Friends, wealth, job: All are network phenomena∗ Roman Chuhay, Higher School of Economics June 17, 2011

Abstract The present paper builds a framework that relates labor market, wealth formation mechanism and social network into one process. This framework allows us to obtain a distribution of wealth ranging from lognormal, observed in lower-wealth region, to Pareto type, peculiar for high levels of wealth. The parameter that governs resulting wealth distribution has a clear sociological interpretation - homophily of the society. The paper sheds some light on the question of occupational mismatch in the case of employment through personal contacts. Particularly, we show that a worker may be underpaid or receive some bonuses in the case of employment through contacts depending on the status and prevailing wealth distribution.

1

Introduction

While examining wealth distribution one typically finds two distinct regions. At the low-end range, wealth distribution can be approximated by a lognormal distribution, while at the high-end range distribution follows the Pareto law. To explain origins of the Pareto distribution at the high-wealth range Solomon (1999) and Levy (2003) employ multiplicative stochastic process (MSP hereafter) with a lower bound. Another approach uses neoclassical growth model with uninsurable idiosyncratic earning shocks to generate empirically observed wealth and income distributions at the whole wealth range (see Aiyagari, 1994; Castaneda et al., 1997; Huggett, 1996; Krusell, 1998). The main difficulty of matching the whole range of wealth distribution arises due to differences of key factors that shape wealth accumulation of a person at lower and higher ranges of wealth distribution. At the lower range main factors are labor income and consumption, while at the high-wealth range the accumulation is mostly shaped by capital gains (e.g. see Levy, 2003). Thus, while wealth accumulation at ∗I

would like to thank Fernando Vega-Redondo, Hubert Kiss, Gergely Horvath, and Alexander Otenko for useful comments.

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the lower range is driven by the process similar to additive, evolution at the high levels is governed mostly by multiplicative processes. This is one of the reasons why neoclassical models typically produce distribution of wealth significantly different from the observed, see e.g. Quadrini and R´ıos-Rull (1997). In the present paper we study the effect of social ties on the wage, employment status and wealth of workers. At the same time we allow wealth distribution to shape the formation of links between workers. This framework allows us to obtain distributions of wealth close to the one observed in reality. In our model a network evolution is driven by processes of links formation and destruction. Thus, at an exogenously given rate a worker is given an opportunity to establish link to another worker who possesses a similar level of wealth. The process of links creation coexists with the decay of existing links. We interpret the maximal wealth distance within which a link can be formed as a homophily level of the society. Homophily is a tendency of people to interact more with those who are similar to them, which has been documented at least since Aristotle’s time.1 , 2 In terms of employment, worker can be in one of two states - employed or unemployed. Independently of the employment status, a worker may hear about a job offer directly from an employer. If an unemployed worker receives an offer she immediately accepts it. In case when an offer arrives to an employed worker, she passes it to one of her randomly chosen contacts. We assume that the wage offered for the job is proportional to the wealth level of the worker who receives the offer. In the absence of consumption, the wealth level of a worker summarizes her employment history. That is why in the model the level of wealth has an additional function representing workers experience accumulated during the periods of employment. In the case of direct employment the assumption implies that the wage offered to a worker is proportional to her work experience. In the case of indirect employment it represents an idea that contacts hear about a job similar to their current employment. In the model an increase of homophily augments the correlation of wealth level and wage of connected workers, generating a wealth accumulation process close to multiplicative. In the case of extremely high homophily the wealth accumulation is described by a multiplicative stochastic process (MSP), which as shown in Solomon (1999) leads to the Pareto distribution of wealth. On the other hand, if homophily of the society is extremely low, wage of the worker who finds a job through contacts does not depend on her wealth and is drawn from the same distribution for all workers. In this case the additive part of the process gains higher weigh, which produces a lognormal distribution. The model also sheds some light on the occupational mismatch in the case of employment through contacts. Numerous studies examined the effect of employment 1 In Aristotles Rhetoric and Nichomachean Ethics, he noted that people “love those who are like themselves” (Aristotle 1934, p. 1371). 2 The term homophily appeared in the sociological literature for the first time in Lazarsfeld and Mertons (1954) who also quoted the proverbial expression - “birds of a feather flock together,” which has summarized homophily ever since.

