e r e x p e c t s t h a t t h e p r o d u c e r i s w i l l i n g t o ke e p
wo r k i n g w i t h h i m , w h e n t h e p r o d u c e r h a s n o t f o u n d a n e w g o o d m a t ch a f t e r t h e y s t a r t
e d wo r k i n g t o g e t h e r f o r t h e r s t t i m e . T h i s a s s u m p t i o n i s i n n o c u o u s i n t h e n o i n n ova t
i o n c a s e , a n d we c o u l d a s we l l a s s u m e t h a t c o o p e r a t i o n i n t h e p a i r c e a s e s f o r e ve r i f t
h e p r o d u c e r s w i t ch e s , w h e t h e r h e n d s a n e w g o o d m a t ch o r n o t .
A n a l t e r n a t i ve s t o r y c o u l d b e o n e w h e r e a f r a c t i o n o f t h e s u p p l i e r s a r e i r r a t i o n a l l
y b a d , w h i ch i s t h e k i n d o f a s s u m p t i o n o f t e n u s e d i n m o d e l s o f r e p u t a t i o n [ M a i l a t h
a n d S a mu e l s o n , 2 0 0 6 ] .
punishing a pro ducer w ho has deviated.1 7 1 8Overall, the equilibrium we describ e features the
highest p ossible le vel of co op eration in go o d matches, with the ercest comp etition
amongst suppliers. In App endix A, we prove:
Prop osition 1 Under conditions C1, C2, C3 and C4 payo/ s in a symmetric SPNE are uniquely
de ned on equilibrium path. In particular, normalized investment is equal to nin bad matches,
and normalized investment is constant, equal to a unique x 2(n;m], in good matches.
We sp ecify b elow the level of investment x . App e ndix A provides a full formal description of a
SPNE that satis e s C1, C2, C3 and C4. Inversely, as shown in app endix A, any equilibrium
satisfying C1, C2, C3 and C4 has the follow ing structure:
1. there i s Bertrand comp etition amongst suppliers: when a new relationship is formed, pro
ducers capture the entire exp ected value of the relationship, and when a s upplier is a go o d
match he captures the entire surplus of the relationsh ip over any other re lation ship
2. pro ducers keep switching suppliers until they nd a go o d match supplier, once they have
found one, the supplier co op erates and the pro duce r keeps working with him;
3. if a go o d match sup plier invests less than x 1 9, then, either the pro ducer lo oks for a new
supplier in the following p erio d, or the pair keeps working together but normalized
investment remains at n.
Whether, once a deviation has o ccurred, it is b etter for the pro ducer to stick with a non co
op eratin g go o d match, or to lo ok for a new supplie r, dep ends on param eter values. We
now describ e in more details the equilib rium in the later case. In particular, we show how the
equilibrium normalized l evel of investment in go o d matches, x 2 0, is de termined (App endix A
deals with the form er case, which is very sim ilar).We use Vz 0(resp ectively, Vz 1), to denote th
e value of a pro ducer ( z= p), of a supplie r (z= s), or their joint value (z= T), when they are in a
new relationship, b efore the typ e of the match is reve aled, (resp ectively, when they are in a
go o d
1 70T
h e r e c u r s i ve d e n i t i o n o f c o n d i t i o n C 4 i s a c t u a l l y n o t n e c e s s a r y i n t h e n o i n n ova t
i o n c a s e ( s o t h a t o n e c a n i g n o r e t h e r e s t r i c t i o n t h a t jht+1a l s o s a t i s e s C 4 i n t h e d e n i t
i o n o f c o n d i t i o n C 4 ) , b u t s i m p l i e s t h e ch a r a c t e r i z a t i o n o f t h e e q u i l i b r i u m i n t h e i n n
ova t i o n c a s e .
1 8A c o n d i t i o n e q u i va l e nt t o c o n d i t i o n C 4 , i s t h a t t h e e q u i l i b r i u m i s r e n e g o t i a t i o n p r
o o f e x c e p t a f t e r a t e r m i n a l d e v i a t i o n , a n d t h e a m o u nt o f i nve s t m e nt i s i n d e p e n d e nt o
n t h e e x - a nt e t r a n s f e r t h a t h a s b e e n p a i d ( i t i s p o s s i b l e t o b u i l d o t h e r e q u i l i b r i a r e n
e g o t i a t i o n p r o o f e x c e p t a f t e r a t e r m i n a l d e v i a t i o n , b u t t h e y f e a t u r e l owe r c o o p e r a t
i o n , a n d r e q u i r e s t h a t i f t h e s u p p l i e r s e x - a nt e t r a n s f e r i s h i g h e r t h a n i t i s s u p p o s e
d t o , t h e s u p p l i e r wo u l d r e d u c e s t h e l e ve l o f c o o p e r a t i o n , b a s i c a l l y p u n i s h i n g h i m s
elf...).
1 9 I f a p r o d u c e r d o e s n o t ke e p a g o o d m a t ch s u p p l i e r , b u t n d s a n e w g o o d m a t ch s u p p l i
e r , h e ke e p s wo r k i n g w i t h t h e n e w g o o d m a t ch s u p p l i e r , w h o s e n o r m a l i z e d i nve s t m e
nt i s g i ve n by x.
2 0T h e f o l l ow i n g i s n o t a p r o o f o f p r o p o s i t i o n 1 , t h e p r o o f i s i n A p p e n d i x A . H e r e we t a
ke a s g i ve n t h e d e s c r i p t i o n o f t h e S P N E a n d r e d e r i ve t h e I C c o n s t r a i nt .
17
match relationship). Since investment is constant in go o d matche s, the value of a
relationship remains constant after the rst p erio d. As already m entioned, Bertrand comp
etition drives away any rent for the supp lier at the b eginning of a new relationship, but,
ensures that a supplier in a go o d relationship captures the whole surplus of the go o d
relationship over the second b est option for the pro ducer (starting a new relationship). Th
erefore , we get:
Vs = 0 and Vp = VT
0
VT
0
0
s =
1 1
1
= Vp 0: (6)
+ V
; (5) V
The total value of a new relationship can b e expressed as:
T ; (7)
= (1 b)
(x ) + 1 D1
+b
(n) + 1 D
0
+ VT 1
with probability (1 b), the match turns out to b e go o d, pro ts are given by (x) and the value
of the relationship next p erio d is VT 1; with p robability b, the relationship turns out to b e bad,
p er p erio d pro ts are only given by (n), the value of the supplie r in th e next p erio d is 0,
but the value of the pro ducer is once again equal to the total value of a ne w relationship. The
total value of a go o d match is given by:
= (x ) + 1 D1 + VT ;1 (8)
VT 1
0
VT VT 0 and Vp
1
pro ts are always (x
Vs 1
D
s1
’(x) 1 D1 + Vs 1
18
), and the re lati on ship will continue in the following p erio d. Combining
(6), (7) and (8), we get:
= (1 + )b( (x) (n)) 1 + b(1 ) ; (9)hence, the value of a go o d match supplier is the p er p erio
d di/erence b etwee n pro ts in a go o d match and exp ected pro ts in a new relationship
b( (x) (n)), prop erly discounted. When the supplier makes his investment decision, the ex-ante
transfer is already paid,
however, he is yet to receive the ex-p ost pro ts from the investment, that is a share of the
revenues minus the cost. Therefore, the su pplier has an incentive to deviate from investing an
amount x, by investing the Nash level n, which maximiz es his ex p ost pro ts , his gain would
then b e given by ’(x), where:
’(x) ( R(n) n) ( R(x) x): (10)
However, if the supplier carries out the investment x, the pro ducer will keep him in the following
p erio d, guaranteeing the supplier a value V, whereas, if he do es not, he will lose the supplier
(this is the case we are considering here) and the value he gets out of the relationship will b e 0.
The supplier would then b e willing to carry on the investment xonly as long as:
; (11)
In an equilibrium where co op eration is as high as p ossible, investment should maximize joint
pro ts unde r this in centive constraint. Note, however, that Vs 1itself dep ends on the equilibrium
level of investment in go o d matches x , hence x is a solution to a xed p oint problem . In App
endix A, we prove that this xed p oint problem de nes a unique level of co op eration x :
Prop osition 2 When the value of a new relationship exceeds the value of a good match where
the supplier does not cooperate, the equilibrium amount of invest ment x in good matches is
given by the rst best level mif
’(m) 1 D1 + b(1 D) b( (m) (n));
and otherwise, by the unique solution in (n;m) to: ) b( (x ) (n)):
’(x ) = 1 D1 + b(1 D
DPro ts in a good match weakly increase with the number of bad matches, b, and decrease with
the relative productivity of bad matches, , the discount rate, , and the probability of death .
DHow
much suppliers co op erate de p ends on how bad the alternative option is. Therefore if
the probability of a bad match, bis higher, or if th ey are more severe (low ), a go o d
relationship will have more value, and the p otential for co op eration is highe r. A higher value
of the future (lower and 2 1) have the same e/ect. As shown in App end ix A, our analy sis do
es not dep end on the assumption that the value of a new relationship exceeds the value of a
nonco op erative go o d match.
In this equilibrium co op eration arises in go o d matches, b ecause, in the following p erio d,
the supplier captures the additional value of the ongoing relationship over a random new
relationship. This sp eci c features comes from the assumption of Bertrand comp etition,
however, this equilibrium can b e generalized to the case of ex-ante Nash Bargaining b etwee
n suppliers and pro ducers, or even to the case where pro ducers make take it or leave it o/ers
to suppliers, by letting pro ducers o/er a b onus to the supplier when they deliver the
appropriate amount of go o d quality inputs. Pro duc ers would themselves face an incentive
compatibility constraint, and would pay the b onus only if it remains smaller than the cost
of nding a ne w supplier. Equilibrium level of investment are then exactly identical, but the
description of the equilibrium is slightly more complicated.
2 1M
o r e s p e c i c a l l y t h e l e ve l o f e q u i l i b r i u m d e r i ve d s o f a r b( + D) (1 )
(1b)(1+ )
h o l d s a s l o n g a s 1 +1 + DD
1
( (x) (n)). I n t h i s c a s e , x
)
19
=
(n)
(x
+
D
D
). O t h e r w i s e xi s a t t h e r s t b e s t i f ’(m) ( (m) (n));a n d o t h e r w i s e t h e u n i q u e s o l u t i o n i n
(n;m) t o ’(xi s d e t e r m i n e d by t h e d i /e r e n c e i n p r o t s b e twe e n a g o o d m a t ch o n p a t h a n d w
h e n a d e v i a t i o n h a s o c c u r r e d , a n d d o e s n o t d e p e n d o n t h e nu mb e r o r s e ve r i ty o f b a d
m a t ch e s .
2 2The
incumb ent supplier has the informational advantage that the nature of the match has b
een revealed. This informational advantage acts as a xed cost that pushes the pro ducer to
stick to the same supplier, which, in return, incentivizes the suppli er to co op erate. Crucially,
this xed cos t interacts naturally with the incomplete contractibility : in a situation with
incomplete c ontractibility, there is no co op eration in bad matches, as bad matches have no
prosp ect, which makes bad matches even worse relatively to go o d matches than in the
complete contractibility case or the no co op eration case .This feature will b e cruc ial when we
intro d uce innovation.
4 Stickiness of relationships and innovation
The previous section established that co op eration in long term relationships mitigates the
under-inve stment problem asso ciated with contractual incompletene ss. We now turn to the
downside of this, namely that co op eration makes relationships to o rigid , which creates
dynamic ine¢ ciencies. Dynamic ine¢ ciencies could arise for a large variety of macro economic
issues (for example when th e economy has to adapt to business cycles sho cks), but, in this
pap er we sp eci cally fo c us on innovation, and consider an endogenous growth mo del in
which suppliers engage in research to improve technology. In the rst subsection, we analyze
whether a pro ducer in a go o d relations hip would b e willing or not to switch to a ne w
innovator, and establish that co op eration makes it harder for an innovator to break into the
market; in the second s ubsection, we analyze how innovation itself a/ects the degree of co op
eration; in the third subsection, we show that the incentive to innovate and therefore growth, are
reduced with noncontractibility, and may b e further reduced by co op eration, b ecause of the
rigidity of relationships; nally, we show through s imulations, that welfare might b e reduced by
co op eration.
