Social Capital and Investment in R&D: New Externalities

We introduce social capital in an endogenous growth model with physical capital, human capital and varieties, and we compare the market with the ef-ﬁcient solutions. As social capital is not tradable in the market and it favours research networks, it introduces new externalities in this framework. These externalities induce the market to invest less in social capital than a social planner would do and decrease the tendency to underinvestment in R&D. We quantify the distortions in the model. In some conditions, the new distortions are strong enough to overcome the usual result of underinvestment in R&D.


Introduction
There are several recent articles that evaluate distortions in the decentralized market economy in models of endogenous economic growth. These models evaluate the differences that arise between the decentralized markets and the social planner choices due to several types of externalities: spillovers in the R&D process, duplications in the R&D process, gains from specialization due to new varieties, and erosion effects from R&D into human capital accumulation. However, social capital has not been considered in these models, despite its increasing interest in the economic growth literature. Social capital can introduce distortions in market allocations mainly due to two features of social capital: the failure of a market for social capital and the impact it can have in R&D due to research networks. The first reason is justified as firms do not pay for social capital when they contract workers; they pay for hours of work and at most for the level of qualifications. This may happen because the features usually included in social capital (confidence, truth, informal networks) are more difficult to evaluate and monitorize than degrees or years of schooling. The second reason is based on the importance of social networks between researchers in R&D productivity. Social networks can help knowledge sharing between researchers in an informal way, simply by proximity at work, "cheap talk" at lunch, etc. An example usually given is the importance of networks of researchers in Silicon Valley. Another can be the proximity of research staff in universities.
Social capital is a sociological concept that has been introduced recently in the economic growth literature. The definition of Putnam [20] refers to this concept as "features of social organization, such as trust, norms, and networks, that can improve the efficiency of society by facilitating coordinated actions". Most of the empirical literature has found a positive influence of social capital on economic growth, although it varies substantially (examples include Knack and Keefer [17]; Temple and Johnson [29], Whiteley [31], and Rupasingha et al. [24]). The introduction of social capital in growth models is still very scarce, but a good example is Beugelsdijk and Smulders [5], who also test the model against empirical data using the European Values Survey. Economic agents like to socialize (bonding), which they do by losing consumption, since participation in social networks is time-consuming and reduces time available to work.
Hence, higher levels of social capital may decrease economic growth. However, partic-ipation in community networks (bridging) reduces the incentive for rent seeking and cheating, and so through this channel, higher level of social capital produces positive effects on economic growth.
The positive connection between social capital and human capital accumulation was first described in Coleman [6] and in Teachman et al. [28] in sociological studies about high school dropouts. Grafton et al. [10] test a theoretical growth model against empirical data to explain international country differences in productivity and find a positive impact of people's knowledge connections on productivity. Dinda [8] uses an AK-type growth model to study the role of social capital in the production of human capital and in economic growth and compares theoretical results with empirical results finding a positive effect of social capital. Sequeira and Ferreira-Lopes [25] in an endogenous growth model framework also study the interactions between human and social capital and document the decline in social capital found by Putnam [21].
Literature about the connections of social capital, R&D, and economic growth is also very recent, scarce, and empirical. For example, Landry et al. [15], De Clercq and Dakhli [7] and Lee at al. [16] test empirically if there is indeed a connection and find positive effects of social capital in R&D and in innovation activities, although estimates vary widely.
No previous attempt that we know off brought the positive connection between social capital and R&D to an endogenous growth model. Our main contribution to the literature is to evaluate for the first time the impact of externalities caused by the presence of social capital in an endogenous growth model. Specifically, we want to determine the influence of social capital in the distortions between the decentralized equilibrium and the efficient solution. As already mentioned, social capital can introduce two types of new distortions in the market economy: first, because there is not a market for social capital, the social planner may choose higher levels of social capital than decentralized agents; second, because social capital influence R&D through research networks, which R&D firms do not consider, the social planner may also choose higher levels of social capital than decentralized agents. This is an important issue as opens the possibility of designing policies to enhance accumulation of social capital.
We also want to contribute to the discussion on "Too Much of a Good Thing?" or on the optimality of R&D investments. 1 Thus, this paper is also inserted in the literature on the macroeconomic efficiency of R&D investments within endogenous growth models without scale effects, whose first contributions were Jones [12] and Jones and Williams [14]. The most common result in the literature tends to indicate that underinvestment in R&D happens in the real world (Romer [23]; Grossman and Helpman [11], and Aghion and Howitt [1] in endogenous growth models with scale effects and also Jones[12] and Jones and Williams [14] in models without scale effects, and Jones and Williams [13] in an empirical article). The exceptions are Stockey [26] and Benassy [4] that, in models with scale effects, discovered that for more general preferences or production, overinvestment in R&D can occur. Most recently, Reis and Sequeira [22] and Strulik [27] showed that overinvestment in R&D can be more plausible than has been thought before.
We build an increasing varieties model with different production sectors. A model with increasing varieties is appropriate as it always predicts more overinvestment in R&D than quality-ladders models or those that combine both increasing variety and quality-ladders. We argue that in this type of model the presence of social capital decreases the scope for underinvestment.
Section two presents the model and sections three and four present, respectively, optimal growth and the decentralized equilibrium. Section 5 compares the shares of human capital allocation in the social planner and in the decentralized equilibrium and discusses distortions in the decentralized equilibrium. In Section 6 we implement a calibration exercise to answer the question how much social capital influences the distortions between the efficient and the decentralized solution. Finally in section 7 we conclude.