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through contacts on wage, but failed to reach a consensus. Bentolila et al. (2004), measures a wage discount of at least 2.5% for jobs found through personal contacts as compared to the wage for jobs found through formal methods. The authors attribute wage discount to the occupational mismatch, which arises due to the impatience of workers that accept job offers passed by contacts in sectors where they are not productive. In contrast, Granovetter (1974), Corcoran et al. (1980), Simon and Warner (1992) and Kugler (2003) provide evidence for positive wage premia for jobs found through contacts. Our model proposes different from Bentolila et al. (2004) point of view on the occupational mismatch in the case of employment through contacts. In our model there is only one sector, but jobs differ in the required level of skills and consequently in the offered wage. Our model produces simultaneously both wage discount and wage premia in the case of employment through contacts. The intuition is the following. In our model workers create links with those who posses similar wealth and work experience. A worker’s ratio of number of links with poorer workers to the number of links with richer workers depends on a worker’s wealth and prevailing wealth distribution. For example, a Pareto-like distribution of wealth implies that in the neighborhood of a sufficiently rich worker the proportion of poorer workers is higher and becomes even higher as homophily level falls. Note, according to the Pareto distribution each time we double the amount of wealth, the number of people possessing it falls by a constant factor. That is why a rich worker in more than half of the cases of employment through contacts gets a lower wage as compared to direct employment. On the contrary, the neighborhood of workers close to the lower bound, mainly consists of workers who are richer than they. Hence, the poorest workers are better-off in the case of employment through referrals. The paper is organized as follows: Section 2 presents the theoretical model; Section 3 elaborates on calibration and simulations procedures; Section 4 presents quantitative results of simulations and discusses the intuition behind them; Section 5 concludes.

2 2.1

Model Network dynamics

We consider a set N = {1, . . . , n} of heterogeneous workers who differ in the amount of possessed wealth ai . At each time period the network of workers’ contacts is represented by a non-directed graph g(t). Hence, we think that interaction among workers i and j takes place if ij ∈ g(t). At rate η worker i is given an opportunity to establish a new link to a worker j, uniformly drawn from the set of potential neighbors Nip (t) of worker i. When there is no link between i and j (i.e. ij ∈ / g(t)), the link ij is formed g(t + 1) = g(t) ∪ ij. On the other hand, if the link ij is already a part of the network (i.e. ij ∈ g(t)), the link remains unaltered. The set of potential neighbors of worker i consists of all workers whose wealth level is sufficiently close to the wealth of worker i:

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¯ Nip (t) = {∀j ∈ N s.t. | log aj (t) − log ai (t)| ≤ log d}, where d¯ ≥ 1 is the maximal distance in terms of logarithms of wealth within which a link between workers can be formed. We assume that a link is beneficial for workers only if they have a sufficiently similar position in the society and can exchange information on relevant job-offers. We measure the distance in terms of logarithms to reflect the importance of the ratio of wealth levels of two workers as compared to the absolute difference. To illustrate the idea let us consider two workers who possess 10 and 11 dollars. If we think that they are close enough in terms of wealth then we would like to think that workers possessing 100 and 110 are close too. Recall that a common multiplier in the case of logarithmic difference cancels out. We are not the first to incorporate wealth level of workers as a measure of proximity. Chantarat and Barrett (2008) use wealth as one of two parameters in a modified Euclidian distance function to explore the role of social capital in escaping from poverty trap. Mogues and Carter (2005) employ a similar distance function to investigate the polarization of the society and evolution of wealth distribution. There is also a process of link destruction and the existing links decay at rate λ. This component of the process may have various interpretations. It may be conceived as the obsolescence of the existing links. For example, neighbor j of worker i may diverge too much in terms of work experience. This would make job offers coming from j not desirable for i and job offers coming from i not accessible for j due to the lack of work experience. It also may be understood as a result of capacity constraints in the ability of any particular worker to sustain many links. This formulation of the network evolution process implies that the expected number of links that a worker can simultaneously sustain is finite. The network formation mechanism presented in the paper is close in spirit to Ehrhardt et al. (2008). In their paper workers at rate 1 receive an opportunity to establish a new link with an uniformly drawn worker from the society. However, the link is formed only if workers are close enough according to some distance function. In the case of a Pareto type distribution the mechanism proposed by Ehrhardt et al. (2008) would imply that few rich workers in general are under-connected as compared to poor workers. In contrast, in our model a future contact is drawn from the set of potential neighbors Nip (t). Our mechanism is more consistent with an idea that people with the same social status attend the same events where they can meet each other. One can find many examples of such pre-screening in real life. For example, rich people attend private clubs, which facilitate search of potential business partners, similarly a lot of business exhibitions are devoted to the same aim. Poor people are more likely to meet peers at the place of their current employment or at the street they live. The main consequence of this assumption is that irrespectively of wealth distribution, the workers connectivity follows the distribution close to Poisson with average connectivity z ' 2η λ .

