the role of social capital in enhancing factor productivity: does its erosion depress per capita...

The Journal of Socio-Economics 37 (2008) 1539–1553

The role of social capital in enhancing factorproductivity: Does its erosion

depress per capita GDP?

Stefano Bartolini a, Luigi Bonatti b,∗a University of Siena, Italyb University of Trento, Italy

Received 9 May 2006; received in revised form 23 January 2007; accepted 15 March 2007

Abstract

We aim at reconciling Putnam’s claim that social capital has declined in the U.S. in the last decades withthe satisfactory growth performance of the U.S. economy over the same period. This puzzle originates fromthe fact that – according to most literature – social capital enhances factor productivity (mainly by reducingdefiant and opportunistic behavior). We model the hypotheses that the expansion of market activities weakenssocial capital formation, and that society reacts to the decline in social capital by spending more to protectproperty and enforce contracts. We show that this process may lead to a higher GDP level.© 2007 Elsevier Inc. All rights reserved.

JEL classification: O13; O41; Q20; Z13

Keywords: Transaction costs; Substitution process; Growth; Externalities; Defensive expenditures

1. Introduction

In accordance with a definition of social capital as those features of social organizations thatfacilitate coordination and cooperation (specifically, values and norms of reciprocity inhering inone’s social networks), Putnam (2000) uses three broad measures of social capital, namely theintensity of political participation, the density of voluntarily associational activity and survey onlevels of trust and mutual cooperation, to document the marked decline in social capital that hasoccurred in the United States in the last decades. As some critics have emphasized (Durlauf,

∗ Corresponding author.E-mail addresses: [email protected] (S. Bartolini), [email protected] (L. Bonatti).

1053-5357/$ – see front matter © 2007 Elsevier Inc. All rights reserved.doi:10.1016/j.socec.2007.03.005

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1540 S. Bartolini, L. Bonatti / The Journal of Socio-Economics 37 (2008) 1539–1553

2002b), Putnam explains the decline in the U.S. social capital by invoking a few large exogenousevents such as the rapid diffusion of the television or the passing of the World War II generation,rather than by attempting to formulate a consistent causal theory.

In this paper, we take for granted that the decline in the U.S. social capital is a stylized fact,1

and we note that – especially in the 1990s – this fall in social capital does not appear to havebeen paralleled by a decline in the U.S. potential for economic growth. This observation seemsat odds with Putnam’s statements that “norms and networks of civic engagement contribute toeconomic prosperity and are in turn reinforced by that prosperity” (1993, p. 180), and that socialcapital produces “aggregate economic growth” (2000, pp. 322–323).2 However, some strikingevidence presented by Putnam (2000) to support his claim that social trust has steadily declinedin the U.S. in the last decades, i.e. the evidence that documents the explosive increase in thesociety’s expenditures in formal activities of social control and dispute resolution, is consistentwith our hypothesis that points at the existence of a positive link between the decline in socialcapital and the increase in GDP. Indeed, one may argue that the erosion of social capital stimulatesthe rapid growth of entire sectors of the economy, which are the sectors providing those servicesthat are used to protect the economic agents against increasing opportunistic and defiant behaviorby others.3 In the same time, we claim that the progressive “marketization” of social life, theprocess through which market relationships become more pervasive, contributes to the diffusionof values, attitudes and behaviors that do not favor the formation of social capital. Therefore, onecannot take for granted that measures of social capital and of market activities move in the samedirection.

In spite of the U.S. stylized facts discussed above, cross-sectional studies using internationaldata appear to show the existence of a positive relation between social capital (generally mea-sured in terms of generalized trust and associational activity) and economic growth (Knack andKeefer, 1997; La Porta et al., 1997; Zak and Knack, 2001; Beugelsdijk et al., 2004; Beugelsdijk

1 The general thesis on the decline in social capital in the United States, documented by Skocpol (1999) and Putnam(2000), was already presented in Putnam (1995), raising a critical debate. Some researchers (see e.g., Ladd, 1996; Paxton,1999) contested Putnam’s conclusions. Subsequent studies tend to confirm the main Putnam’s claim (see e.g., Costa andKahn, 2003; Kolodinsky et al., 2003; Keele, 2004).

2 This point is stressed by Durlauf and Fafchamps (2004): “Putnam (2000), focusing on the U.S. experience since the1950s, argues that social capital, defined as membership in formal and informal clubs, has declined monotonically sincethe 1950s. This is true for all states, all decades and all measures of social capital. However, he finds no relationshipbetween the speed of the decline of social capital and economic performance across U.S. states or across time periods.Further, the relationship between social capital and socioeconomic outcomes is even harder to characterize when onelooks at subperiods. For example, the 1990s were a period of rapid economic growth in the U.S. yet it is also a periodof rapid decline in social capital, at least based on the sorts of measures he uses. To be clear, Putnam does attempt toassociate higher social capital with better socioeconomic outcomes, our point is that the relationship between the two forthe United States is even at first glance relatively complicated” (p. 12).