Model
In this model we combine different types of capital: physical capital, human capital, social capital, and technological capital. Physical capital is used in the production of 1 When applied to the economics of investment in R&D this expression has been first used by Jones and Williams (2000) as part of the article's title. the final good. Human capital has different uses: it is employed in the production of differentiated goods, in schools, where it is the main input to new human capital; it is used in the accumulation of social capital, as suggested by previous literature, and is also used in the innovation process. Social capital is used in the production of the final good, in facilitating the accumulation of embodied knowledge (human capital), in facilitating the research networks that increase R&D productivity, and in its own accumulation. In assuming these interactions between different capital types, we based on different bodies of economic literature, revised above. A crucial feature in the model is that there is not a market for social capital. Social capital is produced because it makes families happier. This follows the notion of bonding in Beugelsdijk and Smulders [5]. However, firms (both firms in the final good and in the R&D market) benefit from social capital, which follow the notion of bridging in the same article. As firms benefit from social capital without paying for it, this carries out externalities with less social capital in the market than in the efficient solution. The distortions caused by social capital act on the opposite direction of gains from specialization and spillovers in the R&D process.

Capital Accumulation
The accumulation of physical capital (K P ), arises through production that is not consumed, and is subject to depreciation: where Y denotes production of final goods, C is consumption, and δ P represents depreciation.
As in the literature that began with Arnold [3], in this model human capital is the 'ultimate' source of growth. To have endogenous growth, one should have nondecreasing returns in the human capital production function, regardless of the con-sidered inputs to human capital. Following most of the literature that deals with externalities in the last generation of endogenous growth models and also the evidence according to which human capital is the main input to human capital production, we propose that human capital K H is produced using human capital allocated to schooling as well as the total amount of social capital, Ks, according to: where H H are school hours, ξ > 0 is a parameter that measures productivity inside We based the choice of the functional form for the dynamic evolution of social capital on the literature that suggests a strong link between human capital and social capital. Also, some empirical literature on social capital has already calculated an economic payoff from it (e.g., Knack and Keefer [17] and Temple and Johnson [29]). Hence, social capital accumulation requires human capital to be allocated to its production but at each point in time it will also depend on the current stock of social capital, i.e.: where ω measures the productivity of human capital in the production of social capital and Ω ≶ 0 measures the dynamic effect of social capital on its own production. If Ω > 0 existing social networks are strong enough to keep growing without additional human capital. Some types of social capital (such as cultural norms or values) are given by the family, which mean that people do not have to make efforts to acquire it. An alternative way of thinking about a positive Ω is that people with stronger social networks find it easier to continue improving networks than people with fewer.
If Ω < 0, on the other hand, there is a net depreciation effect. 2