4.1 To switch or not to switch
IIn
this subsection we study the trade-o/ face d by a pro ducer in a go o d match when an
innovator comes into the market with a sup erior technology: should he stay with his current go
o d match, or should he switch to the innovator, b earing the risk that the innovator will b e a bad
match. We ass ume that every p erio d, an outside supplier has the opp ortunity to innovate. For
the moment, we take the innovation dec ision as given an d assum e that an innovation happ
ens w ith probability (we show how Iis determined i n subsection (4.3)). When an innovation o
ccurs, the innovator has access to a technology times more pro ductive than the previous
frontier technology, however, we assume that after a single p erio d, the innovator is imitated ,
and all
2 2E
ve n i f c o o p e r a t i o n we r e t o a r i s e i n a s p e c i c b a d m a t ch , t h e l e ve l o f c o o p e r a t i o n - d
e n e d by t h e n o r m a l i z e d i nve s t m e nt l e ve l - wo u l d b e l owe r t h a n i n g o o d m a t ch e s , a n d s o
e ve n t h i s c o o p e r a t i ve b a d m a t ch wo u l d b e r e l a t i ve l y wo r s e , t h a n i n t h e c o m p l e t e c o
nt r a c t i b i l i ty c a s e o r t h e n o - c o o p e r a t i o n c a s e .
20
suppliers have acces s to his tech nology (in subsection (5.4), we analyze what happ ens when
patents last for more than one p erio d). We denote by Athe frontier technology level, so that, in
p erio ds without innovation all suppliers use technology A, and, in p erio ds with innovation only
the innovator u ses the techn ology A, while the other suppliers use 12 3A).
The overall structure of the game remains the same as in section 3. I n the contractible case,
the description of the equilibrium remains simple. Normalized investme nt is alway s at the rst
b est level m. Pro ducers switch suppliers until they nd a go o d match, if they have not found
a go o d match and if an innovation o ccurs, they try out the innovator rst (he has the same
probability of b eing a go o d match as any other suppliers but he o/ers a more pro ductive
technology). If they have found a go o d match, they face a trade-o/, switching to the innovator
entails a b etter technology but the risk of facing a bad match. However, the technological
advantage of the innovator lasts for only one p erio d and, if the innovator turns out to b e a
bad match, the pro ducer can always revert to his old supp lier (who remains a go o d match).
Therefore, the pro ducer switches to the innovator if and only if the exp ected pro ductivity of
the innovator is higher than the pro ductivity of an outdated go o d match, that is, if and only if:
>0, or eq uivalently, >C
1 (1 b+ b )
=C
1 b+ b 1
21
Nash
24
23
I
24
; (12) with
probability (1 b) the innovator is a go o d match, with probability bhe is a bad match,but the
technology of the old go o d match supplier is times less pro du ctive. The Nash case is exactly
identical except that normalized investment are always at the Nash level n, so th at pro ducer
previously in a go o d match switch to the innovator if and only if >. For the c o op erative case,
we extend naturally the requirement on the SPNE: conditions C2,
C3 and C4 hold unchanged, condition C1 should allow for strategies to dep end on whether the
p erio d is one with innovation or not. Note, howeve r, that the fact that a go o d match
supplier forgives a pro ducer who switches s uppliers but do es not nd a new go o d match, is
no longer inno cuous in p erio ds where innovation o ccurs. I ndeed, in this s et-up, co op eration
b etwee n a pro ducer and an old go o d match supplier, resumes after one p erio d if the pro
ducer s witches to the innovator for one p erio d, and the innovator turns out to b e a bad match.
As shown in subsection (5.2), the main results of the section are actually reinforced in the
alternative cas e, where suppliers do not forgive pro ducers who switch, no matter whether they
have found go o d or bad matches, but the current ve rs ion allows for simpler
expressions.Therefore, pro ducers
O f c o u r s e , o n c e g r ow t h i s i nt r o d u c e d t h e d i /e r e nt i a t e d s e c t o r w i l l e ve nt u a l l y b e c
o m e s o p r o d u c t i ve , t h a t t h e c o n s u m p t i o n o f t h e h o m o g e n o u s g o o d w i l l b e d r i ve n t o 0
. Te ch n i c a l l y, w h a t we p r e s e nt h e r e i s a n a p p r ox i m a t i o n , w h i ch i s va l i d o n l y a s l o n g a s t
h e p r o d u c t i v i ty o f t h e d i /e r e nt i a t e d s e c t o r r e m a i n s s u ¢ c i e nt l y l ow . A l t e r n a t i ve l y, we
c a n a s s u m e t h a t t h e p r o d u c t i v i ty o f t h e h o m o g e n o u s g o o d g r ow s a t t h e r a t e o f t h e t e
ch n o l o g i c a l f r o nt i e r ( t h r o u g h k n ow l e d g e e x t e r n a l i ty ) , i n w h i ch c a s e , w h a t we p r e s e
nt i s n o t a n a p p r ox i m a t i o n b u t t h e e x a c t s o l u t i o n . We a l s o a s s u m e t h a t a n d a r e s u ¢ c i
e nt l y s m a l l , t h a t t h e va l u e s o f r m s c o nve r g e .
T h e r e c u r s i ve f o r mu l a t i o n o f c o n d i t i o n C 4 n ow p l ay s a r o l e . U n d e r s o m e p a r a m e t e r
va l u e s , a t s o m e n o d e , a h i g h e r l e ve l o f c o o p e r a t i o n a t c o u l d b e r e a ch e d i f t h e s u p p l
i e r s s t r a t e g y wa s t o p u n i s h a p r o d u c e r w h o s w i t ch e s t o t h e i n n ova t o r e ve n i f t h e i n n
ova t o r we r e a b a d m a t ch : s o t h a t t h e r e wo u l d n o t b e a ny S P N E s a t i s f y i n g C 1 ,
2 5switch
suppliers, cho osing the innovator whenever in novation o ccurs, until they nd a go o d
match. Once they have found a go o d match, they keep working with the same supplier as long
as no innovation o ccurs. If an innovation o ccurs, they face a trade-o/ b etween switching or
not, knowing that, if they switch, and the innovator turns out to b e a bad match, they can go
back to their ol d go o d match supplier in the following p erio d, and the old go o d match
supplier will resume co op eration.
Because of Bertrand comp etition, a pro ducer previously in a go o d match relationship
switches to the innovator, if and only if the exp ected value of joint pro ts with the innovator is
higher than wi th the old supplier. As the old supplier imitates th e innovator s technology after
one p erio d, and as the p ro ducer can resume co op eration with the old supplier if the
innovator turns out to b e a bad match, the decision to switch dep ends only on the di/erenc e in
exp ected pro ts in the rst p erio d. The innovator is a go o d match with probability (1 b), in
which case he invests x , and a bad match with probability b, in which case he invests n, while
the old go o d match invests y and his technology is times less pro d uctive, therefore:
Lemma 1 Producers previously in a good match switch to the innovator if and only if
(1 b) (x) + b (n) 1 (y ) >0: (13)
( (y )= (x )) >0: (14)
Pro of. See App endix C
We can rewrite (13) as:
1 b+ b ( (n)= (x)) 1
Now as shown in App endix C, co op eration o ccurs in go o d matches so that x>n,
moreover, as explained b elow, y x, therefore (n)= (x ) <1 and, (y )= (x ) 1, so that (12) is
more easily satis ed than (14), which gives us:
NCProp
osition 3 (i) The parameter set for which innovators capture the whole market in t he
cooperative case is strictly smal ler t han the parameter set for which innovators capture the
whole market in the contractible or the Nash cases. (ii) In part icular, the minimum
technological leap required for an innovator t o capture the whole market in t he cooperative
case (C) is higher than that in the contractible or Nash cases (;Nash): NC>C= Nash2 6.
Moreover, as shown in App endix D:
C 2 , C 3 a n d a n o n r e c u r s i ve C 4 .
2 5A s i n t h e p r e v i o u s s e c t i o n , w h e n a s u p p l i e r d e v i a t e s a n d i nve s t s l e s s t h a n h e i s s
u p p o s e d t o , t h e r e a r e two p o s s i b l e c a s e s d e p e n d i n g o n p a r a m e t e r va l u e s , e i t h e r t
h e p r o d u c e r s t a r t s l o o k i n g f o r a n e w s u p p l i e r , o r t h e p r o d u c e r s t ay s w i t h t h e s a m e
s u p p l i e r w h o d o e s n o t c o o p e r a t e a ny m o r e .
2 6Te ch n i c a l l y, NCi s d e n e d a s t h e i n n i mu m o n t h e r a n g e o f f o r w h i ch a n i n n ova t o r c a n
c a p t u r e t h e e nt i r e m a r ke t a n d m a ke s t r i c t l y p o s i t i ve p r o t s .
22
Remark 1 Consider the case where the value of a new relationship is larger than the value of a
good match relationship with no cooperation. Then, the levels of investments (x ;y NC) are
uniquely de ned. For <, innovation is not adopted by producers previously in a good match, and
there is an >0, such that for 2by al l producers, for = NC NC;NC+
, innovation must be
adopted, the share of producers previously in a good match who adopt the innovation is
undetermined.
Prop osition (3) delivers the rst imp ortant message of the pap er: in a context of weak
contractibility, co op eration makes it more di¢ cult to break down e xisting relationships.
Because of the existence of bad matches, for su¢ ciently close to 1, innovations are not adopted
by suppliers in go o d matches, but the threshold for adoption is higher in the co op erative case
than in th e contractible or Nash cases.
The intuition b ehind this result aris es from two e /ects. The rst - and most robust e/ect is a
worse bad matches e/ ect : a bad match is more costly relative to a go o d match in the co op
erative case, indeed, in this case, bad matches do not only involve an inherently lower pro
ductivity level, they als o involve l ess investment, this e/ect is re ected by the term (n)= (x ) in
(14). The second e/ect is an encouragement e/ ect, namely the fact that co op eration is
(weakly) higher when using the outdated technology, than it is when using the frontier
technology (that is, y , which a/ects the decision to switch or not th rough the term (y )= (x x)
in (14)). The reason is that the incentive to deviate for a given investment level is higher for a
supplier using the frontier technology than for a supplier using the outdated technology,
whereas the reward from co op eration is the same. Indeed, the incentive to deviate is
scalable by th e technology used by the supplier, so that it is given by ’(:)12 7Afor suppliers
using the outdated technology, w hereas it is given by ’(:)Afor suppliers. The reward from co
op eration is the s am e b ecause, in the following p erio d, the outdated suppliers have
access to the frontier te chnology as well, so that the values of all suppliers are the same,
irresp ective of w hether they were outdated or not in the previous p erio d, and the value of a
supplier is precisely the reward from co op eration. In oth er words, the opp ortunity to imitate
the frontier technology in the following p erio d encourage outdate d suppliers to provide a
larger e/ort, partly comp ensating the fact that they are using an outdated technology.
tment x
A
s
t
h
e
l
e
v
e
l
s
o
f
i
n
v
e
s
a
n
dy
NC
I
;y ), but b ecause
23
I
do es not, prop osition 3 (ii)
would
27
are endogenous, our description of the equilibrium
could leave ro om for p otentially multiple equilibria (under the set of strategies considered so
far), remark (1) shows, that this do es not o ccur when pro ducers would rather n d a ne w
supplier than stay with a nonco op erative go o d match. Moreover, note that dep ends on the
rate of innovation through (x
T h i s l a t t e r e /e c t i s ve r y s t r o n g h e r e a s p a t e nt s l a s t f o r o n l y o n e p e r i o d , h owe ve r , s o
m e f o r m o f t h i s e /e c t w i l l a l way s b e p r e s e nt a s l o n g a s t h e s u p p l i e r h a s a p o s i t i ve p r o
b a b i l i ty t o e ve nt u a l l y g e t a c c e s s t o t h e f r o nt i e r t e ch n o l o g y.
remain true with endogenous innovation and di/erent rates of innovation for the co op erative,
Nash and contractible cases.
2 8That go o d match suppliers may b e more reluctant to adopt the technology of the innovator
when co op eration arises in l on g-term relationships, do es not directly re duce welfare, it d o
es so only through the impact on the decision to innovate (see subsection (4.3)).Nevertheless ,
prop osition (3) (ii) states that innovation would b e i mmediately adopted by all rms when 2
C;NC in the contractible and Nash cases but not the co op erative cas e, so that we ge
t:Corollary 1 When innovation 2 C;NC rms in the cooperative case than in the cont ractible or
Nash cases. , there is more technological di/ erences across
CThe
mo del predicts that there s hould b e more technological di/erences and/or that
innovations s pread slower in countries with p o or contractibility in stitutions (as long as co op
eration arises in equilibrium), b ecause is indep endent of I.