3 Putnam (2000) emphasizes that during the 1980s spending on security rose rapidly as a share of U.S. GDP. Moreover,he observes that by 1995 America had 40% more police and guards and 150% more lawyers and judges than would havebeen projected in 1970, even given the growth of population and the economy (see Putnam, 2000, p. 146). As a matter offact, in the last three decades of the 20th century, the U.S. tort costs (inclusive of the payments to compensate victims)grew at a compound annual rate of 9.1%, while during the same period U.S. population and nominal GDP grew at a rateof, respectively, 1.1% and 7.6% (see Tillinghast-Towers Perrin, 2003). Bowles and Jayadev (2006) present data showingthe significant secular increase in the resources devoted in the United States to the execution of contracts and defenceof property rights. In particular, they document that supervisors and guards (police, corrections officials and securitypersonnel) were 17.9% of the U.S. labor force in 2002, while the corresponding figure was 10.8% in 1966.

S. Bartolini, L. Bonatti / The Journal of Socio-Economics 37 (2008) 1539–1553 1541

and van Schaik, 2005).4 However, these studies have been subject to severe criticism (Durlauf,2002a; Durlauf and Fafchamps, 2004).5 Moreover, both Durlauf (2002b) and Bovenberg (2003)complained about the absence of theoretical models that define precisely the mechanisms throughwhich endogenous and exogenous variables interact and co-determine the time profile of socialcapital and of other indicators of economic performance.6 We contribute to fill this void by pre-senting a growth model linking social capital formation to the decisions by which economicagents determine their working time and accumulate physical capital, in a context where thepublic authorities devote resources to monitoring possible misconduct, protecting property andproviding security. A simplifying assumption of our model, indeed, is that those services neces-sary to protect the economic agents from opportunistic and defiant behavior are provided entirelyby the state.

Our analysis focuses on the long-term impact of social capital on the level of GDP rather thanon its growth rate, since – except for some relevant exogenous event – social capital tends to changevery slowly: as Dasgupta (2000) noticed, the difference between the total factor productivity oftwo countries may be due to their different endowment of social capital and remain roughly thesame for long periods of time, thus explaining the difference between their GDP levels, evenif one does not observe social capital in growth accounting. Our model allows us to show thateconomies where social capital tends to decline in response to increases in GDP may exhibithigher steady-state levels of per capita output than economies where social capital is positivelyaffected by increases in GDP. This result depends on the society’s tendency to substitute market-produced goods and services for the declining endowment of social capital, which may dominatethe depressing impact that its erosion has on factor productivity.

By recognizing that a larger endowment of social capital allows to use the existing resourcesmore efficiently, we accept the approach of the so-called “new economic sociology” (seeWoolcock, 1998), according to which the members of communities with high stocks of socialcapital tend to be more able to costlessly monitor one another’s behavior, reach informal under-standing and agreements, enforce contracts, resolve disputes amicably. In such communities, one

4 Consistently with these studies, Rupasingha et al. (2002) find evidence that higher levels of social capital – togetherwith other economic and social factors – have systematic and positive effects on growth rates among U.S. counties.However, the existence of a systematic relationship between social capital and economic performance across U.S. statesis questioned by Casey and Christ (2002): their regression analysis incorporates state-level measures of physical andhuman capital, showing that connection between social capital index measures and output indicators is either negative orstatistically insignificant.

5 Even admitting the existence of a positive cross-sectional relationship between social capital and income, other studiesfind evidence of a reverse causation going from economic growth to social capital, since they do not find that high initiallevels of social capital are good predictors of future economic development (see Miguel et al., 2001). One may argue thatalso the theoretical relationship between social capital and economic growth is ambiguous because of the “downside” ofsocial capital (see Olson, 1982; Portes and Landolt, 1996; Stolle and Rochon, 1998; Annen, 2003; Knack, 2003): strongand long-standing groups may hinder growth by their rent-seeking activities aiming at controlling a disproportionatelylarge share of domestic resources, or by placing heavy obligations on members that make more difficult for them to increasetheir economic opportunities by joining larger social networks. With regard to this point, one can fruitfully distinguishbetween “bridging” social capital – that is generated in networks spanning different communities and may be positivefor growth – and “bonding” social capital – that arises among close friends or members of the same family and that isgenerally negative for growth. On the relationship between economic growth and social capital see also Dasgupta (2000).

6 A recent theoretical model focusing on the relation between social capital and economic growth is Beugelsdijk andSmulders (2004), which accounts for the possible trade off between social capital formation and GDP growth. Indeed, theauthors assume that the participation in intercommunity networks (“bridging” social capital) enhances growth by reducingthe incentives for rent seeking and cheating, and it depresses growth by reducing the time devoted by people to marketactivities.


Table 1Countries differences in patterns and trends of market work time per 15–64 year olds (annual hours worked per person)

France West Germany Italy United States

1970a 1295 1127 1122 11692004 905 934 910 1299

Source: Tabulated from OECD, Employment Outlook Database and OECD Employment Outlook, 2005.a West German figure is for 1979.

would expect a low incidence of litigations, corruption, conflicts and crime. However, we alsoaccount for the fact that state intervention may be considered as a substitute for communities’social capital, since the latter is probably the less expensive but not the only possible way ofreducing the inefficiency caused by opportunistic and defiant behavior (Durlauf and Fafchamps,2004).