R&D Technology
Technological capital, or new varieties, K R , is produced in a R&D sector with human capital employed in R&D labs (H R ), by the stock of disembodied knowledge (K R ), and is also influenced by the stock of social capital: where ε > 0 measures the productivity in the production of technological capital, υ measures duplication effects, 0 < φ < 1 measures the degree of spillover externalities in R&D across time, as in Jones [12], and 0<χ < 1 measures the positive effect of social networks in R&D productivity. 3 The parameter χ measures an externality from social capital to R&D. Since agents, when deciding how much to invest in social capital, do not take into account the effect this have in the R&D firms, they invest less in social capital than it would be socially optimal. This externality acts in the same direction as the duplication effects (the parameter υ in the equation) and in the opposite direction of spillovers (parameter φ in the equation), since it acts in favor of overinvestment in R&D. 2 We choose to model social capital based on the yet scarce literature about it, i.e., we do not consider higher bounds for the stock of social capital. This is the same assumption we do for human capital, following the recent literature on human capital. There is no reason to consider that human capital grows without bounds, at a constant long-run rate, and to consider the opposite for social capital. 3 Theoretically, one could generally assume that χ could have negative values and values above unity. However, for simplification we assume that it varies between 0 (no effect of researchers networks) and 1 (proportional effect of researchers networks). In fact in the calibration exercise, we reasonably assume that the effect of researchers networks is smaller than the effect of past technological knowledge (χ < φ). If χ = 0, we would obtain the formulation for R&D technology used in Jones (1995).

Final Good Production
The final good is a differentiated one, produced with a Cobb-Douglas technology: 4 D is an index of intermediate capital goods and is produced using the following Dixit-Stiglitz CES technology: The elasticity of substitution between varieties is measured by 0 < α < 1. x i is the intermediate capital good i and is produced in a differentiated goods sector using physical capital: This means that (6) can be re-written as: In what follows we will see that σ measures an externality from the household' to the firms' choice of social capital. Although households choose social capital comparing its marginal utility with its opportunity cost in terms of human capital, firms do not choose social capital, they face social capital as a 'gift' embodied in workers. This leads to another externality, which implies less social capital in the decentralized equilibrium than in the optimum and also tend to increase the scope for overinvestment in R&D. 4 Using σ > 1 instead does not change our main results. 5 We modelled taste for variety in this specific manner in order to isolate the gains of specialization (η ) from the mark-up (1/ α) and from the share of physical capital in the final good production (β ). This specification follows Alvarez-Pelaez and Groth [2] and allows us to separate important externalities in comparison to what happens in the standard specification.

Consumers
To capture the multi-faceted character of social capital the model for household preferences specifies it, along with consumption, as arguments of the intertemporal utility function: where ψ represents the preference for social capital and ρ is the utility discount rate. 6

Optimal Growth
It is clear that when an economy faces the externalities already mentioned, the decentralized equilibrium will not maximize aggregate welfare. Thus we must solve a social planner's problem. In this section we derive the conditions associated with the maximization of (9) subject to the production function (6)  The problem gives rise to the following Hamiltonian function: where the λ j are the co-state variables for each stock K j , with j = P, H, S, R. Consider- (3)), the first order conditions yield: as well as:λ representing the marginal utilities of consumption and social capital respectively.

Optimal Growth Rates
Growth rates will, by definition, be constant, so equation (1) tells us that K P , Y , and C all grow at the same rate. Furthermore, K S and K H components will also be growing at that same rate, respecting equations (2) to (4). 8 Denote the growth rate of technological capital as g K R and the growth rate of human capital as g K H . From equation (5) we can see that these two growth rates have to respect this relation: In the steady-state, we can obtain the human capital growth rate as follows. From (12) we find g λ H = g λ P + g Y − g K H and using equation (16) we can then replace the (11). Then, using equations (5) and (8) we get: χ+ν g K R we solve for the growth rate of technological capital: By substituting (18) in the previous equality we find: While the impact of the social capital share (σ) is positive in growth rates, the impact of preference for social capital (ψ) is negative as it has a trade-off with consumption. This has a parallel with the effects of bonding and bridging in growth rates in the article from Beugelsdijk and Smulders [5]. Optimal growth rates depend on parameters of the model as usual in non-scale models of endogenous growth.

