Basic evidence from China

The Longitudinal Survey on Rural Urban Migration in China (RUMiC) ^{Footnote 1} emphasises social networks such as relatives and friends as an important method for rural–urban migrants to seek jobs. Table 1 reports an interesting finding from the RUMiC: approximately 60% of employed rural migrants found their jobs in urban centres through referrals of their relatives and friends, and only 12.5% obtained their jobs by direct application. This shows that social networks play an important role in migrants’ job seeking.

Table 1. Job-seeking methods used by rural–urban migrants.

Source: The Longitudinal Survey on Rural Urban Migration in China (RUMiC) in 2008. ^{Footnote 1}

Another survey ^{Footnote 2} conducted in the Pearl River Delta of China in 2008 also attests to the importance of social networks in migrants’ job seeking. Table 2 summarises the roles of relatives and friends in job seeking by rural–urban migrants in the survey. It indicates that 86.3% of the interviewed migrants obtained employment information from relatives or friends, 85.4% gained their jobs with the recommendation of relatives and friends, 88.0% were accompanied by relatives and friends for job interviews and 70.2% directly arranged employment with the help of their relatives and friends. In one of our previous surveys, we also found that the social network was one of the main channels through which rural–urban migrant workers were successfully employed. ^{Footnote 3}

Table 2. Basic statistics for migrants’ current jobs.

Source: Survey of migrants’ behaviour at the Pearl River Delta of China, 2008. This survey was organised by a research team from Sun Yet-Sen University, China. ^{Footnote 2,Footnote 3}

Chinese migrant workers nowadays depend strongly on social ties embedded in labour markets for their employment in urban areas. Using data from a detailed household survey in Northeast China, Reference Zhang and LiZhang and Li (2003) show that social networks (guanxi) have a significant impact on an individual’s probability of securing a nonfarm job. Furthermore, other studies show that a social network arising from an individual’s friends or relatives who have worked in urban areas is more likely to reduce the individual’s job-searching cost and to improve his or her wage (Reference Bridges and VillemezBridges and Villemez, 1986; Reference Caliendo, Schmidl and UhlendorffCaliendo et al., 2010; Reference ColemanColeman, 1988; Reference Durlauf and FafchampsDurlauf and Fafchanps, 2004; Reference GranovetterGranovetter, 1985). In one of our recent studies, we show empirically that social capital has a significant and positive effect on rural migrants’ income (Reference Wang and ZhouWang and Zhou, 2013). ^{Footnote 4}

Why the majority of migrant workers rely on social networks for seeking jobs and how social networks influence those migrants’ strategic behaviour and their wages, are pressing issues. Some previous studies suggest that social networks significantly contribute to the income of rural migrant workers in China and other developing economies (Reference Amuedo-Dorantes and MundraAmuedo-Dorantes and Mundra, 2007; Reference Giles, Park and CaiGiles et al., 2006; Reference Knight and YuehKnight and Yueh, 2008; Reference Zhang and LiZhang and Li, 2003), while others make counter conclusions from the US or European evidence (Reference MouwMouw, 2002, Reference Mouw2003). In addition, Reference PellizzariPellizzari (2010) finds large cross-country and cross-industry variation in income premiums associated with the role of informal social networks. The aforementioned studies help us to understand the relationship between social networks and labourers’ wages, but do not provide a detailed mechanism to explain why and how social networks impact on Chinese migrant workers’ wages. Furthermore, little of the literature concentrates on the benefit-shifting of both migrants and firms due to the extension of social networks over time. This article attempts to undertake a detailed analysis of decision-making as social networks are extended for both firms and workers, when there exists within these networks a mechanism for narrowing the information gap regarding work compensation.

This article focuses on how social networks influence the strategic behaviours of individuals and the wage of migrant workers. It is structured as follows. This first section provides the background and basic evidence from China, and the section ‘Literature review’ provides a review of the theoretical literature. In the section titled ‘The model’, a theoretically novel framework is proposed, and section ‘The determination of the wage’ provides a solution of the model. ‘The effects of the social networks on the offer wage’ then draws out the theoretical relationship between social networks and wages, with the section ‘Conclusion and remarks’ providing a conclusion.