4.2 Impact of innovation on the level of co op eration
As in the no innovation case, a lower discount rate, , a lower p robabili ty of death D, a larger
probability of a bad match b, and a l ower pro ductivity in bad matches favor co op eration (that
is, inc rease the levels of normalized investments x and y I). We now brie y lo ok at the impact of
innovation itself (the size , and the rate ) on the level of co op eration, and we get:
Remark 2 Consider the case where the value of a new relationships is higher then the value of a
relationship with a nonco op erative go o d match, and where the innovator captures the entire
market, then the level of co op eration (x ;y ) is increas ing in the size of innovations , and, for
su¢ ciently s mall (b 1 D 2 I <1 + ), is decreasing in the rate of innovation I.Pro of. See App
endix E Therefore, when the innovator captures the entire market, scarce but large innovations
(large , high ) favor co op eration in go o d matches. Larger innovations lead to a higher growth
rate, w hich increases the exp ecte d value a supplier can capture by co op erating, favoring
more inve stment in go o d matches. More frequent innovations have three e/ects on the
investment levels: (i) a p os itive e/ect through a higher growth rate, (ii) a negative e/ect through
a higher probability of ending the relationship, and (iii) a further negative e /ect that comes from
the fact that the b ene t of b eing in a go o d match over a random match is higher in p erio ds
with no innovation, than in p erio ds wi th innovation, knowing that, this b ene t is precisely what
drives the incentive to co op erate. For s u¢ ciently s mall innovations, the e/ect (ii) dominates
2 8A
c t u a l l y, f r o m a we l f a r e p o i nt o f v i e w , a n d a t g i ve n r a t e o f i n n ova t i o n , p r o d u c e r s s w
i t ch t o t h e i n n ova t o r t o o mu ch . I n d e e d , b a d m a t ch e s a r e e ve n m o r e d e t r i m e nt a l t o we l
f a r e t h a n t o p r o t s , a s n a l g o o d p r o d u c e r s a r e m o n o p o l i s t s ( t h e l e ve l o f n o r m a l i z e d
i nve s t m e nt t h a t m a x i m i z e s we l f a r e i s h i g h e r t h a n m) , a n d s w i t ch i n g t o t h e i n n ova t o r i
n e v i t a b l y i nvo l ve m o r e b a d m a t ch e s .
24
2 9the
e/ect (i), and therefore more freque nt innovations will lower the l evel of co op eration. We
can compare this result to [Francois et al., 2003], who show that an inc re as e in innovation can
push rms towards provid ing short-term contract arrangements in stead of implicit guarantees
of lifetime employment to their workers. In our mo del, the same idea is captured by the
decrease in co op eration following an increase in the rate of innovation.
4.3 Endogenous rate of innovation
ISubsection
(4.1) has showed that co op eration creates rigidities in long-term relationships, we
now show th at these rigidities are the source of dynamic ine¢ ciencies, by studying how the
equilibrium rate of innovation Iis determined. Every p e rio d one supplier gets a new idea. This
idea turns into a useful innovation with probability if the the p otential innovator inve sts
I is a
convex function and where lim A(where Ais the frontier technological level b efore innovation o
ccurs), where I!1 0 I = 1(the size of innovation is a constant).Because the probability that the p
otential innovator has already made a successful innovation is in nitesimal, the market share of
the p otential innovator is in nitesimal, so that, for all purp oses the p otential innovator is an
entrant. In this subsection, we compare th e rate of innovation in the three di/erent cases:
contractibl e, Nash and co op erative.
3 0Because of Bertrand comp etition, when the relationship with the innovator is the hi gh est p
ossible, the innovator captures the entire b ene t of this relationship over the second b est option
of the pro ducer.Recall that patents last for only one p erio d, and that, in the co op erative
noncontractible case, go o d match s uppliers re sume co op eration if the pro duc er switches to
the innovator and the innovator turns out to b e a b ad match. The di/erence b etween th e value
of a relationship with th e innovator and the val ue of a relationship with the b est alternative, is
then s imply equal to the di/erence in pro ts in the rst p erio d. We denote by Vs;t I;K
+the value captured by the innovator (normalized by the after innovation frontier pro ductivity
level) from a re lation ship with a pro ducer, who knows a go o d match supplier ( t= g), or who
do es not know any go o d match supplier ( t= b), for the contractible case (K= C), the Nash
case ( K= Nash) and for the co op erative case (K= NC). De ning X= max(X;0), we get that, in
the contractible case, the value captured by the innovator from a relationship with a pro ducer
previously not in a go o d match is given by:
= (1 b+ b ) 1 1 (m); (15)
25
Vs;b
I;C
29
30
while the value captured by the innovator from a relationship with a pro ducer previously in a
I n t h e i r m o d e l t h e s i z e o f i n n ova t i o n va r i e s a c r o s s s e c t o r s , a n d s e c t o r s w i t h t h e s
m a l l e s t s i z e o f i n n ova t i o n a r e t h e o n e s s u p p o r t i n g l o n g - t e r m i m p l i c i t c o nt r a c t s .
I f i n s t e a d o f B e r t r a n d c o m p e t i t i o n , we h a d a s s u m e d e x - a nt e N a s h B a r g a i n i n g , t h
e i n n ova t o r wo u l d c a p t u r e o n l y p a r t o f t h e d i /e r e n c e , b u t a s l o n g a s h e c a p t u r e s a p o
s t i ve p a r t , t h e r e s u l t s o f t h i s s u b s e c t i o n c a r r y t h r o u g h .
go o d match is given by: Vs;g I;C = 1 b+ b 1 +
= (1 b+ b ) 1 1 (n) and V
= 1 b+ b 1 +
s;g 1 (y )
= (1 b) (x= (1s;b I;NC
I;C
Vs;b I;NC
b) (x
(n): (17)
+ b ) + b (n) 1 1 (n) (18)
s;b I;C
and Vs;g I;NC
(m)
: (16) The situation of pro ducers previously in a go o d
match has b een analyzed in (12), for pro ducers
previously not in a go o d match, the reasoning is similar:
joint exp ected pro ts are the same with the innovator and
any oth er supp lier, exce pt in the rst p erio d where they
are times higher with the innovator, Bertran d comp etition
allows the innovator to capture all the surplus of a
relationship with him over any other relationship.
Similarly, for the Nash case, we get:
s;g
Vs;b
I;Nash
I;Nash
and V
Finally, in the co op erative case, we get:
)
>V
26
1 (y ) +
s;g I;NC
: (19)
The case of pro ducers previously in go o d matches was
analyzed in (13). The case of pro ducers previously not in
go o d matches follows the same logic, if they cho ose a
random outdated supplier in stead of the innovator, exp
ected joint pro ts w ill b e the sam e except in the rst p erio
d where the outdated suppl ier will b e times less pro
ductive, and will invest yinstead of xif he is a go o d match,
b ecause of Bertrand comp etition, the innovator captures
the di/erence in exp ected joint pro ts. It can then b e
shown:
Lemma 2 The value captured by the innovator from a
relationship is lower in the cooperative case than in the
contractible case: V V, however, it may be higher or lower
than in the Nash case.
Three e/ects explain this result: the worse bad match
e/ect (bad matches in the co op erati ve noncontractible
case feature lower investment in addition to lower pro
ductivity compared to go o d matches), the
encouragement e/ect (in the co op erative noncontractible
cas e, outdated suppliers are willing to co op erate more
than suppliers at the technological frontier), and a scale e/
ect. By scale e/ect, we refer to the fact that thanks to co
op eration, underinvestment in go o d matches is less s
evere than in the nonco op erative case, therefore, pro ts
are higher than without co op eration, but they remain
(weak ly) lower than in the contractible case, that is: (n)
< (x) (m). The rs t two e/ects lower the value that the
innovator captures in the co op erati ve case compared to
the contractible and Nash cases. The third e/ect
increases the value captured by the innovator compared
to the Nash case, but further decreases it compared to
the contractible case. Therefore in the comparison b
etween the contractible and co op erative cases, the three
e/ects work in the same direction, whereas in the
comparison b etween the contractible and the Nash
cases, the sc al e e/ect works in the opp osite direction to
the worse bad matches and encouragement e/ects.
max b
equilibrium, the steady-state fraction of rms previously not in a go o d match is constant,b
indep endent of the rate of innovation and given by != = 1 1 D b .3 1Hence, assuming
that the steady state has b een reached, th e innovator solves the problem:
DIn
I
+ (1 !)Vs;g I;K
but do so in the contractible case, if >NC, innovations break up relationships in b oth cases, but
as innovations are more frequent in the contractible case, relationships still last longer in the co
op erative case, therefore:
Corollary 2 Relationships last longer in the cooperative case than in the contractible case
Therefore the mo del predicts that as long as co op eration o ccurs, relationships should last
longer in c ou ntries with p o or contractibility institutions.
4.4 Numerical exercise
To illustrate prop os ition 4, we simulate our mo del and show one case where co op eration
raises welfare, and one where it do es not. Throughout we keep th e following parameters:
Ddiscount rate
0:03 elasticity of demand 2 pro ductivity of bad matches 0:9 death rate of
pro duc er 0:01 initial technological level A00:01 bargaining share 0:5
We consider two values for th e size of innovation : L = 1:1 and H
C
L I = 0:1 and bH
for b oth values of b. For b= bH
NC
N
;
C
C=
and H
0:11, when = H , NC
33
L
,
33
I
I ’1:03
2 CNC
Now, note that if 2
and as long
as >NC
C
;NC
C
I
H
>L
, where is adjusted so that the rate of
L, C
>
, relationships ne ver break down in the co op erative
case
= 1:25, and two value s for
the sh are of bad matches b: bI= 0:33. The cost of innovation function takes the
form
=
2the contractible c ase is given by = 1 , we get 2 0; is nearly constant equal
to 1:12, therefore . Similarly, for b= b’1:01 and for th e relevant range of is nearly constant
equal to 1:02, so that C. The following table c om pare s the three cases : contractible, Nash and
co op erative. Th e measure we us e to express the welfare cost i s the equivalent reduction in
units of homogenous go o d consumed eve ry p erio d from th e full contractible case (recall that
every p erio d a mass one of homogenous go o d is pro duce d). For the Nash and co op erative
cases, we rs t compute the welfare cost assuming that the rate of innovation is
exogenously xed at , and then endogenize the growth rate an d compute the corresp onding
welfare cost.
Fo r t h e p a r a m e t e r va l u e s ch o s e n , t h e r a t e o f i n n ova t i o n i n t h e c o o p e r a t i ve c a s e i s
a l way s u n i q u e , a n d n o s i n g l e p a i r i n a b a d m a t ch wo u l d b e w i l l i n g t o s t a r t c o o p e r a t i n g
.
28
H
Table 1
bH
H
units and b= bL
H
Prob. of bad matches b bL
Size of innovation L
Contractible case Rate
of innovation C
L
C
Nash
Nash
C
NC
NC
2
and b= bH , <NC
29
L
5 Extensions
4:59 11 3:8 11 Nash
case Welfare cost, = 0:35 1:8 0:36 3:83 Rate of
innovation 3:5 8:59 2:89 8:59 Welfare
cost, = 0:39 7:66 0:38 7:64 Co op erative case
Welfare cost, = 0:08 0:46 0:08 1:01 Rate of
innovation 3:2 8:74 0:08 6:3 Welfare
cost, = 0:28 7:22 0:17 7:76 All rates of
innovation and welfare c osts are in 10
The most interesting part in table 1 is the last row. In this case, co op eration
dramatically reduces the welfare cost of incomplete contractibility when innovation is
exogenous (it decreases from 3:83% to 1:01%). However, when innovation is endogenized,
the rate of innovation decreases a lot, and do es so even more in the co op erative case
than in the Nash cas e. As a consequence, the welfare costs of incomplete contractibility
are much larger, and b ecome even larger in the co op erative case than in the Nash case:
co op eration reduces innovation so much that the static gains are dominated by the
dynamic losses. When = , the probability of a bad match is su¢ ciently low that the worse
bad match e/ect (and the encouragement e/ect) are dominated by the scale e/ect, so that
the rate of innovation is higher in the co op erative than in the Nash cases (and welfare of
course is higher in the co op erative than in the Nash case). When = , and innovation do es
not break down existing go o d match relationships in the co op erative case, and
innovation nearly disapp ears.
In the previous section, we s howed that co op eration in long-term relationships turns the s
tatic problem of under-investment into a dynamic problem of insu¢ cient innovation. We
now turn to di/erent extensions of our basic mo del to assess the robustness of our results
and get additional predictions.