Decentralized Equilibrium
In the decentralized equilibrium both consumers and firms make choices that maximize, respectively, their own felicity or profits. 9 Consumers maximize their intertemporal utility function: subject to the budget constraint: where a represents the family physical assets, r is the return on physical capital, and W H is the market wage. The market price for the consumption good is normalized to 1.
Since it is making an intertemporal choice, the family also takes into account equations (2) and (4), which represent human and social capital accumulation, respectively. 10 The markets for purchased production factors are assumed to be competitive. However, we assume that the firm cannot buy social capital, as there is, in effect, no market for it. Social capital is treated here as exogenous, although it affects the firm's production. Hence, consumer decisions will carry social capital externalities.
From this problem we know that returns on production are as follows: where p D represents the price for the index of intermediate capital goods.
Each firm in the intermediate goods sector owns an infinitely-lived patent for selling 9 In this section we are working with individual variables. 10 See Appendix A and B respectively for the FOC and growth rates in the decentralized equilibrium.
its variety x i . Producers of differentiated goods act under monopolistic competition in which they sell their own variety of the intermediate capital good x i and maximize operating profits, π i : With identical technologies and symmetric demand, the quantity supplied is the same for all goods, x i = x. Hence, equation (7) can be written as: From p D D = pxK R , together with (23) and (25), we obtain: After insertion of equations (25) and (27) into (24), profits can be rewritten as: Let υ denote the value of an innovation, defined by: Taking into account the cost of an innovation as determined by eq.(5), free-entry in R&D implies that, Finally, the no-arbitrage condition requires that investing in patents has the same return as investing in bonds: This fully describes the economy.

Optimality of Human Capital Allocations
Using the FOC obtained from the social planner solution (11) to (17) The equations that were presented in this section provide a basis for a complete analysis of all the relationships between the different capital stocks and also a basis for the comparison with the decentralized equilibrium solution. Using the restriction that u * Y + u * S + u * H + u * R = 1, we obtain the social to human capital ratio: As growth rates and the consumption to output ratio are equal in the social planner and decentralized equilibrium solutions, the differences from the two solutions are spillovers in R&D, duplication effect in R&D, the specialization gains, and the externalities from social capital. As in Alvarez-Palaez and Groth [2], the social gains from specialization (η) compare with the private gain from an innovation (β(1−α)). 11 From the comparison of (33)-(36) to (39)-(42) taking into account the comparison between (44) and (38), it is possible to write the following proposition. As usual in the studies that intend to evaluate distortions between the social planner and decentralized equilibrium solutions, this evaluation is a quantitative issue.
Thus we now implement a calibration exercise to evaluate the distortions. parameter Ω that can measure a positive effect of social capital in its accumulation or a depreciation of social capital, we use alternatively 0.01 and -0.01. For each of these exercises, we set the steady-state economic growth rate to be equal to 1.85%, which gives us a value for ξ. This procedure yields us values in the range used by human capital literature (e.g. Funke and Strulik [9]). For the impact of social capital in economic growth, we use a lower bound estimate for σ of 0.08 as in Knack and Keefer [17], for whom a 10% increase in trust implied a 0.8% increase in the economic growth rate. We also use a high bound for σ = 1 − β − σ = 0.32, suggested by the evidence in Whiteley [31] that points out an effect of social capital as big as the effect of human capital and approximating the evidence on World Bank [30] that pointed out a share of 0.78 to intangible capital, which included both human and social capital.
For the consumption-output ratio (C/Y ) we use the reasonable value of 0.6. we set spillovers to 0.4 (see Reis and Sequeira [22]), duplication to 0.5 (see Pessoa [19]), specialization gains to 0.196 (see Jones and Williams [14]) and the social capital share to a value between 0.08 and 0.32 (see above).