Literature review

As mentioned above, the empirical literature has reached diverse conclusions relating to the issue of social networks and migrant workers’ wage. In resolving this debate, the first priority is to investigate the mechanism in operation. That is, why and how do social networks benefit jobseekers as well as employers?

Reference MontgomeryMontgomery (1991) develops a simple two-period model categorising workers into two groups with high and low ability. Assuming that high-ability workers have a higher probability of holding a tie to high-ability workers of the next period, Montgomery derives that an employing firm will seek to hire second-period workers by using referrals from their high-ability workers in the first period, instead of hiring from the open labour market. This is consistent with the finding by Reference ReesRees (1966): hiring by networks is an efficient channel for firms to gain more information about worker quality than hiring directly from labour markets.

Classification of social networks is another important issue. Existing literature generally categorises social networks into two types: those based on strong ties and those characterised by weak ties. The first type refers to close relationships such as with relatives and good friends, while the latter refers to a more distant relationship such as that with colleagues or common acquaintances. Reference GranovetterGranovetter (1973, Reference Granovetter1975) argues that weak ties can convey more useful job information to jobseekers than strong ties. Reference Lin, Castiglione, Van Deth and WollebLin (2008) believes that weak ties facilitate sending more new job information to jobseekers, since strong ties such as those among relatives and good friends may involve no more than a sharing of information that is the same as, or similar to that already known to the jobseekers. Based on search theory, Reference MontgomeryMontgomery (1992) advances a framework that involves the ideas of both Granovetter and Lin. In this model, a higher offer rate embodies Granovetter’s view and a better wage distribution reflects Lin’s idea. Montgomery’s model implies that jobseekers’ minimum acceptable wage is likely to increase with the rate of weak ties in the total networks. However, this model also shows that only in very special cases will the expected wage gained through weak ties be higher than the expected wage gained through strong ties. Montgomery’s study nevertheless indicates that the structure of networks (the rate of weak ties in the total networks) should be valued.

A related issue of interest in the literature is the impact of social networks on inequality of employment opportunities and wages. Reference MontgomeryMontgomery (1994) advances a simplified Markov model that involves weak ties, strong ties and formal channels in job seeking. In equilibrium, the existence of weak ties may reduce inequality of employment opportunities. Moreover, under some conditions, more weak ties are likely to increase the employment rate. Reference Calvó-Armengol and JacksonCalvó-Armengol and Jackson (2003, Reference Calvó-Armengol and Jackson2004) develop a general Markov model that does not depend on any specific network (though the network should be unchanging). They prove that the employment situation of heterogeneous individuals in different periods is positively related in the steady state. Allowing for search costs and the possibility of dropping out of the labour market, they also suggest that, even though the structure of two network groups may be identical, the group that starts with the better wage or employment situation will end up with a better wage or employment situation in the steady state. Reference LouryLoury (2006) indicates that a better match and limited-choice mechanism can account for differences in the wage effects of different types of job contacts. The above findings suggest that the existence of social networks contributes to the probability of inequality, which is opposite to Reference MontgomeryMontgomery’s (1994) conclusion.

Typically, three main methods are used to model the mechanism of networks’ impact on wages. The first method (e.g. Reference MontgomeryMontgomery, 1992) is based on the recursive equation of search theory. That is, the discounted utility of accepting the offer at this stage equals the subsequent discounted utility of finding a job in the next stage. Here, the social network is assumed to change the offer rate and wage directly. This kind of analysis focuses on how jobseekers make the offer-acceptance decision and determine the reserve wage, and so it cannot derive the impact of networks on the actual wage.

The second method (Reference Calvó-Armengol and JacksonCalvó-Armengol and Jackson, 2003; Reference MontgomeryMontgomery, 1994) analyses the transition probability based on the Markov model. It is suitable for all kinds of network structures, as different kinds of network can be interpreted by different transition probability matrices. However, since the probability is exogenously set, the method cannot illustrate how the transition probability matrix forms, nor can it interpret how the wage changes as the network structure shifts. Moreover, it merely analyses employers’ behaviour. But firms’ behaviour is actually impacted by social networks.

The third method describes both the workers’ and firms’ behaviour directly. Because different networks and different model settings may directly differ in workers’ and firms’ behaviour, this type of model depends on the model setting per se and is not universal. The Reference MontgomeryMontgomery (1991) two-period model is such a typical example. Nevertheless, in these models, the wage is determined by the endogenous behaviour of workers and firms. If uncertainty exists, the distribution of wage levels would be a deductive result rather than an assumption, which should be considered cautiously.

