Academic Research, SocialInteractions and Economic Growth

Maria Rosaria Carillo* and Erasmo Papagni*** Università degli Studi di Napoli, �Parthenope�,

Via Medina 40, 80133, Napoli, Italy.tel:39081/8284044,

e-mail:carillo@unina.it, carillo@uninav.it** Seconda Università di Napoli,

Corso Gran Priorato di Malta, 81043, Capua, Italy,tel. 390823274023; fax 390823274042

e-mail: papagni@unina.it

7th November 2009

Abstract

In this paper we investigate the aggregate implications of social interactionsfor basic research and economic growth. Basic research is organized by thestate and is modelled as a contest with endogenous entrance. A CES indexof the distribution of both talent and e¤ort summarizes the features of theinteractions of the scienti�c community. The interaction between the e¤ectof competition and that of social exchange among researchers brings abouta nonlinear relation between interactions and performance in the scienti�csector. We show that, while keeping the number of researchers constant,an increase in the intensity of interactions between researchers has posit-ive e¤ects on sector productivity, while if the number of researchers varies,then positive e¤ects will be obtained only if there is already a favorable en-vironment for social interaction and if the scienti�c community is not verylarge. The model shows multiple equilibria, among which a poverty trapwith zero knowledge production and zero growth may emerge. Sensitivityanalysis of the model and simulation results show that greater social ex-change in a scienti�c community increases the employment size of science.However, the relation between social interactions and the rates of innovationand economic growth is non-monotonic. A possible pattern is described byan inverted U where the discovery rate increases with social interactions forthe small science sector until a saturation point is reached.

1 Introduction

The production of knowledge is widely recognized as a key factor behindthe economic growth which has occurred in western countries since the In-dustrial Revolution. Economic historians place the institutional evolution ofscience at the center of this long-running growth process (e.g., Rosenbergand Birdzell,1986; 1990; Bekar and Lipsey, 2001; and Mokyr, 2005). Ac-cording to this view, in the nineteenth century a new organization of basicresearch emerged, characterized by the di¤usion and free communication ofknowledge, and the connection between scienti�c knowledge and technologybecame tighter1. This type of science organization, known as �Open Sci-ence�(David, 1998), is characterized by several forms of interactions amongresearchers that greatly in�uence their productivity. In this paper we invest-igate the implications of such social interactions on science productivity andon economic growth.The science sector produces knowledge under a set of particular collect-

ive norms such as the full disclosure of scienti�c information, peer evaluationand the rule of priority. Several of these norms imply knowledge exchangeamong researchers and facilitate interactions during the work of research.For example, the full disclosure of new �ndings and the awarding of priorityfor a discovery involve the whole community in the process of research eval-uation. Researchers work in environments made by department colleagues,whose characteristics are very important (Allison and Long, 1990; Stephanand Levin, 1992; Carayol and Matt, 2005), and by �eld colleagues parti-cipating in the scienti�c enterprise of �Invisible college�(e.g., David, 1998;Adams et. al., 2005). In the last decade major innovations in informationand communication technologies have expanded the opportunities for inter-actions between scientists so that today research in any �eld is carried outby communities of scientists directly or indirectly connected (e. g., David,1998; Newman, 2001). However, although organization of the science sectoris so heavily based on social interactions and knowledge exchange among re-searchers, the consequences of such organization patterns on science sectorproductivity and on economic growth still remain an unexplored issue.In this paper we seek to address this lacuna. We put forward a general

equilibrium model of growth with two sectors: goods and science. Agents

0The authors thank for their helpful comments the participants at the conferences:"Economic Growth and Distribution", 2004 in Lucca; European Economic Associationannual conference in Madrid.

1Since then, the linkages between the world of science and the economy have becomestronger, as clari�ed by several works on the connections between scienti�c advance andtechnical progress (e.g., Ja¤e, 1989; Adams, 1990; Mans�eld, 1991; 1995).

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are endowed with heterogeneous talent and basic research is modeled as acontest. Social interactions in science are captured by a CES index of thedistribution of both talent and e¤ort of scientists. Entrance in the race isendogenous. The interaction between the e¤ect of competition and thatof social exchange among researchers lies at the center of the model andbrings about a nonlinear relation between interactions and performance in thescienti�c sector. Indeed, social interactions developing between researchersmay have both a positive e¤ect and a negative e¤ect on productivity in thescience sector and hence on economic growth. The result depends both onthe intensity already reached by interactions between agents, and on the sizeof the scienti�c network within which such relations develop. In particularwe show that, if the number of researchers is constant, an increase in theintensity of interactions between researchers always has positive e¤ects onsector productivity, while, if the number of researchers varies, the positivee¤ects will be obtained only when the scienti�c community is not very large.The relationship between social interactions and science performance be-

comes even more complex if the analysis is extended to a context of generaleconomic equilibrium in which the number of researchers and their e¤ort areendogenously determined. Actually in such a context we obtain other inter-esting results concerning multiple equilibria. Indeed, if social interactions arevery low, we have a unique equilibrium with positive scienti�c production.As social interactions become more intense, increasing returns to the numberof scientists and multiple equilibria may emerge. Interestingly, one of theseequilibria could be a poverty trap with zero knowledge production and zerogrowth. However, the latter case arises only if social interactions have anintermediate intensity, but it does not arise if they are strong enough. Fromthis narrow point of view, it can be maintained that the two polar envir-onments of low and high social interactions could produce better economicoutcomes than the intermediate case. When we analyze the relation betweensocial interactions and the rates of innovation and economic growth, we �ndthat it is non-monotonic. A possible pattern is described by an invertedU where the discovery rate increases with social interactions for the smallscience sector until a saturation point is reached.What are the policy implications of our analysis? One important con-

sideration is that an expansion of the science sector is not always positivefor sector e¢ ciency and, in order to avoid the negative e¤ect due to strongercompetition among researchers, it must be followed by an institutional re-design aimed at increasing scienti�c collaboration and knowledge exchangeamong researchers.. Moreover we show that policies aimed at improving sci-ence sector e¢ ciency di¤er according to the size of the sector and the stageof evolution it has already reached. Indeed, when the science sector is small

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and in the �rst stage of development a policy which increases interactionsand knowledge exchange among researchers is undoubtedly bene�c insofaras it increases both sector size, as more researchers enter, and their averageproductivity. When the research sector is large and has reached a maturedevelopment phase, then economic policy interventions have to be more com-plex, in that they must take the quality rather the quantity of researchersinto account. Indeed, one must adopt measures that increase knowledge ex-change between them without increasing competition, which would reducee¤ort. This may be achieved by controlling the number of researchers, mak-ing access to the sector more di¢ cult, and facilitating entry only for those ofhigh ability since the latter, as we show below, are also those more inclinedto commit more e¤ort to research and hence are less a¤ected by competition.The picture is further complicated by the fact that in the transition from

one development phase to another the science sector may remain caught in a"trap equilibrium", where sector productivity could decrease so much as tobe almost zero. The implication of this is that, as occurs in all cases wherethere are traps, the transition from one stage to another must not be gradualbut must occur with a "leap" which nevertheless involves heavy resource use.

This paper provides a theoretical framework for the explanation of sev-eral phenomena highlighted by the applied literature on science productivityand economic growth. Indeed, in recent years, in the �eld of network ana-lysis, several studies have empirically investigated the relationship betweenconnectivity within a network and its productivity, reaching con�icting con-clusions: some (Fleming and Marx, 2006; Smith, 2006) �nd that there is norelation between network performance and intensity of interaction betweenagents; according to others (Shilling and Phelps, forthcoming) this relationexists and is positive, while Uzzi and Spiro (2005) and Guimera et al. (2005)�nd a parabolic relation, which increases when interactions are low, and de-creases when intense. The parabolic relation could explain, as suggested byUzzi and Spiro (2005), the variety of results found in the literature sincethe e¤ect of interactions depends on the degree of intensity already reached.However, our results suggest another explanation: due account may not havebeen taken of the e¤ect of social interactions upon the incentive of enteringthe network and of the e¤ect that this entry in turn has upon network per-formance. We �nd that an increase in social interactions leads to an increasein the number of individuals making up the scienti�c community, which willhave further e¤ects, both positive and negative, upon sector productivityaccording to the type of social interactions involved.The relationship between the amount of resources invested in the scienti�c

sector and sector performance has been also analyzed by science historians.De Solla Price (1963) observed that there was a logistical relationship between