Distortions from Social Capital
In the next lines we present the differences between the decentralized equilibrium and the optimal solution when there are only distortions from social capital. Thus, we apply benchmark calibration values. In Figure 1 we present the different values of the K S /K H ratio through different values for the share of social capital in production. In   Figures 4 to 6 we can see that increasing the effect of social capital in R&D technology also increases distortions but less than the rise in distortions caused by the final good social capital share. In this case a change from χ = 0 to χ = 1 implies a rise in K S /K H by 0.05, the change caused in u S is about 1% and finally the change implied in u R is near 0.001%. We have also a tendency for underinvestment in social capital and overinvestment to R&D. The next step is to evaluate how much decrease in the distortion can social capital impose when all other distortions are present.

Taking all Distortions Together
In this section, we present results from quantitative exercises in which we apply the calibration values depicted as 'Reasonable' in Table 1. Tables 2 to 4 compare the social planner allocations to the decentralized equilibrium ones. We show three different exercises: the first eliminates the distortions due to social capital and sets other distortions at reasonable levels, given by parameter values discussed above; the second considers a lower limit for the social capital share in production and a reasonable value for the externality of social capital in the R&D technology and keeping other values for parameters at the level used in the first exercise; the third exercise is equal to the second, except for the share of social capital in production that increases to 0.32. The unique difference in Table 3 (from Table 2) is that it uses a depreciation for human capital of δ H = 0.05. The unique difference in Table 4 (also from Table 2) is to consider a depreciation of social capital (Ω = −0.01). In all these exercises g K H oscillate from 1.36% to 1.47% and g K R from 1.23% to 1.59%.   Without the distortions introduced in this article (σ = 0, χ = 0), we note that the tendency for underinvestment in R&D is high, as previous literature also predicted (see e.g. Jones and Williams [14]). Human capital allocation in the decentralized economy is almost at the optimal level and there are over-allocations to the final good production and to the social capital sector. It is worth noting that due to the distortions from social gains from specialization, spillovers, and duplication, there is a relatively higher social capital to human capital ratio in the market economy when compared to the social planner choice.
When we consider positive values for σ and χ we note that now the social to human capital ratio is higher in the efficient solution than in the market economy, which is due to the distortions introduced in this article. The absence of a market for social capital is responsible for having relatively lower social capital in the market than in the case in which social welfare would be taken into account. This is reflected in the allocation of human capital to social capital production, which should also be higher than it is in the market economy. Because of that, allocation of human capital to the human capital accumulation sector is above the optimal level and the level of underinvestment in R&D is reduced. In the third exercise, the distortion in the social capital sector is so high that the social planner would allocate to that sector nearly twice the human capital allocated by the market economy (from 4.98% to 9.96% in Table 2; from 2.62% to 5.17% in Table 3, and from 9.77% to 16.31% in Table 4).
Thus in these scenarios the social planner would reallocate human capital from final good production and schools to social capital accumulation sectors and to R&D firms.
This means that some policies can be designed to enhance the production of social capital. Considering a positive depreciation for human capital as we do in Table 3, introduces almost no differences in distortions from the social planner allocations.
Such a high depreciation in human capital predicts however an implausible share of human capital allocated to the human capital accumulation sector. Nevertheless, when we introduce a depreciation in social capital accumulation, as in Table 4, we note some important differences. The share of human capital allocated to social capital production sector is higher than in previous exercises because human capital allocated to that sector must compensate the depreciation effect, while in previous exercises this did not happen because social capital could grow by itself (exogenously). The most important implication is that underinvestment in R&D is much reduced (from u * R /u R = 1.15 to u * R /u R = 0.96), opening the possibility to overinvestment in R&D. This means that the threshold level for the share of social capital in production (σ) above which there is overinvestment in R&D is below 0.32, which is in the range of plausible values, according to World Bank [30]. The higher the depreciation for social capital, the lower the threshold value for its share in production above which overinvestment in R&D occurs. In fact, in the case of the third exercise in Table 2, we can observe that considering a 1% depreciation for social capital, we can obtain overinvestment to R&D and keeping reasonable values for the allocations through sectors in the economy, with the highest allocation to the final good production.