By admitting that the greatest danger to the social capital, which is so important for marketefficiency, arises from the market system itself, we refer to an idea that has a long history behind.Some authors went so far as to claim that capitalism contains within itself the mechanism of itsown destruction (see Hirsch, 1976; Hirschman, 1982): the decline of the values (honesty, businessethics, trust, . . .) that prevent the spread of the opportunism generated by a market society willend up by destroying the latter.7 In other words, they argued that even the survival of the marketsystem can be jeopardized by that progressive weakening of its cultural and ethical base which is aconsequence of its evolution and success, since the individualistic and competitive values systemconnected with the expansion of a market economy is the greatest threat to the efficient functioningof markets. Also the socio-economic transformations that according to Putnam (2000) may havenegatively affected the formation of social capital in the United States in the last decades canbe considered a by-product of the process of marketization. Indeed, he identifies some possibledeterminants of the decline in the U.S. social capital in the rising female participation in the labormarket, in the increase in geographical mobility, in “the replacement of the corner grocery by thesupermarket” and in the “privatizing” or “individualizing” of the leisure time (mainly due to theTV and more in general to the diffusion of home-entertainment technologies). In particular, thedocumented increase in per capita hours of market work that has occurred in the United Statesin the last 30 years – in contrast with almost all other industrialized countries (see Table 1) –is probably important for explaining both the decrease in social capital and the acceleration ofgrowth that has characterized the U.S. economy in recent years. With this regard, it is significantthat in the model presented here the erosion of social capital may lead to a higher steady-statelevel of per capita working time.

Finally, it is worth to note that our model also admits an environmental interpretation of theasset enhancing factor productivity, which can be viewed as natural capital whose quality tendsto be lowered by the expansion of production. With this regard, one could be surprised by thefact that the degradation of natural assets may have a net positive effect on the level of marketactivities. Indeed, although it is well known that – as far as man-made goods are close substitutes

7 Fukuyama (1995) fully embraces the idea that capitalism tends to erode social capital but offers an optimistic viewof its ability to regenerate that capital. The perception that there is a conflict between a development strategy advocatinga stronger role for social capital and an agenda emphasizing market incentives and material values are present also inthe current policy debates (see e.g., Heyer et al., 2002). In the same time, the idea that any development process bringsdestruction of social capital has been recently challenged by studies focusing on specific episodes and experiences (seee.g., Miguel et al., 2001).


for the declining quality of environmental assets – the erosion of the stock of natural capital doesnot constitute an impassable limit to output growth, it is relatively new the notion that a moredetrimental impact of market activities on renewable resources may even boost the steady-statelevel of market production.

The paper is organized as follows: Section 2 presents the model, Section 3 characterizes theequilibrium trajectories of the economy and Section 4 concludes.

2. The model

We consider an economy in discrete time with an infinite time horizon. In this economy thereare firms, households and the government.

2.1. The firms

For simplicity and without loss of generality, it is assumed that there are a fixed and largenumber (normalized to be one) of perfectly competitive firms that are identical and produce thesingle good existing in this economy. The representative firm produces its output Yt according tothe technology

Yt = Lαt K1−α

t Aβt , 0 < α < 1, 0 < β < 1, (1)

where Lt is the labor, Kt the physical capital and At is the variable affecting factor productivity.Note that Yt is the numeraire of the system and that its price is set to be one.

The variable affecting factor productivity is given by

At = Rt + φXt, φ > 0, (2)

where Rt is the stock of social capital existing in the economy in period t and Xt is the measureof the resources that the government devotes to the enforcement of contracts and the defense ofproperty rights. Following Bowles and Jayadev (2006), we may interpret Rt as a measure of thelevel of trust, work ethics, honesty, effective protection from confiscation and the like existingin the society at time t. In Eq. (2) it is assumed that a higher level of social capital raises totalfactor productivity and that Xt can be used as a (perfect) substitute for Rt, where the efficiencyof Xt as a substitute for Rt is measured by the parameter φ: if φ > 1 (φ < 1), the effect on At ofa unit loss of social capital is more (less) than offset by a unit rise in the public resources spentfor enforcing contacts and protecting property. In other words, one may think that a large (small)value of φ captures a situation in which a decline in peer monitoring and informal sanctioning canbe counterbalanced by a modest (massive) increase in the amount of resources invested in formalmechanisms of social control and dispute resolution.

If one interprets Rt as an environmental asset, one may suppose that an increasing amount ofpublic resources has to be devoted to preserve factor productivity as environmental quality worsens(for instance, more public works and defensive expenditures are needed to protect productivefacilities as global warming accelerates, more medical care is needed to preserve labor productivityas air quality deteriorates).