In this article, we develop a model that fits the third type, which is based on Reference MontgomeryMontgomery’s (1991) framework. This study makes a positive contribution to the previous literature in the following respects: First, it extends Montgomery’s model by generalising the assumption that a high-ability worker is more likely to hold a tie to a high-ability worker in the next period. In our model, types of firms and workers are continuous random variables distributed on [0, 1]. The match of types between firms and workers is pursued; thus, the types are no longer limited to the diverse quality of workers. These different types in our model can be further interpreted as indication of diverse sectors of industry. Second, we derive the equilibrium wage and make a further comparative static analysis of the positive effect of social networks on the wage instead of conducting a partial analysis as in Reference MontgomeryMontgomery’s (1991) model. Finally, in contrast with Reference MontgomeryMontgomery (1991)’s model, ours assumes that each firm provides an offer to the one worker who is the closest match to the firm’s type from among the workers with network links to the firm’s first-period workers, and each worker chooses to accept the highest offer wage. The mechanism in our model, whereby the employer hires through referrals in informal channels, is more consistent with reality.

The model

Rural–urban jobseekers often find jobs through acquaintances such as friends and relatives who provide job information and referrals. Meanwhile, firms usually rely on employee referrals to screen new migrant worker job applicants for ability and skills. The information gap between employers and migrant workers would otherwise often lead to an imperfect labour market, especially in developing countries such as China (Reference Knight, Quheng and ShiKnight et al., 2011; Reference Lee, Meng, Meng and ManningLee and Meng, 2010; Reference MengMeng, 2001). Social networks such as employee referrals may play an important role in delivering information and facilitating the proper matching between jobseekers and firms. Our model below investigates why social networks help narrow the information gap and affect the going wage.

Assumptions

Labour market

To be consistent with Reference MontgomeryMontgomery’s (1991) assumptions, we assume that there is a two-period economy, where there are N firms and N workers in each period. Firms exist in two periods, while workers participate for only one period, that is, the first-period workers will all retire from the labour market at the end of the first period. In the second period, N new workers will enter the labour market. For simplicity, we assume that each firm can only hire one worker in each period. Denote the type of firm $\theta$ as a random variable representing the variety of industries. Correspondingly, assume the type of migrant worker is also a random variable $\theta$ matching the firm to which the worker is most suited. As there are N firms and N workers in each period, labour demand always equals labour supply, namely, the economy is always in equilibrium.

Information

Each firm knows exactly its own type $\theta$ , but does not know other firms’ types. The distribution of firm types is common knowledge, for simplicity, denoted as $\theta ~U[0,1]$ . Before recruiting, firms do not know migrant workers’ types but only their distribution $\theta ~U[0,1]$ . However, after recruitment, firms become cognisant of their employees’ types. A worker only knows his or her own type and the type of the matching employer, and the distribution of other workers and firm types is common knowledge.

Two-period model

In Period 1, firms are established. Denote the type of a representative firm i by ${\theta }_i,i=1,2,\dots ,N$ . The firm hires a worker from the labour market stochastically. After hiring, the firm knows exactly its employee’s type ${\theta }_{1i}$ , but does not know the types of employees in other firms, only their distribution: $\theta ~U[0,1]$ .

In Period 2, workers in the first period retire and firms recruit new workers from the labour market directly or through referrals provided by their first-period employees. If a firm hires workers through the labour market, the only information about new workers available to the firm is that the type of new workers (denoted by ${\theta }_{2\mathrm{Mi}}$ ) is distributed uniformly on $[{\theta }_i,{\theta }_{1i}]$ or $[{\theta }_{1i},{\theta }_i]$ . Alternatively, the firm often hires second-period workers on the basis of first-period employees’ referrals (i.e. via a social network), whereby the worker recommends a worker of the type most suited to the type of firm. Assume that the density of social networks of each first-period employee is $m$ , that is, each employee in the first period knows $m$ candidate workers in the second period. The type of each candidate is uniformly distributed on $[{\theta }_i,{\theta }_{1i}]$ or $[{\theta }_{1i},{\theta }_i]$ . The firm will determine the offer wage to maximise its expected profit and then make the offer to the new worker. The firm can only send one offer. The workers who have received more than one offer will accept the highest offer and refuse others. Firms that have been refused and workers who receive no offer will continue to search in the labour market and pay the market wage.