5.1 Co op eration and distance to the technological frontier
In the version of the mo del presented so far, te chnological progress in a country o
ccurred only through innovation. However, w hen a country is far from the world
technological frontier,
the su p erior technology can progressively di/use in the economy through imitation. One can
argue that imitating an existing technology is a less risky activity than innovation and requires
less sp ecial skills (see for instance [Ace moglu et al., 2006]), in this subsection we take the
extreme view that at the b eginning of every p erio d all suppliers can freely imitate world the
technological frontier. We then show that far from the world te chnological frontier co op eration
in long-term relationships is a much b etter alternative to contractibility than it is close to it. The
reason is twofold: rst imitation is very imp ortant far from the world technological frontier and
innovation b ecomes relatively less imp ortant in the overall growth pro cess (if one country
innovates but the other one do es not, the second country will imitate m ore in the n ext p erio d,
partly comp ensating for the absence of innovation); second far from the technological frontier
co op eration in long-term relationships is enhanced, that is the level of normalized investments
in equilibrium are not constant any more and tend to b e higher when a c ou ntry is far from the
world technological frontier. Indeed, the incentive to deviate in a given p erio d is scaled by the
current technology, while the reward from c o op eration in the following p erio d is scaled up by
the future technologies, if growth is larger, the reward from co op eration b ecomes large relative
to the incentive to deviate, allowing to sustain a higher level of c o op eration in the rst place. In
e/ect, the e/ect of a highe r growth rate is similar to the e/ect of a lower dis count rate.
To illu strate this result we simulate numerically the economy when imitation from a
technological fronti er. To simplify things we consider a xed technological frontier, A, and we
assume that at the b eginning of every p erio d, b efore any p otential innovation o ccurs, all
suppliers have access to a technology Ab t, where Ab tsatis es:
H,
= At1 t + " AAt1
= Ab
Ab t
t
0
=A
t
I
H
34
C
is the rate of im itation and At1
is" the nal l evel of technology in p erio d t1. The nal level of technology in
by Ab tif innovation o ccurs, and by Aif no innovation o ccurs, where the cos
given by
Ab t. For the numerical simulation we cho ose the same param
with = and b= b, exc ept for the initial level of technology that we x at A= 0
ose the level of the technological frontier at A= 0:01 and the rate of imitatio
1 shows the normalized level of investment w hen using the most advance
o d matches ( x t3 4) ( gure 1A), the rate of innovation ( gure 1B), the level o
( gure 1C), and the utility ow minus 1 ( gure 1D), for the three cases: contr
erative and Nash.Figures 1C and
To m o d e l i n n ova t i o n i n a c o n s i s t e n way, we d r aw a r a n d o m nu mb e r
o r m a l l y o n ( 0 , 1 ) , i f t h e nu mb e r i s s m a l l e r t h a n , i n n ova t i o n o c c u r s
l e c a s e , i f i t i s s m a l l e r t h a n Nash, i t a l s o o c c u r s i n t h e N a s h c a s e a n d
a n NC ti t o c c u r s i n t h e c o o p e r a t i ve c a s e .
30
1D are on a logarithmic scale.
Figure 1: Innovation and Imitation
Figure 1A shows that the level of co op e ration in the co op erative case is higher at the b
eginning, w hen the technology is far from the world frontier, the rst b est is even reached in
the rst p erio d. Figure 1B shows that the rate of innovation itself is slightly higher, however,
this is not a general result, it dep ends on the relative imp ortance of a stronger scale e/ect (as
co op eration is enhanced) and a weaker encouragement e/ect (the e ncourage ment e/ect
disapp ears initially as the x tand y tare b oth close to the rst b est) pushin g towards a larger
rate of innovation, versus a larger worse bad match e/ect pushing towards a smaller rate of
innovation. Figure 1C shows that in the imitation phase the technologies of the three regimes
are very close, the world technological frontier is ach ieved after 26 p erio ds in the co op erative
case and slightly so oner in the two other cases , thanks to an innovation that takes place in
these two cases but not in the co op erative case . Afterwards technology grows the fastest in
the contractible case and faster in the Nash than the co op erative case, according to th e
pattern implied by the steady-state rates of innovation. Figure 1D shows that the utility ow in
the co op erative case an d in the contractible case are very similar early on (far from the
technological frontier), while the utility ow in the Nash case is much lower. However, far
31
from the technological frontier, co op eration in long term relations hips ceases to b e a go o d
substitute to contractibility, and in this case where the equilibrium rate of innovation is lower in
the co op erative than in the Nash case, eventually turns out to b e a burde n (the utility ow in
the Nash cases ends up higher than in the co op erative case after p erio d 150).
Therefore, this subsection showed that far from the world technological frontier, co op
eration in long-term relationships can b e a go o d subs titute to contractibility, as long as
imitation takes place easily in existing relationships. This reconciles our motivational evidence
that s howed the strong impact of contractibility on innovation with [Acemoglu and Johnson,
2005] who showed that contractibility seemed to have a mo derate impact on the level of
development (their fo cus b eing on former Europ e an colonies, the sample is comp osed of
mostly de veloping countries, far from the technological frontier).
5.2 Case where co op eration ends forever when a pro ducer switches supplier
So far, we made the assumption that a supplier forgive s the pro ducer if the pro ducer switches
supplier but only nd a bad match. In this subsection we assume that a supplier stops co
operating if a pro ducer switches to another supplier, no matter whether this other s upplier is a
go o d or a bad match. In particular, this means that if a pro duc er switches to the innovator,
and the innovator turns out to b e a bad match, the old go o d match supplier would not b e
willing to re sume co op eration if the pro ducer comes back to work with him again .
Loss of co op eration e/ect. Therefore, the pro ducer will su/er additional losses in the p erio
ds following innovation as he would have to either stick with a nonco op erative go o d match
or keep lo oking for a go o d match. This loss of cooperation e/ ect is an additional force that
pushes towards more rigid relationships and eventually a lower rate of innovation in the co op
erative case. More sp eci cally, as shown in App endix G, in the case where pro duce rs would
rather lo ok for a new supplier than stay with a go o d match supplier who invests the Nash
level, pro ducers previously in a go o d match switch to the innovator if and only if:
3 5T
h e t e r m i s e a s y t o i nt e r p r e t : w h e n a p r o d u c e r i s l o o k i n g f o r a n e w s u p p l i e r ( w h i ch h
a p p e n s o n l y i f t h e i n n ova t o r t u r n e d t o b e a b a d m a t ch , t h a t i s w i t h p r o b a b i l i ty b) , t h e r e
i s a p r o b a b i l i ty bt h a t t h e n e w s u p p l i e r
32
)).
)
1
(x1b+b
(n) )
1 (y ) (x
)
1
b
2
D
I
I
1
+ b(1
D)(1
)
taying with a nonco op erative
go o d match supplier. It is
equal to the loss in exp ected
pro ts from the risk of having
to lo ok for a new supplier in
the subsequent p erio ds ,
scaled by pro ts in a go o d
match when no innovation
arises ( (xOur main results
completely carry through
under this assumption: one
(n) (x
(y
)
I
3 6 3 7needs
a larger in the co op erative case to break up existing re lations hips than in the
Nash or contractible c ases, innovation is lower in the co op erative case than in the contractible
case , and may b e even lower than in the Nash case.
3 8Multiple equilibria in the co op erative case. Furthermore, in the co op erative case, when
innovations are su¢ ciently large that all pro ducers sw itch to the innovator, !, the share of pro
duc ers who are not in an ongoing go o d match relationship (without deviation), dep ends p
ositively on the rate of innovation: as more innovation me an s that more go o d match
relationships break up de nitely.But, as the value of an innovator is higher from relationships
with pro ducers previously not in a go o d match relations hips than from pro ducers previously
in an ongoing go o d match relationship, the larger is !, the larger is the incentive to innovate.
Assuming that innovators do not observe the current share of pro ducers in an ongoing go o d
match relationship b efore mak ing their investment in innovation, innovation still arises at a
constant rate, but there is typically ro om for multiple equilibria in the co op erative case. As an
equivalent to remark (2) holds, there could b e one equilibrium where innovation is scarce, so
that most pro ducers have found a go o d match supplier and co op eration is high, and another
equilibrium, where innovation is frequent, many pro ducers are not engaged in a go o d match
relationship and co op eration is low. Now, when innovation breaks down relationships in the
Nash or contractible cas e but not in th e co op erative case, the numb er of pro ducers
previously in a go o d match is larger in the co op erative case, which is yet another force
pushing towards lower growth in the co op erative case.
Waves of innovations in the co op erative case. If, on the contrary, innovators were able to
observe the share of pro ducers previously not in an ongoing go o d match relationship, the rate
of innovation (and therefore the investment levels ) will not b e constant overtime any more: the
incentive to innovate would b e larger just after an innovation has o ccurred. Innovations would
then to o ccur in waves, very much in the s pirit of [Stein, 1997].
i s a b a d m a t ch , i n w h i ch c a s e p r o t s a r e g i ve n by (n) i n s t e a d o f (xI) , a n d i n s t e a d o f (y ) i
n a g o o d m a t ch i n a p e r i o d w i t h n o i n n ova t i o n ( w h i ch h a p p e n s w i t h p r o b a b i l i ty 1 I) i n a g
o o d m a t ch i n a p e r i o d w i t h i n n ova t i o n ( w h i ch h a p p e n s w i t h p r o b a b i l i ty ) , t h i s d i /e r e n
c e i n p r o t s i s a p p r o p r i a t e l y d i s c o u nt e d i n ( 2 1 ) .
3 6I t i s d i ¢ c u l t t o m a ke a c l a i m c o m p a r i n g t h e t h r e s h o l d NCNCi n t h i s c a s e r e l a t i ve t o t h e b
a s e l i n e . I n d e e d , t h e l o s s o f c o o p e r a t i o n e /e c t p u s h e s t owa r d s a mu ch h i g h e r h e r e , h
owe ve r , c o o p e r a t i o n i n g e n e r a l i s l owe r t h a n i n t h e b a s e l i n e r e d u c i n g t h e i m p a c t o f t h
e wo r s e b a d m a t ch e /e c t .
3 7Ye t a n o t h e r p o s s i b l e a s s u m p t i o n wo u l d b e o n e w h e r e t h e n a t u r e o f a m a t ch wo u l d b
e f o r g o t t e n w h e n a s u p p l i e r a n d a p r o d u c e r s t o p wo r k i n g t o g e t h e r . I n t h i s c a s e , a n e
/e c t e q u i va l e nt t o t h e l o s s o f c o o p e r a t i o n e /e c t wo u l d a r i s e b o t h i n t h e N a s h a n d c o nt r
a c t i b l e c a s e s . H owe ve r , t h e e /e c t wo u l d r e m a i n s t r o n g e r i n t h e c o o p e r a t i ve c a s e , a n
dourresultsstillcarrythrough.
3 8W h e n p r o d u c e r s wo u l d r a t h e r l o o k f o r a n e w s u p p l i e r t h a n s t ay w i t h a n o n c o o p e r a t
i ve g o o d m a t ch , t h e s h a r e o f p r o d u c e r s p r e v i o u s l y n o t i n a g o o d m a t ch i s g i ve n by1 b D+ b (
1 DI( 1 ) b+ b ID)( 1 D) w h e n >NCb u t by1 b D( 1 NCD) w h e n <. T h i n g s a r e a b i t m o r e c o m p l i c a t e d w h e n p
r o d u c e r s wo u l d r a t h e r s t i ck t o a n o n c o o p e r a t i ve g o o d m a t ch t h a n l o o k f o r a n e w s u p p
l i e r , a s , i n e q u i l i b r i u m , t h e r e w i l l ty p i c a l l y b e t h r e e k i n d s o f p r o d u c e r s , t h e o n e s i n
c o o p e r a t i ve g o o d m a t ch e s , t h e o n e s i n n o n c o o p e r a t i ve g o o d m a t ch e s a n d t h e o t h e r
o n e s w h o h ave n e ve r b e e n i n a g o o d m a t ch .
33
5.3 Cho osing the typ e of innovation
We revert back to the baseline equilibrium where go o d match suppliers resume co op eration if
a pro ducer sw itches supplier but only nds a bad match; and we inve stigate more formally the
intuition that co op eration in the noncontractible case may pus h innovators to pursue larger
innovations. To illustrate in the simplest p ossible framework und er which circumstances this is
the case, we fo cus on a discrete choice of two innovation regimes: regime 1 (1; 1) features
small but frequent innovations, while regime 2 features large but rare innovations ( 3 9); and we
investigate when regime 1 is an equilibrium, that is when it is the case that when in regime 1, an
innovator has no incentive to switch to regime 2, for the contractible, Nash and co op erative
cases.We denote by (x1;y12>1; 2< 1) the equilibrium level of investme nt in regime 1 in the co op
erative case.