In each t, the representative firm chooses Lt and Kt in order to maximize its profits, which aregiven by

πt = Yt − wtLt − rtKt, (3)

where rt is the capital rental rate and wt is the wage rate.


2.2. The law of motion of the social capital

Across the social sciences, a recurring hypothesis is that the expansion of market activities mayundermine the society’s ability to regenerate its social assets. According to this thesis, the levelof generalized trust, civic engagement, public ethics and personal honesty may suffer because ofthe increased “marketization” of social life, which brings about as a by-product the diffusion ofattitudes and values like greed, cynicism and opportunism. However, in modeling the dynamicsof the social capital, we do not rule the opposite hypothesis that the expansion of market activitieshas a positive impact of the formation of social capital. In both cases, we use the volume of goodsand services that are produced for monetized exchange by profit-maximizing firms as a proxy ofthe degree of marketization. Hence, we may model the evolution in time of the stock of socialcapital as follows:

Rt+1 = γRt + S − ηYt, 0 < γ < 1, S > 0, η≥<0, Rt ≥ 0 ∀t, R0 given, (4)

where η > 0 captures the hypothesis that social capital is negatively affected by increases in GDPand η < 0 is consistent with the opposite hypothesis. It is worth to emphasize that Yt is the aggregatemarket output in period t. Therefore, a single firm has only a negligible effect on the evolutionof Rt. Indeed, the effect of the producers’ activities on the future state of Rt is significant onlybecause of the large number of firms populating the economy. Thus, each single firm acting infull autonomy can ignore the impact of its productive activity on the future stock of social capital.

If one interprets Rt as an environmental asset affecting productivity, Eq. (4) models the neg-ative effect of production on the nature’s absorption capacity (η > 0), namely on its capacity ofpreserving a certain level of environmental quality. Consistently with this interpretation, ηYt mayrepresent the pollution generated by the total production taking place in t and affecting Rt+1 (η > 0is a parameter capturing the “dirtiness” of the technology). Moreover, S − (1 − γ)Rt is nature’sabsorption capacity, that is, the amount of pollution that can be assimilated without a change inenvironmental quality (see Smulders, 2000). A high level of environmental quality can be pre-served either if the level of production is low (small Yt) or if the technology is “clean” (η close tozero).

2.3. The households

For simplicity and without loss of generality, it is assumed that the population is constantand that each household contains one adult, working member of the current generation. Thus,there are a fixed and large number (normalized to be one) of identical adults who take accountof the welfare and resources of their actual and prospective descendants. Following Barro andSala-i-Martin (1995) we model this intergenerational interaction by imagining that the currentgeneration maximizes utility and incorporates a budget constraint over an infinite future. Thatis, although individuals have finite lives, we consider immortal extended families (“dynasties”).8

Finally, we assume that agents’ expectations are rational, in the sense that they are consistentwith the real processes followed by the relevant variables. In this framework, in which there is nosource of random disturbances, this implies perfect foresight.

8 As Barro and Sala-i-Martin (1995, p. 60) point out, “this setting is appropriate if altruistic parents provide transfers totheir children, who give in turn to their children, and so on. The immortal family corresponds to finite-lived individualswho are connected via a pattern of operative intergenerational transfers that are based on altruism”.


In each period t, the utility of the representative household is an increasing function of con-sumption and leisure:

Ut = ln(Ct) + ϕ ln(1 − Lt), ϕ > 0, Lt ≤ 1, (5)

where Ct is consumption in t and 1 − Lt is the time devoted to leisure by the representativehousehold (the total amount of time available to the household in each t is normalized to be one).

The period budget constraint of the representative household is the following:

It + Ct = πt + rtKt + wtLt − Tt, (6)

where It is gross investment in physical capital by the representative household and Tt are the taxesthat each household must pay to the government. It is assumed that each household is entitled toreceive an equal share of the (possible) firms’ profits.

The stock of physical capital evolves according to

Kt+1 = It + Kt(1 − δ), 0 < δ < 1, K0 given, (7)

where δ is a parameter capturing capital depreciation.The problem of the representative household amounts to choose {It}∞0 , {Ct}∞0 and {Lt}∞0 in

order to maximize∞∑t=0

θtUt, 0 < θ < 1, (8)

subject to Eq. (6), where θ is a time-preference parameter.

2.4. The government

We assume that the government is benevolent: its scope is to maximize (8) by choosing theamount of public good Xt to provide in each t subject to a balanced budget constraint:

Xt = Tt. (9)

2.5. Market-clearing conditions

Equilibrium in the market for the product implies:

Kt+1 + Ct + Xt = Kt(1 − δ) + Yt. (10)

Equilibrium in the markets for labor and for physical capital implies, respectively

Ldt = Ls

t (11)

and

Kdt = Ks

t . (12)

3. Characterization of the equilibrium trajectories of the economy

The social capital is a common, in the sense that it can be considered a productive asset towhich all producers have free access, with a public good as its substitute in the production function.


Within this framework, we consider two possible cases: the first deals with the situation in whichthe government’s decision process ignores the impact of production on the evolution of Rt, whilein the second case this impact is fully internalized.