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2.2

Job offer flow and wages

Now we turn to the labor market dynamics. We assume that each worker could be in one of two states, namely employed or unemployed. The employment status of a worker is represented by si (t) such that si (t) = 1 if i is employed at time period t and si (t) = 0 otherwise. We employ the same process of job offer flow as in CalvoArmengol (2004). At rate α, each worker receives information about new job directly from an employer (direct employment). If she is currently unemployed she accepts the offer. If she is employed she passes the information to one of her uniformly drawn unemployed acquaintance (indirect employment). If there are none (all neighbors are employed) the information is lost. At rate β, an employed worker loses job and becomes unemployed. Hence, each worker could receive a job information directly from an employer or indirectly from one of neighbors. In the case when the network is empty, namely all workers are disconnected (e.g. η ¯ λ → 0, or d = 0), employment depends only on rates that govern labor market dynamics. By appealing to simple balance of job creation and destruction we obtain α . In a more general case, however, employment depends also on the that e(t) = α+β average connectivity in a non trivial way. At the rate γ a randomly drawn individual i receives a payoff from her labor activity si (t)wi (t), where wi (t) is the wage of worker i. In the present model this event could be treated as a payday. Despite the fact that workers receive payoffs not regularly, but at the rate γ, on average they are paid the same number of times. The last thing left to specify is wage wi of worker i. If worker i is unemployed and receives an offer directly, the offered wage is equal to a fixed part ρ of her wealth level wi = ρai (t). The wage of a worker who receives a direct job offer is proportional to her wealth, due to the twofold role assigned to the amount of wealth of a worker. On the one hand, wealth represents the quantity of assets possessed by a worker. On the other hand, in the absence of consumption, the wealth level encompasses all information about the past work experience of worker. Thus we assume that offered wage in the case of direct employment is proportional to the accumulated work experience of worker. In the case when unemployed worker i receives a job offer from one of her acquaintances j, we assume that she is offered the same wage as would be offered to the worker who passes information to her, thus the wage of worker i becomes wi (t) = ρaj (t). The main assumption is that an employer by contacting a particular worker reveals information about the position and wage3 . We assume that d¯ is chosen in such a way that both an unemployed worker and a firm prefer to be matched today rather than wait for a better match in future. We assume that the underlying network of social contacts evolves much slower than a labor market. Thus, at each time period we allow the labor market to evolve 3 In

the present formulation we do not explicitly model firm behavior. However, we assume that in the equilibrium supply equals demand and that is why number of job offers with say a wage w is proportional to the number of workers whose experience(wealth) is w . ρ

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sufficient number of sub-iterations to get to the equilibrium on the labor market. The assumption represents the idea that social contacts have a longer life-period as compared to job positions.

2.3

Evolution of wealth distribution

We assume that a worker wealth is accumulated by means of labor income. Further, we assume a lower bound of the wealth accumulation process that moves with the average wealth of workers. Thus if a worker’s wealth is less than ω proportion of the average wealth in the society, the worker is compensated by the government to maintain minimum standards of living. Thus the motion of workers wealth is governed by the following process: ai (t + 1) = ai (t) + wi (t)si (t) (1) s.t. ai (t + 1) ≥ ω¯ a(t), where a ¯(t) is the average wealth possessed by workers4 .

3

The effect of homophily on wealth distribution

In our model homophily level has an important impact on the network dynamics and ¯ consequently on the wealth accumulation of workers. By varying maximal distance d, which defines homophily, we can reproduce the Pareto distribution, which is typical for high-wealth levels and lognormal distribution observed at lower levels of wealth. ¯ Thus These distributions materialize for the extreme values of maximal distance d. ¯ if homophily is extremely high (d → 1) we get a Pareto type of distribution. If homophily is extremely low (d¯ → ∞) we get a lognormal type of distribution. For the sake of exposition, let us consider both extreme values of d¯ separately and establish some analytical results for them. The value of d¯ → ∞ generates the network represented by the classical random graph, since now worker i chooses worker j randomly from the whole population Nip (t) ≡ N . In this case network dynamics is unaffected by wealth distribution. Thus, if we consider a finite, but sufficiently large number of workers n → ∞ the expected connectivity of a worker is determined by the ratio λη . Moreover, since workers are connected randomly, job offers coming to them from acquaintances are independent of their wealth. The wage offered in the case of employment through contacts is drawn from the distribution of wages of currently employed workers in the society. On the contrary, the wage offered in the case of direct employment is proportional to 4 Note that this specification could be easily extended to include consumption of workers. For example, given that we do not model explicitly worker’s intertemporal choice we can assume that workers devote a fixed part φ of their wage to consumption purposes. This modification will not affect neither network dynamics nor wealth accumulation process, since we can redefine a share of wealth offered as a wage in the case of employment ρ to ρ0 = ρφ.