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the number of scientists and the rate of scienti�c advances. His explanationis in line with our results, since he believes that this logistical pattern isexplained with a relationship which is in turn logistical between sector per-formance and sector size. Also the literature on basic research and economicgrowth (see Aizenman and Noy, 2007) highlights the presence of a nonlinearrelationship, which �rst rises and is then constant, between the amount ofresources invested and science sector performance. Using a model that allowsfor the main characteristics of the institution of "Open Science", we obtain aresult which is in line with �ndings elsewhere and which allows joint analysisboth of the e¤ects of size of the scienti�c community and the intensity ofsocial interactions among the individuals involved.Finally, international data on size and productivity of science (e.g., Cole

and Phelan, 1999; Schofer, 2004) highlight large inequalities across countries,that are even wider than economic inequalities. Indeed, these papers showthat many medium and low income countries have a negligible scienti�c in-frastructure that sharply contrasts with that of a very small club with animportant scienti�c production. Hence, economic and scienti�c inequalit-ies seem part of the same phenomenon which �nds an interpretation in thispaper.Our paper relates to di¤erent strands of the literature on the theory of

economic growth. In particular the Schumpeterian strand (e.g., Romer, 1990;Grossman and Helpman, 1991; Aghion and Howitt, 1992) where innovation isthe outcome of races among �rms for a patent, shares a similar structure withthe scienti�c contests of our paper, but does not consider any kind of socialinteractions. These are the key topic of a di¤erent strand of literature whichanalyzes the e¤ects of several forms of social interactions upon economicgrowth. This is the case of �Social Capital� studies such as Knack andKeefer (1997), Temple and Johnson (1998), Helliwell and Putnam (2001),and Zak and Knack (2001), where indicators of social trust and civic virtuesperform well in growth equations. Another important channel of the in�uenceof social interactions analyzed by the economic growth literature is humancapital accumulation, since neighborhood signi�cantly a¤ects human capitalformation (e.g., Benabou, 1996a and 1996b; Temple 2001). Such a pervasivein�uence of social interactions can also be found in the world of science whereknowledge production occurs under a set of collective rules that magnifythe importance of relations among researchers. Since basic research spurstechnological innovation and economic growth, the extension of this approachto the scienti�c production, strongly a¤ected by social interactions, agreeswith Brock and Durlauf (2005) who stress the important role of social factorsfor the explanation of cross-country growth di¤erences.Another line of research to which our paper is related is the literature on

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scienti�c networks (Albert and Barabasi, 2002; Goyal et al., 2006; Labandand Tollison, 2000; Newman, 2001). This line of literature focuses on theformation of scienti�c networks and their features. In particular, some (Goyalet al., 2006; Newman, 2001) analyze the characteristics of scienti�c networksand their changes over time. However none of these papers consider theaggregate e¤ects of such evolution, which is the aim of our paper.The paper is organized as follows. Section 2 deals with the main forms

of social interactions in science. Section 3 sets out the theoretical model.Section 4 analyzes the model�s equilibrium solution. In section 5 we presentthe results of sensitivity analysis. The conclusions follow in section 6.

2 The organization of �Open Science� andsocial interactions among researchers

It is widely recognized that social interactions among researchers are a keyfactor for scienti�c knowledge production (e.g., David, 1998; Howitt, 2000).Several features of the science sector a¤ect the probability that two or moreagents interact, thereby intensifying the exchange of knowledge among sci-entists. We summarize them in four categories: collective norms, academicnetwork size, academic network structure and communication technology.

Collective norms in science Social interactions a¤ect scienti�c know-ledge production not only due to unintentional spillover e¤ects but also, andmaybe above all, because of the norms which regulate the institution of �openscience�. Such norms make scienti�c production not only the result of e¤ortof single researchers, but rather the outcome of a cognitive process whichinvolves the whole scienti�c community.It is widely held (e. g., David, 2004; Dasgupta and David, 1987; Merton,

1957; Ziman, 1994) that scientists conform to the norm of communalism,which means that researchers identify with the community of scientists andshould always share knowledge with others researchers. One consequence ofcommunalism is the full disclosure of �ndings and methods which is a keyaspect of the social program of research. Indeed, full disclosure makes dis-seminating and publishing the results of research particularly intense suchthat new ideas are di¤used into the whole scienti�c community. Two fur-ther norms, disinterest and universalism, reinforce the communal characterof scienti�c production, since the former lays down that a researcher hasto pursue only the general interest of science, while the latter establishesthat the scienti�c community is open to all persons of competence regardless

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of their personal characteristics. Finally, both the norm of scepticism, bywhich all claims will be subjected to trials of replications and veri�cations,and the attainment of scienti�c closure, by which a new scienti�c proposi-tion is accepted only when a preponderant consensus concerning its validityemerges, show the collective nature of scienti�c inquiry and point out thatthe production of new knowledge is essentially a social process2.However, norms which regulate academia entail not only collaborative

relations among peers, but also competitive ones. Indeed, another funda-mental norm is originality which ensures that only new knowledge is of valueto the scienti�c community. This norm makes scienti�c production a �winnertakes all�contest and gives scientists powerful incentive to innovate becauserewards, both in terms of recognition and resources, invariably accrue onlyto those who discover things �rst (Dasgupta and David, 1987 and 1994; Mer-ton, 1957). This method of assigning rewards engenders particularly intensecompetition among scientists, given that the higher the number of other re-searchers, the lower the probability of a researcher being the �rst to arriveat an innovation and hence to obtain the reward for it.

E¤ects of academic network size Some forms of social interactionswhich arise in the scienti�c community imply direct interactions among re-searchers. People meet at conferences and seminars, visit other departments,exchange comments on papers, collaborate on the same project, and so on.In these cases the size of scienti�c networks may reduce the costs of dir-ect interactions and hence it may increase the intensity of social exchange.This reduction may occur because in larger communities it is easier to meetpeople. Moreover, interaction costs are greater when researchers are hetero-geneous 3. Large networks, by increasing the probability of matching withsimilar individuals, reduce the costs which derive from heterogeneity. Thisis due to the well known �thick market externalities�, which is common insearch problems, but it can also be applied to the case in hand of searchingfor partners in basic research or more simply for informal contacts amongresearchers.The above considerations imply that the size of the scienti�c network

may have positive e¤ects on the social interactions among scientists andon the amount of knowledge exchange among them, which, in turn, a¤ectsthe average productivity in the sector. However, this conclusion does not

2Scientists themselves recognize the importance of the cumulative research e¤ort whichoccurs within the scienti�c community, as noted in statements such as Newton�s �if I haveseen further, it is standing on the shoulders of giants�.

3Researchers are heterogeneous in many respects: national culture, language, interests,psychological attributes.

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mean that there is a well-de�ned positive relation between size and averageproductivity of the research sector, since the positive e¤ects of size throughsocial interactions may be countervailed by the negative e¤ects due to thecontest to become the �rst discoverer.

Evolution of the academic network structure Recently, several pa-pers have analyzed the joint evolution of scienti�c networks for di¤erentscienti�c disciplines in terms of their size and structure4. Papers by Albertand Barabasi (2002), Rosenblat and Mobius (2004), and Goyal et al. (2006)maintain that scienti�c networks show an apparent trend toward a markedincrease in size and a reduction in �social distance�between researchers (thelatter is equivalent to an increase in social interaction). They interpret thisphenomenon as evidence of scienti�c networks becoming similar to a �smallworld�5. A small world is characterized both by the presence of a large com-ponent, which has a high degree of clustering and a very small social distanceamong agents, and by the presence of other components, very small in abso-lute and relative terms, with a lower degree of clustering and a higher socialdistance among agents. However, it is not clear what are the e¤ects of the�small world� evolution of the scienti�c networks on the whole amount ofsocial exchange among researchers: on the one hand, the average social dis-tance between people within the giant component is small, and this increasesthe social exchange among agents; on the other, the social distance withinthe small components and between the latter and the giant component islarge, and this reduces the amount of social exchange within the network.Nevertheless, the literature shows that the average social distance is alwaysreduced and social interaction is always increased within the small worldnetwork if the giant component is su¢ ciently large.

E¤ects of communication technology Another important factor thata¤ects interactions among agents is the technology of communication thatmay reduce or increase the costs of interacting. Some scholars (e.g., Kim,Morse and Zingales, 2006 and Rosenblat and Mobius, 2004) argue that thesharp reduction in communication costs, made possible by new informationtechnologies, has undoubtedly had a great in�uence on the nature and fre-

4See Albert and Barabasi (2002), Ravasz and Barabasi (2003), Newman (2001) andWatts and Strogaz (1998) for medicine, physics and computer science, Grossman (2002),for mathematics, Goyal et al. (2006), Rosenblat and Mobius (2004), Kim, Morse andZingales (2006), for economics.

5The idea of a �small world�has a long history in sociology. Theoretical work on itwas initiated by Pool and Kochen (1978) and empirical work by Milgram (1967). For aninteresting general survey see Watts (1999) and Kochen (1989).

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quency of interactions among agents, and hence on the amount of socialexchange possible within a given network. However, according to Rosenblatand Mobius (2004), this phenomenon could also have ambiguous e¤ects onthe amount of social exchange in science. These authors study a model ofa network by identifying two types of separations among agents: group sep-aration and individual separation. The former is de�ned as the separationbetween types of agents, while the latter describes the social distance betweentwo randomly chosen individual agents. Rosenblat and Mobius (2004) arguethat while the reduction in communication costs increases separation betweengroups of di¤erent types, it reduces the social distance between agents of thesame type. Thus the overall e¤ect on the individual separation and on socialexchange may be ambiguous. However, they conclude that if the networksize is su¢ ciently large social interactions always increase.To summarize the main arguments of this section, we can assert that:a) Basic research is ruled by norms which involve strong forms of interac-

tions among scientists which allow knowledge exchange, even if the originalityrule entails sti¤ competition among them;b) The recent evolution of scienti�c network structure and the reduction

in communication costs seem to have enhanced the intensity of social inter-actions among scientists, even if this is certainly true only when the networkis large.In the next section we present a model where we analyze the e¤ects of

social interactions on the productivity of the science sector and on the growthrate of the economy, when the size of science network is endogenously de-termined.