3.1. Case 1: no internalization of production’s impact on the evolution of Rt

This case occurs typically when the policy maker does not take into account that social capitalis a resource connected with group membership and social networks (see Bourdieu, 1986), whichtends to deteriorate as market activities become more pervasive (assuming that η > 0). In otherwords, the policy-maker’s decision process does not internalize the negative effect of the progres-sive marketization on social cohesion and general trust. Therefore, society’s faces the tendency ofsocial capital to decline only by raising its expenditure aimed at protecting people and propertiesfrom increased opportunism and defiant behavior.

In case 1, the equilibrium path of the market economy can be characterized by maximizing

∞∑i=0

θi{ln[Lαt+iK

1−αt+i (Rt+i + φXt+i)

β − It+i − Xt+i] + ϕ ln(1 − Lt+i)

+ λt+i[It+i + (1 − δ)Kt+i − Kt+i+1]} (13)

with respect to It, Lt, Xt, Kt+1 and λt, where λt is a Lagrangean multiplier and the motion of Rt

is governed by Eq. (4).9 From this maximization and from Eq. (4) one can derive the system ofdifference equations in Rt, It and Xt governing the motion of the economy (see Appendix A):

Ω(Rt+1, It+1, Xt+1, Rt, It, Xt)

= θ

[(((1 − α)(Rt+1 + φXt+1))/k(Rt+1, It+1, Xt+1)) + (1 − δ)βφ

Rt+1 + φ(1 − β)Xt+1 − βφIt+1

]

− βφ

Rt + φ(1 − β)Xt − βφIt

= 0, (14)

Ψ (Rt+1, It+1, Xt+1, Rt, It, Xt) = k(Rt+1, It+1, Xt+1) − It − (1 − δ)k(Rt, It, Xt) = 0,

(15)

Λ(Rt+1, Rt, Xt) = Rt+1 − S − γRt + η

βφ(Rt + φXt) = 0, (16)

where k(Rt, It, Xt) = Kt = {(Rt + φXt)1−β/βφ[l(Rt, It, Xt)]α}1/(1−α) and l(Rt, It,Xt) = Lt = α(Rt + φXt)/[(α + ϕ)(Rt + φXt) − ϕβφ(It + Xt)].10

By setting Rt+1 = Rt = R, It+1 = It = I and Xt+1 = Xt = X in Eqs. (14)–(16), one can characterizethe steady state of the economy for the case in which the production’s impact on Rt is ignored by

9 Equivalently, one can characterize the equilibrium trajectory of the economy by decentralizing the solution, namelyby solving the maximization problems of firms and households for a given path of Xt, and then by solving the optimizationproblem of the policy maker, thus determining the optimal path of Xt.10 We consider equilibrium paths consistent with Xt > 0 ∀t (along an equilibrium path, the policy maker’s problem of

choosing Xt has always an interior solution).


the decision makers:

R = f (X, η) = φ(Sβ − ηX)

βφ(1 − γ) + η, (17)

I = g(X, η) = δθ(1 − α)[f (X, η) + φX]

βφ[1 − θ(1 − δ)], (18)

where X must satisfy the following equation:

ω(X, η) = α[f (X, η) + φX]β/α

ϕ

[θ(1 − α)

1 − θ(1 − δ)

](1−α)/α

− [f (X, η) + φX](ϕ + α)

βφϕ

+X + g(X, η) = 0. (19)

One can check that there may exist at most two steady-state triple (R, I, X) such that X > 0.11

Moreover, at steady state one has also:

Y = q(X, η) = S + φ(1 − γ)X

βφ(1 − γ) + η. (20)

By comparing Eqs. (17) and (20), one can see that at steady state a larger amount of X isaccompanied – other things being equal – by both a lower R and a higher Y. Moreover, one maywonder whether steady-state output increases or decreases as market activities tend to have a moredetrimental (less beneficial) impact on the future endowment of R, namely as η becomes larger.Indeed, it is apparent in Eq. (20) that an increase in η has a direct negative effect on Y, reflectingthe reduced productivity due to the decline in R. However, it may be the case that this effect isdominated by the society’s tendency to counterbalance this decline by augmenting the productionof Y devoted to substitute for the declining R, namely by increasing X:

Proposition 1. Without internalization of the effects of market production on the evolutionof a resource which positively affects productivity, steady-state output may be higher as marketproduction has stronger negative (weaker positive) effects on the future endowment of the resource.