Using (15), (16) and (17), the inn ovator has no incentive to switch to regime 2 as long as:
21) + (1 !)(1
1 !(1 b+ b )(1
+(1 b+ b ) 1) (22)(1 b+ b ) 1)
!(1 b+ b )(
+ ;
2
2
1)
+
(1 !)(2
2
2
1
;y
1
m;’1
1
1
in the contractible and Nash cases. In the co op erative case, note that if the innovator
switches to regime 2, the level of investment remains (x) except in th e p e ri o d whe re he is
innovating. Indeed, in this case, the investment that go o d match sup pliers with the outdated
technology undertake (denoted ey) should re ect that in the followi ng p erio d, the old supplier
will have access to a te chnology (and not ) times more pro ductive than his current
technology ( ey= min(2’(y)=1switch to regime 2 as long as:) ). Therefore, us ing (18) and
(19), the innovator has no incentive to
1 (x1) (y
1
1)) + b (1) (n)) + (1 !)(
((1 b) (x
) + b (n)) (y1
2
1
1 2
1
(23)
2
+))
+
2
!
(
(
1
2
b
)
(
!((1 b)(
)
(x
C2
1;
C2
)) + b (1) (n)) + (1 !)(
(
e
y
NC 2
1
;2
in the co op erative case. For given () we compute the highest innovation rate in regime 2, 2,
that would make the innovator stick to the regime 1, in the contractible ( ), the Nash ( ) and the
co op erative ( ) cases. The inequality > NC 2can then b e interpreted as the innovator having a
higher incentive to pursue larger but scarcer innovations in the co op erative case than in the
((1 b) (x1) +
Nash 2
contractible case . We easily get Nash 2= C 2, the reward from innovation is just scaled up by a
factor (o)= (n) in the full contractible case, so the choice of regime by the innovator is not
a/ected. Hence, it is only the existence of co op eration in long-term relationships that a/ects the
relative reward from di/erent regimes of innovations.
3 9I
n a f u l l y e n d o g e n o u s s e t t i n g , t h e ch o i c e o f t h e s i z e a n d r a t e o f i n n ova t i o n wo u l d d e
p e n d o n h ow c o s t f u n c t i o n d e p e n d o n b o t h a n d . C o n s i d e r i n g t h e ch o i c e b e twe e n 2 r e
g i m e s a l l ow s u s t o a b s t r a c t f r o m t h i s i s s u e .
34
Let us cons ider that 1 2
NC
;NC
C
such that for 2 >eNC
. Then, in the co op erative case, the innovator do
es
C NC = ((1 b) (x1
1)= (m) + b (n)= (m))
;NC
1
4
0.
1
not capture the entire market unde r regime 1, but, as shown in App endix F, there is a
(i) for
threshold e, the innovator could capture the entire market by switching to regime 2. If yWe
then get the following prop osition.= m, this threshold is de ned by e
Prop osition 5 Assume that
2=
> , it is also a necessary condition when x1NC C
2
N
a
s
h
Pro of. See App endix F
Hence for 2
2
, a s i nve s t m e nt l e ve l x
i s n o t d e c r e a s i n g i n f o r >1, t h a t we h ave e
2
2
=
2
=
times les
s pro ducti
ve.
1
NC
NC
NC
1
35
NC
4 0I
ngeneral
eNC
1
1
= m, 2
(ii) if y
NC Nash 2+
!
>e
>1
NC 2
C
1
=
eNC
ecause when su¢ ciently large, the
innovator has a higher incentive to switch
to regime 2 in the co op erative case than
in the contractible or Nas h cases. The
intuition is that in the co op erative case,
switching allows him to capture the whole
market, whereas in the two other cases,
he captures the whole market even in
regim e 1 (the threshold is higher than eis
slightly ab ove e, the pro ts captured by
the innovator from pro ducers previously
in go o d matche s in the co op erative
case are very small). The expression of
the threshold when co op eration is su¢
ciently high ( y1= m), eNC+ ! 1 =
interesting by itself. It is increasing in !: if
most pro ducers are in a go o d match
relationsh ip, (!low), s witching in the co
op erative case involves capturing most of
th e market size, whereas it is already
captured in the contractible c ase. The
threshold is also decreasin g in 1: if ,
switching to the regime 2 b ecomes more
interesting in the co op erative case than
in the contractible case.is close to
eTherefore, intro ducing co op eration in
long-term relationships may lead to the
surprising result, that when contractibility
is weak, although the rate of innovation is
reduced, the size of an in cremental
innovation may b e larger, as long as co
op eration arises .
5.4 Substitutability b etween patenting
institutions and contractual institutions
In this subsection, we brie y investigate
C
is a su¢ cient condition for
1
b
, isNC
C
1
1 eNC
Cd o e s n o t a d j u s
ywhenx
what happ ens if technology is not
imitated after one p erio d, for instance
if rms patent their innovations and
patents are well enforced, or if the
technology is to o comple x to b e so on
imitated by comp etitors. More s p
eci cally, we assume that the innovator
has a monop oly over his technology until
the next innovation. Hence, at any p oint
in time, th ere is a monop olist who has
access to the technological frontier, while
all other suppliers have access to an
outdated technology
6=
= ma n d t h e e q u i l i b r i u m l e ve l x= (1 b+
b (n)= (m))
The main result from this section is that in this case, the arrival of an innovation weakens co op
eration in establishe d relationships, s o that co op eration in long-term relationships may not b e
as much a barrier to entry as b efore; in fact, prop osition 3 can b e reversed. Therefore, there
is substitutability b etween patenting institutions and contractual institutions: p o or
enforcement of contracts reduces innovation more when patenting is p o orly enforced.
To avoid a taxonomy of cases, we fo cus on the case where pro ducers would rather not
stick to a bad match (e ven the innovator), and, for the co op erative case, where they would
rather lo ok for a new supplier than stick with a nonco op erative supplier (even the innovator).
This will actually b e the case if b ad matches are su¢ ciently bad, but not to o numerous and
innovation is not to o large. However, it should b e clear that the intuition develop ed is more
general.
4 1In App end ix H, we show that in the contrac tible and Nash cases , a pro ducer previously
in a go o d match s witches to the innovator if and only if:
D(1 b)(1 + ) 1 + (1 )(1 I) 1
+b
1 >0: (24)
1
Unsurprisingly, the threshold on the size of innovation is lower than in the case where patents
lasted only for one p erio d: a pro ducer who do es not switch to the innovator ends up using the
outdated technology until the next innovation arrives.
For the co op erative case, we rst need to make the natural extension of the s trate gie s
playe d so far. We keep assuming that as long as no dev iation o ccurred, suppli ers are willing
to co op erate as much as p ossible. We also assume that in a pair pro ducer/supplier, if a
supplier has invested less than he was supp osed to, or if the pro ducer has switched supplier
and found out a new go o d match, co op eration in the pair ceases forever. We als o consider
that there is no co op eration in bad matches (so that pro ducers do not stick to a bad match
even if he is the innovator as sp eci ed ab ove). Hence, co op eration arises only when pro
ducers and suppliers are in a go o d match. The level of co op eration (that is the amount of
normalize d inve stment) dep ends on whether the supplier is the monop olist or an outdated
supplier. As in the previous section, we denote the amount of normalized investment by the
innovator by x . Moreover, the level of co op eration of an outdated supp lier in a go o d match
also dep ends on whether the pro duce r knows whether the innovator is a bad match or not, as
this a/ec ts the outside option of the pro ducer. We denote the amount of normalized investment
by z
when the pro ducer knows that the innovator is a bad match for him, and by y the amount of
normalized investment in the other case.
4 1I
n f a c t w i t h t h e a s s u m p t i o n t h a t i m i t a t i o n d o e s n o t o c c u r u nt i l t h e f o l l ow i n g i n n ova
t i o n , t h e r e a r e s e ve r a l e q u i l i b r i a i n t h e c o nt r a c t i b l e o r N a s h c a s e s a s we l l . We f o c u s
o n t h e e q u i l i b r i u m w h e r e t h e i n n ova t o r c a p t u r e s a l l t h e b e n e c e f r o m h i s i n n ova t i o n ,
t h a t i s t h e B e r t r a n d c o m p e t i t i o n e q u i l i b r i u m ( w h i ch i s t h e e q u i l i b r i u m s t u d i e d i n t h
e c o o p e r a t i ve c a s e a s we l l ) .
36
4 2App
endix H shows that a pro ducer previously in a go o d match switches to the innovator
if and only if:
(25)
(1 b)(1 + ) 1 + (1 + b
(n) (x D)(1 ) 1 (yI ) ) (x 1 1) (y ) (x ) !
+1 (y ) (x ) 1 D 1 1 + (1 DI )(1 I) (z ) (x )
>0:
We then get:
Lemma 3 Consider parameter values such that a producer would rather switch to a new supplier
than staying with a bad mat ch supplier or a noncooperative good match supplier, and such in
the vicinity of the parameter, there is an equilibrium where producers previously in a good match
are indi/ erent between switching to the innovator and staying with their old supplier. The level of
cooperation by outdated suppliers is higher when producers know that the innovator is a bad
match for them than when they do not: z y ; and, as long as 1 + >b 1 D 1 I+ Iwith an
outdated supplier who does not know the type of the innovator: x , the level of cooperation is
higher with the innovator than y 4 3.
As a consequence, it is not clear anymore that pro ducers stick to their old supplier for larger
innovation size in the co op erative case than in the contractible or Nash cases. Indeed, under
the hyp otheses of the lemma, only the worse bad match e/ect (the fact that (n)= (x ) <1)
would make (24) more easily satis ed than (25), having (y )= (x ) 1 and (z ) (y ) b oth push
towards (25) b eing more easily satis ed than (24).
y
The intuition for why z
37
42
43
is easily understo o d. When a pro ducer works with an
outdated supplier, his outside option in the next p erio d is b etter when there is a chance that
the innovator is a go o d match than when it is known that the innovator is a bad match. The
reward from co op eration for an outdated supplier is then higher in the later case, which justi es
that the outdated supplier co op erates more in the later case. Knowing that outdated suppliers
Te ch n i c a l l y, t h i s h o l d s f o r p a r a m e t e r va l u e s c l o s e t o t h e o n e s s u p p o r t i n g a n e q u i
l i b r i u m w h e r e p r o d u c e r s p r e v i o u s l y i n g o o d m a t ch e s a r e i n d i /e r e nt b e twe e n s w i t ch i
n g o r n o t , t h a t i s f o r p a r a m e t e r va l u e s w h e r e t h e l e f t h a n d s i d e o f ( 2 5 ) i s c l o s e t o 0 . A s
we a r e i nt e r e s t e d i n t h e m i n i m a l s i z e o f i n n ova t i o n f o r w h i ch t h e i n n ova t o r c a p t u r e s t h
e e nt i r e m a r ke t , t h i s i s t h e i nt e r e s t i n g r a n g e o f p a r a m e t e r va l u e s .
Te ch n i c a l l y, we m a ke ye t a n o t h e r a s s u m p t i o n t h a t a n o u t d a t e d s u p p l i e r w h o i s k n ow
n t o b e a g o o d m a t ch i s a b e t t e r o p t i o n f o r t h e p r o d u c e r t h a n a n e w o u t d a t e d s u p p l i e
r . A s s h ow n i n A p p e n d i x H , t h i s d o e s n o t n e c e s s a r i l y h o l d b e c a u s e t h e e x p e c t e d va
l u e o f t h e g o o d m a t ch o u t d a t e d s u p p l i e r w h e n t h e p r o d u c e r s w i t ch e s t o t h e i n n ova t
o r m ay b e s u ¢ c i e nt l y l a r g e - b e c a u s e o f t h e i n c r e a s e d c o o p e r a t i o n i n c a s e t h e i n n
ova t o r i s a b a d m a t ch - t h a t w h a t h e o /e r s t h e p r o d u c e r t o s t ay w i t h h i m i s l owe r t h a n w h
a t a n o t h e r o u t d a t e d s u p p l i e r wo u l d o /e r t o t h e p r o d u c e r . H owe ve r , i f t h i s i s t h e c a s
e , t h e p r o d u c e r wo u l d s w i t ch t o t h e i n n ova t o r e ve n f o r ve r y c l o s e t o 1 .
are willing to co op erate more if the innovator turns out to b e a bad match, gives the pro
ducer an additional inc entive to switch in order to learn the typ e of the innovator.