3 The model

3.1 Environment and technology

We consider an economy consisting of non-overlapping generations. Time isdiscrete and each period and generation are denoted by t. In each generationthere is a continuum of agents of measure one, indexed by i 2 [0; 1], who livefor one period and are endowed with heterogeneous levels of ability. Thereare two productive sectors: goods and science. The �nal good is consumedby all agents, is produced with labor input by �rms and is sold in competitivemarkets. Output is produced according to the following production function:

Yt = atLt (1)

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where at is the parameter measuring the level of the technology available inperiod t, Lt is the labor factor in the goods sector. Workers are allocatedbetween two sectors: basic research and consumption goods.

3.2 The basic research sector and technology

The science sector produces new knowledge, a public good which translatesone-to-one into new technology at. In each period the number of potentialdiscoveries is limited6. An implication of this assumption is that a number ofresearchers greater than one participates in the race for the same innovation.A race of this kind arises only when the number of innovations is limited;otherwise each researcher would seek to produce a di¤erent innovation inorder to maximize his/her chances of obtaining the reward. Moreover, mostresearch is not generic but focused on speci�c issues. Hence, the researcheridenti�es what advances are possible and concentrates his/her research e¤ortonly on those. This type of contest seems to capture what happens in the sci-enti�c world, where most innovations or advances in a scienti�c discipline arelimited in a given period, and scienti�c communities are often engaged withproblems on which there is consensus as to their importance for advancingscienti�c knowledge.Assuming that t is the number of the potential innovations that increase

the productivity of labor by a constant factor � > 0, the dynamics of pro-ductivity over time can be described as follows:

at = at�1(1 + t�) (2)

For the sake of simplicity we assume that the number of potential dis-coveries is constant and equal to one, so that = 1 if the innovation isintroduced, = 0 if it is not.The arrival rate of an innovation in the research sector is uncertain. We

denote with S(It) the probability of the event It : �an innovation occursin the research sector in the period t; i.e. = 1�, while 1 � S (It) is theprobability of = 0. The expected level of technology in period t will begiven by7:

eat = at�1 [1 + S(It)�] (2.b)

6This assumption, which is at variance with that usually made in the literature oninnovation-driven growth (Aghion and Howitt 1992, Romer 1990), was �rst introduced ina growth model by Zeira (2003).

7Since the rest of the model is concerned with equilibrium in each time period, belowwe omit the time index in notation of variables until we discuss economic growth.

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In our model research is carried out by individual scientists within sci-enti�c organizations, such as universities or other scienti�c institutions, thatare regulated by norms typical of academia discussed above, such as: a)originality; b) communalism; c) full disclosure.To capture the originality rule, we assume that each researcher particip-

ates in a collective contest where only the �rst to obtain the innovation hasthe reward. Success of a single researcher is uncertain for two reasons, �rstlybecause the arrival of a new idea in the whole scienti�c sector is not certain,secondly because it is uncertain who is the �rst discoverer.By denoting with Ii the event �agent i is the �rst to obtain an innov-

ation�, the probability that an innovation occurs and it is obtained by theith researcher is given by the probability of the joint event Ii and I. Thisprobability enters the calculation made by each agent of the expected rewardfrom research involvement and can be expressed as8:

p(Ii \ I) = S(I)p(IijI): (3)

Equation (3) establishes that the probability that the researcher i obtainsan innovation is given by the probability that the science community asa whole obtains an innovation, S(I), and by the conditional probability,p(IijI); that the winner of the innovation race is the ith researcher. The �rstprobability summarizes the role of the scienti�c network and captures thee¤ect of the environment and of the social interactions which occur withinthe scienti�c community on the productivity of a single scientist, while theconditional probability captures the e¤ect of the race among researchers.Therefore it captures a feature of the complex relation among researchers,that we de�ne as the �competition e¤ect�.

3.2.1 The competition e¤ect

The conditional probability that the researcher i is the winner of the racefor innovation is assumed to depend directly on the resources that the in-dividual researcher devotes to her research, denoted by hi, and inversely onthe resources that all other researchers devote to that same research activity,denoted by H. This is a consequence of the originality rule which lays down

8From the de�nition of conditional probability we have

p(IijI) =p(Ii \ I)S(I)

;

from which equation (3) is easily derived.

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that only the �rst discoverer gains the reward. In other words, we assume aTullock contest success function (Tullock, 1980) given by:

p(IijI) =hiH

(4)

Resources that scientists devote to research consist of e¤ort, ei, and skills,�i. The latter are heterogeneous in the population and distributed accordingto a distribution function f(�); with � 2 � = f� 2 R j 0 � � � 1g. Theamount of resources devoted to research by scientist i is given by hi = �iei:The community of scientists is made by agents with ability in the set �R;with � = �R [ �O, and �O denotes the set of abilities of workers in thegoods sector. With such notation, the amount of resources applied by allresearchers in the sector is given by:

H =

Z�R

e�f(�)d� (5)

By substituting equation (5) in eq. (4) we have:

p(IijI) =ei�iR

�Re�f(�)d�

(6)

Hence, the greater the size of the scienti�c sector, the lower the probabilityof a single researcher being the winner of a scienti�c contest. This negativerelation captures the strength of the competition e¤ect in basic research.

3.2.2 Social interactions e¤ect

The probability that an innovation occurs in the science sector, S(I), capturesthe productivity of the whole scienti�c community. This probability dependsboth on the total amount of resources that are invested in the science sectorand on social interactions among scientists.In particular, we assume that the larger is the amount of scienti�c know-

ledge, the more likely is an innovation to be attained. Formally:

S (I) = s

where s is a parameter which captures the e¢ ciency of the sector and is theamount of scienti�c knowledge of the whole sector that increases not only dueto investment in science but also due to knowledge exchange among research-ers arising from social interactions. But what factors facilitate (or hinder)social interactions among scientists? As we have already discussed, the size

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of the scienti�c community, communication technology and the structure ofscienti�c networks strongly a¤ect the likelihood and the way in which agentsinteract. In our model, communication costs or the network structure are notexplicitly considered but their e¤ects on the intensity of social interactionsand on the amount of social exchange are summarized by an aggregate indexof scienti�c knowledge. Formally, we assume that scienti�c knowledge of thewhole scienti�c community results from aggregation of the knowledge of allagents according to the following CES index9:

=

24Z�R

(e�)(1�1{ )f(�)d�

35 1

1� 1{

(7)

where { is the parameter that captures the intensity of social interactions andallows us to take into account the e¤ects on scienti�c knowledge of changesin the intensity of social interactions among researchers whatever their cause.As a matter of fact, when 1

{ > 0, we can say that individual resources inves-ted in research are complements. This means that the scienti�c network ischaracterized by few social interactions and hence by low knowledge exchangeamong agents. In this case the high h individuals cannot easily transfer theirknowledge to less talented scientists, thereby reducing the average level of h.When 1

{ < 0, individuals are substitutes, which means that social interactionsare strong and hence knowledge exchange among them is high. In this casethe most talented scientists may more easily transfer their knowledge to thelow h types, thereby pulling the average level of h upward. Hence this indic-ator of social knowledge can summarize several features of the environmentof basic research such as network structure, communication technologies, oralso shared norms and values which we do not model explicitly but have thee¤ect of enlarging or reducing the knowledge exchange among agents in thenetwork10. It can be noted that the index also depends on the size of thescienti�c community which is endogenous to our model and interacts withthe substitutability parameter ( 1{ ) in nontrivial ways. For the sake of clarity,below we refer to 1

{ as a unique indicator of social exchange and interaction.

9This indicator was used by Benabou (1996a and 1996b) to capture the in�uence ofsocial interactions on the formation of local human capital.10For example parameter { may also capture e¤ects on knowledge exchange of di¤erent

behavior or shared rules such as conformist behavior and status-seeking behavior. Inthe �rst case 1={ assumes positive values, while in the case of status seeking behavior itassumes negative values (see Benabou, 1996a and 1996b).

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3.2.3 Preferences

Every scientist participates in a contest for a new �nd which awards prizem to the winner of the race, while the rest of the participants receive noth-ing. The researcher chooses the amount of e¤ort by maximizing expectedutility that we assume to be the sum of two components. One is utility fromconsumption and the other is disutility from e¤ort. In this simple model, sci-entist income equals m or zero according to the outcome of the race. Belowwe simplify notation with: pi � p(Ii \ I) and S � S(I). Hence, income ism with probability pi and 0 with probability 1 � pi. Every agent spends allher income on consumption and we assume also that utility is linear in con-sumption. Disutility of e¤ort is represented by an increasing convex functionwhich depends on the expected level of technology.Hence the expected utility of a scientist is:

ui;R = mpi �deae1+�i

1 + �, (8)

where d > 0, � > 0 are two parameters which capture the disutility derivingfrom e¤ort.