Proof. Proposition 1 holds if for some admissible set of parameter values one has

∂Y

∂η= − S + φ(1 − γ)X

[βφ(1 − γ) + η]2 + φ(1 − γ)

βφ(1 − γ) + η

∂X

∂η> 0, (21)

where ∂X/∂η = −ωη/ωX

∣∣ω(X,η)=0. By computing the inequality (21), one can check that it holds

if and only if

ωX|ω(X,η)=0 < 0. (22)

Numerical examples can be displayed for which the inequality (22) is true.12

11 More precisely, one may have only one value of X > 0 satisfying ω(X, η) = 0 if β ≥ α, and at most two if β < α. Indeed,with β ≥ α, (i) ωXX ≥ 0 and (ii) ω(X, η) ≥ 0 at X = 0 entails ωX > 0 at X = 0. Thus, with β ≥ α, one cannot have any valueof X > 0 satisfying ω(X, η) = 0 when the parameter values are such that ω(X, η) ≥ 0 at X = 0, and one may have only onevalue of X > 0 satisfying ω(X, η) = 0 when ω(X, η) < 0 at X = 0. In contrast, with β < α, one has ωXX < 0. Thus, with β < α,one may have only one value of X > 0 satisfying ω(X, η) = 0 when the parameter values are such that ω(X, η) ≥ 0 at X = 0,and at most two when ω(X, η) < 0 at X = 0.12 For instance, the inequality (22) holds for φ = ϕ = 1, α = 0.6, β = 0.3, δ = γ = 0.05, θ = 0.98, S = 0.1 and η = 0.


More in general, a necessary condition for the inequality (22) to hold is β < α: only if thepartial elasticity of output with respect to A is strictly lower than the partial elasticity of outputwith respect to labor, it is possible to have ∂Y/∂η > 0. In other words, the output must be moresensitive to the increase in labor whereby households tend to react facing a reduction in R than tothe reduction in R itself.

An implication of Proposition 1 is that under certain conditions, the long-run level of marketproduction and the long-run endowment of social (or environmental) assets are linked by a negativerelationship: changes or events that tend to depress the steady-state level of R may end up boostingeconomic activity (∂R/∂η < 0 and ∂Y/∂η > 0). The reason is to be found in the increased amountof output devoted by society to prevent factor productivity from decreasing in the face of alower level of R. Another implication of Proposition 1 is that – ceteris paribus – economieswhere social capital tends to decline in response to increases in GDP (i.e., economies whose η

is strictly positive) may exhibit higher steady-state levels of per capita output than economieswhere social capital is positively affected by increases in GDP (i.e., economies whose η is strictlynegative).

3.2. Case 2: full internalization of production’s impact on the evolution of Rt

This case applies to a situation where the policy maker is aware of the impact of marketactivities on the formation of social capital and has the policy instruments whereby to induce theprivate agents to fully internalize the externalities generated by their activities.

In this case, one can solve the decision-maker’s problem by maximizing

∞∑i=0

θi{ln[Lαt+iK


β − It+i − Xt+i] + ϕ ln(1 − Lt+i)

+λt+i[It+i + (1 − δ)Kt+i − Kt+i+1] + μt+i[S + γRt+i − ηLαt+iK


β

−Rt+i+1]}, (23)

with respect to It, Lt, Xt, Kt+1, Rt+1, λt and μt, where μt is a multiplier that measures the marginalincrement in discounted utility due to a marginal increment in the stock of Rt along an optimalpath.13 From this maximization one can derive the conditions that an optimal path must satisfy(see Appendix A), which can be used to obtain the system of difference equations in Rt, Kt, Lt

and Xt governing the motion of the economy:

Γ (Rt+1, Kt+1, Xt+1, Lt+1, Rt, Kt, Xt, Lt)

= θ(1 − α)(Rt+1 + φXt+1)

βφKt+1+ θ(1 − δ)

−[

K1−αt+1 Lα

t+1(Rt+1 + φXt+1)β − Xt+1 − i(Rt+1, Kt+1, Xt+1, Lt+1)

K1−αt Lα

t (Rt + φXt)β − Xt − i(Rt, Kt, Xt, Lt)

]= 0, (24)

13 Again, one can equivalently characterize the equilibrium trajectory of the economy by decentralizing the solution,namely by solving the maximization problems of firms and households for a given path of Xt, and then by solving theoptimization problem of the policy maker, thus determining the optimal path of Xt.


Σ(Rt+1, Kt+1, Xt+1, Lt+1, Rt, Kt, Xt, Lt)

= K1−αt Lα

t

(Rt + φXt)1−β

[θ(γφ + η)β − θγ(Rt+1 + φXt+1)1−β

K1−αt+1 Lα

t+1

]

−[

K1−αt+1 Lα

t+1(Rt+1 + φXt+1)β − Xt+1 − i(Rt+1, Kt+1, Xt+1, Lt+1)

K1−αt Lα

t (Rt + φXt)β − Xt − i(Rt, Kt, Xt, Lt)

]

×[

βφK1−αt Lα

t

(Rt + φXt)1−β− 1

]= 0, (25)

Θ(Kt+1, Rt, Kt, Xt, Lt) = Kt+1 − i(Rt, Kt, Xt, Lt) − (1 − δ)Kt = 0, (26)

Ξ(Rt+1, Rt, Kt, Xt, Lt) = Rt+1 − S − γRt + ηK1−αt Lα

t (Rt + φXt)β = 0, (27)

where i(Rt, Kt, Xt, Lt) = It = K1−αt Lα

t (Rt + φXt)β − Xt − ((α(Rt + φXt)(1 −Lt))/ϕφβLt).14

By setting Rt+1 = Rt = R, Kt+1 = Kt = K, Xt+1 = Xt = X and Lt+1 = Lt = L in Eqs. (24)–(27), one cancharacterize the steady-state of the economy for the case in which production’s impact on Rt isfully internalized in the decision-making process of the policy maker:

R = h(X, η) = Sβ[φ(1 − γθ) − ηθ] − ηφ(1 − γθ)X

β(1 − γ)[φ(1 − γθ) − ηθ] + η(1 − γθ), (28)

K = n(X, η) = θ(1 − α)[h(X, η) + φX]

βφ[1 − θ(1 − δ)], (29)

L = p(X, η) = n(X, η)

[[1 − θ(1 − δ)]φ(1 − γθ)

θ(1 − α)[h(X, η) + φX]β[φ(1 − γθ) − ηθ]

]1/α

, (30)

where X must satisfy the following equation:

σ(X, η) = α[h(X, η) + φX]β/α

ϕ

[θ(1 − α)

1 − θ(1 − δ)

](1−α)/α [1 − ηθ

φ(1 − γθ)

]− [h(X, η)+φX]

βφϕ

×[

ϕφ(1 − γθ)

[φ(1 − γθ) − ηθ]+ α

]+ X + δn(X, η) = 0. (31)

As in the previous case, one can easily check that there may exist at most two steady-states.Moreover, at steady state one has also:

Y = m(X, η) = [S + φ(1 − γ)X](1 − γθ)

βφ(1 − γ)(1 − γθ) + η[1 − γθ + βγθ − βθ]. (32)

14 Again, we consider optimal paths consistent with Xt > 0 ∀t (along an optimal path, the problem of choosing Xt hasalways an interior solution).


3.3. A numerical example

Considering a limiting situation in which production has neither a positive impact nor a neg-ative impact on the motion of Rt (η = 0), it is straightforward that Eqs. (17)–(20) and (28)–(32)characterize the same steady-state: R* = R◦, K* = K◦, L* = L◦ and X* = X◦, entailing Y* = Y◦,where “*” and “◦” denote, respectively, the steady-state value of a variable in the absence ofinternalization of production’s impact on the motion of Rt and under full internalization. Inparticular, let φ = ϕ = 1, α = 0.6, β = 0.3, δ = γ = 0.05, θ = 0.98, S = 0.1 and η = 0. Given theseparameter values, one obtains: R* = R◦ ≈ 0.10526, K* = K◦ ≈ 4.7763212, L* = L◦ ≈ 0.5257867and X* = X◦ ≈ 0.1469559, entailing Y* = Y◦ ≈ 0.84073 and C* = C◦ ≈ 0.454958. One can checkthat in the neighborhood of this steady state the economy is saddle-path stable.15 Onecan also check that ∂K∗/∂η|η=0 > 0, ∂L∗/∂η|η=0 > 0, ∂X∗/∂η|η=0 > 0, ∂Y∗/∂η|η=0 >

0, ∂C*/∂η < 0 and ∂U*/∂η < 0; while ∂K◦/∂η|η=0 < 0, ∂L◦/∂η|η=0 < 0, ∂X◦/∂η|η=0 < 0,∂Y◦/∂η|η=0 < 0, ∂C◦/∂η < 0 and ∂U◦/∂η < 0. For instance, by setting η = 0.01 (negative impact),one obtains R* ≈ 0.0962 < R◦ ≈ 0.0967, K* ≈ 4.891 > K◦ ≈ 4.57, L* ≈ 0.532 > L◦ ≈ 0.5233 andX* ≈ 0.162 > X◦ ≈ 0.14466, entailing Y* ≈ 0.86 > Y◦ ≈ 0.8129 and C* ≈ 0.454295 > C◦ ≈ 0.4397;while by setting η = −0.01 (positive impact), one obtains R* ≈ 0.1139 < R◦ ≈ 0.1144,K* ≈ 4.665 < K◦ ≈ 4.986, L* ≈ 0.5196 < L◦ ≈ 0.5281 and X* ≈ 0.1324 < X◦ ≈ 0.1489, entailingY* ≈ 0.821178 < Y◦ ≈ 0.868768 and C* ≈ 0.455469 < C◦ ≈ 0.4705.

From this example, one can draw this conclusion:

Proposition 2. As private production generates negative (positive) externalities affecting theevolution of a resource which positively influences productivity, the shift toward an institutionalframework implying the full internalization of these externalities may reduce (raise) the long-termlevels of output and working time.

4. Conclusions

In this paper we have insisted on an interpretation of social capital as a resource connected withgroup membership and social networks (Bourdieu, 1986) which tends to deteriorate as marketactivities become more pervasive. The deterioration of this resource can be interpreted as a declinein social cohesion and general trust that forces the public authorities to raise the expenditures aimedat protecting the economic agents from increased opportunism and defiant behavior. To shed lighton the dynamics of an economy where social capital has these characteristics, we have augmenteda Solow–Ramsey growth model by including: (i) social capital enhancing factor productivity, (ii)negative externalities affecting social capital formation and increasing with the level of economicactivity, (iii) the possibility for the policy makers to substitute public goods for social capital, and(iv) a labor-leisure choice.