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the current wealth of a worker. This mixture of two types of accumulating processes gives rise to lognormal distribution. On the other extreme, when d¯ → 1, the link formation mechanism becomes extremely sensitive to the difference in wealth and only workers with the similar wealth levels can establish a link. A continuous distribution of wealth implies that the network is represented by an empty graph and all job offers come to a worker directly from an employer. In this case the wage offered to a worker is proportional to her wealth, which generates the following multiplicative stochastic process (MSP) with a lower bound: ai (t + 1) = ξ(t)ai (t) (2) s.t. ai (t + 1) ≥ ω¯ a(t), ( α 1 + ρ, p = α+β where ξ(t) = 1, otherwise Note that in an empty network the average employment equals

α α+β .

Proposition 1 In the case of extreme homophily level (d¯ = 1) or zero rate of link creation (α = 0) the wealth accumulation process is governed by the MSP with a lower bound. The result holds in case of any non-degenerate initial wealth distribution. In this case we can apply the result obtained by Solomon (1999), which states that any MSP with a lower bound eventually converges to the Pareto distribution. The result holds irrespectively of a distribution that governs the stochastic process. The prediction is in accordance with the results coming form simulations, which we report in Section 4.

4

Calibration and simulation procedure

4.1

Calibration

In order to perform quantitative analysis for the case when 1 < d¯ < ∞ we move to calibration of the model. The following 8 parameters should to be calibrated: • Network evolution: rate of link creation η, rate of link destruction λ and maxi¯ mal distance d; • Labor market dynamics: arrival rate of job offers α, destruction rate of existing jobs β, rate γ at which workers receive payoff from their labor activity; • Wealth accumulation process: fraction of wealth ρ offered to workers as a wage in the case of receiving a direct offer, fraction of average wealth of society ω below which workers are compensated by the government to maintain basic standards of living. 7

Using the assumption that the underlying network of social contacts evolves much slower than the labor market we can model these processes separately. By an appropriate time rescaling we eliminate one rate in each process. Thus, we assume two destruction rates to be equal to 1, namely the rate of link destruction λ and rate of job destruction β. In simulations we are going to vary maximal distance parameter d¯ and observe its implications on the wealth distribution. These considerations bound the set of model parameters to the following five parameters {α, η, γ, ρ, ω}. Despite some empirical studies e.g. Bernard et al. (1989, 2001) that are devoted to the evaluation of number of acquaintances in the framework of labor market, there is no agreement on its value. Fontaine (2005) argues that number of contacts reported by these surveys crucially depend on the definition of relevant ties and methods used for its evaluation. That is why our calibration strategy of parameters governing labor market and evolution of social network is to match unemployment rate and probability to get a job offer through personal contacts. We match unemployment rate of 6.7% in U.S. during 1972-1992 years to be able to use the data on inequality from Quadrini and Rios-Rull (1997). We also use field-study by Granovetter (1973), which reports that 56% of workers found a job through personal contacts. β(1 − u) = αu + α(1 − u)

∞ X 

 1 − (1 − u)k p(k|z)

(3)

k=0

Expression (3) describes equilibrium unemployment in the model, when there is a balance between job destruction and job creation. At rate β(1 − u) workers lose their jobs. At rate αu workers hear about a job directly from an employer. The last term on right hand side is a rate at which unemployed workers hear about offer from contacts. Indeed the probability that a worker with k contacts has at least one unemployed contact is 1 − (1 − u)k . Thus the probability that a randomly chosen  P∞  employed worker has at least one unemployed contact is k=0 1 − (1 − u)k p(k|z). Recall that in the model nodes connectivity is distributed according to the Poisson k −z e distribution with average connectivity z, thus p(k|z) = z k! . Let us now find the probability that an unemployed worker hears about a job from one of her employed contacts. First, we find the probability that employed contact j with k contacts passes a spare offer to unemployed worker i. The probability that a randomly chosen link of j leads to another unemployed worker is u. Given that there is a link between j and i, the probability that the worker i receives the offer is: r(k, u) =

k−1 X m=0



 k−1 1 um (1 − u)k−m−1 = m+1 m

1 1 1 − (1 − u)k = (1 − u)k−1 + (k − 1) u(1 − u)k−2 + ... + un−1 = 2 k uk A number of connections of worker’s contact is distributed differently from the connectivity of a randomly chosen worker. The intuition is the following: the more contacts worker has the higher is the probability that a randomly selected link leads 8

to her. That is why the correct distribution of number of links of worker’s contact is given by z1 kp(k|z). Weighing the probability to get an offer from a contact with k links by the connectivity distribution of a contact we obtain the expected probability to get an offer from a contact. Finally, the probability to hear about a job offer from a contact in the case of worker’s employment is given by the following expression: 1