3.2.4 Research �nancing

The state levies taxes on wages of workers in the consumption sector in orderto �nance the production of knowledge by the research sector. The �nancingconsists in an amount of real income which is distributed only to those whowin the innovation contest11. At the beginning of each period the governmentsets up scienti�c contests and announces both the prize that will be awardedfor a discovery and the amount of taxes that it will levy: �eaL, such that:

m = �eaL. (9)

where 0 < � < 1 is the �at rate of income tax. The government collectstaxes, �eaL, on consumption production both in the event of innovation, i.e. = 1, and in the opposite case, = 0. When in the scienti�c race nobodyis awarded the prize, the government spends tax revenues on consumptiongoods. Hence, the government budget constraint is always balanced in eachperiod.11Actually, the most common system of scientist reward is made by income which accrues

only to innovators and a salary related to routine academic activities such as teaching,which shelters researchers against the risk of not producing any innovation. Below, weassume for the sake of simplicity that only the former type of income is assured for re-searchers, although extension to the more general case is straightforward and does notsigni�cantly alter the results.

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3.3 The consumption goods sector

To simplify the analysis, we assume that in the consumption goods sectorlabor productivity does not depend on individual skills nor on e¤ort. Further-more, work in the sector does not cause disutility, and each worker suppliesinelastically one unit of labor input. Given these assumptions, we specify thefollowing function of the expected utility obtainable by every worker in theproduction of goods:

uy = c (10)

which is maximized under the constraint c � wy, where c stands for theexpected consumption and wy for the expected wage in consumption goodsproduction net of workers�taxes. Pro�t maximization under perfect compet-ition determines the expected wage: wy = (1� �)ea:4 Equilibrium

The model economy works in a simple way. At the beginning of everyperiod the government organizes a scienti�c contest. Agents decide the kindof job they will do in their lifetime and, if scientists, the amount of e¤ortthey will devote to research. If an innovation occurs in the science sector thegovernment collects taxes from wages and pays a prize to the winner of thescienti�c race. In the next period, the knowledge base available for goodsproduction is greater due to the innovation and the economy grows. In theopposite case of unsuccessful research, the economy is stationary.At the core of the model is the scienti�c contest that can be thought of

as de�ning a two-stage game where in the �rst stage agents decide to workeither in research or in good production, and in the second stage scientistscompete for a prize by choosing optimal e¤ort. As usual, we approach thesolution of the game backward by deriving �rst the equilibrium optimal e¤ortin science, then the size of the sector.

4.1 Optimal e¤ort

Scientists choose the level of e¤ort by maximizing their expected utility.After substitution of the expressions we assumed for pi and for S into ui;R,the expected utility of researchers becomes:

ui;R = �seaL ei�iR�R(e�)f(�)d�

0@Z�R

(e�)(1�1{ )f(�)d�

1A 1

1� 1{

� deae1+�i

1 + �. (11)

14

Every scientist maximizes utility ui;R with respect to ei given her/hisbeliefs about the choice of e¤ort and talent of the community of researchers.Hence, the �rst order condition is:

@ui;R@ei

= �sL�iR

�R(ei�)f(�)d�

0@Z�R

(e�)(1�1{ )f(�)d�

1A 1

1� 1{

� de�i = 0.

From the set of �rst order conditions the reaction function of researchersparticipating in the race can be derived:

ei =

264 �Ls�idR�R(e�)f(�)d�

0@Z�R

(e�)(1�1{ )f(�)d�

1A 1

1� 1{

3751�

; 8i : �i 2 �R: (12)

To derive the Nash equilibrium of a research contest, which is a function eithat describes equilibrium choices, we assume rational expectations. From theNash equilibrium the functions of S and pi can be obtained, as maintainedin the following proposition:

Proposition 1 Given the size of the research sector, there exist the optimumlevel of individual e¤ort, ei; the individual probability, pi; and the averageprobability of innovation, S; that in each time period are equilibrium solutionsto the contest for a discovery and are given by:

ei =

��Ls�id

� 1�

24Z�R

�1+�� f(�)d�

35� 1�24Z�R

�(1� 1

{ )(1+�)� f(�)d�

35 1

�(1� 1{ )

; (13)

pi = �1+ 1

�i s

��Ls

d

� 1�

24Z�R

�1+�� f(�)d�

35�(1+ 1� ) 24Z

�R

�(1� 1

{ )(1+�)� f(�)d�

35(1+�)

�(1� 1{ )

;

(14)

S = s

��Ls

d

� 1�

24Z�R

�1+�� f(�)d�

35� 1�24Z�R

�(1� 1

{ )(1+�))� f(�)d�

35(1+�)

�(1� 1{ )

. (15)

15

Proof. See the appendix.Simple inspection of the optimal level of both ei and pi reveals a �rst

interesting result: scientists endowed with greater ability �i choose highere¤ort and have greater productivity. Therefore the more able researchersare more likely to be the �rst to produce an innovation, not only becausetheir ability directly in�uences that probability but also because they investmore in the research activity (Lazear, 1997). Actually, since Lotka (1926),several empirical studies have investigated inequality in scienti�c productivityshowing that a small share of scientists produces a large share of papers(Stephan, 1996). Our result may explain this acclaimed empirical evidence.Another important property of the equilibrium of a scienti�c race is that

the population is divided into two parts: workers and researchers, accordingto innate talent. To �nd the equilibrium number of researchers we substitutethe equilibrium functions (13), (14), (15) in equation (8) and obtain themaximum level of utility in research, given by:

umi;R=dea�1 + �

�1+��i

0B@ �Ls

dR�R�1+�� f(�)d�

1CA1+�� 0@Z

�R

�(1� 1

{ )(1+�)� f(�)d�

1A(1+�)

�(1� 1{ )

(16)

Every agent makes a decision over the sector where he/she will work bycomparing the two payo¤s. If umi;R < uy then he/she will join the goodsector, while in the opposite case: umi;R � uy he/she will join research, asmaintained in the following proposition:

Proposition 2 In the model economy there exists a threshold level of abilityz > 0 such that: umR (z) = uy. If �i < z, the individual i will work in goodproduction, while if �i � z, he/she will choose to enter the research sector.

Proof. See the appendix.Proposition 2 says that only the ablest individuals will choose the science

sector. From this proposition we derive also the number of workers in thegood sector, given by: L = F (z) =

R z0f(�)d�; and the share of the population

working in basic research, equals to R � 1 � F (z). The equilibrium of thesecond stage of the game provides a picture of the functioning of a researchcommunity of a given size. Hence, we can derive some interesting resultson the e¤ects of social interactions and of the enlargement of the scienti�ccommunity on individual and aggregate performances.

E¤ects of social interactions in a research contest of a given size

16

Intuition would suggest that scientists working in environments whereknowledge �ows are stronger would improve their productivity and �ndgreater incentive to make an e¤ort. This intuition �nds con�rmation un-der fairly general hypotheses in the following proposition.

Proposition 3 In a scienti�c race with a given number of participants, anincrease in the intensity of social interactions (i.e. lower 1={) increases boththe e¤ort of a single researcher and the average productivity of the sciencesector.Proof. See the appendix.

This proposition implies that, when the number of scientists remains con-stant, the e¤ect of an increase in social interactions on the basic researchproductivity is positive. This is an important result since one policy implica-tion is that by making the science environment more favorable to knowledgeexchange among scientists, for example by favouring researchers�mobilityor scienti�c collaborations, it is possible to rise the innovation rate withoutemploying more resources. However, the empirical literature does not reachclear-cut conclusions about the e¤ects of social interactions on network pro-ductivity. Indeed, Shilling and Phelps (forthcoming), on analyzing the net-work of innovative �rms, �nd the more a �rm is embedded in a networkwith intense social exchange among �rms, the higher is the �rm�s rate ofpatenting. By contrast, Fleming and Marx (2006), who study inventors inSilicon Valley and Route 128 in Boston, �nd no relationship between theintensity of social interactions in the inventors networks and patenting ratesin these regions. Smith (2006) arrives at a similar conclusion on analyzingthe network of rappers in the United States. Uzzi and Spiro (2005) obtaina rather di¤erent and interesting result: in analyzing the network of Broad-way actors, they �nd that the relationship between actor performance andthe intensity of social interactions is positive for the low level of the latter,while after a threshold, too much connectivity can undermine the bene�tsit creates and the relationship between social interactions and performancebecomes decreasing. The nonlinearity of this relation is also con�rmed bythe results of Guimera et al. (2005) who, in analyzing the networks of sci-enti�c collaborations in four academic disciplines, �nd that the performanceof scientists increases as the connectivity of the network increases. However,as social interactions within the network become very strong, performancedeclines.Hence, according to the applied literature on networks characteristics,

there is evidence of a non-monotonic relation between social interactions andnetwork performance. At �rst sight, this evidence seems to con�ict with the

17

arguments in Proposition 3, since we found that an increase in the intensityof social interactions always rises the output of the science sector. However,it must be noted that the cited papers do not consider the endogenous vari-ation in the size of the social network which could substantially change thephenomenon. Indeed, the intensity of social interactions and its e¤ect onperformance may depend on network size. Such e¤ects can be appreciatedby allowing the entrance of other researchers in a race, as we do below.The e¤ects of entrance into the scienti�c raceIf other researchers enter into the science sector, the competition among

scientists increases and this may reduce their e¤ort and their productivity,but the e¤ect of this entrance on the individual and average productivitydepends also on the intensity of social interactions in a non trivial way.In this regard, the main results of model analysis are summarized in thefollowing proposition:

Proposition 4 Greater R always decreases scientists�productivity in com-munities where there is high complementarity (i.e. low intensity of social in-teractions 1 < 1={ < +1). The shape of this relation cannot be determinedunder general conditions on parameters when there is substitutability and lowcomplementarity (i.e. high and medium social interactions -1 < 1={ � 1).However, in this case, under the hypothesis of uniform distribution of tal-ent12, the function of average productivity S with respect to R is shaped likean inverted U13.Proof. In appendix.