Firm

Assume that in both of the two periods, there is only one product and all firms produce the same product, and a representative firm i’s production function is

$q_i=1-|{\theta }_i-{\theta }_k|$

where $q_i$ is the output, ${\theta }_i$ and ${\theta }_k$ are the firm’s and its employees’ types, respectively, and $|{\theta }_i-{\theta }_k|$ is the distance between the firm’s and the employees’ types, which is a measure of their matching cost. This specification of the production function makes it easy to show that the distance between the two types – of the firm and the worker – constitutes the cost to the firm, in reduced output, that results from a poor match. It is this matching cost that the firm seeks to reduce, by hiring through employee referrals (or social networks).

The profit of firm i is $y_i=1-|{\theta }_i-{\theta }_k|-w$ , where $w$ is the wage. Denote the market wages in the first and second periods as $w_{1M}$ and $w_{2M}$ , respectively, which are both assumed to be exogenous variables.

The firm’s behaviour

In Period 1, firm i hires a worker from the labour market stochastically with its profit function being $y_{1i}=1-|{\theta }_i-{\theta }_{1i}|-w_{1M}$ . In Period 2, it determines whether to hire through the labour market or through a social network. In the former case, the type of workers is

${\theta }_{2Mi}~\{\begin{array}{c}U[\theta i,\theta 1i],if\theta i<\theta 1i\\ U[\theta \text{1}i,\theta i],if\theta i>\theta 1i\end{array}$

and the firm’s profit function is $y_{2i}=1-|{\theta }_i-{\theta }_{2M}|-w_{2M}$ . In the latter case, firms have to determine the offer wage.

Denote the type of the $m$ candidates at the end of Period 1 as ${\theta }_{2i}^k$ ( $k=1,2,\dots ,m$ ) with

${\theta }_{2i}^k~\{\begin{array}{c}U[\theta i,\theta 1i],if\theta i<\theta 1i\\ U[\theta \text{1}i,\theta i],if\theta i>\theta 1i\end{array}$

The cumulative distribution function is denoted by $H_{1i}(·)$ . As illustrated above, each new candidate is recommended by a first-period employee according to his or her type and the firm’s type. As the firm does not know the specific type ${\theta }_{2i}^k$ exactly, this hiring through referrals can be seen as if the firm randomly samples m times from the distribution $U[{\theta }_i,{\theta }_{1i}]$ or $U[{\theta }_{1i},{\theta }_i]$ , and chooses the closest candidate type denoted by ${\theta }_{2\mathrm{Ri}}$ . If the firm’s offer is accepted, its expected profit in Period 2 will be $y_{2i}=1-|{\theta }_i-{\theta }_{2\mathrm{Ri}}|-w_{2\mathrm{Ri}}$ ; otherwise, it has to hire through labour market and the expected profit is $y_{2i}=1-|{\theta }_i-{\theta }_{2\mathrm{Mi}}|-w_{2M}$ .

The determination of the wage

We now derive the firm’s optimal offer wage which is related to social networks $m$ and $n$ . Firm i provides its offer wage $w_{2M}$ to maximise its expected profit:

$\underset{w_{2\mathrm{Ri}}}{max}E\left(y_2\right|{\theta }_i,{\theta }_{1i})\equiv {\pi }_i(w_{2\mathrm{Ri}}\left)\right[E\left(q_{2\mathrm{Ri}}\right|{\theta }_i,{\theta }_{1i})-w_{2\mathrm{Ri}}]+[1-{\pi }_i(w_{2\mathrm{Ri}}\left)\right]\left[E\right(q_{2\mathrm{Mi}}|{\theta }_i,{\theta }_{1i})-w_{2M}]$