θ =1− 1 + (1 − u)

∞ P k=0

(4)

1−(1−u)k kp(k|z) uk

Taking into account that β is 1 and substituting u = 0.067 and θ = 0.56 we can solve numerically the system of equations (3) and (4). The solution gives us α = 6.12 and z = 1.44. Using the fact that z = 2η/λ we obtain η = 0.72. We calibrate the rate at which a worker receives payoff from the labor activity γ to 12. If we think about time period being a year then worker on average loses a job once per 48 weeks, hears directly about a job offer once per 8 weeks and receives salary each 4 weeks. We are left to specify two parameters ρ and ω related to the process of wealth formation. In the case of employment through contacts parameter ρ is the fraction of neighbor’s wealth proposed to the acquaintance as the wage. We assume a benchmark value ρ = 0.1. In general, the proportion of wealth offered as a salary should be of no importance, since it applies to all workers. A number of simulations performed with different values of ρ and the benchmark parameterization show robustness of results with respect to this parameter. Finally, we borrow the lower bound of wealth level ω = 0.2 from Levy (2003). Table 1 summarizes the benchmark parameters.

TABLE 1. Benchmark parameters of the model. Parameters: Values:

4.2

α 6.12

η 0.72

γ 12

ρ 0.1

ω 0.2

Simulation procedure

We carried out extensive numerical simulations for number of workers n = 5000 with each simulation having T = 500000 iterations. At time period zero, the wealth level of each worker is drawn from the uniform distribution U [0, 1] and the network of social contacts is represented by empty graph. Simulations were conducted using the discrete time Markov chain to approximate the continuous time process of network dynamics. At each period of time one and only one of the following events occurs: creation of a new link or the destruction of existing link. The probability that the occurring event is the destruction of existing link is proportional to the number of existing links multiplied by the link destruction

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rate: λ zn 2 . The probability that the event is the link creation is proportional to the number of workers multiplied by the link creation rate: nη. To capture the idea that the underlying network of social contacts evolves at a rate much slower than the labor market, at each time period we allow labor market to evolve 100 sub-iterations to achieve equilibrium. At each sub-iteration one and only one of the following events occurs: a randomly drawn worker receives a direct offer from an employer, one of the existing job positions is destroyed, or a randomly chosen worker receives payoff from the labor activity. Hence, the probability of receiving a direct job offer is proportional to the number of workers multiplied by the job arrival rate: αn. The probability that the event is the destruction of an existing job position is proportional to the number of employed workers multiplied by the job-destruction Pn rate: β i=1 si (t). And eventually at rate nγ a sub-iteration is payday for a uniformly drawn worker.

5

Results

In this section we present quantitative implications of the model. First, we focus on the shape of wealth distribution that emerges in the model. After that we turn to the implications of a Pareto-like distribution of wealth on the network structure and occupational choice of workers. More precisely, we explore the relationship between clustering coefficient and wealth. In addition, we investigate the occupational mismatch that arises in the case of employment through contacts. We start numerical analysis by examining the two extreme cases of maximal distance parameter that we have considered before. The first, workers are extremely selective in their linking (d¯ = 1), which as we have seen implies an empty network. The second, workers have no restrictions in selecting contacts (d¯ → ∞). In this case the induced network structure corresponds to the case of generalized random network with Poisson degree distribution. A commonly employed technique to verify whether the obtained distribution is of a Pareto type is to use a double logarithmic plot of empirical pdf function. However, there is a growing consensus that a simple observation of pdf function on the loglog plot is not an appropriate method. In the case of distributions with heavy tails, such as Pareto and lognormal, partitioning of data into bins puts the majority of observations into a few bins, which makes these distributions indistinguishable for many parameters. To use all available observations, we employ a complementary CDF (hereafter CCDF) of a distribution, which is simply 1 − F (a). On the doublelogarithmic plot CCDF of the Pareto distribution should appear as a straight line. Figure 1a shows the logarithm of CCDF against the logarithm of wealth level for the case when d¯ = 1. The logarithm of the empirical CCDF fits the straight line well, which supports our finding. To get a better understanding of the shape of wealth distributions we use a non-linear optimization technique to match empirical CDF with a CDF from the Pareto distribution family. This procedure gives us R2 equal to 0.9989 and match is depicted at Figure 1b. 10