Interpretation of the �rst part of this proposition is straightforward: ifresearchers work in isolation, increasing the number of participants in the

12The choice of the uniform distribution of individuals� abilities entails the following

assumption: 1{ <

�1+2�1+�

�. This assumption is required to ensure that equations that

describe the model equilibrium solution are de�ned on R.13This result can be obtained with general f(�) under more restrictive assumptions.

Indeed, the shape of the function S(R) depends on the shape of the distribution functionf(�). S(R) has an inverted U shape if two sets of su¢ cient conditions on parameters andtalent distribution hold:i)Agent low substitution ( �1 < 1

{ < 0):f(z)

df(z)=dz <�(�+ 1

{ )1{ (1+�)(1�

1{ );

ii) Agent low complementarity ( 0 < 1{ < 1):

f(z)df(z)=dz <

�(�+ 1{ )

1{ (1+�)(1�

1{ )and 1

{ (1�1{ ) < �.

A proof is available from the authors on request. The restriction on the shape of f(�) at thethreshold value z is easily satis�ed if the function decreases at z. Indeed, most distributionfunctions have a bell shape which means that this requirement is satis�ed if z is big enough(small R) and we actually observe small research sectors in any country . The condition onparameters requires a fairly high value of the parameter of e¤ort disutility � (for examplegreater than 1, since 1

{ 2 (0; 1)).

18

race reduces individual incentives and the probability of success. The secondpart states that the same phenomenon, in environments which are more fa-vorable to social exchange, increases scientist productivity until the scienti�ccommunity has reached a critical size. Indeed, interactions among a growingnumber of researchers strengthen individual incentives and bring about sig-ni�cant increases in productivity that outweigh the negative e¤ects due tohigher competition among researchers. However, increasing returns gradu-ally vanish and become decreasing after employment in the science sectorhas reached a critical size, after which productivity declines. This result un-derlines the role of social interactions, actually the above proposition saysthat if an increase in researchers is not followed by policy measures aimedat increasing knowledge exchange and collaborations among them, this risecould reduce the e¢ ciency of the science sector. Thus an enlargement of thescience sector must be accompanied by an institutional re-design of it.This �nding is interesting also because it seems consistent with the hy-

pothesis advanced by several historians of science (Cole and Cole 1972, DeSolla Price, 1963) and its evolution. In particular, on analyzing the histor-ical pattern of science in the USA and in some European countries, De SollaPrice (1963) found that the �relationship between the number of scientistsand the rate of scienti�c advances is described by an S-shaped curve, i.e. �rstexponentially increasing and then converging to a point of saturation, afterwhich the rate of advances diminishes�14. This result is also consistent withthe evidence found by the empirical literature on basic research. Indeed, theeconometric analysis of Aizenman and Noy (2007) suggests that the relationbetween the lagged GDP, a proxy of resources invested in basic research, andthe share of major scienti�c prizes (e.g., Nobel, Field) can be approximatedby a logistic function.The results of this section may o¤er an explanation for the empirical evid-

ence on the non linear e¤ect of social interactions on network productivityshowed in the previous section. We found that networks which are morefavorable to knowledge exchange entail greater productivity in a given com-munity, but changes in the number of researchers participating in the racecan have negative e¤ects on the average probability of a discovery. Hencethe e¤ect of social interactions on innovation rate can be non-monotonic. Ofcourse, the question of determining the number of researchers must be ad-dressed in the context of the general equilibrium of the model to which weturn our attention.14De Solla Price (1963) explains this phenomenon with the �Galton hypothesis�, by

which, as the number of scientists relative to the population increases, the ability of thoseentering science declines since the distribution of scienti�c talent in the population is givenand follows a Gaussian distribution.

19

4.2 Employment size of basic research

In the previous section we solved the second stage of the sequential gamedescribing research contests. In this section, we approach the determinationof equilibrium with respect to the share of the population who enter the race.Substitution of equilibrium values of ei; S and pi in equation (11) yields themaximum level of utility umR given by:

umR;i(z) =dea�1 + �

�F (z)s�i

dR 1z�1+�� f(�)d�

! 1+�� �Z 1

z

�(1� 1

{ )(1+�)� f(�)d�

� (1+�)

�(1� 1{ )

(17)

The function umR;i(z) increases with scientist skill and is non-monotonicwith respect to the ability level above which individuals enter the researchsector z. More precisely, the maximum utility of researchers increases withz if the competition e¤ect is strong enough. In fact, an increase in z is equi-valent to a reduction in the number of individuals engaged in research, suchthat there is less competition and hence a greater probability of success. Bycontrast, if the social interactions e¤ect predominates, shrinking the researchsector (i.e. an increase in z) lowers the maximum utility obtainable therein.The equilibrium value of z; hence of R; derives from the ability level at

which the marginal worker is indi¤erent between the two sectors. Substitut-ing c = wy and equation (17) into uy = umR (z) and rearranging the termsyields:

�(1� �)� + 1

�d

� �1+�

=�sF (z)z

dR 1z�1+�� f(�)d�

�Z 1

z

�(1� 1

{ )(1+�)� f(�)d�

� 1

(1� 1{ )

(18)

The right-hand side still represents the utility that the marginal researcherderives from choosing the research sector. The left-hand side of this equationrepresents the opportunity cost of choosing the research sector. In equilib-rium the marginal worker must be indi¤erent between the two choices. Whilethe l.h.s. of eq. (18) is constant, its r.h.s. is a complex function of z whichdepends on the shape of the distribution of ability F (�). Our analysis of theequilibrium condition is summarized in the following proposition:

Proposition 5 The ability threshold z is determined by the equilibrium con-dition Eq. (18) according to the following cases:a) Under high agent complementarity (i.e. 1 < 1={ < +1), there is one

equilibrium value of z which is interior and stable;

20

b) Under agent substitutability or low complementarity (i.e. �1 < 1={ �1); there can be multiple equilibria in the interval 0 < z � 1.Proof. See the appendix.

This proposition characterizes the equilibrium in the general case, andhighlights some features of the economy that rely on the strength of socialinteractions among scientists. This e¤ect depends on the degree of comple-mentarity or substitutability of scientists�abilities. When there is low socialexchange among scientists, the economy is characterized by a unique equi-librium with a positive science sector. In such an environment sources ofexternalities are weak, and competition produces a standard unique equilib-rium.Economies where basic research is made under intense social exchange

can be characterized by multiple equilibria. Agents may coordinate theirexpectations on one or another equilibrium threshold z that determines thesize of employment in science, the opportunities for innovation and economicgrowth. Multiple equilibria have strong aggregate implications since inequal-ities among economies can be magni�ed and initial conditions are relevantin the selection of equilibrium. Network structures that facilitate social ex-change and reduce social distance among researchers, or a reduction in com-munication costs, mean that initial conditions are important.A signi�cant feature of the case which allows for multiple equilibria is the

certain presence of one equilibrium without employment in research, that canbe stable or unstable, as maintained in the following:

Proposition 6 When -1 < 1={ � 1, z = 1 is always an equilibrium. Thisequilibrium is unstable in the case of agent substitutability, while it is stablein the case of low complementarity (e.g. 0 < 1={ � 1).

Proof. See the appendix.This proposition sheds more light on our results: it states that when the

model economy allows for multiple equilibria there is always an equilibriumat z = 1: However, only if agents�complementarity is low, this equilibriumis stable and implies a steady state with null technological progress and nullexpected rate of growth of per capita income. Hence, this stable equilibriumcan be termed a poverty trap. The case of low complementarity is character-ized by the e¤ects of competition and knowledge exchange in the scienti�ccommunity that do not clearly prevail over the other. Hence, incentive toresearch can be low and agents might coordinate their expectations on anequilibrium without scientists. Under the other two polar environments ofhigh complementarity and substitutability (respectively low and high social

21

interactions), incentives to engage in basic research are strong for oppositereasons and there is no room for an equilibrium in which every agent thinksnobody will make research.

4.2.1 Example: uniform distribution of skills

Since the previous propositions on the number of equilibria thresholds arevery general, we assume that ability is distributed uniformly across the pop-ulation in order to derive some insights into the number of equilibria valuesof z under di¤erent con�gurations of the parameter 1={, which is pivotal inthe model. Under this hypothesis Eq. (18) becomes:

�(1� �)� + 1

�d

� �1+� �

2� + 1

�1� z 2�+1�

�= (19)

=s

dz2�

�

(1� 1{ )(1 + �) + �

� 1

(1� 1{ )�1� z

(1� 1{ )(1+�)+�

�

� 1

(1� 1{ )

The left-hand side of this equation, LS(z), is always decreasing and concavein z, while the shape of the right-hand side, RS(z) depends on the valuesassumed by 1={. It may always be increasing (case 1 < 1={ < +1), or itmay have an inverted U-shape. Two simulations of the model with values of1={ in the range of substitutability and in the range of low complementarityare depicted respectively in Figures 1 and 215.