where ${\pi }_i\left(w_{2\mathrm{Ri}}\right)$ is the probability that a worker will accept the firm’s offer $w_{2\mathrm{Ri}}$ , and $q_{2\mathrm{Ri}}=1-|{\theta }_i-{\theta }_{2\mathrm{Ri}}|$ and $q_{2\mathrm{Mi}}=1-|{\theta }_i-{\theta }_{2\mathrm{Mi}}|$ are the firm’s second-period output when hiring workers through social networks and through the labour market, respectively. Denote $A_i\equiv q_{2\mathrm{Ri}}-q_{2\mathrm{Mi}}$ . This is the extra output due to the reduction of the matching cost from hiring through networks instead of through the market. It can also be seen as the extra common benefit, since the firm has to share $A_i$ with the worker who accepts the wage offer. Denote $E{A}_i\equiv E(q_{2\mathrm{Ri}}-q_{2\mathrm{Mi}}|{\theta }_i,{\theta }_{1i})$ . From Appendix 2, $\mathrm{EAi}=(m-1)/2(m+1)|{\theta }_i-{\theta }_{1i}|$ . The firm’s optimal offer wage $w_{2\mathrm{Ri}}^{*}$ satisfies the following first-order condition

(1) ${\pi }_i\left(w_{2\mathrm{Ri}}^{*}\right)={\pi }_i\prime \left(w_{2\mathrm{Ri}}^{*}\right)[{\mathrm{EA}}_i-(w_{2\mathrm{Ri}}^{*}-w_{2M}\left)\right]$

Hence $w_{2\mathrm{Ri}}^{*}$ is a function of $E{A}_i$ , denoted as

(2) $w_{2\mathrm{Ri}}^{*}=w_i\left(E{A}_i\right)$

where $w_i(·)$ is the optimal response function of firm i conditional on ${\theta }_i$ and ${\theta }_{1i}$ , the distribution of ${\theta }_{2\mathrm{Mi}}$ and ${\theta }_{2\mathrm{Ri}}$ , and the specific form of ${\pi }_i(·)$ . This shows that firm i will set the optimal wage offer $w_{2\mathrm{Ri}}^{*}$ according to the realisation of $E{A}_i$ . The uncertainty of $w_{2\mathrm{Ri}}^{*}$ is due to the randomness of $E{A}_i$ . Appendix 2 derives the distribution of $E{A}_i$ , which is related to the social network $m$ .

The probability ${\pi }_i\left(w_{2\mathrm{Ri}}^{*}\right)$ reflects the response of the worker who has received firm i’s offer $w_{2\mathrm{Ri}}^{*}$ . Whether the worker of type ${\theta }_{2\mathrm{Ri}}$ accepts firm i’s wage offer $w_{2\mathrm{Ri}}^{*}$ depends on whether he or she has received a higher wage from other firms. Suppose the worker of type ${\theta }_{2\mathrm{Ri}}$ also receives firm j’s offer $w_{2\mathrm{Rj}}^{*}$ . Then

$\begin{array}{c}{\pi }_i\left(w_{2Ri}^{*}\right)=P\left\{{\theta }_{2Ri}\text{acceps}{w}_{2Ri}^{*}\right\}\prod _{j=1,j\ne i}^{N}P\left\{{\theta }_{2Ri}\text{receives no offer withwage}{w}_{2Rj}^{*}>w_{2Ri}^{*}\right\}\\ =\prod _{j=1,j\ne i}^N\left(1-P\left\{{\theta }_{2Ri}\text{receives}{w}_{2Rj}^{*}>w_{2Ri}^{*}\right\}\right)\end{array}$

where $P\{{\theta }_{2Ri}receives{w}_{2Rj}^{*}>w_{2Ri}^{*}\}$ can be written as

(3) $P\left\{{\theta }_{2Ri}\text{ties to}{\theta }_{1j}\right\}\cdot P\left\{{\theta }_{1j}\text{make offer to}{\theta }_{2Ri}|{\theta }_{2Ri}\text{ties to}{\theta }_{1j}\right\}\cdot P\left\{w_{2Rj}^{*}>w_{2Ri}^{*}|{\theta }_{2Ri}\text{ties to}{\theta }_{1j},{\theta }_{1j}\text{make offer to}{\theta }_{2Ri}\right\}$