Within this framework, it is shown that (i) as the expansion of market activities has a strongernegative impact on the formation of social capital, the steady-state level of GDP may be higher, (ii)economies where social capital tends to decline in response to increases in GDP may exhibit higher

15 With η = 0, the system of difference Eqs. (14)–(16) can be split into two autonomous subsystems (14)–(15) and (16). Inparticular, one can see immediately from Eq. (16) that Rt converges monotonically to R* = R◦ = S/(1 − γ) ≈ 0.10526, since0 < γ < 1. The characteristic equation of the system obtained by linearizing Eqs. (14) and (15) around (I* = I◦, X* = X◦,R* = R◦) is the following: χ2 − 2.0239451χ + 1.0204079 = 0, which can be solved for the characteristic roots χ1 ≈ 0.9513and χ2 ≈ 1.0726, implying saddle-path stability.


steady-state levels of per capita output than economies where social capital is positively affectedby increases in GDP, and (iii) as private production generates negative externalities affecting theevolution of a resource which positively influences productivity, the shift toward an institutionalframework implying the full internalization of these externalities may reduce the steady-statelevels of output and working time.

The harmful impact of the destruction of social capital reduces individual welfare but notnecessarily the steady-state level of GDP, which may be enhanced. As social capital declines, alarger share of GDP is used to prevent the erosion of social capital from seriously lowering factorproductivity. Following Weitzman (1976), the cost of maintaining total factor productivity associal and environmental assets are eroded should be subtracted from gross output in calculatinga welfare relevant concept of net national product (see Bowles and Jayadev, 2006).

The model presented here is consistent with the view that capitalism tends to erode the socio-cultural sediment on which it rests, but it does not share the view that this erosion imperilsnecessarily the growth prospects of the economy. However, it should be noticed that our model isindependent of the answer that one may give to the question concerning the effects of the expansionof market activities on social capital. Indeed, the model can be applied both to a situation in whichthe formation of social capital is negatively affected by increases in output and to a situation inwhich it is positively affected by such increases, since the mechanism of substitution that is criticalfor our results works in both directions. In any case, we need systematic empirical evidence inorder to measure the impact of the expansion of market activities on social capital formation andto assess the effects of changes in social capital on the society’s demand for the services that cansubstitute for it.

Appendix A

A.1. Case 1: conditions to be satisfied along an optimal path

By maximizing (13) with respect to It, Lt, Xt, Kt+1 and λt, one can derive the optimalityconditions

1

Lαt K1−α

t (Rt + φXt)β − It − Xt

− λt = 0, (A1a)

αLα−1t K1−α

t (Rt + φXt)β

Lαt K1−α


− ϕ

(1 − Lt)= 0, (A1b)

βφLαt K1−α

t (Rt + φXt)β−1 − 1

Lαt K1−α


= 0, (A1c)

θ(1 − α)Lαt+1K

−αt+1(Rt+1 + φXt+1)β

Lαt+1K

1−αt+1 (Rt+1 + φXt+1)β − It+1 − Xt+1

+ θ(1 − δ)λt+1 − λt = 0 (A1d)

and

It + (1 − δ)Kt − Kt+1 = 0. (A1e)

Moreover, an optimal path must satisfy the transversality condition:

limt→∞θtλtKt = 0. (A1f)


A.2. Case 2: conditions to be satisfied along an optimal path

By maximizing (23) with respect to It, Lt, Xt, Kt+1, Rt+1, λt and μt, one can derive the conditions(A1a), (A1e),

αLα−1t K1−α

t (Rt + φXt)β

Lαt K1−α


− ϕ

(1 − Lt)− ημtαLα−1

t K1−αt (Rt + φXt)

β = 0, (A2a)

βφLαt K1−α

t (Rt + φXt)β−1 − 1

Lαt K1−α


− ημtβφLαt K1−α

t (Rt + φXt)β−1 = 0, (A2b)

θ(1 − α)Lαt+1K

−αt+1(Rt+1 + φXt+1)β

Lαt+1K

1−αt+1 (Rt+1 + φXt+1)β − It+1 − Xt+1

+ θ(1 − δ)λt+1 − λt − ημt+1θ(1 − α)

×(

Lt+1

Kt+1

)α

(Rt+1 + φXt+1)β = 0, (A2c)

θβLαt+1K

1−αt+1 (Rt+1+φXt+1)β−1

Lαt+1K

1−αt+1 (Rt+1+φXt+1)β − It+1 − Xt+1

+θμt+1[γ − βηLαt+1K

1−αt+1 (Rt+1 + φXt+1)β−1]

−μt = 0 (A2d)

and

S + γRt − ηLαt K1−α

t (Rt + φXt)β − Rt+1 = 0. (A2e)

Moreover, an optimal path must satisfy the transversality conditions (A1f) and

limt→∞θtμtRt = 0. (A2f)

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