Whether x is greater or smaller than y
38
) (y
results from two e/ects. On one hand, as in the
one-p erio d patent cas e, there is an encouragement e/ect: if in the following p erio d yet
another innovation o ccurs, the current innovator and the current outdated supplier will have
access to the same technology, so the reward from co op eration will b e the same, whereas the
current incentive to deviate is higher for the innovator. On the other hand, if no innovation o
ccurs in the nex t p erio d, the monop olist will still have access to a b etter technology than the
outdated supplier. In this case, the outside option of a pro ducer who has not tried out the
innovator yet, will b e to do so; whereas the outside option for a pro ducer who has already
worked with the innovator, will to start a new relationship with an outdated supplier, which is
necessarily worse. As a consequence, the value captured in the following p erio d by the
innovator will b e h igh er than the value captured by an outdated supplier, leading to a higher
level of co op eration in the rst place. If is su¢ ciently low the rst e/e ct is dominated, and (x),
which pushes towards innovation b eing more easily adopted in the co op e rative case than in
the contractible or Nash cases. Therefore the lasting prese nce of a monop olist using the
frontier technology can weaken co op eration in existing relationship.
Of course, it is not always the case that prop osition 3 is reversed when patents last until the
next innovation, or technology is easily imitated. Fu rthe rm ore , in the case whe re suppliers
stop co op eratin g with a pro ducer who switches no matter w hether the pro ducer has found a
new go o d match or not (that is in the case of subsection 5.2), the loss of c o op eration e/ect
would b e another forc e pushing towards more stickiness in relationships in the co op erative
case than in the two other cases. Rather, one should interpret the res ult of this subsection as
showing that there may b e substitutability b etween patenting and contractual institutions , in
the sense that it is when patents are p o orly enforced, that having bad contracting institutions is
particularly bad. It is l ikely to b e the case that countries with p o or contracting institutions also
features p o or patenting enforcement...
6 Conclusion
In this pap er we showed that the development of implicit contracts in a context of p o or
contractibility is a very p o or substitute for strong institutions, particularly for economies that
have reached a certain level of d evelopment. In a nutshell, our argument went as follows:
incomplete contractibility leads rms to engage in co op erative long-term relationships, which
can help overcoming the classic underinvestment issue asso ciated with the lack of
contractibility; however these relationships are very rigid, and slow down the pro cess of
creative destruction. Once innovation decisions are endogenized, the dynamic growth costs
can b e su¢ ciently large to overcome the static gains of co op eration. Our mo del predicts
that with p o or formal institu-
tions but informal contracting, relationships are more rigid and last longer, innovation sp re ad s
slowe r and is scarcer than w ith go o d formal in stitutions, in addition innovations are more
likely to arrive in waves and may b e larger when they happ en. The ne gative e/ects of co op
eration can b e attenuated in countries far from the technological frontier, or if rms manage to
avoid imitation of their innovations by comp etitors (for instance if they pate nt their innovations
and patenting enforcement is of go o d quality). We provided rst empi ric al evidence p ointing
towards more rigid relationships and towards les s innovation in c ountries with weaker legal
institutions.
In the sp eci c context of long-term relationship and innovation , it woul d b e interesting to
add to our mo del th e p ossibility for pro ducers and suppliers to develop technologies together.
Weak contractibility would typically increase the b ene t from relationship-sp ec i c innovation
develop ed in partnerships b etween pro ducers and suppliers. Innovation e /orts would b e
redirected away from general purp ose innovations towards relationship-sp eci c innovations, w
hich would further reinforce the rigidity of relationships. Another way to e xtend our current an
aly sis would b e to include foreign outsourcing, issues of incomple te contractibility may b e
even more stringent when a rm is dealing with a supplier in a di/erent country, as the rm may
b e less familiarized with the lo c al judicial system, therefore the development of long-term
relationships could b e even more stringent in this context.
More generally, the ide a that the co op eration in long-term relationships can lead to dy
namic ine¢ ciencies can b e ex ploited in di/erent contexts, varying b oth the re as on for
why rms develop long-term relationships and the source of the dynamic ine¢ cie ncy. For
instance, in the context of the relationship b etween banks and rms, the reason to engage in
long-term relationships could b e moral hazard or adverse selection. Another sources of
dynamic ine¢ ciency could b e a to o slow reallo cation of resources from one s ector to another
in the pres ence of macro economics sho cks.
39
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7 App endix 0
Table 0.1 rep orts the distribution of patents at the USPTO from 1992 to 2006 according to the
country of the assignee, and the two measures of the quality of contractual enforcement in the
corresp onding countries, while table 0.2 rep orts the distribution of patents at the USPTO from
1992 to 2006 according to the corresp onding industrial sector and the (two) measures of
contractual intensity asso ciated.
43
44
Table 0.1
45
Table 0.2
46
8 App endix A
In this app endix we prove the results containe d in subsection 3.4. First, we show prop osition 1
and the de scription of the equilibrium structure; second, we derive strategies supp orting an
equilibrium satisfying conditions C1, C2, C3 and C4; nally, we justify condition C3 by show ing
the imp ossibility of an equilibrium featuring C1, C 2 and C4 extended to bad matches, and by
deriving a condition under which n o pair in a bad match could renegotiate their s trate gie s to
achieve co op eration.
Part 1: Proof of proposition 1 The pro of is done in 7 steps : A) we show that investment is
constant in go o d matches B) we show that suppliers are engaged in Bertrand comp etition
C) we derive the IC constraint D) we show that condition C4 is s atis ed F) we show the
existence of a unique level of investment G) we derive the corresp onding p ossible ex-ante
trans fers In the following we will refer to a supplier with whom a terminal deviation has o
ccurred as
a deviation s upplier. Also b ecause the strategies are indep endent of what happ ens with other
pro ducers, we fo cus on one pro ducer and ignores the other pro ducers (so we drop the index
j).
Part 1. A: Constant joint values and investment in go o d matches Condition C4 s tipulates
that once a go o d match supplier with whom no terminal deviation
o ccurred, k, has b een picked up by the pro ducer, strategies must maximize the joint value
under the condition that at any history b elonging to Hg t(k) strategies must always b e such that
the joint value of the pro ducer and the supplier is the highest p ossible (know ing that it must b
e also b e the case in all subsequent his torie s where supplier kis chosen and as long as no
terminal de viation has o ccurred). In the same time, the strategies of the other suppliers are
indep endent of the particular history of the game b etween the pro ducer and the supplier k,
and of the particular time p erio d. However, the situation may b e di/erent dep endent on
whether the pro ducer knows another go o d match or not, as this a/ects th e outside option of
the pro d ucer - it do es not matter however if the pro ducer knows more than one other go o d
match and on whether these other go o d matches were de viation suppliers or not, b ecause
once the pro ducer would have worked with the suppl ier k, all the othe r suppliers b e come
deviation suppliers and their strategies are then given by condition C2-. Therefore, the situation
is totally symmetric at any history in Hg t(k), at any time t, if the pro ducer d o es not know any
other go o d match and s imilarly if the pro ducer knows at least another go o d match. We
denote the joint value in the rst case by VT 1and in the second case by VT0 1. The joint value is
the sum of the exp ected payo/ of the pro ducer and the chosen supplier just b efore the
investment o ccurs, after the nature of the match has b een revealed.
47
Note that th e joint value VT must satisfy the law of motion:
1
VT = (xt) + 1
1
+ VT 1
D1
;
, so th at VT 1
is
the amount of investment, b ecause VT(x is constant,
1
+ D
x
tmust b e constant to o, equal to some level x
=1+
):
Similarly the level of investment in go o d match relationship
when the pro ducer knows a deviation supplier is also constant in
equilibrium equal to some level x 0, and we have
=1
x0
;
+ + D
VT0 1
where xt
T0
T0
s0
Vp 0
p0
VT 0
T0
T1
T1
1+
1+
V
D
Vp 0
4
8
Part 1.B: Bertrand comp etition. Suppliers make take it or leave
it o/er to pro ducers, b ecause of Part 1.A, the joint value
of a pair pro ducer - supplier is indep endent of the o/er the supplier
has made if the pro duc er chose the supplier, this leads to
Bertrand Comp etition.
First let us consider the case where the highest joint value is
obtained from a relationship with a new supplier. Then all new
suppliers should o/er the entire joint value to the pro ducer.
Indeed, if one of the suppliers o/e rs more and is chosen, his exp
ected payo/ is negative, which is not subgame p erfect. If all
suppliers were to o/er less than the entire joint value, one
supplier could o/er slightly more than the b est o/er so far, he
would b e sure to capture the pro ducer and would have a p
ositive exp ected value instead of 0. Therefore, when a new
relationship is the op tion with the highest joint value, the pro
ducer captures the entire b ene t of the new relationship.
Therefore, if we denote by Vthe exp ected joint value of a new
relationship when the pro ducer do es not know any go o d match
supplier (b efore or after the ex-ante transfer has b een paid, as
the ex-ante transfer is just a monetary exchanged b etween the
pro ducer and the supplier), we get:
and Vs 0
T
=V
0
are th e value of the pro ducer (and
V
+ b (n) + b 1
+ b (n) + b 1
D
= (1 b)V
= (1 b)V
= 0; where Vof
the supplier), just b efore the ex-ante transfer is paid. Note that
Vmust then ob ey the law of motion:
as in the next p erio d the pro ducer would again c ap ture the
joint valu e of the relationship. The same holds when the pro
ducer knows some deviation supplier(s), and when the joint value
of
starting a new relationship i s higher than the joint value of staying with a bad match. Denoting
by VT0 0the total value of a new relationship in this case we get:
Vp = VT0 and
= 0; (26)
00 0
Vs0 0
VT0 = (1 b)VT0 1 + b (n) + b 1 D1 + VT0 0: (27)
0
4 4The
condi ti on that the value of a new relationship is higher than the valu e of a relationship
with a go o d match with whom a terminal deviation has o ccurred can b e expressed as:
+ 1 + b(1 )1 b (n) <VT0 1: (29)
+ D
Now the same reasoning applies as so on as th ere are at least two s uppliers o/ering the
highest joint value. This can b e the case when the pro ducer knows at least two deviation
suppliers (but none go o d matches with whom no terminal deviation has o ccurred). In this
case, the value of the suppli ers is always equal to 0, the pro ducer capture the entire value of
the relationship. Note that the condition under which the joint value of a new relationship is
indeed smaller than the joint value of a relationship with one the deviation suppliers writes as:
(1 b)VT0 1 + b (n) + b 1 D1
<VT = 1
(n);
N
+ VT N
+ + D
which is equivalent to the op p osite of (29). Let us consider the case where the highest joint
value is obtained with a si ngl e supplier.
Then in equilibrium, the b est s upplier makes an ex-ante transfer that makes the pro ducer just
indi/erent b etween cho osing him or the second b est supplier: if he o/ers less, he loses the pro
ducer in th e coming p erio d (and therefore lose s the asso ciated payo/s), if he o/ers more he
can always just lower his o/er slightly and increase his payo/. In return, the second b est
supplier must make an o/er that leaves him with 0 exp ected surplus if he is chosen, otherwise,
in equilibrium the o/er of the rst b est supplier would leave the pro ducer wi th what the second
b est supplier o/e rs as well, so the second b est supplier could make a slightly higher monetary
transfer, ensuring that he is chosen and giving him a non z ero exp ected value. In equilibrium,
the strategy of the pro ducer is then to cho ose the suppl ier making him the b est o/er and if two
of them are making eq ually go o d o/ers, he should cho ose the one with the highest joint value
. There are three scenarios following into that case.
First, consider the case where the supplier k nows a go o d match with whom no deviation o
ccurs and no deviation suppliers. To make the pro ducer indi/erent b etween switching or
4 4D
enoting
by VT N
VT t h e j o i nt va l u e w h e n t h e p r o d u c e r wo r k s w i t h a d e i va t i o n s u p p l i e r
N , we g e t :
= (n) + 1 D1 + VT0 0; ( 2 8 )
a s , by a s s u m p t i o n o n t h e e q u i l i b r i u m , i n t h e n e x t p e r i o d t h e p r o d u c e r s h o u l d s t a r t a
n e w r e l a t i o n s h i p w h e r e h e wo u l d c a p t u r e t h e t o t a l va l u e . C o mb i n i n g ( 2 7 ) a n d ( 2 8 )
we g e t ( 2 9 ) .