Fig.1 and �g. 2 here

In Figure 1 high social exchange implies the existence of a unique stableequilibrium. In the case of low complementarity, Figure 2 shows the presenceof two stable equilibria: one with a positive research sector, the other withnil basic research. When complementarity is high the function RS(z) startsfrom the origin and is increasing, hence crossing from below LS(z) only onceat a value of z which represents a stable equilibrium.The overall picture of equilibrium thresholds z shows that even in the

case of the most simple distribution of talent our model produces multipleequilibria if the parameter 1={ lies in some ranges. Hence, this feature of themodel seems robust with respect to the distribution of talent in the economy.

15Parameters assumed the following values: Figure 1: � = 0:05;� = 0:5; d = 1; s =10;� = �2; Figure 2: � = 0:05;� = 1:5; d = 1; s = 10 � = 2:

22

5 E¤ects of changes in the intensity of socialinteractions on growth

The previous discussion of the sharp di¤erences in the con�gurations of equi-libria would suggest that economic growth can be very sensitive to socialinteractions in the environment of research. In this model economic growthdepends on the occurrence of innovations in each time period. Since this isan uncertain event, growth of per capita income is described by a stochasticprocess with discrete leaps. The expected growth rate of per capita incomebetween two time periods is:

g = E

�Yt � Yt�1Yt�1

�= S� (20)

where S, in equilibrium, is given by:

S = s

�F (z)s

dR 1z�1+�� f(�)d�

! 1� �Z 1

z

�(1� 1

{ )(1+�))� f(�)d�

� (1+�)

�(1� 1{ )

.

Economic growth depends on the probability of success of researchers whocompete to be the �rst discoverer of new knowledge and gain a real prize.In turn, this probability depends on the distribution of the talent of agentswho work in science and on the characteristics of the research environmentthat may facilitate or hinder knowledge exchange through social interactions.The probability S is also in�uenced by the employment size of basic research.Indeed, the growth rate is a function of the threshold value of talent z, henceof R;which is an endogenous variable.In this theoretical framework, changes in the social environment of basic

research have signi�cant consequences on economic growth in directions noteasily guessed. Comparative statics analysis is complex for two reasons.First of all, the set of equilibrium con�gurations described in the last sectionshows that di¤erent types of social interactions in the research sector implysigni�cant di¤erences in equilibrium con�guration. This implies that majorchanges in the parameter 1

{ may entail real changes in the growth regime.Moreover we may have a case where there are multiple equilibria, which implythat there is no function of the threshold z; hence of R; with respect to theparameters. Accordingly, our analysis of small changes in 1={ will be limitedto local e¤ects in the neighborhood of stable equilibrium values of z. Insightsinto the e¤ects of wide changes of 1={ will come from model simulations:Another problem that can further complicate comparative statics analysis

is that the e¤ects of changes in parameter 1={ on the equilibrium growth

23

rate also pass through the e¤ects of parameter changes on R:

dg

d�= �

�@S

@�+@S

@R

@R

@�

�; � = 1={: (21)

As we can see from equation (21), the parameter 1={ a¤ects the growth rateby modifying the advantages of knowledge exchanges (direct e¤ect throughS) and the incentive to join basic research (indirect e¤ect through R). The�nal e¤ect on the growth rate depends on the interplay of these two phe-nomena. As regards the direct e¤ect through @S

@�, we know from Proposition

3 that an increase in the intensity of social interactions always raises averageproductivity in the science sector. The indirect e¤ect (i.e @S

@R@R@�) depends on

the sign of @R@�that is provided in the following:

Proposition 7 An increase in social interactions (i.e. lower 1={) increasesthe employment size of basic research, R.

Proof. In appendix.Hence an increase in social interactions between researchers ensures that

more agents join the research sector, this rise however does not imply a pos-itive e¤ect on economic growth, since, according to the previous analysis,a larger number of researchers does not mean a greater probability of in-novation. Indeed, in the case of high complementarity the opposite e¤ect isobtained. Hence the direction of the net e¤ect on growth cannot be obtained.That said, Proposition 4 states that, if F (�) is uniform, an increase in R

raises productivity S when interactions are signi�cant and the science sectoris small. Then, in this case, greater interaction in the scienti�c communitybrings about higher expected rates of innovation and growth in per capitaproduct. These results are summarized in the following:

Proposition 8 Under the assumption that the distribution F (�) is uniformthe following statements hold:i) in the case of substitutability and low complementarity; if R is lower

than a threshold, an increase in social interactions increases the expectedgrowth rate since both the direct and indirect e¤ect of social interactions para-meter are positive;ii) in the case of high complementarity or with a R higher than a threshold,

the direct e¤ect of an increase in social interactions is positive while theindirect e¤ect is negative, and the net result on the expected growth rateremains ambiguous.Proof. This derives from the application of the results of Propositions

3,4 and 7 to Equation (21).

24

According to this proposition, when the basic research community is char-acterized by substitutability and it is lower than a critical size, policies aimedat increasing social interaction among scientists are successful at promotingeconomic growth. However if the number of scientists is already high or thereis high complementarity, the same types of policies have ambiguous e¤ectson the growth rate since the direct e¤ect on average productivity is positivewhile the indirect e¤ect, via the number of researchers, is negative. Thus,though it might seem counter-intuitive, an increase in human resources al-located to the science sector is not always bene�cial for the system�s rateof innovation. It follows that policies aimed at increasing social interactionamong researchers must be coupled with policies aimed at controlling theentrance of new researchers, since the consequent rise in the competition ef-fect could be so strong as to reduce the e¤ort of scientists and hence theirproductivity. Limiting the entrance of new researchers is not the only wayto escape from diminishing returns, there are in fact other policies that canpreserve the strength of increasing returns when employment in science ap-proaches the critical size. This can be obtained by increasing scienti�c awardsor by fragmenting major �elds of research into di¤erent sub�elds, which isequivalent to a rise in scienti�c awards. If this policy measure is also associ-ated to an increase in social interactions among researchers, it is possible toescape from diminishing returns and obtain a new phase of growth, where sci-ence increases both in the number of scientists and in its e¢ ciency. Actuallythis scenario ably depicts the modern development of science which has seenboth a rise in the number of scientists and in the number of scienti�c awardsand of numerous new disciplines and sub�elds of research16. Recently, in themore developed countries there has been a large increment in the number ofscienti�c awards. Zuckerman (1992) reports that in North America there are3000 scienti�c awards, �ve times more than thirty years ago.

5.1 From the high complementarity to the substitut-ability regime

Another question that arises when we analyze the e¤ects of social interac-tions is the variety of equilibrium con�gurations that arises according to thetype of social interactions regime which prevails: with high complementaritythere is one equilibrium with a small but a positive science sector, while in

16This is also what De Solla Price predicted, according to whom �the reaching of thesaturation point does not necessarily lead to a senility of science, but it may induce areorganization of science from which a new logistic curve rises phoenixlike on the ashes ofthe old�(De Solla Price, 1963 p. 25).

25

the two cases of substitutability and low complementarity there are multipleequilibria and in the latter case the economy can be trapped in a low equilib-rium where technical progress is nil. This implies that, if by increasing socialinteractions the economy gradually shifts from a high complementarity toa substitutability regime, the number of scientists could be �rst increasing,then it can collapse to zero, if in the low complementarity the poverty trapequilibrium prevails, to rise again in the substitutability case. The growthrate may show the same course, if under high complementarity, the positivee¤ect due to a more intense social exchange prevails, on the contrary it canbe �rst decreasing and then increasing as the 1

{ parameter goes in the substi-tutability range. Conditions under which such outcomes can arise are worthof further investigation. We provide an answer to this question by simulatingthe model under the hypothesis of uniform distribution of talent (see �g. 4and 5).Table 1 presents the equilibrium values ofR and g when social interactions

vary from complementarity to substitutability regime, under di¤erent valuesof the parameters � and s. These simulations represent the whole set ofcon�gurations of equilibria implied by the model. Starting from values inthe high complementarity range (i.e., 1 < 1={ < +1), we see that there isone equilibrium value of R and g that increases with a reduction in 1

{ . Hencesimulation results indicate that for the given values of parameters � and s;also in the case of high complementarity policies enhancing social interactionshave a positive e¤ect on the growth rate of the economy.When 1

{ is in the range of low complementarity (i.e., 0 < 1={ � 1)the model shows two di¤erent equilibrium con�gurations: a) the case of oneequilibrium with null employment in science (a poverty trap); b) the caseof two stable equilibria, one with positive R and the other with R = 0.Hence, in the simulated economy knowledge production may disappear evenin the case of one equilibrium. When the model economy has two equilibria,innovation may again become positive if agents coordinate their expectationstoward the higher stable equilibrium. Otherwise, the economy converges toa poverty trap even in the presence of a positive alternative.Further reductions in 1

{ drive the parameter into the range of substitut-ability. Having assumed a uniform distribution of talent, we �nd a uniquestable equilibrium of employment in science17. This research environmentseems the most favorable to knowledge production and economic growth. In-deed, simulations show that greater social interactions increase both R and g;as in the case of low social exchange. However, the most important evidence

17Notice that in this case; with general distribution F (�) ; there may be multiple equi-libria.