To relate the response of the worker to the density of social networks $n$ and $m$ , we study each term in equation (3). Under the assumption of equal probability, $P\left\{{\theta }_{2\mathrm{Ri}}\text{ties}\text{to}{\theta }_{1j}\right\}=n/N$ . Since both firm j and the worker know nothing about the type of firm i except the distribution of the type, the second term $\tau \equiv P\left\{{\theta }_{1j}\text{send}\text{offer}\text{to}{\theta }_{2\mathrm{Ri}}\right|{\theta }_{2\mathrm{Ri}}\text{ties}\text{to}{\theta }_{1j}\}$ is irrelevant to ${\theta }_{1i}$ and ${\theta }_i$ . Also, since the firm’s decision to make an offer only depends on the distance between its type and the worker’s type, $\tau$ is irrelevant to $n$ . Moreover, $\tau$ is a function of the social network density $m$ . We can assume that $\partial \tau /\partial m<0$ since the probability of ${\theta }_{2\mathrm{Ri}}$ receiving an offer from another firm will decrease with the size of the social network of first-period workers. The last term in equation (3) is $1-K_i\left(w_{2\mathrm{Ri}}^{*}\right)$ , where $K_i\left(w\right)=P(w_{2\mathrm{Rj}}^{*}<w,j\ne i)$ is the distribution function of firm j’s offer wage $w_{2\mathrm{Rj}}^{*}$ to the worker ${\theta }_{2\mathrm{Ri}}$ . Furthermore, as $n$ , $N$ , $\tau$ and $K(·)$ are all the same for all firms, we can write $\pi \left(w_{2\mathrm{Ri}}^{*}\right)={\pi }_i\left(w_{2\mathrm{Ri}}^{*}\right)$ . Hence

(4) $\pi \left(w_{2\mathrm{Ri}}^{*}\right)=\underset{\frac{j=1}{j\ne i}}{\overset{N-1}{\Pi }}(1-\frac{n}{N}·\tau ·[1-K_i\left(w_{2\mathrm{Ri}}^{*}\right)])={(1-\frac{n}{N}·\tau ·[1-K\left(w_{2\mathrm{Ri}}^{*}\right)])}^{N-1}$

The specific form of $K\left(w\right)$ is determined by the distribution of $E{A}_j$ (from the viewpoint of firm i). It follows from the monotonicity of $w\left(E{A}_i\right)$ and the distribution of $E{A}_i$ that

(5) $K\left(w_{2\mathrm{Ri}}^{*}\right)=P\{w_{2\mathrm{Rj}}^{*}\le w_{2\mathrm{Ri}}^{*}\}=P\left\{w\right(E{A}_j)\le w(E{A}_i\left)\right\}=P\{E{A}_j\le E{A}_i\}=A\left(E{A}_i\right)$

Since $0\le E{A}_i\le \left(\right(m-1)/2(m+1\left)\right)$ and $E{A}_i=w^{-1}\left(w_{2Ri}^{*}\right)$ , substituting equation (5) into equation (4), we have

(6) $\pi \left(w_{2\mathrm{Ri}}^{*}\right)={[1-\frac{n\tau }{N}{(1-\frac{2(m+1)}{m-1}{w}^{-1}\left(w_{2\mathrm{Ri}}^{*}\right))}^2]}^{N-1}$

Denote $\delta \equiv N/n\tau$ and $c\equiv 2(m+1)/m-1$ . Differentiating (6) with respect to $w_{2\mathrm{Ri}}^{*}$ and solving the differential equation (1) with the initial condition: $w_{2\mathrm{Ri}}^{*}=w_{2M}$ as $E{A}_i=0$ , we have

(7) $w_{2\mathrm{Ri}}^{*}=w_{2M}+E{A}_i-S$

where $S\equiv {\int }_0^{E{A}_i}{\left(\right(\delta -{(1-\mathrm{cx})}^2)/(\delta -{(1-\mathrm{cE}{A}_i)}^2\left)\right)}^{N-1}\mathrm{dx}$ , $0\le S\le E{A}_i$ , $\delta =N/n\tau$ and $c=\left(2\right(m+1\left)\right)/(m-1)$ .