Figure 1: a) Logarithm of complementary CDF function of empirical distribution as a function of logarithm of wealth level; b) The empirical cdf of the wealth distribution (circles) and best match of theoretical cdf (line) coming from the Pareto distribution. Inset shows the fit of the theoretical cdf at the values close to the bound. The obtained empirical results are in accordance with Proposition 1. As we mentioned before, in the case of d¯ = 1 the network of labor contacts is empty. Hence, we have a MSP with a lower bound, which leads to the Pareto distribution. Quite a different picture is obtained if we consider the case when d¯ → ∞. In this case the distribution of wealth is close to lognormal. A typical distribution obtained in simulations is depicted at Figure 2a. An increase in d¯ lowers the correlation between workers wealth and wage in the case of indirect employment. This brings an additive component to the wealth accumulation process. Recall that in the case of d¯ → ∞ the link formation process is independent of wealth distribution, since Nip (t) ≡ N . That is why a wage in the case of indirect employment is proportional to the wealth of a randomly chosen worker. Particularly, d¯ → ∞ implies that as compared to the wage offered in the case of direct employment, poor workers are generally better-off in the case of indirect employment. In contrast, rich workers are worse-off, since the proportion of poorer workers in their neighborhood is higher. This effectively shrinks the distribution and suppresses formation of fat tails. To test the obtained distribution we take the logarithm of wealth. Recall, that if a variable is distributed according to the lognormal distribution its logarithm should be distributed normally. Figure 2a shows the distribution of logarithm of wealth and the best fit of the normal distribution to its empirical counterpart. The skewness and kurtosis are .03375 and 3.1158 respectively, which are very close to 0 and 3 as in the case of normal distribution. Figure 2b shows log-log plot of CCDF of the wealth distribution. To check more formally whether the obtained distribution is sufficiently close to lognormal, we use normality tests of the logarithm of wealth. Particularly, we employ Jarque-Berra, Shapiro-Wilk, and Shapiro-Francia tests to check our hypothesis. The p-values of all tests imply that we can not reject null hypothesis that the logarithm

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Figure 2: a) The distribution of logarithm of wealth (bars) and theoretical fit of normal distribution (red line) with benchmark parameterization and d¯ → ∞(d¯ = 10000000); b) The log-log plot of complementary CDF of wealth distribution and theoretical fit. of wealth in the case of d¯ → ∞ is distributed normally at any standard level of significance, see Table 2. TABLE 2. Normality tests result of logarithm of wealth in the case of d¯ → ∞. Test Jarque-Bera (S-K test) Shapiro-Wilk Shapiro-Francia

N 500 500 500

Stat. χ2 W W

Dist. χ2 (2) -

Val. 0.55 0.99740 0.99756

p-value 0.7595 0.62589 0.61618

Below we study the robustness of obtained results to the change of parameters ¯ We already have seen that in the case when d¯ = 1 we obtain MSP other than d. with a lower bound. This process gives rise to the Pareto distribution as long as the multiplier of MSP is distributed independently and identically for all workers. This condition holds under all parameters of the model and thus the resulting Pareto distribution is robust to the parameterization. The situation, however, differs in the case of lognormal distribution. The resulting distribution in the case of d¯ → ∞ is sensitive to the probability that recently employed worker got a job indirectly. This result is not surprising, since by varying this probability we change the importance of additive and multiplicative parts of the wealth accumulation process. The probability itself is not a parameter of our model, but it depends on the average connectivity of workers. The more contacts worker has, the more often she receives job offers from neighbors and consequently the more important is the additive part of the process. General case 1 ≤ d¯ ≤ ∞ Let us now turn to the case when d¯ is in between of two extremes. Indeed, the network of professional contacts is neither empty as implies d¯ = 1, nor absolutely 12