26

arising from simulations is that the transition from high complementarityto substitutability means dramatic increases in both variables. Also, underhigh social interactions, marginal reductions in 1={ cause greater e¤ects onthe expected growth rate than under low social interactions. Accordingly,numerical analysis of the model seems to suggest policies which modify theresearch environment through a discrete jump from low to high social ex-change. These policies would successfully place the economy on a path ofhigh knowledge production and economic growth. However, the transitionfrom low to high growth equilibria involves the risk of being trapped in astable equilibrium without growth. Indeed, this may be the case of gradualimprovements in social interactions in science which end in the intermediatecase of low complementarity (e.g. 0 < 1={ � 1).The whole set of simulation results lends itself to the interpretation of

the large di¤erences in scienti�c production and economic growth which weobserve across the world. The main distinction between successful sciencesystems and peripheral systems seems to rely on the capacity of taking ad-vantage of the huge opportunities for research productivity improvementso¤ered by several channels of social interactions in science. The interplaybetween individual objectives and the social environment shapes every re-search system, and causes strong nonlinearity in its aggregate productivity.These nonlinear e¤ects of social interactions can enhance di¤erences amongcountries and make the transition to a better organization of scienti�c re-search a very challenging task for economic policy. In this respect, accordingto our simulations, policies that increase the resources devoted to fundingbasic research have positive e¤ects on employment in science and economicgrowth in most of the parameter con�gurations of Table 1. The same argu-ments apply to positive changes in the parameter s which can be thoughtof as approximating the e¢ ciency of the translation of aggregate scienti�cknowledge into the probability of the occurrence of discoveries, such as theICT infrastructure of a country.

6 Conclusions

In this paper we put forward a model of economic growth through scienti�cproduction of knowledge. The main aims of the paper were to determine theaggregate e¤ects of social interactions in the scienti�c community and ac-count for the broad di¤erences in the development of a scienti�c structure inthe world. At the core of our model is a science sector made up by agents en-dowed with heterogeneous talent, where scientists participate in contests forpriority and a real prize. Social interactions in science are summarized by an

27

index of researcher talent and e¤ort. The equilibrium solution of the modelshows the relevance of increasing returns and the possibility of multiple equi-libria. Comparative statics suggests that scienti�c communities where socialexchange is signi�cant bene�t from stronger social interactions with a higherexpected rate of economic growth if the employment in the science sector hasnot reached a critical size. Our model represents a preliminary attempt toextend growth theory with the consideration of social interactions in science.Even though we are able to draw from the model several potential proposi-tions, further work still remains to clarify the economic consequences of theparticular collective rules and norms that prevail in the scienti�c communitywhich still constitute a largely unexplored research topic.

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AppendixProof of Proposition 1.

This proposition concerns the existence and uniqueness of an equilibriumsolution to the second stage of the game. This equilibrium is a function e (�)that solves the reaction functions:

32

ei =

264 �Ls�idR�R(e�)f(�)d�

0@Z�R

(e�)(1�1{ )f(�)d�

1A 1

(1� 1{ )

3751�

; 8i : �i 2 �R:

Let us start from these equations and multiply both sides by �if(�): Then,after taking integrals of both sides we have:

Z�R

(e�)f(�)d� =

Z�R

�1+�� f(�)d�

264 �Ls

dR�R(e�)f(�)d�

0@Z�R

(e�)(1�1{ )f(�)d�

1A 1

(1� 1{ )

3751�

;

from which the following can be easily derived:

24Z�R

(e�)f(�)d�

35 1+��

=

��Ls

d

� 1�Z�R

�1+�� f(�)d�

2640@Z�R

(e�)(1�1{ )f(�)d�

1A 1

(1� 1{ )

3751�

;

such that the integral on the left hand side of this equation can be expressedas:

Z�R

(e�)f(�)d� =

��Ls

d

� 11+�

24Z�R

�1+�� f(�)d�

35 �1+�24Z�R

(e�){�1{ f(�)d�

35 {({�1)(1+�)

:

(A1)Now we can substitute this integral in equation (12) and obtain:

e = �1�

��Ls

d

� 11+�

24Z�R

�1+�� f(�)d�

35� 11+�24Z�R

(e�)(1�1{ )f(�)d�

35 1

(1� 1{ )(1+�)

:

If we multiply both sides of this equation by � and follow similar steps asbefore, then we get the following expression for the function :

33

=

24Z�R

(e�)(1�1{ )f(�)d�

35 1

(1� 1{ )

=

��Ls

d

� 1�

24Z�R

�1+�� f(�)d�

35� 1�24Z�R

�(1�1{ )f(�)d�

35(1+�)

(1� 1{ )�

:

(A2)If we raise both sides of equation A2 by 1

1+�and plug the result in equation

A1, we get the following integral:

Z�R

(e�)f(�)d� =

��Ls

d

� 1�

24Z�R

�1+�� f(�)d�

351� 1�24Z�R

�(1� 1

{ )(1+�)� f(�)d�

35 1

(1� 1{ )�

:

(A3)At this stage of the proof it is clear that substitution of equations A2 and A3in equation (12) provides the equilibrium function ei which solves the secondstage of the game:

ei = �1�i

��Ls

d

� 1�

24Z�R

�1+�� f(�)d�

35� 1�24Z�R

�({�1)(1+�)

{� f(�)d�

35 {({�1)�

: (A4)

The rest of the proposition easily follows. In order to derive the equilib-rium function of the individual probability pi = p(IijI)S; we substitute Eqs.A3 and A4 in p(IijI) and get:

p(IijI) = �1+��i

24Z�R

�1+�� f(�)d�

35�1 :The product between this equation and Eq. A2 provides:

pi = �1�+1

i s

��Ls

d

� 1�

24Z�R

�1+�� f(�)d�

35�(1+ 1�) 24Z�R

�(1� 1

{ )(1+�)� f(�)d�

35(1+�)

(1� 1{ )�

:

The last function in the proposition can be obtained by substitution of Eq.A2 in S = s. �

Proof of Proposition 2.

34

The function umi;R(z) starts from the origin, and is monotone increasing,while uy is a positive constant. Hence, umi;R(z) crosses uy from below. Henceonly more able individuals undertake research activity - they will have ahigher relative pay-o¤ from it - while less able workers enter goods produc-tion. �

Proof of Proposition 3.Proposition 3 concerns the relation between the equilibrium functions e,

S and the parameter 1={. Indeed, it maintains the existence of an inverserelation between those functions and the parameter which represents thedegree of complementarity among agents.Let us consider equation (13):

ei =

��F (z)s�i

d

� 1��Z 1

z

�1+�� f(�)d�

�� 1��Z 1

z

�(1� 1

{ )(1+�)� f(�)d�

� 1

�(1� 1{ )

and focus on the last term of the product. To prove that�R 1z�(1� 1

{ )(1+�)� f(�)d�

� 1

�(1� 1{ )

decreases with 1={, we rely on Schl::omilch inequality. Indeed, it maintainsthat for a positive random variable X, the function

M (p) = E (Xp)1p

de�ned for all real p, increases with p when M (p) exists and is �nite. Aproof of this result can be found in Lange (2003), pp. 61-62 (Diewert, 2008provides an alternative proof of Schl

::omilch inequality and a discussion of

applications in economics).Now we apply Schl

::omilch inequality to derive the monotonicity of:�R 1

z�(1� 1

{ )(1+�)� f(�)d�

� 1

�(1� 1{ )with respect to 1

{ . In particular, we de�ne

the function of 1{ :

r �(1� 1

{ )(1 + �)

�� r

�1

{

�;

such that we can write the following:�Z 1

z

�(1� 1

{ )(1+�)� f(�)d�

� 1

�(1� 1{ )

=

�Z 1

z

�rf(�)d�

� (1+�)r�2

; (A5)

where the right side increases with r by Schl::omilch inequality.

Then, the sign of the derivative of A5 with respect to 1{ can be easily

obtained by applying the chain rule of di¤erentiation:

35

@

�R 1z�(1� 1

{ )(1+�)� f(�)d�

� 1

�(1� 1{ )

@( 1{ )=

(A6)

(1 + �)

�2

8<:�Z 1

z

�rf(�)d�

� 1r9=;(1+�)

�2�1

�@hR 1z�rf(�)d�

i 1r@r

�@r�1{

�@( 1{ )

< 0:

This result is clearly su¢ cient to prove the inverse relation of ei withrespect to 1

{ .Now let us prove the proposition for the function:

S(z) = s

��F (z)s

d

� 1��Z 1

z

�1+�� f(�)d�

�� 1��Z 1

z

�(1� 1

{ )(1+�))� f(�)d�

� (1+�)

�(1� 1{ )

.

A glance at this function reveals that to derive the sign of the derivative with

respect to 1={ we have to study the function�R 1

z�(1� 1

{ )(1+�))� f(�)d�

� (1+�)

�(1� 1{ ).