This is the expression of the optimal response $w_{2\mathrm{Ri}}^{*}=w\left(E{A}_i\right)$ . It can also be written as

(8) $w_{2\mathrm{Ri}}^{*}=w_{2M}+\frac{m-1}{2(m+1)}|{\theta }_i-{\theta }_{1i}|-\frac{m-1}{2(m+1)}\underset{0}{\overset{|{\theta }_i-{\theta }_{1i}|}{\int }}{\left(\frac{N-n\tau {(1-x)}^2}{N-n\tau {(1-|{\theta }_i-{\theta }_{1i}\left|\right)}^2}\right)}^{N-1}\mathrm{dx}$

Note that $w_{2M}\le w_{2Ri}^{\ast }\le E{A}_i+w_{2M}$ . This implies that the optimal offer wage should be greater than the market wage; otherwise, workers will refuse the offer. Also, $w_{2\mathrm{Ri}}^{*}$ should not be higher than the sum of the market wage and extra common benefit; otherwise, firms will not hire through referrals. ^{Footnote 5}

The optimal wage offer, as shown in equation (8), is composed of three parts. The first part is the market wage, $w_{2M}$ , which is the benchmark for pricing labour in any wage offer. The second part is $E{A}_i$ , the difference in expected production from these two hiring channels. It is due to the existence of the extra common benefit that the firm is inclined to hire through networks. The third part is a positive number $S$ deducted from the wage offer. S and $E{A}_i-S$ constitute the extra common benefit and belong to the firm and the worker, respectively.

By the definition of S, the division $S$ from extra common benefit is determined by the total number of firms and workers $N$ , the social networks of first-period workers $m$ , the social networks of second-period workers $n$ , the probability of the worker receiving another offer $\tau$ and the extra common interest $E{A}_i$ (also influenced by $m$ ). These factors are all taken into account by the firm in determining its wage offer. Meanwhile, the division of the extra common interest is determined.

In fact, the division results from the interaction of the ‘power’ of firms and workers. First, the social network of first-period workers $m$ is the ‘power’ of firms, and hence can increase $E{A}_i$ , S and $w_{2\mathrm{Ri}}^{*}$ . Second, $n$ is the ‘power’ of second-period workers, which can increase the probability of the workers receiving wage offers higher than $w_{2\mathrm{Ri}}^{*}$ and refusing the firm’s offer. As the firm being refused cannot earn the extra common benefit, it seeks to increase the offer wage to ensure that the probability of being accepted is not too low. Third, $\tau$ , the probability of receiving an offer from other firms, to some extent can represent the intensity of competition among firms. The more intense the competition, the more ‘power’ second-period workers have. This will certainly increase the offer wage and decrease the extra common benefit S to the firm. All these results imply that the social network is one of the significant determinants of the offer wage.

The effects of social networks on the offer wage

The influence of social network of the first-period worker on the offer wage

As stated above, first-period workers help their employers find the most suitable second-period employees. Accordingly, the social network of first-period workers can be seen as that of the firms in which first-period workers worked. It follows from equations (7) and (8) that $0\le S\le {\mathrm{EA}}_i$ and $\partial w_{2\mathrm{Ri}}^{*}/\partial m>0$ . That is, the social network of firms plays a positive role in enhancing the offer wage for second-period workers. By $\partial w_{2\mathrm{Ri}}^{*}/\partial m=\partial {\mathrm{EA}}_i/\partial m-\partial S/\partial m>0$ , we know that the firm can benefit by the extra common interest when it hires through networks so that the matching cost decreases. However, the second-period workers have the right to reject the offer. So, the firm is inclined to share some benefit (by means of a setting higher wage) with its target employee to increase the probability for its wage offer to be accepted. It is this mechanism that makes both the firm and second-period workers benefit more when the firm hires through networks. Furthermore, ${\partial }^{2}S/\partial m^2<0$ and ${\partial }^2{w}_{2\mathrm{Ri}}^{*}/\partial m^2<0$ . This implies that the positive effect of the social network on wage is marginally diminishing.

Moreover, it follows from

$\frac{{\mathrm{EA}}_i-S}{{\mathrm{EA}}_i}=1-\frac{1}{|{\theta }_i-{\theta }_{1i}|}\underset{0}{\overset{|{\theta }_i-{\theta }_{1i}|}{\int }}{\left(\frac{N-n\tau {(1-x)}^2}{N-n\tau {(1-|{\theta }_i-{\theta }_{1i}\left|\right)}^2}\right)}^{N-1}\mathrm{dx}$

that the proportion of the extra common benefit which the firm and second-period workers share does not change with $m$ . Thus, the social networks of firms influence the wage only by increasing the extra common benefit. The benefit will not change the sharing structure on either side.