49
not, it must b e that the value Vp of the pro ducer and the value Vp
1t
1t+1
T0
1
Vp
+ b (n) + b 1
p
D
1t+1:
of the pro ducer in the following p erio d if he go es back to the supplier
after having sw itched supplier but only to nd a bad match (note that
this value is conceptually di/erent from his value on path in the next p
erio d if he stays with the same supplier all the way) fol low:
= (1 b)V
1
+ V
Indeed, if the pro ducer switch, he captures the entire joint value , with
probability (1 b) the new supplier is a go o d match and the joint value
is given by VT0 1(as the pro ducer now knows a deviation supplier: the
previous go o d match supplier), with probability bthe new supplier is a
bad match, pro ts in the current p erio d are only (n), the c ontinuation
value of the suppl ier is 0, while the continuation value of the pro ducer
is the value he captures by going back to his old go o d match. Now
de ning
V = (1 b)VT0 11 D1+ +
(30
b (n) 1 b
)
1+ b(1 D)(Vp V ), and we know that
V
=
, using (27), we get that Vp 1t+1
=V
1t
1t
1+ b(1
D)
s 1t
p 1t
=V
Vp 1
and V
and Vp 1t+2
, Vp
T
0
p 1t
1t
0
p
1t+1
>1. Of course V(the val ue of the pro ducer if he go es back to his old
go o d match in p erio d t+ 2, after having switched suppliers b oth in p
erio d tand t+ 1, but only to nd bad matches), should follow the s am e
relation. Therefore, if Vshould explo de, which is imp ossible. Therefore
V;the value of the supplier, are constant such that:
and Vs = VT VT0 0. (31)
1
1
Similarly if the pro ducer knows exactly on e deviation supplier and the
joint value of a relationship with him is higher than the value of starting
a new relationship, we get that the value of a pro ducer Vp Nand the
value of a supplier Vs Nmust satisfy:
Vp = VT0 = V and Vs = VT VT0 0 with VT = 1
(n): (32)
N 0
N
N
N
+ + D
It is then straightforward to show that the condition under w hich the joint
value of a relationship with a deviation supplier is lower than the value of
a new relationship is ex actly the reverse of (29). Note that we have now
prove d that whether (29) holds or not determines in all cases uniquely
whether the value of a new relationship is higher or lower than the value
of a relationship with a deviation suppl ier.
Similarly if the sup plier knows a go o d match w ith whom no deviation
o ccurs, and at least one othe r go o d match, we get that Vp0 1tand Vp0
1t+1de ned in the same way as in the previous paragraph must follow:
= max (1 b)VT0 1 + b (n) + b 1 D1 + Vp0 1t+1; (n) + 1 D1
VT0 0; 1
+ max
Vp0 1t
50
+ + D
(n)
:
In this case the b est alternative for the pro ducer is either to switch to a new supplier or to go
back to one of the deviating supplier, if it is the second option in the following p erio d he will
have the ch oic e b etween starting a new relationship or still staying with the deviation
supplier. Let us rst consider the case where VT0 0>1+ + D (n), that is (29) holds. Then as Vp0
1t+1 VT0 0T0 1+b (n)+b1 D1+ Vp0 1t+1(a pro ducer could always cho ose to start a new relationship
w hen he has switched supplier and the supplier turned out to b e a bad match instead of
going back to his old go o d match) we get that (1 b)V (n)+1 D1+ VT0 0. Therefore the previous
argument applies and we get:
Vp = VT0 and
= VT0 VT0 0
01 0
1
Vs0 1
T0 0 1+
<
1
(n). Then we get
(n) ;
T0
+D
+
V
+
D
1
Vp0 1t
and
the
same
holds for Vp0
1t+1
. Now ass ume on the contrary that V
= max (1 b)V
+ b (n) + b 1 D
p0 1t+1;
1
+
T0 1+b
T0 1+
p0 1t
(n)+b
b (n) + b
and V
>
1 D1+ Vp0 <
1
D p0
1t+2
1+ + +b
D
(n)+b (n), so that= < 1+
1+
Vp0 1 11 D1+
+D
+D
p
0
1
1 D1+ Vp0 1 1+ + and
so on. If for all >t, it was the case that (1 b)V (n), then the same logic as
b efore would apply and Vp0 1t+1= V but this would contradict the reverse of (29). Th en there
must b e some , for which (1 b)VT0 1V (n), in which case the reverse of (29) directly leads to
(1 b)V (n) for all < , so that V=1+ + D (n). Therefore we then get
Vp = 1
(n) and Vs0 1 = VT0 + 1
(n). (33)
01 + + D
1
+ D
We have exhausted all p ossible cases and showed that Bertrand comp etition holds. Note
that the equilibrium conditions do not uniquely pin down the value of the pro ducer and the
supplier when the pro ducer decides to match with a supplier with whom the joi nt value is
strictly lower than the joint value with two other suppliers. This situation however, never arises in
equilibrium.
Part 1.C, IC constraint Let us consider a pair pro duce r j- supplier kwho know that they are
in a go o d m atch. In
a SPNE, the equilibrium investment level must b e such that the gain from deviating from th e
equilibrium investment (for the sup plier) is lower than the reward from making the investment
that consists of a higher continuation payo/. The maximal short-run gain from deviating can b e
expressed as
’(x) = R(n) n( R(x) x) the di/erence in ex-p os t pro ts when
investing the Nash level and when investing the level x.
The continuation value is given by the value that the supplier kwould capture from b eing in a go
o d relationship in the fol lowing p erio d, which is Vs 1(or Vs0 1), if he do es not dev iate and
51
is 0 if he d eviates except when the pro ducer do es not know any other go o d match supplier
and the reverse of (29) holds, in which case the supplier would still capture Vs Nif he dev iate
s. Therefore the IC constraint writes as
= 1 D1 + VT V
’(x) 1 D1 + Vs 11
’(x)
1 D1 + Vs0= 1 D1 + VT0V
1
1
’(x) 1 D1
Vs N ) = 1 D1
+ 1
(n)
+ (Vs 1
+ VT 1
+ D
if (29) holds and the pro ducer do es not know any other go o d match; as
if (29) holds and the pro ducer knows another go o d match;
as
if the reverse of (29) hol ds and the pro ducer do es not know any other go o d match; and as
’(x) 1 D1
= 1 D1
+ 1
(n)
+ Vs0 1
+ VT0 1
+ D
if the reverse of (29) holds and the pro ducer kn ows at least another go o d match. Now
condition C4 stipu late s that the strategies must b e such that the joint value is maximized amongst the set of strategies for which the joint value is maximized in all subsequent
histories in Hg t+ (k). As we already said, if such an equilibrium exist, the investment leve l in a
go o d match relationship must b e constant. Knowing that VT 1and VT0 1are th e discounted
value of the joint total pro ts derived from the investment undertaken by the supplier (while
’(x) 1 D
(x 0
1
D1+
wher
eV
=
max
V;
52
1+
1
+
dep ends only on the investment level o
+
D
(n)
V
1+ +
D
, we denote that element ex(which therefore dep ends on the
equilibrium level of investment through its dep endence on V). Part 1D. The equilibrium satis es
C4 Now we need to che ck that the equilibrium actually exists, that is we need to check that
there are no strategies still satisfying C4 in the next p erio d, and such that the entire pro le is
still a SPNE, which would improve the joint value of the relationship. Note, that no matter what
these alternative strategies could b e, as long as they form a S PN E with the (given)
strategies played by the other supplier, Vremains the s am e constant. Further note that if the
pro ducer randomly switches suppliers, it would not increase the incentive to invest of the
supplier and cannot increase the joint value of the relationship; moreover, if the supp lier
1 D1+ underinvests,
it decreases the joint pro t in the p erio d of deviation, and the punishment
sp eci ed by C 2 is the harshes t one. The only way an alternative strategy could do b etter
then is by cho osing an alternative pro le of investment, however any alternative pro le must
still satisfy C4 and b e SPNE, and so investment is b ound to b e given by exin future p erio
ds. Yet, the highest element in (n;m] satisfying ’(x) 1+ + D1 D1+ (ex) V is precisely ex: if the
constraint is not binding, ’(m) 1+ + D (ex) V ex= m= x, otherwise the constraint is binding
and ’(x) =1 D1+ 1 D1+ 1+ + D1+ + D (m) V , s o that (ex) V = ’(ex),which implie s x= exas ’is
strictly increasing on (n;m]. Therefore the strategies describ e so far satisfy C 4.Part 1E.
Equilibrium level of investment Note that (34) do es not change whether the pro ducer knows
another go o d match or not,
therefore, in equilibrium, we ge t
= VT0 and x = x 0 . x must then satisfy either:
1
VT 1
x = mand ’(m) 1 D1 +
+ 1
(m) max
(1 b) 1+ + D1 D1+ (m) +
;1
(n) !!
+ D
b (n) 1 b
+ +
D
;1+
(n) !!
:
(x
1 D1+
or 2(n;m) and ’(x
) = 1 D1 + 1
(x ) max
(1 b) 1+ ) + b (n) 1 b
+ D
+D
x
+
+ D
! (n) (x)! ;0 !
:
Let us de ne
g(x) ’(x) 1 D1 + b(1 + max 1 DD) b( (x) (n)) (1 + )(1 b)(1
+ b(1 D))( + D) 1 +b + D (1 )(1 b)(1 + )
Then, all the p ossible equilibrium levels of investment ne ed to satisfy either x = m and
g(m) 0 or x2(n;m) and g(x) = 0. Now, g(n) 0, and gis convex as ’(x) is convex, (x) concave
and max of a convex function is itself convex, so that if g(m) 0 there is no solution to g(x) = 0
in (n;m), and if g(m) >0, there is a unique solution to g(x) = 0. Therefore, there is a unique
candidate eq uilibrium level of investment. All we n eed to do now is to e xhibit strategies
sustaining such a SPNE. In case of a terminal deviation, the pro ducer would (no matter the
circumstances) lo ok for a new supplier rather than working with one of the deviation suppliers
if (29) holds that is if:
1 + b + D (1 )(1 ! (n) < (x ) (35)
b)(1 + )
if the reverse inequality holds, he will not, and if there is equality, he can do so with any
probability.
53
;
Part 1 G: Corresp onding ex-ante transfers Before describing the equ ilibrium strategies in
the next section, however, we compute the
ex-ante transfe rs corresp onding to the equili bri um.4 5
If (35) holds, or if the reverse of (35) holds but the pro ducer do es not know any other go o d
match, th e value of a supplier in a go o d match Vs 1(or Vs0 1) is given by (31), that is
= (1 + )b( (x
D) ,
(x
1 D1+
1+ ) + b (n) 1 b
Vs = 1
(x ) (1 b)
) (n)) 1 + b(1
+D
1 + + D
so that the ex-ante transfer t1 paid must b e such
that:
= (1 + )b( (x
D) )
t1 + R(x ) x
+ 1 D1
= Vs
) (n)) 1 + b(1
1
+ Vs 1
) b( (x) (n)) :
t1 = R(x ) x
+ D
) (n)) + b ( R(n) n):
t
1 + b(1 D
0
s0
s1
= (1 b)
D)
D
b( (x
When the pro duc er do es not know any go o d match, or when the pro d ucer knows s ome
go o d matches but (35) holds, V= 0 and in case of a go o d match the value of the supplier
would b e V. Therefore the ex-ante m on etary transfer t0o/ered by suppliers who have never
worked with the nal go o d pro ducer must b e:
R(x ) x + 1 1 + b(1
+ 1 D1 + Vs 1 + b ( R(n) n)
)
x
= (1 b)
R(x
When the reverse of (35) holds, and the pro ducer knows a deviation supplier, using (33), the
value of a s upplier in a go o d match (with whom no d eviation has o ccurred) is given by:
Vs = 1
( (x ) (n));
01 + + D
so that the ex-ante transfer t0 is given
1
by
t0 1 + R(x ) x + 1 D1
= Vs0 = 1
( (x ) (n)) )
1
+ Vs0 1
+ + D
( (x
t0 1= R(x ) x
45
) (n)): Now, still when the reverse of (35)
holds, if the pro ducer knows exactly one go o d match
supplier, the value of s uppliers who have never worked with the pro ducer must b e null if they
We d e s c r i b e t h e e x - a nt e t r a n s f e r w h e n e i t h e r ( 3 5 ) o r i t s r e ve r s e h o l d , w h e n t h e r e
i s a c t u a l l y e q u a l i ty i n ( 3 5 ) , t h e t r a n s f e r s mu s t ( a n d c a n ) s a t i s f y b o t h s e t s o f c o n s t r a
i nt s .