In this respect, we apply again the Schl::omilch inequality. Indeed, we can

write �Z 1

z

�(1� 1

{ )(1+�)� f(�)d�

� (1+�)

�(1� 1{ )

=

�Z 1

z

�rf(�)d�

� (1+�)2�2r

;

and take the derivative:

@

�R 1z�(1� 1

{ )(1+�)� f(�)d�

� (1+�)

�(1� 1{ )

@( 1{ )=

(A/)

(1 + �)2

�2

8<:�Z 1

z

�rf(�)d�

� 1r9=;(1+�)2

�2�1

�@hR 1z�rf(�)d�

i 1r@r

�@r�1{

�@( 1{ )

< 0:

Given A7, the decreasing monotonicity of the function S with respect to1{ easily derives, which completes the proof of Proposition 3. �Proof of Proposition 4.Since the size of the science sector 1�F (z) is a decreasing function of z,

Proposition 4 is proved if the function S(z) decreases with the variable z.

36

Case of high complementarity (1 < 1{ < +1):

In order to simplify the notation, we de�ne the following functions:

�(z) =

Z 1

z

�1+�� dF (�); (z) =

Z 1

z

�({�1)(1+�))

{� dF (�);

such that we can write:

S = s��sd

� 1�F (z)

1��(z)

�1� (z)

(1+�){�({�1) ;

With this notation we take the derivative of S(z):

@S

@z= S(z)

�1

�F (z)�1Fz(z)�

1

��(z)�1�z(z) +

(1 + �){�({ � 1)(z)

�1z(z)

�:

In the case 1{ > 1 we have

{({�1) < 0. Since Fz(z) = f(z) � 0, and

�z(z) = �hz1+�� f (z)

i� 0;

z(z) = �hz({�1)(1+�))

{� f (z)i� 0:

we have @S@z� 0:

In the case 1{ � 1; the parameter

{({�1) is positive and the derivative

@S@z

is made by the addition of two terms with positive sign to another term withnegative sign. Hence, its sign is not determinate.Case of a uniform distribution of talent and substitutability and low com-

plementarity (�1 < 1{ � 1):

In the case of uniform distribution, the equilibrium function S is givenby:

S = sh�zsd

i 1�

��

1 + 2�

�� 1� h1� z 1+2��

i� 1�

�1

b

� (1+�){�({�1) �

1� zb� (1+�){�({�1) .

where b = ({�1)(1+�)+{�{� :

By deriving the above expression with respect to z, we obtain the follow-ing expression:

@S

@z= Sz�1

1

�

�1 +

1 + 2�

�z1+2��

�1� z 1+2��

��1� {(1 + �)z

bb

({ � 1)�1� zb

��1�37

which provides the following condition concerning the sign

@S

@zR 0 () z

1+��{

�zb(1 + �)(2{ � 1)

{ � 1 + z�1+2�� +

(� + 1)

�

���1 +

(� + 1){b{ � 1

�R 0

The above condition is composed of two terms: one positive, that wedenote as P (z); and one negative constant, denoted by N . When there islow complementarity (i.e., 0 < 1

{ � 1), the function P (z) is U-shaped, with

limz!0P (z) = +1;

limz!1P (z) = �N .

This implies that there is a value of z, z = z such that P (z)+N = 0 and thederivative of S is null. These results imply that @S

@zis in a �rst trait positive,

then it is negative and becomes again zero in z = 1. From these results wecan derive the shape of the function S(R); where R = 1� z. In fact,

�@S (z)

@z> 0 if 0 < z < z

�=)

�@S (R)

@R< 0 if R < R < 1

�;�

@S (z)

@z< 0 if z < z < 1

�=)

�@S (R)

@R> 0 if 0 < R < R

�:

Hence, S(R) increases for R starting from the origin, then it reaches a max-imum at R = 1� z, and beyond R the function decreases.When the parameter 1

{ is negative, the positive term is U-shaped with

limz!0P (z) = +1;

limz!1P (z) = �N;

if the following condition holds: (3� 1

{ )( 1{ )

2� 1{

> 1+��2: This is satis�ed for values

of � su¢ ciently high. Then the function S has the same shape as the lowcomplementarity case.�Proof of Proposition 5.This proposition characterizes the existence and number of stable equi-

librium values of z under two di¤erent con�gurations of the parameter 1={.Here we base our proof on the analysis of the equilibrium condition:

�(1� �)� + 1

�d

� �1+�

=�sF (z)z

dR 1z�1+�� f(�)d�

�Z 1

z

�({�1)(1+�)

{� f(�)d�

� {{�1

(A8)

38

It is easy to verify that the left side is a positive constant, while the rightside is a complex function of z. The r.h.s. shows limit values at the twoextremes given by:

limz!0

�sF (z)z

dR 1z�1+�� f(�)d�

�Z 1

z

�({�1)(1+�)

{� f(�)d�

� {{�1

= 0

and:

limz!1

�sF (z)z

dR 1z�1+�� f(�)d�

�Z 1

z

�({�1)(1+�)

{� f(�)d�

� {{�1

=n1 if {

{�1<0

0 if {{�1>0

Hence, we know that the r.h.s. of the equilibrium condition always startsfrom the origin, but it may end at two opposite values. Indeed, it goes toin�nity when z ! 1 if 1 < 1={ < +1 (high complementarity of agents),while the r.h.s.. goes to 0 if -1 < 1={ � 1.Let us consider the �rst case and denote the r.h.s. of equation A8 as r(z).

The derivative of the r.h.s. with respect to z is:

@r(z):@z

= r(z)

�f(z)z+F (z)F (z)z

+ z1+�� f (z)

�R 1z�1+�� f(�)d�

��1� {

{�1

�R 1z�({�1)(1+�)

{� f(�)d���1�

>

0.Hence the r.h.s. of Eq. (A8) is monotone increasing, and it crosses only

once the l.h.s. at the equilibrium value of threshold z. Since an equilibrium zis stable if uy = umR (z; z), and u

mR (z; z) crosses uy from below, monotonicity

of r.h.s. ensures stability of the unique equilibrium value of z. The proof ofcase a) is complete.Under the case - 1 < 1={ � 1 we have {

{�1 > 0 such that the sign

of @r(z):@z

is indeterminate. Since the r.h.s. takes nonnegative values, it cancross the l.h.s. line more than once before reaching 0 at z = 1, and thesearguments complete the proof of part b) of Proposition 5.�Proof of Proposition 6.This proposition deals with the case of multiple equilibrium thresholds z

and maintains that if 1{ 2 (0; 1) then the economy can be characterized by

a stable equilibrium at z = 1, while the same equilibrium value is not stablein the case 1

{ < 0. In order to prove this proposition we concentrate on thestability of the equilibrium at z = 1. Let us rewrite the equilibrium equation(A8) as:

�(1� �)� + 1

�d

� �1+�Z 1

z

�1+�� f(�)d� =

�szR z0f(�)d�

d

�Z 1

z

�({�1)(1+�)

{� f(�)d�

� {{�1

39

and de�ne the left-hand side as C(z) and the right-hand side as D(z). It iseasy to verify that the condition for equilibrium stability translates into

D0(z) > C

0(z):

The derivatives of the functions D(z) and C(z) are:

D0(z) =

�s

d

�Z 1

z

�({�1)(1+�)

{� f(�)d�

� {{�1�Z z

0

f(�)d� + zf(z)

�+

+�s{

d(1� {)

�Z 1

z

�({�1)(1+�)

{� f(�)d�

� 1{�1

z({�1)(1+�)

{� f(z);

and

C0(z) = �

�(1� �)� + 1

�d

� �1+�

z1+�� f(z)

In order to verify the stability of the equilibrium z = 1 we take the limits ofderivatives at z = 1.Case 0<1={ � 1:

limz!1D

0(z) = 0.

Case (1={) < 0:limz!1D

0(z) = �1.

While in any case:

limz!1C

0(z) = �

�(1� �)� + 1

�d

� �1+�

f(1) < 0

Hence, under the case of low complementarity ( 1{ � 1 ) the stability condi-

tion D0(1) > C

0(1) is veri�ed, and under the case of substitutability 1

{ < 0

it is not: D0(1) < C

0(1). �

Proof of Proposition 7.This proposition summarizes the e¤ects of comparative statics of the

model.

Let us consider the equilibrium condition:

40

�(1� �)� + 1

�d

� �1+�

=�sF (z)z

dR 1z�1+�� dF (�)

Z 1

z

�(1� 1

{ )(1+�)� f(�)d�

! 1

(1� 1{ )

and de�ne the left hand side as A(�), and the right hand side as B(z; �),where � = 1={. Comparative statics implies:

@z

@�=

@A(�)@�

� @B(z;�)@�

@B(z;�)@z

:

Given that @B(z;�)@z

> 0 is a su¢ cient condition for the stability of anequilibrium threshold of talent z, we can concentrate on the derivatives ofthe two sides of the equilibrium condition with respect to the parameters:

@A(�)

@(�)= 0;

@B(z; �)

@�=

�sF (z)z

dR 1z�1+�� f(�)d�

@

��R 1z�(1��)(1+�)

� f(�)d�� 11���

@�:

The proof of Proposition 3 shows that by applying Schl::omilch inequality

to the function�R 1

z�(1��)(1+�)

� f(�)d�� 11��

it can be proved that the derivative@B(z;�)@�

is negative. It follows that @z@�> 0. �

41