Figure 3: Relationship of clustering coefficient and logarithm of possessed wealth level of workers in the case of d¯ close to 1. random as in the case of d¯ → ∞. One of the explanations is that job offers passed by much less experienced contacts offer a rather small wage and workers do not accept them. On the contrary, if a contact is not experienced enough a firm would not hire her. That is why we assume that only links between workers with a sufficiently similar work experience are mutually beneficial. In general as d¯ departures from 1 wages offered in the case of direct employment and employment through contacts start to diverge. However, the divergence depends on the position of a worker in the society and the prevailing wealth distribution. To facilitate the analysis, assume that d¯ is higher, but still close to 1. In this case the network of professional contacts is not empty and in the same time all worker’s contacts possess very similar wealth levels. As we have seen before this gives rise to a Pareto distribution of wealth. Recall that in the case of Pareto distribution, doubling the amount of wealth leads to a fall of proportion of people possessing it by a constant factor. One of the implications of Pareto distribution is that the cardinality of the set of potential neighbors decreases in worker wealth. In particular this implies that two connected workers more probably have a common contact. In the network theory the degree to which nodes in a network tend to cluster together is measured by a clustering coefficient. Figure 3 shows the clustering coefficient of worker as a function of wealth level in the case of d¯ close to 1. High clustering equalizes wages in the case of indirect employment by increasing the probability that connected workers are employed by the same contact. This leads to a higher correlation of wealth levels of connected workers and reinforces the correlation of wages offered in the case of indirect employment. At the same time as wages of connected workers become more correlated, wages offered in the case of direct and indirect employment become closer. This makes the process of accumulation of

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Figure 4: Log-log plot of complementary cdf of realized wealth distribution with benchmark parameterization and d¯ = 1.1. The lines depict fitted lognormal and Pareto distributions. wealth by wealthy workers closer to MSP and leads to a fat right tail of wealth distribution. The opposite situation happens with workers from the lower range of wealth. The Pareto distribution implies that 90% of poor workers occupy a small region of wealth, which leads to a high cardinality of the set of potential neighbors. The probability that connected workers have a common neighbor goes to zero, which implies a zero clustering coefficient. In this case wages offered to poor workers in the case of direct and indirect employment differ more than those of wealthy workers. As we have seen before, the process with both additive and multiplicative components leads to lognormal distribution, but now it unfolds only at the low-wealth range. The former considerations imply that the main force governing the wealth accumulation of rich is of multiplicative nature, while accumulation of wealth of poor has both components. Figure 4 depicts the log-log plot of CCDF of the obtained wealth distribution. The figure has two distinct regions: one with low-middle values of wealth and another with high wealth values. The shape of the low-middle wealth region is closer to lognormal, while the shape of the top wealth region perfectly fits to the straight line (as should be in the case of the Pareto distribution). This result is in agreement with the empirical paper by Clementi and Gallegati (2006). The paper shows that income distribution is consistent with a lognormal function for the low-middle income group 97%-99%, and with a Pareto function for the high income group 1%-3%. Now let us examine how well our model reproduces statistics of observed wealth and income distributions. Table 3 reports statistics of households wealth and earnings distribution in United States and model predictions. Note that statistics for U.S. earnings are for households heads 35-50 years old. Unfortunately, the statistics of

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wealth distribution for working age is not available, that is why we consider both of them. TABLE 3. Actual U.S. Earnings and Wealth distribution in 1992 and Distribution generated by the model. Source Actual U.S. data

Model

1

Type Earnings1 Wealth d¯ = 1 ¯ d = 1.1 d¯ = 1.5 ¯ d = 10.0 d¯ → ∞

Gini coefficient 0.51 0.76 0.73 0.60 0.43 0.2 0.21

Bottom 40% 10.3 2.2 8.38 9.3 15.77 26.81 26.5

Top 20% 53.6 77.1 77.94 67.5 51.59 32.52 32.54

10-5% 10.7 12.6 5.43 15.47 12.47 8.31 8.43

5-1% 13.5 23.1 8.59 24.1 16.43 8.19 8.3

1% 14.1 28.2 58.41 14.99 8.66 3.24 3.0

The U.S. data on earnings is for household aged 35-50 years.

Source of actual U.S. data: 1992 Survey of Consumer Finances; Quadrini & Rios-Rull 1997

Perhaps the most noticeable feature of Table 3 is that all examined model parameterizations over-predict the wealth share of the bottom 40% of people as compared to the data. It happens due to several reasons. First of all, U.S. data of wealth is not corrected for the working age. The empirical evidence from Cagetti and De Nardi (2005) suggests that the Gini coefficient for the old people is much higher than one for the society as a whole, that is why their inclusion increases inequality. The second reason is that we abstract from consumption and bequests. It is a well-established fact that the average savings rate of the poor is lower than that of the rich, e.g. see Dynan et al. (2004). All these considerations increase inequality observed in the real data. Let us look how well statistics from lognormal and Pareto distributions are in accordance with the real data. In the case of the Pareto distribution (d¯ = 1) the model predicts Gini coefficient close to the one observed for U.S. wealth. The prediction of the wealth share of the top 20% is in accordance with the data as well. However, the model does not predict correctly the right and left tails of the distribution. It over-predicts the wealth share of top 1% (58.4 as compared to 28.2) and the wealth of the bottom 40% (8.38 as compared to 2.2). Thus, despite the fact that the Pareto distribution fits well the distribution of wealth of the rich, it has obvious shortcomings in fitting the distribution of the whole society. The statistics predicted by the lognormal distribution, which appears for d¯ → ∞ is also quite far from the one observed in reality. The lognormal distribution substantially under-predicts the Gini coefficient The wealth shares of bottom 40% and top 1% are correspondingly over-predicted and under-predicted. Let us turn to the wealth distribution generated by the model with intermediate ¯ The value of maximal distance d¯ = 1.1 performs much better in fitting values of d. actual U.S. data on earnings and wealth distribution. The fact that the emerged