Conclusion
We motivate this article from the increasing importance of social capital in the economic growth analysis, both in theoretical and empirical perspectives. The interaction between social capital and R&D has been pointed out essentially due to research networks. We build the production side of the model taking into account the interactions between the different types of capital that have been pointed out in previous literature. In particular, we note the importance of the use of human capital in the social capital accumulation and the importance of this last factor in the production of the final good, and also in the discovery of new ideas.
Social capital can introduce distortions in market allocations mainly due to two features of social capital: the failure of a market for social capital and the impact it can have in R&D due to research networks. The first reason is justified as firms do not pay for social capital when contract workers; they pay for hours of work and at most for the level of qualifications. This may happen because the features usually classified as social capital (confidence, truth, networks) are more difficult to evaluate and monitorize than degrees or years of schooling. The second reason is based on the importance of social networks between researchers in the R&D productivity. This means that social capital is produced because agents like to socialize but it is used in production side in a way that enhance firms productivity without any payment for it.
In the model, we also consider the most important distortions present in previous models: the social benefit from specialization, spillovers, and duplication in R&D. We implement a calibration exercise in order to evaluate the strength of the new distortions from social capital. First we show that both new distortions lead to underinvestment in social capital, both when we compare the social to human capital ratio and when we compare allocations of human capital to the social capital accumulation sector.
Second, we also show that the presence of these distortions decrease the tendency to underinvestment in R&D. However, quantitatively, these distortions are not strong enough to get overinvestment in R&D when social capital has a positive effect in its own accumulation. The opposite result is obtained when social capital depreciates.
In fact, in this case, the social capital externalities introduced in this article are capable of generating overinvestment in R&D. This complements the recent literature (Strulik [27] and Reis and Sequeira [22]) that presented more arguments in favor of overinvestment. Moreover, our results pointed out a share of human capital in social capital accumulation which oscillates in the decentralized equilibrium from 2% to 10% which is an additional quantitative reason to integrate social capital in an endogenous growth model with R&D and human capital accumulation, as we did in this article. as well as:λ a λ a = ρ + δ P − r (48) where λ a is the co-state variable for the budget constraint, and λ H and λ S are co-state variables for the stocks of human and social capital, respectively.

B Growth Rates in the Decentralized Equilibrium
In the steady-state, we can obtain the human capital growth rate of the decentralized equilibrium as follows. By using equation(49) and replacing it in g λ H = g λa + g W which we get by (46), we findλ a λ a = ρ + δ H − ξ − g W . From (22)we get g W = g Y − g K H . Substituting this last equation in the previous one and introducing both in − 1 τ g y + ψ 1 − 1 τ g K H =λ a λa which we find in (45) and using equations (5) and (8) we get: Using the fact that g K H = (1−φ) ν+χ g K R we solve for the growth rate of technological capital: From (8) and g K H = (1−φ) ν+χ g K R we find g Y = g K H η ( ν+χ 1−φ ) 1−β + 1 . By substituting (18) in the previous equality we find: These three growth rates are equal to the ones that we found in the social planner problem.
Now we also show that the consumption to output ratio is equal in the decentralized equilibrium and in the social planner solution.
From equation (1) we get:K Also, from this equation we get g K P = g Y = g C because the growth rates have to be constant in steady state. Since we have shown that (g K P = g Y = g C ) * = (g K P = g Y = g C ) DE the left hand-side of equation (54) is equal in the social planner and in the decentralized equilibrium and δ P is a constant.
In the social planner problem by transforming equation (11) we obtain -λ P λ P = − 1 τ g C + ψ(1 − 1 τ )g K S . Replacing this last equation in equation (15) we get -βY K P = ρ + δ P + 1 τ g C − ψ(1 − 1 τ )g K S . Since we have shown that g * . Using the fact that C K P can be written has C Y Y K P and putting Y K P in evidence, we obtain that (27) we note that Y K P DE = r αβ and that r = αP mg K , thus Y