54
are chosen - Vs0 0= 0 -, if these suppliers turn out to b e a go o d match, their value will b e
given by Vs0 1, so the ex-ante transfer they o/er t0 0, must b e:
R(x ) x +
( (x
+ b ( R(n) n):
1 + DD
t0 = (1 b) = R(x ) x + 1 D1
+ b ( R(n)
0 (1 b)
+ Vs0 1
n)) (n))
If the pro ducer knows at least two go o d matches (with whom a deviation has o ccurred or not),
the value of the relationship with a new supplier is no longer the second highes t one, so the
only restriction on the transfer is that it has to b e at leas t as s mall as t0 0. However in condition
C1, we required that the strategies do not dep end on whethe r the pro ducer knew more than
one deviation supplier or not, so we have to x the transfer at t0 0there to o. If the reverse of (35)
holds, the value of a dev iati on supplier when he i s in a relationship
with a pro ducer who do es not know any other go o d match is given by (using (32)):
Vs = 1
(n) VT 0
n + + D
2
= (1 + )D(1 b) ( + )(1
1 + b(1 ) + D(1 ! (n) (x ) ! ;
+ b(1 D))
+ )(1 b)
therefore the ex-ante transfer he o/ers must b e:
tn2 = R(n) n D+ 1 +
Vs
n
= R(n) n (1 + )(1 b)(1 + b(1
D))
1+
b(1 ) + D(1 ! (n) (x ) ! :
+ )(1 b)
If the reverse of (35) holds and the pro ducer knows several deviation suppl ier, the joint value
in a relationship with one of them is at least the second highest value, so the value of one of
thes e deviation suppliers must b e 0, therefore they o/er an ex-ante transfer tn1:
tn1 = ( R(n) n);
However, if (??) is not satis ed, th e value of a relationship with a nonco op erative go o d match
is lower than the value of a new relationship so any transfer equal or lower than tn1is p ossible in
equilibrium.
Finally the value of a relationship with a bad match s upplier is neve r the highest or second
highest value, so bad match supplier should o/er ex-ante transfer that are lower or equal to:
tnb = ( R(n) n):
This achieves the description of equilibria satisfying C1, C2, C3 and C4
55
Part 2: St rategies to sustained a SPNE sat isfy ing C1, C2, C3 and C4 We call "new"
suppliers, suppliers with whom the pro ducer has never worked b efore, and
we keep calling "deviation" supplier, a go o d match supplier with whom a terminal deviation has
o ccurred.
Part 2.A: Case 1: x satis es (35). A new pro ducer starts in phase P0
Phase P0: the pro ducer is not currently in a go o d match relationship
We call
1. New suppliers o/er t0, those who know they are bad matches o/er t, deviation suppliers
o/er tn1nb
4 62.
The pro ducer cho oses a supplier amongst the set Sof those with whom his exp ected
value is the highes t p ossible, cho osing preferably amongst the new s uppliers.
3. If the pro ducer picks up a new supplier, and the supplier turns out to b e a go o d match,
the supplier invests x. In any other cases a bad match supplier invests n, and a deviation
supplier invests n.
4. If the pro ducer has worked with a new supplier, who turned out to b e a go o d matched
and invested at least x , move to phase P1; otherwise stay in phas e P0.
Phase P1: the pro ducer is currently in a go o d match relationship We call "ongoing"
supplier the last go o d match supplier wh o has worked with the pro ducer,
we call "deviation" supplier any other go o d match supplier who has worked with the pro duc er
(so that a deviation must have o ccurred).
1. The old supplier o/ers t1, new suppliers o/er t, those who kn ow they are bad matches o/er
tnb, deviation suppliers o/er tn10
2. The pro d ucer cho oses a supplier amongst the set Sof those with whom his exp ected val
ue is the highest p ossible, cho osing preferably the ongoing supplier, and if he is not in the
set S, ch o osing preferably a new sup plier.
3. If the pro ducer picked up the ongoing supplier or if he picked a new supplier who turns out
to b e a go o d match, the supplier invests x . Otherwise the suppli er invests nif he is a bad
match and nif he is a deviation sup plier.
4 6O
nequilibriumpaththesetSconsistspreciselyofthesetofnewsuppliers,thisf
o r mu l a t i o n a l l ow s t o c o n s i d e r o / p a t h c a s e s w h e r e a b a d m a t ch o r a g o o d m a t ch w i t h w
h o m a d e v i a t i o n h a s o c c u r r e d o /e r s s u ch a h i g h e x - a nt e t r a n s f e r t h a t t h e p r o d u c e r s h
o u l d wo r k w i t h o n e o f t h e m f o r o n e p e r i o d .
56
4. If the pro ducer has kept working with the ongoing supplier, and the ongoing supplier has
invested at least x , or if the pro ducer has worke d with a new supplier, who turned out to b e
a go o d m atched and has invested at l east x 4 7, or if the supplier has worked with a bad
match, stay in phase P1; otherwise move to phase P0.
Thanks to the discussion ab ove it is direct th at this is SPNE. On path the foll owing will o
ccur: a pro ducer switches suppliers till he nd a go o d match, once he has found a go o d
match, the go o d match inves ts x , and the pro ducer stays with him. If a pro ducer deviates but
ends up with a bad match the supplier forgives him, if the pro du cer go es back to a previous go
o d match or if the supplier deviates invests less than he ought to, the pro ducer moves back to
the initial phase.
Part 2B Case 2: x do es not satisfy (35) A new pro ducer starts in
phase P0 Phase P0: the pro ducer has never found a go o d
match supplier
1. New suppliers o/er t0, those who know they are bad matches o/er tnb.
2. The pro ducer cho oses a supplier amongst the set Sof those with whom his exp ected
value is the highes t p ossible, cho osing preferably amongst new suppliers.
3. If the supplier turns out to b e a go o d match, the supplier invests x . If the supplier is a
bad match he invests n.
4. If the pro duce r has worked with a go o d match supplier who inves ted at least x move to
phase P1, if he has worked with a go o d match who invested less than x move to phase P2,
otherwise stay in phas e P0.
Phase P1: the pro ducer is in an ongoing relationship and has never worked with any other
go o d match supplier b e fore
We call ongoing supplier the supplier with whom the pro d ucer was working in the previous
p erio d.
1. The ongoing supplier o/ers t1, new suppliers o/er t, those who know they are bad matches
o/er tnb.0
2. The pro ducer cho oses a supplier amongst the set Sof those with whom his exp ected
value is the highest p ossible, cho osing preferably the ongoing supplier, and if he is not in
the set S, cho osing preferably a new su pplier.
4 7T
h i s i n c l u d e s t h e c a s e s w h e r e t h e s u p p l i e r d e v i a t e d a n d w h e r e t h e p r o d u c e r wo r
ke d w i t h a g o o d m a t ch w h o i s n e i t h e r t h e o n g o i n g g o o d m a t ch o r a n e w g o o d m a t ch .
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3. If the supplier is a go o d match, the supplier invests x . If the supplier is a bad match, the
supplier invests n.
4. If the pro ducer has kept working with the ongoing supplier, and the ongoing supplier has
invested at least x , or if the pro ducer has worked with a bad match supplier, stay in phase
P1; if the pro ducer has kept workin g with the ongoing supplier, but the ongoing supplier has
invested less than x , move to phase P2; if the pro ducer has s witched to a new supplie r, the
n ew supplier turned out to b e a go o d match and has inve sted at least x , move to phase
P3; if the pro ducer has switched to a new supplier who turned out to b e a go o d match, but
investe d less than x , move to phase P4. Phase P2: the pro ducer knows exactly one go o d
match supplier, but a deviation o ccurred
with him. We c al l "deviation" supplier the supplier who is a go o d match but with whom a
deviation
has o ccurred. 1. The deviation supplier o/ers tn2, new suppliers o/er t0 0, those who know they
are bad matches o/er tnb. 2. The pro ducer cho oses a supplier amongst the set Sof those with
whom his exp ected
value is the highest p ossible, cho osing preferably the deviation supp lier, and if he is not
in the set S, cho osing preferably a new su pplier.
3. If the deviation supplier was chosen, he invests n; if the supplier is a new supplier and a go
o d match he invests x ; if the supplier is a bad match he invests n. 4. If the pro ducer has
switched to a new supplier, the new supplier turned out to b e a go o d
match and invested at least x , move to phase P3; if the pro ducer has switched to a new
supplier who turned out to b e a go o d match but invested less than x , move to phase P4;
in any other cases stay in p has e P2.
Phase P3: the pro ducer is in an ongoing relationship but he knows other go o d match pro
ducers with whom a deviation has o ccurred.
We c all ongoing supplier the last go o d m atch s upplier with whom the pro duce r has b
een working; and "deviation" suppliers the other go o d match suppliers.
1. The ongoing supp lier o/ers t0 1, deviation suppliers o/er tn1, new suppliers o/er t0 0, those
who know they are bad matches o/er tnb. 2. The pro ducer cho oses a supplier amongst the
set Sof those with whom his exp ected
value is the highest p ossible, cho osing preferably the ongoin g supplier; if he is not in the
set S, cho osing preferably a deviation su pplier; if neither of them are in the se t, cho
osing preferably a new supplier.
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3. If the supplier is either the ongoing s upplier or a new su pplier who turned out to b e a go
o d match, he inves ts x ; if the supplier is a deviation supplier, he invests n; if he is a bad
match, he invests n.
4. If the pro ducer has kept working with the ongoing supplier, and the ongoing supplier has
invested at least x , or if the pro ducer has switched to a new supplier, the new supplier
turned out to b e a go o d match and invested at least x , or if the pro ducer has worked with a
bad match supplie r, stay in phase P3; otherwise move to phase P4.
Phase P4: the pro ducer knows at least two go o d match suppliers with whom a deviation
has o ccurred and is not in an ongoing relationship.
We call "deviation" suppliers the go o d match suppliers, the pro du cer has b een working
with.
1. Deviation suppliers o/er tn1, new s uppliers o/er t0 0, those wh o know they are bad m
atches o/er tnb.
2. The pro d ucer cho oses a supplier amongst the set Sof those with whom his exp ected
value is the highes t p ossible, cho osing preferably a deviation supplier; if neither of them are
in the set, cho osing preferably a new supplier.
3. If the supplier is a deviation supplier, he invests n; if the supplier is a new supplier who
turned out to b e a go o d match, he invests x ; if he is a bad match, he invests n.
4. If the pro duc er has worked with a new supplier who turned out to b e a go o d match and
invested at least x , move to phase P3, otherwise stay in ph as e P4.
Once again it should b e pretty obvious that th ese strategies give rise to a SPNE satisfying
C1, C2 and C3. On path the following will o ccur: a pro ducer switches suppliers till he nd a go
o d match, once he has found a go o d match, the go o d match invests x , and the pro ducer
stays with him. If th e pro ducer deviates but nds a bad match he is forgiven, if the pro ducer
deviates and nds a go o d match, the new go o d match co op erates. If the supplier deviates
and invests less than x , the pro ducer keeps working with him but the supplier only invest nfrom
then on.
Part 3: Ju st ifying the choice of this particular SPNE Part 2.A: There is no symmetric SPNE
satisfying C1, C2 and C4 also after histories
b elonging to Hb t(j;k). If that were the case , investment in bad matches would also b e given by
a constant xb>n, and the supplier would have to b e kept with p ositive probability. To get co op
eration in go o d and bad matches, b oth the value of a relationship with a go o d match and the
value of a
59
relationship with a bad match need to b e strictly greater than the value of a new relationship,
which is a weighted average of the two of them: hence there is a contradiction.
Part 2.B: Deriving condition under which no co op e ration would b e p ossible in a pair pro
ducer - supplier in a bad match.
Let us assume that a pair pro ducer - supplier in a bad match co op erates. If this is p ossible,
it is p ossible to co op erate with a weakly increasing amount of co op eration, let us call x, the
limit level of investment. Then xbshould satisfyb
’(xb) 1 + DD (xb) VT 0
)
))( + D) (1 + )(1 b)( (xb) (x)) + b + D ( (xb) (n)) :If this has no solution in (n
’(xb) 1 D(1 + b(1 D
bad match would like to renegotiate their strategies and to start co op erating.
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