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Figure 5: The average ratio of wage in the case of indirect employment to wage, which worker could claim at the labor market versus logarithm of wealth. distribution fits the earnings data as well should not be surprising. The data on earnings represent the working population, which actively participates in the labor market. The other two parametrizations, namely d¯ = 1.5 and d¯ = 10.0 significantly over-predict the wealth share of bottom 40% and under-predict the wealth share of top 1%. High values of d¯ implies that poor workers may escape poverty just by establishing links to wealthy workers, while rich and experienced workers start to accept job offers with inadequately small wages. In some sense this result supports our main assumption that workers should have similar status to form a link. Occupational mismatch The positive role of social contacts in decreasing unemployment spells is well recognized in the literature. Personal contacts serve as a reliable source of information about job offers. Nevertheless, some job offers coming from contacts do not perfectly match worker’s skills. Thus, the availability of social contacts and the opportunity to find a job more easily may tempt workers to undertake a career in professions, sectors, or locations where their abilities are not fully exploited. As a consequence, a worker could be underpaid as compared to the case of direct employment. The empirical papers on this topic do not come to a consensus whether the offered wage in the case of employment through contacts is higher or lower than the one which a worker can claim on the labor market directly. Recent studies by Simon and Warner (1992) and Kugler (2003) found a positive wage premia for jobs found through contacts, while Bentolila et al. (2004) measure a wage discount of at least 2.5% for jobs found through acquaintances. Our model can shed some light on the source of disagreement in the empirical literature. The model implies that employment through contacts depending on worker’s status has a different effect on the wage. Thus, a worker is generally better-off if her set 16

of potential neighbors has a higher proportion of richer workers than poorer. That is why the effect of referral employment on the wage depends on the realized distribution and worker’s position in the society. In the society as a whole this effect could be ambiguous for many groups of workers. Let us consider a Pareto-like distribution, which we obtain with parameter d¯ close to 1. Poor workers at the lower bound are generally better-off in the case of indirect employment, since all their contacts have the same or higher wealth level. On the other hand the wealthiest workers are generally worse-off in the case of referral employment. Note that Pareto distribution implies that all sufficiently rich workers have in the neighborhood high proportion of poorer workers. The same situation occurs in the case of the lognormal distribution. Thus, considering the whole society one can observe simultaneously underpaying the rich and overpaying the poor in the case of indirect employment. Figure 5 represents the ratio of the offered wage in the case of indirect employment to the wage, which a worker could claim directly at the labor market. The figure shows that workers with a wealth level close to the lower bound are overpaid, while those who are close to the right tail are generally worse-off. Thus depending on the group of workers under consideration the ratio of wages may be higher or lower than one. We think that this phenomenon can contribute to the indecisive conclusion of the empirical literature concerning the wage offered in the case of indirect employment.

6

Conclusions

In the paper, we study a general framework, which combines labor market, wealth formation and social network. The framework allows us to get the distribution of wealth, which ranges from the lognormal to Pareto. The parameter that governs the resulting wealth distribution a has clear sociological interpretation - homophily of the society. We find that for moderate values of the homophily level the wealth distribution obtained in the model consists of two parts. At lower wealth levels, the wealth is distributed lognormally, while at higher levels distribution is more consistent with a Pareto type. This result is in an accordance with the empirical paper by Clementi and Gallegati (2006), which showed that the U.S. income distribution is consistent with a lognormal function for the bottom 97%-99%, and with the Pareto function for the top 1%-3%. We use our model to address the question of occupational mismatch in the case of employment through neighbors. Particularly, we find that a worker is underpaid or receives a bonus in the case of employment through contacts depending on the worker’s status and the prevailing wealth distribution in the society. We think this prediction could contribute to the indecisive conclusion of the empirical literature concerning the question: whether the offered wage in the case of employment through contacts is higher or lower than the one which a worker could claim in the labor market directly. 17

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